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

, ,22n 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 atomsgeneralization 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

phot

on n

umbe

r

atom

ic in

vers

ion

normalnormal

super-radiant

super-radiant

photon number

atomic inversion

parameters:

:c :c

† /a a j

Thermodynamic limitHolstein-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

1 162

real 0 / 2 c † †i a a b b

e

parity operator: , 0H

ground state has positive parity

Super-radiant phasemacroscopic 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

22 1 , 12jA B j

where

2

2c

22 22 2 2 2 20 0

02 2

1 42

mean photon number: † 2 ( )a a A O j

global symmetry becomes broken

new local symmetries: † †(2) i c c d d

e

Phase transition

parameters:

0 1, 0.5c

second-order phasetransition

0 :E ground-state energy

critical exponents: 0

photon number grows linearly nearc12

cA 1 1, 32

mean field exponents

Experimental realization

even sitesodd 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 workDefinition: 0W E E

:f iE E difference of final and initial ground-state energiesprobability density function: 0

|,

m n m nn 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 phaseHeisenberg 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( ) ,

uiG 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

12

E W t t

variance: 2 2 2 2 2212

D W t t t t

12

iuWP W e G u du

inverse Fourier-transform

simple special case: adiabatic process

,f iiu E Ef 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

arac

teri

stic

tim

esca

les

adiabatic regime

diab

atic

reg

ime

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