Nonequilibrium theory of exciton-polariton condensates. Michiel Wouters

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1 Noequilibrium theory of excito-polarito codesates Michiel Wouters

2 Collaboratio with Iacopo Carusotto Treto, theory) Vicezo Savoa EPFL theory) Cristiao Ciuti Paris 7, theory) Kostatios Lagoudais, Barbara Pieta, Augusti Baas, Beoit Deveaud EPFL experimet) Maxime Richard, Le Si Dag Greoble, experimet)

3 Outlie Itroductio Mea field model Elemetary excitatios Spectral shape of ihomogeeous codesates Vortices Sychroizatio Beyod mea field Fluctuatios

4 Relatio with Laser ad BEC Plaar Laser Polarito codesate/laser Atomic BEC Coheret field light> light> matter> matter> Iteractio eergy < 10-5 mev 1meV 1 Hz Decay rate 1 mev 1 mev 10-4 Hz

5 The mea field model [M.W. ad I. Carusotto, Phys. Rev. Lett. 99, )] pump relaxatio from free carriers phoo emissio Rate-diffusio equatio: pol-pol scatterig reservoir codesate Geeralized Gross-Pitaevsii equatio: Ispired by model for atom laser by Keer et al. [PRA 58, )]. Related to the Complex Gizburg-Ladau equatio from o-equilibrium patter formatio [M.C. Cross ad P.C. Hoheberg, RMP 1993], used by Cambridge group for polaritos [J. Keelig ad N. Berloff, arxiv: ].

6 Steady state 0 R = 0 R t) R ψ 0 P-P th )/γ 1 P/P th ψ t) μ T μ = g = ψ e ψ 0 0 iμtt ~ = μ g th R codesate iteractios oly

7 Elemetary excitatio spectrum ω iγ ) = ± ω ) Γ 4 ± bog Goldstoe mode diffusive at small recovers spectrum of equilibrium codesate at large : ω bog )>>Γ. form also foud by Littlewood group with Keldysh Gree fuctio techique [M.Szymasa et al. PRL 006] geeral, model idepedet

8 Negative frequecies Bogoliubov theory predicts egative frequecies i the elemetary excitatio spectrum. Should be visible for polaritos Not see so far Try with FWM?

9 Trasmissio vs FWM

10 Outlie Itroductio Mea field model Elemetary excitatios Spectral shape of ihomogeeous codesates Vortices Sychroizatio Beyod mea field Fluctuatios

11 Ihomogeeous system M.W, I. Carusotto ad C. Ciuti, Phys. Rev. B 77, ). Experimet Large pump spot Small pump spot M. Richard et al. PRB 7, ) M. Richard et al. PRL 94, ) Geeralized Gross-Pitaevsii equatio

12 Ihomogeeous system: ituitive picture Blue shifts/quatum pressure create atitrappig potetial Potetial eergy coverted i ietic eergy Depedig o spot size: small spot : more polaritos with large ietic eergy = o dispersio, fiite large spot : more polaritos with large potetial eergy = blue shifted from dispersio, small

13 Outlie Itroductio Mea field model Elemetary excitatios Spectral shape of ihomogeeous codesates Vortices Sychroizatio Beyod mea field Fluctuatios

14 K. Lagoudais et al. arxiv: Vortices Some correlatio measuremet images show a dislocatio i the frige patter Sigularity i relative phase φx)-φ-x) π phase widig

15 Uliely explaatios Thermally activated BKT physics): these vortices would move ad be ot visible after averagig over may experimets, but give a decrease of frige cotrast. Kibble-Zure defects formed whe goig through the phase trasitio): could be pied to disorder, but why always the same sig?

16 Phase sigularities i ggpe No rotatio of the sample! Not at equilibrium GP equatio has real groud state) Not foud with flat exteral potetial Iterplay betwee pumpig-losses-potetial Also foud i Complex Gizburg Ladau equatio [J. Keelig ad N. Berloff, PRL08]

17 A simple cofiguratio? small potetial well Illumiated with big Excitatio laser eergy coservatio forbids this solutio. A state with a vortex is formed istead

18 Outlie Itroductio Mea field model Elemetary excitatios Spectral shape of ihomogeeous codesates Vortices Sychroizatio Beyod mea field Fluctuatios

19 Oe frequecy? Equilibrium: chemical potetial μ is costat. What i oequilibrium polarito codesates? Experimet: Depeds o the positio o the sample A. Baas et al. Phys. Rev. Lett. 100, ).

20 Theory for coupled wells P P J Josephso Oscillatios i the desychroized state Δε M. Wouters, Phys. Rev. B 77, 1130R) 008)

21 Spatially overlappig states P Icreasig pumpig E x P.R. Eastham PRB ).

22 Outlie Itroductio Mea field model Elemetary excitatios Spectral shape of ihomogeeous codesates Vortices Sychroizatio Beyod mea field Fluctuatios

23 Beyod mea field Why? Occupatio of excited states Decay of first order spatial correlatio fuctio BKT??) Secod order correlatio fuctio desity fluctuatios) How? Iclude oise term i Gross-Pitaevsii equatio by iterpretatio as stochastic motio equatio for Wiger distributio fuctio idψ x, t) = dt h m i R ) γ 4 R R R ) γ ) gψ gr ψ x, t) dwx dw x is a oise term that comes from the quatum ature of the field ψ Mai physical process: shot oise of gai ad losses

24 Li with Boltzma I) Icoheret limit of GP equatio = Boltzma without 1 ), ), t x g g m t x dt d i R ψ ψ ψ = h If ', * ) ) ', ), t t δ ψ ψ = it follows that ) ) g dt d = ) ε ε ε δ ε ε ε δ π GP equatio Simplest model for occupatio of excited states = Boltzma ca almost) be reduced to Boltzma if we See e.g. Y. Kaga i Bose-Eistei codesatio, A. Griffi, D. Soe ad S. Strigari eds.), 1994

25 Li with Boltzma II) Boltzma with 1 I the icoheret limit, it follows that Boltzma col out i I R R dt d = ) ) 1) ) γ ε ε Iclude gai/losses/fluctuatios: ) x R out R i R R out R i dw R R t x g g R R i m dt t x id 4 ) ) ), ) ) ), γ ψ ψ γ ψ = h ] )[... g ε ε ε ε δ π Quatum Classical Spurious, but OK if i) ΔV >>1 ii) γ >> g/δv

26 Prelimiary umerical results Built up of log rage spatial coherece for icreasig pump itesity

27 Coclusios Perspectives simple ad geeric model for oequilibrium codesatio elemetary excitatios: diffusive Goldstoe mode explaatio for differet codesate states for small ad large pump spots iterpretatio of experimetally observed vortices due to drivig, dissipatio ad exteral potetial picture of sychroizatio formalism for fluctuatios uderstad fluctuatios study time depedece of codesate formatio