Out of time ordered effective dynamics of a quartic oscillator

Abstract

We study the dynamics of a quantum Brownian particle weakly coupled to a thermal bath. Working in the Schwinger-Keldysh 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 non-linear Langevin equation. The Schwinger-Keldysh effective theory, or the equivalent non-linear 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 Schwinger-Keldysh framework. We determine the additional quartic couplings in this OTO effective action and show their dependence on the bath's 4-point 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 fluctuation-dissipation relation between the non-Gaussianity 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. Contents SciPost PhysicsSubmission [22,[27][28][29]. Nevertheless, it is interesting to explore the corrections to this effective dynamics by including the cubic and higher degree terms in the analysis. A simple way to obtain such corrections is to extend the Caldeira-Leggett model by introducing a non-linear combination of the bath oscillators' positions in the operator that couples to the particle [29,30].One such extension was discussed in [30] where the oscillators in the bath were divided into two sets (labelled by X and Y). Then apart from the bilinear interactions (which are present in the Caldeira-Leggett model), small cubic interactions were introduced between the particle (q) and pairs of bath oscillators. Each such pair consisted of one oscillator drawn from the set X and the other from the set Y. Due to these cubic interactions, this model was called the 'qXY model'.Introduction of these cubic interactions leads to cubic and higher degree terms in the effective action of the particle. In [30], this effective theory was studied up to the cubic terms in a Markovian limit. It was shown to be dual to a non-linear Langevin dynamics which has perturbative corrections over the linear Langevin dynamics in the Caldeira-Leggett model. The additional parameters in this non-linear dynamics receive contributions from the 3-point correlators of the bath. These 3-point correlators lead to an anharmonicity in the particle's dynamics and a non-Gaussianity in the distribution of the thermal noise experienced by the particle. In addition, they also give rise to thermal jitters 1 in both the frequency and the damping of the particle. The thermal jitter in the damping coefficient and the non-Gaussianity in the noise are connected by a generalised fluctuation-dissipation relation [30...

Nonlinear Langevin dynamics via holography

“…These symmetries drastically reduce the number of non-linear corrections to the linear theory that can appear in general. This is in contrast to [9,10], where a non-linear Brownian oscillator moving in a potential was studied. As we will argue below, the above equation then represents the most general non-linear Langevin equation to quartic order in q and upto first order in time derivatives.…”

“…Our goal in this section is to introduce the non-linear Langevin theory for the Brownian particle and its description within Schwinger-Keldysh formalism. The discussion here is based on the results of [9,10] which we will refer the reader for a more detailed description of the models under consideration. The reader will find in those works a detailed justification of how the non-linear Langevin effective theory arises generically out of weakly coupled baths along with examples drawn from simple bath models.…”

“…In the previous subsection, we mentioned that the effects of the particle-bath interaction on the quark's dynamics are encoded in its influence phase. Such influence phases have been studied in detail for several microscopic models where a Brownian particle is weakly coupled to a thermal bath [4,9,10,46,47]. In such scenarios, one can perturbatively expand the influence phase as a power series in the particle-bath coupling.…”

“…Recently, the Onsager-Casimir reciprocal relations were generalised to the cubic and quartic couplings of a Brownian particle in [9] and [10]. These papers demonstrated that under time reversal certain Schwinger-Keldysh correlators of the bath get mapped to its Out of Time Order Correlators (OTOCs) [71].…”

“…These papers demonstrated that under time reversal certain Schwinger-Keldysh correlators of the bath get mapped to its Out of Time Order Correlators (OTOCs) [71]. As argued in these works, a convenient way to study the effects of such relations between the bath's correlators on the particle's dynamics is to extend its effective theory by including the contributions of the bath's OTOCs [10,51]. Such an extension of the effective theory requires the introduction of some new OTO couplings [10,51] which encode information about the bath's OTOCs.…”