Parity-Time Symmetry Breaking beyond One Dimension: The Role of Degeneracy

Open Access

Parity-Time Symmetry Breaking beyond One Dimension: The Role of Degeneracy

Li Ge and A. Douglas Stone

Phys. Rev. X 4, 031011 – Published 21 July 2014

Abstract

We consider the role of degeneracy in parity-time ( $P T$ ) symmetry breaking for non-Hermitian wave equations beyond one dimension. We show that if the spectrum is degenerate in the absence of $T$ breaking, and $T$ is broken in a generic manner (without preserving other discrete symmetries), then the standard $P T$ symmetry-breaking transition does not occur, meaning that the spectrum is complex even for infinitesimal strength of gain and loss. However, the realness of the entire spectrum can be preserved over a finite interval if additional discrete symmetries $χ$ are imposed when $T$ is broken, if $χ$ decouples all degenerate modes. When the decoupling holds only for a subset of the degenerate spectrum, there can be a partial $P T$ transition in which this subset remains real over a finite interval of $T$ breaking. If the spectrum has odd degeneracy, a fraction of the degenerate spectrum can remain in the symmetric phase even without imposing additional discrete symmetries, and they are analogous to dark states in atomic physics. These results are illustrated by the example of different $T$ -breaking perturbations of a uniform dielectric disk and sphere, and a group-theoretical analysis is given in the disk case. Finally, we show that multimode coupling is capable of restoring the $P T$ -symmetric phase at finite $T$ breaking. We also analyze these questions when the parity operator is replaced by another spatial symmetry operator and find that the behavior can be qualitatively different.

Received 26 January 2014

DOI:https://doi.org/10.1103/PhysRevX.4.031011

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Li Ge^*

Department of Engineering Science and Physics, College of Staten Island, CUNY, Staten Island, New York 10314, USA, and The Graduate Center, CUNY, New York, New York 10016, USA

A. Douglas Stone

Department of Applied Physics, Yale University, New Haven, Connecticut 06520-8482, USA

^*li.ge@csi.cuny.edu

Popular Summary

Time-reversal symmetry describes invariance under a reversal of the direction of time. Systems that do not obey this symmetry are particularly interesting because their particle number changes with time, such as the increase in photon number (i.e., light intensity) in a light amplifier. However, it has been shown mathematically that such an increase (or decrease) in particle number need not occur if another symmetry is present, such as parity-time ( $P T$ ) symmetry that leaves the system unchanged under a combined mirror-reflection and time-reversal operation. Studies of $P T$ symmetry have been conducted primarily using one-dimensional and quasi-one-dimensional setups. We study $P T$ symmetry in higher dimensions and find that the presence of degeneracy and additional discrete symmetries dictates the spontaneous symmetry-breaking transition.

Systems with $P T$ symmetry lack the hermicity required by standard quantum Hamiltonians. Nevertheless, such $P T$ -symmetric systems can be realized physically in optical devices by introducing a balanced arrangement of optical elements with amplifying (gain) and absorptive (loss) media. In a typical system with only gain or only loss, the eigenvalues of the wave equation are complex, corresponding to amplification or attenuation. However, in $P T$ -symmetric systems studied previously, these eigenvalues remained real up to a threshold degree of (balanced) gain and loss, above which they underwent a spontaneous symmetry-breaking transition to complex (amplifying and attenuating) eigenvalues. This surprising behavior has been tested and confirmed in a number of experiments in one-dimensional and quasi-one-dimensional $P T$ systems. In higher dimensions with degenerate eigenvalues before gain and loss is added, we find that there is no longer a region of purely real eigenvalues if no additional discrete symmetries are present other than $P T$ . Instead, all degenerate eigenvalues become complex as soon as an infinitesimal amount of gain and loss is present. However, if some additional discrete symmetries are preserved, then a fraction or all of the eigenvalues can remain real over a finite interval before displaying the $P T$ symmetry-breaking transition.

Key Image

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 4, Iss. 3 — July - September 2014

Subject Areas

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Absence of $P T$ transitions in a $P T$ -symmetric disk with no additional discrete symmetry. The gain and loss gradient is given by $g (r, ϕ) = ϕ / π (r > R / 5; - π < ϕ < π), 0 (r \leq R / 5)$ and shown schematically as the inset in (a). (a) Evolution of the imaginary part of two pairs of initially degenerate eigenfrequencies as a function of $τ$ . Solid and dashed lines represent modes with a dominant angular momentum $m = 2, 3$ , respectively. These numerical results cannot be distinguished from the approximation (4). (b) Intensity profiles of the four marked modes in (a) at $τ = 0.2$ . Dashed lines show the designated $P$ axes. The refractive index $n = 3.3$ is used in all examples.
Reuse & Permissions
Figure 2
Protected $P T$ transitions and an entirely real spectrum at small $τ$ with additional discrete symmetries. (a) Three pairs of broken-symmetry modes are shown, and complex eigenvalues only appear above the lowest $τ_{TH} ≃ 0.047$ . Inset: The symmetries of the system are described by the generalized dihedral group ${D T}_{2 v}$ with $v = 6$ (see Sec. 3c). It includes, for example, $P_{π / 2}$ and $R_{2 π / 3}$ in addition to the $P T$ symmetry. Here, $P_{ϕ}$ denotes the parity operation about the $ϕ, ϕ + π$ axis, and $R_{ϕ}$ denotes clockwise rotation about the origin by $ϕ$ . (b) Intensity profiles of the four broken-symmetry modes marked in (a) at $τ = 0.16$ . The broken-symmetry modes can be nondegenerate (e.g., modes $A$ and $B$ ), in which case they are simultaneous eigenfunctions of $P_{π / 2}$ and $R_{2 π / 3}$ ; they can also be degenerate (e.g., modes $C 1$ and $C 2$ ), in which case they can either be eigenfunctions of $P_{π / 2}$ or $R_{2 π / 3}$ but not both. Modes $C 1$ and $C 2$ plotted are eigenfunctions of $P_{π / 2}$ , and they can be linearly combined to be eigenfunctions of $R_{2 π / 3}$ [26].
Reuse & Permissions
Figure 3
Partial transitions with additional discrete symmetries. (a) Same as Fig. 1 for the $m = 2, 3$ modes but with uniform gain or loss in each of the four quadrants, which is described by the generalized dihedral group ${D T}_{2 v}$ with $v = 4$ (see Sec. 3c). Dashed lines show the absence of the standard $P T$ phase transition for the $m = 3$ modes. Solid lines show the standard $P T$ phase transition for the odd-parity $m = 2$ mode coupled to a nearby $m = 0$ mode. Their threshold $τ_{TH} = 0.498$ is well approximated by Eq. (6), which gives $τ_{TH} = 0.483$ . The pairing even-parity $m = 2$ mode stays in the $P T$ symmetry phase in the range shown. (b) Intensity profiles of the four modes at $τ = 1$ . The $m = 3$ modes are slightly perturbed from those in Fig. 1 due to their weak coupling to a pair of $m = 1$ modes nearby as $τ$ becomes larger.
Reuse & Permissions
Figure 4
A finite transition threshold always appears in an $R T$ -symmetric disk, with or without additional symmetries. The one shown in the inset of (a) belongs to the former, and the gain and loss gradient is given by $g (r, ϕ) = ϕ / π (r > R / 5; 0 < ϕ < π), - (ϕ + π) / π (r > R / 5; - π < ϕ < 0), 0 (r \leq R / 5)$ . In (a), only the two eigenfrequencies with the lowest $τ_{TH}$ are shown. (b) The intensity profile of the marked mode at $τ = 0.233$ . The intensity profile of its $R T$ -symmetric partner is obtained by rotating (b) by $π$ .
Reuse & Permissions
Figure 5
Examples of the ${M T}_{2 v}$ group with (a) $v = 3$ and (b) $v = 4$ . Similarly to the schematics in Figs. 1 and 4, black and red regions represent loss ( $L$ ) and gain ( $G$ ), and the transparency of the colors shows their strength.
Reuse & Permissions
Figure 6
Restoration of the $P T$ -symmetric phase due to multimode coupling. (a) Evolution of the imaginary part of two $m = 2$ and one $m = 0$ eigenfrequencies as a function of the gain and loss strength $τ$ (solid lines) in the $P T$ -symmetric disk studied in Fig. 1. $k_{0} R = 2.62$ and $k_{2} R = 2.55$ at $τ = 0$ . Dotted lines show the three-mode approximation (7). Dashed lines show the case if the coupling between the $m = 0$ mode and the odd-parity $m = 2$ mode increased by 30%, which exhibits a restored $P T$ -symmetric phase in $τ \in [0.152, 0.166]$ . (b) Top: Intensity profiles of the $m = 0$ mode and the odd-parity $m = 2$ mode at $τ = 0$ . Bottom: The $P T$ symmetry-broken modes $A$ and $B$ at $τ = 0.2, 0.3$ [marked by black dots in (a)]. Only the central regions of $r < R / 2$ are shown.
Reuse & Permissions

Physical Review X