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arXiv:1710.11266 · Physical Review A

Spectrum and normal modes of non-Hermitian quadratic boson operators

Javier Gonzalo Garcia, R. Rossignoli

Published Dec 26, 2017• 12 citations
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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 a harmonic regime where both $H$ and ${H}^{\ifmmode\dagger\else\textdagger\fi{}}$ have a discrete spectrum with biorthogonal eigenstates, and a coherent-like regime where either $H$ or ${H}^{\ifmmode\dagger\else\textdagger\fi{}}$ have a continuous complex twofold degenerate spectrum, while its adjoint has no convergent eigenstates. These regimes reflect the nature of the pertinent normal boson operators. Nondiagonalizable cases as well critical boundary sectors separating these regimes are also analyzed. The extension to $N$-dimensional quadratic systems is also discussed.

DOI
10.1103/physreva.96.062130

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We analyze the spectrum and normal-mode representation of general quadratic bosonic forms HHH not necessarily Hermitian. It is shown that, in the one-dimensional case, such forms exhibit either a harmonic regime where both HHH and H\ifmmode†\else\textdagger\fi{H}^{\ifmmode\dagger\else\textdagger\fi{}}H\ifmmode†\else\textdagger\fi have a discrete spectrum with biorthogonal eigenstates, and a coherent-like regime where either HHH or H\ifmmode†\else\textdagger\fi{H}^{\ifmmode\dagger\else\textdagger\fi{}}H\ifmmode†\else\textdagger\fi have a continuous complex twofold degenerate spectrum, while its adjoint has no convergent eigenstates. These regimes reflect the nature of the pertinent normal boson operators. Nondiagonalizable cases as well critical boundary sectors separating these regimes are also analyzed. The extension to NNN-dimensional quadratic systems is also discussed.
We discuss the necessity of using nonstandard boson operators for diagonalizing quadratic bosonic forms which are not positive definite and its convenience for describing the temporal evolution of the system. Such operators correspond to non-Hermitian coordinates and momenta and are associated with complex frequencies. As application, we examine a bosonic version of a BCS-like pairing Hamiltonian, which, in contrast with the fermionic case, is stable just for limited values of the gap parameter and requires the use of the present extended treatment for a general diagonal representation. The dynamical stability of such forms and the occurrence of nondiagonalizable cases are also discussed.
Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.
We propose a unified approach to exactly diagonalize generalized non-Hermitian so(3,2) models. This approach is a series of similarity transformations, which is constructed by some similarity transformation operators associated with su(1,1) and su(2) subalgebras of so(3,2) Lie algebra. During this diagonalization, it is worth noting that a key step is to get rid of the terms Eˆ± and Fˆ± together via the proper similarity transformations first. In this way, exact solutions of the non-Hermitian so(3,2) models are obtained. Meanwhile we give the corresponding eigenstates, which are regarded as Lie algebra so(3,2) coherent-like number states. The results can cover the generic form of the eigenvalues and eigenstates to the generalized non-Hermitian two-mode boson systems with the discrete spectrum, including 2D PT-symmetric and non-PT-symmetric oscillators as the special cases. Also they are true for the Hermitian case.
Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.
Abstract Unlike their fermionic counterparts, the dynamics of Hermitian quadratic bosonic Hamiltonians are governed by a generally non-Hermitian Bogoliubov-de Gennes effective Hamiltonian. This underlying non-Hermiticity gives rise to a dynamically stable regime, whereby all observables undergo bounded evolution in time, and a dynamically unstable one, whereby evolution is unbounded for at least some observables. We show that stability-to-instability transitions may be classified in terms of a suitably generalized P T symmetry , which can be broken when diagonalizability is lost at exceptional points in parameter space, but also when degenerate real eigenvalues split off the real axis while the system remains diagonalizable. By leveraging tools from Krein stability theory in indefinite inner-product spaces, we introduce an indicator of stability phase transitions, which naturally extends the notion of phase rigidity from non-Hermitian quantum mechanics to the bosonic setting. As a paradigmatic example, we fully characterize the stability phase diagram of a bosonic analogue to the Kitaev–Majorana chain under a wide class of boundary conditions. In particular, we establish a connection between phase-dependent transport properties and the onset of instability, and argue that stable regions in parameter space become of measure zero in the thermodynamic limit. Our analysis also reveals that boundary conditions that support Majorana zero modes in the fermionic Kitaev chain are precisely the same that support stability in the bosonic chain.
The relevance in physics of non-Hermitian operators with real eigenvalues is being widely recognized not only in quantum mechanics but also in other areas, such as quantum optics, quantum fluid dynamics, and quantum field theory. In this note, a quantum system described by a non-Hermitian Hamiltonian, which is constituted by two types of interacting bosons, is investigated. The real eigenvalues of the Hamiltonian are explicitly determined, as well as complete biorthogonal sets of eigenfunctions of the Hamiltonian and its adjoint. The diagonal representation of H is obtained using pseudo-bosonic operators.