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Realizing the $XY$ Hamiltonian in polariton simulators
Natalia G. Berloff, Kirill Kalinin, Matteo Silva, Wolfgang Langbein, Pavlos G. Lagoudakis
TL;DR
The paper investigates finding the global minimum of the XY Hamiltonian for polariton-graph sample configurations. It estimates coupling coefficients from condensate wavefunctions and evaluates phase-minimizing configurations across geometries, including square and triangular arrangements.
Problem
The study addresses how to find the global minimum of the XY Hamiltonian for polariton-graph sample configurations.
Method
The approach estimates coupling coefficients from the Hankel-transformed condensate wavefunction and uses numerical simulations with experimentally matched condensate profiles and outflow wavenumbers.
Results
The analysis identifies ferromagnetic, antiferromagnetic, and frustrated phase configurations, including geometry-dependent minima for square and triangular arrangements.
Takeaways & Limitations
Polariton graphs can directly realize and evaluate diverse XY-model phase configurations for the sample geometries considered.
Abstract
from arXiv · showhide
Several platforms are currently being explored for simulating physical systems whose complexity increases faster than polynomially with the number of particles or degrees of freedom in the system. Defects and vacancies in semiconductors or dielectric materials, magnetic impurities embedded in solid helium \cite{lemeshko13}, atoms in optical lattices, photons, trapped ions and superconducting q-bits are among the candidates for predicting the behaviour of spin glasses, spin-liquids, and classical magnetism among other phenomena with practical technological applications. Here we investigate the potential of polariton graphs as an efficient simulator for finding the global minimum of the $XY$ Hamiltonian. By imprinting polariton condensate lattices of bespoke geometries we show that we can simulate a large variety of systems undergoing the U(1) symmetry breaking transitions. We realise various magnetic phases, such as ferromagnetic, anti-ferromagnetic, and frustrated spin configurations on unit cells of various lattices: square, triangular, linear and a disordered graph. Our results provide a route to study unconventional superfluids, spin-liquids, Berezinskii-Kosterlitz-Thouless phase transition, classical magnetism among the many systems that are described by the $XY$ Hamiltonian.
Microcavity Sample
The study uses a planar, strain-compensated 2λ GaAs microcavity with embedded InGaAs quantum wells. Its multilayer design provides high reflectance and supports polariton condensation at sufficiently high densities.
- The sample is a planar, strain-compensated 2λ GaAs microcavity containing embedded InGaAs quantum wells.The cavity includes three pairs of quantum wells at field antinodes and additional wells at the first and last nodes.
- The distributed Bragg reflectors contain 26 bottom and 23 top pairs of GaAs and AlAs0.98P0.02, yielding reflectance above 99.9% in the stop-band.
- The quantum-well design increases Rabi splitting while keeping exciton density per well below the Mott density at polariton-condensation densities.
Experimental setup
The experiments use cryogenic, non-resonant optical excitation whose spatial profile is shaped into a graph of approximately equal-sized Gaussian spots with a spatial light modulator.
- The sample is held at approximately 6 K in a cold-finger cryostat and excited continuously by a Ti:Sapphire laser.
- Non-resonant excitation enters from the epi side, while emission is detected from the substrate side so the GaAs substrate filters the excitation.
- A reflective spatial light modulator shapes the excitation into a graph with approximately equal-diameter Gaussian spots at each vertex.A high-numerical-aperture objective focuses the modulated beam to approximately 1–2 µm diameter spots.
Wavevector Tomography
Two-dimensional Fourier-space tomography measures the condensate wavevector and distinguishes condensate outflow from self-diffraction fringes in Ising-chain configurations.
- Fourier-space tomography uses a tunable Fabry–Perot etalon and CCD camera to measure the condensate wavevector kc.The etalon has approximately 20 µeV FWHM bandwidth.
- The outer ring in the Fourier-space images corresponds to kc, while inner fringes arise from self-diffraction from the Ising chain.
- For the anti-ferromagnetic chain kc ≈ 1.79 µm^-1, compared with kc ≈ 1.67 µm^-1 for the ferromagnetic chain.
- Figure S1 compares false-colour normalised photoluminescence Fourier-space tomography for two Ising-chain configurations at condensate energy.
Finding the expression for the coupling coefficients
The coupling coefficients are estimated from the Hankel-transform width of an individual condensate, which is controlled by the pumping-profile width and produces a phase-shifted sign-switching criterion.
- The coupling coefficients Jij are estimated from the width of the Hankel transformation of an individual condensate wavefunction.
- The Hankel-transform density peaks at k = kc, with width ϵ inversely proportional to the condensate-density width set by the pumping profile.
- In the limit ϵ → 0, Jij = kc|bΨ(kc)|2J0(kcdij)/π, while finite transform width induces a phase shift φ.The phase-switching criterion is therefore approximated by sign changes of cos(kcdij + φ).
Minimization of the XY Hamiltonian for sample configurations
The paper evaluates XY-Hamiltonian minima for square, rhombus, triangular, and multi-triangle configurations. Ferromagnetic, antiferromagnetic, and frustrated arrangements produce distinct phase patterns, including non-trivial winding.
- Square: For the square, ferromagnetic couplings minimize the Hamiltonian when all phases equal the reference phase, θi0 = 0.
- Square: For antiferromagnetic square couplings, neighboring sites differ by π, with θ10 = π, θ20 = 0, and θ30 = π when |δ| < 1.Here δ is the ratio of diagonal to neighboring-site coupling strengths.
- Rhombus and triangle: For a rhombus, antiferromagnetic coupling is minimized by θ10 = θ30 = π and θ20 = 0, unlike the ±2π/3 minimum of an isolated equilateral triangle.
- Multiple triangles: Three connected equilateral triangles can support non-trivial winding around sites.
- Frustrated configuration: In a frustrated configuration, antiferromagnetic coupling yields θ10 = −θ40 = ±0.73π and θ20 = −θ30 = ∓0.54π, producing alternating winding around each equilateral triangle.Small positional deviations near ferro- to antiferromagnetic switching points can produce more complex configurations.
Parameters of the numerical simulations
The numerical simulations use Gaussian pumping calibrated to match the experimental condensate width and outflow wavenumber. Common integration parameters are fixed across the simulated geometries.
- Simulation setup: The simulations use a Gaussian pumping profile producing an experimental condensate width of FWHM 2.6µm.
- Simulation setup: Pumping intensity is chosen to obtain the correct outflow wavenumber for a single condensate.
- Simulation setup: Common integration parameters are g = 0.1, b = 1, γ = 0.3, η = 0.4, and p = 9.5 exp(−0.4r2).