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arXiv:quant-ph/0301057 · American Journal of Physics

Wave packet spreading: Temperature and squeezing effects with applications to quantum measurement and decoherence

G. W. Ford, Robert F. O’Connell

Published Mar 1, 2002• 43 citations
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Abstract

A localized free particle is represented by a wave packet and its motion is discussed in most quantum mechanics textbooks. Implicit in these discussions is the assumption of zero temperature. We discuss how the effects of finite temperature and squeezing can be incorporated in an elementary manner. The results show how the introduction of simple tools and ideas can bring the reader into contact with topics at the frontiers of research in quantum mechanics. We discuss the standard quantum limit, which is of interest in the measurement of small forces, and decoherence of a mixed (“Schrödinger cat”) state, which has implications for current research in quantum computation, entanglement, and the quantum-classical interface.

DOI
10.1119/1.1447540

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A localized free particle is represented by a wave packet and its motion is discussed in most quantum mechanics textbooks. Implicit in these discussions is the assumption of zero temperature. We discuss how the effects of finite temperature and squeezing can be incorporated in an elementary manner. The results show how the introduction of simple tools and ideas can bring the reader into contact with topics at the frontiers of research in quantum mechanics. We discuss the standard quantum limit, which is of interest in the measurement of small forces, and decoherence of a mixed (“Schrödinger cat”) state, which has implications for current research in quantum computation, entanglement, and the quantum-classical interface.
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.
Much of the discussion of decoherence has been in terms of a particle moving in one dimension that is placed in an initial superposition state (a Schrödinger "cat" state) corresponding to two widely separated wave packets. Decoherence refers to the destruction of the interference term in the quantum probability function. Here, we stress that a quantitative measure of decoherence depends not only on the specific system being studied but also on whether one is considering coordinate, momentum or phase space. We show that this is best illustrated by considering Wigner phase space where the measure is again different. Analytic results for the time development of the Wigner distribution function for a two-Gaussian Schrodinger "cat" state have been obtained in the high-temperature limit (where decoherence can occur even for negligible dissipation) which facilitates a simple demonstration of our remarks.
Distribution functions defined in accordance with the quantum theory of measurement are combined with results obtained from the quantum Langevin equation to discuss decoherence in quantum Brownian motion. Closed form expressions for wave packet spreading and the attenuation of coherence of a pair of wave packets are obtained. The results are exact within the context of linear passive dissipation. It is shown that, contrary to widely accepted current belief, decoherence can occur at high temperature in the absence of dissipation. Expressions for the decoherence time with and without dissipation are obtained that differ from those appearing in earlier discussions.
The behaviour of localized quantum wave packet has lately become an object of much interest to forge a link between classical and quantum mechanics. Wave packet is often taken as the best approximation of quantum theory to a classical particle. If the wave packet is regarded as a point, with zero amplitude everywhere except in a very small region, then its motion can be interpreted as the motion of a single particle. The advent of short-pulsed lasers made it possible to produce and detect superposition of electron states for many physical systems. Such superposition leads to the formation of localized electron wave packets. The localized wave packet solution disperses or spreads in time, however, at a later time, it reforms and appears as its initial state. Revivals in quantum systems were first studied in the Jaynes–Cumming model, describing a two-level atom interacting with a resonant monochromatic field.
Localized quantum wave packets can be produced in a variety of physical systems and are the subject of much current research in atomic, molecular, chemical, and condensed-matter physics. They are particularly well suited for studying the classical limit of a quantum-mechanical system. The motion of a localized quantum wave packet initially follows the corresponding classical motion. However, in most cases the quantum wave packet spreads and undergoes a series of collapses and revivals. We present a generic treatment of wave-packet evolution, and we provide conditions under which various types of revivals occur in ideal form. The discussion is at a level appropriate for an advanced undergraduate or first-year graduate course in quantum mechanics. Explicit examples of different types of revival structure are provided, and physical applications are discussed.
Efforts to achieve quantum computation, teleportation, and communication depend on minimizing decoherence, which is the destruction of a quantum interference pattern. Here, we examine effects arising from the universal zero-point (temperature T = 0) oscillations of the electromagnetic field on a free electron in a Schrödinger cat superposition state. A unique conclusion is that the spreading of an electron wavepacket and the rate of decay of decoherence depend on the bare mass m of the electron. However, only for m = 0 does decoherence occur and the fact that it occurs almost instantly is ruled out by electron interference experiments. For m ≠ 0, the electron essentially behaves as a free particle.