Alan A. Dzhioev, D. S. Kosov
We use super-fermion representation of quantum kinetic equation to develop nonequilibrium perturbation theory for inelastic electron current through quantum dot. We derive Lindblad type kinetic equation for an embedded quantum dot (i.e. a quantum dot connected to Lindblad dissipators through a buffer zone). The kinetic equation is converted to non-Hermitian field theory in Liouville-Fock space. The general nonequilibrium many-body perturbation theory is developed and applied to the quantum dot with electron-vibron and electron-electron interactions. Our perturbation theory becomes equivalent to Keldysh nonequilibrium Green’s functions perturbative treatment provided that the buffer zone is large enough to alleviate the problems associated with approximations of the Lindblad kinetic equation.
http://arxiv.org/abs/1201.1230
Mesoscale and Nanoscale Physics (cond-mat.mes-hall)
E.A.Rakhmanov
This paper is devoted to a study of S-curves, that is systems of curves in the complex plane whose equilibrium potential in a harmonic external field satisfies a special symmetry property (S-property). Such curves have many applications. In particular, they play a fundamental role in the theory of complex (non-hermitian) orthogonal polynomials. One of the main theorems on zero distribution of such polynomials asserts that the limit zero distribution is presented by an equilibrium measure of an S-curve associated with the problem if such a curve exists. These curves are also the starting point of the matrix Riemann-Hilbert approach to srtong asymptotics. Other approaches to the problem of strong asymptotics (differential equations, Riemann surfaces) are also related to S-curves or may be interpreted this way. Existence problem S-curve in a given class of curves in presence of a nontrivial external field presents certain challenge. We formulate and prove a version of existence theorem for the case when both the set of singularities of the external field and the set of fixed points of a class of curves are small (in main case — finite). We also discuss various applications and connections of the theorem.
http://arxiv.org/abs/1112.5713
Complex Variables (math.CV); Mathematical Physics (math-ph)
Vincenzo Grecchi, André Martinez
We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian,\(H(\beta)=-d^2/dx^2+x^2+i\sqrt{\beta}x^3\),for \(\beta\) in the cut plane \(\C_c:=\C\backslash (-\infty, 0)\). Moreover, we prove that the spectrum consists of the perturbative eigenvalues \(\{E_n(\beta)\}_{n\geq 0}\) labeled by the constant number $n$ of nodes of the corresponding eigenfunctions. In addition, for all \(\beta\in\C_c\), \(E_n(\beta)\) can be computed as the Stieltjes-Pad\’e sum of its perturbation series at \(\beta=0\). This also gives an alternative proof of the fact that the spectrum of \(H(\beta)\) is real when \(\beta\) is a positive number. This way, the main results on the repulsive PT-symmetric and on the attractive quartic oscillators are extended to the cubic case.
http://arxiv.org/abs/1201.2797
Mathematical Physics (math-ph); Spectral Theory (math.SP)
Johann Foerster, Alejandro Saenz, Ulli Wolff
We represent low dimensional quantum mechanical Hamiltonians by moderately sized finite matrices that reproduce the lowest O(10) boundstate energies and wave functions to machine precision. The method extends also to Hamiltonians that are neither Hermitian nor PT symmetric and thus allows to investigate whether or not the spectra in such cases are still real. Furthermore, the approach is especially useful for problems in which a position-dependent mass is adopted, for example in effective-mass models in solid-state physics or in the approximate treatment of coupled nuclear motion in molecular physics or quantum chemistry. The performance of the algorithm is demonstrated by considering the inversion motion of different isotopes of ammonia molecules within a position-dependent-mass model and some other examples of one- and two-dimensional Hamiltonians that allow for the comparison to analytical or numerical results in the literature.
http://arxiv.org/abs/1112.5294
Quantum Physics (quant-ph); High Energy Physics – Lattice (hep-lat); Mathematical Physics (math-ph)
Li Ge, Y. D. Chong, A. D. Stone
We analyze the optical properties of one-dimensional (1D) PT-symmetric structures of arbitrary complexity. These structures violate normal unitarity (photon flux conservation) but are shown to satisfy generalized unitarity relations, which relate the elements of the scattering matrix and lead to a conservation relation in terms of the transmittance and (left and right) reflectances. One implication of this relation is that there exist anisotropic transmission resonances in PT-symmetric systems, frequencies at which there is unit transmission and zero reflection, but only for waves incident from a single side. The spatial profile of these transmission resonances is symmetric, and they can occur even at PT-symmetry breaking points. The general conservation relations can be utilized as an experimental signature of the presence of PT-symmetry and of PT-symmetry breaking transitions. The uniqueness of PT-symmetry breaking transitions of the scattering matrix is briefly discussed by comparing to the corresponding non-Hermitian Hamiltonians.
http://arxiv.org/abs/1112.5167
Optics (physics.optics)
Clinton Thompson, Yogesh N. Joglekar, Gautam Vemuri
We investigate the effects of disorder on single particle time-evolution and two-particle correlations in an array of evanescently coupled waveguides with position-dependent tunneling rates. In the clean limit, the energy spectrum of such an array is widely tunable. In the presence of a Hermitian on-site or tunneling disorder, we find that the localization of a wave packet is highly sensitive to this energy spectrum. In particular, for an input confined to a single waveguide, we show that the fraction of light localized to the original waveguide depends on the tunneling profile. We compare the two-particle intensity correlations in the presence of Hermitian, tunneling disorder and non-Hermitian, parity-and-time-reversal \{\mathcal{PT}\} symmetric, on-site potential disorder. We show the two-particle correlation function in both cases is qualitatively similar, since both disorders preserve the particle-hole symmetric nature of the energy spectrum.
http://arxiv.org/abs/1112.4720
Quantum Physics (quant-ph); Optics (physics.optics)
Hamidreza Ramezani, Tsampikos Kottos, Vassilios Kovanis, Demetrios N. Christodoulides
We theoretically investigate the flow of electromagnetic waves in complex honeycomb photonic lattices with local PT symmetries. Such PT structure is introduced via a judicious arrangement of gain or loss across the honeycomb lattice, characterized by a gain/loss parameter \{\gamma\}. We found a new class of conical diffraction phenomena where the formed cone is brighter and travels along the lattice with a transverse speed proportional to \{\sqrt{\gamma}\}.
http://arxiv.org/abs/1112.4734
Optics (physics.optics); Quantum Physics
T. K. Jana, P. Roy
We show that the non Hermitian Black-Scholes Hamiltonian and its various generalizations are eta-pseudo Hermitian. The metric operator eta is explicitly constructed for this class of Hamitonians. It is also shown that the e?ective Black-Scholes Hamiltonian and its partner form a pseudo supersymmetric system.
http://arxiv.org/abs/1112.3217
General Finance (q-fin.GN)
Miloslav Znojil
A non-unitary version of quantum scattering is studied via an exactly solvable toy model. The model is merely asymptotically local since the smooth path of the coordinate is admitted complex in the non-asymptotic domain. At any real angular-momentum-like parameter the reflection R and transmission T are shown to change with the winding number (i.e., topology) of the path. The points of unitarity appear related to the points of existence of quantum-knot bound states.
http://arxiv.org/abs/1112.2644
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Syo Kamata, Hidekazu Tanaka
The one-loop Wilsonian renormalization group flows of the two dimensional two-flavors Gross-Neveu model with minimal doubling fermion are calculated numerically. The off-diagonal mass components which are non-Hermiticity are generated in the flows. We considered a relation among $\gamma_{5}$Hermiticity, R-Hermiticity and PT symmetry, satisfied two of the three conditions is a sufficient condition for satisfied another condition but not a necessary condition. Because of kinetic terms which do not have R-Hermiticity, non-Hermiticity effective masses appear.
http://arxiv.org/abs/1111.4536
High Energy Physics – Lattice (hep-lat)