Alexander G. Anderson, Carl M. Bender
This paper examines the complex trajectories of a classical particle in the potential \(V(x)=-\cos(x)\). Almost all the trajectories describe a particle that hops from one well to another in an erratic fashion. However, it is shown analytically that there are two special classes of trajectories \(x(t)\) determined only by the energy of the particle and not by the initial position of the particle. The first class consists of periodic trajectories; that is, trajectories that return to their initial position \(x(0)\) after some real time \(T\). The second class consists of trajectories for which there exists a real time \(T\) such that \(x(t+T)=x(t) \pm2 \pi\). These two classes of classical trajectories are analogous to valence and conduction bands in quantum mechanics, where the quantum particle either remains localized or else tunnels resonantly (conducts) through a crystal lattice. These two special types of trajectories are associated with sets of energies of measure 0. For other energies, it is shown that for long times the average velocity of the particle becomes a fractal-like function of energy.
http://arxiv.org/abs/1205.3330
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Samuel Kalish, Zin Lin, Tsampikos Kottos
The scattering properties of randomly layered optical media with \({\cal PT}\)-symmetric index of refraction are studied using the transfer-matrix method. We find that the transmitance decays exponentially as a function of the system size, with an enhanced rate \(\xi_{\gamma}(W)^{-1}=\xi_0(W)^{-1}+\xi_{\gamma} (0)^{-1}\), where \(\xi_0(W)\) is the localization length of the equivalent passive random medium and \(\xi_{\gamma}(0)\) is the attenuation/amplification length of the corresponding perfect system with a \({\cal PT}\)-symmetric refraction index profile. While transmitance processes are reciprocal to left and right incident waves, the reflectance is enhanced from one side and is inversely suppressed from the other, thus allowing such \({\cal PT}\)-symmetric random media to act as unidirectional coherent absorbers.
http://arxiv.org/abs/1205.1849
Optics (physics.optics)
Sanjib Dey, Andreas Fring, Laure Gouba
We provide a systematic procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing noncommutative spaces. The large number of possible free parameters in these calculations is reduced to a manageable amount by imposing various different versions of PT-symmetry on the underlying spaces, which are dictated by the specific physical problem under consideration. The representations for the corresponding operators are in general non-Hermitian with regard to standard inner products and obey algebras whose uncertainty relations lead to minimal length, areas or volumes in phase space. We analyze in particular one three dimensional solution which may be decomposed to a two dimensional noncommutative space plus one commuting space component and also into a one dimensional noncommutative space plus two commuting space components. We study some explicit models on these type of noncommutative spaces.
http://arxiv.org/abs/1205.2291
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Hamidreza Ramezani, J. Schindler, F. M. Ellis, Uwe Guenther, Tsampikos Kottos
The beat time \({\tau}_{fpt}\) associated with the energy transfer between two coupled oscillators is dictated by the bandwidth theorem which sets a lower bound \({\tau}_{fpt}\sim 1/{\delta}{\omega}\). We show, both experimentally and theoretically, that two coupled active LRC electrical oscillators with parity-time (PT) symmetry, bypass the lower bound imposed by the bandwidth theorem, reducing the beat time to zero while retaining a real valued spectrum and fixed eigenfrequency difference \(\delta\omega\). Our results foster new design strategies which lead to (stable) pseudo-unitary wave evolution, and may allow for ultrafast computation, telecommunication, and signal processing.
http://arxiv.org/abs/1205.1847
Classical Physics (physics.class-ph)
Zin Lin, Joseph Schindler, Fred M. Ellis, Tsampikos Kottos
We investigate experimentally parity-time \({\cal PT}\) symmetric scattering using \(LRC\) circuits in an inductively coupled \({\cal PT}\)- symmetric pair connected to transmission line leads. In the single-lead case, the \({\cal PT}\)-symmetric circuit acts as a simple dual device – an amplifier or an absorber depending on the orientation of the lead. When a second lead is attached, the system exhibits unidirectional transparency for some characteristic frequencies. This non-reciprocal behavior is a consequence of generalized (non-unitary) conservation relations satisfied by the scattering matrix.
http://arxiv.org/abs/1205.2176
Mesoscale and Nanoscale Physics (cond-mat.mes-hall)
Emanuela Caliceti, Sandro Graffi, Michael Hitrik, Johannes Sjoestrand
It is established that a PT-symmetric elliptic quadratic differential operator with real spectrum is similar to a self-adjoint operator precisely when the associated fundamental matrix has no Jordan blocks.
http://arxiv.org/abs/1204.6605
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Jun-Qing Li, Qian Li, Yan-Gang Miao
Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones. Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged. In order to give the positive definite inner product for the PT-symmetric systems, a new operator V, instead of C, can be introduced. The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics, however, it has the advantage that V can be constructed directly in terms of Hamiltonians. The spectra of the two non-Hermitian PT-symmetric systems are obtained, which coincide with that given in literature, and in particular, the Hilbert spaces associated with positive definite inner products are worked out.
http://arxiv.org/abs/1204.6544
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)
Jiangbin Gong, Qing-hai Wang
A possibly fruitful extension of conventional random matrix ensembles is proposed by imposing symmetry constraints on conventional Hermitian matrices or parity-time- (PT-) symmetric matrices. To illustrate the main idea, we first study 2*2 complex Hermitian matrix ensembles with O(2) invariant constraints, yielding novel level-spacing statistics such as singular distributions, half-Gaussian distribution, distributions interpolating between GOE (Gaussian Orthogonal Ensemble) distribution and half Gaussian distributions, as well as gapped-GOE distribution. Such a symmetry-reduction strategy is then used to explore 2*2 PT-symmetric matrix ensembles with real eigenvalues. In particular, PT-symmetric random matrix ensembles with U(2) invariance can be constructed, with the conventional complex Hermitian random matrix ensemble being a special case. In two examples of PT-symmetric random matrix ensembles, the level-spacing distributions are found to be the standard GUE (Gaussian Unitary Ensemble) statistics or “truncated-GUE” statistics.
http://arxiv.org/abs/1204.6126
Quantum Physics (quant-ph)
Miloslav Znojil
For the description of quantum evolution, the use of a manifestly time-dependent quantum Hamiltonian \(\mathfrak{h}(t) =\mathfrak{h}^\dagger(t)\) is shown equivalent to the work with its simplified, time-independent alternative \(G\neq G^\dagger\). A tradeoff analysis is performed recommending the latter option. The physical unitarity requirement is shown fulfilled in a suitable ad hocrepresentation of Hilbert space.
http://arxiv.org/abs/1204.5989
Quantum Physics (quant-ph); General Relativity and Quantum Cosmology (gr-qc)
D. A. Zezyulin, V. V. Konotop
By rearrangements of waveguide arrays with gain and losses one can simulate transformations among parity-time (PT-) symmetric systems not affecting their pure real linear spectra. Subject to such transformations, however, the nonlinear properties of the systems undergo significant changes. On an example of an array of four waveguides described by the discrete nonlinear Schr\”odinger equation with dissipation and gain, we show that the equivalence of the underlying linear spectra implies similarity of neither structure nor stability of the nonlinear modes in the arrays. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of a newly obtained system. Neither the stability is directly related to the PT symmetry: stable nonlinear modes exist even when the spectrum of the linear array is not purely real. We use graph representation of PT-symmetric networks allowing for 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)