# Entropy and Uncertainty of Squeezed Quantum Open Systems

## Author

**Don Koks**,
**Andrew Matacz** and
**B.L. Hu**

## Status

Research Report 96-41

Date: 29 November 1996

## Abstract

We define the entropy S and uncertainty function of a squeezed system
interacting with a thermal bath, and study how they change in time by
following the evolution of the reduced density matrix in the influence
functional formalism. As examples, we calculate the entropy of two
exactly solvable squeezed systems: an inverted harmonic oscillator
and a scalar field mode evolving in an inflationary universe.
For the inverted oscillator with weak coupling to the bath,
at both high and low temperatures, S -> r, where
r is the squeeze parameter. In the de Sitter case, at high temperatures,
S -> (1-c)r where c = gamma_0/H, gamma_0 being the coupling to the bath
and H the Hubble constant.

These three cases confirm previous results
based on more ad hoc prescriptions for calculating entropy. But at
low temperatures, the de Sitter entropy S -> (1/2-c)r is noticeably
different. This result, obtained from a more rigorous approach,
shows that factors usually ignored by the
conventional approaches, i.e., the nature of the environment and the coupling
strength betwen the system and the environment, are important.

## Key phrases

Quantum Brownian Motion. Non-equilibrium Statistical Physics.

## Content

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Sydney Mathematics and Statistics