Transcript Slide 1

Pontificia Universidad Católica de Chile
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Entanglement and
quantum phase
transitions in the Dicke
model
dicke model
Vladimír Bužek
SLOVAK ACADEMY
OF SCIENCE
Miguel Orszag
PONT.UNIV.CATO
LICA DE CHILE
Marián Roško
SLOVAK ACADEMY
OF SCIENCE
52 years of Dicke model
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COHERENCE IN SPONTANEOUS RADIATION
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Interaction between quantum objects lead to correlations
that have no classical analogue. These purely quantum
Correlations, known as entanglement, play a fundamental
role in modern physics and have already found
their applications in quantum information processing
And communications.
-(Criptography with EPR correlations,Eckert)
-(Nielsen and Chuang, Quantum Computation and
Quantum communication(Cambridge U.Press,2000)
Also, quantum Systems in a pure state tend to exhibit more
Pronounced entanglement between their constituents
than statistical mixtures.
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I am presenting here the study of the ground state of
the Dicke Model.
The Dicke Model was introduced by him, describing
the interaction of one mode of the radiation field
with a collection of two level atoms.
It is a well known radiation-matter interaction model.
and it triggered numerous investigatons of various
Physical effects described by the model.
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He described how a collection of atoms prepared in
a certain initial state could decay “COLLECTIVELY”
Like a hughe dipole, with the emission of radiation not
proportionally to N, as one would suspect from
Independent radiators, but to N^2.
This radiation pulses proportional to the square of the
Number of atoms were demonstrated experimentally
In the 80’s by various groups.
Also in the 70’s people started talking about a
phase transition between a “normal” and a
“Superradiant state”.(Hepp,Lieb;Narducci,et al)
This turned out a more controversial subject
(Wodkiewicz et al)
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The Dicke Hamiltonian is derived from the well known
Radiation matter interaction:

e
 1
2
H   F a a   j 1 
( p j  A(rj ))  V (rj )
c
 2m


a, a 
V (rj )
N
Are the annihilation and
creation oper For the field
Binding potential, including
longitudinal
Components of the field
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assumptions
Dipole approximation
A^2 term negligible
Resonance between atom and field
RWA
The model
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 Hamiltonian
H
A
2
N
N
  F a a    (e
j 1
z
j
†
j 1
irj k j
a  e
†

j
 irj k j
a j )
interaction
Different point of view
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Eigensystem of the Hamiltonian
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Integral of motion P
 [ P, H ]  0
N
P  a† a    j  j
j 1
- total excitation number
Subspace of p excitations spanned by p+1 vectors:
g N
p , {e, g ( N 1) } p  1 , {e2 g ( N  2) } p  2 ,...,
{e k , g ( N  k ) } p  k ,..., {e p , g  ( N  p ) } 0
Solutions
GROUND STATE
One excitation
No excitation
eigenstate
energy
E (0)  g  N 0
N
E ( )   
2
(0)
eigenstate
E  A0 g  N 1  A1 {e, g ( N 1) } 0
H E E E
 2 N
 2 
U 
  N


energy
2 N
E ( ) 
  N
2
(1)
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E
(1)

UA  EA

 N 

2 N 

2

1

g  N 1  {e, g ( N 1) } 0
2

Solutions
GROUND STATE
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Two excitations
eigenstate
energy
E (2) 
E (2) ( ) 
N
1
N 1
g N 2 
{e, g ( N 1) } 1 
{e2 , g ( N 2) } 0
4N  2
4N  2
2
4 N
   2(2 N  1)
2
Arbitrary (p) number of excitations
(p+1)x(p+1)
matrix
 (2 p  N ) / 2

 pN
0
0


(2 p  N ) / 2
 ( p  1)2( N  1)
0
  pN



0
 ( p  1)2( N  1)
(2 p  N ) / 2
 ( p  2)3( N  2)


U





0
 2( p  1)( N  p  2)
(2 p  N ) / 2
 p( N  p  1) 



0
0
 p( N  p  1)
(2 p  N ) / 2 

Quantum phase transitions
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Transition points
1st
2nd
 / 
 / 
1
N
1
4N  2  N
3rd
 / 
4th
 / 
1
5( N  1)  (4 N  5) 2  8 N  4 N  2
1
10 N  15  3 17  12 N  4 N 2  5( N  1)  (4 N  5) 2  8 N
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N

Low en p=0
2
2 N
E1 
   N , Low en p=1
2
4 N
E2 
   2( 2 N  1) Low en p=2
2
As we increase k, the lowest energy has one excitation…
E0  
E1  E 0



1
N
1st phase transition
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Total ground state energy as a function of a scaled coupling
Constant for 12 atoms. We see explicitly 12 quantum phase
transitions.
Entanglement
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TTwo particles of spin 1/2
Density matrix
Pauli matrix
 AB
*
 AB  ( y   y )  AB
( y   y )
M   AB  AB
M   m
'
j
'
j
i   

' 2
i
 0 i 
y 

i
0


Concurrence – measure of entanglement
CAB  max{1  2  3  4 ,0}
Entanglement in the Dicke model
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No excitation
GS
Procedure
1. Trace over
bosonic field
2.Trace over
remaining N-2
particles
3.Calculate
concurrence
Product state => C = 0
One excitation
Reduced matrix
 N 1
m,n  
 N
g 2
g 2 
Concurrence
C (1) 
1
N
1

{e, g} {e, g} 
N

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Higher Excitation
For p=2, for example we get:
N
N  2 ( N  2)(N  3)


4N  2
2N
N (4 N  2)
1
2( N  2)


2 N N (4 N  2)
2
 
N (4 N  2)

 mn

0

0

0
0


0
0
 0 
 0

0 
0
C has 12 diff regions,
The largest one corresponds to p=1
For large  ,C is not cero
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N=12
P=
Boundaries
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“total atomic
entanglement”
N ( N  1)
A  C
2
2
N
p
Total atomic Bi-partite concurrence of the ground state,
as a Function of N and p
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The largest entanglement
Corresponds to p=1.
Even for large coupling
P=N, arbitrary pairs of
Atoms are still entangled
(diagonal line)
Field-atom entanglement I
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((p)-excitation Eigenstate
GS
E ( p )  A0 g  N
p  A1 {e, g ( N 1) } p  1  A2 {e2 g ( N  2) } p  2  ... 
 Ap k {e k , g ( N  k ) } p  k  ...  Ap {e p , g ( N  p ) } 0
Field matrix (fock states)
 F  A0
 Ap  k
p
2
2
p p  A1
pk
p  1 p  1  A2
p  k  ...  Ap
 kj p  j p  j
j 0
2
2
0 0 
2
p  2 p  2  ... 
Field-atom entanglement II
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Entropy
S p   p j Log ( p j )
j
Maximal entropy of
p+1 dimensional
Hilbert space:
Smax  Log ( p  1)
Pj is the probability for example of j photons
Red:field entropy ;
system
E
N
T
R
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O
P
Y
Reflec
E
N
T
A
N
G
L
E
M
E
N
T
blue:maximal entropy for p+1 dim
CKW inequalities
COFFMAN,KUNDU,WOOTERS
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For p=1 the GS of the
Dicke Model
saturates
The CKW
Inequalities
NO MULTIPARTITE
ENTANGLEMENT
 Inequality
N

k 1; k  j
C 2j , k  C 2j , j

Where the sum on the left hand side is taken over
all qubits except for the qubit j and C J , J denotes the tangle
Between j and the rest of the system

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For a pair of qubits A and B
C AB  2 det  A
 A  TrB  AB
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If we assume that the qubit j represents the field mode,
Then we can find the tangle between the field and the
System of atoms(for p=1 is a qubit)
C j , j  4det  F  1
C

j,
j
 4Det(  F )  1
And while the concurrences between each atom and
The field is
C
1

N
1
Cjj ,,kk
N
Notice the CKW inequality becomes an equality in this case
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From the analysis it follows that for the ground state
of the Dicke Model and p=1, the
Coffman-Kundu_Wooters is saturated(equality), which
Proves that the atom-Field interaction, as described
By the D.Model with small coupling (coupling
Constant between k1 and k2), does induce only
bi-partide entanglement and doesnot result in
Multipartide quantum correlations.
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For the moment, it is impossible to make the analysis
And generalize this result for p>1, that is for a qudit
(field mode for p>1) and a set of qubits(atoms).
No generalization of the CKW inequalities are known.
Photon statistics
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P=N
Distribution
Peak: 2/3N
5N
Dispersion:  
26
Entropy:
1
1
S p  Log[ N  1]  Smax
2
2
In the high kappa limit, the photon number distribution is peaked
At 2N/3=n. The distribution Pn is sharply peaked (sub Poissonian)
NON RESONANT CASE
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Here we assume that the field frequency  Fis different from the
atomic one  A .We define    F   A ,     
2
The eigensystem is modified
F
A
Energy versus coupling constant for different
Excitation numbers.
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Colours correspond to diff.number of excitations.
The energy is not linear with coupling con
The first derivative not continuous
Phase Diagram of the GS energy for various
excitation numbers(colours) versus detuning and
coupling constant
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

N=5
-2
p=0 yellow
P=1 green
P=2 light blue
P=3 blue
P=4 pink
P=5 red
0
2
0

1
2
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Entanglement Phase Transition for the case  A  F
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Entangle
ment
bigger
than in
resonant
case.
Steps
bigger
and
variable
with
coupling
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C
0.
0
0
0
1

2

FINITE TEMPERATURE EFFECTS
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Until now,all this work was done at T=0.
We put now the system in contact with a reservoir, at
Temperature T, but kT small.
Concurrence is a smooth function
Of kappa and T, except for very near kT=0
Where the steps are noticeable.
Also, as temperature increases,
The entanglement between the
atoms decreases
FINITE TEMPERATURE EFFECTS….
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Ground State Energy
around the first phase
transition. Only near
kT=0, the slope
changes
discontinuously.
In the rest of the
parameter space, E is a
smooth function of
kappa and kT
Conclusion
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Phase transitions in Dicke model
Entanglement
Strong coupling limit
Detuning, Finite kT
Dicke
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" I have long believed that an experimentalist should not be
unduely inhibited by theoretical untidyness. If he insists on
having every last theoretical t crossed before he starts his
research the chances are that he will never do a significant
experiment. And the more significant and fundamental the
experiment the more theoretical uncertainty may be tolerated.
By contrast, the more important and difficult the experiment the
more that experimental care is warranted. There is no point in
attempting a half-hearted experiment with an inadequate
apparatus."
References
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•Dicke,R.H. Coherrence is spontaneous process, Phys. Rev.
93,99 (1954)
•Tavis, M. and Cummings, F.W. Exact Solution for an Nmolecule-radiation-field Hamiltonian, Phys. Rev. 170, 379
Approximate solutions for an N-molecule-radiation field
Hamiltonian, Phys. Rev. 188, 692 (1969)
•Narducci, L.M., Orszag M. and Tuft, R. A. On the ground state
instability of the Dicke Hamiltonian. Collective Phenomena 1,
113, (1973)
•Narducci, L.M., Orszag M. and Tuft, R. A. Energy spectrum of
the Dicke Hamiltonian. Phys. Rev. A 8, 1892 (1973)
•Hepp, K. and Lieb, E. On the superradiant phase transitions for
molecules in a quantized radiation field: the Dicke maser model.
Ann. Phys. (NY) 76, 360 (1973)
•Koashi, M., Bužek, V. and Imoto, N., Entangled webs: Tight
bounds for symmetric sharing of entanglement. Phys. Rev. A
62, 05030 (2000)
Experimental References …
Experiments
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1.N.Skribanowitz,I.P.Herman,J.C.McGillivray,M.Feld,Phys.Rev.Lett,30,
309(1973),”Observation of Dicke Superradiance in Optically Pumped
HF Gas.
2.M.Gross,C.Fabre,P.Pillet,S.Haroche,Phys.Rev.Lett,36,1035(1976),”O
bservation of Near Infrarred Dicke Superradiance on Cascading
transition in Atomic Sodium”
3.I.Kaluzni,P.Goy,M.Gross,J.M.Raymond,S.Haroche,
Phys.Rev.Lett,51,1175(1983),”Observation of Self-Induced Rabi
Oscillations in Two-Level atoms excited inside a resonant cavity:the
ringing regime of Superradiance”
4.D.J.Heinzen,J.E.Thomas,M.S.Feld,Phys.Rev.Lett,54,677(1985),
“Coherent ringing in Superfluorescence”
5.C.Greiner,B.Boggs,T.W.Mossberg,Phys.Rev.Lett,85,3793(2000)”Sup
erradiant Dynamics in an optically thin material…”
6.E.M.Chudnovsky,D.A.Garanin,Phys.Rev.Lett,89,157201(2002)
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V.Buzek,M.Orszag,M.Rosko,PRL(PRL,94(2005)
V.Buzek,M.Orszag,M.Orszag,PRA,(in press)
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