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arXiv:2106.07509 · Physical review. E

Quantum counterpart of energy equipartition theorem for a dissipative charged magneto-oscillator: Effect of dissipation, memory, and magnetic field

Jasleen Kaur, Aritra Ghosh, Malay Bandyopadhyay

Published Dec 10, 2021• 16 citations
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Abstract

In this paper, we formulate and study the quantum counterpart of the energy equipartition theorem for a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a passive quantum heat bath through coordinate variables. The bath is modeled as a collection of independent quantum harmonic oscillators. We derive closed form expressions for the mean kinetic and potential energies of the charged dissipative magneto-oscillator in the forms ${E}_{k}=\ensuremath{\langle}{\mathcal{E}}_{k}\ensuremath{\rangle}$ and ${E}_{p}=\ensuremath{\langle}{\mathcal{E}}_{p}\ensuremath{\rangle}$, respectively, where ${\mathcal{E}}_{k}$ and ${\mathcal{E}}_{p}$ denote the average kinetic and potential energies of individual thermostat oscillators. The net averaging is twofold; the first one is over the Gibbs canonical state for the thermostat, giving ${\mathcal{E}}_{k}$ and ${\mathcal{E}}_{p}$, and the second one, denoted by $\ensuremath{\langle}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\rangle}$, is over the frequencies $\ensuremath{\omega}$ of the bath oscillators which contribute to ${E}_{k}$ and ${E}_{p}$ according to probability distributions ${\mathcal{P}}_{k}(\ensuremath{\omega})$ and ${\mathcal{P}}_{p}(\ensuremath{\omega})$, respectively. The relationship of the present quantum version of the equipartition theorem with that of the fluctuation-dissipation theorem (within the linear-response theory framework) is also explored. Further, we investigate the influence of the external magnetic field and the effect of different dissipation processes through Gaussian decay and Drude and radiation bath spectral density functions on the typical properties of ${\mathcal{P}}_{k}(\ensuremath{\omega})$ and ${\mathcal{P}}_{p}(\ensuremath{\omega})$. Finally, the role of system-bath coupling strength and the memory effect is analyzed in the context of average kinetic and potential energies of the dissipative charged magneto-oscillator.

DOI
10.1103/physreve.104.064112

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In this paper, we formulate and study the quantum counterpart of the energy equipartition theorem for a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a passive quantum heat bath through coordinate variables. The bath is modeled as a collection of independent quantum harmonic oscillators. We derive closed form expressions for the mean kinetic and potential energies of the charged dissipative magneto-oscillator in the forms Ek=\ensuremath⟨Ek\ensuremath⟩{E}_{k}=\ensuremath{\langle}{\mathcal{E}}_{k}\ensuremath{\rangle}Ek​=\ensuremath⟨Ek​\ensuremath⟩ and Ep=\ensuremath⟨Ep\ensuremath⟩{E}_{p}=\ensuremath{\langle}{\mathcal{E}}_{p}\ensuremath{\rangle}Ep​=\ensuremath⟨Ep​\ensuremath⟩, respectively, where Ek{\mathcal{E}}_{k}Ek​ and Ep{\mathcal{E}}_{p}Ep​ denote the average kinetic and potential energies of individual thermostat oscillators. The net averaging is twofold; the first one is over the Gibbs canonical state for the thermostat, giving Ek{\mathcal{E}}_{k}Ek​ and Ep{\mathcal{E}}_{p}Ep​, and the second one, denoted by \ensuremath⟨\ifmmode⋅\else\textperiodcentered\fi\ensuremath⟩\ensuremath{\langle}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\rangle}\ensuremath⟨\ifmmode⋅\else\textperiodcentered\fi\ensuremath⟩, is over the frequencies \ensuremathω\ensuremath{\omega}\ensuremathω of the bath oscillators which contribute to Ek{E}_{k}Ek​ and Ep{E}_{p}Ep​ according to probability distributions Pk(\ensuremathω){\mathcal{P}}_{k}(\ensuremath{\omega})Pk​(\ensuremathω) and Pp(\ensuremathω){\mathcal{P}}_{p}(\ensuremath{\omega})Pp​(\ensuremathω), respectively. The relationship of the present quantum version of the equipartition theorem with that of the fluctuation-dissipation theorem (within the linear-response theory framework) is also explored. Further, we investigate the influence of the external magnetic field and the effect of different dissipation processes through Gaussian decay and Drude and radiation bath spectral density functions on the typical properties of Pk(\ensuremathω){\mathcal{P}}_{k}(\ensuremath{\omega})Pk​(\ensuremathω) and Pp(\ensuremathω){\mathcal{P}}_{p}(\ensuremath{\omega})Pp​(\ensuremathω). Finally, the role of system-bath coupling strength and the memory effect is analyzed in the context of average kinetic and potential energies of the dissipative charged magneto-oscillator.
Abstract We reveal a new face of the old clichéd system: a dissipative quantum harmonic oscillator. We formulate and study a quantum counterpart of the energy equipartition theorem satisfied for classical systems. Both mean kinetic energy Ek and mean potential energy Ep of the oscillator are expressed as Ek = 〈εk〉 and Ep = 〈εp〉, where 〈εk〉 and 〈εp〉 are mean kinetic and potential energies per one degree of freedom of the thermostat which consists of harmonic oscillators too. The symbol 〈...〉 denotes two-fold averaging: (i) over the Gibbs canonical state for the thermostat and (ii) over thermostat oscillators frequencies ω which contribute to Ek and Ep according to the probability distribution Pk(ω){{\mathbb{P}}}_{k}(\omega )Pk​(ω) ℙ k ( ω ) and Pp(ω){{\mathbb{P}}}_{p}(\omega )Pp​(ω) ℙ p ( ω ) , respectively. The role of the system-thermostat coupling strength and the memory time is analysed for the exponentially decaying memory function (Drude dissipation mechanism) and the algebraically decaying damping kernel.
We obtain the quantum Langevin equation (QLE) of a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a quantum heat bath through momentum variables. The bath is modeled as a collection of independent quantum harmonic oscillators. The QLE involves a random force which does not depend on the magnetic field, and a quantum-generalized classical Lorentz force. These features are also present in the QLE for the case of particle-bath coupling through coordinate variables. However, significant differences are also observed. For example, the mean force in the QLE is characterized by a memory function that depends explicitly on the magnetic field. The random force has a modified form with correlation and commutator different from those in the case of coordinate-coordinate coupling. Moreover, the coupling constants, in addition to appearing in the random force and in the mean force, also renormalize the inertial term and the harmonic potential term in the QLE.
We investigate the quantum dissipative dynamics of a charged magneto-oscillator(CMO) coupled to a heat bath of harmonic oscillators in the presence of an additional field-bath interaction. The problem is formulated within the framework of the generalized quantum Langevin equation, where the external confining potential modifies the random force correlations and the memory kernel of the environment, and the effective spring constant. Using the corresponding non-Markovian master equation, we examine the influence of the field-bath coupling on the temporal decay of the reduced density matrix of the system. We show that the additional interaction leads to an enhancement of the decoherence rate by altering the dissipative response of the bath. Thus our results shed light on the role of field-bath coupling in the quantum to classical transition of a CMO. The resulting fluctuation-dissipation relation is analyzed, and the position autocorrelation, position-velocity correlation and velocity autocorrelation are obtained explicitly. The dependence of these experimentally accessible quantities on the field-bath coupling parameter is discussed. We outline an experimental proposal for testing our theoretical predictions. The study sheds light on reservoir engineering which is central to quantum technologies.
We study the quantum counterpart of the theorem on energy equipartition for classical systems. We consider a free quantum Brownian particle modeled in terms of the Caldeira-Leggett framework: a system plus thermostat consisting of an infinite number of harmonic oscillators. By virtue of the theorem on the averaged kinetic energy Ek{E}_{k}Ek​ of the quantum particle, it is expressed as ${E}_{k}=\ensuremath{\langle}{\mathcal{E}}_{k}\ensuremath{\rangle}$, where Ek{\mathcal{E}}_{k}Ek​ is the thermal kinetic energy of the thermostat per 1 degree of freedom and $\ensuremath{\langle}\ensuremath{\cdots}\ensuremath{\rangle}$ denotes averaging over the frequencies $\ensuremath{\omega}$ of thermostat oscillators which contribute to Ek{E}_{k}Ek​ according to the probability distribution $\mathbb{P}(\ensuremath{\omega})$. We explore the impact of various dissipation mechanisms, via the Drude, Gaussian, algebraic, and Debye spectral density functions, on the characteristic features of $\mathbb{P}(\ensuremath{\omega})$. The role of the system-thermostat coupling strength and the memory time on the most probable thermostat oscillator frequency as well as the kinetic energy Ek{E}_{k}Ek​ of the Brownian particle is analyzed.
One of the fundamental laws of classical statistical physics is the energy equipartition theorem which states that for each degree of freedom the average kinetic energy equals Ek=kBT/2E_k=k_B T/2Ek​=kB​T/2, where kBk_BkB​ is the Boltzmann constant and TTT is temperature of the system. Despite the fact that quantum mechanics has already been developed for more than 100 years still there is no quantum counterpart of this theorem. We attempt to fill this far-reaching gap and formulate the quantum law for equipartition of energy in the appealing form Ek=⟨Ek⟩E_k = \langle \mathcal E_k \rangleEk​=⟨Ek​⟩, where Ek\mathcal E_kEk​ is thermal kinetic energy per one degree of freedom of the thermostat consisting of harmonic oscillators and ⟨...⟩\langle ...\rangle⟨...⟩ denotes averaging over frequencies ω\omegaω of those thermostat oscillators which contribute to EkE_kEk​ according to the probability distribution P(ω)\mathbb P(\omega)P(ω). We derive it for two paradigmatic and exactly solvable models of quantum open systems: a free Brownian particle and a harmonic oscillator. We formulate conditions for validity of the relation Ek=⟨Ek⟩E_k = \langle \mathcal E_k \rangleEk​=⟨Ek​⟩ for other quantum systems.
Abstract One of the fundamental laws of classical statistical physics is the energy equipartition theorem which states that for each degree of freedom the mean kinetic energy E k equals , where is the Boltzmann constant and T is the temperature of the system. Despite the fact that quantum mechanics has already been developed for more than 100 years, still there is no quantum counterpart of this theorem. We attempt to fill this far-reaching gap and consider the simplest system, i.e. the Caldeira–Leggett model for a free quantum Brownian particle in contact with a thermostat consisting of an infinite number of harmonic oscillators. We prove that the mean kinetic energy E k of the Brownian particle equals the mean kinetic energy per one degree of freedom of the thermostat oscillators, i.e. . We show that this relation can be obtained from the fluctuation–dissipation theorem derived within the linear response theory and is universal in the sense that it holds true for any linear and non-linear systems in contact with a bosonic thermostat.