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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Dissipative effects on the localization of a charged oscillator in a magnetic field

X. L. Li, G. W. Ford, Robert F. O’Connell

Published Apr 1, 1996• 23 citations
Publisher

Abstract

We consider the effect of dissipation on a charged, quantum harmonic oscillator coupled to a heat bath, in the presence of an external magnetic field. General conclusions about the equal-time position autocorrelation functions can be reached by using only those properties of the generalized susceptibility tensor imposed by fundamental physical principles. Explicit calculations are made for an Ohmic heat bath at zero temperature, enabling us to show that (unlike the problem without the magnetic field where dissipation always leads to enhanced localization) when the magnetic field is stronger than a certain critical value, weak dissipation actually delocalizes the oscillation of the charged particle.

DOI
10.1103/physreve.53.3359

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Charged oscillator in a heat bath in the presence of a magnetic field

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We obtain the generalized susceptibility for the motion of a charged oscillator in a harmonic potential well in the presence of a uniform external magnetic field, and linearly coupled to a heat bath, by using the generalized quantum Langevin equation we obtained in an earlier paper. The result is then used to calculate the autocorrelation function of the position operator of the oscillator by the fluctuation-dissipation theorem. The free energy of the oscillator is then obtained in terms of the determinant of the susceptibility matrix. Applications are made to two types of heat baths: the Ohmic case and the case of blackbody radiation. As a special case (no heat bath) of our general formalism, a well-known eigenspectrum result is obtained but in a simple and rather novel fashion.

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Magnetic-field effects on the motion of a charged particle in a heat bath

X. L. Li, G. W. Ford, Robert F. O’Connell

In a recent paper, Ford, Lewis, and O'Connell [Phys. Rev. A 37, 4419 (1988)] considered a charged quantum particle moving in an arbitrary potential and linearly coupled to a heat bath, and they showed that the macroscopic equation describing the time development of the particle motion is in the form of a quantum generalized Langevin equation. We generalize these results to include the presence of an external magnetic field. We find that the magnetic field manifests itself in the presence of an additional term in the Langevin equation, which is the quantum generalization of the Lorentz force, but the magnetic field does not affect the memory function nor the random force appearing in the quantum Langevin equation. It follows that the noise-noise autocorrelation function, as well as the nonequal time commutator of the noise, is the same as that in the absence of a magnetic field. The case of a blackbody radiation heat bath is shown to be easily analyzed as a special case of our general formalism.

Physical Review A•May 1, 1990•48 citations
We consider the effect of dissipation on a charged, quantum harmonic oscillator coupled to a heat bath, in the presence of an external magnetic field. General conclusions about the equal-time position autocorrelation functions can be reached by using only those properties of the generalized susceptibility tensor imposed by fundamental physical principles. Explicit calculations are made for an Ohmic heat bath at zero temperature, enabling us to show that (unlike the problem without the magnetic field where dissipation always leads to enhanced localization) when the magnetic field is stronger than a certain critical value, weak dissipation actually delocalizes the oscillation of the charged particle.
We obtain the generalized susceptibility for the motion of a charged oscillator in a harmonic potential well in the presence of a uniform external magnetic field, and linearly coupled to a heat bath, by using the generalized quantum Langevin equation we obtained in an earlier paper. The result is then used to calculate the autocorrelation function of the position operator of the oscillator by the fluctuation-dissipation theorem. The free energy of the oscillator is then obtained in terms of the determinant of the susceptibility matrix. Applications are made to two types of heat baths: the Ohmic case and the case of blackbody radiation. As a special case (no heat bath) of our general formalism, a well-known eigenspectrum result is obtained but in a simple and rather novel fashion.
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.
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 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 $\ensuremath{\langle}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\rangle}$, is over the frequencies $\ensuremath{\omega}$ of the bath oscillators which contribute to Ek{E}_{k}Ek​ and Ep{E}_{p}Ep​ 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.
The quantum thermodynamic behaviour of a charged oscillator in the presence of a magnetic field and coupled to a heat bath through different coupling schemes is obtained analytically. It is shown that finite dissipation substitutes the zero-coupling result of exponential decay of entropy by a power law behaviour at low temperature. For the coordinate-coordinates coupling scheme the low temperature explicit results for the case of Ohmic, exponentially correlated and more generalized heat bath models are derived. In all the above mentioned cases free energy and entropy vanish linearly with temperature (TTT) as T→0T\to 0T→0 in conformity with Nernst's theorem. It is seen that coordinate (velocity)-velocities (coordinates) coupling is much more beneficial than the coordinate-coordinates coupling to ensure third law of thermodynamics. The case of radiation heat bath shows T3T^3T3 decay behaviour for entropy as T→0T\to 0T→0. It is observed that at low temperature free energy and entropy decay faster for the velocity-velocities scheme than any other coupling schemes. This implies velocity-velocities coupling scheme is the most advantageous coupling scheme in restoring the third law of thermodynamics. It is shown that the low temperature thermodynamic functions are independent of magnetic field for all the above mentioned cases except the without dissipation case.
We present a comprehensive study of the effect of a transverse magnetic field on the macroscopic quantum tunneling of two identical coupled particles with dissipation. Two coupled particles of identical masses but opposite charges in the plane, in the presence of a constant transverse external magnetic field and an external potential interacting with a bath of harmonic oscillators are studied. In this case, the problem cannot be mapped to a one-dimensional problem, it strictly remains two-dimensional. We calculate the exact effective action both for the case of linear coupling to the bath and without a linear coupling to the bath using imaginary time path integral and Leggett's prescription at T=0. In the limit of zero magnetic field we recover a two dimensional version of the result found in \cite{em1} for the case of two identical particles. We find that in the limit of weak and strong dissipation, the effective action reduces to a two dimensional version of the Caldeira-Leggett form in terms of the reduced mass and magnetic field. The case of Ohmic dissipation with the motion of the two particles damped by the Ohmic frictional constant η\etaη is studied in detail.
In a recent paper, Ford, Lewis, and O'Connell [Phys. Rev. A 37, 4419 (1988)] considered a charged quantum particle moving in an arbitrary potential and linearly coupled to a heat bath, and they showed that the macroscopic equation describing the time development of the particle motion is in the form of a quantum generalized Langevin equation. We generalize these results to include the presence of an external magnetic field. We find that the magnetic field manifests itself in the presence of an additional term in the Langevin equation, which is the quantum generalization of the Lorentz force, but the magnetic field does not affect the memory function nor the random force appearing in the quantum Langevin equation. It follows that the noise-noise autocorrelation function, as well as the nonequal time commutator of the noise, is the same as that in the absence of a magnetic field. The case of a blackbody radiation heat bath is shown to be easily analyzed as a special case of our general formalism.