Nonlinear waves in $\mathcal{PT}$-symmetric systems

Recent progress on nonlinear properties of parity-time ($\mathcal{PT}$)-symmetric systems is comprehensively reviewed in this article. $\mathcal{PT}$ symmetry started out in non-Hermitian quantum mechanics, where complex potentials obeying $\mathcal{PT}$ symmetry could exhibit all-real spectra. This concept later spread out to optics, Bose-Einstein condensates, electronic circuits, and many other physical fields, where a judicious balancing of gain and loss constitutes a $\mathcal{PT}$-symmetric system. The natural inclusion of nonlinearity into these $\mathcal{PT}$ systems then gave rise to a wide array of new phenomena which have no counterparts in traditional dissipative systems. Examples include the existence of continuous families of nonlinear modes and integrals of motion, stabilization of nonlinear modes above $\mathcal{PT}$-symmetry phase transition, symmetry breaking of nonlinear modes, distinctive soliton dynamics, and many others. In this article, nonlinear $\mathcal{PT}$-symmetric systems arising from various physical disciplines are presented, nonlinear properties of these systems are thoroughly elucidated, and relevant experimental results are described. In addition, emerging applications of $\mathcal{PT}$ symmetry are pointed out.

Figure 1

Real eigenvalues of potential (10) for different $N$. Adapted from [47].

Figure 2

(a) Superpotential (25) with $c=1+i$; (b) periodic superpotential (26) with $c=0.5-2i$ and $V_0=1$. From [425].

Figure 3

Left panel: details of two-waveguide layered structure which includes the Cr stripe and features complex refractive index; right panel: scanning electron microscopy picture of the finalized device. Adapted from [163].

Figure 4

Measured density distribution (upper panels) and relative phase difference between the two components (lower panels) of the $\mathcal{PT}$-symmetric coupler in (a) unbroken and (b) broken $\mathcal{PT}$ symmetries. Below the $\mathcal{PT}$-symmetry-breaking threshold the phase difference lies in the interval $[0,\pi ]$, depending on the magnitude of gain, while above the threshold this value is fixed at $\pi /2$. Adapted from [354].

Figure 5

Mapping of the pulse propagation in loops consisting of dispersion compensating fibers into a 2D lattice corresponding to (a) locally $\mathcal{PT}$-symmetric and (b) globally $\mathcal{PT}$-symmetric settings. Light (red) and dark (blue) segments correspond to pulses traveling during the paths with gain and with losses. Alternating gain and losses are shown by purple rings in (b). From [408].

Figure 6

(a) Schematic of the $\mathcal{PT}$ microring laser. The diameter and width of the microring resonator are $8.9\mu \mathrm{m}$ and 900 nm, respectively. (b) Multimode lasing spectrum observed from the typical microring WGM laser, showing a series of lasing modes corresponding to different azimuthal orders. (c) Single-mode lasing spectra of the $\mathcal{PT}$ microring lasers operating at the $m=53$ and 55 azimuthal orders. Adapted from [140].

Figure 7

(a) Spectrum obtained from an evenly pumped pair of microrings. (b) The intensity pattern shows that both resonators equally contribute. (c) Single-moded spectrum under $\mathcal{PT}$-symmetric conditions. (d) Lasing exclusively occurs in the active resonator. Adapted from [180].

Figure 8

(a) Isotopes of $\mathrm{\Lambda }$ atoms and Raman transitions. (b) A geometry for the atomic cell and fields applied. (c) Locally parabolic ([170]) and (d) two-channel ([169]) spatial distributions of the real (solid lines) and imaginary (dashed lines) parts of the refractive index.

Figure 9

(a) Antisymmetric and symmetric modes in a long-range plasmonic waveguide. The color map shows $y$-electric field components of the modes. (b) Cross section of a single hybrid waveguide. (c), (d) Even and odd modes in coupled waveguides, respectively. Adapted from [61].

Figure 10

(a) A $\mathcal{PT}$-symmetric dimer of SSRs, (b) schematic illustration of a 1D $\mathcal{PT}$ metamaterial, and (c) domains of broken and unbroken $\mathcal{PT}$ symmetry for the array shown in (b); the phase-transition line ${\gamma }_{\mathcal{PT}}$ is computed in Sec. 4b3. Based on ideas of [233].

Figure 11

Upper panels: Polarizations corresponding to unbroken and broken $\mathcal{PT}$-symmetric phases and to the exceptional point. Lower panel: Photograph of a $\mathcal{PT}$ symmetric metasurface on silicon substrate composed of 300 nm thick silver [light gray (yellow)] and lead [dark gray (turquoise)] SRRs. From [231].

Figure 12

(a) Semiconductor pillars pumped on the one side form a polariton Josephson junction. (b) The model for the junction: a continuous-wave laser excites high-energy excitons pumping to the reservoir. A resonant laser is used to create the required initial conditions. Adapted from [253].

Figure 13

(a) Spectrum (57) for $\omega =0.5$ and $\kappa =2$. Crossing eigenvalues occur at ${\nu }_m$, $m=1,2,\dots$. (b) Real parts of eigenvalues for $\gamma =0.2$, $\omega =0.5$, and $\kappa =2$, which merge pairwise near ${\nu }_m$, and the corresponding imaginary parts become nonzero at the same time. (c) Domains of unbroken (shadowed regions $d_0,d_1,\mathrm{...}$) and broken (white regions) $\mathcal{CPT}$ symmetries shown in the plane $(\kappa ,\nu )$ at $\gamma =0.2$ and $\omega =0.5$. Dashed lines correspond to $\nu ={\nu }_m$. From [206].

Figure 14

(a) Electronic $\mathcal{PT}$-symmetric dimer. The left circuit contains the gain element due to feedback from a voltage-doubling buffer. (b) Theoretical (solid line) and experimental (circles) values of the eigenfrequency $\omega$ [open circles are reflections of the data with respect to the $\mathrm{Im}(\omega )=0$ axis]. Here ${\omega }_0=2\times {10}^5{\mathrm{s}}^{-1}$ and $\mu =0.2$. Adapted from [360].

Figure 15

Left column: the system composed of two whispering-gallery-mode resonators coupled with each other and with two fiber-taper waveguides. The resonators are silica toroids of approximately $30\mu \mathrm{m}$ radius. One microcavity ($\mu R_1$) is active due to $E^{3+}$ dopants while the other ($\mu R_2$) is passive. The ions in the active microcavity are pumped by the laser in the $1.417\mu \mathrm{m}$ band providing gain in the $1.55\mu \mathrm{m}$ band. The second, third, and fourth columns show different operating regimes of the coupler. The upper and lower lines correspond to left and right incidences of light. The fourth column illustrates nonreciprocity of light propagation. From [332].

Figure 16

(a) Infinite $\mathcal{PT}$-symmetric chain with alternating coupling defined by $\kappa$, $\epsilon$ and gain-loss coefficients ${\gamma }_1$, ${\gamma }_2$. (b) A clustered closed $\mathcal{PT}$-symmetric chain of six sites. (c) $\mathcal{PT}$-symmetry-breaking “phase diagram” of the open $\mathcal{PT}$-symmetric quadrimer ([436]) which corresponds to the network shown in (a) truncated at $q_{-1}$ and $q_2$, with $\kappa =\epsilon =1$.

Figure 17

1D and 2D arrays of identical $\mathcal{PT}$-symmetric dimers. Interdimer and intradimer couplings are characterized by $\kappa$ and $C$. Open red and filled blue circles correspond to the sites with gain and losses, respectively. Based on ideas of [59] (1D), [87] (2D), and [108] (1D, staggered).

Figure 18

(a) Bifurcations of discrete $\mathcal{PT}$-symmetric solitons from the ACL which corresponds to $\epsilon =0$. Solid blue and dashed red fragments correspond to stable and unstable solitons, respectively. (b) Profile of an unstable soliton on the homogeneous lattice $\epsilon =\kappa$. Open (red) and filled (blue) circles correspond to the sites with gain and losses, respectively. For both panels, $\kappa =1$, $\gamma =0.1$, and $\mu =10$. Adapted from [222].

Figure 19

(a) Array of waveguides that supports discrete compactons. (b) Bifurcation diagram of branches of compactons and solitons for $\kappa =1$, $\tilde{\kappa }=0.25$, and $\chi =1$. The point where the branches intersect is indicated by a dot. Adapted from [431].

Figure 20

(a) Array of $N$ waveguides with one active ($n=1$) and one lossy ($n=N$) waveguide. Phase circulation direction is indicated by anticlockwise (red, $m>0$) and clockwise (blue, $m<0$) arrows. (b) Linear propagation constants $\beta$ vs $C$ for a conservative ($\gamma =0$, upper panel) and $\mathcal{PT}$-symmetric ($\gamma =0.2<{\gamma }_{\mathcal{PT}}$, lower panel) ring of $N=4$ waveguides. Degenerate vortex modes occur at the intersection marked by the black dot. Curves labeled with “$+1$” and “$-1$” consist of the modes with the respective TC. Stable (unstable) nonlinear modes of $N=3$ ring are shown with solid (dashed) lines for (c) $C=1.3$, $\gamma =0.2<{\gamma }_{\mathcal{PT}}$ and (d) $C=1.3$, $\gamma =0.6>{\gamma }_{\mathcal{PT}}$. TCs are indicated next to the curves, $m=+1$ in red (purple), $m=-1$ in blue (brown), and $m=0$ in black (gray). Adapted from [239].

Figure 21

(a) Real and (b) imaginary parts, and the (c) phase structure, of field $u_{m,n}$ for a typical stable on-site $\mathcal{PT}$-symmetric vortex soliton with $(C,\kappa ,\gamma )=(-0.03,-1,0.4)$. From [87].

Figure 22

Two scenarios of discrete soliton scattering on a $\mathcal{PT}$-symmetric dimer defect. (a), (b) Feature the same model parameters, but different amplitudes of the incident discrete soliton: $A=0.2$ vs $A=0.5$. In both panels, $n$ runs from $n=-75$ to $n=75$. Adapted from [379].

Figure 23

Solitons in locally $\mathcal{PT}$-symmetric lattices. (a) The linear dispersion relation (105) with $G=1.1$. Black and red lines with labels “Re” and “Im” correspond to real and imaginary parts of $\theta$, respectively. (b) Family of solitons in gap of the spectrum (shaded blue domain) on the diagram $\theta$ vs $E$. (c) Formation and propagation of a discrete soliton as the input power $P\approx 120\mathrm{mW}$. The color bar shows ${\log }_{10}$ (intensity). Adapted from [408].

Figure 24

Band-gap spectrum (106) for (a) the broken $\mathcal{PT}$ symmetry at $G=1.4$, ${\varphi }_0=0$, and (b) unbroken $\mathcal{PT}$ symmetry at $G=1.4$, ${\varphi }_0=0.4\pi$. Black and red lines with labels Re and Im show real and imaginary parts of $\theta$. (c) Formation of a broad single-hump soliton for $G=1.4$ and ${\varphi }_0=-0.4\pi$. The color bar shows ${\log }_{10}$ (intensity). Adapted from [408].

Figure 25

Intensity evolution of the field components $|{\psi }_1{|}^2$ and $|{\psi }_2{|}^2$ (left and right columns) of the plane wave with (a) $\rho =1.604$, $\chi =0.5$, and $\tilde{\chi }=-1$; (b) $\rho =0.76$, $\chi =-1.5$, and $\tilde{\chi }=1$; and (c) $\rho =0.98$, $\chi =0.25$, and $\tilde{\chi }=1$. For all panels, $k=0$, $\delta =\pi /4$. Adapted from [67].

Figure 26

(a) Collision of antisymmetric (initially left) and symmetric (initially right) solitons for $\gamma =0.5$ and $a=0.3$. A pair of breathers emerge from the collision. (b) Inelastic collision of two breathers for $\gamma =0.3$ and $a=0.3$. Adapted from [40].

Figure 27

(a) Symmetric soliton passing the defect at ${\kappa }_0=2$, $\ell =1$ (in this case ${\ell }_{\text{cr}}\approx 7$). The interaction of an antisymmetric soliton with the defect at ${\kappa }_0=4$, (b) $\ell =1.1$, (c) $\ell =2.2$, and (d) $\ell =2.7$ (in this case ${\ell }_{\text{cr}}\approx 3.4$). In (b) broadening is repeated with period $\approx 10$ while the breather period $\approx 0.8$. In all panels the initial conditions are taken in the form (116) with $a=1/\sqrt{2}$, the defect is centered at $z=0$, and ${\kappa }_{\min }=1$. Only the first component $|{\psi }_1{|}^2$ is shown; the behavior of $|{\psi }_2{|}^2$ is similar. Adapted from [66].

Figure 28

(a) Soliton switching between arms by the $\mathcal{PT}$-symmetric segment ${\gamma }_2(z)=-{\gamma }_1(z)=\mathrm{\Gamma }{\mathbf{(}\arctan [5(z-z_a)]-\arctan [5(z-z_b)]\mathbf{)}}^2$ with $\mathrm{\Gamma }=0.065$, $z_a=1.5$, and $z_b=3.0$. In the inset of (a) a gain-loss segment is added at $z\sim 10$. The parameters were engineered in order to keep $Q(z)\approx 1$. (b) The same as in (a) but with the only dissipative segment included in the first arm [${\gamma }_2(z)\equiv 0$] and with $\mathrm{\Gamma }=0.135$. The inset shows the relative phase. In both panels ${\psi }_1(0)=20\mathrm{sech}(10\sqrt{2}x)$ and ${\psi }_2(0)=5\mathrm{sech}(5x/\sqrt{2})$ [i.e., $F(0)=3/5]$. Adapted from [3].

Figure 29

(a) Vector Peregrine soliton (127) with $\rho =1$, $\chi =0.5$, and $\tilde{\chi }=-1$. (b), (c) Intensities of the Peregrine solitons for $\rho =1.604$, $\chi =0.5$, $\tilde{\chi }=-1$, and $\delta =\pi /4$, initiated with slightly perturbed initial conditions at $z=z_{\text{ini}}=-4$. The shown scenario of the evolution corresponds to the scenario of MI shown in Fig. 25. Adapted from [65].

Figure 30

(a) Families of nonlinear modes of a SO BEC in a parabolic trap $V(x)={\nu }^2{x}^2/2$ bifurcating from the linear mode ${\mu }_2$ (see Fig. 13) for $\kappa =1$, $\nu =2$, $\gamma =0.2$, and $\omega =0.5$. Solid blue (dashed red) curves correspond to stable (unstable) modes. Here $N={\int }_{-\infty }^{\infty }{\mathbf{\Psi }}^{†}\mathbf{\Psi }dx$, and curves with labels $N_a$ and $N_r$ correspond to attractive and repulsive nonlinearities, respectively. The circle in (a) corresponds to a two-hump nonlinear mode shown in (b). Here $n_{\uparrow \downarrow }=|{\psi }_{\uparrow \downarrow }{|}^2$. Adapted from [206].

Figure 31

Scattering of a soliton by the potential $U(x)=-V_0^2\mathrm{sech}(V_{0}x)+i{W}_{0}x\mathrm{sech}(V_{0}x)$ with $V_0=W_0=-2$. (a) Almost perfect reflection of the soliton moving from the left. (b) Almost perfect transmission of the soliton moving from the right. In both cases the velocity of the incident soliton is $|v_0|=0.25$, and the initial position is at $|x_0|=10$. From [22].

Figure 32

Families of nonlinear modes in the conservative ($\alpha =0$) and $\mathcal{PT}$-symmetric ($\alpha =1$) parabolic potential (8) and focusing ($g=1$) and defocusing ($g=-1$) nonlinearities. Bold segments on the power curves correspond to stable nonlinear modes. Adapted from [435].

Figure 33

The power $P$ as a function of the strength $\gamma$ of the imaginary potential in Eq. (134) with $V_0=0$. The bifurcation diagram shows the merging of different solution branches through saddle-node bifurcations. Solid (dashed) lines indicate dynamically stable (unstable) solutions. Here $g=-1$, $\mathrm{\Omega }\approx 0.07$, and $\mu =-3$. Adapted from [12].

Figure 34

(a) Inverted plot of the $\mathcal{P}\mathcal{T}$-symmetric potential (137) generated by Eq. (138) with $A_{-}=A_{+}=2$, $x_0=1.2$, $x_0=1.2$, and $\alpha =1$. (b) Power diagram for nonlinear modes with $g=1$ (solid blue: stable solitons; dashed red: unstable solitons). (c), (d) $\mathcal{P}\mathcal{T}$-symmetric and asymmetric solitons corresponding to points c and d on the power diagram with $\mu =4.3$. (e), (f) Evolution of the solitons shown in (c) and (d) under 1% random-noise perturbations. Adapted from [427].

Figure 35

(a) Inverted plot of the asymmetric potential $U(x)$ for $w(x)$ defined by Eqs. (137) and (138) with $A_{-}=2.3$, $A_{+}=2$, $x_0=1.5$, and $\alpha =0$. (b) Power diagram for nonlinear modes. Solid blue (dashed green) lines correspond to defocusing (focusing) nonlinearity. (c) Intensity ${\rho }^2$ and current $j=v{\rho }^2$ for the nonlinear mode marked as c in (b). Adapted from [225].

Figure 36

Diffraction relations of 1D $\mathcal{PT}$ lattice (143) for three values of $W_0$ ($V_0=6$). The inset in the lower right panel is an amplification of the small boxed region near $k=1$ and $\mathrm{Im}[\mu ]=0$ of the same panel. From [313].

Figure 37

1D solitons in the semi-infinite gap of Eq. (143) under focusing nonlinearity ($g=1$) for $V_0=6$ and $W_0=0.45$. (left) Power curves of these solitons; solid blue and dashed red lines represent stable and unstable solitons, respectively; the shaded region is the first Bloch band. (right) A fundamental soliton at $\mu =-3.5$ (marked by a dot on the lower curve of the left panel); the solid blue line is for the real part and the dashed pink line for the imaginary part. From [313].

Figure 38

Two left panels: dipole solitons at the marked points of the lower power curve in Fig. 37. Right panel: linear-stability spectrum for the dipole soliton in the middle panel. From [313].

Figure 39

(Left) Power curve of fundamental 2D solitons in the semi-infinite gap under focusing nonlinearity ($g=1$) for $V_0=6$ and $W_0=0.3$. The inset is amplification of the power curve near the first Bloch band in the same panel. Solid blue lines indicate stable solitons, while dashed red lines indicate unstable solitons. (Right) Real and imaginary parts of the soliton $\psi (x,y)$ at $\mu =-8.5$ (marked by a dot on the power curve). From [313].

Figure 40

(a) Linear unidirectional wave packet and (b) linear wave packet splitting at the phase-transition point $W_0=1$. Other parameters are $\epsilon =0.1$, $\mu =1$, and $V_0=\sqrt{6}$. From [318].

Figure 41

Nonlinear wave packet solutions below the phase-transition point ($c=1$) under self-defocusing nonlinearity. From left to right: a blowup solution in the envelope equation and the full equation; a periodic bound state in the envelope equation and the full equation. From [318].

Figure 42

(Left) Diffraction relation near intersection point $(k_x,k_y,\mu )=(1,1,2)$ (marked by red dot). (Right) Linear pyramid diffraction of initial Gaussian envelope at phase-transition point in the envelope equation (151). From [314].

Figure 43

Nonlinear dynamics of wave packets below phase transition. Upper row: envelope solutions in Eq. (151) at $T\approx 2$ for three values of $A_0$ in Eq. (152). Lower row: solutions of the full equation (141) for the initial wave packet with $A_0=6$ (left) at later times under focusing (middle) and defocusing (right) nonlinearities. From [314].

Figure 44

Upper panels: families of fundamental solitons in the semi-infinite gap. Left, center, and right panels correspond to cases 1, 2, and 3 with $\beta =-0.316$, 0, and $-0.5134$, respectively. Thick and thin lines correspond to stable and unstable solitons. Shaded regions denote bands of the FF and/or SH. Insets show powers of the FF ($P_1$) and SH ($P_2$) fields close to the band edge. Lower panels: intensities of stable fundamental solitons indicated by black circles in the respective upper panels: $b=1.25$ (case 1), 1.43 (case 2), and 1.21 (case 3). Other parameters are $V_0={\tilde{V}}_0=2$ and $W_0=0.35$. Adapted from [287].

Figure 45

Families of nonlinear modes in ${\chi }^{(2)}$ coupler (160) for (a) $k_1=1$, $k_2=2$, $\beta =0.5$, and ${\gamma }_{1,2}=0$; and (b) ${\gamma }_1=0.1$ and ${\gamma }_2=0.9$. Solid blue and dashed red segments correspond to stable and unstable modes, respectively. Adapted from [249].

Figure 46

(a) Power diagram of soliton families in the potential (162) under focusing nonlinearity (solid blue segments are stable and dashed red unstable); (b), (c) amplitude and phase fields of the soliton at the marked point of the power curve. Adapted from [426].