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Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians. I linear amplification

by L. F. Alves da Silva, R. A. Dourado, and M. H. Y. Moussa

This is not the current version.

Submission summary

As Contributors: Rodrigo Dourado · Miled Moussa
Preprint link: scipost_202110_00013v1
Date submitted: 2021-10-11 17:03
Submitted by: Moussa, Miled
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Quantum Physics
Approach: Theoretical

Abstract

In this work we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for a TD non-Hermitian Hamiltonian H(t). Assuming H(t) to be simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t)=Ξ⁻¹(t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J. Math, Phys. 43, 3944 (2002)] for the time-independent (TI) scenario. We apply our method for the derivaton of the symmetry associated with a TD non-Hermitian linear Hamiltonian: a cavity field under linear amplification. The computed TD symmetry operator is then particularized for the equivalent TI linear Hamiltonian and its PT-symmetric restriction. In this TI scenario we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator: I^{2k}=1 with k odd [J. Phys. A 35, L467 (2002)]. The results here derived together with those in the sequel, where we extend our analysis for a TD non-Hermitian quadratic Hamiltonian, enables us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians.

Current status:
Has been resubmitted


Submission & Refereeing History

Resubmission scipost_202110_00013v2 on 1 December 2021

Reports on this Submission

Anonymous Report 1 on 2021-11-12 (Invited Report)

Strengths

This is the first attempt to extend two basic theorems on time-independent non-Hermitian operators to time-dependent Hamiltonian operators.

Weaknesses

Lacks physical motivation. Its presentation may be substantially improved. The paper can made much shorter.

Report

The current version does not meet the standards of the journal. But I think the authors can be given a chance to respond to the criticisms I have listed in my report. An improved version may qualify for publication in SciencePost and other more specialized journals.

Requested changes

See the attached PDF.

Attachment


  • validity: ok
  • significance: ok
  • originality: good
  • clarity: ok
  • formatting: acceptable
  • grammar: below threshold

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Comments

Anonymous on 2021-11-17  [id 1949]

The paper first presents briefly a formulation of a time-dependent anti-linear symmetric operator. Then the paper describes in details the computation of the time-dependent non-Hermitian Hamiltonian of a cavity field.

I don't recommend its publication.
First of all, the paper is difficult to understand because of bad presentation. I didn't quite understand the derivation of the time-dependent non-linear symmetric operator because the derivation in Sec. III is not described in detail. Then the calculation for the cavity field in Secs. V and VI is too much in detail.

After all, I don't see what is achieved by the authors' method. The authors frequently emphasize that their formulation is consistent with results in other papers, but I don't see if the authors' formulation is the unique extension. As another point, I don't see if the calculation in Secs. V and VI for a physical model is only achieved by the authors' formulation. The bottom line is "so what?"