Spectrum and normal modes of non-hermitian quadratic boson operators

Abstract

We analyze the spectrum and normal mode representation of general quadratic bosonic forms $H$ not necessarily hermitian. It is shown that in the one-dimensional case such forms exhibit either an harmonic regime where both $H$ and $H^{†}$ have a discrete spectrum with biorthogonal eigenstates, and a coherent-like regime where either $H$ or $H^{†}$ have a continuous complex two-fold degenerate spectrum, while its adjoint has no convergent eigenstates. These regimes reflect the nature of the pertinent normal boson operators. Non-diagonalizable cases as well critical boundary sectors separating these regimes are also analyzed. The extension to $N$-dimensional quadratic systems is as well discussed.

arXiv comment10 pages, 2 figures. Final version, to be published in Phys. Rev. A

References in corpus (46)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Unidirectional Invisibility induced by PT-Symmetric Periodic Structures
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• PT-symmetric laser-absorber
• Pseudo-Hermitian Representation of Quantum Mechanics
• Experimental Study of Active LRC Circuits with PT-Symmetries
• Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• PT-Symmetric Phonon Laser
• Biorthogonal Quantum Mechanics
• PT symmetry and spontaneous symmetry breaking in a microwave billiard
• Observation of Defect States in PT-Symmetric Optical Lattices
• On Pseudo-Hermitian Hamiltonians and Their Hermitian Counterparts
• Observation of Fast Evolution in Parity-Time-Symmetric System
• Twofold Transition in PT-Symmetric Coupled Oscillators
• Bifurcation diagram and pattern formation in superconducting wires with electric currents
• Time-dependent version of cryptohermitian quantum theory
• Bypassing the bandwidth theorem with PT symmetry
• PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues
• Stimulation of the fluctuation superconductivity by the PT-symmetry
• Consistency of PT-symmetric quantum mechanics
• The non-Hermitian Swanson model with a time-dependent metric
• $\mathcal{P}\mathcal{T}$-symmetric wave guide system with evidence of a third-order exceptional point
• Examples of Pseudo-bosons in quantum mechanics
• Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians
• Complex modes in unstable quadratic bosonic forms
• Non-Hermitian interaction representation and its use in relativistic quantum mechanics
• Non-Hermitian anisotropic XY model with intrinsic rotation-time reversal symmetry
• Tilted optical lattices with defects as realizations of PT symmetry in Bose-Einstein condensates
• Moyal products -- a new perspective on quasi-hermitian quantum mechanics
• Vortices in Bose-Einstein condensates with $\mathcal{P}\mathcal{T}$-symmetric gain and loss
• Generalized Swanson models and their solutions
• Non-hermitean hamiltonians with unitary and antiunitary symmetry
• PT-symmetric spectral singularity and negative frequency resonance
• Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation
• From Pseudo-bosons to Pseudo-Hermiticity via multiple generalized Bogoliubov transformations
• $\mathcal{P}\mathcal{T}$ symmetric phase transition and single-photon transmission in an optical trimmer system
• Dynamics of entanglement between two harmonic modes in stable and unstable regimes
• Chern number in Ising models with spatially modulated real and complex fields
• Is space-time symmetry a suitable generalization of parity-time symmetry?
• Stability, complex modes and non-separability in rotating quadratic potentials
• Non-Hermitian oscillators with $T_d$ symmetry
• Non-Hermitian description of the dynamics of inter-chain pair tunnelling
• PT-symmetric interpretation of the electromagnetic self-force
• Generalized Bogoliubov transformations versus $\mathcal{D}$% -pseudo-bosons

Cited by in corpus (11)

• Deconstructing effective non-Hermitian dynamics in quadratic bosonic Hamiltonians
• Quantum quench in PT-symmetric Luttinger liquid
• Non-equilibrium Floquet steady states of time-periodic driven Luttinger liquids
• Restoring number conservation in quadratic bosonic Hamiltonians with dualities
• Rotation-time symmetry in bosonic systems and the existence of exceptional points in the absence of $\mathcal{P}\mathcal{T}$ symmetry
• Broken Hermiticity phase transition in Bose-Hubbard model
• Wavefunction structure in quantum many-fermion systems with $k$-body interactions: conditional $q$-normal form of strength functions
• Out-of-equilibrium full counting statistics in Gaussian theories of quantum magnets
• A New Boson realization of Fusion Polynomial Algebras in Non-Hermitian Quantum Mechanics : $\gamma$-deformed $su(2)$ generators, Partial $\mathcal{P}\mathcal{T}$-symmetry and Higgs algebra
• Algebraic analysis of non-Hermitian quadratic Hamiltonians
• Bosonic Holes in Quadratic Bosonic Systems