Paolo Amore, Francisco M. Fernández, Javier Garcia, German Gutierrez
We study both analytically and numerically the spectrum of inhomogeneous strings with \(\mathcal{PT}\)-symmetric density. We discuss an exactly solvable model of \(\mathcal{PT}\)-symmetric string which is isospectral to the uniform string; for more general strings, we calculate exactly the sum rules \(Z(p) \equiv \sum_{n=1}^\infty 1/E_n^p\), with \(p=1,2,\dots\) and find explicit expressions which can be used to obtain bounds on the lowest eigenvalue. A detailed numerical calculation is carried out for two non-solvable models depending on a parameter, obtaining precise estimates of the critical values where pair of real eigenvalues become complex.
http://arxiv.org/abs/1306.1419
Mathematical Physics (math-ph)
Boyan T. Torosov, Giuseppe Della Valle, Stefano Longhi
A non-Hermitian shortcut to adiabaticity is introduced. By adding an imaginary term in the diagonal elements of the Hamiltonian of a two state quantum system, we show how one can cancel the nonadiabatic losses and perform an arbitrarily fast population transfer, without the need to increase the coupling. We apply this technique to two popular level-crossing models: the Landau-Zener model and the Allen-Eberly model.
http://arxiv.org/abs/1306.0698
Quantum Physics (quant-ph)
S. Longhi, G. Della Valle
Scattering of a quantum particle from an oscillating barrier or well does not generally conserve the particle energy owing to energy exchange with the photon field, and an incoming particle-free state is scattered into a set of outgoing (transmitted and reflected) free states according to Floquet scattering theory. Here we introduce two families of oscillating non-Hermitian potential wells in which Floquet scattering is fully suppressed for any energy of the incident particle. The scattering-free oscillating potentials are synthesized by application of the Darboux transformation to the time-dependent Schr\”{o}dinger equation. For one of the two families of scattering-free potentials, the oscillating potential turns out to be fully invisible.
http://arxiv.org/abs/1306.0675
Quantum Physics (quant-ph)
S. Longhi, G. Della Valle
We show that invisible localized defects, i.e. defects that can not be detected by an outside observer, can be realized in a crystal with an engineered imaginary potential at the defect site. The invisible defects are synthesized by means of supersymmetric (Darboux) transformations of an ordinary crystal using band-edge wave functions to construct the superpotential. The complex crystal has an entire real-valued energy spectrum and Bragg scattering is not influenced by the defects. An example of complex crystal synthesis is presented for the Mathieu potential.
http://arxiv.org/abs/1306.0667
Quantum Physics (quant-ph)
Hadiseh Alaeian, Jennifer A. Dionne
We theoretically investigate the optical properties of parity-time (PT)-symmetric three dimensional metamaterials composed of strongly-coupled planar plasmonic waveguides. By tuning the loss-gain balance, we show how the initially isotropic material becomes both asymmetric and unidirectional. The highly tunable optical dispersion of PT -symmetric metamaterials provides a foundation for designing an entirely new class of three-dimensional bulk synthetic media, with applications ranging from sub-diffraction-limited optical lenses to non-reciprocal nanophotonic devices.
http://arxiv.org/abs/1306.0059
Optics (physics.optics)
Eva-Maria Graefe, Chiara Liverani
A generalised mean-field approximation for non-Hermitian many-particle systems has been introduced recently for a Bose-Hubbard dimer with complex on-site energies. Here we apply this approximation to a Bose-Hubbard dimer with a complex particle interaction term, modelling losses due to interactions in a two mode Bose-Einstein condensate. We derive the mean-field equations of motion leading to nonlinear dissipative Bloch dynamics, related to a nontrivial complex generalisation of the nonlinear Schrodinger equation. It is shown that depending on the parameter values there can be up to six stationary states. Further, for small values of the interaction strength the dynamics shows limit cycles.
http://arxiv.org/abs/1305.7160
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Carl M. Bender, Mariagiovanna Gianfreda
The inspiration for this theoretical paper comes from recent experiments on a PT-symmetric system of two coupled optical whispering galleries (optical resonators). The optical system can be modeled as a pair of coupled linear oscillators, one with gain and the other with loss. If the coupled oscillators have a balanced loss and gain, the system is described by a Hamiltonian and the energy is conserved. This theoretical model exhibits two PT transitions depending on the size of the coupling parameter \epsilon. For small \epsilon the PT symmetry is broken and the system is not in equilibrium, but when \epsilon becomes sufficiently large, the system undergoes a transition to an equilibrium phase in which the PT symmetry is unbroken. For very large \(\epsilon\) the system undergoes a second transition and is no longer in equilibrium. The classical and the quantized versions of the system exhibit transitions at exactly the same values of \(\epsilon\).
http://arxiv.org/abs/1305.7107
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Jiteng Sheng, Mohammad-Ali Miri, Demetrios N. Christodoulides, Min Xiao
We demonstrate that a coherently-prepared four-level atomic medium can provide a versatile platform for realizing parity-time (PT) symmetric optical potentials. Different types of PT-symmetric potentials are proposed by appropriately tuning the exciting optical fields and the pertinent atomic parameters. Such reconfigurable and controllable systems can open up new avenues in observing PT-related phenomena with appreciable gain/loss contrast in coherent atomic media.
http://arxiv.org/abs/1305.4908
Optics (physics.optics)
Miloslav Znojil
The practical use of non-Hermitian (i.e., typically, PT-symmetric) phenomenological quantum Hamiltonians is discussed as requiring an explicit reconstruction of the ad hoc Hilbert-space metrics which would render the time-evolution unitary. Just the N-dimensional matrix toy models Hamiltonians are considered, therefore. For them, the matrix elements of alternative metrics are constructed via solution of a coupled set of polynomial equations, using the computer-assisted symbolic manipulations for the purpose. The feasibility and some consequences of such a model-construction strategy are illustrated via a discrete square well model endowed with multi-parametric close-to-the-boundary real bidiagonal-matrix interaction. The degenerate exceptional points marking the phase transitions are then studied numerically. A way towards classification of their unfoldings in topologically non-equivalent dynamical scenarios is outlined.
http://arxiv.org/abs/1305.4822
Quantum Physics (quant-ph)
Yogesh N. Joglekar, Clinton Thompson, Derek D. Scott, Gautam Vemuri
Over the last two decades, advances in fabrication have led to significant progress in creating patterned heterostructures that support either carriers, such as electrons or holes, with specific band structure or electromagnetic waves with a given mode structure and dispersion. In this article, we review the properties of light in coupled optical waveguides that support specific energy spectra, with or without the effects of disorder, that are well-described by a Hermitian tight-binding model. We show that with a judicious choice of the initial wave packet, this system displays the characteristics of a quantum particle, including transverse photonic transport and localization, and that of a classical particle. We extend the analysis to non-Hermitian, parity and time-reversal (\(\mathcal{PT}\)) symmetric Hamiltonians which physically represent waveguide arrays with spatially separated, balanced absorption or amplification. We show that coupled waveguides are an ideal candidate to simulate \(\mathcal{PT}\)-symmetric Hamiltonians and the transition from a purely real energy spectrum to a spectrum with complex conjugate eigenvalues that occurs in them.
http://arxiv.org/abs/1305.3565
Optics (physics.optics); Disordered Systems and Neural Networks (cond-mat.dis-nn); Quantum Physics (quant-ph)