Derek D. Scott, Yogesh N. Joglekar
We study the properties of an N-site tight-binding ring with parity and time-reversal (PT) symmetric, Hermitian, site-dependent tunneling and a pair of non-Hermitian, PT-symmetric, loss and gain impurities \(\pm i\gamma\). The properties of such lattices with open boundary conditions have been intensely explored over the past two years. We numerically investigate the PT-symmetric phase in a ring with a position-dependent tunneling function \(t_\alpha(k)=[k(N-k)]^{\alpha/2}\) that, in an open lattice, leads to a strengthened PT-symmetric phase, and study the evolution of the PT-symmetric phase from the open chain to a ring. We show that, generally, periodic boundary conditions weaken the PT-symmetric phase, although for experimentally relevant lattice sizes \(N \sim 50\), it remains easily accessible. We show that the chirality, quantified by the (magnitude of the) average transverse momentum of a wave packet, shows a maximum at the PT-symmetric threshold. Our results show that although the wavepacket intensity increases monotonically across the PT-breaking threshold, the average momentum decays monotonically on both sides of the threshold.
http://arxiv.org/abs/1207.1945
Quantum Physics (quant-ph); Optics (physics.optics)
Holger Cartarius, Daniel Haag, Dennis Dast, Günter Wunner
The time-independent nonlinear Schrodinger equation is solved for two attractive delta-function shaped potential wells where an imaginary loss term is added in one well, and a gain term of the same size but with opposite sign in the other. We show that for vanishing nonlinearity the model captures all the features known from studies of PT symmetric optical wave guides, e.g., the coalescence of modes in an exceptional point at a critical value of the loss/gain parameter, and the breaking of PT symmetry beyond. With the nonlinearity present, the equation is a model for a Bose-Einstein condensate with loss and gain in a double well potential. We find that the nonlinear Hamiltonian picks as stationary eigenstates exactly such solutions which render the nonlinear Hamiltonian itself PT symmetric, but observe coalescence and bifurcation scenarios different from those known from linear PT symmetric Hamiltonians.
http://arxiv.org/abs/1207.1669
Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Chaotic Dynamics (nlin.CD)
Henning Schomerus
I review how methods from mesoscopic physics can be applied to describe the multiple wave scattering and complex wave dynamics in non-hermitian PT-symmetric resonators, where an absorbing region is coupled symmetrically to an amplifying region. Scattering theory serves as a convenient tool to classify the symmetries beyond the single-channel case and leads to effective descriptions which can be formulated in the energy domain (via Hamiltonians) and in the time domain (via time evolution operators). These models can then be used to identify the mesoscopic time and energy scales which govern the spectral transition from real to complex eigenvalues. The possible presence of magneto-optical effects (a finite vector potential) in multichannel systems leads to a variant (termed PTT’ symmetry) which imposes the same spectral constraints as PT symmetry. I also provide multichannel versions of generalized flux-conservation laws.
http://arxiv.org/abs/1207.1454
Quantum Physics (quant-ph); Optics (physics.optics)
A.S. Rodrigues, K. Li, V. Achilleos, P.G. Kevrekidis, D.J. Frantzeskakis, Carl M. Bender
In this work we analyze PT-symmetric double-well potentials based on a two-mode picture. We reduce the problem into a PT-symmetric dimer and illustrate that the latter has effectively two fundamental bifurcations, a pitchfork (symmetry-breaking bifurcation) and a saddle-center one, which is the nonlinear analog of the PT-phase-transition. It is shown that the symmetry breaking leads to ghost states (amounting to growth or decay); although these states are not true solutions of the original continuum problem, the system’s dynamics closely follows them, at least in its metastable evolution. Past the second bifurcation, there are no longer states of the original continuum system. Nevertheless, the solutions can be analytically continued to yield a new pair of branches, which is also identified and dynamically examined. Our explicit analytical results for the dimer are directly compared to the full continuum problem, yielding a good agreement.
http://arxiv.org/abs/1207.1066
Pattern Formation and Solitons (nlin.PS); Quantum Gases (cond-mat.quant-gas); Mathematical Physics (math-ph)
Carl M. Bender, Sergii Kuzhel
It is shown that if the C operator for a PT-symmetric Hamiltonian with simple eigenvalues is not unique, then it is unbounded. Apart from the special cases of finite-matrix Hamiltonians and Hamiltonians generated by differential expressions with PT-symmetric point interactions, the usual situation is that the C operator is unbounded. The fact that the C operator is unbounded is significant because, while there is a formal equivalence between a PT-symmetric Hamiltonian and a conventionally Hermitian Hamiltonian in the sense that the two Hamiltonians are isospectral, the Hilbert spaces are inequivalent. This is so because the mapping from one Hilbert space to the other is unbounded. This shows that PT-symmetric quantum theories are mathematically distinct from conventional Hermitian quantum theories.
http://arxiv.org/abs/1207.1176
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Anjana Sinha
We apply the factorization technique developed by Kuru et. al. [Ann. Phys. {\bf 323} (2008) 413] to obtain the exact analytical classical trajectories and momenta of the continuum states of the non Hermitian but \(\cal{PT}\) symmetric Scarf II potential. In particular, we observe that the strange behaviour of the quantum version at the spectral singularity has an interesting classical analogue.
http://arxiv.org/abs/1206.6987
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Miloslav Znojil
The bound-state spectrum of a Hamiltonian H is assumed real in a non-empty domain D of physical values of parameters. This means that for these parameters, H may be called crypto-Hermitian, i.e., made Hermitian via an ad hoc choice of the inner product in the physical Hilbert space of quantum bound states (i.e., via an ad hoc construction of the so called metric). The name of quantum catastrophe is then assigned to the N-tuple-exceptional-point crossing, i.e., to the scenario in which we leave domain D along such a path that at the boundary of D, an N-plet of bound state energies degenerates and, subsequently, complexifies. At any fixed \(N \geq 2\), this process is simulated via an N by N benchmark effective matrix Hamiltonian H. Finally, it is being assigned such a closed-form metric which is made unique via an N-extrapolation-friendliness requirement.
http://arxiv.org/abs/1206.6000
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Carl M. Bender, David J. Weir
Non-Hermitian PT-symmetric quantum-mechanical Hamiltonians generally exhibit a phase transition that separates two parametric regions, (i) a region of unbroken PT symmetry in which the eigenvalues are all real, and (ii) a region of broken PT symmetry in which some of the eigenvalues are complex. This transition has recently been observed experimentally in a variety of physical systems. Until now, theoretical studies of the PT phase transition have generally been limited to one-dimensional models. Here, four nontrivial coupled PT-symmetric Hamiltonians, \(H=p^2/2+x^2/2+q^2/2+y^2/2+igx^2y\), \(H=p^2/2+x^2/2+q^2/2+y^2+igx^2y\), \(H=p^2/2+x^2/2+q^2/2+y^2/2+r^2/2+z^2/2+igxyz\), and \(H=p^2/2+x^2/2+q^2/2+y^2+r^2/2+3z^2/2+igxyz\) are examined. Based on extensive numerical studies, this paper conjectures that all four models exhibit a phase transition. The transitions are found to occur at \(g\approx 0.1\), \(g\approx 0.04\), \(g\approx 0.1\), and \(g\approx 0.05\). These results suggest that the PT phase transition is a robust phenomenon not limited to systems having one degree of freedom.
http://arxiv.org/abs/1206.5100
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)
Carl M. Bender, Bjorn K. Berntson, David Parker, E. Samuel
If a Hamiltonian is PT symmetric, there are two possibilities: Either the eigenvalues are entirely real, in which case the Hamiltonian is said to be in an unbroken-PT-symmetric phase, or else the eigenvalues are partly real and partly complex, in which case the Hamiltonian is said to be in a broken-PT-symmetric phase. As one varies the parameters of the Hamiltonian, one can pass through the phase transition that separates the unbroken and broken phases. This transition has recently been observed in a variety of laboratory experiments. This paper explains the phase transition in a simple and intuitive fashion and then describes an extremely elementary experiment in which the phase transition is easily observed.
http://arxiv.org/abs/1206.4972
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)
Eva-Maria Graefe
The understanding of nonlinear PT-symmetric quantum systems, arising for example in the theory of Bose-Einstein condensates in PT-symmetric potentials, is widely based on numerical investigations, and little is known about generic features induced by the interplay of PT-symmetry and nonlinearity. To gain deeper insights it is important to have analytically solvable toy-models at hand. In the present paper the stationary states of a simple toy-model of a PT-symmetric system are investigated. The model can be interpreted as a simple description of a Bose-Einstein condensate in a PT-symmetric double well trap in a two-mode approximation. The eigenvalues and eigenstates of the system can be explicitly calculated in a straight forward manner; the resulting structures resemble those that have recently been found numerically for a more realistic PT-symmetric double delta potential. In addition, a continuation of the system is introduced that allows an interpretation in terms of a simple linear matrix model.
http://arxiv.org/abs/1206.4806
Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas)