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Results in Physics

Linkage between thermodynamic quantities and the uncertainty relation in harmonic oscillator model

Shoichi Nagata

Published Jan 1, 2016• 7 citations
Publisher

Abstract

The Heisenberg uncertainty relation, ΔxΔpx ⩾ ℏ/2 (ℏ ≡ h/2π), is well known. The purpose of this paper is to study a linkage between macroscopic thermodynamic quantities and the uncertainty relation. To simplify matters, we consider one-dimensional harmonic oscillator model. The calculations are carried out by two steps. First, the thermal average of the uncertainty relation is obtained for canonical system in thermal equilibrium at temperature T, which is denoted by (Δx)T(Δpx)T. Second, the thermodynamic quantities are expressed analytically as a function of (Δx)T(Δpx)T within the harmonic oscillator model. Finally, a connection between the energy fluctuations and the uncertainty relation is clarified for the Einstein’s heat capacity model. The analysis is made on the basis of fluctuation theory. The uncertainty relation is valid not only in quantum state but also at finite temperature.

DOI
10.1016/j.rinp.2016.11.007

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The Heisenberg uncertainty relation, ΔxΔpx ⩾ ℏ/2 (ℏ ≡ h/2π), is well known. The purpose of this paper is to study a linkage between macroscopic thermodynamic quantities and the uncertainty relation. To simplify matters, we consider one-dimensional harmonic oscillator model. The calculations are carried out by two steps. First, the thermal average of the uncertainty relation is obtained for canonical system in thermal equilibrium at temperature T, which is denoted by (Δx)T(Δpx)T. Second, the thermodynamic quantities are expressed analytically as a function of (Δx)T(Δpx)T within the harmonic oscillator model. Finally, a connection between the energy fluctuations and the uncertainty relation is clarified for the Einstein’s heat capacity model. The analysis is made on the basis of fluctuation theory. The uncertainty relation is valid not only in quantum state but also at finite temperature.
Abstract It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we include such interactions and, using quantum estimation theory, derive a generalised thermodynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system’s state and its effective energy operator gives rise to quantum fluctuations that increase the temperature uncertainty. Surprisingly, these additional fluctuations are described by the average Wigner-Yanase-Dyson skew information. We demonstrate that the temperature’s signal-to-noise ratio is constrained by the heat capacity plus a dissipative term arising from the non-negligible interactions. These findings shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics and will inform the design of optimal nanoscale thermometers.
It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we include such interactions and, using quantum estimation theory, derive a generalised thermodynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system's state and its effective energy operator gives rise to quantum fluctuations that increase the temperature uncertainty. Surprisingly, these additional fluctuations are described by the average Wigner-Yanase-Dyson skew information. We demonstrate that the temperature's signal-to-noise ratio is constrained by the heat capacity plus a dissipative term arising from the non-negligible interactions. These findings shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics and will inform the design of optimal nanoscale thermometers.
The paper addresses physics of thermodynamic fluctuations in temperature and energy. These fluctuations are interrelated and, hence, can affect various micro- and macro systems. It is shown that the thermodynamic uncertainty relation must be taken into account in the physics of superconductivity, in quantum computations and other branches of science, where temperature and energy fluctuations play a critical role. One of the most important applications of quantum thermodynamics is quantum computers. It is assumed that in the near future the state structures will create a specific quantum cryptocurrency obtained using quantum computing. The quantum cryptocurrency exhibits two main features: the maximum reliability (quantum protection against hacking threats) and the possibility of state control (at the moment, only large scientific state centers have quantum computers). The paper reviews the studies aimed to theoretically prove the validity of the thermodynamic uncertainty relation. This relation connects fluctuations in temperature and energy of a system. Other similar relations are considered, including the relationship between fluctuations in pressure and volume, in entropy and temperature, and others. The main purpose of the paper is to validate the thermodynamic analogue of the uncertainty relation that interconnects temperature and energy fluctuations. Experimental data was obtained on the basis of the study of the transport properties of semiconductor devices – transistors. In the experiment, the transport properties of a pair of semiconductor transistors placed on a single silicon crystal were studied. In this system, one transistor was used to determine temperature fluctuations, and the other one was employed to estimate energy fluctuations. The key role of the thermodynamic uncertainty relation in modern thermodynamics has been clarified. The performed experimental studies confirm the validity of the thermodynamic uncertainty relation.
Temperature uncertainty of a quantum system in canonical ensemble is inversely determined by its energy fluctuation, which is known as the temperature-energy uncertainty relation. No such uncertainty relation was discovered for a non-equilibrium open quantum system. In this article, we derive a universal temperature uncertainty relation for general non-equilibrium processes. We find that it is the fluctuation of heat, which is defined as the change in bath energy, determines the temperature uncertainty in non-equilibrium case. Specifically, the heat is divided into trajectory heat and backaction heat, which are associated with the system's trajectory of evolution and the backaction of partial measurement on system, respectively. Based on this decomposition, we reveal that both correlations between system and bath's process function and state function are the resources for enhancing temperature precision. Our findings are conductive to design ultrahigh sensitive quantum thermometer.
The solutions for a combination of the isotropic harmonic oscillator plus the inversely quadratic potentials and a combination of the pseudo-harmonic with inversely quadratic potentials has not been reported, though the individual potentials have been given attention. This study focuses on the solutions of the combination of the potentials, as stated above using the parametric Nikiforov–Uvarov (PNV) as the traditional technique to obtain the energy equations and their corresponding unnormalized radial wave functions. To deduce the application of these potentials, the expectation values, the uncertainty in the position and momentum, and the thermodynamic properties, such as the mean energy, entropy, heat capacity, and the free mean energy, are also calculated via the partition function. The result shows that the spectra for the PHIQ are higher than the spectra for the IHOIQ. It is also shown that the product of the uncertainties obeyed the Heisenberg uncertainty relation/principle. Finally, the thermal properties of the two potentials exhibit similar behaviours.
Thermodynamic uncertainty principles make up one of the few rare anchors in the largely uncharted waters of nonequilibrium systems, the fluctuation theorems being the more familiar. In this work we aim to trace the uncertainties of thermodynamic quantities in nonequilibrium systems to their quantum origins, namely, to the quantum uncertainty principles. Our results enable us to make this categorical statement: For Gaussian systems, thermodynamic functions are functionals of the Robertson-Schrödinger uncertainty function, which is always non-negative for quantum systems, but not necessarily so for classical systems. Here, quantum refers to noncommutativity of the canonical operator pairs. From the nonequilibrium free energy, we succeeded in deriving several inequalities between certain thermodynamic quantities. They assume the same forms as those in conventional thermodynamics, but these are nonequilibrium in nature and they hold for all times and at strong coupling. In addition we show that a fluctuation-dissipation inequality exists at all times in the nonequilibrium dynamics of the system. For nonequilibrium systems which relax to an equilibrium state at late times, this fluctuation-dissipation inequality leads to the Robertson-Schrödinger uncertainty principle with the help of the Cauchy-Schwarz inequality. This work provides the microscopic quantum basis to certain important thermodynamic properties of macroscopic nonequilibrium systems.