Observation of collective-emission-induced cooling of atoms in an optical cavity

Subradiant-to-Subradiant Phase Transition in the Bad Cavity Laser

We show that the onset of steady-state superradiance in a bad cavity laser is preceded by a dissipative phase transition between two distinct phases of steady-state subradiance. The transition is marked by a nonanalytic behavior of the cavity output power and the mean atomic inversion, as well as a discontinuity in the variance of the collective atomic inversion. In particular, for repump rates below a critical value, the cavity output power is strongly suppressed and does not increase with the atom number, while it scales linearly with atom number above this value. Remarkably, we find that the atoms are in a macroscopically entangled steady state near the critical region with a vanishing fraction of unentangled atoms in the large atom number limit.

The Panopticon device: An integrated Paul-trap-hemispherical mirror system for quantum optics

We present the design and construction of a new experimental apparatus for the trapping of single Ba⁺ ions in the center of curvature of an optical-quality hemispherical mirror. We describe the layout, fabrication, and integration of the full setup, consisting of a high-optical access monolithic "3D-printed" Paul trap, the hemispherical mirror, a diffraction-limited in-vacuum lens (NA = 0.7) for collection of atomic fluorescence, and a state-of-the art ultra-high vacuum vessel. This new apparatus enables the study of quantum electrodynamics effects such as strong inhibition and enhancement of spontaneous emission and achieves a collection efficiency of the emitted light in a single optical mode of 31%.

Cavity Cooling of a Levitated Nanosphere by Coherent Scattering

We report three-dimensional (3D) cooling of a levitated nanoparticle inside an optical cavity. The cooling mechanism is provided by cavity-enhanced coherent scattering off an optical tweezer. The observed 3D dynamics and cooling rates are as theoretically expected from the presence of both linear and quadratic terms in the interaction between the particle motion and the cavity field. By achieving nanometer-level control over the particle location we optimize the position-dependent coupling and demonstrate axial cooling by two orders of magnitude at background pressures of 6×10^{-2}  mbar. We also estimate a significant (>40  dB) suppression of laser phase noise heating, which is a specific feature of the coherent scattering scheme. The observed performance implies that quantum ground state cavity cooling of levitated nanoparticles can be achieved for background pressures below 1×10^{-7}  mbar.

Lasing by driven atoms-cavity system in collective strong coupling regime

The interaction of laser cooled atoms with resonant light is determined by the natural linewidth of the excited state. An optical cavity is another optically resonant system where the loss from the cavity determines the resonant optical response of the system. The near resonant combination of an optical Fabry-Pérot cavity with laser cooled and trapped atoms couples two distinct optical resonators via light and has great potential for precision measurements and the creation of versatile quantum optics systems. Here we show how driven magneto-optically trapped atoms in collective strong coupling regime with the cavity leads to lasing at a frequency red detuned from the atomic transition. Lasing is demonstrated experimentally by the observation of a lasing threshold accompanied by polarization and spatial mode purity, and line-narrowing in the outcoupled light. Spontaneous emission into the cavity mode by the driven atoms stimulates lasing action, which is capable of operating as a continuous wave laser in steady state, without a seed laser. The system is modeled theoretically, and qualitative agreement with experimentally observed lasing is seen. Our result opens up a range of new measurement possibilities with this system.

Cavity Cooling of Many Atoms

We demonstrate cavity cooling of all motional degrees of freedom of an atomic ensemble using light that is far detuned from the atomic transitions by several gigahertz. The cooling is achieved by cavity-induced frequency-dependent asymmetric enhancement of the atomic emission spectrum, thereby extracting thermal kinetic energy from the atomic system. Within 100 ms, the atomic temperature is reduced from 200 to 10  μK, where the final temperature is mainly limited by the linewidth of the cavity. In principle, the technique can be applied to molecules and atoms with complex internal energy structure.

Coupling to modes of a near-confocal optical resonator using a digital light modulator

Digital micromirror devices (DMD) provide a robust platform with which to implement digital holography, in principle providing the means to rapidly generate propagating transverse electromagnetic fields with arbitrary mode profiles at visible and IR wavelengths. We use a DMD to probe a Fabry-Pérot cavity in single-mode and near-degenerate confocal configurations. Pumping arbitrary modes of the cavity is possible with excellent specificity by virtue of the spatial overlap between the incident light field and the cavity mode.

Supercooling of Atoms in an Optical Resonator

We investigate laser cooling of an ensemble of atoms in an optical cavity. We demonstrate that when atomic dipoles are synchronized in the regime of steady-state superradiance, the motion of the atoms may be subject to a giant frictional force leading to potentially very low temperatures. The ultimate temperature limits are determined by a modified atomic linewidth, which can be orders of magnitude smaller than the cavity linewidth. The cooling rate is enhanced by the superradiant emission into the cavity mode allowing reasonable cooling rates even for dipolar transitions with ultranarrow linewidth.

Superradiance for Atoms Trapped along a Photonic Crystal Waveguide

We report observations of superradiance for atoms trapped in the near field of a photonic crystal waveguide (PCW). By fabricating the PCW with a band edge near the D(1) transition of atomic cesium, strong interaction is achieved between trapped atoms and guided-mode photons. Following short-pulse excitation, we record the decay of guided-mode emission and find a superradiant emission rate scaling as Γ̅(SR)∝N̅Γ(1D) for average atom number 0.19≲N̅≲2.6 atoms, where Γ(1D)/Γ'=1.0±0.1 is the peak single-atom radiative decay rate into the PCW guided mode, and Γ' is the radiative decay rate into all the other channels. These advances provide new tools for investigations of photon-mediated atom-atom interactions in the many-body regime.

Cavity cooling a single charged levitated nanosphere

Optomechanical cavity cooling of levitated objects offers the possibility for laboratory investigation of the macroscopic quantum behavior of systems that are largely decoupled from their environment. However, experimental progress has been hindered by particle loss mechanisms, which have prevented levitation and cavity cooling in a vacuum. We overcome this problem with a new type of hybrid electro-optical trap formed from a Paul trap within a single-mode optical cavity. We demonstrate a factor of 100 cavity cooling of 400 nm diameter silica spheres trapped in vacuum. This paves the way for ground-state cooling in a smaller, higher finesse cavity, as we show that a novel feature of the hybrid trap is that the optomechanical cooling becomes actively driven by the Paul trap, even for singly charged nanospheres.

Steering matter wave superradiance with an ultranarrow-band optical cavity

A superfluid atomic gas is prepared inside an optical resonator with an ultranarrow bandwidth on the order of the single photon recoil energy. When a monochromatic off-resonant laser beam irradiates the atoms, above a critical intensity the cavity emits superradiant light pulses with a duration on the order of its photon storage time. The atoms are collectively scattered into coherent superpositions of discrete momentum states, which can be precisely controlled by adjusting the cavity resonance frequency. With appropriate pulse sequences the entire atomic sample can be collectively accelerated or decelerated by multiples of two recoil momenta. The instability boundary for the onset of matter wave superradiance is recorded and its main features are explained by a mean field model.

Cavity cooling of free silicon nanoparticles in high vacuum

Laser cooling has given a boost to atomic physics throughout the last 30 years, as it allows one to prepare atoms in motional states, which can only be described by quantum mechanics. Most methods rely, however, on a near-resonant and cyclic coupling between laser light and well-defined internal states, which has remained a challenge for mesoscopic particles. An external cavity may compensate for the lack of internal cycling transitions in dielectric objects and it may provide assistance in the cooling of their centre-of-mass state. Here we demonstrate cavity cooling of the transverse kinetic energy of silicon nanoparticles freely propagating in high vacuum (<10⁻⁸ mbar). We create and launch them with longitudinal velocities down to v≤1 m s⁻¹ using laser-induced ablation of a pristine silicon wafer. Their interaction with the light of a high-finesse infrared cavity reduces their transverse kinetic energy by up to a factor of 30.

Laser cooling has been a successful technique to cool atoms and diatomic molecules to very low temperatures. Here, using an external cavity for an improved light coupling, Asenbaum et al. achieve the cooling of much larger objects, silicon nanoparticles, and reduce their transverse kinetic energy by up to a factor of 30.

Bose-Einstein condensates with cavity-mediated spin-orbit coupling

We propose a novel scheme to generate the spin-orbit coupling for a condensate placed inside an optical cavity by using a standing wave and a traveling wave. It is shown that the interplay of the laser lights and the cavity gives rise to rich quantum phases. Our scheme also generates a large synthetic magnetic field for the dressed spin state, which may facilitate the study of the quantum Hall effect in ultracold atomic gases.

Cavity-mediated near-critical dissipative dynamics of a driven condensate

We investigate the near-critical dynamics of atomic density fluctuations in the nonequilibrium self-organization transition of an optically driven quantum gas coupled to a single mode of a cavity. In this system cavity-mediated long-range interactions between atoms, tunable by the drive strength, lead to softening of an excitation mode recently observed in experiments. This phenomenon has previously been studied within a two-mode approximation for the collective motional degrees of freedom of the atomic condensate, which results in an effective open-system Dicke model. Here, including the full spectrum of atomic modes we find a finite lifetime for a rotonlike mode in the Bogoliubov excitation spectrum that is strongly pump dependent. The corresponding decay rate and critical exponents for the phase transition are calculated explaining the nonmonotonic pump-dependent atomic damping rate observed in recent experiments. We compute the near-critical behavior of the intracavity field fluctuations that has been previously shown to be enhanced with respect to the equilibrium Dicke model in a two-mode approximation. We highlight the role of the finite size of the system in the suppression of it below the expectations of the open Dicke model.

Experimental methods of molecular matter-wave optics

We describe the state of the art in preparing, manipulating and detecting coherent molecular matter. We focus on experimental methods for handling the quantum motion of compound systems from diatomic molecules to clusters or biomolecules.Molecular quantum optics offers many challenges and innovative prospects: already the combination of two atoms into one molecule takes several well-established methods from atomic physics, such as for instance laser cooling, to their limits. The enormous internal complexity that arises when hundreds or thousands of atoms are bound in a single organic molecule, cluster or nanocrystal provides a richness that can only be tackled by combining methods from atomic physics, chemistry, cluster physics, nanotechnology and the life sciences.We review various molecular beam sources and their suitability for matter-wave experiments. We discuss numerous molecular detection schemes and give an overview over diffraction and interference experiments that have already been performed with molecules or clusters.Applications of de Broglie studies with composite systems range from fundamental tests of physics up to quantum-enhanced metrology in physical chemistry, biophysics and the surface sciences.Nanoparticle quantum optics is a growing field, which will intrigue researchers still for many years to come. This review can, therefore, only be a snapshot of a very dynamical process.

Self-organization threshold scaling for thermal atoms coupled to a cavity

We make a detailed experimental study of the threshold for the self-organization of thermal 87Rb atoms coupled to a high-finesse cavity over a range of atom numbers and cavity detunings. We investigate the difference between probing with a traveling wave and a retroreflected lattice. These two scenarios lead to qualitatively different behavior in terms of the response of the system as a function of cavity detuning with respect to the probe. In both cases, we confirm a N(-1) scaling of the threshold with atom number.

Cavity Cooling Below the Recoil Limit

Conventional laser cooling relies on repeated electronic excitations by near-resonant light, which constrains its area of application to a selected number of atomic species prepared at moderate particle densities. Optical cavities with sufficiently large Purcell factors allow for laser cooling schemes, avoiding these limitations. Here, we report on an atom-cavity system, combining a Purcell factor above 40 with a cavity bandwidth below the recoil frequency associated with the kinetic energy transfer in a single photon scattering event. This lets us access a yet-unexplored regime of atom-cavity interactions, in which the atomic motion can be manipulated by targeted dissipation with sub-recoil resolution. We demonstrate cavity-induced heating of a Bose-Einstein condensate and subsequent cooling at particle densities and temperatures incompatible with conventional laser cooling.

Correlated photon emission from multiatom Rydberg dark States

We consider three-level atoms driven by two resonant light fields in a ladder scheme where the upper level is a highly excited Rydberg state. We show that the dipole-dipole interactions between Rydberg excited atoms prevents the formation of single particle dark states and leads to strongly correlated photon pairs from atoms separated by distances large compared to the emission wavelength. For a pair of atoms, this enables realization of an efficient photon-pair source with on average one pair every 30 μs.

Optomechanical cavity cooling of an atomic ensemble

We demonstrate cavity sideband cooling of a single collective motional mode of an atomic ensemble down to a mean phonon occupation number ⟨n⟩(min⁡)=2.0(-0.3)(+0.9). Both ⟨n⟩(min) and the observed cooling rate are in good agreement with an optomechanical model. The cooling rate constant is proportional to the total photon scattering rate by the ensemble, demonstrating the cooperative character of the light-emission-induced cooling process. We deduce fundamental limits to cavity cooling either the collective mode or, sympathetically, the single-atom degrees of freedom.

Fast cavity-enhanced atom detection with low noise and high fidelity

Cavity quantum electrodynamics describes the fundamental interactions between light and matter, and how they can be controlled by shaping the local environment. For example, optical microcavities allow high-efficiency detection and manipulation of single atoms. In this regime, fluctuations of atom number are on the order of the mean number, which can lead to signal fluctuations in excess of the noise on the incident probe field. Here we demonstrate, however, that nonlinearities and multi-atom statistics can together serve to suppress the effects of atomic fluctuations when making local density measurements on clouds of cold atoms. We measure atom densities below 1 per cavity mode volume near the photon shot-noise limit. This is in direct contrast to previous experiments where fluctuations in atom number contribute significantly to the noise. Atom detection is shown to be fast and efficient, reaching fidelities in excess of 97% after 10 μs and 99.9% after 30 μs.

Single atoms can be detected using optical resonators that extend the lifetime of the photon. Here, the authors demonstrate fast, high-fidelity detection of very low atom densities using a microfabricated optical cavity to couple the detection light with the atoms.

In situ characterization of an optical cavity using atomic light shift

We report the precise characterization of the optical potential obtained by injecting a distributed-feedback erbium-doped fiber laser at 1560 nm to the transverse modes of a folded optical cavity. The optical potential was mapped in situ using cold rubidium atoms, whose potential energy was spectrally resolved thanks to the strong differential light shift induced by the 1560 nm laser on the two levels of the probe transition. The optical potential obtained in the cavity is suitable for trapping rubidium atoms and eventually to achieve all-optical Bose-Einstein condensation directly in the resonator.

Doppler cooling a microsphere

Doppler cooling the center-of-mass motion of an optically levitated microsphere via the velocity-dependent scattering force from narrow whispering gallery mode resonances is described. Light that is red detuned from the whispering gallery mode resonance can be used to damp the center-of-mass motion in a process analogous to the Doppler cooling of atoms. The scattering force is not limited by saturation but can be controlled by the incident power. Cooling times on the order of seconds are calculated for a 20 μm diameter silica microsphere trapped within optical tweezers.

Dramatic reductions in inelastic cross sections for ultracold collisions near Feshbach resonances

We show that low-energy inelastic cross sections can decrease as well as increase in the vicinity of a zero-energy Feshbach resonance. When an external field is used to tune across such a resonance, the real and imaginary parts of the scattering length show asymmetric oscillations with both peaks and troughs. In favorable circumstances, the inelastic collision rate can be reduced to almost zero. This may be important for efforts to achieve evaporative and sympathetic cooling for molecules.

Cavity sideband cooling of a single trapped ion

We report a demonstration and quantitative characterization of one-dimensional cavity cooling of a single trapped (88)Sr(+) ion in the resolved-sideband regime. We measure the spectrum of cavity transitions, the rates of cavity heating and cooling, and the steady-state cooling limit. The cavity cooling dynamics and cooling limit of 22.5(3) motional quanta, limited by the moderate coupling between the ion and the cavity, are consistent with a simple model [Phys. Rev. A 64, 033405 (2001)] without any free parameters, validating the rate equation model for cavity cooling.

Prospects for a millihertz-linewidth laser

We propose a new light source based on having alkaline-earth atoms in an optical lattice collectively emit photons on an ultranarrow clock transition into the mode of a high Q resonator. The resultant optical radiation has an extremely narrow linewidth in the mHz range, even smaller than that of the clock transition itself due to collective effects. A power level of order 10;{-12} W is possible, sufficient for phase locking a slave optical local oscillator. Realizing this light source has the potential to improve the stability of the best clocks by 2 orders of magnitude.

Experimental realization of an optical one-way barrier for neutral atoms

We demonstrate an asymmetric optical potential barrier for ultracold 87Rb atoms using laser light tuned near the D2 optical transition. Such a one-way barrier, where atoms incident on one side are transmitted but reflected from the other, is a realization of Maxwell's demon and has important implications for cooling atoms and molecules not amenable to standard laser-cooling techniques. In our experiment, atoms are confined to a far-detuned dipole trap consisting of a single focused Gaussian beam, which is divided near the focus by the barrier. The one-way barrier consists of two focused laser beams oriented almost normal to the dipole-trap axis. The first beam is tuned to present either a potential well or barrier, depending on the state of the incident atoms. On the reflecting side of the barrier, the second beam optically pumps the atoms to the reflecting (barrier) state, thus producing the asymmetry.

Vibrational cooling of spin-stretched dimer states by He buffer gas: quantum calculations for Li2(a 3Sigma(u)+) at ultralow energies

The interaction between the triplet state of the lithium dimer, (7)Li(2), with (4)He is obtained from accurate ab initio calculations where the vibrational dependence of the potential is newly computed. Vibrational quenching dynamics within a coupled-channel quantum treatment is carried out at ultralow energies, and large differences in efficiency as a function of the initial vibrational state of the targets are found as one compares the triplet results with those of the singlet state of the same target.

Collective atomic recoil lasing with a partially coherent pump

We investigate the effect of pump phase noise on the collective backscattering of light by a cold, collisionless atomic gas. We show that for a partially coherent pump field, the growth rate of the backscattered field is reduced relative to that for a coherent pump, but the backscattered intensity can be increased. Our results demonstrate that fluctuations and noise can play a counterintuitive role in nonlocally coupled many-body systems.

Cavity nonlinear optics at low photon numbers from collective atomic motion

We report on Kerr nonlinearity and dispersive optical bistability of a Fabry-Perot optical resonator due to the displacement of ultracold atoms trapped within. In the driven resonator, such collective motion is induced by optical forces acting upon up to 10(5) 87Rb atoms prepared in the lowest band of a one-dimensional intracavity optical lattice. The longevity of atomic motional coherence allows for strongly nonlinear optics at extremely low cavity photon numbers, as demonstrated by the observation of both branches of optical bistability at photon numbers below unity.

Deterministic loading of individual atoms to a high-finesse optical cavity

Individual laser-cooled atoms are delivered on demand from a single atom magneto-optic trap to a high-finesse optical cavity using an atom conveyor. Strong coupling of the atom with the cavity field allows simultaneous cooling and detection of individual atoms for time scales exceeding 15 s. The single atom scatter rate is studied as a function of probe-cavity detuning and probe Rabi frequency, and the experimental results are in qualitative agreement with theoretical predictions. We demonstrate the ability to manipulate the position of a single atom relative to the cavity mode with excellent control and reproducibility.

Large velocity capture range and low temperatures with cavities

There are interesting modifications to the Doppler force when atoms strongly couple to an optical cavity. In particular, there is the possibility to increase the velocity capture range while maintaining a final temperature close to the Doppler limit. The mechanism is based on the multiple absorption emissions of each cavity photon. A previously reported counterintuitive Doppler effect is clarified.

Normal mode splitting and mechanical effects of an optical lattice in a ring cavity

A novel regime of atom-cavity physics is explored, arising when large atom samples dispersively interact with high-finesse optical cavities. A stable far-detuned optical lattice of several million rubidium atoms is formed inside an optical ring resonator by coupling equal amounts of laser light to each propagation direction of a longitudinal cavity mode. An adjacent longitudinal mode, detuned by about 3 GHz, is used to perform probe transmission spectroscopy of the system. The atom-cavity coupling for the lattice beams and the probe is dispersive and dissipation results only from the finite photon-storage time. The observation of two well-resolved normal modes demonstrates the regime of strong cooperative coupling. The details of the normal mode spectrum reveal mechanical effects associated with the retroaction of the probe upon the optical lattice.

Cooling trapped atoms in optical resonators

We derive an equation for the cooling dynamics of the quantum motion of an atom trapped by an external potential inside an optical resonator. This equation has broad validity and allows us to identify novel regimes where the motion can be efficiently cooled to the potential ground state. Our result shows that the motion is critically affected by quantum correlations induced by the mechanical coupling with the resonator, which may lead to selective suppression of certain transitions for the appropriate parameters regimes, thereby increasing the cooling efficiency.

Compression of atomic phase space using an asymmetric one-way barrier

We show how to construct asymmetric optical barriers for atoms. These barriers can be used to compress phase-space of a sample by creating a confined region in space where atoms can accumulate with heating at the single photon recoil level. We illustrate our method with a simple two-level model and then show how it can be applied to more realistic multilevel atoms.

Four-wave mixing with self-phase matching due to collective atomic recoil

We describe a method for nondegenerate four-wave-mixing in a cold sample of four-level atoms. An integral part of the four-wave-mixing process is a collective instability which spontaneously generates a periodic density modulation in the cold atomic sample with a period equal to half of the wavelength of the generated high-frequency optical field. Because of the generation of this density modulation, phase matching between the pump and scattered fields is not a necessary initial condition for this wave-mixing process to occur; rather the density modulation acts to "self-phase match" the fields during the course of the wave-mixing process. We describe a one-dimensional model of this process, and suggest a proof-of-principle experiment which would involve pumping a sample of cold Cs atoms with three infrared pump fields to produce blue light.

Suppression of Bragg scattering by collective interference of spatially ordered atoms with a high-Q cavity mode

When N driven atoms emit in phase into a high-Q cavity mode, the intracavity field generated by collective scattering interferes destructively with the pump driving the atoms. Hence atomic fluorescence is suppressed and cavity loss becomes the dominant decay channel for the whole ensemble. Microscopically, 3D light-intensity minima are formed in the vicinity of the atoms that prevent atomic excitation and form a regular lattice. The effect gets more pronounced for large atom numbers, when the sum of the atomic decay rates exceeds the rate of cavity losses and one would expect the opposite behavior. These results provide new insight into recent experiments on collective atomic dynamics in cavities.

Cavity cooling of a single atom

All conventional methods to laser-cool atoms rely on repeated cycles of optical pumping and spontaneous emission of a photon by the atom. Spontaneous emission in a random direction provides the dissipative mechanism required to remove entropy from the atom. However, alternative cooling methods have been proposed for a single atom strongly coupled to a high-finesse cavity; the role of spontaneous emission is replaced by the escape of a photon from the cavity. Application of such cooling schemes would improve the performance of atom-cavity systems for quantum information processing. Furthermore, as cavity cooling does not rely on spontaneous emission, it can be applied to systems that cannot be laser-cooled by conventional methods; these include molecules (which do not have a closed transition) and collective excitations of Bose condensates, which are destroyed by randomly directed recoil kicks. Here we demonstrate cavity cooling of single rubidium atoms stored in an intracavity dipole trap. The cooling mechanism results in extended storage times and improved localization of atoms. We estimate that the observed cooling rate is at least five times larger than that produced by free-space cooling methods, for comparable excitation of the atom.

Vacuum-field level shifts in a single trapped ion mediated by a single distant mirror

A distant mirror leads to a vacuum-induced level shift in a laser-excited atom. This effect has been measured with a single mirror 25 cm away from a single, trapped barium ion. This dispersive action is the counterpart to the mirror's dissipative effect, which has been shown earlier to effect a change in the ion's spontaneous decay [Nature (London) 413, 495 (2001)]]. The experimental data are well described by eight-level optical Bloch equations which are amended to take into account the presence of the mirror according to the model in Phys. Rev. A 66, 023816 (2002)]. Observed deviations from simple dispersive behavior are attributed to multilevel effects.

Observation of collective friction forces due to spatial self-organization of atoms: from Rayleigh to Bragg scattering

We demonstrate that emission-induced self-organization of two-level atoms can effect strong damping of the sample's center-of-mass motion. When illuminated by far-detuned light, cold cesium atoms assemble into a density grating that efficiently diffracts the incident light into an optical resonator. We observe random phase jumps of pi in the emitted light, confirming spontaneous symmetry breaking in the atomic self-organization. The Bragg diffraction results in a collective friction force with center-of-mass deceleration up to 1000 m/s(2) that is effective even for an open atomic transition.

Collective atomic motion in an optical lattice formed inside a high finesse cavity

We report on collective nonlinear dynamics in an optical lattice formed inside a high finesse ring cavity in a so far unexplored regime, where the light shift per photon times the number of trapped atoms exceeds the cavity resonance linewidth. We observe bistability and self-induced squeezing oscillations resulting from the retroaction of the atoms upon the optical potential wells. We can well understand most of our observations within a simplified model assuming adiabaticity of the atomic motion. Nonadiabatic aspects of the atomic motion are reproduced by solving the complete system of coupled nonlinear equations of motion.