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Physical Review A

Nonlinear dynamics of a passive, coherently driven mesoscopic micromaser

C. Balconi, Federico Casagrande, Alessandro Cesare Mondini, W. Lange

Published Jul 1, 1996• 12 citations
Publisher

Abstract

A mesoscopic beam of two-level Rydberg atoms, prepared in the lower state, interacts with a coherently driven cavity mode. The dynamics is described by Maxwell-Bloch equations with atomic propagation. On resonance, the system was predicted to exhibit multistability, upper-branch instabilities, and self-pulsing. In the presence of atomic detuning its dynamical behavior is shown to include in addition period doubling, frequency locking, quasiperiodicity, and chaos, thus providing a remarkable example of nonlinear dynamics of a passive autonomous atomic system with propagation. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.898

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Physical Review A•Jan 1, 1989•49 citations
A mesoscopic beam of two-level Rydberg atoms, prepared in the lower state, interacts with a coherently driven cavity mode. The dynamics is described by Maxwell-Bloch equations with atomic propagation. On resonance, the system was predicted to exhibit multistability, upper-branch instabilities, and self-pulsing. In the presence of atomic detuning its dynamical behavior is shown to include in addition period doubling, frequency locking, quasiperiodicity, and chaos, thus providing a remarkable example of nonlinear dynamics of a passive autonomous atomic system with propagation. \textcopyright{} 1996 The American Physical Society.
We consider a mesoscopic maser, in which a beam of excited two-level Rydberg atoms interacts with a single radiation mode during propagation through a resonant cavity. The number of atoms in the cavity ranges from a few tens to a few hundreds. Steady-state and nonlinear dynamics are investigated both with and without a coherent driving field. At steady state, atomic cooperation can induce multistability and temporal instabilities. A novel type of bistability is predicted for the amplitude of the coherently driven cavity field as a function of atomic density. The system dynamics exhibits a variety of regimes, including chaos. We have explored them by varying the experimentally accessible parameters: atomic density, cavity linewidth and driving field amplitude.
We describe a system of N two-level Rydberg atoms which travel through a resonant cavity and interact with a single mode of the resonator. N is assumed to be on the order of a few tens or hundreds of atoms. In particular, we analyze the case where the atoms are injected to the lower state (absorber) and the system is driven by an external coherent radiation field exactly resonant with the atoms and with the cavity. In the case of a monoenergetic atomic beam, the picture that arises is substantially different from that of standard optical bistability. The switching behavior is not governed by saturation, but by the Rabi cycles of the atoms in the cavity. The hysteresis cycle of the population of the upper level is butterfly shaped. When the bistability parameter exceeds an appropriate second threshold, the system displays multistability rather than simple bistability, and under proper conditions it develops domains of instability in positive-slope regions of the steady-state curve. We analyze the effects of a velocity distribution (thermal or Gaussian) in the atomic beam, and of inhomogeneous broadening caused by stray electric fields. Finally, we discuss the feasibility of an experimental observation of the effects predicted by our analysis.
The interaction of a mesoscopic system of two-level Rydberg atoms with a coherently driven single cavity mode is treated outside the rotating wave approximation (RWA). This yields a system of non-autonomous Maxwell–Bloch equations that can be solved using Fourier decomposition of both field and atomic variables up to first harmonics and to O ( Q −1 ), where Q is the cavity quality factor. For the absorber and amplifier regimes, the first harmonic component of the output field (outside the RWA) exhibits closed hysteresis loops compared with hysteresis cycles for the non-oscillatory (fundamental) component within the RWA. A cooperative switching diagram (output field versus cooperative atomic parameter) outside the RWA shows overlapping branches in the resonant case and closed loops in the non-resonant case.
The steady state behaviour describing the interaction of a mesoscopic system of coherently injected two-level Rydberg atoms with a coherently driven single-mode cavity is investigated in both resonant and non-resonant regimes. Multiple-switching effects between the output field states are predicted with the atomic coherent excitation (θ) and relative coherent phase parameter (Φ r ) of the atomic coherent state . The relation between the transmitted field ( x ) and the atomic cooperation parameter ( C ) shows multistable behaviour for some values of the polar and azimuthal angles (θ,ϕ). The transverse field feature in the form of a Gaussian beam profile tends to degrade the multistable structure in the ( C – x ) and the input–output field ( x – Y ) relations. Micromaser action is analysed for the system with atomic coherence within and without the plane wave approximation.
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 compare theoretical predictions and experimental data for the multimode instability of optical bistability, which arises from the coherent dynamics of a system of two-level molecules contained in an optical cavity and driven by an external stationary laser field. Considerable insight into this phenomenon is provided by two ``rules of thumb'' that govern the behavior of the self-pulsing frequency, and establish its relation with the Rabi frequency of the intracavity field. We describe in detail the experimental apparatus and the main features of the experimental findings. The theoretical results are based on the Maxwell-Bloch equations for a ring cavity in the plane-wave approximation. Despite the crudeness of this model, the numerical data display a satisfactory qualitative agreement with the experimental results. Additional period-doubling and chaotic phenomena, predicted by the theory and not observed in the experiment, are presumably washed out by the longitudinal and radial variation of the electric field in the cavity.