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).
http://arxiv.org/abs/1203.6590
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
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.
http://arxiv.org/abs/1203.5702
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
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.
http://arxiv.org/abs/1203.5011
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)
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.
http://arxiv.org/abs/1203.2110
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
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.
http://arxiv.org/abs/1203.1885
Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Chaotic Dynamics (nlin.CD)
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.
http://arxiv.org/abs/1203.1862
Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)
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.
http://arxiv.org/abs/1203.1345
Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas)
D. A. Zezyulin, V. V. Konotop
By similarity transformations a parity-time (PT-) symmetric Hamiltonian can be reduced to a Hermitian or to another PT-symmetric Hamiltonian having the same linear spectrum. On an example of a PT-symmetric quadrimer we show that the spectral equivalence of different PT-symmetric and Hermitian systems implies neither similarity of the nonlinear modes nor their stability properties. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of the underlying linear system. Neither the stability is directly related to the PT symmetry: stable nonlinear modes exist beyond the PT symmetry breaking threshold. A “phase diagram” of a general PT-symmetric quadrimer allows for existence of “triple” points, where three different phases meet. We use graph representation of PT-symmetric networks giving simple illustration of linearly equivalent networks and indicating on their possible experimental design.
http://arxiv.org/abs/1202.3652
Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)
Ray J. Rivers
An analysis of classical mechanics in a complex extension of phase space shows that a particle in such a space can behave in a way redolant of quantum mechanics; additional degrees of freedom permit ‘tunnelling’ without recourse to instantons and lead to time/energy uncertainty. In practice, ‘classical’ particle trajectories with additional degrees of freedom have arisen in several different formulations of quantum mechanics. In this talk we compare the extended phase space of the closed time-path formalism with that of complex classical mechanics, to suggest that $\hbar$ has a role in our understanding of the latter. However, differences in the way that trajectories are used make a deeper comparison problematical. We conclude with some thoughts on quantisation as dimensional reduction.
http://arxiv.org/abs/1202.4117
Quantum Physics (quant-ph); Classical Physics (physics.class-ph)
Li Chen, Rujiang Li, Na Yang, Da Chen, Lu Li
We investigate the unique properties of various analytical optical modes, including the fundamental modes and the excited modes, in a double-channel waveguide with parity-time (PT) symmetry. Based on these optical modes, the dependence of the threshold values for the gain/loss parameter, i.e., PT symmetry breaking points, on the structure parameters is discussed. We find that the threshold value for the excited modes is larger than that of the fundamental mode. In addition, the beam dynamics in the double-channel waveguide with PT symmetry is also investigated.
http://arxiv.org/abs/1202.2956
Optics (physics.optics)