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Physical Review A

Infinite set of relevant operators for an exact solution of the time-dependent Jaynes-Cummings Hamiltonian

J. L. Gruver, J. Aliaga, Hilda A. Cerdeira, Araceli N. Proto

Published Dec 1, 1994• 23 citations
Publisher

Abstract

The dynamics and thermodynamics of a quantum time-dependent field coupled to a two-level system, well known as the Jaynes-Cummings Hamiltonian, is studied, using the maximum entropy principle. In the framework of this approach we found three different infinite sets of relevant operators that describe the dynamics of the system for any temporal dependence. These sets of relevant operators are connected by isomorphisms, which allow us to consider the case of mixed initial conditions. A consisent set of initial conditions is established using the maximum entropy principle density operator, obtaining restrictions to the physically feasible initial conditions of the system. The behavior of the population inversion is shown for different time dependencies of the Hamiltonian and initial conditions. For the time-independent case, an explicit solution for the population inversion in terms of the relevant operators of one of the sets is given. It is also shown how the well known formulas for the population inversion are recovered for the special cases where the initial conditions correspond to a pure, coherent, and thermal field.

DOI
10.1103/physreva.50.5274

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The Relevant Operators for the generalized time-dependent m-photon Jaynes-Cummings Hamiltonian

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The m-photon Jaynes-Cummings Hamiltonian is a natural generalization of the much studied Jaynes-Cummings Hamiltonian. In this short note we give the relevant operators for the time-dependent generalized m-photon Jaynes-Cummings Hamiltonian. The dynamical equations for these operators are also given. These operators are needed and indeed are the basic building blocks for performing calculations in the context of the Maximum Entropy Formalism.

•Feb 5, 1997•1 citation
The dynamics and thermodynamics of a quantum time-dependent field coupled to a two-level system, well known as the Jaynes-Cummings Hamiltonian, is studied, using the maximum entropy principle. In the framework of this approach we found three different infinite sets of relevant operators that describe the dynamics of the system for any temporal dependence. These sets of relevant operators are connected by isomorphisms, which allow us to consider the case of mixed initial conditions. A consisent set of initial conditions is established using the maximum entropy principle density operator, obtaining restrictions to the physically feasible initial conditions of the system. The behavior of the population inversion is shown for different time dependencies of the Hamiltonian and initial conditions. For the time-independent case, an explicit solution for the population inversion in terms of the relevant operators of one of the sets is given. It is also shown how the well known formulas for the population inversion are recovered for the special cases where the initial conditions correspond to a pure, coherent, and thermal field.
Jaynes–Cummings (JC) models are by far the most widely used to study the dynamics of two-level systems, such as atoms or molecules under suitable conditions, coupled to quantized electromagnetic fields. Nowadays, analytical solutions of JC models are more useful than ever for understanding and controlling radiation-matter interactions relevant to quantum information processing, quantum computing, and other quantum technologies. This study presents the analysis and generalization of a method for the solution of the dynamical problem in the time-independent JC model (Carbonaro et al., 1979) that enables an exact description of the atom-field dynamics in the resonant time-dependent JC model and in a nontrivial class of off-resonance time-dependent JC models. Our analysis produces the ready-to-use forms of the evolution operator in the resonant and non-resonant cases, which are then used to derive the evolution of the atomic population inversion for some types of atom-field coupling. The central idea behind the proposed approach offers opportunities for its extension to other time-dependent cases and its generalization to JC-related models.
The time-evolution operator corresponding to the fractional-time Schrödinger equation is nonunitary because it fails to preserve the norm of the vector state in the course of its evolution. However, in the context of the time-dependent non-Hermitian quantum formalism applied to the time-fractional dynamics, it has been demonstrated that a unitary evolution can be achieved for a traceless two-level Hamiltonian. This is accomplished by considering a dynamical Hilbert space embedding a time-dependent metric operator concerning which the system unitarily evolves in time. This allows for a suitable description of a quantum system consistent with the standard quantum mechanical principles. In this work, we investigate the Jaynes-Cummings model in the fractional-time scenario taking into account the fractional-order parameter α and its effect in unitary quantum dynamics. We analyze the well-known dynamical properties, such as the atomic population inversion and the atom-field entanglement, when the atom starts in its excited state and the field in a coherent state.
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.
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.