Out of Time Ordered Quantum Dissipation

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 fluctuation-dissipation relations connect the non-Gaussianity of the thermal noise to the thermal jitter in the damping constant of the Brownian particle.


Motivation
The dynamics of a Brownian particle interacting with a thermal bath is a topic that has been studied for over a hundred years. A systematic understanding of a quantum Brownian particle emerged in the 1960s with the works of Schwinger [1] and Feynman-Vernon [2]. In these works, 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. The bath degrees of freedom can be exactly integrated out to get a non-local, non-unitary theory describing the evolution of the particle. In this simple model, Caldeira and Leggett managed to show the following interesting result: First, if the distribution of the bath oscillator frequencies is chosen appropriately then the bath correlators decay with time. Consequently, 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.
In this work, we seek to generalise Caldeira-Leggett like models. We extend such models in two directions : one extension is to go beyond the linear Langevin description to non-linear generalisations of Langevin equation. While the non-linear versions have indeed been considered in different contexts before (see references above), the authors of this work are not aware of a systematic classification of all possible terms at the leading level of non-linearity for a Brownian particle coupled to a generic bath. Such a systematic study along with the concomitant generalisation of the famous fluctutation dissipation theorems and Onsager's reciprocal relations is one of our objectives.
The power of Caldeira-Leggett model lies in its ability to relate the effective couplings of a dissipative/open quantum system description to the underlying microscopic physics. At a superficial level, the idea is the one familiar in usual effective theories : one computes correlators in the effective description and in the microscopic description and then matches the two. But, in open quantum systems, the integrated out degrees of freedom are not quite the heavy degrees of freedom. Thus, the integrated degrees of freedom can go onshell resulting in the non-unitarity of the effective description. For this reason, the matching is not merely one of comparing low frequency behaviour.
At high temperatures, a whole band of frequency domain contributes right upto the thermal scale. Further, this contribution is modulated by how effectively the particle couples to the various states of the bath. We will take these aspects into account by writing the effective couplings as frequency integrals over complex contours with appropriate iǫ prescription that picks up the correct causal response. The integrands would be the spectral functions of the bath which encode the effective number of bath states accessible to the particle in a given process. From a mathematical point of view, we will write down expressions that relate the effective couplings of the Langevin description to generalised discontinuities of the appropriate (in general, out of time ordered) bath correlators.
A second related objective is the generalisation of Caldeira-Leggett like models to take into account out of time order correlations(OTOCs) and how they get transmitted from the bath to the Brownian particle. Two of us [9] recently considered the general formalism to tackle the question of how the OTOCs of a probe records information about the OTOCs of a system. This work is motivated by trying to find a concrete realisation of the ideas considered in [9]. The perspective provided by extending the ideas about open systems to include out of time order correlations is crucial to our work. As we shall show explicitly, the fluctuationdissipation theorems and Onsager's reciprocal relations relate parameters governing timeordered correlations to those controlling out of time ordered correlations. The straightforward way to understand the relations between time-ordered couplings is via the relations between out of time-ordered correlations and time-ordered correlations. This is the macroscopic counterpart to the observation that including out of time ordered correlations simplify the structure of thermal correlators in a general quantum field theory [10][11][12].
We illustrate our results about non-linear corrections to Langevin theory by considering a simple extension of Caldeira-Leggett model. Our model consists of a Brownian particle coupled to two sets of oscillators at the same temperature. Apart from the usual linear coupling, we will also turn on weak three-body interactions involving the particle and two other bath oscillators, one drawn from each set. We show that under appropriate distribution of couplings, the bath continues to be Markovian. We derive the non-linear corrections to Langevin equation that result from such an interaction in the Markovian limit and check that these couplings indeed satisfy the correct generalisations of fluctuationdissipation/Onsager reciprocal relations.
In the following sections, we will elaborate on these ideas and summarise our results. We will begin by reviewing the model by Caldeira-Leggett in section §2. This is followed by a detailed description of a non-linear generalisation of Langevin theory in section §3, where we also summarise our main results on such non-linear corrections for a general bath. This is followed by a description of the particular model we work with. A complete specification of the qXY model is given in section §4. The derivation of the non-linear Langevin theory from qXY model is the subject of section §5. We give a general analysis of both the generalised Onsager and fluctuation-dissipation relations in the non-linear Langevin theory in the section §6, before our concluding remarks in the section §7.
Appendix A enumerates the dimension of various physical quantities that appear in this work. In Appendix B, we give a brief review of the rules that constrain the form of effective theories that can arise by integrating out the 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.

Review of Caldeira-Leggett model
We will now begin with a brief review of the salient features of the quantum Brownian particle and later, how these get generalised in the context of our work. We will sketch the basic structural features postponing detailed derivations for later. The reader familiar with this material is encouraged to quickly skim over this description, making note of our notation.

Review of Langevin theory and Fluctuation-Dissipation theorem
Consider, for definiteness, a Brownian particle whose evolution is described by Langevin equation : Here γ is the damping constant of the particle. The term f 2 N (t) is the fluctuating force ('the noise term') which is commonly approximated to be Gaussian and delta-correlated, viz., its only non-zero cumulant is taken to be of the form Here N (t) is normalised to have the inverse dimension of velocity for later convenience, so that f 2 measures the statistical variance of the fluctuating force per unit mass (and has the dimension of acceleration 2 ×time). The cumulant statement above can also be recast into a statement about the probability distribution governing the noise ensemble : 3) The significance of Caldeira-Leggett model is that it shows how such a system of equations can arise from an underlying quantum mechanical model, under appropriate approximations, via integrating out the effects of a harmonic bath. Another triumph of Caldeira-Leggett model is that it can reproduce the well-known fluctuation-dissipation relation [13][14][15][16] between the parameter f 2 characterising thermal fluctuations and the parameter γ characterising dissipation. Let us remind the reader the classical argument why such a relation should be expected : the thermal fluctuations in the bath which source the noise also cause the dissipative effects of the bath. On one hand, the rate of damping γ is roughly determined by the effective number of degrees of freedom of the bath with which it interacts. On the other hand, the thermal noise is produced by the energy fluctuations in the bath which is roughly k B T times the number of degrees of freedom (here, T denotes the temperature of the bath and k B is the Boltzmann constant). This suggests a relationship of the form f 2 ∼ γ k B T mp between the fluctuation and the dissipation parameters. Here, we have introduced the mass of the particle m p to match the dimensions.
A more precise argument is as follows : at the level of classical stochastic dynamics, one can integrate the Langevin equation to obtain We square this equation and average over the noise under the assumption that the initial velocity distribution and the fluctuating force are uncorrelated. At large times, this yields for the long-time average kinetic energy for a particle of mass m p : Demanding that this average kinetic energy approach the thermal equipartition value 1 2 k B T , we obtain the fluctuation-dissipation relation where we have introduced the rms thermal velocity v th of the Brownian particle defined via The physics behind such a classical stochastic argument is clear : the kicks of the fluctuating force should, on average, replenish the energy lost due to dissipation so that the eventual balance is achieved at the Maxwell-Boltzmann value for the average kinetic energy.
As pointed out by Kubo, Martin and Schwinger [17,18], the origins of this relation can be traced back, in the underlying quantum description, to the mathematical structure of thermal correlators. In the linear response theory, both the fluctuating force and damping rate felt by the Brownian particle can be computed from a characteristic spectral function of the bath denoted by ρ(ω), related to the Fourier domain commutator of bath operators that couple to the particle (see below). We then have the frequency integrals (often termed the sum rules): where β ≡ k B T is the periodicity of the thermal circle and f ω ≡ 1 e βω −1 is the Bose-Einstein distribution. These relations make precise the afore-mentioned intuition that the same effective degrees of freedom (encoded in the single function ρ(ω) of the bath) determine both f 2 and γ.
Note the factor (1 + 2f) whose three terms describe spontaneous emission, stimulated emission and absorption of the fluctuation of frequency ω, all of which add incoherently into the variance of noise. At high temperature limit (i.e., small β) one then recovers Eq.(2.6). The iǫ prescription for the frequency integral is the frequency domain analogue of step functions that appear in time domain retarded Green functions. For example, the formula above for γ can also be written in the form which can be thought of as coming in turn from the approximation of a non-local retarded Green function : While such time domain expressions clearly exhibit the causality properties, we will find it convenient to work in frequency domain where the structure of thermal correlators can be examined clearly. All our frequency domain contour integrals can, if needed, be readily converted into time domain integrals with appropriate step functions. Let us examine in some more detail how this works : consider a probe particle coupled to a bath in a general time-independent state and assume that at long enough time scales, a local autonomous description is still possible for the particle (i.e., Markovian approximation can be justified and any effect of the memory of the bath can be ignored). Under these assumptions, the effect of the bath can be integrated out following the method of Schwinger [1] and Feynman-Vernon [2] to get a local Schwinger-Keldysh effective action (or equivalently a local Feynman-Vernon influence functional). We can match this local effective action against the generating functional for the Langevin correlators following the method of Martin-Siggia-Rose [19]-De Dominicis-Peliti[20]-Janssen [21] (See [22] for a recent review). This results in a general expression for the Langevin couplings : (2.12) Here the integrands are the Fourier transformed expectation values of the commutator/anticommutator (2.13) In the above, O is the bath operator that couples linearly to the Brownian particle position q and the expectation values are evaluated in an appropriate time-independent state of the bath. 1 The reader should also note the specific energy-conserving iǫ prescription needed to write the sum-rules above : . (2.14) The above expressions relating the microscopic dynamics of the bath degrees of freedom to the effective couplings can be obtained by standard procedures of effective theory : for instance by matching the correlators predicted by the effective theory against the microscopic computations. The iǫ prescription then naturally appears when comparing appropriately retarded correlators. Another important property of the complex contour C 2 is the following : it remains invariant under a simultaneous complex conjugation and the reversal of frequencies. It follows then that, if the integrands are similarly invariant (as the above integrands indeed are), then it follows that the resultant answers are real.
The expressions above imply that, from the point of view of thermal correlators, whereas the damping constant γ is related to the commutator of bath operators, the fluctuation f 2 is related to the anti-commutator. If the bath state is thermal, the Kubo-Martin-Schwinger(KMS) conditions relate these frequency domain functions as [12] . (2.15) Using time translation invariance to write ρ [12] = ρ(ω 1 ) 2πδ(ω 1 + ω 2 ), we then recover Eq.(2.9). The Eq.(2.15) can be motivated by the following relation between thermal averages in a harmonic oscillator of frequency ω : KMS showed that such a relation continues to hold true for a general quantum system, thus giving rise to Eqs. (2.9) and (2.6) under very general assumptions. To summarise, the microscopic justification of fluctuation-dissipation theorem lies in the following steps : • In the first step, one identifies the relevant effective couplings of the system (here f 2 and γ of the Brownian particle) and connects it with the appropriate bath correlators (via sum rules like the ones in Eq.(2.12)). This step already assumes the emergence of a Markovian description which can be checked explicitly in a simple model like that of Caldeira-Leggett.
• In the next step, one uses thermality to derive KMS relations akin to Eq.(2.15). Note that this step, by itself, does not immediately result in a simple relation between the effective couplings of the local Markovian description.
KMS condition merely says that that two couplings are related to two different moments of the same spectral function (see Eq.(2.9)). These two moments are a priori two independent numbers provided by the theory behind the bath. Thus, the KMS condition alone is of limited utility to an experimentalist probing the local dynamics of the Brownian particle. 2 • In the last step, we take a high temperature limit to get a fluctuation-dissipation relation of the form Eq.(2.6). It is in this step that one obtains the fluctuationdissipation relation between the effective couplings.
Apart from the fluctuation-dissipation relations, there are other useful relations that can be derived when one has more than one Brownian degree of freedom and when one can assume additional symmetries. An example of such a symmetry is the microscopic timereversal invariance whose consequences were explored by Onsager [29,30] and Casimir [31] (we refer the reader to the monograph by Stratonovich [27] for extensions and a detailed exposition). Say we had many degrees of freedom denoted by the coordinate q A , which undergo coupled Langevin dynamics governed by a matrix of Langevin couplings γ AB and f 2 AB . In that case, the sum rules of the discussion above generalise to Here O A is the bath operator that couples to the coordinate q A and we have used a matrix of spectral functions Say the dynamics and the initial state of the bath are microscopic time-reversal invariant and we will assume that the Hermitian operators {O A } have definite time-reversal parities {η A }. The microscopic time-reversal then acts by the simultaneous exchange of ω 1 and ω 2 along with complex conjugation. 3 One then obtains the famous Onsager reciprocal relations [29,30] : In this work, we will mainly confine ourselves to studying the dynamics of a single degree of freedom, which can in principle couple to many bath operators {O A } with different couplings g A . Then, the Onsager reciprocal relations can be interpreted as statements relating the contributions proportional to g A g B to the damping constant and the noise variance. We will later meet an example of similar relations, when we study non-linear corrections to the dynamics of a single degree of freedom (see the discussion in section 6.2).

Model of the harmonic bath in Caldeira-Leggett like models
We will now move from the general description to particular models of the bath. The equation (2.12) for the damping constant γ naively suggests the following: if we can engineer a bath of harmonic oscillators with a linear spectral function of the form 4 a naive residue integral seems to pickup the required contribution. This is however misleading. In fact, the integral for γ with the linear spectral function is UV divergent and needs to be regulated appropriately. A simple and commonly used regulator is to assume a Lorentz-Drude form for the spectral function, with the linear growth at low frequencies : which does give back γ after the contour integral is performed. How can such a distribution be obtained from the underlying microscopic dynamics of the harmonic bath ? Say we are interested in the thermal harmonic bath where q(t) of the Brownian particle is coupled linearly to the bath oscillators via (2.22) We want to now examine what set of couplings g x,i and masses m x,i for the bath oscillators will result in a Lorentz-Drude spectral function. Computing the commutator of the bath operator in the thermal state and Fourier transforming yields where we have defined a function of µ x : A spectral function ρ(ω) as in Eq.(2.21) is then obtained, if we take a continuum of oscillators whose couplings add up to give (2.25) Thus, the continuum approximation with an infinite set of oscillators gives the required smooth form for the spectral function. Only in this limit can the set of bath oscillators be idealised as a perfect thermal bath into which the Brownian particle can irreversibly dissipate into. It is also only in this limit that the information about the particle is quickly forgotten by the thermal bath, thus allowing us to ignore any memory effects. This last point can be explicitly checked by computing the bath correlators and confirming that they indeed decay at time scales set by Ω −1 . Thus, we expect a local description to be good when there is a hierarchy of frequency scales : The continuum approximation and the resultant irreversibility are good approximations at time scales much smaller than the inverse of the typical gap in the bath spectrum. This ends our brief review of the standard Langevin theory. One main goal of this work is to see many of these ideas and expressions generalise to higher point functions, out of time order correlations and to non-linear Langevin dynamics.

Introduction to Non-linear Langevin equation
It is a natural question to ask how these results generalise once we go beyond the linear Langevin description. A particular non-linear generalisation is of relevance to this work, which we shall now describe. Consider a non-linear generalisation of Langevin theory described by the following stochastic equation : Here, we will take N to be a random noise drawn from the non-Gaussian probability distribution We will assume that the corrections to the Langevin equation are small : this is equivalent to assuming that the parameters {ζ γ , ζ µ , λ 3 , λ 3γ , ζ N , Z I } are small. The physical meaning of these non-linear parameters should be evident : ζ γ , ζ µ characterise the thermal jitter in the damping constant γ and the (renormalised) natural frequencyμ 2 of the Brownian oscillator. The parameters λ 3 , λ 3γ control the anharmonicity in the model whereas ζ N characterises the non-Gaussianity of the thermal noise. The above equation includes all terms upto quadratic in amplitudes of q and N and upto one time derivative acting on q (except for the inertial term d 2 q dt 2 which was already present before the bath came into picture). In this sense, this is indeed the most generic leading non-linear correction to the linear Langevin theory.
Another equivalent way to define these couplings would be to state how they occur in the long-time three point cumulants of the Brownian particle which starts off from the harmonic oscillator vacuum at an initial time t = t 0 . While specification in terms of cumulants does not have the immediate intuitive appeal of the description above, it is a very useful characterisation for computing these couplings from a microscopic model. Consider long enough time scales such that the memory effects of the bath can be ignored and the Markovian approximation is valid. We will however be interested in the time scales much smaller than γ −1 , the time scale at which damping effects become substantial. In this time window, one can write down universal expressions for the vacuum cumulants of the Brownian particle in terms of the effective Langevin couplings. In the semi-classical limit (i.e., ignoring O( ) terms ), they take the following form [9] : for t 1 > t 2 > t 3 , we get In the above, we have defined the function with t ij ≡ t i − t j and we have used the square bracket notation to indicate nested commutators (with a + subscript indicating anticommutator). For example, The six correlators given above cover all possible time orderings with three operators. In the above, we have divided by a factor of for every commutator in LHS, so that the commutators smoothly go to Poisson brackets in the classical limit.
The reader should also note that apart from the non-linear Langevin/1-OTO couplings 5 , three new 'out of time ordered' 2-OTO couplings κ 3 , κ 3γ and κ 3γ are needed to fit the long-time behaviour of arbitrarily ordered cumulants. On general grounds, we expect correlators with four out of the six time orderings to be computable via standard Schwinger-Keldysh/Feynman-Vernon influence functionals : these are the 1-OTO correlators in the classification of [10]. A basis of such correlators is provided by all of which can be written down in terms of the standard non-linear Langevin/1-OTO couplings, as is evident from the expressions above. The two other remaining time-orderings are however genuinely out of time ordered which can neither be captured by the standard Schwinger-Keldysh/Feynman-Vernon influence functionals nor can they be written solely in terms of the standard non-linear Langevin couplings. A historical aside may be in order : many authors have studied non-linear generalisation of Langevin equations in a variety of contexts (See, for example [33][34][35][36]). However, as far as the authors are aware, there is no systematic discussion in the literature including all leading nonlinear corrections allowed on general grounds, nor a microscopic model within which Markovian approximation is justified and all couplings derived. Similarly, despite a long and rich literature on non-linear fluctuation-dissipation/Onsager relations [16,[37][38][39][40][41][42][43][44][45][46], the relations we derive here for non-linear Langevin theory are new, as far as we know.
In the rest of this section, we will give a detailed summary of our results listing the integrals that relate the couplings that appear in the above equation to the bath correlators in the microscopic theory. The reader desirous of a briefer summary is encouraged to consult the subsection § §3.7 at the end of this section.

Linear Langevin couplings
We will now summarise our results regarding how the above coefficients could be computed starting from a non-linear generalisation of Caldeira-Leggett like setup. Assume the Langevin degree of freedom q(t) is still linearly coupled to some bath operator O(t). As in the discussion above Eq.(2.12), take the bath to be in a general time-independent state and assume a Markovian description at long enough time scales.
Under these assumptions, a general expression can be written for the Langevin couplings in leading order causal perturbation theory in particle-bath coupling (or equivalently by matching the local influence functional to MSRDPJ action of the above stochastic equation to leading order in perturbation theory) : where as before, we have an integral over the causal contour Here ∆m p and ∆μ 2 denote the renormalisation of mass and frequency of the Brownian oscillators.
The spectral functions appearing in the integrand are defined via Fourier integrals Further, in writing the second equality for the couplings above, we have assumed that the bath state is thermal and consequently, we have used the Kubo-Martin-Schwinger (KMS) relation to convert the anti-commutator to commutator.
The complex contour C 2 has the following reality property : it remains invariant under a simultaneous complex conjugation and the reversal of frequencies ; where ω c i denotes the complex conjugate of the frequency ω i . Further, the hermiticity of bath operator O(t) yields the result Thus, integrating the conjugated, frequency-reversed spectral function over the conjugated, frequency-reversed contour gives the same answer as the original spectral function integrated over the original contour. Since the relabelling of the integration variables ω c i → −ω i should not change the value of the integral, the above assertion is equivalent to the statement of reality of the Langevin couplings.
The complex contour C 2 also has the following time-reversal property 6 : it remains invariant under a simultaneous complex conjugation and the exchange of frequencies ω 1 and ω 2 . Say the spectral function has a specific time-reversal parity η 12 inherited from the time parity of operators that define it. We then have thus guaranteeing that the couplings derived above obey Onsager reciprocal relations using a similar argument to the above argument for reality.

Anharmonicity parameters : time ordered and out of time ordered
A similar exercise can be carried out for 3-point functions : for the anhormonic couplings we have As before, we have defined here the spectral function (3.14) and the frequency domain contour which picks up the causal part of three point function : where ǫ 1 , ǫ 3 > 0. We will find it convenient to not fix a particular ordering between ǫ 1 and ǫ 3 : the ordering will not matter, provided we take care that our integrands do not have poles/branch cuts in ω 2 near the real axis. For example, the above expression for the effective couplings is valid provided the spectral function has no discontinuities or branch cuts in the real axis : In cases where there are discontinuities, they are known to lead to long time memory effects which, in turn, lead to a breakdown of Markovian approximation [32]. Markovian approximation also needs the following constraint on the spectral function : lim The complex contour C 3 has the following reality property : it remains invariant under a simultaneous complex conjugation and the reversal of frequencies (similar to our discussion above for the two-point causal contour C 2 ) . It follows then that, if the spectral function This along with the invariance of the contour guarantees reality of the couplings. The complex contour C 3 also has the following time-reversal property : it remains invariant under a simultaneous complex conjugation and the exchange of frequencies ω 1 and ω 3 (with the concomitant exchange of their imaginary parts ǫ 1 and ǫ 3 ) . If there are no poles/branch cuts in ω 2 near the real axis, we can deform the contour back to C 3 : it is in this sense that the frequency contour C 3 is time-reversal invariant. This is similar to the time-reversal invariance of the two-point causal contour C 2 . However, unlike the two point function case, the action on the integrand cannot be simply described : the spectral function ρ[123] gets mapped to a new function ρ[321] where η 123 is the time parity of the spectral function inherited from the time parities of the underlying operators. In the above, the function ρ[321] can be thought of as the time-reversed/out of time order (OTO) spectral function : While the expression for ρ[321] looks formally similar to that of ρ[123], note that these are two different functions on the C 3 contour due to the different iǫ prescriptions. For a bath with no microscopic time-reversal invariance, these two spectral functions are a priori unrelated. On the other hand, in the presence of time-reversal invariance, the analogue of Onsager relations for cubic couplings relate the Langevin anharmonic couplings to anharmonic couplings in the time-reversed stochastic dynamics. The above statement can be made precise by introducing the out of time order (or time-reversed) anharmonic couplings Say the bath operators O k (t) with a definite time-reversal parity η k couple to the single degree of freedom of the Brownian particle. For simplicity, assume that the relevant spectral function gets contribution only from the product of three operators O k=1,2,3 . We can then write down the non-linear, OTO analogue of the Onsager reciprocal relations for the resultant couplings as where η O ≡ η 1 η 2 η 3 is the total time-reversal parity of the given spectral function. Thus, rather than constraining the cubic couplings in the non-linear Langevin dynamics, the time-reversal invariance ends up relating the standard anharmonic couplings to the OTO anharmonic couplings.

Frequency Noise parameter ζ µ
We now move on to quote the results for the other non-linear couplings that appear in the non-linear Langevin equation. We begin with the parameter ζ µ that governs the thermal jitter in the frequency of the Brownian oscillator: (3.23) Here, the spectral functions are defined as before by Fourier transforming the appropriate nested commutators/anti-commutators. 7 For example, we have where the subscript + indicates the anti-commutator. The integral is over the same causal contour C 3 as before. The first line gives the appropriate causal Green function from which the effective coupling ζ µ can be extracted. The expression in the second line above arise from generalised Jacobi identities of the form The discussion on the reality properties is similar to before : the causal contour C 3 and the commutator spectral functions are invariant under simultaneous conjugation and frequency reversal (assuming no ω 2 discontinuities on the real axis). The anti-commutator spectral functions are odd under simultaneous conjugation and frequency reversal : This, then ensures the reality of ζ µ . The reality can also be shown using the expression in terms of Bose-Einstein distributions, however the argument involved is slightly more subtle : the following property of Bose-Einstein functions should be used along with the assumption that the potential discontinuity due to f 2 near ω 2 → 0 is cancelled among the two terms. 8 As discussed in the last subsection, time-reversal acts on the causal contour C 3 via a simultaneous complex conjugation and the exchange of frequencies ω 1 and ω 3 . If the bath 7 including a factor of inverse for every commutator to guarantee smooth classical limit. 8 One can explicitly prove that this indeed happens in the Markovian models of the bath we use.
operator O(t) has a definite time-reversal parity η O , we can then write down the non-linear Onsager reciprocal relation for ζ µ as i.e., one obtains a trivial relation for time-reversal even bath operators and for time-reversal odd bath operators, ζ µ = 0 . At large temperatures (or small β), one can approximate We then get a high temperature formula for ζ µ of the form where, in the last line, we have used the fact that the spectral functions are proportional to δ(ω 1 + ω 2 + ω 3 ) for time-independent state of the bath. We now turn to possible fluctuation-dissipation type relation involving ζ µ . While the integrand which appears in the first line above has structural similarities with the integrands in the sum rules of the couplings λ 3γ and κ 3γ , we have not succeeded in establishing any general fluctuation-dissipation type relation between these couplings. Nevertheless, in many explicit models with time-reversal invariance 9 , we find the following relation : where the last equality is the relation λ 3γ = κ 3γ which holds for even time-reversal bath operators. More generally, we find that the following integral vanishes in many explicit examples : thus resulting in a relation of the form

Dissipative noise parameter ζ γ and its OTO counterpart
We will now move on to the parameter ζ γ which describes the jitter in the dissipative constant γ of the Brownian particle. The sum rule(s) for computing ζ γ is given by In the first line, we have the representation in terms of the causal correlator. The second line comes from generalised Jacobi identities and the last line from Kubo-Martin-Schwinger (KMS) relations. Following arguments very similar to the ones sketched in the previous subsection, one can argue that ζ γ produced by the above integral is real.
Next we turn to examine the behaviour of the integrand under the exchange of frequencies ω 1 and ω 3 (which would be relevant for the action of time-reversal on this coupling). We have the following identity : where we have defined a new OTO coupling κ 3γ via We conclude that if η O = 1, one has the constraint κ 3γ = 2ζ γ whereas for η O = −1, one has the constraint κ 3γ = 0. Thus, κ 3γ can be thought of as the part of ζ γ even under time-reversal, whereas the part odd under time-reversal is given by (3.39) At high temperature, these expressions become 40) or equivalently and (3.42) In simplifying these expressions, we have used the identity which holds because of the δ(ω 1 + ω 2 + ω 3 ) inside the spectral functions.

Non-Gaussianity ζ N and its fluctuation-dissipation relation
We finally turn our attention to the last parameter of the non-linear Langevin model which is the non-Gaussianity parameter ζ N . We find the sum rule(s) for computing ζ N as where we have used the condition for the nested anti-commutator to write the second equality. The relevant spectral function is It can be checked that the integral above does give a real coupling as an answer using methods described before. In order to study time-reversal, consider the following timereversed/OTO counterpart of the spectral function appearing above : With this in mind, we examine the integral (3.48) Here, we have used the sum rules quoted before to write a combination of couplings with good time-reversal properties. Microscopic time-reversal invariance thus implies Thus, the non-Gaussianity in thermal noise is quantum suppressed for time-reversal odd coupling to the bath. At high temperature limit, we get (3.51) where we have used Eq. (3.43). Comparing this against the high temperature limit of κ 3γ , we obtain the fluctuation-dissipation relation : We note that, in this way of writing both sides of these equations have the same timereversal property.

Force term at cubic order
We will now state the result for the force per unit mass induced by the bath. This is an effective displacement of the minima of the Brownian oscillator due to the coupling to the bath. This force is given by the sum rule Note that we have separated out a quantum correction 4μ m p λ 3 due to the anharmonic coupling for convenience. The discussion on the reality of this effective coupling is akin to the discussions in the previous subsections. At high temperature limit, we get (3.54) While there does not seem to be any fluctutaion-dissipation type or Onsager type relations involving the above integral, we find that the following relation is satisfied in our model described in the next section : It would be good to study other models to check whether such a relation holds more generally.

Summary of relations in non-linear Langevin theory
We will now summarise all the Onsager type relations between the couplings of the nonlinear Langevin theory : (3.56) In addition, we have the consequence of KMS relations : Note that, in general, each of these conditions relate the time-ordered Langevin couplings to OTO Langevin couplings. These relations can be combined to solve for 5 of the non-linear Langevin couplings in terms of the 3 remaining couplings {λ 3 , λ 3γ , ζ γ } : These relations are one of the main results of this work. We collect in Table 4, the Onsager pairs related to each other by time-reversal. The following tables summarise the integrands I[g] associated with each coupling g. The tables 1, 2 and 3 summarise the integrals for quadratic couplings that are induced, in the leading order in perturbation theory, from a bath in a general time-independent state, by a general thermal bath and by a bath at high temperature respectively. The couplings are given by expressions of the form (3.59) Similarly, the tables 4, 5 and 6 summarise the integrals for cubic couplings that are induced, in the leading order in perturbation theory, from a bath in a general time-independent state, by a general thermal bath and by a bath at high temperature respectively. In this case, the couplings are given by (3.60)

Introduction to the qXY model
In this section, we 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. Our motivation here is to construct a physical microscopic description in which one can check the Markovian assumption and the relation between the effective couplings that emerge therein.

Model of the bath
We will now begin with the Caldeira-Leggett model and then modify it to suit our requirements. As described before, the model is that of a single system oscillator (denoted by a degree of freedom q) coupled to a bath of oscillators (denoted by degrees of freedom X). One starts with a distribution of couplings and masses of bath oscillators specified by a characteristic distribution function, defined by This distribution function multiplied by the spectral contribution of each bath oscillator can be summed to give the Caldeira-Leggett spectral function : To obtain a Lorentz-Drude spectral function, we consider a continuum of oscillators whose couplings add up to give a distribution of the form where γ x denotes the contribution of X oscillators to the damping constant γ of the particle. The contribution to the noise is determined by fluctutation-dissipation relation To get a simple nonlinear generalisation, we will double the number of oscillators into two kinds of bath both at same temperature. We imagine them to be two different sets of bath oscillators distinguished by the letters X and Y . For simplicity, we will assume that the Y type oscillators also have similar coupling distribution as X type oscillators and they add to the damping due to X type oscillators. Thus, γ = γ x + γ y and the noise contributions from the two sets of oscillators also add up. Till now, this is merely a relabelling of the original model, and the model is hence exactly solvable and yields linear Langevin theory.
We will now introduce non-linearity into this theory by introducing a very small 3-body interaction term of the form qXY . More precisely, one considers a system of oscillators with the Lagrangian where we denote the oscillator's position by q and L B [X, Y ] is the free Lagrangian of the harmonic bath. The operator that acts on the Hilbert space of the bath and couples to q When we integrate out the effects of the bath degrees of freedom, the effective couplings are now induced for the oscillator and there are corrections to the linear Langevin theory. These Langevin effective couplings have their microscopic origin in the thermal correlators/spectral functions of the operator above. One can think of the above as a toy model for say an atom coupled to photons whereby, apart from the standard, dominant linear dipole coupling responsible for single photon emission/absorption processes, one also has two photon processes involving two photons of two different frequencies. The physics here is familiar one say from Brillouin scattering of photons against phonons or the inelastic Raman scattering of photons against molecules.
At the level of linear Langevin couplings, both the noise and the damping constant receive contributions from the inelastic 3-body scattering : first, there is an effect due to two 'photon' emission and absorption into/from the thermal bath which gives a contribution proportional to in the spectral function. The second effect is due to inelastic scattering whose contribution is proportional to Putting these effects together, we get a spectral function 10 (4.10) Here we have introduced the cubic coupling distribution  This cubic coupling distribution should be judiciously chosen so that its dynamics does not destroy the Markovian approximation. We find it convenient to choose a distribution of couplings such that More precisely, we take (4.13) 10 The correction to the spectral function appearing in the last two lines comes from computing the Fourier transform of the thermal commutator (4.9) As we will show later, this distribution function is sufficient to give a fast decay of correlator at timescales larger than Ω −1 . In the large temperature limit i.e. β → 0, this distribution function gives the following form of ρ [12] upto O(β 0 ) : (4.14) Thus, the damping constant acquires a correction of the order Γ xy Ω v 2 th . This is a small correction provided we have a condition we will assume from now on.
We can now turn to the three point spectral functions obtained by Fourier transforming the nested commutators of the bath operator O defined in Eq.(4.6). This yields and where we have defined the distribution function g x g y g xy m x m y ≡ ij g x,i g y,j g xy,ij m x,i m y,j 2πδ(µ x − µ i )2πδ(µ y − µ j ) . (4.17) We will find it convenient to assume More precisely, we take Here, the parameter Γ 3 can roughly be thought of as an inverse penetration depth for the three body scattering. This induces a small correction to the usual Langevin dynamics if a condition we will assume from now on. In the next subsection, we will examine the correlation functions of the bath and to what extent they justify a Markovian appproximation.

KMS relations and decay of bath correlations
Consider the model described in the above section of a single oscillator coupled to two harmonic baths. In this subsection, we will be interested in the evolution starting from an initial time t 0 and whether and how the bath correlators decay with time.
We will assume that the bath and the oscillator are unentangled before t 0 . Moreover, the initial state of the oscillator is assumed to be its ground state |0 . Therefore, the density matrix of the oscillator and the bath at time t 0 is given by where H B is the Hamiltonian of the bath, and Z B is its partition function. The effective theory of the Brownian particle is obtained after integrating out the degrees of freedom of the thermal bath. In the process, the bath correlation functions (in general out-of-time-order) imprint themselves on the effective couplings of the Brownian particle. In this section we are interested in studying the out-of-time-order bath correlators of the operator O. We will use these correlators later to determine the effective couplings of the particle.
Since  At the level of the two point function, the statement implies that there is only one independent two-point correlator of O. We choose that to be ρ [12]. Then, using the KMS relations, the expectation value of the anticommutator is given by where f 1 =  Hence the spectral functions ρ [12], ρ[123] and ρ[321] are sufficient to determine all two-point and three-point bath correlators.
For the two-point function the correlator of the anticommutator in equation (4.23) provides a measure of the thermal noise arising from the thermal fluctuations in the bath whereas the correlation function of the commutator in that equation gives a measure of the dissipation/damping in the bath due to the motion of the Brownian particle [2]. We denote the connected part of nested commutators of operators in time domain by a tilde in the following : (4.25) For the cumulants of the nested anticommutators we use a similar notation with an extra '+' sign inside the square bracket indicating the position of the anticommutator as follows (4.26) We can use the forms of the spectral functions to get the bath correlators in time domain. The bath correlators decay with increase in separation between any two insertions. In the following, we provide the two-point and three-point cumulants for the slowest decaying modes with frequency Ω.
For our model, the two point cumulants decay as i [12] = Ω 2 m p (γ x + γ y ) exp(−Ωt 12 ) + Γ xy Ω 2 exp(−2Ωt 12 ) cot βΩ 2 ,    The other cumulants can also be computed to yield  Thus, given the decay of the memory in the bath at time-scales much larger than ( 1 Ω ) we expect to obtain a local effective theory for the particle at long time-scales. In the next section, we will describe how such an effective theory be obtained starting from the microscopic description.
5 Effective theory of the Brownian particle

OTO Influence Phase
We will begin by describing the procedure we employ to derive the effective theory for the particle at long time-scales, closely following [9]. Our aim is to obtain the couplings in the effective action of the Brownian particle (after systematically integrating out the degrees of freedom of the bath) in a way that keeps track of out-of-time order correlations. Thus, we are interested in an effective theory which has sufficient number of effective couplings that can compute along with the time-ordered correlators, also the out-of-time order correlators (OTOCs) of the particle.
We use the generalised Schwinger-Keldysh path integral formalism to arrive at an effective action/generalised influence phase for the Brownian particle. The path integral representation of 2-OTO correlators (correlators with two insertions whose immediate neighbours lie to their pasts) 11 is then defined on a contour with two time-folds as shown in figure 1. There are four copies of the degrees of freedom of the particle and bath , Y j4 } living on the four legs of the contour. The action that enters in the path integral is  The degrees of freedom of the particle are identified at the turning points of the contour while performing the path integral. After integrating out the degrees of freedom of the bath, one obtains an out-of-time-order generalised influence phase W for the particle which can be expanded in powers of the particle-bath coupling: The n-th term in this perturbative expansion is given by Here the subscripts denote the contour legs and the expectation values are contour ordered cumulants computed in the initial state of the bath.

Markovian approximation and effective action
The correlators of the Brownian particle calculated from the generalised influence phase can be obtained from a 1-PI effective action. The 1-PI effective action is generally nonunitary and non-local. However since in our model the cumulants of O(t) decay sufficiently fast, with an increase in separation between the insertions compared to the natural time scale of the particle, we can work in a Markovian limit [6]. In this limit we get a local, non-unitary 1-PI effective action. We assume that in the action the terms with two or more time derivatives on q are negligible. Such a local 1-PI effective Lagrangian for the Brownian particle has the following form [9,55] 1PI and L 1PI correspond to terms that are linear, quadratic and cubic in q's respectively. The linear term is given by The quadratic term is given by The cubic term L 1PI can be split into 2 parts: One part, which reduces to the cubic terms in the Schwinger-Keldysh effective theory under identification of the degrees of freedom on any two successive legs, is given by The other part, which vanishes under such identifications, is given by The effective action constructed from this Lagrangian has to satisfy certain conditions as elaborated in the appendix B. The conditions arise due to the fact that the Brownian particle and the bath together comprise a closed unitary quantum system.

Comparison with the Nonlinear Langevin system
We will now relate the description in terms of generalised influence phase and generalised Schwinger Keldysh effective action to the classical stochastic description in terms of nonlinear Langevin equation. This relation, familiar in the stochastic quantisation literature, can be studied at various levels : we can for example, study the correlators predicted by the two descriptions and match them against each other. We will instead derive the dictionary between the quantum and stochastic descriptions by deriving a path integral which generates the non-linear Langevin correlators and then matching its terms against the influence phase obtained by ignoring out of time ordered contributions. We are interested in the non-linear Langevin theory described by the stochastic equation : Here, we will take N to be a random noise drawn from the non-Gaussian probability distribution We will assume that the corrections to the Langevin equation are small : this is equivalent to assuming the parameters {ζ γ , ζ µ , λ 3 , λ 3γ , ζ N , Z I } are small.
The equation above is a non-linear stochastic ODE with multiplicative noise, i.e., the noise variable N appears in the equation multiplied by the functions of the fundamental stochastic variable q of the differential equation. In the theory of stochastic ODEs, such ODEs need a definite prescription for equal time stochastic products to be well-defined. In this work, we will adopt a time-symmetric (or Stratonovich) prescription for equal time stochastic products. But we will only need leading order corrections due to the multiplicative noise terms, where the subtleties regarding various prescriptions for stochastic products will not matter.
To study this non-linear Langevin theory in the context of path integrals, one can employ the following method (often attributed to Martin-Siggia-Rose [19]-De Dominicis-Peliti[20]-Janssen [21]) : We start by thinking about the functional integral over noise realisations along with the imposition of the non-linear Langevin equation on a variable q a (t): where we have given the standard functional integral representation of the delta function.
We can now discretise the noise integral, add appropriate counterterms and perform the path integral perturbatively in the small parameters {ζ γ , ζ µ , λ 3 , λ 3γ , ζ N , Z I }. This exercise yields ∂N .

(5.12)
This MSRDPJ effective action can be connected to Schwinger Keldysh effective action by identifying q d = q 1 + q 2 , q a = 1 2 (q 1 − q 2 ). We will refer the reader to [56] for a textbook level discussion of why this is the correct identification that maps the Schwinger Keldysh boundary conditions on the quantum side to the causal boundary conditions of the stochastic path integral. Using this map, we can write the above effective action in the form : 1PI,SK = − where the Schwinger Keldysh effective couplings are given in terms of Langevin couplings This two point spectral function obeys the following integral identities which yield the following quadratic couplings at high temperature : Similarly, the cubic spectral functions are given by These three point spectral functions obeys the following integral identities at high temperature : which yields the following values for cubic couplings (5.23) We provide more details about these integrals and the poles which contribute via their residues in the appendix C of this work. It is evident from the expressions above that many of the couplings in the effective theory are related to each other by a series of relations. As we will elaborate in the next section, quite a few of these relations can be explained on general grounds using the fact that the bath correlators exhibit microscopic time-reversal invariance and obey KMS conditions.

Relations between effective couplings
In this section we discuss the origin of the relations between the cubic effective couplings given in Eq.(5.23). As we show in the following two subsections, most of these relations are based on the following two general features of our model: 1. Microscopic time-reversal invariance in the bath,

KMS relations between bath correlators.
While discussing the consequences of these features, we will first give a general proof of the relations between the couplings, and then describe why our particular model satisfies the conditions that go into the proof. The arguments in the following two subsections will show that most of the relations between the effective couplings in our model are not just particular features of our model alone. Rather, they are generally valid for a broad class of systems, whenever the two conditions mentioned above are satisfied.

Consequences of time-reversal invariance
First, let us discuss the consequence of microscopic time-reversal invariance in the bath. As we mentioned in the introduction, the implications of such microscopic time-reversal invariance for systems with multiple degrees of freedom were analysed by Onsager in [29,30] where he showed that the quadratic effective couplings such as γ AB and f 2 AB are symmetric under the exchange of the indices. The derivation of such reciprocal relations relied on the operators {O A } being invariant under time-reversal. These relations were later generalised by Casimir [31] to the case where the operators {O A } have the parities {η A } under timereversal. The corresponding generalisation of the Onsager relations is as follows: Here, we generalise the Onsager-Casimir reciprocal relations to cubic couplings in the OTO effective theory. We find that microscopic time-reversal invariance in the bath leads to certain relations between the 2-OTO couplings and the 1-OTO couplings which are derived below. We would like to point out that, unlike the scenario considered by Onsager and Casimir, our system (the Brownian particle) has a single degree of freedom. Nevertheless, the relations that we obtain between the couplings are based on principles similar to those for the reciprocal relations.
To keep the discussion precise, let us note that the operator O(t) is defined with respect to some particular reference point in time when it coincides with O (the Schrodinger-picture operator). While calculating the contribution of the correlators of this operator to the particle's dynamics, this reference point must be the instant t 0 at which the particle starts interacting with the bath. However, if the bath's initial state (described by the density then such correlators are independent of the choice of the reference point and depend only on the intervals between the insertions. This is true, for instance, in our model where the bath is assumed to be in a thermal state. For such an initial state of the bath, one can shift the reference point to t = 0 which can be chosen well into the domain of validity of the particle's effective theory. With respect to this new reference point, the bath's correlators with insertions at both positive and negative values of time are relevant for the particle's dynamics. In the following discussion we are going to assume time-translation invariance of the initial state of the bath and choose the reference point for the operators to be at the origin of time t = 0.
Let us now assume that the bath's dynamics has time-reversal invariance and the initial state of the bath respects this symmetry. Then, there exists an anti-linear and anti-unitary time-reversal operator T such that 12 At the level of correlators, this symmetry implies Now, say the operator O has a definite time parity i.e., where η O = ±1. The fact that T is an anti-linear operator which commutes with H B , implies Inserting this transformation of O(t) into the equation (6.4) and imposing the timereversal invariance of the initial state, we get As the operator O is Hermitian, the complex conjugation in the RHS of the above equation implies reversing the order of the insertions. So we have Such relations, in general, imply that correlators with different OTO numbers (i.e., the number of minimum time-folds required to compute the correlators [10]) get related to each other. In case of 3-point functions, as was mentioned earlier, we have at most 2-OTO correlators. For three time instants t 1 > t 2 > t 3 , the 2-OTO correlators are: In both these correlators, we have 2 future turning point insertions: O(t 1 ) and O(t 2 ), and hence they are 2-OTO correlators. From (6.8) we can see that these correlators are related to their time-reversed counterparts as follows: The correlators in the RHS of the above equations are 1-OTO correlators. So we see that all 2-OTO 3-point correlators of the bath get related to 1-OTO correlators. It is natural to ask whether such relations between the bath's correlators lead to any relation between the 2-OTO couplings and the 1-OTO couplings in the effective theory of the particle. To answer this question, first notice that the relation (6.7) implies that, when all the frequencies are real, correlators of O in frequency space are either purely real or purely imaginary i.e.
This reality property gets carried over to the connected parts i.e. the cumulants of these correlators. Equivalently, the spectral functions satisfy relations of the form We use this property of the cumulants in frequency space to derive relations between the 2-OTO and 1-OTO couplings. From the expressions of the cubic couplings, we see that the couplings can be divided into doublets and and singlets as shown in the table 7 . We will show that the pair of couplings in each doublet are related to each other due to time-reversal invariance. On the other hand, time-reversal maps the singlets to themselves upto a factor of η O . So when η O = 1, these relations are trivial. But when η O = −1, these relations imply that these singlets vanish.
As the couplings mentioned in table 7 are all real, their complex conjugates are equal to them. This can be used to obtain alternative expressions for these couplings by complex conjugating the integrals. Such a complex conjugation in the frequency space maps the contour of integration from C 3 : to C * 3 where the frequencies run over the following values: . (6.14) Note that the integration over ω c 1 in the C * 3 contour runs just above the real axis, exactly like the integration over ω 3 in the C 3 contour. Similarly, integration over ω c 3 in the C * 3 contour runs just below the real axis exactly like the integration over ω 1 in the C 3 contour. Therefore, under the following redefinitions: the contour of integration gets mapped back to C 3 with ǫ 1 and ǫ 3 exchanged (and the imaginary part of ω 2 reversed). This exchange of ǫ 1 and ǫ 3 and the concomitant reversal of the imaginary part of ω 2 can be undone by a contour deformation, provided our integrands have no ω 2 discontinuities near real axis (as required for the validity of Markovian approximation). To conclude, assuming appropriate analyticity in ω 2 , the complex conjugation and the above redefinition leave C 3 contour invariant.
Now, let us turn to how the integrands are modified under the above operation. Notice that each term in the integrand has the following form : it is a product of a rational function of the frequencies and the bath cumulants. The modification of the rational functions is simple : the rational functions are modified by complex conjugating them and then performing the above frequency redefinition. This has the effect of replacing any explicit i by (−i) and exchanging ω 1 and ω 3 .
Turning to the cumulants, time-reversal invariance implies that, when the frequencies are real, these cumulants are either purely real or purely imaginary depending on their time-reversal parity η O . Thus, the cumulants can be complex-conjugated by conjugating the frequencies in the argument of the cumulants followed by a multiplication by η O . The frequency redefinition above then results in the exchange of ω 1 and ω 3 in the arguments. To summarise, the modified integrands are obtained from the original ones by the following rules: • Replace i by −i in the coefficients, • Exchange ω 1 and ω 3 in the rational functions and the cumulants, After re-expressing the couplings in terms of these modified integrals over the same contour C 3 , one can compare them with the expression given in table 7 and find the following relations : This, in turn, implies

Time-reversal invariance of the bath in qXY model
We found in the preceding discussion that the bath needs to satisfy the following conditions for the relations given in equation (6.18) to hold true: 1. Time-translation invariance of the initial state, 2. Time-reversal invariance in the dynamics, 3. Time-reversal invariance of the initial state, 4. Time-reversal invariance of the operator that couples to the particle.
Let us check whether these conditions are satisfied in our model one by one.
As we have already mentioned, the initial state of the bath is a thermal state and hence it is invariant under time-translations.
To see the time-reversal invariance in the bath's dynamics, first note that the bath consists of two sets of harmonic oscillators. We can denote the lowering and raising operators of these oscillators by a i and a † i for the X-type oscillators and by b j and b † j for the Y-type oscillators. Therefore, the Hamiltonian of the bath is given by Now, the action of the time-reversal operator on the raising and lowering operators is as follows: which means that time-reversal is a symmetry of the dynamics. Now, the initial state is a thermal state i.e.
Therefore, the commutation of the time-reversal operator with the Hamiltonian also implies i.e. the initial state is invariant under time-reversal. Finally, the operator that couples to the particle is Here the positions of the oscillators are give by Therefore, using the transformations in (6.21) we have Example of a time-reversal odd operator coupling to the particle: We can slightly modify the qXY model to introduce a piece in the operator O that is odd under time-reversal. For this, consider the following particle-bath interaction: Integrating by parts, we see that the operator that couples to the particle's position is Here, the operatorsẊ i andẎ j are odd under time-reversal i.e.
The thermal correlators of 3 point functions O receive contributions only from terms of the form: All such terms are correlators with 3 position operators (which are time-reversal even) and one velocity operator (which is time-reversal odd). Therefore, the overall correlator satisfies the following relation: This, in turn, means that the relations in equation (6.19) are satisfied in this model.
A brief comment on couplings of quartic and higher degree terms: The fact that all the 2-OTO couplings get related to 1-OTO couplings is not generally true for higher degree terms in the particle's effective action. To see this consider the quartic couplings which receive contributions from 4-point cumulants of the operator O. Such 4-point functions can again be at most 2-OTO. But not all of these 2-OTO correlators are related to 1-OTO correlators by time-reversal. For instance, consider the following correlator: where t 1 > t 2 > t 3 > t 4 . This is a 2-OTO correlator as there are 2 future-turning point insertions in it, viz., O(t 1 ) and O(t 2 ). Under time-reversal, this gets related to the another correlator as follows: Now the correlator on RHS of the above equation also has two future-turning point insertions, viz., O(−t 4 ) and O(−t 3 ). Thus, this is a genuine 2-OTO correlator which is not related to a 1-OTO correlator under time-reversal. Such 2-OTO correlators may lead to genuinely new 2-OTO quartic couplings in the effective dynamics of the particle, unrelated to the 1-OTO coupling [59].

Consequence of KMS relations: Generalised Fluctuation-Dissipation Relations
We have already seen that, at the high temperature limit, the KMS relations between thermal 2-point functions of the operator O leads to a relation between the damping coefficient γ of the particle and the strength of the additive noise f 2 that it experiences: This is the fluctuation-dissipation relation that was originally discovered through the studies of Brownian motion by Einstein, Smoluchowski and Sutherland. An analogous relation in electrical circuits was discovered by Johnson [13] and was theoretically derived by Nyquist [14]. A general proof of such relations was worked out by Callen and Welton in [15] which was further generalised by Stratonovich [27]. All these fluctuation-dissipation relations are relations between what we now understand to be 1-OTO couplings and are rooted in the KMS relations between the 1-OTO correlators of the bath. However, in [11] it was pointed out that the KMS relations can also relate 2-OTO correlators to 1-OTO correlators. For example, consider the following 2-OTO correlator: where t 1 > t 2 > t 3 . By KMS relations, this 2-OTO correlator is related via analytic continuation to a 1-OTO correlator as follows: It is natural to wonder what imprint do such relations have on the effective dynamics of the particle. Do they lead to generalisations of the relation in (6.36)? If so, then do such relations connect 2-OTO couplings with the 1-OTO ones?
In section §3, we indeed saw such a relation between the 2-OTO coupling κ 3γ and the 1-OTO coupling ζ N at high temperature: As we mentioned there, this followed from using the KMS relations between the 3-point thermal correlators of the bath to express the couplings in terms of the spectral functions ρ[123] and ρ[321], and keeping only the leading order term in β-expansion. Thus, this relation can be considered to be a generalisation of the fluctuation-dissipation relation between the quadratic couplings in (6.36). Microscopic time-reversal invariance with η O = 1 implies λ 3 = κ 3 and κ 3γ = 2ζ γ . In this case, the above relation reduces to This is hence a relation between the coefficient ζ N of the cubic non-gaussian term in the probability distribution of the noise and the jitter ζ γ in the damping coefficient of the particle.
On the other hand, if η O = −1, then κ 3γ = 0 along with This implies that the relation in (6.39) is trivially satisfied. Apart from these relations, we also see another analogous fluctuation-dissipation relation in our model which is given by As we discussed in the last subsection, the relation between κ 3γ and λ 3γ in the second half of the above equation is a consequence of time-reversal invariance in the bath. This equation relates the cubic 1-derivative anharmonicity in the particle's motion to the jitter in its frequency. As yet, we do not know how generic this relation is. However, we suspect that, with some assumptions about the properties of the spectral functions, a general relation of the following form can be proven: to zero and hence the relation (6.43) is trivially true. Moreover, we find it to be satisfied in the qXY model and a variety of related models. Hence, we expect it to be true for a broad class of models than the one studied in this work. It will be interesting to check this expectation and determine the exact conditions which are required for this relation to hold.

Conclusion and discussion
In this paper, we have constructed an effective theory of a Brownian particle which goes beyond the standard Langevin dynamics. We remind the reader that the standard Langevin theory describes a Brownian particle subject to linear damping and a Gaussian thermal noise. The effective theory described in this paper includes in addition anharmonic couplings λ 3 and λ 3γ along with a thermal jitter ζ µ in the frequency and a jitter ζ γ in the damping constant. Apart from these parameters and the usual Langevin couplings, this theory also contains a parameter ζ N which is the strength of the non-Gaussianity in the thermal noise experienced by the particle. When out of time ordered correlations (or more specifically, 2-OTO correlators) transmitted from the bath are kept track of, one has to add three more OTO couplings κ 3 , κ 3γ and κ 3γ which are related by time-reversal to the standard (1-OTO) couplings λ 3 , λ 3γ and ζ γ respectively. To demonstrate how these couplings affect the dynamics of the particle, we have expressed the classical limits of the particle's correlators in terms of these couplings. We find that the OTO couplings show up in the out of time ordered nested Poisson brackets of the particle.
We explore the constraints imposed by microscopic time-reversal invariance of the (par-ticle+bath) combined system on the effective theory of the particle. Such an invariance of the overall dynamics of the combined system under time-reversal holds when the interaction term between the particle and the bath is even under this transformation. This fixes the OTO cubic effective couplings in terms of the parameters in the nonlinear Langevin theory via the relations κ 3 = λ 3 , κ 3γ = λ 3γ , κ 3γ = 2ζ γ .
These relations between the cubic couplings are the generalisations of the well known Onsager-Casimir reciprocal relations that originate from the microscopic time-reversal invariance of the combined system.
Since the bath is in a thermal state, the bath correlators satisfy the KMS relations. In the high temperature limit, these further give rise to a generalised fluctuation-dissipation relation between the 2-OTO cubic derivative coupling κ 3γ and the 1-OTO cubic nonderivative coupling ζ N . Combining time-reversal invariance and the generalised fluctuationdissipation relation, the coefficient of the thermal jitter ζ γ in the damping term of the nonlinear Langevin equation gets related to the coefficient of non-Gaussianity ζ N of thermal noise.
To provide a concrete model where these general results are justified, we have constructed an OTO-effective theory of a Brownian particle interacting with a dissipative thermal bath composed of two sets of harmonic oscillators. To this, we add a small 3-body interaction coupling the particle to two other oscillators, one from each set. For this model, we show that all the bath correlators decay exponentially at late times, leading to a local non-unitary effective theory for the particle.
Working out the effective couplings of the particle in this model, we find that the above mentioned relations between these couplings are indeed satisfied. Furthermore, in this model, we find another fluctuation-dissipation type relation between the strength (ζ µ ) of the thermal jitter in the frequency and the coefficient (λ 3γ ) of the anharmonic term with a single time derivative.
An immediate future direction would be to check whether the relations between the effective couplings (arising from time-reversal invariance and thermality of the bath) are valid at higher orders in the system-bath coupling. Moreover, it will be interesting to explore quartic and higher order terms in the effective action of the particle. At the quartic order, there will be genuinely new 2-OTO effective couplings that are not fixed in terms of the Schwinger-Keldysh 1-OTO effective couplings by time-reversal invariance.
It will be useful to understand the generalisation of fluctuation-dissipation relations and the consequence of time reversal invariance in the quartic case. Furthermore, such an extension to quartic terms in the effective theory may lead to observing chaotic motion of the particle 13 . If that is indeed the case, then it may be insightful to connect the already existing studies on quantum chaos based on OTOCs with this example.
Our techniques can potentially be extended to the context of quantum optics [60,61]. As a toy model one can consider an atom that is interacting electromagnetically with a gas of photons [62][63][64][65]. One can write down an effective theory for the atom and calculate its OTOCs. Such predictions for the behaviour of OTOCs may be verified with development in experimental techniques to measure them [66][67][68][69][70][71]. Moreover, our prediction of a generalised fluctuation-dissipation relation between the thermal jitter in the damping and the nongaussianity in the noise may be testable in such a setup.
Another possible extension is to 1-D spin chains in a thermal environment [72][73][74]. It will be interesting to explore the possibility of writing a similar effective theory of the chain (or a part of it) which would aid in studying the thermalisation of its OTOCs and comparing them with those of time-ordered correlators. Such comparisons may be useful in classifying systems according to the behaviour of their OTOCs.
In this work, we saw that the OTO couplings are not just a feature of the quantum mechanical theory of the particle, but they show up in out of time ordered Poisson brackets in the classical limit as well (Similar classical limits of OTOCs have been discussed in [75][76][77][78]). A stochastic interpretation of this classical OTO behaviour would be useful in understanding the significance of such OTOCs in the quantum mechanical framework (where OTO dynamics has been mostly studied up till now) as well as devising experiments to measure them. In this context, it will also be interesting to extend the idea of decoherence of quantum systems to their out of time ordered dynamics.
One potentially useful way to study the OTO dynamics of open systems connected to holographic baths would be to use the AdS/CFT correspondence for mapping this to a problem in gravity. It would be nice to systematically derive the spectral functions from gravity following [79][80][81] and calculate OTOCs in AdS gravity theories.
Another interesting direction to pursue would be to systematically develop diagrammatic methods for OTO perturbation theory [57]. This would provide a simpler way of computing OTOCs of the system. It would also be nice to derive cutting rules for OTOCs along the lines of [53,82,83].

B Structure of 1PI effective action
The dynamics of the combined system of particle and bath obeys unitary time evolution. Consequently the 1-PI effective action of the Brownian particle has to satisfy the following conditions [9,10,12,55,84].

Collapse Conditions :
The contour-ordered correlator of the particle remains unchanged when one slides an operator insertion across a turning point of the time contour. This is true, provided as we move from one leg to another, we return to the same real time instant and there are no operator insertions in between. At the level of the 1-PI effective action this implies that under identification of the degrees of freedom on any two consecutive legs, the effective action becomes independent of the common degree of freedom of these two legs i.e. the effective action becomes independent ofq under any one of the following identifications [9,10,12,84]: Under any of these collapses, the 1-PI effective action reduces to the Schwinger-Keldysh 1-PI effective action for the degrees of freedom on the remaining two legs. These rules further impose the following relations between the couplings [9,55]:

Reality condition :
A correlation function of hermitian operators gets complex conjugated when the operators are inserted in the reverse order within the correlation function. This is assured when the 1PI effective action picks up a negative sign under simultaneous complex conjugation of all the effective couplings and exchange of the degrees of freedom in the following way: This constraint is satisfied when the effective couplings F , γ and κ 3 are real.

C Contour integrals and Poles for the Effective Couplings
In this appendix, we will present some more details about the contour integrals that have to be evaluated to obtain the effective couplings in our model. For each coupling, we write down the explicit integrals over which double contour integration 14 needs to be performed. We will also tabulate the poles in each integrand whose residues add to give the final coupling, as we perform the contour integral first over ω 1 and then over ω 3 . We will write our integrands performing an integral over the frequency delta function: (C.1) The explicit integrals for the cubic couplings are given by the following expressions :