A Polariton Graph Simulator

We discuss polariton graphs as a new platform for simulating the classical XY and Kuramoto models. Polariton condensates can be imprinted into any two-dimensional graph by spatial modulation of the pumping laser. Polariton simulators have the potential to reach the global minimum of the XY Hamiltonian in a bottom-up approach by gradually increasing excitation density to the threshold or to study large-scale synchronisation phenomena and dynamical phase transitions when operating above the threshold. We consider the modelling of polariton graphs using the complex Ginzburg-Landau model and derive analytical solutions for a single condensate, the XY model, two-mode model and the Kuramoto model establishing the relationships between them.


Introduction
Engineering a physical system to reproduce a many-body Hamiltonian has been at the heart of Feynman's idea of an analogue Hamiltonian simulator [1]. Such simulators could address problems that cannot be solved by a conventional Turing classical computer, which would allow us to test various models of lattice systems or discover new states of matter. Analogue Hamiltonian simulations led to the observation of a superfluidinsulator phase transition in ultracold Bose gases that is closely related to metal-insulator transition in condensed-matter materials [2]. In the last decade various other systems have been proposed as classical or quantum simulators: ultracold bosonic and fermionic atoms and molecular gases in optical lattices [3][4][5][6], photons [7], trapped ions [8,9], superconducting q-bits [10], network of optical parametric oscillators [11,12], and coupled lasers [13] among other systems.
The design of an analogue Hamiltonian simulator consists of several important ingredients [14]: (i) mapping of the Hamiltonian of the system to be simulated into the elements of the simulator and the interactions between them; (ii) preparation of the simulator in a state that is relevant to the physical problem of interest: one could be interested in finding the ground or excited equilibrium state at a finite temperature; (iii) performing measurements on the simulator with the required precision. One of the platforms that have recently been explored is based on exciton-polariton lattices. Exciton-polaritons (or polaritons) are the composed lightmatter bosonic quasi-particles formed in the strong exciton-photon coupling regime in semiconductor microcavities [15]. Under non-resonant optical excitation, free carriers relax, scatter, emit phonons and when the particle density reaches quantum degeneracy threshold, polaritons condense in the same state [16], driven by bosonic stimulation [17]. The steady state of such a condensate is characterized by the balance of pumping and dissipation of photons that decay through the Bragg reflectors carrying all information of the corresponding polariton state wavefunction such as the energy, momentum, density, phase and spin. This information allows for the in situ characterization of a polariton condensate in its steady state and during any dynamical transition. The non-equilibrium nature of polariton condensates gives rise to pattern formation in these systems that has been a subject of many investigations, for review see [18,19]. Several methods for imprinting polariton lattices have been proposed. To introduce a photonic trap a partial [20] or complete etching [21] can be used. A thinmetal film technique on a grown wafer has been used to weakly modulate in-plane one-dimensional photon lattice [22]. Exciton trap states have been explored by introducing a mechanical strain in a sample [23] or by Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
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applying electric or magnetic fields that change the exciton energy [24]. Polariton condensates can be created at the vertices of any two-dimensional graph by spatial modulation of the pumping source. The first theoretical proposal [25] and its experimental realization [26,27] to imprint polariton condensates in multi-site configurations were focused on the states, such as vortex lattices, created by the outflowing polaritons from the condensate sites. Next question concerned the way the coherence is established between the various condensates. As the excitation intensity is increased from below polaritons at the lattice site i start to condense with the wavefunction y r , characterized by the number density r ( ) r i and a phase ( ) S r i with the relative phase-configuration that carries the highest polariton occupation due to the bosonic stimulation during the condensate formation [28]. By controlling the pumping intensity and profile, the graph geometry and the separation distance between the lattice sites one can control the couplings between the sites and realize various phase configurations that minimize the XY model as was shown in [29]. This gives rise to the use of the polariton graph as an analogue XY Hamiltonian simulator. The search for the global minimum of the XY Hamiltonian is via a bottom-up approach which has an advantage over classical or quantum annealing techniques, where the global ground state is reached through either a transition over metastable excited states or via tunnelling between the states in time that depends on the size of the system. Figure 1(a) shows the schematics of the classical thermal annealing, quantum annealing via tunnelling between the metastable states and the bosonic stimulation.
The XY model has been previously simulated by other physical systems: ultracold atomic optical lattices [30] and coupled photon lasers network [13]. Polariton graphs are as scalable as these platforms regarding the number of nodes it can involve. In [29] we have shown the graphs that consist of 45 nodes. Polariton graphs enjoy higher flexibility in engineering any geometrical configuration of nodes, since in optical lattices and laser networks it is harder going beyond a regular lattice configuration for the arrangement of the vertices. In a microcavity used in [29] there is only one longitudinal mode resonant to the exciton energy and thus the system can operate at threshold more stably than photonic laser which contains a large number of longitudinal modes within the gain bandwidth. Also, the large number of modes competing for lasing necessitates pumping much above a threshold for stable operation. This suggests that polariton graphs have an advantage over other systems for finding the ground state of XY models. Figure 1(b) shows the schematical difference between operating at the threshold and well above the threshold for finding the global minimum of the energy landscape which in our case is represented by the XY Hamiltonian.
The interest in using an analogue simulator for finding the global minimum of the XY Hamiltonian is motivated by the recent result in the theory of quantum complexity: there exist universal Hamiltonians such that all other classical spin models can be reproduced within such a model, and certain simple Hamiltonians such as the 2D Ising model with fields on a square grid with only nearest neighbour interactions are already universal [31]. This suggests that hard computational problems that can be formulated as a universal Hamiltonian can be solved by a simulator that is designed for finding the global minimum of such Hamiltonian. In particular, the XY model is a quadratic constrained optimization model, which is an NP-hard problem for non-convex and sufficiently dense matrices [29,32].
A more traditional use of analogue Hamiltonian simulators has been based on modelling electrons as they move on a lattice generated by the periodic array of atomic cores. To elucidate such a behaviour ultracold atomic condensates were loaded into optical lattices such as cubic-type lattices [33], superlattice structures [34,35], triangular [36], hexagonal [36,37] and Kagome [38] lattices. Polariton graphs can easily produce such or any other ordered or disordered lattice. It was recently shown that a linear periodic chain of exciton-polariton condensates demonstrate not only various classical regimes: ferromagnetic, antiferromagnetic and frustrated spiral phases, but also at higher pumping intensities bring about novel exotic phases that can be associated with spin liquids [39]. Relationship between the energy spectrum of the XY Hamiltonian and the total number of condensed polariton particles has been established in [39] where it was shown that 'particle mass residues' of successive polariton states (defined as the difference between the masses of the individual condensates and the total mass) that occur with increasing excitation density above condensation threshold are an accurate approximation of the XY Hamiltonian's energy spectrum. Therefore, polariton graph condensate system may represent not only the ground state but also the spectral gap of the XY model.
Our paper is organized as follows. In section 2 we consider the mean-field model of polariton condensates and derive analytical solutions for a single condensate. We establish the phase mapping of a polariton graph into the XY model in section 3. In section 4 we derive the Kuramoto model that describes the dynamics of the phases of the polariton condensates and show its relevance to the XY model. We conclude with the discussions in section 5.

An approximate analytical solution for a single condensate
The mean field of polariton condensates can be modelled [40,41] in association with atomic lasers by writing a driven-dissipative Gross-Pitaevskii equation (aka the complex Ginzburg-Landau equation (cGLE)) for the condensates wavefunction y ( ) t r, : coupled to the rate equation for the density of the hot exciton reservoir, ( ) t r, : In these equations m is polariton effective mass, U 0 and g R are the strengths of effective polariton-polariton and polariton-exciton interactions, respectively, h d is the energy relaxation coefficient specifying the rate at which gain decreases with increasing energy, R R is the rate at which the reservoir excitons enter the condensate, g C is the rate of the condensate losses, g R is the redistribution rate of reservoir excitons between the different energy levels, and P is the pumping into the reservoir. In the limit g g  R C one can replace equation (2) with the stationary state for the reservoir  g y To non-dimensionalize the cGLE we use y  ℓ mU 2 The dimensionless form of equations (1) and (2) becomes the cGLE with a saturable nonlinearity By taking the Taylor expansion for small Y | | 2 in the expression for the reservoir n R we arrive at the more standard cGLE where, in the view of smallness of η we dropped h Y | | g 2 term. We can compare the relative strength of nonlinearities in equations (3)-(5) depending of the physical quantities that define b. By taking the values the system parameters typically accepted for GaAs microcavities [26,42,43] p s 2 4 R . With g R on the order of 1 ps −1 we have b of the order of the real nonlinearity.
We consider the asymptotics and approximations of the steady state solutions for a single radially symmetric Gaussian pumping profile  In the view of their asymptotic behaviour the condensate density and velocity can be approximated by c c utilizing their behaviour at the origin and infinity and introducing parameters x a a , , 0 3 , and l that define the parametric family of solutions. Their values should be found from the governing equations via matching asymptotics, as shown below.
We neglect η in the view of its smallness and substitute the expressions for the density, velocity and the pumping profile into equations (6) and (7). By expanding the resulting expressions about r=0 and setting the term to the order ( ) r 2 to zero we obtain the equations that define the unknown parameters x a a l , , , 0 3 and k c in terms of the system parameters g s g b p , , , , 0 . The leading order expansion of equation (6) and the first order expansion of equation (7) fix ξ and a 3 as The expansion to ( ) r 2 of equation (6) determines k c as g g g g g g g s g Equations (8) and (9) with the parameters defined by equations (10)-(13) for the given system parameters g g b , , and the pumping parameters p 0 and σ fully specify the approximate analytical solution of equations (6) and (7). Figure 2 shows the comparison of the approximate analytical solutions (solid lines) given by equations (8)

Mapping of phases into the classical XY model
In the previous section we obtained solutions of the governing equation equation (5) for a single pumping Gaussian spot. Spatial light modulator can be used to pump condensates at the vertices of a distributed graph via where p i stands for the pumping intensity at the centre of the spot at position = r r i . In what follows we shall assume that all vertices are pumped identically, so that = p p i 0 , s s = i for all = ··· i N 1, , . To the leading order and assuming that all condensates are well-separated we can approximate the resulting condensate wavefunction, is the solution of the stationary equation (5) for a single localized radially symmetric condensate pumped by is given by equation (8). To find the total amount of matter  we write and J 0 is the Bessel function. The total mass becomes Since the system maximizes the total number of particles given by equation (18), this is equivalent to minimizing the XY Hamiltonian functional  q = -å < J cos ij [29]. The main contribution to the integral defining Ỹ ( ) k 0 is from = k k c , where k c is the outflow wavevector from the pumping site fully determined by the pumping profile [28,29]. Figure 3 shows the density contour plots (normalized) and the spin orientations (arrows) representing the relative phases, with the incoherent pump spots located at the vertices. The coupling between the adjacent polariton sites can be made ferromagnetic  [30].
The ultimate advantage of polariton graphs for analog Hamiltonian simulations is the potential to control both the sign and the strength of any coupling, J ij , by tuning the distance between polariton sites, or the characteristics of the pumping spots (the intensity, p 0 , or the inverse width of the Gaussian, σ) leading to more exotic phases. We illustrate the control over an individual J ij on the seven-vertex graphs of figures 3(e), (f). In figure 3(e) we utilize control over an individual J ij by tuning the distance between two vertices and introduce a single ferromagnetic coupling (defect edge) into an otherwise antiferromagnetic configuration. In figure 3(f) we control the coupling of a single polariton site (central vertex) to all its neighbours by tuning the intensity of its pumping spot and switch them to ferromagnetic in an otherwise antiferromagnetically coupled graph (star configuration). Finally, polariton graphs allow for fully disordered systems to be addressed as shown in figure 3(g).

Kuramoto model
The cGLE can be reformulated as the Kuramoto model which is a paradigm for a spontaneous emergence of collective synchronization and that has been widely used to understand the topological organization of real complex systems from neural networks to power grids [44]. In this context the polariton lattice describes collective dynamics of N coupled phase oscillators with phases q ( ) t i , characterized by the natural frequencies w i which are associated with the chemical potential of individual condensates.
To show this we adopt a two-mode model that neglects the spacial variations and represents the network of interacting polariton condensates via the radiative couplings  ij between ith and jth condensates We neglect the blueshift g and without loss of generality let g = 1. The radiative coupling  ij is due to the interference of the condensates from different pumping spots [45], and are such that  r  ij i for any j. First, we shall assume that the density number dynamics is faster than the phase dynamics, so that the densities acquire the instantaneous steady state values r r = = - so that the Kuramoto model (22) describes the gradient flow to the minima of q ( ) V and, therefore, minimizes the XY Hamiltonian.
Next, we will allow for the density variations and consider two spots only. For the system with just two condensates we introduce the average density operation [46] and condensation under electrical pumping [47], polariton graph based simulators offer unprecedented opportunities in addressing NP-complete and hard problems, quantum information processing and the study of exotic phase transitions. Finally, we would like to emphasize that the word 'quantum' could be attached to our proposal for a simulator to reflect the statistical nature of polariton condensates. The process of Bose-Einstein condensation is inherent to quantum statistics where a large fraction of bosons occupies the lowest quantum state, at which point macroscopic quantum phenomena become apparent. The use of the classical mean-field equations to describe the kinetics of the condensate does not negate the quantum statistic nature of its existence. At the same time, the proposed simulator has a quantum speed-up which is associated with the stimulated process of condensation i.e. an accelerated relaxation to the global ground quantum state.