Collective Dynamics of a Generalized Dicke Model
J. Keeling, J. A. Mayoh, M. J. Bhaseen, B. D. Simons
Harvard, January 2012
Funding:
Jonathan Keeling Collective dynamics Harvard, January 2012 1 / 25 New relevance Superconducting qubits
Quantum dots
Nitrogen-vacancies in diamond
Cavity κ κ Ultra-cold atoms Pump
Pump Rydberg atoms
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 2012 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
Cavity κ κ Ultra-cold atoms Pump
Pump Rydberg atoms
Jonathan Keeling Collective dynamics Harvard, January 2012 2 / 25 If |r − r | λ, use P S → S dρ Γ i j i i = − S+S−ρ − S−ρS+ + ρS+S− Collective decay: dt 2
N/2 2 ΓN /2 z
= | | = / /dt
If S S N 2 initially: 〉 z 〉 S z
z 2 0 〈 S d dhS i ΓN ΓN 〈 I ∝ −Γ = sech2 t Γ dt 4 2 I=- -N/2 0 tD tD Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972]
Dicke effect: Enhanced emission
X † − −ik·ri Hint = gk ψk Si e + H.c. k,i
Jonathan Keeling Collective dynamics Harvard, January 2012 3 / 25 N/2 2 ΓN /2 z
= | | = / /dt
If S S N 2 initially: 〉 z 〉 S z
z 2 0 〈 S d dhS i ΓN ΓN 〈 I ∝ −Γ = sech2 t Γ dt 4 2 I=- -N/2 0 tD tD Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972]
Dicke effect: Enhanced emission
X † − −ik·ri Hint = gk ψk Si e + H.c. k,i
If |r − r | λ, use P S → S dρ Γ i j i i = − S+S−ρ − S−ρS+ + ρS+S− Collective decay: dt 2
Jonathan Keeling Collective dynamics Harvard, January 2012 3 / 25 Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972]
Dicke effect: Enhanced emission
X † − −ik·ri Hint = gk ψk Si e + H.c. k,i
If |r − r | λ, use P S → S dρ Γ i j i i = − S+S−ρ − S−ρS+ + ρS+S− Collective decay: dt 2
N/2 2 ΓN /2 z
= | | = / /dt
If S S N 2 initially: 〉 z 〉 S z
z 2 0 〈 S d dhS i ΓN ΓN 〈 I ∝ −Γ = sech2 t Γ dt 4 2 I=- -N/2 0 tD tD
Jonathan Keeling Collective dynamics Harvard, January 2012 3 / 25 Dicke effect: Enhanced emission
X † − −ik·ri Hint = gk ψk Si e + H.c. k,i
If |r − r | λ, use P S → S dρ Γ i j i i = − S+S−ρ − S−ρS+ + ρS+S− Collective decay: dt 2
N/2 2 ΓN /2 z
= | | = / /dt
If S S N 2 initially: 〉 z 〉 S z
z 2 0 〈 S d dhS i ΓN ΓN 〈 I ∝ −Γ = sech2 t Γ dt 4 2 I=- -N/2 0 tD tD Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972]
Jonathan Keeling Collective dynamics Harvard, January 2012 3 / 25 If Sz = |S| = N/2 initially:
1600 __ __ 1600 T=2ln(√N)/√N N=2000 1400 1400 1200 1200 2 2 1000 1000 (t)| (t)| 800 800 ψ ψ | | 600 600 400 400 __ 200 1/√N 200 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 5 Time Time
Collective radiation with a cavity: Dynamics
X † − + Single cavity mode: oscillations Hint = ψ Si + ψSi i
[Bonifacio and Preparata PRA ’70; JK PRA ’09]
Jonathan Keeling Collective dynamics Harvard, January 2012 4 / 25 1600 N=2000 1400 1200
2 1000
(t)| 800 ψ | 600 400 200 0 0 1 2 3 4 5 Time
Collective radiation with a cavity: Dynamics
X † − + Single cavity mode: oscillations Hint = ψ Si + ψSi z i If S = |S| = N/2 initially:
1600 __ __ T=2ln(√N)/√N 1400 1200
2 1000
(t)| 800 ψ | 600 400 __ 200 1/√N 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time [Bonifacio and Preparata PRA ’70; JK PRA ’09]
Jonathan Keeling Collective dynamics Harvard, January 2012 4 / 25 Collective radiation with a cavity: Dynamics
X † − + Single cavity mode: oscillations Hint = ψ Si + ψSi z i If S = |S| = N/2 initially:
1600 __ __ 1600 T=2ln(√N)/√N N=2000 1400 1400 1200 1200 2 2 1000 1000 (t)| (t)| 800 800 ψ ψ | | 600 600 400 400 __ 200 1/√N 200 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 5 Time Time [Bonifacio and Preparata PRA ’70; JK PRA ’09]
Jonathan Keeling Collective dynamics Harvard, January 2012 4 / 25 ω ⇓ SR
0 0 g-√N
2 Spontaneous polarisation if: Ng > ωω0
Dicke model: Equilibrium superradiance transition
† z † − + H = ωψ ψ + ω0S + g ψ S + ψS .
† + Coherent state: |Ψi → eλψ +ηS |Ωi Energy: ω N |η|2 − 1 η∗λ + λ∗η E = ω|λ|2 + 0 + gN 2 |η|2 + 1 1 + |η|2 Small g, min at λ, η = 0
[Hepp, Lieb, Ann. Phys. ’73]
Jonathan Keeling Collective dynamics Harvard, January 2012 5 / 25 ω ⇓ SR
0 0 g-√N
2 Spontaneous polarisation if: Ng > ωω0
Dicke model: Equilibrium superradiance transition
† z † − + H = ωψ ψ + ω0S + g ψ S + ψS .
† + Coherent state: |Ψi → eλψ +ηS |Ωi Energy: ω N |η|2 − 1 η∗λ + λ∗η E = ω|λ|2 + 0 + gN 2 |η|2 + 1 1 + |η|2 Small g, min at λ, η = 0
[Hepp, Lieb, Ann. Phys. ’73]
Jonathan Keeling Collective dynamics Harvard, January 2012 5 / 25 ω ⇓ SR
0 0 g-√N
2 Spontaneous polarisation if: Ng > ωω0
Dicke model: Equilibrium superradiance transition
† z † − + H = ωψ ψ + ω0S + g ψ S + ψS .
† + Coherent state: |Ψi → eλψ +ηS |Ωi Energy: ω N |η|2 − 1 η∗λ + λ∗η E = ω|λ|2 + 0 + gN 2 |η|2 + 1 1 + |η|2 Small g, min at λ, η = 0
[Hepp, Lieb, Ann. Phys. ’73]
Jonathan Keeling Collective dynamics Harvard, January 2012 5 / 25 Dicke model: Equilibrium superradiance transition
† z † − + H = ωψ ψ + ω0S + g ψ S + ψS .
† + Coherent state: |Ψi → eλψ +ηS |Ωi
ω ⇓ Energy: SR ω N |η|2 − 1 η∗λ + λ∗η E = ω|λ|2 + 0 + gN 2 |η|2 + 1 1 + |η|2 0 0 Small g, min at λ, η = 0 g-√N 2 Spontaneous polarisation if: Ng > ωω0 [Hepp, Lieb, Ann. Phys. ’73]
Jonathan Keeling Collective dynamics Harvard, January 2012 5 / 25 No go theorem:. Minimal coupling (p − eA)2/2m
X e X A2 − A · p ⇔ g(ψ†S− + ψS+), ⇔ Nζ(ψ + ψ†)2 m i 2m i i For large N, ω → ω + 2Nζ. (RWA) 2 Need Ng > ω0(ω + 2Nζ). 2 But Thomas-Reiche-Kuhn sum rule states: g /ω0 < 2ζ. No transition
No go theorem and transition
2 Spontaneous polarisation if: Ng > ωω0
[Rzazewski et al PRL ’75]
Jonathan Keeling Collective dynamics Harvard, January 2012 6 / 25 For large N, ω → ω + 2Nζ. (RWA) 2 Need Ng > ω0(ω + 2Nζ). 2 But Thomas-Reiche-Kuhn sum rule states: g /ω0 < 2ζ. No transition
No go theorem and transition
2 Spontaneous polarisation if: Ng > ωω0 No go theorem:. Minimal coupling (p − eA)2/2m
X e X A2 − A · p ⇔ g(ψ†S− + ψS+), ⇔ Nζ(ψ + ψ†)2 m i 2m i i
[Rzazewski et al PRL ’75]
Jonathan Keeling Collective dynamics Harvard, January 2012 6 / 25 2 Need Ng > ω0(ω + 2Nζ). 2 But Thomas-Reiche-Kuhn sum rule states: g /ω0 < 2ζ. No transition
No go theorem and transition
2 Spontaneous polarisation if: Ng > ωω0 No go theorem:. Minimal coupling (p − eA)2/2m
X e X A2 − A · p ⇔ g(ψ†S− + ψS+), ⇔ Nζ(ψ + ψ†)2 m i 2m i i For large N, ω → ω + 2Nζ. (RWA)
[Rzazewski et al PRL ’75]
Jonathan Keeling Collective dynamics Harvard, January 2012 6 / 25 2 But Thomas-Reiche-Kuhn sum rule states: g /ω0 < 2ζ. No transition
No go theorem and transition
2 Spontaneous polarisation if: Ng > ωω0 No go theorem:. Minimal coupling (p − eA)2/2m
X e X A2 − A · p ⇔ g(ψ†S− + ψS+), ⇔ Nζ(ψ + ψ†)2 m i 2m i i For large N, ω → ω + 2Nζ. (RWA) 2 Need Ng > ω0(ω + 2Nζ).
[Rzazewski et al PRL ’75]
Jonathan Keeling Collective dynamics Harvard, January 2012 6 / 25 No go theorem and transition
2 Spontaneous polarisation if: Ng > ωω0 No go theorem:. Minimal coupling (p − eA)2/2m
X e X A2 − A · p ⇔ g(ψ†S− + ψS+), ⇔ Nζ(ψ + ψ†)2 m i 2m i i For large N, ω → ω + 2Nζ. (RWA) 2 Need Ng > ω0(ω + 2Nζ). 2 But Thomas-Reiche-Kuhn sum rule states: g /ω0 < 2ζ. No transition [Rzazewski et al PRL ’75]
Jonathan Keeling Collective dynamics Harvard, January 2012 6 / 25 Cavity κ κ
Pump
Pump
Dicke phase transition: ways out
2 Problem: g /ω0 < 2ζ for intrinsic parameters. Solutions: Non-solution Ferroelectric transition in D · r gauge. [JK JPCM ’07 ] See also [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11] Grand canonical ensemble: z † I If H → H − µ(S + ψ ψ), need only: 2 g N > (ω − µ)(ω0 − µ) I Incoherent pumping — polariton condensation.
Dissociate g, ω0, e.g. Raman scheme: ω0 ω. [Dimer et al. PRA ’07; Baumann et al. Nature ’10. Also, Black et al. PRL ’03 ]
Jonathan Keeling Collective dynamics Harvard, January 2012 7 / 25 Cavity κ κ
Pump
Pump
Dicke phase transition: ways out
2 Problem: g /ω0 < 2ζ for intrinsic parameters. Solutions: Non-solution Ferroelectric transition in D · r gauge. [JK JPCM ’07 ] See also [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11] Grand canonical ensemble: z † I If H → H − µ(S + ψ ψ), need only: 2 g N > (ω − µ)(ω0 − µ) I Incoherent pumping — polariton condensation.
Dissociate g, ω0, e.g. Raman scheme: ω0 ω. [Dimer et al. PRA ’07; Baumann et al. Nature ’10. Also, Black et al. PRL ’03 ]
Jonathan Keeling Collective dynamics Harvard, January 2012 7 / 25 Cavity κ κ
Pump
Pump
Dicke phase transition: ways out
2 Problem: g /ω0 < 2ζ for intrinsic parameters. Solutions: Non-solution Ferroelectric transition in D · r gauge. [JK JPCM ’07 ] See also [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11] Grand canonical ensemble: z † I If H → H − µ(S + ψ ψ), need only: 2 g N > (ω − µ)(ω0 − µ) I Incoherent pumping — polariton condensation.
Dissociate g, ω0, e.g. Raman scheme: ω0 ω. [Dimer et al. PRA ’07; Baumann et al. Nature ’10. Also, Black et al. PRL ’03 ]
Jonathan Keeling Collective dynamics Harvard, January 2012 7 / 25 Cavity κ κ
Pump
Pump
Dicke phase transition: ways out
2 Problem: g /ω0 < 2ζ for intrinsic parameters. Solutions: Non-solution Ferroelectric transition in D · r gauge. [JK JPCM ’07 ] See also [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11] Grand canonical ensemble: z † I If H → H − µ(S + ψ ψ), need only: 2 g N > (ω − µ)(ω0 − µ) I Incoherent pumping — polariton condensation.
Dissociate g, ω0, e.g. Raman scheme: ω0 ω. [Dimer et al. PRA ’07; Baumann et al. Nature ’10. Also, Black et al. PRL ’03 ]
Jonathan Keeling Collective dynamics Harvard, January 2012 7 / 25 Dicke phase transition: ways out
2 Problem: g /ω0 < 2ζ for intrinsic parameters. Solutions: Non-solution Ferroelectric transition in D · r gauge. [JK JPCM ’07 ] See also [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11] Grand canonical ensemble: z † I If H → H − µ(S + ψ ψ), need only: 2 g N > (ω − µ)(ω0 − µ) I Incoherent pumping — polariton condensation.
Dissociate g, ω0, Cavity e.g. Raman scheme: ω0 ω. κ κ [Dimer et al. PRA ’07; Baumann et al. Nature Pump
’10. Also, Black et al. PRL ’03 ] Pump
Jonathan Keeling Collective dynamics Harvard, January 2012 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 2012 8 / 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 2012 9 / 25 g2 Feedback: U ∝ 0 ωc − ωa
˙ − 2 − ∗ z Semiclassical EOM S = −i(ω0+U|ψ| )S + 2ig(ψ + ψ )S (|S| = N/2 1) S˙ z = ig(ψ + ψ∗)(S− − S+) ψ˙ = − [κ + i(ω+USz )] ψ − ig(S− + S+) √ ω0 ∼ kHz ω, κ, g N ∼ MHz.
Extended Dicke model [Baumann et al. Nature ’10] 2 Level system, | ⇓i, | ⇑i: ⇓: Ψ(x, z) = 1 gψ 0 κ κ 0 ⇑: Ψ(x, z) = P eik(σx+σ z) Ω σ,σ0=± z ω0 = 2ωrecoil x Pump 2 Level System
† z † − + † H = ωψ ψ + ω0S + g(ψ + ψ )(S + S )+USz ψ ψ. † † † ∂t ρ = −i[H, ρ]−κ(ψ ψρ − 2ψρψ + ρψ ψ)
Jonathan Keeling Collective dynamics Harvard, January 2012 10 / 25 g2 Feedback: U ∝ 0 ωc − ωa
˙ − 2 − ∗ z Semiclassical EOM S = −i(ω0+U|ψ| )S + 2ig(ψ + ψ )S (|S| = N/2 1) S˙ z = ig(ψ + ψ∗)(S− − S+) ψ˙ = − [κ + i(ω+USz )] ψ − ig(S− + S+) √ ω0 ∼ kHz ω, κ, g N ∼ MHz.
Extended Dicke model [Baumann et al. Nature ’10] 2 Level system, | ⇓i, | ⇑i: ⇓: Ψ(x, z) = 1 gψ 0 κ κ 0 ⇑: Ψ(x, z) = P eik(σx+σ z) Ω σ,σ0=± z ω0 = 2ωrecoil x Pump 2 Level System
† z † − + † H = ωψ ψ + ω0S + g(ψ + ψ )(S + S )+USz ψ ψ. † † † ∂t ρ = −i[H, ρ]−κ(ψ ψρ − 2ψρψ + ρψ ψ)
Jonathan Keeling Collective dynamics Harvard, January 2012 10 / 25 ˙ − 2 − ∗ z Semiclassical EOM S = −i(ω0+U|ψ| )S + 2ig(ψ + ψ )S (|S| = N/2 1) S˙ z = ig(ψ + ψ∗)(S− − S+) ψ˙ = − [κ + i(ω+USz )] ψ − ig(S− + S+) √ ω0 ∼ kHz ω, κ, g N ∼ MHz.
Extended Dicke model [Baumann et al. Nature ’10] 2 Level system, | ⇓i, | ⇑i: ⇓: Ψ(x, z) = 1 gψ 0 κ κ 0 ⇑: Ψ(x, z) = P eik(σx+σ z) Ω σ,σ0=± z ω0 = 2ωrecoil x Pump g2 2 Level System Feedback: U ∝ 0 ωc − ωa
† z † − + † H = ωψ ψ + ω0S + g(ψ + ψ )(S + S )+USz ψ ψ. † † † ∂t ρ = −i[H, ρ]−κ(ψ ψρ − 2ψρψ + ρψ ψ)
Jonathan Keeling Collective dynamics Harvard, January 2012 10 / 25 ˙ − 2 − ∗ z Semiclassical EOM S = −i(ω0+U|ψ| )S + 2ig(ψ + ψ )S (|S| = N/2 1) S˙ z = ig(ψ + ψ∗)(S− − S+) ψ˙ = − [κ + i(ω+USz )] ψ − ig(S− + S+) √ ω0 ∼ kHz ω, κ, g N ∼ MHz.
Extended Dicke model [Baumann et al. Nature ’10] 2 Level system, | ⇓i, | ⇑i: ⇓: Ψ(x, z) = 1 gψ 0 κ κ 0 ⇑: Ψ(x, z) = P eik(σx+σ z) Ω σ,σ0=± z ω0 = 2ωrecoil x Pump g2 2 Level System Feedback: U ∝ 0 ωc − ωa
† z † − + † H = ωψ ψ + ω0S + g(ψ + ψ )(S + S )+USz ψ ψ. † † † ∂t ρ = −i[H, ρ]−κ(ψ ψρ − 2ψρψ + ρψ ψ)
Jonathan Keeling Collective dynamics Harvard, January 2012 10 / 25 √ ω0 ∼ kHz ω, κ, g N ∼ MHz.
Extended Dicke model [Baumann et al. Nature ’10] 2 Level system, | ⇓i, | ⇑i: ⇓: Ψ(x, z) = 1 gψ 0 κ κ 0 ⇑: Ψ(x, z) = P eik(σx+σ z) Ω σ,σ0=± z ω0 = 2ωrecoil x Pump g2 2 Level System Feedback: U ∝ 0 ωc − ωa
† z † − + † H = ωψ ψ + ω0S + g(ψ + ψ )(S + S )+USz ψ ψ. † † † ∂t ρ = −i[H, ρ]−κ(ψ ψρ − 2ψρψ + ρψ ψ)
˙ − 2 − ∗ z Semiclassical EOM S = −i(ω0+U|ψ| )S + 2ig(ψ + ψ )S (|S| = N/2 1) S˙ z = ig(ψ + ψ∗)(S− − S+) ψ˙ = − [κ + i(ω+USz )] ψ − ig(S− + S+)
Jonathan Keeling Collective dynamics Harvard, January 2012 10 / 25 Extended Dicke model [Baumann et al. Nature ’10] 2 Level system, | ⇓i, | ⇑i: ⇓: Ψ(x, z) = 1 gψ 0 κ κ 0 ⇑: Ψ(x, z) = P eik(σx+σ z) Ω σ,σ0=± z ω0 = 2ωrecoil x Pump g2 2 Level System Feedback: U ∝ 0 ωc − ωa
† z † − + † H = ωψ ψ + ω0S + g(ψ + ψ )(S + S )+USz ψ ψ. † † † ∂t ρ = −i[H, ρ]−κ(ψ ψρ − 2ψρψ + ρψ ψ)
˙ − 2 − ∗ z Semiclassical EOM S = −i(ω0+U|ψ| )S + 2ig(ψ + ψ )S (|S| = N/2 1) S˙ z = ig(ψ + ψ∗)(S− − S+) ψ˙ = − [κ + i(ω+USz )] ψ − ig(S− + S+) √ ω0 ∼ kHz ω, κ, g N ∼ MHz.
Jonathan Keeling Collective dynamics Harvard, January 2012 10 / 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 2012 11 / 25 Sz
Sy
S x Small g: ⇑, ⇓ only. Larger g: SR too. (ω = 30MHz, UN = −40MHz)
Fixed points (steady states)
ψ = 0, S = (0, 0, ±N/2) 2 − ∗ z 0 = i(ω0+U|ψ| )S + 2ig(ψ + ψ )S always a solution. ∗ − + 0 = ig(ψ + ψ )(S − S ) If g > gc, ψ 6= 0 too y − 0 = − [κ + i(ω+USz )] ψ − ig(S− + S+) A S = −=[S ] = 0 B ψ0 = <[ψ] = 0
Jonathan Keeling Collective dynamics Harvard, January 2012 12 / 25 Larger g: SR too.
Fixed points (steady states)
ψ = 0, S = (0, 0, ±N/2) 2 − ∗ z 0 = i(ω0+U|ψ| )S + 2ig(ψ + ψ )S always a solution. ∗ − + 0 = ig(ψ + ψ )(S − S ) If g > gc, ψ 6= 0 too y − 0 = − [κ + i(ω+USz )] ψ − ig(S− + S+) A S = −=[S ] = 0 B ψ0 = <[ψ] = 0
Sz
Sy
S x Small g: ⇑, ⇓ only. (ω = 30MHz, UN = −40MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 12 / 25 Fixed points (steady states)
ψ = 0, S = (0, 0, ±N/2) 2 − ∗ z 0 = i(ω0+U|ψ| )S + 2ig(ψ + ψ )S always a solution. ∗ − + 0 = ig(ψ + ψ )(S − S ) If g > gc, ψ 6= 0 too y − 0 = − [κ + i(ω+USz )] ψ − ig(S− + S+) A S = −=[S ] = 0 B ψ0 = <[ψ] = 0
Sz
Sy
S x Small g: ⇑, ⇓ only. Larger g: SR too. (ω = 30MHz, UN = −40MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 12 / 25 40 ⇓ SR(A): Sy = 0 UN=0
20 ⇓ SR
0 (MHz) ω ⇓ + ⇑ SR(B): ψ0 = 0 -20 ⇑ SR
-40 0 0.5 1 1.5 g√N (MHz)
Steady state phase diagram
UN=0, κ=0
2 − ∗ z 0 = i(ω0+U|ψ| )S + 2ig(ψ + ψ )S ω ⇓ SR 0 = ig(ψ + ψ∗)(S− − S+) 0 = − [κ + i(ω+USz )] ψ − ig(S− + S+) 0 0 g-√N
See also Domokos and Ritsch PRL ’02, Domokos et al. PRL ’10
Jonathan Keeling Collective dynamics Harvard, January 2012 13 / 25 ⇓ + ⇑ SR(B): ψ0 = 0
Steady state phase diagram
UN=0, κ=0
2 − ∗ z 0 = i(ω0+U|ψ| )S + 2ig(ψ + ψ )S ω ⇓ SR 0 = ig(ψ + ψ∗)(S− − S+) 0 = − [κ + i(ω+USz )] ψ − ig(S− + S+) 0 0 g-√N 40 ⇓ SR(A): Sy = 0 UN=0
20 ⇓ SR
0 (MHz) ω -20 ⇑ SR
-40 0 0.5 1 1.5 g√N (MHz) See also Domokos and Ritsch PRL ’02, Domokos et al. PRL ’10
Jonathan Keeling Collective dynamics Harvard, January 2012 13 / 25 Steady state phase diagram
UN=0, κ=0
2 − ∗ z 0 = i(ω0+U|ψ| )S + 2ig(ψ + ψ )S ω ⇓ SR 0 = ig(ψ + ψ∗)(S− − S+) 0 = − [κ + i(ω+USz )] ψ − ig(S− + S+) 0 0 g-√N 40 ⇓ SR(A): Sy = 0 UN=-10
20 ⇓ SRA
0 ⇓+⇑ SRB (MHz) ω ⇓ + ⇑ SR(B): ψ0 = 0 -20 ⇑ SRA
-40 0 0.5 1 1.5 g√N (MHz) See also Domokos and Ritsch PRL ’02, Domokos et al. PRL ’10
Jonathan Keeling Collective dynamics Harvard, January 2012 13 / 25 Steady state phase diagram
UN=0, κ=0
2 − ∗ z 0 = i(ω0+U|ψ| )S + 2ig(ψ + ψ )S ω ⇓ SR 0 = ig(ψ + ψ∗)(S− − S+) 0 = − [κ + i(ω+USz )] ψ − ig(S− + S+) 0 0 g-√N 40 ⇓ SR(A): Sy = 0 UN=-10 SRA+⇑ 20 ⇓ SRA SRA 0 ⇓+⇑ SRB (MHz) ω ⇓ + ⇑ SR(B): ψ0 = 0 -20 ⇑ SRA SRA+⇓
-40 0 0.5 1 1.5 g√N (MHz) See also Domokos and Ritsch PRL ’02, Domokos et al. PRL ’10
Jonathan Keeling Collective dynamics Harvard, January 2012 13 / 25 Steady state phase diagram
UN=0, κ=0
2 − ∗ z 0 = i(ω0+U|ψ| )S + 2ig(ψ + ψ )S ω ⇓ SR 0 = ig(ψ + ψ∗)(S− − S+) 0 = − [κ + i(ω+USz )] ψ − ig(S− + S+) 0 0 g-√N 40 ⇓ SR(A): Sy = 0 UN=-20
20 ⇓ SRA
0 ⇓+⇑ SRB (MHz) ω ⇓ + ⇑ SR(B): ψ0 = 0 -20 ⇑ SRA
-40 0 0.5 1 1.5 g√N (MHz) See also Domokos and Ritsch PRL ’02, Domokos et al. PRL ’10
Jonathan Keeling Collective dynamics Harvard, January 2012 13 / 25 Steady state phase diagram
UN=0, κ=0
2 − ∗ z 0 = i(ω0+U|ψ| )S + 2ig(ψ + ψ )S ω ⇓ SR 0 = ig(ψ + ψ∗)(S− − S+) 0 = − [κ + i(ω+USz )] ψ − ig(S− + S+) 0 0 g-√N 40 ⇓ SR(A): Sy = 0 UN=-40 ⇓ SRA 20
0 ⇓+⇑ SRB (MHz) ω ⇓ + ⇑ SR(B): ψ0 = 0 -20 ⇑ SRA
-40 0 0.5 1 1.5 g√N (MHz) See also Domokos and Ritsch PRL ’02, Domokos et al. PRL ’10
Jonathan Keeling Collective dynamics Harvard, January 2012 13 / 25 Steady state phase diagram
UN=0, κ=0
2 − ∗ z 0 = i(ω0+U|ψ| )S + 2ig(ψ + ψ )S ω ⇓ SR 0 = ig(ψ + ψ∗)(S− − S+) 0 = − [κ + i(ω+USz )] ψ − ig(S− + S+) 0 0 g-√N 40 ⇓ SR(A): Sy = 0 SRA+⇑ UN=-40 ⇓ SRA 20 SRB+⇑ SRB 0 ⇓+⇑ +⇓+⇑ SRB (MHz) ω SRB+⇑ ⇓ + ⇑ SR(B): ψ0 = 0 -20 ⇑ SRA SRA+⇓ -40 0 0.5 1 1.5 g√N (MHz) See also Domokos and Ritsch PRL ’02, Domokos et al. PRL ’10
Jonathan Keeling Collective dynamics Harvard, January 2012 13 / 25 Comparison to experiment
0
-10 MHz) π -20 ) (2 c ω - p
ω -30 (
-40 0 0.5 1 1.5 2 2.5 g2 N (MHz)2
UN = −10MHz [Baumann et al Nature ’10 ] Adapted from: [Bhaseen et al. PRA ’12] 2 5 g0 ω = ωc − ωp + UN, UN = − 2 4(ωa − ωc)
Jonathan Keeling Collective dynamics Harvard, January 2012 14 / 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 2012 15 / 25 100 (i) SR(A) 80 √ √ 0 0 1 2 2 | ψ Gray: S = ( N, N, −N/2) | 40 Black: Wigner distribution of S, ψ 0 0 20 40 60 80 t (ms)
(ii) SR(B) 200 2 | ψ | 100
0 0 0.1 0.2 0.3 0.4 t (ms)
(iii) SR(A) 120 50 40150 151 2
| 80 ψ Oscillations: ∼ 0.1ms | 40 Decay: 20ms, 0.1ms, 20ms 0 0 100 200 t (ms)
Dynamics: Evolution from normal state
40 (i) UN=-40 ⇓ SRA 20 (ii)
0 ⇓+⇑ SRB (MHz) ω
-20 ⇑ (iii) SRA
-40 0 0.5 1 1.5 g√N (MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 16 / 25 (ii) SR(B) 200 2 | ψ | 100
0 0 0.1 0.2 0.3 0.4 t (ms)
(iii) SR(A) 120 50 40150 151 2
| 80 ψ | 40 , 0.1ms, 20ms 0 0 100 200 t (ms)
Dynamics: Evolution from normal state
100 (i) SR(A) 80 √ √ 0 0 1 2 2 | ψ Gray: S = ( N, N, −N/2) | 40 Black: Wigner distribution of S, ψ 40 0 (i) UN=-40 0 20 40 60 80 t (ms) ⇓ SRA 20 (ii)
0 ⇓+⇑ SRB (MHz) ω
-20 ⇑ (iii) SRA
-40 0 0.5 1 1.5 g√N (MHz)
Oscillations: ∼ 0.1ms Decay: 20ms
Jonathan Keeling Collective dynamics Harvard, January 2012 16 / 25 (iii) SR(A) 120 50 40150 151 2
| 80 ψ | 40 , 20ms 0 0 100 200 t (ms)
Dynamics: Evolution from normal state
100 (i) SR(A) 80 √ √ 0 0 1 2 2 | ψ Gray: S = ( N, N, −N/2) | 40 Black: Wigner distribution of S, ψ 40 0 (i) UN=-40 0 20 40 60 80 t (ms) ⇓ SRA 20 (ii) (ii) SR(B) 200 2 0 ⇓ ⇑ |
+ SRB ψ |
(MHz) 100 ω
-20 ⇑ (iii) SRA 0 0 0.1 0.2 0.3 0.4 t (ms) -40 0 0.5 1 1.5 g√N (MHz)
Oscillations: ∼ 0.1ms Decay: 20ms, 0.1ms
Jonathan Keeling Collective dynamics Harvard, January 2012 16 / 25 Dynamics: Evolution from normal state
100 (i) SR(A) 80 √ √ 0 0 1 2 2 | ψ Gray: S = ( N, N, −N/2) | 40 Black: Wigner distribution of S, ψ 40 0 (i) UN=-40 0 20 40 60 80 t (ms) ⇓ SRA 20 (ii) (ii) SR(B) 200 2 0 ⇓ ⇑ |
+ SRB ψ |
(MHz) 100 ω
-20 ⇑ (iii) SRA 0 0 0.1 0.2 0.3 0.4 t (ms) -40 0 0.5 1 1.5 g√N (MHz) (iii) SR(A) 120 50 40150 151 2
| 80 ψ Oscillations: ∼ 0.1ms | 40 Decay: 20ms, 0.1ms, 20ms 0 0 100 200 t (ms)
Jonathan Keeling Collective dynamics Harvard, January 2012 16 / 25 Dynamics: Evolution from normal state
100 (i) SR(A) 80 √ √ 0 0 1 2 2 | ψ Gray: S = ( N, N, −N/2) | 40 Black: Wigner distribution of S, ψ 40 0 (i) UN=-40 0 20 40 60 80 t (ms) ⇓ SRA 20 (ii) (ii) SR(B) 200 2 0 ⇓ ⇑ |
+ SRB ψ |
(MHz) 100 ω
-20 ⇑ (iii) SRA 0 0 0.1 0.2 0.3 0.4 t (ms) -40 0 0.5 1 1.5 g√N (MHz) (iii) SR(A) 120 50 40150 151 2
| 80 ψ Oscillations: ∼ 0.1ms | 40 Decay: 20ms, 0.1ms, 20ms 0 0 100 200 t (ms)
Jonathan Keeling Collective dynamics Harvard, January 2012 16 / 25 Starting from ⇓ 40 103 Asymptotic state SRA 20 102 ⇓ 0 SRB 101 2 | (MHz) ψ | ω
-20 100 ⇑ SRA
-40 10-1 0 0.5 1 1.5 g√N (MHz)
Asymptotic state: Evolution from normal state
(Near to experimental UN = −13MHz). All stable attractors: 40 UN=-10
20 ⇓ SRA
0 ⇓+⇑ SRB (MHz) ω -20 ⇑ SRA
-40 0 0.5 1 1.5 g√N (MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 17 / 25 Asymptotic state: Evolution from normal state
(Near to experimental UN = −13MHz). Starting from ⇓ All stable attractors: 40 103 40 Asymptotic state UN=-10 SRA 20 2 20 ⇓ SRA 10 ⇓ ⇓ ⇑ 0 SRB 101 0 + SRB 2 | (MHz) (MHz) ψ | ω ω
-20 ⇑ SRA -20 100 ⇑ SRA
-40 -40 -1 0 0.5 1 1.5 10 g√N (MHz) 0 0.5 1 1.5 g√N (MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 17 / 25 40 10s Asymptotic decay time SRA 1s 20 100ms ⇓ 10ms 0 SRB (MHz)
ω 1ms
-20 100µs Expand in ω /κ: SRA 0 ⇑ 10µs Oscillations: ∼ ω0, -40 2 0 0.5 1 1.5 Decay: ∼ ω0 or ω0/κ g√N (MHz)
Timescales for dynamics: What are they? 3 40 10s 40 10 Initial growth time Asymptotic state No unstable 1s SRA directions 2 20 20 10 100ms ⇓ 10ms 0 SRB 101 0 One unstable direction 2 | (MHz) (MHz) ψ ω |
ω 1ms 0 -20 10 -20 100µs ⇑ SRA Two unstable directions 10µs -40 10-1 0 0.5 1 1.5 -40 0 0.5 1 1.5 g√N (MHz) g√N (MHz)
Growth Most unstable eigenvalues near S = (0, 0, −N/2) Decay Slowest stable eigenvalues near final state
Jonathan Keeling Collective dynamics Harvard, January 2012 18 / 25 40 10s Asymptotic decay time SRA 1s 20 100ms ⇓ 10ms 0 SRB (MHz)
ω 1ms
-20 100µs SRA ⇑ 10µs -40 0 0.5 1 1.5 g√N (MHz)
Timescales for dynamics: What are they? 3 40 10s 40 10 Initial growth time Asymptotic state No unstable 1s SRA directions 2 20 20 10 100ms ⇓ 10ms 0 SRB 101 0 One unstable direction 2 | (MHz) (MHz) ψ ω |
ω 1ms 0 -20 10 -20 100µs ⇑ SRA Two unstable directions 10µs -40 10-1 0 0.5 1 1.5 -40 0 0.5 1 1.5 g√N (MHz) g√N (MHz)
Growth Most unstable eigenvalues near S = (0, 0, −N/2) Decay Slowest stable eigenvalues near final state
Expand in ω0/κ: Oscillations: ∼ ω0, 2 Decay: ∼ ω0 or ω0/κ
Jonathan Keeling Collective dynamics Harvard, January 2012 18 / 25 Timescales for dynamics: What are they? 3 40 10s 40 10 Initial growth time Asymptotic state No unstable 1s SRA directions 2 20 20 10 100ms ⇓ 10ms 0 SRB 101 0 One unstable direction 2 | (MHz) (MHz) ψ ω |
ω 1ms 0 -20 10 -20 100µs ⇑ SRA Two unstable directions 10µs -40 10-1 0 0.5 1 1.5 -40 0 0.5 1 1.5 g√N (MHz) g√N (MHz) 40 10s Asymptotic decay time Growth Most unstable eigenvalues SRA 1s 20 near S = (0, 0, −N/2) 100ms ⇓ 10ms Decay Slowest stable eigenvalues 0 SRB (MHz) near final state ω 1ms -20 100µs Expand in ω /κ: SRA 0 ⇑ 10µs Oscillations: ∼ ω0, -40 2 0 0.5 1 1.5 Decay: ∼ ω0 or ω0/κ g√N (MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 18 / 25 60 10ms sweep 3 10
40 102 20 1 (MHz) 10 ω 0
0 -20 10
-40 10-1 0.0 0.5 1.0 1.5 2.0 2.5 g2 N (MHz2)
60 200ms sweep 3 10
40 102 20 1 (MHz) 10 ω 0
0 -20 10
-40 10-1 0.0 0.5 1.0 1.5 2.0 2.5 g2 N (MHz2)
Timescales for dynamics: Consequences for experiment
40 103 Asymptotic state SRA 20 102 ⇓ 0 SRB 101 2 | (MHz) ψ | ω
-20 100 ⇑ SRA
-40 10-1 0 0.5 1 1.5 g√N (MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 19 / 25 60 200ms sweep 3 10
40 102 20 1 (MHz) 10 ω 0
0 -20 10
-40 10-1 0.0 0.5 1.0 1.5 2.0 2.5 g2 N (MHz2)
Timescales for dynamics: Consequences for experiment
60 10ms sweep 3 10
40 102 20 1 (MHz) 10 ω 0 40 103 Asymptotic state 100 SRA -20 20 102 -40 10-1 ⇓ 0.0 0.5 1.0 1.5 2.0 2.5 1 0 SRB 10 g2 N (MHz2) 2 | (MHz) ψ | ω
-20 100 ⇑ SRA
-40 10-1 0 0.5 1 1.5 g√N (MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 19 / 25 Timescales for dynamics: Consequences for experiment
60 10ms sweep 3 10
40 102 20 1 (MHz) 10 ω 0 40 103 Asymptotic state 100 SRA -20 20 102 -40 10-1 ⇓ 0.0 0.5 1.0 1.5 2.0 2.5 1 0 SRB 10 g2 N (MHz2) 2 | (MHz) ψ | ω -20 0 10 60 200ms sweep 3 ⇑ SRA 10
-1 40 -40 10 2 0 0.5 1 1.5 10 g√N (MHz) 20 1 (MHz) 10 ω 0
0 -20 10
-40 10-1 0.0 0.5 1.0 1.5 2.0 2.5 g2 N (MHz2) Jonathan Keeling Collective dynamics Harvard, January 2012 19 / 25 δg = g0 − g, 2g¯ = g0 + g
40 40 UN=-10 g-√N=1 20 ⇓ SRA 20
0 ⇓+⇑ SRB 0 (MHz) (MHz) ω ω -20 ⇑ SRA -20
-40 -40 0 0.5 1 1.5 -0.01 -0.005 0 0.005 0.01 g√N (MHz) δg/g-
Timescales for dynamics: Why so slow and varied? Suppose co- and counter-rotating terms differ
∆ a∆ b
Ω † − + 0 † + − Ω a b H = ... + g(ψ S + ψS ) + g (ψ S + ψS ) + ... g0ψ g0ψ
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 2012 20 / 25 δg = g0 − g, 2g¯ = g0 + g
40 g-√N=1 20
0 (MHz) ω
-20
-40 -0.01 -0.005 0 0.005 0.01 δg/g-
Timescales for dynamics: Why so slow and varied? Suppose co- and counter-rotating terms differ
∆ a∆ b
Ω † − + 0 † + − Ω a b H = ... + g(ψ S + ψS ) + g (ψ S + ψS ) + ... g0ψ g0ψ
2 Level System
40 UN=-10
20 ⇓ SRA
0 ⇓+⇑ SRB (MHz) ω -20 ⇑ SRA
-40 0 0.5 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 2012 20 / 25 Timescales for dynamics: Why so slow and varied? Suppose co- and counter-rotating terms differ
∆ a∆ b
Ω † − + 0 † + − Ω a b H = ... + g(ψ S + ψS ) + g (ψ S + ψS ) + ... g0ψ g0ψ
0 0 2 Level System δg = g − g, 2g¯ = g + g
40 40 UN=-10 g-√N=1 20 ⇓ SRA 20
0 ⇓+⇑ SRB 0 (MHz) (MHz) ω ω -20 ⇑ SRA -20
-40 -40 0 0.5 1 1.5 -0.01 -0.005 0 0.005 0.01 g√N (MHz) δg/g-
SR(A) near phase boundary at small δg → Critical slowing down SR(A), SR(B) continuously connect
Jonathan Keeling Collective dynamics Harvard, January 2012 20 / 25 Timescales for dynamics: Why so slow and varied? Suppose co- and counter-rotating terms differ
∆ a∆ b
Ω † − + 0 † + − Ω a b H = ... + g(ψ S + ψS ) + g (ψ S + ψS ) + ... g0ψ g0ψ
0 0 2 Level System δg = g − g, 2g¯ = g + g
40 40 UN=-10 g-√N=1 20 ⇓ SRA 20
0 ⇓+⇑ SRB 0 (MHz) (MHz) ω ω -20 ⇑ SRA -20
-40 -40 0 0.5 1 1.5 -0.01 -0.005 0 0.005 0.01 g√N (MHz) δg/g-
SR(A) near phase boundary at small δg → Critical slowing down SR(A), SR(B) continuously connect
Jonathan Keeling Collective dynamics Harvard, January 2012 20 / 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 2012 21 / 25 1200
1000 1200
800 800 2 | 600 ψ | 400 400
200 0 0 0 5 10 15 0 2 4 6 8 10 12 14 16 18 t (ms)
Regions without fixed points
ψ g0 g2 Changing U: Ω U ∝ 0 ωc − ωa
2 Level System
40 UN=-40 ⇓ SRA 20
0 ⇓+⇑ SRB (MHz) ω
-20 ⇑ SRA
-40 0 0.5 1 1.5 g√N (MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 22 / 25 1200
1000 1200
800 800 2 | 600 ψ | 400 400
200 0 0 0 5 10 15 0 2 4 6 8 10 12 14 16 18 t (ms)
Regions without fixed points
g0ψ g2 Changing U: Ω U ∝ 0 ωc − ωa
2 Level System
40 UN=-40 ⇓ SRA 20
0 ⇓+⇑ SRB (MHz) ω
-20 ⇑ SRA
-40 0 0.5 1 1.5 g√N (MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 22 / 25 1200
1000 1200
800 800 2 | 600 ψ | 400 400
200 0 0 0 5 10 15 0 2 4 6 8 10 12 14 16 18 t (ms)
Regions without fixed points
g0ψ g2 Changing U: Ω U ∝ 0 ωc − ωa
2 Level System
40 UN=0
20 ⇓ SR
0 (MHz) ω -20 ⇑ SR
-40 0 0.5 1 1.5 g√N (MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 22 / 25 1200
1000 1200
800 800 2 | 600 ψ | 400 400
200 0 0 0 5 10 15 0 2 4 6 8 10 12 14 16 18 t (ms)
Regions without fixed points
g0ψ g2 Changing U: Ω U ∝ 0 ωc − ωa
2 Level System
40 UN=20
20 ⇓ SRA
0 (MHz) ω -20 ⇑ SRA
-40 0 0.5 1 1.5 g√N (MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 22 / 25 1200
1000 1200
800 800 2 | 600 ψ | 400 400
200 0 0 0 5 10 15 0 2 4 6 8 10 12 14 16 18 t (ms)
Regions without fixed points
g0ψ g2 Changing U: Ω U ∝ 0 ωc − ωa
2 Level System
40 UN=40 ⇓ SRA 20
0 (MHz) ω
-20 ⇑ SRA
-40 0 0.5 1 1.5 g√N (MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 22 / 25 Regions without fixed points
g0ψ g2 Changing U: Ω U ∝ 0 ωc − ωa
2 Level System
40 UN=40 1200 ⇓ SRA 1000 1200 20 800 800 2 | 600 ψ 0 Persistent Oscillations | (MHz)
ω 400 400
-20 200 ⇑ SRA 0 0 0 5 10 15 -40 0 2 4 6 8 10 12 14 16 18 0 0.5 1 1.5 t (ms) g√N (MHz)
Jonathan Keeling Collective dynamics Harvard, January 2012 22 / 25 Get: Fix ω + USz = 0 if ψ0 = 0. ˙ 2 r θ = ω0 + U|ψ| N2 S− = re−iθ r = − (Sz )2 ψ˙ + κψ = −2igr cos(θ) 1200 4 |ψ|2 S 0.4 1000 x Sy S 800 z 0.2 z , S 2 | y
ψ 600 0 | , S x S 400 -0.2 200 -0.4 0 18.00 18.02 18.04 18.06 18.08 t(ms)
Persistent (optomechanical) oscillations 1200
1000 1200
800 800 ˙ − 2 − ∗ z 2 | 600 S = −i(ω + U|ψ| )S + 2ig(ψ + ψ )S ψ 0 | 400 400 ˙ z ∗ − + 200 S = ig(ψ + ψ )(S − S ) 0 0 0 5 10 15 0 2 4 6 8 10 12 14 16 18 z − + t (ms) ψ˙ = − [κ + i(ω + US )] ψ − ig(S + S )
Jonathan Keeling Collective dynamics Harvard, January 2012 23 / 25 Get:
˙ 2 r θ = ω0 + U|ψ| N2 S− = re−iθ r = − (Sz )2 ψ˙ + κψ = −2igr cos(θ) 1200 4 |ψ|2 S 0.4 1000 x Sy S 800 z 0.2 z , S 2 | y
ψ 600 0 | , S x S 400 -0.2 200 -0.4 0 18.00 18.02 18.04 18.06 18.08 t(ms)
Persistent (optomechanical) oscillations 1200
1000 1200
800 800 ˙ − 2 − ∗ z 2 | 600 S = −i(ω + U|ψ| )S + 2ig(ψ + ψ )S ψ 0 | 400 400 ˙ z ∗ − + 200 S = ig(ψ + ψ )(S − S ) 0 0 0 5 10 15 0 2 4 6 8 10 12 14 16 18 z − + t (ms) ψ˙ = − [κ + i(ω + US )] ψ − ig(S + S )
Fix ω + USz = 0 if ψ0 = 0.
Jonathan Keeling Collective dynamics Harvard, January 2012 23 / 25 Get:
˙ 2 r θ = ω0 + U|ψ| N2 S− = re−iθ r = − (Sz )2 ψ˙ + κψ = −2igr cos(θ) 1200 4 |ψ|2 S 0.4 1000 x Sy S 800 z 0.2 z , S 2 | y
ψ 600 0 | , S x S 400 -0.2 200 -0.4 0 18.00 18.02 18.04 18.06 18.08 t(ms)
Persistent (optomechanical) oscillations 1200
1000 1200
800 800 ˙ − 2 − ∗ z 2 | 600 S = −i(ω + U|ψ| )S + 2ig(ψ + ψ )S ψ 0 | 400 400 ˙ z ∗ − + 200 S = ig(ψ + ψ )(S − S ) 0 0 0 5 10 15 0 2 4 6 8 10 12 14 16 18 z − + t (ms) ψ˙ = − [κ + i(ω + US )] ψ − ig(S + S )
Fix ω + USz = 0 if ψ0 = 0.
Jonathan Keeling Collective dynamics Harvard, January 2012 23 / 25 1200 |ψ|2 S 0.4 1000 x Sy S 800 z 0.2 z , S 2 | y
ψ 600 0 | , S x S 400 -0.2 200 -0.4 0 18.00 18.02 18.04 18.06 18.08 t(ms)
Persistent (optomechanical) oscillations 1200
1000 1200
800 800 ˙ − 2 − ∗ z 2 | 600 S = −i(ω + U|ψ| )S + 2ig(ψ + ψ )S ψ 0 | 400 400 ˙ z ∗ − + 200 S = ig(ψ + ψ )(S − S ) 0 0 0 5 10 15 0 2 4 6 8 10 12 14 16 18 z − + t (ms) ψ˙ = − [κ + i(ω + US )] ψ − ig(S + S ) Get: Fix ω + USz = 0 if ψ0 = 0. ˙ 2 r θ = ω0 + U|ψ| N2 S− = re−iθ r = − (Sz )2 ψ˙ + κψ = −2igr cos(θ) 4
Jonathan Keeling Collective dynamics Harvard, January 2012 23 / 25 Persistent (optomechanical) oscillations 1200
1000 1200
800 800 ˙ − 2 − ∗ z 2 | 600 S = −i(ω + U|ψ| )S + 2ig(ψ + ψ )S ψ 0 | 400 400 ˙ z ∗ − + 200 S = ig(ψ + ψ )(S − S ) 0 0 0 5 10 15 0 2 4 6 8 10 12 14 16 18 z − + t (ms) ψ˙ = − [κ + i(ω + US )] ψ − ig(S + S ) Get: Fix ω + USz = 0 if ψ0 = 0. ˙ 2 r θ = ω0 + U|ψ| N2 S− = re−iθ r = − (Sz )2 ψ˙ + κψ = −2igr cos(θ) 1200 4 |ψ|2 S 0.4 1000 x Sy S 800 z 0.2 z , S 2 | y
ψ 600 0 | , S x S 400 -0.2 200 -0.4 0 18.00 18.02 18.04 18.06 18.08 t(ms)
Jonathan Keeling Collective dynamics Harvard, January 2012 23 / 25 Possible realization: Hyperfine levels
σ− Atoms σ+
m =+1 F B mF =0
mF =−1
0 Tuning g, g , U [Dimer et al. Phys. Rev. A. (2007)]
∆ a∆ b
Ω a Ω b Separate pump strength/detuning g ψ g ψ 0 0 g Ω g Ω g 2 g 2 g ∼ 0 b , g0 ∼ 0 a , U ∼ 0 − 0 ∆ ∆ ∆ ∆ 2 Level System b a a b
Jonathan Keeling Collective dynamics Harvard, January 2012 24 / 25 Atoms
B
0 Tuning g, g , U [Dimer et al. Phys. Rev. A. (2007)]
∆ a∆ b
Ω a Ω b Separate pump strength/detuning g ψ g ψ 0 0 g Ω g Ω g 2 g 2 g ∼ 0 b , g0 ∼ 0 a , U ∼ 0 − 0 ∆ ∆ ∆ ∆ 2 Level System b a a b Possible realization: Hyperfine levels
σ−
σ+
mF =+1
mF =0
mF =−1
Jonathan Keeling Collective dynamics Harvard, January 2012 24 / 25 0 Tuning g, g , U [Dimer et al. Phys. Rev. A. (2007)]
∆ a∆ b
Ω a Ω b Separate pump strength/detuning g ψ g ψ 0 0 g Ω g Ω g 2 g 2 g ∼ 0 b , g0 ∼ 0 a , U ∼ 0 − 0 ∆ ∆ ∆ ∆ 2 Level System b a a b Possible realization: Hyperfine levels
σ− Atoms σ+
m =+1 F B mF =0
mF =−1
Jonathan Keeling Collective dynamics Harvard, January 2012 24 / 25 Summary
Wide variety of dynamical phases 40 40 40 40 UN=40 UN=0 UN=-40 g-√N=1 ⇓ SRA ⇓ SRA 20 20 ⇓ SR 20 20
0 0 0 ⇓+⇑ SRB 0 (MHz) (MHz) (MHz) (MHz) ω ω ω ω -20 -20 ⇑ SR -20 -20 ⇑ SRA ⇑ SRA
-40 -40 -40 -40 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 -0.01 -0.005 0 0.005 0.01 g√N (MHz) g√N (MHz) g√N (MHz) δg/g- Slow dynamics 40 10s 40 10s Initial growth time Asymptotic decay time 50 No unstable 120 directions 1s SRA 1s 20 20 40 100ms 100ms 150 151 ⇓ 2
| 80 10ms 10ms
ψ One unstable direction SRB | 0 0 (MHz) (MHz) 40 ω 1ms ω 1ms -20 100µs -20 100µs SRA Two unstable directions 0 10µs ⇑ 10µs -40 -40 0 100 200 0 0.5 1 1.5 0 0.5 1 1.5 t (ms) g√N (MHz) g√N (MHz) Persistent oscillations if U > 0 1200 1200 |ψ|2 1200 S 0.4 1000 1000 x Sy 800 S 0.2 800 800 z z 2 | , S
600 2 ψ | | y
ψ 600 0 | , S
400 400 x S 400 -0.2 200 0 200 0 0 5 10 15 -0.4 0 2 4 6 8 10 12 14 16 18 0 t (ms) 18.00 18.02 18.04 18.06 18.08 t(ms)
JK et al. PRL ’10, Bhaseen et al. PRA ’12
Jonathan Keeling Collective dynamics Harvard, January 2012 25 / 25 Jonathan Keeling Collective dynamics Harvard, January 2012 26 / 30 Extra slides
5 Ferroelectric phase transition
6 Lipkin-Meshkov-Glick
7 Tricritical point
Jonathan Keeling Collective dynamics Harvard, January 2012 27 / 30 Two-level systems — dipole-dipole coupling
z † + − † † 2 + − 2 H = ω0S + ωψ ψ + g(S + S )(ψ + ψ ) + Nζ(ψ + ψ ) −η(S − S )
Ferroelectric polarisation if ω < 2ηN (nb g2, ζ, η ∝ 1/V ). 0 Gauge transform to dipole gauge D · r
z † + − † H = ω0S + ωψ ψ + g¯(S − S )(ψ − ψ )
2 “Dicke” transition at ω0 < Ng¯ /ω ≡ 2ηN But, ψ describes electric displacement
Ferroelectric transition Atoms in Coulomb gauge
X † X 2 H = ωk ak ak + [pi − eA(ri )] + Vcoul i
Jonathan Keeling Collective dynamics Harvard, January 2012 28 / 30 Ferroelectric polarisation if ω0 < 2ηN Gauge transform to dipole gauge D · r
z † + − † H = ω0S + ωψ ψ + g¯(S − S )(ψ − ψ )
2 “Dicke” transition at ω0 < Ng¯ /ω ≡ 2ηN But, ψ describes electric displacement
Ferroelectric transition Atoms in Coulomb gauge
X † X 2 H = ωk ak ak + [pi − eA(ri )] + Vcoul i Two-level systems — dipole-dipole coupling
z † + − † † 2 + − 2 H = ω0S + ωψ ψ + g(S + S )(ψ + ψ ) + Nζ(ψ + ψ ) −η(S − S )
(nb g2, ζ, η ∝ 1/V ).
Jonathan Keeling Collective dynamics Harvard, January 2012 28 / 30 Gauge transform to dipole gauge D · r
z † + − † H = ω0S + ωψ ψ + g¯(S − S )(ψ − ψ )
2 “Dicke” transition at ω0 < Ng¯ /ω ≡ 2ηN But, ψ describes electric displacement
Ferroelectric transition Atoms in Coulomb gauge
X † X 2 H = ωk ak ak + [pi − eA(ri )] + Vcoul i Two-level systems — dipole-dipole coupling
z † + − † † 2 + − 2 H = ω0S + ωψ ψ + g(S + S )(ψ + ψ ) + Nζ(ψ + ψ ) −η(S − S )
Ferroelectric polarisation if ω < 2ηN (nb g2, ζ, η ∝ 1/V ). 0
Jonathan Keeling Collective dynamics Harvard, January 2012 28 / 30 Ferroelectric transition Atoms in Coulomb gauge
X † X 2 H = ωk ak ak + [pi − eA(ri )] + Vcoul i Two-level systems — dipole-dipole coupling
z † + − † † 2 + − 2 H = ω0S + ωψ ψ + g(S + S )(ψ + ψ ) + Nζ(ψ + ψ ) −η(S − S )
Ferroelectric polarisation if ω < 2ηN (nb g2, ζ, η ∝ 1/V ). 0 Gauge transform to dipole gauge D · r
z † + − † H = ω0S + ωψ ψ + g¯(S − S )(ψ − ψ )
2 “Dicke” transition at ω0 < Ng¯ /ω ≡ 2ηN But, ψ describes electric displacement
Jonathan Keeling Collective dynamics Harvard, January 2012 28 / 30 Timescales for dynamics: U = 0 and Lipkin-Meshkov-Glick
Since κ ω0, can consider eliminating ψ
˙ − 2 − ∗ z S = −i(ω0+U|ψ| )S + 2ig(ψ + ψ )S S˙ z = ig(ψ + ψ∗)(S− − S+) ψ˙ = − [κ + i(ω+USz )] ψ − ig(S− + S+)
If U = 0, simple result:
2 2 ∂t S = {S, H} − ΓS × (S × zˆ), H = ω0Sz − Λ+Sx − Λ−Sy
ω 0 2 2κ 02 2 with: Λ± ≡ κ2+ω2 (g ± g ) , Γ ≡ κ2+ω2 (g − g ) NB, g0 = g, no dissipation. Dissipation restored by finite κ.
Jonathan Keeling Collective dynamics Harvard, January 2012 29 / 30 Timescales for dynamics: U = 0 and Lipkin-Meshkov-Glick
Since κ ω0, can consider eliminating ψ
˙ − 2 − ∗ z S = −i(ω0+U|ψ| )S + 2ig(ψ + ψ )S S˙ z = ig(ψ + ψ∗)(S− − S+) ψ˙ = − [κ + i(ω+USz )] ψ − ig(S− + S+)
If U = 0, simple result:
2 2 ∂t S = {S, H} − ΓS × (S × zˆ), H = ω0Sz − Λ+Sx − Λ−Sy
ω 0 2 2κ 02 2 with: Λ± ≡ κ2+ω2 (g ± g ) , Γ ≡ κ2+ω2 (g − g ) NB, g0 = g, no dissipation. Dissipation restored by finite κ.
Jonathan Keeling Collective dynamics Harvard, January 2012 29 / 30 Tricritical point 40 UN=-40 20 ⇓ SRA ⇓+⇑ SRA+⇑ 20 19 SRB
0 ⇓ ⇑ + SRB 19.70 ⇑ (MHz) (MHz) SRA+ ω ω 18 2SRA -20 ⇑ SRB+⇓+⇑ SRA 19.68 17 SRB SRB ⇓ -40 + 1.60 1.65 0 0.5 1 1.5 0 0.5 1 1.5 g√N (MHz) g√N (MHz)
1500 (a) g√N =1.0 (MHz) (b) 800 1000 600 2 If −ωu = −UN/2 > κ, crossing of | 2SRA ψ | SRA SRA SRB SRA 400 500 boundaries at: 200 SRB 0 0 π 9π/16 (d) q (c) ∗ 2 2 π/2 ω = ω − κ )
u ψ 0 π/2 Arg( √ r -π/2 −ω0UN ∗ -π 7π/16 g N = 0.5 0.5 4 (e) (f) /N
z 0 0.4 For g > g∗, 2SRA region exists at S edge of SRB. -0.5 0.3 -20 -10 0 10 20 4.1 4.2 ω (MHz) ω (MHz) Jonathan Keeling Collective dynamics Harvard, January 2012 30 / 30