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arXiv:2608.26646

A thermometric quantum Brownian model for low-temperature electronics

Harry T. H. Fung, Thomas M. Stace

quant-phSubmitted Aug 27, 2026• Announced Aug 28, 2026• 0 citations
v1 · Aug 27, 2026v2 · Aug 28, 2026
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Abstract

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.

Comments
11 pages, 9 figures

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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.
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.
Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.
We analyze quantal Brownian motion in ddd dimensions using the unified model for diffusion localization and dissipation, and Feynman-Vernon formalism. At high temperatures the propagator possess a Markovian property and we can write down an equivalent Master equation. Unlike the case of the Zwanzig-Caldeira-Leggett model, genuine quantum mechanical effects manifest themselves due to the disordered nature of the environment. Using Wigner picture of the dynamics we distinguish between two different mechanisms for destruction of coherence. The analysis of dephasing is extended to the low temperature regime by using a semiclassical strategy. Various results are derived for ballistic, chaotic, diffusive, both ergodic and non-ergodic motion. We also analyze loss of coherence at the limit of zero temperature and clarify the limitations of the semiclassical approach. The condition for having coherent effect due to scattering by low-frequency fluctuations is also pointed out. It is interesting that the dephasing rate can be either larger or smaller than the dissipation rate, depending on the physical circumstances.
We revisit the model of a system made up of a Brownian quantum oscillator linearly coupled to an environment made up of many quantum oscillators at finite temperature. We show that the HPZ master equation for the reduced density matrix derived earlier [B.L. Hu, J.P. Paz, Y. Zhang, Phys. Rev. D 45, 2843 (1992)] has incorrectly specified coefficients for the case of nonlocal dissipation. We rederive the QBM master equation, correctly specifying all coefficients, and determine the position uncertainty to be free of excessive cutoff sensitivity. Our coefficients and solutions are reduced entirely to contour integration for analytic spectra at arbitrary temperature, coupling strength, and cut-off. As an illustration we calculate the master equation coefficients and solve the master equation for ohmic coupling (with finite cutoff) and example supra-ohmic and sub-ohmic spectral densities. We determine the effect of an external force on the quantum oscillator and also show that our representation of the master equation and solutions naturally extends to a system of multiple oscillators bilinearly coupled to themselves and the bath in arbitrary fashion. This produces a formula for investigating the standard quantum limit which is central to addressing many theoretical issues in macroscopic quantum phenomena and experimental concerns related to low temperature precision measurements. We find that in a dissipative environment, all initial states settle down to a Gaussian density matrix whose covariance is determined by the thermal reservoir and whose mean is determined by the external force. We specify the thermal covariance for the spectral densities we explore.