Algebraic analysis of non-Hermitian quadratic Hamiltonians

Francisco Marcelo Fernández

Paulo E G Assis

A class of non-Hermitian quadratic su (2) Hamiltonians having an anti-linear symmetry is constructed. This is achieved by analysing the possible symmetries of such systems in terms of automorphisms of the algebra. In fact, different realizations for this type of symmetry are obtained, including the natural occurrence of charge conjugation together with parity and time reversal. Once specified, the underlying anti-linear symmetry of the Hamiltonian, the former, if unbroken, leads to a purely real spectrum and the latter can be mapped to a Hermitian counterpart by, amongst other possibilities, a similarity transformation. Here, Lie-algebraic methods which were used to investigate the generalized Swanson Hamiltonian ( Assis and Fring 2009 J. Phys. A: Math. Theor . 42 015203 ) are employed to identify the class of quadratic Hamiltonians that allow for such a mapping to the Hermitian counterpart. Whereas for the linear su (2) system, every Hamiltonian of this type can be mapped to a Hermitian counterpart by a transformation which is itself an exponential of a linear combination of su (2) generators, the situation is more complicated for quadratic Hamiltonians. Therefore, the possibility of more elaborate similarity transformations, including quadratic exponents, is also explored in detail. The existence of finite-dimensional representations for the su (2) Hamiltonian, as opposed to the su (1, 1) studied before, allows for comparison with explicit diagonalization results for finite matrices. Finally, the similarity transformations constructed are compared with the analogue of Swanson's method for exact diagonalization of the problem, establishing a simple relation between both approaches.

Junqing Li, Yan-Gang Miao, Zhao Xue

Nimrod Moiseyev

Mark S. Swanson

The non-Hermitian quadratic Hamiltonian H=ωa†a+αa2+βa†2 is analyzed, where a† and a are harmonic oscillator creation and annihilation operators and ω, α, and β are real constants. For the case that ω2−4αβ⩾0, it is shown using operator techniques that the Hamiltonian possesses real and positive eigenvalues. A generalized Bogoliubov transformation allows the energy eigenstates to be constructed from the algebra and states of the harmonic oscillator. The eigenstates are shown to possess an imaginary norm for a large range of the parameter space. Finding the orthonormal dual space allows the inner product to be redefined using the complexification procedure of Bender et al. for non-Hermitian Hamiltonians. Transition probabilities governed by H are shown to be manifestly unitary when the complexification procedure is followed. A specific transition element between harmonic oscillator states is evaluated for both the Hermitian and non-Hermitian cases to identify the differences in time evolution.

Alessandro Sergi

In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg's or the Schrödinger's picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized (non-Hamiltonian) commutators. These results provide a general structure (a template) for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.