strongly interacting matter and light far from equilibrium · strongly interacting matter and light...
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Strongly Interacting Matter and Light Far From Equilibrium
Marco Schiro’ CNRS-IPhT Saclay
Center For Optical Quantum Technologies, Hamburg April 2015
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Condensed Matter
Ultra-Cold Atoms
Quantum Many Body Physics with “Light” (?)
Light used to probe phases or (more recently) to ‘’induce’’
phases of matter
Light used to trap (optical lattices) to probe or to excite atoms
Non-Linear Optics with Single Photons at Finite Density
Bloch group, Munich
Cavalleri group H
amburg
Houck group, Princeton
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Outline
Coupling Light and Matter at the Quantum Level
Many Body Physics with “Atoms” and “Photons”: Perspectives and Challenges
Quantum Phase Transitions of Light in Arrays Coupled CQED Units
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Exploring Light-Matter Interaction at
the Quantum Level
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How Strong Can the Coupling Between Light and Matter Be?
M. Devoret, S. Girvin&R. Schoelkopf, Ann Phys 16 767 (2007)
g = dat E0/~
Fine Structure Constant Limit
g
Coupling via current (i.e. magnetic field) can be even larger!
dat eL~r
4 0
2E2
0 V
g r
L
r
r2
‘’Poor-Man’’ Estimate of Light-Matter Dipole Coupling
g eL
rr
2 ~0 V
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Cavity Quantum ElectroDynamics
Trapping photons and atoms into cavities!
Light-Matter coupling (Dipole)
J. M. Raimond, M. Brune and S. Haroche, Rev. Mod. Phys. 73, 565 (2001)
Hint d ·E
Alkali or Rydberg Atoms trapped into the cavity
Optical/Microwave 3D cavities supporting discrete modes
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A Solid-State Analogue: Circuit QED
Microwave Transmission Line Resonators
Circuit QED
A. Blais et al, Phys. Rev. A 69 062320 (2004)
Artificial Atoms: Superconducting Qubits
Superconducting-based to avoid dissipation
Mesoscopic SC system with ‘‘atom-like’‘ spectrum
Capacitive/Inductive Coupling
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Energy Scales in CQEDCircuit QEDCavity QED
Photon Losses and Atomic Decay Rates (Dissipation)
g , Strong-Coupling Regime of cQED
Photon Frequency/Atom splitting r,q
Light-Matter Coupling (Vacuum Rabi Splitting)
,
g r,q
Typical Circuit QED:
g 1Ghz , 500 kHz!r 10Ghz
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Effective Photon-Photon Interactions
by Light-Matter Coupling
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Quantum Models of Light and Matter
I. Rabi (1936)
Harmonic Mode
(Photon)
Anharmonic Mode (Atom)
Linear coupling (dipole)
Circuit QEDCavity QED
Often in Rotating Wave Approximation (requires )g min(r,q)
Jaynes,Cumming (1963)
X
HRabi = r a†a+ q
+ + ga† + a
+ +
HRabi = r a†a+ q
+ + ga† + hc
+ g
a†+ + hc
HJC = r a
†a+ q+ + g
a† + hc
The Rabi Model
The Jaynes-Cumming Model
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Elementary Light-Matter Excitations: Polaritons
Npol
= a†a+ +
[HJC
, Npol
] = 0
|n,+ = sinn |n | + cosn |n 1 | |n, = cosn |n | sinn |n 1 |
En± = r n+
2±
p2/4 + ng2
Weakly Anharmonic Spectrum
Atom-Induced Photon Non Linearity
Exp Realization with Circuit QED under drivingL. Bishop et al, Nat. Phys 5, 109 (2009)
HJC = r a†a+ q
+ + ga† + hc
tan n = 2gpn/( +
p2/4 + ng2)
= !r !q
/2
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Photon-Blockaded Transport
A.J Hoffmann et al, Phys. Rev. Lett 107, (2011)
HJC r a†a+ Ueff
a†a
2g |r q|
Ueff g4/|r q|3
Strongly Dispersive Regime:
Effective Photon Hamiltonian
Steps in the Photon Transmission through a driven cavity
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Many Body Physics with Strongly Interacting Light and
Matter
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Exciton-Polariton Condensate Cold-Atoms in Optical Cavities
H.Ritsch, P. Domokos, F. Brennecke, T. Esslinger, RMP 85 (2013)I. Carusotto, C. Ciuti, RMP 85 (2013)
Driven-Dissipative Superfluid Self-Organization/Dicke Transition
Exp: ETH (Esslinger), Hamburg (Hemmerich)Exp: Paris (J.Bloch), Stanford (Yamamoto),..
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Coupled Arrays of CQED UnitsExperiments at Princeton: (@ A. Houck’s Lab)
Hopping between adjacent resonators/cavities
Effective Photon-Photon Interactions
Delocalized bosonic particles with an effective onsite interaction...
Bose-Hubbard Physics(?)
HJCL =X
ij
Jij a†i aj +
X
i
HiJC
Jaynes-Cumming Lattice Model
J. Koch and K. Le Hur PRA 80 023811 (2009)
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The importance of being a Photon
µph =E
N= 0
Unlike massive particles Photons are not conserved in Nature
Photons have zero chemical potential!
The density of a Photon Gas is set by drive-dissipation or interaction with matter!
Think of QED!
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Can we drive photons into steady states with non trivial
quantum correlations?
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Two possible directions
Balancing Drive and DissipationCritical Behavior for Driven-Dissipative Quantum Systems?
Exotic Non-Equilibrium Steady States?
External Drive? Coherent vs Incoherent?
Increase Light-Matter Coupling: effective driving Counter-Rotating Terms
g r,q
Hint = HJC + ga†+ + a
‘Ultra-Strong’ Coupling Regime
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Photon Circuit QED Lattices with Rabi Non Linearity
M. Schiro, M. Bordyuh, B. Oztop and H. Tureci, Phys. Rev. Lett. 109, 053601 (2012)
M. Schiro, M. Bordyuh, B. Oztop and H. Tureci, J. Phys. B: At. Mol. Opt. Phys. 46 224021(2013)
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Ultra-Strong Coupling of Light and Matter
Ultra-Strong Coupling Regime with Superconducting Circuits
Devoret et al (2010)
T. Niemczyk et al, Nat Phys (2010)
Inductively Coupled Flux Qubit
g/r ' 0.12
Is the ‘‘standard’’ CQED picture working?S. Nigg et al, Phys. Rev. Lett 108 240502 (2012)
Many Other Platforms: Quantum Dots, 2DEG,WG Superlattices,..Todorov et al, PRL (2010), G. Scalari et al Science (2012), A.Crespi et al PRL (2012),..
Deviations from JC Physics
P. Nataf & C. Ciuti, Nat. Comm (2010), O. Viehnmann et al, PRL (2011)
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No Chemical Potential!
Light-Matter interaction fix the average number of excitations in the ground state
Can we have a non-trivial ground-state due to an ‘‘effective’’ drive?
HiRabi = r a
†iai + q
+i
i + g
a†i + ai
+i +
i
The Rabi Lattice Model
HRL =X
ij
Jij a†i aj +
X
i
HiRabi
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Insights from the Rabi Model
Z2 symmetry enough for an ‘‘exact’’ solution!
D. Braak, Phys Rev Lett 107 100401 (2011)
Symmetries: Parity † a = a
† x = x
= eiNpolZ2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
g/tr
-1
0
1
2
3
4
< N >< a >< a2 >
Finite Number of Excitations, Squeezed, Symmetric
0 2 4 6 8 10n0
0.1
0.2
0.3
0.4
0.5
P n- = | c
n- |2
g = 1.5 tr
Non Trivial Ground-state Properties = !r !q = 0
|Rabi =X
n,=±cn |n
HRabi = r a†a+ q
+ + ga† + a
+ +
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A Low Energy Doublet at Strong Coupling
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
g/tr-1
-0.5
0
0.5
1
1.5
2
Energy
W = + 1
W = < 1
W = + 1
W = < 1
Low-energy spectrum
First excited state, with opposite parity, is almost degenerate for
= !r !q = 0
Ground-state has always the same (even) parity
g r
= r e2(g/r)
2
Exponentially Small Splitting!
HRabi = r a†a+ q
+ + ga† + a
+ +
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Small Hopping: Disordered Phase, Gapped, Insulating
Large Hopping: Ordered Phase, Gapped
h aii 6= 0hx
i
i 6= 0
hx
i
i = 0 h aii = 0
Qualitative Phase Diagram in the J vs g plane:
What happens at finite (small) Hopping?
HRL =X
ij
Jij a†i aj +
X
i
HiRabi
Z2 Parity Symmetry Breaking
Quantum Phases and QPT are different from the physics of massive quantum particles on a lattice!
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Equilibrium Phase Diagram
‘’Para-electric’’ Insulator
h aii = 0
hx
i
i = 0
h aii 6= 0
0 0.2 0.4 0.6 0.8
J/ω0
0.5
1
1.5
2g
/ω0
δ = − 0.5δ = 0.0δ = 0.5δ = 1.0δ = 2.0
Negative detuning favors the ordering phase
= !q !r
Ising Universality
Z2 Parity Symmetry Breaking
‘‘Ferro-electric’’ Insulator
hx
i
i 6= 0
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Tunable Open Rabi Hubbard Model with Engineered Driving
M. Schiro, C. Joshi, M. Bordyuh, R. Fazio, J. Keeling and H. Tureci, arXiv: 1503.04456
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Driven 4-level Atom in a CavityH0 = !cav a
†a+X
j=0,1,r,s
Ej | jih j|
Hg = gr | rih 0| a+ gs | sih 1| a+ hc
Hdrive(t) =
r
2ei!p
r t | rih 1|+ s
2ei!p
s t | sih 0|+ hc
Cavity Photon mediate excited state transition
Two-pump scheme driving, Raman assisted transitions
For large detunings eliminate excited states and get an effective Hamiltonian for the 0,1 manifold
|0ih1| |1ih0| +
r/s gr/s,r/s,!pr/s
F. Dimer et al, PRA (2007) M. Baden et al, PRL (2014)
“Rotating-Frame”: time-independent Hamiltonian in a non-equilibrium, Markovian Bath
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Tunable Light-Matter Couplings
0 0.2 0.4 0.6 0.8 10.5
0.6
0.7
0.8
0.9
1
1.1
g/ω
R
0 0.2 0.4 0.6 0.8 1
Ωs/Ω
r
0
0.2
0.4
0.6
0.8
1
g’/
g
Heff = !R a†a+ !qz + g
a† + hc
+ g0
a†+ + hc
g =grr
2r
g0 =gss
2s
0 0.2 0.4 0.6 0.8 1
Ωs/Ω
r
-8
-6
-4
-2
0
δ
α = 0.010α = 0.012α = 0.013α = 0.014
Parameters of the effective model in the regime
= !q !R
g/!R g/g0 1
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Steady State Patterns? Broken Symmetry? Open Quantum Criticality?
Driven-Dissipative Cavity Array
κ κ κ κ
Pump
PumpCavity
Cavity
HRL =X
ij
Jij a†i aj +
X
i
HiRabi
@t = i[HRL, ] + L[, ] + L[a, ]Dissipative vs Unitary Dynamics
L[J, ] = 2OJ† J†J J†J Jumps Operator
(t) =Y
i
i(t) Time-Dependent Gutzwiller Ansatz
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Steady State Phase Diagram (I)
g
0
0.5
1
1.5
2
0 π/2 π
(a)Most unstable k
g
⟨σx
nσx
n+1⟩
J=0.45J=0.9
0
0.5
1
1.5
2-0.5 0 0.5
(b) ⟨σx
nσx
n+1⟩ vs g
⟨σx nσx n+
1⟩
J
g=1.5g=0.5-0.5
0
0.5
0 0.25 0.5 0.75
(c) ⟨σx
nσx
n+1⟩ vs J
⟨σx nσx n+l⟩
l
J=0.4J=0.6
J=0.8
0 4 8 12
-0.5
0
0.5(d) ⟨σ
x
nσx
n+l⟩vs l, g=1.5
g = g0
Commensurate-to-Incommensurate Order as J,g varied
Critical Hopping Jc pushed to finite value in the open case
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Steady State Phase Diagram (II)
g
J
0
1
2 0 0.25 0.5 0.75
(a) Most unstable kg'/g=0.5
J
0 0.25 0.5 0.75
0
π/2
π(b) Most unstable k
g'/g=0.25⟨σ
x nσ
x n+l⟩
g
l=1l=2
l=3l=4
-0.2
0
0.2
0 0.6 1.2 1.8
(c) ⟨σxnσ
xn+l⟩ vs g
l
g=0.16g=0.32
g=0.64g=0.80
0 4 8
(d) ⟨σxnσ
xn+l⟩ vs l
g'/g=0.25, J=0.5
g 6= g0
Upon decreasing of g’/g: suppression of order, then transition to AF ordering
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0 2 4 6 8 10
r-10
-5
0
5
J eff(r
,ω)
ω = 0.0ω = 0.5 ω
0
ω = 2ω0
ω = 4 ω0
Photon-Induced Qbit InteractionsPhotons always appear quadratically
and can be ‘‘traced’’ out
Effective exchange interaction between quits
Exchange depends from photon band-structure/
population
Short-distance component from “ferro” to “anti-ferro” as frequency increases
Sg = g
Zdt
X
i
a†i (t)
i (t) + hc
Sint
=
Zdtdt0
X
ij
x
i
(t)Jeff
ij
(t t0)x
j
(t0)
Jeffij (!) = g2
X
k
cosk (Ri Rj)! !k
(! !k)2+ 2
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From Rabi single site: low-energy doublet with a tiny splitting
Universal Behavior at the Rabi QPT
Opposite Parity: ‘‘Isospin’’
x|± = ±|±
Anisotropic XY model ! Quantum Ising Universality Class
Heff
=X
h iji
Jx x
i
x
j
+ Jy y
i
y
j
+
2
X
i
z
i
Project onto this low energy doublet:
Rabi Hamiltonian in this subspace
HiRabi !
2z
i
“Renormalized” Dissipation D[] = +L[+, ] + L[, ]
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Ferro-Antiferro-Ferro Transition
Suppression of ordering (Ferro/Normal/Antiferro): Energy Splitting
vanishes at gc
Antiferro Z2 ordering coincides with population inversion!
Infinite-MPO Simulation of Effective Dissipative Spin Model
confirms the picture!
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Summing UpDriven-Dissipative Light Matter Systems
Promising Interface Quantum Optics/Condensed Matter
Perspectives:
• Real Circuit QED architecture (flux qubits?) • Effect of Disorder
Tunable Open Rabi Hubbard Array by Engineered Driving
Platform for non-equilibrium quantum many body physics
• Coupling strongly correlated electrons to cavity fields?Y. Laplace, S. Fernandez-Pena, S. Gariglio, J.M. Triscone, A. Cavalleri, arXiv:1503.06117
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Acknowledgements
In collaboration with:
Discussions with:
Andrew Houck, D.Underwood, D. Sadri, D. Huse (Princeton) Takis Kontos(ENS), Nicolas Roch (Grenoble), K. Le Hur(Polytechnique)
Hakan Tureci, Mykola Bordyuh, Baris Oztop (Princeton)
Chatanya Joshi, Jonathan Keeling (St. Andrews)
Saro Fazio (Pisa)
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Thanks!
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Interplay of Drive and DissipationTime-Dependent Drive can be “often” gauged away by
going to a “rotating-frame”
(t) = †(t)(t)(t)
H(t) = i†(t)@t(t) †(t)H(t)(t)
@t = i[H(t), (t)]
In this rotating frame Hamiltonian becomes time-independent..
…but occupation of bath modes become non-equilibrium!
GKbath(!) = ImGR
bath(!) coth
! + drive
2
Violation of Fluctuation-Dissipation Theorem
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J=0 -- Polaritons
J<Jc -- Mott Insulator of Polaritons
J>Jc -- ‘’Superfluid’’
Hartmann et al, Greentree et al, Schmidt et al,Rossini et al, Tomadin et al,..
[HJC
, Npol
] = 0
haii 6= 0
U(1) Spontaneous Symmetry Breaking J. Koch and K. Le Hur PRA 80 023811 (2009)
Mott to Superfluid Transition of Polaritons
HJCL
=X
ij
Jij
a†i
aj
+ h.c.+
X
i
Hi
JC
µX
i
N i
pol
Jaynes-Cumming Lattice Model
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The Role of Counter-Rotating Terms: From Jaynes-Cumming to Rabi
M. Schiro et al, (in preparation)
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From Jaynes-Cumming to Rabi
Structure of the Ground State vs g0/g
Pn = | cn|2 Probabilty of having n polaritons| =
X
n,=±cn |n
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Pn-
g’/g = 0
0 2 4 6 8 100
0.2
0.4
0.6
0.8g’/g = 0.25
0 2 4 6 8 10n0
0.1
0.2
0.3
0.4
Pn-
g’/g = 1.0
0 2 4 6 8 10n0
0.1
0.2
0.3
0.4
0.5g’/g = 0.75
0 0.2 0.4 0.6 0.8 1
g’/g
0
0.5
1
1.5
2
2.5
3
3.5
4
g/ω
0
Π = +1
Π = −1
Π = +1
Π = −1
Level Crossings between different parity sectors at finite g0/g
HRabi = r a† a+ q
+ + g (a† + hc) + g(a†+ + hc)
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0 0.25 0.5 0.75 10
0.2
0.4
0.6
0.8
Jc/ω
0
g = 0.5 ω0
0 0.25 0.5 0.75 10
0.05
0.1
Jc/ω
0
g = 1.1 ω0
0 0.25 0.5 0.75 1
g’/g
1e-12
1e-10
1e-08
1e-06
0.0001
0.01
Jc/ω
0
g = 3.5 ω0
0 0.25 0.5 0.75 1
g’/g
1e-06
0.0001
0.01
Jc/ω
0
g = 2.5 ω0
Π = +1
Π = +1
Π = −1
Π = −1
Π = +1Π = −1
< a > = 0 < σx >=0
Π = +1Π = −1
Π = +1
The Fate of Mott Lobes
Lobes at fixed Parity survive at any finite but shift upg0/g 6= 1
What happens to the Jaynes-Cumming Mott Lobes as we tune g’/g?
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The Role of Counter-Rotating Terms
Effective Action Z = Tr eH =
Zi Di D
i eSeff [i ,i]
Seff [i ,i] =
Z
0d
X
h iji
i J
1ij j +
X
i
[i ,i]
“Local” Physics (Rabi Non-Linearity)
M. Fisher et al, PRB(1989) for Bose-Hubbard
Standard Field Theory of U(1) Superfluid-to-Mott Transition...
.....+ Relevant Operators (explicit breaking down to Z2)
K. Damle & S. Sachdev, PRL(96)
[,] = logh eR 0 d(() a()+a†()())i
Seff =
Z
0d
K1
+K2||2 +K3||2 + r||2 + u||4
SCRT =
Z
0d g0 (+)
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Basics of Superconducting Circuits (I)
Transmission Line Resonator
Circuit ‘‘Quantization’’
Hr =X
n
n a†n an
n = n
dplc
Superconducting-based to give dissipation-less currents
Hr =
Z d
0dx
q2(x)
2c+
2(x)
2l
q(x) =X
n
An(x)an + a
†n
(x) = iX
n
Bn(x)an a†n
M. Devoret, Les Houches (1995)
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Qubit: Mesoscopic SC system with ‘‘atom-like’’ spectrum
The Josephson Junction is the crucial (non-linear) ingredient!
Qubits come in different flavors (charge, flux, phase qubits)
Basics of Superconducting Circuits (II)
Ex:The Cooper Pair Box
HCPB = 4EC (n ng)2 EJ cos
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Ultra-Strong Coupling of Light and Matter
Ultra-Strong Coupling Regime with Superconducting Circuits
Devoret et al (2010)
How large can be the coupling between atom and photon in circuit QED?
M. Devoret, S. Girvin and R. Schoelkopf,
Ann Phys 16 767 (2007)
T. Niemczyk et al, Nat Phys (2010)
Inductively Coupled Flux Qubit
g/r ' 0.12
Is the ‘‘standard’’ CQED picture working?S. Nigg et al, Phys. Rev. Lett 108 240502 (2012)
Many Other Platforms: Quantum Dots, 2DEG,WG Superlattices,..Todorov et al, PRL (2010), G. Scalari et al Science (2012), A.Crespi et al PRL (2012),..
Deviations from JC Physics
P. Nataf & C. Ciuti, Nat. Comm (2010), O. Viehnmann et al, PRL (2011)
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Disordered Arrays of Coupled Cavities
D. Underwood, W. Shanks, J. Koch and A. Houck, PRA 86, 023837 (2012)
H =X
i
(r + i) a†i ai +
X
ij
tija†i aj + hc
tij = 2Z0 Cij (r + i) (r + j)