Abstract
Nonlinearity and finite signal propagation speeds are omnipresent in nature, technologies, and realworld problems, where efficient ways of describing and predicting the effects of these elements are in high demand. Advances in engineering condensed matter systems, such as lattices of trapped condensates, have enabled studies on nonlinear effects in manybody systems where exchange of particles between lattice nodes is effectively instantaneous. Here, we demonstrate a regime of macroscopic matterwave systems, in which ballistically expanding condensates of microcavity excitonpolaritons act as picosecond, microscale nonlinear oscillators subject to timedelayed interaction. The ease of optical control and readout of polariton condensates enables us to explore the phase space of two interacting condensates up to macroscopic distances highlighting its potential in extended configurations. We demonstrate deterministic tuning of the coupledcondensate system between fixed point and limit cycle regimes, which is fully reproduced by timedelayed coupled equations of motion similar to the LangKobayashi equation.
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Introduction
Time delay is widespread in nature and occurs when the constituents of a given system interact via signals with a finite propagation time^{1}. If the characteristic timescale of the system, such as the period of a simple pendulum, is much longer than the signal propagation time, then one arrives at familiar examples such as Huygens clock synchronisation described by instantaneous interactions^{2}. When propagation times are appreciably long, the role of the system’s history is enhanced and the interactions are said to be time delayed. Such systems dictate the neurological function of our brains, affect traffic flow, influence economic activities, define population dynamics of biological species, regulate physiological systems, determine the stability of lasers, and have application in control engineering^{3,4}. Besides their ubiquity in nature and science, coupled systems with continuous timedelayed interactions exhibit interesting mathematical properties such as an infinite dimensional state space, i.e. for a fixed timedelay \(\tau\) there are infinitely many initial conditions of the system in the time interval \(\tau \le t\le 0\) needed to predict the dynamics for \(t \, > \, 0\) (ref. ^{5}). Timedelayed interaction or selffeedback is known to greatly increase the dynamical complexity of a system, giving rise to chaotic motion, chimera states^{6}, as well as being able to both stabilise and destabilise fixed point and periodic orbit solutions^{4}.
Technological applications of delaycoupled systems appear in diverse areas such as control engineering^{7}, high speed randombit generation^{8}, secure chaos communication^{9}, but have also recently emerged in machine learning, where the demand for neuroinspired computing units has led to various hardware realisations of artificial neural networks^{10}. The intrinsic highdimensional state space of delaycoupled systems enables an efficient platform for tasks such as pattern recognition, speech recognition, and time series prediction as demonstrated in electronic^{11}, photonic^{12,13,14}, and optoelectronic^{15,16,17} systems.
In this work, we demonstrate the prospect of an ultrafast microscale platform for engineering timedelayed coupled oscillator networks based on condensates of microcavity exciton–polaritons (from here on polaritons)^{18,19}. Polaritons are bosonic quasiparticles that can undergo a powerdriven quantum phase transition to a macroscopically occupied state with longrange phase coherence^{20}. This phase transition is associated with the formation of a polariton condensate, a nonlinear matterwave quantum fluid, which differs from classical cold atom Bose–Einstein condensates, and liquid light droplets^{21,22} due to its nonequilibrium nature. One of the greatest advantages of polariton condensates for optoelectronic applications is the easy implementation of arbitrary geometries, or graphs, using adaptive optical elements for the excitation laser beam^{23}. We show controllable tuning between steady state (singlecolour) and limitcycle (twocolour) regimes of the two coupled condensate system and explain our observations through timedelayed equations of motion.
Results
Timedelayed coupled matterwave condensates
Conventionally, networks and lattices of condensates consisting of cold atoms^{24}, Cooper pairs^{25}, photons^{26}, or polaritons^{27,28} are studied in trap geometries, weakly coupled via tunnelling currents, that allow for the study of solid state physics phenomena such as superfluidMott phase transitions^{29}, magnetic frustration^{30}, \(PT\)symmetric nonlinear optics^{31}, and Josephson physics^{28,32}. Here we investigate the inverse case, wherein polariton condensates are freely expanding from small (pointlike) sources experiencing dynamics reminiscent to macroscopic systems such as timedelayed coupled semiconductor lasers^{33}. Ballistic expansion extends over two orders of magnitude beyond the pump beam waist, and occurs due to the repulsive potential formed by the uncondensed exciton reservoir injected by the nonresonant pump^{34}. In the case of spatially separated polariton condensates, propagation from one condensate centre to another results in a substantial phase accumulation, interpreted as a retardation of information flow between the condensates. To date, coupled polariton condensates were restricted to distances \(d \, \lesssim \, 40\ \upmu {\rm{m}}\)^{23,35,36}, wherein coupling was assumed to be instantaneous although propagation time was comparable or longer than the characteristic timescale of the systems. To unravel the role of timedelayed interactions, we demonstrate the synchronisation between two tightly pumped condensates separated by up to \(d\approx 114\, \upmu {\rm{m}}\), as shown in Fig. 1a. Figure 1b, c compares schematically the conventional regime of coupled condensates separated by a potential barrier and described by a tunnelling current \(J\), to the macroscopically coupled drivendissipative matterwave condensates interacting via radiative particle transfer subject to finite propagation time \(\tau\).
Dynamics of two interacting condensates
We utilise a strain compensated semiconductor microcavity to enable uninhibited ballistic expansion of polaritons over macroscopic distances^{37}. We inject a polariton dyad with identical Gaussian spatial profiles, with fullwidthathalfmaximum (FWHM) of \(\approx \!2\, \upmu {\rm{m}}\), using nonresonant optical excitation at the first Bragg minima of the reflectivity stopband, and employ spatial light modulation to continuously vary the separation distance from \(6\) to \(94\, \upmu {\rm{m}}\), while keeping constant the excitation density of each pump beam at \({P}_{1,2}\approx 1.5\times {P}_{{\rm{thr}}}^{(1)}\), where \({P}_{{\rm{thr}}}^{(1)}\) is the condensation threshold power of an isolated condensate. For each separation distance \(d\) (more than 400 positions), we record simultaneously the spatial profile of the photoluminescence (realspace), the dispersion (energy vs inplane momentum in the direction of propagation), and the photoluminescence in reciprocal (Fourier) space (see Supplementary Movie 1). We observe that opposite to the twofold hybridisation of two evanescently coupled condensates^{28}, the macroscopically coupled system is characterised by a multitude of accessible modes of even and odd parity (i.e., 0 and \(\uppi\) phase difference between condensate centres) that alternate continuously between opposite parity states with increasing dyad separation distance. For a range of separation distances only one resonant mode is present in the gain region of the dyad, wherein the polariton dyad is stationary, occupying a single energy level. Between the separation distances, wherein only one mode is present, we observe the coexistence of two resonant modes of opposite parity resulting in nonstationary periodic states.
In Fig. 2 we present an example of the two different regimes, stationary and nonstationary states. Figure 2a, b shows realspace photoluminescence, Fig. 2c, d shows Fourierspace photoluminescence and Fig. 2e, f shows the polariton dispersion along the axis of the dyad with separation distances of (a, c, e) \(d=12.7\, \upmu {\rm{m}}\) and (b, d, f) \(d=37.3\ \upmu {\rm{m}}\). Figure 2g depicts the integrated spectra of Fig. 2e, f using black dots and red squares, respectively. Absolute values of the energy levels are given as a blueshift with respect to the groundstate of the lower polariton branch. Although from the clear fringe pattern in Fig. 2a one could infer that only one mode of odd parity is present, it is through the simultaneous recording of either the Fourierspace or the dispersion that the coexistence of an even parity mode becomes apparent. Each of the two modes has a well defined but opposite parity to the other. We note that as long as we maintain the mirror symmetry of the system by pumping both of the two condensate centres with the same power \({P}_{1}={P}_{2}\), we do not observe formation of nontrivial phase configurations \({\phi }_{1}{\phi }_{2} \, \ne \, 0,\pi\).
To unravel the dynamics of the coupling on the separation distance between the two condensates, we obtain the spectral position, parity, and spectral weight of both energy levels for pump spot separation distances from \(d=5\, \upmu {\rm{m}}\) to \(d=66\ \upmu {\rm{m}}\). We have recorded configurations with more than two occupied energy levels for several distances \(d\), but with the relative spectral weight of the third peak less than a few percent. In the following, we focus our analysis to the two brightest energy levels for each configuration. The measured normalised spectral weights and spectral positions of the two states versus pump spot separation distance \(d\) are illustrated in Fig. 3a, b using black dots for even parity states and red hollow squares for odd parity states, respectively. Figure 3a shows that the system follows an oscillatory behaviour in the spectral weights of the two parity states giving rise to continuous transitions between even and odd parity states interleaved by configurations exhibiting both parity states. Interestingly, each period of these oscillations in relative intensity (starting and ending with a vanishing spectral weight) displays an ‘energy branch’ featuring a notable reduction in energy with increasing pump spot separation distance \(d\) as is shown in Fig. 3b. The energy of an isolated condensate is measured in the same sample space and its value \(\approx {\!}2.22\ {\rm{meV}}\), above the ground state energy of the lower polariton dispersion, is illustrated with a blue dashed horizontal line. We observe that the phasecoupling of two spatially separated condensate centres is dominated by a spectral redshift with respect to the energy of an isolated condensate. The spectral size of each ‘energy branch’, i.e., the measurable redshift, is decaying from branchtobranch with increasing pump spot separation distance \(d\). The two dyad configurations exhibiting singlecolour and twocolour states shown in Fig. 2 are indicated with grey vertical dashed lines in Fig. 3. Interestingly, it has been shown that the observation of phaseflip transitions, accompanied with changes in oscillation frequency to another mode is a universal characteristic of timedelayed coupled nonlinear systems^{5,38}. Such dynamics can be associated with neuronal systems^{39,40,41}, and coupled semiconductor lasers^{33,42}, but have also been demonstrated experimentally for other types of timedelayed coupled systems such as nonlinear electronic circuits^{38,43}, living organisms^{44}, chemical oscillators^{45}, and candleflame oscillators^{46}.
Time averaged measurements over \(\sim \!1000\) realisations of the system as presented in Fig. 2a, c, e cannot reveal whether the system of two coupled polariton condensates exhibits periodic dynamics in the form of twocolour states or whether it stochastically picks one of the two parity states. However, it is well known that the power spectrum of a timedependent signal is related to its autocorrelation function by means of Fouriertransform (Wiener–Khinchin theorem). A twocolour solution, as shown in Fig. 2g, results in the periodic disappearance and revival of the firstorder temporal coherence function \({g}^{(1)}(\bar{\tau })\). To characterise the temporal evolution of \({g}^{(1)}(\bar{\tau })\), we perform interferometric measurements of the two coupled condensates and extract the fringe visibility \({\mathcal{V}}\) which—up to a normalisation factor—is proportional to the magnitude of the coherence function \(\left{g}^{(1)}\right\). Assuming two coexisting states of opposite parity but equal spectral weight, one may simplify the complex amplitudes of the two coupled condensates \({\Psi }_{1}\) and \({\Psi }_{2}\) as
with timeindependent complex amplitude \({\psi }_{0}\) and relative phase \(\phi\) between the two modes of opposite parity and energy \({\mu}_{{\rm{e}}},\, {\mu }_{{\rm{o}}}\) for the even and odd mode respectively. The mixture of two modes, with energy splitting \(\hslash \Delta =\left{\mu }_{{\rm{e}}}{\mu }_{{\rm{o}}}\right\), causes an antiphase temporal beating in the intensities \({\left{\Psi }_{1,2}\right}^{2}\) similar to oscillatory population transfer in coupled bosonic Josephson junctions. Therefore, while the occupation amplitude of both condensate centres oscillates with a period \(T=2\pi/\Delta\), they feature a relative phase shift of \(\pi\) due to mixture of even and odd parity states. We use a Michelson interferometer, comprising a retroreflector mounted on a translational stage, and measure both the time averaged interference of the photoluminescence of the same condensate \({I}_{11}(\bar{\tau })=\langle{\left{\Psi }_{1}(t)+{\Psi }_{1}(t+\bar{\tau })\right}^{2}\rangle\) as well as the time averaged interference of the photoluminescence of opposite condensates \({I}_{12}(\bar{\tau })=\langle{\left{\Psi }_{1}(t)+{\Psi }_{2}(t+\bar{\tau })\right}^{2}\rangle\), where \(\bar{\tau }\) is the relative time delay controlled by the adjustable position of the retroreflector. Examples of the interferometric images are illustrated in Fig. 4a for two condensates with a separation distance of \(d=10.3\, \upmu{\rm{m}}\), which demonstrates condensation in both an even and an odd parity state as shown in Fig. 4b. Assuming the spectral composition as noted in Eqs. (1) and (2) the corresponding interferometric visibilities \({\mathcal{V}}\) can be written as
The experimentally extracted visibilities \({\mathcal{V}}_{11}(\bar{\tau })\) and \({\mathcal{V}}_{12}(\bar{\tau })\) versus relative time delay \(\bar{\tau }\) are illustrated in Fig. 4c with blue circles and red squares, respectively. In agreement with the predicted correlations [Eqs. (3) and (4)] we observe high fringe visibility \({\mathcal{V}}_{11}(0) \, > \, 0.8\) for the interference of one condensate with itself and almost vanishing fringe visibility \({\mathcal{V}}_{12}(0) \, < \, 0.1\) for the interference of opposite condensates at zero time delay. Furthermore, we find periodic disappearance and revival of fringe visibility with period \(T\approx 15.3\ {\rm{ps}}\), which is consistent with the observed energy splitting of the two modes \(\hslash \Delta =270\, \upmu {\rm{eV}}\) (see Fig. 4b). The relative phase shift between the two visibilities \({\mathcal{V}}_{11}(\bar{\tau })\) and \({\mathcal{V}}_{12}(\bar{\tau })\) is a direct result of the periodic population transfer between the two condensates. For comparison the inset in Fig. 4c shows the expected visibilities (Eqs. (3) and (4)) multiplied with an exponential decay accounting for the finite coherence time of the system.
In Fig. 4d we show the decay of temporal coherence for two coupled condensates in a singlecolour state (\(d=20\, \upmu {\rm{m}}\)), a twocolour state (\(d=20.5\, \upmu {\rm{m}}\)), and for an isolated condensate excited with the same pump power density. We note that the coherent exchange of particles in both regimes of the coupled condensate system results in an enhanced coherence time. For the singlecolour state (\(d=20\, \upmu {\rm{m}}\)) and the isolated condensate the coherence time \({\bar{\tau }}_{{\rm{c}}}\) can be extracted from exponential fits yielding \(25.5\) and \(10.2\ {\rm{ps}}\), respectively.
Numerical analysis
The dynamics of polariton condensates can be modelled via the mean field theory approach where the condensate order parameter \(\Psi ({\bf{r}},t)\) is described by a twodimensional (2D) semiclassical wave equation often referred as the generalised Gross–Pitaevskii equation coupled with an excitonic reservoir which feeds noncondensed particles to the condensate^{47}. In Supplementary Note 1 we give numerical results of the spatiotemporal dynamics of the dyad reproducing semiquantitatively the experimentally observed results. We also provide a supplemental animation showing the calculated evolution of the twocolour condensate (Supplementary Movie 2).
In the following, we show that our experimental observations are described as a system of timedelayed coupled nonlinear oscillators. For simplicity we consider the onedimensional (1D) Schrödinger equation corresponding to the problem of \(\mathrm s\)wave scattering of the condensate wavefunctions with \({\rm{\delta }}\)shaped complexvalued, pump induced, potentials. We start by characterising the energies of the timeindependent nonhermitian single particle problem:
where \(V(x)\) describes complexvalued \(\delta\)shaped potentials separated by a distance \(d\), \(V(x)={V}_{0}\left(\delta (x+d/2)+\delta (xd/2)\right)\), and \({V}_{0}\) lies in the first quadrant of the complex plane, i.e. repulsive interactions and gain. The eigenfunctions of Eq. (5) describing normalisable solutions of outwards propagating waves from the potential (nonresonant pump) centres are written as
where \(k\) also belongs to the first quadrant of the complex plane. This problem is well known for the case of lossless attractors (\(\Re ({V}_{0}) \, < \, 0\), \(\Im ({V}_{0})=0\)) describing electron states in a 1D diatomic Hydrogen molecule ion. The resonance condition of the system is written as
The solutions of Eq. (7) can explicitly be written as
with \(\tilde{V}=m{V}_{0}/{\hslash }^{2}\). Equation (8) describes infinitely many solutions of the system of even \((+)\) and odd \(()\) parity, where \({W}_{n}\) are the branches of the Lambert \(W\) function. The corresponding complexvalued eigenvalues,
are illustrated in Fig. 5 and qualitatively reproduce the experimental findings of the multiple energy branches shown in Fig. 3. We interpret the experimental occurrence of predominantly two lasing modes with the behaviour of the imaginary values of \({E}_{n}\) for the simplistic 1Dtoy model. While there are periodically alternating regions of even and oddparity solutions dominating the gain, we expect distances at which two modes (of opposite parity) have equal gain and thus can operate with equal intensity. It is worth noting that the Lambert \(W\) function naturally arises for problems involving delay differential equations^{48}.
The timedependent problem can be formulated as a superposition of two displaced, normalisable, and ballistically propagating waves \({\psi }_{1,2}(x)\), each emerging from one of the condensate centres,
Here, in analogy with Eq. (6), the normalised ansatz is written as \({\psi }_{1,2}(x)=\sqrt{\kappa }{\mathrm{e}}^{ik {x}\!\pm\!{d/2}}\), with a complexvalued wavevector \(k={k}_{{\rm{c}}}+i\kappa\). When the coupling between condensates is weak, i.e. small \(\xi =\exp(\kappa{d})\), one can omit all terms of order \({\mathcal{O}}({\xi }^{2})\) and higher. Then plugging Eq. (10) into the timedependent form of Eq. (5) and integrating out the spatial degrees of freedom, assuming that Eq. (7) is satisfied, one gets (see Supplementary Note 2),
where \(j=3i\) and \(i=1,2\) are the condensate indices. When setting \({c}_{i}=\pm \!{c}_{j}\), and solving for stationary states, the above equation recovers the exact resonant solutions dictated by Eq. (8). Equation (11) then shows that intercondensate interaction is in the form of a coherent influx of particles from condensate \(j\) onto the condensate centre of condensate \(i\) (and vice versa), with a phase retardation of \({k}_{{\rm{c}}}d\). When \({c}_{i}\) and \({c}_{j}\) oscillate at a fixed frequency \(\omega\) we can transform the phaseshifting term \(\exp (i{k}_{{\rm{c}}}d)\) into an effective time delay,
The time delay \(\tau\) corresponds to an interaction lag between condensate centres \(i\) and \(j\) caused by their spatial separation and is given by \(\tau ={k}_{{\rm{c}}}d/\omega\). In the case of weak coupling, i.e. small changes in oscillation frequency \(\omega\) compared to the frequency \({\omega }_{0}\) of a single unperturbed condensate, the timedelay is approximately proportional to the dyad separation distance \(d\), i.e. \(\tau \approx {k}_{{\rm{c}},0}d/{\omega }_{0}\) where we also use the notation \({k}_{0}={k}_{{\rm{c}},0}+i{\kappa }_{0}\) for the wavevector of the single unperturbed condensate. The exponentially decaying term on the righthand side of Eq. (12) accounts for the 1D spatial decay of particles propagating in between the two condensate centres. Introducing local nonlinear interactions, reservoir gain, and blueshift one can write the full nonlinear equation of motion for the two coupled condensates as^{47}
Here, \({n}_{i}\) correspond to the pump induced exciton reservoirs providing blushift and gain into their respective condensates, \(g\) is the polaritonreservoir interaction strength, \(R\) is the rate of stimulated scattering of polaritons into the condensate from the active reservoir, \(\alpha\) is the interaction strength of two polaritons in the condensate, \(v={\omega }_{0}/{k}_{{\rm{c}},0}\) is the phase velocity of the polariton wavefunction, and \({\Gamma }_{{\rm{A}}}\) is the radiative decay rate of the bottleneck reservoir excitons. The parameter \(\Omega ={\Omega }_{0}i\Gamma\) captures the selfenergy of each condensate which will, in general, have a contribution from a background of optically inactive dark excitons generated at the pump spot, and \(\Gamma\) denotes the effective linewidth of the polaritons expanding away from the pump spot. The complex valued coupling is written \(J\exp (i\beta )={V}_{0}\kappa \exp (\kappa d)\) for brevity. Equation (13) is then in the form of a discretised complex Gross–Pitaevskii equation but with timedelayed interaction between the bosonic particle ensembles: which greatly increases the dimensionality of phasespace and complexity of the coupled system. Our system then has strong similarity with the famous Lang–Kobayashi equation^{49,50} where in our case each condensate acts as a radiating antenna of symmetrically expanding waves. The two spatially separated antennas interfere and maximise their gain by adjusting both their common frequency and their relative phase difference \(\phi =\arg ({c}_{i}^{* }{c}_{j})\). Similarities of the dynamics of coupled polariton condensates to equations of motion in the form of the Lang–Kobayashi equation was discussed in theoretical works recently for instantaneously coupled condensates with complexvalued couplings^{51}. Unlike trapped ground state bosonic systems, such as cold atoms, the ballistically expanding polariton matterwave condensates necessarily experience timedelayed coupling since \({k}_{{\rm{c}},0}d\gg 1\) similar to intercavity coupling of semiconductor lasers^{33}.
In order to accurately reproduce experimentally observed spectra using Eqs. (13) and (14) we fit the distance dependence of the coupling amplitude \(J(d)\) to the spatial envelope of a single condensate. It is known that in a 2D system the 0order Hankel function of the first kind describes the cylindrically symmetric radial outflow of particles in the linear regime^{34}, therefore we choose
where \({J}_{0}\) is a parameter describing the coupling strength (see Supplementary Note 3). Numerical integration of Eqs. (13) and (14) is computed for separation distances \(d \, > \, 10\, \upmu {\rm{m}}\) using Gaussian white noise as initial conditions which, in close analogy to the experimental findings, reproduces periodic parityflip transitions accompanied by cyclic solutions in the transition region. Figure 6a depicts the experimentally measured and normalised emission spectra of the polariton dyad versus pump spot separation distance \(d\) on a greyscale colourmap, for which the extracted two most dominant spectral peaks are depicted in Fig. 3. The red dots represent the numerically calculated spectral peak from Eq. (13) when in single mode operation, or the two most dominant spectral peaks when multiple spectral components exist, showing excellent agreement with experiment. As an example, we illustrate a continuous transition of the system from an antiphase to inphase state described by the timedelayed coupled model in Fig. 6b and c showing the corresponding spectral decomposition and phasespace diagrams of the system for an increasing set of distances \(d\) from \(20\) to \(21\, \upmu {\rm{m}}\). Similar stationary and oscillatory behaviour, for separation distances \(d=20\) and \(20.5\, \upmu {\rm{m}}\) respectively, is shown in Fig. 4d. The numerically simulated phasespace diagrams depict periodic orbits in the transition region, involving periodic oscillations of the phase difference \(\phi =\arg ({c}_{1}^{* }{c}_{2})\) and population imbalance \(z=( {c}_{1}{ }^{2} {c}_{2}{ }^{2})/( {c}_{1}{ }^{2}+ {c}_{2}{ }^{2})\), which is also confirmed by 2Dsimulations of the generalised Gross–Pitaevskii equation (see Supplementary Note 1). We have verified through numerics that timedelay physics are indeed an accurate representation of the coupled condensate dynamics (see Supplementary Note 2).
We note that in the limit of fast active reservoir relaxation, \({\Gamma }_{{\rm{A}}}^{1}\ll {\Gamma }^{1}\), one can adiabatically eliminate Eq. (14) and introduce an effective nonlinear term to Eq. (13), \(({\alpha }_{\text{eff}}i\sigma ) {c}_{i}{ }^{2}\), accounting for polariton–polariton and polariton–reservoir interactions, as well as condensate gain saturation. The dynamical equations of two coupled condensates are then described by timedelayed coupled Stuart–Landau oscillators,
Numerical analysis of Eq. (16) is given in Supplementary Note 4 showing qualitative agreement with experiment.
Discussion
We present an extensive experimental and theoretical study of a system of two ballistically expanding (untrapped) interacting polariton condensates, the fundamental building block of polariton graphs with higher connectivity. We demonstrate a regime for coupled matterwave condensates, wherein the coupling is not instantaneous but mediated by a particle flow inherently connected with time retardation effects. We observe deterministic selection of steady state (singlecolour) or dynamical (twocolour) modes of the system by controlling the separation distance between the condensates. Timedelay polaritonics potentially offers an ultrafast platform for simulating the dynamics of realworld systems of timedelayed coupled nonlinear oscillators that appear in photonics, electronics, and neural circuits. Given the high nonlinearities of polaritons and the ease of optically imprinting multiple condensates of arbitrary geometries on planar microcavities, the system offers promising applications for neuromorphic devices based on lattices of history dependent, nontrapped, strongly interacting, polariton condensates with a wide range of coupling strengths, fast optical operations (input), dynamics (processing), and readout.
Methods
Microcavity sample and experimental methods
The microcavity sample used is a strain compensated \(2\lambda\) GaAs microcavity with three pairs of 6 nm InGaAs quantum wells embedded at the antinodes of the electric field^{37}. The intracavity layer contains a wedge and the position on the sample is chosen such that the cavity photonic mode is redshifted from the excitonic mode at zero inplane wavevector (\( {\bf{k}} =0\)) by \(\approx 5.5\) meV. For all experiments the microcavity is held in a cold finger cryostat (temperature \(T\approx 6\, {\mathrm{K}}\)) and is optically pumped with a circularly polarised continuous wave monomode laser bluedetuned above the cavity stopband (\(\lambda \approx 785\ {\rm{nm}}\)). To prevent heating of the sample, an acoustooptic modulator is used to generate square wave packets at a frequency of 10 kHz and duty cycle of \(5 \%\). A liquid crystal spatial light modulator (SLM) imprints a phase pattern such that when the beam is focused through the 0.4 numerical aperture microscope objective lens it excites the sample with the desired spatial geometry. The phase patterns are carefully designed so that when changing the pump separation distance \(d\) of the twospot excitation pattern, the diffraction efficiency of the SLM remains constant and both pump spots retain equal excitation power and width. The photoluminescence is collected in reflection geometry and spectrally resolved using an 1800 grooves/mm grating in a \(750\ {\rm{mm}}\) spectrometer, which is equipped with a chargecoupled device (CCD). Realspace, Fourierspace and dispersion images are acquired using exposure times in the order of milliseconds.
Interferometry
A modified Michelson interferometer, where one mirror is replaced with a retroreflector mounted on a translational stage, is used for measuring the interference and temporal coherence of the emission. By tilting the angle of the emission entering the interferometer, we select to spatially overlap the emission of either opposing condensates or the same condensates onto a CCD camera. Interference fringe visibilities are extracted from the normalised \({{\rm{first}}}\) diffraction order of the computed discrete Fourier transform of each interference image.
Image processing
Image displayed in Fig. 1a has been digitally processed with a lowpass filter to increase the visibility of interference fringes.
Data availability
The data that support the findings of this study are openly available from the University of Southampton repository (https://doi.org/10.5258/SOTON/D1149)^{52}.
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Acknowledgements
The authors are grateful to N. G. Berloff for fruitful discussions and acknowledge the support of the UK’s Engineering and Physical Sciences Research Council (grant EP/M025330/1 on Hybrid Polaritonics).
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P.G.L. led the research project. P.G.L., J.D.T., and L.P. designed the experiment. J.D.T. and L.P. carried out the experiments and analysed the data. J.D.T. and H.S. developed the theoretical modelling. H.S. performed numerical simulations. All authors contributed to the writing of the manuscript.
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Töpfer, J.D., Sigurdsson, H., Pickup, L. et al. Timedelay polaritonics. Commun Phys 3, 2 (2020). https://doi.org/10.1038/s4200501902710
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DOI: https://doi.org/10.1038/s4200501902710
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