Generating maximal entanglement between non-interacting atoms by collective decay and symmetry breaking
Abstract
A simple scheme is presented for achieving effectively maximal pure-state entanglement between non-interacting atoms through purely collective decay and controlled symmetry breaking. The scheme requires no measurements or feedback or even knowledge of the initial states of the atoms. It relies on breaking the symmetry of the system Hamiltonian to ensure the existence of a unique attractive steady state and minimal control to achieve almost perfect overlap of this steady state with the maximally entangled singlet state. We demonstrate how our scheme can be implemented for two qubits encoded in hyperfine levels of atoms such as Rubidium in a lossy microwave cavity using only small magnetic field gradient. Error analysis suggests considerable robustness with regard to many imperfections including atomic decay, asymmetric atom-cavity coupling and frequency offsets.
pacs:
03.67.Hk,03.67.Lx,7510.Pq,78.67.LtEntanglement is an important resource in quantum physics, especially quantum information, and many schemes to generate entanglement have been proposed, using various tools including coherent control, measurements and feedback. In coherent control schemes (e.g., (1)) dissipation tends to destroy entanglement created by Hamiltonian evolution and reduce the control effectiveness, but it can also be a resource for creating entanglement. Some interacting systems have entangled ground states and relaxation processes can drive the system into such states, known as cooling into entangled states (2). More generally, careful design of the Hamiltonian and interaction with the environment, i.e., Hamiltonian and reservoir engineering, can ensure decay to a certain state (3); (4); (5). Reservoir engineering by measurements has been used to drive systems into entangled states (6), and schemes relying on direct or indirect feedback have been proposed (7). Entanglement induced purely by interaction with an environment has also been explored (8), but the entanglement achieved this way is usually low-grade mixed-state entanglement and often transient, i.e., the system relaxes to a separable steady state eventually.
Ideally, we would like to engineer systems that relax to a stable, maximally entangled, pure steady state regardless of the initial state. For open systems described by Lindblad master equation this can be achieved by engineering the dynamics to render the steady state unique as uniqueness implies that any initial state “collapses” to this state (4); (5). Based on this idea, we show that effectively maximal pure-state entanglement can be created for atoms in a damped cavity driven by a classical field, even if there is no interaction between the atoms and no measurements are performed, i.e., purely by dissipation, and the initial state is unknown. Of course, the singlet state for two identical atoms in a cavity is maximally entangled, but it is not attractive, i.e., if the system is initially in any other state, it will not evolve into the latter. However, breaking the symmetry in the model, e.g., by varying the energy level splitting of the atoms slightly, results in a unique attractive steady state, and surprisingly, this is a pure state whose concurrence can be made arbitrarily close to by suitable choice of the parameters, and which can be reached in a short time from an unknown initial state simply by adiabatically decreasing the induced asymmetry.
There are many possible physical realizations for the type of system described but to fix the ideas and demonstrate the feasibility of the scheme we consider two trapped ultracold atoms such as Rb, situated in a damped microwave cavity driven by a classical field as shown in Fig. 1 as a particularly promising implementation. A magnetic field is applied to provide a quantization axis and photon polarization direction, and the qubit states are encoded in two hyperfine magnetic sublevels such as and . These basis states are stable with very long coherence times, i.e., once the atoms are prepared in a state such as the singlet state, we can expect them to remain in this state for a long time. The system could be prepared by loading the ultracold atoms into optical lattices to ensure the atoms are well localized and stationary. A detuning of the atomic transition frequencies of from the cavity field frequency can be achieved via the Zeeman effect by applying a small magnetic field gradient. The total Hamiltonian of the system is , where the Hamiltonians for the two atoms and the cavity are and and are the annihilation operators for the th atom and intra-cavity field, respectively. In a typical setup with neutral atoms at least several microns apart direct interactions are negligible, . The atom-cavity interaction is given by and the classical coherent driving field adds the term where is the coupling strength. Including spontaneous emission and decay of the cavity field, the evolution of the total density operator of the driven atom-cavity system is given by
(1) |
with , which can be simplified by transforming to a rotating frame (RFT) with The RFT leaves the Lindblad terms invariant and gives
(2) |
In the small detuning limit, , we can neglect terms rotating with frequency , obtaining
(3) |
where is the Rabi frequency of the driving field, the collective atom-cavity interaction, and with incorporates the detuning of the atoms from the cavity field. The RFT and above differ slightly from the usual interaction picture transformation and definition of to allow for off-resonant and asymmetric interactions. We further eliminate by the standard displacement transformation with . commutes with the atomic operators and the standard commutation relations give for or (e.g., using Hadamard’s lemma), and thus
with . Setting cancels driving field term and gives the simplified master equation
(4) |
where the Lindblad operator for the atomic subsystem is
(5) |
and , . If the cavity is heavily damped following the standard procedure for adiabatic elimination of the cavity mode (e.g. (7); (10)) leads to the equation of motion for the atomic subsystem
(6) |
with . If the population of the first excited cavity mode , and thus , remains small and the atomic spontaneous emission rates then the contributions of and will be negligible, yielding the Dicke-type master equation for the atomic subsystem with a detuning term
(7) |
If the detuning of the atoms from the cavity frequency and their interaction with the cavity field are symmetric, i.e., and , then the detuning term , and it is easy to verify that with is an eigenstate with eigenvalue of the effective Hamiltonian and the Lindblad operator and thus a steady state of the dynamics (7). For (no detuning), the populations of the and eigenspaces of are conserved, i.e., the system is decomposable and there are infinitely many other steady states, rendering non-attractive (5), i.e., if the atoms are initially in a state other than the maximally entangled singlet state (MES) , they will not evolve into the MES but a low-concurrence mixed state (9), i.e., the MES is protected against the decay but cannot be created starting from another state this way.
To be able to create a MES, or a state arbitrarily close to it, we must ensure that the target state is the only steady state of the system. This can be accomplished by noting that even a small detuning breaks the symmetry and renders the unique globally attractive steady state (5). Moreover, as the steady state is pure, any stochastic quantum trajectory of the system must converge to this state. Setting , the concurrence of the steady sate is , showing that attains only for , but can be made arbitrarily close to for by increasing . Fig. 2(a) shows the dependence of on the detuning parameter ; 99% concurrence is achievable with . The steady state concurrence is independent of the decay rate but determines the rate of convergence to the steady state. Fig. 2(b) shows the average time required for the concurrence of the system state to reach 99% of . This time is virtually independent of the initial state, i.e., the variation for different initial states is minimal, but depends strongly on , decreasing substantially with increasing . The graphs also suggest that there is an optimum value of between and , depending on , with being close to optimal for a wide range of . This suggests that the best strategy to achieve high concurrence in the shortest possible time is to choose near the optimum value and the largest possible that still allows us to reach the desired concurrence, or better yet, start with a large and gradually decrease it to , e.g., linearly or exponentially.
To assess whether we can achieve high entanglement for the setup in FIG. 1 with this scheme, we turn to simulations. For Rb we choose GHz, and for atoms a few microns apart, detunings up to MHz are attainable via a magnetic field gradient. Recent experimental results (11) suggest that realistic values for the coupling constant and effective Rabi frequency for the driving field in a lossy microwave cavity are on the order of a few hundred kHz, and the cavity damping rate can easily be made about one order of magnitude larger. We choose kHz and , which gives , kHz and kHz, and , which is in the optimal range to minimize the convergence time according to Fig. 2(b). Our previous results show that to achieve a steady state concurrence of % requires or kHz. Fig. 3 shows that with a constant detuning kHz we can reach 99% concurrence from an arbitrary initial state in less than 20 ms. The figure also shows that this result can be significantly improved if we start with a larger detuning and gradually decrease it to . E.g., starting with kHz, which is still well below the limit of MHz, we can obtain over 99% concurrence in less than ms if the detuning is decreased exponentially.
Next we verify the validity of the approximations made in deriving (7), i.e., adiabatic elimination and negligibility of the terms and . Comparison of the exact solution given by (4) and the solution of the approximate model (7) suggests that the latter is a very good approximation in the parameter regime above. For , the reduced density operator for the atomic subsystem obtained by solving (4) and tracing over the cavity modes, , and the approximate solution of (7) are in excellent agreement with the norm error for kHz (const) and for , and we have verified that the concurrences predicted by the exact and approximate models are in very good agreement for all the simulations below.
For the systems considered here should be easily achievable under current experimental conditions, as the lifetimes of hyperfine states of ultracold atoms such as Rb are on the order of seconds, suggesting Hz, while our effective cavity decay rate is kHz, but the effect of non-zero atomic decay rates on the attainable concurrence deserves investigation. Fig. 4 shows the concurrence of the system state as function of time and the atomic decay rates for constant and exponentially decreasing detuning. For kHz, the effect of the atomic decay is virtually imperceptible on the timescales considered but for kHz, atomic decay decreases the attainable concurrence substantially. The exponentially decaying achieves a higher peak concurrence in the presence of atomic decay, but as the detuning approaches , atomic decay takes over and drives the concurrence down again, while for constant detuning the concurrence reaches a lower steady state value but remains near constant on the timescales considered. This might be explained in terms of inhibition of spontaneous emission by detuning (12) resulting in reduced effective decay rates. This suggests that in the presence of non-negligible atomic decay the best strategy is to start with a large initial detuning to maximize the rate of convergence, but to decrease it not to zero but a non-zero asymptotic value (offset), chosen such that is close to the peak value for the concurrence. Indeed, Fig. 5(left) shows that this strategy appears highly effective in maximizing the steady-state entanglement and the rate at which it is reached.
Another possible source of error is asymmetry in the couplings of the atoms to the cavity. If the atoms are identical and interact in phase with the cavity field, i.e., if they are well localized in the cavity, should be a good approximation even in the presence a small magnetic field gradient as is proportional to the magnetic dipole moment, which is not affected by a small magnetic field gradient. If there is an asymmetry, say , the system no longer has a pure steady state but a unique mixed state in the interior, which has lower concurrence than the MES, as illustrated in Fig. 6. Again, as for atomic decay, the exponential detuning profile allows us to reach higher concurrences fast but the concurrence decreases as the detuning vanishes. Fig. 5 (right) shows that the results can again be substantially improved by adding an offset to the exponential detuning to stabilize the concurrence near its peak value.
Finally, we consider the effect of a frequency-offset that might result if the atoms are not positioned symmetric with respect to the magnetic field gradient, leading to asymmetric detuning and an error term . Surprisingly high entanglement is attainable even for large . A numerical fit to the simulation data shows that the concurrence after ms is well described by a quadradic model with for kHz and for . For the constant detuning is preferable as the large initial amplitude of the exponential detuning magnifies the error, but in both cases substantial entanglement, % and %, is still attainable even if the offset is on the order of the detuning, .
We have shown that effectively maximal entanglement of non-interacting atoms can be realized by putting the atoms in a lossy cavity driven by a classical field and applying a magnetic field gradient. While there are many schemes for entangling qubits, this scheme exploits a novel concept of environment-induced entanglement to create high-grade entanglement and offers exceptional simplicity, eliminating the need for direct coupling between the atoms, control lasers to manipulate internal atomic states, and complicated measurement setups and problems due to limited detection efficiency. Global attractivity of the target state ensures that the atoms converge to this state for any initial state, eliminating the need for initial state preparation. Error analyis shows that high-grade entanglement can be still achieved with simple detuning profiles in the presence of atomic decay, asymmetric atom-cavity coupings or frequency offsets.
We thank M. Atatüre, C. Lu, C. Zipkes, C. Sias from the Cavendish Laboratory, and S. Bose and L.-C. Kwek for valuable discussions, and acknowledge funding from EPSRC ARF Grant EP/D07192X/1 and Hitachi.
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