Download Open-System Quantum Simulation with Atoms and Ions

Open-System Quantum Simulation with
Atoms and Ions
P. Zoller,1,2 J. T. Barreiro2,3 and M. Müller1,2
1
2
Institute for Theoretical Physics, University of Innsbruck, Innsbruck, Austria
Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck,
Austria
3
Institute for Experimental Physics, University of Innsbruck, Innsbruck, Austria
Abstract We summarize recent work on the engineering of open-system dynamics of many particles by a
controlled coupling to an environment. We discuss these aspects in the context of quantum optical
systems of cold atoms and ions.
1 Introduction
A universal quantum simulator is a controlled quantum device that reproduces the dynamics of
any other many-particle quantum system with short-range interactions [1, 2]. These dynamics
can refer to both coherent Hamiltonian and dissipative open-system evolution. While impressive
progress has been reported in isolating the physical systems from the environment and coherently
controlling their many-body dynamics in both quantum computing and quantum simulation, we
will focus here on engineering the open-system dynamics of many particles by a controlled coupling
to an environment. We will discuss both the basic theoretical concepts as well as their physical
implementation with cold Rydberg atoms [3] and ions [4, 5]. In particular, for a system of trapped
ions a realization of a toolbox for simulating an open quantum system with up to five qubits has
been reported in Ref.[4]. We note that dissipation by entanglement in atomic ensembles and a
dissipative version of quantum computing and memories have been recently discussed in Refs. [6, 7]
and [8, 9], respectively.
2 Entanglement via Dissipation
A standard way of generating deterministic entanglement in a system of N quits (or spins) is
based on unitary evolution, |ψ0 i → |ψt i = Ut |ψ0 i, where according to the quantum logic network
model we decompose the unitary time evolution operator Ut into a sequence of one- and two–qubit
quantum gates. Remarkable progress in implementing such ideas has been reported during recent
years with trapped ions and atoms, where present experimental challenges include the realization
of high-fidelity quantum gates and quantum memories, and the scaling of these systems to a large
number of qubits. A basic requirement is the complete isolation from the environment to avoid
decoherence. For systems of atoms and ions this decoherence will include, for example, spontaneous
emission as fundamental quantum noise, and various sources of technical noise. In contrast, we will
discuss below a scenario, where we engineer a controlled coupling to the environment to achieve
preparation of a desired entangled state.
The time evolution of an open quantum system, which is initially uncorrelated from its environment, can be described by a completely positive map
X
ρ 7→ Eρ =
Ek ρEk† .
k
Here ρ is the reduced density operator of the system, and Ek are the so-called Kraus operators.
This evolution is in general non-unitary. The Markovian limit of these open-system dynamics can
be written as a master equation
X 1
1
ρ̇ = −i[H, ρ] +
γα cα ρc†α − c†α cα ρ − ρ c†α cα
(1)
2
2
α
with a system Hamiltonian H and Lindblad (or quantum jump) operators cα and γα dissipative
rates reflecting the coupling of the system to the environment. Dissipation and decoherence in
quantum optical systems is usually formulated in the master-equation language, as illustrated by
the optical Bloch equations for two-level atoms coupled to the vacuum modes of the radiation
field. While the above is a familiar language to describe the dissipative dynamics of the system of
interest coupled to an environment, and thus a description of decoherence, we are interested here
in a situation where we engineer the environment coupling to achieve a “cooling” of our system, i.e.
ρ −→ |ψihψ|.
Here |ψi is the desired pure state ψ of our system, and in particular the desired entangled state in
a many-particle system. For the master equation, necessary conditions to achieve such dissipative
dynamics which contracts to a pure state is given by the conditions H |ψi = E |ψi, and ∀α
cα |ψi = 0. As the last condition illustrates, |ψi is a state decoupled from the environment, which
in quantum optics is called “dark state”. We note that this is a dissipative, albeit deterministic
preparation of the state |ψi. On a single-particle level the concept of preparing a pure state by
dissipation is well-known from optical pumping of internal electronic states of atoms and laser
cooling of motional states.
Dissipative preparation of N-particle states by engineering Kraus operators, or quantum jump
operators is best illustrated by the conceptually simplest example of “Bell state cooling”. Consider
the four Bell states |Φ+ i = √12 (|00i + |11i), |Φ− i = √12 (|00i − |11i), |Ψ+ i = √12 (|01i + |10i), and
|Ψ− i = √12 (|01i − |10i). We note that these states are common eigenstates of the two commuting
stabilizer operators X1 X2 and Z1 Z2 with eigenvalues ±1, where the Xi and Zi denote Pauli
operators. Thus it is straightforward to formulate a master equation with Hamiltonian H = 0 and
Lindblad operators c1 = X1 12 (1 + Z1 Z2 ) and c2 = Z1 12 (1 + X1 X2 ) which achieves ρ → |Ψ− i hΨ− |.
We note that these jump operators act on two particles simultaneously, and thus their physical
realization requires an engineered system-bath coupling.
As discussed in detail by Barreiro et al. [4, 5], systems of cold trapped ions allow the realization
of such master equations, and thus of dissipative entangled-state preparation. The idea is to divide
the system of ions into “system” and “environment” ions. The ancilla ion is coupled via optical
pumping to the vacuum modes of the radiation field. In combination with the usual single quit and
entangling gates of the ions, this provides a complete toolbox to build a given dissipative dynamics
of Kraus operators or a master equation [10]. This is illustrated in Figs. 1a and b for the two
examples of Kraus map engineering with a “cooling” probability of p = 1, and the master equation
limit, respectively, for the example of Bell state cooling discussed above. Barreiro et al. have also
demonstrated an extension of these ideas to 4 + 1 ions showing four-particle stabilizer pumping to
a GHZ state (see figure 2 and text below).
In the language of quantum control theory the above corresponds to open-loop dynamics. We
can also perform a measurement on the ancilla qubits, however, and thus implement a quantum
feedback algorithm on the system based on the measurement outcome, i.e., closed-loop dynamics. In
the case of Barreiro et al. this amounts to implementing a Quantum Non-Demolition measurement
Figure 1: Experimental signatures of Bell-state pumping (results according to Ref. [4]).
Evolution of the Bell-state populations |Φ+ i (down triangles), |Φ− i (circles), |Ψ+ i (squares) and
|Ψ− i (up triangles) of an initially mixed state under a pumping process with probability a, p = 1 or
deterministic and b, p = 0.5. Error bars, not shown, are smaller than 2% (1σ).
of a set of stabilizer operators [4].
While deterministic dissipative preparation of entangled states is interesting from a conceptual
point of view as a novel approach to engineer entangled states, and even as a means to perform
quantum computation [8], we will focus in the following section on aspects of open-system quantum
simulation. In fact, some of the ideas presented above for ions were originally developed in the
context of cold atoms in optical lattices as a “Rydberg Quantum Simulator” [3].
3 Open-System Quantum Simulation
Quantum simulation of many-particle physics is usually discussed for Hamiltonian systems,
i.e. closed systems with unitary time evolution. Quantum simulation is of interest both for
equilibrium systems, e.g. to determine the phase diagram of an interacting many-particle system,
and for non-equilibrium many-particle dynamics, when the wave function of interest involves
large-scale entanglement which cannot be represented efficiently by a classical device. There are
two paths to realize quantum simulations: as so-called analog or as digital (see Ref.[11] for a
recent overview). The familiar example of analog simulation in atomic physics is cold atoms in
optical lattices which faithfully represent Hubbard and spin Hamiltonians with external control
parameters (or with trapped ions[12, 13]). In digital quantum simulations the time evolution of a
many spin system is represented as a sequence of single– and many–qubit gates according to a
Trotter decomposition of the time evolution operator, reminiscent of quantum computing with
a quantum logic network model. In contrast to the remarkable achievements in analog quantum
simulation with atoms and ions, digital quantum simulations are, from an experimental point of
view, much less explored. The reason is the requirement to perform a large number of high-fidelity
gates. For systems involving up to six spins this has become possible only very recently with
trapped ions [14].
The concept of quantum simulation can be extended to dissipative systems, for example those
described by the master equation as in Eq. (1) [2]. Again, we can simulate the master equation
either in analog or digital form. Weimer et al. have described a digital quantum simulator for
open many-particle spin systems, and have proposed an implementation with atoms in optical
lattices involving long-range Rydberg interactions [3]. The key idea is to introduce besides the
spins of interest (representing the system), a set of auxiliary spins on the lattice whose role is to
(i) mediate n-particle interactions between systems spins, and (ii) to mimic n-particle quantum
jump operators in the master equation, similar to what we discussed in the previous section. The
simplest example described in [3] is the open-system dynamics of Kitaev’s toric-code Hamiltonian
Figure 2: Four-qubit stabilizer pumping (experimental results according to Ref.
√ [4]). a,
Schematic of the four system qubits to be pumped into the GHZ state (|0000i + |1111i)/ 2, which is
uniquely characterized as the simultaneous eigenstate with eigenvalue +1 of the shown stabilizers. b,
Reconstructed density matrices (real part) of the initial mixed state ρmixed and subsequent states
ρ1,2,3,4 after sequentially pumping the stabilizers Z1 Z2 , Z2 Z3 , Z3 Z4 and X1 X2 X3 X4 . Populations in
the initial mixed state with qubits i and j antiparallel, or in the -1 eigenspace of the Zi Zj stabilizer,
disappear after pumping this stabilizer into the +1 eigenspace. For example, populations in dark blue
dissappear after Z1 Z2 -stabilizer pumping. A final pumping of the stabilizer X1 X2 X3 X4 builds up the
coherence between |0000i and |1111i, shown as red bars in the density matrix of ρ4 .
[15] as a 2D spin model on a square lattice:
X
X
x
z
H = −h
S
−h
S+
+
x
z
with commuting stabilizer operators S
= X1 X2 X3 X4 and S+
= Z1 Z2 Z3 Z4 involving four spin in
a and + lattice configuration on the lattice. Kitaev’s toric code is a paradigmic example for a
complex spin system involving 4-body interaction terms on a plaquette. We note that the Kitaev
model supports excitations in the form of Abelian anyons. Engineered dissipation in the form of
a stabilizer pumping can, according to Ref. [3], provide cooling to the ground state of Kitaev’s
toric-code Hamiltonian.
An ion-trap experiment demonstrating the basic ingredients of open-system digital quantum
simulation of the Kitaev model following the theoretical ideas outlined by Weimer et al. was
presented in a recent publication [4]. The experiment considered a single plaquette of four spins
and one auxiliary particle (see Fig. 2a) as the minimal instance of the Kitaev model described
above, which was mapped to five ions in a linear ion-trap. A key result of the experiment was
the demonstration of stabilizer pumping for the four commuting operators Z1 Z2 , Z2 Z3 , Z3 Z4 and
X1 X2 X3 X4 into the
√ common eigenspace of eigenvalue +1, which corresponds to a GHZ state
(|0000i + |1111i)/ 2 (see Fig. 2b). Besides engineering the dissipative processes, the experiment
was also able to show coherent toric-code four-body interactions in a single digital-simulation step.
While present ion-trap experiments have demonstrated a complete toolbox of open-system
quantum simulation, the combination of these elements for a complete quantum simulation, and in
particular a proper assessment of errors, remains a challenge for future work (see, however, [14]).
Furthermore, in its present form these experiments are not scalable to large systems, although 2D
traps arrays are currently being developed. We remark, however, that cold atom experiments in
optical lattices do provide an a priori scalable way to implement the above ideas [3]. With the
recent achievement of Rydberg gates [16, 17] and single site addressing in optical lattices [18, 19]
all the essential ingredients seem to exist in the laboratory to implement scalable open-system
quantum simulation. However, it remains to be seen if neutral atoms will be able to achieve the
remarkable fidelities of quantum gate operations demonstrated with ions.
4 Conclusions
There are various questions of open-system dynamics of many-body systems which we have not
touched in the above discussion. In particular, we point out the possibility of realizing an analog
simulation of many-body master equations by coupling atoms in optical lattices to a bath of
Bogoliubov excitations in a driven dissipative quantum system [20]. Non-equilibrium dynamics of
atoms in lattices have been discussed in the context of dynamical quantum phase transitions, where
Liouvillian competes with Hamiltonian dynamics [21], and BCS-type pairing of fermions induced
by engineered dissipation [22]. Very recently, Majorana modes in a 1D atomic quantum wires
in a purely dissipative setting were discussed in Ref. [23]. This work provides a first illustration
of the emergence of topological properties of many-body systems by dissipative couplings to an
engineered environment.
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