Non-equilibrium quantum many-body physics with optical systems

Transcription

1 Non-equilibrium quantum many-body physics with optical systems Iacopo Carusotto BEC CNR-INFM and Università di Trento, Italy

2 Many experimental signatures of polariton BEC 1 Narrowing of the momentum distribution Bose-Einstein condensate of exciton-polaritons Kasprzak et al., Nature 443, 409 (2006) The first atomic BEC M. H. Anderson et al. Science 269, 198 (1995)

3 2 First order coherence: Young two-slit experiment No BEC BE C Interference pattern of emitted light from a polariton BEC M. Richard et al., PRL 94, (2005) Interference pattern of two expanding atomic BECs M. R. Andrews, Science 275, 637

4 3 Threshold behaviour Coherence suddenly appears at threshold M. Richard et al., PRL 94, (2005) Bose condensed fraction J. R. Ensher et al. PRL 77, 4984 (1996)

5 4 Long coherence time Condensate emission: phase coherent for long times what determines decoherence? D. N. Krizhanovskii, D. Sanvitto, A. P. Love, M. S. Skolnick, D. M. Whittaker, and J. S. Roberts, PRL 97, (2006) role of interactions beyond standard Schawlow-Townes laser linewidth? analogies and differences with equilibrium system (cfr Alice's talk)?

6 5 Noise reduction in the condensed phase Razor-blade experiment to mesure g(2) A. Baas et al., PRL 96, (2006) Suppressed density fluctuations by 3! reduced 3-body recombination rate E. A. Burt et al. PRL (1997)

7 6 Quantized vortices Vortices in polariton BEC: K. G. Lagoudakis, M. Wouters, M. Richard, A. Baas, IC, R. André, Le Si Dang, B. Deveaud-Pledran, Quantised Vortices in an Exciton Polariton Fluid, Nature Physics 4, 706 (2008). dislocated fringes in interference pattern pinned to defects quantization of phase circulation by 2π

8 More than standard BEC: non-equilibrium steady state Optical injection Relaxation: polariton-polariton and polariton-phonon scattering Stimulation of scattering to lowest states Losses: particle number NOT conserved NO thermodynamical equilibrium Steady-state determined by dynamical balance of driving and dissipation (Figure from Kasprzak et al., Nature 2006) Standard concepts of equilibrium statistical mechanics are not applicable Physics might be different from usual equilibrium BEC

9 We therefore wonder... What new non-equilibrium physics can be learnt from polaritons that was not possible with liquid Helium and ultracold atoms? What more in polariton BEC than standard laser operation? strong interactions, significant fluctuations, continuum of modes along cavity plane, peculiar dispersion relation of polaritons... What are the consequences on applications to optoelectronic devices? Can it lead to completely new states of matter under strong interactions?

10 The physical system: DBR microcavity with QWs Photon 0 Exciton 2 C k = C 1 k / k z Polaritons DBR: stack λ/4 layers (e.g. GaAs/AlAs) Cavity layer confined photonic mode, delocalized along 2D plane: 2 e and h confined in QW layer (e.g. InGaAs) e-h pair: sort of H atom. Exciton Excitons bosons if nexc a2bohr «1 Excitons delocalized along cavity plane. Flat exciton dispersion ωx(k) ωx Exciton radiatively coupled to cavity photon at same in-plane k Bosonic superpositions of exciton and photon, called polaritons Polariton mass mpol 10-4 me BEC at high temperature!!

11 Part I The shape of a non-equilibrium condensate

12 Intruiguing experimental data for the BEC shape below at above below above wide pump spot: 20 μm narrow pump spot: 3 μm M. Richard et al., PRB 72, (2005) M. Richard et al., PRL 94, (2005) Experimental observations: condensate formation under non-resonant pump condensate shape depends on pump spot size visible interference fringes: condensate coherent and not fragmented

13 A generalized GPE for non-resonantly pumped BECs Condensate : mean-field approx., GPE with losses / amplification macroscopic wavefunction ψ(x), loss rate γ, amplification R(nB) Incoherent reservoir : rate equation for density nb(x) pumping rate P, spatial diffusion D, thermalization rate γb Fast reservoir limit: reduces to a Complex-Ginzburg Landau Equation M. Wouters, IC, Excitations in a Nonequilibrium Bose-Einstein Condensate of Exciton Polaritons, PRL 99, (2007) Including (quantum) fluctuations: IC, C. Ciuti, PRB 72, (2005); M. Wouters, V. Savona, PRB (09)

14 Numerical integration of non-equilibrium GPE Equilibrium, harmonic trap: Thomas-Fermi parabolic profile Non-equilibrium: dynamics affects shape. Stationary flow possible related work on flow patterns : J. Keeling, N. G. Berloff, PRL 100, (2008) Narrow pump spot: σ = 5μm Emission on ring at finite k Wide pump spot: σ = 20 μm Broad emission centered at k=0 Good agreement with experiments!! M. Wouters, IC, and C. Ciuti, Spatial and spectral shape of inhomogeneous non-equilibrium exciton-polariton condensates, PRB 77, (2008)

15 Physical interpretation of condensation at k 0 Repulsive interactions outward radial acceleration energy conservation σ= 20 μm E=k2/2m + Uint(r) radially increasing local flow velocity coherent ballistic flow Narrow spot: ballistic free flight outside pump spot Uint(r)=0 emission mostly on free particle dispersion =5 μm

16 Part II The elementary excitation spectrum of a non-equilibrium condensate

17 Bogoliubov theory of non-equilibrium BEC Linearize GPE around steady state Reservoir R mode at -i γr Condensate density and phase modes at: with: density (-) and phase (+) oscillations decoupled around k=0 Goldstone phase (+) mode is diffusive M. Wouters and IC, Excitations in a non-equilibrium polariton BEC, Phys. Rev. Lett. 99, (2007) Similar results in: M. H. Szymanska, J. Keeling, P. B. Littlewood, PRL 96, (2006)

18 Consequences on superfluidity Naïf Landau argument: Landau critical velocity vl=mink[ω(k) / k] = 0 at non-equilibrium BEC Any moving defect expected to emit phonons

19 But nature is always richer than expected... Low v : emitted k purely imaginary no real propagating phonons localized perturbation around defect Critical velocity vc< c: corresponds to bifurcation point decreases with Γ / μ High v: v/c=0.7 v/c=1.3 v/c=2.0 emitted propagating phonons: Cerenkov cone parabolic precursors spatial damping of Cerenkov cone M. Wouters and IC, Excitations and superfluidity in non-equilibrium Bose-Einstein condensates of exciton-polaritons, Superlattices and microstructures 43, 524 (2008).

20 Part III Experimental study of Landau superfluidity (under coherent pump)

21 Simplest configuration: coherent, resonant pump Direct injection of polaritons by resonant pump coherence inherited from pump condensate dynamics fully described by non-equilibrium GPE no incoherent dynamics to be modeled, almost ab initio calculation next order: Bogoliubov theory, interesting quantum fluctuation effects (quantum images, analog Hawking radiation...) resonant Rayleigh scattering on defect, suppressed when superfluid near- and far-field profiles observed in reflection and/or transmission IC and C.Ciuti, PRL 93, (2004) Related proposals: Chiao, Boyce, PRA Phys. Rev. A 60, 4114 (1999); Tanzini, Sorella, PLA 263, 43 (1999). Bolda, Chiao, Zurek, PRL 86, 416 (2001).

22 Theoretical predictions for homogeneous system Polaritons injected at finite kp: flow towards right incident on defect Non-interacting polaritons Non-superfluid (a kind of vp > cs) IC and C.Ciuti, PRL 93, (2004) Superfluid (a kind of vp < cs)

23 Experimental data: superfluid behaviour superfluid flow scattering on defect increase polariton density Figure from LKB-P6 group: J.Lefrère, A.Amo, S.Pigeon, C.Adrados, C.Ciuti, IC, R. Houdré, E.Giacobino, A.Bramati, Observation of Superfluidity of Polaritons in Semiconductor Microcavities, arxiv: , to appear on Nature Physics Theory: IC and C. Ciuti, PRL 93, (2004).

24 More experimental data: Cerenkov wake Experiment Theory Expt with atomic BEC Expt. image from JILA (P. Engels, E. Cornell). Theory IC, Hu, Collins, Smerzi, PRL 97, (2006) Super-sonic flow hitting a defect: Cerenkov conical wave, aperture cos(φ) = cs / v single-particle-like parabolic precursors Expt with duck J.Lefrère, A.Amo, S.Pigeon, C.Adrados, C.Ciuti, IC, R. Houdré, E.Giacobino, A.Bramati, arxiv: ; IC and C. Ciuti, PRL 93, (2004).

25 Review of polariton superfluidity expts Paris 6 (Bramati-Giacobino group, Nature Phys. in the press) Resonant laser directly injecting coherent polaritons forming BEC, no spontaneous BEC transition Relatively weak perturbation in quasi-homogeneous flow, direct implementation of Landau criterion Quantitative comparison to theory Madrid I (Sanvitto-Viña group, Nature 2009) TOPO configuration: pump around threshold to spontaneous parametric BEC Long-lived moving BEC injected by pulsed probe on parametric idler Bogoliubov theory with naïve Landau criterion would predict non-superfluid (yet dispersionless) behaviour Possible interpretation of observed bullet propagation: robust solitonic wavepacket bound by nonlinearity Madrid II (Sanvitto-Viña group, arxiv/09) TOPO configuration, vortices imprinted by probe pulse with angular momentum Long-lived vortices: metastable supercurrents. Related to gyrolaser operation Other experiments Sonic dispersion observed in luminescence, but no negative-energy Bogoliubov feature (Yamamoto) Quantized phase in spontaneous vortices, but no immediate relation to superfluidity (Deveaud) Still to experimentally investigate Diffusive collective excitation spectrum of spontaneous BEC under incoherent or OPO-like pump, negative energy features Consequences of diffusive diffusive (flat) dispersion on superfluidity

26 The new frontier: non-equilibrium strongly correlated quantum gases Array of coupled nonlinear cavities: driven-dissipative Bose-Hubbard model Equilibrium system: Mott superfluid to insulator transition Pumping and losses likely to destabilize commensurate Mott phase Other correlations robust against non-equilibrium if U» J» γ Tonks-Girardeau states under coherent pump: Impenetrable bosons for U» J : Girardeau mapping onto Fermi wavefunction Fermionization of polaritons apparent in spectral position of transmission peaks Strongly correlated real- and far-field luminescence Possible implementations: photonic crystal cavities, micropillars, microdisk, polariton boxes, superconducting circuits,... IC, D. Gerace, H. E. Türeci, S. De Liberato, C. Ciuti, A. Imamoglu, PRL 103, (2009)

27 Work done in collaboration with: Michiel Wouters Cristiano Ciuti (EPFL Lausanne) (Univ. Paris 7) Davide Sarchi Simon Pigeon Arnaud Verger (BEC-Trento) (Paris 7) (Paris 7, now St. Gobain) Atac Imamoglu Hakan Türeci Dario Gerace Rosario Fazio (ETH Zürich) (ETH Zürich) (Univ. Pavia) (SNS-Pisa) Taofiq Paraïso Konstantinos Lagoudakis Augustin Baas Benoit Deveaud (EPFL Lausanne) Alberto Amo Alberto Bramati Elizabeth Giacobino (LKB, Paris 6) Daniele Sanvitto Luis Viña (UAM Madrid) Le Si Dang Maxime Richard (Grenoble) PhD and PostDoc positions soon available at BEC-Trento Brave young researchers: PLEASE APPLY!! Photo: Fabrice Neyret And thanks to our experimental friends: