SciPost Submission Page
Emergence of PT-symmetry breaking in open quantum systems
by Julian Huber, Peter Kirton, Stefan Rotter, Peter Rabl
This is not the current version.
|As Contributors:||Julian Huber · Peter Rabl|
|Date submitted:||2020-08-17 10:45|
|Submitted by:||Huber, Julian|
|Submitted to:||SciPost Physics|
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.
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 2 on 2020-9-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202008_00009v1, delivered 2020-09-23, doi: 10.21468/SciPost.Report.2019
proposal of an interesting new concept (Liouvillian PT symmetry - of the steady state)
perspective with respect to existing literature/concepts could be more accurate
This is no doubt a very interesting paper and in my opinion deserves publication in SciPost. It proposes a new concept of the Liouvillean PT symmetry, which can be defined in particular with respect to the steady state. This symmetry (in the steady state) can be spontaneously broken, which can happen as a crossover - or in a suitable thermodynamic limit - as a particular non-equilibrium phase transition.
I also fully agree with the Authors' response to the criticism of the first referee.
The paper refers to Ref.  which proposed an alternative Liouvillian PT symmetry (see also a companion paper Phys. Rev. A 86, 044103 to Ref.  which includes general types of physical examples and which could also be cited). The relation between the concept of Ref.  and the current one is perhaps not clearly presented here. The point is that Ref. discusses the PT symmetry of the Liouvillean superoperator, and as a consequence finds a general dihedral symmetry of the Liouvillian spectrum and a spontaneous symmetry breaking transition when the spectrum leaves the "cross", while the current manuscript discusses the PT symmetry of the steady state density operator (which inherits the equilibrium physics of some corresponding closed system Hamiltonian) and hence is not (so) relevant for Liouvillian dynamics.
This point of difference should be made clear. I think simply saying that Ref. has no "immediate physical interpretation or classical correspondence" (as oposed to a concept proposed here?) is misleading.
Anonymous Report 1 on 2020-8-21 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202008_00009v1, delivered 2020-08-21, doi: 10.21468/SciPost.Report.1929
The paper maintains its strengths from the original submission
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.
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?
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.