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arXiv:1107.5846 · American Journal of Physics

Cavity quantum electrodynamics of a two-level atom with modulated fields

Uday Pisipati, Ismail M. Almakrami, Amitabh Joshi, Juan D. Serna

Published Jun 18, 2012• 6 citations
arXivHTMLPDF

Abstract

We present a numerical study of the interaction of a two-level atom with a frequency-modulated cavity mode described by the Jaynes-Cummings Hamiltonian. In particular, we explore how the population of the atomic levels changes for monochromatic and bichromatic modulations of the cavity field. The primary goal is to find sets of boundary conditions for which the evolution of the system exhibits nonlinear dynamic behavior, by solving the equations of motion of the atomic system using embedded and semi-implicit numerical algorithms. We find that the system shows quasiperiodicity when the monochromatic frequencies or ratio of the bichromatic frequencies of the field are irrational (incommensurate) numbers. In addition, when the modulated frequencies are overtones of the Rabi frequency a nearly single dominant frequency becomes apparent for the evolution of the atomic level population of the system.

DOI
10.1119/1.3703016

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We present a numerical study of the interaction of a two-level atom with a frequency-modulated cavity mode described by the Jaynes-Cummings Hamiltonian. In particular, we explore how the population of the atomic levels changes for monochromatic and bichromatic modulations of the cavity field. The primary goal is to find sets of boundary conditions for which the evolution of the system exhibits nonlinear dynamic behavior, by solving the equations of motion of the atomic system using embedded and semi-implicit numerical algorithms. We find that the system shows quasiperiodicity when the monochromatic frequencies or ratio of the bichromatic frequencies of the field are irrational (incommensurate) numbers. In addition, when the modulated frequencies are overtones of the Rabi frequency a nearly single dominant frequency becomes apparent for the evolution of the atomic level population of the system.
We study the nonlinear dynamics of the interaction of two-level atoms and a selected mode of a high- Q cavity with frequency modulation analytically and numerically. In the absence of modulation, the corresponding semiclassical Heisenberg equations for the expectation values of the collective atomic observables and the field-mode amplitudes allow, in the rotating wave approximation and in the strong-coupling limit, an exact solution with arbitrary detuning. Using this solution, we detect the coherent effect of trapping of the population of atomic levels and of trapping of the number of photons in the cavity. The explanation for this effect lies in the destructive interference of the atomic dipoles and the field mode. The integrable version of the system of equations exhibits a separatrix near which a stochastic layer is formed when modulation is introduced. The width of the layer is found to gradually increase with degree of modulation, and finally it fills the entire energy-permissible volume of the phase space. We show that the rotating wave approximation does not hinder the formation of Hamiltonian chaos in cavity semiclassical electrodynamics. The calculation of the maximum Lyapunov indices of nonlinear (in this approximation) equations of motion as functions of the modulation frequency δ and the frequency of natural Rabi oscillations of the atom-field system, Ω, suggests that Hamiltonian chaos appears first in the area of the fundamental parametric resonance, δ /2Ω≃1. Parametric instability increases with increasing modulation and decreasing detuning from the atom-field resonance, generating at exact resonance new areas of chaos corresponding to multiple parametric resonances. The results of numerical experiments and estimates of the characteristic parameters show that Rydberg atoms placed in a high- Q microwave cavity are possible objects for observing parametric instability and dynamical chaos.
We extend the standard Jaynes-Cummings model to include transient effects due to a modulation of the atom-field coupling. The exact solution for such a generalized model is found in terms of Bessel functions for the atom-cavity mode interaction modulated by the exponential function. This function is incorporated into the basic Hamiltonian phenomenologically for modelling the processes of adiabatic switching on or switching off the interaction of a two-level atom (at rest) with a quantized mode. The dynamical behaviour of the atomic inversion for such processes is plotted and compared with the atomic dynamics for the standard model. If we take into account that, in micromaser experiments, Rydberg atoms move through a cavity with a spatially inhomogenous field, we deal with the Jaynes-Cummings model with a spatial modulation the form of which is defined by the mode-shape function depending, in turn, on the geometrical form of resonator. In the cases of a rectangular and a spherical cavity the analytical solutions for atomic inversion are found in the explicit form in the weak-coupling limit. We consider in detail the time evolution of the atomic inversion while the atom passes through a rectangular cavity. Contrary to the standard model the initial inversion can be exactly restored during the atomic transit depending on the relation between the detuning, the atomic velocity and the number of half-wavelengths of the chosen mode. The numerical solutions of the basic equations are also obtained and plotted for rectangular and spherical resonators.
Using the Jaynes-Cummings model,the dynamical properties of the fidelity of quantum states in the system that consists of a two-level atom interacting with a single-mode field are studied.The frequency of the field varying with time in the form of sine is considered.The results obtained by numerical calculations show that the time-evolution behavior of the fidelities of field and system are not influenced by the frequency variation of field,but the time-evolution behavior of the fidelity of atom is modulated by the frequency variation of field.The larger the amplitude of the field frequency variance u is,the stronger modulation interaction of the field frequency variance is.When the amplitude u is larger than a fixed value,the time-evolution behavior of the fidelity of atom will display periodic oscillation behavior.Its oscillation period is in agreement with the period of frequency varying.
We study the optical bistability and dynamics in a hybrid optomechanical system, where the cavity mode is coupled to the mechanical resonator via radiation pressure and to a two-level atom via the Jaynes–Cummings coupling. The nonlinearity due to optomechanical coupling and the addition of an atom leads to bistable behaviors of the steady-state intracavity photon number and atomic population inversion. We show that the optical bistability can be effectively controlled by the Rabi frequency of the atomic driving field, the cavity–atom coupling strength, and the amplitude and frequency of the cavity pump field. Moreover, the effects of different initial conditions of the cavity field and the amplitude of the cavity pump field on the dynamics of the system also have been discussed.
Using the multiphoton Jaynes-Cummings model,the dynamical properties of the fidelity of quantum states in the system that consists of a two-level atom interacting with a single-mode field via multiphoton transition are studied.The frequency of the field varying with time in the form of sine is considered.The results obtained by numerical calculations show that the time-evolution behavior of the fidelities of field and system is not influenced by the frequency variation of field,but the time-evolution behavior of the fidelity of atom is modulated by the frequency variation of field.The larger the amplitude of the field frequency variance u is,the stronger modulation interaction of the field frequency variance is.When the amplitude u is larger than a fixed value,the time-evolution behavior of the fidelity of atom will display quasiperiodic oscillation behavior.Its oscillation period is in agreement with the period of frequency varying.The initial state of atom influences the time-evolution behavior of the fidelity of atom.
The interactions between a two-level atom and a coherent field with a time-varying frequency in a Kerr-like medium have been investigated. It is found that the average of the atomic population inversion is deformed as the frequency changing with time in sine form. It is interesting that when the amplitude of field modulation and third-order nonlinear susceptibility are adjusted to certain values, the collapse–revival phenomena of population inversion in the standard Jaynes–Cumming model disappear and are replaced by a nearly Rabi-oscillation-like evolution. In other words, population inversion in the initial coherent field behaves like that in the initial Fock state. This provides a new way to manipulate the interaction between the atom and the field, and has potential applications in quantum information and quantum communication.