Pontificia Universidad Católica de Chile

1. VladimírBužek Miguel Orszag MariánRoško Pontificia Universidad Católica de Chile grenoble06 Entanglement and quantum phase transitions in the Dicke model dicke model SLOVAK ACADEMY OF SCIENCE SLOVAK ACADEMY OF SCIENCE PONT.UNIV.CATO LICA DE CHILE

2. 52 years of Dicke model grenoble06 COHERENCE IN SPONTANEOUS RADIATION

3. 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.

4. 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.

6. grenoble06 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)

7. grenoble06 The Dicke Hamiltonian is derived from the well known Radiation matter interaction: Are the annihilation and creation oper For the field Binding potential, including longitudinal Components of the field

8. assumptions Dipole approximation A^2 term negligible Resonance between atom and field RWA

9. interaction The model grenoble06 • Hamiltonian

10. Different point of view grenoble06

11. - total excitation number Eigensystem of the Hamiltonian grenoble06 • Integral of motion P • Subspaceofp excitationsspannedbyp+1vectors:

12. eigenstate eigenstate energy energy Solutions grenoble06 GROUND STATE • One excitation • No excitation

13. eigenstate energy (p+1)x(p+1) matrix Solutions grenoble06 GROUND STATE • Two excitations • Arbitrary (p) number of excitations

14. Quantum phase transitions 1st 2nd 3rd 4th grenoble06 • Transition points

15. grenoble06 Low en p=0 Low en p=1 Low en p=2 As we increase k, the lowest energy has one excitation… 1st phase transition

16. Total ground state energy as a function of a scaled coupling Constant for 12 atoms. We see explicitly 12 quantum phase transitions.

17. Density matrix Entanglement grenoble06 • TTwo particles of spin 1/2 • Pauli matrix • Concurrence – measure of entanglement

18. Product state => C = 0 Reduced matrix Concurrence Entanglement in the Dicke model grenoble06 • No excitation GS • Procedure • Trace over bosonic field • 2.Trace over remaining N-2 particles • 3.Calculate concurrence • One excitation

19. Higher Excitation For p=2, for example we get:

20. C has 12 diff regions, The largest one corresponds to p=1 For large ,C is not cero grenoble06 N=12 C P=

21. Boundaries “total atomic entanglement” grenoble06 p N

22. Total atomic Bi-partite concurrence of the ground state, as a Function of N and p grenoble06 The largest entanglement Corresponds to p=1. Even for large coupling P=N, arbitrary pairs of Atoms are still entangled (diagonal line)

23. Field-atom entanglement I • ((p)-excitation Eigenstate grenoble06 GS • Field matrix (fock states)

24. Maximal entropy of p+1 dimensional Hilbert space: Field-atom entanglement II grenoble06 • Entropy Pj is the probability for example of j photons

25. E N TROPYReflects ENTANGLEMENT Red:field entropy ; blue:maximal entropy for p+1 dim system INSET:FIELD ENTROPY as a func. of N for p=N

26. CKWinequalities COFFMAN,KUNDU,WOOTERS grenoble06 For p=1 the GS of the Dicke Model saturates The CKW Inequalities NO MULTIPARTITE ENTANGLEMENT • Inequality 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 system

27. For a pair of qubits A and B

28. 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 and The field is Notice the CKW inequality becomes an equality in this case

29. 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.

30. 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.

31. Peak: 2/3N Dispersion: Entropy: Photon statistics grenoble06 • Distribution P=N In the high kappa limit, the photon number distribution is peaked At 2N/3=n. The distribution Pn is sharply peaked (sub Poissonian)

32. NON RESONANT CASE grenoble06 Here we assume that the field frequency is different from the atomic one .We define The eigensystem is modified

33. Energy versus coupling constant for different Excitation numbers. grenoble06 Colours correspond to diff.number of excitations. The energy is not linear with coupling con The first derivative not continuous

34. Phase Diagram of the GS energy for various excitation numbers(colours) versus detuning and coupling constant grenoble06 N=5 -2 0 p=0 yellow P=1 green P=2 light blue P=3 blue P=4 pink P=5 red 2 0 1 2

35. grenoble06

36. Entanglement Phase Transition for the case grenoble06 Entanglement bigger than in resonant case. Steps bigger and variable with coupling

37. grenoble06 C 0. 0 0 0 1 2

38. FINITE TEMPERATURE EFFECTS 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

39. FINITE TEMPERATURE EFFECTS…. 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

40. Phase transitions in Dicke model • Entanglement • Strongcoupling limit • Detuning, Finite kT Conclusion grenoble06

41. Dicke grenoble06 " 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."

42. References grenoble06 • 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 Phenomena1, 113, (1973) • Narducci, L.M., Orszag M. and Tuft, R. A. Energy spectrum of the Dicke Hamiltonian. Phys. Rev.A8, 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. A62, 05030 (2000)

43. Experimental References … grenoble06 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)

44. grenoble06 V.Buzek,M.Orszag,M.Rosko,PRL(PRL,94(2005) V.Buzek,M.Orszag,M.Orszag,PRA,(in press)

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