Bose-Einstein Condensation and Superfluidity of Exciton-Polaritons ...

Semiconductor Cavity QED in Weak Coupling RegimeY. Yamamoto et al., in Coherence and Quantum Optics VI ( Plenum, New York, 1989) p.1249DBRSQW in λ/2 cavityDBREnhanced conicalspontaneous emissionEnhancedspontaneous emissionInhibitedspontaneous emissionRed detuning (ω c < ω e ) On-resonance (ω c = ω e ) Blue detuning (ω c > ω e )2

Quantum Degeneracy at Thermal Equilibrium ConditionsGaAs: H. Deng et al., Phys. Rev. Lett., 97 146402 (2006)[ CdTe: J. Kasprzak et al., Nature 443, 409 (2006) ]quantum degeneracy thresholdBE distribution observed at blue detuning regime(exciton-like polariton: Δ=6.7 meV, t=40ps)T LP=4.4K, μ=−0.04 meV < BEC threshold (− 0.35 meV)Δ (meV)10 1N LP(E)10 010 -1lattice temperature0 0.2 0.4 0.6 0.8 1E LP− E LP0(meV)The experimental results are reproduced byBoltzmann-Master equation:T.D.Doan, H.T. Cao, D.B.T. Thoai and H. Haug,Solid State Commun. 145, 48 (2008)8

Spatial Coherence (Order Parameter)-Off-diagonal long range order (L. Onsager, R. Penrose, C. N. Yang)-H. Deng et al., Phys. Rev. Lett. 99, 126403 (2007)[ R. Balili et al., Science 316, 1007 (2007)]Interference Pattern through Young’s Double Slit InterferometerSolid lines:Fourier transform ofthe experimentalmomentum distributionTheory based onBoltzmann-Masterequation:T. Doan, H. Cao,D.T. Thoai andH. Haug,Phys. Rev. B 78,205306 (2008)9

Second and Third Order Coherence− H. Deng et al., Science 298,199 (2002)−A coherent state is all-order coherent: g (1) (0)=g (2) (0)=·····=1.A thermal state features photon bunching: g (n) (0)=n!(R. Glauber, 1963)thermal depletion+qquantum depletion-q +qg (2) (0), g (3) (0)g (3) (0)g k=0(2) (0)with scattering (BEC)k=0without scattering (laser)• Photon bunching effect due to bosonic final statestimulation observed only above threshold, whereτ pulse < τ noise is satisfied.• g (2) (0) > 1 and g (3) (0) > 1 at well above thresholdsuggests the excess intensity noise in the condensate.Third order coherence: T. Horikiri et al., in preparationTheory: P. Schwendimann and A. Quattropani,Phys. Rev. B 77, 085317 (2008)10

Outline• Recap on Bose-Einstein condensation of exciton-polaritons:MotivationE vs. k dispersionThermodynamicsCoherence• New Experiments on SuperfluidityBogoliubov excitationsMean field energy shift U(n)Universal dispersion relation E/U(n) vs. kξ(n)Quantized vortex-pairVortex-pair with pinned orientationVortex-pair with switching orientation• Future prospectsQuantum simulation of Bose-Hubbard modelsQuantum computation by BECBEC-BCS crossover11

Brief History of BEC and SuperfluidityA. EinsteinP.L. KapitsaL.D.LandauR.D. FeynmanA. EinsteinBEC of non-interacting ideal gas(1925)P.L. KapitsaDiscovery of superfluid He (1937)London,InterpretationTiszaof superfluid He interms of BEC of ideal gas(1938)L. D. LandauPhenomenological model based onexcitation spectrum (1941)N.N. BogoliubovQuantum field theory of BECofinteracting particles (1947)R.P. FeynmanAtomic theory of Landau’s twofluid model (1955)Excitation spectrum of superfluid He12

Bogoliubov Theory of Weakly Interacting Bose Particles: RevisitSystem Hamiltonian and Gross-Pitaevskii equation:kinetic termInteraction termorder parameterBogoliubov transformation:real particlecollective excitationsDiagonalization ofExcitation spectrum of atomic BECD.M. Stamper-Kurn et al.,PRL 83, 2876 (1999)Bogoliubovspectrum: mean fieldenergy shift: Bogoliubov excitation spectrum: linear dispersionat small p: quadratic dispersionat large p13

Repulsive Interaction Energy U(n)=gngE0= N2VdE0U ( n)=dN2= gn(Mean-field energy shift of the condensate)(Can be measured as blue shift of emitted photon energy at k=0=12Δωexc+g202Δ+4exciton energy blue shiftdue to exchange interaction−( g0+ Δg)2( Δ − Δω+4excreduced polariton splittingdue to phase space filling)2g 0: polariton splittingΔ: detuning parameter(=ω c−ω ex)Experiment:S. Utsunomiya et al., Nat.Phys. 4,700,2008Theory:V. Savona et al.,private communication14

Universal Dispersion Relation of Bogoliubov Excitation Spectrumin Exciton-Polariton CondensatesS. Utsunomiya et al., Nature Physics 4, 700 (2008)ε (p)E/Uat far belowthreshold(p/p th =0.001)• A: Δ=1.41 (meV), P=4P th• B: Δ=0.82 (meV), P=8P th• C: Δ=4.2 (meV), P=4P th• D: Δ=-0.23 (meV), P=24P thHealinglength:Linear dispersion at low momentum regimeSound velocity: c=dE(p)/dp~10 8 cm/s(c~1cm/s for atomic BEC, c~10 4 cm/s for superfluid 4 He)15

Energy Shift vs. Interaction Energy U in Free Particle Regime(S. Utsunomiya et al., Nature Physics 4, 700 (2008))Indistinguishable bosons Direct + Exchange terms2ˆ p + g 2 1++ +H = ∑ aˆˆ +0+ ∑ ( 2 ˆ ˆ p + ˆ ˆ + ˆ ˆpa p N gn apa apa−)2 2 2− papapm Vpp ≠ 0Energy shift for condensate particleU=gn 0Energy shift for non-condensed particleadditionalU=gn0Untrapped systemTrapped system(5-7microns)E B-E LP(meV)10.1A BC DE B −E LP =2UE k(P>P th)-E k(PaP th)=2UE B-E LP(meV)10.1E FG HE B −E LP =2U0.1 1Interaction energy U (meV)0.1 1Interaction energy U (meV) 16

Outline• Recap on Bose-Einstein condensation of exciton-polaritons:MotivationE vs. k dispersionThermodynamicsCoherence• New Experiments on SuperfluidityBogoliubov excitationsMean field energy shift U(n)Universal dispersion relation E/U(n) vs. kξ(n)Quantized vortex-pairVortex-pair with pinned orientationVortex-pair with switching orientation• Future prospectsQuantum simulation of Bose-Hubbard modelsQuantum computation by BECBEC-BCS crossover17

Observation of Vortex-Pairs in Exciton-Polariton Condensatesusing Michelson InterferometerBEC-BKT crossover in a 2D system:Key: Macroscopic phase stabilization by formation of vortex-pairsA. Posazhennikova, Rev. Mod. Phys. 78, 1111 (2006) and refs in it.Atomic BKT phase study based on population of free vortices:Z. Hadzibabic et al., Nature 441, 1118 (2006)Polariton free vortex pinned to crystal defects:K.G. Lagoudakis et al., Nat. Phys. 4, 706 (2008)phase distributionmeasurementvisibility measurement18

Phase Mapping for Vortex-Pair with Pinned Orientationanti-vortexPhase distribution (Theory)vortexInterference pattern (Theory)cf. free vortexfoldedRotation of the phase by 2π around a vortex, and by -2π around an anti-vortexPhase distribution (Experiment)cdInterference pattern (Experiment)Vortex – antivortex distancehealing lengthsubtractconstantphase slope19

Phase Mapping for Vortex-Pairs with Switching OrientationsIf the vortex and anti-vortex can flip mutual positions from shot to shot, areas withπ-phase shift and surrounded by the minimum fringe visibility should be still observed.TheoryExperimentfolded between vortexand anti-vortexexact cancellation 2017

Outline• Recap on Bose-Einstein condensation of exciton-polaritons:MotivationE vs. k dispersionThermodynamicsCoherence• New Experiments on SuperfluidityBogoliubov excitationsMean field energy shift U(n)Universal dispersion relation E/U(n) vs. kξ(n)Quantized vortex-pairVortex-pair with pinned orientationVortex-pair with switching orientation• Future prospectsQuantum simulation of excited states in Bose-Hubbard modelsComputation of NP-complete problems by BECBEC-BCS crossover in exciton-polariton condensates21

Quantum Simulation of Bose-Hubbard Models3D Optical Lattice Experiment: M. Greiner et al., Nature 419, 6901 (2002)s -orbital p -orbital d -orbital(bonded s waves) (anti-bonded p xwaves) (anti-bonded d xywaves)2D Square LatticeTrap for PolaritonsNear FieldAmplitudeFar FieldIntensityd-waveS-wavep-wavek y’k x’22C.W. Lai et al., Nature 450, 529 (2007) N.Y. Kim et al., in preparation

Computing NP-complete problems by Bose-Einstein condensation- T. Byrnes et al., quant-ph/09092530-Objective: Find ground state ofIsing type HamiltonianIsing Hamiltonian can be implementedby measurement-feedback control circuit.ofEquilibration time:(ε: error rate, α: single particle cooling rate)23

BEC-BCS BCS Crossover in Exciton-Polariton CondensatesA.J. Leggett, J. Phys. (Paris) Colloq. 41, C7 (1980)P. Nozieres and S. Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985)P=0.1PthP=1.2PthBelow thresholdUPLPP≈ 60PthP 160Pth≈Above threshold: Goldstone modeJ. Keeling, P. R. Eastham, M.H. Szymanska and P.B. Littlewood,Phys. Rev. B72, 115320 (2005)24

Exciton-Polariton BEC vs. Photon LaserExciton-Polariton BECThreshold carrier densityPhoton LaserThreshold carrier densityexciton-polaritondispersionEexternal injectionelectron-holepair dispersionEstimulatedcoolingspontaneouscoolingk = 0 LPvia multiple phonon emission& polariton-polariton scatteringpopulationinversionstimulated emissionof photonsexternal pumpingfinal bosonic modeleakage of photonsvia cavity mirrorcavity photonfinal bosonic modecrystal ground statekcrystal ground statek25

Exciton-Polariton Condensation vs. Photon Laser: ExperimentH. Deng et al., Proc. Natl. Acad. Sci., 100, 15318 (2003)Polaritons per Mode at k || ∼0polariton condensationE10 3 LP =1.6166 eVphoton laserE CAV =1.6477 eV10 210 110 0polaritoncondensationQuantum degeneracythresholdphotonlaserPhotons per Mode at k || ∼010 -1no inversioninversion10 9 10 10 10 11 10 12injected exciton density (cm -2 )Polariton condensation threshold observed without population inversionElectrically pumped exciton-polariton BEC is under experimental study at several laboratoriesaround the world.26

ConclusionEvidence for Bose-Einstein condensationDramatic change of E vs. k dispersion relationQuantum degeneracy at thermal equilibrium conditionFormation of spatial coherence (ODLRO) and persistent photon bunchingBogoliubov excitation spectrumUniversal dispersion relation E/U vs. kξ experimentally confirmedSound velocity (cm/s)Quantized vortex-pairVortex-pairs with pinned and switching orientations observedVortex-antivortex separation is roughly equal to healing length27