The dynamics of a discretized atom-field interaction model with a physically relevant form factor is analyzed. It is shown that after some short time interval only a small fraction of eigenvalues and eigenstates (belonging to the close vicinity of the excited atomic state energy E = ω0/2) contributes to the nondecay probability amplitudes in the long-time regime, whereas the contribution of all other eigenstates and eigenvalues is negligible. Nevertheless, to describe correctly the non-Markovian dynamics in the short-time regime the contribution of all eigenstates and eigenvalues must be taken into account.
The dynamics of a discretized atom-field interaction model with a physically relevant form factor is analyzed. It is shown that after some short time interval only a small fraction of eigenvalues and eigenstates (belonging to the close vicinity of the excited atomic state energy E = ω0/2) contributes to the nondecay probability amplitudes in the long-time regime, whereas the contribution of all other eigenstates and eigenvalues is negligible. Nevertheless, to describe correctly the non-Markovian dynamics in the short-time regime the contribution of all eigenstates and eigenvalues must be taken into account.
We examine the dynamics of bipartite quantum entanglement between a two-level atom and the electromagnetic field. We begin with the Jaynes-Cummings model with a single field mode and examine in detail the exact time evolution of entanglement, including cases where the atomic state is initially mixed and the atomic transition is detuned from resonance. We then explore the effects of other nearby modes by calculating the exact time evolution of entanglement in more complex systems with two, three, and five field modes. For these cases we can obtain exact solutions which include the strong-coupling regime. Finally, we consider the entanglement of a two-level atom with the infinite collection of modes present in the intracavity field of a Fabre-Perot cavity. In contrast to the usual treatment of atom-field interactions with a continuum of modes using the Born-Markov approximation, our treatment in all cases describes the full non-Markovian dynamics of the atomic subsystem under general couplings. Only when an analytic expression for the infinite mode case is desired do we need to make a weak coupling assumption which at late times approximates Markovian dynamics.
We perform a first-principles derivation of the general master equation to study the non-Markovian dynamics of a two-level atom (2LA) interacting with an electromagnetic field (EMF). We use the influence functional method, which can incorporate the full back reaction of the field on the atom, while adopting Grassmannian variables for the 2LA and the coherent-state representation for the EMF. We find exact master equations for the cases of a free quantum field and a cavity field in the vacuum. In response to the search for mechanisms to preserve maximal coherence in quantum computations in ion trap prototypes, we apply these equations to analyze the decoherence of a 2LA in an EMF, and find that decoherence time is close to relaxation time. This is at variance with the claims by authors who studied the same system but used a different coupling model. We explain the source of difference and argue that, contrary to common belief, the EMF, when resonantly coupled to an atom, does not decohere it as efficiently as a bath does on a quantum Brownian particle. The master equations for non-Markovian dynamics derived here are expected to be useful for exploring new regimes of 2LAEMF interaction, which is becoming physically important experimentally.
The nonlinear-oscillating system in action-angle variables is characterized by the dependence of frequency of oscillation $\ensuremath{\omega}(I)$ on action I. Periodic perturbation is capable of realizing in the system a stable nonlinear resonance at which the action I adapts to the resonance condition $\ensuremath{\omega}({I}_{0})\ensuremath{\simeq}\ensuremath{\omega}$, that is, ``sticking'' in the resonance frequency. For a particular physical problem there may be a case when $I\ensuremath{\gg}\ensuremath{\hbar}$ is the classical quantity, whereas its correction $\ensuremath{\Delta}I\ensuremath{\simeq}\ensuremath{\hbar}$ is the quantum quantity. Naturally, dynamics of $\ensuremath{\Delta}I$ is described by the quantum equation of motion. In particular, in the moderate nonlinearity approximation $\ensuremath{\varepsilon}\ensuremath{\ll}(d\ensuremath{\omega}/dI)(I/\ensuremath{\omega})\ensuremath{\ll}1/\ensuremath{\varepsilon}$, where $\ensuremath{\varepsilon}$ is the small parameter, the description of quantum state is reduced to the solution of the Mathieu-Schr\"odinger equation. The state formed as a result of sticking in resonance is an eigenstate of the operator $\ensuremath{\Delta}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{I}$ that does not commute with the Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$. Expanding the eigenstate wave functions in Hamiltonian eigenfunctions, one can obtain a probability distribution of energy level population. Thus, an inverse level population for times lower than the relaxation time can be obtained.
In a previous paper we developed a method, Filter-Diagonalization, for extracting eigenvalues and eigenstates of a given operator at any desired energy range. In essence, the method eliminates correlation between distant eigenstates through a short-time filter while correlations between closely lying states are eliminated by diagonalization. Here we extend Filter-Diagonalization. When used to extract eigenvalues for a given operator H, we show that all eigenvalue information is directly extracted from a short segment of the correlation function C(t)=(ψ(0)‖e−iHt‖ψ(0)), or alternately from a small number of residues (ψ(0)‖Rn(H)‖ψ(0)), where ψ(0) is a random initial function and Rn(H) is any desired polynomial expansion in H. The implications of this feature are twofold. First, in contrast to the previous version the wave packet needs only to be propagated once (to prepare C(t)), and eigenstates at all desired energy windows can then be extracted with negligible extra computation time (and negligible storage requirements). In a simulation presented here, accurate eigenvalues are extracted using propagation times which are only a 0.0041 fraction of the ‘‘natural’’ time, i.e., the time by which the relative phase of the two closest eigenstates reaches 2π. The second and more important feature is that the method is automatically suitable for extracting eigenvalues (or normal modes) using a short-time segment of any signal C(t) which is a sum of (unknown) Fourier components (C(t)=∑ndne−iεnt) regardless of its origin. In addition to its use for determining eigenvalues of known operators, the method may also be utilized to extract normal modes from classical-dynamics simulations, eigenstates from real-time Quantum Monte-Carlo studies, frequencies from experimental optical or electrical signals, or be utilized in any other circumstance where a correlation function or general signal is only known for short times (or expensive to generate at long times).
A model of a coupled atom---quantum-field system is presented, consisting of a time-independent Hamiltonian and a realistic dynamical initial condition. The Hamiltonian describes a fixed bound atom coupled to the complete electromagnetic field. The initial condition is a consistent quantum-mechanical description of an atom in the exact ground state of the coupled atom-field system, on which is superimposed an approaching quasiclassical excitation of the field which will cause the interesting dynamical processes later on. Tools are presented for the manipulation of this model, with emphasis towards series expansions of all relevant quantities suitable for small strength $\ensuremath{\lambda}$ of the atom-field coupling and (potentially) large magnitude $\ensuremath{\alpha}$ of the coherent field excitation, such that the "influence of the coherent excitation on the atom" (i.e., the product $\ensuremath{\lambda}\ensuremath{\alpha}$) remains constant as $\ensuremath{\lambda}$ tends to zero. The following results are obtained, at the lowest nontrivial order in $\ensuremath{\lambda}$, in an approximation suitable for times much shorter than any radiative lifetime of the atom: (a) The change in free-field energy due to the interaction with the atom is exactly compensated at all times by the change in bare-atom energy plus the atom-field coupling energy. This checks the consistency of the proposed scheme. (b) The frequency distribution of this change in free-field energy has a (somewhat unexpected) component, with a smooth spectrum, spreading over the whole width of the spectrum of the exciting pulse, in addition to $\ensuremath{\delta}$ and dispersive singularities centered on the atomic transition frequencies. This frequency distribution is discussed in detail and its features are illustrated by a number of figures, in a simple case, numerically tractable. (c) The heterodyne detection of the radiation emitted by the atom is investigated, with emphasis on the issue of causality and propagation of light. Strict causality is obtained, without further approximation, in a model with an atom-field coupling involving the field at a single point and a detector similarly sensitive to properties of the field at a single point. This result breaks down if one uses nonlocal approximations such as the rotating-wave approximation or a detection observable ${\mathbf{E}}^{\ensuremath{-}}(\mathbf{r})\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{E}}^{+}(\mathbf{r})$. Finally, an appendix is devoted to the use, in quantum mechanics, of bases moving with respect to each other.