SUPA Superradiance and self-organisation of cold atoms in optical cavities Jonathan Keeling

Superradiance and self-organisation of cold
atoms in optical cavities
Jonathan Keeling

SUPA
University of
St Andrews
1413-2013
Strathclyde, September 2014
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
1
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
Superradiance of atoms in cavities
Strathclyde, September 2014.
2
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 & NV centres
Ultra-cold atoms
κ
κ
Cavity
Pump
Pump
Rydberg atoms/polaritons
Microcavity Polaritons
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
2
Dicke effect: Superradiance
Hint ¸
gk pψk eikri
H.c.qpSi
Si q
k,i
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
3
Dicke effect: Superradiance
Hint ¸
gk pψk eikri
H.c.qpSi
Si q
k,i
If |ri rj | ! λ, use
Collective decay:
Jonathan Keeling
°
i
Si
ÑS
dρ
dt
2Γ
S S ρ S ρS
Superradiance of atoms in cavities
ρS S Strathclyde, September 2014.
3
Dicke effect: Superradiance
Hint ¸
gk pψk eikri
H.c.qpSi
Si q
k,i
If |ri rj | ! λ, use
Collective decay:
°
i
Si
ÑS
dρ
dt
|S| N {2 initially: ΓN 2
ΓN
d xS z y
sech2
t
I9 Γ
dt
4
2
2Γ
S S ρ S ρS
ρS S N/2
ΓN2/2
z
I=-Γd〈S 〉/dt
z
〈S 〉
If S z
0
-N/2
0
tD
Jonathan Keeling
Superradiance of atoms in cavities
tD
Strathclyde, September 2014.
3
Dicke effect: Superradiance
Hint ¸
gk pψk eikri
H.c.qpSi
Si q
k,i
If |ri rj | ! λ, use
Collective decay:
°
i
Si
ÑS
dρ
dt
|S| N {2 initially: ΓN 2
ΓN
d xS z y
sech2
t
I9 Γ
dt
4
2
2Γ
S S ρ S ρS
ρS S N/2
ΓN2/2
z
z
I=-Γd〈S 〉/dt
If S z
〈S 〉
0
-N/2
0
tD
tD
Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
3
Dicke model and Dicke-Hepp-Lieb transition
H
ωψ:ψ
¸
ω0 Siz
g pψ
ψ : qpSi
Si q
i
Coherent state: |Ψy Ñ eλψ
Small g, min at λ, η
0
:
ηS
|Ωy
[Hepp, Lieb, Ann. Phys. ’73]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
4
Dicke model and Dicke-Hepp-Lieb transition
H
ωψ:ψ
ωψ:ψ
¸
g pψ
ω0 Siz
i
ω0 S z
g pψ
ψ : qpS
Coherent state: |Ψy Ñ eλψ
Small g, min at λ, η
0
Si q
ψ : qpSi
:
ηS
Sq
|Ωy
[Hepp, Lieb, Ann. Phys. ’73]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
4
Dicke model and Dicke-Hepp-Lieb transition
H
ωψ:ψ
ωψ:ψ
¸
g pψ
ω0 Siz
i
ω0 S z
g pψ
ψ : qpS
Coherent state: |Ψy Ñ eλψ
Small g, min at λ, η
0
Si q
ψ : qpSi
:
ηS
Sq
|Ωy
[Hepp, Lieb, Ann. Phys. ’73]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
4
Dicke model and Dicke-Hepp-Lieb transition
ô
H
ωψ:ψ
ωψ:ψ
¸
g pψ
ω0 Siz
i
ω0 S z
g pψ
ψ : qpS
Coherent state: |Ψy Ñ eλψ
Small g, min at λ, η
:
0
Non-zero cavity field if: 4Ng 2
Jonathan Keeling
Si q
ψ : qpSi
Sq
ηS
|Ωy
¡ ωω0
Superradiance of atoms in cavities
[Hepp, Lieb, Ann. Phys. ’73]
Strathclyde, September 2014.
4
Dicke model and Dicke-Hepp-Lieb transition
ô
ωψ:ψ
ωψ:ψ
¸
g pψ
ω0 Siz
i
ω0 S z
g pψ
ψ : qpS
Coherent state: |Ψy Ñ eλψ
Small g, min at λ, η
:
0
Non-zero cavity field if: 4Ng 2
Jonathan Keeling
Si q
ψ : qpSi
Sq
ηS
⇓
ω
H
|Ωy
¡ ωω0
Superradiance of atoms in cavities
SR
0
0
g-√N
[Hepp, Lieb, Ann. Phys. ’73]
Strathclyde, September 2014.
4
No go theorem for Dicke-Hepp-Lieb transition
ô
Spontaneous polarisation if: 4Ng 2
¡ ωω0
[Rzazewski et al PRL ’75]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
5
No go theorem for Dicke-Hepp-Lieb transition
ô
Spontaneous polarisation if: 4Ng 2
No go theorem:.
¸e
i
m
A pi
¡ ωω0
Minimal coupling pp eAq2 {2m
ô g pψ :
ψ qpS S
q,
¸ A2
i
2m
ô
Nζ pψ
ψ : q2
[Rzazewski et al PRL ’75]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
5
No go theorem for Dicke-Hepp-Lieb transition
ô
Spontaneous polarisation if: 4Ng 2
No go theorem:.
¸e
i
m
For large N, ω
Minimal coupling pp eAq2 {2m
ô g pψ :
A pi
¡ ωω0
ψ qpS S
q,
¸ A2
i
Ñω
2m
ô
Nζ pψ
ψ : q2
4Nζ. (RWA)
[Rzazewski et al PRL ’75]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
5
No go theorem for Dicke-Hepp-Lieb transition
ô
Spontaneous polarisation if: 4Ng 2
No go theorem:.
¸e
i
m
For large N, ω
Minimal coupling pp eAq2 {2m
ô g pψ :
A pi
¡ ωω0
ψ qpS S
¸ A2
q,
i
Ñω
2m
ô
Nζ pψ
ψ : q2
4Nζ. (RWA)
Need 4Ng 2
¡ ω0pω
4Nζ q.
[Rzazewski et al PRL ’75]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
5
No go theorem for Dicke-Hepp-Lieb transition
ô
Spontaneous polarisation if: 4Ng 2
No go theorem:.
¸e
i
m
For large N, ω
Minimal coupling pp eAq2 {2m
ô g pψ :
A pi
¡ ωω0
ψ qpS S
¸ A2
q,
i
Ñω
2m
ô
Nζ pψ
ψ : q2
4Nζ. (RWA)
Need 4Ng 2
But f -sum rule states: g 2 {ω0
¡ ω0pω
4Nζ q.
ζ. No transition
[Rzazewski et al PRL ’75]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
5
Ways around the no-go theorem
Problem: g 2 {ω0
1
ζ for intrinsic parameters. Solutions:
Ferroelectric transition in D r gauge.
[JK JPCM ’07, Vukics & Domokos PRA 2012 ]
Ÿ
2
Grand canonical ensemble:
Ÿ
Ÿ
3
Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]
If H Ñ H µpS z ψ : ψ), need only:
g 2 N ¡ pω µqpω0 µq
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
Superradiance of atoms in cavities
Strathclyde, September 2014.
6
Ways around the no-go theorem
Problem: g 2 {ω0
1
ζ for intrinsic parameters. Solutions:
Ferroelectric transition in D r gauge.
[JK JPCM ’07, Vukics & Domokos PRA 2012 ]
Ÿ
2
Grand canonical ensemble:
Ÿ
Ÿ
3
Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]
If H Ñ H µpS z ψ : ψ), need only:
g 2 N ¡ pω µqpω0 µq
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
Superradiance of atoms in cavities
Strathclyde, September 2014.
6
Ways around the no-go theorem
Problem: g 2 {ω0
1
ζ for intrinsic parameters. Solutions:
Ferroelectric transition in D r gauge.
[JK JPCM ’07, Vukics & Domokos PRA 2012 ]
Ÿ
Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]
2
1
Grand canonical ensemble:
Ÿ
Ÿ
3
If H Ñ H µpS z ψ : ψ), need only:
g 2 N ¡ pω µqpω0 µq
Incoherent pumping — polariton
condensation.
Dissociate g, ω0 ,
e.g. Raman scheme: ω0
unstable
0
(µ-ω)/g
2
-1
SR
-2
-3
-4
-5
-4
-3
-2
-1
0
(ω0 - ω)/g
1
2
! ω.
[Dimer et al. PRA ’07; Baumann et al. Nature
’10. Also, Black et al. PRL ’03 ]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
6
Ways around the no-go theorem
Problem: g 2 {ω0
1
ζ for intrinsic parameters. Solutions:
Ferroelectric transition in D r gauge.
[JK JPCM ’07, Vukics & Domokos PRA 2012 ]
Ÿ
Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]
2
1
Grand canonical ensemble:
Ÿ
Ÿ
3
If H Ñ H µpS z ψ : ψ), need only:
g 2 N ¡ pω µqpω0 µq
Incoherent pumping — polariton
condensation.
Dissociate g, ω0 ,
e.g. Raman scheme: ω0
unstable
0
(µ-ω)/g
2
! ω.
-1
SR
-2
-3
-4
-5
-4
-3
-2
-1
0
(ω0 - ω)/g
κ
κ
[Dimer et al. PRA ’07; Baumann et al. Nature
’10. Also, Black et al. PRL ’03 ]
Jonathan Keeling
Superradiance of atoms in cavities
1
2
Cavity
Pump
Pump
Strathclyde, September 2014.
6
Outline
1
Dicke model, superradiance and no–go theorem
2
Superradiance and self-organisation
Raman scheme
Hierarchies of approximation
Equilibrium theory of Dicke
3
Fermionic self organisation
Equilibrium phase diagrams
Landau theory and microscopics
Open system?
4
Open system dynamics of Bososn
Attractors of open Dicke model
Bosons beyond Dicke
5
Conclusions
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
7
Outline
1
Dicke model, superradiance and no–go theorem
2
Superradiance and self-organisation
Raman scheme
Hierarchies of approximation
Equilibrium theory of Dicke
3
Fermionic self organisation
Equilibrium phase diagrams
Landau theory and microscopics
Open system?
4
Open system dynamics of Bososn
Attractors of open Dicke model
Bosons beyond Dicke
5
Conclusions
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
8
Raman scheme, decoupling g, ω0
∆
∆
H
g0ψ
ω0Sz
ω0
2 level system
Imbalanced case (internal states):
H ω0 S z g pψS
ψ : S q g 1 pψS Imbalance: g
S
2 Level system, | óy, | òy
g0 Ω
Coupling g 2∆
Rotating frame of pump, ω
Ω
Ω
ψ : qpS g pψ
ψ:S
q
q
ωψ : ψ
ωcavity ωpump
ωψ : ψ
Uψ : ψS z
0 Ωb
0 Ωa
g2∆
g 1 g2∆
a
b
New “feedback” term U
g02
2
g0
2∆ 2∆
b
a
[Dimer et al. PRA ’07 ]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
9
Raman scheme, decoupling g, ω0
∆
∆
H
g0ψ
ω0
2 level system
Imbalanced case (internal states):
H ω0 S z g pψS
ψ : S q g 1 pψS Imbalance: g
S
2 Level system, | óy, | òy
g0 Ω
Coupling g 2∆
Rotating frame of pump, ω
Ω
Ω
ψ : qpS g pψ
ω0Sz
ψ:S
q
q
ωψ : ψ
ωcavity ωpump
ωψ : ψ
Uψ : ψS z
0 Ωb
0 Ωa
g2∆
g 1 g2∆
a
b
New “feedback” term U
g02
2
g0
2∆ 2∆
b
a
[Dimer et al. PRA ’07 ]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
9
Raman scheme, decoupling g, ω0
∆
∆
H
g0ψ
ω0
2 level system
Imbalanced case (internal states):
H ω0 S z g pψS
ψ : S q g 1 pψS Imbalance: g
S
2 Level system, | óy, | òy
g0 Ω
Coupling g 2∆
Rotating frame of pump, ω
Ω
Ω
ψ : qpS g pψ
ω0Sz
ψ:S
q
q
ωψ : ψ
ωcavity ωpump
ωψ : ψ
Uψ : ψS z
0 Ωb
0 Ωa
g2∆
g 1 g2∆
a
b
New “feedback” term U
g02
2
g0
2∆ 2∆
b
a
[Dimer et al. PRA ’07 ]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
9
Raman scheme, decoupling g, ω0
∆
∆
H
g0ψ
S
2 Level system, | óy, | òy
g0 Ω
Coupling g 2∆
Rotating frame of pump, ω
Ω
Ω
ψ : qpS g pψ
ω0Sz
ω0
2 level system
Imbalanced case (internal states):
H ω0 S z g pψS
ψ : S q g 1 pψS ψ:S
q
q
ωψ : ψ
ωcavity ωpump
ωψ : ψ
∆b
∆a
Imbalance: g
g0ψ
0 Ωb
0 Ωa
g2∆
g 1 g2∆
a
b
New “feedback” term U
g02
g02
2∆ 2∆
b
Uψ : ψS z
a
Ωb
Ωa
ω0
2 level system
[Dimer et al. PRA ’07 ]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
9
Raman scheme, decoupling g, ω0
∆
∆
H
g0ψ
S
2 Level system, | óy, | òy
g0 Ω
Coupling g 2∆
Rotating frame of pump, ω
Ω
Ω
ψ : qpS g pψ
ω0Sz
ω0
2 level system
Imbalanced case (internal states):
H ω0 S z g pψS
ψ : S q g 1 pψS ψ:S
q
q
ωψ : ψ
ωcavity ωpump
ωψ : ψ
∆b
∆a
Imbalance: g
g0ψ
0 Ωb
0 Ωa
g2∆
g 1 g2∆
a
b
New “feedback” term U
g02
g02
2∆ 2∆
b
Uψ : ψS z
a
Ωb
Ωa
ω0
2 level system
[Dimer et al. PRA ’07 ]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
9
Transversely pumped cavity
Internal state Ñ momentum states
1
Full description
»
H0 ωcavity ψ : ψ
zx
Pump
Jonathan Keeling
ωatom ce: ce
Superradiance of atoms in cavities
2
d r
¸
α e,g
c:
α
E px, z, t qpce: cg
∇ 2 c
2m
cg: ce q
Strathclyde, September 2014.
α
10
Transversely pumped cavity
Internal state Ñ momentum states
1
Full description
»
H0 ωcavity ψ : ψ
zx
ωatom ce: ce
Pump
2
d r
¸
c:
α
∇ 2 c
2m
α e,g
E px, z, t qpce: cg
cg: ce q
α
No cavity field
With cavity field
2
Eliminate e state
Ÿ
H
Rotating frame ω
ωψ:ψ
V prq »
d 2 rc : prq
ωcavity ωpump Nδ
g2 :
ψ ψ cosp2qx q
2∆
Jonathan Keeling
∇2
2m V prq c prq
gΩ
pψ
∆
0
-1
V(r) -2
-3
-40
5
10
x (cavity)
ψ : q cospqx q cospqz q
Superradiance of atoms in cavities
15
0
5
15
10
z (pump)
Ω2
cosp2qz q
2∆
Strathclyde, September 2014.
10
Transversely pumped cavity
Internal state Ñ momentum states
1
Full description
»
H0 ωcavity ψ : ψ
zx
2
d r
c:
α
∇ 2 c
2m
α e,g
ωatom ce: ce
Pump
¸
E px, z, t qpce: cg
cg: ce q
α
No cavity field
With cavity field
Eliminate e state
2
Ÿ
H
ωψ:ψ
V prq 3
Rotating frame ω
»
d 2 rc : prq
g2 :
ψ ψ cosp2qx q
2∆
∇2
-1
V(r) -2
2m V prq c prq
gΩ
pψ
∆
-3
-40
5
10
x (cavity)
ψ : q cospqx q cospqz q
Dicke: project to atomic states φpx, z q9
Jonathan Keeling
0
ωcavity ωpump Nδ
#
Superradiance of atoms in cavities
15
0
5
15
10
z (pump)
Ω2
cosp2qz q
2∆
1
cospqz q cospqz q
Strathclyde, September 2014.
10
Mapping transverse pumping to Dicke model
κ
κ
g0ψ
Ω
z
Pump
x
2 Level System
Reduced#basis:
1
φpx, z q9
cospqz q cospqz q
H
ωψ:ψ
ω0 S z
g pψ
ψ : qpS ó
ò
S
q
USz ψ : ψ.
[Baumann et al Nature ’10 ]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
11
Mapping transverse pumping to Dicke model
κ
g0ψ
κ
Ω
z
Pump
x
2 Level System
Reduced#basis:
1
φpx, z q9
cospqz q cospqz q
H
ωψ:ψ
ω0 S z
g pψ
ψ : qpS ó
ò
S
q
USz ψ : ψ.
2
“Feedback” due to extra states U
Jonathan Keeling
g0
4∆
Superradiance of atoms in cavities
[Baumann et al Nature ’10 ]
Strathclyde, September 2014.
11
Experimental phase diagram
?
Pump power g 9 Power
Pump-cavity detuning ω
Jonathan Keeling
∆
[Baumann et al Nature ’10 ]
Superradiance of atoms in cavities
Strathclyde, September 2014.
12
Phase diagram of extended Dicke model
?
Ground state energy, λ xψ y{ N:
E
N
b
ωλ2 N2 pω0
Superradiant
transition:
UN
2
4g N ¡ ω ω0
2
Jonathan Keeling
UNλ2 q2
16g 2 Nλ2
Stability,
λ Ñ 8
|U |N λ2
E ω
2
Superradiance of atoms in cavities
...
Strathclyde, September 2014.
13
Phase diagram of extended Dicke model
?
Ground state energy, λ xψ y{ N:
E
N
b
ωλ2 N2 pω0
Superradiant
transition:
UN
2
4g N ¡ ω ω0
2
Jonathan Keeling
UNλ2 q2
16g 2 Nλ2
Stability,
λ Ñ 8
|U |N λ2
E ω
2
Superradiance of atoms in cavities
...
Strathclyde, September 2014.
13
Phase diagram of extended Dicke model
?
Ground state energy, λ xψ y{ N:
E
N
b
ωλ2 N2 pω0
Superradiant
transition:
UN
2
4g N ¡ ω ω0
2
Jonathan Keeling
UNλ2 q2
16g 2 Nλ2
Stability,
λ Ñ 8
|U |N λ2
E ω
2
Superradiance of atoms in cavities
...
Strathclyde, September 2014.
13
Phase diagram of extended Dicke model
?
Ground state energy, λ xψ y{ N:
b
ωλ2 N2 pω0
E
N
UNλ2 q2
16g 2 Nλ2
Stability,
λ Ñ 8
|U |N λ2
E ω
2
Superradiant
transition:
UN
2
4g N ¡ ω ω0
2
...
40
0.1
UN=-10
UN=0
UN=10
ωD
0.06
20
Normal
SR
Normal
SR
Normal
SR
0.04
10
Cavity field, λ
0.08
30
0.02
Unstable
0
0
0.5
Unstable
1
__
g√N
Jonathan Keeling
1.5
0
0.5
1
1.5
__
g√N
Superradiance of atoms in cavities
0
0.5
0
1
1.5
__
g√N
Strathclyde, September 2014.
13
Outline
1
Dicke model, superradiance and no–go theorem
2
Superradiance and self-organisation
Raman scheme
Hierarchies of approximation
Equilibrium theory of Dicke
3
Fermionic self organisation
Equilibrium phase diagrams
Landau theory and microscopics
Open system?
4
Open system dynamics of Bososn
Attractors of open Dicke model
Bosons beyond Dicke
5
Conclusions
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
14
Fermions in optical cavities
H
ωψ:ψ
V prq »
d 2 rc : prq
g2 :
ψ ψ cosp2qx q
2∆
∇2
No cavity field
With cavity field
2m V prq c prq
gΩ
pψ
∆
0
-1
V(r) -2
-3
-40
5
10
x (cavity)
ψ : q cospqx q cospqz q
15
0
5
15
10
z (pump)
Ω2
cosp2qz q
2∆
[Domokos & Ritsch, PRL ’03; Black et al. PRL ’03; Piazza, Strack, Zwerger, Ann. Phys
’13]
Pauli blocking
Commensurability effects
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
15
Fermions in optical cavities
H
ωψ:ψ
V prq »
d 2 rc : prq
g2 :
ψ ψ cosp2qx q
2∆
∇2
No cavity field
With cavity field
2m V prq c prq
gΩ
pψ
∆
0
-1
V(r) -2
-3
-40
5
10
x (cavity)
ψ : q cospqx q cospqz q
15
0
5
15
10
z (pump)
Ω2
cosp2qz q
2∆
[Domokos & Ritsch, PRL ’03; Black et al. PRL ’03; Piazza, Strack, Zwerger, Ann. Phys
’13]
Pauli blocking
Commensurability effects
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
15
Fermions in optical cavities
H
ωψ:ψ
V prq »
d 2 rc : prq
g2 :
ψ ψ cosp2qx q
2∆
∇2
No cavity field
With cavity field
2m V prq c prq
gΩ
pψ
∆
0
-1
V(r) -2
-3
-40
5
10
x (cavity)
ψ : q cospqx q cospqz q
15
0
5
15
10
z (pump)
Ω2
cosp2qz q
2∆
[Domokos & Ritsch, PRL ’03; Black et al. PRL ’03; Piazza, Strack, Zwerger, Ann. Phys
No cavity field
With cavity field
M
’13]
q
Pauli blocking
Commensurability effects
kz
X
0
Γ
M
-q
-q
Jonathan Keeling
Superradiance of atoms in cavities
0
kx
Strathclyde, September 2014.
q
15
Fermions in optical cavities
H
ωψ:ψ
V prq »
d 2 rc : prq
g2 :
ψ ψ cosp2qx q
2∆
∇2
No cavity field
With cavity field
2m V prq c prq
gΩ
pψ
∆
0
-1
V(r) -2
-3
-40
5
10
x (cavity)
ψ : q cospqx q cospqz q
15
0
5
15
10
z (pump)
Ω2
cosp2qz q
2∆
[Domokos & Ritsch, PRL ’03; Black et al. PRL ’03; Piazza, Strack, Zwerger, Ann. Phys
No cavity field
With cavity field
M
’13]
q
Pauli blocking
Commensurability effects
kz
X
0
Γ
M
[JK, Bhaseen, & Simons; Piazza & Strack; Chen et
al. All PRL ’14.]
-q
-q
Jonathan Keeling
Superradiance of atoms in cavities
0
kx
Strathclyde, September 2014.
q
15
Dimensionless variables and free energy
Rescale with
Ÿ
N { NL
?
nF
Free energy f
f pω̃, η, nF
2q, ωr
Ÿ
~2q 2{2m, Dimensionless variables:
ω
Ñ ω̃
F {NLωr
Ñ µ; φq ω̃φ2
µnF
β1
k,n from ĥ ∇2 V pη, φ; rq
Momentum space: hk,k1 k 2 δk,k1
vk,k1
φ2
¸
δk,k1
s
ηφ
ΩÑη
Ÿ
»
d 2k
BZ
Ÿ
¸ ln 1
xψ y Ñ φ
eβ pk,n µq
n
vk,k1
?
s 2x̂
¸
s,s1
δk,k1 ?s x̂ ?s1 ẑ
2
2
η2
¸
δk,k1
?
s 2ẑ
s
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
16
Dimensionless variables and free energy
Rescale with
Ÿ
N { NL
?
nF
Free energy f
f pω̃, η, nF
2q, ωr
Ÿ
~2q 2{2m, Dimensionless variables:
ω
Ñ ω̃
F {NLωr
Ñ µ; φq ω̃φ2
µnF
β1
k,n from ĥ ∇2 V pη, φ; rq
Momentum space: hk,k1 k 2 δk,k1
vk,k1
φ2
¸
δk,k1
s
ηφ
ΩÑη
Ÿ
»
d 2k
BZ
Ÿ
¸ ln 1
xψ y Ñ φ
eβ pk,n µq
n
vk,k1
?
s 2x̂
¸
s,s1
δk,k1 ?s x̂ ?s1 ẑ
2
2
η2
¸
δk,k1
?
s 2ẑ
s
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
16
Dimensionless variables and free energy
Rescale with
Ÿ
N { NL
?
nF
Free energy f
f pω̃, η, nF
2q, ωr
Ÿ
~2q 2{2m, Dimensionless variables:
ω
Ñ ω̃
F {NLωr
Ñ µ; φq ω̃φ2
µnF
β1
k,n from ĥ ∇2 V pη, φ; rq
Momentum space: hk,k1 k 2 δk,k1
vk,k1
φ2
¸
δk,k1
s
ηφ
ΩÑη
Ÿ
»
d 2k
BZ
Ÿ
¸ ln 1
xψ y Ñ φ
eβ pk,n µq
n
vk,k1
?
s 2x̂
¸
s,s1
δk,k1 ?s x̂ ?s1 ẑ
2
2
η2
¸
2
δk,k1
?
s 2ẑ
s
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
16
Phase diagram
Free energy f
f pω̃, η, nF
nF
F {NLωr
Ñ µ; φq ω̃φ
2
µnF
1
β
»
d 2k
BZ
¸ ln 1
eβ pk,n µq
n
Ñ 0, Dicke, expect SR.
Instability, φ Ñ 8,
k,n Ñ 2φ2
f pω̃ 2nF qφ2
First order at low η
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
17
Phase diagram
Free energy f
f pω̃, η, nF
nF
F {NLωr
Ñ µ; φq ω̃φ
2
µnF
1
β
»
d 2k
BZ
¸ ln 1
eβ pk,n µq
n
Ñ 0, Dicke, expect SR.
Instability, φ Ñ 8,
k,n Ñ 2φ2
f pω̃ 2nF qφ2
First order at low η
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
17
Phase diagram
Free energy f
f pω̃, η, nF
nF
F {NLωr
Ñ µ; φq ω̃φ
2
µnF
1
β
»
d 2k
BZ
¸ ln 1
eβ pk,n µq
n
Ñ 0, Dicke, expect SR.
Instability, φ Ñ 8,
k,n Ñ 2φ2
f pω̃ 2nF qφ2
First order at low η
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
17
Phase diagram
Free energy f
f pω̃, η, nF
nF
F {NLωr
Ñ µ; φq ω̃φ
2
µnF
1
β
»
d 2k
BZ
¸ ln 1
eβ pk,n µq
n
Ñ 0, Dicke, expect SR.
Instability, φ Ñ 8,
k,n Ñ 2φ2
f pω̃ 2nF qφ2
First order at low η
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
17
Phase diagram
Free energy f
f pω̃, η, nF
nF
F {NLωr
Ñ µ; φq ω̃φ
2
µnF
1
β
»
d 2k
BZ
¸ ln 1
eβ pk,n µq
n
Ñ 0, Dicke, expect SR.
Instability, φ Ñ 8,
k,n Ñ 2φ2
f pω̃ 2nF qφ2
First order at low η
aφ2 bφ4 cφ6
b 0 at small η.
f
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
17
Origin of first order transition
k,n from ĥ k 2 δk,k1
Vk,k1
φ
¸
2
ηφ
η2
s
¸
s,s1
¸
δk,k1
Vk,k1
?
s 2x̂
δk,k1 ?s x̂ ?s1 ẑ
2
2
δk,k1
?
s 2ẑ
s
Landau expansion: f
apω̃, η, nF qφ2
bpη, nF qφ4
c pη, nF qφ6
Second order perturbation theory,
φ4|mk,k1 |2{pEk1 Ekq
Larkin-Pikin like mechanism
Survives to low nF : Bosons!
Ÿ
Ÿ
?
But needs state φpx, z q cosp 2x q
Missed by Dicke model
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
18
Origin of first order transition
k,n from ĥ k 2 δk,k1
Vk,k1
φ
¸
2
ηφ
η2
s
¸
s,s1
¸
δk,k1
Vk,k1
?
s 2x̂
δk,k1 ?s x̂ ?s1 ẑ
2
2
δk,k1
?
s 2ẑ
s
Landau expansion: f
apω̃, η, nF qφ2
bpη, nF qφ4
c pη, nF qφ6
Second order perturbation theory,
φ4|mk,k1 |2{pEk1 Ekq
Larkin-Pikin like mechanism
Survives to low nF : Bosons!
Ÿ
Ÿ
?
But needs state φpx, z q cosp 2x q
Missed by Dicke model
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
18
Origin of first order transition
k,n from ĥ k 2 δk,k1
Vk,k1
φ
¸
2
ηφ
η2
s
¸
s,s1
¸
δk,k1
Vk,k1
?
s 2x̂
δk,k1 ?s x̂ ?s1 ẑ
2
2
δk,k1
?
s 2ẑ
s
apω̃, η, nF qφ2
bpη, nF qφ4
Second order perturbation theory,
φ4|mk,k1 |2{pEk1 Ekq
Larkin-Pikin like mechanism
Survives to low nF : Bosons!
Ÿ
Ÿ
?
But needs state φpx, z q cosp 2x q
Missed by Dicke model
Jonathan Keeling
Superradiance of atoms in cavities
c pη, nF qφ6
2
b(η,nF)/nF
Landau expansion: f
nF → 0
1
0
-1
-2
0
0.2
0.4
0.6
ηc
Pump field, η
Strathclyde, September 2014.
0.8
18
Origin of first order transition
k,n from ĥ k 2 δk,k1
Vk,k1
φ
¸
2
ηφ
η2
s
¸
s,s1
¸
δk,k1
Vk,k1
?
s 2x̂
δk,k1 ?s x̂ ?s1 ẑ
2
2
δk,k1
?
s 2ẑ
s
apω̃, η, nF qφ2
bpη, nF qφ4
Second order perturbation theory,
φ4|mk,k1 |2{pEk1 Ekq
Larkin-Pikin like mechanism
Survives to low nF : Bosons!
Ÿ
Ÿ
?
But needs state φpx, z q cosp 2x q
Missed by Dicke model
Jonathan Keeling
Superradiance of atoms in cavities
c pη, nF qφ6
2
b(η,nF)/nF
Landau expansion: f
nF → 0
1
0
-1
-2
0
0.2
0.4
0.6
ηc
Pump field, η
Strathclyde, September 2014.
0.8
18
Higher fillings
f
aφ2
bφ4
cφ6
Phase diagram unchanged
for nF 1
2nd order line a 0
Tricritical
at a b 0
2nd band, new structure.
Ÿ
Ÿ
Critical end-point
a 0 line cut by 1st order
SR–SR phase boundary
Ÿ
Ÿ
Jonathan Keeling
No symmetry breaking
Liquid–gas type
(metamagnetic)
Superradiance of atoms in cavities
Strathclyde, September 2014.
19
Higher fillings
f
aφ2
bφ4
cφ6
Phase diagram unchanged
for nF 1
2nd order line a 0
Tricritical
at a b 0
2nd band, new structure.
Ÿ
Ÿ
Critical end-point
a 0 line cut by 1st order
SR–SR phase boundary
Ÿ
Ÿ
Jonathan Keeling
No symmetry breaking
Liquid–gas type
(metamagnetic)
Superradiance of atoms in cavities
Strathclyde, September 2014.
19
Higher fillings
f
aφ2
bφ4
cφ6
Phase diagram unchanged
for nF 1
2nd order line a 0
Tricritical
at a b 0
2nd band, new structure.
Ÿ
Ÿ
Critical end-point
a 0 line cut by 1st order
SR–SR phase boundary
Ÿ
Ÿ
Jonathan Keeling
No symmetry breaking
Liquid–gas type
(metamagnetic)
Superradiance of atoms in cavities
Strathclyde, September 2014.
19
Why liquid–gas transition?
~ =5.2
ω
~ =5.3
ω
~ =5.4
ω
~ =5.5
ω
~ =5.6
ω
Free energy, f
1
0
-1
0
nF=1.5, η=0.8
0.5
f pφq Ñ multiple minima
η
1
1.5
Cavity field, φ
Plot bands infk rk,n s
Contribution of 2nd band
Non-trivial form:
Ÿ
Ÿ
px , pz orbitals cross at η φ
n ¡ 1 bands initially go up
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
20
Why liquid–gas transition?
~ =5.2
ω
~ =5.3
ω
~ =5.4
ω
~ =5.5
ω
~ =5.6
ω
Free energy, f
1
0
-1
Ÿ
Ÿ
px , pz orbitals cross at η φ
n ¡ 1 bands initially go up
1
1.5
1
0
-1
-2
η=0.8
-3
0
Jonathan Keeling
η
2
Atomic energy bands
Contribution of 2nd band
Non-trivial form:
nF=1.5, η=0.8
0.5
Cavity field, φ
f pφq Ñ multiple minima
Plot bands infk rk,n s
0
0.2
Superradiance of atoms in cavities
0.4
0.6
0.8
Cavity field, φ
1
1.2
Strathclyde, September 2014.
1.4
20
Why liquid–gas transition?
~ =5.2
ω
~ =5.3
ω
~ =5.4
ω
~ =5.5
ω
~ =5.6
ω
Free energy, f
1
0
-1
Ÿ
Ÿ
px , pz orbitals cross at η φ
n ¡ 1 bands initially go up
1
1.5
1
0
-1
Filled bands, nF=1.5
-2
η=0.8
-3
0
Jonathan Keeling
η
2
Atomic energy bands
Contribution of 2nd band
Non-trivial form:
nF=1.5, η=0.8
0.5
Cavity field, φ
f pφq Ñ multiple minima
Plot bands infk rk,n s
0
0.2
Superradiance of atoms in cavities
0.4
0.6
0.8
Cavity field, φ
1
1.2
Strathclyde, September 2014.
1.4
20
Why liquid–gas transition?
~ =5.2
ω
~ =5.3
ω
~ =5.4
ω
~ =5.5
ω
~ =5.6
ω
Free energy, f
1
0
-1
Ÿ
Ÿ
px , pz orbitals cross at η φ
n ¡ 1 bands initially go up
1
1.5
1
0
-1
Filled bands, nF=1.5
-2
η=0.8
-3
0
Jonathan Keeling
η
2
Atomic energy bands
Contribution of 2nd band
Non-trivial form:
nF=1.5, η=0.8
0.5
Cavity field, φ
f pφq Ñ multiple minima
Plot bands infk rk,n s
0
0.2
Superradiance of atoms in cavities
0.4
0.6
0.8
Cavity field, φ
1
1.2
Strathclyde, September 2014.
1.4
20
Phase diagram vs density
Phase topology change:
Fix η, plot vs nF
SR–SR after critical point
Peak in 2nd order line 0 apω̃, nF , η q ω̃
Susceptibility χ asymptote
η Ñ 8
1 nF χ 16η 2 ln 1 nf Jonathan Keeling
Superradiance of atoms in cavities
χpη, nF q
Strathclyde, September 2014.
21
Phase diagram vs density
Phase topology change:
Fix η, plot vs nF
SR–SR after critical point
Peak in 2nd order line 0 apω̃, nF , η q ω̃
Susceptibility χ asymptote
η Ñ 8
1 nF χ 16η 2 ln 1 nf Jonathan Keeling
Superradiance of atoms in cavities
χpη, nF q
Strathclyde, September 2014.
21
Phase diagram vs density
Phase topology change:
Fix η, plot vs nF
SR–SR after critical point
Peak in 2nd order line 0 apω̃, nF , η q ω̃
Susceptibility χ asymptote
η Ñ 8
1 nF χ 16η 2 ln 1 nf Jonathan Keeling
Superradiance of atoms in cavities
χpη, nF q
Strathclyde, September 2014.
21
Phase diagram vs density
Phase topology change:
Fix η, plot vs nF
SR–SR after critical point
Peak in 2nd order line 0 apω̃, nF , η q ω̃
Susceptibility χ asymptote
η Ñ 8
1 nF χ 16η 2 ln 1 nf Jonathan Keeling
Superradiance of atoms in cavities
χpη, nF q
Strathclyde, September 2014.
21
Phase diagram vs density
Phase topology change:
Fix η, plot vs nF
SR–SR after critical point
Peak in 2nd order line 0 apω̃, nF , η q ω̃
Susceptibility χ asymptote
η Ñ 8
1 nF χ 16η 2 ln 1 nf At nF 1, nesting
° of
vk,k1 . . . ηφ s,s1 δk,k1 ?s x̂ ?s1 ẑ . . ..
2
Jonathan Keeling
χpη, nF q
2/ 2
1/ 2
2
Superradiance of atoms in cavities
Strathclyde, September 2014.
21
Open system vs ground state phase diagram
Open system, ρ9 i rH, ρs κLrψ s. Stable attractors
What survives — Normal-SR boundary
Ÿ
Ÿ
Ÿ
Fluctuations δφ ueiνt
v eiν t ,
Secular equation:
pi ω̃r ν κ̃q2 ω̃rω̃ χpν, η, nF qs 0
Stable if Imrν s ¡ 0. Boundary:
ω̃ 2
κ̃2
ω̃
χpη, nF q
What must change
Ÿ
Unstable region Ñ new attractors
Known unkowns:
Ÿ
Limit cycles? Multistability? Spinodal lines?
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
22
Open system vs ground state phase diagram
Open system, ρ9 i rH, ρs κLrψ s. Stable attractors
What survives — Normal-SR boundary
Ÿ
Ÿ
Ÿ
Fluctuations δφ ueiνt
v eiν t ,
Secular equation:
pi ω̃r ν κ̃q2 ω̃rω̃ χpν, η, nF qs 0
Stable if Imrν s ¡ 0. Boundary:
ω̃ 2
κ̃2
ω̃
χpη, nF q
What must change
Ÿ
Unstable region Ñ new attractors
Known unkowns:
Ÿ
Limit cycles? Multistability? Spinodal lines?
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
22
Open system vs ground state phase diagram
Open system, ρ9 i rH, ρs κLrψ s. Stable attractors
What survives — Normal-SR boundary
Ÿ
Ÿ
Ÿ
Fluctuations δφ ueiνt
v eiν t ,
Secular equation:
pi ω̃r ν κ̃q2 ω̃rω̃ χpν, η, nF qs 0
Stable if Imrν s ¡ 0. Boundary:
ω̃ 2
κ̃2
ω̃
χpη, nF q
What must change
Ÿ
Unstable region Ñ new attractors
Known unkowns:
Ÿ
Limit cycles? Multistability? Spinodal lines?
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
22
Open system vs ground state phase diagram
Open system, ρ9 i rH, ρs κLrψ s. Stable attractors
What survives — Normal-SR boundary
Ÿ
Ÿ
Ÿ
Fluctuations δφ ueiνt
v eiν t ,
Secular equation:
pi ω̃r ν κ̃q2 ω̃rω̃ χpν, η, nF qs 0
Stable if Imrν s ¡ 0. Boundary:
ω̃ 2
κ̃2
ω̃
χpη, nF q
What must change
Ÿ
Unstable region Ñ new attractors
Known unkowns:
Ÿ
Limit cycles? Multistability? Spinodal lines?
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
22
Open system vs ground state phase diagram
Open system, ρ9 i rH, ρs κLrψ s. Stable attractors
What survives — Normal-SR boundary
Ÿ
Ÿ
Ÿ
Fluctuations δφ ueiνt
v eiν t ,
Secular equation:
pi ω̃r ν κ̃q2 ω̃rω̃ χpν, η, nF qs 0
Stable if Imrν s ¡ 0. Boundary:
ω̃ 2
κ̃2
ω̃
χpη, nF q
What must change
Ÿ
Unstable region Ñ new attractors
Known unkowns:
Ÿ
Limit cycles? Multistability? Spinodal lines?
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
22
Outline
1
Dicke model, superradiance and no–go theorem
2
Superradiance and self-organisation
Raman scheme
Hierarchies of approximation
Equilibrium theory of Dicke
3
Fermionic self organisation
Equilibrium phase diagrams
Landau theory and microscopics
Open system?
4
Open system dynamics of Bososn
Attractors of open Dicke model
Bosons beyond Dicke
5
Conclusions
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
23
Experimental phase diagram
?
Pump power g 9 Power
Pump-cavity detuning ω
Jonathan Keeling
∆
[Baumann et al Nature ’10 ]
Superradiance of atoms in cavities
Strathclyde, September 2014.
24
Dicke model classical dynamics
Open dynamical system:
ωψ:ψ ω0Sz g pψ ψ:qpS S q
Bt ρ i rH, ρsκpψ:ψρ 2ψρψ: ρψ:ψq
H
Fixed points: S9 0, ψ9
USz ψ : ψ.
0
Limit cycles?
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
25
Dicke model classical dynamics
Open dynamical system:
ωψ:ψ ω0Sz g pψ ψ:qpS S q
Bt ρ i rH, ρsκpψ:ψρ 2ψρψ: ρψ:ψq
H
Classical EOM
(|S| N {2 " 1)
USz ψ : ψ.
S9 i pω0 U |ψ|2qS 2ig pψ ψqSz
S z ig pψ ψ qpS S q
ψ rκ i pω US z qs ψ ig pS S q
9
9
Fixed points: S9 0, ψ9
0
Limit cycles?
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
25
Dicke model classical dynamics
Open dynamical system:
ωψ:ψ ω0Sz g pψ ψ:qpS S q
Bt ρ i rH, ρsκpψ:ψρ 2ψρψ: ρψ:ψq
H
Classical EOM
(|S| N {2 " 1)
USz ψ : ψ.
S9 i pω0 U |ψ|2qS 2ig pψ ψqSz
S z ig pψ ψ qpS S q
ψ rκ i pω US z qs ψ ig pS S q
9
9
Long-time behaviour:
Fixed points: S9 0, ψ9
0
Limit cycles?
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
25
Equilibrium Dicke vs open phase diagram, UN
40
0.1
40
UN=0
ω0=0.05
30
UN=0
0.08
20
Normal
SR
0.04
10
0
0
0.5
1
ω (MHz)
20
0.06
ωD
0
SR
⇑
SR
0
0.02
-20
0
-40
1.5
⇓
0
__
0.5
1
g√N (MHz)
1.5
g√N
Shift boundary pκ2
Allow negative ω
Jonathan Keeling
ω 2 q{ω
χpω0q
Ñ inverted
Superradiance of atoms in cavities
Strathclyde, September 2014.
26
Equilibrium Dicke vs open phase diagram, UN
40
0.1
40
UN=0
ω0=0.05
30
UN=0
0.08
20
Normal
SR
0.04
10
0
0
0.5
1
ω (MHz)
20
0.06
ωD
0
SR
⇑
SR
0
0.02
-20
0
-40
1.5
⇓
0
__
0.5
1
g√N (MHz)
1.5
g√N
Shift boundary pκ2
Allow negative ω
Jonathan Keeling
ω 2 q{ω
χpω0q
Ñ inverted
Superradiance of atoms in cavities
Strathclyde, September 2014.
26
Equilibrium Dicke vs open phase diagram, UN
40
0.1
40
UN=0
ω0=0.05
30
UN=0
0.08
20
Normal
SR
0.04
10
0
0
0.5
1
ω (MHz)
20
0.06
ωD
0
SR
⇑
SR
0
0.02
-20
0
-40
1.5
⇓
0
__
0.5
1
g√N (MHz)
1.5
g√N
Shift boundary pκ2
Allow negative ω
Jonathan Keeling
ω 2 q{ω
χpω0q
Ñ inverted
Superradiance of atoms in cavities
Strathclyde, September 2014.
26
. . . Dicke . . . UN
10MHz
40
0.1
40
UN=-10
ω0=0.05
30
20
Normal
SR
0.04
10
Unstable
0
0
0.5
1
1.5
ω (MHz)
20
0.06
ωD
UN=-10
0.08
0
0.02
-20
0
-40
0
__
g√N
⇓
SRA
⇓+⇑
SRB
⇑
SRA
0.5
1
g√N (MHz)
1.5
Coexistence regions
Unstable Ñ SRB
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
27
. . . Dicke . . . UN
10MHz
40
0.1
40
UN=-10
ω0=0.05
30
⇓
20
Normal
SR
0.04
10
Unstable
0
0
0.5
1
1.5
ω (MHz)
20
0.06
ωD
UN=-10
0.08
SRA+⇑
SRA
SRA
⇓+⇑
0
0.02
-20
0
-40
⇑
0
__
g√N
SRB
SRA+⇓
0.5
1
g√N (MHz)
SRA
1.5
Coexistence regions
Unstable Ñ SRB
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
27
. . . Dicke . . . UN
10MHz
40
0.1
40
UN=-10
ω0=0.05
30
⇓
20
Normal
SR
0.04
10
Unstable
0
0
0.5
1
1.5
ω (MHz)
20
0.06
ωD
UN=-10
0.08
SRA+⇑
SRA
SRA
⇓+⇑
0
0.02
-20
0
-40
⇑
0
__
g√N
SRB
SRA+⇓
0.5
1
g√N (MHz)
SRA
1.5
Coexistence regions
Unstable Ñ SRB
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
27
. . . Dicke . . . UN
40MHz
40
0.1
UN=40
ω0=0.05
30
g0ψ
Changing U:
0.08
ωD
0.06
20
Normal
Ω
10
SR
0.04
Unstable
0.02
0
2 Level System
0
0
0.5
1
1.5
__
g√N
40
UN=40
ω (MHz)
20
0
-20
-40
0
0.5
Jonathan Keeling
1
g√N (MHz)
1.5
Superradiance of atoms in cavities
Strathclyde, September 2014.
28
. . . Dicke . . . UN
40MHz
40
30
g0ψ
0.08
0.06
ωD
Changing U:
0.1
UN=40
ω0=0.05
20
Normal
Ω
10
SR
0.04
Unstable
0.02
0
2 Level System
0
0
0.5
1
1.5
__
g√N
40
UN=40
ω (MHz)
20
0
-20
-40
0
0.5
Jonathan Keeling
1
g√N (MHz)
1.5
Superradiance of atoms in cavities
Strathclyde, September 2014.
28
. . . Dicke . . . UN
40MHz
40
30
g0ψ
0.08
0.06
ωD
Changing U:
0.1
UN=40
ω0=0.05
20
Normal
Ω
SR
10
0.04
Unstable
0.02
0
2 Level System
0
0
0.5
1
1.5
__
g√N
40
UN=40
⇓
1000 1200
SRA
20
800
|ψ|2
ω (MHz)
1200
Persistent Oscillations
0
800
600
400 400
-20
⇑
SRA
200
0
0
0
-40
0
0.5
Jonathan Keeling
1
g√N (MHz)
1.5
Superradiance of atoms in cavities
0
2
5
4
10
6
15
8 10
t (ms)
12
14
16
Strathclyde, September 2014.
18
28
Outline
1
Dicke model, superradiance and no–go theorem
2
Superradiance and self-organisation
Raman scheme
Hierarchies of approximation
Equilibrium theory of Dicke
3
Fermionic self organisation
Equilibrium phase diagrams
Landau theory and microscopics
Open system?
4
Open system dynamics of Bososn
Attractors of open Dicke model
Bosons beyond Dicke
5
Conclusions
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
29
Open system and Beyond Dicke?
10MHz figures
40
0.1
40
UN=-10
ω0=0.05
30
20
0.06
ωD
UN=-10
0.08
20
Normal
SR
0.04
10
Unstable
0
0
0.5
1
1.5
ω (MHz)
UN
0
0.02
-20
0
-40
0
__
g√N
Jonathan Keeling
Superradiance of atoms in cavities
⇓
SRA
⇓+⇑
SRB
⇑
SRA
0.5
1
g√N (MHz)
1.5
Strathclyde, September 2014.
30
Open system and Beyond Dicke?
10MHz figures
40
0.1
40
UN=-10
ω0=0.05
30
⇓
20
0.06
ωD
UN=-10
0.08
20
Normal
SR
0.04
10
Unstable
0
0
0.5
1
1.5
ω (MHz)
UN
SRA+⇑
SRA
SRA
⇓+⇑
0
0.02
-20
0
-40
⇑
0
__
g√N
SRB
SRA+⇓
0.5
1
g√N (MHz)
SRA
1.5
From fermions, found:
Survives to low nF : Bosons!
?
But needs state φpx, z q cosp 2x q
Missed by Dicke model
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
30
Bosons beyond Dicke
°
BEC self organisation: |Ψatoms y p
i B t χk
k χk ak
qN |0y, χk obeys:
ωr |k|2δk,k1 Vk,k1 pφq χk
¸ p1q
¸ p2q
χk mk,k1 χk1
χk mk,k1 χk1 iκqφ ηE0
i Bt φ pω E0
k,k1
k,k1
Truncate |k| nM , nM
1 Ñ Dicke
Boundary moves
ω0 2ωr
Hysteresis –
Larkin-Pikin
2nd order at large ω̃
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
31
Bosons beyond Dicke
°
BEC self organisation: |Ψatoms y p
i B t χk
k χk ak
qN |0y, χk obeys:
ωr |k|2δk,k1 Vk,k1 pφq χk
¸ p1q
¸ p2q
χk mk,k1 χk1
χk mk,k1 χk1 iκqφ ηE0
i Bt φ pω E0
k,k1
k,k1
Truncate |k| nM , nM
1 Ñ Dicke
Boundary moves
ω0 2ωr
Hysteresis –
Larkin-Pikin
2nd order at large ω̃
Cavity intensity, |φ|2
3
Dicke
nM=2
nM=3
nM=5
nM=10
2.5
2
~ =2.6
ω
1.5
1
0.5
0
0.4
0.5
0.6
0.7
0.8
Pump field, η
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
31
Bosons beyond Dicke
°
BEC self organisation: |Ψatoms y p
i B t χk
k χk ak
qN |0y, χk obeys:
ωr |k|2δk,k1 Vk,k1 pφq χk
¸ p1q
¸ p2q
χk mk,k1 χk1
χk mk,k1 χk1 iκqφ ηE0
i Bt φ pω E0
k,k1
k,k1
Truncate |k| nM , nM
1 Ñ Dicke
Boundary moves
ω0 2ωr
Hysteresis –
Larkin-Pikin
2nd order at large ω̃
Cavity intensity, |φ|2
3
Dicke
nM=2
nM=3
nM=5
nM=10
2.5
2
~ =2.7
ω
1.5
1
0.5
0
0.4
0.5
0.6
0.7
0.8
Pump field, η
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
31
Bosons beyond Dicke
°
BEC self organisation: |Ψatoms y p
i B t χk
k χk ak
qN |0y, χk obeys:
ωr |k|2δk,k1 Vk,k1 pφq χk
¸ p1q
¸ p2q
χk mk,k1 χk1
χk mk,k1 χk1 iκqφ ηE0
i Bt φ pω E0
k,k1
k,k1
Truncate |k| nM , nM
1 Ñ Dicke
Boundary moves
ω0 2ωr
Hysteresis –
Larkin-Pikin
2nd order at large ω̃
Cavity intensity, |φ|2
3
Dicke
nM=2
nM=3
nM=5
nM=10
2.5
2
~ =2.8
ω
1.5
1
0.5
0
0.4
0.5
0.6
0.7
0.8
Pump field, η
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
31
Bosons beyond Dicke
°
BEC self organisation: |Ψatoms y p
i B t χk
k χk ak
qN |0y, χk obeys:
ωr |k|2δk,k1 Vk,k1 pφq χk
¸ p1q
¸ p2q
χk mk,k1 χk1
χk mk,k1 χk1 iκqφ ηE0
i Bt φ pω E0
k,k1
k,k1
Truncate |k| nM , nM
1 Ñ Dicke
Boundary moves
ω0 2ωr
Hysteresis –
Larkin-Pikin
2nd order at large ω̃
Cavity intensity, |φ|2
3
Dicke
nM=2
nM=3
nM=5
nM=10
2.5
2
~ =2.9
ω
1.5
1
0.5
0
0.4
0.5
0.6
0.7
0.8
Pump field, η
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
31
Bosons beyond Dicke
°
BEC self organisation: |Ψatoms y p
i B t χk
k χk ak
qN |0y, χk obeys:
ωr |k|2δk,k1 Vk,k1 pφq χk
¸ p1q
¸ p2q
χk mk,k1 χk1
χk mk,k1 χk1 iκqφ ηE0
i Bt φ pω E0
k,k1
k,k1
Truncate |k| nM , nM
1 Ñ Dicke
Boundary moves
ω0 2ωr
Hysteresis –
Larkin-Pikin
2nd order at large ω̃
Cavity intensity, |φ|2
3
Dicke
nM=2
nM=3
nM=5
nM=10
2.5
2
~ =3.0
ω
1.5
1
0.5
0
0.4
0.5
0.6
0.7
0.8
Pump field, η
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
31
Acknowledgements
G ROUP :
C OLLABORATORS :
F UNDING :
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
32
Summary
Fermions self organisation, liquid gas, and multicritical points
Atomic energy bands
2
1
0
-1
Filled bands, nF=1.5
-2
η=0.8
-3
0
0.2
0.4
0.6
0.8
Cavity field, φ
1
1.2
1.4
First order transitions for bosons, outside Dicke model
b(η,nF)/nF
2
nF → 0
1
0
-1
-2
0
0.2
0.4
0.6
ηc
Pump field, η
0.8
JK, Bhassen, Simons PRL ’14
Bosons: Dicke model shows many dynamical phases
0
-20
⇑
SRA
-40
40
UN=0
⇓
20
SR
0
⇑
-20
SR
-40
0
0.5
1
g√N (MHz)
1.5
0
SRA
SRB
-20
⇑
SRA
-40
0
0.5
1
g√N (MHz)
1.5
40
UN=-40
⇓
⇓+⇑
20
g-√N=1
20
ω (MHz)
SRA
ω (MHz)
ω (MHz)
40
UN=40
⇓
20
ω (MHz)
40
0
-20
-40
0
0.5
1
g√N (MHz)
1.5
-0.01
-0.005
0
δg/g-
0.005
0.01
JK et al. PRL ’10, Bhaseen et al. PRA ’12
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
33
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
34
6
Liquid gas bistability
7
Confined Fermi gas
8
Classical dynamics
Dicke model timescales
9
Ferroelectric transition
10
Grand canonical
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
35
Bistability, signatures
~ =5.2
ω
~ =5.3
ω
~ =5.4
ω
~ =5.5
ω
~ =5.6
ω
Free energy, f
1
0
-1
0
nF=1.5, η=0.8
0.5
Narrow bistable region
2
1
1.5
η=0.8, nF=1.5
Global min
Extrema
Unstable
Cavity field, φ
1.5
η
Cavity field, φ
1
0.5
0
0
2
4
6
8
10~
Cavity-pump detuning, ω
Jonathan Keeling
12
14
Superradiance of atoms in cavities
Strathclyde, September 2014.
36
Bistability, signatures
~ =5.2
ω
~ =5.3
ω
~ =5.4
ω
~ =5.5
ω
~ =5.6
ω
Free energy, f
1
0
-1
Narrow bistable region
2
nF=1.5, η=0.8
0.5
FS distortion
0.5
kz
η
1
1.5
Cavity field, φ
~ =5.1, φ=1.09
ω
~ =5.5, φ=0.63
ω
nF=1.50
η=0.8, nF=1.5
Global min
Extrema
Unstable
Cavity field, φ
1.5
0
0
1
-0.5
b) SRhi
0.5
-0.5
c) SRlo
0
kx
0.5
-0.5
0
kx
0.5
0
0
2
4
6
8
10~
Cavity-pump detuning, ω
Jonathan Keeling
12
14
Superradiance of atoms in cavities
Strathclyde, September 2014.
36
Fermi gas in a trap
Trapped gas, V pr q ER pr {r0 qα
Rescale via A πr02
Commensuration visible if flat
enough (α ¡ 4)
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
37
Classical dynamics of the extended Dicke model
Open dynamical system:
ωψ:ψ ω0Sz g pψ ψ:qpS S q
Bt ρ i rH, ρsκpψ:ψρ 2ψρψ: ρψ:ψq
H
USz ψ : ψ.
Neglects quantum fluctuations
Linearisation about fixed point Ñ stability, spectrum
[JK et al. PRL ’10, Bhaseen et al. PRA ’12]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
38
Classical dynamics of the extended Dicke model
Open dynamical system:
ωψ:ψ ω0Sz g pψ ψ:qpS S q
Bt ρ i rH, ρsκpψ:ψρ 2ψρψ: ρψ:ψq
H
Classical EOM
(|S| N {2 " 1)
USz ψ : ψ.
S9 i pω0 U |ψ|2qS 2ig pψ ψqSz
S z ig pψ ψ qpS S q
ψ rκ i pω US z qs ψ ig pS S q
9
9
Neglects quantum fluctuations
Linearisation about fixed point Ñ stability, spectrum
[JK et al. PRL ’10, Bhaseen et al. PRA ’12]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
38
Classical dynamics of the extended Dicke model
Open dynamical system:
ωψ:ψ ω0Sz g pψ ψ:qpS S q
Bt ρ i rH, ρsκpψ:ψρ 2ψρψ: ρψ:ψq
H
Classical EOM
(|S| N {2 " 1)
USz ψ : ψ.
S9 i pω0 U |ψ|2qS 2ig pψ ψqSz
S z ig pψ ψ qpS S q
ψ rκ i pω US z qs ψ ig pS S q
9
9
Neglects quantum fluctuations
Linearisation about fixed point Ñ stability, spectrum
[JK et al. PRL ’10, Bhaseen et al. PRA ’12]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
38
Classical dynamics of the extended Dicke model
Open dynamical system:
ωψ:ψ ω0Sz g pψ ψ:qpS S q
Bt ρ i rH, ρsκpψ:ψρ 2ψρψ: ρψ:ψq
H
Classical EOM
(|S| N {2 " 1)
USz ψ : ψ.
S9 i pω0 U |ψ|2qS 2ig pψ ψqSz
S z ig pψ ψ qpS S q
ψ rκ i pω US z qs ψ ig pS S q
9
9
Neglects quantum fluctuations
Linearisation about fixed point Ñ stability, spectrum
[JK et al. PRL ’10, Bhaseen et al. PRA ’12]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
38
Fixed points (steady states)
0 i pω0 U |ψ |2 qS 0 ig pψ
0 rκ
2ig pψ
ψ qpS S
i pω US
Jonathan Keeling
z
ψ qS z
q
qs ψ ig pS
S
q
ψ 0, S p0, 0, N {2q
always a solution.
If g ¡ gc , ψ 0 too
Superradiance of atoms in cavities
A S y =rS s 0
B ψ 1 <r ψ s 0
Strathclyde, September 2014.
39
Fixed points (steady states)
0 i pω0 U |ψ |2 qS 2ig pψ
ψ qpS S
0 ig pψ
0 rκ
i pω US
z
ψ qS z
q
qs ψ ig pS
S
q
ψ 0, S p0, 0, N {2q
always a solution.
If g ¡ gc , ψ 0 too
A S y =rS s 0
B ψ 1 <r ψ s 0
Sz
Sy
Sx
Jonathan Keeling
Small g: ò, ó only.
(ω 30MHz, UN
40MHz)
Superradiance of atoms in cavities
Strathclyde, September 2014.
39
Fixed points (steady states)
0 i pω0 U |ψ |2 qS 2ig pψ
ψ qpS S
0 ig pψ
0 rκ
i pω US
z
ψ qS z
q
qs ψ ig pS
S
q
ψ 0, S p0, 0, N {2q
always a solution.
If g ¡ gc , ψ 0 too
A S y =rS s 0
B ψ 1 <r ψ s 0
Sz
Sy
Sx
Jonathan Keeling
Small g: ò, ó only.
(ω 30MHz, UN
Larger g: SR too.
40MHz)
Superradiance of atoms in cavities
Strathclyde, September 2014.
39
Outline
6
Liquid gas bistability
7
Confined Fermi gas
8
Classical dynamics
Dicke model timescales
9
Ferroelectric transition
10
Grand canonical
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
40
Comparison to experiment: UN
10MHz
(ωp -ωc) (2π MHz)
0
-10
-20
-30
-40
0
0.5
1
1.5
g2 N (MHz)2
2
2.5
UN 10MHz
[Baumann et al Nature ’10 ]
Adapted from: [Bhaseen et al. PRA ’12]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
41
Timescale to reach steady state
40
Asymptotic state
⇑
SRA
103
102
0
SRB
101
|ψ|2
-ω (MHz)
20
⇓
-20
100
SRA
-40
0
0.5
1
g√N (MHz)
Jonathan Keeling
1.5
10-1
Superradiance of atoms in cavities
Strathclyde, September 2014.
42
Timescale to reach steady state
40
10ms sweep
103
20
-ω (MHz)
2
10
0
-20
101
-40
100
-60
0.0
0.5
1.0
1.5
2
2
2.0
2.5
10-1
g N (MHz )
40
Asymptotic state
⇑
SRA
103
102
0
SRB
101
|ψ|2
-ω (MHz)
20
⇓
-20
100
SRA
-40
0
0.5
1
g√N (MHz)
Jonathan Keeling
1.5
10-1
Superradiance of atoms in cavities
Strathclyde, September 2014.
42
Timescale to reach steady state
40
10ms sweep
103
20
-ω (MHz)
2
10
0
-20
101
-40
100
-60
0.0
0.5
1.0
1.5
2
2
2.0
2.5
10-1
g N (MHz )
40
Asymptotic state
⇑
SRA
40
103
200ms sweep
1
10
⇓
-20
100
SRA
-1
-40
0
0.5
1
g√N (MHz)
1.5
10
-ω (MHz)
SRB
103
20
102
0
|ψ|2
-ω (MHz)
20
102
0
-20
101
-40
100
-60
0.0
0.5
1.0
1.5
2
2
2.0
2.5
10-1
g N (MHz )
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
42
Timescales for dynamics: Why so slow and varied?
Suppose co- and counter-rotating terms differ
∆a
∆b
Ωb
Ωa
g0 ψ
g0 ψ
H
...
g pψ : S ψS
q
g 1 pψ : S
ψS q
...
2 Level System
SR(A) near phase boundary at small δg
Ñ Critical slowing down
SR(A), SR(B) continuously connect
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
43
Timescales for dynamics: Why so slow and varied?
Suppose co- and counter-rotating terms differ
∆a
∆b
Ωb
Ωa
g0 ψ
H
g0 ψ
...
g pψ : S ψS
q
g 1 pψ : S
ψS q
...
2 Level System
40
UN=-10
ω (MHz)
20
0
-20
⇓
SRA
⇓+⇑
SRB
⇑
SRA
-40
0
0.5
1
g√N (MHz)
1.5
SR(A) near phase boundary at small δg
Ñ Critical slowing down
SR(A), SR(B) continuously connect
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
43
Timescales for dynamics: Why so slow and varied?
Suppose co- and counter-rotating terms differ
∆a
∆b
Ωb
Ωa
g0 ψ
H
g0 ψ
...
g pψ : S δg
2 Level System
40
-20
SRA
⇓+⇑
SRB
⇑
SRA
2ḡ
g1
g
-0.005
0
δg/g-
ψS q
...
0.005
0.01
40
g-√N=1
20
ω (MHz)
0
⇓
g 1 pψ : S
q
g 1 g,
UN=-10
20
ω (MHz)
ψS
0
-20
-40
-40
0
0.5
1
g√N (MHz)
1.5
-0.01
SR(A) near phase boundary at small δg
Ñ Critical slowing down
SR(A), SR(B) continuously connect
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
43
Timescales for dynamics: Why so slow and varied?
Suppose co- and counter-rotating terms differ
∆a
∆b
Ωb
Ωa
g0 ψ
H
g0 ψ
...
g pψ : S δg
2 Level System
40
-20
SRA
⇓+⇑
SRB
⇑
SRA
2ḡ
g1
g
-0.005
0
δg/g-
ψS q
...
0.005
0.01
40
g-√N=1
20
ω (MHz)
0
⇓
g 1 pψ : S
q
g 1 g,
UN=-10
20
ω (MHz)
ψS
0
-20
-40
-40
0
0.5
1
g√N (MHz)
1.5
-0.01
SR(A) near phase boundary at small δg
Ñ Critical slowing down
SR(A), SR(B) continuously connect
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
43
Ferroelectric transition
Atoms in Coulomb gauge
H
¸
ωk ak: ak
¸
rpi eApri qs2
Vcoul
i
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
44
Ferroelectric transition
Atoms in Coulomb gauge
H
¸
ωk ak: ak
¸
rpi eApri qs2
Vcoul
i
Two-level systems — dipole-dipole coupling
H
ω0Sz
ωψ : ψ
g pS
S qpψ
ψ:q
Nζ pψ
ψ : q2 η pS
S q2
(nb g 2 , ζ, η 91{V ).
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
44
Ferroelectric transition
Atoms in Coulomb gauge
H
¸
ωk ak: ak
¸
rpi eApri qs2
Vcoul
i
Two-level systems — dipole-dipole coupling
H
ω0Sz
ωψ : ψ
(nb g 2 , ζ, η 91{V ).
Jonathan Keeling
g pS
S qpψ
ψ:q
Nζ pψ
Ferroelectric polarisation if ω0
Superradiance of atoms in cavities
ψ : q2 η pS
S q2
2ηN
Strathclyde, September 2014.
44
Ferroelectric transition
Atoms in Coulomb gauge
H
¸
ωk ak: ak
¸
rpi eApri qs2
Vcoul
i
Two-level systems — dipole-dipole coupling
H
ω0Sz
ωψ : ψ
g pS
S qpψ
ψ:q
Nζ pψ
Ferroelectric polarisation if ω0
(nb g 2 , ζ, η 91{V ).
Gauge transform to dipole gauge D r
H
ω0 S z
“Dicke” transition at ω0
ωψ : ψ
ḡ pS
ψ : q2 η pS
S q2
2ηN
Sqpψ ψ:q
N ḡ 2{ω 2ηN
But, ψ describes electric displacement
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
44
Grand canonical ensemble
Grand canonical ensemble:
If H
Ñ H µ pS z
ψ : ψ), need only: g 2 N
Fix density / fix µ ¡ 0 — pumping
¡ pω µq|ω0 µ|
Transition at:
g 2 N ¡ pω µqpω0 µq
µ hits lowest mode
[Eastham and Littlewood, PRB ’01]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
45
Grand canonical ensemble
Grand canonical ensemble:
If H
Ñ H µ pS z
ψ : ψ), need only: g 2 N
Fix density / fix µ ¡ 0 — pumping
¡ pω µq|ω0 µ|
Transition at:
g 2 N ¡ pω µqpω0 µq
µ hits lowest mode
[Eastham and Littlewood, PRB ’01]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
45
Grand canonical ensemble
Grand canonical ensemble:
If H
Ñ H µ pS z
ψ : ψ), need only: g 2 N
Fix density / fix µ ¡ 0 — pumping
¡ pω µq|ω0 µ|
2
1
unstable
(µ-ω)/g
0
-1
SR
Transition at:
g 2 N ¡ pω µqpω0 µq
-2
-3
µ hits lowest mode
-4
-5
-4
-3
-2
-1
(ε - ω)/g
0
1
2
[Eastham and Littlewood, PRB ’01]
Jonathan Keeling
Superradiance of atoms in cavities
Strathclyde, September 2014.
45