Vortex formation of coherent waves in nonseparable mesoscopic billiards.

The connection between the wavefunctions and the classical periodic orbits in a 2D harmonic oscillator is analytically constructed by using the representation of SU(2) coherent states. It is found that the constructed wavefunction generally corresponds to an ensemble of classical trajectories and its localization is extremely efficient. With the constructed wavefunction, we also analyse the property of the probability current density associated with the classical periodic orbit. The appearance of vortex structure in the quantum flow is clearly found to arise from the wave interference.

The localization of eigenfunctions around classical periodic orbits is studied numerically for the H-atom in a strong magnetic field by calculating their Husimi distribution in phase space. In contrast to the configuration space representation, the phase space distributions are simply structured: about 90% of eigenstates may be unambigously related to fixed points and invariant manifolds of periodic orbits, indicating that scars are the rule rather than the exception. In order to measure the influence of one particular orbit, we calculate the integrals of the energetically lowest 500 Husimi distributions along the orbit. Their incoherent superposition defines the scar strength distribution for the particular periodic orbit which is analyzed by Fourier transformation. The Husimi distribution at (q, p) in phase space may be represented as a scalar product of the wavefunction with a coherent state of the unperturbed system, i.e., a radial Gaussian wave packet located at (q, p) in the (regularized) Coulomb system. This simplifies the actual calculation of the Husimi distribution and allows to treat their incoherent superposition within Gutzwillers theory extended to matrix elements of an operator A, if we choose A to be the projector on a coherent state.

Anderson localization is related to exponential localization of a particle in the configuration space in the presence of a disorder potential. Anderson localization can be also observed in the momentum space and corresponds to quantum suppression of classical diffusion in systems that are classically chaotic. Another kind of Anderson localization has been recently proposed, i.e. localization in the time domain due to the presence of {\it disorder} in time. That is, the probability density for the detection of a system at a fixed position in the configuration space is localized exponentially around a certain moment of time if a system is driven by a force that fluctuates in time. We show that an electron in a Rydberg atom, perturbed by a fluctuating microwave field, Anderson localizes along a classical periodic orbit. In other words the probability density for the detection of an electron at a fixed position on an orbit is exponentially localized around a certain time moment. This phenomenon can be experimentally observed.

Vortex formation and vortex lattices in a Bose-Einstein condensate with Lee-Huang-Yang (LHY) correction

Path integration over closed loops and Gutzwiller's trace formula

We first give a short outline of the semiclassical periodic orbit theory that allows one to relate quantum shell fluctuations in a mean-field system to the periodic orbits of the corresponding classical system. Earlier applications to gross-shell effects in nuclei (ground-state deformations), metal clusters (ground-state deformations and supershells), and quantum dots (conductance oscillations in weak magnetic fields) are briefly summarized. We then report on a recent simple and transparent semiclassical interpretation of the origin of the mass asymmetry in the fission of heavy nuclei. Using only a few classical periodic orbits and a cavity model for the nuclear mean field, we explain the onset of left-right asymmetric shapes at the fission isomer minimum by the principle of constant action of the shortest orbits.

An oscillatory pattern in the smoothed quantum spectrum, which is unique to single-particle motions in a reflection-asymmetric superdeformed oscillator potential, is investigated by means of the semiclassical theory of shell structure. Clear correspondence between the oscillating components of the smoothed level density and the classical periodic orbits is found. It is shown that an interference effect between two families of the short periodic orbits, called supershell effect, becomes more significant with increasing reflection-asymmetric deformations. Possible origins of this enhancement phenomena as well as quantum signatures of period-multipling bifurcations are discussed in connection with stabilities of the classical periodic orbits.

The relationship between quantum degeneracies and the emergence of classical periodic orbits in the Fock-Darwin (FD) system is systematically explored by using the ladder operators in the Cartesian coordinates. The quantum-classical connection is analytically developed with the time-dependent coherent state that is theoretically verified to correspond to the Gaussian wave packet state with unitary transformation. The time-dependent coherent state is further used to derive the stationary coherent state that can be expressed as a superposition of degenerate eigenstates. More importantly, the stationary coherent state can alternatively be expressed as an integral of the Gaussian wave packet over a periodic orbit. With the integral representation, the quantum vortex structures of the stationary coherent states can be precisely characterized up to extremely high order. The vortex arrays are generally observed when the phase structures are near the crossing points and cusps of the classical trajectories. Finally, the quantum Fourier transform is employed to verify that the circularly symmetric eigenstate physically corresponds to an ensemble of classical hypotrochoid orbits. This work can provide a pedagogical insight into the formation of Laguerre-Gaussian eigenmodes and hypotrochoid geometric modes in degenerate spherical cavities.

The quantum form-factor K( tau )-the Fourier transform of the spectral auto-correlation function-may be represented semiclassically in terms of a sum over classical periodic orbits. We consider the problem of how this approximation behaves in the limit of long (scaled) time tau . It is shown that whilst K itself tends to unity, the periodic-orbit sum typically grows exponentially as tau to infinity . This behaviour is related to the fact that leading-order semiclassical quantization methods yield complex eigenvalues with imaginary parts that are of higher order in Planck's constant. Divergence from the quantum limit begins when tau = tau *(h(cross)), which, for typical two-degrees-of-freedom systems and maps, is shown to be independent of h(cross) as h(cross) to 0. In the case of the baker's map, however, quantum diffraction from the classical discontinuity instead causes the analogue of tau * to tend to zero like N-12/, Where N is the integer that corresponds to the inverse of Planck's constant. This is in agreement with recent numerical studies. Finally, we consider the implications of the semiclassical divergence studied here for the method developed by Argaman et al. (1993) of investigating correlations between the periodic orbits of chaotic systems.

It was recently shown (Keating J P and Prado S D 2001 Proc. R. Soc. A 457 1855–72) that, in the semiclassical limit, the scarring of quantum eigenfunctions by classical periodic orbits in chaotic systems may be dramatically enhanced when the orbits in question undergo bifurcation. Specifically, a bifurcating orbit gives rise to a scar with an amplitude that scales as ℏα and a width that scales as ℏω, where α and ω are bifurcation-dependent scar exponents whose values are typically smaller than those (α = ω = ½) associated with isolated and unstable periodic orbits. We here analyse the influence of bifurcations on the autocorrelation function of quantum eigenstates, averaged with respect to energy. It is shown that the length-scale of the correlations around a bifurcating orbit scales semiclassically as ℏ1−α, where α is the corresponding scar amplitude exponent. This imprint of bifurcations on quantum autocorrelations is illustrated by numerical computations for a family of perturbed cat maps.

The arbitrary trajectory quantization method (ATQM) is a time dependent approach to quasiclassical quantization based on the approximate dual relationship that exists between the quantum energy spectra and classical periodic orbits. It has recently been shown however, that, for polygonal billiards, the periodicity criterion must be relaxed to include closed almost-periodic (CAP) orbit families in this relationship. In light of this result, we reinvestigate the ATQM and show that at finite energies, a smoothened quasiclassical kernel corresponds to the modified formula that includes CAP families while the delta function kernel corresponding to the periodic orbit formula is recovered as E-->infinity. Several clarifications are also provided.

Within the framework of the Ginzburg–Landau theory, we study the features of the localized nucleation of the order parameter in superconducting systems with inhomogeneous effective mass m of the Cooper pairs, which is due to the spatial modulation of the diffusion coefficient and/or fluctuations in the local anisotropy axis in the sample. In the asymptotics of the weak magnetic fields H, for which the magnetic length [Φ0/(2πH)]1/2, where Φ0 is the magnetic-flux quantum, is much shorter than the inhomogeneity scale, the spatial scale of the order parameter is determined by the sample-average coherence length and the regular lattice of the Abrikosov vortices is formed in the superconductor. In sufficiently strong magnetic fields H, the order parameter is localized near the minima of the coherence length ξ ∝ m −1/2, which results in an increase in the critical temperature and destruction of the regular lattice of the Abrikosov vortices. Therefore, competition between the two superconductivity-nucleation types is observed during a gradual increase in the magnetic field, which leads to the positive curvature of the phase-transition line. We have also studied the features of the temperature dependences of the upper critical magnetic field for some model spatial mass profiles of the Cooper pairs. The obtained results are in good agreement with direct numerical calculations.

A microwave driven five-level X-type atomic system is proposed to diffract weak probe light into higher-order directions via the phenomenon of electromagnetically induced grating. In the proposed scheme, the effect of various system and field parameters on its higher-order diffraction efficiency is studied. The present atomic scheme offers excellent control over higher-order diffraction intensities by utilizing the microwave induced quantum interference effect. It is observed that the desired first-order diffraction efficiency can be attained through optimal selection of microwave field strength and relative phase factor.

In this paper, we introduce two classes of superposed states generated by a superposition of two single-mode para-Bose coherent states (CS) with arbitrary relative phase factors. The first class is superposition of two opposite para-Bose CS and second class consists of two para-Bose CS, [Formula: see text] out of phase with each other. These states are reduced to the well-known superposed single-mode CS of the simple harmonic oscillator when the deformation parameter tends to one. We study the nonclassical properties of the introduced states by evaluating their para-Bose Fock state distribution, second-order squeezing, Klyshko’s and Mandel’s parameters. We investigate the essential role of coherent and deformation parameters on their nonclassicality. We show that some nonclassical properties of these states are different from those of the even and odd coherent states. Finally, we present a simple scheme for the physical generation of the introduced states.

We examine the vortex formation and vortex lattices in a rotating dipolar Bose-Einstein condensate (BEC) with two- and three-body interactions. For this aim, we numerically solve the quasi two-dimensional, cubic-quintic Gross-Pitaevskii equation with nonlocal dipole-dipole nonlinearity. We find the critical rotation frequency for vortex formation in a pure dipolar BEC and also in a dipolar BEC with presence of both two- and three-body interactions. We also show the vortex lattices in a dipolar BEC with two- and three-body interactions for large rotation frequencies.