Abstract
We consider a quantum Brownian particle interacting with two harmonic baths, which is then perturbed by a cubic coupling linking the particle and the baths. This cubic coupling induces non-linear dissipation and noise terms in the influence functional/master equation of the particle. Its effect on the Out-of-Time-Ordered Correlators (OTOCs) of the particle cannot be captured by the conventional Feynman-Vernon formalism.We derive the generalised influence functional which correctly encodes the physics of OTO fluctuations, response, dissipation and decoherence. We examine an example where Markovian approximation is valid for the OTO dynamics.If the original cubic coupling has a definite time-reversal parity, the leading order OTO influence functional is completely determined by the couplings in the usual master equation via OTO generalisation of Onsager-Casimir relations. New OTO fluctuationdissipation relations connect the non-Gaussianity of the thermal noise to the thermal jitter in the damping constant of the Brownian particle.
Highlights
Introduction to the qXY modelwe will begin by describing a microscopic model of an oscillator coupled to bath oscillator degrees of freedom in a way that results in an effective non-linear Langevin equation for the original oscillator
Its effect on the Out-of-Time-Ordered Correlators (OTOCs) of the particle cannot be captured by the conventional Feynman-Vernon formalism.We derive the generalised influence functional which correctly encodes the physics of OTO fluctuations, response, dissipation and decoherence
We have constructed an effective theory of a Brownian particle which goes beyond the standard Langevin dynamics
Summary
The dynamics of a Brownian particle interacting with a thermal bath is a topic that has been studied for over a hundred years. An effective theory was derived for a quantum Brownian particle by tracing out the thermal bath’s degrees of freedom These analyses were later extended by Caldeira and Leggett [3] to a concrete model of a particle linearly coupled to a harmonic bath. At sufficiently long time-scales, one obtains a local effective theory for the particle which in classical limit reduces to the standard Langevin dynamics This model of Caldeira-Leggett and its generalisations [4, 5] (see [6,7,8] for textbook level discussion) have been crucial in understanding dissipation and decoherence in quantum systems. This is followed by a detailed description of a non-linear generalisation of Langevin theory, where we summarise our main results on such non-linear corrections for a general bath. This is followed by appendix C, where we have summarised various contour integrals that are useful in computation of the effective Langevin couplings from our microscopic model
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