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Physical Review A

Light-shift-induced chaos in a nonlinear optical resonator

C. Boden, M. Dämmig, Fedor M. Mitschke

Published May 1, 1992• 8 citations
Publisher

Abstract

The interaction of spin-1/2 atoms in an optical resonator with a near-resonant cw light field and an external static magnetic field gives rise to self-sustained spin precession. This entails a modulation on the transmitted light at roughly the Larmor frequency. We show that under suitable conditions this oscillation can become irregular. A detailed analysis reveals a deterministically chaotic process. Dimensions, entropies, and Lyapunov exponents are determined; cross-checks demonstrate their mutual consistency. To describe the observed dynamics, a three-dimensional model is presented that resembles the well-known Bloch equations. Numerical simulations compare well with the experimental results. Of all the nonlinear terms in the model, the one describing the light shift can be singled out as responsible for the chaotic behavior. This allows an intuitive understanding of the onset of chaos as a result of the interaction between spin precession and light shift.

DOI
10.1103/physreva.45.6829

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The interaction of spin-1/2 atoms in an optical resonator with a near-resonant cw light field and an external static magnetic field gives rise to self-sustained spin precession. This entails a modulation on the transmitted light at roughly the Larmor frequency. We show that under suitable conditions this oscillation can become irregular. A detailed analysis reveals a deterministically chaotic process. Dimensions, entropies, and Lyapunov exponents are determined; cross-checks demonstrate their mutual consistency. To describe the observed dynamics, a three-dimensional model is presented that resembles the well-known Bloch equations. Numerical simulations compare well with the experimental results. Of all the nonlinear terms in the model, the one describing the light shift can be singled out as responsible for the chaotic behavior. This allows an intuitive understanding of the onset of chaos as a result of the interaction between spin precession and light shift.
We describe a simple nonlinear optical device which transforms an ingoing cw light beam into a periodically modulated one. The mechanism of operation is due to a spin precession in the ground state of optically pumped atoms with a J=(1/2)\ensuremath{\rightarrow}J'=(1/2) transition in the presence of a static transverse magnetic field. With optical feedback from a resonator, this magnetically induced spin precession can be self-sustained and then can give rise to a self-pulsing of the transmitted light at roughly the Larmor frequency. A detailed theoretical description of the system is presented and stability criteria are considered. In contrast to earlier work, our calculations take into account the resonator round-trip time, an optical detuning from the atomic resonance, and absorptive losses within the resonator. New signal features are predicted: These include a complicated structure of the initial transient as well as a precipitation to a stationary state. Our theoretical model is confirmed by measurements which are performed by means of a Fabry-Perot resonator containing sodium vapor. The behavior of the device is studied for a wide range of experimental parameters; threshold powers for oscillation (\ensuremath{\gtrsim}5 mW) and the oscillation frequency and its tuning range (140 kHz--13 MHz) are investigated as well as the dynamics of the system following a step input of light.
The dynamical behavior of a Fabry-Perot cavity filled with atoms having Zeeman sub-levels in the ground state is investigated. Such a system is known to exhibit optical tristability, where the atomic spins are oriented parallel or antiparallel to the optical axis even for the linearly polarized incident light. It is found that, when a static magnetic field is applied transversely to the optical axis, self-sustained precession of the spin polarization occurs. Correspondingly, the ${\ensuremath{\sigma}}_{+}$ and ${\ensuremath{\sigma}}_{\ensuremath{-}}$ components of the transmitted light are modulated at about the Larmor frequency. We also show that the Bloch equation describing the motion of the spin polarization in the cavity can be reduced to van der Pol's equation.
Magnetically induced optical self-pulsing has been observed in a Fabry-Perot resonator containing sodium vapor. Experiments are performed on optically pumped Zeeman sub-levels of the Na ground state in the presence of a static transverse magnetic field BBB. When BBB exceeds a critical value Bcr{B}_{\mathrm{cr}}Bcr​, self-sustained spin precession occurs, and the transmitted light is found to be modulated at about the Larmor frequency. The use of this tunable current-controlled oscillator in optical FM signal transmission is demonstrated.
The nonlinear dynamics of a mechanical resonator in an optomechanical system with linear, quadratic and cubic photon-vibration interactions (with respect to mechanical displacements) in a modulated driving field under conditions of adiabatic elimination of the optical field is studied. Based on the constructed bifurcation diagrams of the mechanical coordinate and the largest Lyapunov exponent as a function of the modulation amplitude, as well as power spectra, phase portraits and Poincare sections, regions of regular and chaotic dynamics of the optomechanical system are identified. It is also shown that for a certain modulation amplitude in the presence of all three types of interactions, chaotic dynamics of the mechanical resonator (oscillator) is realized, which is replaced by quasi-periodic oscillations in the absence of cubic interaction, and the system returns to chaotic behavior if only linear interaction remains. This non-monotonic dependence of chaotic dynamics on the order of nonlinearity originates from the interplay between parametric driving and effective potential reshaping and manifests that nonlinearity does not always enhance chaos. For an optomechanical system in a membrane-in-the-middle configuration, where only quadratic photon-vibration interaction is present, it is demonstrated that at small modulation amplitudes the mechanical oscillator exhibits quasi-periodic motion in each of the wells of a symmetric two-minimum potential, whereas large modulation amplitudes lead to chaotic motion, involving interwell transitions.
The dynamical behavior of counterpropagating light waves interacting with a nonlinear medium is studied numerically. It is found that the wave and the medium form a system that becomes unstable under certain conditions and exhibits self-oscillation. Because the interacting medium is modeled by an ensemble of two-level systems, we recognize the self-oscillation frequency as the Rabi-frequency of the constituent systems. The physical mechanisms responsible for this self-oscillation are the gain and the distributed feedback of the combined lightwave-medium system. When the environment changes, in particular when the intensity is increased, this system becomes more unstable and we find that the oscillation exhibits more complex behavior, including quasi-periodic motion and chaos. We analyze the output fields by using Fourier spectra, phase portraits, and autocorrelation functions. In the chaotic regime, the Lyapunov exponents and dimensions are also calculated. A physical interpretation of the quasiperiodic motion is given by an exact calculation of the absorption spectrum of our two-level medium. The negative absorption (gain) peaks are found at the frequencies of the quasi-periodic motions, thus implying that the gain of the combined light-medium system is responsible at least in part for the observed complex behavior. In addition, we investigate the stability of the propagating plane wave when a transverse wave is added to the system as a perturbation. Instabilities are analyzed by linearizing the nonlinear equations which model the lightwave-medium system. The results show that the instability is highly-correlated with the four-wave mixing phase conjugation.
Within the wide field of nonlinear dynamics we investigate temporal and spatial behavior of electromagnetic systems. A strange type of laser, the nuclear magnetic resonance (NMR) laser, shows truly chaotic behavior and is therefore ideally suited to analyze experimentally and theoretically a variety of temporal nonlinear effects. Of particular interest is the analysis of its strange attractors in terms of unstable periodic orbits. For the extension of our research from the NMR laser (representing a purely temporal system described by the Bloch-Kirchhoff ordinary differential equations) to spatial and spatiotemporal systems, we chose, as a three-dimensional dynamic system, polarized laser beams interacting nonlinearly with sodium atoms. Among other effects, we have observed beam bouncing, beam splitting, and beam switching. This can be well described by partial differential equations for beam propagation derived from the Schrödinger equation and the Maxwell equations. An intuitive explanation is given in terms of intensity and polarization patterns formed by optical-pumping-induced mutual refractive index modifications between polarized resonant laser beams.