A vortex is a fundamental topological structure associated with the circulation loops in various vector fields. Quantized vortices present in quantum fluids were discovered as elementary topologically nontrivial excitations of superfluid helium1 and superconductors.2 In scalar quantum fields, the existence of vortices is allowed by the gauge invariance of the wave functions (WFs) with respect to phase shifts of 2πn (n∈Z). The phase ambiguity at the vortex core requires a vanishing WF density, which is a vivid observable signature. Another manifestation is a diverging momentum curl, which results in non-vanishing energy and angular momentum contributions associated with the vortex. Importantly, in many-body systems of interacting particles, described with the Gross–Pitaevskii equation, quantum vortices are stable solutions whose spatiotemporal dynamics governs the quantum turbulence.3^–6 Stable and well-defined quantum vortices are also present in driven-dissipative systems, such as exciton–polariton interacting quantum fluids,7^–10 where they are responsible for the Berezinskii–Kosterlitz–Thouless transition.11 The polaritonic spin-orbit coupling due to the TE-TM splitting 12^,13 allows new topologically nontrivial excitations such as half-integer vortices14 and polarization vortices15 in semiconductor microcavities.

Along with the real space vortices, similar topological configurations exist in reciprocal (momentum) space. Such vortices may stem from the nontrivial topology in band structures, manifesting Berry curvature sources. In graphene, momentum space vortices of the eigenstates induced by pseudospin exist at Dirac points16^–18, and their topological charges have opposite signs in the two nonequivalent valleys in the valence and conduction bands. In general, pseudospin (sublattice) and valley (K and K′) degrees of freedom lead to rich optical behaviors in artificial graphene and related photonic lattices for orbital angular momentum (OAM) managing. In photonic honeycomb and Lieb lattices, pseudospin-orbit interaction allowed obtaining the conversion of pseudospin and OAM19 into the angular momentum of the output signal. At the same time, valley-selective excitation by a structured field in photonic graphene leads to the emergence of vortices of topological charge with arbitrary sign in the Bragg-reflected component in the complementary Dirac points.20 Polarization vortices in momentum space were reported in topological honeycomb and Lieb plasmonic lattices,21 and their conversion into real-space vortices was shown for bound states in the continuum.22 The strong spin-orbit coupling was employed to give rise to the appearance of spin antivortices in momentum space in atomically thin metals with graphene-like structure.23 The relation and the interconversion of real and reciprocal space vortices offer large possibilities for beam engineering with broad applications.24^–27

Recently, the artificial photonic lattices based on coherent atomic vapors with refraction index modulated by electromagnetically induced transparency (EIT)28 have shown their advantages for analog physics: the real-time tuning of the lattice potential that allows for studying the dynamics of the wave packets. This technique boosted the investigation and visualization of angle-dependent Klein tunneling29 and edge-state solitons,30 as well as the observation of particle-like dynamics of the vortex cores formed during the wave packet evolution in photonic graphene.31 So far, the conversion between OAM and the pseudospin via the Dirac point involved complicated beam structures to ensure the excitation of a single pseudospin of the lattice, which generates one specific vortex corresponding to the excited pseudospin.20^,31^–33 Therefore, it is essential to find alternative mechanisms for creating vortices with simpler schemes, better efficiency, and richer physics. Meanwhile, as all the above-mentioned systems involve geometrically symmetric lattices that display topological features, it is not clear whether topology is a necessity for the generation of vortices, or whether the vortices simply originate from the lattice symmetry.

In this article, we report the creation of momentum space vortices in atomic-vapor-cell-based honeycomb and hexagonal photonic lattices excited by a nonstructured Gaussian probe beam. Such a scheme allows the easy generation of multiple vortex-antivortex pairs uniformly distributed in momentum space upon a single excitation. Moreover, the resulting OAM textures can be tuned via probe symmetry. Our study reveals the underlying physics between symmetry and vortex generation and shows that such processes are possible even in the absence of any Berry curvature.

The photonic lattice is created by engineering the susceptibility via the EIT effect in a Rb85 vapor cell.28^,31 As sketched in Fig. 1(a), the coupling field E2 is constructed by the interference of three light beams from the same laser, forming a two-dimensional (2D) hexagonal pattern in the x−y plane propagating along z. For simplicity, the optical path of E2 is not shown in the graph, with technical details already introduced in Ref. 31. The probe beam E1 from a second laser, focused by a lens onto the front surface of the atomic vapor cell, propagates through the cell and experiences variation of susceptibility of the medium induced by coupling field E2. Then it is collected by another lens with the same focal length and detected by a charge coupled device (CCD) camera placed at the collection lens’ back focal plane, which shows the Fourier image presenting the momentum space of the probe beam. The phase measurements are made by homodyne detection, i.e., by interfering the probe beam with a reference beam from the same laser at the position of the CCD, where the fork-like structure of the interference stripes indicates the formation of vortices. The susceptibility experienced by E1 is determined by the frequency detunings of E1 (denoted as Δ1) and E2 (denoted as Δ2) via the EIT effect in a three-level atomic configuration [Fig. 1(b)]. When the two-photon detuning Δ1-Δ2 is set to result in higher (lower) refractive index at the bright sites of the hexagonal pattern, the bright (dark) sites form a hexagonal (honeycomb) photonic lattice. Figures 1(c) and 1(d) show the simulated spatial distribution of the refractive indices of the honeycomb and hexagonal lattices, respectively. The Gaussian probe beam is focused to a spot size comparable to that of a honeycomb lattice site. During the propagation along z in the 2D lattice, which can be considered as a matrix of coupled waveguides, the probe beam expands within the x−y plane and interacts with the 2D lattice to form reciprocal space vortices. It should be noted that the coupling field E2 alone induces the EIT effect, but does not allow one to observe and utilize it. This is why we send the probe beam, and we study its evolution under the EIT conditions. However, when performing the measurements with a coupling field only, one observes maxima at three equivalent K points of the induced honeycomb lattice, which allows one to locate the positions of vortices in momentum space in the configuration with probe beam E1 in further experiments; see Fig. S1 in the Supplementary Material.

We first set the frequency detunings as Δ1=−80MHz and Δ2=−100MHz, which yields a honeycomb lattice in the Rb cell with a lattice constant of ∼76μm, as shown in Fig. 1(c). The probe beam is centered at one of the honeycomb lattice sites. After propagating through the Rb cell, the interference pattern between the momentum space image and the reference beam shows six evenly distributed fork-like features in the interference fringes, indicating the presence of six vortices, as shown in Figs. 2(a) and 2(b). The vortices are located near the Dirac points, and their signs depend on the neighboring valley type, K or K′. The probe beam, which is focused to a small spot, contains a large range of incident angles (determined by the lens numerical aperture) that correspond to a continuum of photon momenta. The intensity of the high-momentum probe components is enough to excite sufficiently not only the first Brillouin zone (BZ), but also its nearest replicas (reciprocal lattice cells) and thus to generate well-resolved vortices near the Dirac points in momentum space, providing an easy technology of generating multiple and spatially well-separated vortices using nonstructured light.

Although the vortices can be originated from the Berry curvature sources associated with the Dirac points, as previously reported,19 such an interpretation does not apply to our configuration. Instead, we find that creation of multiple vortices is very sensitive to the initial position of the focused probe beam on the honeycomb lattice, indicating that the process occurs only under special situations highly restricted by symmetry. This indicates that the vortex formation in such systems is strongly associated with the symmetry of the excited wave packet and the properties of Fourier transformation of the finite size signals in periodic lattices.

To perform a quantitative theoretical analysis of the beam propagation in the photonic graphene lattice, we have used the paraxial approximation for the probe field profile in the vapor cell with spatial modulation of the refraction index, equivalent to Schrödinger equation with the honeycomb-type potential with the correspondence between the z coordinate in the cell and time.31 Emergence of the momentum space quantized vortices was reproduced with the initial WF state defined as a narrow Gaussian profile with a width comparable to the size of a single site. It was centered on one of the honeycomb lattice sites, as in experiment. The details of calculations, including initial conditions are given in the Supplementary Material. In this excitation configuration, the real space WF of the probe beam exhibits a C3v symmetry during time evolution, as shown in Fig. 3(a). The calculated momentum space profiles, as depicted in Figs. 3(b)–3(d) for the probability density, phase, and interference patterns, show the quantum vortices in the vicinity of the Dirac points, robustly determined by the automatic vortex detection algorithm.6 These results agree very well with the experimental measurements. The real part of the WF Fourier image (as well as its absolute value) has a sixfold rotation symmetry, whereas the imaginary part obeys the C3v symmetry, in accordance with Refs. 34–36. This result is similar to the one given in Fig. 1(a) from Ref. 34 with the exchanged real and momentum space.

The time-dependent tight-binding-type model (TBM) calculations ascertain insensitivity of the results to the particular shape of the site potential. The initial condition with a single excited site was used, which corresponds to the numerical solution of the full Schrödinger equation and the experimental design. Figure 3(e) shows the probability density in real space, and Fig. 3(f) depicts the interference pattern in momentum space with better visible BZ replicas.

The 2D Fourier transform ψ˜(k)=∬ψ(r)e−ikrdr can be evaluated analytically within the TBM in the approximation of delta function sites at the time given by inverse tunneling constant t=πℏ/J. At this moment, WF ψ(r) is dominantly located at the first nearest neighbors (NNs) of the initially excited site and momentum space signal has a form ψ˜(k)=∑j=1,2,3e−ikdj≡f(k),(1)with dj the radius-vectors of three NNs. The phase of ψ˜(k) manifests the phase factors ±2π/3 in the vicinity of Γ points of BZ replicas and ±2π winding (vortices) around Dirac points in the same way as demonstrated in Refs. 16 and 18 for the relative phase of the sublattices in the eigenstates of graphene TBM Hamiltonian, Hk=−J(0fk+fk0),(2)which is besides defined via f(k) in momentum representation.

However, such direct correspondence with a tight-binding Hamiltonian is only obtained in this particular case. More generally, the Dirac point vortices exist in the basic element of the C3v symmetry WF in honeycomb lattice, a tetramer, formed of one central site with Ψ0=1 and three neighbors with Ψ1,2,3=aeiφ, depending on parameters a and φ; see Fig. S3 in the Supplementary Material. Qualitatively, the tunneling from the initial site or sites to their neighbors in real space leads to the accumulation of a phase for various wave vectors in agreement with the Fourier shift theorem. The gained phase difference of the momentum space points depends on the symmetry of WF in real space and leads to a nonzero winding around the Dirac points, resulting in the reciprocal space vortices we observe. According to the Fourier shift theorem, for site separation distances of the order of lattice constant a0, the wave vectors ≳π/a0 corresponding to higher BZ replicas, as in the performed experiments, must be involved to gain a phase difference ∼π required to observe the vortices.

Other types of the WF symmetry that can be realized in the honeycomb lattice are the C6v and C2v symmetries. Figures 4(a) and 4(b) show the WF in real space and its phase in momentum space for initial excitation of six sites belonging to the same hexagon. For a C6v-object, both real and imaginary parts of the momentum space WF inherit its symmetry, which forbids the presence of single opposite-signed vortices near the Dirac points due to the mirror reflection symmetry. Figures 4(c) and 4(d) demonstrate the vortices in the momentum space for initial excitation between two NN sites. In this case, the vortices in momentum space are positioned at the rectangle corners in agreement with C2v symmetry.

We have tested the configurations described above based on differing symmetries experimentally; the corresponding data is presented in Fig. 5. Figures 5(a) and 5(b) show the momentum space interference pattern and the result of phase extraction for the case where the honeycomb lattice was excited at the center of the hexagonal ring, thus providing the C6v symmetry. In accordance with theoretical predictions, this symmetry violates the presence of momentum space vortices at Dirac points. For the case of C2v symmetry in a honeycomb lattice obtained by excitation between two lattice sites, one can observe the two vortex-antivortex pairs; see Figs. 5(c) and 5(d).

The WF of C2v symmetry can be also realized in the hexagonal lattice lacking the sources of Berry curvature. Figures 4(e) and 4(f) show the WF in real space and its phase image in momentum space, which exhibits rectangular vortex patterns very similar to Figs. 4(c) and 4(d).

To test the corresponding configuration experimentally, we have established the required topologically trivial hexagonal lattice, with a lattice constant of ∼131μm, by setting the frequency detunings to Δ1=−190MHz and Δ2=−100MHz, as simulated in Fig. 1(d). The probe beam was focused at the middle point between two lattice sites. Four vortices, composed of two vortex and anti-vortex pairs, appear at the corners of a slightly skewed rectangle, as shown in Figs. 5(e) and 5(f). This indicates that the generation of multiple vortices can still occur in topologically trivial structures, demonstrating that symmetry plays the main role instead of topology.

Thus one can conclude that Berry curvature sources do not play a key role for generation of vortices in momentum space in the present configuration, but the symmetry does. Fourier transform maps the symmetry of the WF in real space to momentum space. At the same time, the photonic lattices are a very flexible and simple way to implement symmetries that will be inherited by the evolving wave packet. In addition, the lattice defines the size of the wave packet in real space, and also consequently defines the profile of the wave packet in momentum space, including the positions of vortices. The particular symmetry of the WF as well as its radial distribution can be shaped by position and profile of the excitation beam. We benefit from advanced application of discrete nondiffracting beams34 by usage of real and momentum space equivalence and control of the signal symmetry. Even the simplest Gaussian beams carrying no OAM (cf. Refs. 19, 37, and 38) allow flexible engineering of the WF profile and realization of momentum space vortices with multiple configurations. Moreover, in contrast with, e.g., Refs. 20, 31, and 32, selective addressing sublattice and valley in the honeycomb lattice are also not needed. The described experimental scheme does not require structured light for probing and dramatically simplifies the techniques for creating vortices employing a honeycomb photonic lattice.

The symmetries were always understood as being absolutely fundamental in physics.39 While in previous decades the topological invariants have appeared in solid-state physics as something seeming to go beyond symmetries, recent works have started to reestablish their relation to symmetries via the so-called symmetry indicators,40^–43 allowing one to determine the topology from the eigenstates at high symmetry points. Our work makes an important step in this direction, demonstrating that formation of topological defects such as momentum space vortices in periodic potentials is not necessarily determined exclusively by topology of the lattice, while the lattice symmetries can be an origin of complex OAM textures.

Reciprocal space vortices and their pairs can be studied as a part of a global reciprocal space pattern, but they can also be used for reciprocal space engineering of beams for further real-space applications. Indeed, different wave vectors in the reciprocal space correspond to different propagation angles, and the parts of the beam described by different wave vector components split in space while propagating after the vapor cell. This allows one to select a certain region of the reciprocal space and filter away everything else, as has already been done in previous works.31^,33 For example, if three pairs of opposite-sign vortices are formed at the Dirac points (Fig. 2), one can select only a region of wave vectors (propagation angles) corresponding to a given Dirac point with a single vortex. The Fourier transform of this filtered beam from reciprocal to real space necessarily gives a vortex in real space (see, e.g., Ref. 44 as an example of vortex conservation in Fourier conversion), and such vortices have numerous applications,45 including data encoding,46 tractor beams,47 and holographic optical tweezers.45^,48 Generation of multiple vortices in real or momentum space is important for multiplexing49 and high-angular momentum states.50 Investigation of vortex pair dynamics is important for studies of quantum turbulence in the framework of point-like vortex gas models51^–55, and appropriate methods are required to create the vortex pairs to trace further evolution. The problem of initial excitation (quantum fluid “stirring”) is also important for the studies of quantum turbulence4^–6^,56, and the proposed scheme of simultaneous generation of multiple vortices can be beneficial for engineering the initial WF by vortex imprinting.57^,58 Finally, the obtained momentum space phase patterns can serve as starting point for spatial light modulator phase profiles to selectively address the photonic lattice sites in real space.59

In conclusion, we have shown the presence of vortices in momentum space aligned in the vicinity of Dirac points in photonic graphene based on the atomic vapor cell with spatially modulated susceptibility after the scattering of a narrow probe beam. This configuration is an analog of the wave packet evolution in honeycomb lattice in the framework of the Schrödinger equation with single-site initial excitation. In the honeycomb lattice, C3v symmetry of the wave packet supports, whereas C6v symmetry cancels, the presence of vortices near the Dirac points. The vortices can exist regardless the topological properties of the lattice. For C2v symmetry WF, vortices also form a rectangular pattern with a lower (mirror reflection) symmetry of the spreading wave packet, both in hexagonal and honeycomb lattices. These results show that the creation of multiple momentum space vortices is strongly linked with the wave-packet symmetry, but not necessarily with the topological nature of the lattice. The resulting pattern of vortices is determined by the product of the probe and the lattice symmetries. Our results provide a novel method to generate well-distinguished vortex arrays in the simplest manner that requires no structured probe light, and reveal the basic rules of how to achieve the desired spatial distribution of vortices by designing the initial symmetry of the lattice and of the probe beam.

Feng Li is a professor at Xi’an Jiaotong University, China. He received the bachelor’s and master’s degrees at Tianjin University, China, in 2006 and 2008, respectively. He obtained his PhD at CNRS and the University of Nice Sophia Antipolis, France, in 2013, supported by the European Marie Curie ITN project CLERMONT4. Subsequently, he worked as a research associate at the University of Sheffield, United Kingdom, from January 2014 to May 2017. He joined Xi’an Jiaotong University as a professor in June 2017, with a main research interest in light–matter interaction in microcavities and micro/nanostructures.

Sergei V. Koniakhin received his BS and MS degrees in Peter the Great St.Petersburg Polytechnic University in 2012 and worked as a junior researcher at Ioffe Institute, St.Petersburg, Russia from 2010 to 2015; he defended his PhD thesis in 2016. From 2015 to 2021, he worked as a researcher and senior researcher in Zhores Alferov University, St. Petersburg. From 2017 to 2020, he was a PhD student at Institut Pascal, Clermont-Ferrand, France. He became a Young Scientist Fellowship Team Leader at the Physics of Complex Systems Center, Institute for Basic Science, Daejeon, Republic of Korea starting in 2021.

Anton V. Nalitov obtained his master’s degree in condensed matter physics from Saint-Petersburg Academic (Zhores Alferov) University, Russia, in 2012, and his PhD from Université Blaise Pascal, Clermont-Ferrand, France, in 2015. His professional path includes Ioffe Institute, University of Southampton, University of Iceland, and University of Wolverhampton. Currently, he is a senior research assistant at Abrikosov Center for Theoretical Physics, Moscow Institute of Physics and Technology.

Evgeniia Cherotchenko obtained her master’s degree from Peter the Great St.Petersburg Polytechnic University in 2012 and her PhD from the University of Southampton in 2017. Her research experience includes working in ITMO University and currently she conducts her research in Ioffe Institute of the Russian Academy of Sciences.

Dmitry D. Solnyshkov is a professor at the University Clermont Auvergne, France. He wrote his PhD thesis in France, in 2007, became an assistant professor in 2008, and obtained his HDR (second thesis) in 2012, to become a full professor in 2019. He was a junior member of Institut Universitaire de France in 2017 to 2022. He is a theoretician working on Bose–Einstein condensation, quantum fluids, strong light–matter coupling, and topology.

Guillaume Malpuech is CNRS senior researcher at the Institut Pascal in Clermont Ferrand, France. He received his PhD in 2001 from the University Blaise Pascal and became CNRS researcher at the Institut Pascal in 2002. He is a theoretician working on exciton-polaritons, quantum fluids of light, and topological photonics. His major contributions are the proposals of room temperature polariton lasers, spin-orbit coupling of light based on TE-TM splitting, polariton Chern insulators, topological lasers, and quantum geometric tensor measurement.

Zhaoyang Zhang received his doctoral degree in electronic science and technology from Xi’an Jiaotong University. He is currently a professor at the School of Electronic Science and Engineering at Xi’an Jiaotong University. His research focuses mainly on atomic physics and nonlinear optics.

Biographies of the other authors are not available.