Transcript 1. dia - Budapest University of Technology and Economics

Non-equilibrium dynamics
in the Dicke model
Izabella Lovas
Supervisor: Balázs Dóra
Budapest University of Technology and Economics
2012.11.07.
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
The Rabi model
f field
bozonic
interaction between a bosonic field and a single two-level atom
H   a † a  E1S11  E2 S 22    a †  a   S12  S 21 
2
Ei : energies of the atomic states
 : vacuum Rabi frequency
Sij : transition operators between atomic states j and i
The Jaynes-Cummings model
†
a
rotating-wave approximation: S21 , aS12 are neglected
H JC   a † a  E1S11  E2 S22    a † S12  aS 21 
2
†
conservation of excitation: a a  S22
H JC is exactly solvable: infinite set of uncoupled two-state
Schrödinger equations
for n excitations: n
1, n 1  2 basis states
  n 
 0
E2  E1
H n  
  ,   n   n
,  
2    n  2 
if the initial state is a basis state, we get sinusoidal changes in
populations: Rabi oscillations
The Dicke model
N atoms
bosonic field
generalization of the Rabi model: N atoms, single mode field
N
J    S(i ) ,
i 1
N
J z   S z(i )
collective atomic operators
i 1
H  0 J z   a a 
†

a

N
†
 a  J  J 
N  1 -level system
pseudospin vector of length
j N /2
Thermodynamic limit
c  0 / 2
  c : super-radiant phase
photon number
  c :
normal phase
super-radiant
normal
†

a
a / j
photon number
atomic inversion  J z / j
atomic inversion
QPT at critical coupling strength
super-radiant
normal
parameters:
0    1, c  0.5
Thermodynamic limit
b, b†   1
Holstein-Primakoff representation:
J   b † 2 j  b †b ,
J   2 j  b †b b ,
J z  b†b  j
Normal phase:
H  0b b   a a    a  a  b  b   j0
†
†
†
†
two coupled harmonic oscillators
1 2
     02 
2
2


real

e
i a† a b†b


2

2 2
0


 16 0 

2
  0 / 2  c
parity operator:
ground state has positive parity
, H   0
Super-radiant phase
macroscopic occupation of the field and the atomic ensemble
a†  c†  A, b†  d †  B
or
a†  c†  A, b†  d †  B
linear terms in the Hamiltonian disappear
A
2

j
2
1


, B


2
j 1   
where


2
2 2
 2 0 
1  2 0
2
2 2 
    2     2   4 0

2

 



†
2
mean photon number: a a  A  O( j )
global symmetry

becomes broken
new local symmetries:

(2)

e
i c†c  d †d

c2
 2

Phase transition
E0 : ground-state energy
parameters:
0    1, c  0.5
second-order phase
transition
critical exponents:
 0
photon number grows linearly near
c
A
1
     1,   3 mean field exponents
2
  c
1
2
Experimental realization
even sites
spontaneous symmetry-breaking
at Pcr critical pump power
odd sites
K. Baumann, et al. Nature 464, 1301 (2010)
•constructive interference
•increased photon number
in the cavity
Experimental results
The relative phase of the pump and cavity field depends on the
population of sublattices:
Statistics of work
E : eigenvalue of H    
P(W)
i  ground state
Definition: W  E  E 0
E 0 : eigenvalue of H    0
W   E f  Ei 
E f  Ei : difference of final and initial ground-state energies



0
probability density function: P W     W  Em  En
n ,m
Fourier-transform
G u    e
iuW
 p
m|n
characteristic function:
P W  dW  e
iuHH
eiuH0
P W  appears in fluctuation relations:
M. Campisi, et al. Rev. Mod. Phys. 83, 771 (2011)
Jarzynski-inequality
Tasaki-Crooks relation
Determination of G(u) for the normal phase
effective Hamiltonian:
H  0  b†b  a † a     a †  a  b†  b   j0
diagonalization with Bogoliubov-transformation:
ab
a †  b†
c  cosh r
 sinh r
,
2
2
2
eigenfrequencies:   0 1 
 t      t 
  0
tanh r 
  0
0
protocol:
the Hamiltonian contains only the following terms:
c c , c c , c
†
 
†
 

† 2

, c , c
2


† 2

, c2
Determination of G(u) for the normal phase
Heisenberg equation of motion:
c  t   i        t  e
2 r
 c t 

i  t  e
2 r
c t 
†

c t    t  c  0   t  c†  0
differential equations for the coefficients with initial conditions
  0  1,   0  0
i
G(u)  e
G  u  
u
      
2
G  u  G  u  , where
1
i   t 
cos     t   u 
sin     t   u
    t  


   t  can be expressed in terms of  t  ,  t 


The characteristic function
cumulant expansion:
iu 

ln G  u     n
n!

n
 n : nth cumulant
of the distribution
n 1
1
expected value: E W   1      t      t            
2
1 2
2
variance: D W    2     t    2  t   2    t    2    t  
2

1
P W  
2

 iuW
e
 G u  du inverse Fourier-transform

simple special case: adiabatic process
G u   e

iu E f  Ei


, P W    W   E f  Ei 

E f , Ei : final and initial ground state energies

Sudden quench
        :
  0  1,   0  0
0
        
   
position of peaks:
2k    2l  
k, l 
parameters:
0    1,   0,
   0.495
      1.41
      0.1
diabatic regime
characteristic timescales
Linear quench
adiabatic regime
  t     
t

transition between
adiabatic and diabatic
limit
  0 diabatic limit: sudden      quench
  adiabatic limit: P W  consists of a single Dirac-delta
Small  , far from c
approximate formula for the solution of the differential equation
cumulant expansion
 
adiabatic limit:
nth cumulant, expected value, variance
1  E f  Ei , n  0 n  2
0  1,   0.3,   0.005
approximate formula
numerical result
approximate formula
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