SciPost Phys. 7, 013 (2019) ·
published 29 July 2019

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We study the dynamics of a quantum Brownian particle weakly coupled to a
thermal bath. Working in the SchwingerKeldysh formalism, we develop an
effective action of the particle up to quartic terms. We demonstrate that this
quartic effective theory is dual to a stochastic dynamics governed by a
nonlinear Langevin equation.
The SchwingerKeldysh effective theory, or the equivalent nonlinear Langevin
dynamics, is insufficient to determine the out of time order correlators
(OTOCs) of the particle. To overcome this limitation, we construct an extended
effective action in a generalised SchwingerKeldysh framework. We determine the
additional quartic couplings in this OTO effective action and show their
dependence on the bath's 4point OTOCs.
We analyse the constraints imposed on the OTO effective theory by microscopic
reversibility and thermality of the bath. We show that these constraints lead
to a generalised fluctuationdissipation relation between the nonGaussianity
in the distribution of the thermal noise experienced by the particle and the
thermal jitter in its damping coefficient.
The quartic effective theory developed in this work provides extension of
several results previously obtained for the cubic OTO dynamics of a Brownian
particle.