(Non) equilibrium dynamics: a (broken) symmetry of the Keldysh generating functional

We are much grateful to both Referee 1 and Referee 2 for the time they spent writing a thorough, positive, review of our work. The appreciations of both Referees being quite similar, we answer them in a single reply.

While both Referees acknowledged the significance and the originality of our work, they also pointed out that, given the rather formal nature of our approach, additional efforts could be made to better present the physical findings. We fully agree with the Referees, and we thank them for their suggestions in that respect, which we followed (see the List of changes).

We hope that the referees will find the revised version of our manuscript suitable for publication in SciPost without further delay.


Camille Aron
Giulio Biroli
Leticia F. Cugliandolo

- We revised the introductory Section 1.1 "Motivations" to better appeal to the community of quantum thermodynamics by articulating one of our main findings, namely the identification of a quantum version of the irreversible entropy production rate, in the context of what has already been achieved in the classical realm (stochastic thermodynamics), and of the questions that are still debated in the quantum realm.

- We revised the introductory Section 1.3 "Main results", paragraph "Out-of-equilibrium dynamics", by announcing a later discussion on a concrete and simple model where our findings can be worked out and discussed explicitly, numerically or even experimentally.

- We added the corresponding discussion in Sect. 4.2 "Symmetry breaking term", where we consider a two-level system (TLS) which is driven through an avoided crossing at different rates. Despite its simplicity, the time-dependent driven TLS has two virtues:
(i) it belongs to a different `class' of systems than the bosonic and fermionic fields that are treated in the rest of the manuscript. The TLS is an SU(2) symmetric object with a more cumbersome path-integral representation, however the quantum thermodynamic operators that were derived using bosonic and fermionic fields are readily applicable to any quantum mechanical system,
(ii) the excitations created when ramping across the avoided crossing at a finite rate are of purely quantum mechanical origin (contrary to the case of, say, a driven quantum harmonic oscillator which comes with a 'classical' intuition/phenomenology). While the evolution is purely unitary, it is clear that these tunnel excitations can be associated to the production of irreversibility in the system.
The novel quantity $\langle \Sigma \rangle$ is studied numerically in different regimes, and compared to the work $\langle \mathcal{W} \rangle$. This simple model is used again in Section 4.3 "Quantum fluctuation theorems", paragraph "Jarzynski equality", to illustrate our proposed definition of irreversibility production, $\mathcal{S}_{irr}$, given in Eqs. (100) and (104).