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arXiv:cond-mat/0202044

Elementary explanation of the inexistence of decoherence at zero temperature for systems with purely elastic scattering

Y. Imry

Published Feb 4, 2002• 1 citation
arXivHTMLPDF

Abstract

This note has no new results and is therefore not intended to be submitted to a "research" journal in the foreseeable future, but to be available to the numerous individuals who are interested in this issue. Several of those have approached the author for his opinion, which is summarized here in a hopefully pedagogical manner, for convenience. It is demonstrated, using essentially only energy conservation and elementary quantum mechanics, that true decoherence by a normal environment approaching the zero-temperature limit is impossible for a test particle which can not give or lose energy. Prime examples are: Bragg scattering, the Mössbauer effect and related phenomena at zero temperature, as well as quantum corrections for the transport of conduction electrons in solids. The last example is valid within the scattering formulation for the transport. Similar statements apply also to interference properties in equilibrium.

DOI
10.48550/arxiv.cond-mat/0202044

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This note has no new results and is therefore not intended to be submitted to a "research" journal in the foreseeable future, but to be available to the numerous individuals who are interested in this issue. Several of those have approached the author for his opinion, which is summarized here in a hopefully pedagogical manner, for convenience. It is demonstrated, using essentially only energy conservation and elementary quantum mechanics, that true decoherence by a normal environment approaching the zero-temperature limit is impossible for a test particle which can not give or lose energy. Prime examples are: Bragg scattering, the Mössbauer effect and related phenomena at zero temperature, as well as quantum corrections for the transport of conduction electrons in solids. The last example is valid within the scattering formulation for the transport. Similar statements apply also to interference properties in equilibrium.
A popular model of decoherence based on the linear coupling to harmonic oscillator heat baths is analysed and shown to be inappropriate in the regime where decoherence dominates over energy dissipation, called pure decoherence regime. The similar mechanism essentially related to the energy conservation implies that, on the contrary to some recent conjectures [21], chaotic environments can be less efficient decoherers than regular ones. Finally, the elastic scattering mechanism is advocated as the simplest source of pure decoherence.
We point out that even at the absolute zero of temperature environmental decoherence limits the destructive interference between time-reversed paths for an electron in a disordered metal, and thus causes the leading (`weak localization') quantum correction to the conductivity to saturate at T=0. Our calculation, which is intended to be illustrative rather than complete, uses a model in which an electron interacts with the fluctuations of the mean voltage in the sample. The average of the fluctuations produces the steady damping well known in Brownian motion, introduces a direction of time, ensures that arbitrarily long time-reversed electron paths lose phase coherence, and is consistent with the experimental observation of a finite low temperature phase-breaking lifetime.
Abstract In the interactions of electromagnetic radiation with matter which we have considered so far, we have almost invariably known accurately the state of the system before interaction. As a result the notion of an open system with an external environment did not have to be systematically discussed. In the quantum electrodynamics of elementary particles, the scattering events are predominantly affected by only few-body interactions and the concepts of temperature and environment are irrelevant. (The resolution of the problem of infra-red divergences is an exception where the environment of soft photons is crucial.) For very small numbers of atoms it may be possible to know the initial state of the system, but in a macroscopic device this is essentially impossible. Although for transitions at optical frequencies the temperature of the environment may be negligible, this will not necessarily be the case for microwave transitions. We have already seen how atomic linewidths can be produced by interactions with the bath of photons at zero temperature. In the general case a quantum-mechanical system will show not only fluctuations due to the non-commutative nature of quantum theory but also thermal fluctuations and dissipation due to interactions with the environment. For complicated systems it will not be possible to know with certainty the initial state of the system, and so the approach of studying the evolution of a state vector through the Schrodinger equation has to be replaced by a consideration of the time evolution of the density operator introduced in Chapter 4. We will formulate a systematic way of treating the dynamics of a quantum-mechanical open system interacting with a heat bath, which will be illustrated with important examples.
We start by reviewing some interesting results in mesoscopic physics illustrating nontrivial insights on Quantum Mechanics. We then review the general principles of dephasing (sometimes called “decoherence”) of Quantum-Mechanical interference by coupling to the environment degrees of freedom. A particular recent example of dephasing by a current-carrying (nonequilibrium) system is then discussed in some detail. This system is itself a manifestly Quantum Mechanical one and this is another illustration of detection without the need for “classical observers” etc. We conclude by describing briefly a recent problem having to do with the orbital magnetic response of conduction electrons (another manifestly Quantum Mechanical property): The magnetic response of a normal layer (N) coating a superconducting cylinder (S). Some recent very intriguing experimental results on a giant paramagnetic component of this response are explained using special states in the normal layer. It is hoped that these discussions illustrate not only the vitality and interest of mesoscopic physics but also its extreme relevance to fundamental issues in Quantum Mechanics.
Elastic electron scattering is one of the primary means of investigating materials on the atomic scale. It is usually described in a one-particle approach (the probe electron's evolution in a perturbative static potential), whereas we are dealing here with a two-body between the probe and the sample, both described by separate quantum states, inducing entanglement. In this work, we present a quantum treatment of elastic electron scattering. We find that the entanglement between probe and scatterer can have far-reaching consequences, particularly on coherence and image contrast. As a timely example, we discuss decoherence in Bragg scattering on nanoparticles. We find that conventional scattering theory is recovered in most cases. The situation changes dramatically for freely evolving nanoparticles as, e.g., levitated motional ground states, an active field of research.
Most discussions of decoherence in the literature consider the high temperature regime but it is also known that, in the presence of dissipation, decoherence can occur even at zero temperature. Whereas most previous investigations all assumed initial decoupling of the quantum system and bath, we consider that the system and environment are entangled at all times. Here, we discuss decoherence for a free particle in an initial Schrödinger cat state. Memory effects are incorporated by use of the single relaxation time model.