March 2012
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Month March 2012

Pseudo-Hermitian Quantum Mechanics with Unbounded Metric Operators

Ali Mostafazadeh

We extend the formulation of pseudo-Hermitian quantum mechanics to eta-pseudo-Hermitian Hamiltonian operators H with an unbounded metric operator eta. In particular, we give the details of the construction of the physical Hilbert space, observables, and equivalent Hermitian Hamiltonian for the case that H has a real and discrete spectrum and its eigenvectors belong to the domain of eta and its positive square root.
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Negative-energy PT-symmetric Hamiltonians

Carl M. Bender, Daniel W. Hook, S. P. Klevansky

The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian \(H=p^2+x^2(ix)^\epsilon\) has real, positive, and discrete eigenvalues for all \(\epsilon\geq 0\). These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues \(E_n=2n+1\) (n=0, 1, 2, 3, …) at \(\epsilon=0\). However, the harmonic oscillator also has negative eigenvalues \(E_n=-2n-1\) (n=0, 1, 2, 3, …), and one may ask whether it is equally possible to continue analytically from these eigenvalues. It is shown in this paper that for appropriate PT-symmetric boundary conditions the Hamiltonian \(H=p^2+x^2(ix)^\epsilon\) also has real and {\it negative} discrete eigenvalues. The negative eigenvalues fall into classes labeled by the integer N (N=1, 2, 3, …). For the Nth class of eigenvalues, \(\epsilon\) lies in the range \((4N-6)/3<\epsilon<4N-2\). At the low and high ends of this range, the eigenvalues are all infinite. At the special intermediate value \(\epsilon=2N-2\) the eigenvalues are the negatives of those of the conventional Hermitian Hamiltonian \(H=p^2+x^{2N}\). However, when \(\epsilon\neq 2N-2\), there are infinitely many complex eigenvalues. Thus, while the positive-spectrum sector of the Hamiltonian \(H=p^2+x^2(ix)^\epsilon\) has an unbroken PT symmetry (the eigenvalues are all real), the negative-spectrum sector of \(H=p^2+x^2(ix)^\epsilon\) has a broken PT symmetry (only some of the eigenvalues are real).
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

WKB Analysis of PT-Symmetric Sturm-Liouville problems. II

Carl M. Bender, Hugh F. Jones

In a previous paper it was shown that a one-turning-point WKB approximation gives an accurate picture of the spectrum of certain non-Hermitian PT-symmetric Hamiltonians on a finite interval with Dirichlet boundary conditions. Potentials to which this analysis applies include the linear potential \(V=igx\) and the sinusoidal potential \(V=ig\sin(\alpha x)\). However, the one-turning-point analysis fails to give the full structure of the spectrum for the cubic potential \(V=igx^3\), and in particular it fails to reproduce the critical points at which two real eigenvalues merge and become a complex-conjugate pair. The present paper extends the method to cases where the WKB path goes through a pair of turning points. The extended method gives an extremely accurate approximation to the spectrum of \(V=igx^3\), and more generally it works for potentials of the form \(V=igx^{2N+1}\). When applied to potentials with half-integral powers of \(x\), the method again works well for one sign of the coupling, namely that for which the turning points lie on the first sheet in the lower-half plane.
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

The Pauli equation with complex boundary conditions

D. Kochan, D. Krejcirik, R. Novak, P. Siegl

We consider one-dimensional Pauli Hamiltonians in a bounded interval with possibly non-self-adjoint Robin-type boundary conditions. We study the influence of the spin-magnetic interaction on the interplay between the type of boundary conditions and the spectrum. A special attention is paid to PT-symmetric boundary conditions with the physical choice of the time-reversal operator T.
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)

Lax-Phillips scattering theory for PT-symmetric ρ-perturbed operators

Petru A. Cojuhari, Sergii Kuzhel

The S-matrices corresponding to PT-symmetric \(\rho\)-perturbed operators are defined and calculated by means of an approach based on an operator-theoretical interpretation of the Lax-Phillips scattering theory.
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Model of a PT symmetric Bose-Einstein condensate in a delta-functions double well

Holger Cartarius, Günter Wunner

The observation of PT symmetry in a coupled optical wave guide system that involves a complex refractive index has been demonstrated impressively in the experiment by Ruter el al. (Nat. Phys. 6, 192, 2010). This is, however, only an optical analogue of a quantum system, and it would be highly desirable to observe the manifestation of PT symmetry and the resulting properties also in a real, experimentally accessible, quantum system. Following a suggestion by Klaiman et al. (Phys. Rev. Lett. 101, 080402, 2008), we investigate a PT symmetric arrangement of a Bose-Einstein condensate in a double well potential, where in one well cold atoms are injected while in the other particles are extracted from the condensate. We investigate, in particular, the effects of the nonlinearity in the Gross-Pitaevskii equation on the PT properties of the condensate. To study these effects we analyze a simple one-dimensional model system in which the condensate is placed into two PT symmetric delta-function traps. The analysis will serve as a useful guide for studies of the behaviour of Bose-Einstein condensates in realistic PT symmetric double wells.
Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Chaotic Dynamics (nlin.CD)

A variational approach to Schroedinger equation with parity-time symmetry Gaussian complex potential

Sumei Hu, Guo Liang, Shanyong Cai, Daquan Lu, Qi Guo, Wei Hu

A variational technique is established to deal with the Schrodinger equation with parity-time(PT) symmetric Gaussian complex potential. The method is extended to the linear and self-focusing and defocusing nonlinear cases. Some unusual properties in PT systems such as transverse power flow and PT breaking points can be analyzed by this method. Following numerical simulations, the analytical results are in good agreement with the numerical results.
Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

PT-symmetry breaking and universal chirality in a PT-symmetric ring

Derek D. Scott, Yogesh N. Joglekar

We investigate the properties of an \(N\)-site tight-binding lattice with periodic boundary condition (PBC) in the presence of a pair of gain and loss impurities \(\pm i\gamma\), and two tunneling amplitudes \(t_0,t_b\) that are constant along the two paths that connect them. We show that the parity and time-reversal PT-symmetric phase of the lattice with PBC is robust, insensitive to the distance between the impurities, and that the critical impurity strength for PT-symmetry breaking is given by \(\gamma_{PT}=|t_0-t_b|\). We study the time-evolution of a typical wave packet, initially localized on a single site, across the PT-symmetric phase boundary. We find that it acquires chirality with increasing \(\gamma\), and the chirality reaches a universal maximum value at the threshold, \(\gamma=\gamma_{PT}\), irrespective of the initial location of the wave packet or the lattice parameters. Our results imply that PT-symmetry breaking on a lattice with PBC has consequences that have no counterpart in open chains.
Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas)