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arXiv:quant-ph/0410183 · The European Physical Journal D

Berry phase in open quantum systems: a quantum Langevin equation approach

Gabriele De Chiara, Artur Łoziński, G. Massimo Palma

Published Aug 17, 2006• 17 citations
arXivHTMLPDF
DOI
10.1140/epjd/e2006-00186-y

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We introduce a Langevin unravelling of the density matrix evolution of an open quantum system over matrix product states, which we term the time-dependent variational principle-Langevin equation. This allows the study of entanglement dynamics as a function of both temperature and coupling to the environment. As the strength of coupling to and temperature of the environment is increased, we find a transition where the entanglement of the individual trajectories saturates, permitting a classical simulation of the system for all times. This is the Hamiltonian open system counterpart of the saturation in entanglement found in random circuits with projective or weak measurements. If a system is open, there is a limit to the advantage in simulating its behaviour on a quantum computer, even when that evolution harbours important quantum effects. Moreover, if a quantum simulator is in this phase, it cannot simulate with quantum advantage.
Phase-space methods are powerful tools for the treatment of Markovian open quantum systems: they map the reduced dynamics of a system S, in interaction with an environment E, exactly onto Langevin equations for c-number stochastic variables, as opposed to Heisenberg-Langevin equations for operators. Langevin equations provide analytical insight in key regimes and excel at handling strong nonlinearities and couplings, where other methods often falter. Extending phase-space methods to non-Markovian dynamics, however, has remained a long-standing challenge. Here we address this gap by applying phase-space representations to the full S+E system; integrating out the environmental degrees of freedom then yields a general Langevin framework for S that incorporates both deterministic and stochastic contributions from E. Normally ordered representations, such as the Glauber-Sudarshan P representation and its positive variant due to Drummond and Gardiner, lead to Langevin equations in which (i) non-Markovian effects emerge exclusively in the deterministic terms, via a memory kernel, and (ii) noise contributions vanish when E is initially in the vacuum state. To demonstrate the power of this framework, we address the paradigmatic problem of delayed coherent feedback, in which the system is driven by its own past state, and study its impact on the laser linewidth: we recover the narrowing observed well above threshold and predict an enhanced narrowing just above it. Crucially, the number of stochastic variables scales linearly with the system size, making the framework suitable for problems ranging from a few degrees of freedom to genuinely many-body systems. This opens the way to the systematic study of non-Markovian driven-dissipative quantum systems using the same analytical and numerical tools that have long made phase-space methods so successful in the Markovian regime.