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