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Communications in Theoretical Physics

Entropy and Entanglement in Master Equation of Effective Hamiltonian for Jaynes–Cummings Model

H. A. Hessian, F.A Mohammed, A.‐B. A. Mohamed

Published Apr 1, 2009• 0 citations
Publisher

Abstract

In this paper, we analytically solve the master equation for Jaynes–Cummings model in the dispersive regime including phase damping and the field is assumed to be initially in a superposition of coherent states. Using an established entanglement measure based on the negativity of the eigenvalues of the partially transposed density matrix we find a very strong sensitivity of the maximally generated entanglement to the amount of phase damping. Qualitatively this behavior is also reflected in alternative entanglement measures, but the quantitative agreement between different measures depends on the chosen noise model. The phase decoherence for this model results in monotonic increase in the total entropy while the atomic sub-entropy keeps its periodic behaviour without any effect.

DOI
10.1088/0253-6102/51/4/27

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In this paper, we analytically solve the master equation for Jaynes–Cummings model in the dispersive regime including phase damping and the field is assumed to be initially in a superposition of coherent states. Using an established entanglement measure based on the negativity of the eigenvalues of the partially transposed density matrix we find a very strong sensitivity of the maximally generated entanglement to the amount of phase damping. Qualitatively this behavior is also reflected in alternative entanglement measures, but the quantitative agreement between different measures depends on the chosen noise model. The phase decoherence for this model results in monotonic increase in the total entropy while the atomic sub-entropy keeps its periodic behaviour without any effect.
We study how intrinsic decoherence leads to growing entropy and a strong degradation of the maximal generated entanglement in the multiquanta Jaynes–Cummings model. We find an exact solution of the Milburn equation in multiquanta precesses and calculate the partial entropy of the particle (atom or trapped ion) and field subsystem as well as total entropy. As the total entropy is not conserved, and it is shown to increase as time develops, one cannot use the partial field or atomic entropy as a direct measure of particle–field entanglement. For a good entropy measure, we also calculate the negativity of the eigenvalues of the partially transposed density matrix. We find that, at least qualitatively, the difference of the total entropy to the sum of field and atom partial entropies can be also used as an entanglement measure. Our results show that the degree of entanglement is very sensitive to any change in the intrinsic decoherence parameter.
Abstract: We investigate the dynamics of the two-mode Jaynes–Cummings model with generalized nonlinear coupling including intrinsic noise as described by the Milburn equation. Based on a general analytic solution of the corresponding master equation, we calculate the growth of entropy in the system at the special case of two Raman–coupled long lived atomic states as well as the partial entropies of the atom and field subsystem. We see that the system is very sensitive to phase noise, if both fields are initially in a coherent state. The dynamics then rapidly leads to suppression of quantum revivals and a steady entanglement between the fields and the atom. This hints for serious limitations for the use of Raman induced transitions in quantum information processing setups. For an alternative entanglement measure we calculate the negativity of the eigenvalues of the partially transposed density matrix and compare it to the difference of the total entropy to the sum of field and atom partial entropies.
On the basis of a representation in terms of photon-number states we derive an analytically solvable set of ordinary differential equations for the matrix elements of the density operator belonging to the Jaynes-Cummings model. We allow for atomic detuning, spontaneous emission, and cavity damping, but we do not take into account the presence of thermal photons. The exact results are employed to perform a careful investigation of the evolution in time of atomic inversion and von Neumann entropy. A factorization of the initial density operator is assumed, with the privileged field mode being in a coherent state. We invoke the mathematical notion of maximum variation of a function to construct a measure for entropy fluctuations. In the undamped case the measure is found to increase during the first few revivals of Rabi oscillations. Hence, the influence of the surroundings on the atom does not decrease monotonically from time zero onwards. A further non-Markovian feature of the dynamics is given by the strong dependence of our measure on the initial atomic state, even for times at which damping brings about irreversible decay. For weak damping and high initial energy density the atomic evolution exhibits a crossover between quasireversible revival dynamics and irreversible Markovian decay. During this stage the state of maximum entropy acts as an attractor for the trajectories in atomic phase space. Subsequently, all trajectories follow a unique route to the atomic ground state, for which the off-diagonals of the atomic density matrix equal zero. From our entropy studies one learns what kind of difficulties must be overcome in establishing formulae for entropy production, the use of which is not limited to semigroup-induced dynamics.
The Jaynes-Cummings model (JCM) is the simplest fully quantum model that describes the interaction between light and matter. We extend a previous analysis by Phoenix and Knight [Ann. Phys. 186, 381 (1988)] of the JCM by considering mixed states of both the light and matter. We present examples of qualitatively different entropic correlations. In particular, we explore the regime of entropy exchange between light and matter, i.e., where the rate of change of the two are anticorrelated. This behavior contrasts with the case of pure light-matter states in which the rate of change of the two entropies are positively correlated and in fact identical. We give an analytical derivation of the anticorrelation phenomenon and discuss the regime of its validity. Finally, we show a strong correlation between the region of the Bloch sphere characterized by entropy exchange and that characterized by minimal entanglement as measured by the negative eigenvalues of the partially transposed density matrix.
In this paper, we investigate the temporal evolution of various measures of degree of entanglement and total correlation of a single-mode light field interacting with a two-level atom. The phase-damped cavity is taken into account. The effects of phase damping on a temporal of partial entropies for the atom and the field, total entropy, the entropy difference and the sum of negative eigenvalues of the partially transposed density matrix as a quantitative entanglement measure for the master equation in the dressed-state approximation are taken into account.