Table of contents

By definition, an ensemble of quantum trajectories reproduces the density matrix of an open system after ensemble averaging. We present a general class of stochastic Schrödinger equations for quantum trajectories which unifies and extends existing methods. Various quantum jump and state-diffusion methods are recognized as opposite limiting cases. The inherent freedom of choice can be used to significantly reduce the calculation time in specific cases. Examples of this are methods with adaptive noise, for which a specific observable becomes noise free to first order. Linear stochastic equations can also be constructed, which allow analytical solutions. As an illustration, the decoherence of Schrödinger cat states of a cavity mode is discussed.

Second-harmonic generation in the no-energy-transfer regime can be a source of quasi-stationary sub-Poissonian light as was recently shown by Bajer et al (Bajer J, Haderka O and Perina J 1999 J. Opt. B: Quantum Semiclass. Opt.1 529). We generalize their results for higher-harmonic generation by applying the numerical method of Hamiltonian diagonalization and the analytical semiclassical description of classical trajectories. The quasi-stationary behaviour of the sub-Poissonian photocount noise in the no-energy-transfer regime is explained. An approximate formula for the Fano factor is derived for arbitrary harmonics. It is predicted that the deepest quasi-stationary reduction of photocount noise in the no-energy-transfer regime is achieved in the third-harmonic generation.

A system comprising two nonlinear thin films irradiated from both sides by monochromatic light is considered. It is shown that interplay of a subcritical pitchfork symmetry-breaking bifurcation and a pattern-forming instability gives rise to the formation of symmetry-breaking soliton-like structures below the onset of the above instabilities.

This review paper is devoted to amplification and lasing without population inversion involving atomic transitions in gas media. We start by discussing the main motivation in inversionless lasing research, namely, the generation of short-wavelength laser light. Then, we review the basic physics of inversionless lasing in two-level and, eventually, in three- and multilevel atomic configurations. Finally, we summarize the current state of the art of LWI experiments and indicate the main difficulties with respect to short-wavelength laser generation.

In this paper, the disentanglement-free state is prepared by using the cavity QED technique. Through this disentanglement-free state we have obtained a long-distance Einstein-Podolsky-Rosen (EPR) pair. By using n EPR pairs of distance-atom and n distance-cavity, the entanglement of the n distance-cavity has been obtained. This entanglement can then be used in quantum communications.

A lower bound on the amount of noise that must be added to a GHZ-like entangled state to make it separable (also called the random robustness) is found using the transposition condition. The bound is applicable to arbitrary numbers of subsystems, and dimensions of Hilbert space, and is shown to be exact for qubits. The new bound is compared with previous such bounds on this quantity, and found to be stronger in all cases. It implies that increasing the number of subsystems, rather than increasing their Hilbert space dimension, is a more effective way of increasing entanglement. An explicit decomposition into an ensemble of separable states, when the state is not entangled, is given for the case of qubits.

A multiple Schrödinger cat state can be defined as a quantum superposition of an arbitrary number of coherent states. In this paper, we study the process of decoherence of such multiple states in interaction with their environments, using the model of a harmonic oscillator coupled to a thermal bath. The evolution of a superposition of various coherent states into a statistical mixture is described with the Wigner distribution function. The decoherence effect is expressed in analytic form, and it is seen that the damping time is very sensitive to the temperature. Multiple superpositions of states can be conveniently described using crystallographic group techniques; numerical calculations of the Wigner functions for some typical groups are also presented.

An experimental comparison of several operational phase concepts is presented. In particular, it is shown that statistically motivated evaluation of experimental data may lead to a significant improvement in phase fitting upon the conventional procedure of Noh et al (1993 Phys. Rev. Lett.71 2579). The analysis is extended to the asymptotic limit of large intensities, where a strong evidence in favour of multi-dimensional estimation procedures has been found.

We consider how the conventional spectroscopic and interferometric schemes can be rearranged to serve for reconstructing quantum states of physical systems possessing SU(2) symmetry. The discussed systems include a collection of two-level atoms, a two-mode quantized radiation field with a fixed total number of photons, and a single laser-cooled ion in a two-dimensional harmonic trap with a fixed total number of vibrational quanta. In the proposed rearrangement, the standard spectroscopic and interferometric experiments are inverted. Usually one measures an unknown frequency or phase shift using a system prepared in a known quantum state. Our aim is just the inverse, i.e., to use a well-calibrated apparatus with known transformation parameters to measure unknown quantum states.

We show that for appropriate choices of parameters it is possible to achieve photon blockade in idealized one-, two- and three-atom systems. We also include realistic parameter ranges for rubidium as the atomic species. Our results circumvent the doubts cast by recent discussion in the literature (Grangier et al 1998 Phys. Rev. Lett.81 2833, Imamoglu et al 1998 Phys. Rev. Lett.81 2836) on the possibility of photon blockade in multi-atom systems.

We propose the use of a feedback mechanism to control the level of quantum noise in a radiation field emerging from a pendular Fabry-Perot cavity. It is based on the possibility of performing quantum-nondemolition measurements by means of optomechanical coupling.

Using the truncated Wigner function approach and the Bloembergen solutions for nondegenerate down-conversion we calculate the conversion efficiency of spontaneous parametric down-conversion. In addition we determine higher moments of the pump and signal photon number. We find the upper bound for the efficiency of energy transfer and give a physically intuitive explanation for its existence. The depletion of the pump mode is immediately characterized by a strong destruction of its coherence.

We introduce the quantitative measures characterizing the rates of decoherence and thermalization of quantum systems. We study the time evolution of these measures in the case of a quantum harmonic oscillator whose relaxation is described in the framework of the standard master equation, for various initial states (coherent, `cat', squeezed and number). We establish the conditions under which the true decoherence measure can be approximated by the linear entropy 1-Tr <sup>2</sup>. We show that at low temperatures and for highly excited initial states the decoherence process consists of three distinct stages with quite different time scales. In particular, the `cat' states preserve 50% of the initial coherence for a long time interval which increases logarithmically with the increase of the initial energy.

Nonclassical properties of a particular class of quantum superpositions of two coherent states of a single-mode quantized electromagnetic field are brought to light and discussed. A new and simple way of engineering such states is presented. Our proposal, developed in the context of micromaser theory, exploits the passage of one atom only through a high-Q bimodal cavity supporting two electromagnetic modes of different frequencies.

The Ginibre ensemble of complex random matrices is studied. The complex-valued random variable of the second difference of complex energy levels is defined. For the N = 3 dimensional ensemble, we calculate distributions of the second difference real and imaginary parts, as well as its radius and of its argument (angle). For the generic N-dimensional Ginibre ensemble an exact analytical formula for the second difference's distribution is derived. Comparison with the real-valued random variable of the second difference of adjacent real-valued energy levels for a Gaussian orthogonal, unitary and symplectic ensemble of random matrices, as well as for a Poisson ensemble, is provided.

A Cauchy-Schwarz inequality in photon noise spectra is derived and discussed for macroscopic fields of light. Violation of this Cauchy-Schwarz inequality is demonstrated experimentally in quantum noise-correlated light beams radiated from a pair of electrically coupled light-emitting diodes. Our measurements directly demonstrate that the violation of this Cauchy-Schwarz inequality can be observed in the absence of sub-shot noise fluctuations in either the single-beam intensity or the dual-beam intensity difference. Therefore, the dual-beam quantum correlation condition required for the violation of this Cauchy-Schwarz inequality is not associated with sub-shot noise fluctuations.

Here we investigate some general properties of the so-called even circular state (ECS) produced in a cavity, consisting of a superposition of N coherent states |α_k⟩ (k = 1,2,...,N) with the same |α_k|. Several special states may emerge from this kind of superposition: in particular, when 1<<(e|α_k|²/N)^N<<4^N a Fock state with N quanta is produced, whereas when (e|α_k|²/N)^N<<1 one gets a vacuum state. We analyse the atomic scattering the ECSs produce when two-level atoms go through the cavity and also the non-classical depth of these states.

We propose three schemes to engineer a circular state (superposition of the harmonic oscillator coherent states on a circle in phase space) for the centre-of-mass motion of a trapped ion. We analyse the necessary duration of each laser pulse for constructing such states, and calculate the probability of obtaining the subtle superposition. We also show that it is possible to engineer Fock number states as a result of the interference effects in phase space of the coherent states superposition.

We show that the Wigner fuction of a cavity field can be constructed from the spontaneous emission spectrum in driven three-level atomic systems. We consider two configurations, the upper- and lower-level couplings, in which the driving field is the quantized field inside a cavity. In the proposed method, the Wigner function of the driving field is recovered in a straightforward manner, without much mathematical manipulation of the experimental data, from the spontaneous emission spectra.

In this paper, we study the influence of atomic decay on the linewidth of a two-photon micromaser using an exact Hamiltonian approach. We consider a three-level cascade atomic system (where each level has a finite lifetime) interacting with a single mode of the radiation field. The intermediate level is considered to be off resonant with the field. We calculate the linewidth for both small- and large-detuning limits. In both cases, the well known oscillatory behaviour of the linewidth vanishes for large decay rates. We also find a reduction in the linewidth in certain regions of interaction times. In the limit of small decay rates, we recover the linewidth of the ordinary two-photon micromaser. We also compare the results under the large-detuning limit with the results obtained using a phenomenological Hamiltonian approach.

A theory of intracavity fourth-harmonic generation has been developed in the constant intensity approximation with account of changes in the interacting wave phases. The effect both of the phase mismatch and of the absorption of all the interacting waves in a nonlinear medium on the frequency conversion efficiency has been clarified. It has been found that unlike the constant field approximation, in the constant intensity approximation the selection of optimal problem parameters such as the pump intensity, the crystal length and the relationship of the interacting wave phases can considerably increase the frequency conversion efficiency.

Interference between waves can yield quite counterintuitive results. Some time ago, Elizur and Vaidman (1993 Found. Phys.23 987) proposed a way to detect the presence or absence of a perfectly absorbing object by probing it with light, without any photons being absorbed. Here we present experimental results where a photographic film's shape is imaged on an identical piece of film without exposing the first film. This is possible by using resonant imaging, i.e. by holding the film to be imaged inside an optical resonator. Using the same technique, it is also possible to image nearly transparent objects. The resonant imaging will `amplify' small transmission variations rendering the nearly invisible clearly visible.

This issue contains a collection of papers devoted to dynamics and pattern formation in nonlinear optical systems. The articles originate from contributions to a conference on control of complex behaviour in optical systems and applications (COCOS) held in Münster, Germany, in October 1999. It was the second of a series of Euroconferences on Trends in Optical Nonlinear Dynamics. Physical Problems and Applications funded by the European Union through its TMR Programme and co-sponsored by various institutions. The topics of the conference covered all kinds of complex self-organization phenomena in nonlinear optics and this diversity is also reflected in the papers. The contributions cover a great variety of material systems (atomic vapours, nonlinear crystals, photorefractives, liquid crystal light valves, fibres, semiconductors, solid-state lasers, ...) and interaction schemes (nonlinear resonators, single-mirror feedback schemes, nonlinear beam propagation, four-wave mixing).

Though there is emphasis on the characterization and control of spatial structures including extended patterns as well as localized states, domain walls and solitons, several papers also discuss temporal instabilities and develop strategies for stabilization or optimal steering. In both fields there are a number of contributions considering semiconductor systems and their control. This is a strong indication that nonlinear dynamics in optics is on its way to making more and more useful contributions to applications. Some papers are also providing connections between two fields, either between patterns and localized structures or between purely temporal and spatial phenomena. They analyse e.g. the dynamics of coupled lasers, pulse trains in nonlinear fibres or spatiotemporal chaos. Last but not least, papers investigating vectorial effects or coherence phenomena (e.g. in inversionless amplification) demonstrate that there are features of optical systems that have no obvious counterpart in nonlinear systems of other kinds.

The diversity of the papers is even more remarkable, since the collection of papers presented here does not form `proceedings' in a comprehensive sense, but covers about twenty percent of the contributions to COCOS only. Nevertheless, it is our opinion that they are somehow representative of the European activities in the field of nonlinear dynamics in optics. It is our hope that the papers give an idea of how nonlinear optics can help in understanding general features of self-organization common to all kinds of complex systems on the one hand, and on the other hand how it can provide well-controlled, high performance devices for the photonic applications of tomorrow.

Finally, we thank the Editors of the journal for providing the forum for this presentation and the Editorial Office for all the work done.

The kinetics of the domain walls that occur in the degenerate optical parametric oscillator are studied within the propagation model. The formation of large intensity peaks for null and positive signal mistuning is shown to be associated with a dynamical scaling law ~t^1/3. In the parameter range where the degenerate optical parametric operator reduces to potential systems, the growth law ~t^1/2 is observed. It is also obtained for negative mistuning of order unity, and up to three times above the threshold, i.e. beyond the validity range of the Swift-Hohenberg model equation. In addition, we describe the labyrinth formation which displays self-similar growth with the law ~t^1/5.

We study analytically the evolution of the beam parameters in a system of layered nonlinear graded index fibres within the framework of the paraxial wave equation. It is shown that the fibre parameters can be chosen to induce resonant behaviour, a phenomenon which could be used to achieve pulse compression or to decouple the beam from the fibre.

We study propagation effects on lasing without population inversion in a Doppler-broadened V-type three-level system. In particular, we focus our analysis on frequency up-conversion lasing without inversion in atomic rubidium. In an atomic beam configuration, we show that it is possible to increase notably the probe gain per single pass through the active medium by detuning driving and probe fields out of one-photon resonance but maintaining the two-photon resonance condition. In a vapour cell configuration, we show that due to propagation effects the probe gain per single pass depends strongly on the driving field intensity at the entrance of the active medium. In fact, for appropriate parameter values, it is possible to reach values of the probe gain per single pass of around 40%. For this last case, we have considered the feedback of a ring laser cavity.

Recently it has been shown that pulse reshaping in a synchronously driven fibre ring resonator can give rise to the formation of a soliton array which may exist either in a `fluid' or `solid' state, called a soliton gas and soliton crystal, respectively. Here, we demonstrate in numerical simulations that a soliton gas can be cooled to a soliton crystal by a procedure analogous to evaporative cooling.

We show that the presence of cylindric metallic objects, such as waveguides and apertures, inside lasers with Fabry-Perot cavities modifies considerably the symmetry of these systems. This, in turn, changes the structure of the patterns that can be observed and could lead to the use of patterns as a way of studying properties of the scattering objects.

We report the first investigations of the dynamical behaviour of a system of three coupled Nd:YVO₄ microchip lasers, operating under stationary conditions near the threshold of phase locking. We observe a number of different types of intensity and phase correlation instabilities. These include synchronous intensity pulsations of two or three lasers as well as states where one intensity pulsation has a different amplitude and phase. This constitutes the first observation of localized synchronization in a system of three unmodulated coupled lasers.

We discuss various nonfeedback techniques, based on the knowledge of the phase space, to steer the turn-on of class B lasers. The features and pitfalls of single-step steering are analysed and the effectiveness of two-driving-step ballistic steering is demonstrated on the basis of a realistic model for an edge emitting semiconductor laser. Continuous steering is then introduced; simple criteria for its use are presented and applied numerically to the turn-on of a CO₂ laser. The potential applications resorting from this nonlinear dynamical analysis are addressed and a brief review of the relevant technical literature is presented.

In this paper we investigate spatio-temporal disorder in a liquid crystal light valve with two-dimensional feedback and modulated input. We focus on the specific case of an optical feedback with a 180° rotation. By modulating the input field with appropriate amplitude and frequency, chaotic domains are formed in different locations of the transverse space [1, 2]. Because of diffusion, a global description of the spatio-temporal dynamics requires a partial differential equation. However, the local dynamics of the chaotic domains can be embedded in low-dimensional manifolds. The detection of a variety of different unstable periodic orbits (UPOs) at different spatial locations and the presence of UPOs for different detector sizes are strong indications that the dimensionality of the model of spatio-temporal chaos in our system can be drastically reduced.

The interaction of pattern formation with an intrinsic polarization instability is studied in a modified single feedback mirror experiment. We report on hexagonal patterns in an inversion symmetric system which occur due to a symmetry-breaking polarization instability and on domain patterns which are stabilized by the modulational instability. For parameter values for which no symmetry breaking bifurcation occurs, we observe rhombic and triangular patterns.

Spontaneous pattern formation and competition in a nonlinear ring cavity are studied. Complex patterns arising from static and Hopf bifurcations and Hopf-static interactions have been numerically observed and analysed. Among them are flower-like patterns, alternating rolls and oscillating hexagonal structures.

We compare the morphology of the spatial structures displayed by an optical system consisting of a liquid crystal light valve (LCLV) in a feedback configuration, as the feedback is gradually tuned from purely diffractive to mixed interferential and diffractive. Different kinds of spatially coherent structures (e.g. hexagons, rolls), as well as localized structures and space-time turbulent patterns are observed. The features of the localized structures change with the parameter setting, and certain regions of the parameter space provide stable clusters of isolated spots (`molecules'). Numerical simulations based on a Kerr-like model of the LCLV are in agreement with the experimental observations. We analyse the links between the observed behaviours and the results of a linear-stability analysis of the underlying homogeneous stationary states.

The spatial mode structure of circular broad-area bottom-emitting vertical-cavity surface-emitting lasers is investigated close to threshold. The devices display very high-order flower-like Laguerre modes as well as structures with a Cartesian symmetry in spite of the original O(2) symmetry. In some cases the latter resemble tilted waves more than Hermite-Gaussian modes. Spatial structures appear for both sides of the detuning between the gain peak and the cavity resonance. The results indicate the necessity to refine the theoretical models to include realistic boundary conditions and inhomogeneities.

We present extensive studies on feedback-induced instabilities in semiconductor lasers (SLs) subject to delayed optical feedback. We demonstrate that a sufficient reduction of the linewidth enhancement factor α changes the dynamical structure of the system such that permanent emission in a stable emission state is achieved. This behaviour can be well understood on the basis of the Lang-Kobayashi rate equation model. We give first experimental evidence for its major theoretical predictions concerning the stable emission state and investigate the robustness of this stable state against external perturbations. We demonstrate that noise-induced escape from the basin of attraction of the stable state shows similarities to the classical problem of thermally induced escape from a potential well. Thus, we have developed and realized experimentally an efficient concept to avoid and stabilize feedback-induced instabilities in SLs.

We study the selection between hexagonal and square polarization patterns in a sodium vapour cell with optical feedback provided by a single mirror. Scanning an external magnetic field, a transition sequence from squares via negative hexagons, squares, and positive hexagons again to squares is observed close to the instability onset. Well above threshold the hexagons give way to squares in a secondary bifurcation. These observations demonstrate how the macroscopic symmetries of optical patterns can be controlled by manipulating the microscopic symmetries of the light-matter interaction. The key to this control is the vectorial degrees of freedom of the optical pattern-forming system.

Recent experimental work demonstrates a rich transition scenario between patterns of different symmetry (hexagons and squares) close to and beyond threshold in a sodium vapour cell with optical feedback dependent on the magnitude of an external magnetic field. We reproduce the observations in numerical simulations and explain them by calculating the quadratic coupling coefficient between inhomogeneous modes which determines the pattern selection close to threshold. The observed scenario is in accordance with expectations based on general symmetry principles and should occur in many systems, but apparently only in optical systems is it possible to tune the parameters continuously over a sufficiently large area to observe the complete bifurcation scenario.

We investigate the formation of transverse travelling waves corresponding to tilted (off-axis) emission in cascade lasers with plane-parallel resonators. The interplay between transverse wavenumber selection and the dynamic Stark effect is discussed. When a relationship between these two phenomena exists for large intermediate detunings, the wavenumber value can be used to distinguish one-photon from two-photon emission processes.

When diffraction operates together with chromatic dispersion, the type II second-harmonic generation exhibits a three-dimensional (3D) modulational instability leading to the formation of 3D periodic solutions travelling at the group velocity of light within the cavity. Numerical analysis reveals the occurrence of 3D localized structures in the form of isolated light drops or cylinders. They are specific to the parameter range where 3D stable periodic patterns coexist with the single homogeneous steady state.

We show the first evidence of patterns in a passive resonator made of GaAlAs vertical-cavity microresonators. Rolls, rhombs and hexagons could be observed depending on the wavelength detuning. We show and discuss how cavity resonance fluctuations affect the mechanisms of pattern selection.

We present experimental results of the dynamics in a semiconductor laser operating on several longitudinal modes subject to external feedback. Both low-frequency fluctuations (LFFs) and locked states are studied through observing the outputs from the dominant mode, all modes except the dominant mode, and all modes. Synchronized dropout events of LFFs are observed among each of the solitary modes while energy competition and trading occurs among these modes when the total output exhibits the locked state.

Recording of localized transmission gratings by four-wave mixing in non-local photorefractive media with strong response is considered. The input conditions to record dynamic gratings with a stable grating amplitude distribution as well as grating amplitude oscillations are defined. They are determined both by intensities and by a phase difference of input waves. The influence of fluctuations of wave phases on stability of the recorded gratings is considered. Methods of optical control of both the grating amplitude localization degree and the grating amplitude location are defined.