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Journal of Physics A Mathematical and General

Entropy studies for the damped and the undamped Jaynes-Cummings model

F. Farhadmotamed, A. J. van Wonderen, K. Lendi

Published Apr 17, 1998• 28 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/31/15/006

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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.
Employing projection methods from kinetic theory, we study the non-Markovian quantum evolution of a two-level atom that is coupled to a single mode of the electromagnetic field. The interaction between the atom and field mode is described by the damped Jaynes–Cummings model. The general case is considered, in which dissipation is generated by both a photonic and an atomic reservoir of finite temperature. Only one special choice is made. The frequencies of the atom and field mode are in the same ratio as the temperatures of the atomic and the photonic reservoirs. Making use of Laplace transformation, we show that the atomic density matrix evolves to the state of maximum von Neumann entropy if the time, the cube of the initial electromagnetic energy density, the inverse of the photonic damping parameter and the inverse of the atomic damping parameter tend to infinity equally fast. We propose a large class of states from which the full density operator for the atom and field may start. This class includes entangled states. Expansion of the time-dependent exponential of the Laplace backtransform enables us to derive the limit of maximum entropy directly, without explicit evaluation of the atomic density matrix. We interchange limits and sums without proof, so our derivation is not entirely rigorous. Next, we remove the photonic reservoir. Then the damping process gets a sequential character. The field mode is assumed to start from a photon-number state. For the special choice of zero temperature and detuning we verify that the limit of maximum entropy survives the qualitative modification of the model. The only consequence is a slightly different scaling between the parameters that become large. Finally, we argue that our case study could be of value in finding out to what extent quantum dissipative processes obey the general principles of thermodynamics.
Operating in Laplace language and making use of a representation based on photon-number states, we find the exact solution for the density operator that belongs to the Jaynes-Cummings model with cavity damping. The detuning parameter is set equal to zero and the optical resonator does not contain any thermal photons. It is shown that the master equation for the density operator can be replaced by two algebraic recursion relations for vectors of dimension 2 and 4. These vectors are built up from suitably chosen matrix elements of the density operator. By performing an iterative procedure, the exact solution for each matrix element is found in the form of an infinite series. We demonstrate that all series are convergent and discuss how they can be truncated when carrying out numerical work. With the help of techniques from function theory, it is proved that our solutions respect the following conditions on the density operator: conservation of trace, Hermiticity, convergence to the initial state for small times, and convergence to the ground state for large times. We compute some matrix elements of the density operator for the case of weak damping and find that their analytic structure becomes much simpler. Finally, it is shown that the exact atomic density matrix converges to the state of maximum von Neumann entropy if the time, the square of the initial electromagnetic energy density, and the inverse of the cavity-damping parameter tend to infinity equally fast. The initial condition for the atom can be chosen freely, whereas the field may start from either a coherent or a photon-number state.
By combining iterative methods with Laplace transformation, we construct the solution of a dissipative Jaynes–Cummings model. The dissipative part of the model is based on the standard Markovian master equation for a harmonic oscillator that is coupled to a heat bath of nonzero temperature. Besides photon loss, we also take into account frequency detuning between atom and field. Before commencing the iteration, we subject the matrix elements of the density operator to a transformation that depends on temperature. As a result, the pole structure of all Laplace transformed matrix elements is improved. It becomes manifest which poles do not contribute to the asymptotic behavior of the density operator. In proving that our iterative process yields convergent results, we assume upper bounds on: the matrix elements of the density operator, the matrix elements of the initial density operator, the damping parameter of the heat bath, and the temperature of the heat bath. All of these bounds are physically acceptable. The photon field may start from a coherent state or a number state. For experiments in a microwave cavity, temperatures of the order of 0.1 [K] are allowed. As an application, the evolution of the atomic density matrix is studied. We propose a limit for which this matrix converges to the state of maximum von Neumann entropy. The time, the cubed initial energy density, and the inverse of the damping parameter must tend to infinity equally fast. The photon field is assumed to be in a number state at time zero, whereas the initial state of the atom can be chosen freely.
An exact density matrix of a phase-damped Jaynes-Cummings model (JCM) with entangled Bell-like initial states formed from a model two-state atom and sets of adjacent photon number states of a single-mode radiation field is presented. The entanglement of the initial states and the subsequent time evolution is assured by finding a positive lower bound on the concurrence of local $2\ifmmode\times\else\texttimes\fi{}2$ projections of the full $2\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ JCM density matrix. It is found that the time evolution of the lower bound of the concurrence systematically captures the corresponding collapse and revival features in atomic inversion, relative entropies of atomic and radiation, mutual entropy, and quantum deficit. The atom and radiation subsystems exhibit alternating sets of collapses and revivals in a complementary fashion due to the initially mixed states of the atom and radiation employed here. This is in contrast with the result obtained when the initial state of the dissipationless system is a factored pure state of the atom and radiation, where the atomic and radiation entropies are necessarily the same. The magnitudes of the entanglement lower bound and the atomic and radiation revivals become larger as both the magnitude and phase of the Bell-like initial state contribution increase. The time evolution of the entropy difference of the total system and that of the radiation subsystem exhibit negative regions called ``supercorrelated'' states which do not appear in the atomic subsystem. Entangled initial states are found to enhance this supercorrelated feature. Finally, the effect of phase damping is to randomize both the subsystems for asymptotically long times. It may be feasible to experimentally investigate the results presented here using the Rabi oscillation methods of microwave and optical cavity quantum electrodynamics since pure photon number states have recently been produced and observed.
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.
Abstract Quantum dissipation is studied for a discrete system that linearly interacts with a reservoir of harmonic oscillators at thermal equilibrium. Initial correlations between system and reservoir are assumed to be absent. The dissipative dynamics as determined by the unitary evolution of system and reservoir is described by a Kraus map consisting of an infinite number of matrices. For all Laplace-transformed Kraus matrices exact solutions are constructed in terms of continued fractions that depend on the pair correlation functions of the reservoir. By performing factorizations in the Kraus map a perturbation theory is set up that conserves in arbitrary perturbative order both positivity and probability of the density matrix. The latter is determined by an integral equation for a bitemporal matrix and a finite hierarchy for Kraus matrices. In the lowest perturbative order this hierarchy reduces to one equation for one Kraus matrix. Its solution is given by a continued fraction of a much simpler structure as compared to the non-perturbative case. In the lowest perturbative order our non-Markovian evolution equations are applied to the damped Jaynes–Cummings model. From the solution for the atomic density matrix it is found that the atom may remain in the state of maximum entropy for a significant time span that depends on the initial energy of the radiation field.