Fourier treatment of near-field synchrotron radiation theory

In this paper, we couple synchrotron radiation (SR) theory with a branch of physical optics, namely laser beam optics. We show that the theory of laser beams is successful in characterizing radiation fields associated with any SR source. Both radiation beam generated by an ultra-relativistic electron in a magnetic device and laser beam are solutions of the wave equation based on paraxial approximation. It follows that they are similar in all aspects. In the space-frequency domain SR beams appear as laser beams whose transverse extents are large compared with the wavelength. In practical situations (e.g. undulator, bending magnet sources), radiation beams exhibit a virtual “waist” where the wavefront is often plane. Remarkably, the field distribution of a SR beam across the waist turns out to be strictly related with the inverse Fourier transform of the far-field angle distribution. Then, we take advantage of standard Fourier Optics techniques and apply the Fresnel propagation formula to characterize the SR beam. Altogether, we show that it is possible to reconstruct the near-field distribution of the SR beam outside the magnetic setup from the knowledge of the far-field pattern. The general theory of SR in the near-zone developed in this paper is illustrated for the special cases of undulator radiation. Using known analytical formulas for the far-field pattern and its inverse Fourier transform we find analytical expressions for near-field distributions in terms of far-field distributions.     

Realization of a single-beam mini magneto-optical trap: A candidate for compact CPT cold atom-clocks

We have demonstrated the experimental realization of a single-beam mini magneto-optical trap of ⁸⁷Rb atoms, originally designed for a cold atom-clock with coherent population trapping (CPT). Only one beam is used as cooling, trapping and repumping beams rather than the three pairs of orthogonal beams of the standard magneto-optical trap. The core optics, which consists of a modified pyramidal funnel type mirror, a quarter-wave plate and a retroreflect mirror, is installed inside a mini titanium cubic chamber. The vacuum system, rubidium source, magnetic field coils and beam expander are designed in a compact geometry. As many as 1.1 × 10⁷ rubidium atoms are cooled and trapped, and thus the mini trap is ready for the implementation of a novel compact coherent population trapping cold atom-clock.     

Time reparameterization in Bianchi type I spinor cosmology

The problem of time reparameterization is addressed at both the classical and quantum levels in a Bianchi-I universe in which the matter source is a massive Dirac spinor field. We take the scale factors of the metric as the intrinsic time and their conjugate momenta as the extrinsic time. A scalar character of the spinor field is identified as a representation of the extrinsic time. The construction of the field equations and quantization of the model is achieved by solving the Hamiltonian constraint after time identification has been dealt with. This procedure leads to a true Hamiltonian whose exact solutions for the above choices of time are presented.     

Quantum treatment of the time-dependent coupled oscillators under the action of a random force

In this communication we introduce the problem of time-dependent frequency converter under the action of external random force. We have assumed that the coupling parameter and the phase pump are explicitly time dependent. Using the equations of motion in the Heisenberg picture the dynamical operators are obtained, however, under a certain integrability condition. When the system is initially prepared in the even coherent states the squeezing phenomenon is discussed. The correlation function is also considered and it has been shown that the nonclassical properties are apparent and sensitive to any variation in the integrability parameter. Furthermore, the wave function in Schrödinger picture is calculated and used it to derive the wave function in the coherent states. The accurate definition of the creation and annihilation operators are also introduced and employed to diagonalize the Hamiltonian system.     

Exact solution to the problem of nonlinear pulse propagation through random layered media and its connection with number triangles

An energy pulse refers to a spatially compact energy bundle. In nonlinear pulse propagation, the nonlinearity of the relevant dynamical equations could lead to pulse propagation that is nondispersive or weakly dispersive in space and time. Nonlinear pulse propagation through layered media with widely varying pulse transmission properties is not wave-like and a problem of broad interest in many areas such as optics, geophysics, atmospheric physics and ocean sciences. We study nonlinear pulse propagation through a semi-infinite sequence of layers where the layers can have arbitrary energy transmission properties. By assuming that the layers are rigid, we are able to develop exact expressions for the backscattered energy received at the surface layer. The present study is likely to be relevant in the context of energy transport through soil and similar complex media. Our study reveals a surprising connection between the problem of pulse propagation and the number patterns in the well known Pascal’s and Catalan’s triangles and hence provides an analytic benchmark in a challenging problem of broad interest. We close with comments on the relationship between this study and the vast body of literature on the problem of wave localization in disordered systems.     

The quantum acousto-optic effect in Bose-Einstein condensate

We investigate the interaction between a single mode light field and an elongated cigar shaped Bose-Einstein condensate (BEC), subject to a temporal modulation of the trap frequency in the tight confinement direction. Under appropriate conditions, the longitudinal sound like waves (Faraday waves) in the direction of weak confinement acts as a dynamic diffraction grating for the incident light field analogous to the acousto-optic effect in classical optics. The change in the refractive index due to the periodic modulation of the BEC density is responsible for the acousto-optic effect. The dynamics is characterised by Bragg scattering of light from the matter wave Faraday grating and simultaneous Bragg scattering of the condensate atoms from the optical grating formed due to the interference between the incident light and the diffracted light fields. Varying the intensity of the incident laser beam we observe the transition from the acousto-optic effect regime to the atomic Bragg scattering regime, where Rabi oscillations between two momentum levels of the atoms are observed. We show that the acousto-optic effect is reduced as the atomic interaction is increased.     

ABCD rule for Gaussian beam propagation in the context of quantum optics derived by the IWOP technique

The development of technique of integration within an ordered product (IWOP) of operators extends the Newton-Leibniz integration rule, originally applying to permutable functions, to the non-commutative quantum mechanical operators composed of Dirac’s ket-bra, which enables us to obtain the images of directly mapping symplectic transformation in classical phase space parameterized by [A, B; C, D] into quantum mechanical operator through the coherent state representation, we call them the generalized Fresnel operators (GFO) since they correspond to Fresnel transforms in Fourier optics. Based on GFO we find the ABCD rule for Gaussian beam propagation in the context of quantum optics (both in one-mode and two-mode cases) whose classical correspondence is just the ABCD rule in matrix optics. The entangled state representation is used in discussing the two-mode case.     

Spin gauge fields: From Berry phase to topological spin transport and Hall effects

The paper examines the emergence of gauge fields during the evolution of a particle with a spin that is described by a matrix Hamiltonian with n different eigenvalues. It is shown that by introducing a spin gauge field a particle with a spin can be described as a spin multiplet of scalar particles situated in a non-Abelian pure gauge (forceless) field U (n). As the result, one can create a theory of particle evolution that is gauge-invariant with regards to the group Uⁿ (1). Due to this, in the adiabatic (Abelian) approximation the spin gauge field is an analogue of n electromagnetic fields U (1) on the extended phase space of the particle. These fields are force ones, and the forces of their action enter the particle motion equations that are derived in the paper in the general form. The motion equations describe the topological spin transport, pumping, and splitting. The Berry phase is represented in this theory analogously to the Dirac phase of a particle in an electromagnetic field. Due to the analogy with the electromagnetic field, the theory becomes natural in the four-dimensional form. Besides the general theory, the article considers a number of important particular examples, both known and new.     

Topological defects, geometric phases, and the angular momentum of light

Recent reports on the intriguing features of vector vortex bearing beams are analyzed using geometric phases in optics. It is argued that the spin redirection phase induced circular birefringence is the origin of topological phase singularities arising in the inhomogeneous polarization patterns. A unified picture of recent results is presented based on this proposition. Orbital angular momentum (OAM) shift within the light beam has exact equivalence with the angular momentum holonomy associated with the geometric phase consistent with our conjecture.     

The dynamic natures of microscopic particles described by nonlinear Schrödinger equation

There are a lot of difficulties and troubles in quantum mechanics, when the linear Schrödinger equation is used to describe microscopic particles. Thus, we here replace it by a nonlinear Schrödinger equation to investigate the properties and rule of microscopic particles. In such a case we find that the motion of microscopic particle satisfies classical rule and obeys the Hamiltonian principle, Lagrangian and Hamilton equations. We verify further the correctness of these conclusions by the results of nonlinear Schrödinger equation under actions of different externally applied potential. From these studies, we see clearly that rules and features of motion of microscopic particle described by nonlinear Schrödinger equation are greatly different from those in the linear Schrödinger equation, they have many classical properties, which are consistent with concept of corpuscles. Thus, we should use the nonlinear Schrödinger equation to describe microscopic particles.     

Doughnut-like beam generation of Laguerre-Gaussian mode with extremely high mode purity

An apparatus to achieve a single doughnut-like Laguerre-Gaussian (LG) mode with extremely high mode purity higher than 99.6% is reported. The proposed apparatus is a conceptual combination of a high-order HG mode selector and an astigmatic mode converter, which consists of only four simple and easily-made optical elements: three lens elements and one binary phase element. Theoretical simulations show that the apparatus has a good tolerance for both system production and alignment errors. With such high mode purity, the doughnut-like LG beam generated by the apparatus carrying an orbital angular momentum will be beneficial to numbers of studies including the trapping and rotating of micro and nano-particles, non-linear optics, and atom-light interaction.     

Complete Hamiltonian Description of Wave-Like Features in Classical and Quantum Physics

The analysis of the Helmholtz equation is shown to lead to an exact Hamiltonian system describing in terms of ray trajectories, for a stationary refractive medium, a very wide family of wave-like phenomena (including diffraction and interference) going much beyond the limits of the geometrical optics (“eikonal”) approximation, which is contained as a simple limiting case. Due to the fact, moreover, that the time independent Schrödinger equation is itself a Helmholtz-like equation, the same mathematics holding for a classical optical beam turns out to apply to a quantum particle beam moving in a stationary force field, and leads to a system of Hamiltonian equations providing exact and deterministic particle trajectories and dynamical laws, and containing the laws of Classical Mechanics in the eikonal limit.     

Spectropolarised ray-tracing simulations in densely packed particulate medium

Light scattering from particulate medium is simulated using the Monte Carlo ray-tracing technique. The medium is modelled as a randomly packed medium of ellipsoidal grains with stochastically rough surfaces, with an optional thin coating. Optical properties are modelled using a wavelength-dependent complex refractive index and taking Fresnelian reflections and refractions from the interfaces. The size and shape of the grains are assumed to be large and smooth enough for geometric optics to apply reasonably well.The ray-tracing technique uses parallel, weighted rays for computing simultaneously over a wide wavelength spectrum and a small roughness range, and scaling to obtain a large range of sizes and absorbities simultaneously. Polarisation is fully accounted for. The multiobservation technique is effectively used at each scattering point. The scattering from thinner sample layers is also received as a subresult.Simulations are run for a set of model samples to study the effects and sensitivities regarding the values of certain parameters. It has been found that the size and composition of the grains affect the scattering in a unique and invertible way. The shape of the grain causes similar significant effects that must certainly be taken into account if any accuracy is required, although inverting for the shape is difficult without further constraints. The packing density has a small but observable effect. Polarisation can be used to study the composition of low-albedo objects.     

Quantum field theoretical description of unstable behavior of trapped Bose-Einstein condensates with complex eigenvalues of Bogoliubov-de Gennes equations

The Bogoliubov-de Gennes equations are used for a number of theoretical works on the trapped Bose-Einstein condensates. These equations are known to give the energies of the quasi-particles when all the eigenvalues are real. We consider the case in which these equations have complex eigenvalues. We give the complete set including those modes whose eigenvalues are complex. The quantum fields which represent neutral atoms are expanded in terms of the complete set. It is shown that the state space is an indefinite metric one and that the free Hamiltonian is not diagonalizable in the conventional bosonic representation. We introduce a criterion to select quantum states describing the metastablity of the condensate, called the physical state conditions. In order to study the instability, we formulate the linear response of the density against the time-dependent external perturbation within the regime of Kubo’s linear response theory. Some states, satisfying all the physical state conditions, give the blow-up and damping behavior of the density distributions corresponding to the complex eigenmodes. It is qualitatively consistent with the result of the recent analyses using the time-dependent Gross-Pitaevskii equation.     

Decoherence in atom–field interactions: A treatment using superoperator techniques

Decoherence is a subject of great importance in quantum mechanics, particularly in the fields of quantum optics, quantum information processing and quantum computing. Quantum computation relies heavily in the unitary character of each step carried out by a quantum computational device and this unitarity is affected by decoherence. An extensive study of master equations is therefore needed for a better understanding on how quantum information is processed when a system interacts with its environment. Master equations are usually studied by using Fokker–Planck and Langevin equations and not much attention has been given to the use of superoperator techniques. In this report we study in detail several approaches that lead to decoherence, for instance a variation of the Schrödinger equation that models decoherence as the system evolves through intrinsic mechanisms beyond conventional quantum mechanics rather than dissipative interaction with an environment. For the study of the dissipative interaction we use a correspondence principle approach. We solve the master equations for different physical systems, namely, Kerr and parametric down conversion. In the case of light-matter interaction we show that although dissipation destroys the quantumness of the field, information of the initial field may be obtained via the reconstruction of quasiprobability distribution functions.     

Hamiltonian quantum dynamics with separability constraints

Schroedinger equation on a Hilbert space H, represents a linear Hamiltonian dynamical system on the space of quantum pure states, the projective Hilbert space PH. Separable states of a bipartite quantum system form a special submanifold of PH. We analyze the Hamiltonian dynamics that corresponds to the quantum system constrained on the manifold of separable states, using as an important example the system of two interacting qubits. The constraints introduce nonlinearities which render the dynamics nontrivial. We show that the qualitative properties of the constrained dynamics clearly manifest the symmetry of the qubits system. In particular, if the quantum Hamilton’s operator has not enough symmetry, the constrained dynamics is nonintegrable, and displays the typical features of a Hamiltonian dynamical system with mixed phase space. Possible physical realizations of the separability constraints are discussed.     

High-order resonances in a parametrically excited magneto-optical trap

We present analytical and numerical study of high-order parametric resonance in a driven magneto-optical trap of cold atoms. We have obtained the general solutions for parametric resonance of arbitrary order. In particular, the amplitude and phase of atomic limit-cycle motion is expressed as a function of the modulation amplitude and frequency. Moreover, the atomic dynamics for high-order parametric resonance is investigated in terms of the Hamiltonian approach, which is useful in studying transitions between attractors. We find that the analytical results are in good agreement with the numerical calculations.     

A Hamilton-Jacobi formalism for thermodynamics

We show that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory, analogous to mechanics. Even though the thermodynamic variables come in conjugate pairs such as pressure/volume or temperature/entropy, the phase space is odd-dimensional. For a system with n thermodynamic degrees of freedom it is 2n+1-dimensional. The equations of state of a substance pick out an n-dimensional submanifold. A family of substances whose equations of state depend on n parameters define a hypersurface of co-dimension one. This can be described by the vanishing of a function which plays the role of a Hamiltonian. The ordinary differential equations (characteristic equations) defined by this function describe a dynamical system on the hypersurface. Its orbits can be used to reconstruct the equations of state. The ‘time’ variable associated to this dynamics is related to, but is not identical to, entropy. After developing this formalism on well-grounded systems such as the van der Waals gases and the Curie-Weiss magnets, we derive a Hamilton-Jacobi equation for black hole thermodynamics in General Relativity. The cosmological constant appears as a constant of integration in this picture.