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arXiv:1604.07744

Exceptional Points in a non-Hermitian extension of the Jaynes-Cummings Hamiltonian

Fabio Bagarello, Francesco Gargano, Margherita Lattuca, Lucia Rizzuto, Roberto Passante, Spagnolo, Salvatore

Published Apr 26, 2016• 0 citations
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Abstract

We consider a generalization of the non-Hermitian ${\mathcal PT}$ symmetric Jaynes-Cummings {Hamiltonian, recently introduced for studying optical phenomena with time-dependent physical parameters, that includes environment-induced decay}. In particular, we investigate the interaction of a two-level fermionic system (such as a two-level atom) with a single bosonic field mode in a cavity. The states of {the} two-level system are allowed to decay because of the interaction with the environment, {and this is included phenomenologically in our non-Hermitian Hamiltonian by introducing complex energies for the fermion system. We focus our attention} on the occurrence of exceptional points in the spectrum of the Hamiltonian, clarifying its mathematical and physical meaning.

DOI
10.48550/arxiv.1604.07744

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PT\mathcal{PT}PT-Symmetric Spin--Boson Model with a Continuous Bosonic Spectrum: Exceptional Points and Dynamics
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Quantum physics can be extended into the complex domain by considering non-Hermitian Hamiltonians that are PT\mathcal{PT}PT-symmetric. These exhibit exceptional points (EPs) where the eigenspectrum changes from purely real to purely imaginary values and have useful properties enabling applications such as accelerated entanglement generation and the delay of the sudden death of entanglement in noisy systems. An interesting question is whether similar beneficial effects can be achieved away from EPs, since this would extend the available parameter space and make experiments more accessible. We investigate this by considering a PT\mathcal{PT}PT-symmetric optomechanical system but also consider what happens when two-mode squeezing interactions are included, taking us into the pseudo-Hermitian regime. The addition of squeezing is motivated by an attempt to extend the lifetime of the system's entanglement. While this does not prove to be the case, rich dynamics are nonetheless observed in both the pseudo-Hermitian and PT\mathcal{PT}PT-symmetric systems, including the sudden death and revival of entanglement under certain conditions. In both cases, we find that the sudden disappearance of entanglement can be mitigated at EPs, and also show that the revival of entanglement is quite robust to thermal noise in a group of parameters away from the EPs. This investigation extends our understanding of non-Hermitian systems and opens a new perspective for the development of quantum devices in non-Hermitian systems even away from EPs.
Abstract We provide the first solution of a time-dependent metric operator for the non-Hermitian Jaynes–Cummings Hamiltonian. We use this solution to calculate the entanglement between two identical isolated such Hamiltonians. The presence of a non-Hermitian interaction term leads to a spontaneously broken PT -symmetric regime which manifests itself in the exotic time-evolution of entanglement. When the symmetry is broken, oscillatory modes transition into decay. As such, there is a drastic difference in behaviour between the broken and unbroken regimes.
We study a non-Hermitian PT−PT-PT−symmetric generalization of an NNN-particle, two-mode Bose-Hubbard system, modeling for example a Bose-Einstein condensate in a double well potential coupled to a continuum via a sink in one of the wells and a source in the other. The effect of the interplay between the particle interaction and the non-Hermiticity on characteristic features of the spectrum is analyzed drawing special attention to the occurrence and unfolding of exceptional points (EPs). We find that for vanishing particle interaction there are only two EPs of order N+1N+1N+1 which under perturbation unfold either into [(N+1)/2][(N+1)/2][(N+1)/2] eigenvalue pairs (and in case of N+1N+1N+1 odd, into an additional zero-eigenvalue) or into eigenvalue triplets (third-order eigenvalue rings) and (N+1)mod  3(N+1)\mod 3(N+1)mod3 single eigenvalues, depending on the direction of the perturbation in parameter space. This behavior is described analytically using perturbational techniques. More general EP unfoldings into eigenvalue rings up to (N+1)(N+1)(N+1)th order are indicated.