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Pontificia Universidad Católica de Chile
Entanglement and quantum phase transitions in the Dicke model
dicke model
Vladimír Bužek Miguel Orszag Marián RoškoSLOVAK ACADEMYOF SCIENCE
SLOVAK ACADEMY OF SCIENCE
PONT.UNIV.CATOLICA DE CHILE
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52 years of Dicke model
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 entanglemententanglement, play a fundamental role in modern physics and have already found
their applications in quantum information processingAnd 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 morePronounced 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 notproportionally to N, as one would suspect from
Independent radiators, but to N^2.This radiation pulses proportional to the square of theNumber 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:
)(
,
)())((21
12
j
N
j jjjF
rV
aa
rVrAcep
maaH
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
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The model
Hamiltonian† †
1 1
( )2
j j j jN N
ir k ir kzAj F j j
j j
H a a e a e a
interaction
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Different point of viewgrenoble06
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- total excitation number
Eigensystem of the Hamiltonian
[ , ] 0P H Integral of motion P
Subspace of p excitations spanned by p+1 vectors:
†
1
N
j jj
P a a
( 1) 2 ( 2)
( ) ( )
, { , } 1 , { } 2 ,...,
{ , } ,..., { , } 0
N N N
k N k p N p
g p e g p e g p
e g p k e g
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eigenstate
energy
eigenstate
energy
Solutions
No excitation One excitation
(0) 0NE g
(0) ( )2NE
( 1)0 11 { , } 0N NE A g A e g
H E E E UA EA
22
22
N NU
NN
(1) ( 1)1 1 { , } 02
N NE g e g (1) 2( )
2NE N
GROUND STATEgrenoble06
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energy
eigenstate
(p+1)x(p+1) matrix
Solutions
Two excitations
Arbitrary (p) number of excitations
(2) ( 1) 2 ( 2)1 12 { , } 1 { , } 04 2 4 22
N N NN NE g e g e gN N
(2) 4( ) 2(2 1)2NE N
(2 ) / 2 0 0
(2 ) / 2 ( 1)2( 1) 0
0 ( 1)2( 1) (2 ) / 2 ( 2)3( 2)
0 2( 1)( 2) (2 ) / 2 ( 1)
0 0 ( 1) (2 ) / 2
p N pN
pN p N p N
p N p N p NU
p N p p N p N p
p N p p N
GROUND STATE grenoble06
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1st
3rd
2nd
4th
Quantum phase transitions
Transition points
1/N
1/4 2N N
2
1/5( 1) (4 5) 8 4 2N N N N
2 2
1/10 15 3 17 12 4 5( 1) (4 5) 8N N N N N N
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N
EE
NNE
NNE
NE
1
)12(22
4
,2
22
01
2
1
0
As we increase k, the lowest energy has one excitation…
Low en p=0
Low en p=1
Low en p=2
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.
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Density matrix
Entanglement
Concurrence – measure of entanglement
TTwo particles of spin 1/2
AB ABM
2'i i
AB
00y
ii
*( ) ( )AB y y AB y y
1 2 3 4max{ ,0}ABC
Pauli matrix
' 'j jM m
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Reduced matrix
Concurrence
Product state => C = 0
Entanglement in the Dicke model
Procedure
1. Trace over bosonic field
2.Trace over remaining N-2 particles
3.Calculate concurrence
No excitation
One excitation
2 2,
1 1 { , } { , }m nN g g e g e gN N
(1) 1CN
GS
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)24(2
)24()2(2
21
)24()3)(2(
22
24
NN
NNN
N
NNNN
NN
NN
Higher Excitation
For p=2, for example we get:
0000000000
mn
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CP=
C has 12 diff regions,The largest one corresponds to p=1
For large ,C is not cero
N=12
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Boundaries
“total atomic entanglement”
2 ( 1)2A
N NC
N p
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The largest entanglementCorresponds to p=1.
Even for large couplingP=N, arbitrary pairs of
Atoms are still entangled(diagonal line)
Total atomic Bi-partite concurrence of the ground state, as a Function of N and p
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Field-atom entanglement I((p)-excitation Eigenstate
Field matrix (fock states)
( ) ( 1) 2 ( 2)0 1 2
( ) ( )
{ , } 1 { } 2 ...
{ , } ... { , } 0
p N N N
k N k p N pp k p
E A g p A e g p A e g p
A e g p k A e g
2 2 20 1 2
2 2
0
1 1 2 2 ...
... 0 0
F
p k p
p
jj
A p p A p p A p p
A p k p k A
k p j p j
GS
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Maximal entropy of p+1 dimensional Hilbert space:
Field-atom entanglement II
Entropy
( )p j jj
S p Log p
max ( 1)S Log p
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Pj is the probability for example of j photons
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Red:field entropy ;
ENTROPYReflectsENTANGLEMENT
systemINSET:FIELD ENTROPY as a func. of N for p=N
blue:maximal entropy for p+1 dim
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CKW inequalities
2 2, ,
1;
N
j k j jk k j
C C
Inequality
COFFMAN,KUNDU,WOOTERS
For p=1 the GS of theDicke Model
saturatesThe CKW
InequalitiesNO MULTIPARTITEENTANGLEMENT
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Where the sum on the left hand side is taken over all qubits except for the qubit j and denotes the tangle
Between j and the rest of the systemJJ
C,
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For a pair of qubits A and B
ABBA
AAB
Tr
C
det2
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, 4det 1Fj jC
,1
j kCN
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)
And while the concurrences between each atom andThe field is
Notice the CKW inequality becomes an equality in this case
1)(4,
Fj
Detj
C
NC kj
1,
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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 (couplingConstant 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 analysisAnd 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.
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Peak: 2/3N
Dispersion:
Entropy:
Photon statistics
Distribution
526N
max1 1[ 1]2 2pS Log N S
In the high kappa limit, the photon number distribution is peakedAt 2N/3=n. The distribution Pn is sharply peaked (sub Poissonian)
P=N
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NON RESONANT CASE
Here we assume that the field frequency is different from the atomic one .We define
FA
AFAF
,
2
The eigensystem is modified
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Colours correspond to diff.number of excitations.The energy is not linear with coupling con
The first derivative not continuous
Energy versus coupling constant for different Excitation numbers.
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-2
0
20 21
Phase Diagram of the GS energy for various excitation numbers(colours) versus detuning and coupling constant
p=0 yellowP=1 greenP=2 light blueP=3 blueP=4 pinkP=5 red
N=5
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Entanglement Phase Transition for the case FA
Entanglement bigger than in resonant case. Steps bigger and variable with coupling
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1
C
0.
00
2
0
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FINITE TEMPERATURE EFFECTS
Concurrence is a smooth functionOf kappa and T, except for very near kT=0Where the steps are noticeable.Also, as temperature increases,The entanglement between the atoms decreases
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.
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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
FINITE TEMPERATURE EFFECTS….
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Conclusion
Phase transitions in Dicke model
Entanglement
Strong coupling limit
Detuning, Finite kT
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Dicke
" 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."
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References
•Dicke,R.H. Coherrence is spontaneous process, Phys. Rev. 93,99 (1954)•Tavis, M. and Cummings, F.W. Exact Solution for an N-molecule-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)
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Experimental References …Experiments1.N.Skribanowitz,I.P.Herman,J.C.McGillivray,M.Feld,Phys.Rev.Lett,30,3
09(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),”Observation 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)”Superradiant 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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