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Definition of Current Density in Non-Hermitian Quantum Systems

Hiroto Oka

TL;DR

This work addresses the breakdown of the continuity equation in non-Hermitian quantum systems by redefining the current density to flow inside the system, ensuring CE holds under particle-number conservation. By introducing electromagnetic coupling and deriving modified Maxwell equations, the author identifies a unique current density $\widetilde{\bm{j}}$ that preserves CE and governs electromagnetic responses in NHQS. The approach highlights that non-Hermiticity originates from measurements and post-selection rather than particle gain/loss, with potential extensions to non-Abelian gauges as shown in the BRST-based appendix. The framework provides a pathway to observable electromagnetic effects in NHQS and sets the stage for further theoretical and experimental exploration.

Abstract

In recent years, non-Hermitian quantum systems (NHQS) have been actively studied. In conventional quantum mechanics, Hermiticity is a fundamental property of Hamiltonians. In NHQS, however, states evolve under non-Hermitian Hamiltonians and novel physical phenomena are predicted due to the non-Hermiticity. One difference from Hermitian systems is that the continuity equation (CE) does not hold in NHQS even when the number of particles is conserved. In this study, I extended the definition of current density so that CE holds also in NHQS. The newly defined current density does not only satisfy CE but also has physical meanings in the sense that it affects physical observables in the same manner as conventional current density.

Definition of Current Density in Non-Hermitian Quantum Systems

TL;DR

This work addresses the breakdown of the continuity equation in non-Hermitian quantum systems by redefining the current density to flow inside the system, ensuring CE holds under particle-number conservation. By introducing electromagnetic coupling and deriving modified Maxwell equations, the author identifies a unique current density that preserves CE and governs electromagnetic responses in NHQS. The approach highlights that non-Hermiticity originates from measurements and post-selection rather than particle gain/loss, with potential extensions to non-Abelian gauges as shown in the BRST-based appendix. The framework provides a pathway to observable electromagnetic effects in NHQS and sets the stage for further theoretical and experimental exploration.

Abstract

In recent years, non-Hermitian quantum systems (NHQS) have been actively studied. In conventional quantum mechanics, Hermiticity is a fundamental property of Hamiltonians. In NHQS, however, states evolve under non-Hermitian Hamiltonians and novel physical phenomena are predicted due to the non-Hermiticity. One difference from Hermitian systems is that the continuity equation (CE) does not hold in NHQS even when the number of particles is conserved. In this study, I extended the definition of current density so that CE holds also in NHQS. The newly defined current density does not only satisfy CE but also has physical meanings in the sense that it affects physical observables in the same manner as conventional current density.

Paper Structure

This paper contains 9 sections, 55 equations, 4 figures.

Table of Contents

  1. I. Introduction
  2. Non-Hermitian quantum systems in the quantum trajectory
  3. Conservation of particle number
  4. Violation of the continuity equation and current density flowing in the system
  5. I I. the method to define current density in NHQS
  6. Maxwell's equations and current density in NHQS
  7. I I I. conclusion and discussion
  8. Acknowledgement
  9. Appendix

Figures (4)

  • Figure 1: Although the interaction can change the number of particles in the system of interest, only cases where it is conserved are extracted by post-selection.
  • Figure 2: In this research, instead of incoming and outgoing current density, current density flowing in the system was introduced so that CE holds in NHQS.
  • Figure 3: Measured quantities, or classical EM field are affected by the non-Hermiticity-induced current density flowing in quantum EM filed region.
  • Figure 4: Current density corresponding to discontinuous particle density change by wave function collapse. solid and dotted line: particle density before and after a measurement.