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ICSSUR 05 Pontificia Universidad Católica de Chile Entanglement and quantum phase transitions in the Dicke model dicke model Vladimír Bužek Miguel Orszag Marián Roško SLOVAK ACADEMY OF SCIENCE SLOVAK ACADEMY OF SCIENCE PONT.UNIV.CATO LICA DE CHILE grenoble06

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Vladimír Bužek. Miguel Orszag. Mari án Roško. Pontificia Universidad Católica de Chile. grenoble06. E ntanglement and q uantum p hase t ransitions in the Dicke model. dicke model. SLOVAK ACADEMY OF SCIENCE. SLOVAK ACADEMY OF SCIENCE. PONT.UNIV.CATO LICA DE CHILE. - PowerPoint PPT Presentation

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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:

)(

,

)())((2

11

2

j

N

j jjjF

rV

aa

rVrAc

ep

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 j

N Nir k ir kzA

j F j jj j

H a a e a e a

interaction

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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 excitationOne excitation

(0) 0NE g

(0) ( )2

NE

( 1)0 11 { , } 0N NE A g A e g

H E E E UA EA

2

22

2

NN

UN

N

(1) ( 1)11 { , } 0

2N NE g e g

(1) 2( )

2

NE N

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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 { , } 0

4 2 4 22N N NN N

E g e g e gN N

(2) 4( ) 2(2 1)

2

NE 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

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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

NN

E

NN

E

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

0

0y

i

i

*( ) ( )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 n

Ng g e g e g

N N

(1) 1C

N

GS

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)24(

2

)24(

)2(2

2

1

)24(

)3)(2(

2

2

24

NN

NN

N

N

NN

NN

N

N

N

N

Higher Excitation

For p=2, for example we get:

000

00

00

000

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 2

0 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

,

1j kC

N

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

5

26

N

max

1 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 …

Experiments

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),”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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