Ground-state energy fluctuations of a system coupled to a bath

Abstract

It is often argued that a small non-degenerate quantum system coupled to a bath has a fixed energy in its ground state since a fluctuation in energy would require an energy supply from the bath. We consider a simple model of a harmonic oscillator (the system) coupled to a linear string and determine the mean squared energy fluctuations. We also analyze the two time correlator of the energy and discuss its behavior for a finite string.The coupling of two subsystems is a central problem in many areas of physics. The interaction of atoms with the radiation field has been central in the development of quantum mechanics 1 . The division into subsystems often depends on the questions asked: For instance dephasing in mesoscopic systems investigates the quantum motion of a single electron in a conductor viewing all other conduction electrons as the bath 2 . In this work we are interested in the energy fluctuations in a test system coupled to another system. Of particular interest is the zero-temperature limit. It is often argued that in the zero-temperature limit two subsystems can not exchange energy 3,4 . The argument is based on the assumption that both subsystems are in their separate ground state (which they would assume in the absence of any coupling). Since both the test system and the bath are in their ground state neither of the two can supply an energy to the other 3,4 . Below we consider a simple system of an oscillator (the test system) coupled to a linear string (the bath) and investigate the energy fluctuations of the test system. This is an exactly solvable model which demonstrates the existence of energy fluctuations in the zero-temperature limit. These fluctuations are a consequence of the finite coupling energy between the test system and the bath.Often the interaction of two subsystems is treated in terms of the Einstein coefficients for the absorption and the spontaneous and stimulated emission of the test system. In such an approach rates of transitions of the test system to change energy form one eigenstate E n to another eigenstate E m are investigated. These are inelastic transition rates which vanish when both systems are in the ground state. The resulting probabilities are used in a classical stochastic equation for the probabilities of occupation of the test system. Only probabilities enter in the description of the reservoir system interaction and this approach again yields no energy fluctuations in the ground state. Such a description is challenged both in Laser physics 1 and in the more recent discussions of macroscopic quantum coherence 5,6 .The role of zero-point fluctuations in mesoscopic conductors is a hotly debated issue and we cite here only Ref. 7 as an entry to the literature. The point of view taken here has been applied to investigate the persistent current of a small mesoscopic loop with a quantum dot capacitively coupled to a transmission line 8 . The persistent current is a measure of the quantum coherence of the ground state. It was found that the ground state undergoes a cro...

Loss of quantum coherence in a system coupled to a zero-temperature environment

“…37, so, for a continuous environment, we can explicitly compute q 2 and p 2 for any mass distribution µ(ω). These results have already been obtained (see [16,10]), following a different approach, namely the fluctuation-dissipation theorem.…”

“…3, see [10,14,15].T (T ) can be calculated explicitly (see appendix A). It is plotted for different values of the coupling η between the particle and the environment in fig.…”

“…The subsequent theoretical debate [6,7,8] mainly focused on zero-temperature decoherence induced by Coulomb interactions in disordered electron systems, but as a spin-off has led to the more general question "Can a zerotemperature environment induce decoherence?". Recently, this issue was discussed in [10,11] in the framework of a well-known, simple model : a harmonic oscillator (the "particle") coupled to a chain of harmonic oscillators (the "environment"). It was shown that the particle exchanges energy with the environment, even at zero temperature.…”

Untitled

“…If that were the case, it would imply that the usual instanton methods, when applied to a system interacting with an environment, downplay the role of the backreaction of the environment on the system dynamics. This can be qualitatively interpreted in the following way: while, roughly speaking, the tunneling effect for an isolated system can be regarded as a consequence of the energy fluctuations implied by Heisenberg's uncertainty principle, the interaction with an environment would induce fluctuations on the system due to the quantum fluctuations of the environment itself (Nagaev and Büttiker, 2002), which would enhance the tunneling rate.…”

Ground state entanglement energetics

“…For instance for a harmonic oscillator with frequency ω0 coupled to an ohmic bath of harmonic oscillators, we find that the energy fluctuations [6,7] are given by…”