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Out of time ordered effective dynamics of a quartic oscillator
by Bidisha Chakrabarty, Soumyadeep Chaudhuri
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Submission summary
Authors (as Contributors):  Soumyadeep Chaudhuri 
Submission information  

Arxiv Link:  https://arxiv.org/abs/1905.08307v5 (pdf) 
Date accepted:  20190717 
Date submitted:  20190711 02:00 
Submitted by:  Chaudhuri, Soumyadeep 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
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.
Published as SciPost Phys. 7, 013 (2019)
Author comments upon resubmission
Dear Editor,
We thank the referee for his/her comments.
We will begin by addressing the referee's question on whether we could have obtained the generalised fluctuationdissipation relation between the quartic SchwingerKeldysh couplings without including the OTO couplings in the analysis. Our response is as follows:
The generalised fluctuationdissipation relation essentially relies on certain relations between the bath's correlators due to a combination of microscopic reversibility and thermality. Therefore, in principle, one could use these relations between the bath's correlators to derive the generalised fluctuationdissipation relation without introducing the OTO couplings. However, as we have argued in our paper, studying these relations between the correlators of the bath requires one to include the bath's OTOCs in the analysis. As the OTO couplings of the particle encode the effects of the bath's OTOCs on the particle's dynamics, introducing them simplifies the analysis. As we have shown, these OTO couplings allow one to study the constraints imposed by the bath's microscopic reversibility and thermality separately. Combining these constraints, one can easily get the generalised fluctuationdissipation relation.
Now, we would like to mention that we have fixed the typos pointed out by the referee in his/her report. In addition, we noticed two more minor errors in the previous version which we enumerate below:
1) In equations (5), (7) and (9), we made a mistake while writing the quartic combinations of the couplings between the particle and the bath oscillators which contribute to the cumulants of the 4point correlators of O (the bath operator that couples to the particle). We have corrected them in the current version. One can verify that these corrected combinations indeed contribute to the cumulants of the 4point correlators of O by looking at the Feynman diagrams given in Figure 9 of appendix A.2.
2) We also made a mistake while showing the positions of the insertions in figure 3. We have corrected this in the current version. It is now consistent with the correlator mentioned in the caption of the same figure.
Sincerely, the Authors
List of changes
1) In the last line of the first paragraph of the introduction, we have replaced the phrase "degrees of freedom of the system" by "infrared (lowfrequency) degrees of freedom of the system".
2) In the line just below equation (19) in page 10, we have replace the phrase "Plugging these expression" by "Plugging these expressions".
3) In the the caption of figure 5, we have replaced "t?0" by "t>0".
4) In equations (5), (7) and (9), we have corrected the combination of the quartic couplings between the particle and the bath oscillators which contribute to the cumulants of the 4point correlators of O (the bath operator that couples to the particle).
5) In figure 3, we have corrected the positions of the insertions.
6) We have updated the references.