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arXiv:2308.07423 · AAPPS bulletin

Fokker-Planck equations for a trapped particle in a quantum-thermal Ohmic bath: general theory and applications to Josephson junctions

Koichiro Furutani, Luca Salasnich

Published Sep 19, 2023• 3 citations
arXivHTMLPDF

Abstract

Abstract We consider a particle trapped by a generic external potential and under the influence of a quantum-thermal Ohmic bath. Starting from the Langevin equation, we derive the corresponding Schwinger-Keldysh action. Then, within the path-integral formalism, we obtain both the semiclassical Fokker-Planck equation and the quantum Fokker-Planck equation for this out-of-equilibrium system. In the case of an external harmonic potential and in the underdamped regime, we find that our Fokker-Planck equations contain an effective temperature $$T_{\text {eff}}$$ T eff , which crucially depends on the interplay between quantum and thermal fluctuations in contrast to the classical Fokker-Planck equation. In the regime of high temperatures, one recovers the classical Fokker-Planck equation. As an application of our result, we also provide the stationary solution of the semiclassical Fokker-Planck equations for a superconducting Josephson circuit and for a Bose Josephson junction, which are experimentally accessible.

DOI
10.1007/s43673-023-00087-2

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Abstract We consider a particle trapped by a generic external potential and under the influence of a quantum-thermal Ohmic bath. Starting from the Langevin equation, we derive the corresponding Schwinger-Keldysh action. Then, within the path-integral formalism, we obtain both the semiclassical Fokker-Planck equation and the quantum Fokker-Planck equation for this out-of-equilibrium system. In the case of an external harmonic potential and in the underdamped regime, we find that our Fokker-Planck equations contain an effective temperature TeffT_{\text {eff}}Teff​ T eff , which crucially depends on the interplay between quantum and thermal fluctuations in contrast to the classical Fokker-Planck equation. In the regime of high temperatures, one recovers the classical Fokker-Planck equation. As an application of our result, we also provide the stationary solution of the semiclassical Fokker-Planck equations for a superconducting Josephson circuit and for a Bose Josephson junction, which are experimentally accessible.
This article has to do with the derivation and solution of the Fokker-Planck equation associated to the momentum-integrated Wigner function of a particle subjected to a harmonic external field in contact with an ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung, and Grabert of interpreting the operators as c numbers to derive the quantum master equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm, and Ingold. By means of a single Wigner transformation, a simpler equation than that mentioned before is found. The Wigner function reproduces the known results of the classical limit. This allowed us to rewrite the underdamped classical Langevin equation as a first-order stochastic differential equation with time-dependent drift and diffusion terms.
Using the quantum effective action in the Schwinger-Keldysh formalism, we derive quantum-thermal corrections to the semiclassical dynamics of a dissipative system described by a macroscopic degree of freedom. We discuss the connection with the Ehrenfest theorem and, within the local potential approximation, obtain the corrections in closed form for arbitrary temperature and damping, showing that as the temperature increases, they cross over from purely quantum to purely thermal. In the low-temperature and weak-damping regime, corrections are set by the zero-point energy of fluctuations evaluated at the classical underdamped frequency, closely paralleling the conservative case, which allows us to go to higher order in the derivative expansion. We apply these general results to the resistively and capacitively shunted superconducting Josephson junction and to an elongated bosonic junction, where corrections can reach the percent level under realistic conditions.
For a dissipative quantum system in the low-damping regime, we show that the master-equation description of the reduced density matrix reduces, in the semiclassical limit, to the energy-diffusion version of the Fokker-Planck equation. This result can be generalized to include non-Ohmic damping and an external harmonic perturbation. Results of calculations based on the master equation show good agreement with experiments on the resistanceless-to-resistive transition in Josephson junctions in the classical regime and with results of purely classical calculations.
This article has to do with the derivation and solution of the Fokker--Planck equation associated to the momentum-independent Wigner function, of a particle subjected to a harmonic external field and immersed in a ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung and Grabert of interpreting the operators as c-numbers to derive the adjoint equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm and Ingold. By means of a single Wigner transformation, it is found a simpler master quantum equation than that of mentioned before. The Wigner function reproduces the known results of the classical limit. This allowed to rewrite the underdamped classical Langevin equation as a first order stochastic differential equation with time-dependent drift and diffusion terms.
We develop a generalized quantum Fokker-Planck theory in a non-Gaussian-Markovian model bath. The semiclassical bath adopted in this work is charactered by three parameters. One denotes the strength of system-bath coupling and the other two are chosen to interpolate smoothly the solvation dynamics between the long- and short-time regimes. The fluctuation-dissipation relation in this model bath is analyzed in detail. Based on this model bath, we derive two sets of coupled Fokker-Planck equations. These two equation sets are equivalent in the second order of system-bath coupling but different in the higher orders. The corresponding reduced Liouville equation in one set of the Fokker-Planck formulation is characterized by a memory relaxation kernel, while that in the other is by a local-time relaxation tensor. Each resulting set of Fokker-Planck equations involves only the reduced density operator and a series of well-characterized Hilbert-space relaxation operators. The present theory is valid for arbitrary time-dependent Hamiltonians and is applicable to the study of quantum coherence and relaxation in various dynamic systems.
This chapter is devoted to the classical limit of the quantum dissipative action obtained in Chapter 3. We show how it yields Langevin, Fokker–Planck and optimal path descriptions of classical stochastic systems. These approaches are used to discuss activation escape, fluctuation relation, reaction models and other examples. Classical dissipative action In Section 3.2 we derived the Keldysh action for a quantum particle coupled to an Ohmic environment, Eq. (3.20). If only linear terms in the quantum coordinate X q (t) are kept in this action, it leads to a classical Newtonian equation with a viscous friction force, Eq. (3.21). Such an approximation completely disregards any fluctuations: both quantum and classical . Our goal now is to do better than that and to keep classical thermal fluctuations, while still neglecting quantum effects. To this end it is convenient to restore the Planck constant ħ in the action and then take the limit ħ → 0. For dimensional reasons, the factor ħ -1 should stay in front of the entire action. To keep the part of the action responsible for the classical equation of motion (3.21) free from the Planck constant it is convenient to rescale the quantum component as X q → ħ X q . Indeed, when this is done all terms linear in X q do not contain ħ. Finally, to have the temperature in energy units, one needs to substitute T with T /ħ.