Alessandro Sergi, Konstantin G. Zloshchastiev
We consider a non-Hermitian Hamiltonian in order to effectively describe a two-level system coupled to a dissipative environment. The total Hamiltonian of the model is obtained by adding a general anti-Hermitian part, depending on four parameters, to the Hermitian Hamiltonian of a tunneling two-level system. The time evolution is formulated and derived in terms of the density matrix of the model, different types of decays are revealed and analyzed. In particular, the population difference and coherence are defined and calculated analytically. We have been able to mimic various physical situations with different properties, such as dephasing and vanishing population difference.
http://arxiv.org/abs/1207.4877
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
Tomaz Prosen
We discuss a combination of unitary and anti-unitary symmetry of quantum Liouvillian dynamics, in the context of open quantum systems, which implies a D2 symmetry of the complex Liovillean spectrum. For sufficiently weak system-bath coupling it implies a uniform decay rate for all coherences, i.e. off-diagonal elements of the system’s density matrix taken in the eigenbasis of the Hamiltonian. As an example we discuss symmetrically boundary driven open XXZ spin 1/2 chains.
http://arxiv.org/abs/1207.4395
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
R. Driben, B. A. Malomed
Dynamics of symmetric and antisymmetric 2-solitons and 3-solitons is studied in the model of the nonlinear dual-core coupler and its PT-symmetric version. Regions of the convergence of the injected perturbed symmetric and antisymmetric N-solitons into symmetric and asymmetric quasi-solitons are found. In the PT-symmetric system, with the balanced gain and loss acting in the two cores, borders of the stability against the blowup are identified. Notably, in all the cases the stability regions are larger for antisymmetric 2-soliton inputs than for their symmetric counterparts, on the contrary to previously known results for fundamental solitons (N=1). Dynamical regimes (switching) are also studied for the 2-soliton injected into a single core of the coupler. In particular, a region of splitting of the input into a pair of symmetric solitons is found, which is explained as a manifestation of the resonance between the vibrations of the 2-soliton and oscillations of energy between the two cores in the coupler.
http://arxiv.org/abs/1207.3917
Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)
Sanjib Dey, Andreas Fring
The two dimensional set of canonical relations giving rise to minimal uncertainties previously constructed from a q-deformed oscillator algebra is further investigated. We provide a representation for this algebra in terms of a flat noncommutative space and employ it to study the eigenvalue spectrum for the harmonic oscillator on this space. The perturbative expression for the eigenenergy indicates that the model might possess an exceptional point at which the spectrum becomes complex and its PT-symmetry is spontaneously broken.
http://arxiv.org/abs/1207.3303
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Sanjib Dey, Andreas Fring
We provide an explicit construction for Gazeau-Klauder coherent states related to non-Hermitian Hamiltonians with discrete bounded below and nondegenerate eigenspectrum. The underlying spacetime structure is taken to be of a noncommutative type with associated uncertainty relations implying minimal lengths. The uncertainty relations for the constructed states are shown to be saturated in a Hermitian as well as a non-Hermitian setting for a perturbed harmonic oscillator. The computed value of the Mandel parameter dictates that the coherent wavepackets are assembled according to sub-Poissonian statistics. Fractional revival times, indicating the superposition of classical-like sub-wave packets are clearly identified.
http://arxiv.org/abs/1207.3297
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Ali Mostafazadeh
We show that, contrary to a claim made in arXiv:1011.0645, the von Neumann-Winger bound states that lie in the continuum of the scattering states are fundamentally different from Naimark’s spectral singularities.
http://arxiv.org/abs/1207.2278
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)
Boris F. Samsonov
One of the simplest non-Hermitian Hamiltonians first proposed by Schwartz (1960 Commun. Pure Appl. Math. 13 609) which may possess a spectral singularity is analyzed from the point of view of non-Hermitian generalization of quantum mechanics. It is shown that \(\eta\) operator, being a second order differential operator, has supersymmetric structure. Asymptotic behavior of eigenfunctions of a Hermitian Hamiltonian equivalent to the given non-Hermitian one is found. As a result the corresponding scattering matrix and cross section are given explicitly. It is demonstrated that the possible presence of the spectral singularity in the spectrum of the non-Hermitian Hamiltonian may be detected as a resonance in the scattering cross section of its Hermitian counterpart. Nevertheless, just at the singular point the equivalent Hermitian Hamiltonian becomes undetermined.
http://arxiv.org/abs/1207.2525
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Boris F. Samsonov
Being chosen as a differential operator of a special form, metric \(\eta\) operator becomes unitary equivalent to a one-dimensional Hermitian Hamiltonian with a natural supersymmetric structure. We show that fixing the superpartner of this Hamiltonian permits to determine both the metric operator and corresponding non-Hermitian Hamiltonian. Moreover, under an additional restriction on the non-Hermitian Hamiltonian, it becomes a superpartner of another Hermitian Hamiltonian.
http://arxiv.org/abs/1207.2522
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Vladimir V. Konotop, Valery S. Shchesnovich, Dmitry A. Zezyulin
The combination of the interference with the amplification of modes in a waveguide with gain and losses can result in a giant amplification of the propagating beam, which propagates without distortion of its average amplitude. An increase of the gain-loss gradient by only a few times results in a magnification of the beam by a several orders of magnitude.
http://arxiv.org/abs/1207.1792
Optics (physics.optics)
Ananya Ghatak, Bhabani Prasad Mandal, Zafar Ahmed
We bring out the existence of at most one spectral singularity (SS) and deep multiple minima in the reflectivity of the non-Hermitian (complex) Ginocchio potential. We find a parameter dependent single spectral singularity in this potential provided the imaginary part is emissive (not absorptive). The reflectionlessness of the real Hermitian Ginocchio’s potential at discrete positive energies gives way to deep multiple minima in reflectivity when this potential is perturbed and made non-Hermitian (complex). A novel co-existence of a SS with deep minima in reflectivity is also revealed wherein the first reflectivity zero of the Hermitian case changes to become a SS for the non-Hermitian case.
http://arxiv.org/abs/1207.1979
Quantum Physics (quant-ph)