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arXiv:quant-ph/0112042 · Physical Review A

Entanglement in the steady state of a collective-angular-momentum (Dicke) model

Sara Schneider, Gerard J Milburn

Published Mar 18, 2002• 265 citations
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Abstract

We model the behavior of an ion trap with all ions driven simultaneously and coupled collectively to a heat bath. The equations for this system are similar to the irreversible dynamics of a collective angular momentum system known as the Dicke model. We show how the steady state of the ion trap as a dissipative many-body system driven far from equilibrium can exhibit quantum entanglement. We calculate the entanglement of this steady state for two ions in the trap and in the case of more than two ions we calculate the entanglement between two ions by tracing over all the other ions. The entanglement in the steady state is a maximum for the parameter values corresponding roughly to a bifurcation of a fixed point in the corresponding semiclassical dynamics. We conjecture that this is a general mechanism for entanglement creation in driven dissipative quantum systems.

DOI
10.1103/physreva.65.042107

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We model the behavior of an ion trap with all ions driven simultaneously and coupled collectively to a heat bath. The equations for this system are similar to the irreversible dynamics of a collective angular momentum system known as the Dicke model. We show how the steady state of the ion trap as a dissipative many-body system driven far from equilibrium can exhibit quantum entanglement. We calculate the entanglement of this steady state for two ions in the trap and in the case of more than two ions we calculate the entanglement between two ions by tracing over all the other ions. The entanglement in the steady state is a maximum for the parameter values corresponding roughly to a bifurcation of a fixed point in the corresponding semiclassical dynamics. We conjecture that this is a general mechanism for entanglement creation in driven dissipative quantum systems.
Quantum many-body systems driven far from equilibrium can exhibit phase transitions, chaos and entanglement, but probing these phenomena in large, nearly closed systems remains challenging. The Dicke model, which couples an ensemble of two-level systems collectively to a single bosonic mode, provides a fundamental framework for exploring these phenomena in light-matter interactions. Here we show that a two-dimensional crystal of approximately 100 trapped ions realizes this model and accesses integrable dynamical phase transitions, signatures of chaotic dynamics and fluctuation-driven spin-phonon excitation growth. Adiabatic elimination of the vibrational mode reveals ferromagnetic and paramagnetic dynamical spin phases, whereas strong spin-phonon coupling produces irregular trajectories and local relaxation consistent with quantum thermalization. Following a quench from an unstable fixed point, correlated excitation growth precedes generalized vacuum Rabi collapses and revivals; calibrated calculations identify two-mode squeezing 2.6 dB below the standard quantum limit, supported by reconstruction from spin measurements. By connecting collective dynamical instabilities to experimentally accessible signatures of quantum correlations, these results establish large ion crystals as a scalable platform for investigating non-equilibrium light-matter dynamics and generating resources for quantum-enhanced sensing.
Driven dissipative steady state entanglement schemes take advantage of coupling to the environment to robustly prepare highly entangled states. We present a scheme for two trapped ions to generate a maximally entangled steady state with fidelity above 0.99, appropriate for use in quantum protocols. Furthermore, we extend the scheme by introducing detection of our dissipation process, significantly enhancing the fidelity. Our scheme is robust to anomalous heating and requires no sympathetic cooling.
We propose a dissipative method for preparation of a maximally entangled steady state of two trapped ions in the Lamb-Dicke limit. By addressing the trapped-ion system with a monochromatic standing-wave laser pulse of frequency resonant with the ionic transition and a microwave field coupled to the ground-state transitions, we obtain an effective coupling between two particles, which is independent of the phonon-number fluctuations. Meanwhile, the controlled spontaneous emission of trapped ions is implemented via pumping the metastable states upwards to the short-lived ionic states by an auxiliary laser field. Combining the unitary processes with the engineered dissipation, a deterministic Bell state can be produced irrespective of the initial states of systems. Moreover, our result shows that the Clauser-Horne-Shimony-Holt inequality can be violated for a wide range of decoherence parameters, even at finite temperature.
We investigate the dynamics of two trapped ions interacting with a common thermal reservoir, focusing on how cross-correlated dissipation influences heating, steady-state behavior, and parameter sensitivity. Starting from a microscopic system--reservoir model, we derive the corresponding Heisenberg--Langevin equations and show that reservoir-induced correlations generate collective decay channels and, when the cross-damping rate matches the local damping, a decoherence-free normal mode that preserves memory of the initial excitations. Using the Fisher information associated with motional population measurements, we identify the parameter regimes in which cross-damping enhances the estimability of both system and reservoir properties. For nonclassical initial states, we also show that reservoir-mediated correlations can generate or maintain entanglement, with the strongest effects occurring near the decoherence-free condition.
This article investigates dissipative preparation of entangled nonequilibrium steady states (NESS). We construct a collision model where the open system consists of two qubits which are coupled to heat reservoirs with different temperatures. The baths are modeled by sequences of qubits interacting with the open system. The model can be studied in different dynamical regimes: with and without environmental memory effects. We report that only a certain bath temperature range allows for entangled NESS. Furthermore, we obtain minimal and maximal critical values for the heat current through the system. Surprisingly, quantum memory effects play a crucial role in the long-time limit. First, memory effects broaden the parameter region where quantum correlated NESS may be dissipatively prepared and, second, they increase the attainable concurrence. Most remarkably, we find a heat current range that does not only allow, but even guarantees that the NESS is entangled. Thus, the heat current can witness entanglement of nonequilibrium steady states.