April 2012
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Month April 2012

2*2 random matrix ensembles with reduced symmetry: From Hermitian to PT-symmetric matrices

Jiangbin Gong, Qing-hai Wang

A possibly fruitful extension of conventional random matrix ensembles is proposed by imposing symmetry constraints on conventional Hermitian matrices or parity-time- (PT-) symmetric matrices. To illustrate the main idea, we first study 2*2 complex Hermitian matrix ensembles with O(2) invariant constraints, yielding novel level-spacing statistics such as singular distributions, half-Gaussian distribution, distributions interpolating between GOE (Gaussian Orthogonal Ensemble) distribution and half Gaussian distributions, as well as gapped-GOE distribution. Such a symmetry-reduction strategy is then used to explore 2*2 PT-symmetric matrix ensembles with real eigenvalues. In particular, PT-symmetric random matrix ensembles with U(2) invariance can be constructed, with the conventional complex Hermitian random matrix ensemble being a special case. In two examples of PT-symmetric random matrix ensembles, the level-spacing distributions are found to be the standard GUE (Gaussian Unitary Ensemble) statistics or “truncated-GUE” statistics.

Quantum Physics (quant-ph)

Crypto-unitary quantum evolution operators

Miloslav Znojil

For the description of quantum evolution, the use of a manifestly time-dependent quantum Hamiltonian \(\mathfrak{h}(t) =\mathfrak{h}^\dagger(t)\) is shown equivalent to the work with its simplified, time-independent alternative \(G\neq G^\dagger\). A tradeoff analysis is performed recommending the latter option. The physical unitarity requirement is shown fulfilled in a suitable ad hocrepresentation of Hilbert space.

Quantum Physics (quant-ph); General Relativity and Quantum Cosmology (gr-qc)

Nonlinear modes in finite-dimensional PT -symmetric systems

D. A. Zezyulin, V. V. Konotop

By rearrangements of waveguide arrays with gain and losses one can simulate transformations among parity-time (PT-) symmetric systems not affecting their pure real linear spectra. Subject to such transformations, however, the nonlinear properties of the systems undergo significant changes. On an example of an array of four waveguides described by the discrete nonlinear Schr\”odinger equation with dissipation and gain, we show that the equivalence of the underlying linear spectra implies similarity of neither structure nor stability of the nonlinear modes in the arrays. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of a newly obtained system. Neither the stability is directly related to the PT symmetry: stable nonlinear modes exist even when the spectrum of the linear array is not purely real. We use graph representation of PT-symmetric networks allowing for simple illustration of linearly equivalent networks and indicating on their possible experimental design.

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

Nonlinear PT-symmetric plaquettes

Kai Li, P. G. Kevrekidis, Boris A. Malomed, Uwe Guenther

We introduce four basic two-dimensional (2D) plaquette configurations with onsite cubic nonlinearities, which may be used as building blocks for 2D PT -symmetric lattices. For each configuration, we develop a dynamical model and examine its PT symmetry. The corresponding nonlinear modes are analyzed starting from the Hamiltonian limit, with zero value of the gain-loss coefficient. Once the relevant waveforms have been identified (chiefly, in an analytical form), their stability is examined by means of linearization in the vicinity of stationary points. This reveals diverse and, occasionally, fairly complex bifurcations. The evolution of unstable modes is explored by means of direct simulations. In particular, stable localized modes are found in these systems, although the majority of identified solutions is unstable.

Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th)

PT-Symmetric Quantum Electrodynamics and Unitarity

Kimball A. Milton, E. K. Abalo, Prachi Parashar, Nima Pourtolami, J. Wagner

More than 15 years ago, a new approach to quantum mechanics was suggested, in which Hermiticity of the Hamiltonian was to be replaced by invariance under a discrete symmetry, the product of parity and time-reversal symmetry, \(\mathcal{PT}\). It was shown that if \(\mathcal{PT}\) is unbroken, energies were, in fact, positive, and unitarity was satisifed. Since quantum mechanics is quantum field theory in 1 dimension, time, it was natural to extend this idea to higher-dimensional field theory, and in fact an apparently viable version of \(\mathcal{PT}\)-invariant quantum electrodynamics was proposed. However, it has proved difficult to establish that the unitarity of the scattering matrix, for example, the Kallen spectral representation for the photon propagator, can be maintained in this theory. This has led to questions of whether, in fact, even quantum mechanical systems are consistent with probability conservation when Green’s functions are examined, since the latter have to possess physical requirements of analyticity. The status of \(\mathcal{PT}\)QED will be reviewed in this report, as well as the general issue of unitarity.

High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

Generation of families of spectra in PT-symmetric quantum mechanics and scalar bosonic field theory

Steffen Schmidt, S. P. Klevansky

This paper explains the systematics of the generation of families of spectra for the PT-symmetric quantum-mechanical Hamiltonians \(H=p^2+x^2(ix)^\epsilon\), \(H=p^2+(x^2)^\delta\), and \(H=p^2-(x^2)^\mu\). In addition, it contrasts the results obtained with those found for a bosonic scalar field theory, in particular in one dimension, highlighting the similarities and differences to the quantum-mechanical case. It is shown that the number of families of spectra can be deduced from the number of noncontiguous pairs of Stokes’ wedges that display PT-symmetry. To do so, simple arguments that use the WKB approximation are employed, and these imply that the eigenvalues are real. However, definitive results are in most cases presently only obtainable numerically, and not all eigenvalues in each family may be real. Within the approximations used, it is illustrated that the difference between the quantum-mechanical and the field-theoretical cases lies in the number of accessible regions in which the eigenfunctions decay exponentially. This paper reviews and implements well-known techniques in complex analysis and PT-symmetric quantum theory.

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Weak Measurements in Non-Hermitian Systems

A. Matzkin

“Weak measurements” — involving a weak unitary interaction between a quantum system and a meter followed by a projective measurement — are investigated when the system has a non-Hermitian Hamiltonian. We show in particular how the standard definition of the “weak value” of an observable must be modified. These studies are undertaken in the context of bound state scattering theory, a non-Hermitian formalism for which the Hilbert spaces involved are unambiguously defined and the metric operators can be explicitly computed. Numerical examples are given for a model system.

Quantum Physics (quant-ph)

Scattering states of a particle, with position-dependent mass, in a \({\cal{PT}}\) symmetric heterojunction

Anjana Sinha

The study of a particle with position-dependent effective mass (pdem), within a double heterojunction is extended into the complex domain — when the region within the heterojunctions is described by a non Hermitian \({\cal{PT}}\) symmetric potential. After obtaining the exact analytical solutions, the reflection and transmission coefficients are calculated, and plotted as a function of the energy. It is observed that at least two of the characteristic features of non Hermitian \({\cal{PT}}\) symmetric systems — viz., left / right asymmetry and anomalous behaviour at spectral singularity, are preserved even in the presence of pdem. The possibility of charge conservation is also discussed.


Quantum Physics (quant-ph)

PT-symmetric deformations of integrable models

Andreas Fring

We review recent results on new physical models constructed as PT-symmetrical deformations or extensions of different types of integrable models. We present non-Hermitian versions of quantum spin chains, multi-particle systems of Calogero-Moser-Sutherland type and non-linear integrable field equations of Korteweg-de-Vries type. The quantum spin chain discussed is related to the first example in the series of the non-unitary models of minimal conformal field theories. For the Calogero-Moser-Sutherland models we provide three alternative deformations: A complex extension for models related to all types of Coxeter/Weyl groups; models describing the evolution of poles in constrained real valued field equations of non linear integrable systems and genuine deformations based on antilinearly invariant deformed root systems. Deformations of complex nonlinear integrable field equations of KdV-type are studied with regard to different kinds of PT-symmetrical scenarios. A reduction to simple complex quantum mechanical models currently under discussion is presented.

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Coulomb potential and the paradoxes of PT-symmetrization

Miloslav Znojil

Besides the standard quantum version of the Coulomb/Kepler problem, an alternative quantum model with not too dissimilar phenomenological (i.e., spectral and scattering) as well as mathematical (i.e., exact-solvability) properties may be formulated and solved. Several aspects of this model are described. The paper is made self-contained by explaining the underlying innovative quantization strategy which assigns an entirely new role to symmetries.

Quantum Physics (quant-ph)