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arXiv:2210.03531 · JOURNAL OF PHYSICS AND CHEMISTRY RESEARCH

Thermal and Quantum Fluctuations of Harmonic Oscillator

Shigenori Tanaka, Yuto Komeiji

Published Dec 3, 2022• 0 citations
arXivHTMLPDF

Abstract

We discuss the positional fluctuations of a quantum harmonic oscillator in a heat bath. Analytic expressions are given for the probability distribution functions of the oscillator position in general and limiting (classical and ground state) cases.
DOI
10.36266/jpcr/151

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We consider a thermal quantum harmonic oscillator weakly coupled to a heat bath at a different temperature. We analytically study the quantum heat exchange statistics between the two systems using the quantum-optical master equation. We exactly compute the characteristic function of the heat distribution and show that it verifies the Jarzynski-W\'ojcik fluctuation theorem. We further evaluate the heat probability density in the limit of long thermalization times, both in the low- and the high-temperature regimes, and investigate its time evolution by calculating its first two cumulants.
The nonzero ground-state energy of the quantum mechanical harmonic oscillator implies quantum fluctuations around the minimum of the potential with the mean-square value proportional to Planck's constant. On the other hand, thermal fluctuations occur when the oscillator is coupled to a heat bath. Using quantum statistical mechanics, this paper describes the transition from pure quantum fluctuations at zero temperature to classical thermal fluctuations in the high-temperature limit. It was early pointed out by Peierls that the classical mean-square thermal fluctuations in a harmonic chain—a linear chain of coupled harmonic oscillators—increase linearly with the distance of the masses from the fixed end of the chain, destroying long-range crystalline order. As shown here, the corresponding quantum fluctuations lead to a much slower logarithmic increase with the distance from the end. It is also shown that this behavior implies, for example, the absence of sharp Bragg peaks in x-ray scattering in an infinite chain at zero temperature, which instead shows power-law behavior typical for one-dimensional quantum liquids (called Luttinger liquids).
In the framework of the Lindblad theory for open quantum systems, we derive closed analytical expressions of the Heisenberg and Schr\"odinger generalized uncertainty functions for a particle moving in a harmonic-oscillator potential. The particle is initially in an arbitrary correlated coherent state, and interacts with an environment at finite temperature. We describe how the quantum and thermal fluctuations contribute to the uncertainties in the canonical variables of the system and analyze the relative importance of these fluctuations in the evolution of the system. We show that upon contact with the bath, the system evolves from a quantum-dominated to a thermal-dominated state in a time that is of the same order as the decoherence time calculated in other models in the context of transitions from quantum to classical physics.
We formulate exact generalized nonequilibrium fluctuation relations for the quantum mechanical harmonic oscillator coupled to multiple harmonic baths. Each of the different baths is prepared in its own individual (in general nonthermal) state. Starting from the exact solution for the oscillator dynamics we study fluctuations of the oscillator position as well as of the energy current through the oscillator under general nonequilibrium conditions. In particular, we formulate a fluctuation-dissipation relation for the oscillator position autocorrelation function that generalizes the standard result for the case of a single bath at thermal equilibrium. Moreover, we show that the generating function for the position operator fullfills a generalized Gallavotti-Cohen-like relation. For the energy transfer through the oscillator, we determine the average energy current together with the current fluctuations. Finally, we discuss the generalization of the cumulant generating function for the energy transfer to nonthermal bath preparations.