arXiv subjects

H. F. Jones

Publications and source records attributed to H. F. Jones.

At least 19 recordsLinked to original sources

arXiv2015

Intrinsically single-mode cavities: towards PT-symmetric surface-emitting lasers

We explore a new class of Distributed Feedback (DFB) structures that employ the recently-developed concept of Parity-Time (PT) symmetry in optics. We demonstrate that, based on PT-symmetric pure reflective volume gratings, vertical cavity surface-emitting lasers can be constructed. We provide a detailed analysis of the threshold conditions as well as the wavelength and angular spectral characteristics using the Kogelnik coupled-wave approximation, backed up by an exact solution of the Helmholtz equation. We demonstrate that such a PT-symmetric cavity can be configured to support one and only one longitudinal mode, leading to inherently single-mode lasing.

physics.optics↗

arXiv2015

An angle on invisibility

The $PT$-symmetric optical grating with index profile $e^{iβz}$ has been shown to have the interesting property of being essentially invisible for light incident from one side, while possessing greatly enhanced reflection at a particular wavelength for light incident from the other side. We extend a previous analysis of this grating to obtain an analytic solution for the case when the grating is embedded on a substrate, with different refractive indices on either side. We also generalize the previous case of normal incidence to incidence at an arbitrary angle. In that case the enhanced reflection occurs at a particular angle of incidence for a given wavelength. Finally we discuss how the grating may be used to give lasing.

physics.optics↗

arXiv2015

Analysis of unidirectional non-paraxial invisibility of purely reflective PT-symmetric volume gratings

We study the diffraction produced by a slab of purely reflective PT-symmetric volume Bragg grating that combines modulations of refractive index and gain/loss of the same periodicity with a quarter-period shift between them. Such a complex grating has a directional coupling between the different diffraction orders, which allows us to find an analytic solution for the first three orders of the full Maxwell equations without resorting to the paraxial approximation. This is important, because only with the full equations can the boundary conditions, allowing for the reflections, be properly implemented. Using our solution we analyze unidirectional invisibility of such a grating in a wide variety of configurations.

physics.optics↗

arXiv2014

Analytic Solution for $PT$-Symmetric Volume Gratings

We study the diffraction produced by a $PT$-symmetric volume Bragg grating that combines modulation of refractive index and gain/loss of the same periodicity with a quarter-period shift between them. Such a complex grating has a directional coupling between the different diffraction orders, which allows us to find an analytic solution for the first three orders of the full Maxwell equations without resorting to the paraxial approximation. This is important, because only with the full equations can the boundary conditions, allowing for reflections, be properly implemented. Using our solution we analyze the properties of such a grating in a wide variety of configurations.

physics.optics↗

arXiv2014

Singular Mapping for a $PT$-Symmetric Sinusoidal Optical Lattice at the Symmetry-Breaking Threshold

A popular $PT$-symmetric optical potential (variation of the refractive index) that supports a variety of interesting and unusual phenomena is the imaginary exponential, the limiting case of the potential $V_0[\cos(2πx/a)+iλ\sin(2πx/a)]$ as $λ\to 1$, the symmetry-breaking point. For $λ<1$, when the spectrum is entirely real, there is a well-known mapping by a similarity transformation to an equivalent Hermitian potential. However, as $λ\to 1$, the spectrum, while remaining real, contains Jordan blocks in which eigenvalues and the corresponding eigenfunctions coincide. In this limit the similarity transformation becomes singular. Nonetheless, we show that the mapping from the original potential to its Hermitian counterpart can still be implemented; however, the inverse mapping breaks down. We also illuminate the role of Jordan associated functions in the original problem, showing that they map onto eigenfunctions in the associated Hermitian problem.

physics.optics↗

arXiv2014

Unique optical characteristics of a Fabry-Perot resonator with embedded PT-symmetrical grating

We explore the optical properties of a Fabry-Perot resonator with an embedded Parity-Time (PT) symmetrical grating. This PT-symmetrical grating is non diffractive (transparent) when illuminated from one side and diffracting (Bragg reflection) when illuminated from the other side, thus providing a unidirectional reflective functionality. The incorporated PT-symmetrical grating forms a resonator with two embedded cavities. We analyze the transmission and reflection properties of these new structures through a transfer matrix approach. Depending on the resonator geometry these cavities can interact with different degrees of coherency: fully constructive interaction, partially constructive interaction, partially destructive interaction, and finally their interaction can be completely destructive. A number of very unusual (exotic) nonsymmetrical absorption and amplification behaviors are observed. The proposed structure also exhibits unusual lasing performance. Due to the PT-symmetrical grating, there is no chance of mode hopping; it can lase with only a single longitudinal mode for any distance between the distributed reflectors.

physics.optics↗

arXiv2012

The Floquet Method for $PT$-symmetric Periodic Potentials

By the general theory of $PT$-symmetric quantum systems, their energy levels are either real or occur in complex-conjugate pairs, which implies that the secular equation must be real. However, for periodic potentials it is by no means clear that the secular equation arising in the Floquet method is indeed real, since it involves two linearly independent solutions of the Schrödinger equation. In this brief note we elucidate how that reality can be established.

math-ph↗

arXiv2011

Analytic Results for a PT-symmetric Optical Structure

Propagation of light through media with a complex refractive index in which gain and loss are engineered to be $PT$ symmetric has many remarkable features. In particular the usual unitarity relations are not satisfied, so that the reflection coefficients can be greater than one, and in general are not the same for left or right incidence. Within the class of optical potentials of the form $v(x)=v_1\cos(2βx)+iv_2\sin(2βx)$ the case $v_2=v_1$ is of particular interest, as it lies on the boundary of $PT$-symmetry breaking. It has been shown in a recent paper by Lin et al. that in this case one has the property of "unidirectional invisibility", while for propagation in the other direction there is a greatly enhanced reflection coefficient proportional to $L^2$, where $L$ is the length of the medium in the direction of propagation. For this potential we show how analytic expressions can be obtained for the various transmission and reflection coefficients, which are expressed in a very succinct form in terms of modified Bessel functions. While our numerical results agree very well with those of Lin et al. we find that the invisibility is not quite exact, in amplitude or phase. As a test of our formulas we show that they identically satisfy a modified version of unitarity appropriate for $PT$-symmetric potentials. We also examine how the enhanced transmission comes about for a wave-packet, as opposed to a plane wave, finding that the enhancement now arises through an increase, of $O(L)$, in the pulse length, rather than the amplitude.

physics.optics↗

arXiv2011

PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold

The $PT$ symmetric potential $V_0[\cos(2πx/a)+iλ\sin(2πx/a)]$ has a completely real spectrum for $λ\le 1$, and begins to develop complex eigenvalues for $λ>1$. At the symmetry-breaking threshold $λ=1$ some of the eigenvectors become degenerate, giving rise to a Jordan-block structure for each degenerate eigenvector. In general this is expected to result in a secular growth in the amplitude of the wave. However, it has been shown in a recent paper by Longhi, by numerical simulation and by the use of perturbation theory, that for a broad initial wave packet this growth is suppressed, and instead a saturation leading to a constant maximum amplitude is observed. We revisit this problem by explicitly constructing the Bloch wave-functions and the associated Jordan functions and using the method of stationary states to find the dependence on the longitudinal distance $z$ for a variety of different initial wave packets. This allows us to show in detail how the saturation of the linear growth arises from the close connection between the contributions of the Jordan functions and those of the neighbouring Bloch waves.

physics.optics↗

arXiv2011

Perturbation Theory for PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold

The $PT$ symmetric potential $V_0[\cos(2πx/a)+iλ\sin(2πx/a)]$ has a completely real spectrum for $λ\le 1$, and begins to develop complex eigenvalues for $λ>1$. At the symmetry-breaking threshold $λ=1$ some of the eigenvectors become degenerate, giving rise to a Jordan-block structure for each degenerate eigenvector. In general this is expected to give rise to a secular growth in the amplitude of the wave. However, it has been shown in a recent paper by Longhi, by numerical simulation and by the use of perturbation theory, that for an initial wave packet this growth is suppressed, giving instead a constant maximum amplitude. We revisit this problem by developing the perturbation theory further. We verify that the results found by Longhi persist to second order, and with different input wave packets we are able to see the seeds in perturbation theory of the phenomenon of birefringence first discovered by Makris et al.

physics.optics↗

arXiv2010

Use of Equivalent Hermitian Hamiltonian for $PT$-Symmetric Sinusoidal Optical Lattices

We show how the band structure and beam dynamics of non-Hermitian $PT$-symmetric sinusoidal optical lattices can be approached from the point of view of the equivalent Hermitian problem, obtained by an analytic continuation in the transverse spatial variable $x$. In this latter problem the eigenvalue equation reduces to the Mathieu equation, whose eigenfunctions and properties have been well studied. That being the case, the beam propagation, which parallels the time-development of the wave-function in quantum mechanics, can be calculated using the equivalent of the method of stationary states. We also discuss a model potential that interpolates between a sinusoidal and periodic square well potential, showing that some of the striking properties of the sinusoidal potential, in particular birefringence, become much less prominent as one goes away from the sinusoidal case.

physics.optics↗

arXiv2010

Green Functions for the Wrong-Sign Quartic

It has been shown that the Schwinger-Dyson equations for non-Hermitian theories implicitly include the Hilbert-space metric. Approximate Green functions for such theories may thus be obtained, without having to evaluate the metric explicitly, by truncation of the equations. Such a calculation has recently been carried out for various $PT$-symmetric theories, in both quantum mechanics and quantum field theory, including the wrong-sign quartic oscillator. For this particular theory the metric is known in closed form, making possible an independent check of these approximate results. We do so by numerically evaluating the ground-state wave-function for the equivalent Hermitian Hamiltonian and using this wave-function, in conjunction with the metric operator, to calculate the one- and two-point Green functions. We find that the Green functions evaluated by lowest-order truncation of the Schwinger-Dyson equations are already accurate at the (6-8)% level. This provides a strong justification for the method and a motivation for its extension to higher order and to higher dimensions, where the calculation of the metric is extremely difficult.

hep-th↗

arXiv2009

Quantum and Classical Statistical Mechanics of a Class of non-Hermitian Hamiltonians

This paper investigates the thermodynamics of a large class of non-Hermitian, $PT$-symmetric oscillators, whose energy spectrum is entirely real. The spectrum is estimated by second-order WKB approximation, which turns out to be very accurate even for small quantum numbers, and used to generate the quantum partition function. Graphs showing the thermal behavior of the entropy and the specific heat, at all regimes of temperature, are given. To obtain the corresponding classical partition function it turns out to be necessary in general to integrate over a complex "phase space". For the wrong-sign quartic, whose equivalent Hermitian Hamiltonian is known exactly, it is demonstrated explicitly how this formulation arises, starting from the Hermitian case.

quant-ph↗

arXiv2009

Which Green Functions Does the Path Integral for Quasi-Hermitian Hamiltonians Represent?

In the context of quasi-Hermitian theories, which are non-Hermitian in the conventional sense, but can be made Hermitian by the introduction of a dynamically-determined metric $η$, we address the problem of how the functional integral and the Feynman diagrams deduced therefrom "know" about the metric. Our investigation is triggered by a result of Bender, Chen and Milton, who calculated perturbatively the one-point function $G_1$ for the quantum Hamiltonian $H=\half(p^2+x^2)+igx^3$. It turns out that this calculation indeed corresponds to an expectation value in the ground state evaluated with the $η$ metric. The resolution of the problem turns out be that, although there is no explicit mention of the metric in the path integral or Feynman diagrams, their derivation is based fundamentally on the Heisenberg equations of motion, which only take their standard form when matrix elements are evaluated with the inclusion of $η$.

hep-th↗

arXiv2009

Gauging non-Hermitian Hamiltonians

We address the problem of coupling non-Hermitian systems, treated as fundamental rather than effective theories, to the electromagnetic field. In such theories the observables are not the $\bs{x}$ and $\bs{p}$ appearing in the Hamiltonian, but quantities $\bs{X}$ and $\bs{P}$ constructed by means of the metric operator. Following the analogous procedure of gauging a global symmetry in Hermitian quantum mechanics we find that the corresponding gauge transformation in $\bs{X}$ implies minimal substitution in the form $\bs{P}\to \bs{P} - e\bs{A}(\bs{X})$. We discuss how the relevant matrix elements governing electromagnetic transitions may be calculated in the special case of the Swanson Hamiltonian, where the equivalent Hermitian Hamiltonian $h$ is local, and in the more generic example of the imaginary cubic interaction, where $H$ is local but $h$ is not.

quant-ph↗

arXiv2008

Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation

We consider the interaction between the Hermitian world, represented by a real delta-function potential $-αδ(x)$, and the non-Hermitian world, represented by a PT-symmetric pair of delta functions with imaginary coefficients $iβ(δ(x-L)-δ(x+L))$. In the context of standard quantum mechanics, the effect of the introduction of the imaginary delta functions on the bound-state energy of the real delta function and its associated wave-function is small for L large. However, scattering from the combined potentials does not conserve probability as conventionally defined. Both these problems can be studied instead in the context of quasi-Hermiticity, whereby quantum mechanics is endowed with a new metric $η$, and consequently a new wave-function $Ψ(x)$, defined in terms of the original wave-function $ψ(x)$ by means of $η$. In this picture, working perturbatively in $β$, the bound-state wave-function is actually unchanged from its unperturbed form for $|x|<< L$. However, the scattering wave-function, for $|x|>>L$, is changed in a significant manner. In particular there are incoming and outgoing waves on both sides of the potential. One can then no longer talk in terms of reflection and transmission coefficients, but the total right-moving flux is now conserved.

hep-th↗

arXiv2008

Comment on the Quantum Brachistochrone Problem

In this brief comment we attempt to clarify the apparent discrepancy between the papers [1] and [2] on the quantum brachistochrone, namely whether it is possible to use a judicious mixture of Hermitian and non-Hermitian quantum mechanics to evade the standard lower limit on the time taken for evolution by a Hermitian Hamiltonian with given energy dispersion between two given states.

quant-ph↗

arXiv2007

Equivalent Hamiltonian for Lee Model

Using the techniques of quasi-Hermitian quantum mechanics and quantum field theory we use a similarity transformation to construct an equivalent Hermitian Hamiltonian for the Lee model. In the field theory confined to the $V/Nθ$ sector it effectively decouples $V$, replacing the three-point interaction of the original Lee model by an additional mass term for the $V$ particle and a four-point interaction between $N$ and $θ$. While the construction is originally motivated by the regime where the bare coupling becomes imaginary, leading to a ghost, it applies equally to the standard Hermitian regime where the bare coupling is real. In that case the similarity transformation becomes a unitary transformation.

hep-th↗