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arXiv:quant-ph/0106082 · Physical Review A

Entanglement and quantal coherence: Study of two limiting cases of rapid system-bath interactions

Nicole F. Bell, R. F. Sawyer, Raymond R. Volkas

Published Apr 16, 2002• 13 citations
arXivHTMLPDF

Abstract

In this paper, we consider the dynamics of a system coupled to a thermal bath, going beyond the standard two-level system through the addition of an energy excitation degree of freedom. Further extensions are to systems containing many fermions, with the master equations modified to take Fermi-Dirac statistics into account, and to potentials with a time-dependent bias that induce resonant avoided crossing transitions. The limit $\stackrel{\ensuremath{\rightarrow}}{Q}\ensuremath{\infty},$ where the interaction rate with the bath is much greater than all free oscillation rates for the system, is investigated. Two behaviors are possible: freezing (quantum Zeno effect) or synchronization (motional narrowing). We clarify the conditions that give rise to each possibility, making an explicit connection with quantum-measurement theory. We compare the evolution of quantal coherence for the two cases as a function of Q, noting that full coherence is restored as $\stackrel{\ensuremath{\rightarrow}}{Q}\ensuremath{\infty}.$ Using an extended master equation, the effect of system-bath interactions on entanglement in bipartite system states is computed. In particular, we show that the sychronization case sees bipartite system entanglement fully preserved in the large Q limit.

DOI
10.1103/physreva.65.052105

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In this paper, we consider the dynamics of a system coupled to a thermal bath, going beyond the standard two-level system through the addition of an energy excitation degree of freedom. Further extensions are to systems containing many fermions, with the master equations modified to take Fermi-Dirac statistics into account, and to potentials with a time-dependent bias that induce resonant avoided crossing transitions. The limit Q\ensuremath→\ensuremath∞,\stackrel{\ensuremath{\rightarrow}}{Q}\ensuremath{\infty},Q\ensuremath→​\ensuremath∞, where the interaction rate with the bath is much greater than all free oscillation rates for the system, is investigated. Two behaviors are possible: freezing (quantum Zeno effect) or synchronization (motional narrowing). We clarify the conditions that give rise to each possibility, making an explicit connection with quantum-measurement theory. We compare the evolution of quantal coherence for the two cases as a function of Q, noting that full coherence is restored as Q\ensuremath→\ensuremath∞.\stackrel{\ensuremath{\rightarrow}}{Q}\ensuremath{\infty}.Q\ensuremath→​\ensuremath∞. Using an extended master equation, the effect of system-bath interactions on entanglement in bipartite system states is computed. In particular, we show that the sychronization case sees bipartite system entanglement fully preserved in the large Q limit.
A master equation has been constructed for a global system-bath interaction both in the absence as well as presence of non-Markovian noise. The master equation has been exactly solved for a special class of two qubit X states (which contains Bell diagonal and Werner states). The l1l_1l1​ norm of coherence has been calculated to observe the dynamics of quantum coherence in the presence of a global system-bath interaction. We show that the global part of the system-bath interaction compensates for the decoherence, resulting in the slow down of coherence decay. The concurrence and the Fisher information, explicitly calculated for a particular two qubit Werner state, indicate that the decay of these quantum features also slow down under a global system-bath interaction. We also show that the coherence is the most robust of all the three non-classical features under environmental interaction. Entanglement is shown to be the most costly of them all. For an appropriately defined limiting case, all the three quantities show freezing behaviour. This limiting condition is attainable when the separations between the energy levels of both the atomic qubits are small.
We investigate a system of two interacting qubits having one of them isolated and the other coupled to a thermal reservoir. We consider two different models of system-reservoir interaction: i) a model, in which the master equation is derived taking into account the interaction between the two subsystems (qubits), ii) a naive model, in which the master equation consists of a dissipative term added to the unitary evolution term. We study the dynamics of quantities such as bipartite entanglement, quantum discord and the linear entropy of the isolated qubit for both strong and weak coupling regimes (qubit-qubit interaction) as well as for different temperatures of the reservoir. We find significant disagreements between the results obtained from the two models even in the weak coupling regime. For instance, we show that according to the phenomenological model, the isolated qubit would approach a maximally mixed state more slowly for higher temperatures (unphysical result), while the microscopic model predicts the opposite behaviour (correct result).
The evolution of a system coupled to baths is commonly described by a master equation that, in the long-time limit, yields a steady-state density matrix. However, when the same evolution is unraveled into quantum trajectories, it is possible to observe a transition in the scaling of entanglement within the system as the system-bath coupling increases - a phenomenon that is invisible in the trajectory-averaged reduced density matrix of the system. Here, we go beyond the paradigm of trajectories from master equations and explore whether a qualitatively analogous entanglement-scaling transition emerges in a single unitary evolution of the combined system-bath setup, without monitoring the dynamics of the system. We investigate the scaling of entanglement in a unitary quantum setup composed of a two-dimensional lattice of free fermions, where each site is coupled to a fermionic bath. As the system-bath coupling increases, the logarithmic fermionic negativity reveals an entanglement transition from logarithmic-law to area-law scaling. This occurs while the system's steady-state properties are trivial, highlighting that the signatures of these different scalings are within the bath-bath correlations. Evidence of the transition is also found in the mutual information and the correlations of the full system-bath setup, suggesting that the entanglement transition is underpinned by a change in the spatial structure of quantum information.
We investigate the behavior of entanglement between a single fermionic level and a fermionic bath in three distinct thermodynamic regimes. First, in thermal equilibrium, we analyze the dependence of entanglement on the considered statistical ensemble: for the grand canonical state, it is generated only for a sufficiently strong system-bath coupling, whereas it is present for arbitrarily weak couplings for the canonical state with a fixed particle number. The threshold coupling strength at which entanglement appears is shown to strongly depend on the bath bandwidth. Second, we consider the relaxation to equilibrium. In this case a transient entanglement in a certain time interval can be observed even in the weak-coupling regime, when the reduced dynamics and thermodynamics of the system can be well described by an effectively classical and Markovian master equation for the state populations. At strong-coupling strengths, entanglement is preserved for long times and converges to its equilibrium value. Finally, in voltage-driven junctions, a steady-state entanglement is generated for arbitrarily weak system-bath couplings at a certain threshold voltage. It is enhanced in the strong-coupling regime, and it is reduced by either the particle-hole or the tunnel coupling asymmetry.
The second law of thermodynamics places a limitation into which states a system can evolve into. For systems in contact with a heat bath, it can be combined with the law of energy conservation, and it says that a system can only evolve into another if the free energy goes down. Recently, it's been shown that there are actually many second laws, and that it is only for large macroscopic systems that they all become equivalent to the ordinary one. These additional second laws also hold for quantum systems, and are, in fact, often more relevant in this regime. They place a restriction on how the probabilities of energy levels can evolve. Here, we consider additional restrictions on how the coherences between energy levels can evolve. Coherences can only go down, and we provide a set of restrictions which limit the extent to which they can be maintained. We find that coherences over energy levels must decay at rates that are suitably adapted to the transition rates between energy levels. We show that the limitations are matched in the case of a single qubit, in which case we obtain the full characterization of state-to-state transformations. For higher dimensions, we conjecture that more severe constraints exist. We also introduce a new class of thermodynamical operations which allow for greater manipulation of coherences and study its power with respect to a class of operations known as thermal operations.
We propose a method to suppress decoherence in entangled systems by applying the series of joint measurements. While the standard Quantum Zeno Effect approach can only compensate for the slow enough Hamiltonian dynamics of the quantum system, the characteristic frequencies of the thermal bath contacting system may be greater than the maximal achievable frequencies of measurements. In this case, they cannot cancel out non-Hamiltonian evolution leading to the decoherence. We propose to use entanglement between the qubits of a quantum register to mitigate the negative effects of the interaction with the thermal bath, even with the lower frequencies of measurement. We show that even though this protocol does not lead to a complete freezing of the initial state, it still allows effective preservation of the entanglement in the system.