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arXiv:quant-ph/0503235 · Journal of Physics A Mathematical and General

Solvability and -symmetry in a double-well model with point interactions

Miloslav Znojil, Vít Jakubský

Published May 18, 2005• 32 citations
arXivHTMLPDF

Abstract

The concept of point interactions offers one of the most suitable guides towards a quantitative analysis of properties of certain specific non-Hermitian (so-called -symmetric) quantum-mechanical systems. This is illustrated with a double-well model, the easy solvability of which is shown to lead to a clear picture of the mechanisms of the unavoided level crossing and of the spontaneous -symmetry breaking at a certain strong-non-Hermiticity boundary. Below this limit the model is shown to be suitable for an explicit illustration of technicalities related to the standard probabilistic physical interpretation of bound states in -symmetric quantum mechanics.

DOI
10.1088/0305-4470/38/22/024

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The concept of point interactions offers one of the most suitable guides towards a quantitative analysis of properties of certain specific non-Hermitian (so-called -symmetric) quantum-mechanical systems. This is illustrated with a double-well model, the easy solvability of which is shown to lead to a clear picture of the mechanisms of the unavoided level crossing and of the spontaneous -symmetry breaking at a certain strong-non-Hermiticity boundary. Below this limit the model is shown to be suitable for an explicit illustration of technicalities related to the standard probabilistic physical interpretation of bound states in -symmetric quantum mechanics.
Within the framework of 𝒫𝒯-symmetric quantum mechanics of bound states (which works with parity-pseudo-Hermitian Hamiltonians H = 𝒫 H †𝒫 and real spectra) we mimic some effects of the double-well structure of potentials by a pair of δ functions with mutually complex conjugate strengths. The model is solvable by the standard matching technique and exhibits several interesting features. We observe an amazingly robust absence of any 𝒫𝒯-symmetry breaking. A quasi-degeneracy which occurs in the high-energy domain is interpreted as a manifestation of certain ‘quantum beats’.
Abstract We study the dynamical evolution and symmetry breaking of the non-Hermitian double Jaynes–Cummings model with symmetry. The results indicate that there exist symmetry maintained and breaking phase in this model, and the non-Hermitian terms lead to different behaviors in two phases. In the symmetry phase, there exists stable Rabi oscillation caused by the interaction between photons and atoms. At the exceptional point, the system shows critical property that the probabilities in each state over time tend to be equal. The system will undergo a transition from the symmetry phase to the symmetry breaking phase, as increasing the coupling constant. In the symmetry breaking phase, complex eigenvalues appear, and the probability in each state increases monotonically in an exponential form. This study can help us to understand the effect of symmetry on the interaction between photons and atoms in non-Hermitian systems.
Our goal is to find a model for the phenomenon of spontaneous symmetry breaking arising in one dimensional quantum mechanical problems.For this purpose we consider boundary value problems related with two interior points of the real line, symmetric with respect to the origin.This approach can be treated as a presence of singular potentials containing shifted Dirac delta functions and their derivatives.From mathematical point of view we use a technique of selfadjoint extensions applied to a symmetric differential operator which has a domain containing smooth functions vanishing in two mentioned above points.We calculate the resolvent of corresponding extension and investigate its behavior if the interior points change their positions.The domain of these extensions can contain some functions that have non differentiability or discontinuity at the points mentioned above, the latter can be interpreted as an appearance of singular potentials centered at the same points.Next, broken-symmetry bound states are discovered.More precisely, for a particular entanglement of boundary conditions, there is a ground state, generating a spontaneous symmetry breaking, stable under the phenomenon of decoherence provoked from external fluctuations.We discuss the model in the context of the "chiral" broken-symmetry states of molecules like N H 3 .We show that within a Hilbert space approach a spontaneous symmetry breaking disappears if the distance between the mentioned above interior points tends to zero.
It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT\mathcal{PT}PT-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of PT\mathcal{PT}PT-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of PT\mathcal{PT}PT symmetry have been applied in the context of many different branches of physics. This review introduces the concepts of PT\mathcal{PT}PT symmetry by focusing on elementary one-dimensional PT\mathcal{PT}PT-symmetric quantum and classical mechanics and relies, in particular, on oscillator models to illustrate and explain the basic properties of PT\mathcal{PT}PT-symmetric quantum theory.
It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of PT-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this Review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of PT symmetry have been applied in the context of many different branches of physics. This Review introduces the concepts of PT symmetry by focusing on elementary one-dimensional PT-symmetric quantum and classical mechanics and relies in particular on oscillator models to illustrate and explain the basic properties of PT-symmetric quantum theory.
One of the most characteristic mathematical features of the PT\mathcal{PT}PT-symmetric quantum mechanics is the explicit Hamiltonian dependence of its physical Hilbert space of states H=H(H)\mathcal{H}=\mathcal{H}(H)H=H(H). Some of the most important physical consequences are discussed, with emphasis on the dynamical regime in which the system is close to phase transition. Consistent perturbation treatment of such a regime is proposed. An illustrative application of the innovated perturbation theory to a non-Hermitian but PT\mathcal{PT}PT-symmetric user-friendly family of JJJ-parametric ``discrete anharmonic'' quantum Hamiltonians $H=H(\stackrel{P\vec}{\ensuremath{\lambda}})$ is provided. The models are shown to admit the standard probabilistic interpretation if and only if the parameters remain compatible with the reality of the spectrum, $\stackrel{P\vec}{\ensuremath{\lambda}}\ensuremath{\in}{\mathcal{D}}^{(\mathrm{physical})}$. In contradiction to conventional wisdom, the systems are then shown to be stable with respect to admissible perturbations, inside the domain D(physical){\mathcal{D}}^{(\mathrm{physical})}D(physical), even in the immediate vicinity of the phase-transition boundaries $\ensuremath{\partial}{\mathcal{D}}^{(\mathrm{physical})}$.