Decoherence without dissipation

The prototypical Schrödinger cat state, i.e., an initial state corresponding to two widely separated Gaussian wave packets, is considered. The decoherence time is calculated solely within the framework of elementary quantum mechanics and equilibrium statistical mechanics. This is at variance with common lore that irreversible coupling to a dissipative environment is the mechanism of decoherence. Here, we show that, on the contrary, decoherence can in fact occur at high temperature even for vanishingly small dissipation.  2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.Bz; 05.30.-d; 05.40.+j Quantum teleportation [1], quantum information and computation [2,3], entangled states [4], Schrödinger cats [5], and the classical–quantum interface [6]: topics at forefront of research embracing quantum physics, information science and telecommunications and all depending on an understanding of decoherence [7], i.e., how a quantum interference pattern is destroyed. In an introduction to the contents of a recent book devoted wholly to this subject, Joos surveys the current situation and, in discussing the mechanism of decoherence, states that “. . . irreversible coupling to (a dissipative) environment seems to have become widely accepted . . .”. Here, while we agree that coupling to the environment is necessary to establish thermal equilibrium, we show that at high temperature decoherence occurs even for vanishingly small dissipation. The situation is like that for an ideal gas: colli* Corresponding author. E-mail address: [email protected] (R.F. O’Connell). sions are necessary to bring the gas to equilibrium but do not appear in the equation of state, nor in the velocity distribution. Much of the discussion of decoherence [8–11] has been in terms of the simple problem of a particle moving in one dimension that is placed in an initial superposition state (Schrödinger “cat” state) corresponding to two widely separated wave packets. The motivation for this choice is that it can be applied, say, to describe the interference pattern arising in Young’s twoslit experiment [8] or that arising from a quantum measurement involving a pair of “Gaussian slits” [12,13]. Of primary interest is the question of the classical– quantum interface, i.e., how the interference pattern is destroyed with the evolution of a classical state corresponding to two separately propagating packets. Decoherence refers to this destruction of the interference pattern and key questions are what is the origin of decoherence and what is the time scale for loss of coherence. The maintenance of coherence is an essential element in quantum teleportation, etc. Thus, an 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S0375-9601(01) 00 41 94 88 G.W. Ford, R.F. O’Connell / Physics Letters A 286 (2001) 87–90 understanding of all physical phenomena which can cause decoherence is essential. Our purpose here is to give an elementary calculation showing that at high temperature (kT h̄γ , where γ is the dissipative decay rate) decoherence occurs in a very short time that is, contrary to widely held belief, independent of the strength of coupling to the environment. Our starting point is the prototypical Schrödinger cat state, i.e., an initial state corresponding to two separated Gaussian wave packets. The corresponding wave function has the form ψ(x,0)= 1 [2(1+ e−d/8σ )]1/2 (1) × ( exp {− (x−d/2)2 4σ 2 + i mv h̄ x }