* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Ultrafast geometric control of a single qubit using chirped pulses
Matter wave wikipedia , lookup
Quantum fiction wikipedia , lookup
Renormalization group wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Dirac equation wikipedia , lookup
Wave function wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Algorithmic cooling wikipedia , lookup
Hydrogen atom wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Scalar field theory wikipedia , lookup
EPR paradox wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Probability amplitude wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Quantum computing wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Path integral formulation wikipedia , lookup
Hidden variable theory wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
History of quantum field theory wikipedia , lookup
Quantum decoherence wikipedia , lookup
Quantum machine learning wikipedia , lookup
Quantum key distribution wikipedia , lookup
Coherent states wikipedia , lookup
Density matrix wikipedia , lookup
Bra–ket notation wikipedia , lookup
Canonical quantization wikipedia , lookup
Quantum group wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Quantum state wikipedia , lookup
Ultrafast laser spectroscopy wikipedia , lookup
Home Search Collections Journals About Contact us My IOPscience Ultrafast geometric control of a single qubit using chirped pulses This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2012 Phys. Scr. 2012 014013 (http://iopscience.iop.org/1402-4896/2012/T147/014013) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 173.54.221.108 The article was downloaded on 18/02/2012 at 03:03 Please note that terms and conditions apply. IOP PUBLISHING PHYSICA SCRIPTA Phys. Scr. T147 (2012) 014013 (4pp) doi:10.1088/0031-8949/2012/T147/014013 Ultrafast geometric control of a single qubit using chirped pulses Patrick E Hawkins1 , Svetlana A Malinovskaya1 and Vladimir S Malinovsky1,2 1 Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA 2 ARL, 2800 Powder Mill Road, Adelphi, MD 20783, USA E-mail: [email protected] Received 13 August 2011 Accepted for publication 6 September 2011 Published 17 February 2012 Online at stacks.iop.org/PhysScr/T147/014013 Abstract We propose a control strategy to perform arbitrary unitary operations on a single qubit based solely on the geometrical phase that the qubit state acquires after cyclic evolution in the parameter space. The scheme uses ultrafast linearly chirped pulses and provides the possibility of reducing the duration of a single-qubit operation to a few picoseconds. PACS numbers: 03.67.−a, 42.50.Ex, 42.50.Hz In this paper, we address a single-qubit manipulation solely based on the geometrical effect. We demonstrate how to use the geometric phase, which is controllable by the relative phase between external fields, to perform ultrafast arbitrary unitary operations on a single qubit. Moreover, we show that femtosecond laser systems can be used for quantum computing, allowing us to perform unitary operations on a single qubit in a few picoseconds. The proposed method combines adiabaticity and pulse area control. In more detail, we demonstrate that by controlling the pulse area and the relative phase of the chirped pulses, we can prepare the resonant qubits in an arbitrary coherent superposition, while adiabaticity of the excitation guarantees to keep off-resonant qubits unexcited. Keeping in mind that the decoherence rate of the qubit transitions in a quantum register should be small enough to allow many quantum operations [6, 19], we restrict our consideration by choosing the two lowest levels in the three-level system as the qubit states. External addressing of the qubit is done by using the Raman excitation scheme through a third ancillary state. We assume that the ancillary state is far off-resonance with external fields to ensure that decoherence on the ancilla–qubit transitions can be neglected. 1. Introduction In the last few decades, many universal sets of quantum gates have been proposed for quantum computation [1–3]. Various combinations of quantum gates have been intensively discussed in the literature related to the universality in quantum computation [1, 4–6]. To perform quantum computation, one must have two major building blocks at their disposal: (i) any arbitrary unitary operations on a single qubit and (ii) a controlled-NOT operation on two qubits. The most common method for implementing single-qubit quantum gates is based on the Rabi solution for a two-level system excited by an external field [1]. There is a very simple reason for this choice. To construct a quantum gate, one needs to know the exact form of the evolution operator of the qubit under the external field excitation. The evolution operator has an easily interpreted form in the case of the Rabi solution [7]. Essentially, the whole dynamics of the qubit is governed by the pulse area. There have been several proposals to design quantum gates in a robust fashion, i.e. to use adiabatic methods for a coherent superposition preparation [8, 9]. The ingenious idea of using the geometric phase [10–12] to manipulate the qubit state was recently developed in a new direction called geometric quantum computing [13–16]. The adiabatic manipulation of a quantum system using only geometric phase has some advantages since it reduces the requirements for perfect tuning of control field parameters, and also the geometric operations can be significantly more robust against noise [13, 17, 18]. 0031-8949/12/014013+04$33.00 2. The qubit evolution operator in the adiabatic approximation Let us consider the dynamics of a single qubit presented by the two lowest levels in the three-level λ-system (see figure 1): the two lowest quantum states |0i and |1i have energies 0 and 1 © 2012 The Royal Swedish Academy of Sciences Printed in the UK Phys. Scr. T147 (2012) 014013 P E Hawkins et al |e > Δ ωS ωP δ |1 > |0 > Figure 2. Rotation trajectory of the Bloch vector representing qubit states |0i, |ii in panel (a) and states |1i, | − ii in panel (b) resulting from the application of the sequence of the resonant π and π/2 pulses with relative phase ϕ + π . Figure 1. Schematic diagram of the qubit states as the two lowest states in the three-level system. 1 , respectively, and the excited ancillary state |ei has energy e . To manipulate in time by the total wave function of the quantum system |9(t)i = a0 (t)|0i + a1 (t)|1i + b(t)|ei, where a0,1 (t) and b(t) are the probability amplitudes to be in state |0i, |1i or |ei, we use the external fields, which in general can (0) be described as E P,S (t) = E (0) P,S (t) cos φ P,S (t), where E P,S (t) are the pulse envelopes and φ P,S (t) are the time-dependent phases. We consider the case of linearly chirped pulses so that φ P,S (t) = φ P,S + ω P,S t + α P,S t 2 /2, where φ P,S are the initial phases, ω P,S are the center frequencies and α P,S are the chirps of the pulses [20, 21]. Using the rotating wave approximation (RWA) and assuming large detunings of the pump and Stokes’ field frequencies from the transition frequencies to the ancillary state, 1 = ω P − (e − 0 )/h̄ = ω S − (e − 1 )/h̄, we utilize the adiabatic elimination of the ancillary state and obtain the following form of the Hamiltonian in the field interaction representation: H = −δ(t)σ z h̄/2 − (ei1φ σ + + e−i1φ σ − )e (t)h̄/2 , qubit, δ(t) = δ = 0, we have θ (t) = θ (0) = π/4 and the unitary evolution operator for the wave function of the resonant qubit takes the form U (t) = cos(S(t)/2) I + i sin(S(t)/2) (ei1φ σ + + e−i1φ σ − ), (3) Rt where S(t) = 0 dt 0 e (t 0 ) is the effective pulse area. Note that this solution of the Schrödinger equation in the adiabatic approximation is the exact solution, since the nonadiabatic coupling is exactly zero for the resonant qubit. 3. Geometric manipulation of the qubit states To demonstrate a pure geometric operation on a single qubit, we employ the Bloch vector representation by using the Pauli matrix decomposition, % = (I + B · σ)/2, of the qubit density matrix, % = |ΨihΨ|, where B = (u, v, w) = (%01 + %10 , i(%01 −%10 ), %00 −%11 ), %i j = ai (t)a ∗j (t) and i, j = {0, 1}. In this representation, the Bloch equation Ḃ = Ω × B describes a precession of the Bloch vector B about the pseudo-field vector, Ω = (−e (t) cos 1φ, e (t) sin 1φ, −δ), with components determined by the effective Rabi frequency two-photon detuning, and the relative phase between the pump and Stokes pulses. Arbitrary unitary operations on a single qubit are equivalent to the rotation of the Bloch vector B. It is proven that any unitary rotation of the Bloch vector can be decomposed as U = eiα0 Rz (α1 )R y (α2 )Rz (α3 ), where Ri = eiασi (i = y, z) are the rotation operators [1, 3]. Therefore, to prepare an arbitrary state of a single qubit we need to demonstrate rotations of the qubit Bloch vector about the z- and y-axes by applying various sequences of the external pulses. Below, we demonstrate how this can be done by controlling the parameters of the external pulses that are defined by the explicit form of the evolution operator (see equation (3)). To implement the rotation of the Bloch vector about the z-axis (the phase gate) based on the geometrical phase, we can use the evolution operator of the resonant qubit, equation (3). The sequence of two π pulses with relative phase 1φ = ϕ + π gives U π;ϕ+π U π ;0 = cos ϕI + i sin ϕσ z ≡ Rz (ϕ), where the first subindex of U indicates the pulse area S(T ) and the second one indicates the relative phase 1φ. Figure 2 shows the Bloch vector trajectories: initially the qubit is in the state |0i (|1i), which corresponds to the (1) where δ(t) = δ + (α S − α P )t + 1S(t), 1S(t) = (2P0 (t) − 2S0 (t))/(41) is the effective ac-Stark shift, e (t) = P0 (t) S0 (t)/(21) is the effective two-photon Rabi frequency, P0,S0 (t) = µ0e,e1 E (0) are the Rabi P,S (t)/h̄ frequencies, µ0e,e1 are the dipole moments, δ is the two-photon detuning, 1φ = φ P − φ S , σ ± = (σ x ± iσ y )/2 are the Pauli raising and lowering operators and σ x,y,z are the Pauli matrices. The Hamiltonian in equation (1) controls the dynamics of the qubit wave function as long as the approximation of the adiabatic elimination is valid. Here we consider the adiabatic excitation of the qubit. In this case, the solution of the Schrödinger equation with the Hamiltonian in equation (1) has the form 1φ 2 σz 3(t) 1φ e−iθ(t)σ y ei 2 σ z eiθ(0)σ y e−i 2 σ z |9(0)i, (2) p Rt where 3(t) = 0 dt 0 λ(t 0 ), λ(t) = δ 2 (t) + 2e (t) and tan[2θ (t)] = e (t)/δ(t). Note that the general form of the evolution operator in equation (2) is obtained in the adiabatic approximation when the the nonadiabatic coupling has been neglected. This approximation is well justified if the condition ˙ e (t)δ(t)| λ3 (t) is valid. |e (t)δ̇(t) − In the case of completely overlapped pulses, P0 (t) = S0 (t), with identical chirp rates, α P = α S , for the resonant |9(t)i = ei 2 Phys. Scr. T147 (2012) 014013 P E Hawkins et al Bloch vector pointing in the z (−z)-direction, while the vector Ω1 = (−e , 0, 0) points in the −x-direction (for simplicity we chose 1φ = 0 for the first π-pulse). The first π-pulse flips the population to the state |1i (|0i), correspondingly the Bloch vector turns about the effective field vector Ω1 (about the x-axis); it stays in the y, z-plane all the time and points in the −z (z)-direction at the end of the pulse. Due to the second π-pulse, the population is transferred back to the initial state |0i (|1i); therefore the Bloch vector returns to its original position pointing along the z (−z)-axis. However, since we chose 1φ = ϕ + π for the second π-pulse, the pseudo-field vector is rotated counterclockwise by the angle ϕ + π in the x, y-plane, Ω2 = (e cos ϕ, −e sin ϕ, 0), and the Bloch vector moves in the plane perpendicular to the x, y-plane and has the angle π/2 − α (−π/2 − α) with the x, z-plane. The Bloch vectors representing a pair of orthogonal basis states |0i and |1i follow a path that encloses, respectively, solid angles of 2ϕ and −2ϕ. The geometrical phase is equal to one half of the solid angle, which means the basis states |0i and |1i gain phases ϕ and −ϕ and the evolution operator takes the form of the Rz (ϕ) rotation. Therefore, by applying the sequence of two π-pulses with the relative phase 1φ = ϕ + π, we implement the rotation operator about the z-axis with the relative phase controlling the rotation angle. The rotation operator about the y-axis controlled by the relative phase and determined by the geometrical phase, which the Bloch vector gains along the cyclic path, can be constructed by using three pulses. The first and third pulses are π/2-pulses with 1φ = 0, while the second pulse is the π-pulse with the relative phase π + ϕ. It is easy to show using equation (3) that this three-pulse sequence results in the rotation operator about the y-axis, U π2 ;0 U π ;π +ϕ U π2 ;0 = eiϕσ y ≡ R y (ϕ), and the relative phase controls the rotation angle. To demonstrate the geometrical nature of the operation R y (ϕ), we note that this operation creates the relative phase √ between two basis states | ± ii = (|0i ± i|1i)/ 2. In the Bloch representation, these states have the form | ± ii = cos π4 |0i + e±iπ/2 sin π4 |1i, which are two vectors pointing in the y- and −y-directions (see figure 2). The trajectory of the Bloch vector representing the states | ± ii is shown in figure 2. The pseudo-field vectors Ω1 and Ω3 are defined by the effective Rabi frequencies of the first and third pulses and point in the −x-direction since we chose 1φ = 0. The second pseudo-field vector Ω2 is rotated counterclockwise by the angle ϕ + π in the x, y-plane, the same as in the case above. The initial Bloch vector points in the y (−y)-direction. The first π/2-pulse rotates the Bloch vector about Ω1 to the position pointing in the −z (z)-direction. The second pulse flips the direction of the Bloch vector. The third π/2-pulse returns the Bloch vector to its original position. During the whole evolution, the Bloch vector and the pseudo-field vector are orthogonal. As we observe, similar to the previous case, the basis states |ii and | − ii follow a path enclosing, respectively, solid angles of 2ϕ and −2ϕ. Therefore, they gain the relative phase 2ϕ, which is the geometrical phase controllable by the relative phase between the external pulses. This phase gate in the | ± ii basis is equivalent to the R y (ϕ) rotation operator in the |0i, |1i basis. 4. Conclusions We have demonstrated ultrafast arbitrary operations on a single qubit, which are solely based on the geometrical phase. The operations are robust with respect to external noise, since the final states of the qubit do not depend on the dynamical phase [13, 17, 18]. Our proposal combines the pulse area control with adiabaticity by using chirped pulses. An analytic expression for the evolution operator for the resonant qubit, which utilizes the Raman excitation of the three-level λ-system with a large single-photon detuning, provides a clear geometrical interpretation of the qubit dynamics. Note that the general solution, equation (2), of the Schrödinger equation in the adiabatic approximation also provides a description of the off-resonant qubit dynamics. It can be easily shown that for the off-resonant qubit, δ 6= 0, the evolution operator (in the adiabatic limit) resulting from the adiabatic return at the end of the pulse excitation, T , has the form U (T ) = cos[3(T )/2]I + sin[3(T )/2]σ z . Therefore, the off-resonant qubits in the quantum register acquire only the additional phase determined by the value of the dynamical phase, 3(T ). This phase can be accounted for and corrected later at the end of quantum computation. To estimate the time scale of the proposed geometric operations, we can use 100 fs pulses with intensity of the order of 1012 W cm−2 . Pulse duration increases to a few picoseconds by applying a linear chirp, α P,S , of the order of 10−5 fs−2 . This amount of chirping is sufficient to provide the adiabatic excitation of the three-level system [20–22] and can be readily produced experimentally using commercially available pulse-shaping systems. Note that the pump and Stokes Rabi frequencies, P0,S0 (t), depend on the chirp rate [20, 21] and this gives us a way to control the value of the nonadiabatic coupling to satisfy the adiabaticity conditions. At the same time, the effective pulse area, S(T ) = RT 0 0 0 dt e (t ), is independent of the chirp rate [22], which provides additional flexibility for a potential experimental realization. Acknowledgments The authors acknowledge partial financial support from DARPA (HR0011-09-1-0008) and NSF (PHY-0855391). References [1] Nielsen M A and Chuang I L 2006 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) [2] Benenti G, Casati G and Strini G 2005 Principles of Quantum Computation and Information (Singapore: World Scientific) [3] Nakahara M and Ohmi T 2008 Quantum Computing (Boca Raton, FL: CRC Press) [4] Lloyd S 1995 Phys. Rev. Lett. 75 346 [5] Deutsch D, Barenco A and Ekert A 1995 Proc. R. Soc. Lond. A 449 669 [6] DiVincenzo D P 2000 Fortschr. Phys. 48 771 [7] Berman P R and Malinovsky V S 2011 Principles of Laser Spectroscopy and Quantum Optics (Princeton, NJ: Princeton University Press) [8] Lacour X, Sangouard N, Guérin S and Jauslin H R 2006 Phys. Rev. A 73 042321 3 Phys. Scr. T147 (2012) 014013 P E Hawkins et al [9] Wunderlich Ch et al 2007 J. Mod. Opt. 54 1541 [10] Berry M V 1984 Proc. R. Soc. Lond. A 392 45 [11] Aharonov Y and Anandan J 1987 Phys. Rev. Lett. 58 1593 [12] Bohm A, Mostafazadeh A, Koizumi H, Niu Q and Zwanziger J 2003 The Geometrical Phase in Quantum Systems (Berlin: Springer) [13] Zanardi P and Rasetti M 1999 Phys. Lett. A 264 94 [14] Jones J A, Vedral V, Ekert A and Castagnoli G 2000 Nature 403 869 [15] Falci G, Fazio F, Palma G M, Siewert J and Vedral V 2000 Nature 407 355 [16] Unanyan R G and Fleischhauer M 2004 Phys. Rev. A 69 050302 [17] De Chiara G and Palma G M 2003 Phys. Rev. Lett. 91 090404 [18] Lupo C and Aniello P 2009 Phys. Scr. 79 065012 [19] Ladd T D et al 2010 Nature 464 45 [20] Malinovsky V S and Krause J L 2001 Phys. Rev. A 63 043415 [21] Malinovsky V S and Krause J L 2001 Eur. Phys. J. D 14 147 [22] Malinovskaya S A and Malinovsky V S 2007 Opt. Lett. 32 707 4