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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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Presentation on theme: "Non-equilibrium dynamics in the Dicke model Izabella Lovas Supervisor: Balázs Dóra Budapest University of Technology and Economics 2012.11.07."— Presentation transcript:

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

2 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

3 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

4 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

5 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

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

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

8 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:

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

10 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)

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

12 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

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

14 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

15 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

16 Sudden quench position of peaks: parameters:

17 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

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

19 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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