Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Optical Engineering

Using nuclear spins, radio waves, sodium atoms, and laser light to investigate spatiotemporal nonlinear systems

Reto Holzner, Josef Simonet, L. Flepp, B. Röhricht, Patrik Eschle, S. Dangel, U. Rusch, H. Schmid, Ernst Brun

Published Aug 1, 1995• 1 citation
Publisher

Abstract

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.

DOI
10.1117/12.200618

Related papers

Observation of polarization instabilities and chaos

Daniel J. Gauthier, Michelle S. Malcuit, Robert W. Boyd

We have observed that the polarizations and intensities of counterpropagating laser beams in sodium vapor are temporally unstable. The output from a stabilized dye laser tuned near to the 3S1/2 → 3P1/2 atomic transition was split to form the two counterpropagating beams. These beams were polarized with the same linear polarization and were weakly focused to a spot size of ~750 μm in sodium vapor having a density of ~3 × 1013 cm−3. We monitored the light generated in the polarization orthogonal to that of the pump beams. For a detuning of 1.01 GHz to the high-frequency side of the 3S1/2(F = 2) → 3P1/2(F = 2) transition and a power of 160 mW in one of the beams, we found that the threshold for the instability occurred at a power of 15 mW in the other beam. Slightly above threshold, the instability was found to be periodic in nature. As the intensity in the counterpropagating beam was increased far above the threshold value, wild fluctuations in the polarizations and intensities of the transmitted beams were observed. We have found that these fluctuations are chaotic and that the system evolves on a strange attractor whose dimension increases with increasing laser power. These observations are in qualitative agreement with the theoretical predictions of Gaeta et al.,1 which predict that the polarizations of counterpropagating beams in a Kerr medium are unstable to the growth of temporal fluctuations.

Annual Meeting Optical Society of America•Jan 1, 1988•0 citations

Polarization instabilities of counterpropagating laser beams in sodium vapor

Daniel J. Gauthier, Michelle S. Malcuit, Robert W. Boyd

We have observed temporal instabilities in the polarizations of counterpropagating laser beams in atomic sodium vapor. For intensities slightly above the instability threshold, the polarizations fluctuate periodically. For higher intensities, the fluctuations are chaotic and the system evolves on a strange attractor whose fractal dimension increases with increasing laser intensity.

Physical Review Letters•Oct 17, 1988•60 citations

Optical pumping-induced spatio-temporal modifications to propagation, polarization and intensity of laser beams in sodium vapour

R. Holzner, S. Dangel

Circularly polarized laser beams propagating through sodium vapour and tuned to the buffer-gas-broadened atomic transition can optically pump sodium atoms into a non-absorbing ground state. This causes an intensity-dependent refractive index gradient along as well as transverse to the laser beam propagation direction, giving rise to a number of nonlinear spatio-temporal intensity and polarization pattern creating processes. In the case of a single circularly polarized laser beam we have observed self-focusing and defocusing, the transformation of the incident Gaussian beam intensity profiles into ring profiles, a large shift of about 5 GHz of the maximum of the absorption profile when suitable magnetic fields are applied and the deflection of a beam by the inhomogeneous transverse magnetic field of a current-carrying wire. When two beams of opposite circular polarization are superimposed, astonishing effects such as the mutual deflection of both beams (beam bouncing), the mutual extinction of both beams (beam switching), the separation of initially overlapping beams (beam splitting) and the mutual attraction of both beams (beam attraction) can be observed. While most of the effects can be well described for the stationary state by a to atomic transition model, the correct description of the dynamics requires the consideration of all hyperfine states.

Quantum and Semiclassical Optics Journal of the European Optical Society Part B•Feb 1, 1998•5 citations

Light-shift-induced chaos in a nonlinear optical resonator

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

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.

Physical Review A•May 1, 1992•8 citations

Ruby NMR laser: A phenomenon of spontaneous self-organization of a nuclear spin system

Peter Boesiger, Ernst Brun, D. Meier

Radio wave amplification by stimulated emission of radiation (NMR laser) was observed in ${\mathrm{Al}}_{2}$${\mathrm{O}}_{3}$: ${\mathrm{Cr}}^{3+}$ (ruby) with the $^{27}\mathrm{Al}$ spin system pumped to an inverted Zeeman state by means of dynamic nuclear polarization (DNP) and with an LC circuit tuned to the NMR transitions. A theory is developed which combines standard Bloch equations with the dynamic equations of a heat-reservoir model for the spin dynamics. Experimental data concerning the relaxation and polarization mechanisms in the nonradiant state are presented, and the parameters of the reservoir model are determined. In accord with these results, dynamic NMR laser equations for different modes of operation (continuous or pulsed) are derived from which the threshold of the NMR laser, its critical behavior near threshold, and its transient response after $Q$-switch tuning can be calculated. The agreement between theory and experiments in the superradiant state reflects unambiguously the interplay between nuclear-spin relaxation, DNP, spin fluctuations, and the self-induced ordering of the $^{27}\mathrm{Al}$ spins perpendicular to the applied field. In particular, the results highlight the decisive role of the electronic dipole-dipole system in the spin and NMR laser dynamics of ruby. Special emphasis is placed on the close relationship between the NMR laser, as a system far from thermal equilibrium which shows the phenomenon of cooperative self-organization, and a system at thermal equilibrium which undergoes a second-order phase transition. Thus, the rotating nuclear-spin magnetization is treated as an order parameter.

Physical Review A•Aug 1, 1978•46 citations

Coherent Laser-Atom Interactions With Partially Coherent Laser Fields

Daniel S. Elliott

Abstract : In this research program, the investigators have explored the role of fluctuations of a laser field on coherent interactions between the laser field and simple atomic systems. A key feature of these experiments is their ability to provide measurements of complex nonlinear optical interactions which are sufficient for direct, quantitative comparison with theory. The main thrust of the present work has been the study of non-Markovian field statistics, present in experiments that use multiple, time-delayed input beams to drive the nonlinear system. The specific interaction studied was Phase-Conjugate Four-Wave Mixing in an optically-pumped beam of atomic sodium, which the researchers measured as a function of laser bandwidth, Rabi frequency, and delay time between the pump and probe beams. The investigators have also calculated the expected signal strength through numerical integration of the Optical Bloch Equations. Experimental and numerical results are in very good agreement. The investigators also studied theoretically two-photon absorption with correlated, time-delayed input fields, and provided the first experimental confirmation of the fundamental theory of Phase-Conjugate Four-Wave Mixing with monochromatic fields in a simple atomic system.

•Aug 30, 1999•0 citations
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.
We have observed that the polarizations and intensities of counterpropagating laser beams in sodium vapor are temporally unstable. The output from a stabilized dye laser tuned near to the 3S1/2 → 3P1/2 atomic transition was split to form the two counterpropagating beams. These beams were polarized with the same linear polarization and were weakly focused to a spot size of ~750 μm in sodium vapor having a density of ~3 × 1013 cm−3. We monitored the light generated in the polarization orthogonal to that of the pump beams. For a detuning of 1.01 GHz to the high-frequency side of the 3S1/2(F = 2) → 3P1/2(F = 2) transition and a power of 160 mW in one of the beams, we found that the threshold for the instability occurred at a power of 15 mW in the other beam. Slightly above threshold, the instability was found to be periodic in nature. As the intensity in the counterpropagating beam was increased far above the threshold value, wild fluctuations in the polarizations and intensities of the transmitted beams were observed. We have found that these fluctuations are chaotic and that the system evolves on a strange attractor whose dimension increases with increasing laser power. These observations are in qualitative agreement with the theoretical predictions of Gaeta et al.,1 which predict that the polarizations of counterpropagating beams in a Kerr medium are unstable to the growth of temporal fluctuations.
Circularly polarized laser beams propagating through sodium vapour and tuned to the buffer-gas-broadened atomic transition can optically pump sodium atoms into a non-absorbing ground state. This causes an intensity-dependent refractive index gradient along as well as transverse to the laser beam propagation direction, giving rise to a number of nonlinear spatio-temporal intensity and polarization pattern creating processes. In the case of a single circularly polarized laser beam we have observed self-focusing and defocusing, the transformation of the incident Gaussian beam intensity profiles into ring profiles, a large shift of about 5 GHz of the maximum of the absorption profile when suitable magnetic fields are applied and the deflection of a beam by the inhomogeneous transverse magnetic field of a current-carrying wire. When two beams of opposite circular polarization are superimposed, astonishing effects such as the mutual deflection of both beams (beam bouncing), the mutual extinction of both beams (beam switching), the separation of initially overlapping beams (beam splitting) and the mutual attraction of both beams (beam attraction) can be observed. While most of the effects can be well described for the stationary state by a to atomic transition model, the correct description of the dynamics requires the consideration of all hyperfine states.
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.
Radio wave amplification by stimulated emission of radiation (NMR laser) was observed in Al2{\mathrm{Al}}_{2}Al2​O3{\mathrm{O}}_{3}O3​: Cr3+{\mathrm{Cr}}^{3+}Cr3+ (ruby) with the 27Al^{27}\mathrm{Al}27Al spin system pumped to an inverted Zeeman state by means of dynamic nuclear polarization (DNP) and with an LC circuit tuned to the NMR transitions. A theory is developed which combines standard Bloch equations with the dynamic equations of a heat-reservoir model for the spin dynamics. Experimental data concerning the relaxation and polarization mechanisms in the nonradiant state are presented, and the parameters of the reservoir model are determined. In accord with these results, dynamic NMR laser equations for different modes of operation (continuous or pulsed) are derived from which the threshold of the NMR laser, its critical behavior near threshold, and its transient response after QQQ-switch tuning can be calculated. The agreement between theory and experiments in the superradiant state reflects unambiguously the interplay between nuclear-spin relaxation, DNP, spin fluctuations, and the self-induced ordering of the 27Al^{27}\mathrm{Al}27Al spins perpendicular to the applied field. In particular, the results highlight the decisive role of the electronic dipole-dipole system in the spin and NMR laser dynamics of ruby. Special emphasis is placed on the close relationship between the NMR laser, as a system far from thermal equilibrium which shows the phenomenon of cooperative self-organization, and a system at thermal equilibrium which undergoes a second-order phase transition. Thus, the rotating nuclear-spin magnetization is treated as an order parameter.
Abstract : In this research program, the investigators have explored the role of fluctuations of a laser field on coherent interactions between the laser field and simple atomic systems. A key feature of these experiments is their ability to provide measurements of complex nonlinear optical interactions which are sufficient for direct, quantitative comparison with theory. The main thrust of the present work has been the study of non-Markovian field statistics, present in experiments that use multiple, time-delayed input beams to drive the nonlinear system. The specific interaction studied was Phase-Conjugate Four-Wave Mixing in an optically-pumped beam of atomic sodium, which the researchers measured as a function of laser bandwidth, Rabi frequency, and delay time between the pump and probe beams. The investigators have also calculated the expected signal strength through numerical integration of the Optical Bloch Equations. Experimental and numerical results are in very good agreement. The investigators also studied theoretically two-photon absorption with correlated, time-delayed input fields, and provided the first experimental confirmation of the fundamental theory of Phase-Conjugate Four-Wave Mixing with monochromatic fields in a simple atomic system.