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Applied Physics Letters

Quantum description of the resistor–capacitor circuit and Brownian motion

Wayne H. Richardson

Published Jan 21, 2002• 2 citations
Publisher

Abstract

An equation for the reduced density matrix which describes a capacitor, that is coupled to a resistor, is derived using the total Hamiltonian, and without resorting to any artificial model. A Master equation is also obtained for the dissipative medium. The theory erases the notion of the reservoir. It is shown that the dynamical interaction with the medium is not completely determined by the resistance (or the corresponding fluctuation–dissipation relation). Unexpectedly, through the charge density–density correlation, the longitudinal dielectric function is shown to play a critical role.

DOI
10.1063/1.1432761

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An equation for the reduced density matrix which describes a capacitor, that is coupled to a resistor, is derived using the total Hamiltonian, and without resorting to any artificial model. A Master equation is also obtained for the dissipative medium. The theory erases the notion of the reservoir. It is shown that the dynamical interaction with the medium is not completely determined by the resistance (or the corresponding fluctuation–dissipation relation). Unexpectedly, through the charge density–density correlation, the longitudinal dielectric function is shown to play a critical role.
An equation for the reduced density matrix which describes a free particle, that is interacting with a linearly dissipative medium, is derived using the total Hamiltonian, and without resorting to any artificial model. A Master equation is also obtained for the dissipative medium. The theory erases the notion of the reservoir. It is shown that the dynamical interaction with the medium is not completely determined by the friction constant (or the corresponding fluctuation-dissipation relation). Unexpectedly, through the density-density correlation, the longitudinal dielectric function is shown to play a critical role.
Recently, the kinetic equation for the reduced density matrix that describes a capacitor coupled to a resistor was derived from a fundamental hamiltonian. In this work, a solution of the Markoffian Master equation that describes the capacitor is presented. A preliminary look at a solution that takes into account the nonMarkoffian terms is also included. The total solution reveals for the first time the quantum dynamical features of the RC circuit and the results are dramatic. The solution for the quasi probability distribution shows that even at zero temperature, the expectation value of the square of the phase, varies linearly with time, for times longer than the characteristic time constant of the coupled system. This yields a finite value for the diffusion constant at zero temperature. Using the formula it is now possible to answer a question that has been asked by researchers since 1905: is there a relationship between diffusion and mobility at absolute zero temperature.
In Chap. 16 we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the reduced density matrix concept and worked in an interaction picture. In this chapter we also consider the system-reservoir problem, but work with quantum oper­ ator equations of motion (see Sec. 7-4). Here too we eliminate the reservoir variables and find that the operator equations of motion for the system ac­ quire damping terms and “noise” operators which produce fluctuations. The resulting equations resemble the classical Langevin equations which describe, for example, the Brownian motion of a particle suspended in a liquid or the behavior of a current as it passes through a resistor.
A non-Markovian master equation is derived for the reduced probability density matrix of a subsystem interacting with a general reservoir of coupled anharmonic modes. Relaxation of the subsystem is mediated through resonant interactions with the reservoir. These interactions correspond to local vibrational motions with finite lifetimes due to diffusion/dephasing. The derivation assumes that the density of these interaction modes is very large so that they can be treated using statistical methods. The resulting master equation is shown to preserve probability, Hermiticity, and translational invariance, and conditions are stated which guarantee the positivity of the reduced density. At long time the master equation reduces to a Markovian equation of dynamical semigroup type. In the high temperature limit the theory is parameter free, and shows good agreement with the exact master equation of a spin-boson system.
A fully quantum treatment of Einstein's Brownian motion is given, stressing in particular the role played by the two original requirements of translational invariance and connection between dynamics of the Brownian particle and atomic nature of the medium. The former leads to a clearcut relationship with a generator of translation-covariant quantum-dynamical semi-groups recently characterized by Holevo, the latter to a formulation of the fluctuation-dissipation theorem in terms of the dynamic structure factor, a two-point correlation function introduced in seminal work by Van Hove, directly related to density fluctuations in the medium and therefore to its atomistic, discrete nature. A microphysical expression for the generally temperature dependent friction coefficient is given in terms of the dynamic structure factor and of the interaction potential describing the single collisions. A comparison with the Caldeira-Leggett model is drawn, especially in view of the requirement of translational invariance, further characterizing general structures of reduced dynamics arising in the presence of symmetry under translations.
Quantum mechanical models of resistors in quantum electronics are often based on the quantum optical master equation (QOME). The QOME overlooks several fundamental properties, limiting its ability to model certain superconducting phenomena. Here we present a thermometric model of resistors, adapted from the quantum Brownian motion equation (QBME), that facilitates practical use of the QBME in modelling dissipative electronics at low temperatures. We compare the thermometric QBME presented here with predictions of the QOME, in the simple example of a transmon shunted by a resistor. We show that both the QBME and QOME yield comparable but physically distinct predictions, and discuss potential experimental tests with which to discriminate their use in modelling practical experiments.