non-equilibrium dynamics in the dicke model izabella lovas supervisor: balázs dóra budapest...
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![Page 1: Non-equilibrium dynamics in the Dicke model Izabella Lovas Supervisor: Balázs Dóra Budapest University of Technology and Economics 2012.11.07](https://reader035.vdocuments.site/reader035/viewer/2022062713/56649cee5503460f949bc3c8/html5/thumbnails/1.jpg)
Non-equilibrium dynamics in the Dicke model
Izabella Lovas
Supervisor: Balázs Dóra
Budapest University of Technology and Economics2012.11.07.
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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
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† †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
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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
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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
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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
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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
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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
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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
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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)
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Experimental results
The relative phase of the pump and cavity field depends on thepopulation of sublattices:
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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
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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
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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
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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
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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
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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
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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
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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