Exact Solution of Damped Cummings Tavis Model and Entanglement

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Exact Solution of Damped Cummings - Tavis Model
and Entanglement

• A.V. Gorokhov,
• I.E. Sinaysky
• Samara State University
• gorokhov_at_ssu.samara.ru

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Contents

• Introduction
• Rydberg atoms and one-atom maser experiments
• Jaynes Cummings model
• Exact solution for JCM in nonideal cavity
• Two atoms in cavity with damping and entanglement
  (generalized Cummings Tavis model)
• Summary and outlook

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Introduction
It is considered the dynamics of two identical
two-level atoms in non-ideal cavity. Using
approach developed before by us for case of one
two-level atom, the analytical expression for
density matrix of the system is presented.
Dynamics of level populations, mean number of
photons and correlation function for photons are
calculated. Using Peres - Horodecki and Wootters
criteria exact expression for entanglement
dynamics of two identical two-level atoms in
cavity is given. Influence of detuning and
damping constants are investigated.
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Rydberg atoms and one-atom maser experiments
Herbert Walther (1935-2006)
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• Rabi oscillations between atomic levels
• Cummings collapse and revivals

The one atom maser setup view (Max Planck
Institute for Quantum Optics)
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E.T. Jaynes F.W. Cummings, "Comparison of
Quantum and Semiclassical Radiation Theories with
application to the Beam Maser", IEEE, 51, 89
(1963).
Frederick W. CummingsProfessor of Physics
(Emeritus), Berkeley, Ca.,
Edwin T. Jaynes, 1922-1998
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Jaynes Cummings and Cummings Tavis models
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Rabi oscillations, collapses and revivals
coherent state for photon mode
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JCM with photon damping as one-atom maser theory
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Exact solution for JCM in nonideal cavity
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Time dependence of atomic population inversion.
Initial photon mode prepared in coherent state
with
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Time dependence of mean photon numbers in cavity
with initial 10 photon state.
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Mandel Q-factor time dependence for different
initial states of photon mode
(3) shows
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Two atoms in cavity with damping and
entanglement(generalized Cummings Tavis model)
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Master equation for density matrix
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Initial conditions
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Peres-Horodecki criterion of entanglement
The Peres-Horodecki matrix
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Entanglement
Wootters concurrence measures degree
of entanglement
Where are the
eigenvalues of the matrix
Here denotes the complex conjugate of
and is the Pauli matrix expressed in the same
basis as
W. K. Wootters, PRL, 80, 2245 (1997)
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Dynamics of entanglement parameters
Concurrence and negativity dynamics for
different initial states of atomic subsystem and
photon CS state.
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Dynamics of entanglement parameters
Concurrence and negativity dynamics for
different initial states of atomic subsystem and
photon CS state.
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Dynamics of entanglement parameters
Concurrence and negativity dynamics for
different initial states of atomic subsystem and
photon CS state.
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Dynamics of entanglement parameters
Concurrence and negativity dynamics for
different initial states of atomic subsystem and
photon CS state.
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Summary and outlook

• We have found that time dependencies of Peres
  Horodecki and Wootters parameters are very
  similar.
• Essential dependencies on initial field state,
  damping constant and atomic distance have been
  observed.
• Non-trivial behavior of negativity dynamics for
  initial exited atoms is observed only for initial
  coherent photon states. In cases of initial
  thermal and n-photon states the previously exited
  subsystem of two atoms is separable (non
  entangled) for all studied values of the
  parameters.
• We are planning to take into account the
  dipole-dipole interaction between atoms and
  counter-rotating terms in interaction Hamiltonian
  for strong value of atom field coupling
  constant g.

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