Non-Hermitian model for asymmetric tunneling

Abstract

We present a simple non-hermitian model to describe the phenomenon of asymmetric tunneling between two energy-degenerate sites coupled by a non-reciprocal interaction without dissipation. The system was described using a biorthogonal family of energy eigenvectors, the dynamics of the system was determined by the Schrödinger equation, and unitarity was effectively restored by proper normalization of the state vectors. The results show that the tunneling rates are indeed asymmetrical in this model, leading to an equilibrium that displays unequal occupation of the degenerate systems even in the absence of external interactions.

Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians

“…Hopefully, the analysis of this situation will be undertaken soon. 4. S ϕ and S Ψ are bounded, strictly positive, self-adjoint, and invertible.…”

“…As a matter of fact, more than one such normalization will now be considered, each one related to a possible different definition of the transition probability in the present context. We should also stress that the necessity of introducing some normalization in similar contexts was already clear to many authors, already several years ago, [26], and recently reconsidered by other authors, [4], as well as by many others. In practice, going from self-adjoint to non self-adjoint hamiltonians opens the possibility of having several possible (apparently) inequivalent definitions of transition probabilities, all of which appear to be absolutely reasonable.…”

“…The system we will discuss here was considered recently in [4], and it is described, in its simplified version, by the non self-adjoint hamiltonian…”

“…This is, in fact, the core of our paper since it could be used, in principle, to deduce which are the correct Hilbert space, scalar product, norm and adjoint, or, more explicitly, which definitions reproduce the experimental data. This proposal is made more precise in Section 4, with the aid of an explicit example, originally introduced in [4] and discussed here adopting a simple and general pseudo-fermionic representation. Section 5 contains our conclusions.…”

Macroscopic quantum tunneling escape of Bose-Einstein condensates

“…For example, the use of TEBD to simulate the Bose-Hubbard model for superfluid decay, has confirmed numerical limits on instanton computations [19,20]. Non-Hermitian quantum mechanics, which is frequently used in scattering problems and now applied to tunneling problems as an effective model, is broadening the view of tunneling phenomena, such as its use in asymmetric tunneling and interchain pair tunneling [21,22]. Another possible many-body method, the time-adaptive MCTDHB, was used in our Josephson example above [12].…”