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|
|Arxiv Link:||https://arxiv.org/abs/2003.02265v2 (pdf)|
|Date submitted:||2020-05-29 02:00|
|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.
Submission & Refereeing History
You are currently on this page
Reports on this Submission
Anonymous Report 1 on 2020-6-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2003.02265v2, delivered 2020-06-14, doi: 10.21468/SciPost.Report.1761
1- Addresses a topical question on quantum dynamics with gain and loss
2- Provides a clear definition of a physically interesting symmetry
3- Illustrates the consequences in meaningful examples
1- The symmetry employed in this paper is generally known as a particle-hole/charge-conjugation symmetry, which is a well-established symmetry in physics. It should not be confused with a PT symmetry.
2- Some details of the discussion and context, as outlined in the report.
PT symmetric quantum systems were originally introduced as a non-Hermitian generalization of conventional quantum mechanics built upon Hermitian operators. While no fundamentally PT symmetric quantum systems are known, it was later realized that the symmetry occurs naturally in open systems, such as optical systems with gain and loss. However, the underlying dynamics of these systems is classical, so that at this stage the concept was transferred only on a purely phenomenological level.
In this paper the authors consider the question of how to define the PT symmetry in the corresponding quantum setting of these open systems. This question can be pursued along two routes: quantization around a still PT-symmetric mean-field description, or, as here, based on the non-Hermitian dynamical matrix that occurs in a Lindblad master equation, which itself does not obey PT symmetry. The authors address the interesting question if a useful definition of PT symmetry can still be given in the latter case, and suggest that this can be achieved by a transformation that interchanges creation and annihilation operators. By well-selected examples, they demonstrate that this is a physically interesting symmetry, which results in a clear phenomenology.
On the technical level, this is a well-carried out study. However, the paper suffers from a fundamental flaw, namely, that the suggested symmetry has already a well-established place in physics as a particle-hole or charge-conjugation symmetry, which should be seen as clearly distinct from PT symmetry. Indeed, besides being well established for conventional quantum systems, where this particle-hole/charge-conjugation symmetry is crucial for their 10-fold classification, it also features prominently as an independent symmetry (again distinct from any generalized time reversal symmetry such as PT) in recent classifications on non-Hermitian systems. The dressing of this symmetry with the P operator is an interesting twist, but does not change the fundamental nature of the symmetry, in the same way that fundamentally, a PT symmetry is still a time-reversal symmetry.
I consider this a major flaw that has to be addressed before this work can be considered for publication.
Minor remarks: The authors describe PT-symmetry breaking as a transition where eigenvalues become purely imaginary (they mention this twice in the introduction). However, PT-symmetry breaking generally results in complex-conjugated eigenvalues. As a matter of fact, their notion seems to be a symptom of the confusion with the particle-hole or charge-conjugation symmetry, which on the mean-field level gives rise to an anti-PT symmetric Hamiltonian. The latter makes the spectrum symmetric to the imaginary axis; if PT and anti-PT symmetry occur at the same time, eigenvalues are on the real axis, imaginary axis, or form quadruplets in the complex plane.
In the introduction, the point about finite-dimensional Hilbert spaces is unclear; some of the cited PT-quantum optics papers address resonators that have an infinite Hilbert space.
The discussion of the examples could benefit from a brief comparison with the phenomenology of superradiance.
I sketch one possible resolution of this problem; however, there may be other ways to place this work into the proper context:
1- Change the focus of this paper to the consideration of symmetry-breaking transitions in systems with a particle-hole/charge conjugation symmetry. This would indeed be very interesting as quantum noise has been much less studied in this setting.
2-This context would require to expand the references with works on non-Hermitian particle-hole and charge-conjugation symmetry, their interplay with PT symmetry, and their role in present classifications of non-Hermitian systems.
3-If still relevant after such revisions, I would also encourage to mention the mean-field version of the effective Hamiltonian in the Lindblad equation, which can be PT symmetric so that from this perspective there is no conceptual problem to transfer this symmetry to the quantum setting.
4- In the discussion of the examples, the authors should briefly describe the relation or distinction between the phenomenological effects of breaking the symmetry in question, and the general phenomenon of superradiance.
5-Clarify the statement in the introduction about the dimensionality of Hilbert space.