Entanglement in non-Hermitian quantum theory

Entanglement is one of the key features of quantum world that has no classical counterpart. This arises due to the linear superposition principle and the tensor product structure of the Hilbert space when we deal with multiparticle systems. In this paper, we will introduce the notion of entanglement for quantum systems that are governed by non-Hermitian yet PT-symmetric Hamiltonians. We will show that maximally entangled states in usual quantum theory behave like non-maximally entangled states in PT-symmetric quantum theory. Furthermore, we will show how to create entanglement between two PT qubits using non-Hermitian Hamiltonians and discuss the entangling capability of such interaction Hamiltonians that are non-Hermitian in nature.     

Fractional statistics in one dimension: view from an exactly solvable model

One-dimensional fractional statistics is studied using the Calogero-Sutherland model (CSM) which describes a system of non-relativistic quantum particles interacting with an inverse-square two-body potential on a ring. The inverse-square exchange can be regarded as a pure statistical interaction and this system can be mapped to an ideal gas obeying the fractional exclusion and exchange statistics. The details of the exact calculations of the dynamical correlation functions or this ideal system is presented in this paper. An effective low-energy one-dimensional “anyon” model is constructed; and its correlation functions are found to be in agreement with those in the CSM; and this agreement provides an evidence for the equivalence of the first- and the second-quantized construction of the 1D anyon model at least in the long-wavelength limit. Furthermore, the finite-size scaling applicable to the conformally invariant systems is used to obtain the complete set of correlation exponents for the CSM.     

The C Operator in iϕ 3 Field Theory

The C operator defines a dynamically-determined positive-definite metric in PT-symmetric theories. We show how the operator formalism for the perturbative calculation of C can be extended from quantum mechanics to quantum field theory with a cubic self interaction.     

The different effect of electron-electron interaction on the spectrum of atoms and quantum dots

Though atoms and quantum dots typically contain a comparable number of electrons, the number of discrete levels resolved in spectroscopy experiments is very different for the two systems. In atoms, hundreds of levels are observed while in quantum dots that number is usually smaller than 10. In the present work, this difference is traced to the different confining potentials in these systems. In atoms, the soft confining potential leads to large spatial extent of the excited electron's wave function and hence to weak Coulomb interaction with the rest of the atomic electrons. The resulting level broadening is smaller than the single particle level spacing and decreases as the excitation energy is increased. In quantum dots, on the other hand, the sharp confining potential results in electron-electron scattering rates that grow rapidly with energy and fairly quickly exceed the approximately constant single particle level spacing. The number of discrete levels in quantum dots is hence limited by electron-electron interaction, whose effect is negligible in atoms. Received 3 April 2000 and Received in final form 7 August 2000     

Classical Motion of Complex 2-D Non-Hermitian Hamiltonian Systems

Classical motion of complex 2-D non-Hermitian Hamiltonian systems is investigated to identify periodic, unbounded and chaotic trajectories. The caustic curves, the Lyapunov exponents, and spectral analysis have been used to identify periodic and chaotic trajectories. Using classical trajectories, we were able to predict quantum transition frequaencies of pseudo-Hermitian non-PT symmetric systems accurately. This indicates that there exists a correspondence between classical mechanics and quantum mechanics for certain non-Hermitian Hamiltonians as in the case of real Hermitians.     

Asymptotic Properties of Solvable $\mathcal{PT}$-Symmetric Potentials

The asymptotic region of potentials has strong impact on their general properties. This problem is especially interesting for PT\mathcal{PT}-symmetric potentials, the real and imaginary components of which allow for a wider variety of asymptotic properties than in the case of purely real potentials. We consider exactly solvable potentials defined on an infinite domain and investigate their scattering and bound states with special attention to the boundary conditions determined by the asymptotic regions. The examples include potentials with asymptotically vanishing and non-vanishing real and imaginary potential components (Scarf II, Rosen-Morse II, Coulomb). We also compare the results with the asymptotic properties of some exactly non-solvable PT\mathcal{PT}-symmetric potentials. These studies might be relevant to the experimental realization of PT\mathcal{PT}-symmetric systems.     

Nonlinear quantum shock waves in fractional quantum Hall edge states

Using the Calogero model as an example, we show that the transport in interacting nondissipative electronic systems is essentially nonlinear and unstable. Nonlinear effects are due to the curvature of the electronic spectrum near the Fermi energy. As is typical for nonlinear systems, a propagating semiclassical wave packet develops a shock wave at a finite time. A wave packet collapses into oscillatory features which further evolve into regularly structured localized pulses carrying a fractionally quantized charge. The Calogero model can be used to describe fractional quantum Hall edge states. We discuss perspectives of observation of quantum shock waves and a direct measurement of the fractional charge in fractional quantum Hall edge states.     

PT Symmetry in Quantum Field Theory

The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian. The Hermiticity of H guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). In this talk we investigate an alternative formulation of quantum mechanics in which the mathematical requirement of Hermiticity is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. We show that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are H=p ²+ix ³ and H=p ²-x ⁴. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry that we call C. Using C, we show how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.     

Permutation invariant algebras, a Fock space realization and the Calogero model

We study permutation invariant oscillator algebras and their Fock space representations using three equivalent techniques, i.e. (i) a normally ordered expansion in creation and annihilation operators, (ii) the action of annihilation operators on monomial states in Fock space and (iii) Gram matrices of inner products in Fock space. We separately discuss permutation invariant algebras which possess hermitean number operators and permutation invariant algebras which possess non-hermitean number operators. The results of a general analysis are applied to the -extended Heisenberg algebra, underlying the M-body Calogero model. Particular attention is devoted to the analysis of Gram matrices for the Calogero model. We discuss their structure, eigenvalues and eigenstates. We obtain a general condition for positivity of eigenvalues, meaning that all norms of states in Fock space are positive if this condition is satisfied. We find a universal critical point at which the reduction of the physical degrees of freedom occurs. We construct dual operators, leading to the ordinary Heisenberg algebra of free Bose oscillators. From the Fock-space point of view, we briefly discuss the existence of a mapping from the Calogero oscillators to the free Bose oscillators and vice versa. Received: 26 July 2001 / Revised version: 9 January 2002 / Published online: 12 April 2002     

Supersymmetry Without Hermiticity

The PT symmetry requirement of a potential defines an inhomogeneous system of first-order differential equations for the real/imaginary and even/odd components of the relevant superpotential. By identifying the general solutions of this system we search for non-trivial supersymmetric partner potentials and analyze whether they both possess PT symmetry. As an illustrative example we present the case of the Rosen-Morse I potential.     

supersymmetric 3-particles Calogero model

We constructed the most general N=4 superconformal 3-particles systems with translation invariance. In the basis with decoupled center of mass the supercharges and Hamiltonian possess one arbitrary function which defines all potential terms. We have shown that with the proper choice of this function one may describe the standard, A₂ Calogero model as well as G₂ and BC₂ Calogero models, which, by construction, possess N=4 superconformal symmetry. The main property of all these systems is that even with the coupling constant equal to zero they still contain nontrivial interactions in the fermionic sector. In other words, there are infinitely many non-equivalent N=4 supersymmetric extensions of the free action depending on one arbitrary function. We also considered quantization and explicitly showed how the supercharges and Hamiltonian are modified. In the quantum case the constructed systems exhibit only invariance with respect to N=4 Poincaré supersymmetry.     

Derivation and classical limit of the mean-field equation for a quantum Coulomb system: Maxwell-Boltzmann statistics

The mean-field density matrix of a changed plasma of quantum particles with Maxwell-Boltzmann statistics in a confining external potential is obtained as a limit of theN-body canonical states for suitably scaled charges. Also, it is shown that the density profile of the quantum mean-field theory converges to the solution of the classical mean-field equation when the Planck's constant tends to zero.     

Higgs interchange and bound states of superheavy fermions

Hypothetical superheavy fourth-generation fermions with a very small coupling with the rest of the Standard Model can give rise to long enough lived bound states. The production and the detection of these bound states would be experimentally feasible at the LHC. Extending, in the present study, the analysis of other authors, a semirelativistic wave equation is solved using an accurate numerical method to determine the binding energies of these possible superheavy fermion-bound states. The interaction given by the Yukawa potential of the Higgs boson exchange is considered; the corresponding relativistic corrections are calculated by means of a model based on the covariance properties of the Hamiltonian. We study the effects given by the Coulomb force. Moreover, we calculate the contributions given by the Coulombic and confining terms of the strong interaction in the case of superheavy quark bound states. The results of the model are critically analysed.     

Quantization and conformal properties of a generalized Calogero model

We analyze a generalization of the quantum Calogero model with the underlying conformal symmetry, paying special attention to the two-body model deformation. Owing to the underlying SU(1,1) symmetry, we find that the analytic solutions of this model can be described within the scope of the Bargmann representation analysis, and we investigate its dynamical structure by constructing the corresponding Fock space realization. The analysis from the standpoint of supersymmetric quantum mechanics (SUSYQM), when applied to this problem, reveals that the model is also shape invariant. For a certain range of the system parameters, the two-body generalization of the Calogero model is shown to admit a one-parameter family of self-adjoint extensions, leading to inequivalent quantizations of the system. PACS 02.30.Ik; 03.65.Fd; 03.65.-w     

Near-horizon states of black holes and Calogero models

We find self-adjoint extensions of the rational Calogero model in the presence of the harmonic interaction. The corresponding eigenfunctions may describe the near-horizon quantum states of certain types of black holes.     

Quantum hydrodynamics, the quantum benjamin-ono equation, and the Calogero model

Collective field theory for the Calogero model represents particles with fractional statistics in terms of hydrodynamic modes--density and velocity fields. We show that the quantum hydrodynamics of this model can be written as a single evolution equation on a real holomorphic Bose field--the quantum integrable Benjamin-Ono equation. It renders tools of integrable systems to studies of nonlinear dynamics of 1D quantum liquids.     

How to test for diagonalizability: the discretized PT-invariant square-well potential

Given a non-Hermitian matrix M, the structure of its minimal polynomial encodes whether M is diagonalizable or not. This note explains how to determine the minimal polynomial of a matrix without going through its characteristic polynomial. The approach is applied to a quantum mechanical particle moving in a square well under the influence of a piece-wise constant PT-symmetric potential. Upon discretizing the configuration space, the system is described by a matrix of dimension three which turns out not to be diagonalizable for a critical strength of the interaction. The systems develops a three-fold degenerate eigenvalue, and two of the three eigenfunctions disappear at this exceptional point, giving a difference between the algebraic and geometric multiplicity of the eigenvalue equal to two. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Quantum Hall Effect on the Hyperbolic Plane in the Presence of Disorder

We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in Comm. Math. Phys. 190 (1998), 629–673, to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian.