Collective dynamics of Bose–Einstein condensate in optical cavities Munich, April 2011

Collective dynamics of Bose–Einstein condensate in
optical cavities
J. Keeling, J. A. Mayoh, M. J. Bhaseen, B. D. Simons
Munich, April 2011
Jonathan Keeling
Collective dynamics
Munich, April 2011
1 / 22
Acknowledgements
People:
Funding:
Jonathan Keeling
Collective dynamics
Munich, April 2011
2 / 22
Coupling many atoms to light
Old question: What happens to radiation when many atoms interact
“collectively” with light.
Jonathan Keeling
Collective dynamics
Munich, April 2011
3 / 22
Coupling many atoms to light
Old question: What happens to radiation when many atoms interact
“collectively” with light.
New relevance
Rydberg atoms
Superconducting qubits
Quantum dots
(excitons, polaritons, ...)
Nitrogen-Vacancies in diamond
Mechanical oscillators, ...
Ultra-cold atoms
Jonathan Keeling
Collective dynamics
Munich, April 2011
3 / 22
Coupling many atoms to light
Old question: What happens to radiation when many atoms interact
“collectively” with light.
New relevance
Rydberg atoms
Superconducting qubits
Quantum dots
(excitons, polaritons, ...)
Nitrogen-Vacancies in diamond
Mechanical oscillators, ...
Ultra-cold atoms
Superradiance — dynamical and steady state.
Jonathan Keeling
Collective dynamics
Munich, April 2011
3 / 22
Dicke effect: Enhanced emission
Hint =
X
P
gk ψk† Si− + H.c. . Use i Si → S
k,i
Emission: I ∝
X
|hfinal|ψk† S − |initiali|2
final
For: |S| = N/2, |Sz | N/2, rate I ∝ N 2 .
Jonathan Keeling
Collective dynamics
Munich, April 2011
4 / 22
Dicke effect: Enhanced emission
Hint =
X
P
gk ψk† Si− + H.c. . Use i Si → S
k,i
Emission: I ∝
X
|hfinal|ψk† S − |initiali|2
final
For: |S| = N/2, |Sz | N/2, rate I ∝ N 2 .
Jonathan Keeling
Collective dynamics
Munich, April 2011
4 / 22
Dicke effect: Enhanced emission
Hint =
X
P
gk ψk† Si− + H.c. . Use i Si → S
k,i
Emission: I ∝
X
|hfinal|ψk† S − |initiali|2
final
For: |S| = N/2, |Sz | N/2, rate I ∝ N 2 .
Jonathan Keeling
Collective dynamics
Munich, April 2011
4 / 22
Dicke effect: Enhanced emission
Hint =
X
P
gk ψk† Si− + H.c. . Use i Si → S
k,i
Emission: I ∝
X
|hfinal|ψk† S − |initiali|2
final
For: |S| = N/2, |Sz | N/2, rate I ∝ N 2 .
Jonathan Keeling
Collective dynamics
Munich, April 2011
4 / 22
Dicke effect and superradiance without a cavity
Hint =
X
gk ψk† Si− e −ik·ri + H.c.
k,i
Jonathan Keeling
Collective dynamics
Munich, April 2011
5 / 22
Dicke effect and superradiance without a cavity
Hint =
X
gk ψk† Si− e −ik·ri + H.c.
k,i
If |ri − rj | λ, use
P
Si → S. Many modes ψk — integrate out:
Γ
dρ
= − S + S − ρ − S − ρS + + ρS + S −
dt
2
Jonathan Keeling
i
Collective dynamics
Munich, April 2011
5 / 22
Dicke effect and superradiance without a cavity
Hint =
X
gk ψk† Si− e −ik·ri + H.c.
k,i
If |ri − rj | λ, use
P
Si → S. Many modes ψk — integrate out:
Γ
dρ
= − S + S − ρ − S − ρS + + ρS + S −
dt
2
ΓN 2
dhS z i
2 ΓN
z
=
sech
t
If S = |S| = N/2 initially: I ∝ −Γ
dt
4
2
i
ΓN2/2
I=-Γd〈Sz〉/dt
〈Sz〉
N/2
0
-N/2
0
tD
Jonathan Keeling
tD
Collective dynamics
Munich, April 2011
5 / 22
Dicke effect and superradiance without a cavity
Hint =
X
gk ψk† Si− e −ik·ri + H.c.
k,i
If |ri − rj | λ, use
P
Si → S. Many modes ψk — integrate out:
Γ
dρ
= − S + S − ρ − S − ρS + + ρS + S −
dt
2
ΓN 2
dhS z i
2 ΓN
z
=
sech
t
If S = |S| = N/2 initially: I ∝ −Γ
dt
4
2
i
ΓN2/2
I=-Γd〈Sz〉/dt
〈Sz〉
N/2
0
-N/2
0
tD
tD
Problem: dipole-dipole interactions dephase.
[Friedberg et al, Phys. Lett. 1972]
Jonathan Keeling
Collective dynamics
Munich, April 2011
5 / 22
Collective radiation with a cavity: Dynamics
Hint =
X
ψ † Si− + ψSi+
Single cavity mode: oscillations
i
[Bonifacio and Preparata PRA 1970; JK PRA 2009]
Jonathan Keeling
Collective dynamics
Munich, April 2011
6 / 22
Collective radiation with a cavity: Dynamics
Hint =
X
ψ † Si− + ψSi+
Single cavity mode: oscillations
If S z = |S| = N/2 initially:
i
__
1600
__
T=2ln(√N)/√N
1400
|ψ(t)|
2
1200
1000
800
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 1970; JK PRA 2009]
Jonathan Keeling
Collective dynamics
Munich, April 2011
6 / 22
Collective radiation with a cavity: Dynamics
Hint =
X
ψ † Si− + ψSi+
Single cavity mode: oscillations
If S z = |S| = N/2 initially:
i
__
__
1600
T=2ln(√N)/√N
1400
1400
1200
1200
1000
1000
|ψ(t)|2
|ψ(t)|
2
1600
800
N=2000
800
600
600
400
400
__
1/√N
200
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 1970; JK PRA 2009]
Jonathan Keeling
Collective dynamics
Munich, April 2011
6 / 22
With a cavity: Superradiance phase transition
With detuning: H = ω0 S z + ωψ † ψ + g ψ † S − + ψS +
[Hepp, Lieb, Ann. Phys. 1973]
Jonathan Keeling
Collective dynamics
Munich, April 2011
7 / 22
With a cavity: Superradiance phase transition
With detuning: H = ω0 S z + ωψ † ψ + g ψ † S − + ψS +
Mean-field wf: |Ψi → e λψ
† +ηS +
|Ωi
Spontaneous polarisation if: Ng 2 > ωω0
[Hepp, Lieb, Ann. Phys. 1973]
Jonathan Keeling
Collective dynamics
Munich, April 2011
7 / 22
With a cavity: Superradiance phase transition
With detuning: H = ω0 S z + ωψ † ψ + g ψ † S − + ψS +
Mean-field wf: |Ψi → e λψ
† +ηS +
|Ωi
Spontaneous polarisation if: Ng 2 > ωω0
[Hepp, Lieb, Ann. Phys. 1973]
Problem: never occurs.
Minimal coupling (p − eA)2 /2m
[Rzazewski et al Phys. Rev. Lett 1975]
Jonathan Keeling
Collective dynamics
Munich, April 2011
7 / 22
With a cavity: Superradiance phase transition
With detuning: H = ω0 S z + ωψ † ψ + g ψ † S − + ψS +
Mean-field wf: |Ψi → e λψ
† +ηS +
|Ωi
Spontaneous polarisation if: Ng 2 > ωω0
[Hepp, Lieb, Ann. Phys. 1973]
Problem: never occurs.
−
X e
A · pi
m
⇔
Minimal coupling (p − eA)2 /2m
g (ψ † S − + ψS + ),
i
[Rzazewski et al Phys. Rev. Lett 1975]
Jonathan Keeling
Collective dynamics
Munich, April 2011
7 / 22
With a cavity: Superradiance phase transition
With detuning: H = ω0 S z + ωψ † ψ + g ψ † S − + ψS +
Mean-field wf: |Ψi → e λψ
† +ηS +
|Ωi
Spontaneous polarisation if: Ng 2 > ωω0
[Hepp, Lieb, Ann. Phys. 1973]
Problem: never occurs.
−
X e
A · pi
m
⇔
Minimal coupling (p − eA)2 /2m
g (ψ † S − + ψS + ),
i
X A2
2m
⇔
Nζ(ψ + ψ † )2
i
[Rzazewski et al Phys. Rev. Lett 1975]
Jonathan Keeling
Collective dynamics
Munich, April 2011
7 / 22
With a cavity: Superradiance phase transition
With detuning: H = ω0 S z + ωψ † ψ + g ψ † S − + ψS +
Mean-field wf: |Ψi → e λψ
† +ηS +
|Ωi
Spontaneous polarisation if: Ng 2 > ωω0
[Hepp, Lieb, Ann. Phys. 1973]
Problem: never occurs.
−
X e
A · pi
m
⇔
Minimal coupling (p − eA)2 /2m
g (ψ † S − + ψS + ),
i
X A2
2m
⇔
Nζ(ψ + ψ † )2
i
For large N, ω → ω + 4Nζ. Need Ng 2 > ω0 (ω + 4Nζ).
But g 2 /ω0 < 4ζ. No transition [Rzazewski et al Phys. Rev. Lett 1975]
Jonathan Keeling
Collective dynamics
Munich, April 2011
7 / 22
Dicke phase transition: ways out
Problem: g 2 /ω0 < 4ζ for intrinsic parameters.
Solutions:
Non-solution Ferroelectric
transition in D · r gauge.
[JK JPCM 2007 ]
Grand canonical ensemble:
I
I
If H → H − µ(S z + ψ † ψ), need only:
g 2 N > (ω − µ)(ω0 − µ)
Incoherent pumping — polaritons.
[JK Semicond. Sci. Technol. 2007]
Dissociate g , ω0 , e.g. Raman
Scheme: ω0 ω.
[Dimer et al PRA 2007; Baumann et al
Nature 2010 ]
See also [Nataf and Ciuti, Nat. Comm. 2010; Viehmann et al 1103.4639]
Jonathan Keeling
Collective dynamics
Munich, April 2011
8 / 22
Dicke phase transition: ways out
Problem: g 2 /ω0 < 4ζ for intrinsic parameters.
Solutions:
Non-solution Ferroelectric
transition in D · r gauge.
[JK JPCM 2007 ]
Grand canonical ensemble:
I
I
If H → H − µ(S z + ψ † ψ), need only:
g 2 N > (ω − µ)(ω0 − µ)
Incoherent pumping — polaritons.
[JK Semicond. Sci. Technol. 2007]
Dissociate g , ω0 , e.g. Raman
Scheme: ω0 ω.
[Dimer et al PRA 2007; Baumann et al
Nature 2010 ]
See also [Nataf and Ciuti, Nat. Comm. 2010; Viehmann et al 1103.4639]
Jonathan Keeling
Collective dynamics
Munich, April 2011
8 / 22
Dicke phase transition: ways out
Problem: g 2 /ω0 < 4ζ for intrinsic parameters.
Solutions:
40
non-condensed
Non-solution Ferroelectric
transition in D · r gauge.
condensed
30
kB T
[JK JPCM 2007 ]
20
Grand canonical ensemble:
I
I
If H → H − µ(S z + ψ † ψ), need only:
g 2 N > (ω − µ)(ω0 − µ)
Incoherent pumping — polaritons.
[JK Semicond. Sci. Technol. 2007]
10
0
0
Cavity
9
1×10
n [cm-2]
9
2×10
Dissociate g , ω0 , e.g. Raman
Scheme: ω0 ω.
[Dimer et al PRA 2007; Baumann et al
Nature 2010 ]
See also [Nataf and Ciuti, Nat. Comm. 2010; Viehmann et al 1103.4639]
Jonathan Keeling
Collective dynamics
Munich, April 2011
8 / 22
Dicke phase transition: ways out
Problem: g 2 /ω0 < 4ζ for intrinsic parameters.
Solutions:
40
non-condensed
Non-solution Ferroelectric
transition in D · r gauge.
condensed
30
kB T
[JK JPCM 2007 ]
20
Grand canonical ensemble:
I
I
If H → H − µ(S z + ψ † ψ), need only:
g 2 N > (ω − µ)(ω0 − µ)
Incoherent pumping — polaritons.
[JK Semicond. Sci. Technol. 2007]
Dissociate g , ω0 , e.g. Raman
Scheme: ω0 ω.
[Dimer et al PRA 2007; Baumann et al
Nature 2010 ]
10
Cavity
0
0
9
9
1×10
n [cm-2]
κ
κ
2×10
Cavity
Pump
Pump
See also [Nataf and Ciuti, Nat. Comm. 2010; Viehmann et al 1103.4639]
Jonathan Keeling
Collective dynamics
Munich, April 2011
8 / 22
Dicke phase transition: ways out
Problem: g 2 /ω0 < 4ζ for intrinsic parameters.
Solutions:
40
non-condensed
Non-solution Ferroelectric
transition in D · r gauge.
condensed
30
kB T
[JK JPCM 2007 ]
20
Grand canonical ensemble:
I
I
If H → H − µ(S z + ψ † ψ), need only:
g 2 N > (ω − µ)(ω0 − µ)
Incoherent pumping — polaritons.
[JK Semicond. Sci. Technol. 2007]
Dissociate g , ω0 , e.g. Raman
Scheme: ω0 ω.
[Dimer et al PRA 2007; Baumann et al
Nature 2010 ]
10
Cavity
0
0
9
9
1×10
n [cm-2]
κ
κ
2×10
Cavity
Pump
Pump
See also [Nataf and Ciuti, Nat. Comm. 2010; Viehmann et al 1103.4639]
Jonathan Keeling
Collective dynamics
Munich, April 2011
8 / 22
Overview
1
Dicke model and collective emission
Ferroelectric transition and gauges
2
Optical lattice realisation and dynamics
Fixed points and phase diagram
Dynamics and critical slowing down
Regions without fixed points
3
Hyperfine levels and extra phases
4
Conclusions
Jonathan Keeling
Collective dynamics
Munich, April 2011
9 / 22
Ferroelectric transition
Atoms in Coulomb gauge
X
X
H=
ωk ak† ak +
[pi − eA(ri )]2 + Vcoul
i
Jonathan Keeling
Collective dynamics
Munich, April 2011
10 / 22
Ferroelectric transition
Atoms in Coulomb gauge
X
X
H=
ωk ak† ak +
[pi − eA(ri )]2 + Vcoul
i
Two-level systems — dipole-dipole coupling
H = ω0 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
Munich, April 2011
10 / 22
Ferroelectric transition
Atoms in Coulomb gauge
X
X
H=
ωk ak† ak +
[pi − eA(ri )]2 + Vcoul
i
Two-level systems — dipole-dipole coupling
H = ω0 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
Ferroelectric polarisation if ω0 < 2ηN
Collective dynamics
Munich, April 2011
10 / 22
Ferroelectric transition
Atoms in Coulomb gauge
X
X
H=
ωk ak† ak +
[pi − eA(ri )]2 + Vcoul
i
Two-level systems — dipole-dipole coupling
H = ω0 S z + ωψ † ψ + g (S + + S − )(ψ + ψ † ) + Nζ(ψ + ψ † )2 −η(S + − S − )2
Ferroelectric polarisation if ω0 < 2ηN
(nb g 2 , ζ, η ∝ 1/V ).
Gauge transform to dipole gauge D · r
H = ω0 S z + ωψ † ψ + ḡ (S + − S − )(ψ − ψ † )
“Dicke” transition at ω0 < N ḡ 2 /ω ≡ 2ηN
But, ψ describes electric displacement
Jonathan Keeling
Collective dynamics
Munich, April 2011
10 / 22
Overview
1
Dicke model and collective emission
Ferroelectric transition and gauges
2
Optical lattice realisation and dynamics
Fixed points and phase diagram
Dynamics and critical slowing down
Regions without fixed points
3
Hyperfine levels and extra phases
4
Conclusions
Jonathan Keeling
Collective dynamics
Munich, April 2011
11 / 22
Extended Dicke model
κ
κ
[Baumann et al. Nature 2010]
2 Level system, | ⇓i, | ⇑i:
⇓: |kx , kz i = |0, 0i,
⇑: |kx , kz i = | ± k, ±ki,
ω0 = 2ωrecoil
g0ψ
Ω
z
x
Pump
2 Level System
H = ωψ † ψ + ω0 S z + g (ψ † S − + ψS + ) + g 0 (ψ † S + + ψS − )+USz ψ † ψ.
N atoms: |S| = N/2
Jonathan Keeling
Collective dynamics
Munich, April 2011
12 / 22
Extended Dicke model
κ
κ
[Baumann et al. Nature 2010]
2 Level system, | ⇓i, | ⇑i:
⇓: |kx , kz i = |0, 0i,
⇑: |kx , kz i = | ± k, ±ki,
ω0 = 2ωrecoil
g0ψ
Ω
z
x
Pump
2 Level System
H = ωψ † ψ + ω0 S z + g (ψ † S − + ψS + ) + g 0 (ψ † S + + ψS − )+USz ψ † ψ.
N atoms: |S| = N/2
Jonathan Keeling
Collective dynamics
Munich, April 2011
12 / 22
Extended Dicke model
κ
κ
[Baumann et al. Nature 2010]
2 Level system, | ⇓i, | ⇑i:
⇓: |kx , kz i = |0, 0i,
⇑: |kx , kz i = | ± k, ±ki,
ω0 = 2ωrecoil
g0ψ
Ω
z
x
Pump
2 Level System
Feedback: U ∝
g02
ωc − ωa
H = ωψ † ψ + ω0 S z + g (ψ † S − + ψS + ) + g 0 (ψ † S + + ψS − )+USz ψ † ψ.
N atoms: |S| = N/2
Jonathan Keeling
Collective dynamics
Munich, April 2011
12 / 22
Extended Dicke model
κ
κ
[Baumann et al. Nature 2010]
2 Level system, | ⇓i, | ⇑i:
⇓: |kx , kz i = |0, 0i,
⇑: |kx , kz i = | ± k, ±ki,
ω0 = 2ωrecoil
g0ψ
Ω
z
Pump
x
2 Level System
Feedback: U ∝
g02
ωc − ωa
H = ωψ † ψ + ω0 S z + g (ψ † S − + ψS + ) + g 0 (ψ † S + + ψS − )+USz ψ † ψ.
N atoms: |S| = N/2
Add decay:
Ṡ − = −i(ω0 +Uψ † ψ)S − + 2i(g ψ + g 0 ψ † )S z
Ṡ z = −ig (ψS + − ψ † S − ) + ig 0 (ψS − − ψ † S + )
ψ̇ = − [κ + i(ω+US z )] ψ − igS − − ig 0 S +
Jonathan Keeling
Collective dynamics
Munich, April 2011
12 / 22
Fixed points at U = 0
H = ωψ † ψ + ω0 S z + g (ψ † S − + ψS + ) + g 0 (ψ † S + + ψS − )
Fixed points Ṡ, ψ̇ = 0.
Jonathan Keeling
S z = ±N/2, ψ = 0 always present
ψ 6= 0 if g , g 0 large.
Collective dynamics
Munich, April 2011
13 / 22
Fixed points at U = 0
H = ωψ † ψ + ω0 S z + g (ψ † S − + ψS + ) + g 0 (ψ † S + + ψS − )
S z = ±N/2, ψ = 0 always present
Fixed points Ṡ, ψ̇ = 0.
ψ 6= 0 if g , g 0 large.
2.0
S4
__
g′√N (MHz)
1.5
1.0
S3
SR
⇑
0.5
SR+⇓
S2
⇓
0.0
0.0
●
0.5
1.0
S1
1.5
2.0
__
g√N (MHz)
Jonathan Keeling
Collective dynamics
Munich, April 2011
13 / 22
Fixed points at U = 0
H = ωψ † ψ + ω0 S z + g (ψ † S − + ψS + ) + g 0 (ψ † S + + ψS − )
S z = ±N/2, ψ = 0 always present
Fixed points Ṡ, ψ̇ = 0.
ψ 6= 0 if g , g 0 large.
2.0
S1
g′√N (MHz)
1.5
__
S2
S4
1.0
S3
SR
⇑
0.5
S4
S2
⇓
0.0
0.0
S3
SR+⇓
●
0.5
1.0
S1
1.5
2.0
__
g√N (MHz)
Jonathan Keeling
Collective dynamics
Munich, April 2011
13 / 22
Slow dynamics near critical g 0 /g
0.5
Sz
2.0
1.5
-0.5
(b)
g′√N (MHz)
1.0
100
SR+⇓
50
⇓
0.0
0.0
(c)
106
|ψ|2
SR
⇑
0.5
|ψ|2
__
(a)
0
0.5
104
●
1.0
__
g√N (MHz)
1.5
2.0
200.0
0
60
80
100
120
140 160
t (ms)
200.2
180
200
200.4
220
240
√
ω, κ, g N ∼ MHz, ω0 ∼ kHz. Much slower decay.
Treating ω0 /κ perturbatively, linear stability gives Im(ν) = −
κω02
κ2 + ω 2
For large κ/ω0 , adiabatically eliminate ψ:
∂t S = {S, H} − ΓS × (S × ẑ), H = ω0 Sz − Λ+ Sx2 − Λ− Sy2
Γ ∝ (g 0 2 − g 2 )
Jonathan Keeling
Collective dynamics
Munich, April 2011
14 / 22
Slow dynamics near critical g 0 /g
0.5
Sz
2.0
1.5
-0.5
(b)
g′√N (MHz)
1.0
100
SR+⇓
50
⇓
0.0
0.0
(c)
106
|ψ|2
SR
⇑
0.5
|ψ|2
__
(a)
0
0.5
104
●
1.0
__
g√N (MHz)
1.5
2.0
200.0
0
60
80
100
120
140 160
t (ms)
200.2
180
200
200.4
220
240
√
ω, κ, g N ∼ MHz, ω0 ∼ kHz. Much slower decay.
Treating ω0 /κ perturbatively, linear stability gives Im(ν) = −
κω02
κ2 + ω 2
For large κ/ω0 , adiabatically eliminate ψ:
∂t S = {S, H} − ΓS × (S × ẑ), H = ω0 Sz − Λ+ Sx2 − Λ− Sy2
Γ ∝ (g 0 2 − g 2 )
Jonathan Keeling
Collective dynamics
Munich, April 2011
14 / 22
Slow dynamics near critical g 0 /g
0.5
Sz
2.0
1.5
-0.5
(b)
g′√N (MHz)
1.0
100
SR+⇓
50
⇓
0.0
0.0
(c)
106
|ψ|2
SR
⇑
0.5
|ψ|2
__
(a)
0
0.5
104
●
1.0
__
g√N (MHz)
1.5
2.0
200.0
0
60
80
100
120
140 160
t (ms)
200.2
180
200
200.4
220
240
√
ω, κ, g N ∼ MHz, ω0 ∼ kHz. Much slower decay.
Treating ω0 /κ perturbatively, linear stability gives Im(ν) = −
κω02
κ2 + ω 2
For large κ/ω0 , adiabatically eliminate ψ:
∂t S = {S, H} − ΓS × (S × ẑ), H = ω0 Sz − Λ+ Sx2 − Λ− Sy2
Γ ∝ (g 0 2 − g 2 )
Jonathan Keeling
Collective dynamics
Munich, April 2011
14 / 22
Finite U phase diagram, g = g 0
H = ωψ † ψ + ω0 S z + g (ψ † + ψ)(S + + S − ) + USz ψ † ψ
Jonathan Keeling
Collective dynamics
Munich, April 2011
15 / 22
Finite U phase diagram, g = g 0
H = ωψ † ψ + ω0 S z + g (ψ † + ψ)(S + + S − ) + USz ψ † ψ
2.0
1.5
SR
g√N (MHz)
1.0
0.5
SR
⇑
SR+⇓
0.5
SR
SR+⇓
Limit Cycle
0.5
SR+⇑+⇓
⇓
0.0
0.0
SR+⇑
1.0
__
__
g′√N (MHz)
1.5
⇓
●
1.0
__
g√N (MHz)
Jonathan Keeling
1.5
2.0
0.0
-80
⇑+⇓
-60
Collective dynamics
-40
-20
0
20
UN (MHz)
40
60
Munich, April 2011
80
15 / 22
Explaining finite U phase diagram
SR
SR+⇑
SR
1.0
__
g√N (MHz)
1.5
SR+⇓
Limit Cycle
0.5
SR+⇑+⇓
0.0
-80
Jonathan Keeling
⇓
⇑+⇓
-60
-40
-20
0
20
UN (MHz)
Collective dynamics
40
60
80
Munich, April 2011
16 / 22
Explaining finite U phase diagram
SR
SR+⇑
SR
1.0
__
g√N (MHz)
1.5
SR+⇓
Limit Cycle
0.5
SR+⇑+⇓
0.0
-80
⇓
⇑+⇓
-60
-40
-20
0
20
UN (MHz)
40
60
80
If g = g 0 , analytic ψ 6= 0 solution. ∂t S z = ig (ψ + ψ † )(S − − S + )
Jonathan Keeling
Collective dynamics
Munich, April 2011
16 / 22
Explaining finite U phase diagram
SR
SR+⇑
SR
1.0
__
g√N (MHz)
1.5
SR+⇓
Limit Cycle
0.5
SR+⇑+⇓
0.0
-80
⇓
⇑+⇓
-60
-40
-20
0
20
UN (MHz)
40
60
80
If g = g 0 , analytic ψ 6= 0 solution. ∂t S z = ig (ψ + ψ † )(S − − S + )
If |UN| < 2ω: S ± = S x
Jonathan Keeling
Collective dynamics
Munich, April 2011
16 / 22
Explaining finite U phase diagram
SR
SR+⇑
SR
1.0
__
g√N (MHz)
1.5
SR+⇓
Limit Cycle
0.5
SR+⇑+⇓
0.0
-80
⇓
⇑+⇓
-60
-40
-20
0
20
UN (MHz)
40
60
80
If g = g 0 , analytic ψ 6= 0 solution. ∂t S z = ig (ψ + ψ † )(S − − S + )
If UN < −2ω Alternate SR
If |UN| < 2ω: S ± = S x
solution
Jonathan Keeling
Collective dynamics
Munich, April 2011
16 / 22
Explaining finite U phase diagram
SR
SR+⇑
SR
1.0
__
g√N (MHz)
1.5
SR+⇓
Limit Cycle
0.5
SR+⇑+⇓
0.0
-80
⇓
⇑+⇓
-60
-40
-20
0
20
UN (MHz)
40
60
80
If g = g 0 , analytic ψ 6= 0 solution. ∂t S z = ig (ψ + ψ † )(S − − S + )
If UN < −2ω Alternate SR
If |UN| < 2ω: S ± = S x
solution
If UN > 2ω No stable fixed
points
Jonathan Keeling
Collective dynamics
Munich, April 2011
16 / 22
Persistent optomechanical oscillations
SR
SR+⇑
∂t S − = −i(ω0 + U|ψ|2 )S − + 2ig (ψ + ψ † )S z
SR
1.0
∂t S z = i(ψ + ψ † )(S − − S + )
__
g√N (MHz)
1.5
SR+⇓
Limit Cycle
0.5
SR+⇑+⇓
0.0
-80
∂t ψ = − [κ + i(ω + US z )] ψ − ig (S − + S + )
⇓
⇑+⇓
-60
-40
-20
0
20
UN (MHz)
Jonathan Keeling
40
60
80
Collective dynamics
Munich, April 2011
17 / 22
Persistent optomechanical oscillations
SR
SR+⇑
∂t S − = −i(ω0 + U|ψ|2 )S − + 2ig (ψ + ψ † )S z
SR
1.0
∂t S z = i(ψ + ψ † )(S − − S + )
__
g√N (MHz)
1.5
SR+⇓
Limit Cycle
0.5
SR+⇑+⇓
0.0
-80
∂t ψ = − [κ + i(ω + US z )] ψ − ig (S − + S + )
⇓
⇑+⇓
-60
-40
-20
0
20
UN (MHz)
40
60
80
Fix S z = −ω/U if Re(ψ) = 0.
Jonathan Keeling
Collective dynamics
Munich, April 2011
17 / 22
Persistent optomechanical oscillations
SR
SR+⇑
∂t S − = −i(ω0 + U|ψ|2 )S − + 2ig (ψ + ψ † )S z
SR
1.0
∂t S z = i(ψ + ψ † )(S − − S + )
__
g√N (MHz)
1.5
SR+⇓
Limit Cycle
0.5
SR+⇑+⇓
0.0
-80
∂t ψ = − [κ + i(ω + US z )] ψ − ig (S − + S + )
⇓
⇑+⇓
-60
-40
-20
0
20
UN (MHz)
40
60
80
Fix S z = −ω/U if Re(ψ) = 0.
Jonathan Keeling
Collective dynamics
Munich, April 2011
17 / 22
Persistent optomechanical oscillations
SR
SR+⇑
∂t S − = −i(ω0 + U|ψ|2 )S − + 2ig (ψ + ψ † )S z
SR
1.0
∂t S z = i(ψ + ψ † )(S − − S + )
__
g√N (MHz)
1.5
SR+⇓
Limit Cycle
0.5
SR+⇑+⇓
0.0
-80
∂t ψ = − [κ + i(ω + US z )] ψ − ig (S − + S + )
⇓
⇑+⇓
-60
-40
-20
0
20
UN (MHz)
40
60
80
Fix S z = −ω/U if Re(ψ) = 0.
Writing
r
N2
−
−iθ
S = re
r=
− (S z )2
4
Get:
∂t θ = ω0 + U|ψ|2
(∂t + κ)ψ = −2igr cos(θ)
Jonathan Keeling
Collective dynamics
Munich, April 2011
17 / 22
Persistent optomechanical oscillations
SR
∂t S − = −i(ω0 + U|ψ|2 )S − + 2ig (ψ + ψ † )S z
SR
1.0
SR+⇓
∂t S z = i(ψ + ψ † )(S − − S + )
Limit Cycle
0.5
SR+⇑+⇓
0.0
-80
∂t ψ = − [κ + i(ω + US z )] ψ − ig (S − + S + )
⇓
⇑+⇓
-60
-40
-20
0
20
UN (MHz)
40
60
80
Fix S z = −ω/U if Re(ψ) = 0.
Writing
r
N2
−
−iθ
S = re
r=
− (S z )2
4
20
Imψ
SR+⇑
__
g√N (MHz)
1.5
0
-20
Get:
∂t θ = ω0 + U|ψ|2
(∂t + κ)ψ = −2igr cos(θ)
S/N
0.5
Sx
Sy
0
-0.5
1100.0
1100.1
1100.2
t (ms)
Jonathan Keeling
Collective dynamics
Munich, April 2011
17 / 22
Comparison to experiment UN = −40MHz
0
(ω − ωU)/2π
10
20
30
0.0
1.0
g2N
[JK et al PRL 2010 ]
Jonathan Keeling
2.0
[Baumann et al Nature 2010 ]
Collective dynamics
Munich, April 2011
18 / 22
Parameters and phases
2.0
1.5
1.0
0.5
⇑
SR+⇓
0.5
SR+⇑
SR
1.0
SR+⇓
Limit Cycle
0.5
S2
SR+⇑+⇓
⇓
0.0
0.0
S3
SR
SR
__
__
g′√N (MHz)
g√N (MHz)
S4
1.5
●
1.0
S1
1.5
2.0
__
g√N (MHz)
Phase
⇓
⇑
SR (ψ 6= 0)
SR + ⇓
Limit cycle
0.0
-80
⇑+⇓
-60
Seen?
X
×
X
?
×
Jonathan Keeling
⇓
-40
-20
0
20
UN (MHz)
40
60
80
(not true TLS)
Need positive U
Collective dynamics
Munich, April 2011
19 / 22
Parameters and phases
2.0
1.5
1.0
0.5
⇑
SR+⇓
0.5
SR+⇑
I
SR+⇓
Limit Cycle
SR+⇑+⇓
●
1.0
S1
1.5
2.0
__
g√N (MHz)
Phase
⇓
⇑
SR (ψ 6= 0)
SR + ⇓
Limit cycle
I
SR
1.0
0.5
S2
⇓
0.0
0.0
S3
SR
Tunable parameters
SR
__
__
g′√N (MHz)
g√N (MHz)
S4
1.5
0.0
-80
I
⇑+⇓
-60
Seen?
X
×
X
?
×
Jonathan Keeling
I
⇓
-40
-20
0
20
UN (MHz)
40
60
80
I
g , g 0 X(g = g 0 )
U ?
ω X
?
κ ×
(not true TLS)
Need positive U
Collective dynamics
Munich, April 2011
19 / 22
Parameters and phases
2.0
1.5
1.0
0.5
⇑
SR+⇓
0.5
SR+⇑
I
SR+⇓
Limit Cycle
SR+⇑+⇓
●
1.0
S1
1.5
2.0
__
g√N (MHz)
Phase
⇓
⇑
SR (ψ 6= 0)
SR + ⇓
Limit cycle
I
SR
1.0
0.5
S2
⇓
0.0
0.0
S3
SR
Tunable parameters
SR
__
__
g′√N (MHz)
g√N (MHz)
S4
1.5
0.0
-80
I
⇑+⇓
-60
Seen?
X
×
X
?
×
Jonathan Keeling
I
⇓
-40
-20
0
20
UN (MHz)
40
60
80
I
g , g 0 X(g = g 0 )
U ?
ω X
?
κ ×
Can we tune g 6= g 0 ?
(not true TLS)
What other phases
occur?
Need positive U
Collective dynamics
Munich, April 2011
19 / 22
Tuning g , g 0 , U
∆a
[Dimer et al. Phys. Rev. A. (2007)]
∆b
Ωb
Ωa
g0 ψ
g0 ψ
2 Level System
Jonathan Keeling
Separate pump strength/detuning
g0 2 g0 2
g0 Ωb 0 g0 Ωa
,g ∼
,U ∼
−
g∼
∆b
∆a
∆a
∆b
Collective dynamics
Munich, April 2011
20 / 22
Tuning g , g 0 , U
∆a
[Dimer et al. Phys. Rev. A. (2007)]
∆b
Ωb
Ωa
g0 ψ
g0 ψ
2 Level System
Separate pump strength/detuning
g0 2 g0 2
g0 Ωb 0 g0 Ωa
,g ∼
,U ∼
−
g∼
∆b
∆a
∆a
∆b
Possible realization: Hyperfine levels
σ−
σ+
mF=+1
B || z
mF=0
mF=−1
Jonathan Keeling
Collective dynamics
Munich, April 2011
20 / 22
Phase diagrams vs g , g 0 , U, ω
2.0
__
g′√N (MHz)
1.5
1.0
0.5
SR
⇑
SR+⇓
⇓
0.0
0.0
0.5
●
1.0
1.5
2.0
__
g√N (MHz)
Jonathan Keeling
Collective dynamics
Munich, April 2011
21 / 22
Phase diagrams vs g , g 0 , U, ω
2.0
__
g′√N (MHz)
1.5
1.0
0.5
SR
⇑
SR+⇓
⇓
0.0
0.0
0.5
●
1.0
1.5
2.0
__
g√N (MHz)
Jonathan Keeling
Collective dynamics
Munich, April 2011
21 / 22
Phase diagrams vs g , g 0 , U, ω
2.0
__
g′√N (MHz)
1.5
1.0
SR
⇑
0.5
SR+⇓
⇓
0.0
0.0
0.5
●
1.0
1.5
2.0
__
g√N (MHz)
1.54
⇓
SR+⇓
⇓
SR
g√N
1.52
1.5
LC
1.48
1.46
SR
-100 -80 -60 -40 -20
⇑
0
ω
SR+⇑
20
Jonathan Keeling
40
⇑
60
80 100
Collective dynamics
Munich, April 2011
21 / 22
Summary
Dynamical vs steady state superradiance
1600
ΓN2/2
1200
|ψ(t)|2
0
N=2000
1400
I=-Γd〈Sz〉/dt
〈Sz〉
N/2
1000
800
600
400
200
-N/2
0
tD
0
0
tD
1
2
3
4
5
Time
Circumventing no-go theorem: (Ferroelectric?), Open systems,
Raman scheme
Realisation of (modified) superradiance transition
κ
Cavity
κ
Pump
Pump
For g 6= g 0 , U 6= 0, wide variety of dynamical phases
2.0
0
1.5
SR+⇓
S2
0.5
●
1.0
__
g√N (MHz)
SR+⇓
SR+⇑+⇓
2.0
20
Limit Cycle
0.5
S1
1.5
10
SR
1.0
__
S3
SR
⇑
⇓
0.0
0.0
SR+⇑
(ω − ωU)/2π
g√N (MHz)
__
g′√N (MHz)
1.0
0.5
SR
S4
1.5
0.0
-80
⇓
30
⇑+⇓
-60
-40
-20
0
20
UN (MHz)
40
60
80
0.0
1.0
g2N
2.0
Slow dynamics and persistent oscillations
Jonathan Keeling
Collective dynamics
Munich, April 2011
22 / 22
Jonathan Keeling
Collective dynamics
Munich, April 2011
23 / 33
Extra slides
5
Polaritons and Dicke model
6
Numerical confirmation of FP
7
Dicke Oscillations
8
Extensions to atomic Dicke realisation
Jonathan Keeling
Collective dynamics
Munich, April 2011
24 / 33
Chemical potential and Dicke model
H = ωψ † ψ +
X
ω0 Siz + g (ψSi+ + H.c.)
i
Cavity
Quantum Wells
Transition occurs at: 1
Ng 2
tanh β (ω0 − µ)
ω−µ=
ω0 − µ
2
Analogy to dynamics:
Pendulum equation in frame rotating at µ
Open system; incoherent pumping.
Polariton condensate
Jonathan Keeling
Collective dynamics
Munich, April 2011
25 / 33
Chemical potential and Dicke model
H = ωψ † ψ +
X
ω0 Siz + g (ψSi+ + H.c.)
i
Cavity
Quantum Wells
Transition occurs at: 1
Ng 2
tanh β (ω0 − µ)
ω−µ=
ω0 − µ
2
How to introduce µ
Analogy to dynamics:
Pendulum equation in frame rotating at µ
Open system; incoherent pumping.
Polariton condensate
Jonathan Keeling
Collective dynamics
Munich, April 2011
25 / 33
Chemical potential and Dicke model
H = ωψ † ψ +
X
ω0 Siz + g (ψSi+ + H.c.)
i
Cavity
Quantum Wells
Transition occurs at: 1
Ng 2
tanh β (ω0 − µ)
ω−µ=
ω0 − µ
2
How to introduce µ
__
1600
__
T=2ln(√N)/√N
1400
1200
|ψ(t)|2
Analogy to dynamics:
Pendulum equation in frame rotating at µ
1000
800
600
Open system; incoherent pumping.
Polariton condensate
Jonathan Keeling
Collective dynamics
400
__
200
1/√N
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time
Munich, April 2011
25 / 33
Chemical potential and Dicke model
H = ωψ † ψ +
X
ω0 Siz + g (ψSi+ + H.c.)
i
Cavity
Transition occurs at: 1
Ng 2
tanh β (ω0 − µ)
ω−µ=
ω0 − µ
2
Quantum Wells
How to introduce µ
__
1600
__
T=2ln(√N)/√N
1400
1200
|ψ(t)|2
Analogy to dynamics:
Pendulum equation in frame rotating at µ
1000
800
600
Open system; incoherent pumping.
Polariton condensate
Ph
Jonathan Keeling
400
__
200
1/√N
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time
Ex
Collective dynamics
Munich, April 2011
25 / 33
Boundaries U = 0
κ 6= 0
2.0
S4
__
g′√N (MHz)
1.5
1.0
S3
SR
⇑
0.5
SR+⇓
S2
⇓
0.0
0.0
●
0.5
1.0
S1
1.5
2.0
__
g√N (MHz)
Jonathan Keeling
Collective dynamics
Munich, April 2011
26 / 33
Boundaries U = 0
κ 6= 0
g0
—, —
=
g
s
(ω + ω0 )2 + κ2
(ω − ω0 )2 + κ2
2.0
S4
__
g′√N (MHz)
1.5
1.0
S3
SR
⇑
0.5
SR+⇓
S2
⇓
0.0
0.0
●
0.5
1.0
S1
1.5
2.0
__
g√N (MHz)
Jonathan Keeling
Collective dynamics
Munich, April 2011
26 / 33
Boundaries U = 0
κ 6= 0
—, —
g0
g
s
=
κ = 0:
)2
κ2
(ω + ω0 +
(ω − ω0 )2 + κ2
—N(g + g 0 )2 = ωω0
2.0
2.0
S4
1.0
S3
SR
⇑
0.5
SR+⇓
S2
⇓
0.0
0.0
__
__
g′√N (MHz)
1.5
g′√N (MHz)
1.5
●
0.5
1.0
1.0
0.5
2.0
0.0
0.0
0.5
1.0
1.5
2.0
__
__
g√N (MHz)
g√N (MHz)
Jonathan Keeling
SR+⇓
⇓
S1
1.5
SR
⇑
Collective dynamics
Munich, April 2011
26 / 33
Numerical confirmation of fixed points
SR
SR+⇑
SR
1.0
__
g√N (MHz)
1.5
SR+⇓
Limit Cycle
0.5
SR+⇑+⇓
0.0
-80
⇓
⇑+⇓
-60
-40
-20
Jonathan Keeling
0
20
UN (MHz)
40
60
80
Collective dynamics
Munich, April 2011
27 / 33
Numerical confirmation of fixed points
0.5
__
SR
Sz
1.5
g√N=0.791
0
SR
500
400
__
g√N (MHz)
-0.5
SR+⇑
1.0
SR+⇓
|ψ|2
Limit Cycle
0.5
0.0
-80
⇓
100
⇑+⇓
-60
300
200
SR+⇑+⇓
-40
-20
Jonathan Keeling
0
20
UN (MHz)
40
60
80
Collective dynamics
0
-80
-60
-40
-20
0
20
UN (MHz)
40
Munich, April 2011
60
80
27 / 33
Numerical confirmation of fixed points
0.5
__
SR
Sz
1.5
g√N=0.791
0
SR
500
400
__
g√N (MHz)
-0.5
SR+⇑
1.0
SR+⇓
|ψ|2
Limit Cycle
0.5
⇓
100
⇑+⇓
-60
-40
-20
0
20
UN (MHz)
40
60
0
-80
80
-60
-40
-20
0
20
UN (MHz)
40
60
80
300
|ψ|2
0.0
-80
300
200
SR+⇑+⇓
200
100
0
-42.75
-42.70
-42.65
UN (MHz)
Jonathan Keeling
Collective dynamics
Munich, April 2011
27 / 33
Numerical confirmation of fixed points
0.5
__
SR
Sz
1.5
g√N=0.791
0
SR
500
400
__
g√N (MHz)
-0.5
SR+⇑
1.0
SR+⇓
|ψ|2
Limit Cycle
0.5
0.0
-80
⇓
100
⇑+⇓
-60
300
200
SR+⇑+⇓
-40
-20
0
20
UN (MHz)
40
60
0
-80
80
-60
-40
-20
0
20
UN (MHz)
40
60
80
300
|ψ|2
T = 360ms
200
100
0
-42.75
-42.70
-42.65
UN (MHz)
Jonathan Keeling
Collective dynamics
Munich, April 2011
27 / 33
How good is semiclassics?
From eigenstates H|Ψq i = Eq |Ψq i:
1600
N=2000
1400
|ψ(t)|2
1200
1000
800
600
400
200
0
0
1
2
3
4
5
Time
Jonathan Keeling
Collective dynamics
Munich, April 2011
28 / 33
How good is semiclassics?
If periodic,
From eigenstates H|Ψq i = Eq |Ψq i:
1600
N=2000
1400
|ψ(t)|2
1200
1000
Overlap intensity
∆Eq = qΩ,
800
√
N
Ω = πg √
ln( N)
50
0
-50
0
20
40
60
80
100
Mode frequency: Ep - Eq
600
400
200
0
0
1
2
3
4
5
Time
Jonathan Keeling
Collective dynamics
Munich, April 2011
28 / 33
How good is semiclassics?
If periodic,
√
N
Ω = πg √
ln( N)
From eigenstates H|Ψq i = Eq |Ψq i:
1600
N=2000
1400
|ψ(t)|2
1200
1000
Overlap intensity
∆Eq = qΩ,
50
0
-50
800
0
20
40
60
80
100
Mode frequency: Ep - Eq
600
400
0.6
0
0
1
2
3
4
Time
Anharmonicity: ∆Eq − q∆E1
5
(∆Eq - q ∆E1)/∆E1
200
0.5
q=2
q=3
q=4
0.4
0.3
0.2
0.1
0
102
103
104
105
106
107
N
Jonathan Keeling
Collective dynamics
Munich, April 2011
28 / 33
Semiclassical approximation: WKB quantisation
Problem is one dimensional; nphot + Sz ≡ N/2, find Ψ(nphot ):
√
√
(E + ∆n)Ψn = gn N + 1 − nΨn−1 + g (n + 1) N − nΨn+1
[Keeling PRA 79 053825; see also Babelon et al. J. Stat. Mech p.07011]
Jonathan Keeling
Collective dynamics
Munich, April 2011
29 / 33
Semiclassical approximation: WKB quantisation
Problem is one dimensional; nphot + Sz ≡ N/2, find Ψ(nphot ):
√
√
(E + ∆n)Ψn = gn N + 1 − nΨn−1 + g (n + 1) N − nΨn+1
0.3
WKB wavefunction:
0.04
0.4
0.02
0.2 0.2
0
0
〈n|ψ〉
-0.02
0.1 -0.2
0
20
40
-0.04
1960
1980
2000
0
-0.1
cos(E Φn + φ + nπ/2)
Ψn ' q
p
(n + 1/2) N − n + 1/2
"s
#
N +1
1
arcosh
Φn ' √
n + 1/2
g N +1
-0.2
0
500
1000
n
1500
2000
Find E , φ by matching assymptotics at n ' 0, n ' N.
[Keeling PRA 79 053825; see also Babelon et al. J. Stat. Mech p.07011]
Jonathan Keeling
Collective dynamics
Munich, April 2011
29 / 33
Semiclassical approximation: WKB quantisation
Problem is one dimensional; nphot + Sz ≡ N/2, find Ψ(nphot ):
√
√
(E + ∆n)Ψn = gn N + 1 − nΨn−1 + g (n + 1) N − nΨn+1
0.3
WKB wavefunction:
0.04
0.4
0.02
0.2 0.2
0
0
〈n|ψ〉
-0.02
0.1 -0.2
0
20
40
-0.04
1960
1980
2000
0
-0.1
cos(E Φn + φ + nπ/2)
Ψn ' q
p
(n + 1/2) N − n + 1/2
"s
#
N +1
1
arcosh
Φn ' √
n + 1/2
g N +1
-0.2
0
500
1000
n
1500
2000
Find E , φ by matching assymptotics at n ' 0, n ' N.
Hard boundary at n = 0: breakdown of Bohr-Sommerfeld quantisation.
[Keeling PRA 79 053825; see also Babelon et al. J. Stat. Mech p.07011]
Jonathan Keeling
Collective dynamics
Munich, April 2011
29 / 33
Scaling with system size
Simple approximation: match WKB soln to eqn for Ψ0 , Ψ1 :
"
√ #
E ln( N)
2√ E
√
N tan
,
1=−
π
g
g
N
Jonathan Keeling
Collective dynamics
Munich, April 2011
30 / 33
Scaling with system size
Simple approximation: match WKB soln to eqn for Ψ0 , Ψ1 :
"
√ #
E ln( N)
2√ E
√
N tan
,
1=−
π
g
g
N
√
√
πg N
3 Cg N
√
Eq = q √
+q
ln( N)
[ln( N)]4
Jonathan Keeling
Collective dynamics
Munich, April 2011
30 / 33
Scaling with system size
Simple approximation: match WKB soln to eqn for Ψ0 , Ψ1 :
"
√ #
E ln( N)
2√ E
√
N tan
,
1=−
π
g
g
N
√
√
πg N
3 Cg N
√
Eq = q √
+q
ln( N)
[ln( N)]4
3
4
10
N
10
5
10
10
Semiclassics controlled by 1/ ln(N).
6
[(∆Eq - q ∆E1)/∆E1]-1/3
5
4
q=2
q=3
q=4
3
2
0.8
1
1.2
__ 1.4
(2/π2) ln[2√N]
Jonathan Keeling
1.6
Collective dynamics
Munich, April 2011
30 / 33
Scaling with system size
Simple approximation: match WKB soln to eqn for Ψ0 , Ψ1 :
"
√ #
E ln( N)
2√ E
√
N tan
,
1=−
π
g
g
N
√
√
πg N
3 Cg N
√
Eq = q √
+q
ln( N)
[ln( N)]4
3
4
10
N
10
5
10
10
Semiclassics controlled by 1/ ln(N).
N
6
103
4
105
106
5
[(∆Eq - q ∆E1)/∆E1]-1/3
[(∆Eq - q ∆E1)/∆E1]-1/3
5
104
q=2
q=3
q=4
3
2
0.8
1
1.2
__ 1.4
(2/π2) ln[2√N]
Jonathan Keeling
4
Exact
n0=0
n0=2
3
2
0.8
1.6
1
1.2
__
1.4
1.6
(2/π2) ln[2√N]
Collective dynamics
Munich, April 2011
30 / 33
Overview
5
Polaritons and Dicke model
6
Numerical confirmation of FP
7
Dicke Oscillations
8
Extensions to atomic Dicke realisation
Jonathan Keeling
Collective dynamics
Munich, April 2011
31 / 33
Many photon modes
(a)
Transition breaks Z2 ⊗ Zn —
crystallisation
(b)
Pump
No cubic mode-mode coupling —
Brazovskii transition
“Supersmectic” phase
Jonathan Keeling
Collective dynamics
Munich, April 2011
32 / 33
Dynamics during/following sweep
Jonathan Keeling
Collective dynamics
Munich, April 2011
33 / 33
Dynamics during/following sweep
2.5
g√N (MHz)
2.0
__
__
g√N (MHz)
2.0
1.5
1.5
1.0
0.5
0.0
1.0
16
18 20 22
ω (MHz)
24
0.5
0.0
-20
Jonathan Keeling
-10
0
ω (MHz)
10
Collective dynamics
20
Munich, April 2011
33 / 33
Dynamics during/following sweep
2.5
1.5
0.5
Sz
1.5
1.0
0.0
1.0
0
-0.5
1200
0.5
1000
16
200
18 20 22
ω (MHz)
24
800
|ψ|2
__
300
__
0
g√N (MHz)
Sz
2.0
-0.5
400
|ψ|2
g√N (MHz)
2.0
0.5
0.5
600
400
100
200
0.0
0
0
0.5
1
t (ms)
Jonathan Keeling
1.5
2
-20
-10
0
ω (MHz)
10
Collective dynamics
20
0
0
0.5
1
t (ms)
Munich, April 2011
1.5
2
33 / 33