## SciPost Submission Page

# Emergence of PT-symmetry breaking in open quantum systems

### by Julian Huber, Peter Kirton, Stefan Rotter, Peter Rabl

### Submission summary

As Contributors: | Julian Huber |

Preprint link: | scipost_202008_00009v1 |

Date submitted: | 2020-08-17 10:45 |

Submitted by: | Huber, Julian |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Quantum Physics |

Approach: | Theoretical |

### Abstract

The effect of PT-symmetry breaking in coupled systems with balanced gain and loss has recently attracted considerable attention and has been demonstrated in various photonic, electrical and mechanical systems in the classical regime. Here we generalize the definition of PT symmetry to finite-dimensional open quantum systems, which are described by a Markovian master equation. Specifically, we show that the invariance of this master equation under a certain symmetry transformation implies the existence of stationary states with preserved and broken parity symmetry. As the dimension of the Hilbert space grows, the transition between these two limiting phases becomes increasingly sharp and the classically expected PT-symmetry breaking transition is recovered. This quantum-to-classical correspondence allows us to establish a common theoretical framework to identify and accurately describe PT-symmetry breaking effects in a large variety of physical systems, operated both in the classical and quantum regimes.

###### Current status:

### List of changes

We added a brief discussion below Eq. (5) to emphasize the difference between our symmetry and particle-hole exchange symmetry.

We added several recent references on the topic of symmetries of non-Hermitian Hamiltonians and master equations for fermionic systems. We now also state explicitly, why our work is different and goes beyond such classification schemes.

We added a new appendix with mean field equations of motion of bosons, fermions and spins, and discuss the symmetry of these effective Hamiltonians.

We corrected a factor of 2 in Eq. 1,2,10,24-26,20.

### Submission & Refereeing History

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## Reports on this Submission

### Anonymous Report 1 on 2020-8-21 Invited Report

### Strengths

The paper maintains its strengths from the original submission

### Weaknesses

The main weakness in the original submission has been partially addressed, and it appears that the remainder can be rectified by straightforward extension of the discussion.

### Report

The reply and revisions address the question of the symmetry to a good extend. As stated earlier in terms of all other aspects this paper deserves publication. However, I strongly advice the authors to fully resolve this issue by further extending the discussion below Eq. (5).

* 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.

* 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.

* 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?

### Requested changes

1- (strongly recommended but optional) Further clarify the definition of the symmetry, in particular regarding the role of different entries in L in eq (4).

2- (optional) Explain the role of P in this construction.

(in reply to Report 1 on 2020-08-21)

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.