# Polariton Simulators

### An international research activity of state-of-the-art experimental laboratories

and theoretical modelling

# Polariton lattices

Polariton lattices

### Quantum gases in optical lattices offer a powerful tool to study condensed matter phenomena in a controllable environment where the geometry and depth of the potential are tuned by optical means. The wide tunability range and control level accuracy of the potential landscape render optical lattices a good candidate for quantum simulation, i.e. for the study of classical magnetism, although the readout process and the single site control remain challenging. Recently, we proposed polariton graphs as a new platform for analogue simulation. Polariton graphs benefit by the continuous optical readout of the phase, energy, momentum and spin of the individual polariton vertices, the strong inter-particle interactions −mediated through the exciton component−, room temperature operation −with appropriate choice of materials−, and even the potential for electrical injection utilizing well-developed semiconductor technologies.

Quantum gases in optical lattices offer a powerful tool to study condensed matter phenomena in a controllable environment where the geometry and depth of the potential are tuned by optical means. The wide tunability range and control level accuracy of the potential landscape render optical lattices a good candidate for quantum simulation, i.e. for the study of classical magnetism, although the readout process and the single site control remain challenging. Recently, we proposed polariton graphs as a new platform for analogue simulation. Polariton graphs benefit by the continuous optical readout of the phase, energy, momentum and spin of the individual polariton vertices, the strong inter-particle interactions −mediated through the exciton component−, room temperature operation −with appropriate choice of materials−, and even the potential for electrical injection utilizing well-developed semiconductor technologies.

### In this project, we investigate the potential of polariton graphs as a platform for analogue simulation and in particular in solving non-deterministic polynomial problems. Unlike a proposal for a quantum computer that is intended as a universal platform, analogue simulators are proposed to solve specialized classes of problems suited for the architecture and capabilities of the underlying physical system. For example, the intensely investigated superconducting quantum bits platform intends to simulate the Ising model with transverse fields through the quadratic unconstrained binary optimization model (QUBO). Trapped ions were used to simulate Ising, XY, and XYZ interactions between effective spins. Another scalable platform that benefits from high temperature operation is the coupled degenerate OPOs Ising Machine, which solves the MAX-CUT. Our polariton platform simulates the XY model, which can be formulated as a quadratic non-convex constrained optimization model (QNCO). The hardest instances of all mentioned problems are in the NP-hard classical complexity class of problems. QUBO or MAX-CUT can be mapped into QNCO and vice versa but with a huge overhead on the number of nodes. Therefore, assuming that all platforms eventually show better than classical computer behavior it is likely that each platform will be used to address its own type of problems.

In this project, we investigate the potential of polariton graphs as a platform for analogue simulation and in particular in solving non-deterministic polynomial problems. Unlike a proposal for a quantum computer that is intended as a universal platform, analogue simulators are proposed to solve specialized classes of problems suited for the architecture and capabilities of the underlying physical system. For example, the intensely investigated superconducting quantum bits platform intends to simulate the Ising model with transverse fields through the quadratic unconstrained binary optimization model (QUBO). Trapped ions were used to simulate Ising, XY, and XYZ interactions between effective spins. Another scalable platform that benefits from high temperature operation is the coupled degenerate OPOs Ising Machine, which solves the MAX-CUT. Our polariton platform simulates the XY model, which can be formulated as a quadratic non-convex constrained optimization model (QNCO). The hardest instances of all mentioned problems are in the NP-hard classical complexity class of problems. QUBO or MAX-CUT can be mapped into QNCO and vice versa but with a huge overhead on the number of nodes. Therefore, assuming that all platforms eventually show better than classical computer behavior it is likely that each platform will be used to address its own type of problems.

## Key researchers:

Key researchers:

### Dr Sergey Alyatkin, Dr Helgi Sigurdsson, Dr Tamsin Cookson

Dr Sergey Alyatkin, Dr Helgi Sigurdsson, Dr Tamsin Cookson

### Title: "Polariton vortex Chern insulator" (INVITED)

Title: "Polariton vortex Chern insulator" (INVITED)

### Authors: S. L. Harrison, A. Nalitov, P. G. Lagoudakis, H. Sigurðsson

Authors: S. L. Harrison, A. Nalitov, P. G. Lagoudakis, H. Sigurðsson

### Title: "Minor embedding with Stuart-Landau oscillator networks"

Title: "Minor embedding with Stuart-Landau oscillator networks"

### Authors: S. L. Harrison, H. Sigurdsson, and P. G. Lagoudakis

Authors: S. L. Harrison, H. Sigurdsson, and P. G. Lagoudakis

### Title: "Enhanced coupling between ballistic exciton-polariton condensates through tailored pumping"

Title: "Enhanced coupling between ballistic exciton-polariton condensates through tailored pumping"

### Authors: Y. Wang, P. G. Lagoudakis, and H. Sigurdsson

Authors: Y. Wang, P. G. Lagoudakis, and H. Sigurdsson

### Title: "Screening nearest-neighbor interactions in networks of exciton-polariton condensates through spin orbit coupling"

Title: "Screening nearest-neighbor interactions in networks of exciton-polariton condensates through spin orbit coupling"

### Authors: D. Aristov, H. Sigurdsson, P. G. Lagoudakis

Authors: D. Aristov, H. Sigurdsson, P. G. Lagoudakis

### Title: "Solving the max-3-cut problem using synchronized dissipative networks"

Title: "Solving the max-3-cut problem using synchronized dissipative networks"

### Authors: S. L. Harrison, H. Sigurdsson, S. Alyatkin, J. D. Töpfer, and P. G. Lagoudakis

Authors: S. L. Harrison, H. Sigurdsson, S. Alyatkin, J. D. Töpfer, and P. G. Lagoudakis

### Title: "Exact Analytical Solution for the Density Matrix of a Nonequilibrium Polariton Bose-Einstein Condensate"

Title: "Exact Analytical Solution for the Density Matrix of a Nonequilibrium Polariton Bose-Einstein Condensate"

### Authors: V. Y. Shishkov, E. S. Andrianov, A. V. Zasedatelev, P. G. Lagoudakis, and Y. E. Lozovik

Authors: V. Y. Shishkov, E. S. Andrianov, A. V. Zasedatelev, P. G. Lagoudakis, and Y. E. Lozovik

### Title: "Reservoir optics with exciton-polariton condensates"

Title: "Reservoir optics with exciton-polariton condensates"

### Authors: Y. Wang, H. Sigurdsson, J. D. Töpfer, and PG Lagoudakis

Authors: Y. Wang, H. Sigurdsson, J. D. Töpfer, and PG Lagoudakis

### Title: "Quantum fluids of light in all-optical scatterer lattices"

Title: "Quantum fluids of light in all-optical scatterer lattices"

### Authors: S. Alyatkin, H. Sigurdsson, A. Askitopoulos, J. D. Töpfer, and P. G. Lagoudakis

Authors: S. Alyatkin, H. Sigurdsson, A. Askitopoulos, J. D. Töpfer, and P. G. Lagoudakis

### Title: "Minor embedding with Stuart-Landau oscillator networks"

Title: "Minor embedding with Stuart-Landau oscillator networks"

### Authors: S. L. Harrison, H. Sigurdsson, and P. G. Lagoudakis

Authors: S. L. Harrison, H. Sigurdsson, and P. G. Lagoudakis

### Title: "Emergence and Ordering of Polygonal Breathers in Polariton Condensates"

Title: "Emergence and Ordering of Polygonal Breathers in Polariton Condensates"

### Authors: S. N. Alperin, N. G. Berloff

Authors: S. N. Alperin, N. G. Berloff

### Title: "XY Neural Networks"

Title: "XY Neural Networks"

### Authors: N. Stroev, N. G. Berloff

Authors: N. Stroev, N. G. Berloff

### Title: "Large-scale Sustainable Search on Unconventional Computing Hardware"

Title: "Large-scale Sustainable Search on Unconventional Computing Hardware"

### Authors: K. P. Kalinin, N. G. Berloff

Authors: K. P. Kalinin, N. G. Berloff

### Title: "Polariton spin jets through optical control "

Title: "Polariton spin jets through optical control "

### Authors: L. Pickup, J. D. Töpfer, H. Sigurdsson, and P. G. Lagoudakis

Authors: L. Pickup, J. D. Töpfer, H. Sigurdsson, and P. G. Lagoudakis

### Title: "Optically controlled polariton condensate molecules"

Title: "Optically controlled polariton condensate molecules"

### Authors: E. D. Cherotchenko, H. Sigurdsson, A. Askitopoulos, and A. V. Nalitov

Authors: E. D. Cherotchenko, H. Sigurdsson, A. Askitopoulos, and A. V. Nalitov

### Title: "Artificial polariton molecules"

Title: "Artificial polariton molecules"

### Authors: A. Johnston, K.P. Kalinin, and N. G. Berloff

Authors: A. Johnston, K.P. Kalinin, and N. G. Berloff

### Title: "Discrete Polynomial Optimization with Coherent Networks of Condensates and Complex Coupling Switching"

Title: "Discrete Polynomial Optimization with Coherent Networks of Condensates and Complex Coupling Switching"

### Authors: N. Stroev and N. G. Berloff

Authors: N. Stroev and N. G. Berloff

### Title: "Engineering spatial coherence in lattices of polariton condensates"

Title: "Engineering spatial coherence in lattices of polariton condensates"

### Authors: J. D. Töpfer, I. Chatzopoulos, H. Sigurdsson, T. Cookson, Y. G. Rubo, and P. G. Lagoudakis

Authors: J. D. Töpfer, I. Chatzopoulos, H. Sigurdsson, T. Cookson, Y. G. Rubo, and P. G. Lagoudakis

### Authors: J. D. Töpfer, H. Sigurdsson, S. Alyatkin, and P. G. Lagoudakis

Authors: J. D. Töpfer, H. Sigurdsson, S. Alyatkin, and P. G. Lagoudakis

### Title: "Simulating the spectral gap with polariton graphs"

Title: "Simulating the spectral gap with polariton graphs"

### Authors: K. Kalinin, P. G. Lagoudakis, and N. G. Berloff

Authors: K. Kalinin, P. G. Lagoudakis, and N. G. Berloff

### Title: "Managing the flow of liquid light"

Title: "Managing the flow of liquid light"

### Authors: N. Stroev and N. G. Berlof

Authors: N. Stroev and N. G. Berlof

### Title: "Synthetic band-structure engineering in polariton crystals with non-Hermitian topological phases"

Title: "Synthetic band-structure engineering in polariton crystals with non-Hermitian topological phases"

### Authors: L. Pickup, H. Sigurdsson, J. Ruostekoski, and P. G. Lagoudakis

Authors: L. Pickup, H. Sigurdsson, J. Ruostekoski, and P. G. Lagoudakis

### Title: "Complexity continuum within Ising formulation of NP problems"

Title: "Complexity continuum within Ising formulation of NP problems"

### Authors: K. P. Kalinin and N. G. Berloff

Authors: K. P. Kalinin and N. G. Berloff

### Title: "Interaction induced point scatterer lattices and flat band condensation of exciton-polaritons"

Title: "Interaction induced point scatterer lattices and flat band condensation of exciton-polaritons"

### Authors: S. Alyatkin, H. Sigurdsson, A. Askitopoulos, J. D. Töpfer, and P. G. Lagoudakis

Authors: S. Alyatkin, H. Sigurdsson, A. Askitopoulos, J. D. Töpfer, and P. G. Lagoudakis

### Title: "Polaritonic XY-Ising Machine"

Title: "Polaritonic XY-Ising Machine"

### Authors: K.P. Kalinin , A. Amo, J. Bloch, and N. G. Berlof

Authors: K.P. Kalinin , A. Amo, J. Bloch, and N. G. Berlof

### Title: "Optical control of couplings in polariton condensate lattices"

Title: "Optical control of couplings in polariton condensate lattices"

### Authors: S. Alyatkin, J. Toepfer, A. Askitopoulos, H. Sigurdsson, and P. G. Lagoudakis

Authors: S. Alyatkin, J. Toepfer, A. Askitopoulos, H. Sigurdsson, and P. G. Lagoudakis

### Title: "Synchronization in optically trapped polariton Stuart-Landau networks"

Title: "Synchronization in optically trapped polariton Stuart-Landau networks"

### Authors: S. L. Harrison, H.Sigurdsson, and P. G. Lagoudakis

Authors: S. L. Harrison, H.Sigurdsson, and P. G. Lagoudakis

### Title: "Time-delay polaritonics"

Title: "Time-delay polaritonics"

### Authors: J. D. Töpfer, H. Sigurdsson, L. Pickup, and P. G. Lagoudakis

Authors: J. D. Töpfer, H. Sigurdsson, L. Pickup, and P. G. Lagoudakis

### Title: "Nonlinear systems for unconventional computing"

Title: "Nonlinear systems for unconventional computing"

### Authors: K. P. Kalinin and N. G. Berloff

Authors: K. P. Kalinin and N. G. Berloff

### Title: "Polaritonic network as a paradigm for dynamics of coupled oscillators"

Title: "Polaritonic network as a paradigm for dynamics of coupled oscillators"

### Authors: K. P. Kalinin and N. G. Berloff

Authors: K. P. Kalinin and N. G. Berloff

### Title: "Toward Arbitrary Control of Lattice Interactions in Nonequilibrium Condensates"

Title: "Toward Arbitrary Control of Lattice Interactions in Nonequilibrium Condensates"

### Authors: K. P. Kalinin and N. G. Berloff

Authors: K. P. Kalinin and N. G. Berloff

### Title: "Global optimization of spin Hamiltonians with gain-dissipative systems"

Title: "Global optimization of spin Hamiltonians with gain-dissipative systems"

### Authors: K. Kalinin and N. G. Berloff

Authors: K. Kalinin and N. G. Berloff

### Title: "Simulating Ising and n-State Planar Potts Models and External Fields with Nonequilibrium Condensates"

Title: "Simulating Ising and n-State Planar Potts Models and External Fields with Nonequilibrium Condensates"

### Authors: K. Kalinin and N. G. Berloff

Authors: K. Kalinin and N. G. Berloff

### Title: "Networks of non-equilibrium condensates for global optimization"

Title: "Networks of non-equilibrium condensates for global optimization"

### Authors: K. Kalinin and N. G. Berloff

Authors: K. Kalinin and N. G. Berloff

### Title: "Gain-dissipative simulators for large-scale hard classical optimisation"

Title: "Gain-dissipative simulators for large-scale hard classical optimisation"

### Authors: K. Kalinin and N. G. Berloff

Authors: K. Kalinin and N. G. Berloff

### Title: "Exotic states of matter with polariton chains"

Title: "Exotic states of matter with polariton chains"

### Authors: K. Kalinin, P. G. Lagoudakis, and N. G. Berloff

Authors: K. Kalinin, P. G. Lagoudakis, and N. G. Berloff

### Title: "Matter wave coupling of spatially separated and unequally pumped polariton condensates"

Title: "Matter wave coupling of spatially separated and unequally pumped polariton condensates"

### Authors: K. Kalinin, P. G. Lagoudakis, and N. G. Berloff

Authors: K. Kalinin, P. G. Lagoudakis, and N. G. Berloff

### Title: "Blockchain platform with proof-of-work based on analog Hamiltonian optimisers"

Title: "Blockchain platform with proof-of-work based on analog Hamiltonian optimisers"

### Authors: K. Kalinin and N. G. Berloff

Authors: K. Kalinin and N. G. Berloff

### Title: "A Polariton Graph Simulator"

Title: "A Polariton Graph Simulator"

### Authors: P.G.Lagoudakis, and N.G. Berloff

Authors: P.G.Lagoudakis, and N.G. Berloff

### Title: "Realizing the classical XY Hamiltonian in polariton simulators"

Title: "Realizing the classical XY Hamiltonian in polariton simulators"

### Authors: N.G. Berloff, M Silva, K Kalinin, A Askitopoulos, J.D. Töpfer, P Cilibrizzi, W Langbein, and P.G. Lagoudakis

Authors: N.G. Berloff, M Silva, K Kalinin, A Askitopoulos, J.D. Töpfer, P Cilibrizzi, W Langbein, and P.G. Lagoudakis

### Title: "Nontrivial Phase Coupling in Polariton Multiplets"

Title: "Nontrivial Phase Coupling in Polariton Multiplets"

### Authors: H. Ohadi, R. L. Gregory, T. Freegarde, Y. G. Rubo, A. V. Kavokin, N. G. Berloff, and P. G. Lagoudakis

Authors: H. Ohadi, R. L. Gregory, T. Freegarde, Y. G. Rubo, A. V. Kavokin, N. G. Berloff, and P. G. Lagoudakis