Conserved quantities in non-Hermitian systems via vectorization method

Abstract

Open classical and quantum systems have attracted great interest in the past two decades. These include systems described by non-Hermitian Hamiltonians with parity-time $(\mathcal{P}\mathcal{T})$ symmetry that are best understood as systems with balanced, separated gain and loss. Here, we present an alternative way to characterize and derive conserved quantities, or intertwining operators, in such open systems. As a consequence, we also obtain non-Hermitian or Hermitian operators whose expectations values show single exponential time dependence. By using a simple example of a $\mathcal{P}\mathcal{T}$-symmetric dimer that arises in two distinct physical realizations, we demonstrate our procedure for static Hamiltonians and generalize it to time-periodic (Floquet) cases where intertwining operators are stroboscopically conserved. Inspired by the Lindblad density matrix equation, our approach provides a useful addition to the well-established methods for characterizing time-invariants in non-Hermitian systems.

arXiv comment7 pages, 2 figure: Proceedings of AAMP XVIII (Prague 2021)

References in corpus (27)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities
• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Visualization of Branch Points in PT-Symmetric Waveguides
• Pseudo-Hermitian Representation of Quantum Mechanics
• Experimental Study of Active LRC Circuits with PT-Symmetries
• Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms
• Physical realization of $\mathcal{P}\mathcal{T}$-symmetric potential scattering in a planar slab waveguide
• Anti-Parity-Time Symmetric Optics via Flying Atoms
• Symmetries and conserved quantities in Lindblad master equations
• Observation of parity-time symmetry breaking in a single spin system
• Exact PT-Symmetry Is Equivalent to Hermiticity
• Must a Hamiltonian be Hermitian?
• Observation of PT phase transition in a simple mechanical system
• Synthetic parity-time symmetry breaking in a single microcavity
• Observation of PT-symmetric quantum interference
• Optical waveguide arrays: quantum effects and PT symmetry breaking
• Light transport in PT-invariant photonic structures with hidden symmetries
• PT spectroscopy of the Rabi problem
• PT-symmetric Rabi model: Perturbation theory
• Construction of a unique metric in quasi-Hermitian quantum mechanics: non-existence of the charge operator in a 2 x 2 matrix model
• Passive $\mathcal{P}\mathcal{T}$-symmetry breaking transitions without exceptional points in dissipative photonic systems
• Observation of slowly decaying eigenmodes without exceptional points in Floquet, dissipative, synthetic circuits
• Floquet exceptional contours in Lindblad dynamics with time-periodic drive and dissipation
• On-demand Parity-Time symmetry in a lone oscillator through complex, synthetic gauge fields

Cited by in corpus (2)

• Formation of Exceptional Points in pseudo-Hermitian Systems
• Linear Response for pseudo-Hermitian Hamiltonian Systems: Application to PT-Symmetric Qubits