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arXiv:quant-ph/0401038 · Physical Review A

Survival of quantum effects for observables after decoherence

G. P. Berman, A. R. Bishop, F. Borgonovi, Diego A. R. Dalvit

Published Jun 23, 2004• 11 citations
arXivHTMLPDF

Abstract

When a quantum nonlinear system is linearly coupled to an infinite bath of harmonic oscillators, quantum coherence of the system is lost on a decoherence time scale ${\ensuremath{\tau}}_{D}$. Nevertheless, quantum effects for observables may still survive environment-induced decoherence and be observed for times much larger than the decoherence time scale. In particular, we show that the Ehrenfest time, which characterizes a departure of quantum dynamics for observables from the corresponding classical dynamics, can be observed for a quasiclassical nonlinear oscillator for times $\ensuremath{\tau}⪢{\ensuremath{\tau}}_{D}$. We discuss this observation in relation to recent experiments on quantum nonlinear systems in the quasiclassical region of parameters.

DOI
10.1103/physreva.69.062110

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When a quantum nonlinear system is linearly coupled to an infinite bath of harmonic oscillators, quantum coherence of the system is lost on a decoherence time scale \ensuremathτD{\ensuremath{\tau}}_{D}\ensuremathτD​. Nevertheless, quantum effects for observables may still survive environment-induced decoherence and be observed for times much larger than the decoherence time scale. In particular, we show that the Ehrenfest time, which characterizes a departure of quantum dynamics for observables from the corresponding classical dynamics, can be observed for a quasiclassical nonlinear oscillator for times \ensuremathτ⪢\ensuremathτD\ensuremath{\tau}⪢{\ensuremath{\tau}}_{D}\ensuremathτ⪢\ensuremathτD​. We discuss this observation in relation to recent experiments on quantum nonlinear systems in the quasiclassical region of parameters.
Environment induced decoherence entails the absence of quantum interference phenomena from the macroscopic world. The loss of coherence between superposed wave packets is a dynamical process the speed of which depends on the packet separation: The farther the packets are apart, the faster they decohere. The precise temporal course depends on the relative size of the time scales of decoherence and other processes taking place in the open system and its environment. We use the exactly solvable model of a harmonic oscillator coupled to a bath of harmonic oscillators to illustrate various decoherence scenarios: These range from exponential golden-rule decay for microscopic superpositions, system-specific decay for larger separations in a crossover regime, and finally the universal interaction-dominated decoherence for ever more macroscopic superpositions investigated in great generality in the accompanying Paper I [W. T. Strunz, F. Haake, and D. Braun, Phys. Rev. A 67,67,67, 022101022101022101 (2003)].
We address the dynamics of nonclassicality for a quantum system interacting with a noisy fluctuating environment described by a classical stochastic field. As a paradigmatic example, we consider a harmonic oscillator initially prepared in a maximally nonclassical state, e.g., a Fock number state or a Schr\"odinger-cat-like state, and then coupled to either a resonant or a nonresonant external field. Stochastic modeling allows us to describe the decoherence dynamics without resorting to approximated quantum master equations and to introduce non-Markovian effects in a controlled way. A detailed comparison among different nonclassicality criteria and a thorough analysis of the decoherence time reveal a rich phenomenology whose main features may be summarized as follows: (i) Classical memory effects increase the survival time of quantum coherence and (ii) a detuning between the natural frequency of the system and the central frequency of the classical field induces revivals of quantum coherence.
The conditions under which an open quantum-mechanical system may be described by mixed quantum-classical dynamics are investigated. Decoherence is studied using influence functional methods in a model composite quantum system comprising two coupled systems, A and C, interacting with a harmonic bath with Ohmic and super-Ohmic spectral densities. Subsystem A is directly coupled to subsystem C, while C is coupled directly to the bath. Calculations are presented for a model where subsystem A is taken to be a two-level system which is bilinearly coupled to a single harmonic oscillator C subsystem. The loss of quantum coherence in each subsystem is discussed in the extreme nonadiabatic regime where the intrinsic dynamics of subsystem A is essentially frozen. Subsystem C is shown to lose its coherence rapidly, while subsystem A maintains coherence for longer time periods since C modulates the influence of the bath on A. Thus, one may identify situations where the coupled AC system evolution effectively obeys mixed quantum-classical dynamics.
We study the quantum decoherence of a bound state interacting with a reservoir of strongly interacting matter within the framework of open quantum systems. The bound state is modeled as a quantum harmonic oscillator whose parameters are tuned to reproduce the root-mean-square radius of J/ΨJ/ΨJ/Ψ particle. The surrounding medium, representing the many degrees of freedom of strongly interacting matter, acts as an environment that induces dissipation and decoherence through system-reservoir coupling. By analyzing the time evolution of the reduced density matrix, we quantify the loss of quantum coherence and its dependence on medium properties. Subsequently, we extend the model by introducing a time dependence in the system-thermal bath coupling, thereby simulating a temperature evolution similar to that occurring during the expansion of a fireball in the central region of heavy-ion collisions. We find that a temperature evolution has a relevant impact on the way the system loses coherence through the coupling with the expanding medium. Finally, we estimate the impact of the time-dependent temperature on the decoherence process, also analyzing a scenario that includes viscous effects without finding a significant change with respect to ideal hydrodynamical evolution.
Environment-induced destruction of quantum coherence is investigated in a simple model where the system is a harmonic oscillator, the environment is a collection of harmonic oscillators, and the interaction between them is linear in the position coordinate of the system. We study decoherence for initial states consisting of coherent superpositions of two Gaussian wave packets in either position or momentum. A new measure of the effectiveness of decoherence appropriate to the model and choice of initial conditions is proposed. By studying the dependence of the decoherence rate on the location of the initial peaks of the Wigner function, we clarify the sense in which position is a preferred observable even though position eigenstates are not the pointer states of this model. We analyze decoherence in the low-temperature regime and show that the usual "high-temperature" approximation is remarkably accurate in its domain of applicability. We also examine the relationship between the decoherence process and the frequency distribution of the environment oscillators (in particular, we focus attention on a specific "supra-Ohmic" environment). Implications of our results for the quantum to classical transition in various contexts are briefly explored.
The threshold for quantum-classical correspondence is D∼ℏ43D \sim \hbar^{\frac43}D∼ℏ34​
In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution after the Ehrenfest time tE∼log⁡(ℏ−1)t_{\mathrm{E}} \sim \log(\hbar^{-1})tE​∼log(ℏ−1), even in the limit ℏ→0\hbar \to 0ℏ→0. Decoherence by the environment is often invoked to explain the persistence of the quantum-classical correspondence at longer timescales. Recent rigorous results for Lindblad dynamics with phase-space diffusion strength DDD show that quantum and classical evolutions remain close for times that are exponentially longer than the Ehrenfest time whenever D≫ℏ43D \gg \hbar^{\frac43}D≫ℏ34​, in units set by the classical Hamiltonian. At the same time, some heuristic arguments have suggested the weaker condition D≫ℏ2D \gg \hbar^{2}D≫ℏ2 always suffices. Here we construct an explicit Lindbladian that demonstrates that the scaling D∼ℏ43D \sim \hbar^{\frac43}D∼ℏ34​ is indeed the threshold for quantum-classical correspondence beyond the Ehrenfest time. Our example uses a smooth time-dependent Hamiltonian and linear Lindblad operators generating homogeneous isotropic diffusion. It exhibits an ℏ\hbarℏ-independent quantum-classical discrepancy at the Ehrenfest time whenever D≪ℏ43D \ll \hbar^{\frac43}D≪ℏ34​, even for ℏ\hbarℏ-independent "macroscopic" smooth observables.