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Non-equilibrium dynamics in the Dicke model
Izabella Lovas
Supervisor: Balázs Dóra
Budapest University of Technology and Economics2012.11.07.
Outline
•Rabi model•Jaynes-Cummings model•Dicke model•Thermodynamic limit•Quantum phase transition•Normal and super-radiant phase•Experimental realization
•General formula for the characteristic function of work•Special cases -Sudden quench -Linear quench
† †1 11 2 22 12 212
H a a E S E S a a S S
The Rabi model
fbozonic field
interaction between a bosonic field and a single two-level atom
:iE energies of the atomic states
: vacuum Rabi frequency
:ijStransition operators between atomic states j and i
The Jaynes-Cummings model
rotating-wave approximation:†21 12,a S aS are neglected
† †1 11 2 22 12 212JCH a a E S E S a S aS
conservation of excitation: † 22a a S
JCH is exactly solvable:infinite set of uncoupled two-state Schrödinger equations
2 10
, ,22
n E EH n n n
n
for n excitations: 1 2, 1n n basis states
if the initial state is a basis state, we get sinusoidal changes inpopulations: Rabi oscillations
The Dicke model
bosonic field N atoms
generalization of the Rabi model: N atoms, single mode field
( ) ( )
1 1
,N N
i iz z
i i
J S J S
collective atomic operators
† †0 zH J a a a a J J
N
1N -level system
pseudospin vector of length / 2j N
Thermodynamic limitQPT at critical coupling strength 0 / 2c
0 1, 0.5c /zJ j
normal phase super-radiant phase
ph
oto
n n
um
ber
ato
mic
in
vers
ion
normalnormal
super-radiant
super-radiant
photon number
atomic inversion
parameters:
:c :c
† /a a j
Thermodynamic limit
Holstein-Primakoff representation:
† † † †2 , 2 , zJ b j b b J j b b b J b b j
†, 1b b
Normal phase:
† † † †0 0H b b a a a a b b j
two coupled harmonic oscillators
22 2 2 2 2 20 0 0
116
2
real 0 / 2 c † †i a a b b
e
parity operator: , 0H
ground state has positive parity
Super-radiant phase
macroscopic occupation of the field and the atomic ensemble
† † † †,a c A b d B † † † †,a c A b d B or
linear terms in the Hamiltonian disappear
221 , 1
2
jA B j
where
2
2c
22 22 2 2 2 20 0
02 2
14
2
mean photon number: † 2 ( )a a A O j
global symmetry becomes broken
new local symmetries: † †
(2) i c c d de
Phase transition
parameters:
0 1, 0.5c
second-order phasetransition
0 :E ground-state energy
critical exponents: 0
photon number grows linearly nearc1
2cA
11, 3
2 mean field exponents
Experimental realization
even sites
odd sites
spontaneous symmetry-breakingat critical pump power crP
•constructive interference•increased photon number in the cavityK. Baumann, et al. Nature 464, 1301 (2010)
Experimental results
The relative phase of the pump and cavity field depends on thepopulation of sublattices:
Statistics of work
Definition: 0W E E
:f iE E difference of final and initial ground-state energies
probability density function: 0|
,m n m n
n m
P W W E E p Fourier-transform characteristic function:
0HiuH iuHiuWG u e P W dW e e
P(W
)
f iW E E
i ground state
M. Campisi, et al. Rev. Mod. Phys. 83, 771 (2011)
:E eigenvalue of H 0 :E eigenvalue of 0H
P W appears in fluctuation relations:Jarzynski-inequalityTasaki-Crooks relation
Determination of G(u) for the normal phase
effective Hamiltonian:
† † † †0 0H b b a a a a b b j
diagonalization with Bogoliubov-transformation:† †
0
0
cosh sinh , tanh2 2
a b a bc r r r
eigenfrequencies: 00
21
protocol: t t the Hamiltonian contains only the following terms:
2 2† † † 2 † 2, , , , ,c c c c c c c c
Determination of G(u) for the normal phase
Heisenberg equation of motion:
2 2 †r rc t i t e c t i t e c t
differential equations for the coefficients with initial conditions
†0 0c t t c t c
0 1, 0 0 2( ) ,
ui
G u e G u G u
where
1
cos sin
G ui t
t u t ut
t can be expressed in terms of ,t t
The characteristic function
1
ln!
n
nn
iuG u
n
cumulant expansion: :n nth cumulant of the distributionexpected value: 1
1
2E W t t
variance: 2 2 2 2 22
1
2D W t t t t
1
2iuWP W e G u du
inverse Fourier-transform
simple special case: adiabatic process
,f iiu E E
f iG u e P W W E E
, :f iE Efinal and initial ground state energies
Sudden quench
: 0 1, 0 0
0
position of peaks:
2 2k l
,k l
parameters:
0 1, 0,
0.495
1.41
0.1
Linear quenchch
ara
cteri
stic
tim
esc
ale
s
adiabatic regime
dia
bati
c re
gim
e
tt
transition between adiabatic and diabaticlimit
0 diabatic limit: sudden quench
adiabatic limit: P Wconsists of a single Dirac-delta
Small far from c,
cumulant expansion nth cumulant, expected value, variance
approximate formula for the solution of the differential equation
adiabatic limit: 1 , 0 2f i nE E n
0 1, 0.3, 0.005
approximate formula approximate formulanumerical result numerical result
Summary
•Quantum-optical models: -Rabi model -Jaynes-Cummings model•Dicke model -Quantum phase transition -Normal and super-radiant phase -Experimental realization•Statistics of work•Characteristic function for the normal phase•Special cases -Sudden quench -Linear quench