Emergence of PT-symmetry breaking in open quantum systems

We would like to thank the referee for his/her overall very positive assessment of our work and for recommending our manuscript for publication. A remaining issue raised by the referee concerns the relation between our work and the alternative definition of PT symmetry introduced by T. Prosen in Ref. [33] (Ref. [30] in the previous version). In brief, we consider the following essential differences:

1) Ref. [30] introduces a symmetry for Liouville operators, which is primarily motivated by mathematical considerations. For example, the T-operation is implemented by taking the adjoint of the Liouville operator, i.e. T: L -> L^\dag. In general, the adjoint of a Liouvillian is no longer a trace-preserving superoperator. Also shifting a Liouvillian by a constant amount does not result in a valid master equation. In contrast, Eq. (4) in our work maps any Liouville operator into another physically consistent Liouvillian and the T-operation has the simple interpretation of replacing gain by loss processes.

2) The symmetry defined in Ref. [30] implies the existence of a cross-like pattern of the Liouvillian eigenvalues for very small \Gamma. For larger \Gamma this pattern can be broken. However, all the eigenvalues still have a negative real part and there is no transition from a purely oscillatory to a damped regime. Further the symmetry of Ref. [30] seems to be restricted to spin ½ systems. In none of our examples with local dimension d greater than 2 we see the behavior discussed in Ref. [30]. Therefore, there seems no apparent connection between the breaking of this pattern and conventional PT-symmetry breaking in classical systems. With our definition this connection can be established in the semiclassical limit.

3) Finally, let us emphasize that while we focus primarily on steady states, we also see PT-symmetry breaking effects in the dynamics of observables. This is illustrated by the example shown in Fig. 4, where we find a transition from an oscillatory to a damped dynamics. It is the purpose of this plot to illustrate that there is no simple relation between the structure of Liouvillian eigenvalues and the dynamics of observables. Unfortunately, such a comparison is not given in Ref. [33], while in the example described in Ref. [47] no qualitative change in the dynamics is observed even for very drastic changes in the eigenvalue patterns.

We agree with the referee that our previous statement concerning this work was too brief and uninformative. In the revised version of the manuscript we have considerable extended this paragraph to explain the differences between this earlier work and the present one. Corresponding statements were also added to the conclusion section.

First of all, we would like to thank the referee for his/her thoughtful second report and for recommending our manuscript for publication. Here below, we address the suggested amendments for the manuscript, which we found very helpful.


Referee: “For the arguments in the reply to be true, it seems crucial that one distinguishes the operators appearing in the jumps and in H; indeed, it seems that the author's example of c^dagger c transform differently depending on whether it is transformed according to the first entry in L in Eq. (4), i.e., part of H, or according to the second and third entry, i.e., is part of the jumps. I suppose this can be done by insisting that H generates the unitary part of the evolution, but the authors should clarify if this does not leave any room for ambiguity, (especially, if the jump operators are dressed by the particle number).
Even better would be a definition of the symmetry discussed here without invoking the precise structure of the Liouvillian, (e.g., by utilizing a clear distinction of the unitary and nonunitary parts of the evolution, only, and then avoiding any possible ambiguities of whether operators are transformed separately or collectively, given that f(Transf(c))/= Transf(f(c)) for the transformations in question). However, I could accept if this possibility is only commented upon, and then left for future consideration.”

Our reply: In the model in Eq. (1) we specify the Liouvillian L in terms of the usual Hermitian Hamiltonian H, which generates the unitary evolution, and the two local jump operators describing dissipation. So given H, c_A and c_B, the Liouvillian is uniquely determined. Based on this specification, the symmetry in Eq. (4) is clearly defined by applying the PT operation in Eq. (3) to the Hamiltonian and to each jump operator separately. The transformed Hamiltonian H’=PT(H), and the transformed jump-operations, c_A’=PT(c_A) and c_B’=PT(c_B), then uniquely define the transformed Liouville operator, L(H’;c_A’,c_B’). This is the procedure as described in the paper that should leave no ambiguity in terms of the transformation.
Please note, in particular, that the Lindblad from of the master equation, as used as a starting point of our paper, represents exactly the distinction between a unitary Hamiltonian evolution and a non-unitary, dissipative part, as the requested by the referee.
In the revised version of the manuscript we now emphasize more clearly below Eq. (1) that H denotes the Hermitian Hamiltonian. Also, we have extended the discussion below Eq. (5), to explain more clearly how to apply the symmetry transformation and that PT acts on the Hamiltonian and the jump operators separately. We hope that these additional details will avoid any further confusion.


Referee: “In describing how their symmetry differ, it would be useful if the authors would make more explicit contact to particle-hole symmetry. Given that in many practical settings jump operators are indeed creation and annihilation operators, it appears to me that the rules in Eq. (5) are nonetheless reminiscent of a particle-hole operation, as adopted in nonhermitian classifications as in PRX 9, 041015 (2019) [Eq. 11], and also in the spirit of the AZ classification as clarified by Z in arXiv:2004.07107. This should be acknowledged a bit more explicitly, simply to further guide the interpretation of the reader.”

Our reply: In the discussion below Eq. (5) we now mention both the differences to PHS, but also point out that in some basic examples our PT symmetry and PHS symmetry is the same (see discussion in App A). Also, the references pointed out by the referee are now included in the bibliography of our manuscript.


Referee: “As a separate, but entirely optional point, I wonder if the authors could clarify the importance of the parity symmetry in this work. I understand that the systems described here have two subsystems that are related by parity. On the other hand, in terms of conventional classifications, PT symmetry is an antiunitary symmetry, with the "P" being essentially irrelevant. A possible way to show this is to adopt a P-invariant basis. Could the systems studied here be rewritten in a similar way?”

Our reply: The aim of our work is to identify the quantum mechanical analogue of PT symmetry as it is known in classical systems. To establish this analogy, including parity symmetry is the natural way forward. Note in this context that an essential point about PT-symmetric systems is the fact that they are neither P nor T-symmetric, but invariant under the combined PT-transformation. Only this feature leads to interesting effects, such as PT-symmetry breaking. The same is true at the quantum level. A master equation can be invariant under a local adjoint operation of the jump operators, but this doesn’t lead to any interesting behavior in the steady states. Of course, one could look for a basis independent way to express the same relation, where P doesn’t appear explicitly. However, in this case both the intuitive physical picture and the connection to conventional PT symmetric systems would be lost. As we believe that this may be confusing for most readers and go against the purpose of the present work, we would prefer to refrain from implementing the referee's optional suggestions.