## Abstract

Artificial lattices of coherently coupled macroscopic states are at the heart of applications ranging from solving hard combinatorial optimization problems to simulating complex many-body physical systems. The size and complexity of the problems scale with the extent of coherence across the lattice. Although the fundamental limit of spatial coherence depends on the nature of the couplings and lattice parameters, it is usually engineering constraints that define the size of the system. Here, we engineer polariton condensate lattices with active control on the spatial arrangement and condensate density that results in near-diffraction limited emission, and spatial coherence that exceeds by nearly two orders of magnitude the size of each individual condensate. We use these advancements to unravel the dependence of spatial correlations between polariton condensates on the lattice geometry.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. INTRODUCTION

Synchronization and the emergence of coherence between coupled elements are universal concepts arising in nature and technology [1]. They dictate collective human behavior [2] and the functioning of neurological systems [3], as well as phase transitions of quantum systems to macroscopically collective entities at low temperatures [4]. For small-sized systems such as two mechanical pendulums coupled through a common support [5] or two coupled laser cavities [6], frequency locking of the two elements depends on the inter-element coupling strength in competition with any inherent dephasing mechanisms. In larger systems consisting of many interacting elements, such as social structures or coupled laser networks, the underlying coupling topology (or network architecture) critically influences the dynamics and coherence formed in these systems [7,8]. Engineering spatial coherence in such large networks is a key element for increasing system performances in power grids [9], novel computational devices for classification tasks [10], or laser arrays for the creation and control of high-power beams [11].

Lattices of coupled condensates are investigated for the simulation and computation of complex tasks in atomic [12–14], photonic [15], and polaritonic [16–18] platforms. Functionality and computational performance of these systems is ultimately limited by the system’s spatial coherence length; i.e., how many condensates can coherently be coupled. From a more fundamental point of view, increasing the coherence allows one to construct a larger many-body system and approach the ideal limit of a homogeneous “infinite” order parameter, which is essential for the study of phase transitions in interacting bosonic systems such as the Berezinskii–Kosterlitz–Thouless transition in two-dimensions (2D) [19].

Macroscopically coherent systems of exciton-polaritons, hybrid light–matter quasiparticles [20], bring together the speed and optical control of photonic platforms and the nonlinearities found in strongly interacting matter systems. As such, polariton lattices offer all-optical control over the reconfigurable non-Hermitian potential landscape and the in-plane particle current provided by optical imprinting of excitonic reservoirs [21–25]. In the case of polariton condensates in semiconductor microcavites, measurements were initially limited to small condensate sizes [26–29] but advancements in sample fabrication and experimental techniques now allow studies on their coherence properties as extended objects far away from the excitation area [21,30], in optical traps [31,32] and photonic microstructures [33]. The emergence of correlations beyond the spatial extension of the laser excitation area has underpinned the creation of networks and lattices of polariton condensates.

Here, we engineer control over the particle density and position of each polariton condensate across a polariton lattice under nonresonant optical excitation. We overcome the challenge of disorder-induced localization and dephasing by employing a feedback scheme for each individual laser spot, balancing the condensate density and allowing us to accurately study the decay of coherence across different coupling topologies. The result is a homogeneous macroscopic, high-energy, condensate lattice with coherence length exceeding multiple lattice cells and near-diffraction-limited emission. Generating such large condensate lattices by using multiple excitation sources allows us to go beyond the standard single-excitation source limit [21,30] overcoming beam size limitations, beam profile inhomogeneities, and condensate fragmentation [20]. We observe that the connectivity of the lattice significantly enhances the system’s coherence length due to increased coherent coupling between the adjacent condensates.

## 2. RESULTS

We optically generate lattices of coupled polariton condensates using nonresonant, pulsed and tightly focused laser excitation spots for each condensate node (see Supplement 1 Section S1 for methods). Exemplary plots of optically engineered homogeneous lattices in square and triangular configurations are illustrated in Fig. 1(a). Polaritons generated at each condensate node convert inherited potential energy into kinetic energy resulting in a radially (ballistically) expanding polariton fluid from an antenna-like source [34,35]. When two or more expanding condensates are brought together, interference effects are revealed, implying phase synchronization between the condensation centers and the emergence of a macroscopic order parameter [17,22,23,35–37]. The striking homogeneity and large visibility of interference fringes in our optically stabilized polariton lattices shown in Fig. 1(a) indicate long-range coherence across the condensate systems.

The spatial coherence in a network (or lattice) of $N$ polariton condensates generated under pulsed excitation is described by the integrated complex coherence factor (see Supplement 1 Section S2),

Here, the correlation function ${\Gamma _{\textit{ij}}}(t) = \langle {{\psi _i}{{(t)}^*}{\psi _j}(t)} \rangle$ denotes the mutual intensity of each pair of condensates averaged over many realizations (pulses) of the system, and ${\psi _i}(t)$ is the complex-valued amplitude of the $i$th condensate. While the modulus of the complex coherence factor $|{\tilde\mu_{\textit{ij}}}| \le 1$ is a normalized measure for the coherence between two condensate nodes, its argument ${\tilde \theta _{\textit{ij}}} = \arg ({\tilde\mu_{\textit{ij}}})$ represents their average phase difference.

#### A. Condensate Density Stabilization

We use a reflective liquid-crystal phase-only spatial light modulator (SLM) to modulate the Gaussian excitation pump beam in the Fourier plane of the optical excitation system to generate desired excitation pump spot geometries at the focal plane of the microscope objective lens [see schematic in Fig. 1(b)]. Phase holograms are calculated using a modified version of the well-known Gerchberg–Saxton (GS) algorithm [38], in which the measured polariton photoluminescence (PL) of the condensate lattice feeds back into the closed-loop sequence (see Supplement 1 Section S3 for a full description). Iterative nonlinear adjustment of the excitation pump profile allows us to create macroscopic lattices of ${\gt}100$ condensates with homogeneous condensate node density [$\le 1\%$ relative standard deviation (RSD)] across the whole lattice.

In Fig. 1(c), we compare the measured emission of all condensate nodes in a triangular lattice of 61 elements and lattice constant $a = 14.9\;{\unicode{x00B5}\text{m}}$ when using 100 iterations of the conventional GS algorithm (blue squares) and the presented condensate density stabilization method (red circles), respectively. Unavoidable effects such as sample disorder, finite accuracy of the GS algorithm, optical aberrations, and missing translational invariance due the finite size of the lattice [39] all contribute to a broad distribution of effective gain for each condensate node and, thus, to a large spread $\approx 37\%$ (RSD) in the distribution of condensate emission powers for the case of no condensate density stabilization. Active stabilization of the lattice using the described closed-loop sequence allows us to compensate for these detrimental elements and yield a reduced spread $\approx 1\%$ (RSD), which is limited by experimental noise in the system.

The near-field (or real space) polariton photoluminescence (PL) measured at pump power $P = 1.2{P_{\text{thr}}}$, where ${P_{\text{thr}}}$ is the system’s condensation threshold pump level, is shown in Figs. 1(d) and (e) for both excitation schemes. The presence of interference fringes with large visibility in-between ballistically expanding condensates demonstrates mutual coherence between nearest neighbor condensates. However, the lack of homogeneity in the spatial distribution of interference fringes in Fig. 1(d) for the case of no density stabilization indicates a broad distribution of relative phase difference ${\theta _{\textit{ij}}}$ between nodes. This is further confirmed by detailed measurements of the integrated complex coherence factor ${\tilde\mu_{\textit{ij}}}$ (see Supplement 1 Sections S4 and S5 for experimental details). In Figs. 1(f) and 1(g), we plot magnitude $|{\tilde\mu_{1j}}|$ and phase ${\tilde \theta _{1j}}$ between the central condensate node 1 and each other condensate node $j$ using false-color and pseudo-spins (black arrows). We find an enhanced and isotropic spatial decay of coherence $|{\tilde\mu_{1j}}|$ and larger homogeneity in relative phase differences ${\tilde \theta _{1j}}$ for the density-stabilized polariton lattice. The noticeable increase of phase differences ${\tilde \theta _{1j}}$––or analogous rotation of pseudo-spins—toward the edges of the lattice is an expected finite-size effect due to a flux of particles escaping the system [39]. In the ideal scenario of an infinite triangular lattice, with homogeneous condensate occupation numbers, one retrieves a homogeneous distribution of phase differences due to the system’s translational invariance (see Supplement 1 Section S6).

The polariton condensate lattice is further probed by far-field (or reciprocal space) measurements analogous to time-of-flight experiments in cold atom systems [19] (see Supplement 1 Section S4 for experimental methods). A zoom into the first Brillouin zones of the recorded far-field emission pattern is shown in Figs. 2(a) and 2(b) for the two lattice realizations with and without node density stabilization pumped at $P = 1.2{P_{\text{thr}}}$. For comparison, we illustrate in Fig. 2(c) the calculated far-field emission of 61 superimposed fully coherent point sources (wavefunctions) in triangular arrangement, which shows good agreement with the emission pattern of the stabilized condensate lattice. We compare the width of the central far-field emission peak at $k = 0$ for the three cases shown in Figs. 2(a)–2(c) by plotting the corresponding extracted intensity profiles along ${k_x} = 0$ in Fig. 2(d). The lobe of the calculated fully coherent system (black line) represents the diffraction-limited interference peak. We extract the central lobe’s full width at half-maximum (FWHM) for varying pump power $P$, as shown in Fig. 2(e), and find near-diffraction-limited far-field emission for the density-stabilized lattice (red dash-dotted line); i.e., a minimum width at $P = 1.17{P_{\text{thr}}}$ that is only $\approx 13\%$ larger than the diffraction limit of a fully coherent system (black horizontal line). Without active condensate density stabilization (blue dashed line) the peak width increases to $\approx 47\%$ at the same pump power. The observed broadening of the far-field emission peak in the physical system is a result of both reduced coherence $|{\tilde\mu_{\textit{ij}}}|$ and nonhomogeneous phase distribution ${\tilde\mu_{\textit{ij}}}$. The system’s pump-power dependencies are summarized in Supplement 1 Section S7.

#### B. Coherence Versus Dimensionality

In this section we begin by investigating the coherence between two ballistic polariton condensates. Previous studies of this system have revealed periodically alternating synchronization patterns of in-phase (${\tilde \theta _{12}} = 0$) and antiphase (${\tilde \theta _{12}} = \pi$) states with an increasing condensate separation distance ${d_{12}}$ [35,36]. While separation distances exist at which the emission of the coupled condensate system is not single mode but exhibits two or more modes of both even and odd parity states [35,36], in the following we only focus on single-mode realiszations. Under this condition, the integrated complex coherence factor ${\tilde\mu_{12}}$ is a measure for the system’s average coherence properties. In Supplement 1 Section S8, we detail on the time-resolved coherence build-up ${\mu _{12}}(t)$ of a polariton dyad.

Recorded near-field and far-field PL of two condensates separated by ${d_{12}} = 12.7\;{\unicode{x00B5}\text{m}}$ and pumped equally at $P = 1.2{P_{\text{thr}}}$ are shown in Figs. 3(a) and 3(b). The interference patterns in both emission images reveal antiphase synchronization between the two condensates (${\tilde \theta _{12}} = \pi$) due to the even number of interference fringes. Far-field emission consists predominantly of PL at large in-plane wavevector $k = 1.8\;{{\unicode{x00B5}\text{m}}^{- 1}}$, demonstrating the ballistic expansion of both condensates due to the strong repulsive interactions between polaritons and the pump-induced exciton reservoirs localized at each condensate center [40]. In Fig. 3(c), we show the recorded far-field interference pattern when spatially filtering the central $2\;{\unicode{x00B5}\text{m}}$ FWHM of each condensate. Analysis of this double-hole interference pattern reveals a large coherence factor $|{\tilde\mu_{12}}| = 0.9$. Increasing the separation distance ${d_{12}}$ while keeping the pump power $P$ constant leads to a decay of coherence $|{\tilde\mu_{12}}|$ between the two condensate centres such that at ${d_{12}} = 89.3\;{\unicode{x00B5}\text{m}}$ no interference pattern can be found in near-field and far-field emissions, as shown in Figs. 3(d)–3(f).

The dependency of the measured coherence $|{\tilde\mu_{12}}|$ on the pump spot separation distance ${d_{12}}$ is shown in Fig. 3(g) as blue circles and is fitted with a Gaussian decay,

The fit parameter ${L_c} = 24.9 \pm 0.4\;{\unicode{x00B5}\text{m}}$ represents a measure for the length over which synchronization of the two-condensate system is possible, and we denote it as the effective coherence length,

We reproduce the experimentally measured decay of mutual coherence between two coupled polariton condensates through 2D numerical simulation using the generalized stochastic Gross–Pitaevskii equation (GPE) shown in Fig. 3(g) as yellow squares (see methods in Supplement 1 Section S1). The dependency of coherence $|{\tilde\mu_{12}}|$ as a function of pump power $P$ between two coupled condensates with a separation distance ${d_{12}} = 12.7\;{\unicode{x00B5}\text{m}}$ is depicted in the inset of Fig. 3(g) for both experimental (blue circles) and numerical simulation (yellow squares). An increase of coherence between the two condensate nodes arising at the condensation threshold is followed by a drop of $|{\tilde\mu_{12}}|$ for larger pump power $P \gt 1.2{P_{\text{thr}}}$. This decrease in $|{\tilde\mu_{12}}|$ is largely attributed to the transition of the system into multimode operation [36], yielding reduced visibility in time-integrated measurements. We point out that the system realizations that are shown in the distance-dependence in Fig. 3(g) exhibit only single-mode emissions. The decrease in coherence with an increasing separation distance is caused by the spatial decay of the wavefunction of ballistically expanding polariton condensates [35]; i.e., a reduced coupling strength between the two condensate nodes. In Supplement 1 Section S9, we compare our results to a dyad under continuous wave excitation.

Next, we increase the number of condensates by investigating a linear chain of 11 equally spaced condensates with nearest-neighbors (NN) distance $a = 12.1\;{\unicode{x00B5}\text{m}}$. The resultant interference patterns in both real and reciprocal space shown in Figs. 4(a) and 4(b) indicate antiphase synchronization; i.e., phase differences between any pair $\{i,j\}$ of condensates according to

with a condensate indexing shown in Fig. 4(a). Such order can be said to be “antiferromagnetic.” It differs from the in-phase synchronization, or “ferromagnetic” order, shown for the triangular lattice in Fig. 1(g) due to the difference in lattice constant [17].We measure and analyze the far-field interference between each pair of condensates $i \ne j$. Extracted magnitude $|{\tilde\mu_{\textit{ij}}}|$ and phase ${\tilde \theta _{\textit{ij}}}$ of the integrated complex coherence factor are illustrated in matrix form in Fig. 4(c) with row and column indices $i,j$ denoting the pair of condensates, where we make use of the hermiticity of the correlation matrix ${\tilde\mu_{\textit{ij}}} = \tilde\mu_{\textit{ji}}^*$. In agreement with Eq. (3) we confirm antiphase synchronization between NNs (antiparallel pseudo-spins). For large condensate pair distances, $|i - j| \gg 1$, the coherence factor ${\tilde\mu_{\textit{ij}}}$ decays in magnitude and its phase deviates from Eq. (3), indicating a loss of long-range antiferromagnetic order. In Fig. 4(d), we show the decay of coherence $|{\tilde\mu_{\textit{ij}}}|$ in the chain as a function of condensate spacing ${d_{\textit{ij}}}$ (i.e., the lattice constant $a$ is fixed). The data are fitted with a single exponential decay (red curve),

which yields a coherence length ${L_c} = 35 \pm 1\;{\unicode{x00B5}\text{m}}$.In the next step, we increase the system size to a 2D square lattice comprising 121 condensates with a lattice constant $a = 12.1\;{\unicode{x00B5}\text{m}}$. The interference patterns in near-field and far-field PL that are shown in Figs. 5(a) and 5(b) reveal antiphase synchronization analogous to the 1D system in Fig. 4. We measure ${\tilde\mu_{1j}}$ between the central condensate (index 1) and all other condensate nodes $j = 2, \ldots ,121$ plotted in Fig. 5(c). The observed antiferromagnetic order (${\tilde\mu_{1j}} \approx \pm 1$) reduces with distance from the center seen from the pseudo-spin rotation toward the edges in Fig. 5(c) due to flux of particles escaping the lattice. However, coherence $|{\tilde\mu_{1j}}|$ between the most central condensate and any other condensate of the lattice does not drop below 0.3 for separation distances as large as ${d_{1j}} = 86\;{\unicode{x00B5}\text{m}}$ toward the corners of the lattice. The spatial decay of coherence $|{\tilde\mu_{1j}}|$ is shown in Fig. 5(d) and fitted with an exponential function [Eq. (4), red curve] with a coherence length ${L_c} = 87 \pm 1\;{\unicode{x00B5}\text{m}}$.

To compare the obtained results, we summarize in Fig. 6(a) the spatial decay of coherence for four different types of networks: two coupled condensates, 1D chain, and 2D square and triangular lattices. For the lattices, the data points for different separation distances ${d_{\textit{ij}}}$ correspond to different pairs of condensate nodes $\{i,j\}$. For the polariton dyad, the physical separation distance between the two condensate nodes is changed. In all cases, the pump power was chosen to maximize the coherence $|{\tilde\mu_{12}}|$ between a pair of NNs in the system. In Fig. 6(c) we illustrate the extracted (effective) coherence length ${L_c}$ versus the average number of NNs in each system. It is apparent that the coherence length ${L_c}$ is increasing for systems with larger connectivity, which we explain in the following. Interestingly, the mutual coherence between NNs in the 1D chain (${d_{\textit{ij}}} = a$) is noticeably lower than in the other systems. Perhaps most surprisingly, the chain’s coherence is lower than the dyad for small ${d_{\textit{ij}}}$ and then overtakes it around $d\gtrsim 25\;{\unicode{x00B5}\text{m}}$. The qualitative difference between the observed decay of coherence in the dyad (Gaussian) compared to the lattice systems (exponential) can possibly be attributed to the fact that the lattices are characterized by Bloch waves moving from unit cell to unit cell, which can experience amplification from the pump spots. In the dyad, however, empty space separates the two condensates and the plane waves traveling between them decay quickly with an increasing separation distance.

We attribute the origin of increased spatial coherence for condensate networks with larger connectivity to their reduced population of reservoir excitons, as we argue in the following. In Fig. 6(b), we compare the dependence of coherence $|{\tilde\mu_{12}}|$ between a pair of NNs as a function of the average excitation pulse energy per condensate in each system. We observe a reduction of threshold pulse energy ${E_{\text{thr}}}$ per condensate node for networks with larger connectivity (larger number of NNs), which is shown in Fig. 6(d) and attributed to the increased gain given by ballistic exchange of particles between neighboring nodes. A lower pump energy generates a smaller reservoir of excitons for each condensate node and since reservoir-condensate interactions play a dominant role in condensate dephasing [41], a lower threshold pump energy will generally result in increased coherence properties.

By discretizing the system into a set of interacting condensates [35], one can show that threshold pump power and coherence length scale approximately linear with the number of condensate NNs. Let us consider the steady state (without phase frustration) of frequency $\nu$ such that the discretized Gross–Pitaevskii equation can be written as

The expected linear dependencies of both threshold pump power ${P_{\text{thr}}} \propto n_x^{\text{(thr)}}$ [Eq. (6)] and coherence length ${L_c}$ [Eq. (8)] are both indicated as gray lines in Figs. 6(c) and 6(d). We point out that similar results (increasing coherence with larger connectivity and dimensionality in networks of coupled elements) have been observed in other technological platforms such as in arrays of coupled VCSELs [43,44], micromechanical oscillator arrays [45], and for coupled fiber lasers [46].

It is instructive to investigate the condensate’s dispersion of elementary excitations (fluctuations), which directly relates to the behavior of space–time correlations in the system. These are also known as Lyapunov exponents in stability analysis and characterize the stability of stationary, orbital, and chaotic solutions of nonlinear differential equations [47]. For simplicity, we will focus on the case of a 2D square lattice like what is displayed in Fig. 5 in a steady state (continuous wave excitation). For our calculation of the Lyapunov exponents we will assume that we are working in the bulk of the condensate and therefore the system is taken to have discrete translational invariance. This allows us to apply Bloch’s theorem to the standard Bogoliubov treatment and solve the Lyapunov exponents in the reduced Brillouin zone of the condensate pump lattice (see Supplement 1 Section S10).

In Figs. 6(e) and 6(f) we plot the Lyapunov exponents around the $\Gamma$-point of the reduced Brillouin zone for two different numbers of particles in the condensate. Yellow and black colored curves correspond to $N \approx 100$ and $\approx 800$ particles, respectively, in the condensate unit cell for a lattice constant of $a = 12\;{\unicode{x00B5}\text{m}}$. The spontaneously broken gauge symmetry of the condensate results in a gapless spectrum $\lambda (0) = 0$, but possesses some qualitative differences to that of thermodynamic equilibrium condensates that possess a phonon-like dispersion $\text{Im}(\lambda)$ at low momenta [48]. Instead, $\text{Im}(\lambda)$ (which corresponds to the oscillatory evolution of the fluctuations) shows a purely diffusive branch $\text{Im}(\lambda) = 0$ at low momenta. This arises due to the openness, or the dissipative character, of our condensate, which modifies the standard Bogoliubov dispersion of elementary excitations, leading to a bifurcation point separating diffusive and dispersive regimes [49] and smoothing the superfluid transition around the Landau critical velocity. It is a general result of open condensed systems and not exclusive to polaritons.

$\text{Re}(\lambda)$ corresponds to the decay (growth) rate of fluctuations (excitations) when negative (positive) valued. In the top branch of the pitchfork displayed in Fig. 6(f), we observe that $\text{Re}(\lambda)$ is more negative for small wavevectors as more particles are in the condensate (black curve). This is in agreement with our experiment where the coherence increases with the pump power (particle number grows and decay of fluctuations increases) up to the point where single-mode (stationary) behavior is lost. Our results are similar to those obtained for spatially uniform systems [42,49,50] underlining that spatial details of the condensate structure are not relevant to long-wavelength fluctuations.

## 3. DISCUSSION

We have demonstrated that the coherence properties in lattices of polariton condensates are enhanced by balancing the condensate emission across the system using a closed-loop feedback scheme to adjust the excitation pump geometry. This scheme reduces the effects of optical aberrations in the experimental system, as well as counteracts mode localization due to sample nonuniformities [51] and nonhomogeneous gain distribution across the coupled condensate network. While actively controlling the condensate lattice uniformity we have accurately determined phase and coherence between any pairs of condensates in different types of networks ranging from two-condensate systems to 1D and 2D periodic structures. The dynamics and synchronization of coupled nonlinear elements is critically influenced by the underlying coupling topology [7]. Here, we have shown that an increase in connectivity (i.e., the number of NNs), significantly reduces the operational pump power per node and increases the coherence length demonstrating a promising route to polaritonic devices with networks of many coupled condensates with potential application in simulators and optical-based computation. The presented measurements and techniques provide a deeper understanding of the coherence properties of coupled light–matter wave fluids in low-dimensional quantum systems, and qualify for other open (dissipative) networks such as laser arrays and photon condensate lattices. Furthermore, our analysis of the lattice condensate fluctuations reveals similar long-wavelength dispersions to those of uniform systems, which are regarded as the ideal case from a theoretical point of view. This suggests access to fundamental long-wavelength physics belonging to uniform systems by designing instead an extended condensate lattice.

## Funding

RFBR (jointly with DFG) (20-52-12026); Russian Foundation for Basic Research (20-02-00919); PAPIIT-UNAM (IN106320); Consejo Nacional de Ciencia y Tecnología (251808); Engineering and Physical Sciences Research Council (EP/M025330/1).

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

The data that support the findings of this study are openly available from Ref. [52].

See Supplement 1 for methods and supporting content.

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