Soton

Non-equilibrium phases of coupled matter-light
systems
Jonathan Keeling
University of
St Andrews
600YEARS
Southhampton, May 2013
Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 1 / 36

Coupling many atoms to light
Old question: What happens to radiation when many atoms interact
“collectively” with light.
Superradiance — dynamical and steady state.

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
Pump
κ
Cavity
Pump
Rydberg atoms/polaritons
Microcavity Polaritons

Dicke effect: Enhanced emission
Hint =
k,i
gk ψ†
k S−
i e−ik·ri + H.c.

Hint =
k,i
gk ψ†
k S−
i e−ik·ri + H.c.
If |ri − rj| λ, use i Si → S
Collective decay:
dρ
dt
= −
Γ
2
S+
S−
ρ − S−
ρS+
+ ρS+
S−

Hint =
k,i
gk ψ†
k S−
i e−ik·ri + H.c.
Collective decay:
dρ
dt
= −
Γ
2
S+
S−
ρ − S−
ρS+
+ ρS+
S−
If Sz = |S| = N/2 initially:
I ∝ −Γ
d Sz
dt
=
ΓN2
4
sech2 ΓN
2
t
-N/2
0
N/2
tD
〈S
z
〉
tD
0
ΓN2
/2
I=-Γd〈S
z
〉/dt

Hint =
k,i
gk ψ†
k S−
i e−ik·ri + H.c.
Collective decay:
dρ
dt
= −
Γ
2
S+
S−
ρ − S−
ρS+
+ ρS+
S−
If Sz = |S| = N/2 initially:
I ∝ −Γ
d Sz
dt
=
ΓN2
4
sech2 ΓN
2
t
-N/2
0
N/2
tD
〈S
z
〉
tD
0
ΓN2
/2
I=-Γd〈S
z
〉/dt
Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972]

Collective radiation with a cavity: Dynamics
Hint =
i
ψ†
S−
i + ψS+
i
Single cavity mode: oscillations
[Bonifacio and Preparata PRA ’70]

Collective radiation with a cavity: Dynamics
Hint =
i
ψ†
S−
i + ψS+
i
0
200
400
600
800
1000
1200
1400
1600
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
|ψ(t)|
2
Time
T=2ln(√N
__
)/√N
__
1/√N
__
Single cavity mode: oscillations
If Sz = |S| = N/2 initially
[Bonifacio and Preparata PRA ’70]

Dicke model: Equilibrium superradiance transition
H = ωψ†
ψ + ω0Sz
+ g ψ†
S−
+ ψS+
.
Coherent state: |Ψ → eλψ†+ηS+
|Ω
Small g, min at λ, η = 0
[Hepp, Lieb, Ann. Phys. ’73]

H = ωψ†
ψ + ω0Sz
+ g ψ†
S−
+ ψS+
.
|Ω
Spontaneous polarisation if: Ng2 > ωω0

H = ωψ†
ψ + ω0Sz
+ g ψ†
S−
+ ψS+
.
|Ω
0
0
ω
g-√N
⇓ SR

No go theorem and transition
[Rzazewski et al PRL ’75]

No go theorem:. Minimal coupling (p − eA)2/2m
−
i
e
m
A · pi ⇔ g(ψ†
S−
+ ψS+
),
i
A2
2m
⇔ Nζ(ψ + ψ†
)2

−
i
e
m
A · pi ⇔ g(ψ†
S−
+ ψS+
),
i
A2
2m
⇔ Nζ(ψ + ψ†
)2
For large N, ω → ω + 2Nζ. (RWA)

−
i
e
m
A · pi ⇔ g(ψ†
S−
+ ψS+
),
i
A2
2m
⇔ Nζ(ψ + ψ†
)2
Need Ng2 > ω0(ω + 2Nζ).

−
i
e
m
A · pi ⇔ g(ψ†
S−
+ ψS+
),
i
A2
2m
⇔ Nζ(ψ + ψ†
)2
Need Ng2 > ω0(ω + 2Nζ).
But Thomas-Reiche-Kuhn sum rule states: g2/ω0 < 2ζ. No transition

Dicke phase transition: ways out
Problem: g2/ω0 < 2ζ for intrinsic parameters. Solutions:
Interpretation
Ferroelectric transition in D · r gauge.
[JK JPCM ’07, Vukics & Domokos PRA 2012 ]
Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]
Grand canonical ensemble:
If H → H − µ(Sz
+ ψ†
ψ), need only:
g2
N > (ω − µ)(ω0 − µ)
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 ]

Dicke phase transition: ways out
Problem: g2/ω0 < 2ζ for intrinsic parameters. Solutions:
Interpretation
Ferroelectric transition in D · r gauge.
[JK JPCM ’07, Vukics & Domokos PRA 2012 ]
Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]
Grand canonical ensemble:
If H → H − µ(Sz
+ ψ†
ψ), need only:
g2
N > (ω − µ)(ω0 − µ)
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 ]
κ
Pump
κ
Cavity
Pump

Outline
1 Introduction: Dicke model and superradiance
2 Dynamics of generalized Dicke model
Summary of experiment and classical dynamcs
Fixed points and dynamical phases
Timescales and consequences for experiment
Persistent oscillating phases
3 Jaynes Cummings Hubbard model
JCHM vv Dicke
Coherently driven array
Disorder

Acknowledgements
GROUP:
COLLABORATORS: Simons, Bhaseen, Schmidt, Blatter, T¨ureci, Kr¨uger
EXPERIMENT: Houck, Wallraff, Fink, Mylnek
FUNDING:

Dynamics of generalized Dicke model
JCHM vv Dicke
Disorder

Reminder of cold-atom extended Dicke model
κ
Pump
κ
2 Level System
x
z
Ω
gψ
0
2 Level system, | ⇓ , | ⇑ :
⇓: Ψ(x, z) = 1
⇑: Ψ(x, z) =
σ,σ =±
eik(σx+σ z)
H = ωψ†
ψ + ω0Sz
+ g(ψ + ψ†
)(S−
+ S+
)+USzψ†
ψ.
∂t ρ = −i[H, ρ]−κ(ψ†
ψρ − 2ψρψ†
+ ρψ†
ψ)
[Baumann et al Nature ’10 ]

κ
Pump
κ
2 Level System
x
z
Ω
gψ
0
⇓: Ψ(x, z) = 1
⇑: Ψ(x, z) =
σ,σ =±
eik(σx+σ z)
Feedback: U ∝
g2
0
ωc − ωa
H = ωψ†
ψ + ω0Sz
+ g(ψ + ψ†
)(S−
+ S+
)+USzψ†
ψ.
∂t ρ = −i[H, ρ]−κ(ψ†
ψρ − 2ψρψ†
+ ρψ†
ψ)

κ
Pump
κ
2 Level System
x
z
Ω
gψ
0
⇓: Ψ(x, z) = 1
⇑: Ψ(x, z) =
σ,σ =±
eik(σx+σ z)
Feedback: U ∝
g2
0
ωc − ωa
H = ωψ†
ψ + ω0Sz
+ g(ψ + ψ†
)(S−
+ S+
)+USzψ†
ψ.
∂t ρ = −i[H, ρ]−κ(ψ†
ψρ − 2ψρψ†
+ ρψ†
ψ)
ω0 ∼ kHz ω, κ, g
√
N ∼ MHz. [Baumann et al Nature ’10 ]

Classical dynamics of the extended Dicke model
Open dynamical system:
H = ωψ†
ψ + ω0Sz
+ g(ψ + ψ†
)(S−
+ S+
)+USzψ†
ψ.
∂t ρ = −i[H, ρ]−κ(ψ†
ψρ − 2ψρψ†
+ ρψ†
ψ)
Neglects quantum ﬂuctuations — restore via Wigner distributed
initial conditions.
Linearisation about ﬁxed point:
Recover Retarded Green’s function (spectrum)
Cannot recover occupations

Classical dynamics of the extended Dicke model
Open dynamical system:
H = ωψ†
ψ + ω0Sz
+ g(ψ + ψ†
)(S−
+ S+
)+USzψ†
ψ.
∂t ρ = −i[H, ρ]−κ(ψ†
ψρ − 2ψρψ†
+ ρψ†
ψ)
Classical EOM
(|S| = N/2 1)
˙S−
= −i(ω0+U|ψ|2
)S−
+ 2ig(ψ + ψ∗
)Sz
˙Sz
= ig(ψ + ψ∗
)(S−
− S+
)
˙ψ = − [κ + i(ω+USz
)] ψ − ig(S−
+ S+
)
Neglects quantum ﬂuctuations — restore via Wigner distributed
initial conditions.
Linearisation about ﬁxed point:
Recover Retarded Green’s function (spectrum)
Cannot recover occupations

Fixed points (steady states)
0 = i(ω0+U|ψ|2
)S−
+ 2ig(ψ + ψ∗
)Sz
0 = ig(ψ + ψ∗
)(S−
− S+
)
0 = − [κ + i(ω+USz
)] ψ − ig(S−
+ S+
)
ψ = 0, S = (0, 0, ±N/2)
always a solution.
If g > gc, ψ = 0 too
A Sy
= − [S−
] = 0
B ψ = [ψ] = 0

0 = i(ω0+U|ψ|2
)S−
+ 2ig(ψ + ψ∗
)Sz
0 = ig(ψ + ψ∗
)(S−
− S+
)
0 = − [κ + i(ω+USz
)] ψ − ig(S−
+ S+
)
ψ = 0, S = (0, 0, ±N/2)
always a solution.
A Sy
= − [S−
] = 0
B ψ = [ψ] = 0
x
Sy
Sz
S
Small g: ⇑, ⇓ only.
(ω = 30MHz, UN = −40MHz)

0 = i(ω0+U|ψ|2
)S−
+ 2ig(ψ + ψ∗
)Sz
0 = ig(ψ + ψ∗
)(S−
− S+
)
0 = − [κ + i(ω+USz
)] ψ − ig(S−
+ S+
)
ψ = 0, S = (0, 0, ±N/2)
always a solution.
A Sy
= − [S−
] = 0
B ψ = [ψ] = 0
x
Sy
Sz
S
Small g: ⇑, ⇓ only. Larger g: SR too.
(ω = 30MHz, UN = −40MHz)

Steady state phase diagram
0 = i(ω0+U|ψ|2
)S−
+ 2ig(ψ + ψ∗
)Sz
0 = ig(ψ + ψ∗
)(S−
− S+
)
0 = − [κ + i(ω+USz
)] ψ − ig(S−
+ S+
) 0
0
ω
g-√N
UN=0, κ=0
⇓ SR
See also Domokos and Ritsch PRL ’02, Domokos et al. PRL ’10

0 = i(ω0+U|ψ|2
)S−
+ 2ig(ψ + ψ∗
)Sz
0 = ig(ψ + ψ∗
)(S−
− S+
)
0 = − [κ + i(ω+USz
)] ψ − ig(S−
+ S+
) 0
0
ω
g-√N
UN=0, κ=0
⇓ SR
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
SR
SR
UN=0
⇓ SR(A): Sy = 0

0 = i(ω0+U|ψ|2
)S−
+ 2ig(ψ + ψ∗
)Sz
0 = ig(ψ + ψ∗
)(S−
− S+
)
0 = − [κ + i(ω+USz
)] ψ − ig(S−
+ S+
) 0
0
ω
g-√N
UN=0, κ=0
⇓ SR
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
SRA
SRA
⇓+⇑ SRB
UN=-20
⇓ SR(A): Sy = 0
⇓ + ⇑ SR(B): ψ = 0

0 = i(ω0+U|ψ|2
)S−
+ 2ig(ψ + ψ∗
)Sz
0 = ig(ψ + ψ∗
)(S−
− S+
)
0 = − [κ + i(ω+USz
)] ψ − ig(S−
+ S+
) 0
0
ω
g-√N
UN=0, κ=0
⇓ SR
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
⇓+⇑
SRA
SRA
SRB
UN=-40
⇓ SR(A): Sy = 0
⇓ + ⇑ SR(B): ψ = 0

0 = i(ω0+U|ψ|2
)S−
+ 2ig(ψ + ψ∗
)Sz
0 = ig(ψ + ψ∗
)(S−
− S+
)
0 = − [κ + i(ω+USz
)] ψ − ig(S−
+ S+
) 0
0
ω
g-√N
UN=0, κ=0
⇓ SR
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
SRA
SRA
⇓+⇑ SRB
SRB+⇑
SRB+⇑
SRA+⇑
SRA+⇓
SRB
+⇓+⇑
UN=-40
⇓ SR(A): Sy = 0
⇓ + ⇑ SR(B): ψ = 0

Comparison to experiment
-40
-30
-20
-10
0
0 0.5 1 1.5 2 2.5
(ωp-ωc)(2πMHz)
g2
N (MHz)2
UN = −10MHz
Adapted from: [Bhaseen et al. PRA ’12]
ω = ωc − ωp +
5
2
UN, UN = −
g2
0
4(ωa − ωc)

JCHM vv Dicke
Disorder

Dynamics: Evolution from normal state
Gray: S = (
√
N,
√
N, −N/2)
Black: Wigner distribution of S, ψ
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
(i)
(ii)
(iii)
⇓
⇑
⇓+⇑
SRA
SRA
SRB
UN=-40
Oscillations: ∼ 0.1ms
Decay: 20ms, 0.1ms, 20ms
(i) SR(A)
0 20 40 60 80
t (ms)
0
40
80
|ψ|2
0 1 2
0
100
(ii) SR(B)
0 0.1 0.2 0.3 0.4
t (ms)
0
100
200
|ψ|
2
(iii) SR(A)
0 100 200
t (ms)
0
40
80
120
|ψ|
2
150 151
40
50

Asymptotic state: Evolution from normal state
(Near to experimental UN = −13MHz).
All stable attractors:
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇑
⇓ SRA
SRA
⇓+⇑ SRB
UN=-10

Asymptotic state: Evolution from normal state
(Near to experimental UN = −13MHz).
All stable attractors:
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇑
⇓ SRA
SRA
⇓+⇑ SRB
UN=-10
Starting from ⇓
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10-1
10
0
10
1
102
103
|ψ|
2
Asymptotic state

Timescales for dynamics: Consequences for
experiment
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10-1
10
0
10
1
102
10
3
|ψ|2
Asymptotic state

experiment
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10-1
10
0
10
1
102
10
3
|ψ|2
Asymptotic state
-40
-20
0
20
40
60
0.0 0.5 1.0 1.5 2.0 2.5
ω(MHz)
g
2
N (MHz
2
)
10
-1
100
10
1
10
2
10310ms sweep

experiment
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10-1
10
0
10
1
102
10
3
|ψ|2
Asymptotic state
-40
-20
0
20
40
60
0.0 0.5 1.0 1.5 2.0 2.5
ω(MHz)
g
2
N (MHz
2
)
10
-1
100
10
1
10
2
10310ms sweep
-40
-20
0
20
40
60
0.0 0.5 1.0 1.5 2.0 2.5
ω(MHz)
g
2
N (MHz
2
)
10
-1
10
0
10
1
10
2
10
3200ms sweep

Timescales for dynamics: What are they?
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10
-1
10
0
10
1
10
2
103
|ψ|
2
Asymptotic state
Growth Most unstable eigenvalues
near S = (0, 0, −N/2)
Decay Slowest stable eigenvalues
near ﬁnal state
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
No unstable
directions
Two unstable directions
One unstable direction
10µs
100µs
1ms
10ms
100ms
1s
10s
Initial growth time

Timescales for dynamics: What are they?
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10
-1
10
0
10
1
10
2
103
|ψ|
2
Asymptotic state
Growth Most unstable eigenvalues
near S = (0, 0, −N/2)
Decay Slowest stable eigenvalues
near ﬁnal state
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
No unstable
directions
10µs
100µs
1ms
10ms
100ms
1s
10s
Initial growth time
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10µs
100µs
1ms
10ms
100ms
1s
10s
Asymptotic decay time

Timescales for dynamics: Why so slow and varied?
Suppose co- and counter-rotating terms differ
2 Level System
Ω
∆
Ω
ψ
b
a b
a
∆
ψg0
g0
H = . . . + g(ψ†
S−
+ ψS+
) + g (ψ†
S+
+ ψS−
) + . . .
SR(A) near phase boundary at small δg → Critical slowing down
SR(A), SR(B) continuously connect

2 Level System
Ω
∆
Ω
ψ
b
a b
a
∆
ψg0
g0
H = . . . + g(ψ†
S−
+ ψS+
) + g (ψ†
S+
+ ψS−
) + . . .
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇑
⇓ SRA
SRA
⇓+⇑ SRB
UN=-10

2 Level System
Ω
∆
Ω
ψ
b
a b
a
∆
ψg0
g0
H = . . . + g(ψ†
S−
+ ψS+
) + g (ψ†
S+
+ ψS−
) + . . .
δg = g − g, 2¯g = g + g
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇑
⇓ SRA
SRA
⇓+⇑ SRB
UN=-10
-40
-20
0
20
40
-0.01 -0.005 0 0.005 0.01
ω(MHz)
δg/g-
g-√N=1

JCHM vv Dicke
Disorder

Regions without ﬁxed points
Changing U:
2 Level System
Ω
gψ
0
U ∝
g2
0
ωc − ωa
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
⇓+⇑
SRA
SRA
SRB
UN=-40

Changing U:
2 Level System
Ω
ψg0
U ∝
g2
0
ωc − ωa
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
⇓+⇑
SRA
SRA
SRB
UN=-40

Changing U:
2 Level System
Ω
ψg0
U ∝
g2
0
ωc − ωa
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
SR
SR
UN=0

Changing U:
2 Level System
Ω
ψg0
U ∝
g2
0
ωc − ωa
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
SRA
SRA
UN=20

Changing U:
2 Level System
Ω
ψg0
U ∝
g2
0
ωc − ωa
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
SRA
SRA
UN=40

Changing U:
2 Level System
Ω
ψg0
U ∝
g2
0
ωc − ωa
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
SRA
SRA
Persistent Oscillations
UN=40
0 2 4 6 8 10 12 14 16 18
t (ms)
0
200
400
600
800
1000
1200
|ψ|
2
0 5 10 15
0
400
800
1200

Persistent (optomechanical) oscillations
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
SRA
SRA
Persistent Oscillations
UN=40
0 2 4 6 8 10 12 14 16 18
t (ms)
0
200
400
600
800
1000
1200
|ψ|
2
0 5 10 15
0
400
800
1200
0
200
400
600
800
1000
1200
18.00 18.02 18.04 18.06 18.08
-0.4
-0.2
0
0.2
0.4
|ψ|
2
Sx,Sy,Sz
t(ms)
|ψ|2
Sx
Sy
Sz

Jaynes Cummings Hubbard model
JCHM vv Dicke
Disorder

Equilibrium: Dicke model with chemical potential
H − µN = (ω − µ)ψ†
ψ + (ω0 − µ)Sz
+ g ψ†
S−
+ ψS+
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω)/g
(ω0 - ω)/g
SR
Transition at:
g2N > (ω − µ)|ω0 − µ|
Reduce critical g
Unstable if µ > ω
Inverted if µ > ω0
[Eastham and Littlewood, PRB ’01]

H − µN = (ω − µ)ψ†
ψ + (ω0 − µ)Sz
+ g ψ†
S−
+ ψS+
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω)/g
(ω0 - ω)/g
unstable
SR
Transition at:
g2N > (ω − µ)|ω0 − µ|
Reduce critical g
Unstable if µ > ω

H − µN = (ω − µ)ψ†
ψ + (ω0 − µ)Sz
+ g ψ†
S−
+ ψS+
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω)/g
(ω0 - ω)/g
⇑
⇓
unstable
SR
Transition at:
g2N > (ω − µ)|ω0 − µ|
Reduce critical g
Unstable if µ > ω

Jaynes-Cummings Hubbard model
H = −
J
z
ij
ψ†
i ψj +
i
∆
2
σz
i + g(ψ†
i σ−
i + H.c.)

H = −
J
z
ij
ψ†
i ψj +
i
∆
2
σz
i + g(ψ†
i σ−
i + H.c.)
-2
-1
0
0.001 0.01 0.1 1
µ/g
J/g
Unstable
Normal
∆/g=1

H = −
J
z
ij
ψ†
i ψj +
i
∆
2
σz
i + g(ψ†
i σ−
i + H.c.)
-2
-1
0
0.001 0.01 0.1 1
µ/g
J/g
Unstable
Normal
∆/g=1
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6

Dicke vs JCHM
JCHM
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6
Dicke
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω)/g
(ω0 - ω)/g
⇑
⇓
unstable
SR
k = 0 mode of JCHM ↔ Dicke photon mode
⇑ ↔ n = 1 Mott lobe

Dicke vs JCHM
JCHM
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6 E
k
UP
Photon
LP
2LS
∆JCHM
∆Dicke
Dicke
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω)/g
(ω0 - ω)/g
⇑
⇓
unstable
SR

Jaynes Cummings Hubbard model
JCHM vv Dicke
Disorder

Coherently pumped JCHM
H = −
J
z
ij
ψ†
i ψj +
i
∆
2
σz
i + g(ψ†
i σ−
i + H.c.)+f(ψieiωLt
+ H.c.)
∂t ρ = −i[H, ρ]−
κ
2
Lψ[ρ] −
γ
2
Lσ− [ρ]

Coherently pumped single cavity [Bishop et al. Nat. Phys ’09]
g H =
∆
2
σz
+ g(ψ†
σ−
+ H.c.)+f(ψeiωpumpt
+ H.c.)
∂t ρ = −i[H, ρ]−
κ
2
Lψ[ρ] −
γ
2
Lσ− [ρ]
Anti-resonance in | ψ |.
Effective 2LS:
|Empty , |1 polariton
IncreasingPumping
Mollow triplet ﬂuorescence
[Lang et al. PRL ’11]

Coherently pumped dimer & array
Chose detuning a la Dicke model
ωpump
ωpump
LP
UP
2g
CavityQubit
Single cavity
2J
LP
LP
UP
E
k
Photon
Qubit
Array
2g
Bistability at intermediate J
More/less localised states
Connects to Dicke limit
[Nissen et al. PRL ’12]

Coherently pumped dimer & array
Chose detuning a la Dicke model
ωpump
ωpump
LP
UP
2g
CavityQubit
Single cavity
2J
LP
LP
UP
E
k
Photon
Qubit
Array
2g
Evolution of anti-resonance vs J.
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a>|
ωpump/g
Bistability at intermediate J
More/less localised states
Connects to Dicke limit

Photon blockade picture J g
Polariton basis
Nonlinearity | 2 − 2 1| ∝ g.
H =
i
2
τz
i + ˜fτx
i
Decouple hopping:
τ+
i τ−
j → ψτ+ + ψ∗τ−
Bistability for
J > Jc =
4
˜f2
2˜f2 + (˜κ/2)2
3
3/2
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a>|
ωpump/g

Photon blockade picture J g
Polariton basis
Nonlinearity | 2 − 2 1| ∝ g.
H =
i
2
τz
i + ˜fτx
i −
˜J
z
ij
τ+
i τ−
j
Decouple hopping:
τ+
i τ−
j → ψτ+ + ψ∗τ−
Bistability for
J > Jc =
4
˜f2
2˜f2 + (˜κ/2)2
3
3/2
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a>|
ωpump/g

Coherently pumped array: correlations & ﬂuorescence
0
0.5
1
g2(t=0)
0.1
0.2
0.3
0.001 0.01 0.1 1 10
|〈a〉|
Hopping zJ/g
Correlations
g2 : 0 → 1 crossover.
Small J: Mollow triplet
Large J: Off resonance
ﬂuorescence
Pump at collective
resonance
Mismatch if J = 0.

Coherently pumped array: correlations & ﬂuorescence
0
0.5
1
g2(t=0)
0.1
0.2
0.3
0.001 0.01 0.1 1 10
|〈a〉|
Hopping zJ/g
Correlations
g2 : 0 → 1 crossover.
Fluorescence
Small J: Mollow triplet
Large J: Off resonance
ﬂuorescence
Pump at collective
resonance
Mismatch if J = 0.

Coherent pumped array – disorder
Effect of disorder, ∆ → ∆i
Distribution of ψ – Washes out bistable jump
Bistability near resonance — phase of ψ depends on ∆i
Complex ψ distribution
Superﬂuid phases in driven system?
-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95
Pump frequency
0
0.1
0.2
0.3
ψ
0
20
40
60
80
100
||
[Kulaitis et al. PRA, ’13]

Coherent pumped array – disorder
Effect of disorder, ∆ → ∆i
Distribution of ψ – Washes out bistable jump
Bistability near resonance — phase of ψ depends on ∆i
Complex ψ distribution
Superﬂuid phases in driven system?
-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95
Pump frequency
0
0.1
0.2
0.3
ψ
0
20
40
60
80
100
||
-0.2
0
0.2
(a) ωp=-0.988 (b) ωp=-0.987 (c) ωp=-0.986
-0.2
0
0.2
(d) ωp=-0.985 (e) ωp=-0.982 (f) ωp=-0.978
-0.2 0 0.2
-0.2
0
0.2
0
20
40
60
80
100
(g) ωp=-0.975
-0.2 0 0.2
(h) ωp=-0.971
-0.2 0 0.2
(i) ωp=-0.968
Re( )
Im()
[Kulaitis et al. PRA, ’13]

Summary
Wide variety of dynamical phases
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
SRA
SRA
UN=40
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
SR
SR
UN=0
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇓
⇑
⇓+⇑
SRA
SRA
SRB
UN=-40
-40
-20
0
20
40
-0.01 -0.005 0 0.005 0.01
ω(MHz)
δg/g-
g-√N=1
Slow dynamics for U < 0 & Persistent oscillations for U > 0
0 100 200
t (ms)
0
40
80
120
|ψ|
2
150 151
40
50
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
No unstable
directions
10µs
100µs
1ms
10ms
100ms
1s
10s
Initial growth time
-40
-20
0
20
40
0 0.5 1 1.5
ω(MHz)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10µs
100µs
1ms
10ms
100ms
1s
10s
Asymptotic decay time
0 2 4 6 8 10 12 14 16 18
t (ms)
0
200
400
600
800
1000
1200
|ψ|
2
0 5 10 15
0
400
800
1200
0
200
400
600
800
1000
1200
18.00 18.02 18.04 18.06 18.08
-0.4
-0.2
0
0.2
0.4
|ψ|2
Sx,Sy,Sz
t(ms)
|ψ|2
Sx
Sy
Sz
JK et al. PRL ’10, Bhaseen et al. PRA ’12
Dicke model and JCHM: connection at J → ∞E
k
UP
Photon
LP
2LS
∆JCHM
∆Dicke
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω)/g
(ω0 - ω)/g
⇑
⇓
unstable
SR
Coherently pumped coupled cavity array
ωpump
ωpump
LP
UP
2g
CavityQubit
Single cavity
2J
LP
LP
UP
E
k
Photon
Qubit
Array
2g
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a>|
ωpump/g
0
0.5
1
g2(t=0)
0.1
0.2
0.3
0.001 0.01 0.1 1 10
|〈a〉|
Hopping zJ/g
-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95
Pump frequency
0
0.1
0.2
0.3
ψ
0
20
40
60
80
100
||
-0.2
0
0.2
(a) ωp=-0.988 (b) ωp=-0.987 (c) ωp=-0.986
-0.2
0
0.2
(d) ωp=-0.985 (e) ωp=-0.982 (f) ωp=-0.978
-0.2 0 0.2
-0.2
0
0.2
0
20
40
60
80
100
(g) ωp=-0.975
-0.2 0 0.2
(h) ωp=-0.971
-0.2 0 0.2
(i) ωp=-0.968
Re( )
Im()
Nissen et al. PRL ’12, Kulaitis et al. PRA ’13

4 Ferroelectric transition
5 Dicke vs JCHM
6 Pumping without symmetry breaking
7 Collective dephasing

Ferroelectric transition
Atoms in Coulomb gauge
H = ωk a†
k ak +
i
[pi − eA(ri)]2
+ Vcoul

H = ωk a†
k ak +
i
[pi − eA(ri)]2
+ Vcoul
Two-level systems — dipole-dipole coupling
H = ω0Sz
+ ωψ†
ψ + g(S+
+ S−
)(ψ + ψ†
) + Nζ(ψ + ψ†
)2
−η(S+
− S−
)2
(nb g2, ζ, η ∝ 1/V).

H = ωk a†
k ak +
i
[pi − eA(ri)]2
+ Vcoul
H = ω0Sz
+ ωψ†
ψ + g(S+
+ S−
)(ψ + ψ†
) + Nζ(ψ + ψ†
)2
−η(S+
− S−
)2
(nb g2, ζ, η ∝ 1/V).
Ferroelectric polarisation if ω0 < 2ηN

H = ωk a†
k ak +
i
[pi − eA(ri)]2
+ Vcoul
H = ω0Sz
+ ωψ†
ψ + g(S+
+ S−
)(ψ + ψ†
) + Nζ(ψ + ψ†
)2
−η(S+
− S−
)2
(nb g2, ζ, η ∝ 1/V).
Ferroelectric polarisation if ω0 < 2ηN
Gauge transform to dipole gauge D · r
H = ω0Sz
+ ωψ†
ψ + ¯g(S+
− S−
)(ψ − ψ†
)
“Dicke” transition at ω0 < N¯g2/ω ≡ 2ηN
But, ψ describes electric displacement

H − µN = (ω − µ)ψ†
ψ + (ω0 − µ)Sz
+ g ψ†
S−
+ ψS+
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω)/g
(ω0 - ω)/g
SR
Transition at:
g2N > (ω − µ)|ω0 − µ|
Reduce critical g
Unstable if µ > ω

H − µN = (ω − µ)ψ†
ψ + (ω0 − µ)Sz
+ g ψ†
S−
+ ψS+
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω)/g
(ω0 - ω)/g
unstable
SR
Transition at:
g2N > (ω − µ)|ω0 − µ|
Reduce critical g
Unstable if µ > ω

H − µN = (ω − µ)ψ†
ψ + (ω0 − µ)Sz
+ g ψ†
S−
+ ψS+
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω)/g
(ω0 - ω)/g
⇑
⇓
unstable
SR
Transition at:
g2N > (ω − µ)|ω0 − µ|
Reduce critical g
Unstable if µ > ω

H = −
J
z
ij
ψ†
i ψj +
i
∆
2
σz
i + g(ψ†
i σ−
i + H.c.)

H = −
J
z
ij
ψ†
i ψj +
i
∆
2
σz
i + g(ψ†
i σ−
i + H.c.)
-2
-1
0
0.001 0.01 0.1 1
µ/g
J/g
Unstable
Normal
∆/g=1

H = −
J
z
ij
ψ†
i ψj +
i
∆
2
σz
i + g(ψ†
i σ−
i + H.c.)
-2
-1
0
0.001 0.01 0.1 1
µ/g
J/g
Unstable
Normal
∆/g=1
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6

Dicke vs JCHM
JCHM
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6
Dicke
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω)/g
(ω0 - ω)/g
⇑
⇓
unstable
SR

Dicke vs JCHM
JCHM
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6 E
k
UP
Photon
LP
2LS
∆JCHM
∆Dicke
Dicke
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω)/g
(ω0 - ω)/g
⇑
⇓
unstable
SR

Raman pumping
How to pump without breaking symmetry
Counter-rotating terms — Raman pumping
Atom proposal [Dimer et al. PRA ’07]
Atom experiment [Baumann et al. Nature ’10]
Qubit — allowed transitions ∆n = 1
Qubit dephasing much bigger than atom
JK, T¨ureci, Houck in progress

Raman pumping
Tunable-coupling-qubit
00
01
10
11
02
20
g
g
0
1
Ω
Ωa
b
Pump
Cavity

Raman pumping
Tunable-coupling-qubit
00
01
10
11
02
20
g
g
0
1
Ω
Ωa
b
Pump
Cavity
0 0.5 1
g0
0
1
2
3
4
Ωa=Ωb=Ω
⇓
SR?

Collective dephasing
Real environment is not Markovian
[Carmichael & Walls JPA ’73] Requirements for correct equilibrium
[Ciuti & Carusotto PRA ’09] Dicke SR and emission
Cannot assume ﬁxed κ, γ
Phase transition → soft modes
Strong coupling → varying decay

Dicke model linewidth:
H = ωψ†
ψ+
N
i=1
i
2
σz
i +g σ+
i ψ + h.c.
+
i
σz
i
q
γq b†
q + bq +
q
βqb†
iqbq.
[Nissen, Fink et al. arXiv:1302.0665]

Dicke model linewidth:
H = ωψ†
ψ+
N
i=1
i
2
σz
i +g σ+
i ψ + h.c.
+
i
σz
i
q
γq b†
q + bq +
q
βqb†
iqbq.
0.008
0.01
0.012
0.014
1 2 3 4 5
linewidth/g
number of qubits, N
experiment
theory
〈a〉
2
(a.u.)
frequency (a.u.)
1
2
3
[Nissen, Fink et al. arXiv:1302.0665]

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