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arXiv:1708.00753

Quantum thermodynamic properties of a cold atom coupled to a heat bath in non-Abelian gauge potentials

Asam Rajesh, Malay Bandyopadhyay, A. M. Jayannavar

Published Aug 2, 2017• 0 citations
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Abstract

In this work, we study different quantum thermodynamic functions (QTFs) of a cold atom subjected to an artificial non-Abelian uniform magnetic field and linearly coupled to a quantum heat bath through either usual coordinate-coordinate coupling or through momentum variables. The bath is modelled as a collection of independent quantum harmonic oscillators. In each of the coupling scheme, the effect of the non-Abelian magnetic field on different QTFs are explicitly demonstrated for a U(2) gauge transformation. In each case, we show that the free energy has a different expression than that for the Abelian case. We consider two illustrative heat bath spectrum (Ohmic bath and Drude model) to evaluate explicit closed form expressions of free energy (F), specific heat (C), and entropy (S) in the low temperature limit for each of the above mentioned coupling scheme. The dependence of different QTFs on the non-Abelian magnetic field are pointed out even if the gauge potential is uniform in space.

DOI
10.48550/arxiv.1708.00753

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In this work, we study different quantum thermodynamic functions (QTFs) of a cold atom subjected to an artificial non-Abelian uniform magnetic field and linearly coupled to a quantum heat bath through either usual coordinate-coordinate coupling or through momentum variables. The bath is modelled as a collection of independent quantum harmonic oscillators. In each of the coupling scheme, the effect of the non-Abelian magnetic field on different QTFs are explicitly demonstrated for a U(2) gauge transformation. In each case, we show that the free energy has a different expression than that for the Abelian case. We consider two illustrative heat bath spectrum (Ohmic bath and Drude model) to evaluate explicit closed form expressions of free energy (F), specific heat (C), and entropy (S) in the low temperature limit for each of the above mentioned coupling scheme. The dependence of different QTFs on the non-Abelian magnetic field are pointed out even if the gauge potential is uniform in space.
In this work, we study the effect of non-Abelian gauge potential on the magnetic moment of a cold atom which is linearly coupled to a quantum heat bath through two different coupling schemes: (i) coordinate-coordinate coupling and (ii) momentum-momentum coupling. The bath is modelled as a collection of independent quantum harmonic oscillators. In each coupling scheme, the effect of the non-Abelian magnetic field on magnetic moment are explicitly demonstrated for a U(2) gauge transformation. For both schemes, we show that the magnetic moment has a different expression than that for the Abelian case. We consider two illustrative heat bath spectrum (Ohmic bath and Drude model) to evaluate the magnetic moment in the low temperature limit.
We derive the quantum Langevin equation (QLE) for a harmonically single trapped cold atom subjected to artificial non-Abelian gauge potentials and linearly coupled to a heat bath. The independent-oscillator (IO) and the momentum-momenta coupling models are studied. In each case, the non-Abelian effect on the QLE is pointed out for a U(2) gauge transformation. For the IO model, only the generalized Lorentz force is modified by the appearance of an additive term. For the momentum-momenta coupling model, the generalized Lorentz force as well as the friction force are subjected to modifications. The dependence of the system on the magnetic field is explicit even if the gauge potential is uniform in space.
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 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.
In this work we expose the role of environment, confinement, and external magnetic field B in determining the low-temperature thermodynamic behavior in the context of cyclotron motion of a charged oscillator with anomalous dissipative coupling involving momentum instead of the much studied coordinate coupling. Explicit expressions for different quantum thermodynamic functions (QTFs) are obtained at low temperatures for different quantum heat baths characterized by the spectral density function μ(ω). The power-law fall of different QTFs is in conformity with the third law of thermodynamics; however, the sensitivity of decay, i.e., the power of the power-law decay, explicitly depends on μ(ω). We also discuss separately the influence of confinement and magnetic field on the low-temperature behavior of different QTFs. In this process we demonstrate how to control the low-temperature behavior of anomalous dissipative quantum systems by varying the confining length a, B, and the temperature T. Momentum dissipation reduces the effective mass of the system and we also discuss its effect on different QTFs at low temperatures.
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.