Stationary Coherent States Research Articles

Transition from eigenmodes to geometric modes characterized by the quantum SU(2) coupled oscillator model: a review

The quantum and classical dynamics of the SU(2) coupled oscillator model are systematically reviewed to provide the quantum eigenstates and stationary coherent states for characterizing laser transverse modes from the analogy with the quantum-classical connection. The integral formula for the representation of the stationary coherent states derived from the evolution of the time-dependent wave packet state is completely reviewed. Several calculated results for the stationary coherent states are illustratively presented to display the spatial distributions for the quantum-classical transition and the plentiful variations of phase singularities. The overall review is believed to provide a comprehensive insight into laser transverse modes characterized by the stationary coherent states of the SU(2) coupled oscillator model.

Manifesting the connection between topological structures of quantum stationary coherent states and bundles of classical Lissajous orbits

Quantum stationary coherent states with spatial intensities localized on Lissajous orbits are theoretically explored by taking the inverse Fourier transform of the time-dependent coherent state. It is analytically verified that the stationary coherent state can be expressed as an integral of the Gaussian wave packet over the classical periodic orbit. With the derived integral, the phase singularities of the stationary coherent state can be precisely manifested from low-order to extremely high-order states. It is found that the phase singularities near the cross-point of the Lissajous orbits are generally arranged as rhombic vortex arrays. Finally, the stationary coherent states can be used as a basis to manifest the connection between topological structures of quantum eigenstates and bundles of classical Lissajous orbits.

Symmetry Breaking Induced Geometric Surfaces with Topological Curves in Quantum and Classical Dynamics of the SU(2) Coupled Oscillators

AbstractIn the three‐dimensional (3D) transversely symmetric oscillator, there are plentiful degeneracies and gaps in the quantum energy spectrum as a function of the ratio of the transverse to longitudinal frequency. It is theoretically verified that while the SU(2) interaction destroys the original degeneracies, numerous new degeneracies and gaps emerge around the original degeneracies to form a similar fine energy spectrum. The classical trajectories at the emergent degeneracies are analyzed to be localized on the 3D parametric surfaces which are constituted by the topologically invariant curves in the transverse tomography. The quantum coherent states are exploited to develop the wave functions that correspond to the 3D geometric surfaces in classical dynamics. Furthermore, the wave structures of the stationary coherent states at small quantum numbers are explored and found to display peculiar patterns with symmetries related to classical trajectories.

Extracting trajectory equations of classical periodic orbits from the quantum eigenmodes in two-dimensional integrable billiards.

The trajectory equations for classical periodic orbits in the equilateral-triangular and circular billiards are systematically extracted from quantum stationary coherent states. The relationship between the phase factors of quantum stationary coherent states and the initial positions of classical periodic orbits is analytically derived. In addition, the stationary coherent states with noncoprime parametric numbers are shown to correspond to the multiple periodic orbits, which cannot be explicable in the one-particle picture. The stationary coherent states are further verified to be linked to the resonant modes that are generally observed in the experimental wave system excited by a localized and unidirectional source. The excellent agreement between the resonant modes and the stationary coherent states not only manifests the importance of classical features in experimental systems but also paves the way to manipulate the mesoscopic wave functions localized on the periodic orbits for applications.

On the late phase of relaxation of two-dimensional fluids: turbulence of unitons

The two-dimensional ideal fluid and the plasma confined by a strong magnetic field exhibit an intrinsic tendency to organization due to the inverse spectral cascade. In the asymptotic states reached at relaxation the turbulence has vanished and there are only coherent vortical structures. We are interested in the regime that precedes these ordered flow patterns, in which there still is turbulence and imperfect but robust structures have emerged. To develop an analytical description we propose to start from the stationary coherent states and (in the direction opposite to relaxation) explore the space of configurations before the extremum of the functional that defines the structures has been reached. We find necessary to assemble different but related models: point-like vortices, its field theoretical formulation as interacting matter and gauge fields, chiral model and surfaces with constant mean curvature. These models are connected by the similar ability to described randomly interacting coherent structures. They derive exactly the same equation for the asymptotic state (sinh-Poisson equation, confirmed by numerical calculation of fluid flows). The chiral model, to which one can arrive from self-duality equation of the field theoretical model for fluid and from constant mean curvature surface equations, appears to be the suitable analytical framework. Its solutions, the unitons, aquire dynamics when the system is not at the extremum of the action. In the present work we provide arguments that the underlying common nature of these models can be used to develop an approach to fluid and plasma states of turbulence interacting with structures.

Synchronization and Bellerophon states in conformist and contrarian oscillators.

The study of synchronization in generalized Kuramoto models has witnessed an intense boost in the last decade. Several collective states were discovered, such as partially synchronized, chimera, π or traveling wave states. We here consider two populations of globally coupled conformist and contrarian oscillators (with different, randomly distributed frequencies), and explore the effects of a frequency–dependent distribution of the couplings on the collective behaviour of the system. By means of linear stability analysis and mean–field theory, a series of exact solutions is extracted describing the critical points for synchronization, as well as all the emerging stationary coherent states. In particular, a novel non-stationary state, here named as Bellerophon state, is identified which is essentially different from all other coherent states previously reported in the Literature. A robust verification of the rigorous predictions is supported by extensive numerical simulations.

Reduced description of exact coherent states in parallel shear flows.

A reduced description of exact coherent structures in the transition regime of plane parallel shear flows is developed, based on the Reynolds number scaling of streamwise-averaged (mean) and streamwise-varying (fluctuation) velocities observed in numerical simulations. The resulting system is characterized by an effective unit Reynolds number mean equation coupled to linear equations for the fluctuations, regularized by formally higher-order diffusion. Stationary coherent states are computed by solving the resulting equations simultaneously using a robust numerical algorithm developed for this purpose. The algorithm determines self-consistently the amplitude of the fluctuations for which the associated mean flow is just such that the fluctuations neither grow nor decay. The procedure is used to compute exact coherent states of a flow introduced by Drazin and Reid [Hydrodynamic Stability (Cambridge University Press, Cambridge, UK, 1981)] and studied by Waleffe [Phys. Fluids 9, 883 (1997)]: a linearly stable, plane parallel shear flow confined between stationary stress-free walls and driven by a sinusoidal body force. Numerical continuation of the lower-branch states to lower Reynolds numbers reveals the presence of a saddle node; the saddle node allows access to upper-branch states that are, like the lower-branch states, self-consistently described by the reduced equations. Both lower- and upper-branch states are characterized in detail.

Geometry of classical periodic orbits and quantum coherent states in coupled oscillators with SU(2) transformations

The geometry of classical dynamics in coupled oscillators with SU(2) transformations is explored and found to be relevant to a family of continuous-transformation orbits between Lissajous and trochoidal curves. The quantum wave-packet coherent states are derived analytically to correspond exactly to the transformation geometry of classical dynamics. By using the quantum wave-packet coherent states derived herein, stationary coherent states are constructed and are shown to possess spatial patterns identical to the transformation geometry between Lissajous and trochoidal orbits.

Commensurate anisotropic oscillator, SU(2) coherent states and the classical limit

We demonstrate a formally exact quantum–classical correspondence between the stationary coherent states associated with the commensurate anisotropic two-dimensional harmonic oscillator and the classical Lissajous orbits. Our derivation draws upon the earlier work of Louck et al (1973 J. Math. Phys. 14 692) wherein they have provided a non-bijective canonical transformation that maps, within a degenerate eigenspace, the commensurate anisotropic oscillator on to the isotropic oscillator. This mapping leads, in a natural manner, to a Schwinger realization of SU(2) in terms of the canonically transformed creation and annihilation operators. Through the corresponding coherent states built over a degenerate eigenspace, we directly effect the classical limit via the expectation values of the underlying generators. Our work completely accounts for the fact that the SU(2) coherent state in general corresponds to an ensemble of Lissajous orbits.

Fractional and integer angular momentum wavefunctions localized on classical orbits: the case of E = 0

In two-dimensional space, where the intermediate statistics between bosons and fermions are possible, we study a localization of some stationary coherent states on classical orbits in the potentials V(r) = −γν/rν, γν > 0 for the total energy E = 0. It is shown that excellent quantum-classical correspondence can be reached for two large classes of classical orbits with ν < −2 (open curves) and with ν > 2 (closed curves). To that purpose, it was necessary to introduce fractional angular momenta for some values of ν. They are a consequence of imposing special boundary conditions on the used wavefunctions.

Elliptic eigenstates for the quantum harmonic oscillator

A new family of stationary coherent states for the two-dimensional harmonic oscillator is presented. These states are coherent in the sense that they minimize an uncertainty relation for observables related to the orientation and the eccentricity of an ellipse. The wavefunction of these states is particularly simple and well localized on the corresponding classical elliptical trajectory. As the number of quanta increases, the localization on the classical invariant structure is more pronounced. These coherent states give a useful tool to compare classical and quantum mechanics and form a convenient basis to study weak perturbations.

Local differential geometry as a representation of the supersymmetric oscillator

The choice of a coordinate chart on an analytical Rn(Rna) provides a representation of the n-dimensional supersymmetric oscillator. The 1-parameter group of dilations provides a Euclidean evolution moving the system through a sequence of charts, that at each instant supply a Hilbert space by Cartan’s exterior algebra endowed with a suitable scalar product. Stationary states and coherent states are eigenstates of the Lie derivatives generating the dilations and the translations, respectively.

Collisional approach to dynamics of resonance atomic states in an external field

The following aspects of the dynamics of an atomic state in an external stationary field are assessed: (i) the rearrangement problem; (ii) the description of the appropriate final-channel wavefunctions; (iii) the analytical properties of the transition amplitude into the continuum. The rearrangement problem was solved by the introduction of the effective Hamiltonian, the eigenstates of which include both the initial state and final states ('modified states of continuum spectrum' MSCS) which describe the potential part of the exact wavefunction of the scattering problem. It is shown that the amplitude of decay and transition into MSCS as functions of time have an exact representation as a sum of resonance terms defined by a set of resonance states and the matrix elements of the shift R-matrix operator. Analysing the following expansion basis sets: stationary states, Volkov-Keldysh states and coherent states it is shown that the most appropriate one is a modified set of Volkov-Keldysh states.

Coherent States and Envelope Kinklike Exciton-Photon Coupled Modes in Nonlinear Frenkel Excitons Interacting with Radiation Fields

Interactions of nonlinear Frenkel excitons with an intense external radiation field (rf) are studied by taking the atomic polarization, the atomic population difference and the electric field as a set of variables to study the coherence and nonlinearity properties of the exciton-rf system. To equations for these quantities, terms are added phenomenologicaly to take care of pumping and dissipative effects. Two kinds of solutions are obtained under certain conditions, one representing the existence of the stationary coherent polarization and the macroscopic number of photons with wave number equal to that of the excitons and another envelope-kink-like, steady exciton-rf coupled modes around the stationary coherent states

Paraxial quantum propagation.

A quantum version of the classical paraxial approximation is established. After expanding the classical free-field wave equation in powers of \ensuremath{\theta} (the characteristic opening angle of the paraxial rays), the familiar time-dependent paraxial equation appears in zeroth order. The arguments employed in deriving this approximation scheme are used in turn to identify the subspace of the photon Fock space consisting of paraxial states of the field. The quantum version of the paraxial approximation is then obtained by restricting the Maxwell field operators to this domain. Exact equal-time commutation relations are recovered as expansions in \ensuremath{\theta}. In zeroth order, the theory yields a quantized analog of the classical paraxial wave equation, and formally resembles a nonrelativistic many-body theory. This formalism is applied to show that Mandel's local-photon-number operator and Glauber's photon-counting operator reduce, in zeroth order, to the same true-number operator. In addition, it is shown that the O(${\mathrm{\ensuremath{\theta}}}^{2}$) difference between them vanishes for experiments described by stationary coherent states.

A strange attractor in stationary coherent states

An example of the realization of a strange attractor in stationary coherent states is given.

Nonlinear Excitations and Coherent States in the Ising Model in a Transverse Field

Nonlinear excitations in the Ising model in a transverse field or an equivalent Frenkel-exciton model for a system of two-level atoms are studied by using a method of diagonal coherent-state representation. It is shown that if the ratio of the Ising interaction to a transverse-field strength exceeds a certain critical value, stationary coherent states giving the ground state and excited states of the model Hamiltonian are characterized by two degenerate structures resulting from spatially uniform coherent states and static domain-wall type excitations, respectively. Particular solutions for the time evolution of spin operators corresponding to atomic polarization are then studied. It is shown that moving domain-wall solutions or moving solitary-wave solutions can exist under certain conditions.

A2 Term, Renormalization of Matter-Photon Interaction and Coherent States in Matter-Photon Systems

A non-relativistic quantum theory of matter-photon interaction is formulated, within the framework of the dipole approximation, by employing a canonical transformation to diago­ nalize the A' term plus the free photon Hamiltonian. The matter-photon interaction is there­ by expressed rigorously in a renormalized A·p form. Using this form for the Dicke model of two-level atoms interacting with a single-mode radiation field, we examine a possibility of the occurrence of stationary coherent states which could arise for a photon and atomic polarization as the ground state of the system, deducing a condition for such coherent states. The condition, expressed as an inequality which imposes a stable nontrivial Bloch angle of the uniform atomic polarization, is identical with that for the occurrence of a second order phase transition, and indicates that the non-existence of a second order phase transition pointed out by Rza2ewski et al. io; in fact a consequence of the present treatment of the A'­ term renormalization. However, a fulfillment of the condition is recovered, if exchange­ type atom-atom interaction is taken into account in the matter system. We also present a method to construct an effective Hamiltonian of atom-atom interaction which is equivalent to the above Dicke model in the thermodynamic limit.