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Transcript

A LBERT-L UDWIGS -U NIVERSIT ÄT F REIBURG Entanglement in periodically driven quantum systems D ISSERTATION zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg im Breisgau vorgelegt von S IMEON S AUER aus Gießen 2013 Dekan: Betreuer der Arbeit: In Zusammenarbeit mit: Referent: Koreferent: Tag der mündlichen Prüfung: Prüfer Prof. Dr. Michael Růžička Prof. Dr. Andreas Buchleitner Dr. Clemens Gneiting Prof. Dr. Andreas Buchleitner Prof. Dr. Gerhard Stock 25.03.2013 Prof. Dr. Andreas Buchleitner Apl. Prof. Dr. Heinz-Peter Breuer Prof. Dr. Tobias Schätz Abstract The entanglement of quantum states is one of the most fundamental aspects of quantum theory, that has led to fertile debates and practical applications. In the present thesis, we investigate how entanglement evolves in the presence of a periodic driving force. Our motivation are recent findings in the literature which indicate that periodic driving can render entanglement more robust against the detrimental influence of the environment, which is present in any realistic quantum system. In particular, we are interested in identifying the conditions for the periodic drive to sustain entanglement in the limit of long times, i.e., in the cyclo-stationary state. This question is discussed from different perspectives: After an introduction and a summary of the theoretical concepts in Chapter 1 and 2, Chapter 3 has its focus on periodically driven quantum systems in which the influence of the environment can be neglected. There, we investigate the entanglement properties of the resulting Floquet states, which define, in essence, periodic solutions of the dynamics. In Chapter 4, we include environment-induced decoherence and dissipation mechanisms in our description and analyze the entanglement of the resulting asymptotic cycle. In this context, we develop a very general approach to determine the optimal periodic driving mechanism for a given target quantity (which can be, inter alia, entanglement). By the end of this work, we will have developed a general understanding of the interrelation of periodic driving and entanglement dynamics in both open and closed quantum systems. Zusammenfassung Die Verschränkung von Quantenzuständen ist ein grundlegender Aspekt der Quantentheorie, der zahlreiche fruchtbare Diskussionen ausgelöst und Anwendungen herbeigeführt hat. In der vorliegenden Arbeit wird untersucht, wie sich Verschränkung in der Gegenwart eines periodischen Antriebs verhält. Unsere Motivation für diese Agenda sind erste Hinweise in der Literatur, nach denen die Verschränkung durch die Gegenwart eines kohärenten, periodischen Antriebs an Robustheit gegenüber den abträglichen Einflüssen der Umgebung gewinnt, welchen jedes realistische Quantensystem ausgesetzt ist. Konkret untersuchen wir, unter welchen Bedingungen ein periodischer Antrieb Verschränkung im Limes langer Zeiten – also im zyklo-stationären Zustand – aufrecht erhalten kann. Diese Frage wird aus unterschiedlichen Blickwinkeln beleuchtet: Nach einer Einleitung und einer Einführung in die theoretischen Grundlagen in den Kapiteln 1 und 2 behandeln wir in Kapitel 3 periodisch getriebene Quantensysteme, in denen der Einfluss der Umgebung vernachlässigt werden kann. Hier untersuchen wir die Verschränkungseigenschaften der resultierenden Floquet-Zustände, die im Wesentlichen periodische Lösungen der Dynamik darstellen. In Kapitel 4 beziehen wir dann umgebungsinduzierte Dekohärenz- und Dissipationseffekte mit in die Untersuchung ein und analysieren die Verschränkung des asymptotischen Zyklus der Dynamik. In diesem Zusammenhang entwickeln wir einen sehr allgemein anwendbaren Zugang zu der Frage, welcher der optimale periodische Antrieb für eine gegebene Zielgröße ist (wie z.B., aber nicht ausschließlich, Verschränkung). An Ende der Arbeit haben wir ein generelles Verständnis der Zusammenhänge zwischen periodischem Antrieb und Verschränkungsdynamik offener und geschlossener Quantensysteme gewonnen. Diese Arbeit ist mit der Unterstützung einer Reihe hilfsbereiter und wohlwollender Menschen entstanden. Ihnen allen gilt mein größter Dank! In erster Linie gilt das natürlich für die Betreuer meiner Arbeit – also Andreas Buchleitner für die Supervision meines Promotionsprojektes, Florian Mintert für die direkte Betreuung während der ersten Phase und Clemens Gneiting während der zweiten Phase. Flo und Clemens gilt mein Dank für die vielen Diskussionen, rehearsals, praktischen Tipps und Ideen – und Clemens ein extra Dank für das zeitaufwendige Durcharbeiten der ersten Version dieser Arbeit. Andreas verdanke ich, dass er 2007 in Dresden meine Begeisterung für Verschränkung geweckt hat, mich von der ersten Stunde an bestärkt hat, dass in diesem Geschäft nicht nur alte Hasen etwas beitragen können, und dass er dafür sorgt, dass die Gruppe stets offen nach innen und außen bleibt. Diese Gruppe ist deshalb als nächstes hervorzuheben: Ohne die Crew im 9. Stock wäre diese Arbeit weder inhaltlich so weit gediehen, noch wäre sie in so guter Atmosphäre entstanden. Deshalb ein großer Dank an Euch alle für viele Tafeldiskussionen, Küchenversammlungen und Partyaktionen. Und um die langjährigsten Mitstreiter persönlich zu nennen: Tobi G., Jochen, Felix, Stefan H. und F., Klaus M. und Z., Schorsch, Dominik, Fede und Mattia, ich wünsche Euch ebenso viel Unterstützung für den Abschluss wie ich sie von Euch erfahren habe! Für viele Anregungen und Hinweise möchte ich mich außerdem bei Ugo, Slava, Malte und Thomas bedanken. Ein großer Dank gilt außerdem der Studienstiftung des deutschen Volkes, die mich mit einem Promotionsstipendium gefördert hat. Die unbürokratische und offene Kultur innerhalb der Stiftung habe ich als äußerst angenehm und inspirierend erlebt. Dafür dass sie Freiburg zu meiner Heimat gemacht haben danke ich Carolina, Ben, Nora und Lion. Meine Eltern Marta und Jürgen und meine Brüder Benjamin und Jonas waren auch während der Promotion – wie bereits während des gesamten Studiums – eine große moralische Unterstützung für mich; erneut ein großes Dankeschön dafür. Beni möchte ich besonders danken für seine riesige Unterstützung als Korrektor und Umzugshelfer! Am Ende danke ich dem wichtigsten Menschen in meinem Leben, Christina, die mich durch alle Aufs und Abs dieser Promotionszeit verständnisvoll, bestärkend und liebend begleitet hat. Contents 1 Introduction 2 Theoretical Concepts 2.1 Floquet theory . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Schrödinger equation and the Floquet theorem 2.1.2 Determining Floquet states and quasi-energies 2.2 Open quantum systems . . . . . . . . . . . . . . . . . 2.2.1 Introduction to open quantum systems . . . . . 2.2.2 The Floquet Born-Markov master equation . . 2.2.3 Master equation with fixed dissipator . . . . . 2.3 Entanglement theory . . . . . . . . . . . . . . . . . . 2.3.1 Definition of entangled states . . . . . . . . . . 2.3.2 Quantifying entanglement . . . . . . . . . . . 2.3.3 Entanglement measures for pure states . . . . . 2.3.4 Entanglement measures for mixed states . . . . 3 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entanglement resonances in closed, driven quantum systems 3.1 Why study the entanglement of Floquet states? . . . . . . . . . . 3.2 Two periodically driven qubits . . . . . . . . . . . . . . . . . . . 3.2.1 Monochromatic driving . . . . . . . . . . . . . . . . . . 3.2.2 Excursion: The Floquet problem of a single, driven qubit . 3.2.3 Different driving profiles . . . . . . . . . . . . . . . . . . 3.2.4 Variation of the qubit-qubit interaction . . . . . . . . . . . 3.3 Beyond two qubits . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Three qubits and GHZ entanglement . . . . . . . . . . . . 3.3.2 Deviations from perfect permutation invariance . . . . . . 3.3.3 Entanglement resonance with an arbitrary number of qubits 7 8 8 11 17 17 22 33 37 37 39 40 44 47 48 53 54 67 74 79 84 84 93 94 4 5 Cyclo-stationary entanglement in open, driven quantum systems 4.1 Driven qubits in Floquet Born-Markov description . . . . . . . . 4.1.1 Setting the stage . . . . . . . . . . . . . . . . . . . . . 4.1.2 Determining entanglement of the asymptotic cycle . . . 4.1.3 Understanding the entanglement of the asymptotic cycle 4.1.4 Varying the ingredients of the model . . . . . . . . . . . 4.2 The optimal stationary state under fixed incoherent dynamics . . 4.2.1 Introductory examples . . . . . . . . . . . . . . . . . . 4.2.2 Systematic derivation of the optimal stationary state . . 4.2.3 The optimal Hamiltonian . . . . . . . . . . . . . . . . . 4.2.4 Relation to dissipative state preparation schemes . . . . 4.3 The optimal asymptotic cycle under fixed incoherent dynamics . 4.3.1 Characterization of asymptotic cycles . . . . . . . . . . 4.3.2 Deriving the optimal asymptotic cycle of a single qubit . 4.3.3 Optimal asymptotic cycle of two qubits . . . . . . . . . 4.3.4 Additional means to further improve the optimum . . . . . . . . . . . . . . . . . . . Conclusion Appendix A.1 Incoherent transition rates for two driven qubits . . A.2 Detailed analysis of the optimal static Hamiltonian A.2.1 The optimal Hamiltonian for two qubits . . A.2.2 Generalization to more than two qubits . . A.3 Proof of inequality (4.110) . . . . . . . . . . . . . A.4 Table of frequently used notations . . . . . . . . . 99 101 101 103 104 111 115 117 122 133 136 143 143 147 156 160 165 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 170 175 175 178 181 183 Chapter 1 Introduction “I would not call [entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.” Erwin Schrödinger (1935), in [1] A periodic driving force of the right frequency can drastically alter the dynamics of a physical system, even if the force is comparatively weak. Such resonance phenomena are observed, e.g., in mechanical systems. Famous examples range from celestial mechanics (e.g., the gravitational force exerted by two planets on each other can destabilize their orbits if the orbital periods are in resonance [2]), over building mechanics (think of the collapse of the Tacoma Narrows Bridge [3]), to the physics of hearing (the ear perceives acoustic waves of different frequencies through resonant oscillations of hair cells [4]). Apart from reinforcing the amplitude of an oscillator, a periodic driving mechanism can qualitatively alter the dynamics in several further aspects. E.g., it can generate chaotic behavior in systems that evolve regularly in the absence of driving, like the periodically kicked rotor [5], or it induces phase transitions in thermodynamic systems, such as the Ising model [6, 7]. Many of such driving-induced phenomena carry over to the realm of quantum physics: the quantum version of the kicked rotor is a paradigmatic example of quantum chaos [8–12]; dynamical quantum phase transitions have been studied both theoretically [13] and experimentally, e.g., with ultra-cold atoms under optical driving [14, 15]; and any spectroscopic scheme to probe the energy levels of a quantum system relies, ultimately, on a resonance phenomenon [16]. Quantum mechanics, however, features also aspects that are unprecedented in the classical world and therefore defy our intuition – inciting even more our curiosity to understand how these genuine quantum effects behave under periodic forcing. 1 2 This has led, e.g., to the investigation of dynamical localization [17, 18] and the coherent destruction of tunneling [19], both of which arise from the interplay of periodic driving and quantum coherence. Arguably the most fascinating trait of quantum mechanics, however, is entanglement. It describes the ability of two spatially separated quantum objects to be in a superposition state; this can lead to non-local correlations between the outcomes of measurement on the individual objects [20]. As expressed by Schrödinger’s quote in the above epigraph, this aspect of quantum mechanics represents the “entire departure from classical lines of thought”. For a long time, it remained undecided whether entanglement is indeed an observable property of nature or just an artifact of our (insufficient) description of it. Only with Bell’s seminal work in 1964, it became clear that entanglement can have experimentally verifiable consequences [21]. Since then, the non-local correlations caused by entanglement have been demonstrated in numerous labs [22–26], providing thus a convincing argument for the correctness of quantum mechanics. Over the last decades, the interest in entanglement has shifted from its fundamental significance to more practical aspects: Entanglement is regarded as a central resource in several quantum technological applications, such as quantum computation [27, 28], quantum cryptography [29], quantum teleportation [30], and quantum metrology [31, 32]. Apart from that, the investigation of entanglement properties of interacting many-body quantum systems has improved the understanding of collective quantum effects in these systems [33]. Despite the central role of entanglement in quantum theory, the understanding of its dynamical behavior in the presence of a periodic driving mechanism is still in its infancy. So far, the following results have been obtained: I Suitable, periodic driving fields can lead to weak, but persistent entanglement between two interacting qubits1 [34] – despite the presence of additional incoherent processes, which are typically detrimental to entanglement [35]. I When a dissipative optomechanical system is driven by a laser that is slightly modulated in time, it eventually approaches a periodic trajectory in the state space (i.e., an asymptotic cycle) that features more entanglement than the corresponding stationary state of the unmodulated system [36, 37] – at least if one regards the peak value during the cycle as the figure of merit. I If the coupling strength between two harmonic oscillators [38] or qubits [39, 40], coupled to a thermal bath, is modulated periodically at the right 1 The notion of a qubit refers to a quantum mechanical two-level system and represents the elementary bit of information in quantum computing [27]. 3 frequency, the asymptotic cycle can feature more entanglement than the respective stationary state of the undriven system at the same bath temperature. While these studies certainly provide valuable insight for particular model systems or driving mechanisms, a comprehensive understanding of the interrelation of entanglement and periodic driving is still lacking to date. This is, hence, the main concern of the present thesis. Specifically, we are guided by the following questions: I Under which conditions does continuous, periodic driving lead to highly entangled solutions of the dynamics and in this sense provoke a “resonant” behavior of entanglement? I How does the influence of periodic driving on the entanglement dynamics change in the presence of decoherence, i.e., in open quantum systems [41]? Does driving always render entanglement more robust against thermal noise, as suggested by the results of Refs. [38–40], or is this a peculiarity of the specific model systems studied in these works? I What is the optimal driving mechanism in the presence of decoherence, i.e., the driving mechanism that entails the asymptotic cycle with the best possible entanglement properties? Is the optimal cycle always superior to the stationary state of the corresponding undriven system? To develop a general understanding of these and the like issues, we elucidate the topic from different perspectives. In all our investigations, however, we always focus on the case of persistent entanglement in the long-term dynamics that is generated under continuous action of the driving mechanism. Thus, we are particularly interested in the entanglement of the asymptotic cycle, in contrast to investigations that study the entanglement dynamics under the influence of a driving field on transient time scales [42, 43]. Our main motivation for focusing on this regime is that entanglement rarely survives on asymptotic time scales in undriven, open quantum systems, since such systems generically relax to thermal equilibrium in the long term, in which entanglement is typically washed out by thermal fluctuations [33]. If entanglement could (at least partially) be upheld in the asymptotic cycle, this would paradigmatically show how a genuine quantum effect can be protected against decoherence if it is constantly driven out of equilibrium by a suitable periodic driving mechanism [44]. Throughout this thesis, the to-be-entangled quantum objects are qubits. Various physical realizations of such quantum mechanical two-level systems are under amazingly accurate experimental control to date. To underpin the 4 universality of our arguments, we avoid the reference to a particular experiment in the main text. Nevertheless, we list the particularly relevant examples for our discussion in Table 1.1, along with the time and energy scales that occur in the different experimental settings. Outline To put our agenda into practice, we proceed as follows: First, we introduce the required concepts in the following Chapter 2. Besides an introduction to Floquet theory – the fundamental framework to describe and analyze periodically driven systems – we discuss ways to account for decoherence effects and provide tools to measure the entanglement content of quantum states. The main part begins in Chapter 3. There, we focus on the case of a closed quantum system, i.e., a system that is isolated from any detrimental influence of decoherence. In this case, any periodically driven quantum system has a set of periodic solutions of the dynamics, namely, its Floquet states. Our main concern is to assess and understand the entanglement properties of these states. It turns out that suitably chosen driving parameters result, indeed, in a resonant behavior of entanglement. This finding enables an adiabatic state preparation scheme based on the control of periodic driving fields, which is more versatile than the conventional adiabatic control ansatz with static fields. In Chapter 4, we additionally take decoherence effects into account. As a consequence, the system evolves into a unique asymptotic cycle, which is typically mixed (in contrast to the Floquet states of the closed system). Its entanglement properties are therefore drastically deteriorated. While we investigate the consequence of decoherence for a particular model system in Section 4.1, the goal in Sections 4.2 and 4.3 is to derive, very generally, the optimal driving protocol; thus, we determine the periodically driven system that leads to the strongest persistent entanglement in the presence of a given decoherence model. In fact, we start out with the optimal undriven system in Section 4.2, to develop the foundations that enable the derivation of the optimal periodically driven scenario in Section 4.3. Finally, we conclude in Chapter 5. 500 kHz [60, 61] 0.1 [62] - 2 Hz [51] 30 Hz [50] phonon-mediated 150 Hz [51] 30 MHz [49] MHz - GHz 10 MHz [51] 15 GHz [50] laser / microwave trapped ions [48–51] 500 kHz [53] 40 kHz [53] dipole-dipole 30 - 400 MHz [52] MHz - GHz 500 MHz - 5 GHz microwave Color centers in diamond [52, 53] 5 kHz [56] 50 MHz [56, 57] dipole-dipole 500 kHz [57] 7 MHz [56] 1014 - 1015 Hz 1014 - 1015 Hz laser Rydberg atoms [54–57] Table 1.1: Summary of energy scales for various experiments with driven, interacting qubits. All numbers are approximate and indicate orders of magnitude. In case of trapped ions, the values refer to microwave driving. The notation (ω0 , ω, F, J, and γ) is kept throughout this thesis. decay rate γ of the excited state 80 MHz [46] 50 - 500 MHz (typical, but also up to F ≈ ω0 [58, 59]) Rabi frequency F (= driving strength) qubit-qubit interaction strength J MHz - GHz driving frequency ω inductively / capacitively, or via cavity 5 - 20 GHz resonance frequency ω0 (of the bare qubit) qubit-qubit interaction mechanism microwave driving source superconducting qubits [45–47] 5 6 Chapter 2 Theoretical Concepts In this chapter, we introduce the theoretical concepts that underlie this thesis. Being interested in the interplay of periodic driving and entanglement, we need to answer the following two questions: I What is the time evolution of a quantum system under periodic driving? I How can one assess the entanglement properties of a quantum state? The first question is addressed by introducing the Floquet picture in Section 2.1. This framework provides a transparent way to describe the time evolution of a periodically driven quantum system that is closed, i.e., isolated from environmental degrees of freedom. The Floquet picture will be of great importance in Chapter 3, where we study entanglement in periodically driven, closed quantum systems. Section 2.2, on the other hand, is dedicated to the description of driven, open quantum systems and provides the basis for studying the interplay of environment-induced decoherence and periodic driving in Chapter 4. Finally, the basic notions of entanglement theory are introduced in Section 2.3. Definitions and approximations of entanglement measures are discussed, which will allow us to assess the entanglement properties of quantum states throughout this thesis. 7 8 2.1 Floquet theory 2.1.1 Schrödinger equation and the Floquet theorem When a quantum system is isolated from any environmental influence, its time evolution is governed by the Schrödinger equation [63] i∂t |ψ(t)i = H(t) |ψ(t)i .1 (2.1) Here, |ψ(t)i is an element of a Hilbert space H , which represents the state of the system at time t, and the Hamiltonian H(t) is a self-adjoint, linear operator on H . The focus of this thesis are quantum systems that are subject to a periodic driving force, to which one can hence assign a frequency ω. This results in a periodicity of T = 2π/ω of the Hamiltonian: H(t) = H(t + T ) (∀t). (2.2) In this case, Eq. (2.1) is a first order differential equation with periodic coefficients, to which the Floquet theorem applies [64–66]. The latter states that any solution |ψ(t)i of Eq. (2.1) is a superposition |ψ(t)i = ∑ ci e−iεit |φi (t)i (2.3) i of the Floquet states |φi (t)i of H(t), garnished by dynamical phase factors that feature the (real-valued) quasi-energies εi of the system.2 The time-independent weighting factors ci of the sum are determined by the initial condition at time t = 0: ci = hφi (0)|ψ(0)i . (2.4) The defining property of the Floquet states |φi (t)i is that they inherit the T -periodicity of the Hamiltonian: |φi (t)i = |φi (t + T )i (∀t). (2.5) It is instructive to consider the situation in which the initial state |ψ(0)i consists of a single Floquet state only. The system remains then in this Floquet state for all times, up to an irrelevant global phase factor: |ψ(0)i = |φ j (0)i ⇒ ci = δi j 1 We set h̄ ≡ 1 throughout this thesis. ⇒ |ψ(t)i = e−iε j t |φ j (t)i . (2.6) Hence, angular frequencies are measured in units of energy. index i labels the different Floquet states. It is either discrete or continuous, depending on the structure of the Hilbert space H . In the following, all formulas refer to a finite dimensional quantum system of dim H ∈ N, which is the relevant case throughout this thesis. 2 The 9 Accordingly, the observable system dynamics is strictly periodic in this case. If one probes the dynamics with a course-grained time resolution (i.e., one averages over a time window of more than one period T ), it seems as if the system does not evolve at all. For this reason, Floquet states are also referred to as the cycle-stationary states of the dynamics – they invoke stationary behavior, up to fast oscillations on a time scale smaller than T . The appeal of the Floquet picture is, from a practical point of view, that the complete dynamics for arbitrary times t and arbitrary initial conditions |ψ(0)i can be determined from the Floquet states and the corresponding quasi-energies alone. In fact, the unitary time evolution operator U(0,t) [63] of the system is easily expressed by the Floquet quantities, U(0,t) = ∑ e−iεit |φi (t)i hφi (0)| ⇒ i |ψ(t)i = U(0,t) |ψ(0)i , (2.7) as derived from Eqs. (2.3) and (2.4). This is a major advantage over the case of a general, time-dependent Hamiltonian H(t), where one typically has to integrate the Schrödinger equation (2.1) numerically, which is cumbersome and accurate only up to finite times t. Analogy to the static case Equations (2.3) to (2.6) suggest that the Floquet picture can be regarded as the generalization of the concept of eigenstates – which applies to undriven systems only – to periodically driven systems. To underpin this point of view, consider an undriven system, described by a static Hamiltonian H. In this case, the Schrödinger equation (2.1) is autonomous, and it is well-known that any solution |ψ(t)i is the superposition of the eigenstates |φi i of H: |ψ(t)i = ∑ ci e−iEit |φi i . i The phase factors contain the energy eigenvalues Ei , and the weighting factors are determined by the initial condition ci = hφi |ψ(0)i . In particular, if the initial state is an eigenstate, the system remains in this state forever (up to an irrelevant global phase): |ψ(0)i = |φ j i ⇒ ci = δi j ⇒ |ψ(t)i = e−iε j t |φ j i . The eigenstates of the Hamiltonian define, hence, the stationary states of the autonomous system, just like the Floquet states define the cycle-stationary states 10 of a periodically driven system. Analogously to Eq. (2.7), the evolution of a general initial state is encoded in the time evolution operator U(0,t), which has the well-known form U(0,t) = ∑ e−iEit |φi i hφi | i in the autonomous case. Periodicity of the Floquet spectrum Up to this point, a strict analogy between the eigenstates of an undriven system and the Floquet states of a periodically driven system holds. There is, however, an aspect of the Floquet picture that has no counterpart in the static case: The Floquet states |φi (t)i and their corresponding quasi-energies εi are not uniquely defined: If |φi (t)i is a Floquet state of H(t) (with periodicity T = 2π/ω) and εi is the corresponding quasi-energy, every pair |φi,n (t)i = einωt |φi (t)i ; εi,n = εi + nω (2.8) – for any integer n – is an equally valid choice of the Floquet state and its quasienergy. This equivalence relies on two facts: 1. When inserted into (2.3), |φi,n (t)i and εi,n lead to the same time evolution as |φi (t)i and εi , since e−iεi,nt |φi,n (t)i = e−i(εi +nω)t einωt |φi (t)i = e−iεit |φi (t)i . 2. |φi,n (t)i inherits the T -periodicity of |φi (t)i: |φi,n (t + T )i = einω(t+T ) |φi (t + T )i = e2inπ einωt |φi (t)i = |φi,n (t)i . As a consequence, the Floquet spectrum of quasi-energies is ω-periodic: Given a quasi-energy εi , εi + nω is likewise a quasi-energy. The corresponding Floquet states are identical, up to an additional periodic phase factor einωt . Formally speaking, one deals with equivalence classes {n ∈ Z : (|φi,n (t)i ; εi,n )} of equivalent pairs of Floquet states. Recognizing that exactly one quasi-energy of each class lies within the first Floquet zone [−ω/2, ω/2), it is convenient to represent the entire class by this element. Hence, with the convention of 11 considering exclusively quasi-energies from the first Floquet zone, one gets rid of the inherent ambiguity of the Floquet picture.3 When quasi-energies are restricted to the first Floquet zone, their number matches the dimension d = dim H of the Hilbert space, and the corresponding Floquet states form at all times an orthonormal basis of the Hilbert space H : hφi (t)|φ j (t)i = δi j , d ∑ |φi (t)i hφi (t)| = 1d (∀t). (2.9) i=1 This is again in complete analogy to the properties of eigenstates of an autonomous system. 2.1.2 Determining Floquet states and quasi-energies So far, we have not discussed how to calculate Floquet states and quasi-energies in practice. By inserting (2.3) into (2.1), and using (2.9), one obtains a defining equation for |φi (t)i and εi : [H(t) − i∂t ] |φi (t)i = εi |φi (t)i . (2.10) This equation is usually solved by Fourier-transforming it to the frequency domain. Due to the T -periodicity of both |φi (t)i and H(t), their Fourier expansions are discrete and contain only integer multiple frequencies of ω: |φi (t)i = H(t) = ∑ |φ̃i (k)i e ikωt with , k∈Z ∑ H̃(k) e ikωt 1 |φ̃i (k)i = T 1 with H̃(k) = T , k∈Z Z T 0 Z T 0 dt |φi (t)i e−ikωt . dt H(t) e−ikωt . (2.11) (2.12) Inserting these expressions into (2.10), and using the relation 1 T 3 This Z T 0 ei(k−k )ωt dt = δkk0 , 0 aspect of the Floquet picture resembles the Bloch picture of a spatially periodic (but time-independent) Hamiltonian H(x) = H(x + L) [67]. There, the Bloch theorem states that the eigenstates of H(x) are Bloch waves, characterized by a wave vector that is unique only up to multiple integers of the spatial frequency 2π/L. To recover uniqueness, one restricts wave vectors to the first Brillouin zone [−π/L, π/L). This analogy is no coincidence; in fact, the Bloch theorem is just the Floquet theorem applied to the stationary Schrödinger equation H(x)ψ(x) = Eψ(x), which is a linear differential equation with periodic coefficients, but this time in the spatial coordinate x instead of the time t. 12 one obtains ∑ k0 ∈Z H̃(k − k0 ) + δkk0 kω |φ̃i (k0 )i = εi |φ̃i (k)i (∀k ∈ Z). (2.13) If one merges all frequency components |φ̃i (k)i of the Floquet state into a single vector and regards H̃(k − k0 ) + δkk0 kω as the (k, k0 )-th block of an (infinite) matrix, Eq. (2.13) can be written in the form of an eigenvalue problem: .. . .. . H̃(0) + ω H̃(+1) H̃(+2) · · · H̃(−1) H̃(0) H̃(+1) H̃(−2) H̃(−1) H̃(0) − ω .. . .. . .. . .. . .. . |φ̃i (+1)i |φ̃i (+1)i · · · |φ̃i ( 0 )i = εi |φ̃i ( 0 )i . |φ̃i (−1)i |φ̃i (−1)i .. .. .. . . . (2.14) Note that each block of this matrix is an operator on the Hilbert space H . In combination, these blocks represents the Floquet Hamiltonian HF . An eigenstate of HF comprises all frequency components |φ̃i (k)i of a Floquet state and can therefore be used to reconstruct the state in the time domain. The eigenvalues of HF are the quasi-energies εi of H(t). They are real, because H̃(−k) = H̃(k)† follows from (2.12), implying that HF is Hermitian. To solve the eigenvalue equation (2.14) numerically, one has to truncate HF to finite size. To do so, one fixes a finite-dimensional basis in H and expresses every frequency component H̃(k) of the Hamiltonian in this basis. (If the dimension of H is not finite, this involves a first truncation step.) Then, one truncates HF to a finite number of frequency components {k ∈ Z : |k| ≤ M}, centered around k = 0. The resulting finite dimensional matrix can be diagonalized, e.g., by numerical means. Of course, one must check the result for convergence with respect to the frequency cut-off M. The influence of the driving parameters on M is discussed in the appendix of Ref. [68]. Noteworthy aspects of the eigenvalue problem (2.14) The reformulation of the original Floquet problem (2.10) as an eigenvalue equation entails several noteworthy aspects: I To begin with, it is again instructive to draw the connection to autonomous systems. There, the stationary states |φi i and the corresponding energies Ei 13 are determined by an eigenvalue problem as well – namely, they obey the eigenvalue equation H |φi i = Ei |φi i (2.15) of the (static) Hamiltonian H. If one inserts the static Hamiltonian into the Floquet eigenvalue equation (2.14), only the blocks on the diagonal appear, since H̃(k) = δ0,k H. Accordingly, the problem decouples into separate eigenvalue equations for each index k: (H + kω) |φ̃i (k)i = εi |φ̃i (k)i . (2.16) For k = 0, this is simply the static eigenvalue problem (2.15). Possible Floquet states are therefore |φ̃i (k)i = δ0,k |φi i, leading to static Floquet states |φi (t)i = |φi i in the time domain. The respective quasi-energies correspond to the energies, εi = Ei . But if one diagonalizes a different k-block instead, say k = 1, one has |φ̃i (k)i = δ1,k |φi i → |φi (t)i = eiωt |φi i and εi = Ei + ω, instead. This reflects the above discussed ambiguity of the Floquet picture, which can be overcome by restricting quasi-energies to the first Floquet zone. I The Floquet picture is semi-classical in the sense that the driven system itself is treated quantum-mechanically, but the driving field enters only via periodically time-dependent terms in the Hamiltonian. The quantum nature of the driving field is thus neglected. This is justified, as long as the modes of the driving field are highly populated. In fact, if one treats the field in a quantized fashion instead, and writes down the “dressed” Hamiltonian of system and field in the limit of large occupation numbers, one exactly reproduces the Floquet Hamiltonian HF [66, 69]. This motivates the interpretation of the Fourier index k in Eqs. (2.11) to (2.14) as the number of quanta exchanged with the driving field – despite the fact that the driving field is not quantized in the Floquet picture. I A further remark regards the relation between the eigenvalue equation (2.14) in the frequency domain and the original Floquet problem (2.10) in the time domain. Being connected via a Fourier transformation, the two equations are equivalent. In fact, both equations are different representations of the same eigenvalue problem. To elaborate on this point, we write the eigenvalue equation (2.14) in an abstract way: HF ||φ i ii = εi ||φ i ii . (2.17) Here, HF is the Floquet Hamiltonian, and the “double ket” ||φ i ii denotes the joint vector of all frequency components |φ̃i (k)i of a Floquet state. 14 Mathematically speaking, ||φ i ii is an element of the Floquet Hilbert space HF ≡ H ⊗ `2 , with `2 being the Hilbert space of complex-valued, square-summable sequences, and HF is a Hermitian operator on HF . But since `2 is isomorphic to the space L2 ([0, T )) of all T -periodic, square-integrable functions [70], it is equally justified to identify the Floquet Hilbert space HF with the space H ⊗ L2 ([0, T )) of all T -periodic trajectories in the original Hilbert space H . In this representation, the Floquet Hamiltonian reads HF = H(t) − i∂t ; the vector ||φ i ii corresponds to the periodic trajectory generated by the Floquet state |φi (t)i in the time domain; and Eq. (2.17) is tantamount to Eq. (2.10). In summary, the eigenvalue equation (2.17) should be regarded as the abstract Floquet problem, and Eq. (2.10) and Eq. (2.14) are its representations in the time and frequency domain, respectively. Table 2.1 summarizes the different representations of the Floquet picture. I The Floquet Hamiltonian HF in frequency representation, Eq. (2.14), may formally be interpreted as the time-independent Hamiltonian of a fictitious, infinite-dimensional quantum system. For this reason, all the tools that have been developed for the solution of the stationary Schrödinger equation (2.15) of autonomous systems can be applied to Eq. (2.14) – or, more generally, to the representation-independent eigenvalue equation (2.17). This argument allows us to make extensive use of time-independent perturbation theory [63] in Chapter 3. The concept is simple: Assume that the Floquet states ||φ j ii of some Floquet Hamiltonian HF are known. The impact of a weak perturbation VF is then determined, to first order, by the matrix elements Ci j = hhφ i ||VF ||φ j ii . (2.18) E.g., in complete analogy to autonomous systems, the unperturbed quasienergies εi of HF are shifted by an amount of Cii ; and if two unperturbed quasi-energies εi and ε j cross under the variation of a system parameter, the perturbation can lift this degeneracy and lead to an avoided crossing with minimal level separation 2|Ci j |. In the frequency domain, expression (2.18) reads Ci j = ∑ hφ̃i (k)|Ṽ (k − k0 )|φ̃ j (k0 )i ; k,k0 in the time domain, it is Ci j = 1 T Z T 0 dt hφi (t)|V (t)|φ j (t)i . (2.19) (V (t) and Ṽ (k) denote, respectively, the time and frequency representations of the perturbation.) ⊗ L2 ([0, T )) H 1 RT T 0 Hilbert space Scalar product ∑k hφ̃ (k)|χ̃(k)i H ⊗ `2 Eq. (2.14) hhφ i ||χ i ii HF Eq. (2.17) hφ |χi H Eq. (2.15) Ei ∈ R |φi i H Autonomous system Table 2.1: Summary of the important quantities of the Floquet picture. Besides the time and frequency notation, all expression are also listed in the representation-independent notation of Eq. (2.17). The corresponding quantities of an autonomous system are quoted in the last column, to highlight the analogy to the time-independent case. dt hφ (t)|χ(t)i Eq. (2.10) εi ∈ R ||φ i ii εi ∈ R |φ̃i (k)i εi ∈ R |φi (t)i HF H̃(k − k0 ) + δkk0 kω H(t) − i∂t Eigenvalue equation (Quasi-) energies (Quasi-) stationary states (Floquet) Hamiltonian Abstract formulation Frequency domain Time domain 15 16 Concluding remarks In summary, the central aspect of the Floquet picture is that the complete dynamics at arbitrary times t and for arbitrary initial conditions are available, once the Floquet states and the corresponding quasi-energies have been determined by solving an eigenvalue problem. In addition, most of the intuitive concepts developed for the autonomous case carry over to the Floquet picture, such as the notion of an energy spectrum, or time-independent perturbation theory. For these reasons, the Floquet picture has been applied to analyze all kinds of periodically driven quantum systems. Examples beyond the simplest case of a driven two-level system [58, 65, 66] are the rapid adiabatic passage technique in atomic physics [71–73], non-dispersive wave packets in Rydberg atoms [74], driven quantum tunneling [75], chaos in driven quantum systems [76–79], or driving-induced phase transitions, e.g., in semiconductor quantum wells [80] or Bose-Einstein condensates [13, 81]. 17 2.2 2.2.1 Open quantum systems Introduction to open quantum systems Throughout the last section, we assumed the quantum system at hand to be isolated from any uncontrolled, external influence. In this case, the time evolution is governed by the Schrödinger equation (2.1). Such a closed system, however, is an idealization that is never perfectly realized, since any quantum system inevitably interacts, to a greater or lesser extent, with surrounding degrees of freedom that are not under experimental control. The latter are usually lumped together under the notion of the environment. Accounting for the coupling to the environment, one speaks of an open quantum system. The most prominent consequence of the coupling to an environment is decoherence, i.e., the reduction of the quantum mechanical character of the system’s state, which is unparalleled in closed systems and can drastically corrupt entanglement properties. Being interested in the influence of driving on entanglement in the present thesis, it is thus essential to add a realistic description of decoherence to our investigation. For this reason, the basic theoretical concepts of open quantum systems are introduced in the present section, with a particular focus on periodically driven, open systems. More details are found in textbooks on open quantum systems [41, 82]. To set the stage, one has to specify the actual open system S and its environment E. The borderline between S and E is not always clear-cut, as one can always include parts of the environment in the definition of an enlarged system S0 with a smaller environment E0 , and vice versa. Reasonably, one draws the line depending on the question under study. E.g., if one is interested in (or has only access to) observables that are defined on a restricted set of particles of a many-body system, then it is reasonable to take this set as the open system S, and to consider the remaining particles as the environment. Mathematically speaking, the open system S and its environment E are defined via their respective Hilbert spaces HS and HE . The combined system S + E lives in the tensor product space HSE = HS ⊗ HE , and its state is described by a density operator ρSE [63, 83] from the space QSE of quantum states, which consists of linear operators on HSE with the properties † ρSE = ρSE , ρSE ≥ 0, The situation is sketched in Fig. 2.1. tr(ρSE ) = 1. (2.20) 18 Total system S+E HSE (t) = HS (t) ⊗ E + S ⊗ HE + Hint Environment E HE Hint Open system S HS (t) Figure 2.1: Sketch of an open quantum system. The open system S comprises the degrees of freedom that one is actually interested in, or that one has access to. It inevitably interacts with uncontrolled degrees of freedom in the environment E, via an interaction mechanism Hint . In general, all contributions to the total Hamiltonian HSE may be time-dependent. The case of interest in this thesis, however, is a periodically time-dependent system Hamiltonian HS (t), which comprises the driving fields, and time-independent contributions HE and Hint . Time evolution of open systems Since the total system S + E is closed, its time evolution is unitary, ρSE (t) = U(t,t0 )ρSE (t0 )U † (t,t0 ), (2.21) with the time evolution operator U(t,t0 ) obeying the Schrödinger equation for the Hamiltonian HSE (t) of the total system S + E, i∂t U(t,t0 ) = HSE (t)U(t,t0 ), U(t0 ,t0 ) = 1SE . In most cases, however, it is not practical to determine the time evolution of the open system by directly solving Eq. (2.21), since the environment comprises all conceivable degrees of freedom that interact with the system, rendering even the storage of the total state ρSE intractable, not to mention the calculation of its evolution. As we are only interested in the properties of S, however, it is not necessary to know the time evolution of the total system, since the reduced state4 ρS = trE (ρSE ) 4 tr E denotes the partial trace over the environment E, rendering ρS a density operator on HS . 19 alone suffices to determine the expectation value of any observable O on HS : hO ⊗ 1E iρSE = tr((O ⊗ 1E ) ρSE ) = trS (O ρS ). The goal in the following is, hence, to boil down the evolution equation (2.21) of the total system S + E to a more practical dynamical equation on the level of the reduced state ρS . The dynamical map By definition, the time-evolved density operator ρS (t) of the open system is ρS (t) = trE (ρSE (t)) = trE (U(t,t0 )ρSE (t0 )U † (t,t0 )). This does not yet define a unique mapping from the initial state of the open system ρS (t0 ) to the final state ρS (t), since ρS (t0 ) can be the reduced state of different initial states ρSE (t0 ) of the total system. Nevertheless, the open system is usually assumed5 to be uncorrelated with the environment at some initial time t0 : ρSE (t0 ) = ρS (t0 ) ⊗ ρE . (2.22) In this case, the assignment V (t,t0 ) ρS (t0 ) 7−→ ρS (t) = trE (U(t,t0 ) (ρS (t0 ) ⊗ ρE ) U † (t,t0 )), with a given initial state ρE of the environment, is unique and defines the two-parameter family {V (t,t0 )} of dynamical maps. The situation is depicted in Fig. 2.2. The dynamical maps V (t,t0 ) are completely positive and trace preserving (CPT), as required for a meaningful time evolution of a quantum state.6 They are, however, in general not invertible, in contrast to the unitary evolution of a closed quantum system. This introduces an “arrow of time” in the dynamics of the open system, which becomes manifest, e.g., in the contraction property of the trace distance7 k.k1 , kV (t,t0 )ρ(t0 ) −V (t,t0 )σ (t0 )k1 ≤ kρ(t0 ) − σ (t0 )k1 , 5 For t ≥ t0 , (2.23) a detailed discussion of this assumption, see Refs. [84, 85] and references therein. Complete positivity implies that not only V (t) is a positive map, but that any extension V (t) ⊗ 1 to larger Hilbert spaces is also positive [41]. 7 In operational terms, the trace distance between two quantum states corresponds to the probability of distinguishing the states by a measurement [27]. 6 20 Total system S+E ρSE (t0 ) = ρS (t0 ) ⊗ ρE U (t, t0 ) [�] U † (t, t0 ) trE [�] Open system S ρS (t0 ) ρSE (t) trE [�] V (t, t0 ) ρS (t) Figure 2.2: Construction of the dynamical map V (t,t0 ). To establish a one-to-one mapping between ρS (t0 ) and ρSE (t0 ), the initial state of the total system has to be uncorrelated. Only with this assumption can a dynamical map V (t,t0 ) be defined via the unitary evolution of the total system. Note that, in general, the inverse of the dynamical map does not exist. This underlines that the dynamics of open quantum systems is generically irreversible, in contrast to closed quantum systems. implying that two initial conditions ρ(t0 ) and σ (t0 ) can only get closer in the course of time [27]. If σ (t0 ) is in particular a fixed point of the dynamics, V (t,t0 )σ (t0 ) = σ (t0 ), any other state can only come closer to it as time evolves. An open system can therefore feature a globally attractive stationary state, into which all initial states evolve in the limit t → ∞ of asymptotic times. This is a qualitative difference to closed systems, which necessarily undergo unitary, and hence distance preserving dynamics. Markov assumption A crucial step in the derivation of a master equation for ρS (t) is the assumption of divisibility of the dynamical map: V (t + s, 0) = V (t + s,t) ◦V (t, 0) (∀ t, s ≥ 0). (2.24) This property resembles the Chapman–Kolmogorov equation defining classical Markov processes [41], inasmuch as it states that the evolution of the system in the time interval [t,t + s) is not conditioned on the entire trajectory in the preceding interval [0,t), but only on the state of the system at time t. This assumption is justified, if the environment is large and “unstructured”, in the sense that it contains a lot of excitable states of all energies (on the scales that are relevant in the system S), such that any excitation the system emits into the environment is unlikely to be reabsorbed. This is often the case in realistic open quantum systems, in particular in the quantum optical context [86]. We therefore assume throughout this thesis that the Markov property (2.24) holds; nonetheless, 21 one should be aware that a vivid discussion about the relevance of non-Markovian effects in open quantum systems has emerged recently [85, 87–89]. Autonomous systems Most textbooks focus on the case of autonomous open quantum systems, featuring a time-independent Hamiltonian HSE . In this case, one has U(t,t0 ) = e−iHSE (t−t0 ) , implying that the dynamical map V (t,t0 ) depends only on the difference t − t0 of the time arguments. One can therefore fix t0 = 0 to obtain a one-parameter family {V (t)} of dynamical maps. The divisibility property (2.24) reduces, in this case, to the semigroup property V (t + s) = V (t) ◦V (s) (∀ t, s ≥ 0). (2.25) Very generally, one can show (under certain mathematical conditions) that any family {V (t)} of CPT maps obeying this semigroup property is generated by a linear operator L that acts on the state space QS of the open system [90, 91]: V (t) = eL t , i.e., ρ̇S (t) = L ρS (t). (2.26) Moreover, L can always be written in Lindblad form [91] L ρS (t) = −i[H, ρS (t)] + d 2 −1 ∑ i=1 1 γi Li ρS (t)Li† − {ρS (t), Li† Li } , 2 (2.27) where d = dim(HS ) denotes the (finite) dimension of the system S and {A, B} = AB + BA is the anti-commutator. The quantities appearing in the Lindblad generator L have the following properties: I H is a self-adjoint operator on HS that generates the coherent evolution of the system; I Li denote the so-called Lindblad operators, responsible for the incoherent dynamics; they can always be chosen traceless and orthonormal, i.e., tr(Li ) = 0 and tr(Li† L j ) = δi j [92]; I γi ≥ 0 are non-negative rates that pertain to each Lindblad operator Li . 22 Vice versa, any operator of the form (2.27) that fulfills these three properties generates, via Eq. (2.26), a family of CPT maps that has the semigroup property (2.25). The incoherent part of the generator L is sometimes referred to as the dissipator D(ρS ): D(ρS ) ≡ d 2 −1 ∑ i=1 1 † † γi Li ρS Li − {ρS , Li Li } . 2 (2.28) Explicitly time-dependent systems In contrast to the autonomous case, we are concerned with periodically driven systems in this thesis, described by a time-dependent Hamiltonian HSE (t). In this case, the Markov assumption is not reflected by the semigroup property (2.25), but by the divisibility property (2.24). Nevertheless, all statements discussed above for autonomous systems can be transferred to the general, time-dependent case [85]. Namely, {V (t,t0 )} is a family of CPT maps with the divisibility property (2.24), if and only if it is generated by a time-dependent L (t): h Rt i L (t 0 )dt 0 V (t,t0 ) = T← e t0 , i.e., ρ̇S (t) = L (t)ρS (t). Here, T← denotes the time-ordering operation, and L (t) is a generator that is at all times of the Lindblad form (2.27). Consequently, all quantities appearing in Eq. (2.27) become time-dependent now, but they still have – at every instance t – the three previously listed properties. This implies, in particular, non-negative rates γi (t) at all times. In conclusion, the most general Markovian8 master equation for the evolution of an open quantum system is the time-dependent Lindblad equation ρ̇S (t) = −i[H(t), ρS (t)] + 2.2.2 d 2 −1 ∑ i=1 1 γi (t) Li (t)ρS (t)Li† (t) − {ρS (t), Li† (t)Li (t)} . 2 (2.29) The Floquet Born-Markov master equation To fill the general, abstract form (2.29) of a Markovian master equation with life, one has to specify a model for system and environment, and derive from it concrete expressions for the quantities H(t), Li (t) and γi (t). 8 If one regards the divisibility property (2.24) as definition of a Markovian quantum evolution. 23 If the coupling between system and environment is much weaker than the typical energy scales of the bare system, the standard approach is to derive the Born-Markov (or weak coupling) master equation [41, 82, 86, 93]. This requirement is typically fulfilled in quantum optical systems [86] and many solid-state realizations of a qubit [94, 95]. In the literature, the derivation of the Born-Markov master equation is often discussed for autonomous systems only. Based on the Floquet picture, however, it can be generalized to periodically driven systems in a straight-forward manner [41, 77, 79, 92, 96–98]. Since Section 4.1 builds on the resulting Floquet Born-Markov master equation, we sketch its derivation in the following. The various approximations and assumptions along the way are exactly the same as in the autonomous case. They are discussed in great detail in Refs. [86, 93]. Setup For our purposes, the system S is a periodically driven, finite-dimensional quantum system, described by a Hamiltonian HS (t) = HS (t + T ). E.g., the case of interest in Chapter 4 are N interacting qubits, driven by an external field of frequency ω, in which case we have d = dim HS = 2N and T = 2π/ω. The environment E is modeled by a collection of harmonic oscillators that provide a continuous range of different frequencies ωn , as described by the Hamiltonian HE = ∑ ωn (a†n an ). (2.30) n (Here, a†n denotes the creation operator of the n-th oscillator.) In order to mimic a heat bath for the system S, we assume all oscillators to be in the thermal state (or Gibbs state) [99] at inverse temperature β :9 ρE = O (n) e−β HE = ρE , −β H E tr e n † (n) with ρE = 1 − e−β ωn e−β ωn an an . (2.31) Apart from the system and environment Hamiltonians HS (t) and HE , the interaction Hint between system and environment contributes to the total Hamiltonian HSE , see Fig. 2.1. A reasonable model that applies to a wide range of physical scenarios is a linear coupling Hint = X ⊗ ∑ gn (a†n + an ) (2.32) n 9 The inverse temperature is defined as β = (kB T )−1 , kB denoting the Boltzmann constant and T the temperature. We prefer to work with β instead of the temperature itself in this thesis, simply because we want to reserve the letter T for the driving period 2π/ω. 24 between the amplitudes a†n + an of the oscillators and a (Hermitian) system operator X, with a (real-valued) coupling strength gn . E.g., if the open system S is a single two-level atom, and the environment is the continuum of electromagnetic √ modes, then X is the dipole operator of the atomic transition, and gn ∝ ωn [86]. For clarity, we consider only a single system observable X that couples to the environment in the following derivation and generalize to multiple coupling operators Xm at the end of this section. Altogether, the total Hamiltonian of system and environment reads HSE (t) = HS (t) ⊗ 1E + 1S ⊗ HE + Hint . (2.33) Interaction picture Since the interaction Hint is by assumption the weakest contribution to the total Hamiltonian, it is advantageous to switch to the interaction picture. The state ρSE of the total system and the interaction Hamiltonian transform like10 ) ρ̃SE (t) = U0† (t)ρSE (t)U0 (t) ˙ SE (t) = −i H̃int (t), ρ̃SE (t) , ⇒ ρ̃ (2.34) H̃int (t) = U0† (t)HintU0 (t) where U0 (t) is the time evolution operator of the uncoupled system S + E, fulfilling U0 (0) = 1SE , with i∂t U0 (t) = H0 (t)U0 (t), H0 (t) = HS (t) ⊗ 1E + 1S ⊗ HE . Accordingly, U0 (t) is the tensor product of the time evolution operators of the bare system and environment, U0 (t) = US (t) ⊗UE (t), defined by h Rt i US (t) = T← e−i 0 HS (τ)dτ , † UE (t) = e−iHEt = e−iωn an ant . A more practicable form of US (t,t0 ) is based on the Floquet states |φi (t)i and quasienergies εi of the periodic system Hamiltonian HS (t), as derived in Eq. (2.7): US (t) = ∑ e−iεit |φi (t)i hφi (0)| . (2.35) i With this, the transformation of the interaction Hamiltonian (2.32) to the interaction picture reads H̃int (t) = X̃(t) ⊗ ∑ gn (a†n e−iωnt + an eiωnt ), n 10 We fix, for simplicity, the initial time of the transformation to t0 = 0. (2.36) 25 with X̃(t) = ∑ ei(εi −ε j )t |φi (0)i hφ j (0)| hφi (t)|X|φ j (t)i . (2.37) i, j It is convenient to employ the Fourier representation |φ̃i (k)i of the Floquet states in the following derivations, see Eq. (2.11). Defining the Floquet transition amplitudes Z T 1 xi j (k) = ∑ hφ̃i (k )|X|φ̃ j (k + k)i = T k0 0 0 0 dt hφi (t)|X|φ j (t)i e−ikωt (2.38) (where ω = 2π/T is the driving frequency of the system Hamiltonian HS (t)), one can then introduce the transition operators X̃(Ω) = ∑ δΩ,ε i, j,k i j +kω xi j (k) |φi (0)i hφ j (0)| , (2.39) (where δx,y denotes the Kronecker symbol and εi j ≡ εi − ε j ), which correspond to frequency representation of X̃(t): X̃(t) = ∑ eiΩt X̃(Ω) = ∑ e−iΩt X̃ † (Ω). Ω (2.40) Ω The sum runs over all possible transition quasi-energies Ω = εi j +kω of the system (with integer k).11 Born approximation To proceed, we formally integrate the evolution equation (2.34) of the total system: ρ̃SE (t) = ρSE (0) − i Z t 0 [H̃int (t 0 ), ρ̃SE (t 0 )]dt 0 . Reinserting this expression into (2.34) and tracing out the environment E leads to Zt ρ̃˙ S (t) = −i trE [H̃int (t), ρSE (0)] − trE [H̃int (t), [H̃int (t 0 ), ρ̃SE (t 0 )]] dt 0 . (2.41) 0 Assuming that system and environment are uncorrelated at the initial time t0 = 0, as expressed by Eq. (2.22), the first term vanishes, because tr(ρE an ) = tr(ρE a†n ) = 0 holds for the thermal state ρE of Eq. (2.31):12 trE [H̃int (t), ρSE (0)] = [X̃(t), ρS (0)] ∑ gn tr ρE a†n e−iωnt + an eiωnt = 0. n 11 E.g., in the context of driven atoms, the transition with k = +1 (k = −1) describes the first blue (red) sideband transition [86]. 12 In fact, one can always eliminate the first term of Eq. (2.41), independently of the interaction model or the environment state ρE , by suitably shifting the contributions HS (t) and Hint in the total Hamiltonian (2.33); see Section 6.6 in Ref. [93]. 26 Based on the premise that the coupling between system and environment is weak, it is reasonable to assume that the correlations between system and environment at later times 0 ≤ t 0 ≤ t do not play a crucial role for the evolution of ρS , either. Moreover, since the environment E is much larger than the system S, its state is hardly affected by the presence of the system S and remains in the thermal state (2.31) at all times. We may therefore set ρ̃SE (t 0 ) ≈ ρ̃S (t 0 ) ⊗ ρE in the second term of Eq. (2.41). Using the form (2.36) of Hint , this leads to: ρ̃˙ S (t) = Z t 0 dt 0 K(t,t 0 ) X̃(t 0 )ρ̃S (t 0 )X̃(t) − X̃(t)X̃(t 0 )ρ̃S (t 0 ) + h.c. (2.42) Here, h.c. denotes the Hermitian conjugate, and we have introduced the bath correlation function 0 0 K(t,t 0 ) = ∑ gn gn0 e−i(ωn0 t −ωnt) tr a†n0 an ρE + ei(ωn0 t −ωnt) tr an0 a†n ρE . (2.43) n,n0 The occurring second order correlations of the bath operators are tr a†n0 an ρE = δnn0 Nth (ωn ) and tr an0 a†n ρE = δnn0 [Nth (ωn ) + 1], where Nth (ωn ) denotes the mean thermal occupation number of an oscillator of frequency ωn at inverse temperature β : Nth (ωn ) = 1 eβ ωn −1 = −[Nth (−ωn ) + 1]. (2.44) Consequently, the double sum over n and n0 in Eq. (2.43) collapses to a single sum, and the bath correlation function depends on the time difference τ = t − t 0 only: K(t,t 0 ) ≡ K(τ) = ∑ g2n eiωn τ Nth (ωn ) − e−iωn τ Nth (−ωn ) . (2.45) n Substituting the integration variable t 0 by τ in Eq. (2.42), we find ρ̃˙ S (t) = Z t 0 dτ K(τ) X̃(t − τ)ρ̃S (t − τ)X̃(t) − X̃(t)X̃(t − τ)ρ̃S (t − τ) + h.c. (2.46) Markov approximation The different phase factors in K(τ) average out quickly as τ grows, since a sufficiently large environment contains all possible frequencies ωn .13 Therefore, 13 More precisely, the relevant assumption at this point is that the spectral density of the bath is sufficiently smooth over the entire frequency range [93]. 27 the correlation function K(τ) decays to zero on a short time scale tcorr , and only times τ < tcorr contribute to the integration in Eq. (2.46). Assuming that the time resolution with which we monitor the evolution of ρS does not resolve the short time scale tcorr , we can safely replace ρ̃S (t − τ) by ρ̃S (t) in the integration. With the same argument, we can furthermore push the upper integration limit from t to +∞. Inserting the frequency representation (2.40) of X̃(t), one arrives at ρ̃˙ S (t) = ∑ W (Ω)ei(Ω−Ω )t 0 Ω,Ω0 X̃(Ω)ρ̃S (t)X̃ † (Ω0 ) − X̃ † (Ω0 )X̃(Ω)ρ̃S (t) + h.c., (2.47) where W (Ω) denotes the one-sided Fourier transform of the correlation function: W (Ω) = Z ∞ 0 dτ e−iΩτ K(τ). (2.48) Secular approximation Finally, one exploits that the summation over Ω and Ω0 involves only the discrete transition quasi-energies Ω = εi − ε j + kω of the system Hamiltonian HS (t), see Eq. (2.39). Since we assume a finite-dimensional system S, different Ω, Ω0 are not arbitrarily close. Therefore, we have (Ω−Ω0 )t 1 already for short times t, so that the phase factors in Eq. (2.47) average out quickly unless Ω = Ω0 . Accordingly, in a secular approximation, one keeps only terms with Ω = Ω0 : ρ̃˙ S (t) = ∑ W (Ω) X̃(Ω)ρ̃S (t)X̃ † (Ω) − X̃ † (Ω)X̃(Ω)ρ̃S (t) + h.c. (2.49) Ω Standard form Next, we split the one-sided Fourier transform W (Ω) into real and imaginary part: 1 W (Ω) = Γ(Ω) + i∆(Ω) 2 (2.50) Inserting Eq. (2.45) into Eq. (2.48), and exploiting that 0∞ dτ e−i(Ω±ωn )τ = πδ (Ω ± 1 ωn ) − iP Ω±ω , with P denoting the Cauchy principle value, one finds n R Γ(Ω) = ∑ 2πg2n (Nth (ωn )δ (Ω − ωn ) − Nth (−ωn )δ (Ω + ωn )) n 1 1 2 ∆(Ω) = − ∑ gn Nth (ωn )P − Nth (−ωn )P Ω − ωn Ω + ωn n 28 Furthermore, we replace the summation over all environment modes n by an integration over all frequencies ω of the environment: ∑ −→ n Z ∞ 1 2π 0 dω S(ω) (2.51) Here, the spectral density S(ω) indicates the number of modes (per frequency interval) at a certain frequency. Using that |Nth (Ω)| = Nth (Ω) for positive Ω, and |Nth (Ω)| = −Nth (Ω) = Nth (|Ω|) + 1 for negative Ω, as derived from Eq. (2.44), one finds Γ(Ω) = S(|Ω|)[g(|Ω|)]2 |Nth (Ω)|, Z ∞ dω ∆(Ω) = P S(|ω|)[g(|ω|)]2 |Nth (ω)|. −∞ 2π(ω − Ω) (2.52) (2.53) Inserting Eq. (2.50) into Eq. (2.49) and going from the interaction picture back to the original Schrödinger picture, one finally arrives at the desired master equation in Lindblad form: ρ̇S (t) = − i[HS (t) + HLS (t), ρS (t)] + ∑ Γ(Ω) X(Ω,t)ρS (t)X † (Ω,t) Ω 1 † − X (Ω,t)X(Ω,t), ρS (t) . 2 (2.54) Here, we have defined the Lamb shift Hamiltonian HLS , † (t), HLS (t) = ∑ ∆(Ω) X † (Ω,t)X(Ω,t) = HLS Ω and the transition operators X(Ω,t) in the Schrödinger picture, (2.39) X(Ω,t) = US (t)X̃(Ω)US† (t) = ∑ δΩ,ε i, j,k i j +kω xi j (k)e−εi j t |φi (t)i hφ j (t)| . (2.55) Pauli rate equation The Floquet Born-Markov master equation (2.54) allows an intuitive interpretation, if one expresses ρS in the adequate basis, namely, the Floquet basis. Hence, we rewrite the master equation in terms of the matrix elements ρi j (t) = hφi (t)|ρS (t)|φ j (t)i . 29 Inserting the definition (2.55) of the transition operators X(Ω,t) into Eq. (2.54), the evolution equation of the diagonal elements ρii (t) (the populations) reads ρ̇ii (t) = ∑ ∑ Γ(Ω)δΩ,εl j +kω δΩ,ε Ω l, j,k l, j,k l j +kω 1 xl j (k)xl∗j (k)(δil δli ρ j j − δll (δ ji + δi j )ρ j j ). 2 Assuming that the transition quasi-energies εl j + kω of the system are non-degenerate, i.e., that εl j + kω = εl j + kω implies l = l, j = j, and k = k, this expression collapses to: ρ̇ii (t) = ∑ Γ(εl j + kω)|xl j (k)|2 (δil ρ j j − δi j ρ j j ). l, j,k Defining the total transition rate γi j from the i-th to the j-th Floquet state, γi j = ∑ Γ(ε ji + kω)|x ji (k)|2 , (2.56) k one obtains the simple Pauli rate equation for the populations: ρ̇ii (t) = ∑(γ ji ρ j j (t) − γi j ρii (t)). (2.57) j The evolution of coherences ρi j (with i 6= j) is derived analogously, and reads # " 1 (2.58) ρ̇i j (t) = − i(εi j + ∆i j ) + ∑(γil + γ jl ) ρi j (t), 2 l with ∆i j = ∑k |x ji (k)|2 ∆(ε ji + kω), and ∆(ε ji + kω) given by Eq. (2.53). The influence of the environment on the dynamics of the open system becomes apparent now: Equation (2.58) reveals that the coherences ρi j (t) between i-th and j-th Floquet state oscillate at a frequency that is modified by the Lamb shift ∆i j , compared to the plain quasi-energy difference εi j (which defines the oscillation frequency in the absence of an environment). The second, more serious consequence of the environment coupling is that the coherences decay in time, with a decay rate 21 ∑l (γil + γ jl ). Equation (2.57), on the other hand, implies that the populations ρii of the different Floquet states increase or decrease due to environment-induced transitions from or to other populations ρ j j . To understand the factors that govern this diffusion process, we recapitulate the definition of the transition rates γi j : γi j (2.56),(2.52) = ∑[g(|ε ji + kω|)]2 S(|ε ji + kω|) |x ji (k)|2 | Nth (ε ji + kω)|. k (2.59) 30 I g(|ε ji + kω|) is the coupling strength between the system and the oscillators of the environment that match the transition quasi-energy ε ji + kω; I S(|ε ji + kω|) is the spectral density of the environment at the transition quasi-energy, reflecting the number of available modes that can trigger the transition; I |Nth (ε ji + kω)| is the thermal occupation number of the oscillators that match the transition quasi-energy; at zero temperature (β → ∞), |Nth (−x)| corresponds to the Heaviside step function θ (x); accordingly, transitions from lower to higher quasi-energy are suppressed in this limit; I x ji (k) is the transition amplitude, reflecting the ability of the operator X – which couples the system to the environment – to mediate between the Floquet states |φi (t)i and |φ j (t)i. The summation over the Fourier index k in Eq. (2.59) is the only essential difference to the “usual” Born-Markov master equation for autonomous systems. Accordingly, in the autonomous case, the transition rates merely depend on the energy differences E j − Ei between the eigenstates |φi i and |φ j i of the (static) system Hamiltonian HS , instead of the transition quasi-energy ε ji + kω. Likewise, the transition matrix element is simply xi j = hφi |X|φ j i in the autonomous case and does not depend on the Fourier index k. Thus, a periodically driven system additionally allows the environment to trigger sideband transitions (with k 6= 0), which are forbidden in the respective undriven system. Due to these additional transitions, the environment is typically more effective in adding noise to the time evolution of the open system. Asymptotic cycle Joining the populations ρii (t) to a probability vector ~w(t), the rate equation (2.57) describes a time-continuous, autonomous (classical) Markov process for ~w(t): ~w˙ (t) = M~w(t). (2.60) The process is governed by a stochastic matrix14 M, defined by its elements (M)i j = γ ji − δi j ∑ γil . (2.61) l 14 Strictly speaking, M is not a stochastic matrix, but M + 1 is. Hence, the Perron-Frobenius theorem [100] guarantees that the largest eigenvalue of M is 0. 31 If M has just one vanishing eigenvalue, the system is ergodic, i.e., all initial conditions ~w(0) eventually evolve into a unique stationary distribution ~w∗ . Namely, ~w∗ is the eigenvector of M that is attributed to the vanishing eigenvalue. A sufficient condition for ergodicity is, e.g., that all γi j are strictly positive. Since all coherences ρi j decay to zero in this case, cf. (2.58), any initial state ρS (0) eventually evolves into the unique, T -periodic cyclo-stationary state (or asymptotic cycle) ρac (t) ≡ ρS (t → ∞) = ∑ w∗i |φi (t)i hφi (t)| . (2.62) i Note that this state is diagonal in the Floquet basis, which is entirely determined by the system Hamiltonian HS (t) alone. Accordingly, the details of the environment can merely alter the stationary weights w∗i , via their impact on the transition rates γi j . In autonomous systems, the stationary state fulfills the detailed balance condition γ ji |Nth (E j − Ei ) + 1| p∗i = = = e−β (Ei −E j ) , ∗ pj γi j |Nth (E j − Ei )| implying that the stationary state is the Gibbs state [99] of the (static) system Hamiltonian HS , i.e., e−β HS . (2.63) ρS (t → ∞) = tr(e−β HS ) In the periodically driven case, however, the stationary weights are no longer Boltzmann-distributed [92], due to the summation over the Fourier index k in Eq. (2.59). The asymptotic cycle of a periodically driven system can therefore be qualitatively different from the stationary state of the respective undriven system. Multiple operators Xm coupling to the environment To conclude, we consider the situation of having several system observables Xm that couple to the environment, instead of the single operator X. Hence, instead of Eq. (2.32), we have Hint = ∑ gm,n Xm ⊗ (a†n + an ). m,n In the scope of this thesis, e.g., we deal with multi-partite systems consisting of N separate subsystems, each of which couples to the environment via its own operator Xm , i.e., the index m runs over the subsystems 1, . . . , N. One must distinguish between two different extremal situations [92, 102], sketched in Fig. 2.3: One possibility is that all atoms are coupled collectively (i.e., in exactly the same way) 32 Figure 2.3: Illustration of collective coupling to a common environment (left) vs. individual coupling to separate environments (right). The quantum system S itself consists of multiple (here: two) subsystems (red); the environment E is built up of harmonic oscillators (blue). Throughout this thesis, we consider the situation of separate environments, which is the more conservative choice for our questions, since separate baths affect entanglement most negatively [101]. (E.g., they prevent the existence of decoherence-free subspaces, see main text.) to the environmental modes, so that gm,n = gn . Defining the collective coupling operator X = ∑m Xm , one then effectively recovers the original situation of a single coupling operator in Eq. (2.32). Thus, the only difference in the final formulae is the definition of the transition rates γi j , which turns into γi j = ∑ Γ(ε ji + kω)| k N ∑ x ji (m) (k)|2 , (2.64) m=1 with xi j (k) = ∑ hφ̃i (k0 )|Xm |φ̃ j (k0 + k)i = (m) k0 1 T Z T 0 dt hφi (t)|Xm |φ j (t)i e−ikωt . Due to the high symmetry of the environment coupling, this case is often connected to the emergence of decoherence-free subspaces [103, 104]. Perfectly identical couplings, however, are rather artificial and difficult to realize in an experiment. The more natural situation is that each subsystem couples differently to the environment – e.g., because the subsystems are separated in space by a distance that allows the environment to distinguish them. For simplicity, we model this situation with subsystems that are coupled to entirely different environments. Hence, every subsystem “sees” a different continuum of oscillators, each of which we have to label by two indices n (for 33 oscillators) and m (for subsystems) now: Hint = N ∑ m=1 † Xm ⊗ ∑ gn,m (an,m + an,m ) . (2.65) n This is similar to the original definition (2.32), but involves an additional summation over the subsystems. Again, in the final master equation, this merely modifies the definition of the transition rates γi j : γi j = ∑ Γ(ε ji + kω) k N ∑ |x ji (m) (k)|2 . (2.66) m=1 Here, the only difference to the result (2.64) for collective environment coupling is that the transition amplitudes attributed to the different coupling operators Xm are summed up incoherently, i.e., after taking the absolute value. 2.2.3 Master equation with fixed dissipator In the Born-Markov approach, the Lindblad operators Li and the rates γi of the abstract Lindblad master equation (2.29) are derived, within several approximations, from a microscopic model. Alternatively, it is common use to simply fix Li and γi “ad hoc”, without a rigorous microscopic derivation (see, e.g., Refs. [34, 105, 106]). The Hamiltonian H(t) in the abstract Lindblad equation is then usually identified with the system Hamiltonian HS (t). Thus, in contrast to the Born-Markov equation, the dissipator D(ρ) of the master equation is independent of HS (t) in this ansatz. We adopt this point of view in Sections 4.2 and 4.3. In the following, we discuss its range of validity and its relation to the Born-Markov equation. Consider, first, the open system S to be a single qubit, HS = C2 . Identifying the computational basis states |0i and |1i with the eigenstates of the Pauli operator σz , a common choice of the Lindblad operators is L1 = σ− , L2 = σ+ , with σ− = (σ+ )† = |0i h1|. Accordingly, the Lindblad master equation (2.29) reads then ρ̇S (t) = −i[HS (t), ρS (t)] + γ− (σ− ρS (t)σ+ − {σ+ σ− , ρS (t)}) + γ+ (σ+ ρS (t)σ− − {σ− σ+ , ρS (t)}). (2.67) This choice is reasonable from a phenomenological point of view: The first incoherent term describes decay from |1i to |0i with rate γ− , the second one the incoherent excitation from |0i to |1i with rate γ+ . In the language of nuclear magnetic resonance (NMR), these incoherent processes describe the amplitude damping [107] of the qubit. 34 If the system Hamiltonian describes a bare two-level system, i.e., if HS = ω0 σz ≡ H0 , 2 the master equation (2.67) coincides15 with the Born-Markov master equation (2.54), because σ+ and σ− precisely induce transitions between the eigenstates |0i and |1i of H0 . In the microscopic description, the incoherent rates γ± attributed to the Lindblad operators σ± are given by Eq. (2.59) and read: γ+ = [g(ω0 )]2 S(ω0 )| h1|X|0i |2 Nth (ω0 ), γ− = [g(ω0 )]2 S(ω0 )| h0|X|1i |2 (Nth (ω0 ) + 1). (2.68) (2.69) Accordingly, any choice of the rates γ+ and γ− is in agreement with a microscopic model at some suitable temperature, as long as it fulfills γ+ ≤ γ− . E.g., γ+ = 0 corresponds to zero temperature (β → ∞), whereas γ+ = γ− marks the limit of infinite temperature (β → 0). The microscopically derived Born-Markov description, on the other hand, is valid as long as γ− ω0 . (2.70) (Due to γ+ < γ− , this implies γ+ ω0 as well.) Beyond this regime, both the weak coupling approximation and the secular approximation break down [93]. However, the condition γ− ω0 is very well satisfied in both atomic and solid-state qubits (cf. Table 1.1). The difference between the two approaches comes to light as soon as additional coherent processes are present. E.g., consider additionally to H0 a driving field HF (t) of amplitude F and frequency ω: HS (t) = H0 + HF (t) = ω0 σz + F cos(ωt)σx . 2 (2.71) To keep the discussion as transparent as possible, we assume resonant driving with ω = ω0 . Hence, the driving term HF (t) induces Rabi oscillations of frequency F between |0i and |1i [108]. The phenomenological approach to describe the open system dynamics in this case is to stick to the master equation (2.67), and to simply add HF (t) to the coherent part of the evolution. The Lindblad operators σ± are kept fixed, despite the additional driving term. The idea is thus to treat the incoherent evolution and the influence of the driving field as independent extensions to the “free” dynamics induced by H0 . This approach is therefore referred to as the approximation of independent rates [86]. It is justified 15 Apart from the Lamb shift HLS , which is usually of negligible magnitude. 35 as long as the original system Hamiltonian H0 dominates both the driving term HF (t) and the incoherent dynamics, i.e., if F ω0 , γ− ω0 . (2.72) Note, however, that there is no constraint on the relative magnitude of driving strength F and decay rate γ− , i.e., the Rabi frequency F can be smaller, larger, or equal to the decay rate γ− . This is different for the Floquet Born-Markov equation (2.54): Since the Floquet states of the driven system Hamiltonian (2.71) do no longer coincide with the computational basis states |0i and |1i, the transition operators X(Ω,t) in Eq. (2.54) are no longer identical to the Lindblad operators σ± of the phenomenological master equation (2.67). For weak driving (F ω0 ) the two quasi-energies εi of HS (t) read:16 ε1 = 1 (ω0 + F) , 2 ε2 = 1 (ω0 − F) 2 ⇒ ε12 = F. (2.73) Hence, the smallest difference between two different transition energies is of the order of F. Therefore, if F is of the order of (or smaller than) any of incoherent rates γi j , the secular approximation fails, and the Floquet Born-Markov master equation becomes invalid. The assumption F ω0 underlying the explicit expressions (2.73), on the other hand, is not crucial for the validity of the master equation. In fact, in this work, we will consider strongly driven systems with F of the order of ω0 or greater. In this case, the minimal difference of transition energies may then be limited by ω0 . In conclusion, the validity of the Floquet Born-Markov master equation for the Hamiltonian (2.71) is bound to the condition that the coherent energy scales F and ω0 dominate over the incoherent rates γi j : γi j F, γi j ω0 . (2.74) This time, there is no constraint of the relative magnitude of ω0 and F. In conclusion, conditions (2.72) for the phenomenological master equation and (2.74) for the Floquet Born-Markov equation show that both equations are justified in different parameter regimes.17 The same discussion holds for any other coherent processes that may be included in the system Hamiltonian HS (t). Consider, e.g., a prototypical interaction of strength J between two qubits: HS = H0 ⊗ 1 + 1 ⊗ H0 + J(σ+ ⊗ σ− + σ− ⊗ σ+ ). 16 This is derived within the rotating wave approximation, see Refs. [68, 86] or Section 3.2.2. do coincide, however, if ω0 F γ− , if one fixes γ− = γ12 and γ+ = γ21 . 17 They 36 The interaction induces a coherent oscillation between the computational states |0i ⊗ |1i and |1i ⊗ |0i with frequency 2J. Again, the phenomenological approach is to simply insert this Hamiltonian into the coherent part of the master equation (2.67), assuming that the incoherent processes and the interaction-induced oscillations are independent.18 This approximation is valid, as long as both of these processes happen on time scales much larger than the “free” dynamics induced by H0 , i.e., if J ω0 , γ− ω0 . (2.75) The validity of the Born-Markov equation for the same interacting system of two qubits, on the other hand, relies on the condition that the transition energies are well-separated (on the scale of the incoherent rates). Since the spectrum of eigenvalues of HS is {±ω0 , ±J}, this is tantamount to γi j J, γi j ω0 . (2.76) In summary, the approximation of independent rates that underlies the phenomenological master equation (2.67) is justified if the system Hamiltonian HS (t) contains a dominant part H0 , and all other coherent and incoherent processes are weak compared to it.19 The (Floquet) Born-Markov master equation, on the other hand, is valid if the incoherent rates are much smaller than all coherent energy scales. The existence of a dominant coherent energy scale is not required in this case. Thus, depending on the different energy scales of the system, both equations have their respective regime of validity. 18 Of course, the phenomenological equation (2.67) must be adjusted in the case of two qubits; instead of the single qubit Lindblad operators σ± , the operators σ± ⊗ 1 and 1 ⊗ σ± describe the (individual) spontaneous decay and excitation of two qubits. 19 The existence of a dominant part H is often obscured, because one commonly works in the 0 interaction picture with respect to H0 , so that H0 does not explicitly appear in the system Hamiltonian. The incoherent part of the phenomenological master equation is invariant under the transformation to the interaction picture, because of eiH0 t σ± e−iH0 t = e±iω0 t σ± . 37 2.3 Entanglement theory After the description of periodically driven, possibly open quantum systems in the previous sections, we turn to the second fundamental ingredient of this thesis: the theory of quantum entanglement. In the following, the definition of entanglement is discussed for both pure and mixed states, and viable ways to quantify entanglement are presented. A more detailed introduction to the subject can be found in Refs. [109–111]. 2.3.1 Definition of entangled states Entanglement is a property that is shared between the individual parts (or subsystems) of a composite quantum system. Accordingly, the Hilbert space H of interest is the tensor product of Hilbert spaces Hi ascribed to the individual parts: H = H1 ⊗ H2 ⊗ · · · ⊗ HN . (2.77) Here, we always consider systems of finite dimension di = dim Hi . Pure state entanglement For a start, consider a pure state |ψi in a bi-partite Hilbert space H = H1 ⊗ H2 . Pure states represent the best possible knowledge one can obtain of the system – in contrast to mixed states, which correspond to a probabilistic mixture of pure states and describe a certain portion of ignorance. The analogue of a pure state in classical mechanics is a perfectly localized distribution in the phase-space of the composite system. In this case, also the (marginal) distributions of the individual parts are perfectly localized and in this sense “pure”. To say it in Schrödingers words: in the classical world, “the best possible knowledge of a whole [. . . ] includes the best possible knowledge of all its parts” [1]. For the pure quantum state |ψi, however, this is not always the case; here, the individual parts are described by the reduced density operators ρ1 and ρ2 , obtained by partially tracing over one subsystem: ρ1 = tr2 |ψi hψ| , ρ2 = tr1 |ψi hψ| . (2.78) If |ψi is a tensor product composed of two states |φ i ∈ H1 and |χi ∈ H2 of the individual parts, the reduced states are pure: |ψi = |φ i ⊗ |χi ⇔ ρ1 = |φ i hφ | , ρ2 = |χi hχ| . (2.79) 38 Conversely, however, if |ψi cannot be written as a single such tensor product, the reduced states are not pure. This situation defies our classical intuition, insofar as the best possible knowledge of a whole does no longer include the best possible knowledge of all its parts. This is, in fact, the defining trait of entanglement. Accordingly, a pure, bi-partite state is by definition entangled, if and only if it cannot be expressed as a tensor product state. Otherwise, it is called separable. This definition is easily extended to the multi-partite case of N > 2 subsystems: |ψi ∈ H is entangled, if and only if it cannot be written like |ψsep i = |φ1 i ⊗ |φ2 i ⊗ · · · ⊗ |φN i . (2.80) With this definition, another trait of entanglement becomes apparent: Separable, pure states can never lead to correlations between measurements on the different parts. E.g., for any two local observables A1 ⊗ 1 and 1 ⊗ A2 of a bi-partite quantum system, we have hψsep |[(A1 ⊗ 1)(1 ⊗ A2 )]|ψsep i = hφ1 |A1 |φ1 i hφ2 |A2 |φ2 i = hψsep |(A1 ⊗ 1)|ψsep i hψsep |(1 ⊗ A2 )|ψsep i , i.e., the measurement outcomes of A1 and A2 are uncorrelated.20 Entangled states, on the other hand, can induce correlations between measurements on different subsystems. These correlations can even become incompatible with any local hidden-variable model [20]; i.e., there are inequalities for the correlation coefficients between local measurements, which must be fulfilled if one assumes a local hidden-variable theory to hold, but which can nevertheless be violated with entangled states [21, 22, 112]. These Bell inequalities have indeed been violated in experiments, proving the predictions of quantum mechanics about the existence of entanglement right [23, 24, 26]. Mixed state entanglement The definition of entanglement must be modified when already the knowledge of the whole system is not described by a pure state, but by a probabilistic mixture, i.e., a density operator ρ ∈ Q on the composite Hilbert space H .21 This is in particular the case when the composite system is an open quantum system, which unavoidably undergoes incoherent processes, as discussed in Section 2.2. A naive extension of definition (2.80) to mixed states is the tensor product of mixed states ρn of the individual subsystems: ρprod = ρ1 ⊗ ρ2 ⊗ · · · ⊗ ρN . 20 In 21 Q fact, the measurement outcomes are not only uncorrelated, but entirely independent. denotes the set of density operators on H , as characterized in Eq. (2.20). (2.81) 39 In such a product state, the different subsystems are uncorrelated and therefore certainly not entangled. However, if one takes several different product states, (i) (i) (i) (i) ρprod = ρ1 ⊗ ρ2 ⊗ · · · ⊗ ρN , and mixes them by a classical mechanism with probabilities pi , one obtains a mixed state of the form ρsep = ∑ pi ρprod = ∑ pi ρ1 ⊗ ρ2 ⊗ · · · ⊗ ρN . (i) (i) (i) (i) (2.82) i i Such a state can very well feature correlations between the subsystems; nevertheless, these correlations are introduced only by the classical mixing procedure and have therefore a purely classical character [113].22 In particular, these correlations cannot violate a Bell inequality. Consequently, Eq. (2.82) defines a separable mixed state. Vice versa, the defining property of an entangled mixed state is that it cannot be written in the separable form (2.82). 2.3.2 Quantifying entanglement For both pure and mixed states, the definition of entanglement is based on a negative statement, since Eq. (2.80) and Eq. (2.82) define what an entangled state is not like. Over the last decades, however, entanglement is more and more regarded as a resource for quantum informational tasks [27] and employed as a tool to study many-body quantum systems [33]. These considerations demand a more fine-grained characterization of entanglement, beyond the rough classification into separable and entangled states. The most practicable approach to this question is to quantify the entanglement amount of a state ρ by a single real number E (ρ). Such an entanglement measure should fulfill at least two basic axioms [109, 110]: I E (ρ) = 0 if and only if ρ is separable, as defined in Eq. (2.82). I E is non-increasing under any separable quantum operation ρ → Φsep (ρ), E (Φsep (ρ)) ≤ E (ρ), (2.83) since the latter can at best create classical correlations between the subsystems [117]. Here, the separable operation Φsep is defined via the separable structure of its Kraus representation [27], ρ → Φsep (ρ) = ∑ A1,i ⊗ . . . ⊗ AN,i ρ A†1,i ⊗ . . . ⊗ A†N,i , (2.84) i 22 Recently, however, the concept of quantum discord has challenged this point of view, regarding also certain classes of separable states as “quantum correlated” [114–116]. 40 with Kraus operator An,i fulfilling ∑i A†1,i A1,i ⊗ . . . ⊗ A†N,i AN,i = 1. In particular, separable operations comprise the important class of local operations and classical communication (LOCC) [110, 118, 119]. An important subclass of LOCC operations, in turn, are local unitary (LU) operations, defined as ΦLU (ρ) = (U1 ⊗U2 ⊗ · · · ⊗UN ) ρ (U1 ⊗U2 ⊗ · · · ⊗UN )† , with all Un† = Un−1 unitary. LU operations describe the time evolution of a closed quantum system in the absence of interactions between the subsystems. Since the inverse operation † Φ−1 LU (ρ) = (U1 ⊗U2 ⊗ · · · ⊗UN ) ρ (U1 ⊗U2 ⊗ · · · ⊗UN ) exists and is also LU (and hence separable), the monotonicity property (2.83) implies that entanglement measures are invariant under LU operations: E (ρ) ≥ E (ΦLU (ρ)) ≥ E (Φ−1 LU ΦLU (ρ)) = E (ρ) ⇒ E (ρ) = E (ΦLU (ρ)). Hence, two LU equivalent states ρ and ρ 0 , connected by ρ 0 = ΦLU (ρ), have identical entanglement measure. Numerous entanglement measures have been proposed in the literature, tailored to tackle different questions [110, 120]. Some of them are based on the geometric distance to the set of separable states (like, e.g., the relative entropy of entanglement [118]); or they have an operational interpretation in terms of which task they can perform (such as the distillable entanglement [117]); others are comparatively easy to compute, but lack an intuitive interpretation (e.g., the negativity [121]). In the following, we present the entanglement measures that become relevant in this thesis. 2.3.3 Entanglement measures for pure states The bi-partite case For the simplest case of a pure, bi-partite quantum state |ψi ∈ HA ⊗ HB , the entanglement properties are reflected in the spectrum of the reduced density matrices ρA and ρB defined in Eq. (2.78). Ordering the subsystems according to their dimension (i.e., dA ≤ dB ), ρA and ρB have dA common eigenvalues λi , the so-called Schmidt coefficients of |ψi; the remaining dB − dA eigenvalues of ρB are zero. Since ρA is a valid quantum state on HA , the Schmidt coefficients λi are non-negative numbers that add up to one. They determine the entanglement 41 properties of |ψi completely, insofar as a state ρ can be transformed into ρ 0 by LOCC (implying E (ρ) ≥ E (ρ 0 )), if and only if the respective Schmidt coefficients λi and λi0 , arranged in descending order, obey the majorization criterion [122] j j i=1 i=1 ∑ λi ≤ ∑ λi0 ∀ j ∈ {1, . . . , dA }. As a consequence, if |ψi has equal Schmidt coefficients λi = 1/dA – tantamount to a completely mixed reduced state ρA –, it can be transformed into any other state by LOCC. This is the case if and only if |ψi is a singlet state, 1 dA |ψs i = √ ∑ |iiA ⊗ |iiB , dA i=1 (2.85) where {|iiA } and {|iiB } form a basis of HA and HB , respectively. Such states define, hence, the top of the entanglement hierarchy and should be regarded as maximally entangled. E.g., for two qubits (dA = dB = 2), the four Bell states 1 |φ ± i = √ (|00i ± |11i) 2 1 |ψ ± i = √ (|01i ± |10i) 2 (2.86) (2.87) are maximally entangled.23 Any state which is LU-equivalent to a Bell state is also of the form (2.85) and likewise maximally entangled. A viable way to condense the Schmidt coefficients λi into a single quantity is the von Neumann entropy of the reduced state ρA [120]: dA S(ρA ) = − ∑ λi ln(λi ). (2.88) i=1 As discussed above, the reduced state is pure if and only if |ψi is separable. In this case, the Schmidt coefficients are {1, 0, . . . , 0}, and the entropy vanishes. On the other hand, if |ψi is not separable, but entangled, then ρA is not pure, and at least two eigenvalues λi are strictly positive. This leads to a strictly positive entropy. Since S(ρA ) fulfills also the monotonicity property (2.83), it defines an entanglement measure for pure states, the entanglement entropy ES : ES (|ψi) = S(ρA ), 23 We denote the basis of the Hilbert space short |00i instead of |0iA ⊗ |0B i. (with ρA ≡ trB |ψi hψ|). (2.89) C2 of a qubit by |0i and |1i from here on, and write 42 It reaches the maximum value of ln(dA ) for the maximally entangled singlet states |ψs i of Eq. (2.85). Instead of the entropy S, one may as well consider the mixedness M of the reduced state ρA : dA M(ρA ) = 1 − trρA2 = 1 − ∑ λi2 . (2.90) i=1 Just like the entropy, it vanishes if and only if ρA is pure. Otherwise, it is strictly positive, and reaches its maximal value (dA − 1)/dA when the reduced state is completely mixed.24 Based on this, one defines the alternative entanglement measure concurrence C (|ψi) for bi-partite pure states |ψi like [117, 123–126] p C (|ψi) = 2M(ρA ). (2.91) With the chosen normalization, the maximally entangled states of two qubits have unit concurrence C (|ψs i) = 1. For the simplest case of two qubits, the first Schmidt coefficient λ1 completely determines the entanglement properties, due to the constraint λ1 + λ2 = 1. For this reason, all entanglement measures for two qubits are equivalent [127], in the sense that the hierarchy of entangled states is the same under any two entanglement measures E1 and E2 : E1 (|ψi) ≥ E1 (|φ i) ⇒ E2 (|ψi) ≥ E2 (|φ i) (∀ |ψi , |φ i). (2.92) This is no longer true in higher dimensions (dA > 2), or for multi-partite systems consisting of N > 2 subsystems, or for mixed states. The multi-partite case In the multi-partite case of N > 2 subsystems, there is a much larger variety of possible entanglement measures, because there is – in contrast to the bi-partite case – no unique notion of a maximally entangled state [128]. Instead, there are various distinct entanglement classes, comprising all states that can be inter-converted by stochastic LOCC protocols.25 Each class has its own maximally entangled state that can be transformed into any state of this class by LOCC. In the definition of multi-partite entanglement measures, one has the freedom to weight the entanglement classes differently by assigning different values to the maximally entangled states of each class. 24 The similar behavior of S and M is no coincidence, as M is obtained from S by replacing the logarithm in (2.88) by its linear approximation ln(λi ) ≈ (λi − 1) for λi ≈ 1. 25 Stochastic LOCC protocols are LOCC protocols that do not necessarily succeed with certainty, but at least with finite probability. 43 E.g., for N = 3 qubits, the Greenberger–Horne–Zeilinger (GHZ) state 1 |GHZi = √ (|000i + |111i) 2 (2.93) and the W state 1 |Wi = √ (|001i + |010i + |100i) (2.94) 3 are the maximally entangled states of distinct entanglement classes [129, 130]. Apart from that, there is the class of separable states, as defined in Eq. (2.80), and three classes of bi-separable states, i.e., states that are separable with respect to one subsystem only, like, e.g., 1 √ (|00i + |11i) ⊗ |1i . 2 (2.95) Beyond the case of three qubits, a general classification of multi-partite entanglement quickly leads to continuous families of entanglement classes [110, 131, 132]. A versatile multi-partite entanglement measure is the N-partite generalization of the concurrence [133]: r C (|ψi) = 21−N/2 ∑ M(ρi ). (2.96) i The index i runs over all 2N − 2 possibilities to compile a subset of the N subsystems (excluding the trivial subsets that comprise all or none of the subsystems), and ρi denotes the reduced density matrix of the subsystems in the compilation i. Hence, C (|ψi) is simply obtained by considering all 2N − 2 non-trivial bi-partite splittings of the N-partite system and adding up the bi-partite concurrence (2.91) for each of them. In this way, the multi-partite concurrence (2.96) neither prefers nor disregards any entanglement class. In contrast, the three-tangle τ is a measure for three qubits that quantifies exclusively entanglement of the GHZ class [134]; i.e., it takes its maximal value of τ = 1 for the GHZ state and vanishes not only for separable states, but also for states of the W and bi-separable classes. (It is hence no entanglement measure in the strict sense.) We will employ the three-tangle in Section 3.3, because W entanglement is omnipresent in the scenario studied there, while GHZ entanglement is rather exceptional. The definition of τ is based on the bi-partite concurrence C (ρ) for mixed states, which will be defined below in Section 2.3.4: τ(|ψi) = [CA:BC (|ψi)]2 − [C (ρAB )]2 − [C (ρAC )]2 . (2.97) 44 Here, ρAB denotes the reduced density matrix of the first and second qubit, and CA:BC is the concurrence for bi-partite pure states, Eq. (2.91), with the three-partite state |ψi ∈ HA ⊗ HB ⊗ HC regarded as a bi-partite state shared between HA and HB ⊗ HC . Hence, the tangle quantifies the “residual” concurrence of the bipartite system A:BC after subtraction of the concurrence that remains in the reduced states ρAB and ρBC . 2.3.4 Entanglement measures for mixed states Any entanglement measure E (|ψi) for pure states |ψi ∈ H can be extended to a measure E (ρ) for mixed states ρ ∈ S (H ) via the convex roof construction [117, 135]: E (ρ) = inf [pn E (|ψn i)] . (2.98) {pn ,|ψn i} The infimum is taken over all pure state decompositions {pn , |ψn i} that are compatible with the mixed state ρ of interest, i.e., over all decompositions for which ρ = ∑ pn |ψn i hψn | (2.99) n holds. It is necessary to take the infimum in Eq. (2.98), because the entire accessible information about a quantum state is encoded in ρ; by no means can one distinguish between different ensembles {pi , |ψi i} that are compatible with ρ. Consequently, if there is a way to write ρ as the mixture of separable pure states, it cannot be entangled, and entanglement measure zero should be assigned to it – independently of how ρ was actually prepared. E.g., if one prepares the two maximally entangled Bell states |φ+ i and |φ− i, Eq. (2.86), and mixes them with equal probability, the resulting mixed state is not entangled, because one can rewrite it as a mixture of separable pure states: 1 1 1 1 |φ+ i hφ+ | + |φ− i hφ− | = |11i h11| + |00i h00| . 2 2 2 2 In order to define a consistent entanglement measure for mixed states, it is therefore necessary to find the decomposition with the poorest entanglement properties. This is precisely the role of the infimum in definition (2.98). The convex roof extension of the entanglement entropy ES (|ψi), Eq. (2.89), is the entanglement of formation EEoF (ρ) [117]. The extension of the pure state concurrence C (|ψi), see Eqs. (2.91) and (2.96), to mixed states, on the other hand, is simply referred to as the concurrence C (ρ). For two qubits, entanglement of formation and (mixed state) concurrence are equivalent in the sense of (2.92), since there is a strictly monotonic functional relation between the two quantities [124]. 45 Computing the infimum of the convex roof In general, the convex roof construction (2.98) renders the evaluation of entanglement measures for mixed states a complex minimization problem that is even numerically hard to cope with. Only in the basic example of entanglement theory – the case of two qubits – can the infimum of the convex roof be computed algebraically, at least for the concurrence, with Wootters’ formula [123, 124]: 4 √ √ C (ρ) = max{0, µ1 − ∑ µi }, (2.100) i=2 Here, µi are the eigenvalues (arranged in descending order) of ρ(σy ⊗ σy )ρ ∗ (σy ⊗ σy ), with ρ ∗ denoting the complex conjugate of ρ in the standard basis, and σy the Pauli matrix. For higher dimensional cases and multi-partite systems, no comparable algebraic tool is available. Here, the simplest attempt to calculate the infimum in the convex roof construction is to run a numerical minimization over all pure state decompositions (2.99) of ρ. The minimization procedure, however, quickly becomes an intractable task as the number N of subsystems or their respective dimensions grow. The problem is further complicated by the fact that a pure state decomposition does not necessarily comprise orthogonal states only. As a consequence, the summation index n in Eq. (2.99) can exceed the rank r of the mixed state ρ.26 Hence, the numerical minimization is likely to get stuck in a local minimum, instead of finding the true infimum. The resulting value can thus only be trusted as an upper bound for E (ρ). Lower bounds of the concurrence More interesting are algebraically computable lower bounds for the concurrence of mixed states [109, 126, 136]. In contrast to upper bounds, they can rigorously witness entanglement: As soon as the lower bound is strictly positive, the exact value of the entanglement measure is certainly also positive, and the state at hand is entangled. The lower bounds rely on a reformulation of the pure state concurrence C (|ψi), Eqs. (2.91) and (2.96), in terms of projection operators A on the twofold copy |ψi⊗2 . Based on this, one defines the tensor √ A jklm = p j pk pl pm hφ j | ⊗ hφk | A |φm i ⊗ |φ j i , 26 It suffices, however, to consider decompositions into not more than r2 pure states for the minimization [135]. 46 where pi and |φi i denote the eigenvalues and eigenvectors of the mixed state ρ of interest. From this tensor, one can derive a family of lower bounds for the mixed state concurrence C (ρ) [109, 126]. In Section 4.1, we employ a particular lower bound: the quasi-pure approximation d √ √ Cqp (ρ) = max{0, ν1 − ∑ νi } ≤ C (ρ), (2.101) i=2 where νi denote the ascending eigenvalues of a d × d dimensional matrix T T † that is defined through the elements of T : Ai11 j Ti j = q . 11 A11 The purer is ρ, i.e., the closer the largest eigenvalue ν1 is to one, the tighter is the lower bound provided by the quasi-pure approximation. In fact, it becomes exact if ρ is a pure state. Moreover, the quasi-pure approximation coincides with the exact formula (2.100) in the case of two qubits. Chapter 3 Entanglement resonances in closed, driven quantum systems With all the necessary ingredients to study the entanglement of periodically driven quantum systems at hand, we are prepared to conduct our first investigation. It regards the relation between periodic driving and entanglement in closed quantum systems, i.e., the influence of a decohering environment is neglected in this chapter. Our motivation here is two-fold: For one, understanding the phenomena in closed systems is an important prerequisite for studying open systems in the next chapter. Moreover, decoherence is often much slower than the coherent dynamics. In this case, the system can be regarded as closed for many driving cycles, and it is thus interesting to study whether suitable driving promotes entanglement in this decoherence-free time window. In a closed quantum system, the dynamics is unitary and does not evolve into a unique asymptotic cycle. Rather, each of the Floquet states represents an equally justified cyclo-stationary state of the system, as explained in Section 2.1. The central question studied in following is therefore: Under which conditions does a periodic driving field render the Floquet states of a multi-partite quantum system entangled? In Section 3.1, we motivate the relevance of this question in more detail. After that, the exemplary case of two periodically driven qubits is studied in-depth in Section 3.2. Most strikingly, we find that at specific values of the driving parameters, the entanglement of Floquet states behaves resonantly. This phenomenon, dubbed entanglement resonance, is then analyzed in a perturbative picture, using concepts well-known from static, interacting quantum systems. The analysis is extended to three or more qubits in Section 3.3. We remark that a significant part of the results presented in this chapter is published in Ref. [68]. 47 48 3.1 Why study the entanglement of Floquet states? The setting in this chapter is a closed, periodically driven system of N qubits, described by a Hamiltonian H(t) = H(t + T ) with period T . As stated in Section 2.1, the dynamics have 2N (= dim H ) cyclo-stationary solutions in this case, |Ψi (t)i = e−iεit |Φi (t)i , (3.1) with T -periodic Floquet states |Φi (t)i. The entanglement of |Ψi (t)i and |Φi (t)i is identical, since any entanglement measure E is invariant under the global phase factor e−iεit : E (|Ψi (t)i) = E (|Φi (t)i) ≡ Ei (t). With this, we have defined the periodically time-dependent Floquet state entanglement Ei (t). Besides a general interest in the entanglement properties of the cyclo-stationary solutions of driven, closed quantum systems, there are two particular motivations for studying the Floquet state entanglement Ei (t). They are presented in the following. Adiabatic preparation of entangled Floquet states First, strongly entangled Floquet states are of interest, because they allow to create persistent entanglement that lasts infinitely long (at least on time scales where decoherence plays no role). To clarify this motivation, suppose that the parameters of a periodically driven system can be tuned such that at least one of its Floquet states |Φi (t)i is strongly entangled, i.e., that Ei (t) is (at all times t) close to the maximal value Emax .1 By initially preparing the system in this state, |Ψ(0)i = |Φi (0)i , (3.2) and keeping all conditions – including the presence of the driving field – fixed at later times, the system remains in this Floquet state and is therefore maximally entangled at all times: E (|Ψ(t)i) = E (|Φi (t)i) = Ei (t) ≈ Emax . (3.3) To prepare the desired Floquet state |Φi (t)i one can make use, e.g., of an adiabatic preparation scheme for Floquet states [74, 137–139]:2 Starting from the ground 1 Typically, entanglement measures are normalized to Emax = 1. the experimental technique of stimulated Raman adiabatic passage (STIRAP) relies on such an adiabatic preparation scheme [140]. 2 E.g., Quasi-energy spectrum 49 2ω 2ω ω ω 0 |Φ0 � 0 0 Field strength |Φi (t)� 0 |Φ0 (t)� Field strength Figure 3.1: Adiabatic preparation of Floquet states. A schematic quasi-energy spectrum is plotted over two Floquet zones, εi ∈ [0, 2ω), as a function of the strength of the driving field. Starting from the field-free ground state |Φ0 i (left panel), the driving strength is ramped up adiabatically, until it has reached the point where the to-be-prepared state |Φi (t)i is strongly entangled. If |Φi (t)i coincides with the adiabatic continuation |Φ0 (t)i of the ground state, the scheme is complete. Otherwise, a coherent excitation pulse has to excite the system from |Φ0 (t)i to the target state |Φi (t)i (right panel). The excitation frequency matches the quasienergy difference εi − ε0 . state in the absence of driving fields, one adiabatically tunes the driving parameters to the desired values at which the Floquet state |Φi (t)i is strongly entangled. If field-free ground state is directly connected to the desired state |Φi (t)i in the quasienergy spectrum, the preparation is complete. If not, an additional control pulse can be applied to coherently excite the system to the desired target state |Φi (t)i. This is sketched in Fig. 3.1. For clarity, we stress that the same arguments equally apply to autonomous systems, described by a static Hamiltonian H: If the parameters of H can be tuned such that one of its energy eigenstates |Φi i becomes strongly entangled, one can create permanent entanglement by preparing the system in this eigenstate. Apart from its conceptual simplicity, this preparation scheme has the advantage that eigenstates are typically more robust against decoherence, compared to superpositions of different eigenstates.3 For this reason, entanglement properties of energy eigenstates (in particular ground states) have been studied extensively, 3 This statement holds for weak coupling to the environment, in which case the pointer basis of the most robust states against decoherence coincides with the energy eigenbasis of the system [104]. 50 e.g., for spin chains [141, 142]. If the control fields, however, are not static but periodic (i.e., AC instead of DC4 ), all these concepts equally apply to Floquet states and quasi-energies instead of eigenstates and energies. Given that AC control fields are present in many experimental setups (like the ones listed in Table 1.1), this motivates the identification of driving parameters which lead to strong Floquet state entanglement Ei (t). In fact, it is often simpler to tune AC control parameters – like, e.g., the frequency of the field – than to adjust DC control fields. Entangled Floquet states as a prerequisite for entangled asymptotic cycles A further motivation for studying Ei (t) is that the Floquet states of a closed quantum system also govern the long-term dynamics of the corresponding open system – at least if the latter is described by the Floquet Born-Markov master equation (cf. Section 2.2.2). Thus, in order to understand how driving affects entanglement in an open system that is subject to decoherence, we first have to understand the decoherence-free situation in the corresponding closed system. In detail, we found with Eq. (2.62) that the Floquet states |Φi (t)i attributed to the system Hamiltonian HS (t) of the open system S determine the basis in which the asymptotic cycle ρac is diagonal: ρac (t) = ∑ w∗i |Φi (t)i hΦi (t)| . (3.4) i The details of the environment (such as, e.g., the temperature), on the other hand, merely determine the weights w∗i of this mixture. This implies that a necessary prerequisite for an entangled asymptotic cycle is at least one entangled Floquet states with finite weight w∗i ; otherwise, Eq. (3.4) describes a mixture of separable pure states, which necessarily results in a separable mixed state. In fact, the convex roof construction (2.98) for an arbitrary entanglement measure E implies E [ρac (t)] ≤ ∑ w∗i E [|Φi (t)i]. (3.5) i Hence, the entanglement of the asymptotic cycle can only reach significant values if all Floquet states with non-negligible weights w∗i are highly entangled. One ingredient to understand under which conditions the asymptotic cycle is entangled is therefore to determine the entanglement properties of the Floquet states |Φi (t)i. For periodically driven systems, this basis consists of the Floquet states (as expressed by Eq. (3.4) below). 4 The acronyms AC and DC refer to alternating current and direct current. 51 Time-averaged Floquet state entanglement The entanglement Ei (t) of a Floquet state is a periodically time-dependent quantity. Since it is often convenient to condense this information to a single number, we consider the time-average over one period, 1 Ei ≡ T Z T 0 Ei (t) dt, (3.6) and simply refer to E i as the entanglement of a Floquet state. While E i contains only partial information about the full time-dependence of Ei (t), it nevertheless unambiguously covers the extremal cases, as illustrated in Fig. 3.2: Since an entanglement measure is bounded to values between 0 and Emax (for separable and maximally entangled states, respectively), we have Ei =0 if and only if E i = Emax if and only if ∀t ∈ [0, T ) : Ei (t) = 0, ∀t ∈ [0, T ) : Ei (t) = Emax . (3.7) Accordingly, the condensed quantity E i contains the most important information about the entanglement properties of a Floquet state, namely, whether it is maximally entangled or separable at all times, or how close it is to either of these situations. Accordingly, E i is our central figure of merit for the remainder of this chapter. 52 Entanglement Ei (t) Entanglement 1 .5 0 0 T Time t Time t 2T Figure 3.2: Exemplary time-dependence of an entanglement measure Ei (t) for three different Floquet states (solid curves, normalized to Emax = 1). Dashed curves indicate the time-average E i . Since 0 ≤ Ei (t) ≤ 1 by definition, the maximal (minimal) value of E i = 1 (E i = 0) is reached if and only if the Floquet state is maximally entangled (separable) at all times. 53 3.2 Two periodically driven qubits We begin our investigation of Floquet state entanglement with the analysis of two qubits. Although this is the smallest quantum system that can feature entanglement, most of the phenomena relevant to our discussion can already be observed and understood within this most elementary setting. In the presence of a time-periodic, classical field that equally drives both qubits, the general structure of the system Hamiltonian is H(t) = h(t) ⊗ 1 + 1 ⊗ h(t) + Hqq . (3.8) Here, the single-qubit Hamiltonian h(t) describes an individual driven qubit, and the static two-qubit operator Hqq incorporates the interaction between the qubits. This interaction is crucial for our considerations: Without Hqq , H(t) would merely consist of single-particle contributions, implying that its Floquet states were simple product states. This comes at no surprise, since the emergence of entanglement is always linked to some form of interaction between the parties. For a start, we model Hqq by an excitation hopping mechanism of strength J: Hqq = J (σ+ ⊗ σ− + σ− ⊗ σ+ ) . (3.9) This mechanism describes, e.g., a dipole-dipole interaction between excitons [143, 144] or Rydberg atoms [55], cf. Table 1.1. If the physical system at hand interacts via a different mechanism, Hqq must in principle be replaced by a different operator. However, the discussion in Section 3.2.4 will show that the precise form of the interaction operator is not crucial for the phenomena encountered in the following. For the single-qubit Hamiltonian h(t), we make the ansatz ω0 h(t) = σz + f (t)σx , (3.10) 2 where ω0 is the energy level splitting of the bare, undriven qubit, and f (t) is the periodically time-dependent amplitude of the driving field: 2π . (3.11) ω Many physical implementations of a driven qubit are well described by this ansatz, such as those listed in Table 1.1. E.g., if the qubits corresponds to two-level atoms, ω0 denotes the transition energy between the two levels, which can be tuned by a static magnetic field through the Zeeman shift [145]. The amplitude f (t), on the other hand, is generated by an external, coherent source of electromagnetic radiation (laser or microwave) in this case, which impacts on the atom via the dipole transition operator σx [86]. f (t) = f (t + T ), with T = 54 3.2.1 Monochromatic driving We begin our investigation with the most common driving profile f (t) = F cos(ωt), (3.12) describing monochromatic radiation of frequency ω and amplitude F. Typical driving amplitudes and frequencies for different implementations of a qubit are listed in Table 1.1. With this choice, the Hamiltonian H(t) of (3.8) depends on four parameters: I the energy level splitting ω0 ; I the driving frequency ω; I the driving amplitude F; I the qubit-qubit interaction strength J. In the following, we mostly regard the driving frequency ω as the unit of energy.5 The remaining parameters ω0 , F, and J are thus measured in units of a single energy quantum ω of the driving field. This choice is somewhat arbitrary; one could equally define, e.g., ω0 as the unit of energy. One reason for picking the driving frequency ω is that we want to study the limit ω0 → 0 in Section 3.2.2. Another reason is that with ω as the unit of energy, the interesting regions in the space spanned by the remaining parameters are well-separated, rendering it more convenient to visually inspect their structure. Nonetheless, the choice of the energy unit has clearly no impact on the final results, since the parameter space for fixed ω can be mapped one-to-one onto the parameter space for fixed ω0 (as long as ω, ω0 6= 0). Numerical findings For different values of the parameters, we solved the Floquet eigenvalue problem (2.10) numerically and calculated the time-averaged entanglement E i of the four6 Floquet states of H(t), as defined in Eq. (3.6). As the specific entanglement measure, the concurrence C was employed, cf. Eq. (2.91). This choice, however, is not crucial for the results, since all entanglement measures for pure states of two qubits are equivalent, as pointed out in Section 2.3.2. For every parameter set (ω0 , F, J), the Floquet states were sorted according to their entanglement, i.e., the least entangled state is labeled by i = 1, and the 5 Mind 6 Two that we set h̄ ≡ 1 throughout this thesis, so ω is both a frequency and an energy. qubits have four (non-equivalent) Floquet states, since dim H = 22 = 4. 55 E2 J = 2ω J = 0.2 ω J = 0.02 ω E1 Figure 3.3: Entanglement E 1 and E 2 of the two less entangled Floquet states of the two-qubit Hamiltonian (3.8) with the monochromatic driving profile (3.12). For each of the plots, the parameter plane is spanned by the driving amplitude F and the qubit energy splitting ω0 , both measured in units of the driving frequency ω. The time-averaged entanglement E i of the Floquet states is visualized by a color code, ranging from white (separable) to black (maximally entangled). The qubitqubit interaction strength J increases from top to bottom row. (Plots taken from [68].) 56 E4 J = 2ω J = 0.2 ω J = 0.02 ω E3 Figure 3.4: (continuation from Fig. 3.3.) Entanglement E 3 and E 4 of the two remaining Floquet states of the two-qubit Hamiltonian (3.9). For symmetry reasons, one Floquet state is always the maximally entangled singlet state |ψ− i, cf. Eq. (3.21), so that E 4 = 1. (Plots taken from [68].) 57 most entangled state by i = 4. Figures 3.3 and 3.4 show the results in several twodimensional plots, in which the parameter plane is spanned by the driving strength F and the energy level splitting ω0 . In the top rows of both figures, the interaction strength between the qubits is weak (J ω), whereas it becomes comparable to the remaining parameters in the central and bottom rows (J = 0.2ω and J = 2ω). The first striking observation is the stark contrast between the less entangled Floquet states (Fig. 3.3) on the one side and the strongly entangled Floquet states (Fig. 3.4) on the other side, at least for the case of weak interaction (upper row): the two less entangled states are separable (white) in large parts of the parameter plane, whereas the two stronger entangled states are almost always maximally entangled (black). The most peculiar features of Fig. 3.3, however, are “ridges”, along which also E 1 and E 2 reach significant values. We dub these features entanglement resonances. While they consist of sharply defined spikes reaching the maximal value of E = 1 in the case of weak interaction (upper row), they grow broader, split up, and eventually overlap as J increases in the central and bottom row. Perturbative analysis in the interaction strength (I): The unperturbed case The rich phenomenology of Figs. 3.3 and 3.4 can be explained by a perturbative analysis in the interaction strength J. This approach is justified, since the case of weak interaction, in which J is the smallest energy scale in the system, is realized in most experiments, cf. Table 1.1. Hence, we start from the non-interacting system with J = 0. In the absence of the interaction term Hqq in Eq. (3.8), the two-qubit Hamiltonian H(t) is simply the single-qubit Hamiltonian h(t) acting on one qubit at a time: H (t) = h(t) ⊗ 1 + 1 ⊗ h(t). (3.13) The Floquet states |Φi (t)i of H (t) are therefore tensor product states, |Φ1 (t)i = |φ+ (t)i ⊗ |φ+ (t)i , |Φ2 (t)i = |φ− (t)i ⊗ |φ− (t)i , |Φ3 (t)i = |φ+ (t)i ⊗ |φ− (t)i , (3.14) |Φ4 (t)i = |φ− (t)i ⊗ |φ+ (t)i , composed of the single-qubit Floquet states |φ+ (t)i and |φ− (t)i of h(t):7 [h(t) − i∂t ] |φ± (t)i = µ± |φ± (t)i . 7 We (3.15) denote Floquet states of two qubits by capital Greek letters in the following, whereas small Greek letters denote single-qubit states. Likewise, the letter ε is reserved for the quasi-energies of two qubits, and µ for those of a single qubit. Moreover, we label the two single-qubit Floquet states and quasi-energies with + and −, instead of using numbers. 58 Accordingly, the quasi-energies of the unperturbed states |Φi (t)i read ε1 = 2µ+ , ε2 = 2µ− , ε3 = µ+ + µ− , (3.16) ε4 = µ− + µ+ . We see that ε3 and ε4 are always degenerate, independently of the system parameters. Any superposition of |Φ3 (t)i and |Φ4 (t)i is, hence, also a Floquet state for the same quasi-energy. Therefore, degenerate perturbation theory [63] has to be employed in order to predict the correct Floquet states for finite interaction strength J > 0. A symmetry argument simplifies this procedure: Since our ansatz for the Hamiltonian H(t) does not distinguish one or the other qubit, it commutes with the permutation operator Π, defined by Π(|ξ i ⊗ |χi) = |χi ⊗ |ξ i . (3.17) This symmetry carries over to the Floquet Hamiltonian HF = [H(t) − i∂t ] acting on the Floquet Hilbert space of two qubits. The Floquet states can thus be chosen as eigenstates of Π. In fact, in case of non-degenerate quasi-energies, they even have to be eigenstates of Π. This is the case when a finite interaction strength J – no matter how weak – lifts the degeneracy of ε3 and ε4 ; then, the two corresponding Floquet states split into the symmetric and antisymmetric combination of the respective product states, as sketched in Fig. 3.5. The correct, symmetry-adapted ansatz for the Floquet states is therefore |Φ1 (t)i = |φ+ (t)i ⊗ |φ+ (t)i , |Φ2 (t)i = |φ− (t)i ⊗ |φ− (t)i , √ |Φ3 (t)i = (|φ+ (t)i ⊗ |φ− (t)i + |φ− (t)i ⊗ |φ+ (t)i) / 2, √ |Φ4 (t)i = (|φ+ (t)i ⊗ |φ− (t)i − |φ− (t)i ⊗ |φ+ (t)i) / 2, (3.18) instead of Eq. (3.14). To recognize the entanglement properties of these states, we introduce the LU transformation ULU (t) = uφ (t) ⊗ uφ (t), with uφ (t) defined as uφ (t) = |0i hφ+ (t)| + |1i hφ− (t)| . (3.19) uφ (t) is unitary, because |φ+ (t)i and |φ− (t)i are orthogonal at every instance t, see Eq. (2.9); therefore, uφ (t) merely describes a time-dependent change of the single-qubit basis from the computational basis {|0i , |1i} to the Floquet basis 59 |11� |01� √1 (|01� + |10�) 2 2J |10� √1 (|01� − |10�) 2 |00� Figure 3.5: The degenerate eigenstates |01i and |10i of two bare, non-interacting two-level systems are split into (anti-)symmetric singlet and triplet states as the interaction (3.9) is turned on. The same “hybridization” mechanism occurs for the Floquet states of the driven system. It explains the general finding of Figs. 3.3 and 3.4, where two Floquet states are separable in large parts of the parameter plane, and the remaining two states are maximally entangled. {|φ+ (t)i , |φ− (t)i}. Since LU transformations do not alter entanglement properties, we can now read off the true nature of the Floquet states (3.18): ULU (t) |Φ1 (t)i = |00i , ULU (t) |Φ2 (t)i = |11i , √ ULU (t) |Φ3 (t)i = (|01i + |10i) / 2, √ ULU (t) |Φ4 (t)i = (|01i − |10i) / 2. (3.20) The first two states are separable at all times; accordingly, they have vanishing entanglement measure, E 1 = E 2 = 0. The third and fourth state, on the other hand, are maximally entangled at all times, E 3 = E 4 = 1. This perfectly corresponds to the big picture observed in Figs. 3.3 and 3.4 for weak interaction, where two Floquet states are hardly entangled in most parts of the parameter plane, and the remaining two states are almost everywhere maximally entangled. The symmetry argument remains valid even beyond perturbation theory. This implies that Floquet states can always be chosen either symmetrically or antisymmetrically with respect to the permutation operator Π. Since the antisymmetric subspace of the two-qubit Hilbert space H = C2 ⊗ C2 comprises only the singlet state 1 (3.21) |ψ− i = √ (|01i − |10i) , 2 this state is always a Floquet state, as long as H(t) commutes with Π. This explains why one of the four Floquet states is always maximally entangled, 60 E 4 = 1, independently of the parameters of H(t). Interesting phenomena occur only in the symmetric subspace, spanned by the triplet states 1 |ψ+ i = √ (|01i + |10i) , |00i , |11i . (3.22) 2 Perturbative analysis in the interaction strength (II): Finite perturbation So far, our argumentation provides an explanation of the generic situation; we still lack, however, an understanding of the fine structure observed in Fig. 3.3, i.e., of the entanglement resonances. To understand their occurrence, we recognize that we only dealt with the systematic degeneracy of the unperturbed quasi-energies ε3 and ε4 , so far, and ignored that also ε1 and ε2 can be degenerate. If this happens, the previous arguments do not hold anymore. As a consequence, also the Floquet states |Φ1 (t)i and |Φ2 (t)i may significantly differ from the product states (3.18), and become entangled. In contrast to the systematic degeneracy ε3 = ε4 , however, ε1 = ε2 holds only if µ+ = µ− . Hence, the degeneracy between ε1 and ε2 occurs only at specific values of the parameters ω0 and F, since these parameters determine the single-qubit Hamiltonian h(t) – and therefore ultimately µ± . Only at these specific parameter values can entanglement resonances emerge. To understand the mechanism that triggers the resonances in detail, we apply first order perturbation theory in the interaction strength J to the Floquet Hamiltonian HF = [H(t) − i∂t ].8 To this end, we represent HF in the unperturbed basis (3.18) and make use of the “double ket” notation |Φi (t)i ≡ ||Φi ii (see Table 2.1): hhΦi ||HF ||Φ j ii = δi j εi +Ci j . (3.23) Here, εi indicates the unperturbed quasi-energies (3.16), and the perturbation matrix elements Ci j are defined in Eq. (2.18): 1 Ci j = hhΦi ||Hqq ||Φ j ii = T (2.19) Z T 0 hΦi (t)|Hqq |Φ j (t)i dt. (3.24) If the diagonal elements εi + Cii of HF are well separated on the scale of the offdiagonal elements Ci j , the Floquet states are hardly affected by the perturbation. The perturbation qualitatively alters Floquet states only if two unperturbed quasienergies εi and ε j are close to degeneracy. In this case, it suffices to consider the 2 × 2 sub-matrix of HF corresponding to the two relevant states: ! ! hhΦi ||HF ||Φi ii hhΦi ||HF ||Φ j ii εi +Cii Ci j = . (3.25) hhΦ j ||HF ||Φi ii hhΦ j ||HF ||Φ j ii Ci∗j ε j +C j j 8 See Ref. [65, 66] for an introduction to the perturbation theory of Floquet Hamiltonians. 61 ||Φ j ��||Φ j �� control parameter control parameter √1 i �� √1 (||Φ (||Φ �� +j ��) ||Φ j ��) + i||Φ 2 2 quasi-energies ||Φi ��||Φi �� (b)JJ>>0J0 > 0 quasi-energies quasi-energies quasi-energies J= (a)0JJ = =00 2|Ci j |2|Ci j | √1 i �� √1 (||Φ (||Φ �� −j ��) ||Φ j ��) − i||Φ 2 2 control parameter control parameter Figure 3.6: Illustration of an avoided crossing between two quasi-energy levels under the variation of an (arbitrary) control parameter. (a) In the absence of a perturbation (J = 0), the quasi-energy levels cross exactly. (b) A finite perturbation strength (J > 0) induces an avoided crossing of width 2|Ci j |. In the center of the avoided crossing, the Floquet states alter their character and turn into the symmetric and antisymmetric superpositions of the original, unperturbed states. The absolute level shifts Cii and C j j are not accounted for in this illustration. They merely lead to an absolute offset of both levels in the perturbed scenario (J > 0). This matrix describes the paradigmatic scenario of an avoided level crossing – a well-known phenomenon in eigenvalue spectra of static Hamiltonians [146], which likewise applies to quasi-energies of periodically driven systems [137, 147]. The paradigm is illustrated in Fig. 3.6: In the unperturbed case J = 0, all perturbation matrix elements Ci j vanish, since Hqq ∝ J, see definition (3.9). Hence, the quasi-energies of the 2 × 2 matrix (3.25) are simply εi and ε j . Suppose that they cross as a function of some parameter λ of the unperturbed system (e.g., in our case, the driving amplitude F). Taking into account a finite perturbation J > 0, the eigenvalues of Eq. (3.25) are shifted by the diagonal elements Cii of the perturbation operator. Moreover, they avoid to cross, due to the off-diagonal elements Ci j . The gap in the center of the avoided crossing amounts to 2|Ci j |. Most interestingly, the eigenstates change their character when λ is swept through the crossing: At some distance from the crossing, the perturbation has little effect, and the eigenstates coincide approximately with the unperturbed states ||Φi ii and ||Φ j ii. As one approaches the center of the crossing, however, they continuously turn into a superposition of the unperturbed states. In the center of the avoided crossing, they reach the balanced superpositions 1 ||φi0 ii = √ (||φi ii + ||φ j ii) , 2 1 ||φ j0 ii = √ (||φi ii − ||φ j ii) . 2 (3.26) 62 For the crossing of ε1 and ε2 , this means that the Floquet states turn from separable into maximally entangled states, because † |Φ1 (t)i = ULU (t) |00i , ⇒ † |Φ2 (t)i = ULU (t) |11i 1 1 † √ (|Φ1 (t)i ± |Φ2 (t)i) = ULU (t) √ (|00i + |11i) . 2 2 Hence, in parameter regions where ε1 and ε2 are degenerate, not only |Φ3 (t)i and |Φ4 (t)i, but also |Φ1 (t)i and |Φ2 (t)i are maximally entangled at all times. This explains why the observed entanglement resonances in Fig. 3.3 reach the maximum possible values E 1 = E 2 = 1. Perturbative analysis in the interaction strength (III): Degeneracy condition So far, we only considered the degeneracy between the levels ε1 and ε2 . This is, however, not the only possible degeneracy that can trigger entanglement resonances. In fact, we must keep in mind that the Floquet spectrum is inherently periodic, as discussed in Section 2.1 (see also Fig. 3.8 below). In fact, Eq. (2.8) implies that if ε1 = ε2 + nω (for integer n), then the Floquet states |Φ1 (t)i and |Φ2,n (t)i = einωt |Φ2 (t)i are in resonance. In this case, the perturbation opens an avoided crossing of strength 1 C12,n = T Z T 0 einωt hΦ1 (t)|Hqq |Φ2 (t)i dt (3.27) between these two levels. In the center of this crossing, the perturbed Floquet states become 1 √ |Φ1 (t)i ± einωt |Φ2 (t)i , (3.28) 2 which are also maximally entangled at all times. This is explicitly shown by applying the local unitary ULU,n (t) = uφ ,n (t) ⊗ uφ ,n (t) to Eq. (3.28), with uφ ,n (t) = |0i hφ+ (t)| + e−inωt/2 |1i hφ− (t)| . (3.29) (The phase factor e−inωt/2 enters twice into ULU,n (t) and ensures, thus, that einωt |Φ2 (t)i is mapped onto |11i.) In conclusion, the proper condition for entanglement resonance is ε1 = ε2 + nω, for some n ∈ N. By means of Eq. (3.16), this is tantamount to n µ+ = µ− + ω. (3.30) 2 This criterion can be further simplified, using the fact that the particular singlequbit Hamiltonian (3.10) is traceless: ω0 tr (h(t)) = tr(σz ) + f (t) tr(σx ) = 0. 2 63 As a consequence, the associated Floquet Hamiltonian hF = [h(t) − i∂t ] is also traceless,9 implying that its eigenvalues µ± are not independent, but always sum up to zero, µ+ + µ− = 0. (Here, we have chosen µ± from the central Floquet zone [−ω/2, ω/2); otherwise, µ+ and µ− sum up to an integer multiple of ω.) Therefore, we will speak of the quasi-energy µ ≡ µ+ = −µ− (3.31) of the single driven qubit problem in the following, instead of explicitly mentioning both quasi-energies µ+ and µ− . With this notion, the degeneracy condition (3.30) reads n µ = ω. 4 (3.32) Let us recapitulate the meaning of this condition: Entanglement resonances can emerge only in parameter regimes where the single-qubit quasi-energy µ fulfills Eq. (3.32) up to a finite tolerance of the order of the matrix element C12,n , because this matrix element determines the width of the avoided crossing between the resonant levels ε1 and ε2 + nω (to first order). Note that a major advantage of the criterion is that it does not require any knowledge of the two-qubit Floquet states or quasi-energies, but relies only on the single-qubit quasi-energy µ. The latter is much simpler to obtain, as discussed in the subsequent Section 3.2.2. This advantage becomes ever more important as we consider more than two qubits in Section 3.3.1. The validity of criterion (3.32) is confirmed in Fig. 3.7: In panel (a), the deviation of the single-qubit quasi-energy µ from condition (3.32) is shown. One recognizes that the white regions, in which the quasi-energy fulfills condition (3.32), exactly reproduce the shape of the entanglement resonances observed in Fig. 3.3. To verify this congruence, we show the upper left panel of Fig. 3.3 once again in Fig. 3.7(b); this time, however, the additional dashed blue lines represent contour lines extracted from (a). The contours are chosen such that they enclose regions in which the degeneracy condition (3.32) is fulfilled up to a tolerance of J. This choice is reasonable, because the matrix elements C12,n that determine the 9 In fact, for hF to be traceless, it suffices if its zeroth Fourier component h̃(0) ≡ traceless, because only this component appears on the diagonal of hF , see Eq. (2.14). 1 RT T 0 h(t)dt is 64 (a) Deviation ∆/ω (b) Entanglement E 1 Figure 3.7: (a) Deviation from the degeneracy condition (3.32) in the parameter space of driving strength F and energy splitting ω0 . Plotted is the distance from the closest degeneracy, ∆ = minn |µ − 4n ω|, measured in units of ω. White color indicates exact resonance, whereas black color corresponds to the largest possible gap between quasi-energies, ∆ = ω/2. (b) Background (as top left of Fig. 3.3): Entanglement E 1 of the first (i.e., least entangled) Floquet state, for weak interaction strength J = 0.02ω. Overlay (dashed blue lines): Contour lines ∆ = J extracted from (a). The regions delimited by these contour lines are congruent with the position of entanglement resonances, apart from the fact that only one out of two of theses regions supports a resonance. (Plots taken from [68].) width of the resonance to first order are of the order of J: (3.27) (3.9) |C12,n | ≤ max | hΦi (t)|Hqq |Φ j (t)i | ≤ max{|spec(Hqq )|} = |J|.10 (3.33) t∈[0,T ) (Here, |spec(Hqq )| refers to the absolute values of the eigenvalues of Hqq .) This inequality implies that the resonances are at most as wide as the “corridors” delimited by the contour lines, but they may also be narrower than this. In fact, they may even vanish completely, if C12,n = 0. 10 The second inequality sign holds, because for any for two normalized states |ψi and |χi and an arbitrary operator A with spectral decomposition A = ∑a a |ai ha|, we have: | hψ|A|χi | = | ∑ a hψ|ai ha|χi | ≤ ∑ |a|| hψ|ai ha|χi | ≤ (max |a|) ∑ | hψ|ai || ha|χi | ≤ max |a|. a a a a Here, we have used both the triangle and the Cauchy-Schwarz inequality. a 65 Parity-suppressed resonances Vanishing matrix elements C12,n occur, indeed, in one out of two crossing scenarios. Consequently, the interaction does not induce an avoided crossing between the respective levels (at least not in first order perturbation theory), and no entanglement resonance occurs. The levels are, in fact, not even coupled to higher order m ≥ 2; otherwise, an avoided crossing – and hence an entanglement resonance – would emerge on the scale of J m . This complete decoupling of the levels in one out of two crossings is the origin of the empty corridors in Fig. 3.7(b). The explanation for this behavior is a generalized11 parity symmetry PF of the Floquet Hamiltonian HF that divides Floquet states into two distinct symmetry classes [19, 65, 68]. In detail, the parity operator PF is defined on the Floquet Hilbert space HF of two qubits as PF |Φ(t)i = σz ⊗ σz |Φ(t + π/ω)i , (3.34) or, equivalently, in the frequency domain through PF |Φ̃(k)i = (−1)k σz ⊗ σz |Φ̃(k)i . Due to (PF )2 = 1, P†F = PF , and [PF , HF ] = 0, one can assign to each of the unperturbed Floquet states ||Φi ii a parity eigenvalue Pi = ±1. Since also the interaction Hqq is conform with the symmetry, [PF , Hqq ] = 0, the coupling matrix element C12,n between the Floquet states ||Φ1 ii, ||Φ2,n ii vanishes, if the respective parity numbers P1 , P2,n are different, because C12,n = hhΦ1 ||Hqq ||Φ2,n ii = hhΦ1 ||Hqq (PF )2 ||Φ2,n ii = hhΦ1 ||P†F Hqq PF ||Φ2,n ii = P1 P2,n hhΦ1 ||Hqq ||Φ2,n ii ⇒ = −C12,n . C12,n = 0. Next, we note that the parity Pi,n of ||Φi,n ii changes its sign when n is increased by one: (3.34) PF ||Φi,n+1 ii = PF (eiωt ||Φi,n ii) = eiω(t+π/ω) PF (||Φi,n ii) ⇒ = −eiωt Pi,n ||Φi,n ii = −Pi,n ||Φi,n+1 ii Pi,n+1 = −Pi,n 11 The parity P is “generalized” in the sense that it acts not just on the Hilbert space H of the F qubits, but also on the time argument, i.e., on the entire Floquet Hilbert space HF . 66 Quasiïenergies ¡i / t 1 ¡2,+2 0.5 ¡1,ï1 ï1 0 ¡2,+3 ¡1,ï2 ¡3,0 0 ï0.5 ¡3,+1 ¡2,+1 ¡2,+2 ¡1,ï2 ¡ ¡3,ï1 0.5 1,ï3 1 1.5 Driving strength F / t 2 Figure 3.8: Spectrum of quasi-energies εi,n of two non-interacting qubits (J = 0) as a function of the driving strength F, at ω0 = ω. (Level ε4,n is not tagged, since it equals ε3,n ≡ nω in the non-interacting case.) Two Floquet zones are plotted, with the central Floquet zone [−ω/2, ω/2) delimited by dotted lines. Red (blue) levels have symmetry eigenvalue Pi,n = +1 (Pi,n = −1) with respect to the generalized parity PF defined in Eq. (3.34). Crossings between levels of opposite parity cannot be turned into avoided crossings by the interaction Hqq , and hence do not trigger an entanglement resonance. For clarity, we note that the apparent avoided crossing of ε1,−3 and ε2,+2 around F ≈ 1.8ω is not linked to the phenomenon of entanglement resonance. It is rather a feature of the single qubit quasi-energy µ [66], linked to the breakdown of the rotating wave approximation (see the subsequent Section 3.2.2); entanglement resonances rather occur at interaction-induced avoided crossings, which do not emerge in this plot at J = 0. (Plot taken from [68].) This is indicated by the red and blue color in the level scheme plotted in Fig. 3.8. Consequently, if C12,n 6= 0 and thus P1 = P2,n , then we have P1 6= P2,n+1 and, accordingly, C12,n+1 = 0. In conclusion, if an entanglement resonance emerges in a corridor where condition (3.32) is fulfilled for some n (i.e., C12,n 6= 0 for this n), then no resonance is observed in the next corridor, where the condition is fulfilled for n + 1. This explains why only every other corridor supports a resonance. Let us, at this point, summarize what was achieved so far: We have identified the mechanism that triggers resonant behavior of Floquet state entanglement. With Eq. (3.32), we have derived a criterion for their emergence. The criterion is, in fact, only a necessary one, as we have seen: Certain resonances predicted by the criterion can be suppressed, e.g., due to a generalized parity symmetry. Apart 67 from this, it allows us to reliably predict the parameter regimes of entanglement resonances, based on the knowledge of the single-qubit quasi-energy µ alone. 3.2.2 Excursion: The Floquet problem of a single, driven qubit So far, our analysis relies on the numerical solution of the single-qubit Floquet problem (3.15). It provides the single-qubit quasi-energy µ as a function of F and ω0 , as shown in Fig. 3.7(a), which determines, in turn, the shape of the entanglement resonances via condition (3.32). To complete the analysis, it would be desirable to have at least an approximate analytical understanding of these patterns. To this end, we study in the following different approaches to (approximately) solve the Floquet problem of a single, monochromatically driven qubit [66]. The aim is, hence, to diagonalize the single-qubit Floquet Hamiltonian hF = h(t) − i∂t . In fact, it suffices to find a single eigenvalue µ, since the remaining eigenvalues are either shifted by integer multiples of the driving frequency ω, or obtained by reversing the sign, see relation (3.31). In the frequency domain, hF reads: . .. .. 0 F F 0 ω + 2ω 0 0 0 −ω0 + 2ω 0 F 1 F 0 hF = 0 0 2 0 0 0 0 0 0 .. . ↑ ↑ |1i ⊗ |1i |0i ⊗ |1i 0 0 0 0 0 0 0 0 0 F 0 0 F 0 0 0 ω0 0 0 F 0 −ω0 F 0 0 F ω0 − 2ω 0 0 0 0 0 0 F 0 ↑ ↑ ↑ ↑ |1i ⊗ |0i |0i ⊗ |0i 0 0 F |1i ⊗ |−1i −ω0 − 2ω F |0i ⊗ |−1i . . .. . (3.35) Here, the gray labels |xi ⊗ |ki indicate the origin of the respective column of hF , where x ∈ {0, 1} refers to the qubit basis, and |ki to the Fourier index, cf. Eq. (2.14). 68 (±) Moreover, we have inked in red and blue two decoupled sub-matrices hF .12 The (+) red sub-matrix hF comprises the excited state |1i of the qubit from blocks with even Fourier index k, and the de-excited state |0i from blocks with odd k; the (−) converse holds for hF . The two sub-matrices are basically identical, apart from a constant shift of ω on the diagonal. Their eigenvalue spectra are thus identical, up to this constant shift. Therefore, we can focus in the following on diagonalizing (+) hF . Leaving out the empty rows and columns, it reads .. . ω̃0 + 4 F̃ F̃ −ω̃0 + 2 F̃ 1 (+) . h̃F = (3.36) F̃ ω̃0 F̃ 2 F̃ −ω̃0 − 2 F̃ F̃ ω̃0 − 4 .. . Here, we have switched to the dimensionless notation X̃ ≡ X/ω for convenience (using ω as the unit of energy, as discussed in the beginning of Section 3.2.1). Rotating wave approximation (+) In a first approach to diagonalize h̃F , we follow a perturbative ansatz in the dimensionless driving strength F̃. By construction, it relies on the assumption of weak driving (F̃ 1), which is often fulfilled in experiments (cf. Table 1.1). The procedure is analogous to the perturbative approach in the interaction strength J that we followed in Eq. (3.25). (Mind, however, that we apply it here to the single-qubit Hamiltonian h(t), and not to the two-qubit Hamiltonian H(t).) Thus, we start from the unperturbed case F̃ = 0, in which the quasi-energies can (+) simply be read off from the diagonal of h̃F . As long as the unperturbed quasi-energies are not close to degeneracy, the perturbation has no effect to first order, because F̃ does not enter on the diagonal of Eq. (3.36). This picture changes, however, if the driving is resonant, i.e., if an integer multiple of the driving frequency ω matches the level splitting parameter ω0 , so that ω̃0 ∈ N. Consider, e.g., the simplest case of resonant driving at the first 12 The origin of this decomposition into decoupled sub-matrices is a generalized parity of h(t) [65, 68], analogously to the two-qubit parity PF of Eq. (3.34). It is also present in the single driven qubit problem with a quantized field (the Rabi problem), cf. Ref. [148]. 69 harmonic, ω̃0 ≈ 1. In this case, the neighboring diagonal elements −ω̃0 + 2 and ω̃0 in Eq. (3.36) are near-degenerate. It is hence justified to consider only the corresponding 2 × 2 sub-matrix: ! 1 −ω̃0 + 2 F̃ . (3.37) 2 F̃ ω̃0 Its eigenvalues are easily found, providing an approximation for the quasi-energy µ in the limit of weak and near-resonant driving: q 1 2 2 µ= ω + (ω0 − ω) + F . 2 (3.38) (Note that we have switched back from the dimensionless to the original notation.) This result can also be derived within the well-known rotating wave approximation (RWA) [86], which is performed in the time domain. There, one transforms the original Hamiltonian h(t) = ω0 ω0 F iωt σz + F cos(ωt) σx = σz + e + e−iωt · (σ+ + σ− ) 2 2 2 via a rotating frame transformation to the interaction picture with respect to h0 = ω 2 σz : ω ω F ω0 e−i 2 σzt σz ei 2 σzt + eiωt + e−iωt · . . . | {z } 2 e−ih0t h(t)eih0t − h0 = =σz ω ω ω ω σ+ ei 2 σzt + e−i 2 σzt σ− ei 2 σzt ) − σz {z } | {z } 2 −i ω2 σz t . . . · (e | =e−iωt σ+ =eiωt σ− ω0 − ω F σz + σx + e−2iωt σ+ + e2iωt σ− . 2 2 = In RWA, one neglects the time-dependent parts in this expression. The justification for this step is analogous to that of the secular approximation in Section 2.2.2: As long as F ω and ω0 ≈ ω, the time-dependent terms induce fast oscillations that average out quickly on the energy/time scales of the static terms. Transforming the truncated Hamiltonian back into the original frame, one has RWA h(t) ≈ ω0 F −iωt σz + e σ+ + eiωt σ− 2 2 (3.39) The corresponding Floquet Hamiltonian decomposes into 2 × 2 blocks which have exactly the structure of (3.37), establishing the result that the quasi-energy µ in 70 Eq. (3.38), derived from first order perturbation theory in F/ω, is precisely the quasi-energy in RWA. The perturbative approach, however, is more general than the RWA: First of all, it seamlessly carries over to multi-photon resonances, for which ω0 ≈ nω with n = 2, 3, . . . [65]. Moreover, it naturally provides higher order corrections in F/ω for the case of strong driving, in which the RWA is not accurate. E.g., the second order correction results in the Bloch-Siegert shift [66, 149]. Beyond driving amplitudes in the regime of F ≈ ω, however, the perturbative approach breaks down as well. Bessel approximation While the RWA result (3.38) is valid in case of weak and near-resonant driving, a different approximation applies when the energy level splitting ω̃0 is small, based on a perturbative approach in ω̃0 . We start again from Eq. (3.36). This time, the unperturbed Floquet Hamiltonian (ω̃0 = 0) is not diagonal, but rather reads .. . +4 F̃ F̃ +2 F̃ 1 (+) h̃F = F̃ 0 F̃ 2 ω̃0 =0 F̃ −2 F̃ F̃ −4 .. . . (3.40) The eigenvalues of this matrix are the integer numbers z ∈ Z [150]. The corresponding eigenvectors vz are given in terms of the Bessel functions of the first kind Jn [151]; namely, the k-th component vz (k) of vz is [150] vz (k) = J(z−k) (F̃). Since the unperturbed eigenvalues are not degenerate, we can use plain, non-degenerate perturbation theory to calculate the first order correction in ω̃0 . 71 Focusing on z = 0, the perturbation matrix element C00 , defined in Eq. (2.18), is: (+) (+) C00 = h̃F − h̃F v0 ω̃0 =0 .. . ω̃0 −ω̃0 (3.36) 1 T = v0 ω̃0 2 −ω̃0 ω̃0 vT0 ω̃0 ω̃0 ∞ = ∑ (−1)k J−k (F̃)J−k (F̃) = 2 2 k=−∞ = .. . v0 J02 (F̃) + 2 ∞ ! ∑ (−1) Jk (F̃)Jk (F̃) k k=1 ω̃0 J0 (2F̃) 2 In the last two steps, we have used identity (9.1.5) and (9.1.78) from Ref. [151]. With this, we have derived the quasi-energy µ of h(t) to first order in ω0 : µ= ω0 J0 (2F/ω). 2 (3.41) Note that this result holds for arbitrary driving strength F̃, as long as ω̃0 1. We remark the same result was obtained in Ref. [65], via a calculation in the time domain. Adiabatic approximation So far, we discussed approximations for the limiting cases of weak driving (F̃ 1) and small energy level splitting (ω̃0 1). In the complementary limit of a slow driving frequency ω, the adiabatic approximation is justified and provides yet another solution of the single driven qubit problem [65]. This time, we work in the time domain, and write down the time-evolution operator u(0,t) that is generated by h(t) in adiabatic approximation [152, 153]: h Rt i Rt u(0,t) = T← e−i 0 h(t)dt ≈ ∑ e−i 0 ei (t)dt |χi (t)i hχi (0)| . i (3.42) 72 Here, |χi (t)i and ei (t) denote the instantaneous eigenvalues and eigenstates of h(t), i.e., the solution of the eigenvalue problem of h(t) at fixed time t: h(t) |χi (t)i = ei (t) |χi (t)i , i = 1, 2. (3.43) The approximation is justified, if [138, 153, 154] hχ1 (t)|χ̇2 (t)i E1 (t) − E2 (t) 1. (3.44) In our case, this corresponds to the condition of slow driving [66]: ω ω0 , ω ω02 /F. (3.45) Comparing (3.42) to the exact time-evolution (2.7) in the Floquet picture, we see that within the adiabatic approximation, the quasi-energies of h(t) simply correspond to the time-average of the instantaneous eigenvalues ei (t). The latter are easily found by inserting the particular Hamiltonian h(t) from Eqs. (3.10) and (3.12) into Eq. (3.43): q E1 (t) = −E2 (t) = (ω0/2)2 + F 2 cos2 (ωt). (3.46) Defining the complete elliptic integral Elliptic(x) = Z π/2 q 0 1 + x2 cos2 (y) dy 1 = 4 Z 2π q 0 1 + x2 cos2 (y) dy, (3.47) we find the following expression for the quasi-energy µ: Z 2π Z 2π q ω dy ω (ωt→y) ω E1 (t) dt = (ω0/2)2 + F 2 cos2 (y) 2π 0 2π 0 ω Z q ω ω0 2π (3.47) 0 1 + (2F/ω0 )2 cos2 (y)dy = Elliptic(2F/ω0 ) = 4π 0 π µ= ⇒ µ= ω0 Elliptic(2F/ω0 ). π (3.48) To conclude this section, Figure 3.9 shows the three approximations (3.38), (3.41), and (3.48) of µ, along with the exact numerical solution. The different regimes of validity are unmistakably identified. 73 (a) Exact result (b) RWA (c) Bessel approximation (d) Adiabatic approximation Figure 3.9: Different approximations to the quasi-energy µ of the single-qubit Hamiltonian h(t). The color code indicates µ, measured in units of ω. Due to Eq. (3.31) and the periodicity of Floquet spectra, we can always chose µ ∈ [0, ω/2). (a) Exact solution, obtained by numerical diagonalization of the Floquet Hamiltonian (3.35). (b) µ in rotating wave approximation (RWA), which is valid for weak driving (F < ω). Lines of constant µ form circles around ω0 = ω and F = 0, as described by Eq. (3.38). (c) Bessel approximation (3.41), which provides accurate results for small energy splitting ω0 < ω. (d) Adiabatic approximation (3.48), valid in the case of slow driving (ω0 /ω > 1 and (ω0 /ω)2 > F/ω). 74 3.2.3 Different driving profiles After this excursion to the single-qubit Floquet problem, we return to the study of the entanglement E i of the Floquet states of the two-qubit Hamiltonian H(t). Equipped with the above developed tools, we are ready to analyze scenarios beyond the monochromatic driving profile (3.12) considered so far. Our motivation here is to show that entanglement resonances are not a peculiarity of monochromatic driving, but rather a universal feature of Floquet states of periodically driven, interacting qubits. Bichromatic driving A straightforward extension of the monochromatic driving profile is bichromatic driving with the second harmonic 2ω: f (t) = F cos(ωt) + F2 cos(2ωt). (3.49) This is a natural scenario in the lab, for both solid-state [60] and quantum optical setups [155]. When discussing Floquet state entanglement under such driving, most of the argumentation of Section 3.2.1 carries over seamlessly. In particular, since the symmetry Π (reflecting permutation invariance of the qubits) is retained, the big picture remains the same: For weak interaction strength J, two Floquet-states are highly entangled, and two are almost always separable, apart from parameter regions where the single-qubit quasi-energy µ is an integer multiple of ω/4, see condition (3.32). There, avoided level crossing can induce entanglement resonances of E 1 and E 2 . They are visible in column (b) of Fig. 3.10 for two different ratios between F and F2 . Compared to the analogous plots for monochromatic driving in Fig. 3.7, there are three major differences: I The shape of the entanglement resonances in parameter space is different. This is due to the different behavior of µ as a function of the parameters, as evident from the left panels of Fig. 3.10. Nevertheless, the prediction scheme for resonances (dashed blue lines) works equally well. I The resonances are narrower. This is explained by the fact that the single-qubit Floquet states |φi (t)i contain more frequency components under bichromatic driving, which typically leads to smaller matrix elements Ci j,n , and consequently to narrower avoided crossings. I All regions enclosed by the dashed, blue contour lines contain a resonance, although some are hardly visible, due to their narrowness. The reason is that the bichromatic driving breaks the generalized parity symmetry PF of 75 (b) Entanglement E 1 F2 = F F2 = F/2 (a) Deviation ∆/ω Figure 3.10: Entanglement resonances for the bichromatic driving profile (3.49). Upper panels: Amplitude F2 of the second harmonic equals half the amplitude F of the first harmonic drive. Lower panels: Equal amplitudes F = F2 . Column (a): Deviation ∆ (in units of ω) from the degeneracy condition (3.32). Column (b): Entanglement E 1 of the first Floquet state, for weak interaction strength J = 0.02ω. See caption of Fig. 3.7 for more details. Eq. (3.34). Hence, this symmetry does no longer systemically suppress one out of two resonances. δ -kicked driving profile A further interesting driving scheme consists of “kicking” the system periodically with a short pulse. If one idealizes the pulse to be infinitesimally short, the driving profile is a δ -function in time:13 f (t) = FT δ (t mod T ). 13 This (3.50) driving scheme is thus closely related to the well-studied quantum kicked rotor [9–12]. 76 Here, T is the period of the pulse train, and FT is the integrated pulse area, reflecting the strength of the kick.14 In contrast to the mono- and bichromatic driving schemes discussed before, the δ -kicked driving scheme can be analyzed fully analytically, because a closed expression for the single-qubit quasi-energy µ is available for δ -kicked driving [68]. It is derived from the general principle that the quasi-energies εi of a Hamiltonian H(t) are determined by the time-evolution operator U(0, T ) over one period T [76]: (2.7) U(0,t) = ∑ e−iε t |φi (t)i hφi (0)| i i (2.5) U(0, T ) = ∑ e−iεi T |φi (0)i hφi (0)| ⇒ i ⇒ spec[U(0, T )] = {e−iε1 T , e−iε2 T , . . .}. (3.51) The quasi-energies εi can thus be read off as the complex argument of the eigenvalues of U(0, T ), divided by the period T . In our case, the time evolution u(0, T ) generated by the single-qubit Hamiltonian h(t) with the δ -kicked driving profile is easily derived. It starts with the instantaneous kick at time t = 0, lim u(0,t) = e−iFT σx , t→0+ followed by a free evolution under ω0 2 σz for the rest of the period: lim u(t, T ) = e−i(ω0 T /2)σz . t→0+ ω0 2 T Defining the angles α = u(0, T ) = e and β = FT , we find −iασz −iβ σx e = + cos(β ) e−iα +i sin(β ) e+iα −i sin(β ) e−iα + cos(β ) e+iα ! . (3.52) The single-qubit quasi-energy µ is now read off from the eigenvalues λ± = exp[±i arccos[cos(α) cos(β )]] of this operator, as stated in Eq. (3.51): µ= 1 1 arg λ+ = arccos [cos(α) cos(β )] . T T (3.53) Expressing all quantities in units of the driving frequency ω = 2π/T , this reads µ 1 ω0 F = arccos cos π cos 2π . (3.54) ω 2π ω ω 14 This parametrization ensures that F has the dimension of a frequency / of an energy, as before. 77 (a) E 1 for δ -kicked driving (b) E 1 for sawtooth driving Figure 3.11: Entanglement resonances (a) for the δ -kicked driving profile (3.50) and (b) for the zero-mean sawtooth driving profile (3.58). In contrast to Figs. 3.7 and 3.10 for monochromatic and bichromatic driving, the dashed blue lines do not rely on the numerical solution of the single-qubit quasi-energy µ, but on the exact analytical expressions (3.54) and (3.59). (Plots taken from [68].) Entanglement resonances for δ -kicked driving are shown in Fig. 3.11(a), in the same fashion as in the respective figures 3.7(b) and 3.10(b) for monochromatic and bichromatic driving. In particular, the contours (dashed blue lines) delimit again regions in which the degeneracy condition (3.32) is approximately fulfilled and describe the position of the resonances accurately. This is expected, since the mechanism that causes the resonances is the same as in the case of mono- or bichromatic driving. Again, only the dependence of the single-qubit quasi-energy µ on the driving parameters is different and leads to differently shaped resonances. The periodicity of the resonances in both F/ω and ω0 /ω is explained by the inherent periodicity of expression (3.54). In physical terms, it reflects the fact that kicks with strength β = FT and β = FT + 2π have the same effect on the evolution of the qubits, as seen from Eq. (3.52). The same argument holds for the free evolution, which generates a unitary rotation by an angle of α = ω20 T during one period T , which has the same effect as α = 3ω2 0 T , α = 5ω2 0 T , etc. Sawtooth driving profile As the last example, we consider a “sawtooth” driving profile, which linearly ramps up the driving field up from f (0) = 0 to f (T ) = F within the period T : f (t) = F · t T mod 1 . (3.55) 78 Like the δ -kicked profile, it can be treated in an analytical fashion. In fact, the time evolution within a single driving period [0, T ) is equivalent to the LandauZener scenario [156, 157], which is typically discussed for a Hamiltonian of the form hLZ (t) = c1t σz + c2 σx . Here, the parameters c1 and c2 correspond to the rate of change of the diabatic energy levels and the coupling matrix element of the Landau-Zener transition, respectively. To explicitly show the equivalence between our original Hamiltonian h(t) and the Landau-Zener Hamiltonian hLZ (t), we introduce the unitary transformation R = eiπσy /4 . (3.56) Identifying c1 ≡ −F/T and c2 ≡ ω0 /2, it unitarily transforms hLZ (t) into h(t): ω0 t R hLZ (t) R† = −c1t σx + c2 σz = σz + F σx 2 T t ω0 σz + F mod 1 σx = h(t) (for 0 ≤ t < T ). = 2 T Hence, instead of considering the eigenvalues of the time evolution u(0,t) under h(t) to determine the quasi-energy µ, one may equally consider the time evolution uLZ (0,t) under hLZ (t). The latter is known explicitly [158]: ! a(t) −b(t) uLZ (0,t) = , b(t)∗ a(t)∗ with a(t) and b(t) defined in terms of Kummer’s function 1 F1 [151]: 2 i(c2 )2 1 2 a(t) = 1 F1 , , −ic1t e−ic1t /2 , 4c1 2 2 1 i(c2 )2 3 2 b(t) = 1 F1 + , , −ic1t ic2t e−ic1t /2 . 2 4c1 2 The quasi-energy µ is finally obtained from the eigenvalues λ± of uLZ (0, T ), as in Eq. (3.53): λ± = e±i arccos[ℜ(a(T ))] ⇒ µ= 1 arccos [ℜ(a(T ))] T Here, ℜ denotes the real part. After inserting the definition of a(t), replacing the parameters c1 and c2 by −F/T and ω0 /2, and expressing all quantities in units of ω = 2π/T , we find µ 1 iπ(ω0 /ω)2 1 F −iπ ωF = arccos ℜ 1 F1 , , −2πi e . (3.57) ω 2π 8(F/ω) 2 ω 79 This is, hence, the exact quasi-energy of a qubit with level spacing ω0 , driven by a field that is periodically ramped up from 0 to F at a repetition rate ω/2π. In the same fashion, one can also treat a sawtooth profile with zero mean value, varying between −F/2 and F/2: t 1 f (t) = F · mod 1 − . (3.58) T 2 Here, the time evolution over one period [0, T ) corresponds to the Landau-Zener scenario in the time interval [−T/2, T/2). Hence, the relevant time evolution operator is ! 2|a(T/2)|2 − 1 −2a∗ (T/2)b(T/2) † uLZ (−T/2, T/2) = uLZ (0, −T/2) uLZ (0, T/2) = , 2a(T/2)b∗ (T/2) 2|a(T/2)|2 − 1 having eigenvalues λ± = exp[±i arccos(2|a(T/2)|2 − 1)]. The single-qubit quasienergy µ is then " # 2 2 1 µ 1 F iπ(ω /ω) 0 −1 . = arccos 2 1 F1 , , −2πi (3.59) ω 2π 8(F/ω) 2 ω The entanglement resonances generated by the zero-mean sawtooth profile (3.58) are shown in Fig. 3.11(b), confirming again that their shape in parameter space can be inferred from the single-qubit quasi-energy µ alone. 3.2.4 Variation of the qubit-qubit interaction We have seen above that the phenomenon of entanglement resonance occurs for all kinds of driving profiles f (t). In fact, any generic driving profile should fulfill the degeneracy condition (3.32) for some parameters and hence trigger an entanglement resonance. Only the position of the resonances in parameter space is influenced by the details of the driving scheme, due to a different behavior of the quasi-energy µ. Moreover, their width may vary (or even be suppressed completely), due to different Floquet matrix elements Ci j,n . The details of the qubit-qubit interaction Hqq , on the other hand, are even less decisive for the phenomenon of entanglement resonance: While the precise form of the operator Hqq does certainly influence the matrix elements Ci j,n , and thus the width of the resonances, their positions are not affected by Hqq ,15 since they are completely determined by the single-qubit quasi-energy µ, which is independent of Hqq . 15 At least not in the limit of weak interaction that we discuss here. 80 (a) σ+ ⊗ σ− + σ− ⊗ σ+ (b) σx ⊗ σx (c) (Rπ/4 σx R†π/4 ) ⊗ (Rπ/4 σx R†π/4 ) (d) Random interaction operator Figure 3.12: Entanglement resonances for four different interactions operators Hqq . The driving profile f (t) is the monochromatic scheme considered throughout Section 3.2.1, and the interaction strength is weak (J = 0.02ω). (a) Excitation hopping mechanism, as considered previously. (b) One-dimensional Ising interaction (3.60). (c) The rotated Ising model (3.62). (d) Random interaction Hamiltonian (3.63). (Plots taken from [68].) 81 To support this argument, four different qubit-qubit interaction mechanisms are compared in Fig. 3.12. The driving profile f (t) is monochromatic in all cases, as considered throughout Section 3.2.1. In the upper left panel (a), Hqq is the familiar excitation hopping mechanism (3.9). The plot is, hence, identical to the upper left panel of Fig. 3.3 and serves as a reference point for the remaining interaction models. In the upper right panel (b), the excitation hopping mechanism is replaced by the one-dimensional Ising model [33]: Hqq = J σx ⊗ σx . (3.60) Apparently, this replacement has little effect on the entanglement resonances; only their width increases slightly. This can be understood by expanding the Ising interaction in terms of the annihilation and creation operators σ+ and σ− : σx ⊗ σx = (σ+ + σ− )⊗2 = σ+ ⊗ σ− + σ− ⊗ σ+ + σ+ ⊗ σ+ + σ− ⊗ σ− .16 (3.61) The excitation hopping mechanism is thus merely a truncated version of the Ising model, in which the terms that do not conserve the number of excitations are neglected. Since both mechanisms are compatible with the generalized parity PF of Eq. (3.34), the suppression of every other resonance is observed in both Fig. 3.12(a) and (b). This is different for the “rotated” Ising model in the lower left panel (c). There, the interaction operator reads ⊗2 1 † ⊗2 (3.56) = J √ (σx + σz ) Hqq = J (Rπ/4 σx Rπ/4 ) . (3.62) 2 The unitary operator Rπ/4 = exp(iπσy /8) generates a π/4-rotation of spin- 12 systems (i.e., qubits) about the y-axis [63]; accordingly, Eq. (3.62) describes a one-dimensional Ising interaction in which the connection line between the two qubits (spins) is rotated 45◦ against the quantization axis (in our notation, the z-axis) that defines the spin-up and spin-down state |0i and |1i. We consider this rather unorthodox interaction mechanism, because it breaks the generalized parity PF . Consequently, an entanglement resonance emerges every time the degeneracy condition (3.32) is fulfilled. This explains why Fig. 3.12(c) features resonances in all regions enclosed by the dashed blue lines in Fig. 3.7(b). On the other hand, the resonances are noticeably narrower, compared to the upper panels, due to systematically smaller matrix elements Ci j,n for this interaction mechanism. 16 The symbolic notation X ⊗n is used for X ⊗ X ⊗ . . . ⊗ X from here on. 82 Finally, we consider a randomly drawn member of the Gaussian Unitary Ensemble (GUE) [159] in the lower right panel (d). In the computation basis {|00i , |01i , |10i , |11i}, the actual numbers are +0.98 −0.82 − 0.56i +0.22 − 1.29i −0.48 − 0.40i −0.82 + 0.56i −0.73 +0.35 − 0.11i −1.43 + 0.23i Hqq = J . (3.63) +0.22 + 1.29i +0.35 + 0.11i −0.44 +0.81 − 0.53i −0.48 + 0.40i −1.43 − 0.23i 0.81 + 0.53i +0.41 It comes as no surprise that this random operator does not commute with PF . Consequently, it suppresses none of the resonances. However, compared to the rotated Ising interaction in panel (c), some resonance are not fully pronounced, but only reach E 1 ≈ 0.5. This is due to the fact that the random Hqq breaks not only the parity symmetry PF , but also the permutation symmetry Π under exchange of the two qubits, which was discussed after Eq. (3.17). Therefore, all four Floquet states of H(t) can couple via Hqq . A more detailed analysis in Fig. 3.13 reveals why this leads to less pronounced resonances in some cases. In conclusion, the results of Fig. 3.12 support our argument that the details of the qubit-qubit interaction Hqq do not alter the overall phenomenon of entanglement resonance, but only have an influence on details, like the symmetry-induced suppression of resonances. However, we emphasize again that a finite (albeit possibly weak) interaction strength J is crucial, because at J = 0, the Hamiltonian H(t) consists of single particle contributions only, and its Floquet states can always be chosen as product states, as seen in Eq. (3.14). 83 (Rπ/4 σx R†π/4 ) ⊗ (Rπ/4 σx R†π/4 ) 1.2 Quasiïenergies ¡ / t 1 i Quasiïenergies ¡i / t Quasi-energies εi /ω 1.2 0.8 0.6 0.4 0.4 0.6 0.8 1 Driving strength F / t 0.4 0.6 0.8 1 Driving strength F / t 0.8 0.6 0.4 0.4 0.6 0.8 1 Driving strength F / t 1.2 0.4 0.6 0.8 1 Driving strength F / t 1.2 1 0.5 0 1 1.2 Floquet state entanglement Floquet state entanglement 1 Entanglement E i Random interaction operator 1.2 0.5 0 Figure 3.13: Detailed analysis of entanglement resonances for the rotated Ising interaction mechanism (3.62) (left column) and the random interaction operator (3.63) (right column). In contrast to the two-dimensional plots of the previous figures, the level splitting is fixed to ω0 = 0.8ω; only the driving strength F is varied. The upper panels show the four quasi-energies εi of the two-qubit Hamiltonian H(t) within one Floquet zone [0, ω) + ω4 (the ω4 -shift makes sure that both crossing regions are adequately depicted). The panels below show the corresponding entanglement E i . It behaves resonantly whenever quasi-energies avoid to cross. The resonance near F/ω ≈ .5 involves only two levels and can therefore be explained in the simple two-state picture of Fig. 3.6. Consequently, it leads to maximal entanglement in the center of the resonance. The situation is different for the resonance near F/ω ≈ 1.1. For the rotated Ising interaction (left), the gray level is decoupled from the rest, because of its antisymmetric character under permutation of the qubits. Since also the blue quasi-energy level is only weakly affected in the resonant region, the two-state picture still holds and explains the value of E i = 1 in the center of the resonance. The random interaction operator (right), on the other hand, breaks the permutation symmetry. Accordingly, it mediates between all four levels. Due to this, the resonance involves not just two, but all four levels, and does not reach E i = 1. This explains why in Fig. 3.12(d), where only the entanglement E 1 of the weakest entangled state is shown, one out of two resonances reaches less than E 1 ≈ 0.5. 84 3.3 Beyond two qubits Now that we have understood the entanglement properties of Floquet states of two periodically driven qubits, we are in the position to extend our investigation to multi-partite systems consisting of N > 2 qubits. 3.3.1 Three qubits and GHZ entanglement As the smallest possible extension, we first consider N = 3 qubits. There are two distinct classes of genuine multi-partite entanglement in this case, as discussed in Section 2.3.3; one is spearheaded by the W state (2.94), the other one by the GHZ state (2.93). In the following, we exclusively quantify entanglement of the latter class, by employing as entanglement measure in Eq. (3.6) the three-tangle τ that was introduced in Eq. (2.97). We do so, on the one hand, because W entanglement is by far less exceptional than GHZ entanglement in the system we are studying here, as we will see below. On the other hand, GHZ entanglement is of great practical interest; e.g., it enables a test of quantum nonlocality that can — in contrast to Bell’s approach [21] — exclude local hidden-variable theories from perfect correlations of the measurement outcomes alone [129]. Moreover, GHZ states can be used to beat the standard quantum limit of metrology [160]. Results for identical qubits A straightforward generalization of the two-qubit Hamiltonian (3.8) to three qubits leads to the ansatz H(t) = 1 ⊗ 1 ⊗ h(t) + 1 ⊗ h(t) ⊗ 1 + h(t) ⊗ 1 ⊗ 1 + Hqq . (3.64) Adopting the general notation a(n) = 1 · · ⊗ 1} ⊗ a ⊗ 1 · · ⊗ 1} | ⊗ ·{z | ⊗ ·{z n−1 times (3.65) N−n times of a single-qubit operator a acting on the n-th qubit of an N-partite system, we can write this as H(t) = 3 ∑ h(n) (t) + Hqq . (3.66) n=1 We focus on the same single-qubit Hamiltonian h(t) as before, Eq. (3.10), but exclusively discuss results for the monochromatic driving profile f (t) of Eq. (3.12) in the following; different driving schemes may be analyzed in complete analogy to the discussion for two qubits in Section 3.2.3. 85 For the inter-qubit interaction Hqq , we exclusively consider pairwise interactions between the qubits, since they typically dominate over three particle contributions. This is, e.g., the natural situation for dipole-dipole interacting Rydberg atoms [161] or chromophores [162], or by construction implemented in engineered systems, like superconducting qubits [163]. A reasonable ansatz is Hqq = J 3 ∑ n<m (n) (m) (n) (m) σ+ σ− + σ− σ+ , (3.67) modeling an excitation hopping mechanism, as in Eq. (3.9). Note that we have assumed identical coupling strength J for all pairwise interactions here; deviations from this idealized situation are discussed at the end of this section. The numerically computed entanglement E i of the 23 = 8 Floquet states of H(t) is presented in Fig. 3.14. The four least entangled states are not shown, since they turn out to have vanishing GHZ entanglement, E i = 0, throughout the entire parameter plane. The entanglement E 5 . . . E 8 of the remaining states vanishes as well in large parts of the parameter plane, apart from the – by now familiar – resonances. The picture is thus, in some sense, complementary to the case of two qubits, where half of the four Floquet states were maximally entangled for virtually any choice of the parameters. Notably, the behavior of the most entangled state of three qubits and of the least entangled state of two qubits is very similar – compare E 8 in Fig. 3.14 to E 1 in the upper row of Fig. 3.3. We will find below that they emerge indeed under exactly the same resonance condition. Detailed analysis of resonances As in the case of two qubits, these observations can be explained by a perturbative approach in the qubit-qubit interaction strength J. The big picture can again be understood by considering the unperturbed Floquet states at J = 0, if one additionally takes into account that H(t) is invariant under permutation of any two qubits. This implies that its Floquet states are eigenstates of the cyclic permutation operator Π, defined by Π(|ξ i ⊗ |χi ⊗ |ζ i) = |ζ i ⊗ |ξ i ⊗ |χi . Since Π3 = 1, there are three symmetry classes now, each corresponding to one of the eigenvalues of Π: spec(Π) = {1, c, c∗ }, with c ≡ e2πi/3 . 86 E5 E6 E7 E8 Figure 3.14: Floquet state entanglement E i of three qubits under monochromatic driving, for weak interaction between the qubits (J = 0.02ω). The remaining four values E 1 . . . E 4 are not shown, because they vanish exactly at any point of the parameter plane. 87 The unperturbed Floquet states with permutation eigenvalue c are 1 † |Φ1 (t)i = ULU (t) √ (|100i + c |010i + c∗ |001i) , 3 1 † ∗ |Φ2 (t)i = ULU (t) √ (|011i + c |101i + c |110i) . 3 Here, ULU (t) = [uφ (t)]⊗3 is a time-dependent local unitary transformation that is defined via the single-qubit operator uφ (t) of Eq. (3.19), just like in the corresponding expression (3.20) for two qubits. Both |Φ1 (t)i and |Φ2 (t)i are at all times LU-equivalent to the W state and, consequently, have no GHZ entanglement, E 1 = E 2 = 0.17 The same situation is encountered in the subspace with symmetry eigenvalue c∗ , where we have 1 † ∗ |Φ3 (t)i = ULU (t) √ (|100i + c |010i + c |001i) , 3 1 † ∗ √ |Φ4 (t)i = ULU (t) (|011i + c |101i + c |110i) . 3 Finally, the symmetric subspace (symmetry eigenvalue 1) comprises the remaining four Floquet states: 1 † √ |Φ5 (t)i = ULU (t) (|100i + |010i + |001i) , 3 1 † |Φ6 (t)i = ULU (t) √ (|011i + |101i + |110i) , 3 † |Φ7 (t)i = ULU (t) |000i , † |Φ8 (t)i = ULU (t) |111i . It is easily read off in this notation that also |Φ5 (t)i and |Φ6 (t)i are W states with E 5 = E 6 = 0, while |Φ7 (t)i and |Φ8 (t)i are separable at all times, resulting thus in E 7 = E 8 = 0 as well. Again, this picture does not change in the presence of a weak interaction J, as long as quasi-energies are non-degenerate. This explains why for a generic choice of the parameters, none of the Floquet states of H(t) carries significant GHZ entanglement, as observed in Fig. 3.14. Only if two (or more) quasi-energy levels are in resonance, the interaction may mediate between these levels in an avoided crossing; then, the corresponding Floquet states of H(t) turn into superpositions 17 Precisely, √1 3 (|100i + c |010i + c∗ |001i) is transformed into |Wi = √1 (|100i + |010i + |001i) 3 by applying (local) phase gates to the qubits [27], i.e., 1 ⊗ [|0i h0| + c∗ |1i h1|] ⊗ [|0i h0| + c |1i h1|]. 88 Quasiïenergies ¡i / t 1 ¡ 8,ï1 (I) ¡6,0 0.5 ¡7,+4 ¡5,+2 ¡7,+3 ¡7,+2 ¡5,+1 0 ¡ 8,ï2 ¡ 7,+1 ï1 0 ¡8,ï3 (II) ¡6,ï1 ï0.5 ¡8,ï4 ¡5,0 0.5 1 1.5 Driving strength F / t ¡6,ï2 2 Figure 3.15: Two Floquet zones of the quasi-energy spectrum of three noninteracting qubits (J = 0) at ω0 = ω. As in the analogous Fig. 3.8 for two qubits, red and blue levels are of opposite symmetry with respect to the generalized parity PF . This results in two different crossing scenarios, referred to as type I and type II in the main text. (Plot taken from [68].) of the above listed unperturbed states |Φi (t)i, yielding the possibility of GHZ entanglement. To understand the emergence of GHZ entanglement in detail, we have to study all possible degeneracies between the unperturbed quasi-energies εi . Similar to Eq. (3.16) for two qubits, they read ε1 =ε3 = ε5 = −µ, ε2 =ε4 = ε6 = +µ, ε7 = − 3µ, (3.68) ε8 = + 3µ, where µ is again the quasi-energy of the single-qubit Hamiltonian h(t). The systematic degeneracies, e.g., between ε1 and ε3 , are irrelevant for the discussion, since the respective levels belong to different symmetry classes; they are thus not coupled via the permutation-invariant interaction Hqq .18 The unperturbed quasi-energies are plotted in Fig. 3.15 over two Floquet zones. Each of the various crossings holds, in principle, the potential for an entanglement resonance – provided that a finite perturbation strength J opens an 18 Put differently, these degeneracies have already been accounted for by choosing the unperturbed Floquet states as eigenstates of the permutation operator Π. 89 avoided crossing at this point. This is, however, not always the case, because H(t) respects the generalized parity symmetry PF that was introduced for two qubits in Eq. (3.34).19 The parity of a level changes between adjacent Floquet zones, as discussed in detail for two qubits and indicated by the red and blue color scheme in Fig. 3.15. Crossings between levels of different parity are not lifted by the perturbation Hqq , but remain exact crossings. Accordingly, they do not lead to entanglement resonances. The remaining crossings between levels of equal parity can be classified into two types. Type I refers to a degeneracy between the levels ε7 and ε8 of the same parity. It occurs if and only if ω ε7 mod ω = ε8 mod ω = . 2 Inserting ε8 from Eq. (3.68), this is tantamount to ω 1 µ= +n (3.69) 3 2 for some integer number n. Under this condition, a finite interaction strength J can open an avoided crossing between ε7 and ε8 and turn the corresponding Floquet states into the balanced superposition of |Φ7 (t)i and |Φ8 (t)i. This results in an exact GHZ state (at all times t): 1 1 † √ (|Φ7 (t)i ± |Φ8 (t)i) = ULU (t) √ (|000i + |111i) , 2 2 Consequently, two maximally entangled Floquet states with E = 1 emerge at a crossing of type I. Crossings of type II, on the other hand, occur between ε5 and ε8 (of the same parity).20 The corresponding resonance condition is read off from Fig. 3.15: ε5 mod ω = ε8 mod ω = 0. By virtue of Eq. (3.68), this is equivalent to [3µ − (−µ)] = nω, i.e., to µ = ω4 n. Taking into account, however, that one out of two crossings involves levels of opposite parity and is therefore incapable of triggering an entanglement resonance, the relevant degeneracy condition becomes: µ= 19 Mind, ω n. 2 (3.70) however, that this symmetry may be violated when a different driving profile f (t) or interaction mechanism Hqq is considered, as discussed in Section 3.2.3. In this case, a more complex structure of entanglement resonances is expected. 20 At the same point of the spectrum, also ε and ε are in resonance. Their crossing is, however, 7 6 independent from the crossing of ε5 and ε8 , because of the different parity of the two pairs. 90 In the center of an avoided crossing of ε5 and ε8 , the corresponding Floquet states turn into the superposition 1 1 † √ (|Φ5 (t)i ± |Φ8 (t)i) = ULU (t) √ (|W i + |111i) . 2 2 In contrast to type I resonances, this superposition does not feature maximal GHZ entanglement, because the three-tangle τ is only r 1 16 τ √ (|Wi ± |111i) = ≈ 0.77. 27 2 When sweeping through the avoided crossing, however, any superposition of |Φ5 (t)i and |Φ6 (t)i is realized at some stage. Therefore, by moving slightly out of the center of the crossing region, the superposition will at some point become "r # r r r 3 1 3 1 † |Φ5 (t)i ± |Φ8 (t)i = ULU (t) |Wi ± |111i . 4 4 4 4 This superposition reaches the maximal three-tangle τ = 1, because it is LU-equivalent to the GHZ state: r r 3 1 1 |Wi ± |111i = √ (|100i + |010i + |001i ± |111i) 4 4 4 1 = √ (|+ + +i ± |− − −i) = (Rπ/2 )⊗3 |GHZi . 2 Here, we have introduced the states |±i, which relate to the rotation operator Rπ/2 of Eq. (3.56) as follows: 1 |+i = √ (|1i + |0i) = Rπ/2 |1i ; 2 1 |−i = √ (|1i − |0i) = Rπ/2 |0i . 2 The discussion for ε6 and ε7 is completely analogous. In conclusion, all four symmetric Floquet states are maximally entangled in type II crossings. Our analysis of type II crossings is confirmed in Fig. 3.16(b), where the Floquet state entanglement E 8 is superimposed with contour lines that indicate the degeneracy condition (3.70). Within regions enclosed by the dashed contours lines, the single-qubit quasi-energy µ, shown in panel (a) of the figure, fulfills (3.70) up to a tolerance of the order of J. These regions perfectly coincide with the observed entanglement resonances. Our previous analysis of type I crossings, however, is at first sight not confirmed in Fig. 3.16: no entanglement resonances are observed in the vicinity of 91 (a) µ (b) E 8 Figure 3.16: Analysis of entanglement resonances for three qubits. (a) Single-qubit quasi-energy µ (measured in units of ω). Dashed blue contour lines indicate that the type I degeneracy condition (3.69) is fulfilled up to a finite tolerance of the order of J, whereas solid blue lines represent the type II condition (3.70). (b) Floquet state entanglement E 8 (as seen in Fig. 3.14), superimposed with the contour lines of panel (a). While entanglement resonances emerge whenever the type II condition is fulfilled (dashed lines), type I resonances are too narrow to be visible at the resolution of this plot. For this reason, the detail indicated by the black arrow is magnified in Fig. 3.17. (Left plot taken from [68].) the solid blue lines that indicate the type I degeneracy condition (3.70). Yet, this is merely a question of scales: Avoided crossings of type I are much narrower than those of type II, because the respective levels ε7 and ε8 are coupled only to second order via the perturbation operator Hqq , resulting in a width of the order of J 2 . The respective entanglement resonances are therefore not visible at the scale of Fig. 3.16(b). In fact, a magnified detail of Fig. 3.16(b) is shown in Fig. 3.17, confirming the existence of a narrow entanglement resonance. The second-order character of the coupling between |Φ7 (t)i and |Φ8 (t)i can be understood by evaluating the respective Floquet perturbation matrix element C78 : C78 = 1 T Z T 0 † h111|ULU (t)HqqULU (t)|000i dt. Recalling the definition ULU (t) = [uφ (t)]⊗3 and inserting our ansatz (3.67) for the interaction term Hqq , we see that the integrand consists exclusively of terms like h111| [uφ (t)σ− u†φ (t)] ⊗ [uφ (t)σ+ u†φ (t)] ⊗ 1 |000i = h1| uφ (t)σ− u†φ (t) |0i · h1| uφ (t)σ+ u†φ (t) |0i · h1|0i , 92 Figure 3.17: Detail of Fig. 3.16(b). At this scale, a narrow entanglement resonance is visible in the vicinity of the solid blue lines. The latter indicate parameter regions where the type I degeneracy condition (3.70) is fulfilled up to J 2 . (Plot taken from [68].) all of which vanish exactly, because of the factor h1|0i = 0. Hence, C78 vanishes, implying that no avoided crossing opens to first order in the qubit-qubit interaction strength J. Note that this is, ultimately, a consequence of our (physically motivated) restriction to pairwise interactions in Hqq . The second order contribution to the avoided crossing, however, is non-vanishing, because an indirect coupling of |Φ7 (t)i and |Φ8 (t)i is possible via the intermediate states |Φ5 (t)i or |Φ6 (t)i – e.g., we have C75 ·C58 6= 0. For completeness, we mention that also the crossing between ε5 and ε6 can trigger a resonance. The corresponding degeneracy condition µ = ω n + 12 , however, is just a special case of the type II condition (3.70) (with odd n) or of the type I condition (3.69) (with n = 1, 4, 7, . . .). It is fulfilled only at isolated points in the parameter plain, near F = 0 and ω0 = nω. The superposition of the corresponding states is † a |Φ5 (t)i + b |Φ6 (t)i = ULU (t) a |Wi + b (Rπ )⊗3 |Wi and has a three-tangle of τ = 43 |ab|2 , reaching thus at most the rather poor value of τ = 13 . With the same argument that was given for type I resonances, we conclude that the coupling between ε5 and ε6 vanishes to first order in J. The resulting resonances are therefore hidden on the scale of Fig. 3.14. The same arguments apply to degeneracies in the non-symmetric subspaces, i.e., between ε1 and ε2 , and between ε3 and ε4 . 93 ±10% variation in ω0 ±10% variation in J Figure 3.18: Floquet state entanglement E 8 for a three-qubit Hamiltonian H(t) that is not perfectly permutation invariant. In panel (a), the level splitting ω0 of the qubits is varied by ±10%, while in panel (b), the pairwise interaction strength J between qubits differs by ±10%. Apparently, both modifications lead only to minor changes in the overall phenomenology of Floquet state entanglement. (Plots taken from [68].) 3.3.2 Deviations from perfect permutation invariance The previous considerations fundamentally rely on the permutation invariance of the three-qubit Hamiltonian H(t). In a realistic setup, however, qubits will never have perfectly identical parameters. This raises the question to what extent small deviations from a perfect permutation invariance affect our results. To this end, we break the symmetry in two different ways: In panel (a) of Fig. 3.18, the level splitting parameter ω0 varies slightly from qubit to qubit. E.g., in the context of NMR or electronic spin qubits, where ω0 determines the Larmor frequency in a static magnetic field, this corresponds to the situation where the local magnetic field strength is not exactly identical for all spins. To mimic this variation, we take the original Hamiltonian (3.64) and increase (decrease) ω0 by 10% at the first (second) qubit, while the third qubit is unaffected. The impact of this symmetry breaking is not as detrimental as one might expect: Compared to the perfectly symmetric case in Fig. 3.14, the broad resonances are only slightly reduced in width, but they still reach maximal GHZ entanglement E 8 = 1; the smaller resonances, on the other hand, are split into “twin arches”. In panel (b), we put a ±10% perturbation on the qubit-qubit interaction strength. Thus, instead of a common interaction strength J as in Hqq in Eq. (3.67), we now assume 1.1 J for the pairwise interaction of the first and second qubit, 94 0.9 J for second and third qubit, and 1.0 J for the remaining pair of first and third qubit. However, hardly any difference to the symmetric case is observed in the numerical findings. This supports, once more, our reasoning that the details of Hqq are not crucial for the phenomenology of entanglement resonances. In summary, we find that even a rather severe deviation from a perfectly permutation-symmetric system does not qualitatively alter the phenomenology of entanglement resonance. 3.3.3 Entanglement resonance with an arbitrary number of qubits Our results for N = 2 and N = 3 can be generalized to an arbitrary number N of qubits. Motivated by the findings of the previous section, we focus on the case of a permutation symmetric system Hamiltonian H(t): H(t) = N ∑ h(n) (t) + Hqq . (3.71) n=1 As discussed for three qubits, it is reasonable to restrict Hqq to pairwise interactions. In the following, we consider, again, the exemplary excitation hopping mechanism Hqq = J N ∑ n<m (n) (m) (n) (m) σ+ σ− + σ+ σ− (3.72) with equal interaction strength J between all pairs (n, m). A more convenient formulation relies on the collective spin operator N ~S = 1 ∑ ~σ (n) , 2 n=1 (n) (n) (n) with ~σ (n) = (σx , σy , σz ). (3.73) Introducing the ladder operator S± = (Sx ± iSy )/2, and inserting our ansatz (3.10) for h(t), Eq. (3.71) can be rewritten in terms of collective operators only: J H(t) = ω0 Sz + 2 f (t)Sx + (S+ S− + S− S+ − N 1). 2 Using the usual commutation relation of angular momentum operators [63], one verifies that H(t) commutes with the total spin operator ~S2 .21 Due to this symmetry, the Floquet states of H(t) can be chosen as eigenstates of ~S2 and classified according to their total spin quantum number, which takes the 21 It is, in fact, precisely the permutation-invariance of H(t) that underlies this symmetry. 95 values N N , − 1, . . . , 0 for even N, 2 2 N N 1 , − 1, . . . , for odd N. 2 2 2 The subspace of maximum total spin N2 contains N + 1 Floquet states that are symmetric under cyclic permutation of the qubits. We focus our analysis on this subspace in the following. This is reasonable, since one typically initiates a system of identical qubits in a symmetric state, e.g., in the de-excited state |0i⊗N , so that the dynamics are restricted to the symmetric subspace in this case. In the non-interacting case, J = 0, the symmetric Floquet states |Φi (t)i of H(t) are simply the symmetrized tensor products of the Floquet states |φ± (t)i of the single-qubit Hamiltonian h(t) [68]: |Φi (t)i = SN |φ− (t)i⊗i ⊗ |φ+ (t)i⊗(N−i) , i ∈ {0, . . . , N}. (3.74) Here, the symmetrization operator SN is defined (up to normalization) through a summation over all N! permutations of N qubits. Making, once more, use of the time-dependent unitary transformation uφ (t), uφ (t) = |0i hφ+ (t)| + |1i hφ− (t)| , we find: [uφ (t)]⊗N |Φi (t)i = SN |1i⊗i ⊗ |0i⊗(N−i) . The right hand side of this expression describes a symmetric superposition of the N i different possibilities to distribute i excited states |1i over N qubits. Such states are known as Dicke states [164]: N − 12 ⊗i ⊗(N−i) |N, ii ≡ SN |1i ⊗ |0i (S+ )i |0i⊗N . = i (3.75) They were first introduced in the context of superradiance, i.e., the cooperative emission of a photon from a collection of atoms [165]. In conclusion, the unperturbed Floquet states are LU-equivalent to Dicke states, |Φi (t)i = [u†φ (t)]⊗N |N, ii , and therefore have the same entanglement properties as the latter. Since |N, ii is converted into |N, N − ii by simply relabeling the local basis (|0i ↔ |1i), these two states have identical entanglement properties. This allows 96 us to reduce the analysis to the lower half of the Dicke ladder, i.e., to i < N2 + 1. The only separable state in this manifold is |N, 0i = |0i⊗N . The next state is the N-qubit W state |WN i: 1 |N, 1i = √ (|00 . . . 01i + |00 . . . 10i + . . . + |10 . . . 00i) ≡ |WN i . N (3.76) The higher excited states |N, ii with 2 < i < N2 + 1 belong to distinct SLOCC classes; all of them have, however, W character [166, 167]. The effect of a finite qubit-qubit interaction Hqq on this picture has been elaborated upon in the previous sections. In short, the Floquet states |Φi (t)i are barely altered as long as the respective quasi-energies εi are far from degeneracy. If, however, two levels εi and ε j are near-degenerate, Hqq may lift this degeneracy and lead to an avoided crossing. Sweeping through this avoided crossing, the corresponding Floquet states change their character from |Φi (t)i and |Φ j (t)i into the superposition a |Φi (t)i ± b |Φ j (t)i = [u†φ (t)]⊗N [a |N, ii ± b |N, ji] (3.77) (with |a|2 + |b|2 = 1), as sketched in Fig. 3.6. Every ratio |a/b| is realized at some point inside the crossing region. This yields the possibility to establish Floquet states with entanglement properties beyond those of the bare Dicke states |N, mi. E.g., by adjusting the driving parameters such that ε0 = εN , the Floquet states become 1 1 √ (|Φ0 (t)i ± |ΦN (t)i) = [u†φ (t)]⊗N √ |0i⊗N ± |1i⊗N , 2 2 which are N-qubit GHZ states. For four qubits, the superposition r r 1 2 |4, 0i + |4, 3i 3 3 belongs to a SLOCC class that is neither equivalent to the single or double excitation Dicke states |4, 1i and |4, 2i, nor to four-qubit GHZ entanglement [168]. Recognizing that the quasi-energy εi of the unperturbed Floquet states in Eq. (3.74) is simply (3.31) εi = i · µ− + (N − i) · µ+ = (N − 2i)µ, the condition for εi and ε j to be degenerate is µ =n ω , 2(i − j) (3.78) 97 for some integer n. We emphasize once more that this condition merely relies on the quasi-energy µ of the single-qubit Hamiltonian h(t). This is a major simplification, compared to the efforts of solving the full Floquet problem of the interacting N-qubit Hamiltonian H(t). Finally, the width of an avoided crossing between εi and ε j – and therefore of the region in which the superposition (3.77) is established – is determined, to first order, by the Floquet matrix element Ci j of the interaction operator Hqq . This was discussed in detail for two and three qubits, see Eq. (3.24). If Hqq contains only pairwise interactions, Ci j vanishes for |i − j| > 2 [68]. Consequently, two levels εi and εi+2m , separated by 2m steps in the Dicke ladder, are coupled only to m-th order. This implies that the width of their avoided crossing is of the order of J m , rendering it experimentally ever more challenging to tune into such resonances, as m grows. On the other hand, this detailed understanding of the width of the resonances may also serve as a tool to investigate an unknown interaction Hqq in a given physical system, e.g., to witness possible contributions beyond the pairwise interaction in a cold Rydberg gas [169]. This concludes our investigation of the entanglement of Floquet states of closed, driven quantum systems. In summary, we have achieved the following: I The entanglement of the Floquet states of the model Hamiltonian (3.8) was determined numerically as a function of the system parameters. It behaves resonantly in well-defined regions of the parameter space. I The phenomenon of entanglement resonance was explained by a perturbative analysis in the qubit-qubit interaction strength J. The key mechanism behind the resonances relies on avoided crossings in the quasi-energy spectrum. At these points in the parameter space, even a weak interaction can qualitatively alter the Floquet states of the non-interacting system and render them entangled. I The phenomenon was shown to be robust against all kinds of variations of the model. This establishes it as a universal feature of (weakly) interacting, periodically driven qubits. I Our perturbative analysis in the Floquet picture culminated in the simple resonance condition (3.32) for two qubits, or, more generally, in condition (3.78) for an arbitrary number of qubits. This condition reliably predicts the position of resonances in the parameter space. I The absence of some of the predicted resonances was attributed to symmetries of the driven system. 98 Chapter 4 Cyclo-stationary entanglement in open, driven quantum systems In the last chapter, we investigated how periodic driving affects entanglement in a closed quantum system. However, as discussed in Section 2.2, any realistic quantum system is open and therefore subject to decoherence. This is taken into account – on different levels of abstraction – in the present chapter. Decoherence adds two new conceptual aspects to our discussion: On the one hand, it renders the time evolution irreversible. This leads, in the generic case, to a unique cyclo-stationary long-term dynamics of the driven, open system; i.e., there is typically a unique periodically time-dependent asymptotic cycle into which every initial state eventually evolves. This is qualitatively different from the situation in closed, driven systems, where each of the d = dim H Floquet states defines a cyclo-stationary state. The second new aspect is that decoherence turns initially pure states into mixtures. Typically, the entanglement properties of the quantum state deteriorate in this process [35]. In particular, in the case of undriven systems, entanglement hardly ever survives on asymptotic times scales in the presence of decoherence; eventually, such systems relax into an equilibrium state in which entanglement is typically washed out by the environment-induced fluctuations [33]. In a periodically driven systems, however, the dynamics is permanently out of equilibrium. This bears the potential for a qualitatively different behavior in the asymptotic cycle, compared to the equilibrium state of the respective undriven system [39, 77]. The central question of this chapter is, hence, to what extent the detrimental effect of decoherence on entanglement can be counteracted by suitable periodic driving. 99 100 The chapter is structured as follows: In Section 4.1, we build upon the results of the previous chapter, but additionally take a weak coupling to the environment into account. In this case, the dynamics of the open system are adequately described by the Floquet Born-Markov master equation that we introduced in Section 2.2.2. Our goal will be to understand the entanglement properties of the resulting asymptotic cycle. In Sections 4.2 and 4.3, we adopt a different perspective: Instead of studying the asymptotic cycle of a particular model system, we aim for the globally optimal driving protocol H(t), i.e., the periodic H(t) that leads to the most entangled asymptotic cycle under a given, fixed incoherent dynamics. With this agenda in mind, the microscopically derived Floquet Born-Markov equation is no longer adequate, since the incoherent dynamics is not independent, but necessarily intertwined with the system Hamiltonian in this description. Therefore, we employ the “phenomenological” master equation (presented in Section 2.2.3) in these final parts of the thesis. As a brief anticipation of the results, we will show that the concurrence of the asymptotic cycle, time-averaged over the whole period of the cycle, can (under realistic environmental conditions) never exceed the value of C = 1/2. This optimal situation is, in fact, achieved with a time-independent cycle, i.e., a stationary state. There is, however, quite some mileage ahead of us to arrive at this result. 101 4.1 Driven qubits in Floquet Born-Markov description 4.1.1 Setting the stage In this section, we investigate the same system of interacting, externally driven qubits as in the last chapter; e.g., in case of two qubits, our ansatz for the system Hamiltonian H(t) is H(t) = h(t) ⊗ 1 + 1 ⊗ h(t) + Hqq , (4.1) into which we insert, for a start, our standard test case ω0 σz + F cos(ωt) σx , 2 Hqq = J (σ+ ⊗ σ− + σ− ⊗ σ+ ) , h(t) = (4.2) (4.3) as seen in Eqs. (3.8) to (3.10) and (3.12). In contrast to the last chapter, each qubit is now also coupled to an (individual) environment,1 modeled by a bath of harmonic oscillators at temperature β −1 = kB T . Throughout this section, we assume this interaction to be much weaker than all energy scales in the system Hamiltonian H(t). The system dynamics are then described by the Floquet Born-Markov master equation (2.54), as discussed in Section 2.2.2. Two-level atoms coupled to the electromagnetic continuum To specify the details of the environment coupling, we focus our investigation on the workhorse model of quantum optics [86]: Qubits are physically represented by two-level atoms, and the harmonic oscillators of the environment correspond to the continuum of quantized modes of the surrounding electromagnetic field. Each two-level atom couples to the electric field via its electric dipole operator. This interaction can be brought into the desired form (2.65), N Hint = ∑ Xm ⊗ ∑ gn,m (a†n,m + an,m ) , (4.4) m=1 n if we identify [41, 77, 86] (m) Xm ≡ eσx , r 2πωn gn,m ≡ i . V 1 Here, (4.5) (4.6) we assume individual environments for each qubit, rather than a collective one. As discussed in the context of Fig. 2.3, this is a conservative assumption, since individual environments are typically less favorable for preserving entanglement [101, 102]. 102 Here, e denotes the elementary charge, V the mode volume, and ωn the frequencies of the bath modes. In addition to the interaction operator Hint , we also need to specify the spectral density S(ω) of the environment, which was defined in Eq. (2.51). Since the number of available modes in three-dimensional free space scales quadratically in ω [86], we have S(ω) = V ω 2, (2πc)3 (4.7) with c the speed of light. Putting all pieces together, the environment-induced transition rates read (2.52) (2.66) γi j = |g̃|2 ∞ ∑ k=−∞ |ε ji + kω|3 |Nth (ε ji + kω)| 1 with = T e and g̃ = i . 2πc3/2 (m) xi j (k) Z T 0 N ∑ |x ji (m) (k)|2 , (4.8) m=1 (m) dt hΦi (t)|σx |Φ j (t)i e−ikωt (4.9) (4.10) Example: A single, undriven qubit To elucidate the meaning of these expressions, let us evaluate them for the simplest case of a single two-level atom (N = 1) in the absence of a driving field (F = 0). The system is autonomous then, and the Floquet states coincide with the eigenstates |0i and |1i of σz . Consequently, the transition matrix elements xi j (k) read 1 T 1 x10 (k) = dt h0|σx |1i e−ikωt = T 0 T x01 (k) = . . . = δk,0 ; Z Z T 0 dt e−ikωt = δk,0 ; (4.11) 1 T dt h0|σx |0i e−ikωt = 0; x00 (k) = T 0 x11 (k) = . . . = 0. Z With the quasi-energies being ε0 = −ω0 /2 and ε1 = +ω0 /2 (implying ε10 = ω0 ), the transition rates read (2.44) γ10 = |g̃|2 |ω0 |3 |Nth (−ω0 )| = |g|2 |ω0 |3 (Nth (ω0 ) + 1), 2 3 (2.44) 2 3 γ01 = |g̃| |ω0 | |Nth (+ω0 )| = |g| |ω0 | Nth (ω0 ). (4.12) Readers with a background in quantum optics will immediately recognize these expressions: They describe the decay of the excited level |1i at rate γ10 , due to 103 spontaneous decay and stimulated emission, while the inverse process of absorption of photons from the environment occurs at rate γ01 . Due to the absence of a driving field, only transitions with k = 0 are allowed; in the driven case, however, the Floquet states are no longer time-independent, and driving-assisted transitions with k 6= 0 can contribute to the incoherent rates γi j . 4.1.2 Determining entanglement of the asymptotic cycle According to our discussion in Section 2.2.2, evolution under the Floquet BornMarkov master equation (2.54) eventually ends up in an asymptotic cycle ρac . The cycle is, in fact, always an incoherent mixture of the Floquet states |Φi (t)i of the system Hamiltonian H(t): ρac (t) = ∑ w∗i |Φi (t)i hΦi (t)| . (4.13) i Having studied thoroughly the Floquet states of H(t) in the last chapter, the remaining task is to determine the stationary weights w∗i . To this end, we recall definition (2.61) of the matrix M that governs the rate equation (2.60) for the weights: (M)i j = γ ji − δi j ∑ γil . (4.14) l The Perron-Frobenius theorem [100] ensures that M has at least one vanishing eigenvalue. The corresponding eigenvector ~w comprises the sought stationary weights w∗i .2 If the nullspace of M has more than one dimension, every vector in it defines a different asymptotic cycle. For all results presented in the following, however, we have numerically verified that this is never the case, so that the dynamics are always completely relaxing to a unique asymptotic cycle. We emphasize that the parameter g̃, as defined in Eq. (4.10), is irrelevant for the asymptotic cycle, because it simply appears as a constant factor |g̃|2 in the incoherent rates γi j (and hence in the definition of M). Consequently, a rescaling of g̃ by a factor does not change the nullspace of M and leaves the stationary state invariant. Only the speed at which the asymptotic state is approached is altered by such a rescaling.3 2 The eigenvector must, of course, be normalized to ∑i w∗i = 1. fact that ρac is independent of g̃ may at first glance be surprising, because g̃ determines the strength of the system-environment coupling. Intuitively, one would expect this strength to have an influence on the asymptotic state. Mind, however, that the assumption of weak environment coupling is always understood when employing the (Floquet) Born-Markov master equation. In this regime, the precise strength of the environment coupling is irrelevant for the asymptotic state. 3 This 104 Similar to the last chapter, we quantify entanglement of the asymptotic cycle by considering the time average of an entanglement measure E : E as ≡ 1 T Z T 0 dt E (ρac (t)). (4.15) Here, we mainly focus on the case of two qubits and employ as entanglement measure Wootters’ formula (2.100) for the concurrence. In summary, a complete analysis of the driven, open system consists of the following steps: I Fix the parameters ω0 , F, ω, and J of the system Hamiltonian H(t) and calculate its Floquet states |Φi (t)i and quasi-energies εi . (m) I Determine the Floquet transition amplitudes xi j according to Eq. (4.9). I Fix the inverse temperature β of the environment, and calculate the transition rates γi j from Eq. (4.8). I Set up the matrix M, Eq. (4.14), and determine its nullspace. Verify that it is one-dimensional, and use the corresponding eigenvector ~w∗ to define the asymptotic cycle ρac (t) by virtue of Eq. (4.13). I Evaluate the time-averaged entanglement E as , Eq. (4.15), of the asymptotic cycle. The numerically obtained results of this procedure for N = 2 qubits are presented in Fig. 4.1. Similar to the analysis of the closed system in the last chapter, we use the driving frequency ω as the unit of energy and focus on weak qubit-qubit coupling (J = 0.02ω); the entanglement E as of the asymptotic cycle is then plotted in the two dimensional parameter plane of F and ω0 , with the bath temperature increasing from β −1 = 0 in the top row to β −1 = ω in the bottom row. At each temperature, two details are shown in panels (a2-d2) and (a3-d3). They highlight the regions of weak driving (F < 0.05ω) around ω0 = ω and ω0 = 0, respectively. 4.1.3 Understanding the entanglement of the asymptotic cycle Most strikingly, the plots largely reproduce the patterns found for Floquet state entanglement in the closed system, cf. Fig. 3.3: E as vanishes in most parts of the parameter plane, but peaks in the same, sharply defined regions where the entanglement resonances of the closed system occur. Yet, this comes at no 105 β −1 = 0 (a1) (a3) β −1 = 0.02ω (b1) (b2) (b3) β −1 = 0.5ω (c1) (c2) (c3) (d1) β −1 = 1ω (a2) (d2) (d3) Figure 4.1: (a1-d1): Entanglement E as of the asymptotic cycle of two driven qubits, for increasing bath temperature β −1 (from top to bottom row). The qubit-qubit interaction Hqq , Eq. (4.3), is weak (J = 0.02ω). Weak, but finite entanglement emerges in regions where also the closed system behaves resonantly, cf. Fig. 3.3. (a2-d2): Details for weak and near-resonant driving (ω0 ≈ ω and F ω). Values up to E as = 0.5 are reached in this region. (a3-d3): Another detail around ω0 ≈ 0, where maximal entanglement E as = 1 is reached at zero temperature. This resonance disappears, however, at temperatures of the order of the qubit-qubit interaction J (i.e., at β −1 = 0.02ω). 106 surprise: We argued earlier (in the beginning of Chapter 3) that entangled Floquet states |Φi (t)i in the closed system are a necessary precondition for an entangled asymptotic cycle ρac of the open system, since ρac is a mixture of Floquet states and therefore certainly separable, if the Floquet states are. Entanglement resonance around ω0 ≈ 0 In contrast to the closed system, however, maximum entanglement E as = 1 is only rarely achieved in the asymptotic cycle of the open system. In fact, it is limited to the region in the lower left corner of the parameter plane (F ω, ω0 ω) and to low temperature (β −1 < 0.02ω), as shown in detail (a3) of Fig. 4.1. The behavior in this parameter region can already be understood by analyzing the undriven system, F = 0. The resulting static system Hamiltonian H is easily diagonalized: ω0 0 J ω0 H= (σz ⊗ 1 + 1 ⊗ σz ) + J (σ+ ⊗ σ− + σ− ⊗ σ+ ) = 2 J 0 −ω0 = ω0 |11i h11| − ω0 |00i h00| + J |ψ + i hψ + | − J |ψ − i hψ − | . (4.16) Here, |ψ ± i are the maximally entangled Bell states, defined in Eq. (2.87), and the matrix representation of H refers to the computational basis. In the absence of driving, the asymptotic cycle ρac degrades to a stationary state. It coincides with the Gibbs state at inverse temperature β , as discussed in Section 2.2.2: −β ω 0 e −β H cosh β J sinh β J 1 (2.63) e = ρac = . tr(e−β H ) 2 (cosh β ω0 + cosh β J) sinh β J cosh β J eβ ω0 Its entanglement, as measured by the concurrence, is:4 sinh(β J) − 1 E (ρac ) = max 0, . cosh(β J) + cosh(β ω0 ) 4 This results follows from a simple formula for the concurrence of states that have “X” form in the computational basis [170]. 107 2ω#0 |11� ω"0 |ψ+ � 0! |ψ− � −ω ï"0 −2ω ï#0 }J − ω0 |00� ω"0 J $%&%t !0 2ω #0 ! Figure 4.2: Spectrum of the undriven system (F = 0), see Eq. (4.16). The resulting stationary state (i.e., the Gibbs state) is maximally entangled if (and only if) the interaction dominates (J > ω0 ) and the temperature is low (β −1 < J − ω0 ). While the former condition ensures that the ground state of the system is |ψ− i, the latter requirement rules out thermal admixing (red arrow) of the first excited state |00i, which spoils the entanglement of the Gibbs state. Since sinh(β J) < 1 for β J < 1, the Gibbs state is separable above temperatures β −1 > J. In the zero-temperature limit β → ∞, on the other hand, the hyperbolic sine and cosine can be approximated by a simple exponential. Then, we have eβ J − 2 E (ρac ) ≈ max 0, β J e + eβ ω0 ! ≈ 1 1 + eβ (ω0 −J) β →∞ −→ ( 0, if ω0 > J 1, if ω0 < J . (4.17) In combination, these statements explain why maximum entanglement E as = 1 emerges in parameter regions in which both the temperature β −1 and the qubit energy splitting ω0 are smaller than the qubit-qubit interaction strength J. The physically interpretation of this result is illustrated in Fig. 4.2: If the qubit-qubit interaction dominates over the local terms of the undriven system (J > ω0 ), the maximally entangled Bell state |ψ − i becomes the energy ground state. At zero temperature, the Gibbs state is precisely the ground state, and thus maximally entangled. This entanglement is lost as soon as the temperature becomes comparable to the energy gap J − ω0 between the ground state |ψ − i and the first excited state |00i. 108 Less pronounced entanglement resonance around ω0 ≈ ω The above arguments are based on the undriven system, and can therefore not explain the behavior of E as at finite driving strength F. E.g., the sharp step-function behavior of E as derived in Eq. (4.17) is only valid in the limit F → 0 in Fig. 4.1(a3). Moreover, we observe non-vanishing entanglement E as in several parameter regions with ω0 and β −1 much larger than J in Fig. 4.1, which are not covered by our analysis so far. These genuine driving-induced entanglement resonances, however, reach only half of the maximum entanglement, E as = 0.5. This behavior can be quantitatively understood for the resonance at weak and near-resonant driving, F ω and ω0 ≈ ω, which is highlighted in Fig. 4.1(a2-d2). There, the driving field is adequately described in RWA, which was introduced in detail in Section 3.2.2. The single-qubit quasi-energies µ± read p 1 µ± = ω ± ∆2 + F 2 (4.18) 2 in RWA, see Eq. (3.38), with ∆ = ω0 − ω denoting the detuning parameter; the corresponding single-qubit Floquet states are |φ+ (t)i = 1 a2 + b2 (a |0i eiωt + b |1i), 1 |φ− (t)i = 2 (b |0i eiωt − a |1i), a + b2 (4.19) √ with a = ∆− ∆2 + F 2 and b = F. However, we find in Fig. 4.1(a3) that the highest value of E as = 0.5 is achieved for very small driving amplitudes, F |∆|. This allows us to set a ≈ 1, b ≈ 0 in our analysis and to assume µ± ≈ (ω ± ∆)/2 for the quasi-energies. Sticking to the recipes developed in Chapter 3, we can write down the twoqubit Floquet states |Φi (t)i and the associated quasi-energies εi in the absence of interaction (J = 0):5 ε1 = −∆, ε2 = 0, ε3 = 0, ε4 = +∆, 5 For |Φ1 (t)i = e+iωt |00i ; 1 (2.87) |Φ2 (t)i = √ (|01i + |10i) = |ψ + i ; 2 1 (2.87) |Φ3 (t)i = √ (|01i − |10i) = |ψ − i ; 2 |Φ4 (t)i = e−iωt |11i . (4.20) convenience, we shift quasi-energies by one Floquet zone in Eq. (4.20), εi → εi − ω, and compensate for this by appending a factor e−iωt to the Floquet states. Moreover, we use a different ordering of the label i here, compared to Chapter 3. 109 (m) From this, one can calculate the transition amplitudes xi j (k), which enter in the transition rates γi j , and finally define the transition matrix M. This is done in appendix A.1. In the limit of zero temperature, β → ∞, we find: 0 0 0 0 0 0 2 3 1 −1 M = |g̃| ω0 (4.21) . 1 0 −1 0 0 1 1 −2 This transition matrix has the stationary distribution ~w∗ = (1, 0, 0, 0), so that the asymptotic cycle – which is, in fact, time-independent in this case – is the fully de-excited state: ρac = |00i h00| . (4.22) Thus, ρac is not entangled at zero temperature. It is intuitively clear that this finding does not improve at higher temperatures. In fact, in the infinite temperature limit (β → 0), we find −2 1 1 0 1 −2 0 1 M = |g̃|2 ω03 Nth (ω0 ) (4.23) 1 0 −2 1 0 1 1 −2 in appendix A.1, and ρac becomes the completely mixed state ρac = 1 1, 4 (4.24) yielding E as = 0 as well. In fact, ρac is separable at any intermediate temperature as well, as discussed in appendix A.1. Similar to our discussion of the closed system, deviations from this analysis may only occur when the Floquet states deviate from Eq. (4.20). This is only the case when the corresponding quasi-energy levels εi become near-degenerate in an avoided level crossing. In fact, a close inspection of Fig. 4.1(a2) reveals that the resonance under investigation is most pronounced around ω0 = 0.98ω, i.e., at −∆ = 0.02ω = J. At this point, the levels ε1 and ε2 cross, due to the following argument: The qubit-qubit interaction Hqq shifts the unperturbed level ε2 by the diagonal perturbation matrix element C22 , which amounts in our case to (2.18) C22 = 1 T Z T 0 dt hΦ2 (t)|Hqq |Φ2 (t)i = J hψ + |σ+ ⊗ σ− + σ− ⊗ σ+ |ψ + i = J. (4.25) 110 On the other hand, ε1 and ε4 are not shifted to first order, since C11 = C44 = 0. Thus, ε1 = −∆ and ε2 = J are degenerate at −∆ = J. In the region of interest, the finite interaction strength J induces an avoided crossing6 between ε1 and ε2 . In its center, the respective Floquet states turn into 1 |Φ1 (t)i = √ e+iωt |00i + |ψ + i , 2 1 |Φ2 (t)i = √ e+iωt |00i − |ψ + i , 2 (4.26) while the remaining states |Φ3 (t)i and |Φ4 (t)i are unaffected, compared to Eq. (4.20). Now, the transition matrix M at zero temperature reads (see appendix A.1): −1 1 0 0 0 1 2 3 1 −1 0 M = |g̃| ω0 (4.27) . 2 4 2 −4 0 2 2 4 −8 This results in stationary weights ~w∗ = 12 (1, 1, 0, 0), and hence in 1 1 1 1 |Φ1 (t)i hΦ1 (t)| + |Φ2 (t)i hΦ2 (t)| = |00i h00| + |ψ + i hψ + | . 2 2 2 2 (4.28) This 50/50 mixture of the separable state |00i and the maximally entangled Bell state |ψ + i yields a concurrence value of C = 21 and explains, hence, the finding E as = 0.5 in Fig. 4.1(a2).7 It is discussed in the appendix that this picture remains valid as long as the temperature β −1 is well below ω0 ; at higher temperatures, one approaches again the completely mixed state ρac = 1/4 with E as = 0. This explains why the resonance under investigation is fading in Fig. 4.1(c3) and vanishes completely in Fig. 4.1(d3). In summary, the genuine driving-related resonance around ω0 ≈ ω can explicitly be derived within an RWA description of the driving. While it reaches only E as = 0.5, it is by far more robust against finite temperatures than the “static resonance” with E as = 1 that we discussed beforehand (β −1 ω0 vs. β −1 J). ρac (t) = 6 Note that the matrix element C12 that determines the width of the avoided crossing vanishes in the limit F → 0. Therefore, the resonance does non exist in the undriven system, but is a unique feature of the driven system. 7 Strikingly, the asymptotic cycle ρ (t) in Eq. (4.28) is time-independent, i.e., it is merely a ac stationary state and not truly a cycle. In the next Section 4.2, we will find that (4.28) can indeed be seen as the stationary state of a static Hamiltonian H. Basically, this static H coincides with the periodically time-dependent H(t) studied here, if one transforms H(t) to a rotating reference frame. 111 We refrain from analyzing the remaining resonances in an explicit calculation, since they are plausibly triggered by similar mechanisms. E.g., the resonance around ω0 = 3ω and F ≈ .5ω also has E as = 0.5 and vanishes at temperatures β −1 ≈ ω0 . 4.1.4 Varying the ingredients of the model So far, all our results were obtained within the quantum optical context introduced in Section 4.1.1, i.e., they refer to two-level atoms coupled to the electromagnetic continuum. In the following, we investigate a number of modifications of both the system Hamiltonian H(t) and the environment model. This will reveal that the previous findings are not a peculiarity of the particular context we studied, but rather a universal feature of driven, interacting two-level systems in the presence of an environment that obeys a Born-Markov description. Figure 4.3 compiles the results for the zero temperature case β −1 = 0. We emphasize, however, that the universality of our findings equally holds at finite temperature. For a start, the dipole operator Xm = eσx , which couples the system to the environment, is replaced by eσz in Fig. 4.3(a). This is a suitable ansatz, e.g., to describe the environment coupling of some types of superconducting qubits [95]. The remaining ingredients and parameters of the system are identical to Fig. 4.1(a1). Only a very close inspection reveals tiny differences between these two plots. In fact, they are restricted to the resonance with maximal entanglement E as = 1 around ω0 ≈ 0 and F ω, which is no longer observed when the coupling operator is σz . Apart from this, the details of the coupling operator have no significant influence on the entanglement of the asymptotic cycle. This is reasonable, because the single-qubit Floquet states |φ± (t)i generically have contributions from both the ground state and the excited state for finite driving strength F, so that the matrix elements hφ+ (t)|σx |φ− (t)i do not differ systematically from hφ+ (t)|σz |φ− (t)i. Since these elements determine, in the end, the transition amplitudes xi j (k) and hence the rates γi j , it is plausible that the asymptotic state is not crucially affected by this modification. In Fig. 4.3(b), the qubit-qubit interaction mechanism (4.3) is replaced by Hqq = σx ⊗ σx , similar to what we studied in Fig. 3.12(b) for the closed system. Compared to our standard case in Fig. 4.1(a1), we observe a slight broadening of entanglement resonances; apart from this, this modification has no major impact on the entanglement properties of the asymptotic cycle. This is plausible, because we argued in Section 3.2.4 that the details of the interaction mechanism do not alter the Floquet states of H(t) significantly. Next, the qubit-qubit interaction strength is increased by one order of magnitude to J = 0.2ω in Fig. 3.12(c). This manifests itself in much broader 112 (a) X = eσz , instead of eσx (b) Hqq = Jσx ⊗ σx (c) J = 0.2ω, instead of 0.02ω (d) N = 3 qubits (e) S(ω) ∝ ω 0 (f) S(ω) ∝ ω 1 Figure 4.3: Several variations of the model assumptions that underlie Fig. 4.1, for vanishing temperature β −1 = 0. The different modifications are described in detail in the main text. None of them changes the findings fundamentally. 113 resonances. This is again expected from our analysis of the closed system; to see this, compare the upper and central row of Fig. 3.3. Nevertheless, all the points made in Section 4.1.3 remain valid. E.g., the entanglement resonance around ω0 ≈ 0 still yields E as = 1 and still extends from ω0 = 0 to ω0 = J. The resonance at F ω and ω0 ≈ ω is also present and reaches, as expected, its maximal value of E as = 0.5 at −∆ = J, i.e., ω0 = 0.8ω. Results for N = 3 qubits are shown in Fig. 3.12(d). Here, entanglement is quantified by the quasi-pure approximation to the generalized concurrence, see (2.101). Thus, in contrast to the investigation of three qubits in the setting of closed systems (Section 3.3), we no longer quantify exclusively GHZ entanglement, but also include the W and bi-separable entanglement classes now. With this choice, the resulting E as resembles closely the two-qubit scenario, but reaches values up to E as ≈ .576. This number could theoretically be even larger, since the quasi-pure approximation is only a lower bound to the exact value. A detailed investigation of the resonances, however, reveals that the underlying asymptotic cycle is a (timeindependent) 50/50 mixture of de-excited and |Wi state, ρac = 1 1 |000i h000| + |Wi hW| , 2 2 (4.29) √ having the exact concurrence value of C = 1/ 3 ≈ 0.577. So far, we assumed a quadratic behavior of the spectral density S(ω), see Eq. (4.7). This corresponds to the mode density of three-dimensional free space. In one or two dimensions, the density scales as ω 0 and ω 1 , respectively. We therefore study these two cases in panel (e) and (f) of Fig. 4.3. In particular, S(ω) ∝ ω 0 corresponds to an Ohmic bath,8 which defines an important scenario for open quantum systems in the solid state context [171, 172]. From visual inspection, we see that the impact of this modification is limited to the parameter region ω0 < J = 0.02ω. Here, the resonance with E as = 1 extends to larger values of the driving strength F, compared to the standard case of Fig. 4.1. Its character, however, is not qualitatively different. In particular, is also vanishes quickly at finite temperatures, i.e., if β −1 6< J (not shown in Fig. 4.3). In conclusion, our findings are robust against all kinds of modifications of the model assumptions, as long as the main ingredients remain unaffected: Weakly interacting qubits, driven by an external field, and coupled to an environment that obeys a Born-Markov description. 8 In the literature, the notion of spectral density often refers to the product ω · S(ω), which is then linear in ω for an Ohmic bath. The √ additional factor ω is due to the system-environment coupling strength, which is proportional to ω in our model and enters quadratically in γi j . 114 Summarizing Section 4.1, we found that maximal entanglement E as = 1 in the asymptotic cycle of the Floquet Born-Markov master equation can be achieved, but only if the bath temperature β −1 is smaller than the qubit-qubit interaction strength J. This resonance, however, can already be generated without a driving field. A unique feature of the driven system, however, is that the state (4.28) with E as = 0.5 (or (4.29) in case of three qubits) can be reached in the asymptotic cycle, if the driving parameters are suitably adjusted. This result holds up to temperatures of the order of the energy splitting ω0 of the individual qubits, which is typically much larger than their interaction strength J. Thus, the 50% loss in entanglement pays off with a significantly increased robustness against thermal noise. 115 4.2 The optimal stationary state under fixed incoherent dynamics All results of the previous section were obtained with the Floquet Born-Markov master equation (2.54). The central assumption underlying its derivation in Section 2.2.2 is a weak coupling between system and environment, leading to incoherent transition rates γi j that are much slower than any coherent dynamics generated by the system Hamiltonian H(t). In Section 2.2.3, we have argued that an alternative approach is to regard the dissipator D(ρ) in the abstract Lindblad master equation (2.29) as fixed, with (more or less) phenomenologically chosen Lindblad operators Li and rates γi . This approach is appropriate, as soon as the incoherent transitions occur at a rate that is comparable to the coherent processes. Within the approximation of independent rates, the coherent dynamics of the system Hamiltonian H(t) are added on top of this, see Eq. (2.67). Formulation of the problem To be specific, we focus on a Lindblad master equation for N qubits, ρ̇(t) = −i[H(t), ρ(t)] + D(ρ(t)), (4.30) with the following phenomenological ansatz for the dissipator: D(ρ) =γ− γ+ γd N ∑ (σ− 1 (m) (m) (m) ρσ+ − {σ+ σ− , ρ}) + 2 ∑ (σ+ 1 (m) (m) (m) ρσ− − {σ− σ+ , ρ}) + 2 ∑ (σz ρσz m=1 N m=1 N m=1 (m) (m) (m) (m) (4.31) 1 (m) (m) − {σz σz , ρ}). {z } 2| =2ρ It describes the three most common incoherent processes: Decay of each qubit from the excited state |1i to the de-excited state |0i at rate γ− , excitation from |0i to |1i at rate γ+ , and dephasing between these two states at rate γd . The microscopic origin of the decay and excitation processes is usually the coupling to a thermal bath, in which case the ratio of their rates is determined by the bath temperature β −1 = kB T , see Eq. (4.12): ) γ− ∝ (Nth (ω0 ) + 1) γ+ ⇒ = e−β ω0 . (4.32) γ− γ+ ∝ Nth (ω0 ) 116 Here, ω0 denotes again the energy splitting between de-excited and excited state of the bare qubits. Dephasing, on the other hand, typically occurs when the energy splitting is not perfectly constant in time, but involves a certain amount of (white) noise, e.g., due to a fluctuating magnetic field, which determines ω0 in atomic and spin qubits. As mentioned above, we regard these three incoherent processes and their rates as fixed and unavoidable in this section. In the absence of any additional coherent dynamics, H(t) = 0, the unique stationary state of (4.30) is the thermal state 1 ρth = e−β ω0 |11i h11| + |10i h10| + |01i h01| + eβ ω0 |00i h00| , 2(cosh(β ω0 ) + 1) (4.33) which is obviously not entangled. Hence, no entanglement can persist in the asymptotic time limit without a suitable system Hamiltonian H(t). At this point, we could proceed similarly to the previous section and insert our model Hamiltonian (4.1) into the dynamics (4.30) to study the parameter regime in which the asymptotic cycle is significantly entangled. However, we pursue a more general goal in this section: Instead of studying a particular ansatz for the Hamiltonian, we rather aim for the optimal driving protocol among all conceivable scenarios. Hence, we seek the periodic Hamiltonian H(t) that leads to the most entangled asymptotic cycle ρac (t). The outcome of this investigation is, a priori, not intuitively clear: Is there a Hamiltonian, such that the resulting asymptotic cycle has maximal entanglement, E as = 1? Or can one at least do better than the value E as = 0.5 that we found in most entanglement resonances of the driven, open system in Born-Markov description in the last section? To achieve our ambitious goal, we divide the investigation into two steps: In the present Section 4.2, we investigate undriven systems, described by a time-independent Hamiltonian H. The extension to driven systems with a periodically time-dependent H(t) follows in Section 4.3. The reason for this division is two-fold: On the one hand, the analysis of undriven systems provides the conceptual foundations that alone enable the investigation of driven systems later on. On the other hand, determining the optimal undriven system is also an investigation of its own merit. The stationary state of autonomous systems With a time-independent Hamiltonian H, the master equation (4.30) becomes autonomous. Hence, instead of an asymptotic cycle ρac (t), an initial state generically evolves into a stationary state ρss , characterized by ρ̇ss = −i[H, ρss ] + D(ρss ) = 0, i.e., i[H, ρss ] = D(ρss ). (4.34) 117 In fact, it was shown in Ref. [173] that the Lindblad master equation with the dissipator (4.31) always has a unique stationary solution ρss and, moreover, that every initial state evolves into this solution in the long time limit t → ∞.9 The stationary state ρss itself, however, depends on the particular choice of H. Hence, the task in this section is to find the optimal H that leads to the most entangled ρss . As an alternative to quantifying entanglement by a genuine entanglement measure E (ρss ), we also consider the simpler Bell state fidelities Fidφ ± (ρ) = hφ ± |ρ|φ ± i and Fidψ ± (ρ) = hψ ± |ρ|ψ ± i (4.35) in this section. They are the quantities of interest if one specifically wants to create one of the four maximally entangled Bell states |φ ± i or |ψ ± i, as defined in Eqs. (2.86) and (2.87). Like the concurrence, they range from zero to one. While a high Bell state fidelity always goes along with strong entanglement, the opposite is not true; hence, the Bell state fidelities do not define full-fledged entanglement measures. Nevertheless, they can be given an operational meaning in terms of the fidelity of a teleportation protocol via the state ρ [174]. Overall, the section is organized as follows: First, we gain some intuition by studying a particular model Hamiltonian for two qubits, as well as “typical” Hamiltonians drawn from a random distribution. After that, we characterize in a systematic way the set of all states that can possibly become stationary and determine the most entangled state in this set. We explicitly provide, hereafter, the Hamiltonian that leads to this optimal situation and discuss its experimental implementation. A generalization to the case of many qubits follows. Finally, we discuss the relation of our findings to the idea of dissipative state preparation that was recently introduced in the literature [175, 176]. 4.2.1 Introductory examples Two qubits with Ising interaction Let us consider the exemplary Hamiltonian of two qubits with a one-dimensional Ising interaction [177, 178] of strength J: H= ω0 (1 ⊗ σz + σz ⊗ 1) + J σx ⊗ σx . 2 (4.36) For starters, we set γ+ = γd = 0 in the dissipator (4.31) and consider, thus, spontaneous decay at a rate γ− ≡ γ only. The condition (4.34) of stationarity can 9 This statement holds as long as either γ− 6= 0 or γ+ 6= 0. 118 then explicitly be solved for the stationary state ρss [40]: 14 x x∗ 1 2 + |x| |00i h00| + |00i h11| + |11i h00| , ρss = 1 + |x|2 4 2 2 (4.37) with x = (ω0 + iγ)/J. Due to the “X” shape of this state in the computation basis, 1 0 0 2x 0 1 0 0 1 ρss = (4.38) , 4(1 + |x|2 ) 0 0 1 0 2x∗ 0 0 1 + 4|x|2 there is a simple expression for its concurrence C [124, 170]: ) ( |x| − 12 . C (ρss ) = max 0, 1 + |x|2 (4.39) Its dependence on the decay rate γ and the interaction strength J is plotted in Fig. 4.4. Maximizing C (ρss ) as a function of |x| leads to an optimal value of 1 √ (4.40) C (ρss ) = ( 5 − 1) ≈ 0.31, 4 which is achieved when |x| is equal to the golden ratio, i.e., if q ω02 + γ 2 1 √ |x| = = (1 + 5). (4.41) J 2 In the limit of strong interaction, J max(ω0 , γ), which is tantamount to |x| 1, the stationary state approaches the completely mixed state ρss = 14 /4. Therefore, we find C (ρss ) = 0 in this limit. In the opposite case of weak interaction, J max(ω0 , γ), or |x| 1, we have ρss = |00i h00|, which is not entangled, either. The Bell state fidelities of the stationary state (4.37), on the other hand, read 1 ±2ℜ(x) − 1 1 + and Fidψ ± (ρss ) = , (4.42) 2 2 4(1 + |x| ) 4(1 + |x|2 ) yielding the optimal values √ 3 + 5 1 Fidφ ± (ρss )x=± 1+√5 = ≈ 0.65 and Fidψ ± (ρss )x=0 = . (4.43) 8 4 2 In summary, with the Ising Hamiltonian (4.36) and in the presence of spontaneous decay, it is not possible to achieve a stationary state ρss that exceeds the values of C (ρss ) ≈ 0.31, Fidφ ± (ρss ) ≈ 0.65 and Fidψ ± (ρss ) = 0.25, no matter how one adjusts the parameters ω0 and J of the Hamiltonian in comparison to the dissipation rate γ. The question is thus: Can one do better than this? Fidφ ± (ρss ) = 119 Figure 4.4: Concurrence C (ρss ) of the stationary state ρss of the one-dimensional Ising Hamiltonian (4.36), as derived in Eq. (4.39). The concurrence value is encoded in the gray scale, while the interaction strength J and the decay rate γ− ≡ γ are varied, on a logarithmic scale, over two orders of magnitude along the axes of the plot. The strongest entanglement of C (ρss ) ≈ 0.31 (black arrow) is reached when the parameters obey expression (4.41). Random Hamiltonians To gain some intuition about the entanglement properties of typical stationary states, we study an ensemble of random Hamiltonians H and investigate the distribution of the concurrence of the resulting stationary states. As before, we consider two qubits undergoing spontaneous decay at rate γ− ≡ γ. The Hamiltonian is drawn from the Gaussian Unitary Ensemble (GUE) [159].10 Thus, we set 1 H = (X + iY ) + h.c. (4.44) 2 and fill the 4 × 4 matrices X and Y with real random entries, independently drawn from a normal distribution with zero mean and standard deviation J. The parameter J defines, hence, the energy scale of the Hamiltonian. Since the condition (4.34) of stationarity is independent of a scaling factor, the ratio J/γ alone determines the stationary state, describing the relative strength of coherent and dissipative dynamics. 10 The choice to draw from the GUE, instead of the orthogonal or symplectic ensemble, is to some extent arbitrary. The GUE, however, is in some sense the most elementary ensemble, since it describes systems that are free of any symmetry. This seems a reasonable property in our context. 120 For different values of J/γ, we have generated 104 realizations of H. For each realization, we determined the corresponding stationary state ρss by numerically solving Eq. (4.34) and calculated both its concurrence and Bell state fidelities. Figure 4.5 shows the results as a function of J/γ on the abscissa. The color code indicates, on a logarithmic scale, the probability density to find a certain value of the quantity of interest in the ensemble. Hence, each column of the plots represents a histogram of the concurrence or the Bell state fidelity at a fixed value of J/γ. At the same time, the red curves visualize the analytical results (4.39) and (4.42) obtained for the Ising Hamiltonian with ω0 = 0 and ω0 = J (solid and dashed curves, respectively). Regarding its ability to generate an entangled stationary state, this Hamiltonian is apparently a generic member of the GUE. E.g., the statistical ensemble achieves the highest concurrence value of C (ρss ) ≈ 0.35 when coherent and dissipative dynamics are of comparable strength, i.e., around J/γ ≈ 1. Likewise, the Ising Hamiltonian reaches its maximum value of C(ρss ) ≈ 0.31, e.g., for ω0 = 0 and J/γ ≈ 0.61. This result appears surprising at first sight; intuitively, one expects weak dissipation to be less detrimental for entanglement than an intermediate dissipation strength in the regime J/γ ≈ 1. To elucidate this behavior, we compare the situation to the well-studied case of a single two-level system, driven by coherent radiation with amplitude F [86], and subject to spontaneous decay with rate γ. For weak driving, F/γ 1, the stationary state is simply the de-excited state ρss = |0i h0|. As F/γ increases, the stationary coherence h0|ρss |1i between ground and excited state grows, but never reaches the highest possible value. In the limit of strong driving F/γ 1, the stationary state turns into the completely mixed state, so the coherence drops to zero in this limit. Hence, not strong, but intermediate strength of the coherent dynamics are optimal for the coherence of the stationary state of a single qubit. This makes our observation plausible that strong coherent dynamics is per se not optimal for entanglement of the stationary state of two qubits, either. In a broader context, this result is reminiscent of the phenomenon of stochastic resonance [179], where coherent and incoherent processes, if suitably synchronized, jointly stimulate a strong response in a quantum system. The limit of strong dissipation, J/γ 1, has a more intuitive consequence: Here, the stationary state approaches the de-excited state ρss = |00i h00|, independently of the Hamiltonian. Hence, the concurrence of both the statistical ensemble and the Ising Hamiltonian vanish in this limit, and the Bell state fidelities approach Fidφ + = 0.5 and Fidψ + = 0. The most important conclusion to be drawn from Fig. 4.5 is that the probability of finding C > 0.35 is vanishingly small, irrespectively of J/γ. The 121 (a) (b) .5 .5 100 a+ = 0 a = a / 100 .25 1 0 .1 1 J/a 10 Concurrence Concurrence + 10 + ï .25 0 0.1 .1 ï a = a / 10 1 10 Probability density (c) 100 (d) .5 .5 FidΦ FidΨ + .75 + .75 .25 0 .25 .1 1 J/ γ 10 0 .1 1 J/ γ 10 Figure 4.5: (a) Distribution of the concurrence C of the stationary state ρss of two qubits that undergo coherent dynamics generated by a random Hamiltonian (from the GUE) of strength J and spontaneous decay of strength γ− ≡ γ. On the abscissa, the ratio J/γ varies over three orders of magnitude. At each value of J/γ, the gray scale indicates, on a logarithmic scale, the probability density to find a certain concurrence value (ordinate) in the ensemble. While the concurrence can in principle reach values up to C = 1, there is zero probability to find values above C > 0.35 in the ensemble. We therefore plot the interval 0 ≤ C ≤ 0.5 only. The red curves visualize the result (4.39) obtained for the Ising Hamiltonian (4.36), with ω0 = 0 (solid curves) and ω0 = J (dashed curves). (b) Probability density function of the concurrence, integrated over all values of J/γ. The plot is rotated by 90◦ , to adapt the abscissa to that of panel (a). Besides the case of pure decay (solid curve), which is considered in (a), results for finite excitation rate γ+ are also plotted (dashed and dotted curve). (c, d) Same as panel (a), but for the Bell state fidelities Fidφ + and Fidψ + , as defined in Eq. (4.35). The plots for the remaining Bell state fidelities Fidφ − and Fidψ − are identical to (c) and (d), apart from the fact that the red curve corresponds to ω0 = −J in this case. 122 value of C ≈ .31 derived for the Ising Hamiltonian is therefore close to the maximal entanglement that can be expected for a generic stationary state. The same holds for the φ ± fidelity: the random ensemble does not exceed the threshold Fidφ ± ≈ 0.65 of the Ising Hamiltonian. Only for the ψ ± fidelity, the ensemble significantly outperforms the Ising Hamiltonian and reaches values up to Fidψ ± ≈ 0.5. If one considers not only spontaneous decay at rate γ− ≡ γ, but also a finite excitation rate γ+ , the achievable entanglement drops to even smaller values. This is shown in Fig. 4.5(b), where the probability distribution of the concurrence, integrated over all values of J/γ, is plotted for three different values of γ+ /γ− . This behavior is expected, since an increasing ratio γ+ /γ− corresponds to increasing temperature, see Eq. (4.32), which is typically associated with decreasing entanglement. 4.2.2 Systematic derivation of the optimal stationary state The previous findings were of statistical nature. While they allow to draw conclusions for typical stationary states, they cannot exclude the existence of atypical states with much better entanglement properties. Therefore, our aim in the following is to systematically derive the globally optimal stationary state ρ ∗ , i.e., the most entangled state for which a Hamiltonian H ∗ exists such that ρ ∗ fulfills the stationarity condition (4.34). More abstractly formulated, the goal is to find the stationary state ρss that maximizes a certain quantity O(ρss ) of interest (in our case, the concurrence or the Bell state fidelity), for a given dissipator D. A natural way to tackle this problem would be the following: 1. Resolve the stationarity condition (4.34) for the stationary state ρss . 2. Express the stationary state as a function of the Hamiltonian: ρss = ρss (H). 3. Optimize O(ρss (H)) over all Hamiltonians H. This procedure has, however, two drawbacks that prevent us from applying it to our problem: Firstly, (4.34) can usually be inverted only by numerical means. Secondly, the set of all conceivable Hamiltonians is unbounded, which makes it difficult to perform an optimization procedure over this set. The set of stabilizable states Therefore, we take a different approach and directly maximize the quantity of interest in the set of stabilizable states S , which was, to our best knowledge, first 123 introduced in Ref. [180]: S = {ρ ∈ Q | ∃H : 0 = −i[H, ρ] + D(ρ)}. (4.45) Here, Q denotes the set of all quantum states, as defined in Eq. (2.20). By definition, S contains all quantum states that can become stationary under a given dissipative dynamics D(ρ). For every state ρ ∈ S , there is a suitable Hamiltonian H that renders this particular state stationary and in this sense “stabilizes” ρ. The set S itself, however, does not depend on a particular Hamiltonian, but it is exclusively determined by the dissipator D(ρ). The advantage of this approach is that S , being a subset of the state space Q, is a bounded set. This facilitates the optimization procedure. Furthermore, for a given state ρ ∈ S , it is possible to solve the stationarity condition (4.34) for the corresponding Hamiltonian H, i.e., to find the mapping ρss → H(ρss ). To derive this mapping, we start from the spectral decomposition of ρ: ρ = ∑ pα |αi hα| , α (4.46) Since ρ ∈ S , there is a H(ρ) such that the stationarity condition (4.34) is fulfilled. “Sandwiching” this condition with hα| from the left and |β i from the right results in i(pβ − pα ) hα|H|β i = hα|D(ρ)|β i . (4.47) For pα 6= pβ , this leads to hα|H|β i = i hα| D(ρ) |β i . pα − pβ (4.48) For pα = pβ , Eq. (4.47) implies that the dissipative matrix element hα| D(ρ) |β i vanishes, and that the Hamiltonian matrix element hα|H|β i can be chosen arbitrarily. This holds, in particular, for all diagonal elements hα|H|αi. Hence, given a stabilizable state ρ ∈ S , the corresponding stabilizing Hamiltonian is H= ∑ pα 6= pβ i hα| D(ρ) |β i |αi hβ | + ∑ Hαβ |αi hβ | , pα − pβ pα =pβ (4.49) with arbitrary11 elements Hαβ , e.g., Hαβ = 0. So far, membership in the set S is defined via the existence of a stabilizing Hamiltonian. This is certainly not a practical characterization to work with, if one 11 Of course, hermiticity of H must be retained, so the minimal requirement is Hαβ = Hβ∗ α . 124 wants to optimize an objective function O(ρ) over S . Therefore, we derive in the following an alternative, less elusive characterization of S . It relies on the fact that the coherent part of the evolution, generated by the term −i[H, ρ] in the master equation, induces strictly unitary dynamics. Accordingly, only the dissipative term D(ρ) can alter the eigenvalues pα of ρ. At a stationary state ρ ∈ S , however, we have D(ρ) = i[H, ρ], implying that the dissipative dynamics compensate for the coherent evolution. Thus, at this particular point in the state space, D(ρ) merely induces unitary dynamics as well, and the evolution under D(ρ) alone must leave the spectrum {pα } of ρ invariant. The spectrum, in turn, is uniquely determined by its leading d = dim H moments µn ,12 defined as µn = tr(ρ n ) = d ∑ (pα )n , n = 1, 2, . . . , d. (4.50) α=1 Therefore, the evolution under D(ρ) alone is only unitary, if it leaves all moments µn invariant, i.e., if ρ̇=D(ρ) d d ↓ 0 = µn = tr(ρ n ) = n tr(ρ n−1 D(ρ)). dt dt (4.51) Recognizing that µ1 = tr(ρ) = 1 is always conserved, this defines (d −1) necessary conditions for ρ ∈ S . In fact, if ρ has non-degenerate eigenvalues, the criterion is also sufficient, as stated by the following proposition: Proposition 1. Let ρ ∈ Q be a d-dimensional quantum state and D(ρ) the dissipative term of the master equation (4.30). If ρ ∈ S , i.e., if there is a Hamiltonian H that renders ρ stationary, then tr ρ n−1 D(ρ) = 0, ∀n = 2, . . . , d. (4.52) If ρ has non-degenerate eigenvalues, then (4.52) is a sufficient condition for ρ ∈ S . Proof. (Necessary condition.) Suppose we have ρ, H and D(ρ) such that ρ is stationary under (4.30), i.e., D(ρ) = i[H, ρ]. (4.53) For all n ∈ N, and in particular for n = 2, . . . , d, this implies −i tr ρ n−1 D(ρ) = tr ρ n−1 [H, ρ] = tr ρ n−1 Hρ − tr ρ n−1 ρH = tr (ρ n H) − tr (ρ n H) = 0. 12 This is due to the fact that there is a one-to-one correspondence between the spectrum of a d-dimensional quantum state ρ and its leading d moments µn , cf. Appendix B of Ref. [181]. 125 (Sufficient condition.) To prove condition (4.52) sufficient for the existence of a Hamiltonian H that renders ρ stationary, we take the spectral decomposition of ρ, (4.54) ρ = ∑ pα |αi hα| , α and suppose that it has non-degenerate eigenvalues pα . In order to prove that ρ is stationary under (4.30), we show that hα|D(ρ)|β i = i hα|[H, ρ]|β i , ∀α, β = 1, . . . , d , (4.55) holds with the Hamiltonian H of Eq. (4.49). Since {|αi} is a complete basis, expression (4.55) is equivalent to (4.53), and hence implies stationarity of ρ. Inserting Eq. (4.49) into (4.55), and using the notation Dαβ ≡ hα|D(ρ)|β i, we find Dαβ =i = ∑ α 0 6=β 0 ∑ α 0 6=β 0 iDα 0 β 0 hα| |α 0 i hβ 0 | ρ − ρ |α 0 i hβ 0 | |β i pα 0 − pβ 0 −Dα 0 β 0 (p 0 − pα 0 )δαα 0 δβ 0 β = (1 − δαβ )Dαβ . pα 0 − pβ 0 β Note that we could replace pα 0 6= pβ 0 by α 0 6= β 0 as the rule for the summation, due to the crucial assumption of non-degenerate eigenvalues pα .13 The above expression is obviously true for α 6= β . It remains to show that Dαα = 0 holds for all α = 1, . . . , d. To this end, we rewrite condition (4.52) using Eq. (4.54): 0 = tr ρ n−1 D(ρ) = d ∑ (pα )n−1 Dαα . (4.56) α=1 By assumption, this holds for all n = 2, . . . , d. In addition, it also holds for n = 1, since any dissipator fulfills tr(D(ρ)) = 0. This follows directly from (2.28). Hence, one can write (4.56) as a matrix equation M~d = 0, with (~d)α ≡ Dαα and (M)αn ≡ (pα )n−1 . Matrices of this form are known as Vandermonde matrices, for which det M = ∏α<β (pα − pβ ) is known [100, 182]. Since the pα are nondegenerate by assumption, we have det M 6= 0, and M is hence invertible. The only solution to M~d = 0 is therefore ~d = 0. Thus, we have shown that Dαα = 0 for all α = 1, . . . , d. Note that Proposition 1 holds for any (not necessarily unique) stationary state of an arbitrary, autonomous master equation of Lindblad form, and not just for the 13 This assumption is indeed crucial; there are ρ’s with degenerate eigenvalues which obey (4.52) and cannot be stabilized, i.e., ρ ∈ / S. 126 particular dissipator D(ρ) of Eq. (4.31). For our purposes, it ensures that the set S of stabilizable states is, in principle, generated by collecting all ρ that obey condition (4.52). Special care must be taken only with degenerate states, for which (4.52) is no sufficient criterion. Nevertheless, this causes no severe problems: Suppose that a degenerate state ρ obeys condition (4.52), but is not in S. For a reasonably well-behaving dissipator D(ρ), there will be a non-degenerate state ρ 0 in the vicinity of ρ that also fulfills (4.52).14 Proposition 1 states then that ρ 0 is in S . Note, however, that it takes a strong Hamiltonian to render ρ 0 stationary, since it has almost degenerate eigenvalues, in which case the denominator in (4.49) becomes small. In conclusion, by collecting all ρ that obey condition (4.52), one might include states that are not in S , but we expect that they will lie infinitely close to S . Our conjecture is, thus, that conditions (4.52) defines the closure S of the set of stabilizable states. Example: Coherence of a single qubit To illustrate our approach, let us apply it to the case of a single qubit first. This analysis will provide us with an intuitive geometric picture that is helpful for the analysis of two qubits later on. Since entanglement is certainly no reasonable category in this setting, our optimization objective O(ρ) in the following is the coherence between ground and excited state:15 Coh(ρ) = 2 | h0|ρ|1i |. (4.57) It reflects the capability of ρ to show interference phenomena in the |0i / |1i basis. This is of interest, e.g., in the field of quantum metrology [160]. Moreover, since quantum coherence is a precursor of entanglement, studying the single-qubit coherence Coh(ρ) also provides, to some extent, insight into the prospects and limitations for entanglement in the stationary state of two qubits. To derive the optimal stabilizable state for this objective, we introduce the three-dimensional Bloch vector~r of a single-qubit state ρ [86]: ~r = tr(~σ ρ). (4.58) The state space Q corresponds to the Bloch ball of all vectors |~r| ≤ 1 in this representation. To derive the geometric shape of the set of stabilizable states S , we rewrite the master equation (4.30) as an evolution equation for the Bloch vector: ~r˙ = ~H ×~r + D~r +~d. (4.59) 14 See also Theorem 2 of Ref. [180] on this issue. 15 A factor of two is included in Def. (4.57) to obtain the convenient normalization Coh(ρ) ∈ [0, 1]. 127 Here, ~H = tr(~σ H) represents the Hamiltonian, while the matrix D and the constant vector ~d reflect the dissipative part of the dynamics: (D)i j = tr (σi D(σ j )) /2, (~d)i = tr (σi D(1)) /2. (4.60) For the particular dissipator (4.31), they read (with the notation Γ± = γ− ± γ+ ): 1 Γ + γ 0 + d 2 1 , ~d = 0 . D = − (4.61) 2 Γ+ + γd Γ+ −Γ− Since the dimension of the Hilbert space of a single qubit is d = 2, the condition (4.52) reduces to a single constraint. In Bloch notation, it reads ~r · (D~r +~d) = 0. (4.62) To interpret this condition, we recognize that the Hamiltonian flux ~H ×~r in (4.59) merely generates a rotation around the axis defined by ~H, but it has no radial component, since~r · (~H ×~r) = 0. Hence, if ~r˙ = 0, then the dissipative flux D~r +~d at this point must generate a rotation in the opposite direction; it has therefore no radial component, either. This is expressed by Eq. (4.62). Let us review the same condition in the language of the density operator ρ, where it reads tr(ρD(ρ)) = 0. This reflects the fact that, if ρ ∈ S , its purity µ2 = tr(ρ 2 ) must be conserved under the dissipative dynamics generated by D(ρ). Since the purity is related to the length of the Bloch vector via µ2 = (|~r|2 + 1)/2, this is tantamount to saying that the dissipative flux must not change the length of ~r. This, in turn, implies that D~r +~d has no radial component, which is just the content of condition (4.62). Geometrically, expression (4.62) defines a quadric surface in R3 that corresponds, by virtue of Proposition 1, to the set of stabilizable states S . For our particular dissipator (4.31), the surface is an ellipsoid, as depicted in Fig. 4.6. It is centered at (0, 0, −Γ− /2Γ+ ); one semi-axis of length Γ− /2Γ+ is aligned with the z-axis, p and the remaining semi-axes in x and y direction are of equal length Γ− / 2Γ+ (2γd + Γ+ ). As mentioned above, degenerate states demand for a separate discussion. For a single qubit, the only degenerate state is the completely mixed state ρ = 12 /2. The corresponding Bloch vector is the center of the Bloch ball,~r = 0. This point fulfills condition (4.62), i.e., it is on the surface of the ellipsoid. However, the dynamic evolution (4.59) reduces to~r˙ = ~d at this point, irrespectively of the Hamiltonian ~H. Hence, the completely mixed state cannot become stationary and is therefore not in 128 γd = γ− /4 γd = γ− /2 γ+ = γ− /2 γ+ = γ− /4 γ+ = 0 γd = 0 Figure 4.6: Bloch representation of the set S of stabilizable states for a single qubit. S corresponds to the surface of the orange ellipsoid. The stationary state ~r∗ with optimal coherence, Eq. (4.64), and the state space Q (gray ball) are also shown. From left to right (top to bottom), the dephasing rate γd (the excitation rate γ+ ) increases, measured in units of the decay rate γ− . As S is contracted from the top left to the bottom right panel, the freedom of controlling the stationary state via a suitable Hamiltonian H is reduced. At γ+ = γ− (not shown), S comprises only the completely mixed state in the center of the Bloch ball. According to Eq. (4.32), this corresponds to the infinite temperature limit, whereas γ+ = 0 defines the zero temperature limit. 129 S .16 However, as discussed above, there are non-degenerate states in the vicinity of~r = 0 that lie on the ellipsoid, and therefore in S . Thus, one can come arbitrarily close to~r = 0, but one has to employ an ever stronger Hamiltonian ~H in this limit. This becomes apparent when expressing in Bloch notation the Hamiltonian H that renders a given state ρ ∈ S stationary, cf. Eq. (4.49): ~ ~H = ~er × (D~r + d) + h0~er . |~r| (4.63) Here, ~er =~r/|~r| is the unit vector in radial direction, and h0 is an arbitrary radial component of ~H, that has no influence on the dynamics. As anticipated, ~H diverges for |~r| → 0. Thus, the completely mixed state is not in S , but in the closure S . Having characterized the set of stabilizable states S for one qubit in this transparent fashion, any quantity of interest can now be optimized in this set. As discussed above, our objective here is the coherence, Eq. (4.57). In Bloch notation, it corresponds to the distance of the Bloch vector~r from the z-axis. It is apparent from Fig. 4.6 that the x and z coordinates of the optimal point~r∗ for this objective correspond to the semi-axes of the ellipsoid; therefore, we have ∗ ~r = Γ− Γ− p , 0, − 2Γ+ 2Γ+ (2γd + Γ+ ) ! . (4.64) The furthest distance from the z-axis is achieved when the only incoherent process is spontaneous decay, see the upper left panel of Fig. 4.6. In this limit, we have √ ~r∗ = (1/ 2, 0, −1/2). Consequently, the coherence between ground and excited state √ can at best become 1/ 2 in the stationary state. The Hamiltonian that renders the optimal state ~r∗ stationary is provided by Eq. (4.63) in Bloch notation, or, equivalently, by Eq. (4.49) in operator notation. Setting the irrelevant radial component h0 to zero and focusing on the case of spontaneous decay only, it reads (in Bloch and operator notation, respectively): γ− ~H∗ = (0, √ , 0) 2 ←→ H∗ = 1 γ− √ σy . 2 2 (4.65) This Hamiltonian is realized, e.g., by resonantly driving a two-level system with √ a Rabi frequency that matches 1/2 2 times the decay rate γ− of the excited level [86].17 16 Only in the infinite temperature limit γ = γ , we have Γ = 0 and thus ~d = 0, rendering the + − − completely mixed state stationary. It is, in fact, the only stabilizable state in this limit. 17 Strictly speaking, H ∗ describes the Hamiltonian in the rotating frame in this case. 130 Two qubits and entanglement Equipped with this intuition, we now turn to the case of two qubits. There, one can similarly define a generalized Bloch vector [120]: (~r)k=4i+ j = tr(σi ⊗ σ j ρ), with σ0 ≡ 12 , σ1 ≡ σx , σ2 ≡ σy , σ3 ≡ σz . (4.66) In this notation, the index pair (i, j) is mapped to the single index k = 4i + j that labels the 16 components k = 0, . . . , 15 of~r. In fact, since (~r)0 = tr(ρ) = 1 is fix, the Bloch vector has only 15 relevant components k = 1, . . . , 15. The state space Q, however, is no longer the unit ball in this representation [120, 183]. Likewise, the set of stabilizable states S has a more complicated structure than its singlequbit counterpart. It is the intersection of three nonlinear hypersurfaces Sn , each of which represents one of the constraints imposed by condition (4.52):18 S = d \ n=2 Sn , with Sn = {ρ ∈ Q | tr ρ n−1 D(ρ) = 0}, (4.67) where d = dim H = 4 for two qubits. The lowest order constraint, n = 2, is again quadratic in the Bloch vector. It can hence be cast into the same form as (4.62). Considering exclusively spontaneous decay (i.e., γ+ = γd = 0 and γ ≡ γ− ),19 the diagonal entries of the resulting 15 × 15 matrix D are γ diag(D) = − (1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 8), (4.68) 2 whereas its non-zero off-diagonal elements read D1,7 = D2,11 = D3,15 = D4,13 = D8,14 = D12,15 = −γ. (4.69) The vector ~d has mostly zero entries, except for (~d)3 = (~d)12 = −γ. The higher order constraints, n = 3, 4, lead to polynomial expressions of third and fourth degree in the Bloch vector. The resulting hypersurfaces S3 and S4 are 18 Strictly speaking, Eq. (4.67) holds only for non-generate states ρ. As discussed before, however, we have the strong conjecture that the equality is exact if one replaces S by its closure S . 19 This choice is motivated by the following argument: With the intuition gained from the singlequbit analysis in Fig. 4.6, it is plausible that adding an excitation or dephasing rate always contracts the set of stabilizable states towards the totally mixed state~r =~0, i.e., every Bloch vector is shortened by some factor x (with 0 ≤ x ≤ 1), so that ~r → x~r. In the language of quantum information, this corresponds to a depolarizing channel [27], which can never improve entanglement properties, since it describes a LOCC (see Section 2.3.2). Thus, whenever spontaneous decay is present (γ− 6= 0), we strongly conjecture that γ+ = γd = 0 is the case that allows for the most entangled optimal stationary state. 131 S3 S3 S4 S S3 S4 S ρ ∗ ρ ∗(2) ρ∗ S4 S3 S ρ∗ ρ ∗(2) S4 S ρ ∗ ρ ∗(2) ρ ∗(2) S2 S2 S2 S2 Figure 4.7: Visualization of relation (4.70). Since S ⊂ S2 , the optimal state ρ ∗(2) in S2 either coincides with the optimum ρ ∗ in S (left) or it has to be more entangled than it (right), i.e., E (ρ ∗ ) ≤ E (ρ ∗(2) ). difficult to grasp, and we refrain from analyzing them in detail. Instead, in order to determine the optimal stabilizable state ρ ∗ in S , we proceed as follows: First, we optimize over S2 alone. Then, we check if the resulting optimum ρ ∗(2) ∈ S2 lies in S . If this is the case, it is also the optimum in S , i.e., ρ ∗ = ρ ∗(2) , since S ⊂ S2 . If ρ ∗(2) is not in S , the procedure provides at least an upper bound for the optimal entanglement E (ρ ∗ ) in S , because E (ρ ∗ ) = sup E (ρ) ≤ sup E (ρ2 ) = E (ρ ∗(2) ). ρ∈S ρ2 ∈S2 (4.70) The situation is illustrated in Fig. 4.7. For simple quantities like the Bell state fidelities Fidφ ± and Fidφ ± , defined in Eq. (4.35), the optimization can now be carried out analytically. In Bloch notation, the objective function becomes FidX (~r) =~rX ·~r, (4.71) where X refers to either of the Bell states φ ± or ψ ± . Applying Lagrange’s method with a multiplier λ , the optimal~r∗ has to satisfy (I) (II) (4.62) ~r∗ · (D~r∗ +~d) = 0, ~∇~r ~rX ·~r − λ~r · (D~r +~d) ~r=~r∗ ! =~rX − λ (D~r∗ +~d) = 0. (4.72) This system of 16 linear equations can explicitly be solved for~r∗ . In case of the ψ ± Bell state fidelity (X = ψ ± ), the density matrix corresponding to~r∗ is ρψ∗ ± = 1 1 |00i h00| + |ψ ± i hψ ± | , 2 2 (4.73) 132 which yields a fidelity of Fidψ ± (ρψ∗ ± ) = 1/2. In the following Section 4.2.3, we derive the Hamiltonian that stabilizes this state; this is explicit proof that ρψ∗ ± lies indeed in S , and not just in S2 . In sum, we have rigorously proved that the upper bound of Fidψ ± ≈ 0.5, which was observed for typical stationary states in Fig. 4.5(d), marks indeed the optimal value among all conceivable stationary states! Optimizing the φ ± Bell state fidelity instead, the solution of (4.72) leads to ρφ∗± = 1 (1 + 9 |00i h00| − |11i h11| ± 3 |00i h11| ± 3 |11i h00|) , 12 (4.74) which has Fidφ ± (ρφ∗± ) = 2/3. However, since the smallest eigenvalue of ρφ∗± is √ (5 − 34)/12 ≈ −0.07, it does not describe a valid quantum state and is therefore not in S . Hence, this analysis provides only an upper bound of 2/3 for the optimal value of the φ ± Bell state fidelity in S . Nevertheless, we have already encountered a valid stabilizable state in Section 4.2.1 that almost perfectly saturates this upper √ bound: The density operator (4.37) has Fidφ ± = (3 + 5)/8 ≈ 0.65 at the optimal √ value x = ±(1 + 5)/2. Since this state is similar to the unphysical state (4.74), we conjecture that it is the true optimal stationary state for the φ ± fidelity in S . The concurrence C cannot be optimized in the same analytical fashion, because its evaluation involves a 4 × 4 eigenvalue problem, rendering the analytical evaluation of the gradient in (4.72) intractable. However, high Bell state fidelity typically corresponds to strong entanglement. It is therefore reasonable to study the concurrence of the fidelity-optimized states from √ above. The state (4.37), which we conjecture to be φ ± -optimal, yields C = ( 5 − 1)/4 ≈ 0.31. The ψ ± -optimal state (4.73), on the other hand, reaches C = 0.5. This exceeds significantly the upper bound of C ≈ 0.35 observed for typical stationary states in Fig. 4.5(a). Moreover, any attempt to further maximize the concurrence by numerical means results in C = 0.5 as well. This strongly suggests that ρψ∗ ± is not only the optimal state with respect to Fidψ ± , but also with respect to the concurrence – and therefore, according to the discussion in Section 2.3.4, with respect to entanglement of formation as well. In conclusion, we may regard ρψ∗ + (and likewise ρψ∗ − ) as the most entangled state ρ ∗ ∈ S that we quested for: (4.73) ρ∗ = 1 1 |00i h00| + |ψ + i hψ + | . 2 2 (4.75) Strikingly, exactly the same state emerges inside the entanglement resonance around ω0 ≈ ω for our particular model Hamiltonian (4.1) with the Floquet Born-Markov master equation, see Eq. (4.28). This is surprising in two regards: Firstly, we work with a different master equation here, which is not 133 microscopically derived, but relies on a fixed choice of the dissipator. Secondly, we have admitted the most general class of all (static) Hamiltonians here, while our investigation in Section 4.1 was focused on a particular (driven) model system. The first paradox is resolved if the recall the discussion in Section 2.2.3: In the parameter regime of interest (ω0 ≈ ω, max (F, J) ω, and all incoherent rates γi j much weaker than the coherent dynamics), the Born-Markov description coincides with our ansatz (4.31) for the dissipator. The second paradox is partially resolved below: We will see that the optimal static Hamiltonian H ∗ which stabilizes ρ ∗ coincides exactly with the model Hamiltonian studied in Section 4.1, if the latter is transformed to the rotating frame. Nevertheless, it is sheer coincidence that we considered in Section 4.1 exactly the model that now turns out to be optimal under a fixed dissipator. 4.2.3 The optimal Hamiltonian So far, we only know that ρ ∗ is the optimal state in S2 ; it remains to show that ρ ∗ ∈ S , i.e., that it can indeed be stabilized. We now demonstrate this by explicitly providing the Hamiltonian that renders ρ ∗ stationary under the Lindblad master equation (4.30) with spontaneous decay of rate γ. Optimal Hamiltonian for two qubits Since ρ ∗ has degenerate eigenvalues pα ∈ {1/2, 1/2, 0, 0}, the simple recipe (4.49) fails to provide a suitable Hamiltonian. One can, however, consider the close-by state 1 3ε 2 ε2 ± (ε) ∗ ρ = ρ + |00i h00| − |ψ i hψ ± | 1 + ε2 2 2 (4.76) p ε→0 + + ∗ 2 −ε 1 − ε |00i hψ | + |ψ i h00| −→ ρ instead, which is also in S2 . Since it has non-degenerate eigenvalues, we can find its stabilizing Hamiltonian via Eq. (4.49). Performing the limit ε → 0, we arrive at the desired Hamiltonian H ∗ that renders ρ ∗ stationary: H∗ = 1 ⊗ ∆ F ∆ F σz + σx + σz + σx ⊗ 1 + J(σ+ ⊗ σ− + σ− ⊗ σ+ ), 2 2 2 2 (4.77) with the following relation between the parameters: ∆ = −J, and |∆| |F| γ. (4.78) 134 More precisely, this means that the stationary state converges to ρ ∗ in the limit |∆/F| → ∞ and |F/γ| → ∞. Hence, ρ ∗ is strictly spoken not in S , but only in the closure S . Nevertheless, with finite ratios as low as |∆/F| = 10 and |F/γ| = 10, the concurrence and the ψ ± fidelity of the stationary state reach more than 98% of their optimal value 0.5. In appendix A.2, we elucidate in physical terms why the Hamiltonian (4.77) renders ρ ∗ stationary. It turns out that the underlying mechanism has the same origin as the entanglement resonance with half-maximum concurrence, which we found for the Born-Markov dynamics in Section 4.1.3. In particular, the optimal state ρ ∗ is precisely the asymptotic state in the center of the resonance, see Eq. (4.28). Experimental realization Let us briefly discuss how to experimentally implement H ∗ . The first two terms of (4.77) describe an external, static field that locally interacts with both qubits. F and ∆, respectively, refer to the field strength in x and z direction. The third term describes an exchange interaction of strength J between the qubits. This situation can directly be realized, e.g., with superconducting qubits [184]. Moreover, it can also be implemented in a way that is viable for almost any experimentally available system of two qubits, be it of quantum optical or solid state nature: Two qubits with identical level splitting ω0 are driven by a monochromatic external field of amplitude F and frequency ω, and interact via a 1D Ising interaction of strength J. The system Hamiltonian is ω ω 0 0 H(t) = 1 ⊗ σz + F cos(ωt)σx + σz + F cos(ωt)σx ⊗ 1 + J σx ⊗ σx . 2 2 (4.79) Performing a rotating frame transformation, and identifying the detuning ω0 − ω with the parameter ∆, it becomes Hrf (t) = e−i 2 (σz ⊗1+1⊗σz ) H(t)ei 2 (σz ⊗1+1⊗σz ) F ∆ 2iωt −2iωt = 1 ⊗ σz + σx + σ− e + σ+ e + 2 2 F ∆ 2iωt −2iωt σz + σx + σ− e + σ+ e ⊗ 1+ 2 2 ω ω (4.80) J σ+ ⊗ σ− + σ− ⊗ σ+ + e2iωt σ− ⊗ σ− + e−2iωt σ+ ⊗ σ+ . As long as the driving amplitude F, the detuning ∆, and the interaction strength J are much smaller than the level splitting ω0 , one can safely neglect the timedependent parts of Hrf (t) in a rotating wave approximation, leading to Hrf = H ∗ . 135 Condition (4.78) can be met in the experiment by adjusting the frequency ω of the driving field such that the detuning ∆ = ω0 − ω is negative and matches −J. The driving amplitude F does not have to be tuned to a specific value; it only has to be much smaller than the detuning and at the same time much larger than the decay rate γ of the qubits. In summary, the desired scenario can be implemented by simply driving two interacting qubits at the right frequency. We stress that this relies not only on the identity Hrf = H ∗ , but also on the fact that the rotating frame transformation (4.80) does not alter the dissipator D(ρ) for spontaneous decay. Furthermore, it is important that (4.80) is a local unitary transformation; consequently, it leaves the entanglement properties of the stationary state invariant, which we are interested in here. Generalization to more than two qubits The Hamiltonian H ∗ has a natural extension to N > 2 qubits: N N ∆ (i) F (i) (i) ( j) (i) ( j) (N) H =∑ σz + σx + ∑ J(σ+ σ− + σ− σ+ ). 2 2 i< j i=1 (4.81) It can be implemented in complete analogy to the two-qubit scenario: Take N qubits with identical level splitting ω0 and drive them by an external field of amplitude F and frequency ω, such that ∆ corresponds to the detuning ω0 − ω. (i) ( j) Moreover, every pair of qubits (i, j) has to interact via a σx σx interaction of equal strength J. In rotating wave approximation, such a setup is described by H (N) . Under the combined action of H (N) and spontaneous decay of each qubit with rate γ, the stationary state ρ (N) is a 50/50 mixture of the de-excited state |0i⊗N and the N-qubit W state |WN i of Eq. (3.76),20 1 1 ρ (N) = (|0i h0|)⊗N + |WN i hWN | , 2 2 (4.82) if the parameters fulfill ∆ = (1 − N) · J and |∆| |F| γ. (4.83) This result is derived in appendix A.2.2. For N = 2, this reproduces the results discussed before: The appropriate detuning parameter is ∆ = −J, and the stationary 20 Note, again, the striking coincidence between the stationary state (4.82) under the master equation with fixed dissipator and the Floquet Born-Markov result (4.29). As discussed before, in the parameter regime of interest, the two master equations are equivalent, and so is the system Hamiltonian H. 136 state (4.82) boils down to ρ ∗ of Eq. (4.75), which has concurrence C (ρ ∗ ) = 1/2. For N > 2, we observed numerically (up to N = 5) that the generalized N-qubit concurrence, Eq. (2.96), of ρ (N) is always half of the concurrence of the pure W state.21 Conjecturing that this result holds for any number of qubits N, this implies that H (N) stabilizes a stationary state with a significant concurrence of 1 C (ρ (N) ) = 2 s 1 2 1− . N (4.84) It should be emphasized that the choice of H (N) in Eq. (4.81) is a heuristic generalization of the optimal two-qubit Hamiltonian (4.77) to N > 2 qubits. Hence, there is no a priori reason for the resulting stationary state ρ (N) to be the optimal state with respect to entanglement. (In particular, since there is no unique notion of maximal entanglement in the multi-partite case of N > 2 qubits.) However, a statistical analysis of random Hamiltonians for N = 3, similar to the one presented in Section 4.2.1, reveals that typical stationary states have rather poor concurrence values in the range of 0 . . . 0.25, whereas ρ (3) yields √ C (ρ (3) ) = 1/ 3 ≈ 0.57. Compare this to the respective values for N = 2, where the advantage of the optimal state (with C (ρ ∗ ) = 1/2) over the typical values (C = 0 . . . 0.35, see Fig. 4.5) is smaller. Hence, even more than in the case of two qubits, H (3) yields a stationary state of exceptionally high concurrence. In summary, N qubits undergoing spontaneous decay at rate γ can be prepared in the stationary entangled state (4.82) in the following way: Implement the Hamiltonian (4.81) as described, make sure to work in the regime of |∆| |F| γ, adjust the detuning parameter ∆ to J(1 − N), and wait for the system to reach its stationary state. 4.2.4 Relation to dissipative state preparation schemes The motivation for the previous investigations was the question to what extent entanglement can be preserved in the asymptotic state of a Lindblad master equation with a fixed dissipator D(ρ), if the Hamiltonian is optimally chosen. This question is conceptionally not far from the idea of dissipative state preparation (DSP) [175, 176]. There, the aim is to prepare a certain target state with some desired property, such as strong entanglement, by making it the unique stationary state of a Lindblad master equation. The major advantage of DSP schemes is that, having properly implemented the respective master equation in a 21 We even observed C (ρ ) = λ C (|W i hW |) for any mixture ρ ≡ (1 − λ )(|0i h0|)⊗N + N N λ λ λ |WN i hWN | with λ ∈ [0, 1], not just for λ = 1/2. 137 physical system, all one has to do is to wait: any initial state will eventually evolve into the desired target state and then remain locked in it [173]. The central difference between DSP and our ansatz is the following: We regard the dissipator in the master equation Eq. (4.30) to be fixed and unavoidably imposed by external conditions of the system at hand. Only the system Hamiltonian H can be manipulated to optimize entanglement in the stationary state. The concept of DSP, in contrast, involves also a manipulation of the dissipative term in order to achieve the desired goal. Hence, instead of the specific ansatz (4.31) for the dissipator D(ρ), DSP schemes start from the most general dissipator of Lindblad form: 1 † † † D(ρ) = ∑ γk Lk ρLk − . (4.85) L Lk ρ + ρLk Lk 2 k k Hamiltonian H and Lindblad operators Lk are then engineered such that the desired target state becomes the unique stationary solution of Eq. (4.30). In practice, however, this agenda can only partially be realized (using rather specific experimental setups [185, 186]), since the manipulation of dissipative processes is an enormous experimental challenge. Dissipative state preparation without restrictions Theoretically, any pure state ρ = |ψi hψ| of an N-qubit quantum register can be turned into the unique stationary state of (4.30), as long as no restriction is imposed on the Lindblad operators Lk and the Hamiltonian H. This can be achieved in the following way [187]: choose the Hamiltonian such that |ψi is an eigenstate, H |ψi = E |ψi, and engineer the incoherent processes such that (k) Lk = Uσ− U † , (4.86) where the unitary transformation U is determined by |ψi = U |0i⊗N . (4.87) The working principle of this scheme becomes apparent in a rotated reference frame, defined through ρ̃ = U † ρU. (4.88) In this frame, the de-excited state |0i⊗N is a stationary state of (4.30), since 1. |0i⊗N is eigenstate of H̃: H̃ |0i⊗N = (U † HU) |0i⊗N = U † H |ψi = E |0i⊗N . This implies [H̃, (|0i h0|)⊗N ] = 0. 138 2. |0i⊗N is annihilated by any L̃k : (k) L̃k |0i⊗N = U † LkU |0i⊗N = σ− |0i⊗N = 0. This leads to D (|0i h0|)⊗N = 0. Since, moreover, the de-excited state is the only stationary state in the rotated frame [173], |ψi = U |0i⊗N is the unique stationary state in the original frame. The major drawback of this scheme is that the involved dissipative processes are not naturally given, but have to be designed artificially. In particular, if the target state |ψi is entangled, it requires the engineering of non-local Lindblad operators: Since, by definition of an entangled state, |ψi is not connected to |0i⊗N via a local unitary transformation, U has to mix at least two subsystems. According to Eq. (4.86), this leads to Lindblad operators Lk that act non-trivially on more than one subsystem and hence correspond to a non-local incoherent process. In fact, the more sophisticated the desired kind of entanglement, the more Lindblad operators must be engineered in a non-local fashion, since all the entanglement properties of |ψi are encoded in U.22 While this requirement can be met in specific experimental situations [185, 186, 188, 189], it poses insurmountable challenges for the vast majority of setups. This severely limits the applicability of such schemes for the preparation of entangled states. A further drawback is that additional, uncontrolled dissipative processes are unavoidable in any realistic setup. They will compete with the engineered processes and spoil the outcome of the preparation scheme. Dissipative state preparation with local Lindblad operators These considerations raise the question whether it is possible to employ naturally occurring incoherent processes in a state preparation scheme, instead of artificially engineered ones. This would have two advantages: On the one hand, no experimental control over the incoherent dynamics is required; on the other hand, the natural incoherent processes do not compete with any expensively engineered ones, and hence do not spoil the outcome of the scheme. 22 E.g., if one wants to entangle only two out of N qubits, say 1 |ψi = √ (|00i + |11i) ⊗ |0i⊗(N−2) , 2 (k) then only L1 and L2 are non-local, while the remaining Lk ’s are just σ− . On the other hand, if one wants to prepare a (linear or 2D) N-qubit cluster state, all N Lindblad operators become non-local (although they are only required to act on a restricted number of neighboring qubits [187]). 139 Since natural incoherent processes are predominantly of local nature, the most severe restriction is that one will have to focus on strictly local dissipators [133]. Thus, the Lindblad operators are restricted to the form Lk = 1 ⊗ · · · ⊗ 1 ⊗ ` ⊗ 1 ⊗ · · · ⊗ 1, (4.89) with ` acting on the k-th subsystem only. Note that we have focused on such natural, local Lindblad operators throughout all our examples in this chapter. In particular, for all the (fixed) Lk ’s that we studied in Section 4.2.2, we found that the most entangled stationary state is always mixed. This implies, in turn, that strongly entangled pure states can never become stationary with these exemplary local Lindblad operators. More generally, one can prove the following statement: Proposition 2. Let ρ = |ψi hψ| be a pure state of N qubits that is stationary under the master equation (4.30). If one of the Lindblad operators Lk is local, as defined in (4.89), then |ψi is separable with respect to the k-th qubit. Proof. Without loss of generality, we assume k = 1, i.e., the local Lindblad operator L1 acts on the first qubit: L1 = ` ⊗ 1 ⊗ · · · ⊗ 1. (4.90) Furthermore, L1 (and therefore also `) can be assumed traceless: If tr L1 6= 0, let L10 ≡ L1 − c1, and H 0 = H + 2i c∗ L1 − 2i cL1† , with c = (tr L1 )/(2N ). The master equation (4.30) is invariant under this transformation, and therefore the proposition also holds for the traceless Lindblad operator L10 , which inherits from L1 the property of being local. Since the stationary state ρ = |ψi hψ| is pure, it must be an eigenstate of all Lindblad operators of the process. (See theorem 1 in ref. [187], or Proposition 4 in ref. [173].) Hence, we have L1 |ψi = z |ψi (4.91) for some eigenvalue z ∈ C. Next, we write down the Schmidt decomposition [27] of |ψi with respect to the bi-partition {1}|{2 . . . N}: |ψi = λa |a1 i ⊗ |a2...N i + λb |b1 i ⊗ |b2...N i . Using (4.90) and (4.91), we find λa (` |a1 i) ⊗ |a2...N i + λb (` |b1 i) ⊗ |b2...N i =λa (z |a1 i) ⊗ |a2...N i + λb (z |b1 i) ⊗ |b2...N i . (4.92) 140 Assume |ψi not to be separable, i.e., both Schmidt coefficients λa and λb to be non-zero. Then, it follows that ` |a1 i = z |a1 i and ` |b1 i = z |b1 i, since the Schmidt decomposition ensures that |a2...N i and |b2...N i are orthogonal. Hence, ` has the two-fold degenerate eigenvalue z. Since ` is a traceless single-qubit operator, we have tr ` = 2z = 0 ⇒ z = 0. (4.93) Hence, ` must be the null operator ` = 0, implying, in turn, L1 = 0. Of course, this case is excluded in the proposition. Consequently, |ψi must to be separable with respect to the bi-partition {1}|{2 . . . N}. This proposition implies, in particular, separability of |ψi with respect to every bi-partition, if there is at least one local Lk for each qubit. Hence, under these circumstances, a stationary state ρ cannot be pure and entangled at the same time – independently of the presence of additional (possibly non-locally engineered) Lindblad operators. Note that this severely limits the applicability of DSP schemes as the one discussed on p. 137f. There, the target state is usually entangled and pure, and the above proposition states that such states can under no circumstances become stationary, as soon as even the weakest local incoherent process is present – which is always the case under realistic conditions.23 With this, we end our excursion to the field of DSP. We have seen that its guiding idea – namely, to employ dissipation for entanglement creation, rather than considering it as its adversary – is lost when considering more realistic, and therefore local incoherent processes. However, the incoherent part of the time evolution has a second important role for DSP, which remains intact even with local dissipation: As it is well known in the theory of open systems [82, 190], it can lead to relaxing quantum dynamics, i.e., to dynamics under which every initial state eventually evolves into the same (stationary) state. This task can never be accomplished by coherent dynamics alone. In this light, the methods developed in the present Section 4.2 can also be seen as a method to derive the optimal DSP scheme for a given (natural) dissipator, such as, e.g., our prime example (4.32), which describes the three most common incoherent processes. We emphasize, however, that our results should not be mistaken as a specific DSP scheme, since it can be applied to many questions beyond this context. In summary, we have derived the following results in Section 4.2: 23 Note, however, that Proposition 2 makes no statement about mixed, stationary states. In fact, it is plausible a DSP scheme will only slightly be spoiled by additional, local incoherent processes, as long as they are weak compared to the artificially designed, nonlocal processes; hence, the intended (pure) target state of the DSP scheme will only become weakly mixed. 141 I We accomplished a practical characterization of the set of stabilizable states S , comprising all states that can become stationary under a given dissipator. Stabilizable states with non-degenerate eigenvalues are fully characterized by condition (4.52), which expresses that the dissipator D(ρ) must generate moment-preserving (i.e., unitary) dynamics only, if it is to be compensated by a suitable Hamiltonian. For degenerate states, (4.52) remains a necessary condition for ρ ∈ S , and we conjecture that it also sufficient if one includes states in the closure of S . I We applied this characterization of stabilizable states to the case of a single qubit, subject to the three incoherent processes of decay, excitation, and dephasing. In this case, S corresponds to the surface of an ellipsoid in the Bloch ball. The optimal stationary state with respect to a certain objective, such as the coherence between ground and excited state, is then easily read off, see Eq. (4.64). The corresponding Hamiltonian (4.65), which stabilizes the optimal state in S , follows straightforwardly. I The same procedure was then applied to determine the most entangled stationary state of two qubits in the presence of spontaneous decay, ρ ∗ in Eq. (4.75). It has half-maximum concurrence C (ρ ∗ ) = 1/2. I We discussed the Hamiltonian (4.77) that prepares the optimal stationary state and its generalization to more than two qubits. It can be implemented with most experimental realizations of a quantum register. I With Proposition 2, we have found a general limitation of DSP schemes: Pure, entangled target states are excluded as soon as local incoherent processes are present. Besides these concrete findings, we stress again that our approach is neither limited to the study of entanglement, nor to the specific dissipator (4.31); it can rather be applied to the optimization of any objective functional O(ρss ) of the stationary state ρss of a finite dimension quantum system that is subject to incoherent dynamics described by an arbitrary (fixed) dissipator D(ρ). Finally, a remark is in place about the relation of our result to the original work of Recht et al., Ref. [180], which introduced the concept of stabilizable states for the first time. Recht et al. define the set S and prove that it is a simply connected manifold. Moreover, the optimal state and Hamiltonian for the coherence of a decaying qubit are derived, which corresponds precisely to our results Eq. (4.64) and (4.65). Also, the case of two qubits and entanglement as the objective is discussed. For a certain model Hamiltonian – which, by chance, coincides exactly which the optimal Hamiltonian H ∗ derived in Eq. (4.77) – the 142 authors study the resulting stationary state and find that it becomes the remarkably entangled state (4.75) in the parameter regime (4.78). Hence, these optimal solutions have been discovered before and are no originary findings of our work. Entirely new to our work, however, are two important aspects: First, we derived Proposition 1, which enables a characterization of S that no longer refers explicitly to the Hamiltonian. Thus, in order to obtain S , one no longer has to cumbersomely parametrize all conceivable Hamiltonians and find, for a given H, the corresponding stationary state ρ. Instead, we obtain S from the dissipator D(ρ) alone and, based on this, determine directly the optimal stabilizable state (w.r.t. a certain objective). This is a great advantage, in particular, for systems beyond a single qubit. In fact, while Recht et al. are able to consider the most general Hamiltonian for a single qubit (and consequently also find optimal stabilizable state in this case), they have to rely on a certain model ansatz for H already in the case of two qubits. Therefore, they are not aware that they have found, by chance, the optimum among all stabilizable states. This is, in fact, the second distinction between our work and Ref. [180]: From our results, we know that one can under no circumstances do better than concurrence 1/2 in the stationary state of two decaying qubits. 143 4.3 The optimal asymptotic cycle under fixed incoherent dynamics Equipped with a concept to determine the optimal state ρ ∗ that can be rendered stationary by a time-independent Hamiltonian, we take a step forward and extend our analysis to driven systems, described by a periodically time-dependent Hamiltonian H(t). Their long-time dynamics are described by an asymptotic cycle ρac (t). Its entanglement properties are the focus of our following investigations. In contrast to most studies on open, driven system in the literature, we do not study a particular model Hamiltonian, but rather aim for the driving protocol H ∗ (t) ∗ (t), i.e., the cycle with the best that yields the globally optimal asymptotic cycle ρac possible entanglement properties at all. This is similar in spirit to our discussion of the undriven system in the previous section. In Section 4.3.1, we gain a practical understanding of the class of possible asymptotic cycles for a fixed, but arbitrary dissipator D(ρ). Based on this class, we develop an approach to derive the optimal driving protocol. This approach is introduced in Section 4.3.2 for the most transparent case of a single qubit and then extended to two qubits in Section 4.3.3. Finally, the mechanism that limits entanglement in the asymptotic cycle is identified in Section 4.3.4, and ideas to overcome these limits are discussed. 4.3.1 Characterization of asymptotic cycles To determine the optimal asymptotic cycle, we have to develop a practical characterization of all asymptotic cycles that are compatible with a given dissipator. In mathematical terms, an asymptotic cycle ρac (t) corresponds to a closed trajectory in the state space Q, parametrized by the time t: ρac : [0, T ] 7→ Q, with ρac (0) = ρac (T ). (4.94) Conversely, however, not every closed trajectory in Q corresponds to an asymptotic cycle of the master equation (4.30) for some suitable periodic Hamiltonian H(t) = H(t + T ). This situation is reminiscent of the autonomous case discussed in the last section: There, only the stabilizable states, forming the set S $ Q, can be rendered stationary by a suitable (static) Hamiltonian. 144 The set of stabilizable cycles Guided by this analogy, we define now the set of stabilizable cycles S . It comprises all closed, differentiable trajectories of arbitrary period T , for which a suitable Hamiltonian H(t) exists that turns the trajectory into a solution of the master equation (4.30): ( ) ρ(0) = ρ(T )∧ [ S = ρ : [0, T ] 7→ Q . ∃H(t) : ρ̇(t) = −i[H(t), ρ(t)] + D(ρ(t)) T ∈R+ (4.95) Just like the set S in the autonomous case, S is independent of a specific Hamiltonian, but depends exclusively on the dissipator D(ρ), which we always consider to be imposed by external circumstances and therefore fix. In contrast to S , however, S is not a subset of the state space Q, because its elements are entire trajectories in Q, and not just density operators. Given an asymptotic cycle ρ(t) ∈ S , it is straightforward to find the corresponding Hamiltonian H(t) that stabilizes this cycle. The procedure is completely analogous to the derivation of the static Hamiltonian (4.49) of the autonomous scenario and leads to H(t) = ∑ pα 6= pβ i hα| (D(ρ) − ρ̇) |β i |αi hβ | + ∑ Hαβ |αi hβ | . pα − pβ pα =pβ (4.96) Again, the elements Hαβ can be chosen freely (e.g., to be zero), and pα and |αi denote the instantaneous eigenvalues and eigenstates of ρ(t) (at the fixed time t).24 The lengthy definition (4.95) of S is of no practical use for determining the optimal asymptotic cycle. In the autonomous case, the way out of this dilemma was paved by Proposition 1, which provided a more practical characterization of S . To our favor, this proposition can be extended to the driven case: Proposition 3. Let ρ : [0, T ] 7→ Q be a closed, differentiable trajectory of period T in the state space Q of a d-dimensional quantum system, and D(ρ) be the dissipative term of the master equation (4.30). If ρ(t) ∈ S , i.e., if there is a Hamiltonian H(t) that renders ρ(t) a solution of the master equation, then n · tr [ρ(t)]n−1 D(ρ(t)) = ∂t tr([ρ(t)]n ), ∀n = 2, . . . , d , (4.97) holds for all t ∈ [0, T ]. If ρ(t) has non-degenerate eigenvalues (for all t), then (4.97) is a sufficient condition for ρ ∈ S . 24 Hence, all quantities appearing in Eq. (4.96) are periodically time-dependent; nevertheless, we suppress the time-arguments here for the sake of a more compact notation. 145 Proof. (Necessary condition.) Suppose that ρ(t) solves (4.30) for some H(t): ρ̇(t) = −i[H(t), ρ(t)] + D(ρ(t)). For all n ∈ N, and in particular for n = 2, . . . , d, this implies tr(ρ n−1 ρ̇) = −i tr ρ n−1 [H, ρ] + tr ρ n−1 D(ρ) = −i tr ρ n−1 Hρ + i tr ρ n−1 ρH + tr ρ n−1 D(ρ) = tr ρ n−1 D(ρ) . (For reasons of briefness, we omit the time argument t.) In combination with ∂t tr(ρ n ) = tr(ρ̇ρ n−1 ) + tr(ρ ρ̇ρ n−2 ) + . . . + tr(ρ n−1 ρ̇) = n · tr(ρ n−1 ρ̇), this leads to Eq. (4.97). (Sufficient condition.) This part of the proof is completely analogous to that of Proposition 1, see p. 124ff.; one must merely replace the static ansatz (4.49) for the Hamiltonian by its time-dependent analogue (4.96). Just like Proposition 1, Proposition 3 relies on the fact that only the dissipative, but not the Hamiltonian dynamics can change the spectrum of the density operator, and therefore its moments µn = tr(ρ n ). More precisely, condition (4.97) states that the temporal change of µn is exclusively governed by the quantity fn (ρ) ≡ n · tr ρ n−1 D(ρ) , (4.98) which is independent of the Hamiltonian. With this definition, the condition reads µ̇n (t) = fn (ρ(t)), ∀n = 2, . . . , d. (4.99) This suggests an interpretation of fn (ρ) as a flux that the dissipator D(ρ) generates for the n-th moment at the specific point ρ in the state space Q. If the flux is positive (negative), µn must increase (decrease) at this stage of the cycle. At points with vanishing flux, fn (ρ) = 0, µn does not change in time; this situation corresponds exactly to the autonomous criterion (4.52). Thus, the sets Sn that we defined in Eq. (4.67) have an additional significance: They separate regions in Q where the n-th moment µn of any consistent trajectory ρ(t) is necessarily decreasing in time from regions where it is continuously growing. To elucidate these concepts, we consider again the simplest case of a single qubit first. Because of d = dim H = 2, Eq. (4.99) imposes only a constraint on the 146 evolution of the purity µ2 = tr(ρ 2 ). In Bloch notation, µ2 has the intuitive meaning of the length of the Bloch vector: 1 µ2 = (|~r|2 + 1). 2 (4.100) The purity flux, on the other hand, reads (4.62) f2 (~r) = ~r · (D~r +~d). (4.101) The contour surfaces of this scalar field, defined by f2 (~r) = const, correspond to quadric surfaces in the Bloch ball Q. For the particular contour value const = 0, this defines the set S of stabilizable states, which corresponds in Bloch representation to the surface of an ellipsoid, as shown in Fig. 4.6. More contour values of f2 (~r) are plotted in Fig. 4.8, to obtain a comprehensive understanding of the value of f2 (~r) in the Bloch ball. In the interior of the ellipsoidal surface S , the flux is positive. Thus, the purity of any consistent trajectories must increase in this region. Vice versa, the purity has to decrease when the trajectory passes through a region exterior to S . Together with Eq. (4.100), this implies that the trajectory has to move away from (towards) the center of the Bloch ball when it evolves inside (outside) the ellipsoidal surface S . The two exemplary red curves (i) and (iii) in Fig. 4.8 do not fulfill this criterion. Consequently, for no Hamiltonian H(t) whatsoever do these trajectories define a solution of the master equation with our given dissipator. The green curves (ii) and (iv), on the other hand, fulfill the criterion. Since they do not pass through the center of the Bloch ball – the only state with degenerate eigenvalues – the criterion is, in fact, already sufficient for these closed curves to be in S , i.e., to be stabilized by the Hamiltonian (4.96). Two general rules can inferred from this discussion: 1. An asymptotic cycle cannot spend its entire period exclusively in the interior or the exterior of S , because it must always pass through both regions of negative and positive purity flux in order to return to its starting point. 2. An asymptotic cycle can at no time be purer than the purest state on the ellipsoidal surface S . The second conclusion follows from the fact that the cycle can gain purity only in the interior of the ellipsoid enclosed by S ; as soon as it leaves this region, it loses purity. Consequently, its purity never exceeds that of the purest state in S . Curve (iii) in Fig. 4.8 violates this rule, because its outer turning point is more distant from the origin than any point in S . Curve (i), on the other hand, violates the first rule, because it exclusively evolves in the exterior of S . 147 (a) γ+ = γd = 0 (b) γ+ = γd = γ− /4 zz z (i)(i) (i) zz z .25.25 .25 00 0 QQB QQB xx x SS S f2f(�2r)(�fr) 2 (�r) (ii)(ii) (ii) (iii) (iii) (iii) (iv) (iv) (iv) SS xx x -1-1 -1 S -2-2 -2 Figure 4.8: Side view of the Bloch ball Q of a single qubit. The color code indicates strength and sign of the purity flux f2 (~r), Eq. (4.101), measured in units of the decay rate γ− , for two exemplary choices of the incoherent rates. (See also the similar Fig. 4.6). The regions of positive (orange) and negative (gray) flux are separated by the set S (dashed line) of states that can be stabilized by a static Hamiltonian. While the green curves (ii) and (iv) are examples of stabilizable asymptotic cycles, the red curves (i) and (iii) do not represent a consistent cycle: At some stage, their purity (i.e., their distance from the center of Q) increases in the gray region of negative flux, in contradiction to condition (4.99). The same rules hold for the higher moments µn and their respective fluxes fn , which become relevant beyond the case of a single qubit; only the role of S has to be replaced by Sn . 4.3.2 Deriving the optimal asymptotic cycle of a single qubit Having understood the restrictions that the dissipator D(ρ) imposes on an ∗ (t) in S . We start, again, asymptotic cycle, we can seek for the optimal cycle ρac with the simplest case of a single qubit. Optimize the average or the peak value? At this point, we have to define what we consider as the optimal asymptotic cycle. In the autonomous scenarios of a time-independent stationary state ρss , this issue was settled with the specification of an objective function O(ρss ). E.g., in case of a single qubit, we considered the coherence between ground and excited state; for 148 two qubits, the concurrence or the Bell fidelities defined the figure of merit. For an asymptotic cycles ρac (t), however, the objective function O(ρac (t)) becomes timedependent and varies over the driving period t ∈ [0, T ). Thus, the figure of merit is no longer some function O(ρss ) of a single state, but a functional O[ρac (t)] of an entire trajectory. E.g., following our definition of entanglement of Floquet states in Chapter 3, one might consider the time-average of the objective function: 1 O[ρac ] ≡ T Z T 0 dt O(ρac (t)). (4.102) In other situations, however, one might be rather interested in achieving a large peak value of an objective O at some stage of the cycle, and disregard the value of O during the rest of the cycle. The relevant quantity of interest then is Omax [ρac ] ≡ sup O(ρac (t)). t∈[0,T ) (4.103) For this latter figure of merit, one can answer the question for the optimal ∗ (t) quickly, thanks to our previous characterization of S . In asymptotic cycle ρac fact, we know that no cycle can enter the region where the purity is larger than the maximal purity in S . In Bloch notation, this means that the cycle is restricted to a ball with radius R = sup |~r|. (4.104) ~r∈S On the other hand, for any point~r0 with |~r0 | ≤ R, one can always construct a cycle that contains this point, as shown below. This implies that the best possible peak value O∗max is just the objective function at the optimal choice for~r0 : O∗max ≡ sup Omax [~r(t)] = sup O(~r0 ). ~r(t)∈S (4.105) |~r0 |≤R In conclusion, an asymptotic cycle that optimizes the peak value Omax is readily found by performing an optimization over all states – not trajectories! – with |~r0 | < R. In particular, in the case of spontaneous decay only, we have R = 1; accordingly, ~r0 may lie anywhere in the state space Q, and there is no limitation for the peak value Omax in this case. The situation is illustrated in Fig. 4.9, where we consider as objective function O(~r) the coherence between ground and excited state, see Eq. (4.57). In Bloch notation, this corresponds to the distance from the z axis. Accordingly, the optimal ~r0 has the maximal possible radial component |~r0 | = R: max [Coh(~r0 )] = R, |~r0 |≤R for~r0 = (R, 0, 0). (4.106) 149 (a) γ+ = γd = 0 (b) γ+ = γd = γ− /4 zz z {{ (i) S zz z RR==1B 1 �r�0r0 (ii) �r�∗r∗ {{ f2f(�2r)(�fr) 2 (�r) .25 .25.25 00 0 B RR xx x (iii) �r�0r0 (iv) �r�∗r∗ S xx x -1-1-1 -2-2-2 Figure 4.9: Sketch of an asymptotic cycle ~r(t) (green curve) that optimizes the peak value Cohmax of the coherence (i.e., the distance to the z axis). As in Fig. 4.8, two different sets of the incoherent rates are shown in (a) and (b), and the color code refers to the purity flux f2 (~r). The red ring in panel (b) indicates the no-go area of |~r| > R that no asymptotic cycle can enter, cf. Eq. (4.104). No such area exists in panel (a), since the ellipsoid of stabilizable states S (dashed line) touches the boundary of the state space, so that R = 1. The optimal cycle passes through the point~r0 with Coh(~r0 ) = R. The optimal point~r∗ of the autonomous system, cf. Fig. 4.6, is plotted as a reference point. The corresponding optimal cycle ~r(t) evolves in the interior of S until it has reached the maximal possible purity |~r(t)| = R. In Fig. 4.9(a), where R = 1, this corresponds to the south pole of the Bloch ball. At this point, a strong, suitably chosen Hamiltonian H(t) generates the unitary transformation – i.e., the rotation in the Bloch ball – which brings the cycle to the desired point~r0 of equal radius. This rotation has to be fast, compared to the incoherent dynamics, so that no purity is lost along the way.25 Subsequently, the cycle immediately begins to lose purity, as~r0 lies in the region of negative flux f2 . It is then driven back into the interior of S , where the purification process starts over. 25 At this point, we rely on the assumption that any Hamiltonian H(t) of arbitrary strength is available. In fact, this assumption underlies our entire discussion in section 4.2 and 4.3, starting from the definition of the set of stabilizable states S , or cycles S . 150 Using the results of our previous calculation of the semi-axes of the ellipsoid S , we find the result Coh∗max = R = Γ− , Γ+ with Γ± = γ− ± γ+ . (4.107) This value always exceeds the coherence of the optimal stationary state~r∗ of the autonomous system, see Eq. (4.64): Γ− Coh(~r∗ ) = p . 2Γ+ (2γd + Γ+ ) (4.108) The difference is particularly pronounced when the dephasing rate γd is large. In conclusion, driving a single qubit with a suitable protocol H(t) can improve its coherence properties significantly, if the figure of merit is the peak value Cohmax of the coherence within the cycle. A more challenging task is to derive the cycle that optimizes the time average O of an objective O(~r), as defined in Eq. (4.102). This will be the aim of the following investigation. The cycle in Fig. 4.9, which optimizes the peak value Cohmax , performs utterly poorly in this game: Its time-averaged coherence vanishes, Coh = 0. The paradoxical result is due to the fact that the cycle must always reach the purest point ~r p in S before it can be rotated to ~r0 . Since the purity flux f2 vanishes, by definition, at any point of S , it becomes arbitrarily small in the neighborhood of ~r p . Hence, the cycle approaches ~r p ever more slowly, as dictated by Eq. (4.99). As a consequence, the cycle spends the vast majority of its time close to ~r p , where the coherence vanishes. This reduces the average coherence Coh drastically. In fact, in the limit |~r0 | → R, the cycle must reach ~r p exactly, and this process takes infinite time. This explains the result Coh = 0 in this limit. Nevertheless, we do know that cycles with Coh > 0 exist: The optimal stationary state ~r∗ is a special, but nonetheless valid asymptotic cycle. Being time-independent, its average coherence Coh is given by Eq. (4.108). This expression provides, hence, a lower bound for the optimal average coherence. Let us see whether one can do better than that. Reduction to elementary cycles of just two points To determine the optimal cycle with respect to the time-averaged objective function O, one has to solve an optimization problem in the set S of stabilizable cycles. Despite the characterization achieved with Proposition 3, S is still so vast and unstructured that a numerical, let alone an analytical treatment seems at first sight utterly hopeless. 151 p S �rp− �bp �r− (p) p1 p p0 p1 pmin pmax S (2) �r+ (p) Figure 4.10: Illustration of a general asymptotic cycle~r(t) ∈ S (green curve), and its purity-parametrization~r± (p) (black dots). The concentric, black rings indicate points of equal purity p = 1/2(|~r|2 +1), ranging from p0 to p1 . (Their circular shape is explained by the fact that states with equal purity have equal radius |~r|.) As in Figs. 4.8 and 4.9, the gray/orange color code indicates the purity flux f2 (~r), and the dashed line illustrates the ellipsoidal surface S of vanishing purity flux, f (~r) = 0. Yet, the complexity of the problem can be reduced drastically. To begin with, we recall one of the consequences of Proposition 3: Any cycle ~r(t) ∈ S undergoes subsequent stages of purity gains or losses, depending on whether it evolves in the interior of S , where f2 (~r(t)) > 0, or in the exterior, where f2 (~r(t)) < 0. The stages are separated by an even number of time steps ti at which the cycle intersects S , i.e., at which f2 (~r(ti )) = 0. In fact, our first general rule on p. 146 implies that the cycle must intersect S at least twice. In the following, we focus on cycles that intersect S exactly twice, as sketched in Fig. 4.10; the analysis of cycles intersecting S more often follows readily from this discussion. Hence, our cycle consists of two stages. The first stage begins at time t0 ≡ 0, when the cycle enters the orange region of positive flux, and ends at t1 , when the cycle leaves this region. During this stage, condition (4.99) dictates that its purity µ2 ≡ p increases strictly monotonically from its minimum value p0 to its maximum value p1 . This implies that one can map the time interval [0,t1 ] one-to-one onto the purity interval [p0 , p1 ]. Accordingly, instead of the time argument t, the cycle can just as well be parametrized by the purity p in this stage; i.e., one can define ~r+ (p) ≡~r(t(p)) for t ∈ [0,t1 ]. The same argument applies to the second, purity-decreasing stage, which begins at time t = t1 and ends at time t = t2 ≡ T : To each purity value p ∈ [p0 , p1 ], one can p0 �ap �rp+ 152 uniquely assign a time t ∈ [t1 , T ], and vice versa; accordingly, the second stage can also be parametrized by the purity, ~r− (p) ≡~r(t(p)) for t ∈ [t1 , T ]. In conclusion, any cycle intersecting S twice can be described by the pair~r± (p): ~r(t) with t ∈ [t0 , T ] ←→ (~r+ (p),~r− (p)) with p ∈ [p0 , p1 ]. (4.109) With this parametrization, the time-averaged objective function O can be bounded from above: O[~r(t)] ≤ sup OTPC (~r+ (p),~r− (p)), (4.110) O(~r+ )| f2 (~r− )| + O(~r− )| f2 (~r+ )| . | f2 (~r+ )| + | f2 (~r− )| (4.111) p∈[p0 ,p1 ] where we defined the quantity OTPC (~r+ ,~r− ) = The inequality (4.110) is proved in appendix A.3. It can be interpreted as follows: The time-averaged objective function O of a general cycle, with purity values in [p0 , p1 ], is smaller than the quantity OTPC , evaluated for the pair~r± (p) at the optimal intermediate value of p. In fact, OTPC has the meaning of the timeaveraged objective function of a minimalistic cycle that rapidly jumps back and forth between ~r+ (p) and ~r− (p). To see this, consider a cycle starting at point ~r− (p), as illustrated in Fig. 4.11. Within a short time δt− , the cycle will lose the purity amount δ p. According to the condition (4.99), which every cycle has to obey, this amount is δ p = ṗ δt− = f2 (~r− (p)) δt− < 0 ⇒ −δ p = | f2 (~r− (p))| δt− > 0. After the short time δt− , we take a “short cut” and instantaneously jump to the point ~r+ (p − δ p) of equal purity p − δ p. This is always possible, since appropriately chosen, strong Hamiltonian dynamics can generate any purity-preserving, unitary “kick”, i.e., any rotation in the Bloch ball. To reach~r+ (p), the purity loss δ p must be re-gained. This takes a time δt+ , determined by δ p = ṗ δt+ = f2 (~r+ (p)) δt+ > 0. After this time, a second unitary rotation to the starting point ~r− (p) closes the cycle. In the limit δ p → 0, this defines a minimalistic two point cycle (TPC). Of 153 �r− (p − δ p) �r− (p) p−δ p p �r+ (p) �r+ (p − δ p) Figure 4.11: Construction of a two point cycle (TPC, blue curve) at an intermediate purity value p of the exemplary asymptotic cycle (green curve) of Fig. 4.10. In the initial step from~r− (p) to~r− (p − δ p), the TPC loses the purity δ p during a short time δt+ ≈ | f2 (~r− (p))|δ p; after that, it is coherently transferred to~r+ (p − δ p) by a unitary, instantaneous “kick”; it regains purity in the third step~r+ (p−δ p) →~r+ (p) (which takes a time δt− ≈ | f2 (~r+ (p))|δ p), and is then again instantaneously rotated back to~r− (p). An exact TPC emerges in the limit δ p → 0. course, the time spans δt± that the cycle spends in the neighborhood of~r± (p) also tends to zero in the TPC limit, and so does the period T = δt+ + δt− . Nevertheless, their ratio remains finite: δt+ δ p/| f2 (~r+ (p))| | f2 (~r− (p))| = = . δt− δ p/| f2 (~r− (p))| | f2 (~r+ (p))| (4.112) Thus, the time averaged objective function of the TPC is O[~rTPC (t)] = 1 T O(~r+ (p))δt+ + O(~r− (p))δt− (4.113) δt+ + δt− δt± →0 0 O(~r+ (p))| f2 (~r− (p))| + O(~r− (p))| f2 (~r+ (p))| | f2 (~r+ (p))| + | f2 (~r− (p))| Z T (4.112) = (4.111) dt O(~rTPC (t)) = lim ≡ OTPC (~r+ (p),~r− (p)). With this, the full benefit of inequality (4.110) comes to light: Any stabilizable cycle is always outperformed by one of the TPCs that it contains. This allows us to restrict the optimization of O over all cycles in S to the much smaller subclass 154 of TPCs: ∗ O ≡ sup O(~r(t)) = sup OTPC (~r+ ,~r− ). S STPC (4.114) This is a tremendous advantage, since a parametrization of the class STPC (S of TPCs becomes a manageable task – in contrast to the original set S . In fact, a TPC is simply defined as a pair (~r+ ,~r− ) of Bloch vectors that obeys the following conditions: I ~r+ and~r− must have equal purity p± = 1/2(1 + |~r± |2 ), i.e., |~r+ | = |~r− |. (4.115) I ~r+ (~r− ) must lie in the purity-increasing (-decreasing) region of the state space, i.e., f2 (~r+ ) > 0 and f2 (~r− ) < 0. (4.116) The only task left is to optimize OTPC (~r+ ,~r− ) under these two constraints. This can be done either numerically or – for simple objective functions O(~r) – possibly even analytically. Before we discuss tangible results, let us emphasize the universality of our arguments: The reduction to simple TPCs is valid for any objective function O(~r) and any purity flux f2 created by a Lindblad dissipator D. Results of the numerical optimization We are now in the position to answer our original question: What is the optimal driving protocol that maximizes the time-averaged coherence Coh? The dissipator be again our standard example (4.31) that describes decay, excitation, and dephasing at rates γ− , γ+ , and γd . Each of the two points~r± of a TPC is described by its modulus r± = |~r± | ≤ 1, its polar angle ϑ± , and its azimuthal angle ϕ± . In principle, this amounts to six parameters that have to be optimized. In our case, however, the optimization quantity OTPC is independent of the azimuthal angles, since both the dissipative flux f2 (~r) and the objective Coh(~r) (the distance from the z axis) are. This allows us to set ϕ± = 0, reducing the number of parameters to four. In addition, condition (4.115) demands r+ = r− ≡ r and thereby reduces this number to three. The second constraint (4.116) is not as simple, but must be accounted for by a side condition in the optimization procedure. The numerical optimization is now a child’s play. In fact, since all three parameters are bounded, one can simply sample the entire three-dimensional parameter space and extract the point of maximal OTPC . This way, one can be sure 155 to find the global maximum, and not to get stuck in local maxima. We have performed this procedure for all possible combinations of the incoherent rates in the range from γ+ = γd = 0 to γ+ = γd = 10 γ− . The result is always the same: The optimal TPC reduces to a single point, i.e., ~r+ =~r− defines the optimum. Rigorously speaking, ~r+ =~r− can never be reached, because ~r+ and ~r− must always lie on opposite sites with respect to the ellipsoidal surface S , due to the condition (4.116); but in the limit ϑ+ → ϑ− , the TPC collapses onto a single point on the ellipsoid. In fact, the optimal TPC collapses exactly onto ~r∗ , the optimal state that we derived in our analysis of the autonomous system, see Eq. (4.64) and Fig. 4.6. In conclusion, the optimal OTPC has exactly the same value as in the autonomous case. The finding that no TPC exceeds the static optimum implies, in turn, that no asymptotic cycle at all achieves this goal. In summary, the definite answer to our initial question is: Given a single qubit that is subject to the incoherent processes of decay, excitation, and dephasing, there is no driving protocol H(t) by which one can generate an asymptotic cycle that outperforms – with respect to the timeaveraged coherence – the optimal stationary state which can be reached by the time-independent Hamiltonian (4.65). We stress, however, that this finding depends strongly on the details of both the dissipator and the objective function. In fact, it is simple to invent an objective O(~r) with respect to which the optimal TPC outperforms any autonomous scenario. Consider, e.g., the “all-or-nothing” objective ( O(~r = (x, y, z)) = 0 if z ≤ 0 1 if z > 0 . (4.117) Sticking to our standard dissipator, the set S of possible stationary states of the autonomous system lies entirely in the lower half of the Bloch ball, where O(~r) = 0; accordingly, the autonomous optimum is O(~r∗ ) = 0. An asymptotic cycle, however, can easily enter the northern hemisphere of the Bloch ball, and therefore reach O(~r(t)) > 0. It turns out that the optimal TPC – and therefore the globally optimal cycle – even reaches OTPC = 1. Despite the rather artificial character of this example, we learn that periodic driving can in principle improve an objective function not only with respect to the peak value Omax , but also with respect to the time-average O. 156 4.3.3 Optimal asymptotic cycle of two qubits Optimal average concurrence C With this background, we are geared up to tackle the case of two qubits, where the figure of merit is the entanglement of the asymptotic cycle, as measured by the time-averaged concurrence C . Compared to the single qubit case, the only new aspect is that Proposition 3 imposes not only a condition on the evolution of the purity µ2 , but also on the third and fourth moment µ3 and µ4 . Consequently, the set of stabilizable cycles is (basically26 ) the intersection of three sets: S = d \ Sn , (4.118) n=2 with d = dim H = 4 and ( Sn = [ T ∈R+ ) ρ(0) = ρ(T ) ∧ ρ : [0, T ] 7→ Q . ∀t : µ̇n (t) = fn (ρ(t)) (4.119) Nevertheless, one can reduce the complexity of the problem by merely optimizing over the set S2 of “purity-compliant” cycles, as sketched in Fig. 4.12. For this set, one can fall back on the concepts developed above for a single qubit. In particular, the rules formulated on p. 146 remain valid, and so does the entire argumentation about TPCs. Relying again on the 15-dimensional, generalized Bloch vector ~r that was introduced in Eq. (4.66), the purity reads p ≡ µ2 = 1 1 + |~r|2 . 4 (4.120) Consequently, the condition of equal purity of the two points~r± of a TPC is again |~r+ | = |~r− |, as in Eq. (4.115). Representing~r± in generalized spherical coordinates [191], this condition is simple to account for and reduces the degrees of freedom of the optimization problem to 2 × 15 − 1 = 29. The purity flux f2 (~r) is given by Eq. (4.101), where one has to insert the respective quantities for two qubits, see Eq. (4.68). It imposes, in form of Eq. (4.116), a side condition on the optimization procedure. A subtle point for the optimization is that the state space Q is no longer the unit ball in the generalized Bloch representation [120, 183]. Accordingly, one cannot 26 This statement holds up to cycles that pass through a state with degenerate eigenvalues, for which Proposition 3 does not provide a sufficient criterion for membership in S . S3� 157 S� ∗ ρac ∗(2) ρac S2� S4� � STPC S3� S� ∗ ρac ∗(2) ρac S2� S4� � STPC S3� S� ∗ ρac ∗(2) ρac S2� S4� � STPC Figure 4.12: Interrelation of the different sets Sn , defined in Eq. (4.119), and ∗(2) their respective optima. The optimal cycle ρac (t) in S2 – with respect to the time-averaged concurrence C – is necessarily a TPC, i.e., it lies within STPC . ∗(2) Two different scenarios are possible: Either ρac (t) ∈ S (left), in which case ∗ (t) in S ; or ρ ∗(2) (t) ∈ / S (right), it coincides with the sought-for optimum ρac ac so that it defines at least an upper bound for the optimum in S . This argument is, hence, similar in spirit to the analogous discussion of the autonomous system, see Fig. 4.7. cover the entire state space by simply restricting the common radius r = |~r± | of the TPC to values between 0 and 1. Rather, the boundary of Q is reached at different radii, depending on the remaining angular degrees of freedom. √ √ In detail, the radial coordinate of the boundary varies between d − 1 and 1/ d − 1, where d = 4 in our case [120]. To account for √ this subtlety in the optimization procedure, we bind r to the outer radius, 0 ≤ r ≤ 3, and introduce an additional side condition that ensures positivity of the density operator. Due to the larger number of variables, the resulting optimization problem for C TPC (~r+ ,~r− ) is not as elementary as in the single qubit case. Nevertheless, it can R be solved numerically, e.g., with a gradient-based algorithm in Matlab [192, 193]. We have performed this procedure for different combinations of the incoherent rates γ+ , γ− , and γd . To avoid convergence to a local rather than the global optimum, the algorithm was repeatedly run with 103 different (random) starting values. Interestingly, the outcome of the procedure is similar to what we observed earlier for the coherence of a single qubit: The optimal TPC always collapses to a single point that corresponds, in fact, to the optimal stationary state~r∗ ∈ S2 . The ∗ optimal value C of the time-averaged concurrence is therefore not improved in the periodically driven system, compared to the autonomous case. The same result is obtained when optimizing the Bell fidelities instead of the concurrence. A selection of numerical values is listed in Table 4.1. S3� S� ∗ ρac ∗(2) ρac S2� S4� � STPC 158 C ∗ Fidφ ± γd = 0 γd = γ− /4 γd = γ− /2 γd = γ− .499 .400 .347 .270 ∗ .656 .600 .569 .544 ∗ .500 .430 .402 .378 Fidψ ± Table 4.1: Entanglement of the numerically determined optimal TPC, quantified by the time-averaged concurrence C or Bell fidelities Fidφ ± and Fidψ ± . The dephasing rate γd varies from 0 to γ− , while the excitation rate is kept fix at γ+ = 0. All values coincide with the results for the optimal stationary state ρ ∗ of the autonomous system, meaning that the optimal TPC collapses to a single point. The same holds true for γ+ > 0 (not listed). In conclusion, the entanglement properties of the asymptotic dynamics of two qubits undergoing decay, excitation and dephasing is under no circumstances improved by widening the scenario from autonomous to periodically driven systems. This is a strong and quite surprising result; after all, the class of periodically time-dependent Hamiltonians outnumbers the class of time-independent Hamiltonians by far (in fact, the latter is contained in the former). Nevertheless, in all investigated cases, the optimal driving protocol H(t) turns out to coincide with the optimal autonomous Hamiltonian H. Optimal peak value Cmax One must emphasize, however, that this conclusion refers to the time-averaged entanglement C . If we consider, instead, the peak value Cmax of the concurrence within the cycle, a suitably driven system can very well outperform the optimal autonomous system. To see this, we recall our analysis of the single qubit case (see Fig. 4.9). There, we argued that any point~r0 can be reached in an asymptotic cycle, as long as it lies within a ball of radius R. The value of R is determined by the largest extension of S , as stated by Eq. (4.104). The crucial argument was that under this condition, ~r0 can still be reached by a rotation (i.e., a unitary transformation) from the purest state in the interior of S . Abstracting from the geometrical picture, the argument can be put as follows: Given a state ρ0 ∈ Q, there is an asymptotic cycle that contains ρ0 , if and only if ρ0 is unitarily equivalent to some state in the interior of the set of stabilizable states S . In this form, the statement holds for quantum systems of any dimension. 159 f2 (�r) |11� |φ− � .5 0 |φ+ � -2 -4 |00� Figure 4.13: Two-dimensional section through the state space of two qubits, illustrating the asymptotic cycle that optimizes the peak value Cmax of the concurrence (green curve). √ During this cycle, the maximally entangled Bell states |φ+ i = (|00i + |11i)/ 2 is reached, so that Cmax = 1. Without any external intervention, the cycle is bound to spend most of its time in the neighborhood of |00i, due to the vanishing flux f2 at this point (indicated by the color code). The ideas presented in Section 4.3.4 aim for an acceleration of this purification process, in order to improve the time-averaged concurrence C of the cycle. Let us, e.g., analyze the case of two qubits that undergo spontaneous decay only (γ+ = γd = 0). In this case, the de-excited state |00i is in S , because it is the stationary state in the absence of any Hamiltonian dynamics, i.e., it is stabilized by the static Hamiltonian H = 0. With |00i being a pure state, this means that any pure state |χi can be reached by a suitable cycle. In particular, |χi can be one of the maximally entangled Bell states, yielding the optimal result Cmax = 1. This situation is illustrated in Fig. 4.13, where a two-dimensional section through the 15-dimensional state space of two qubits is shown. The section represents the projection of the Hilbert space H of two qubits on the subspace spanned by |11i and |00i. It mimics, in this sense, an effective two-level system. The Bell states |φ± i – i.e., the balanced superpositions of |00i and |11i – lie on the equator of the resulting Bloch ball. This renders the situation analogous to our single qubit√ example in Fig. 4.9, where the goal was to reach the state (|0i + |1i)/ 2 of maximal coherence. The main difference in the present case is that the instantaneous rotation from |00i to |φ+ i is a non-local unitary transformation. Its implementation by the driving Hamiltonian H(t) involves, hence, some interaction between the qubits. Specifically, the driving protocol that 160 generates this transformation reads H(t) = π t δ mod 1 σx ⊗ σy . 4 T (4.121) By virtue of its δ -peaked time-dependence, it instantaneously “kicks” the state from |00i to |φ+ i, at stroboscopic times t = nT (n ∈ Z). The period T of this pulse train, however, must be chosen much larger than the relaxation time scale γ −1 , because the cycle has to return to |00i before the next “kick”. (Strictly speaking, this is achieved to full extent only in the limit γT → ∞.) As a consequence, the cycle that optimizes the peak value Cmax spends most of its time close to |00i, where the concurrence vanishes. Hence, the price one has to pay for reaching a maximally entangled state within the cycle is that the time-averaged concurrence C drops to zero. 4.3.4 Additional means to further improve the optimum From the previous investigations, we conclude that there is a bottleneck which prevents the time-averaged concurrence C from exceeding a certain threshold, e.g., C ≤ 1/2 for spontaneous decay alone. In the following, we analyze this bottleneck in more detail and discuss two optional ways to overcome it. To recognize the origin of the limitation to C ≤ 1/2, we rewrite the time-averaged concurrence of a TPC, C TPC , in terms of the infinitesimal “dwell times” δt± that the cycle spends at~r± :27 (4.113) C TPC (~r− ,~r+ ) = C (~r− ) δt− δt+ + C (~r+ ) . δt+ + δt− δt+ + δt− (4.122) This expression is intuitively understood: The time-average of the concurrence over the TPC is the sum of the concurrence at ~r+ and ~r− , weighted by the proportion of the respective dwell times. The (empirically observed) fact that C TPC never exceeds the value of 1/2 implies the following: 1. There is no TPC for which C (~r− ) > 1/2 and C (~r+ ) > 1/2 at the same time. Otherwise, their weighted sum (4.122) would certainly exceed 1/2, independently of the ratio of the dwell times. Thus, either ~r− or ~r+ have concurrence below 1/2. 2. Assume that, say,~r− has concurrence above 1/2, so that C (~r− ) = 1/2 +x (with 0 < x ≤ 1/2). From our argument above, we know that C (~r+ ) = 1/2 − y in 27 For convenience, we omit the limit δt± → 0 in the following discussion. 161 this case (with 0 < y ≤ 1/2). From the observation that altogether C TPC ≤ 1/2 holds, we conclude 1/2 (4.122) (1/2 + x)δt− ≥ δt+ + δt− ⇒ yδ+ ≥ xδ− . + (1/2 − y)δt+ xδt− − yδt+ = 1/2 + δt+ + δt− δt+ + δt− (4.123) Thus, either is the “excess concurrence” x of~r− small compared to the “loss” y at~r+ , or the dwell time δt− is much shorter than δt+ . Apparently, there exists a tradeoff between the two quantities: The higher the concurrence at the one point of the TPC is compared to the other, the smaller is its (relative) dwell time. Since the dwell time is inverse proportional to the modulus of the purity flux | f2 (~r)|, cf. Eq. (4.112), this is tantamount to a tradeoff between concurrence and purity flux: Apparently, a large concurrence value at ~r− always entails a strong purity flux at this point, and hence a short dwell time δt− . This tradeoff can exemplarily be studied for the asymptotic cycle shown in Fig. 4.13, which alternates between the separable state |00i and the maximally entangled Bell state |φ+ i. In the TPC limit, a state near |00i takes the role of the point ~r+ in the purity-increasing (orange) region, and a state adjacent to |φ+ i corresponds to ~r− . The tradeoff is now apparent: While states near |φ+ i easily outclass those around |00i in terms of entanglement (C (|φ+ i) = 1 vs. C (|00i) = 0), their purity flux is by far larger (in absolute value), since the purity flux vanishes completely at |00i, while it is finite at |φ+ i, | f2 (|φ+ i)| = γ− . In the end, this is a consequence of the local dissipation mechanism we are studying here: Entangled states are very fragile to this kind of dissipation, and this is manifested in a strong (negative) purity flux around highly entangled state. In conclusion, our analysis suggests that the bottleneck for the time-averaged entanglement of a TPC (and therefore of any asymptotic cycle) is the capability of the dissipator to efficiently purify the cycle in the less entangled, purity-increasing stage of the cycle.28 From this analysis, we conclude that a possible way to enhance C beyond the threshold 1/2 is to speed up the evolution (i.e., reduce the dwell time) when the cycle is in the poorly entangled stage~r+ of increasing purity. This idea is illustrated by the green arrows in Fig. 4.13. However, since Hamiltonian dynamics just cannot modify the purity flux, this idea necessarily involves additional means beyond a cleverly chosen driving protocol H(t). 28 This conclusion is, in fact, also drawn in Refs. [39, 40], where the authors study the entanglement of two qubits in the asymptotic cycle as well. Hamiltonian and on Cmax as the figure of merit. They focus, however, on a particular model 162 Dedicated amplification of the decay rate One way to speed up the purification is to deliberately amplify the incoherent dynamics when the cycle traverses a purity-increasing region. This means that the incoherent rates γk must be temporarily increased, in a controlled manner, by a factor α > 1.29 According to Eqs. (4.31) and (4.98), the dissipator D(ρ) and hence the purity flux f2 will then be rescaled by the same factor α, so that the purification is increased by the factor α. For a TPC, this means that the purity flux must be increased only while the cycle resides at the purity-increasing point r~+ , so that C (~r+ )| f2 (~r− )| + αC (~r− )| f2 (~r+ )| C TPC (~r− ,~r+ ) = . (4.124) α| f2 (~r+ )| + | f2 (~r− )| In the extreme case of infinite amplification, the cycle is purified instantaneously and spends no time in the poorly entangled region near |00i. This extreme scenario corresponds to a perfect “reset mechanism” that immediately brings the cycle back to |00i from any point in the state space. In this limit, we find α→∞ C TPC (~r− ,~r+ ) −→ C (~r− ). (4.125) This situation is then analogous to the optimization of Cmax . In particular, if spontaneous decay is the only incoherent process (i.e., γ+ = γd = 0), one can reach values up to C TPC = 1 in this way. If the available amplification factor α is limited to some finite value, the ∗ optimal TPC does not reach C TPC = 1, but only a partial improvement over the original threshold 1/2 can be achieved. This effect is studied in Fig. 4.14, where ∗ C TPC is plotted as a function of α. Interestingly, up to α ≈ 2, the amplification does not lead to any improvement; in fact, the optimal TPC still collapses to a single point~r∗ , i.e., the optimal Hamiltonian is still autonomous. From α > 2 on, however, it becomes more advantageous for the TPC to split up into two distinct points. From there on, the time-average entanglement of the cycle increases, and ∗ finally reaches C TPC = 1 in the limit α → ∞. Local projective measurements and feedback An alternative way to achieve instant purification – i.e., a fast reset mechanism to |00i – is based on the idea of quantum feedback loops [197]: If one projectively measures a given quantum state ρ in a complete basis {|ii} and afterwards 29 For the important case of spontaneous decay only (γ = γ = 0), this can achieved by placing + d the qubit in a tunable cavity and increasing the decay rate γ− via the Purcell effect [194–196]. ∗ C TPC Concurrence of optimal TPC 163 1 .75 .5 0 10 1 10 2 _ 10 3 10 α Figure 4.14: Behavior of the time-averaged concurrence C TPC of the optimal TPC (for spontaneous decay only, γ+ = γd = 0), when the decay rate γ− is deliberately amplified by a factor α at the purity-gaining stage ~r+ of the cycle. The data points are obtained from a numerical maximization of expression (4.124) over all TPCs and connected by a line to guide the eye. Above α ≈ 2, the amplification leads an improvement over the original value of C TPC = 1/2. In the limit α → ∞, the purification process is infinitely fast, and the optimal TPC reaches maximal concurrence C TPC = 1. The minor fluctuations at α = 2 and α = 4 are due to the numerical character of the optimization procedure, which ends in a local rather than the global optimum from time to time. performs, depending on the outcome of the measurement, the unitary feedback operations Ui = |χi hi|, one establishes the map measure U i ρ 7−→ |ii hi| 7−→ |χi hχ| . Hence, one always achieves ρ 7→ |χi hχ|, independent of which intermediate state |ii is realized along the way. This way, one can reset any ρ ∈ Q to a desired pure state |χi [198, 199]. This scheme can readily be applied to our scenario: ρ is the state in the neighborhood of |00i that we would like instantaneously reset to |χi = |00i. E.g., one can simply perform a local measurement of both qubits and thereby map ρ on one of the four computational basis states {|00i , |10i , |01i , |11i}. Since ρ is close to |00i, it is most likely to find both qubits in the de-excited state, in which case the reset procedure is complete. If one finds one or both qubits in the excited state, a bit-flipping operation σx has to be applied to the respective qubit(s), in order to map the state back onto |00i. 164 With this, we conclude Section 4.3, where we studied periodically driven, open quantum systems evolving under the master equation (4.30). To summarize, we have achieved the following goals: I We characterized the class of all conceivable asymptotic cycles via Proposition 3, which is a generalization of the respective Proposition 1 for stabilizable states. From this, we derived (on p. 146) two general, intuitive rules that every asymptotic cycle has to obey. I We developed a general framework to determine the optimal asymptotic cycle and the corresponding driving protocol H(t), based on the concept of minimalistic two point cycles. It applies to arbitrary dissipators D(ρ) and arbitrary objective functions O(ρ). Only when discussing specific results, we specified the dissipator by the exemplary form (4.31) and fixed the objective function. I We analyzed the instructive example of a single qubit and determined the cycles that optimize either the peak value or the time-average of the coherence between ground and excited state. The developed concepts were then transferred to the case of two qubits and their entanglement. In both cases, it turns out that periodic driving cannot improve over the values found in the previous Section 4.2 for autonomous systems. I We pointed out how to enhance the observed limits, provided additional control beyond the Hamiltonian driving is available. Chapter 5 Conclusion In the present thesis, we developed a general understanding of the dynamics of entanglement in quantum systems that are subject to periodic driving forces. Our particular focus was the persistent entanglement that survives in the long-term dynamics. Thus, we investigated under which circumstances periodic driving can promote and sustain substantial entanglement in the cycle-stationary state. In the course of the thesis, we studied a variety of approaches to this question, ranging from closed to open quantum systems and from specific model Hamiltonians to the most general class of all periodically time-dependent Hamiltonians. A summary of each of these approaches is given at the end of the respective section. In this final chapter, we highlight the most important results and link them to a bigger picture. The Floquet states of the system Hamiltonian were the objects of interest in Chapter 3, as they define the cyclo-stationary states of the dynamics in a closed quantum system. Specifically, we assumed a physically motivated model Hamiltonian, describing two qubits under external driving and a weak interaction mechanism between them. The entanglement of its Floquet states was determined numerically as a function of the system parameters. It peaks in sharply defined regions in the parameter space. This phenomenon, dubbed entanglement resonance, was explained by a perturbative analysis in the qubit-qubit interaction strength, revealing that it occurs at degeneracy points in the quasi-energy spectrum of the non-interacting system; these degeneracies are lifted and turned into avoided crossings in the presence of a finite interaction strength, triggering thus an entanglement resonance. Finally, the universality of this mechanism was emphasized by studying several modifications to the original model. Our transparent analysis in the Floquet picture has several advantages. First, it allows to predict the position of resonances from the quasi-energies of a single 165 166 driven qubit alone, as expressed by the resonance condition (3.32). In fact, we could solve this problem exactly, e.g., for δ -kicked or sawtooth driving. Secondly, it explains why certain resonances are of second or higher order in the interaction strength, or even completely suppressed, and which modifications of the driving profile or interaction mechanism can activate these resonances. And finally, our analysis can be applied to questions beyond the scope of entanglement: From a broader perspective, it provides a recipe to tune the level of a weakly interacting quantum system into resonance with AC (instead of the conventional DC) driving fields. This recipe can be applied to many problems regarding weakly interacting many-body quantum systems, such as the control of interacting Rydberg atoms by AC fields [54, 169], or the problem of spin diffusion under driving in nuclear magnetic resonance [107, 200]. Chapter 4, on the other hand, was devoted to the study of entanglement and driving in open quantum systems, i.e., in the presence of decoherence. The starting point in Section 4.1 was the same model as in Chapter 3, but coupled to a bath at finite temperature. The dynamics were described by the Floquet Born-Markov master equation, which relies crucially on the assumption of weak environment coupling. In this case, the open system dynamics evolve into an asymptotic cycle that is a mixture of the Floquet states of the system Hamiltonian. We determined its entanglement properties numerically and observed again a resonant behavior in some parts of the parameter space. This phenomenon was studied in more detail. One of the resonances is actually already present in the undriven system; it reaches maximal entanglement, but vanishes already above temperatures of the order of the qubit-qubit interaction strength. The remaining, genuinely driving-related resonances, on the other hand, reach only intermediate entanglement values, but they are far more robust against thermal noise. Nevertheless, it must be stressed that these resonances can already be explained within the RWA, i.e., in an effectively autonomous picture. Stronger or off-resonant driving – provoking dynamics beyond the RWA – neither increases the peak value of the resonances nor their robustness against finite temperature. Once again, we emphasized the universality of these findings by modifying the model assumptions in various ways. To explore whether a periodically driven quantum system can feature even stronger entangled asymptotic cycles, we changed our perspective in Sections 4.2 and 4.3 and asked for the globally optimal, periodic driving protocol; i.e., we sought the periodically time-dependent Hamiltonian that leads to the most entangled among all conceivable asymptotic cycles. Here, we no longer employed the Floquet Born-Markov master equation, but rather imposed a fixed, 167 phenomenologically motivated incoherent dynamics that describes the most important incoherent processes for qubits: decay, excitation, and dephasing. Before we could tackle the vast optimization over all conceivable periodic driving protocols, we had to do a good amount of preparatory work. First, we investigated the corresponding problem for undriven systems in Section 4.2, where the concept of asymptotic cycles is reduced to that of stationary states. After calibrating our expectations by investigating the entanglement of typical asymptotic states, we derived a systematic characterization of the set of stabilizable states, i.e., of states that can be rendered stationary by some static Hamiltonian. The characterization was guided by the fact that coherent (i.e., Hamiltonian) dynamics can only compensate for the spectrum-preserving part of the dissipative dynamics. A state is therefore stabilizable if and only if its spectrum is conserved by the dissipative dynamics. This led to a hierarchy of algebraic conditions for a stabilizable state. An optimization over the set of stabilizable states could then readily be performed and the corresponding optimal Hamiltonian followed straightforwardly, as well. As the main result, we found that the optimal stationary state of two qubits is of finite, but non-maximal entanglement measure, similar to the findings of our specific model system in Section 4.1. Much more generally, however, our approach applies to any optimization objective that one wants to maximize in the stationary state, as well as to any (fixed) dissipative dynamics. This is emphasized in the illustrative example of a single qubit, where the set of stabilizable states corresponds to the surface of an ellipsoid that is embedded in the Bloch ball of all quantum states. These concepts were then generalized to the periodically driven case in Section 4.3. Depending on whether one is interested in the peak or the average value of the objective function in the asymptotic cycle, different approaches were discussed. For the optimization of the peak value, this results in a mere optimization over a single state, instead of an entire trajectory. For the time-averaged objective, a drastic reduction of the problem to elementary cycles with just two points could be achieved, which enables a numerical treatment at least for low-dimensional systems.1 With this, our original question for the optimal driving protocol could be answered. It turned out that the best possible situation is already achieved with the optimal static Hamiltonian studied in 1 In principle, our framework applies to any finite, d-dimensional quantum system. Practically, however, one has to conduct an optimization of a function of two states, Eq. (4.111), with three side conditions, Eqs. (4.115) and (4.116). Accordingly, the biggest numerical challenge with increasing d is the memory required to store two d-dimensional quantum states. In addition, two states provide 2(d 2 − 1) degrees of freedom to be optimized, making it ever more difficult with increasing d to determine the global optimum. 168 Section 4.2 – i.e., the optimal asymptotic cycle is not a proper cycle, but rather a time-independent, stationary state. The reasons for this observation were analyzed at the end of Section 4.3; it turns out that the limiting factor is the capacity of the dissipative dynamics to efficiently purify the quantum system in the respective stage of the cycle. Based on this analysis, we discussed additional means to overcome this bottleneck, which necessarily have to go beyond a periodic, coherent driving mechanism. Again, the scheme applies also to arbitrary optimization objectives beyond entanglement and to arbitrary incoherent dynamics. Altogether, our analysis in Sections 4.2 and 4.3 answers not only our original problem of finding the most entangled asymptotic cycle, but sheds, much more generally, new light on the question which key factors determine the long-term dynamics of open quantum systems. In conclusion, the question whether entanglement can behave resonantly under suitable periodic driving was answered in the affirmative for both closed and open quantum systems in this thesis. While this connection was, in principle, observed before in specific model systems [34, 36, 38–40], we elaborated very generally the conditions that underlie such behavior and systematically analyzed the limitations imposed by decoherence. Along the way, we gained insights and developed recipes that go beyond the scope of entanglement and can be employed to address a variety of different questions in open, (possibly) driven quantum systems. In particular, we characterized in full generality the set of all potential stationary states (for autonomous systems) or asymptotic cycles (for periodically driven systems) that can emerge under a given dissipative dynamics. With this approach, one can determine the maximal amount of a desired (quantum) property that can persist on asymptotic time scales in an open quantum system. This has various applications beyond our original scope: E.g., it allows one to clearly identify the limits of dissipative state preparation, when the dissipative processes cannot be engineered at will, but have to be accepted as fixed by the environment. Very generally, it provides a recipe to devise optimal coherent control mechanisms for open quantum systems that maximize an arbitrary target property on asymptotic time scales. This is certainly relevant to many quantum control scenarios, e.g., to enhance the signal in nuclear magnetic resonance setups [107], or to control of the reaction yield in molecular processes [201]. Appendix 169 170 A.1 Incoherent transition rates for two driven qubits In this appendix, we calculate the incoherent transition rates γi j in the Floquet Born-Markov master equation for the particular example of two periodically driven qubits. The driving field is treated in rotating wave approximation (RWA), see p. 68 ff. The results are used in Section 4.1.3 to understand the entanglement resonance around ω0 ≈ ω. The setting is described in Section 4.1.1; in particular, the sought transition rates are defined in Eq. (4.8): γi j = |g̃|2 ∞ ∑ k=−∞ |ε ji + kω|3 |Nth (ε ji + kω)| 2 ∑ |x ji (m) (k)|2 . (A.1) m=1 Their calculation is simplified by the fact that the transition amplitudes have (m) (m) the property xi j (k) = [xi j (−k)]∗ , see their definition (4.9). Moreover, since all Floquet state are either symmetric or antisymmetric under permutation of the qubits in our case, they have eigenvalues pi = ±1 under the permutation operation Π, implying that (1) (4.9) xi j (k) = Z T 1 T 0 dt hΦi (t)|σx ⊗ 1|Φ j (t)i e−ikωt 1 T = dt hΦi (t)|Π2 σx ⊗ 1|Φ j (t)i e−ikωt T 0 Z 1 T dt hΦi (t)|Π 1 ⊗ σx Π|Φ j (t)i e−ikωt = T 0 Z 1 T (2) = pi p j dt hΦi (t)|1 ⊗ σx |Φ j (t)i e−ikωt = ±xi j (k) T 0 Z ⇒ (1) (4.9) (2) |xi j (k)|2 = |xi j (k)|2 . Due to this, we can replace the summation over the subsystem index m in Eq. (A.1) by evaluating only m = 1 and multiplying the result by two; the superscript (m) is therefore omitted in the following. 171 Transition rates between the non-interacting Floquet states In the absence of qubit-qubit interaction (J = 0), the unperturbed Floquet states in the region of interest (F |∆| ω, with ∆ = ω0 − ω) are provided in Eq. (4.20), |Φ1 (t)i = e+iωt |00i , 1 |Φ2 (t)i = √ (|01i + |10i) = |ψ + i , 2 1 |Φ3 (t)i = √ (|01i − |10i) = |ψ − i , 2 −iωt |Φ4 (t)i = e |11i . (A.2) To calculate the transition amplitudes xi j (k) for these states, we exploit that application of σx merely interchanges the computational basis states |0i and |1i, implying that ( 1 if a 6= c ∧ b = d ∀a, b, c, d ∈ {0, 1} : hab|σx ⊗ 1|cdi = (A.3) 0 else. The non-vanishing transition amplitudes can now be calculated in a straight-forward manner: 1 T Z T 1 dt h00|σx ⊗ 1|ψ + i e−i(k+1)ωt = √ δ−1,k , 0 2 1 x21 (k) = [x12 (−k)]∗ = √ δ+1,k , 2 Z 1 T 1 x13 (k) = dt h00|σx ⊗ 1|ψ − i e−i(k+1)ωt = − √ δ−1,k , T 0 2 1 x31 (k) = [x13 (−k)]∗ = − √ δ+1,k , 2 Z T 1 1 dt hψ + |σx ⊗ 1|11i e−i(k+1)ωt = √ δ−1,k , x24 (k) = T 0 2 1 x42 (k) = [x24 (−k)]∗ = √ δ+1,k , 2 Z 1 1 T dt hψ − |σx ⊗ 1|11i e−i(k+1)ωt = √ δ−1,k , x34 (k) = T 0 2 1 x43 (k) = [x34 (−k)]∗ = √ δ+1,k . 2 x12 (k) = 172 Apparently, there are only transitions with k = ±1. Next, we take care of the transition energies εi j + kω appearing in Eq. (A.1). Since the quasi-energies are ε1 = |∆|, ε2 = ε3 = 0, and ε4 = −|∆| in Eq. (4.20), we have εi j = ±|∆| for all transitions with non-zero amplitude. Due to k = ±1, the relevant transition energies are ±|∆| ± ω; but since ω ∆ and ω ≈ ω0 in the parameter regime under investigation, we can approximate this by ±ω0 . Using property (2.44) of the thermal occupation number Nth (ω), the final result is γ11 = γ22 = γ33 = γ44 = 0 γ14 = γ41 = γ23 = γ32 = 0 γ12 = γ13 = γ24 = γ34 = |g̃|2 ω03 Nth (ω0 ), γ21 = γ31 = γ42 = γ43 = |g̃|2 ω03 [Nth (ω0 ) + 1]. This leads to the following transition matrix (M)i j = γ ji − δi j ∑l γil : 2 M = |g̃| 1 1 −2 ω03 Nth (ω0 ) 0 0 1 1 0 −2 0 1 1 + 1 1 0 1 −2 1 −2 0 0 0 0 −1 0 0 . 0 0 1 −1 1 (A.4) −2 Since the thermal occupation number Nth (ω0 ) vanishes at low temperatures β −1 ω0 , only the second term contributes in this limit. In the opposite case of high temperature β −1 ω0 , the first term dominates. These limits correspond to the two expressions (4.21) and (4.23) that underlie our discussion in Section 4.1.3. At any temperature β −1 , however, M is easily diagonalized and leads to the stationary weight vector ! 2 2 N (ω) + 1 N (ω) + 1 N (ω) [N (ω) + 1] th th th th ~w∗ = , , ,1 Nth (ω) Nth (ω) 4Nth (ω)2 + 4Nth (ω) + 1 Nth (ω)2 Due to w∗2 = w∗3 , |Φ2 (t)i = |ψ + i and |Φ3 (t)i = |ψ − i have equal weight in the asymptotic cycle. Since their balanced mixture is a separable state, (2.87) |ψ + i hψ + | + |ψ − i hψ − | = |01i h01| + |10i h10| , the asymptotic cycle is separable at any temperature. Note that this is, in the end, a consequence of the symmetric coupling of |ψ + i and |ψ − i to the remaining two Floquet states, which is manifest in the relations |x12 (k)| = |x13 (k)| and |x42 (k)| = |x43 (k)|. 173 Transition rates at the avoided crossing of ε1 and ε2 At the avoided crossing of ε1 and ε2 , the Floquet states turn into 1 |Φ1 (t)i = √ e+iωt |00i + |ψ + i , 2 1 |Φ2 (t)i = √ e+iωt |00i − |ψ + i , 2 |Φ3 (t)i = |ψ − i , (A.5) |Φ4 (t)i = e−iωt |11i , as stated in Eq. (4.26). Most of the previous considerations remain valid, but the transition amplitudes change: 1 x11 (k) = T Z T 0 1 dt [h00| e−iωt + hψ + |]σx ⊗ 1[eiωt |00i + |ψ + i]e−ikωt 2 1 = √ (δ+1,k + δ−1,k ) = −x22 (k), 2 2 Z 1 T 1 x12 (k) = dt [h00| e−iωt + hψ + |]σx ⊗ 1[eiωt |00i − |ψ + i]e−ikωt T 0 2 1 = √ (δ+1,k − δ−1,k ) = −x21 (k), 2 2 Z 1 T 1 x13 (k) = dt √ [h00| e−iωt + hψ + |]σx ⊗ 1 |ψ − i e−ikωt T 0 2 1 = − δ−1,k = x31 (−k) = x23 (k) = x32 (−k), 2 Z 1 T 1 x14 (k) = dt √ [h00| e−iωt + hψ + |]σx ⊗ 1 |11i e−iωt e−ikωt T 0 2 1 = δ−1,k = x41 (−k) = −x24 (k) = −x42 (−k), 2 Z 1 T x34 (k) = dt hψ − |σx ⊗ 1|11i e−iωt e−ikωt T 0 1 = √ δ−1,k = x43 (−k). 2 From this, we find the transition rates γ33 = γ44 = 0 γ34 = |g̃|2 |ω03 | Nth (ω0 ), γ43 = |g̃|2 |ω03 | [Nth (ω0 ) + 1], 174 1 γ11 = γ22 = γ12 = γ21 = |g̃|2 |ω03 | [2Nth (ω0 ) + 1], 4 1 2 3 γ13 = γ14 = γ23 = γ24 = |g̃| |ω0 | Nth (ω0 ), 2 1 γ31 = γ41 = γ32 = γ42 = |g̃|2 |ω03 | [Nth (ω0 ) + 1], 2 and the transition matrix −6 2 1 M = |g̃|2 ω03 Nth (ω0 ) 4 2 2 2 2 2 −6 2 2 1 + 4 2 2 2 −8 4 −8 −1 2 1 0 0 −1 0 0 . (A.6) 0 2 2 −4 4 −8 At zero temperature, we have Nth (ω0 ) = 0, and M goes over into expression (4.23). This leads to the entangled stationary state Eq. (4.28). As long as the temperature β −1 is well below ω0 , we have Nth (ω0 ) 1 and effectively remain in the zero temperature regime. Only at temperatures above β −1 ≈ ω0 , the thermal occupation contributes. It dominates, finally, in the limit of infinite temperature, where ~w∗ = 1 4 (1, 1, 1, 1) becomes the null-eigenvector of M; the resulting stationary state of the master equation is, thus, again the completely mixed state 1/4. As an aside, we point out that the completely mixed state is always the stationary state in the infinite temperature limit, if the detailed balance condition γi j Nth (ω0 ) = e−β ω0 = γ ji Nth (ω0 ) + 1 (A.7) holds: Since Eq. (A.7) implies γi j = γ ji for β → 0, M is symmetric and has therefore always the uniform vector ~w∗ = 14 (1, 1, 1, 1) in its nullspace. This implies ρac = 1/4. In the particular case treated above, the detailed balance condition holds, because only transitions with k = ±1 are allowed. Beyond the RWA, this is no longer the case, and one observes deviations from detailed balance [92]. 175 A.2 Detailed analysis of the optimal static Hamiltonian A.2.1 The optimal Hamiltonian for two qubits In the following, we discuss in physical terms why the state (4.73) ρψ∗ ± = 1 1 |00i h00| + |ψ ± i hψ ± | , 2 2 (A.8) becomes the stationary state of the Lindblad master equation (4.30) with spontaneous decay at rate γ and a Hamiltonian ∆ F ∆ F (4.77) σz + σx + σz ± σx ⊗ 1 ± J(σ+ ⊗ σ− + σ− ⊗ σ+ ), H±∗ = 1 ⊗ 2 2 2 2 (A.9) if the parameters obey ∆ = −J and |∆| |F| γ. (A.10) We focus on H+∗ in the following (as we do so in Section 4.2.3), but stress that the discussion for H−∗ is completely analogous. Figure A.1 shows the spectrum of H+∗ as a function of the relative interaction strength J/∆, in the relevant regime of |∆| |F|. It is explained by a perturbative analysis in F: at F = 0, the eigenstates of H+∗ are |00i, |ψ ± i = √12 (|01i ± |10i), and |11i, with corresponding energy levels −∆, ±J, and ∆. These expressions describe the spectrum in Fig. A.1 rather well, apart from the fact that they do not explain the two avoided level crossings at J = ±∆. To derive the first order correction in F, we express the perturbation operator in terms of the unperturbed eigenstates: F F [1 ⊗ σx + σx ⊗ 1] = √ |ψ + i h11| + |ψ + i h00| + h.c. 2 2 (A.11) Since this operator has vanishing diagonal matrix elements in the unperturbed basis, the energy levels are not shifted (to first order). The perturbation alters the spectrum only when the levels |ψ + i and |11i, or |ψ + i and |00i, come close. This is the case for J = ±∆. There, the perturbation lifts the degeneracy and leads to an avoided crossing of width hψ+ | F (A.11) F [1 ⊗ σx + σx ⊗ 1] |11i = √ 2 2 between the corresponding levels. (A.12) 176 |11> ∆ |Ψ+> 0 |Ψ−> |00> −∆ −∆ J ∆ Figure A.1: Spectrum of the Hamiltonian H+∗ , Eq. (A.9), as a function of the interaction strength J, measured in units of the local field strength in z direction (∆). The field strength in x direction is weak (F = ∆/10), as required by (A.10). The avoided crossing at J = −∆ (red circle) is the reason for ρψ∗ + to turn stationary. Along with the energy levels, the eigenstates are also modified in an avoided crossing. In fact, they become the balanced superposition of the participating levels in the center of the avoided crossing. E.g., at J = ∆, the two states with lower energy, |00i and |ψ − i, remain unchanged (since they are not coupled by the perturbation, and hence do not avoid to cross), whereas the states of higher energy, |11i and |ψ + i, are transformed into √12 (|11i ± |ψ + i) at the avoided crossing. The same happens at J = −∆ (see the red circle in Fig. A.1): There, the two levels of lower energy anti-cross and the eigenstates turn into √1 (|00i ± |ψ + i). This is the crucial mechanism that renders ρ ∗ + the stationary ψ 2 state, as revealed by the following analysis. As stated by (A.10), we require strong Hamiltonian dynamics in comparison to the dissipation rate, F, ∆, J γ. In this regime, the right hand side of the master equation (4.30) can only vanish if the Hamiltonian part does so, too – i.e., if [H, ρ] = 0. Thus, the stationary state ρss necessarily commutes with the Hamiltonian, and therefore becomes diagonal in an eigenbasis {|ii} of H, ρss = ∑ wi |ii hi| . i (A.13) 177 Albeit comparatively weak, the incoherent part of the master equation is not irrelevant for the stationary state ρss , but determines the weights wi of the mixture. Inserting (A.13) into (4.34) leads to the rate equation 0 = ẇi = ∑(Γ ji w j − Γi j wi ). (A.14) j Here, the rates Γi j describe the probability flow from |ii to | ji that is induced by the incoherent decay process: Γi j = 2 ∑ γ | h j|σ− (m) m=1 |ii |2 . (A.15) The stationary weights wi are obtained by extracting the eigenvector with zero eigenvalue of a matrix M, defined via (M)i j = Γi j − δi j ∑ Γi j0 . (A.16) j0 The speed of convergence towards the stationary state is determined by spectral gap of M, i.e., by the eigenvalue with the smallest absolute value (excluding, of course, the stationary eigenvalue zero).1 In order to apply this general result to our specific situation, one has to distinguish between being close to or distant from the avoided crossings at J = ±∆. In the former case, the eigenstates of H+∗ are This leads to |1i ≡ |00i , |2i ≡ |ψ + i , |3i ≡ |ψ − i , |4i ≡ |11i . (A.17) Γ21 = Γ31 = Γ42 = Γ43 = γ, and the remaining Γi j = 0. With this, the rate equation (A.14) leads to stationary weights ~w = δi,1 . Thus, the stationary state is ρss = |00i h00|, and the speed of convergence towards this state is γ. At the avoided crossing J = −∆, |10 i ≡ √12 (|1i + |2i) and |20 i ≡ √12 (|1i − |2i) become eigenstates of H+∗ , instead of |1i and |2i. The non-vanishing rates are now: γ (A.18) Γ10 10 = Γ20 20 = Γ10 20 = Γ20 10 = , 4 γ Γ310 = Γ320 = Γ410 = Γ420 = , Γ43 = γ. (A.19) 2 1 Thus, the spectral gap determines how long it takes until even the most remote initial state has approached the stationary state. E.g., a spectral gap δ implies that every initial state approaches the stationary state with precision ε in a finite time of the order of δ −1 log(ε −1 ). The spectral gap is therefore crucial, e.g., in the discussion of dissipative state preparation schemes [176]. In particular, if the spectral gap vanishes, there is no longer a unique stationary state, and not every region of the state space is attracted by the desired target state. 178 This leads to stationary weights w10 = w20 = find the stationary state 1 2 and w3 = w4 = 0. From this, we 1 0 0 1 0 0 1 1 |1 i h1 | + |2 i h2 | = |1i h1| + |2i h2| 2 2 2 2 (A.20) 1 + 1 + = |00i h00| + |ψ i hψ | , 2 2 ∗ which is precisely ρψ + . The speed of convergence towards this state is γ/2. While this is only half as fast as the convergence to |00i h00| found before, it is still a finite value of the order of γ. For completeness, we mention that the same analysis for the avoided crossing between |11i and |ψ + i (at J = +∆) leads to ρss = |00i h00|. We stress that the above calculation resembles closely the analysis for the Floquet Born-Markov master equation in appendix A.1. This is to be expected, because in the relevant parameter regime (weak and near-resonant driving, weak decay rate), the here discussed master equation with fixed dissipator coincides with the Born-Markov description. This equivalence is reflected by Eqs. (A.14) to (A.16), which closely resemble the Born-Markov expressions (2.56), (2.57), and (2.61). ρss = A.2.2 Generalization to more than two qubits Next, we generalize the situation to N qubits, to confirm that 1 1 ρ (N) = (|0i h0|)⊗N + |WN i hWN | 2 2 of Eq. (4.82) becomes stationary, if the Hamiltonian is N N ∆ (i) F (i) (i) ( j) (i) ( j) (N) σz + σx + ∑ J(σ+ σ− + σ− σ+ ) H =∑ 2 2 i< j i=1 (A.21) (A.22) and the parameters obey ∆ = (1 − N) · J and |∆| |F| γ. (A.23) To this end, we analyze the spectrum of H (N) for N = 4 qubits in Fig. A.2. Apart from the fact that it involves more levels, it is very similar to the two qubit spectrum of Fig. A.1. The energy levels depend linearly on J (measured in units of ∆), with different slopes. Some levels avoid to cross, others cross exactly. To understand the spectrum in detail, we make use of the collective spin operator ~S = ∑i 1 ~σ (i) , which we introduced earlier in Eq. (3.73). This leads to 2 N (N) 2 2 ~ H = ∆ · Sz + F · Sx + J S − Sz − . (A.24) 2 179 2∆ 0 −2∆ −∆ J ∆ Figure A.2: Spectrum of the N-qubit Hamiltonian (A.22) for N = 4. The driving strength is F = ∆/10. At the avoided crossing at J = −∆/3 (red circle), the stationary state becomes a 50/50 mixture of the de-excited state |0000i with the four qubit W state |W4 i. At F = 0, H (N) contains only ~S2 and Sz ; its eigenstates are therefore the wellknown angular momentum states |l, mi [63], with l = 0, 1, . . . , N2 , and m = −l, . . . , l. (We assume N to be even, but the case of odd N is completely analogous). The corresponding energy eigenvalues are N 2 Elm = m · ∆ + J l(l + 1) − m − . (A.25) 2 This explains the linear dependence of the eigenvalues on J. To explain the avoided level crossings, we rely again on first order perturbation theory in the driving strength F. The perturbation operator is Sx = (S+ + S− )/2, and its matrix elements in the unperturbed basis |l, mi are [63] p 1 0 0 1 hl , m |(S+ + S− )|l, mi = δll 0 δm0 ,m+1 (l − m)(l + m + 1) + 2 2 p 1 δll 0 δm0 ,m−1 (l + m)(l − m + 1). 2 Hence, only levels with the same quantum number l and neighboring m avoid to cross to first order. For our purposes, the avoided crossing between | N2 , − N2 i and | N2 , − N2 + 1i is most interesting. | N2 , − N2 i is simply the de-excited state |0i⊗N , and 180 | N2 , − N2 + 1i is the N qubit W state |WN i. According to (A.25), they meet when J = ∆/(1 − N), (A.26) as marked by the red circle in Fig. A.2. At the center of their avoided crossing, the balances superpositions 1 √ (|0i⊗N ± |WN i) (A.27) 2 become eigenstates of H (N) . (N) To determine the stationary state ρss in the regime of |∆| |F| γ, one can proceed in complete analogy to the case of two qubits. As long as J is far off the avoided crossing at J = ∆/(1 − N), the analysis yields the de-excited state ρss = (|0i h0|)⊗N . At the avoided crossing, however, the eigenstates are transformed, and the rate equation (A.14) leads to different stationary weights. This results in the stationary state (4.82). 181 A.3 Proof of inequality (4.110) Lemma 1. The time-averaged objective function O from Eq. (4.102) can be bounded from above by O[~r(t)] ≤ sup OTPC (~r+ (p),~r− (p)), (A.28) O(~r+ )| f2 (~r− )| + O(~r− )| f2 (~r+ )| . | f2 (~r+ )| + | f2 (~r− )| (A.29) p∈[p0 ,p1 ] where OTPC is defined as OTPC (~r+ ,~r− ) = Proof. To prove this inequality, we first express the target quantity O in the purity parametrization, see Eq. (4.109): Z t Z Z T 1 1 T 1 O[~r(t)] = dt O(~r(t)) = dt O(~r(t)) + dt O(~r(t)) T 0 T 0 t1 Z p Z p0 1 dp 1 dp O(~r+ (p)) + O(~r− (p)) . = T p0 ṗ p1 ṗ Recalling that p is just a different notation for µ2 , criterion (4.99) allows us to replace ṗ by f2 (~r). Moreover, we exploit that f2 (~r) is strictly positive (negative) in the first (second) stage of the cycle, so that f2 (~r± (p)) = ±| f2 (~r± (p))|. This leads to Z p Z p0 1 1 O(~r+ (p)) O(~r− (p)) O[~r(t)] = dp − dp T | f2 (~r+ (p))| | f2 (~r− (p))| p0 p1 Z O(~r− (p)) 1 p1 O(~r+ (p)) + dp. = T p0 | f2 (~r+ (p))| | f2 (~r− (p))| Expressing also the period T through the purity-parameterized quantities, Z T Z p0 Z p1 Z p1 dp dp 1 1 T= dt = + = + dp, | f2 (~r+ (p))| | f2 (~r− (p))| 0 p0 ṗ p1 ṗ p0 we find the intermediate result R p1 O(~r+ (p)) O(~r− (p)) + p0 | f2 (~r+ (p))| | f2 (~r− (p))| dp . O[~r(t)] = R p1 1 1 p0 | f2 (~r+ (p))| + | f2 (~r− (p))| dp Defining w(p) = R p1 p0 w̃(p) , w̃(p0 )dp0 with w̃(p) = 1 | f2 (~r+ (p))| + 1 | f2 (~r− (p))| , 182 this becomes O[~r(t)] = Z p1 p0 w(p) OTPC (~r+ (p),~r− (p)) dp. (A.30) Since w̃(p) is strictly positive, it can be interpreted as an unnormalized weighting function, and w(p) as its normalized counterpart. Accordingly, Eq. (A.30) states that the time-averaged objective function O of the cycle equals a weighted purityaverage of the quantity OTPC . Clearly, this weighted average is bounded from above by its maximal value within the integration interval [p0 , p1 ], thus proving inequality (A.28). 183 A.4 Table of frequently used notations Notation Meaning Eq. Page ρ Quantum state (2.20) 17 H Hilbert space of a quantum system (2.1) 8 Q Space of (mixed) quantum states (2.20) 17 εi , |Φi (t)i Quasi-energies, Floquet states (of several qubits) (2.3) 8 µ(±) , |φ± (t)i Quasi-energies, Floquet states (of a single qubit) (3.15) 57 ω0 Energy level-splitting of a bare (i.e., undriven) qubit (3.10) 53 ω Driving frequency (3.12) 54 F Driving strength (3.12) 54 J Qubit-qubit interaction strength (3.9) 53 γ or γ− Rate of spontaneous decay (4.31) 115 D(ρ) Dissipative part of the master equation (4.30) 115 S Set of stabilizable states (4.45) 123 S Set of stabilizable cycles (4.95) 144 TPC Two point cycle - 152 O(ρ) (Unspecified) objective function - 122 E (Unspecified) entanglement measure (2.83) 39 C Concurrence (2.91) 42 (2.100) 45 184 Bibliography [1] E. Schrödinger, Math. Proc. Camb. Phil. Soc. 31, 555 (1935), [→] . [2] A. Lemaı̂tre, in Dynamics of Small Solar System Bodies and Exoplanets, edited by J. J. Souchay and R. Dvorak (Springer, 2010), Lect. 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