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Non-equilibrium dynamics in the Dicke model Izabella Lovas Supervisor: Balázs Dóra Budapest University of Technology and Economics 2012.11.07.

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Outline Rabi model Jaynes-Cummings model Dicke model Thermodynamic limit Quantum phase transition Normal and super-radiant phase Experimental realization General formula for the characteristic function of work Special cases -Sudden quench -Linear quench

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The Rabi model f bozonic field interaction between a bosonic field and a single two-level atom energies of the atomic states vacuum Rabi frequency transition operators between atomic states j and i

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The Jaynes-Cummings model rotating-wave approximation: are neglected conservation of excitation: is exactly solvable:infinite set of uncoupled two-state Schrödinger equations for n excitations:basis states if the initial state is a basis state, we get sinusoidal changes in populations: Rabi oscillations

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The Dicke model bosonic field N atoms generalization of the Rabi model: N atoms, single mode field collective atomic operators -level system pseudospin vector of length

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Thermodynamic limit QPT at critical coupling strength normal phasesuper-radiant phase photon number atomic inversion normal super-radiant photon number atomic inversion parameters:

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Thermodynamic limit Holstein-Primakoff representation: Normal phase: two coupled harmonic oscillators real parity operator: ground state has positive parity

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Super-radiant phase macroscopic occupation of the field and the atomic ensemble or linear terms in the Hamiltonian disappear where mean photon number: global symmetrybecomes broken new local symmetries:

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Phase transition parameters: second-order phase transition ground-state energy critical exponents: photon number grows linearly near mean field exponents

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Experimental realization even sites odd sites spontaneous symmetry-breaking at critical pump power constructive interference increased photon number in the cavity K. Baumann, et al. Nature 464, 1301 (2010)

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Experimental results The relative phase of the pump and cavity field depends on the population of sublattices:

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Statistics of work Definition: difference of final and initial ground-state energies probability density function: Fourier-transformcharacteristic function: P(W) M. Campisi, et al. Rev. Mod. Phys. 83, 771 (2011) eigenvalue of appears in fluctuation relations: Jarzynski-inequality Tasaki-Crooks relation

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Determination of G(u) for the normal phase effective Hamiltonian: diagonalization with Bogoliubov-transformation: eigenfrequencies: protocol: the Hamiltonian contains only the following terms:

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Determination of G(u) for the normal phase Heisenberg equation of motion: differential equations for the coefficients with initial conditions where can be expressed in terms of

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The characteristic function cumulant expansion: nth cumulant of the distribution expected value: variance: inverse Fourier-transform simple special case: adiabatic process final and initial ground state energies

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Sudden quench position of peaks: parameters:

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Linear quench characteristic timescales adiabatic regime diabatic regime transition between adiabatic and diabatic limit diabatic limit: suddenquench adiabatic limit: consists of a single Dirac-delta

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Small far from cumulant expansionnth cumulant, expected value, variance approximate formula for the solution of the differential equation adiabatic limit: approximate formula numerical result

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Summary Quantum-optical models: -Rabi model -Jaynes-Cummings model Dicke model -Quantum phase transition -Normal and super-radiant phase -Experimental realization Statistics of work Characteristic function for the normal phase Special cases -Sudden quench -Linear quench

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