February 2013
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Month February 2013

Non-Hermitian Quantum Annealing in the Antiferromagnetic Ising Chain

Alexander I. Nesterov, Gennady P. Berman, Juan C. Beas Zepeda, Alan R. Bishop

A non-Hermitian quantum optimization algorithm is created and used to find the ground state of an antiferromagnetic Ising chain. We demonstrate analytically and numerically (for up to N=1024 spins) that our approach leads to a significant reduction of the annealing time that is proportional to \(\ln N\), which is much less than the time (proportional to \(N^2\)) required for the quantum annealing based on the corresponding Hermitian algorithm. We propose to use this approach to achieve similar speed-up for NP-complete problems by using classical computers in combination with quantum algorithms.

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Bifurcations and exceptional points in dipolar Bose-Einstein condensates

Robin Gutöhrlein, Jörg Main, Holger Cartarius, Günter Wunner

Bose-Einstein condensates are described in a mean-field approach by the nonlinear Gross-Pitaevskii equation and exhibit phenomena of nonlinear dynamics. The eigenstates can undergo bifurcations in such a way that two or more eigenvalues and the corresponding wave functions coalesce at critical values of external parameters. E.g. in condensates without long-range interactions a stable and an unstable state are created in a tangent bifurcation when the scattering length of the contact interaction is varied. At the critical point the coalescing states show the properties of an exceptional point. In dipolar condensates fingerprints of a pitchfork bifurcation have been discovered by Rau et al. [Phys. Rev. A, 81:031605(R), 2010]. We present a method to uncover all states participating in a pitchfork bifurcation, and investigate in detail the signatures of exceptional points related to bifurcations in dipolar condensates. For the perturbation by two parameters, viz. the scattering length and a parameter breaking the cylindrical symmetry of the harmonic trap, two cases leading to different characteristic eigenvalue and eigenvector patterns under cyclic variations of the parameters need to be distinguished. The observed structures resemble those of three coalescing eigenfunctions obtained by Demange and Graefe [J. Phys. A, 45:025303, 2012] using perturbation theory for non-Hermitian operators in a linear model. Furthermore, the splitting of the exceptional point under symmetry breaking in either two or three branching singularities is examined. Characteristic features are observed when one, two, or three exceptional points are encircled simultaneously.

Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Chaotic Dynamics (nlin.CD)

Quantum fragile matter: mechanical excitations of a Reggeon ion chain

Philipp Strack, Vincenzo Vitelli

This paper proposes to study quantum fragile materials with small linear elasticity and a strong response to zero-point fluctuations. As a first model, we consider a non-unitary (but PT-symmetric) massive quantum chain with a Reggeon-type cubic nonlinearity. At the critical point, the chain supports neither the ordinary quantum phonons of a Luttinger liquid, nor the supersonic solitons that arise in classical fragile critical points in the absence of fluctuations. Quantum fluctuations, approximately captured within a one-loop renormalization group, give rise to mechanical excitations with a nonlinear dispersion relation and dissipative spectral behavior. Models of similar complexity should be realizable with trapped ions.

Quantum Gases (cond-mat.quant-gas); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

Hermitian versus non-Hermitian representations for minimal length uncertainty relations

Sanjib Dey, Andreas Fring, Boubakeur Khantoul

We investigate four different types of representations of deformed canonical variables leading to generalized versions of Heisenberg’s uncertainty relations resulting from noncommutative spacetime structures. We demonstrate explicitly how the representations are related to each other and study three characteristically different solvable models on these spaces, the harmonic oscillator, the manifestly non-Hermitian Swanson model and an intrinsically noncommutative model with Poeschl-Teller type potential. We provide an analytical expression for the metric in terms of quantities specific to the generic solution procedure and show that when it is appropriately implemented expectation values are independent of the particular representation. A recently proposed inequivalent representation resulting from Jordan twists is shown to lead to unphysical models. We suggest an anti-PT-symmetric modification to overcome this shortcoming.


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

PT-symmetric lattices with a local degree of freedom

Harsha Vemuri, Yogesh N. Joglekar

Recently, open systems with balanced, spatially separated loss and gain have been realized and studied using non-Hermitian Hamiltonians that are invariant under the combined parity and time-reversal (\(\mathcal{PT}\)) operations. Here, we model and investigate the effects of a local, two-state, quantum degree of freedom, called a pseudospin, on a one-dimensional tight-binding lattice with position-dependent tunneling amplitudes and a single pair of non-Hermitian, \(\mathcal{PT}\)-symmetric impurities. We show that if the resulting Hamiltonian is invariant under exchange of two pseudospin labels, the system can be decomposed into two uncoupled systems with tunable threshold for \(\mathcal{PT}\) symmetry breaking. We discuss implications of our results to systems with specific tunneling profiles, and open or periodic boundary conditions.

Quantum Physics (quant-ph); Optics (physics.optics)

Pseudo-Hermitian random matrix theory

Shashi.C.L. Srivastava, S.R. Jain

Complex extension of quantum mechanics and the discovery of pseudo-unitarily invariant random matrix theory has set the stage for a number of applications of these concepts in physics. We briefly review the basic ideas and present applications to problems in statistical mechanics where new results have become possible. We have found it important to mention the precise directions where advances could be made if further results become available.

Quantum Physics (quant-ph)

New Concept of Solvability in Quantum Mechanics

Miloslav Znojil

In a pre-selected Hilbert space of quantum states the unitarity of the evolution is usually guaranteed via a pre-selection of the generator (i.e., of the Hamiltonian operator) in self-adjoint form. In fact, the simultaneous use of both of these pre-selections is overrestrictive. One should be allowed to make a given Hamiltonian self-adjoint only after an {\em ad hoc} generalization of Hermitian conjugation. We argue that in the generalized, hidden-Hermiticity scenario with nontrivial metric, the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of Hamiltonian) requires a generalization allowing for a non-numerical form of metric. A few illustrative solvable quantum models of this type are presented.

Quantum Physics (quant-ph)

Hermitian four-well potential as a realization of a PT-symmetric system

Manuel Kreibich, Jörg Main, Holger Cartarius, Günter Wunner

A PT-symmetric Bose-Einstein condensate can be theoretically described using a complex optical potential, however, the experimental realization of such an optical potential describing the coherent in- and outcoupling of particles is a nontrivial task. We propose an experiment for a quantum mechanical realization of a PT-symmetric system, where the PT-symmetric currents of a two-well system are implemented by coupling two additional wells to the system, which act as particle reservoirs. In terms of a simple four-mode model we derive conditions under which the two middle wells of the Hermitian four-well system behave exactly as the two wells of the PT-symmetric system. We apply these conditions to calculate stationary solutions and oscillatory dynamics. By means of frozen Gaussian wave packets we relate the Gross-Pitaevskii equation to the four-mode model and give parameters required for the external potential, which provides approximate conditions for a realistic experimental setup.

Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Chaotic Dynamics (nlin.CD)

A generalized family of discrete PT-symmetric square wells

Miloslav Znojil, Junde Wu

N-site-lattice Hamiltonians H are introduced and perceived as a set of systematic discrete approximants of a certain PT-symmetric square-well-potential model with the real spectrum and with a non-Hermiticity which is localized near the boundaries of the interval. Its strength is controlled by one, two or three parameters. The problem of the explicit construction of a nontrivial metric which makes the theory unitary is then addressed. It is proposed and demonstrated that due to the not too complicated tridiagonal-matrix form of our input Hamiltonians the computation of the metric is straightforward and that its matrix elements prove obtainable, non-numerically, in elementary polynomial forms.

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Pseudo Parity-Time Symmetry in Optical Systems

Xiaobing Luo, Honghua Zhong, Jiahao Huang, Xizhou Qin, Qiongtao Xie, Yuri S. Kivshar, Chaohong Lee

We introduce a novel concept of the {\em pseudo} parity-time (\(\mathcal{PT}\)) symmetry in periodically modulated optical systems with balanced gain and loss. We demonstrate that whether the original system is \(\mathcal{PT}\)-symmetric or not, we can manipulate the property of the \(\mathcal{PT}\) symmetry by applying a periodic modulation in such a way that the effective system derived by the high-frequency Floquet method is \(\mathcal{PT}\) symmetric. If the original system is non-\(\mathcal{PT}\) symmetric, the \(\mathcal{PT}\) symmetry in the effective system will lead to quasi-stationary propagation that can be associated with the \emph{pseudo \(\mathcal{PT}\) symmetry}. Our results provide a promising approach for manipulating the \(\mathcal{PT}\) symmetry of realistic systems.

Optics (physics.optics); Quantum Physics (quant-ph)