We describe the detection of the radial motions of a trapped charge in a Penning trap via a fourth-order electrostatic coupling to the axial degree of freedom. Two basic approaches are demonstrated. The first uses a standard resonant drive to determine the cyclotron frequency and has a resolution of 0.1 ppb. The second approach beats the motion due to an off-resonant drive against the motion due to thermal drives to give a continuous phase-coherent monitor of the cyclotron motion with a resolution of 0.01 ppb. The inherent systematic shift of these ‘‘anharmonic-detection’’ schemes is discussed, and approaches are given to reduce or eliminate it.

Laser cooling of trapped ions in a standing- and running-wave configuration is discussed theoretically. The ions are assumed to be spatially localized on the scale provided by the wavelength of the laser (Lamb-Dicke limit). A master equation for the center-of-mass distribution of the ion is derived for a multilevel system, and explicit results are presented for two- and three-level systems and harmonic trapping potentials. For the two-level system located at the node of the standing wave, we find final temperatures that are a factor of 2 lower than the limit for a running wave and cooling rates that do not saturate with the laser intensity. At the point of maximum gradient of the standing wave, blue detuned cooling is found that is analogous to the Sisyphus cooling of free atoms. For a three-level system we compare our results with those of Wineland, Dalibard, and Cohen-Tannoudji [J. Opt. Soc. Am. B 9, 32 (1992)].

Using Lie group methods and prolongation techniques for vector fields, we present heretofore undiscovered symmetries of nonlinear-dynamical equations that describe ion motion in the Paul and in the Penning trap, respectively. The two-ion motion in the trap is investigated in detail. Checking integrability of the nonlinear equations, both Painlevé and modified Painlevé tests failed. However, using symmetry properties discovered by prolongation techniques, we are able to construct integrals of motion and to discover conditions for those parameter combinations that lead to regular motion in the trap. These parameters, which can be related to experimentally adjustable physical quantities, are listed. In the case of rotational invariance, the corresponding analytical solutions are derived from the integrals of motion. Parameter choices different from these values lead to chaotic motion, as demonstrated numerically by the calculation of Poincaré sections as well as by the determination of the Lyapunov spectrum.

The complicated quadratic Zeeman spectrum in a laboratory magnetic field is analyzed with the aid of accurate Coulomb functions. Different Rydberg manifolds are masked by the generalized Gailitis average based on the multichannel quantum-defect theory. Fundamental resonances are thereby accentuated. An efficient and reliable numerical procedure is developed for evaluating the negative-energy Coulomb functions and their derivatives. The method is based on the three-term recursions and employs Steed’s forward-propagation method [A. R. Barnett et al., Comput. Phys. Commun. 8, 377 (1974)] as well as the backward-propagation method of Stegun and Abramowitz [Phys. Rev. 98, 1851 (1955)].

We consider a three-level medium with low-frequency separation between the two upper levels. The medium interacts with an external field that induces a coherence between the two upper levels. We analyze the amplification condition for the probe monochromatic optical field in a general nonresonant case. We show that amplification without population inversion arises either at line center or at the sidebands, depending on the sign of the population difference between the upper sublevels. This amplification occurs in a limited range of the coherent pump intensity.

The nonlinear response of xenon gas due to a near-resonant two-photon excitation at the 5${\mathit{p}}^6$ ${}_{}{}^1{}_{}{}^{}$${\mathit{S}}_0$→6p[1/2${]}_0$ transition with a femtosecond KrF-excimer-laser system is investigated for the intensity range ${10}^{10}$–${10}^{15}$ W/${\mathrm{cm}}^2$. Stimulated hyper-Raman scattering and amplified spontaneous emission of atomic xenon are observed at wavelengths around 823 and 828 nm. These emissions are superimposed by four-wave parametric-oscillation processes leading to strong ultrashort continuum radiation in the visible and near-infrared (650–850 nm), uv (185–400 nm), and vacuum ultraviolet (147–155 nm) spectral ranges with output powers up to 100 MW. From difference-frequency-mixing experiments microscopic nonlinear susceptibilities in the range of ‖${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$‖=${10}^{\mathrm{-}48}$ to ${10}^{\mathrm{-}50}$ ${\mathrm{m}}^5$/${\mathrm{V}}^2$ have been determined, which are in good agreement with calculations.

We develop a theory of near-resonant four-wave mixing induced by broad-bandwidth chaotic fields in a medium composed of two-level atoms. By solving the equations of motion for the elements of the atomic density matrix using an appropriate decorrelation approximation, we derive an analytic expression for the frequency spectrum of the signal wave for the case of the bandwidth of the fluctuations of the pump field exceeding the other relaxation rates in the problem. The probe wave is considered weak and of arbitrary bandwidth. The theory is valid for pump intensities up to and exceeding the bandwidth-dependent saturation value. Finally, we show how the frequency spectrum is modified if account is taken of atomic motion. The theoretical results are of direct relevance to practical experiments involving broad-bandwidth pulsed lasers and employing an atomic vapor as the nonlinear medium.

An approximate analytical analysis of two-dimensional optical beams propagating in saturable self-focusing media is carried out. In particular, the radial shape and the phase shift of the stationary self-trapped fundamental mode are investigated in detail. The dynamic variations of the beamwidth and the phase shift for nonstationary propagation are also analyzed. The predictions of the analytical model are compared with numerical results for stationary propagation and show good agreement.

We report the observation and theoretical interpretation of two-photon double-beam optical bistability, where the nonlinearity is obtained in a two-photon transition in a sodium atomic beam. The experimental results, obtained in the dispersive regime, are compared with a simple crossed-phase-modulation model, and also with a numerical solution of the interaction between the cavity fields and three-level atoms, using the optical Bloch equations. For a wide range of parameters, both models agree with the experimental results. Oscillatory regimes may also appear, in agreement with the predictions of the complete model.

We introduce a phase-space description of quantum phase fluctuations. This description involves the quantum state of light and a reference detector state. It is shown that, if the detector state is the electromagnetic vacuum, this phase-space description is equivalent to Glauber’s Q quasidistribution. We investigate phase fluctuations of a weak coherent state and of a squeezed vacuum. We compare our results with the theory of phase fluctuations based on a unitary and nonunitary phase operators.

We demonstrate an analogy between a two-photon process driven by a chirped pulse and Fresnel diffraction from a slit: in both situations interference between different ‘‘paths’’ leading to the same ‘‘final state’’ determines the resulting ‘‘diffraction pattern.’’ On the basis of this analogy, a spectral Fresnel zone plate was designed for ‘‘focusing of spectral energy’’: at the two-photon level the spectral energy can be concentrated in an effective bandwidth that is much smaller than the bandwidth of the original excitation pulse. To show this effect, two experiments were performed with femtosecond laser pulses with a well-controlled power spectrum and chirp: frequency doubling in a nonlinear crystal and two-photon excitation of Rydberg states in rubidium.

The intensity-noise properties of a constant-current-driven multimode semiconductor laser were studied theoretically and experimentally. It was shown that the total intensity noise of a pump-noise-suppressed multimode semiconductor laser was reduced to below the standard quantum limit even though the intensity noise of each individual mode had large excess intensity noise. This discovery stems from negative quantum correlations between longitudinal-mode intensities.

Scattering of quantum fluctuations by a linear system is entirely characterized by impedance functions that also allow the determination of the system fluctuations. We present a generalization of this linear input-output theory to the case of a nonlinear scattering system. We define response functions that are similar to the susceptibility functions of linear response theory, but behave as noncommuting quantities. These functions contain a ‘‘dynamical’’ part determined by the relaxation of the system and a ‘‘structural’’ one related to the commutators between the system observables. The generalized linear input-output theory provides us with a complete description of the system fluctuations as well as of the output reservoir fields. The consistency of the results is ensured by general relations existing between the response functions and the correlation functions.

We perform detailed calculations of harmonic conversion in a 15-Torr jet of xenon into which a 1064-nm-wavelength 36-ps-pulse-width laser has been tightly focused, so that the peak intensity ranges from 5×${10}^{12}$ to 5×${10}^{13}$ W ${\mathrm{cm}}^{\mathrm{-}2}$. The single-atom emission rates are obtained by integrating the time-dependent Schrödinger equation. We employ an improved atomic model which includes excitation and ionization through states with both low-lying ionic cores. The propagation equations are solved using a general and efficient finite-difference technique. Excellent agreement with experimental data is obtained. We consider the effect of the defocusing of the pump beam by the free electrons in the saturation regime, and find it to be small for the conditions studied here. The spatial, temporal, and spectral profiles of the harmonic emission are presented. Significant blueshifts of the harmonics’ spectral line shapes are observed in the saturation regime. Finally, using model polarizations, we discuss how the harmonic fields build up in the nonlinear medium, through a series of interferences. It allows us to understand why efficient phase matching can be achieved in a strong-field regime for the laser-atom interaction.

In a cavity with photorefractive gain the resonator field is fed by a pump beam through two-wave mixing inside a photorefractive crystal. In this paper we develop a simplified model for such a system which takes into account the transverse dependence of the pump and the resonator fields. We then study the onset of spatiotemporal structures both analytically and numerically.

Spatial variations of the laser mode and pumping rate are incorporated into the theory of conventionally pumped lasers that produce squeezed light. Both a quantum-mechanical theory and a heuristic statistical model are used. While variations in the laser mode are found to have a negligible effect on squeezing, variations in the pumping rate are significant. The maximum attainable squeezing is always reduced compared with the spatially uniform case. However, resonantly enhancing a low-power pump in a Fabry-Pérot cavity, rather than a ring cavity, may give better squeezing.

The response of an Nd-doped optical fiber laser to modulation of the pump power reveals that the polarization of the laser light plays an important role in the linear and nonlinear dynamics of this laser. Experiments have been carried out under pulsed or sinusoidal modulation of the pump power as well as in the continuous-wave regime. Most of the observed phenomena may be interpreted in the framework of a theory of the two-mode laser in which each mode is associated with one polarization eigenstate of the laser. Effects such as the appearance of slow modes and antiphase behavior have also been observed. A model of a two-mode laser including spontaneous emission is theoretically and numerically analyzed. It reproduces well most of the experimental findings.

The effect of radiative decay and collisional damping on the delayed-field population transfer in the three-level system is investigated. The simple analytical steady-state solution of the three-level system, interacting with two intense delayed laser pulses in the dressed-state frame, is obtained and analyzed. The radiative relaxation transfers the system to the ‘‘trapped state,’’ which is shown to be stable relative to both radiative decay and collisional damping. The effect of an additional level ‖m〉, situated close to the final level ‖3〉, on the population-transfer process is considered. The field-induced relaxation rate from intermediate level ‖2〉 to level ‖3〉 is shown to be a very easily controllable parameter of the field intensities and their relative delay. This result leads to selective population transfer to the desired level and may prevent leaking out of the three-level system. Theoretical analysis corresponds entirely to recent experimental data [U. Gaubatz et al., J. Chem. Phys. 92, 1 (1990); 92, 5363 (1990)].

The exact solution of the self-consistent problem of four-wave mixing in a resonant medium of four-level atoms (molecules) is obtained. The pump depletion, level-population movement, and atomic-transition saturation are explicitly taken into account. It is shown that (i) in the process of transformation the generated pulse is split into a train of subpulses, and their number is strongly related to the length of propagation; in the case of saturation, this splitting does not occur; (ii) the conversion-efficiency definition problem is reduced to solution of a third-order polynomial; (iii) by variation of the initial value of the wave-vector mismatch Δ${\mathit{k}}_0$, the processes of conversion may lead to the complete depletion of one of the pump beams; (iv) a prediction is made of the intersection of quasienergetic terms in the process of energy transfer; this phenomenon is shown to be responsible for the experimentally observed disruption of generation.

The formalism that we have previously developed for the phase difference between two quantized electromagnetic fields, which is intimately connected with the measurement process, is explored further, both theoretically and experimentally. We calculate the higher moments of the measured cosine and sine operators for certain two-mode Fock states ‖${\mathit{n}}_1$,${\mathit{n}}_2$〉, and show how the measurement itself exerts a bias on the outcome. We find that the corresponding phase difference becomes uniform over the interval 0 to 2π only in the limit ${\mathit{n}}_1$,${\mathit{n}}_2$→∞. On the other hand, the phase difference associated with the product of a coherent state ‖v〉 with a Fock state can be random for large ‖v‖, because of the availability of an infinite number of photons. Several of our theoretical predictions are compared with predictions based on the Susskind-Glogower and the Pegg-Barnett operators, and they are also tested by experiment. We find that the experimental results confirm our theory in every case, and this includes tests of the higher moments of the measured cosine operator.