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arXiv:quant-ph/0205108 · Physical Review A

Decoherence scenarios from microscopic to macroscopic superpositions

Walter T. Strunz, Fritz Haake

Published Feb 6, 2003• 38 citations
arXivHTMLPDF

Abstract

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 $67,$ $022101$ (2003)].

DOI
10.1103/physreva.67.022102

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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 consider environment induced decoherence of quantum superpositions to mixtures in the limit in which that process is much faster than any competing one generated by the Hamiltonian HsysH_{\rm sys}Hsys​ of the isolated system. This interaction dominated decoherence limit has previously not found much attention even though it is of importance for the emergence of classical behavior in the macroworld, since it will always be the relevant regime for large enough separations between superposed wave packets. The usual golden-rule treatment then does not apply but we can employ a short-time expansion for the free motion while keeping the interaction HintH_{\rm int}Hint​ in full. We thus reveal decoherence as a universal short-time phenomenon largely independent of the character of the system as well as the bath. Simple analytical expressions for the decoherence time scales are obtained in the limit in which decoherence is even faster than any timescale emerging from the reservoir Hamiltonian HresH_{\rm res}Hres​.
We consider environment induced decoherence of quantum superpositions to mixtures in the limit in which that process is much faster than any competing one generated by the Hamiltonian Hsys{H}_{\mathrm{sys}}Hsys​ of the isolated system. This interaction-dominated decoherence limit is of importance for the emergence of classical behavior in the macroscopic domain, since it will always be the relevant regime for large enough separations between the superposed wave packets. The usual golden-rule treatment then does not apply, but we can employ a short-time expansion for the free motion while keeping the interaction Hint{H}_{\mathrm{int}}Hint​ in full. We thus reveal decoherence as a universal short-time phenomenon largely independent of the character of the system as well as the bath and of the basis the superimposed states are taken from. Simple analytical expressions for the decoherence time scales are obtained in the limit in which decoherence is even faster than any time scale emerging from the reservoir Hamiltonian Hres.{H}_{\mathrm{res}}.Hres​.
We consider environment induced decoherence of quantum superpositions to mixtures in the limit in which that process is much faster than any competing one generated by the Hamiltonian HsysH_{\rm sys}Hsys​ of the isolated system. This interaction dominated decoherence limit has previously not found much attention even though it is of importance for the emergence of classical behavior in the macroworld, since it will always be the relevant regime for large enough separations between superposed wave packets. The usual golden-rule treatment then does not apply but we can employ a short-time expansion for the free motion while keeping the interaction HintH_{\rm int}Hint​ in full. We thus reveal decoherence as a universal short-time phenomenon largely independent of the character of the system as well as the bath. Simple analytical expressions for the decoherence time scales are obtained in the limit in which decoherence is even faster than any timescale emerging from the reservoir Hamiltonian HresH_{\rm res}Hres​.
We consider environment induced decoherence of quantum superpositions to mixtures in the limit in which that process is much faster than any competing one generated by the Hamiltonian H(sys) of the isolated system. While the golden rule then does not apply we can discard H(sys). By allowing for couplings to different reservoirs, we reveal decoherence as a universal short-time phenomenon independent of the character of the system as well as the bath and of the basis the superimposed states are taken from. We discuss consequences for the classical behavior of the macroworld and quantum measurement: For decoherence of superpositions of macroscopically distinct states H(sys) is always negligible.
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 interaction of a quantum system with a many degree of freedom environment entails damping and the loss of quantum coherence. In these two lectures we focus on two aspects of open quantum system dynamics. First, we present decoherence as a dynamical phenomenon and determine the corresponding time scales, discuss an experiment and the nature of robust states, and mention universal properties of short-time decoherence of macroscopic superpositions. Secondly, in analogy to the classical description of open systems in terms of Langevin equations, we present a stochastic approach to open quantum system dynamics in terms of a stochastic Schrödinger equation. We give a microscopic derivation that allows us to generalize the stochastic approach to non-Markovian situations. We discuss a convolutionless formulation of our theory and finally, we focus on the quantised version of Brownian motion as an often-employed example.