Collective Dynamics of a Generalized Dicke Model

Collective Dynamics of a Generalized Dicke Model J. Keeling, J. A. Mayoh, M. J. Bhaseen, B. D. Simons Harvard, January 212 Funding: Jonathan Keeling Collective dynamics Harvard, January 212 1 / 25

Coupling many atoms to light Old question: What happens to radiation when many atoms interact collectively with light. Superradiance dynamical and steady state. Jonathan Keeling Collective dynamics Harvard, January 212 2 / 25

Coupling many atoms to light Old question: What happens to radiation when many atoms interact collectively with light. Superradiance dynamical and steady state. New relevance Superconducting qubits Quantum dots Nitrogen-vacancies in diamond Ultra-cold atoms κ κ Cavity Pump Rydberg atoms Pump Jonathan Keeling Collective dynamics Harvard, January 212 2 / 25

Dicke effect: Enhanced emission H int = k,i ) g k (ψ k S i e ik r i + H.c. Jonathan Keeling Collective dynamics Harvard, January 212 3 / 25

Dicke effect: Enhanced emission H int = k,i ) g k (ψ k S i e ik r i + H.c. If r i r j λ, use i S i S Collective decay: dρ dt = Γ 2 [ S + S ρ S ρs + + ρs + S ] Jonathan Keeling Collective dynamics Harvard, January 212 3 / 25

Dicke effect: Enhanced emission H int = k,i ) g k (ψ k S i e ik r i + H.c. If r i r j λ, use i S i S Collective decay: dρ dt = Γ 2 [ S + S ρ S ρs + + ρs + S ] If S z = S = N/2 initially: I Γ d Sz dt = ΓN2 4 sech2 [ ] ΓN 2 t S z N/2 -N/2 t D t D ΓN 2 /2 I=-Γd S z /dt Jonathan Keeling Collective dynamics Harvard, January 212 3 / 25

Dicke effect: Enhanced emission H int = k,i ) g k (ψ k S i e ik r i + H.c. If r i r j λ, use i S i S Collective decay: dρ dt = Γ 2 [ S + S ρ S ρs + + ρs + S ] If S z = S = N/2 initially: I Γ d Sz dt = ΓN2 4 sech2 [ ] ΓN 2 t S z N/2 -N/2 t D t D ΓN 2 /2 I=-Γd S z /dt Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972] Jonathan Keeling Collective dynamics Harvard, January 212 3 / 25

Collective radiation with a cavity: Dynamics H int = i ( ψ S i + ψs + i ) Single cavity mode: oscillations [Bonifacio and Preparata PRA 7; JK PRA 9] Jonathan Keeling Collective dynamics Harvard, January 212 4 / 25

Collective radiation with a cavity: Dynamics H int = i ( ψ S i + ψs + i ) Single cavity mode: oscillations If S z = S = N/2 initially: ψ(t) 2 16 T=2ln( N )/ N 14 12 1 8 6 4 2 1/ N.1.2.3.4.5.6.7.8 Time [Bonifacio and Preparata PRA 7; JK PRA 9] Jonathan Keeling Collective dynamics Harvard, January 212 4 / 25

Collective radiation with a cavity: Dynamics H int = i ( ψ S i + ψs + i ) Single cavity mode: oscillations If S z = S = N/2 initially: ψ(t) 2 16 T=2ln( N )/ N 14 12 1 8 6 4 2 1/ N.1.2.3.4.5.6.7.8 Time [Bonifacio and Preparata PRA 7; JK PRA 9] ψ(t) 2 16 N=2 14 12 1 8 6 4 2 1 2 3 4 5 Time Jonathan Keeling Collective dynamics Harvard, January 212 4 / 25

Dicke model: Equilibrium superradiance transition ( H = ωψ ψ + ω S z + g ψ S + ψs +). Coherent state: Ψ e λψ +ηs + Ω Energy: E = ω λ 2 + ω N η 2 1 2 η 2 + 1 + gn η λ + λ η 1 + η 2 Small g, min at λ, η = [Hepp, Lieb, Ann. Phys. 73] Jonathan Keeling Collective dynamics Harvard, January 212 5 / 25

Dicke model: Equilibrium superradiance transition ( H = ωψ ψ + ω S z + g ψ S + ψs +). Coherent state: Ψ e λψ +ηs + Ω Energy: E = ω λ 2 + ω N η 2 1 2 η 2 + 1 + gn η λ + λ η 1 + η 2 Small g, min at λ, η = ω ḡ N SR Spontaneous polarisation if: Ng 2 > ωω [Hepp, Lieb, Ann. Phys. 73] Jonathan Keeling Collective dynamics Harvard, January 212 5 / 25

No go theorem and transition Spontaneous polarisation if: Ng 2 > ωω [Rzazewski et al PRL 75] Jonathan Keeling Collective dynamics Harvard, January 212 6 / 25

No go theorem and transition Spontaneous polarisation if: Ng 2 > ωω No go theorem:. Minimal coupling (p ea) 2 /2m i e m A p i g(ψ S + ψs + ), i A 2 2m Nζ(ψ + ψ ) 2 [Rzazewski et al PRL 75] Jonathan Keeling Collective dynamics Harvard, January 212 6 / 25

No go theorem and transition Spontaneous polarisation if: Ng 2 > ωω No go theorem:. Minimal coupling (p ea) 2 /2m i e m A p i g(ψ S + ψs + ), For large N, ω ω + 2Nζ. (RWA) i A 2 2m Nζ(ψ + ψ ) 2 [Rzazewski et al PRL 75] Jonathan Keeling Collective dynamics Harvard, January 212 6 / 25

No go theorem and transition Spontaneous polarisation if: Ng 2 > ωω No go theorem:. Minimal coupling (p ea) 2 /2m i e m A p i g(ψ S + ψs + ), For large N, ω ω + 2Nζ. (RWA) Need Ng 2 > ω (ω + 2Nζ). i A 2 2m Nζ(ψ + ψ ) 2 [Rzazewski et al PRL 75] Jonathan Keeling Collective dynamics Harvard, January 212 6 / 25

No go theorem and transition Spontaneous polarisation if: Ng 2 > ωω No go theorem:. Minimal coupling (p ea) 2 /2m i e m A p i g(ψ S + ψs + ), i A 2 2m Nζ(ψ + ψ ) 2 For large N, ω ω + 2Nζ. (RWA) Need Ng 2 > ω (ω + 2Nζ). But Thomas-Reiche-Kuhn sum rule states: g 2 /ω < 2ζ. No transition [Rzazewski et al PRL 75] Jonathan Keeling Collective dynamics Harvard, January 212 6 / 25

Dicke phase transition: ways out Problem: g 2 /ω < 2ζ for intrinsic parameters. Solutions: Non-solution Ferroelectric transition in D r gauge. [JK JPCM 7 ] See also [Nataf and Ciuti, Nat. Comm. 1; Viehmann et al. PRL 11] Grand canonical ensemble: If H H µ(s z + ψ ψ), need only: g 2 N > (ω µ)(ω µ) Incoherent pumping polariton condensation. Dissociate g, ω, e.g. Raman scheme: ω ω. [Dimer et al. PRA 7; Baumann et al. Nature 1. Also, Black et al. PRL 3 ] Jonathan Keeling Collective dynamics Harvard, January 212 7 / 25

Dicke phase transition: ways out Problem: g 2 /ω < 2ζ for intrinsic parameters. Solutions: Non-solution Ferroelectric transition in D r gauge. [JK JPCM 7 ] See also [Nataf and Ciuti, Nat. Comm. 1; Viehmann et al. PRL 11] Grand canonical ensemble: If H H µ(s z + ψ ψ), need only: g 2 N > (ω µ)(ω µ) Incoherent pumping polariton condensation. Dissociate g, ω, e.g. Raman scheme: ω ω. [Dimer et al. PRA 7; Baumann et al. Nature 1. Also, Black et al. PRL 3 ] κ Pump κ Cavity Pump Jonathan Keeling Collective dynamics Harvard, January 212 7 / 25

Outline 1 Introduction: Dicke model and superradiance Rayleigh scheme: Generalised Dicke model 2 Attractors of dynamics (fixed points) Phase diagram: comparison to experiment 3 Approach to attractors: timescales Why slow timescales emerge 4 Attractors of dynamics (oscillations) Reaching other parameter ranges Jonathan Keeling Collective dynamics Harvard, January 212 8 / 25

Extended Dicke model [Baumann et al. Nature 1] z x κ Pump κ gψ Ω 2 Level System 2 Level system,, : : Ψ(x, z) = 1 : Ψ(x, z) = ω = 2ω recoil e ik(σx+σ z) σ,σ =± H = ωψ ψ + ω S z + g(ψ + ψ )(S + S + )+US z ψ ψ. t ρ = i[h, ρ] κ(ψ ψρ 2ψρψ + ρψ ψ) Jonathan Keeling Collective dynamics Harvard, January 212 1 / 25

Extended Dicke model [Baumann et al. Nature 1] z x κ Pump κ gψ Ω 2 Level System 2 Level system,, : : Ψ(x, z) = 1 : Ψ(x, z) = ω = 2ω recoil e ik(σx+σ z) σ,σ =± Feedback: U g2 ω c ω a H = ωψ ψ + ω S z + g(ψ + ψ )(S + S + )+US z ψ ψ. t ρ = i[h, ρ] κ(ψ ψρ 2ψρψ + ρψ ψ) Semiclassical EOM ( S = N/2 1) Ṡ = i(ω +U ψ 2 )S + 2ig(ψ + ψ )S z Ṡ z = ig(ψ + ψ )(S S + ) ψ = [κ + i(ω+us z )] ψ ig(s + S + ) Jonathan Keeling Collective dynamics Harvard, January 212 1 / 25

Extended Dicke model [Baumann et al. Nature 1] z x κ Pump κ gψ Ω 2 Level System 2 Level system,, : : Ψ(x, z) = 1 : Ψ(x, z) = ω = 2ω recoil e ik(σx+σ z) σ,σ =± Feedback: U g2 ω c ω a H = ωψ ψ + ω S z + g(ψ + ψ )(S + S + )+US z ψ ψ. t ρ = i[h, ρ] κ(ψ ψρ 2ψρψ + ρψ ψ) Semiclassical EOM ( S = N/2 1) Ṡ = i(ω +U ψ 2 )S + 2ig(ψ + ψ )S z Ṡ z = ig(ψ + ψ )(S S + ) ψ = [κ + i(ω+us z )] ψ ig(s + S + ) ω khz ω, κ, g N MHz. Jonathan Keeling Collective dynamics Harvard, January 212 1 / 25

Fixed points (steady states) = i(ω +U ψ 2 )S + 2ig(ψ + ψ )S z = ig(ψ + ψ )(S S + ) = [κ + i(ω+us z )] ψ ig(s + S + ) ψ =, S = (,, ±N/2) always a solution. If g > g c, ψ too A S y = I[S ] = B ψ = R[ψ] = Jonathan Keeling Collective dynamics Harvard, January 212 12 / 25

Fixed points (steady states) = i(ω +U ψ 2 )S + 2ig(ψ + ψ )S z = ig(ψ + ψ )(S S + ) = [κ + i(ω+us z )] ψ ig(s + S + ) ψ =, S = (,, ±N/2) always a solution. If g > g c, ψ too A S y = I[S ] = B ψ = R[ψ] = S z S y S x Small g:, only. (ω = 3MHz, UN = 4MHz) Jonathan Keeling Collective dynamics Harvard, January 212 12 / 25

Fixed points (steady states) = i(ω +U ψ 2 )S + 2ig(ψ + ψ )S z = ig(ψ + ψ )(S S + ) = [κ + i(ω+us z )] ψ ig(s + S + ) ψ =, S = (,, ±N/2) always a solution. If g > g c, ψ too A S y = I[S ] = B ψ = R[ψ] = S z S y S x Small g:, only. Larger g: SR too. (ω = 3MHz, UN = 4MHz) Jonathan Keeling Collective dynamics Harvard, January 212 12 / 25

Steady state phase diagram UN=, κ= = i(ω +U ψ 2 )S + 2ig(ψ + ψ )S z ω SR = ig(ψ + ψ )(S S + ) = [κ + i(ω+us z )] ψ ig(s + S + ) ḡ N See also Domokos and Ritsch PRL 2, Domokos et al. PRL 1 Jonathan Keeling Collective dynamics Harvard, January 212 13 / 25

Steady state phase diagram = i(ω +U ψ 2 )S + 2ig(ψ + ψ )S z = ig(ψ + ψ )(S S + ) = [κ + i(ω+us z )] ψ ig(s + S + ) 4 UN= ω UN=, κ= ḡ N SR SR(A): S y = 2 SR -2 SR -4.5 1 1.5 g N (MHz) See also Domokos and Ritsch PRL 2, Domokos et al. PRL 1 Jonathan Keeling Collective dynamics Harvard, January 212 13 / 25

Steady state phase diagram = i(ω +U ψ 2 )S + 2ig(ψ + ψ )S z = ig(ψ + ψ )(S S + ) = [κ + i(ω+us z )] ψ ig(s + S + ) 4 UN=-1 ω UN=, κ= ḡ N SR SR(A): S y = 2-2 + SRB + SR(B): ψ = -4.5 1 1.5 g N (MHz) See also Domokos and Ritsch PRL 2, Domokos et al. PRL 1 Jonathan Keeling Collective dynamics Harvard, January 212 13 / 25

Steady state phase diagram = i(ω +U ψ 2 )S + 2ig(ψ + ψ )S z = ig(ψ + ψ )(S S + ) = [κ + i(ω+us z )] ψ ig(s + S + ) 4 UN=-1 2 + + SRB -2 + -4.5 1 1.5 g N (MHz) See also Domokos and Ritsch PRL 2, Domokos et al. PRL 1 ω UN=, κ= ḡ N SR SR(A): S y = + SR(B): ψ = Jonathan Keeling Collective dynamics Harvard, January 212 13 / 25

Steady state phase diagram = i(ω +U ψ 2 )S + 2ig(ψ + ψ )S z = ig(ψ + ψ )(S S + ) = [κ + i(ω+us z )] ψ ig(s + S + ) 4 UN=-2 ω UN=, κ= ḡ N SR SR(A): S y = 2-2 + SRB + SR(B): ψ = -4.5 1 1.5 g N (MHz) See also Domokos and Ritsch PRL 2, Domokos et al. PRL 1 Jonathan Keeling Collective dynamics Harvard, January 212 13 / 25

Steady state phase diagram = i(ω +U ψ 2 )S + 2ig(ψ + ψ )S z = ig(ψ + ψ )(S S + ) = [κ + i(ω+us z )] ψ ig(s + S + ) 4 2 UN=-4 ω UN=, κ= ḡ N SR SR(A): S y = -2 + SRB + SR(B): ψ = -4.5 1 1.5 g N (MHz) See also Domokos and Ritsch PRL 2, Domokos et al. PRL 1 Jonathan Keeling Collective dynamics Harvard, January 212 13 / 25

Steady state phase diagram = i(ω +U ψ 2 )S + 2ig(ψ + ψ )S z = ig(ψ + ψ )(S S + ) = [κ + i(ω+us z )] ψ ig(s + S + ) 4 2-2 + SRB + + + + SRB+ SRB+ UN=-4 SRB -4.5 1 1.5 g N (MHz) See also Domokos and Ritsch PRL 2, Domokos et al. PRL 1 ω UN=, κ= ḡ N SR SR(A): S y = + SR(B): ψ = Jonathan Keeling Collective dynamics Harvard, January 212 13 / 25

Comparison to experiment (ω p -ω c ) (2π MHz) -1-2 -3-4.5 1 1.5 2 2.5 g 2 N (MHz) 2 UN = 1MHz Adapted from: [Bhaseen et al. PRA 12] [Baumann et al Nature 1 ] ω = ω c ω p + 5 2 UN, UN = g 2 4(ω a ω c ) Jonathan Keeling Collective dynamics Harvard, January 212 14 / 25

Dynamics: Evolution from normal state 4 2 (ii) (i) UN=-4 + SRB -2 (iii) -4.5 1 1.5 g N (MHz) Jonathan Keeling Collective dynamics Harvard, January 212 16 / 25

Dynamics: Evolution from normal state Gray: S = ( N, N, N/2) Black: Wigner distribution of S, ψ 4 2 + (ii) (i) UN=-4 SRB (i) SR(A) ψ 2 8 4 1 1 2 2 4 6 8 t (ms) -2 (iii) -4.5 1 1.5 g N (MHz) Oscillations:.1ms Decay: 2ms Jonathan Keeling Collective dynamics Harvard, January 212 16 / 25

Dynamics: Evolution from normal state Gray: S = ( N, N, N/2) Black: Wigner distribution of S, ψ 4 2-2 + (ii) -4.5 1 1.5 g N (MHz) (i) (iii) Oscillations:.1ms Decay: 2ms,.1ms UN=-4 SRB (i) SR(A) (ii) SR(B) ψ 2 ψ 2 8 4 2 1 1 1 2 2 4 6 8 t (ms).1.2.3.4 t (ms) Jonathan Keeling Collective dynamics Harvard, January 212 16 / 25

Dynamics: Evolution from normal state Gray: S = ( N, N, N/2) Black: Wigner distribution of S, ψ 4 2-2 + (ii) -4.5 1 1.5 g N (MHz) (i) (iii) Oscillations:.1ms Decay: 2ms,.1ms, 2ms UN=-4 SRB (i) SR(A) (ii) SR(B) (iii) SR(A) ψ 2 ψ 2 ψ 2 8 4 2 1 12 8 4 1 1 2 2 4 6 8 t (ms).1.2.3.4 t (ms) 5 4 15 151 1 2 t (ms) Jonathan Keeling Collective dynamics Harvard, January 212 16 / 25

Asymptotic state: Evolution from normal state (Near to experimental UN = 13MHz). All stable attractors: 4 UN=-1 2 + SRB -2-4.5 1 1.5 g N (MHz) Jonathan Keeling Collective dynamics Harvard, January 212 17 / 25

Asymptotic state: Evolution from normal state (Near to experimental UN = 13MHz). Starting from All stable attractors: 4 2 4 Asymptotic state UN=-1 2 + SRB SRB 1 3 1 2 1 1 ψ 2-2 -2 1-4.5 1 1.5 g N (MHz) -4.5 1 1.5 g N (MHz) 1-1 Jonathan Keeling Collective dynamics Harvard, January 212 17 / 25

Timescales for dynamics: What are they? 4 2-2 Asymptotic state SRB -4.5 1 1.5 g N (MHz) 1 3 1 2 1 1 1 1-1 ψ 2 4 2-2 No unstable directions Initial growth time One unstable direction Two unstable directions -4.5 1 1.5 g N (MHz) 1s 1s 1ms 1ms 1ms 1µs 1µs Growth Most unstable eigenvalues near S = (,, N/2) Decay Slowest stable eigenvalues near final state Jonathan Keeling Collective dynamics Harvard, January 212 18 / 25

Timescales for dynamics: What are they? 4 2-2 Asymptotic state SRB -4.5 1 1.5 g N (MHz) 1 3 1 2 1 1 1 1-1 ψ 2 4 2-2 No unstable directions Initial growth time One unstable direction Two unstable directions -4.5 1 1.5 g N (MHz) 1s 1s 1ms 1ms 1ms 1µs 1µs Growth Most unstable eigenvalues near S = (,, N/2) Decay Slowest stable eigenvalues near final state Expand in ω /κ: Oscillations: ω, Decay: ω or ω 2 /κ Jonathan Keeling Collective dynamics Harvard, January 212 18 / 25

Timescales for dynamics: What are they? 4 2-2 Asymptotic state SRB -4.5 1 1.5 g N (MHz) Growth Most unstable eigenvalues near S = (,, N/2) 1 3 1 2 1 1 1 1-1 Decay Slowest stable eigenvalues near final state Expand in ω /κ: Oscillations: ω, Decay: ω or ω 2 /κ ψ 2 4 2-2 No unstable directions Initial growth time One unstable direction Two unstable directions -4.5 1 1.5 4 2-2 g N (MHz) Asymptotic decay time SRB -4.5 1 1.5 g N (MHz) 1s 1s 1ms 1ms 1ms 1µs 1µs 1s 1s 1ms 1ms 1ms 1µs 1µs Jonathan Keeling Collective dynamics Harvard, January 212 18 / 25

Timescales for dynamics: Consequences for experiment 4 Asymptotic state 1 3 2 1 2 SRB 1 1 ψ 2-2 1-4.5 1 1.5 g N (MHz) 1-1 Jonathan Keeling Collective dynamics Harvard, January 212 19 / 25

Timescales for dynamics: Consequences for experiment 6 1ms sweep 1 3 4 2 Asymptotic state SRB 1 3 1 2 1 1 ψ 2 4 2-2 -4..5 1. 1.5 2. 2.5 g 2 N (MHz 2 ) 1 2 1 1 1 1-1 -2 1-4.5 1 1.5 g N (MHz) 1-1 Jonathan Keeling Collective dynamics Harvard, January 212 19 / 25

Timescales for dynamics: Consequences for experiment 6 1ms sweep 1 3 4 2 Asymptotic state SRB 1 3 1 2 1 1 ψ 2 4 2-2 -4..5 1. 1.5 2. 2.5 g 2 N (MHz 2 ) 1 2 1 1 1 1-1 -2 1 6 2ms sweep 1 3-4.5 1 1.5 g N (MHz) 1-1 4 2 1 2 1 1-2 1-4 1-1..5 1. 1.5 2. 2.5 g 2 N (MHz 2 ) Jonathan Keeling Collective dynamics Harvard, January 212 19 / 25

Timescales for dynamics: Why so slow and varied? Suppose co- and counter-rotating terms differ a b Ω a Ωb g ψ g ψ H =... + g(ψ S + ψs + ) + g (ψ S + + ψs ) +... 2 Level System SR(A) near phase boundary at small δg Critical slowing down SR(A), SR(B) continuously connect Jonathan Keeling Collective dynamics Harvard, January 212 2 / 25

Timescales for dynamics: Why so slow and varied? Suppose co- and counter-rotating terms differ a b Ω a Ωb g ψ g ψ H =... + g(ψ S + ψs + ) + g (ψ S + + ψs ) +... 2 Level System 4 UN=-1 2 + SRB -2-4.5 1 1.5 g N (MHz) SR(A) near phase boundary at small δg Critical slowing down SR(A), SR(B) continuously connect Jonathan Keeling Collective dynamics Harvard, January 212 2 / 25

Timescales for dynamics: Why so slow and varied? Suppose co- and counter-rotating terms differ a b Ω a Ωb g ψ g ψ H =... + g(ψ S + ψs + ) + g (ψ S + + ψs ) +... 2 Level System δg = g g, 2ḡ = g + g 4 2 UN=-1 4 2 ḡ N=1 + SRB -2-2 -4.5 1 1.5 g N (MHz) -4 -.1 -.5.5.1 δg/ḡ SR(A) near phase boundary at small δg Critical slowing down SR(A), SR(B) continuously connect Jonathan Keeling Collective dynamics Harvard, January 212 2 / 25

Regions without fixed points Changing U: gψ Ω U g2 ω c ω a 2 Level System 4 2 UN=-4 + SRB -2-4.5 1 1.5 g N (MHz) Jonathan Keeling Collective dynamics Harvard, January 212 22 / 25

Regions without fixed points g ψ Changing U: Ω U g2 ω c ω a 2 Level System 4 2 UN=-4 + SRB -2-4.5 1 1.5 g N (MHz) Jonathan Keeling Collective dynamics Harvard, January 212 22 / 25

Regions without fixed points g ψ Changing U: Ω U g2 ω c ω a 2 Level System 4 UN= 2 SR -2 SR -4.5 1 1.5 g N (MHz) Jonathan Keeling Collective dynamics Harvard, January 212 22 / 25

Regions without fixed points g ψ Changing U: Ω U g2 ω c ω a 2 Level System 4 UN=2 2-2 -4.5 1 1.5 g N (MHz) Jonathan Keeling Collective dynamics Harvard, January 212 22 / 25

Regions without fixed points g ψ Changing U: Ω U g2 ω c ω a 2 Level System 4 2 UN=4-2 -4.5 1 1.5 g N (MHz) Jonathan Keeling Collective dynamics Harvard, January 212 22 / 25

Regions without fixed points g ψ Changing U: Ω U g2 ω c ω a 2 Level System 4 UN=4 2 Persistent Oscillations -2-4.5 1 1.5 g N (MHz) ψ 2 12 112 8 8 6 4 4 2 5 1 15 2 4 6 8 1 12 14 16 18 t (ms) Jonathan Keeling Collective dynamics Harvard, January 212 22 / 25

Persistent (optomechanical) oscillations ψ 2 12 112 8 6 4 2 8 4 5 1 15 2 4 1 6 8 12 14 16 18 t (ms) Ṡ = i(ω + U ψ 2 )S + 2ig(ψ + ψ )S z Ṡ z = ig(ψ + ψ )(S S + ) ψ = [κ + i(ω + US z )] ψ ig(s + S + ) Jonathan Keeling Collective dynamics Harvard, January 212 23 / 25

Persistent (optomechanical) oscillations ψ 2 12 112 8 6 4 2 8 4 5 1 15 2 4 1 6 8 12 14 16 18 t (ms) Ṡ = i(ω + U ψ 2 )S + 2ig(ψ + ψ )S z Ṡ z = ig(ψ + ψ )(S S + ) ψ = [κ + i(ω + US z )] ψ ig(s + S + ) Fix ω + US z = if ψ =. Jonathan Keeling Collective dynamics Harvard, January 212 23 / 25

Persistent (optomechanical) oscillations ψ 2 12 112 8 6 4 2 8 4 5 1 15 2 4 1 6 8 12 14 16 18 t (ms) Fix ω + US z = if ψ =. N S = re iθ 2 r = 4 (Sz ) 2 Ṡ = i(ω + U ψ 2 )S + 2ig(ψ + ψ )S z Ṡ z = ig(ψ + ψ )(S S + ) ψ = [κ + i(ω + US z )] ψ ig(s + S + ) Get: θ = ω + U ψ 2 ψ + κψ = 2igr cos(θ) Jonathan Keeling Collective dynamics Harvard, January 212 23 / 25

Persistent (optomechanical) oscillations ψ 2 12 112 8 6 4 2 8 4 5 1 15 2 4 1 6 8 12 14 16 18 t (ms) Fix ω + US z = if ψ =. N S = re iθ 2 r = 4 (Sz ) 2 ψ 2 12 1 8 6 4 ψ 2 S x S y S z Ṡ = i(ω + U ψ 2 )S + 2ig(ψ + ψ )S z Ṡ z = ig(ψ + ψ )(S S + ) ψ = [κ + i(ω + US z )] ψ ig(s + S + ) Get: θ = ω + U ψ 2 ψ + κψ = 2igr cos(θ) 2 -.4 18. 18.2 18.4 18.6 18.8 t(ms) Jonathan Keeling Collective dynamics Harvard, January 212 23 / 25.4.2 -.2 S x, S y, S z

Tuning g, g, U [Dimer et al. Phys. Rev. A. (27)] Ω a a b Ω Separate pump strength/detuning b g ψ g ψ 2 2 Level System g g Ω b, g g Ω a b, U g g a a 2 b Jonathan Keeling Collective dynamics Harvard, January 212 24 / 25

Tuning g, g, U [Dimer et al. Phys. Rev. A. (27)] Ω a a b Ω Separate pump strength/detuning b g ψ g ψ 2 2 Level System g g Ω b, g g Ω a b Possible realization: Hyperfine levels, U g g a a 2 b σ σ + m = 1 F m = F m =+1 F Jonathan Keeling Collective dynamics Harvard, January 212 24 / 25

Tuning g, g, U [Dimer et al. Phys. Rev. A. (27)] Ω a a b Ω Separate pump strength/detuning b g ψ g ψ 2 2 Level System g g Ω b, g g Ω a b Possible realization: Hyperfine levels, U g g a a 2 b σ σ + Atoms m = 1 F m = F m =+1 F B Jonathan Keeling Collective dynamics Harvard, January 212 24 / 25

Summary Wide variety of dynamical phases 4 2 UN=4 4 2 UN= SR 4 2 UN=-4 4 2 ḡ N=1 + SRB -2-2 SR -2-2 -4.5 1 1.5 g N (MHz) -4.5 1 1.5 g N (MHz) -4.5 1 1.5 g N (MHz) -4 -.1 -.5.5.1 δg/ḡ Slow dynamics ψ 2 12 8 4 5 4 15 151 1 2 t (ms) 4 Initial growth time No unstable directions 2 One unstable direction -2 Two unstable directions -4.5 1 1.5 g N (MHz) 1s 1s 1ms 1ms 1ms 1µs 1µs 4 Asymptotic decay time 2 SRB -2-4.5 1 1.5 g N (MHz) 1s 1s 1ms 1ms 1ms 1µs 1µs Persistent oscillations if U > ψ 2 12 12 1 8 8 6 4 4 2 5 1 15 2 4 6 8 1 12 14 16 18 t (ms) ψ 2 12 ψ 2 S.4 1 x S y S 8 z.2 6 4 -.2 2 -.4 18. 18.2 18.4 18.6 18.8 t(ms) JK et al. PRL 1, Bhaseen et al. PRA 12 S x, S y, S z Jonathan Keeling Collective dynamics Harvard, January 212 25 / 25

Jonathan Keeling Collective dynamics Harvard, January 212 26 / 3

Extra slides 5 Ferroelectric phase transition 6 Lipkin-Meshkov-Glick 7 Tricritical point Jonathan Keeling Collective dynamics Harvard, January 212 27 / 3

Ferroelectric transition Atoms in Coulomb gauge H = ω k a k a k + i [p i ea(r i )] 2 + V coul Jonathan Keeling Collective dynamics Harvard, January 212 28 / 3

Ferroelectric transition Atoms in Coulomb gauge H = ω k a k a k + i [p i ea(r i )] 2 + V coul Two-level systems dipole-dipole coupling H = ω S z + ωψ ψ + g(s + + S )(ψ + ψ ) + Nζ(ψ + ψ ) 2 η(s + S ) 2 (nb g 2, ζ, η 1/V ). Jonathan Keeling Collective dynamics Harvard, January 212 28 / 3

Ferroelectric transition Atoms in Coulomb gauge H = ω k a k a k + i [p i ea(r i )] 2 + V coul Two-level systems dipole-dipole coupling H = ω S z + ωψ ψ + g(s + + S )(ψ + ψ ) + Nζ(ψ + ψ ) 2 η(s + S ) 2 (nb g 2, ζ, η 1/V ). Ferroelectric polarisation if ω < 2ηN Jonathan Keeling Collective dynamics Harvard, January 212 28 / 3

Ferroelectric transition Atoms in Coulomb gauge H = ω k a k a k + i [p i ea(r i )] 2 + V coul Two-level systems dipole-dipole coupling H = ω S z + ωψ ψ + g(s + + S )(ψ + ψ ) + Nζ(ψ + ψ ) 2 η(s + S ) 2 (nb g 2 Ferroelectric polarisation if ω, ζ, η 1/V ). < 2ηN Gauge transform to dipole gauge D r H = ω S z + ωψ ψ + ḡ(s + S )(ψ ψ ) Dicke transition at ω < Nḡ 2 /ω 2ηN But, ψ describes electric displacement Jonathan Keeling Collective dynamics Harvard, January 212 28 / 3

Timescales for dynamics: U = and Lipkin-Meshkov-Glick Since κ ω, can consider eliminating ψ If U =, simple result: Ṡ = i(ω +U ψ 2 )S + 2ig(ψ + ψ )S z Ṡ z = ig(ψ + ψ )(S S + ) ψ = [κ + i(ω+us z )] ψ ig(s + S + ) t S = {S, H} ΓS (S ẑ), H = ω S z Λ + S 2 x Λ S 2 y with: Λ ± ω (g ± g ) 2, Γ 2κ (g 2 g 2 ) κ 2 +ω 2 κ 2 +ω 2 NB, g = g, no dissipation. Dissipation restored by finite κ. Jonathan Keeling Collective dynamics Harvard, January 212 29 / 3

Tricritical point 4 2-2 + UN=-4 SRB -4.5 1 1.5 g N (MHz) 2 19 18 + + 19.7 + 2 SRB SRB+ + 17 SRB 19.68 SRB + 1.6 1.65.5 1 1.5 g N (MHz) If ω u = UN/2 > κ, crossing of boundaries at: ω = ωu 2 κ 2 g ω UN N = 4 For g > g, 2 region exists at edge of SRB. ψ 2 Arg(ψ) S z /N 15 (a) 1 5 π (c) π/2 -π/2 -π.5 (e) g N =1. (MHz) SRB -.5-2 -1 1 2 (b) 2 SRB (d) (f) 8 6 4 2 9π/16 π/2 7π/16.5.4.3 4.1 4.2 Jonathan Keeling Collective dynamics Harvard, January 212 3 / 3