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Non-equilibrium dynamics in the Dicke model

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

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Non-equilibrium dynamics in the Dicke model

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  1. Non-equilibriumdynamicsintheDickemodel Izabella Lovas Supervisor: Balázs Dóra Budapest University of Technology and Economics 2012.11.07.

  2. Outline • Rabi model • Jaynes-Cummingsmodel • Dickemodel • Thermodynamic limit • Quantumphasetransition • Normal and super-radiantphase • Experimentalrealization • General formula forthecharacteristicfunction of work • Specialcases-Suddenquench-Linearquench

  3. The Rabi model f bozonicfield interactionbetween a bosonicfield and a singletwo-level atom energies of theatomicstates vacuum Rabi frequency transitionoperators betweenatomicstates j and i

  4. The Jaynes-Cummingsmodel rotating-waveapproximation: areneglected conservation of excitation: is exactlysolvable: infiniteset of uncoupledtwo-stateSchrödinger equations for n excitations: basisstates iftheinitialstate is a basisstate, wegetsinusoidalchangesin populations: Rabi oscillations

  5. The Dickemodel N atoms bosonicfield generalization of the Rabi model: N atoms, singlemodefield collectiveatomic operators -levelsystem pseudospinvectorof length

  6. Thermodynamic limit QPT atcriticalcouplingstrength normalphase super-radiantphase super-radiant photonnumber atomicinversion super-radiant normal normal parameters: photonnumber atomicinversion

  7. Thermodynamic limit Holstein-Primakoffrepresentation: Normalphase: twocoupledharmonicoscillators real parity operator: groundstate has positiveparity

  8. Super-radiantphase macroscopicoccupation of thefield and theatomicensemble or lineartermsinthe Hamiltonian disappear where meanphotonnumber: globalsymmetry becomesbroken newlocal symmetries:

  9. Phasetransition ground-stateenergy parameters: second-orderphase transition criticalexponents: photonnumbergrowslinearlynear meanfieldexponents

  10. Experimentalrealization spontaneoussymmetry-breaking atcriticalpumppower evensites oddsites • constructiveinterference • increasedphotonnumberinthecavity K. Baumann, et al.Nature 464, 1301 (2010)

  11. Experimentalresults The relativephase of thepump and cavityfielddependsonthe population of sublattices:

  12. Statistics of work Definition: P(W) eigenvalue of eigenvalue of difference of final and initialground-stateenergies probabilitydensityfunction: Fourier-transform characteristicfunction: Jarzynski-inequality Tasaki-Crooksrelation appearsinfluctuation relations: M. Campisi, et al.Rev. Mod. Phys.83, 771 (2011)

  13. Determination of G(u) forthenormalphase effective Hamiltonian: diagonalizationwithBogoliubov-transformation: eigenfrequencies: protocol: the Hamiltonian containsonlythefollowingterms:

  14. Determination of G(u) forthenormalphase Heisenberg equation of motion: differentialequationsforthecoefficientswithinitialconditions where can be expressedinterms of

  15. The characteristicfunction nthcumulant of thedistribution cumulantexpansion: expectedvalue: variance: inverseFourier-transform simplespecialcase: adiabaticprocess final and initialgroundstateenergies

  16. Suddenquench position of peaks: parameters:

  17. Linearquench diabaticregime characteristictimescales transitionbetween adiabatic and diabatic limit adiabaticregime diabatic limit: sudden quench adiabatic limit: consists of a single Dirac-delta

  18. Small far from approximate formula forthesolution of thedifferentialequation cumulantexpansion nthcumulant, expectedvalue, variance adiabatic limit: approximate formula approximate formula numericalresult numericalresult

  19. Summary • Quantum-opticalmodels:-Rabimodel-Jaynes-Cummingsmodel • Dickemodel-Quantumphasetransition-Normal and super-radiantphase-Experimentalrealization • Statistics of work • Characteristicfunctionforthenormalphase • Specialcases-Suddenquench-Linearquench

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