Philip D. Mannheim
We present some interesting connections between \(PT\) symmetry and conformal symmetry. We use them to develop a metricated theory of electromagnetism in which the electromagnetic field is present in the geometric connection. However, unlike Weyl who first advanced this possibility, we do not take the connection to be real but to instead be \(PT\) symmetric, with it being \(iA_{\mu}\) rather than \(A_{\mu}\) itself that then appears in the connection. With this modification the standard minimal coupling of electromagnetism to fermions is obtained. Through the use of torsion we obtain a fully metricated theory of electromagnetism that treats its electric and magnetic sectors completely symmetrically, with a conformal invariant theory of gravity being found to emerge.
http://arxiv.org/abs/1407.1820
High Energy Physics – Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)
Jiun-Yi Lien, Yueh-Nan Chen, Natsuko Ishida, Hong-Bin Chen, Chi-Chuan Hwang, Franco Nori
Exciton-polaritons can condense to a macroscopic quantum state through a non-equilibrium process of pumping and decay. In recent experiments, polariton condensates are used to observe, for a short time, nonlinear Josephson phenomena by coupling two condensates. However, it is still not clear how these phenomena are affected by the pumping and decay at long times and how the coupling alters the polariton condensation. Here, we consider a polariton Josephson junction pumped on one side and study its dynamics within a mean-field theory. The Josephson current is found to give rise to multi-stability of the stationary states, which are sensitive to the initial conditions and incoherent noises. These states can be attributed to either the self-trapping effect or the parity-time (PT) symmetry of the system. These results can be used to explain the emission spectra and the \(\pi\)-phase locking observed in recent experiments. We further predict that the multi-stability can reduce to the self-trapped state if the PT symmetry is broken. Moreover, the polaritons can condense even below the threshold, exhibiting hysteresis.
http://arxiv.org/abs/1407.1271
Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)
Ali Mostafazadeh
We give a complete solution of the problem of constructing a scattering potential v(x) that possesses scattering properties of one’s choice at an arbitrary prescribed wavenumber. Our solution involves expressing v(x) as the sum of at most six unidirectionally invisible finite-range potentials for which we give explicit formulas. Our results can be employed for designing optical potentials. We discuss its application in modeling threshold lasers, coherent perfect absorbers, and bidirectionally and unidirectionally reflectionless absorbers, amplifiers, and phase shifters.
http://arxiv.org/abs/1407.1760
Quantum Physics (quant-ph); Optics (physics.optics)
Asiri Nanayakkara, Thilagarajah Mathanaranjan
In this paper we derive an almost explicit analytic formula for asymptotic eigenenergy expansion of arbitrary odd degree polynomial potentials of the form \(V(x)=(ix)^{2N+1}+\beta _{1}x^{2N}+\beta _{2}x^{2N-1}+\cdot \cdot \cdot \cdot \cdot +\beta _{2N}x\) where \(\beta _{k}^{\prime }\)s are real or complex for \(1\leq k\leq 2N\). The formula can be used to find semiclassical analytic expressions for eigenenergies up to any order very efficiently. Each term of the expansion is given explicitly as a multinomial of the parameters \(\beta _{1},\beta _{2}….\) and \(\beta _{2N}\) of the potential. Unlike in the even degree polynomial case, the highest order term in the potential is pure imaginary and hence the system is non-Hermitian. Therefore all the integrations have been carried out along a contour enclosing two complex turning points which lies within a wedge in the complex plane. With the help of some examples we demonstrate the accuracy of the method for both real and complex eigenspectra.
http://arxiv.org/abs/1407.0191
Mathematical Physics (math-ph)
Katherine Jones-Smith, Connor Wallace
Hofstadter showed that the energy levels of electrons on a lattice plotted as a function of magnetic field form an beautiful structure now referred to as “Hofstadter’s butterfly”. We study a non-Hermitian continuation of Hofstadter’s model; as the non-Hermiticity parameter \(g\) increases past a sequence of critical values the eigenvalues successively go complex in a sequence of “double-pitchfork bifurcations” wherein pairs of real eigenvalues degenerate and then become complex conjugate pairs. The associated wavefunctions undergo a spontaneous symmetry breaking transition that we elucidate. Beyond the transition a plot of the real parts of the eigenvalues against magnetic field resembles the Hofstadter butterfly; a plot of the imaginary parts plotted against magnetic fields forms an intricate structure that we call the Hofstadter cocoon. The symmetries of the cocoon are described. Hatano and Nelson have studied a non-Hermitian continuation of the Anderson model of localization that has close parallels to the model studied here. The relationship of our work to that of Hatano and Nelson and to PT transitions studied in PT quantum mechanics is discussed.
http://arxiv.org/abs/1407.0093
Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)
Özlem Yeşiltaş
Searching for non-Hermitian (parity-time)\(\mathcal{PT}\)-symmetric Hamiltonians with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a \(\mathcal{PT}\)-symmetric non-Hermitian Hamiltonian model which is given as \(\hat{\mathcal{H}}=\omega (\hat{b}^\dagger\hat{b}+\frac{1}{2})+\alpha (\hat{b}^{2}-(\hat{b}^\dagger)^{2})\) where \(\omega\) and \(\alpha\) are real constants, \(\hat{b}\) and \(\hat{b^\dagger}\) are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of \(\mathcal{PT}\) symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian \(\mathcal{H}\) is pseudo-Hermitian, we have obtained the Hermitian equivalent of \(\mathcal{H}\), which is in Sturm- Liouville form, leads to exactly solvable potential models which are effective screened potential and hyperbolic Rosen-Morse II potential. \(\mathcal{H}\) is called pseudo-Hermitian, if there exists a Hermitian and invertible operator \(\eta\) satisfying \(\mathcal{H^\dagger}=\eta \mathcal{H} \eta^{-1}\). For the Hermitian Hamiltonian \(h\), one can write \(h=\rho \mathcal{H} \rho^{-1}\) where \(\rho=\sqrt{\eta}\) is unitary. Using this \(\rho\) we have obtained a physical Hamiltonian \(h\) for each case. Then, the Schr\”{o}dinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics. Mapping function \(\rho\) is obtained for each potential case.
http://arxiv.org/abs/1406.3298
Quantum Physics (quant-ph)
Özlem Yeşiltaş, Nafiye Kaplan
We have obtained the metric operator \(\Theta=\exp T\) for the non-Hermitian Hamiltonian model \(H=\omega(a^{\dag}a+1/2)+\alpha(a^{2}-a^{\dag^{2}})\). We have also found the intertwining operator which connects the Hamiltonian to the adjoint of its pseudo-supersymmetric partner Hamiltonian for the model of hyperbolic Rosen-Morse II potential.
http://arxiv.org/abs/1406.3179
Mathematical Physics (math-ph)
N. Lu, J. Cuevas-Maraver, P.G. Kevrekidis
In this work, we explore a prototypical example of a genuine continuum breather (i.e., not a standing wave) and the conditions under which it can persist in a \(\mathcal{P T}\)-symmetric medium. As our model of interest, we will explore the sine-Gordon equation in the presence of a \(\mathcal{P T}\)- symmetric perturbation. Our main finding is that the breather of the sine-Gordon model will only persist at the interface between gain and loss that \(\mathcal{P T}\)-symmetry imposes but will not be preserved if centered at the lossy or at the gain side. The latter dynamics is found to be interesting in its own right giving rise to kink-antikink pairs on the gain side and complete decay of the breather on the lossy side. Lastly, the stability of the breathers centered at the interface is studied. As may be anticipated on the basis of their “delicate” existence properties such breathers are found to be destabilized through a Hopf bifurcation in the corresponding Floquet analysis.
http://arxiv.org/abs/1406.3082
Pattern Formation and Solitons (nlin.PS)
Vasily N. Rodionov
The modified Dirac-Pauli equations, which are introduced by means of \({\gamma_5}\)-mass factorization of the ordinary Klein-Gordon operator, are considered. We also take into account the interaction of fermions with the intensive homogenous magnetic field focusing attention to their (g-2) gyromagnetic factor. The basis of this approach is developing of methods for study of the structure of regions of unbroken \(\cal PT\) symmetry of Non-Hermitian Hamiltonians which be no studied earlier. For that, without the use of perturbation theory in the external field the exact energy spectra are deduced with regard to spin effects of fermions. We also investigate the unique possible of experimental observability the non-Hermitian restrictions in the spectrum of mass consistent with the conjecture Markov about Maximal Mass. This, in principal will may allow to find out the existence of an upper limit value in spectrum masses of elementary particles and confirm or deny the significance of the Planck mass.
http://arxiv.org/abs/1406.0383
High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)
Naomichi Hatano
Two theoretical methods of finding resonant states in open quantum systems, namely the approach of the Siegert boundary condition and the Feshbach formalism, are reviewed and shown to be algebraically equivalent to each other for a simple model of the T-type quantum dot. It is stressed that the seemingly Hermitian Hamiltonian of an open quantum system is implicitly non-Hermitian outside the Hilbert space. The two theoretical approaches extract an explicitly non-Hermitian effective Hamiltonian in a contracted space out of the seemingly Hermitian (but implicitly non-Hermitian) full Hamiltonian in the infinite-dimensional state space of an open quantum system.
http://arxiv.org/abs/1405.7021
Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Mathematical Physics (math-ph); Nuclear Theory (nucl-th)