Measurement of the Temperature Using the Tomographic Representation of Thermal States for Quadratic Hamiltonians.

Julio A. López-Saldívar,Margarita A. Man’ko,Vladimir I. Man’ko

doi:10.3390/e23111445

Abstract

The Wigner and tomographic representations of thermal Gibbs states for one- and two-mode quantum systems described by a quadratic Hamiltonian are obtained. This is done by using the covariance matrix of the mentioned states. The area of the Wigner function and the width of the tomogram of quantum systems are proposed to define a temperature scale for this type of states. This proposal is then confirmed for the general one-dimensional case and for a system of two coupled harmonic oscillators. The use of these properties as measures for the temperature of quantum systems is mentioned.

Highlights

IntroductionThe description of the state of a system is one of the most important problems of theoretical physics
We presented a small review of the probability representation of quantum states, where the particle states are described by fair tomographic-probability distribution functions
Concrete examples of thermal-equilibrium states of one-mode and two-mode oscillators at temperature T were studied, and explicit expressions for the state symplectic tomograms were calculated. These tomograms are given by Gaussian conditional probability distribution functions

Summary

Introduction

The description of the state of a system is one of the most important problems of theoretical physics. For the particle with Hamiltonian H (q, p) in the environment with temperature T, the thermal equilibrium state is associated with the probability distribution function f (q, p, t) =. The Wigner function known for a quantum state is related by the same Radon transform to the symplectic tomogram of the state with density operator ρ, which determines the quantum particle tomogram, i.e., Wρ ( X | μ, ν) = Tr ρ · δ X 1̂ − μq − ν p. The Wigner and tomographic representations of quantum states have been important for the study of quantum systems They have been used to distinguish between classical or nonclassical behavior of quantum states [18].

Wigner and Tomographic Representations of Unimodal Thermal States

Tomographic Representation of Two-Mode Thermal States

Concluding Remarks