Heat transport along a chain of coupled quantum harmonic oscillators

Abstract

We study the heat transport properties of a chain of coupled quantum harmonic oscillators in contact at its ends with two heat reservoirs at distinct temperatures. Our approach is based on the use of an evolution equation for the density operator which is a canonical quantization of the classical Fokker-Planck-Kramers equation. We set up the evolution equation for the covariances and obtain the stationary covariances at the stationary states from which we determine the thermal conductance in closed form when the interparticle interaction is small. The conductance is finite in the thermodynamic limit implying an infinite thermal conductivity.Fifty years ago, Rieder, Lebowitz, and Lieb [1] introduced and exactly solved a microscopic model for thermal conduction, that consisted of a chain of coupled classic harmonic oscillators with its ends in contact with heat reservoirs at distinct temperatures. Using this model they provided a rigorous proof of the well known result that the thermal conductance, the ratio between the heat current and the temperature difference, is finite regardless of the chain length [2]. This result amounts to say that Fourier's law does not hold because the conductivity, which is the product of the conductance and the chain length, becomes infinite when the length increases without bounds. The reason for the occurrence of a finite conductance is that the excitations in ordered systems with harmonic interactions travel balistically. To get the Fourier's law, new ingredients should be added to the harmonic model in order to transform the ballistic into a diffusive motion. Such ingredients include anharmonic potentials [3][4][5][6][7][8], self-consistent reservoirs [9][10][11][12], energy conserving noise [13][14][15], and others [16][17][18][19][20].In the model studied by Rieder, Lebowitz, and Lieb [1], the oscillators were under the action of conservative forces except the first and the last which in addition were subject to dissipating-fluctuating forces representing the contact with heat reservoirs. They are composed by a dissipative force, proportional to the velocity, and a Gaussian white noise with zero mean and variance proportional to the temperature. The equations of motion are understood as Langevin equations, and the equation governing the time evolution of the probability density is a Fokker-Planck-Kramers (FPK) equation [21][22][23][24].Here, we study a quantum version of the model studied by Rieder, Lebowit, and Lieb [1]. We have exactly calculated the thermal conductance in the regime of small interparticle interaction and reached a similar result that the conductance is finite regardless of the length of the chain. However, as should be expected the conductance is not independent of temperature, as is the case of the classical version. It vanishes in the limit of zero temperature and saturates at the classical value at high temperatures. Our approach is based on a quantum version of the FPK equation coming from a canonical quantization of the ordinary FPK equation, recently in...

Quantum Langevin equation

“…Finally, we remark that the quantum Fokker-Planck-Kramers equation derived here from the Langevin equation is very similar to that obtained by the canonical quanti zation of the classical Fokker-Planck-Kramers equation [22], which has been applied to the calculation of the transport properties of a chain of coupled harmonic oscillators and of a bosonic chain [23,24].…”

Quantum Langevin equation

“…Finally, we remark that the quantum Fokker-Planck-Kramers equation derived here from the Langevin equation is very similar to that obtained by the canonical quantization of the classical Fokker-Planck-Kramers equation [22], which has been applied to the calculation of the transport properties of a chain of coupled harmonic oscillators and of a bosonic chain [23,24].…”

Heat flow and noncommutative quantum mechanics in phase-space

“…Furthermore, supporting the thermal interaction between two general systems, there is a basic statement claiming that for initially uncorrelated systems heat naturally flows from the hotter to the colder system, well know how Clausius statement [23]. Another particularly interesting scenario in which the heat flow has been addressed consist in a chain of coupled harmonic oscillators where the first and last are in contact with thermal reservoirs in distinct temperatures [24][25][26], in cases of several thermal reservoirs in linear quantum lattices [27], and in the presence of time-dependent periodic drivings [28].…”