Non-equilibrium quantum phase transition with ultracold atoms in optical cavity

Transcription

1 Non-equilibrium quantum phase transition with ultracold atoms in optical cavity P. Domokos, D. Nagy, G. Kónya, G. Szirmai J. Asbóth, A. Vukics H. Ritsch (Innsbruck) MTA Szilárdtestfizikai és Optikai Kutatóintézete Ortvay Kollokvium,. április 7., ELTE Fizikai Intézet / 8

2 Many-body physics with ultracold gases / 8

3 Many-body physics with ultracold gases Ultracold atoms weakly interacting many-body system (dilute gas) / 8

4 Many-body physics with ultracold gases Ultracold atoms weakly interacting many-body system (dilute gas) T/T crit Bose-Einstein condensate (BEC) coherent, macroscopic matter wave (Gross-Pitaevskii PDE + Bogoliubov quasi-particle excitations) / 8

5 Many-body physics with ultracold gases Ultracold atoms weakly interacting many-body system (dilute gas) T/T crit Bose-Einstein condensate (BEC) coherent, macroscopic matter wave (Gross-Pitaevskii PDE + Bogoliubov quasi-particle excitations) Internal degrees of freedom / 8

6 Many-body physics with ultracold gases Ultracold atoms weakly interacting many-body system (dilute gas) T/T crit Bose-Einstein condensate (BEC) coherent, macroscopic matter wave (Gross-Pitaevskii PDE + Bogoliubov quasi-particle excitations) 8*+&$'",(9"$4(!$:$&4$&,$ Internal degrees of freedom magnetic field Feshbach resonance tune the interaction strength!b!"##$%$&'()*+&$'",()-)$&'.( /$'$$&(*'-).)-$,3$. a( B) + a (! & n * ' B ) B!" "4'5(-#(%$.-&*&,$ % # $ 6 7 %$.-&*&'()*+&$'",(#"$4 ;<<(=(9$.5/*,5 >$.-&*&,$(?< A5$.".( 5'':BC"*D,--%*4-D$43:3/.'5$.".,*3..$& / 8

7 Many-body physics with ultracold gases Ultracold atoms weakly interacting many-body system (dilute gas) T/T crit Bose-Einstein condensate (BEC) coherent, macroscopic matter wave (Gross-Pitaevskii PDE + Bogoliubov quasi-particle excitations) Internal degrees of freedom 8*+&$'",(9"$4(!$:$&4$&,$ magnetic field Feshbach resonance tune the interaction strength optical dipole potentials change the dimensionality (3D, D, D, optical lattices)!b!"##$%$&'()*+&$'",()-)$&'.( /$'$$&(*'-).)-$,3$. a( B) + a (! & n * ' B ) B!" "4'5(-#(%$.-&*&,$ % # $ 6 7 %$.-&*&'()*+&$'",(#"$4 ;<<(=(9$.5/*,5 >$.-&*&,$(?< A5$.".( 5'':BC"*D,--%*4-D$43:3/.'5$.".,*3..$& / 8

8 Many-body physics with ultracold gases Ultracold atoms weakly interacting many-body system (dilute gas) T/T crit Bose-Einstein condensate (BEC) coherent, macroscopic matter wave (Gross-Pitaevskii PDE + Bogoliubov quasi-particle excitations) Internal degrees of freedom magnetic field Feshbach resonance tune the interaction strength optical dipole potentials change the dimensionality (3D, D, D, optical lattices) strongly correlated matter So far, a field reserved to strongly interacting quantum liquids and nuclear physics (note: table top ke) / 8

9 Many-body physics with ultracold gases Ultracold atoms weakly interacting many-body system (dilute gas) T/T crit Bose-Einstein condensate (BEC) coherent, macroscopic matter wave (Gross-Pitaevskii PDE + Bogoliubov quasi-particle excitations) Internal degrees of freedom magnetic field Feshbach resonance tune the interaction strength optical dipole potentials change the dimensionality (3D, D, D, optical lattices) strongly correlated matter So far, a field reserved to strongly interacting quantum liquids and nuclear physics (note: table top ke) Examples (experiments): QPT from superfluid to Mott-insulator phase [Greiner et al., Nature 45, 39 ()] Tonks-Girardeau hard-core Bose gas in D [Kinoshita et al., Science 35, 5 (4)] 3 Kosterlitz-Thouless crossover in D [Hadzibabic et al., Nature 44, 8 (6)] 4 Fermi degeneracy [DeMarco, Science 85, 73 (999)] 5 molecular BEC-BCS pair crossover [Chin et al., Science 35, 8 (4)] / 8

10 Many-body physics with ultracold gases Ultracold atoms weakly interacting many-body system (dilute gas) T/T crit Bose-Einstein condensate (BEC) coherent, macroscopic matter wave (Gross-Pitaevskii PDE + Bogoliubov quasi-particle excitations) Internal degrees of freedom magnetic field Feshbach resonance tune the interaction strength optical dipole potentials change the dimensionality (3D, D, D, optical lattices) strongly correlated matter So far, a field reserved to strongly interacting quantum liquids and nuclear physics (note: table top ke) Examples (experiments): QPT from superfluid to Mott-insulator phase [Greiner et al., Nature 45, 39 ()] Tonks-Girardeau hard-core Bose gas in D [Kinoshita et al., Science 35, 5 (4)] 3 Kosterlitz-Thouless crossover in D [Hadzibabic et al., Nature 44, 8 (6)] 4 Fermi degeneracy [DeMarco, Science 85, 73 (999)] 5 molecular BEC-BCS pair crossover [Chin et al., Science 35, 8 (4)] This talk is focussed on one particular many-body effect related to non-equilibrium quantum phase transitions / 8

11 Bose-Einstein condensate in an optical resonator Cavity QED experiment [Brennecke,...,Esslinger, Nature 45, 68 (7)] ETH (Zürich), Berkeley, ENS (Paris), MIT, Urbana Champaign, Barcelona, London, Hamburg, Tübingen,... 3 / 8

12 Bose-Einstein condensate in an optical resonator Cavity QED experiment Strong light-matter coupling input laser η, ω α α γ α : atom at nodes α : atom at antinodes output κ [Brennecke,...,Esslinger, Nature 45, 68 (7)] ETH (Zürich), Berkeley, ENS (Paris), MIT, Urbana Champaign, Barcelona, London, Hamburg, Tübingen,... 3 / 8

13 Bose-Einstein condensate in an optical resonator Cavity QED experiment Strong light-matter coupling input laser η, ω α α γ α : atom at nodes α : atom at antinodes output κ atom-photon molecule rendszer paraméterek atom γ A= ω ω A g [Brennecke,...,Esslinger, Nature 45, 68 (7)] ETH (Zürich), Berkeley, ENS (Paris), MIT, Urbana Champaign, Barcelona, London, Hamburg, Tübingen,... rezonátor κ lézer ω C = ω ω C 3 / 8

14 Bose-Einstein condensate in an optical resonator Cavity QED experiment Polarizability U = ω C V χ g A Γ = ω C V χ γ g A atom-photon molecule rendszer paraméterek atom γ A= ω ω A g [Brennecke,...,Esslinger, Nature 45, 68 (7)] ETH (Zürich), Berkeley, ENS (Paris), MIT, Urbana Champaign, Barcelona, London, Hamburg, Tübingen,... rezonátor κ lézer ω C = ω ω C 3 / 8

15 Many-body physics with atoms in optical resonators... in optical lattices 4 / 8

16 Many-body physics with atoms in optical resonators... in optical lattices radiative interaction collisional interaction 4 / 8

17 Many-body physics with atoms in optical resonators... in optical lattices radiative interaction long-range ( infinite, round-trip effect) collisional interaction short range (but: tunability, Feshbach) 4 / 8

18 Many-body physics with atoms in optical resonators... in optical lattices radiative interaction long-range ( infinite, round-trip effect) global coupling collisional interaction short range (but: tunability, Feshbach) pairwise 4 / 8

19 Many-body physics with atoms in optical resonators... in optical lattices radiative interaction long-range ( infinite, round-trip effect) global coupling Kuramoto-model, Lipkin Meshkov Glick-model collisional interaction short range (but: tunability, Feshbach) pairwise Hubbard-type models 4 / 8

20 Many-body physics with atoms in optical resonators... in optical lattices radiative interaction long-range ( infinite, round-trip effect) global coupling Kuramoto-model, Lipkin Meshkov Glick-model driven-damped system (non-equilibrim) collisional interaction short range (but: tunability, Feshbach) pairwise Hubbard-type models conservative system 4 / 8

21 Scattering into the cavity phase difference π pumping cos kz 5 / 8

22 Scattering into the cavity phase difference π pumping cos kz atom-atom coupling by interference x x = (n + ) λ/ destructive interference α = 5 / 8

23 Scattering into the cavity phase difference π pumping cos kz atom-atom coupling by interference x x = (n + ) λ/ destructive interference α = x x = n λ/ constructive interference α 4η t (superradiance) 5 / 8

24 Scattering into the cavity phase difference π pumping cos kz atom-atom coupling by interference x x = (n + ) λ/ destructive interference Full contrast for arbitrary small χ U α = x x = n λ/ constructive interference α 4η t (superradiance) 5 / 8

25 Spatial self-organization of atom clouds homogeneous cloud ηt < ηcrit mode function cos(kx) BEC pump laser 6 / 8

26 Spatial self-organization of atom clouds homogeneous cloud ηt < ηcrit crystalline order ηt > ηcrit mode function cos(kx) outcoupled field κ BEC pump laser 6 / 8

27 Spatial self-organization of atom clouds homogeneous cloud crystalline order ηt < ηcrit ηt > ηcrit mode function cos(kx) outcoupled field κ BEC pump laser [P. Domokos, H. Ritsch, PRL 89, 533 ()] thermal gas of atoms [A.T. Black, H.W. Chan, V. Vuletic, PRL 9, 3 (3)] experimental observation [Nagy, Szirmai, Domokos, EPJD 48, 7 (8))] T=, BEC [K. Baumann, C. Guerlin, F. Brennecke, T. Esslinger, Nature 464, 3 ()] experimental observation 6 / 8

28 Quantized atom field in a single-mode resonator One-dimensional toy model for coupled matter and light fields H = C â â + iη (â â) + [ ˆΨ (x) m d dx + Ng c ˆΨ (x) ˆΨ(x) ] + U â â cos (kx) + iη t cos kx(â â) ˆΨ(x)dx, a, a C cos(kx) κ ˆΨ(x) η t,ω pump laser 7 / 8

29 Quantized atom field in a single-mode resonator One-dimensional toy model for coupled matter and light fields H = C â â + iη (â â) + [ ˆΨ (x) m d dx + Ng c ˆΨ (x) ˆΨ(x) ] + U â â cos (kx) + iη t cos kx(â â) ˆΨ(x)dx, a, a C cos(kx) ˆΨ(x) η t,ω pump laser κ scattering processes (four-wave mixing) absorption and induced emission of cavity photons absorption of a pump photon and emission into the cavity 7 / 8

30 Quantized atom field in a single-mode resonator One-dimensional toy model for coupled matter and light fields H = C â â + iη (â â) + [ ˆΨ (x) m d dx + Ng c ˆΨ (x) ˆΨ(x) ] + U â â cos (kx) + iη t cos kx(â â) ˆΨ(x)dx, a, a C cos(kx) ˆΨ(x) η t,ω pump laser κ scattering processes (four-wave mixing) absorption and induced emission of cavity photons absorption of a pump photon and emission into the cavity dissipation and noise (k B T = ) d dt â = i [â, H] κâ + ˆξ ˆξ(t)ˆξ (t ) = κδ(t t ) 7 / 8

31 Hamiltonian dynamics, quantum criticality Two-mode description of atoms ˆΨ(x) = c + L L c cos kx [ ci, c i ] = i =, Number of particles: c c + c c = N [Nagy, Kónya, Szirmai, Domokos, PRL 4, 34 ()] 8 / 8

32 Hamiltonian dynamics, quantum criticality Two-mode description of atoms Spin representation ˆΨ(x) = L c + [ ci, c i L c cos kx ] = i =, Number of particles: c c + c c = N Ŝ x = (c c + c c ) Ŝ y = i (c c c c ) Ŝ z = (c c c c ) [Nagy, Kónya, Szirmai, Domokos, PRL 4, 34 ()] 8 / 8

33 Hamiltonian dynamics, quantum criticality Two-mode description of atoms Spin representation ˆΨ(x) = L c + [ ci, c i L c cos kx ] = i =, Number of particles: c c + c c = N Ŝ x = (c c + c c ) Ŝ y = i (c c c c ) Ŝ z = (c c c c ) Dicke-type Hamiltonian H/ = δ C a a + ω R Ŝ z + iy(a a)ŝ x / N + ua a ( + Ŝ z /N ) ω R = k /m δ C = C u < u = N U /4 y = Nη t tunable [Nagy, Kónya, Szirmai, Domokos, PRL 4, 34 ()] 8 / 8

34 Hamiltonian dynamics, quantum criticality Two-mode description of atoms Spin representation ˆΨ(x) = L c + [ ci, c i L c cos kx ] = i =, Number of particles: c c + c c = N Ŝ x = (c c + c c ) Ŝ y = i (c c c c ) Ŝ z = (c c c c ) Dicke-type Hamiltonian H/ = δ C a a + ω R Ŝ z + iy(a a)ŝ x / N + ua a ( + Ŝ z /N ) ω R = k /m δ C = C u < u = N U /4 y = Nη t tunable Critical point y crit = δ C ω R [Nagy, Kónya, Szirmai, Domokos, PRL 4, 34 ()] 8 / 8

35 Hamiltonian dynamics, quantum criticality Two-mode description of atoms Spin representation ˆΨ(x) = L c + [ ci, c i L c cos kx ] = i =, Number of particles: c c + c c = N Ŝ x = (c c + c c ) Ŝ y = i (c c c c ) Ŝ z = (c c c c ) Dicke-type Hamiltonian H/ = δ C a a + ω R Ŝ z + iy(a a)ŝ x / N + ua a ( + Ŝ z /N ) ω R = k /m δ C = C u < u = N U /4 y = Nη t tunable Critical point y crit = δ C ω R Originally: photons and atomic excitations [Nagy, Kónya, Szirmai, Domokos, PRL 4, 34 ()] 8 / 8

36 Ground state in the thermodynamic limit Holstein-Primakoff Ŝ + = b N b b Ŝ = N b b b Ŝ z = b b N Spontaneous symmetry breaking for y < y ˆb = β = crit δ Cu δ u y y crit C y u for y > y crit δ C y crit 9 / 8

37 Ground state in the thermodynamic limit Holstein-Primakoff Ŝ + = b N b b Ŝ = N b b b Ŝ z = b b N Spontaneous symmetry breaking for y < y ˆb = β = crit δ Cu δ u y y crit C y u for y > y crit δ C y crit populations.6 β α.5.5 y/y c.5 9 / 8

38 Ground state in the thermodynamic limit Holstein-Primakoff Ŝ + = b N b b Ŝ = N b b b Ŝ z = b b N Spontaneous symmetry breaking for y < y ˆb = β = crit δ Cu δ u y y crit C y u for y > y crit δ C y crit Two boson mode Hamiltonian populations H/ = (δ C uβ )a a + + i Mc (a a)(b + b) + Mx My 4 Mx +My b b (b + b ) β y/y c α β 3 β M c = y ( ) + uα β / β M x = ω R yα β ( ) M β 3/ y = ω R yα β ( ) β / 9 / 8

39 Ground state in the thermodynamic limit Holstein-Primakoff Ŝ + = b N b b Ŝ = N b b b Ŝ z = b b N Spontaneous symmetry breaking for y < y ˆb = β = crit δ Cu δ u y y crit C y u for y > y crit δ C y crit Two boson mode Hamiltonian populations H/ = (δ C uβ )a a + + i Mc (a a)(b + b) + Mx My 4 Mx +My b b (b + b ) b b y/yc a a β 3 β M c = y ( ) + uα β / β M x = ω R yα β ( ) M β 3/ y = ω R yα β ( ) β / 9 / 8

40 Non-equilibrium dynamics (mean field) Mean-field approach â(t) = α(t) + δâ(t) ˆΨ(x, t) = Nϕ(x, t) + δ ˆΨ(x, t) / 8

41 Non-equilibrium dynamics (mean field) Mean-field approach â(t) = α(t) + δâ(t) ˆΨ(x, t) = Nϕ(x, t) + δ ˆΨ(x, t) Gross Pitaevskii-type equation i { } t α = C + NU cos (kx) iκ α + Nη t cos(kx) i t ϕ(x, t) = { m x + α(t) U cos (kx) + Re{α(t)}η t cos(kx) + Ng c ϕ(x, t) }ϕ(x, t) / 8

42 Non-equilibrium dynamics (mean field) Mean-field approach â(t) = α(t) + δâ(t) ˆΨ(x, t) = Nϕ(x, t) + δ ˆΨ(x, t) Gross Pitaevskii-type equation i { } t α = C + NU cos (kx) iκ α + Nη t cos(kx) i t ϕ(x, t) = { m x + α(t) U cos (kx) + Re{α(t)}η t cos(kx) + Ng c ϕ(x, t) }ϕ(x, t) Kuramoto-model Θ i = ω i + K N ( ) sin Θ i Θ j N j= mean-field ρ(θ, ω, t) ρ + { } [ω + Kr sin(ψ Θ)] ρ = t Θ π re iψ = e iθ ρ(θ, ω, t) g(ω)dωdθ π / 8

43 Mean-field self-organization of a BEC in a cavity order parameter Θ = ϕ cos kx ϕ order parameter Θ N η [in units of ωr] 95 [Nagy, Szirmai, Domokos, EPJD 48, 7 (8))] / 8

44 Mean-field self-organization of a BEC in a cavity order parameter steady-states order parameter Θ Θ = ϕ cos kx ϕ N η [in units of ωr] ψ(x) [units of /λ] cavity axis x/λ lattice potential [units of ωr] [Nagy, Szirmai, Domokos, EPJD 48, 7 (8))] / 8

45 Mean-field self-organization of a BEC in a cavity order parameter order parameter Θ threshold Θ = ϕ cos kx ϕ 65 3 N η [in units of ωr] 95 Nηc = δ C + κ δ C steady-states ψ(x) [units of /λ] ωr + Ng c.5.5 cavity axis x/λ lattice potential [units of ωr] [Nagy, Szirmai, Domokos, EPJD 48, 7 (8))] / 8

46 Mean-field self-organization of a BEC in a cavity order parameter order parameter Θ threshold Θ = ϕ cos kx ϕ 65 3 N η [in units of ωr] 95 Nηc = δ C + κ δ C steady-states ψ(x) [units of /λ] ωr + Ng c.5.5 cavity axis x/λ relation to Dicke-H: κ and g c = (applies to the mean-field wavefunction, too) lattice potential [units of ωr] [Nagy, Szirmai, Domokos, EPJD 48, 7 (8))] / 8

47 Mean-field self-organization of a BEC in a cavity order parameter order parameter Θ threshold Θ = ϕ cos kx ϕ 65 3 N η [in units of ωr] 95 Nηc = δ C + κ δ C steady-states ψ(x) [units of /λ] ωr + Ng c.5.5 cavity axis x/λ relation to Dicke-H: κ and g c = (applies to the mean-field wavefunction, too) relation to classical gas: temperature kinetic energy + collision lattice potential [units of ωr] [Nagy, Szirmai, Domokos, EPJD 48, 7 (8))] / 8

48 Experimental mapping of the phase diagram Pump lattice depth (E r) a Pump-cavity detuning (MHz) 3 4 n b Time (ms) c 3 Time (ms) Mean photon number n Pump power (µw) [Baumann, Guerlin, Brennecke, Esslinger, Nature 464, 3 ()] / 8

49 Classes of non-equilibrium systems Thermodynamics Quantum states Equilibrium Non-equilibrium criticality example 3 / 8

50 Classes of non-equilibrium systems Thermodynamics Quantum states Equilibrium Non-equilibrium criticality example Rayleigh-Be nard convection 3 / 8

51 Classes of non-equilibrium systems Thermodynamics Quantum states Equilibrium Non-equilibrium T criticality example Rayleigh-Be nard convection superfluid vortices spin current a la Ra cz 3 / 8

52 Classes of non-equilibrium systems Thermodynamics Quantum states Equilibrium Non-equilibrium T criticality example Rayleigh-Be nard convection superfluid vortices spin current a la Ra cz selforganization BEC in a cavity 3 / 8

53 Spectrum of fluctuations linearized equations ˆR = im ˆR + ˆξ t ˆR [δâ, δâ, δ ˆΨ, δ ˆΨ ] M = M(α, ϕ (x), µ) ˆξ = [ˆξ, ˆξ,, ] 4 / 8

54 Spectrum of fluctuations 6 linearized equations excitation frequencies [ωr] a) ˆR = im ˆR + ˆξ t ˆR [δâ, δâ, δ ˆΨ, δ ˆΨ ] M = M(α, ϕ (x), µ) ˆξ = [ˆξ, ˆξ,, ] N η [in units of ωr] 45 4 / 8

55 Spectrum of fluctuations 6 linearized equations excitation frequencies [ωr] a) ˆR = im ˆR + ˆξ t ˆR [δâ, δâ, δ ˆΨ, δ ˆΨ ] M = M(α, ϕ (x), µ) ˆξ = [ˆξ, ˆξ,, ] N η [in units of ωr] b) decay rates [ωr] N η [in units of ωr] 45 4 / 8

56 Spectrum of fluctuations 6 linearized equations excitation frequencies [ωr] a) ˆR = im ˆR + ˆξ t ˆR [δâ, δâ, δ ˆΨ, δ ˆΨ ] M = M(α, ϕ (x), µ) ˆξ = [ˆξ, ˆξ,, ] N η [in units of ωr] 45 critical point interval.8.7 b) decay rates [ωr] eigenvalues N η [in units of ωr] y/y c 4 / 8

57 Linearized dynamics of quantum fluctuations Normal mode decomposition ˆR = im ˆR t + ˆξ left and right eigenvectors of M ( l (k), r (l) ) = δ k,l normal modes ˆρ k = ( l (k), ˆR) t ˆρ k = iω k ˆρ k + ˆQk projected noise ˆQk ( l (k), ˆξ) 5 / 8

58 Linearized dynamics of quantum fluctuations Normal mode decomposition Second-order correlation functions ˆR = im ˆR t + ˆξ left and right eigenvectors of M ( l (k), r (l) ) = δ k,l normal modes ˆρ k = ( l (k), ˆR) t ˆρ k = iω k ˆρ k + ˆQk projected noise ˆQk ( l (k), ˆξ) ˆξ(t)ˆξ (t ) = κδ(t t ) for T = ˆρk (t)ˆρ l (t) = ˆρ k ()ˆρ l () e i(ω k +ω l )t + κ e i(ω k +ω l )t i(ω k + ω l ) l (k) (l) l 5 / 8

59 Linearized dynamics of quantum fluctuations Normal mode decomposition Second-order correlation functions ˆR = im ˆR t + ˆξ left and right eigenvectors of M ( l (k), r (l) ) = δ k,l normal modes ˆρ k = ( l (k), ˆR) t ˆρ k = iω k ˆρ k + ˆQk projected noise ˆQk ( l (k), ˆξ) ˆξ(t)ˆξ (t ) = κδ(t t ) for T = ˆρk (t)ˆρ l (t) = ˆρ k ()ˆρ l () e i(ω k +ω l )t + κ e i(ω k +ω l )t i(ω k + ω l ) l (k) (l) l κ steady-state i(ω k + ω l ) l(k) (l) l 5 / 8

60 Linearized dynamics of quantum fluctuations Normal mode decomposition Second-order correlation functions ˆR = im ˆR t + ˆξ left and right eigenvectors of M ( l (k), r (l) ) = δ k,l normal modes ˆρ k = ( l (k), ˆR) t ˆρ k = iω k ˆρ k + ˆQk projected noise ˆQk ( l (k), ˆξ) ˆξ(t)ˆξ (t ) = κδ(t t ) for T = ˆρk (t)ˆρ l (t) = ˆρ k ()ˆρ l () e i(ω k +ω l )t + κ e i(ω k +ω l )t i(ω k + ω l ) l (k) (l) l κ steady-state i(ω k + ω l ) l(k) (l) l Quantum depletion of the condensate δ δn(t) = ˆΨ (x, t) δ ˆΨ(x, t) dx = ˆρ k (t)ˆρ l (t) k,l r (k) 4 (x) r (l) (x) dx 3 5 / 8

61 Quantum features of the steady-state: singularity BEC depletion δb δb.5.3. β.5..5 y/yc.5 photon number δa δa α...5 y/yc.5 6 / 8

62 Quantum features of the steady-state: singularity BEC depletion 3.5 Critical exponent = δb δb y/yc β log δb δb log y/yc -6 photon number δa δa α...5 y/yc.5 6 / 8

63 Quantum features of the steady-state: singularity BEC depletion 3.5 Critical exponent = δb δb y/yc β log δb δb log y/yc -6 Analytical result below threshold (ω R κ, C ) photon number δn ( C NU /) + κ (y/yc ) 8ω R ( C + NU /) (y/y c ) δa δa α...5 y/yc.5 6 / 8

64 Quantum features of the steady-state: singularity BEC depletion 3.5 Critical exponent = δb δb y/yc β log δb δb log y/yc -6 Analytical result below threshold (ω R κ, C ) photon number.5 δn ( C NU /) + κ (y/yc ) 8ω R ( C + NU /) (y/y c ) δa δa α two-mode squeezing b b ( y/y c )...5 y/yc.5 6 / 8

65 Quantum features of the steady-state: singularity BEC depletion 3.5 Critical exponent = δb δb y/yc β log δb δb log y/yc -6 Analytical result below threshold (ω R κ, C ) photon number.5 δn ( C NU /) + κ (y/yc ) 8ω R ( C + NU /) (y/y c ).8.4 two-mode squeezing b b ( y/y c ) δa δa α Excess noise depletion.. y δn = O(κ/ω R, C /ω R ).5 y/yc.5 [Szirmai, Nagy, Domokos, PRL, 84 (9)] 6 / 8

66 Evolution towards the steady state System prepared in the mean-field solution: diffuses out towards the steady-state 7 / 8

67 Evolution towards the steady state System prepared in the mean-field solution: diffuses out towards the steady-state Depletion rate. Diffusion [units of κ] y/y c / 8

68 Evolution towards the steady state System prepared in the mean-field solution: diffuses out towards the steady-state Depletion rate. Diffusion [units of κ].5..5 Coarse graining ( δ C + κ) / δt ω R δn(t) δt = κ M c δ C + κ < κω R δ C with M c = y below threshold.5.5 y/y c.5 3 [Nagy, Domokos, Vukics, Ritsch, EPJD 55, 659 (9)] 7 / 8

69 Summary Many-body effects in the motion of atoms in a cavity global coupling (Dicke model) LMG-model (each spin interacts identically with every other) [Lipkin, Meshkov, Glick, Nucl. Phys. 6, 88 (965)] H LMG = hs z y N (S x + γs y ) How to obtain it: [Morrison and Parkins, PRL, 443 (8)] self-organization: a non-equilibrium phase transition (classical vs. quantum) ground state of the Hamiltonian : criticality experimental realization of the Dicke-type quantum phase transition steady-state of the driven-damped system: criticality, other exponent Outlook finite size scaling modeling the non-equilibrium dynamics (beyond linearization) 8 / 8