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We study theoretically the non linear polarization dynamics of Vertical-Cavity Surface-Emitting Lasers in the presence of an external cavity providing delayed optical feedback and cross polarization re-injection. We show that far from the laser threshold, the dynamics remains confined close to the equatorial plane of a Stokes sphere of a given radius and we reduce the dynamics to a dynamical system composed of two phases: the orientation phase of the quasi-linear polarization and the optical phase of the field. We explore the complex modal structure given by the double feedback configuration and recovers as particular cases the Lang-Kobayashi modes and the modes founds by Giudici et al. [1]. We also re-interpret the square waves switching dynamics as phase kinks.

We show that the analog of the geometric phase for non-Hermitian coupled waveguides with $\mathcal{PT}$ symmetry and at least one periodically varying parameter can be purely imaginary, is hence no longer a true phase and instead a real multiplier, and will consequently result in the amplification of Floquet sidebands in the system. The sideband peaks seen in the spectrum of the system's eigenstates after evolution along the waveguides can be directly mapped to the spectrum of the derivative of the geometric function. The sidebands are magnified (becoming virtually unstable) as the exceptional point of the system is approached, and nonadiabatic effects begin to appear. Because the system cannot evolve adiabatically in the vicinity of the exceptional point, $\mathcal{PT}$ symmetry will be observed breaking earlier than theoretically predicted.

The relative phase between two uncoupled Bose-Einstein condensates tends to attain a specific value when the phase is measured. This can be done by observing their decay products in interference. We discuss exactly solvable models for this process in cases where competing observation channels drive the phases to different sets of values. We treat the case of two modes which both emit into the input ports of two beam splitters and of a linear or circular chain of modes. In these latter cases, the transitivity of the relative phase becomes an issue.

Temporal cavity solitons (CSs) enable the generation of coherent Kerr frequency combs with a broad bandwidth. Here we derive analytically and numerically the relationship between the carrier-envelope-phase (CEP) slip of CSs and the pump phase detuning. To preserve the stability of CSs, CEP slips always equal the pump phase detuning. Since the CEP slip for solitons has a contribution from the Kerr effect, the locking between the CEP slip and pump phase detuning results in a universal constraint on the peak power for CSs. The CEP dynamics is also found to affect the stability of CSs. When the CEP slip fails to follow the pump phase detuning, CSs become unstable, exhibiting periodic or chaotic breathing. Our results can be important for the self-referencing of Kerr frequency combs.

We study the phase dynamics of a single-mode ring laser described by the complex Maxwell-Bloch equations. We identify three reference-frame frequencies and determine the properties of the field dynamics observed in these frames. In one of these reference frames, the phase jumps are always equal to π, irrespective of the detuning, while in another reference frame quasiperiodic field portraits reduce to periodic field portraits. We also apply the recent theory of Ning and Haken [Phys. Rev. Lett. 68, 2109 (1992)] to prove that the laser phase can be decomposed into a geometrical component that is frame invariant and a dynamical component that is frame dependent

We study the phase and amplitude distribution of superfluid (SF) order parameters for spin-orbit-coupled two species bosons in a two-dimensional finite-size square lattice using inhomogeneous mean-field analysis. We demonstrate how phase distribution of the SF order parameter evolves as we tune the spin-orbit coupling $\ensuremath{\gamma}$ and $t$, the spin-independent hopping in the strong-coupling limit. For $t\ensuremath{\gg}\ensuremath{\gamma}$, we find the homogeneous superfluid phase where the phase of the SF order parameter is uniform. As we increase $\ensuremath{\gamma}$, spatial inhomogeneity in the phases of the SF order parameter grows leading to a twisted superfluid phase. For $t\ensuremath{\sim}\ensuremath{\gamma}$, competing orderings in the phase distribution are observed. At large $\ensuremath{\gamma}$ limit, a ferromagnetic stripe ordering appears along the diagonal. We explain that this is due to the frustration bought in by the spin-orbit interaction. Isolated vortex formation is also shown to appear. The effect of the detuning field $\ensuremath{\delta}$ on the distribution of phases and amplitudes of the order parameter has also been studied. We also investigate the possible collective modes for this finite-size system. In a deep superfluid regime we derive the Euler-Lagrange equation of motion for the phases and the dynamics of lowest normal modes are discussed.

Theoretical treatment is given to the possibility of the existence of propagating confined states in the nonlinear phase equation by generalizing stationary confined states. The nonlinear phase equation is set forth for the case of propagating patterns with long wavelengths and low-frequency modulation. A large range of parameter values is shown to exist for propagating confined states which have spatially localized regions which travel on a background with unique wavelengths. The theoretical phenomena are shown to correspond to such physical systems as spirals in Taylor instabilities, traveling waves in convective systems, and slot-convection phenomena for binary fluid mixtures.

Confined states in phase dynamics\char21{}the coexistence of states with different wavelengths\char21{}are further studied. In particular, a stable and accurate numerical code is developed to study the dynamic equation of motion for two sets of boundary conditions\char21{}a pinned phase and an unpinned phase. With a pinned phase, stable confined states are found to exist for a wide range of parameter values. With an unpinned phase, the localized states are at best neutrally stable. Also, analytic solutions are found for the stationary confined states.

The basic features characterizing the dynamical evolution of the phase of a detuned-laser field under an unstable regime are physically interpreted in terms of dispersive and dynamical effects. A general method for obtaining any attractor projection containing the phase information is established, which provides evidence for the heteroclinic character of the attractor in the presence of cavity detuning for any emission regime.

We make a geometric study of the phases acquired by a general, pure bipartite two-level system after a cyclic unitary evolution. The geometric representation of the two particle Hilbert space makes use of Hopf fibrations. It allows for a simple description of the dynamics of the entangled state's phase during the whole evolution. The global phase after a cyclic evolution is always an entire multiple of {pi} for all bipartite states, a result that does not depend on the degree of entanglement. There are three different types of phases combining themselves so as to result in the n{pi} global phase. They can be identified as dynamical, geometrical, and topological. Each one of them can be easily identified using the presented geometric description. The interplay between them depends on the initial state and on its trajectory, and the results obtained are shown to be in connection to those on mixed state phases.

We show that the classical dynamics underlying the mean-field description of homogeneous mixtures of spinor $F=1$ Bose-Einstein condensates in an external magnetic field is integrable as a consequence of number conservation and axial symmetry in spin space. The population dynamics depends only on the quadratic term of the Zeeman energy and on the strength of the spin-dependent term of the atom-atom interaction. We determine the equilibrium populations as function of the ratio of these two quantities and the miscibility of the hyperfine components in the ground state spinors are thoroughly discussed. Outside the equilibrium, the populations are always a periodic function of time where the periodic motion can be a libration or a rotation. Our studies also indicate the absence of metastability.

We study the dynamics of a multimode laser diode with delayed optical feedback operating in the low-frequency fluctuations regime. We show that a multimode extension of the Lang-Kobayashi equations that takes into account spontaneous emission noise predicts two qualitatively different behaviors of the laser on the picosecond time scale. Individual modes of the laser can emit pulses in-phase or oscillate out-of-phase, depending on the operating parameters. The corresponding statistical distributions are in good agreement with two recent experiments.

We discuss the long-wavelength, low-frequency excitations of non- equilibrium systems with a helical structure as observed in spiral vortex flow in the Taylor instability in long cylinders. We find that a single-phase equation can give rise to a propagating mode, a situation unknown from hydrodynamics in local thermodynamic equilibrium.

The growth of statistical domains in quenched Ising-like p-state clock models with p = 3 or more is investigated theoretically, reformulating the analysis of Ohta et al. (1982) in terms of a phase variable and studying the dynamics of defects introduced into the phase field when the phase variable becomes multivalued. The resulting defect/phase domain-growth equation is applied to the interpretation of Monte Carlo simulations in two dimensions (Kaski and Gunton, 1983; Grest and Srolovitz, 1984), and problems encountered in the analysis of related Potts models are discussed. In the two-dimensional case, the problem is essentially that of a purely dissipative Coulomb gas, with a sq rt t growth law complicated by vertex-pinning effects at small t.

We analyze the phase properties of a strong cavity field interacting with an ensemble of initially excited N two-level atoms. Using the Pegg-Barnett phase formalism [Phys. Rev. A 39, 1665 (1989)], we calculate the phase probability distribution as well as the phase variance. The phase probability density exhibits a (N+1)-peak structure at the initial stages of the evolution. The phase variance is used to illustrate the progressive randomization of the phase on the long-time evolution. The difference in the phase dynamics for the N even and the N odd case is pointed out.

We study the correlated evolutions of two Rydberg atoms, interacting via a van der Waals (vdW) potential ${V}_{6}$ and driven by a laser field of detunings ${\mathrm{\ensuremath{\Delta}}}_{1}$ and ${\mathrm{\ensuremath{\Delta}}}_{2}$. The two atoms may exhibit the in-phase dynamics with identical Rydberg populations or the antiphase dynamics with complementary Rydberg populations, depending on their initial states. For a moderate vdW potential far from the blockade regime, the in-phase or antiphase dynamics can be attained along two intersecting lines in the parameter space of ${\mathrm{\ensuremath{\Delta}}}_{1}$ and ${\mathrm{\ensuremath{\Delta}}}_{2}$ with an exact or approximate figure of merit, respectively. Note, in particular, that the exact in-phase dynamics is trivial because it requires identical detunings while the approximate in-phase, exact antiphase, and approximate antiphase dynamics are nontrivial because they require distinct detunings. The specific requirements on ${\mathrm{\ensuremath{\Delta}}}_{1}$ and ${\mathrm{\ensuremath{\Delta}}}_{2}$ for both in-phase and antiphase dynamics can be understood by considering the balanced transitions from two initially populated states to two initially empty states in the double-atom state basis.

We show that self-organization occurs in the phase dynamics of soliton modelocking in paramet- ric frequency combs. Reduction of the Lugiato-Lefever equation (LLE) to a simpler set of phase equations reveals that this self-organization arises via mechanisms akin to those in the Kuramoto model for synchronization of coupled oscillators. In addition, our simulations show that the phase equations evolve to a broadband phase-locked state, analogous to the soliton formation process in the LLE. Our simplified equations intuitively explain the origin of the pump phase offset in soliton- modelocked parametric frequency combs. They also predict that the phase of the intracavity field undergoes an anti-symmetrization that precedes phase synchronization, and they clarify the role of chaotic states in soliton formation in parametric combs.

We present a theoretical analysis of spatial correlations in a one-dimensional driven-dissipative non-equilibrium condensate. Starting from a stochastic generalized Gross-Pitaevskii equation, we derive a noisy Kuramoto-Sivashinsky equation for the phase dynamics. For sufficiently strong interactions, the coherence decays exponentially in close analogy to the equilibrium Bose gas. When interactions are small on a scale set by the nonequilibrium condition, we find through numerical simulations a crossover between a Gaussian and exponential decay with peculiar scaling of the coherence length on the fluid density and noise strength., Comment: 5 pages, 2 figures, supplemental material

Recently, we demonstrated that the liquid-solid interface of the isotropic model of lamellar eutectic growth may undergo a parity-breaking transition from a symmetric to a tilted state. We found that the bifurcation to the asymmetric state is supercritical. Now we have solved the full boundary integral equation of the system assuming anisotropic surface tension. We observe that generically the supercritical bifurcation becomes imperfect, as could be expected from a simple phenomenological picture. In order to study whether finite domains of tilted states can exist under these circumstances, we consider a simple model for the coupling between tilt angle and phase dynamics that exhibits an imperfect bifurcation. We find that if the anisotropy is not too strong, the coupling leads to a picture that retains many of the qualitative features of the phenomenological approach given by Coullet, Goldstein, and Gunaratne [Phys. Rev. Lett. 63, 1954 (1989)] for subcritical bifurcations. Furthermore, we offer a natural explanation for the experimental finding that on creation of a tilted state lamellae of a given grain preferentially tilt in one direction, not in the opposite one.

We investigate the relaxation effects on the quantum dynamics in a two-state molecular system interacting with a single-mode strongly amplitude-squeezed coherent field using the second-order Monte Carlo wave-function method. The molecular population inversion (collapse-revival behavior of Rabi oscillations) is known to show the echoes after each revival, which are referred to as ringing revivals, in the case of strongly squeezed coherent fields with oscillatory photon-number distributions due to the phase-space interference effect. Two types of relaxation effects, i.e., cavity relaxation (the dissipation of an internal single mode to outer mode) and molecular coherent (phase) relaxation caused by nuclear vibrations on ringing revivals are investigated from the viewpoint of the quantum-phase dynamics using the quasiprobability (Q function) distribution of a single-mode field and the off-diagonal molecular density matrix {rho}{sub elec1,2}(t). It turns out that the molecular phase relaxation attenuates both the entire revival-collapse behavior and the increase in {rho}{sub elec1,2}(t) during the quiescent region, whereas a very slight cavity relaxation particularly suppresses the echoes in ringing revivals more significantly than the first revival but hardly changes a primary variation in envelope of {rho}{sub elec1,2}(t) in the nonrelaxation case.

A closer examination of Eberly ``revivals,'' Berry ``quantum fractals,'' and Schleich ``carpets'' exposes a simple unitary-cyclic group structure in the dynamics of wave packets for rotorlike systems with quadratic spectra. Quantum quasifractal patterns are revealed to be permeated by self-similar ``X-braid'' structures seen by Hofstadter in Zeeman-perturbed band structures and in other spectra related to circle maps. The patterns are shown to correspond to revival dynamics in the flow of energy or information through these N-level systems with X-braids being directly related to quantum beats or elementary U(2) resonance transfer. ${C}_{N}$-group analysis and Farey arithmetic discretely catalog X-braid hierarchy and suggest the possibility of homocyclic nanostructures or ${C}_{N}$ circuits that are base-N digital quantum computing registers. The phase dynamics corresponds to a fairly sophisticated integer arithmetic that includes an output of all rational numbers resolvable by a given energy input. ${C}_{N}$ circuits automatically give all factors of N. The dynamics of a ${C}_{6}$ device is discussed in detail.

A systematic theoretical and computational investigation of the microscopic factors which determine the phase of the signal field in time-resolved quasidegenerate three-pulse scattering experiments is presented. The third-order phase-matched response is obtained by density-matrix perturbation theory using a Green-function formalism for a system composed of two well-separated sets of closely spaced energy levels. Equations for calculating the electric field of four-wave-mixing signals generated by path-length delayed pulses are given. It is found that the phase of the signal field is determined by the excitation pulse phases, the dynamics of the nonlinear polarization decay, the product of four transition dipole matrix elements, and by a pulse-delay-dependent phase modulation at the frequency of the first dipole oscillation in the four-wave-mixing process. Analytic results for a two-level Bloch model show the phase shift from rapid nonlinear polarization decay. The product of dipole matrix elements is real and positive for three-level processes (bleached ground-state absorption and excited-state emission), but can be real and negative for some four-level Raman processes. The pulse-delay-dependent phase modulation treated here is closely related to the interferometric pulse-delay-dependent amplitude modulation observed in some collinear experiments, and plays a role in producing photon echos in inhomogeneously broadened samples. Numerical calculations of phase-resolved electric fields for finite duration pulses using a Brownian oscillator model appropriate for condensed-phase dynamics are presented. The ability of pulse-delay-dependent phase modulation to encode the frequency of the initially excited dipole onto the phase of the signal field can be exploited to examine energy-level connectivity, reveal correlations hidden under the inhomogeneous lineshape, and probe relaxation pathways in multilevel systems.

We present an experimental and theoretical study of the response of a dipolar supersolid to a Bragg excitation at high-energy defined by the impulse approximation regime. We experimentally observe a continuous reduction of the response when tuning the contact interaction from an ordinary Bose-Einstein condensate to a supersolid state and ultimately to an incoherent array of droplets. Already in the supersolid regime, the observed reduction is faster than the one theoretically predicted by the Bogoliubov--de Gennes theory. By comparing experiments and theories, we are able to attribute this discrepancy to the presence of coherent phase dynamics induced by the crossing of the phase transition. The phase variations are found to change character along the phase diagram and become predominantly incoherent only when reaching the incoherent-droplet regime.