Efficient and tunable photoinduced moiré lattice in an atomic ensemble

Abstract

Photonic moiré lattices (PMLs), with unique twisted periodic patterns, provide a valuable platform for investigating strongly correlated materials, unconventional superconductivity, and the localization−delocalization transition. However, PMLs created either by the misorientation between lattice layers or by twisted van der Waals materials are typically non-tunable and inherently possess immutable refractive indices. Unlike those in the moiré lattices of twisted two-dimensional materials, our work reports a moiré lattice formed by overlapping two identical sublattices with twisted angles in an ultracold atomic ensemble. This photoinduced moiré lattice with two twisted sublattices exhibits high flexibility and rich periodicity through adjustable twisted angles. Our results indicate that both the absorption/dispersion coefficients and the transmission of the photoinduced moiré lattices can be effectively tuned by photon detuning and Rabi frequency, resulting in amplitude- and phase-type moiré lattices. Based on the Fraunhofer diffraction theory, we have demonstrated that the far-field diffraction efficiency can be adjusted via altering photon detuning, and the rotation angle serves as a control knob for modulating the diffracted intensity distribution, thereby optimizing the performance of the photonic lattice. It is also found that the operation domains of the moiré lattices with different rotation angles remain consistent, allowing for seamless conversion between various moiré period structures. Furthermore, a moiré lattice composed of three twisted sublattices is investigated, revealing that the diffraction energy is uniformly dispersed in a circular distribution, which provides excellent agility in the design of optical devices. Such tunable PML offer a powerful tool for studying light propagation control and the intriguing physics of twisted systems in atomic media.

Graphical abstract

Keywords

photonic moiré lattices / far-field diffraction / coherent optical effects / phase shift / electromagnetically induced transparency

1 Introduction

In recent decades, artificial periodic structures (APS), including photonic crystals [1−3], waveguide arrays [4, 5], and metamaterials [6, 7], have garnered considerable attention due to their unprecedented capacities in manipulating the propagation and transmission properties of light [8, 9]. These periodic structures not only enhance our comprehension of natural system’s fundamental physics, but also hold promise in exploring topological photonics [10, 11], non-Hermitian physics [12, 13], and predicting novel physical phenomena, such as strong localization of light [14], inhibited spontaneous emission from atoms [15], optical lattice solitons [16], and photon-atom bound states [17].

Moiré pattern as a novel type of APS has exhibited unique physical properties in condensed matter physics, leading to the rise of an emerging field, twistronics [18]. A prime example is twisted bilayer graphene, which can generate intrinsic unconventional superconductivity or counterintuitive insulator behavior at a certain small magic angle [19−22], triggering extensive theoretical and experimental studies of van der Waals heterojunction structures [23−28]. Recently, the study of moiré pattern has been extended to the regimes of optics and photonics, where photonic moiré lattices (PMLs) have been shown to facilitate the observation of localization-to-delocalization transition of light as well as the formation of optical soliton in linear and nonlinear systems [29−32]. Moreover, the optical properties of bilayer graphene moiré superlattices were revealed [33, 34], and magic-angle lasers with unique confinement mechanisms were fabricated in nanostructured moiré superlattices [35]. PML provides a new platform for the control of the fundamental aspects of light propagation.

The optical implementation of PML using an ultracold atom has opened up new possibilities for exploring the intrinsic physical of associated twisted systems and moiré superlattices in the atomic medium [36−43]. For instance, the simulation of twisted bilayers using cold atoms in state-dependent optical lattices shows Dirac-like physics and band narrowing with larger rotation angles, rendering them suitable for explaining strongly correlated effects [36]. The study of the moiré flat band physics and the associated superfluidity in spin-twisted optical lattices for ultracold atoms has also been investigated [38, 39]. Remarkably, considering electromagnetically inducted grating (EIG) is mature in experiments [44], a scheme for creating electromagnetically induced moiré optical lattice in the atomic ensembles under the regime of electromagnetically induced transparency (EIT) has been proposed [41−43]. The application of externally controlled light in an EIT configuration can generate flexibly tunable and easily reconfigurable photonic lattices in the atomic medium [45, 46], and achieve an optically tunable photonic bandgap [47, 48]. Indeed, the direct tuning of the optical properties of PML in the atomic medium using optical means is highly desirable. However, the photoinduced moiré lattice with tunable optical transmission properties as well as the influence of optical parameters on its discrete diffraction has not been explored yet. In this study, we introduce a photoinduced moiré lattice in an ultracold atomic ensemble generated by the interference of two sets of sublattice-forming fields, and such moiré lattices exhibit high flexibility and rich periodicity via variable twisted angle. We demonstrate that the transmission properties of the photoinduced moiré lattice as well as the corresponding diffraction intensity can be modulated significantly by photon detuning and Rabi frequency. The far-field diffraction results also confirm that the uniform distribution and reorientation of the diffracted energy can be precisely controlled by the adjustable rotation angle. Significantly, the operation domains of the moiré lattices at various rotation angles remain consistent, enabling seamless transitions between various moiré periodic structures. Additionally, the moiré lattices formed by overlapping three twisted sublattices are investigated, and we find that the probe field energy is further dispersed uniformly with a circular distribution. Such photoinduced moiré lattice provides a flexible tool for multi-channel all-optical modulator, low-level all-optical switching, and reconfigurable super-directive beam splitters, and establishes a platform for innovative applications in scalable and functional quantum photonic systems.

It is worth mentioning that our scheme has the following advantages. First, in contrast to the twisted structures in conventional materials, the photoinduced moiré lattice written in the atomic assemble is reconfigurable and can be dynamically tuned, leading to the photonic band structures of moiré lattices being adjustable highly by modulating the frequency detuning. Second, by using the multi-beam interference method, some structure parameters, such as twisted angles, interlayer coupling, and lattice structures, can be precisely controlled in current systems. Third, the resonance enhancement in the EIT configuration as well as the long coherence time and narrow bandwidth within the atomic ensemble allow us to observe optical linear and nonlinear phenomena at weak light. Besides that, the photoinduced moiré lattice structures can be realized in both cold and hot atomic ensembles, and are quite stable as long as the lattice-forming beams are stable. This provides an effective framework for the investigation of optical nonlinear and quantum many-body physics in the atomic ensemble.

2 Theoretical model

Our approach to creating photoinduced moiré lattices involves periodically modulating the refractive index within a cold atomic ensemble. As depicted in Fig.1(a), to construct a 2D photoinduced moiré lattice in a cascade-type three-level ⁸⁵Rb atomic configuration composed of a ground state |0⟩ [5S_1/2 (F = 3)], a metastable |1⟩ [5P_3/2 (F = 3)], and an excited |2⟩ (5D_5/2) [Fig.1(a2)]. The weak probe field E_p (frequency {\omega }_p, wave vector k_p, Rabi frequency {\mathrm{\Omega }}_p) propagates through the atom along the z axis to couple energy levels transition |0⟩\to |1⟩ with frequency detuning {\mathrm{\Delta }}_p={\omega }_p-{\omega }_{10}. The moiré lattice-forming field E_{eff} generated by interfering eight strong control fields E_c (frequency {\omega }_c, wave vector k_c, Rabi frequency {\mathrm{\Omega }}_c) is injected into the atomic vapor cell to drive transition |1⟩\to |2⟩ with frequency detuning {\mathrm{\Delta }}_c={\omega }_c-{\omega }_{21}. {\omega }_{ij}(i,j=1,2,3,4) is the frequency difference between level |i⟩ and level |j⟩.

Fig.1(b) illustrates the configuration of the moiré lattice-forming lasers within the vapor cell. To be specific, four coupling beams (blue) as a set of the sublattice-forming laser are symmetrically placed with respect to the z axis at the small angle {\phi }_x and {\phi }_y, inducing a 2D periodic modulated optical field in the x−y plane [Fig.1(c)], and the effective Rabi frequency can be represented by |{\mathrm{\Omega }}_{eff1}(x,y){|}^2=|{\mathrm{\Omega }}_c\sin (\pi x/a_x){|}^2+|{\mathrm{\Omega }}_c\sin (\pi y/a_y){|}^2. Here, {\mathrm{\Omega }}_c={\mu }_{21}{E}_c/\hslash represents the Rabi frequency of the coupling field, where {\mu }_{ij} is the transition dipole momentum between |i⟩ and |j⟩. a_{x(y)}={\lambda }_c/(2\sin {\phi }_{x(y)}) is the spatial period of this sublattice along x(y) direction, which can be made arbitrarily larger than the wavelength of the coupling fields {\lambda }_c by varying {\phi }_x({\phi }_y). The rest four coupling beams (purple) serve as another set of the sublattice-forming laser with a twist angle \theta relative to the first set [Fig.1(c)], forming sublattice with the spatial period of a_{x^{\prime }(y^{\prime })}={\lambda }_c/(2\sin {\phi }_{x^{\prime }(y^{\prime })}) through the same interference process in the x^{\prime }−y^{\prime } plane, and can be represented by |{\mathrm{\Omega }}_{eff2}(x^{\prime },y^{\prime }){|}^2=|{\mathrm{\Omega }}_c\sin (\pi x^{\prime }/a_{x^{\prime }}){|}^2+|{\mathrm{\Omega }}_c\sin (\pi y^{\prime }/a_{y^{\prime }}){|}^2. Therefore, the effective Rabi frequency of E_{eff} can be written as |{\mathrm{\Omega }}_{eff}(x,y){|}^2=|{\mathrm{\Omega }}_{eff1}(x,y){|}^2+|{\mathrm{\Omega }}_{eff2}(x^{\prime },y^{\prime }){|}^2. The rotated (x^{\prime },y^{\prime }) plane at a rotating angle \theta yields

(1)\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right).

In our model, we superimpose and interfere with these two mutually rotated square sublattices (set a_{x(y)}=a_{x^{\prime }(y^{\prime })}=d) for constructing a 2D photoinduced moiré lattice, as shown in Fig.1(c). Their superposition can form a periodic (commensurable) optical pattern at certain rotation angles and aperiodic (incommensurable) distributions at all other angles. For square lattice, the rotation angles that can form the periodic structure need to satisfy Pythagorean triples, i.e., when \cos \theta =a/c and \sin \theta =b/c, where the positive integers (a, b, c) have no common divisors except 1, constitute a Pythagorean triple. Fig.1(d1) and (d2) show the periodic moiré optical patterns with rotation angles \theta =\arctan (3/4) and \theta =\arctan (5/12), respectively. The blue squares represent the minimum period with translational periodicity and rotational symmetry. The patterns appear as new long periods on the original sublattice period (a_{x(y)}=a_{x^{\prime }(y^{\prime })}=d), that is, the photonic moiré lattice period \mathrm{\Lambda }=d/(\sqrt{2}\sin (\theta /2)). For other rotation angles, the superposition patterns exhibit an aperiodic structure with long-range order and feature rotational symmetry and certain regularity.

In contrast to the conventional periodic structures formed within the EIT system, we introduce a rotational degree of freedom to construct the photoinduced moiré lattice. By adjusting the rotation angle \theta, the moiré pattern can seamlessly transition from a periodic arrangement with varying lattice periods to an aperiodic arrangement, and vice versa, while preserving rotational symmetry. It is worth noting that the simultaneous coupling between energy levels |1⟩ and |2⟩ by applying exactly two sublattice-forming fields to provide such periodic control, without regard to the degree of matching between the two sublattices. In the following discussion, we only consider the periodic moiré patterns formed under the Pythagorean triple condition.

As the atomic transition channel |1⟩\to |2⟩ is periodically dressed by the moiré lattice-forming field E_{eff}, according to the dressed state theory, the state |1⟩ is split into dressed states |+⟩ and |-⟩, and corresponding eigenvalues{\lambda }_{\pm }={\mathrm{\Delta }}_c/2\pm \sqrt{{{\mathrm{\Delta }}_c}^2/4+|{\mathrm{\Omega }}_{eff}(x,y){|}^2}. As a result, the lattice states |\pm ⟩ exhibit periodically and symmetrically changed attributes to the spatial intensity distribution of {\mathrm{\Omega }}_{eff}(x,y) presents moiré lattice, as displayed in Fig.2. Fig.2(a) shows the splitting states |\pm ⟩ caused by the two sublattices at rotation angle \theta =0 within five sublattice periods, and Fig.2(b) and (c) correspond to the splitting states |\pm ⟩ at \theta =\arctan (3/4) and \theta =\arctan (5/12) over ten sublattice periods, respectively, evincing longer moiré periods and rich patterns.

The optical response of the atomic medium to the probe field is governed by a density matrix equation of motion, which in the interaction picture takes the form \mathrm{\partial }\rho /\mathrm{\partial }t=-\mathrm{i}/\hslash \left[\rho ,H_{int}\right]. Under the rotational wave approximation, the interaction Hamiltonian between atomic ensemble and laser fields can be given by H_{int}={\mathrm{\Omega }}_p{\mathrm{e}}^{-\mathrm{i}{\mathrm{\Delta }}_{p}t}|0⟩⟨1|+{\mathrm{\Omega }}_{eff}{\mathrm{e}}^{-\mathrm{i}{\mathrm{\Delta }}_{c}t}|1⟩⟨2|+h.c., where {\mathrm{\Omega }}_p={\mu }_{10}{E}_p/\hslash. Solving the density matrix equation under the assumption of the weak probe field, the induced polarization at {\omega }_p can be written as P({\omega }_p)={\epsilon }_0\chi ({\omega }_p)E_p, where the photoinduced susceptibility can be expressed as

(2)\chi ({\omega }_p)={\displaystyle \frac{\mathrm{i}N|{\mu }_{10}{|}^2}{\hslash {\epsilon }_0}}{\displaystyle \frac{d_{20}}{d_{10}{d}_{20}+|{\mathrm{\Omega }}_{eff}{|}^2}},

where N is the atomic ensemble density, {\epsilon }_0 is the vacuum permittivity, d_{10}={\gamma }_{10}-\mathrm{i}{\mathrm{\Delta }}_p, d_{20}={\gamma }_{20}-\mathrm{i}({\mathrm{\Delta }}_p+{\mathrm{\Delta }}_c), and {\gamma }_{ij} is the dephasing rate between |i⟩ and |j⟩.

The susceptibility \chi ({\omega }_p) is periodically modulated by the moiré field distribution |{\mathrm{\Omega }}_{eff}{|}^2, and can be influenced by {\mathrm{\Omega }}_c, {\mathrm{\Delta }}_p, and {\mathrm{\Delta }}_c within a single period. To elucidate this observation, we display the optical properties of the probe field at the nodes and antinodes of the formed moiré lattice in Fig.3(a1) and (a2). The absorption spectrum [depicted by the blue curve in Fig.3(a)] indicates strong absorption of the probe beam at the nodes, whereas it is predominantly transmitted at the antinodes. On the other hand, the dispersion [red curve in Fig.3(a)] within the EIT window is accompanied by positive at the nodes and negative at the antinodes. These features can lead to a noticeable amplitude modulation and a large phase modulation across the probe beam. It can be seen from Fig.3(b) that {\mathrm{\Delta }}_p and {\mathrm{\Delta }}_c can effectively modulate the displacement and width of the EIT window in the absorption spectrum at the antinodes of the lattice-forming field as well as the dispersion spectrum. Indeed, this is due to the fact that the metastable state |1⟩ is split into |+⟩ and |-⟩.

Given that the absorption and dispersion coefficients of the probe field are significantly influenced by {\mathrm{\Omega }}_{eff}, this enables effective spatially periodic modulation of amplitude and phase in the x−y plane. However, this does not result in modulation along the propagation direction of the probe field; only the diffraction occurring in this direction needs to be considered. Under the slowly varying envelope approximation, the propagation equation of the probe field along the z direction in the atomic medium is described by Maxwell’s wave equation in a self-consistent manner [44]:

(3){\displaystyle \frac{\mathrm{\partial }{E}_p}{\mathrm{\partial }z}}=(-\alpha /2+\mathrm{i}\sigma )E_p,

where \alpha =(4\pi /{\lambda }_p)\mathrm{I}\mathrm{m}[\chi ({\omega }_p)], and \sigma =(2\pi /{\lambda }_p)\mathrm{R}\mathrm{e}[\chi ({\omega }_p)] are the absorption and phase shift coefficients, respectively. {\lambda }_p is the wavelength of the probe field. By solving Eq. (3), the transmission function of the probe beam for an interaction length L can be obtained as

(4)T(x,y)=\exp \left[-{\displaystyle \frac{\alpha (x,y)L}{2}}+\mathrm{i}\sigma (x,y)L\right].

Based on the Fraunhofer diffraction theory, the far-field diffraction pattern of the photoinduced moiré lattice over the diffraction angle \theta with respect to z direction is proportional to the Fourier transform of T(x,y) under the assumption that the incident probe laser is a plane wave. The diffraction intensity distribution is given by following:

(5)\begin{array}{rl}I({\theta }_x,{\theta }_y)=& |E({\theta }_x,{\theta }_y){|}^2\times \frac{{\sin }^2[M\pi {\mathrm{\Lambda }}_x\sin ({\theta }_x)/{\lambda }_p]}{M^2{\sin }^2[\pi {\mathrm{\Lambda }}_x\sin ({\theta }_x)/{\lambda }_p]}\\ & \times \frac{{\sin }^2[N\pi {\mathrm{\Lambda }}_y\sin ({\theta }_y)/{\lambda }_p]}{N^2{\sin }^2[\pi {\mathrm{\Lambda }}_y\sin ({\theta }_y)/{\lambda }_p]},\end{array}

where E({\theta }_x,{\theta }_y)=\iint \mathrm{d}x\mathrm{d}yT(x,y)\exp (-\mathrm{i}2\pi x{\mathrm{\Lambda }}_x\sin ({\theta }_x)/{\lambda }_p)\exp (-\mathrm{i}2\pi y{\mathrm{\Lambda }}_y\sin ({\theta }_y)/{\lambda }_p) represents the Fraunhofer diffraction of a single space period. The diffraction efficiency into any diffraction orders along the x axis and the y axis is defined by the grating equation {\mathrm{\Lambda }}_x\sin {\theta }_x=m{\lambda }_p and {\mathrm{\Lambda }}_y\sin {\theta }_y=n{\lambda }_p, respectively. M and N denote the numbers of spatial periods along x direction and y direction of the moiré lattices which are illuminated by the probe beam, {\mathrm{\Lambda }}_x and {\mathrm{\Lambda }}_y represent the periods along x axis and y axis of moiré lattice with a rotation angle, while {\theta }_x and {\theta }_y are the diffraction angles for x and y directions, respectively.

3 Results and discussion

As detailed in Section 2, we provide a theoretical exploration of constructing a periodic photoinduced moiré lattice through the interference of two sublattice-forming fields within an atomic ensemble. We consider that the trapped ⁸⁵Rb atom could in a regular cold magneto-optical trap is about 1\mathrm{m}\mathrm{m} in diameter, and the temperature of the vapor cell is set at about 4.4\mathrm{m}\mathrm{K}. Eight elliptical-Gaussian shaped coupling beams (with wavelength {\lambda }_c=775.98\mathrm{n}\mathrm{m} and power about p_c=20\mathrm{m}\mathrm{W}) from the same CW Ti: Sapphire laser are split by several polarization beam splitters to specifically drive the upper transition |1⟩\to |2⟩[5S_{3/2}(F=3)\to 5P_{5/2}], and can be detuned by an acousto-optical modulator. Meanwhile, the weak probe beam (with wavelength {\lambda }_p=780\mathrm{n}\mathrm{m} and power p_p=5\mathrm{m}\mathrm{W}) is obtained from an external cavity diode laser and used to connect the lower transition |0⟩\to |1⟩[5P_{1/2}(F=3)\to 5D_{3/2}(F=3)]. By employing a small-angle paraxial arrangement between the coupling and probe beams and two-photon Doppler-free configurations, the clear and discrete diffraction pattern can be observed. This section will discuss and analyze the transmission characteristics and far-field diffraction patterns of the moiré lattices, along with the ability to control diffraction efficiency by adjusting the rotation angle \theta. For the sake of convenience, we assume that M=N=1, a_{x(y)}/{\lambda }_p=a_{x^{\prime }(y^{\prime })}/{\lambda }_p=d/{\lambda }_p=4. Therefore, we infer that the maximum positive and negative diffraction order for each sublattice (or the two sublattices superimposed without rotation angle as \mathrm{\Lambda }=d) is k=\pm 4 through the grating equation d\sin \theta =k{\lambda }_p.

We start from investigating the transmission properties and Fraunhofer diffraction patterns of the two sublattices superimposed at rotation angle \theta =0 with {\mathrm{\Delta }}_p={\mathrm{\Delta }}_c=0\mathrm{M}\mathrm{H}\mathrm{z}. Fig.4(a1) and (a2) show the corresponding transmission profile and the phase modulation over four space periods, respectively. The optical response of the atomic medium to the probe beam around the antinodes is almost transparent but is strongly absorbed at the nodes [Fig.4(a1)], which leads to a periodic amplitude modulation across the profile of the probe beam. On the other hand, the real part of the susceptibility vanishes as {\mathrm{\Delta }}_p={\mathrm{\Delta }}_c=0\mathrm{M}\mathrm{H}\mathrm{z}, and the phase modulation is absent in such lattices [Fig.4(a2)]. Therefore, a phenomenon reminiscent of the pure amplitude-type lattice is implemented in such an atomic lattice, and the normalized Fraunhofer diffraction pattern is shown in Fig.4(a3). The diffraction intensity tends to converge into the center maximum and the higher-order diffraction patterns are limited by this pure amplitude lattice.

The transmission properties and diffraction patterns of photoinduced moiré lattices are specific at rotation angles \theta =\arctan (3/4) and \theta =\arctan (5/12), assuming M=N=1, and these deserve separate discussion. Fig.4(b1) and (b2) show the output profile and the phase of the transmission function T(x,y) for the moiré lattice at rotation angle \theta =\arctan (3/4) with setting {\mathrm{\Delta }}_p=0\mathrm{M}\mathrm{H}\mathrm{z} and {\mathrm{\Delta }}_c=0\mathrm{M}\mathrm{H}\mathrm{z}. It can be found that the transmission of the probe around the antinodes is much higher than that at the nodes [Fig.4(b1)]. Furthermore, since the absence of phase modulation [Fig.4(b2)] contributes to the realization of an amplitude-type moiré lattice, the zero-order diffraction intensity is diffracted finitely to higher-orders as presented in Fig.4(b3), which is consistent with the phenomenon demonstrated shown in Fig.4(a3). However, the diffraction pattern of the moiré lattice with \theta =\arctan (3/4) is much smaller than the one shown in Fig.4(a3). Due to the fact that the large moiré period gives rise to a mini-Brillouin zone in the momentum space. For the moiré lattice at rotation angle \theta =\arctan (3/4), the first Brillouin zone is a square of width 2\pi /(\sqrt{5}d). The width of the first Brillouin zone of the initial square sublattice is 2\pi /d, which means that the formation of the moiré lattice at \theta =\arctan (3/4) will divide the first Brillouin zone of the initial sublattice into five parts. As a result, the diffraction pattern of the moiré lattice at \theta =\arctan (3/4) is confined within its first Brillouin zone causing a reduction of the diffraction pattern. Under the same parameters, the moiré lattice at rotation angle \theta =\arctan (5/12) also exhibits a pure amplitude-type modulation, with strong absorption at the nodes and almost transmission at antinodes [Fig.4(c1)]. The diffracted intensity is greatly retained at zero-order [Fig.4(c3)] due to the absence of phase shift [Fig.4(c2)] and the diffracted pattern decreases with an increase of the moiré lattice period (\mathrm{\Lambda }=\sqrt{13}d).

Subsequently, we optimize the parameters to achieve lossless and efficient phase modulation for the photonic moiré lattices, which direct light into higher-order paths. The coupling fields maintain high intensity and resonant interaction with the |1⟩\to |2⟩ transition, thereby preserving excellent transparency for the probe beams’ profile. The weak probe field is adjusted to deviate from the |0⟩\to |1⟩ transition to implement a \pi phase modulation while still operating within the parameters of the EIT window. The corresponding transmission profile and the phase modulation over four space periods for the two sublattices without rotation angle, with setting {\mathrm{\Delta }}_p=21.21\mathrm{M}\mathrm{H}\mathrm{z} and {\mathrm{\Delta }}_c=0\mathrm{M}\mathrm{H}\mathrm{z}, as displayed in Fig.5(a1) and (a2). Not only does the imaginary part of the susceptibility still carry amplitude modulation on the probe field, but also the real part delivers phase shifts. After introducing frequency detuning {\mathrm{\Delta }}_p=21.21\mathrm{M}\mathrm{H}\mathrm{z}, the absorption and the phase undergo rapid changes at |{\mathrm{\Omega }}_{eff}(x,y)|=|{\mathrm{\Delta }}_p|. Meanwhile, the zero-order diffraction energy is reallocated to higher-order terms; notably, the first-order diffraction along the x axis (and the y axis) or even into the first-order in all four quadrants is significantly enhanced, with the zero-order term attaining the lowest energy, as illustrated in Fig.5(a3). By introducing the detuning {\mathrm{\Delta }}_p in the atomic ensemble, we successfully achieve a phase-type optical lattice.

Similarly, the output profile and the phase of the transmission function T(x,y) for the moiré lattice with rotation angle \theta =\arctan (3/4) are illustrated in Fig.5(b1) and (b2), where a small absorption and a relatively stable phase shift of 1.73\pi are periodically introduced into the transmission profile. The amount of light in zero-order is converted into higher-order diffraction, as shown in Fig.5(b3). In addition, the diffraction intensity of the first-order along the x axis (y axis) is comparable to that of the first-order in the four quadrants, which is different from that in Fig.5(a3). This is attributed to the fact that the diffraction intensity in the four quadrants is redistributed as the first-order along the x′ axis (y′ axis) of the second sublattice with \theta =\arctan (3/4), and the strength of two sublattice-forming fields is the same. Therefore, the moiré lattice possesses a higher diffraction proportion of the first-order in the four quadrants compared to the initial sublattice, the identical proportion to the first-order along the x axis (y axis). Additionally, after the introduction of frequency detuning {\mathrm{\Delta }}_p=21.21\mathrm{M}\mathrm{H}\mathrm{z}, a periodic amplitude modulation [Fig.5(c1)] and a relatively stable phase shift of 1.73\pi [Fig.5(c2)] are also entailed in the transmission function for the moiré lattice with rotation angle \theta =\arctan (5/12). Fig.5(c3) shows the diffraction pattern of the moiré lattice at \theta =\arctan (5/12), where the twisted angle between the first-order along the x axis (y axis) and the first-order in the four quadrants is exactly the rotation angle between the two square sublattices. Moreover, it can be observed from Fig.5(a3−c3) that the first-order diffraction intensity of the moiré lattices with different rotation angles along the x axis (y axis) as well as in the four quadrants is half that of the initial sublattice. Thus, we have successfully implemented both amplitude-type and phase-type moiré optical lattices within the atomic ensemble.

To quantitatively investigate the far-field intensity distribution of photoinduced moiré lattices, the intensities of various diffraction orders, including the zero-order I({\theta }_x^0,{\theta }_y^0), the first-order along the x axis I({\theta }_x^1,{\theta }_y^0), and the first-order in the four quadrants I({\theta }_x^1,{\theta }_y^1), are depicted in Fig.6. In Fig.6(a1−a3), we consider the corresponding normalized diffraction intensity of I({\theta }_x^0,{\theta }_y^0), I({\theta }_x^1,{\theta }_y^0), and I({\theta }_x^1,{\theta }_y^1) for two sublattices without rotation angle by scanning {\mathrm{\Delta }}_p and {\mathrm{\Delta }}_c simultaneously. It can be observed from Fig.6(a1−a3) that a significant Rabi splitting and a clear avoided-crossing shape appear, which manifested as an anti-centrosymmetric structure of two split triangle-like in each plot. This interesting phenomenon clearly illustrates the splitting of state |1⟩ into two dressed states |+⟩ and |-⟩, which is consistent with the modulation spectra of absorption and dispersion exhibited in Fig.3(c).

For the moiré lattice at \theta =\arctan (3/4), the typical avoided-crossing shape is also observed in Fig.6(b1)−(b3). The phase modulation is introduced by setting {\mathrm{\Delta }}_p and {\mathrm{\Delta }}_c in the triangle-like anti-centrosymmetric regions to diffract efficiently the energy of the probe field into higher-order directions. However, the diffraction intensity proportions of I({\theta }_x^1,{\theta }_y^0) and I({\theta }_x^1,{\theta }_y^1) are identical in Fig.6(b2) and (b3). Since the first-order in the four quadrants for the two sublattices at \theta =0 (equivalent to sublattice) belongs to a higher-order diffraction compared to that of the first-order along the x axis, its intensity is lower [Fig.6(a2) and (a3)]. By introducing non-zero rotation angles, we manage to average the first-order diffraction energy of individual sublattice and efficiently redirect it to specific positions in all four quadrants as first-order diffraction energy, rather than to higher orders. As depicted in Fig.6(c), the diffraction intensity distribution of the moiré lattice at \theta =\arctan (5/12) exhibits a typical avoided-crossing shape, which is substantially in line with that of the moiré lattice at \theta =\arctan (3/4) shown in Fig.6(b). It is also the case that the diffraction intensities of I({\theta }_x^1,{\theta }_y^0) and I({\theta }_x^1,{\theta }_y^1) of the moiré lattices with different rotation angles are half that of the first-order of the initial sublattice. The absorption and dispersion properties caused by the multi-beam interference at the same optical parameters are consistent, resulting in evolving similar diffraction intensity distributions for photoinduced moiré lattices with different rotation angles [Fig.6(b) and (c)].

To elucidate the impact of rotation angle \theta (specifically including the Pythagorean angles) on far-field intensity distribution, we present specific diffraction intensity distribution data along the x direction by setting \sin {\theta }_y=0 and -1⩽\sin {\theta }_x⩽1. At probe field resonance, the diffraction energy is predominantly concentrated at the zero-order for the moiré lattices at each Pythagorean angle, as shown in Fig.7(a1). As {\mathrm{\Delta }}_p is switched from 0 to 21.21\mathrm{M}\mathrm{H}\mathrm{z}, it can be observed from Fig.7(b1) that more energy of the probe field is transferred from zero-order to higher-order direction, and the first-order diffraction is distributed over the same diffraction angle {\theta }_x for different rotation angles \theta. Due to the fact that the period in the reciprocal space for moiré lattices with distinct rotation angles have the same common multiple, which is the period of the sublattice in reciprocal space, the diffraction condition will be satisfied at the same diffraction angle {\theta }_x (\sin {\theta }_x\approx \pm 0.25) by following the Ewald construction. For a more intuitive display, the corresponding diffraction distribution of several rotation angles over -1⩽\sin {\theta }_x⩽1 are presented in Fig.7(a2) and (b2). The diffracted pattern is largest at the rotation angle of 0, and as the rotation angle changes, the moiré period changes resulting in a corresponding change in the size of the diffraction pattern. The dependence of the moiré period on the rotation angle \theta is shown in Fig.7(c), where the rotation angle with a small period corresponds to a large diffraction pattern in Fig.7(a) and (b). Therefore, for successive variations of the rotation angles under the case of the same optical parameter, the characteristic of the zero-order energy transfer to higher-order remains unchanged after introducing phase detuning, and the diffraction intensity is roughly constant.

To further confirm the operational domains of the amplitude- and phase-type for moiré lattices, we also examine the effect of {\mathrm{\Delta }}_p on several diffraction orders arising from diverse rotation angles by setting {\mathrm{\Delta }}_c=0\mathrm{M}\mathrm{H}\mathrm{z}. As shown in Fig.8(a) and (b), the normalized diffraction intensity of the zero-order I({\theta }_x^0,{\theta }_y^0) and the first-order along the x axis I({\theta }_x^1,{\theta }_y^0) exhibits a symmetrical distribution with respect to {\mathrm{\Delta }}_p=0\mathrm{M}\mathrm{H}\mathrm{z}. The frequency range in which the diffracted energy is converted from zero-order to higher-order is constant as the rotation angle changes. The first-order along the x axis I({\theta }_x^1,{\theta }_y^0) increases sharply to the maximum at {\mathrm{\Delta }}_p=21.21\mathrm{M}\mathrm{H}\mathrm{z} when {\mathrm{\Delta }}_p is in the scanning region (10\to 21.21\mathrm{M}\mathrm{H}\mathrm{z}), while I({\theta }_x^0,{\theta }_y^0) decreases to the minimum. The modulation range of transmission and phase for the photonic moiré lattices is not affected by the rotation angles but depends on the optical parameters of the lattice-forming field. Thus, the introduction of the same detuning {\mathrm{\Delta }}_p can produce similar energy transfer efficiency for moiré lattices at different rotation angles. This indicates that such photoinduced moiré lattices exhibit excellent stability in creating and switching between these two operating modes, as well as in lossless conversion as successive variations of the rotation angle.

The previously discussed results are universal in that the relative rotation of two identical sublattices facilitates the creation of two-dimensional moiré lattice potentials, enabling a seamless transition between commensurable and incommensurable moiré patterns. This transition also allows for the observation of far-field diffraction in periodic lattices. To demonstrate the broad applicability of this effect, we examine a moiré lattice potential consisting of three rotated square sublattices formed through the interference of twelve coupled beams. The rotation angles of the second and third sublattices with respect to the first sublattice are {\theta }_1 and {\theta }_2, respectively, and their values are in no particular order. For such lattices, the two rotation angles that produce commensurable patterns are still required to satisfy the Pythagorean triangles. Four examples of periodic lattices are presented in Fig.9(a1−d1). The blue squares represent the minimum period \mathrm{\Lambda }=d/[2\sin ({\theta }_1/2)\sin ({\theta }_2/2)] with translational periodicity and rotational symmetry.

In such periodic structures, we set detuning terms at zero ({\mathrm{\Delta }}_p={\mathrm{\Delta }}_c=0\mathrm{M}\mathrm{H}\mathrm{z}) and the amplitude-type moiré optical lattices are realized as shown in Fig.9(a2−d2). We verify that the diffracted intensity is greatly preserved at zero-order due to the absence of phase modulation effect (\mathrm{R}\mathrm{e}[\chi ({\omega }_p)]=0) attributed to the amplitude-type moiré lattices. The magnitude of the diffraction pattern is associated with the moiré period, with longer moiré periods producing smaller diffraction patterns. We display the Fraunhofer diffraction patterns of the moiré lattices when {\mathrm{\Delta }}_p=25.98\mathrm{M}\mathrm{H}\mathrm{z} at different rotation angles [Fig.9(a3−d3)], and observe that most of the probe energy is transferred to the higher orders of diffraction, due to the phase modulation dominates owing to far detuned field. Meanwhile, the zero-order diffraction reaches the lowest, and therefore the phase-type moiré lattices are realized. Here, the twelve diffraction sites around zero-order diffraction (the middle site) are defined as three sets of first-order diffraction along three coordinate planes, just like the studies in the 2D EIG [49−51], which integers are marked in Fig.9(a3) for better description. These three sets of first-order diffraction sites are contributed by three sublattices respectively, whose twist angles correspond to the rotation angles of the three sublattices. By introducing the second rotation angle, the first-order diffraction energy of individual sublattice is further divided into thirds and is available for redistribution to more positions on a circle. It is predictable that the incorporation of more rotation angles will inevitably lead to further energy diversion, which is beneficial for the design of tunable diffractive beam-shaper homogenizers and reconfigurable multiple-beam splitters.

For the twisted structures in conventional materials, the strong interlayer coupling can cause a series of extraordinary properties, such as unconventional superconductivity and the quantum anomalous Hall effect. In the atomic ensemble, the interlayer coupling can be introduced by two methods, namely by modulating the relative strength between two sublattices or by coupling the spin states of the atomic ensemble optical lattices. The former method changes the lattice-forming field intensity of two sublattices directly by optical means, and the latter precisely controls the strength of the interlayer coupling by microwave fields. Further details studies on the interlayer coupling will be presented elsewhere.

4 Conclusions

In summary, we realize 2D nonmaterial moiré lattices by overlapping two identical sublattices with twisted angles in an ultracold atoms ensemble using the photoinduced method. The periodic moiré patterns with different geometric distortion structures are conveniently constructed by varying the Pythagorean rotation angle of two square sublattices. We show how that absorption/dispersion as well as the transmission properties of the photoinduced moiré lattices can be modulated dynamically by photon detuning and Rabi frequency, resulting in amplitude and phase type moiré lattice. Based on Fraunhofer diffraction theory, we demonstrate that the far-field diffraction efficiency can be refined by changing photon detuning, and the tunable rotation angles can efficiently redirect and precisely position diffraction energy, thereby optimizing lattice properties. It is also found that the operational domains of moiré lattices with different rotation angles possess stability that facilitates adjustment and conversion between various moiré periodic structures. Furthermore, more complex moiré lattices formed by interfering three twisted sublattices are investigated, and the results indicate that the diffracted energy is further dispersed uniformly as a circular distribution. Our work promotes the development in the field of all-optical quantum device, and affords a powerful platform for studying intriguing physics properties, such as optical solitons, parity−time-symmetric, and superfluidity in twisted multilayer systems.

References

[1] Y. Akahane , T. Asano , B. S. Song , and S. Noda , High-Q photonic nanocavity in a two-dimensional photonic crystal, Nature 425(6961), 944 (2003)

[2] H. G. Park , S. H. Kim , S. H. Kwon , Y. G. Ju , J. K. Yang , J. H. Baek , S. B. Kim , and Y. H. Lee , Electrically driven single-cell photonic crystal laser, Science 305(5689), 1444 (2004)

[3] J. D. Joannopoulos,S. G. Johnson,J. N. Winn,R. D. Meade, Photonic Crystals: Molding the Flow of Light, Princeton University Press, 2011

[4] D. N. Christodoulides , F. Lederer , and Y. Silberberg , Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature 424(6950), 817 (2003)

[5] H. S. Eisenberg , Y. Silberberg , R. Morandotti , A. R. Boyd , and J. S. Aitchison , Discrete spatial optical solitons in waveguide arrays, Phys. Rev. Lett. 81(16), 3383 (1998)

[6] D. R. Smith , J. B. Pendry , and M. C. K. Wiltshire , Metamaterials and negative refractive index, Science 305(5685), 788 (2004)

[7] T. Han , X. Bai , J. T. L. Thong , B. Li , and C. W. Qiu , Full control and manipulation of heat signatures: Cloaking, camouflage and thermal metamaterials, Adv. Mater. 26(11), 1731 (2014)

[8] M. Bajcsy , A. S. Zibrov , and M. D. Lukin , Stationary pulses of light in anatomic medium, Nature 426(6967), 638 (2003)

[9] J. L. Everett , G. T. Campbell , Y. W. Cho , P. Vernaz-Gris , D. B. Higginbottom , O. Pinel , N. P. Robins , P. K. Lam , and B. C. Buchler , Dynamical observations of selfstabilizing stationary light, Nat. Phys. 13(1), 68 (2017)

[10] L. Lu , J. D. Joannopoulos , and M. Soljačić , Topological photonics, Nat. Photonics 8(11), 821 (2014)

[11] T. Ozawa , H. M. Price , A. Amo , N. Goldman , M. Hafezi , L. Lu , M. C. Rechtsman , D. Schuster , J. Simon , O. Zilberberg , and I. Carusotto , Topological photonics, Rev. Mod. Phys. 91(1), 015006 (2019)

[12] R. El-Ganainy , K. G. Makris , M. Khajavikhan , Z. H. Musslimani , S. Rotter , and D. N. Christodoulides , Non-Hermitian physics and PT symmetry, Nat. Phys. 14(1), 11 (2018)

[13] Z. Zhang , Y. Feng , S. Ning , G. Malpuech , D. D. Solnyshkov , Z. Xu , Y. Zhang , and M. Xiao , Imaging lattice switching with Talbot effect in reconfigurable non-Hermitian photonic graphene, Photon. Res. 10(4), 958 (2022)

[14] J. D. Joannopoulos , P. R. Villeneuve , and S. Fan , Photonic crystals: Putting a new twist on light, Nature 386(6621), 143 (1997)

[15] M. D. Leistikow , A. P. Mosk , E. Yeganegi , S. R. Huisman , A. Lagendijk , and W. L. Vos , Inhibited spontaneous emission of quantum dots observed in a 3D photonic band gap, Phys. Rev. Lett. 107(19), 193903 (2011)

[16] F. Lederer , G. I. Stegeman , D. N. Christodoulides , G. Assanto , M. Segev , and Y. Silberberg , Discrete solitons in optics, Phys. Rep. 463(1−3), 1 (2008)

[17] P. Longo , P. Schmitteckert , and K. Busch , Few-photon transport in low-dimensional systems: Interaction-induced radiation trapping, Phys. Rev. Lett. 104(2), 023602 (2010)

[18] S. Carr , D. Massatt , S. Fang , P. Cazeaux , M. Luskin , and E. Kaxiras , Twistronics: Manipulating the electronic properties of two-dimensional layered structures through their twist angle, Phys. Rev. B 95(7), 075420 (2017)

[19] Y. Cao , V. Fatemi , A. Demir , S. Fang , S. L. Tomarken , J. Y. Luo , J. D. Sanchez-Yamagishi , K. Watanabe , T. Taniguchi , E. Kaxiras , R. C. Ashoori , and P. Jarillo-Herrero , Correlated insulator behaviour at half-filling in magicangle graphene superlattices, Nature 556(7699), 80 (2018)

[20] Y. Cao , V. Fatemi , S. Fang , K. Watanabe , T. Taniguchi , E. Kaxiras , and P. Jarillo-Herrero , Unconventional superconductivity in magic-angle graphene superlattices, Nature 556(7699), 43 (2018)

[21] G. W. Burg , J. Zhu , T. Taniguchi , K. Watanabe , A. H. MacDonald , and E. Tutuc , Correlated insulating states in twisted double bilayer graphene, Phys. Rev. Lett. 123(19), 197702 (2019)

[22] R. Bistritzer and A. H. MacDonald , Moiré bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. USA 108(30), 12233 (2011)

[23] Y. Liu , N. O. Weiss , X. Duan , H. C. Cheng , Y. Huang , and X. Duan , Van der Waals heterostructures and devices, Nat. Rev. Mater. 1(9), 16042 (2016)

[24] B. Y. Jiang , G. X. Ni , Z. Addison , J. K. Shi , X. Liu , S. Y. F. Zhao , P. Kim , E. J. Mele , D. N. Basov , and M. M. Fogler , Plasmon reflections by topological electronic boundaries in bilayer graphene, Nano Lett. 17(11), 7080 (2017)

[25] L. Yuan , B. Zheng , J. Kunstmann , T. Brumme , A. B. Kuc , C. Ma , S. Deng , D. Blach , A. Pan , and L. Huang , Twistangle-dependent interlayer exciton diffusion in WS₂–WSe₂ heterobilayers, Nat. Mater. 19(6), 617 (2020)

[26] G. Niu , J. Lu , J. Geng , S. Li , H. Zhang , W. Xiong , Z. Ruan , Y. Zhang , B. Fu , L. Gao , and J. Cai , Electronic properties of monolayer Copper selenide with one-dimensional moiré patterns, Front. Phys. (Beijing) 18(1), 13303 (2023)

[27] L. Li , M. Wu , and X. Lu , Correlation, superconductivity and topology in graphene moiré superlattice, Front. Phys. (Beijing) 18(4), 43401 (2023)

[28] S. Li , K. Wei , Q. Liu , Y. Tang , and T. Jiang , Twistronics and moiré excitonic physics in van der Waals heterostructures, Front. Phys. (Beijing) 19(4), 42501 (2024)

[29] C. Huang , F. Ye , X. Chen , Y. V. Kartashov , V. V. Konotop , and L. Torner , Localization–delocalization wavepacket transition in Pythagorean aperiodic potentials, Sci. Rep. 6(1), 32546 (2016)

[30] P. Wang , Y. Zheng , X. Chen , C. Huang , Y. V. Kartashov , L. Torner , V. V. Konotop , and F. Ye , Localization and delocalization of light in photonic moiré lattices, Nature 577(7788), 42 (2020)

[31] Q. Fu , P. Wang , C. Huang , Y. V. Kartashov , L. Torner , V. V. Konotop , and F. Ye , Optical soliton formation controlled by angle twisting in photonic moiré lattices, Nat. Photonics 14(11), 663 (2020)

[32] T. Ning , Y. Ren , Y. Huo , and Y. Cai , Efficient high harmonic generation in nonlinear photonic moiré superlattice, Front. Phys. (Beijing) 18(5), 52305 (2023)

[33] W. J. M. Kort-Kamp , F. J. Culchac , R. B. Capaz , and F. A. Pinheiro , Photonic spin Hall effect in bilayer graphene moiré superlattices, Phys. Rev. B 98(19), 195431 (2018)

[34] S. S. Sunku , G. X. Ni , B. Y. Jiang , H. Yoo , A. Sternbach , A. S. McLeod , T. Stauber , L. Xiong , T. Taniguchi , K. Watanabe , P. Kim , M. M. Fogler , and D. N. Basov , Photonic crystals for nano-light in moiré graphene superlattices, Science 362(6419), 1153 (2018)

[35] X. R. Mao , Z. K. Shao , H. Y. Luan , S. L. Wang , and R. M. Ma , Magic-angle lasers in nanostructured moiré superlattice, Nat. Nanotechnol. 16(10), 1099 (2021)

[36] A. González-Tudela and J. I. Cirac , Cold atoms in twistedbilayer optical potentials, Phys. Rev. A 100(5), 053604 (2019)

[37] T. Salamon , A. Celi , R. W. Chhajlany , I. Frérot , M. Lewenstein , L. Tarruell , and D. Rakshit , Simulating twistronics without a twist, Phys. Rev. Lett. 125(3), 030504 (2020)

[38] X. W. Luo and C. Zhang , Spin-twisted optical lattices: Tunable flat bands and Larkin–Ovchinnikov superfluids, Phys. Rev. Lett. 126(10), 103201 (2021)

[39] Z. Meng , L. Wang , W. Han , F. Liu , K. Wen , C. Gao , P. Wang , C. Chin , and J. Zhang , Atomic Bose–Einstein condensate in twisted-bilayer optical lattices, Nature 615(7951), 231 (2023)

[40] X. Liu and J. Zeng , Gap solitons in parity–time symmetric moiré optical lattices, Photon. Res. 11(2), 196 (2023)

[41] R. Tian , S. Li , M. Arzamasovs , H. Gao , Y. C. Zhang , and B. Liu , Localization−delocalization transition in an electromagnetically induced photonic lattice, Phys. Rev. A 108(4), 043711 (2023)

[42] Z. Chen , X. Liu , and J. Zeng , Electromagnetically induced moiré optical lattices in a coherent atomic gas, Front. Phys. (Beijing) 17(4), 42508 (2022)

[43] X. Liu and J. Zeng , Two-dimensional localized modes in nonlinear systems with linear nonlocality and moiré lattices, Front. Phys. (Beijing) 19(4), 42201 (2024)

[44] H. Y. Ling , Y. Q. Li , and M. Xiao , Electromagnetically induced grating: Homogeneously broadened medium, Phys. Rev. A 57(2), 1338 (1998)

[45] S. E. Harris , Electromagnetically induced transparency, Phys. Today 50(7), 36 (1997)

[46] M. Fleischhauer , A. Imamoglu , and P. J. Marangos , Electromagnetically induced transparency: Optics in coherent media, Rev. Mod. Phys. 77(2), 633 (2005)

[47] M. S. Kang , A. Butsch , and P. S. J. Russell , Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre, Nat. Photonics 5(9), 549 (2011)

[48] F. Wen , X. Zhang , H. Ye , W. Wang , H. Wang , Y. Zhang , Z. Dai , and C. W. Qiu , Efficient and tunable photoinduced honeycomb lattice in an atomic ensemble, Laser Photonics Rev. 12(9), 1800050 (2018)

[49] F. Wen , H. Ye , X. Zhang , W. Wang , S. Li , H. Wang , Y. Zhang , and C. W. Qiu , Optically induced atomic lattice with tunable near-field and far-field diffraction patterns, Photon. Res. 5(6), 676 (2017)

[50] J. Yuan , C. Wu , L. Wang , G. Chen , and S. Jia , Observation of diffraction pattern in two-dimensional optically induced atomic lattice, Opt. Lett. 44(17), 4123 (2019)

[51] F. Wen , S. Zhang , S. Hui , H. Ma , S. Wang , H. Ye , W. Wang , T. Zhu , Y. Zhang , and H. Wang , Terahertz tunable optically induced lattice in the magnetized monolayer graphene, Opt. Express 30(2), 2852 (2022)

RIGHTS & PERMISSIONS

Higher Education Press