⇦Chinese Physics Letters, 2023, Vol. 40, No. 4, Article code 044201 Two-Dimensional Gap Solitons in Parity-Time Symmetry Moiré Optical Lattices with Rydberg–Rydberg Interaction Bin-Bin Li (黎斌斌)1,2†, Yuan Zhao (赵元)1,2†, Si-Liu Xu (徐四六)1,2,3*, Qin Zhou (周勤)4, Qi-Dong Fu (傅其栋)5, Fang-Wei Ye (叶芳伟)5*, Chun-Bo Hua (花春波)2,3, Mao-Wei Chen (陈懋薇)1,2, Heng-Jie Hu (胡恒洁)2,3, Qian-Qian Zhou (周倩倩)2,3, and Zhang-Cai Qiu (邱章才)2,3 Affiliations 1School of Biomedical Engineering and Medical Imaging, Xianning Medical College, Hubei University of Science and Technology, Xianning 437100, China 2Laboratory of Optoelectronic Information and Intelligent Control, Hubei University of Science and Technology, Xianning 437100, China 3School of Electronic and Information Engineering, Hubei University of Science and Technology, Xianning 437100, China 4School of Mathematical and Physical Sciences, Wuhan Textile University, Wuhan 430200, China 5School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Received 17 February 2023; accepted manuscript online 4 March 2023; published online 29 March 2023 †These authors contributed equally to this work.
*Corresponding authors. Email: xusiliu1968@163.com; fangweiye@sjtu.edu.cn Citation Text: Li B B, Zhao Y, Xu S L et al. 2023 Chin. Phys. Lett. 40 044201    Abstract Realizing single light solitons that are stable in high dimensions is a long-standing goal in research of nonlinear optical physics. Here, we address a scheme to generate stable two-dimensional solitons in a cold Rydberg atomic system with a parity-time (PT) symmetric moiré optical lattice. We uncover the formation, properties, and their dynamics of fundamental and two-pole gap solitons as well as vortical ones. The PT symmetry, lattice strength, and the degrees of local and nonlocal nonlinearity are tunable and can be used to control solitons. The stability regions of these solitons are evaluated in two numerical ways: linear-stability analysis and time evolutions with perturbations. Our results provide an insightful understanding of solitons physics in combined versatile platforms of PT-symmetric systems and Rydberg–Rydberg interaction in cold gases.
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 DOI:10.1088/0256-307X/40/4/044201 © 2023 Chinese Physics Society Article Text Moiré optical lattices, which are composite lattices when two identical periodic patterns (lattices) are rotated to each other by some angle (twisting angle) and then overlapped, have recently received great theoretical and experimental interests. Using optical induction technology in photorefractive crystals,[1] reconfigurable optical Moiré lattices are created and localization of light in them is reported.[2] Thanks to the suppression of light diffraction in Moiré lattices, nonlinear Moiré lattices are found to support optical solitons,[3,4] second harmonic generation,[5] and Bose–Einstein condensate (BECs)[6] under lower power limits. Moiré optical lattices have potential applications in ultra-cold atomic systems.[7-10] Researchers have found that the periodic optical lattices in ultracold atoms exhibit finite atomic band gaps,[11-13] which is the potential research direction of gap solitons (GSs).[14-19] Among periodic optical lattices, parity-time (PT) symmetric lattices have received much attention in research of nonlinear physics in optical[20-22] and matter-wave[23,24] systems, providing an ideal platform for finding nonlinear waves.[25-27] In research of solitons in nonlinear media, many models have been proposed.[28-32] Theoretical[33] and experimental[34] studies revealed that strong and long-range optical nonlinearities can be built by Rydberg–Rydberg interaction (RRI) between remote atoms.[35] Especially, RRI can be mapped to a nonlocal optical nonlinearity through electromagnetically induced transparency (EIT), which is strong even at the single photon level.[36] This provides an important platform for studies of optical soliton dynamics with tunable parameters.[37] Interestingly, systems with Rydberg atoms are proven to be an effective medium to generate stable solitons with low energies for the strong and long-range optical nonlinearities between Rydberg atoms.[38-40] However, to the best of our knowledge, combination of Rydberg–Rydberg interaction, moiré patterns, and PT symmetry has not yet been reported. Model. We consider a cold lifetime-broadened four-level atomic system with an inverted-Y type configuration, as shown in Fig. 1(a).[40] A standard $\varLambda$-type EIT configuration is constructed by the states $|1\rangle$, $|2\rangle$, $|3\rangle$ and the probe and control fields. The probe/control/auxiliary laser fields are coupled to the transitions $|1\rangle \to |3\rangle$, $|2\rangle \to |3\rangle$, and $|3\rangle \to |4\rangle$, with spontaneous emission decay rate $\varGamma_{13}$, $\varGamma_{23}$, and $\varGamma_{34}$, respectively; $\varDelta_{2}$, $\varDelta_{3}$ and $\varDelta_{4}$ are detunings. This EIT configuration is dressed by a high-lying Rydberg state $|4\rangle$, which is far off-resonantly coupled to $|3\rangle$ through an auxiliary laser field. The scheme of Rydberg-dressed EIT is used to exploit the advantages of both EIT and the Rydberg states.[40] Under the slowly varying amplitude approximation, the wave equation of the probe field following from the Maxwell equations has the following form[25,27] \begin{align} &i\Big({\frac{\partial}{\partial z}+\frac{1}{c}\frac{\partial}{\partial t}}\Big)\varOmega_{\rm p}+\frac{c}{2\omega_{\rm p}}\Big( {\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}}\Big)\varOmega_{\rm p}\notag\\ &+\frac{\omega_{\rm p} }{2c}\chi_{\rm p} \varOmega_{\rm p} =0, \tag {1} \end{align} where $\chi_{\rm p} ={N_{\rm a} (e_{\rm p} \cdot P_{13})^{2}\rho_{31} } / {(\varepsilon_{0} \hslash \varOmega_{\rm p})}$ is the susceptibility of probe field, $N_{\rm a}$ is the atom density, $\rho_{ij}$ is the density-matrix element, $z$ is the propagation direction. $P_{ij}$ represents the electric-dipole matrix element, the subscripts $i$ and $j$ indicate the state $|i\rangle$ and $|j\rangle$. $\varOmega_{\rm p}$ and $\omega_{\rm p}$ are, respectively, the Rabi frequency and angular frequency of probe field. The time derivative in Eq. (1) can be neglected and one is concerned with a stationary state. When the probe field is weak, the population of atomic levels does not change much with the application of the probe field, and consequently, a perturbation expansion can be employed to solve Eq. (1).[27]
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Fig. 1. Excitation scheme of the 4-level atomic Rydberg system with an inverted-Y type configuration and the band structures for the two-dimensional (2D) PT-symmetric moiré lattices. A gain to the probe field has be provided to realize an optical PT symmetry in this weakly absorbed Rydberg-dressed EIT system. Band structures at $\theta =\arctan (3/4)$ with increasing imaginary potential strength $V_{0}$ (b) and lattice strength ratio $p$ (c). The band diagrams of moiré lattices in the reduced zone representation with $V_{0} =0.02$ and $p=1$ at $\theta =\arctan (3/4)$ (d), $\theta =\arctan (5/12)$ (e), and $\theta =\arctan (7/12)$ (f).
The evolution of the probe pulse is governed by the dimensionless (2+1)D nonlinear Schrödinger equation: \begin{align} &i\frac{\partial \psi }{\partial s}+\Big({\frac{\partial^{2}\psi }{\partial \xi^{2}}+\frac{\partial^{2}\psi }{\partial \eta^{2}}} \Big)-\varPi \psi +\gamma \vert \psi \vert^{2}\psi \notag\\ &+\alpha \int {d^{2}} r'G_{2} (r'-r){\rm \vert }\psi {\rm \vert }^{2}\psi =0, \tag {2} \end{align} where $\psi ={\varOmega_{\rm p} } / {\psi_{0}}$, with $\psi_{0}$ being the typical Rabi frequency of the probe pulse, $s=z / {(2L_{\rm diff})}$, $r =(x,y)$, $(\xi,\eta)={(x,y)} / {R_{\bot}}$, $L_{\rm diff} ={2\omega_{\rm p} R_{\bot}^{2}}/c$ and $R_{\bot}$ the typical radius of the probe pulse. Further, $\gamma$ is the local Kerr nonlinear coefficient, and $\alpha$ is the degree of nonlocality of the system, $G_{2} (r -r')=C_{6}/|{r -r'}|^{6}$ is the effective interaction potential contributed by RRI, with $C_{6} < 0$ being the dispersion parameter.[25] The stationary solution of Eq. (2) can be expressed in the form $\psi =qe^{ibs}$, where $b$ is the propagation coefficient and $s$ is the propagation distance. A moiré optical potential is adopted with the form[1-3] \begin{align} &\varPi=V_{1}\{{({\cos^{2}x+\cos^{2}y})+iV_{0}[{\sin (2x)+\sin (2y)}]}\}\notag\\ &+V_{2}\{[{({\cos^{2}{x'}+\cos^{2}{y'}})+iV_{0}[{\sin (2x')+\sin (2y')}]}\}, \tag {3} \end{align} where $V_{1,2}$ are the lattice strengths and $V_{0}$ is the imaginary potential strength. The strength ratio between the two sub-lattices is defined as $p=V_{2} /V_{1}$ for discussion. It is noted that the potential Eq. (3) matches PT symmetry, ${\rm Re}[{V_{\rm pt}(r)}]={\rm Re}[{V_{\rm pt}({-r})}]$ and ${\rm Im}[{V_{\rm pt}(r)}]=-{\rm Im}[{V_{\rm pt}({-r})}]$, and it reduces to the usual (non-PT) moiré lattice at $V_{0} =0$, and to the conventional PT lattice at $V_{2} =0$. The $(x,y)$ plane is related to the rotation plane $(x',y')$ with a twisting angle $\theta$. \begin{align} \begin{pmatrix} {x'} \\ {y'} \\ \end{pmatrix}= \begin{pmatrix} \cos \theta, &-\sin \theta \\ \sin \theta, &\cos \theta\\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix}. \tag {4} \end{align} The spatial distributions of moiré lattices with PT-symmetric structure at different twist angles are shown in Fig. 2. With different twist angles, moiré optical lattices have both periodic and aperiodic characteristics. Only with special twist angles, such as $\theta =\arctan (3/4)$ and $\theta =\arctan (5/12)$, moiré optical lattices have periodic structures. The first and second rows show the periodic structures at $\theta =\arctan (3/4)$ and $\theta =\arctan (5/12)$. When choosing the other twisting angle such as $\theta =\arctan (7/12)$ in Figs. 2(g), 2(h), and 2(i), the spatial distribution of moiré lattices is aperiodic. To take advantage of PT-symmetric structures, the twisting angle is fixed as $\theta =\arctan (3/4)$ in this work. Results and Discussions. The band structures for the 2D PT-symmetric moiré lattice at different twisting angles are shown in Figs. 1(b)–1(g). The curves of propagation coefficient $b$ versus imaginary lattice strength $V_{0}$ and lattice strength ratio $p$ are plotted in Figs. 1(b) and 1(c), respectively. One can find that the band gaps decrease rapidly with the increasing $V_{0}$ at $\theta =\arctan (3/4)$. The band gaps decrease continuously with the increase of $V_{0}$, and there is almost no gap at $V_{0} =0.5$. The bands show flap characteristics with the changing lattice strength ratio $p$. The band structures of the moiré lattice at different twisting angles in the 2D reciprocal space are shown in Figs. 1(d), 1(e), and 1(f). It is found that the bands are localized in the reciprocal space, which is a key factor for the formation of stable solitons. For the periodic case in Figs. 1(d) and 1(e), the band structures are more complex so that there are some band crossings between the first and second gaps; while for the aperiodic structure in Fig. 1(f), the bands are located such that there is no band crossing. A family of solitons, including fundamental GSs, two-pole ones composed of two in-phase or out-of-phase GSs, and vortex GSs, are found in this system. The dynamic properties of these solitons are studied by means of numerical simulations. Their stabilities are evaluated through linear stability analysis and direct perturbed simulation by the fourth-order Runge–Kutta method.
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Fig. 2. Contour plots and spatial distributions of moiré lattices with PT-symmetric structure at $\theta =\arctan (3/4)$ (a)–(d), $\theta =\arctan (5/12)$ (e)–(h), and $\theta =\arctan (7/12)$ (i)–(l). The PT-symmetric moiré potentials are shown in the fourth column, where the real and imaginary (insets) parts of potentials at $y=0$ display even and odd symmetries, respectively.
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Fig. 3. Profiles of zero-vortex solitons. (a)–(c) Fundamental GSs. (d)–(f) Two-pole GSs grouped as two out-of-phase fundamental GSs. (g)–(i) Two-pole GSs grouped as two in-phase fundamental GSs. The first and second columns show the distribution of the real part of wavefunctions $\psi$ and their contour plots. The third column shows the distribution of the imaginary part of $\psi$. Parameters are $b=6$, $\gamma =1$, $\alpha =2.5$ for the fundamental GSs, $b=3.1$, $\gamma =2.5$, $\alpha =5$ for the out-of-phase mode, $b=4$, $\gamma =2.5$, $\alpha =5$ for the in-phase case. Parameters of moiré lattices are $\theta =\arctan (3/4)$, $V_{0} =0.02$, $V_{1} =V_{2} =1$.
Fundamental and Two-Pole GSs. The profiles of fundamental and two-pole GSs are shown in Fig. 3. These solitons all present zero-vortex properties. For the fundamental case, the real part of wavefunction, ${\rm Re}(\psi)$, shows a single peak [Figs. 3(a) and 3(b)], while the imaginary part, ${\rm Im}(\psi)$, has a dipole form [Fig. 3(c)]. By changing the system parameters, two modes of two-pole GSs, the in-phase and out-of-phase ones, are found. Both the real and imaginary parts of wavefunctions have two poles. The main difference between the two modes of two-pole GSs is the directions of the dipoles of ${\rm Re}(\psi)$. For the out-of-phase mode, ${\rm Re}(\psi)$ has one positive and one negative pole [Fig. 3(d)]; while the in-phase case in Fig. 3(g) has two positive poles. The formation of these two kinds of solitons can be modulated and switched by changing the system parameters, such as propagation coefficient $b$, local nonlinearity coefficient $\gamma$, and nonlocal nonlinearity coefficient $\alpha$.
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Fig. 4. Modulation of fundamental and two-pole GSs. [(a), (b), (c), (d)] The light intensity $U$ with respect to the propagation coefficient $b$, the nonlocal nonlinearity coefficient $\alpha$, imaginary potential strength $V_{0}$, and lattice strength ratio $p$, respectively. The blue, red, and green lines represent the fundamental GSs, and two-pole GSs with in-phase and out-of-phase modes, respectively. The solid and dashed lines represent the stable and unstable states. Other parameters are the same as those in Fig. 3.
The modulation of fundamental and two-pole GSs are shown in Fig. 4. The light intensity, $U=\int {|\psi|^{2}{d}\boldsymbol{r}}$, as a function of propagation coefficient $b$ is shown in Fig. 4(a). The $U(b)$ relation does not meet an ‘anti-Vakhitov–Kolokolov’ (anti-VK) criterion, ${{d}U} / {{d}b}>0$, a necessary but not a sufficient condition for the stability of solitons in periodic structures with repulsive local nonlinearity.[41] The stable zone of fundamental GSs is much larger than that of two-pole ones. For the two-pole GSs, the stable zone is divided into three parts: $2.5 < b < 2.99$, $3.0 < b < 3.21$, and $3.21 < b < 5.0$. In the first and third parts (red solid lines), the solitons have the in-phase mode, while in the second part (green dotted line), the solitons take the out-of-phase mode. It is noted that the stable zone of two-pole GSs with out-of-phase mode is very small and nearly depressed to a point. In Fig. 4(b), it is found that $U$ decreases monotonously with $\alpha$ for the fundamental and two-pole GSs with in-phase mode, while only four points, $\alpha =3.4, 3.47, 3.5, 5$, have stable out-of-phase mode. It is relatively hard to obtain the out-of-phase mode as this kind of solitons would collapse even with a subtle derivation of the specific values. This phenomenon is also found when changing other parameters, such as $V_{0}$ and $p$, which are shown in Figs. 4(c) and 4(d). For the out-of-phase mode, due to the interference of the two opposite components, the wavefunction cannot maintain stably easily. Only at finite states, the out-of-phase GS can be obtained. The potential applications of this effect is the encryption and decryption of signals by the parameters of these finite states. The imaginary strength $V_{0}$ and the lattice strength ratio $p$ are tuned in order to study the roles of the PT-symmetric moiré lattices in the system. It is found that the light intensity of solitons increases slowly with the increasing $V_{0}$, as shown in Fig. 4(c). When changing the lattice strength ratio $p$, the fundamental GSs have a large stable zone, while the two-pole GSs, including in-phase and out-of-phase modes, are stable only when $p=1$, as shown in Fig. 4(d).
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Fig. 5. Vortex solitons. [(a), (b)] The real and imaginary parts of the wavefunction $\psi$, respectively. (c) The contour plot of ${\rm Re}(\psi)$. (d) The phase structure of wavefunction. (e)–(h) The modulation of the light intensity $U$ by tuning the propagation coefficient $b$, the nonlocal nonlinearity coefficient $\alpha$, imaginary potential strength $V_{0}$, and lattice strength ratio $p$. The fixed parameters are $b=4$, $\gamma =2.5$, $\alpha =7$, $V_{0} =0.02$, and $p=1$.
According to the above discussions, one finds that the stable zones for the three kinds of zero-vortex solitons are dramatically different. The stable zone of the fundamental GSs is much larger and can be obtained much easier. On the other hand, the stable zones of two-pole GSs are smaller. Especially the stable zone of the out-of-phase mode is so narrow that we can only get it with specific dots. The points $a$ ($b=6$, $\gamma =1$, $\alpha =2.5$), $b$ ($b=3.1$, $\gamma =2.5$, $\alpha =5$), and $c$ ($b=4$, $\gamma =2.5$, $\alpha =5$) give each of the stable states of fundamental GSs, out-of-phase and in-phase modes, respectively. The moiré lattices parameters of the three points are $\theta =\arctan (3/4)$, $V_{0} =0.02$, and $p=1$. Vortex Solitons. Besides the zero-vortex soliton, vortex ones are also found in this work. The profiles of the real part and imaginary parts and the corresponding phase structures are shown in Figs. 5(a), 5(b), 5(c), and 5(d). One finds that both the real and imaginary parts of the vortex solitons have dipole features, one has a positive value and the other has a negative value, which is similar to the two-pole GSs with out-of-phase mode. The main difference between the two cases is the phase structure. Evidently, the phase in Fig. 5(d) has vortex features. The dipole feature of real and imaginary parts of $\psi$ may be explained by the unique structure of PT-symmetric potential, together with the localized and non-localized band structures that arise from the periodic moiré lattices. The stability of vortex solitons is evaluated by the linear stability analysis. The light intensity $U$ with respect to the propagation coefficient $b$ shows a positive slope [Fig. 5(e)]. It is found that the light intensity has simple relations with parameters: decreasing monotonously with the local nonlinear coefficient $\alpha$ [Fig. 5(f)], and increasing monotonously with imaginary lattice strength $V_{0}$ [Fig. 5(g)]. However, when changing the lattice strength ratio $p$, the vortex solitons have two narrow stable zones, $0 < p < 0.06$ and $0.82 < p < 1.0$ [Fig. 5(h)].
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Fig. 6. Profiles of fundamental GSs (a), two-pole GSs with out-of-phase (b) and in-phase (c), vortex solitons (d). The corresponding perturbed evolutions [(e), (f), (g), (h)] and linear eigenvalue spectra for each case are shown in the second and third rows, respectively.
Dynamics of GSs. The stability of all the 2D localized solitons, including the fundamental GSs, two-pole GSs, and vortex solitons, have been performed by the perturbed evolution and linear-stability analysis.[27] The perturbed evolutions of fundamental GSs and two-pole GSs with in-phase mode are shown in Figs. 6(a), 6(e), 6(i) and Figs. 6(c), 6(g), 6(k), respectively. One can see that they are stable, in good agreement with their linear eigenvalue spectra produced by linear-stability analysis (${\rm Re}(\lambda)=0$). However, the two-pole GSs with out-of-phase mode [Figs. 6(c), 6(g), 6(k)] and vortex GSs [Figs. 6(d), 6(h), 6(j)] are unstable during the propagation, agreeing with the linear-stability analysis (${\rm Re}(\lambda)\ne 0$). In summary, we have addressed a scheme to generate stable 2D solitons in a cold Rydberg atomic system with PT-symmetric moiré optical lattices. The formation, properties, and dynamics of fundamental, two-pole with in-phase and out-of-phase modes, and vortical GSs are studied numerically. It is realized that the PT symmetry, the degree of local and nonlocal nonlinearity can be used to control solitons. The stability regions of the localized modes are inspected by linear-stability analysis and perturbed evolution with direct simulations. It is found that the stable domains vary considerably with the changing of system parameters. The GSs in the Rydberg-dressed PT-symmetric moiré lattices system could keep being stable owing to the localized band structures and the existence of band gaps. Our results provide an insightful understanding of solitons physics in combined versatile platforms of PT-symmetric systems and Rydberg–Rydberg interaction in cold gases. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 62275075, 11975172, and 12261131495), the Shanghai Outstanding Academic Leaders Plan (Grant No. 20XD1402000), and the Training Program of Innovation and Entrepreneurship for Undergraduates of Hubei Province (Grant No. S202210927036). References Observation of two-dimensional discrete solitons in optically induced nonlinear photonic latticesLocalization and delocalization of light in photonic moiré latticesOptical soliton formation controlled by angle twisting in photonic moiré latticesLocalization-delocalization wavepacket transition in Pythagorean aperiodic potentialsMultifrequency Solitons in Commensurate-Incommensurate Photonic Moiré LatticesAtomic Bose–Einstein condensate in twisted-bilayer optical latticesCold atoms in twisted-bilayer optical potentialsSimulating Twistronics without a TwistSpin-Twisted Optical Lattices: Tunable Flat Bands and Larkin-Ovchinnikov SuperfluidsElectromagnetically induced moiré optical lattices in a coherent atomic gasDynamics of Bose-Einstein condensates in optical latticesFrontiers in multidimensional self-trapping of nonlinear fields and matterPhotonic band gaps in optical latticesComposition Relation between Gap Solitons and Bloch Waves in Nonlinear Periodic SystemsGap-type dark localized modes in a Bose–Einstein condensate with optical latticesDark matter-wave gap solitons of Bose-Einstein condensates trapped in optical lattices with competing cubic-quintic nonlinearitiesBragg Grating SolitonsGap Solitons in Waveguide ArraysBright Bose-Einstein Gap Solitons of Atoms with Repulsive InteractionMaking sense of non-Hermitian HamiltoniansNon-Hermitian photonics based on parity–time symmetryParity–time symmetry and exceptional points in photonicsObservation of Parity-Time Symmetry in Optically Induced Atomic Lattices P T Symmetry with a System of Three-Level AtomsParity-time symmetry light bullets in a cold Rydberg atomic gasNonlinear waves in PT -symmetric systemsTwo dimensional spacial soliton in atomic gases with PT-symmetry potentialClassification of Four-Qubit States by Means of a Stochastic Local Operation and the Classical Communication InvariantAnalytical Approach to Space- and Time-Fractional Burgers EquationsBound State Solutions of the Schrödinger Equation for a More General Woods—Saxon Potential with Arbitrary l -StateVector Spatiotemporal Solitons and Their Memory Features in Cold Rydberg GasesThree-dimensional solitons in Bose-Einstein condensates with spin-orbit coupling and Bessel optical latticesLong-range interactions and entanglement of slow single-photon pulsesCoherent Optical Detection of Highly Excited Rydberg States Using Electromagnetically Induced TransparencyNonlinear quantum optics mediated by Rydberg interactionsAdvances In Atomic, Molecular, and Optical PhysicsNonlinear light diffraction by electromagnetically induced gratings with PT symmetry in a Rydberg atomic gasSpatiotemporal solitons in cold Rydberg atomic gases with Bessel optical latticesPhoton-Photon Interactions via Rydberg BlockadeTwo-Photon Dynamics in Coherent Rydberg Atomic EnsembleSolitons in combined linear and nonlinear lattice potentials
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