Topological Photonic Integrated Circuits for Controlling Internal Degrees of Freedom of Light

This paper reports the analysis of an electron transfer, oxidation-reduction reaction system. The model electron transfer system consists of a dielectric continuum solvent in which charged electron donor and acceptor reactants are dissolved. The reactants in this model behave as hard sphere entities with internal degrees of vibronic freedom; however, the hard sphere character is important only in connexion with the dynamics of the reactant's encounter during the course of the electron transfer. This model, therefore, apart from the internal vibronic degrees of freedom, is the same as that examined by Levich and Dogonadze. Our analysis of the model system employs the Yamamoto reaction rate theory to obtain electron transfer rate constant expressions. These derived theoretical rate constants are general; they apply to all reactions from those classified as chemically adiabatic to the non-adiabatic cases. The effects of internal vibronic degrees of freedom appear in the rate constant expressions both as additional terms in the energy exponent and in the pre-exponential factor. This manifestation of vibronic contributions is expressed in the pre-exponential factor as a power series in the temperature. The introduction of this additional dynamical dependence on internal degrees of freedom prepares the way to the theoretical consideration of more complicated electron transfer systems.

The theory of the dynamic interaction of the external (translational) and internal (electronic) degrees of freedom of a two‐level atom in the field of a standing light wave in a perfect cavity of the Fabry–Perot type was developed. The theory describes the energy exchange between three subsystems, namely, translational, electronic, and field subsystems, as opposed to the theories of the parametric interaction (in the approximations of Raman–Nath and/or large resonance detuning) and of the atomic motion in free space. In the semiclassical approximation, the corresponding Heisenberg equations of motion were shown to form a closed Hamiltonian dynamic system with two degrees of freedom, namely, translational and collective electron–field degrees of freedom. This system is integrated in terms of the elliptic Jacobian functions in the resonance limit. The solutions obtained describe the effects of trapping of an atom in the periodic potential of the standing light wave, and its cooling and heating, as well as the effect of the dynamic Rabi oscillations. The latter is caused by the interaction of the internal and external atomic degrees of freedom through the radiation field.

In this work we concentrate on phase equilibria in two-dimensional condensed systems of particles where both translational and internal degrees of freedom are present and coupled through microscopic interactions, with a focus on the manner of the macroscopic coupling between the two types of degrees of freedom. First, an unconventional description of the translational degrees of freedom is developed, in which the randomly varying spatial connectivity of the particles is represented by a random lattice whose dynamic structure is given by triangulating the spatial configurations. Based on this random-lattice description, a series of three statistical-mechanical models are then constructed. All of the three models are in essence spin-\textonehalf{} Ising models where the spins, representing internal degrees of freedom, are associated with hard-disk particles and nearest-neighbor particles interact through spin-spin interactions that may have spatial dependence. The fluctuating number of nearest neighbors and the possible spatial dependence of the spin-spin interactions couple microscopically the spin degrees of freedom to the translational degrees of freedom. The first model (I) is a random-lattice Ising model with conventional nearest-neighbor spin-spin interactions. The second model (II) is an extension of this model to include a spatial dependence of the nearest-neighbor spin-spin interactions. The third model (III) is a modification of the second model that accounts for spin states with different internal degeneracy. Monte Carlo simulation techniques, including both a special algorithm for the random-lattice description and histogram and finite-size scaling analysis, are used to investigate the phase behavior of all three models. It is shown that the order-disorder spin transition in model I is decoupled from a first-order singularity---lattice melting---associated with the translational degrees of freedom and remains critical and falls in the universality class of the standard two-dimensional Ising model on regular lattices. Model II is shown to exhibit a phase diagram that has a region where the spin degrees of freedom are slaved by the translational degrees of freedom and develop a first-order singularity in the order-disorder transition that accompanies the lattice-melting transition. The internal degeneracy of the spin states in model III implies that the spin order-disorder singularity can be of first order throughout the phase diagram. It is found that this first-order singularity can be either coupled to or decoupled from the lattice-melting singularity, depending on the strength of the microscopic coupling. The calculated phase diagram and the associated thermodynamic transitional properties for model III are discussed in relation to experiments on planar bilayers of lipid-chain molecules whose properties are determined by a subtle coupling between the translational variables and the intrachain conformational states.

We study the non-equilibrium dissipative dynamics of the center of mass of a particle coupled to a field via its internal degrees of freedom. We model the internal and external degrees of freedom of the particle as quantum harmonic oscillators in 1+1 D, with the internal oscillator coupled to a scalar quantum field at the center of mass position. Such a coupling results in a nonlinear interaction between the three pertinent degrees of freedom -- the center of mass, internal degree of freedom, and the field. It is typically assumed that the internal dynamics is decoupled from that of the center of mass owing to their disparate characteristic time scales. Here we use an influence functional approach that allows one to account for the self-consistent backaction of the different degrees of freedom on each other, including the coupled non-equilibrium dynamics of the internal degree of freedom and the field, and their influence on the dissipation and noise of the center of mass. Considering a weak nonlinear interaction term, we employ a perturbative generating functional approach to derive a second order effective action and a corresponding quantum Langevin equation describing the non-equilibrium dynamics of the center of mass. We analyze the resulting dissipation and noise arising from the field and the internal degree of freedom as a composite environment. Furthermore, we establish a generalized fluctuation-dissipation relation for the late-time dissipation and noise kernels. Our results are pertinent to open quantum systems that possess intermediary degrees of freedom between system and environment, such as in the case of optomechanical interactions.

Usually a finite element with cubic deflection approximation function is applied when evaluating the stress and strain field of bar structures. But such an element only approximately evaluates the actual strain field of the bar affected by a distributed load. The improved finite elements (Fig 1, 2) with fourth and fifth-order deflection approximation functions (1), (6) and (13) are presented in the actual manuscript. The fifth-order deflection approximation function is used for modelling the beams affected by a linearly distributed load (11). The plain bending of the finite element is modelled by 5 and 6 freedom degrees. The additional 5th and 6th freedom degrees are the deflection and deviation of the middle node of element (Fig 2). The element stiffness matrices (Table 1, 2) and node force vectors are presented. The created finite elements exactly modells the stress and strain field of bars, which are affected by distributed load, and also allow to compute directly the middle section displacements of bars. It creates conditions for diminishing the volume of problems and obtaining information, which is necessary to be analysed later. The reduced finite elements (Fig 4) are created by the elimination of the internal freedom degrees. Their number of freedom degrees is decreased up to the number of freedom degrees of a usually applied finite element. But the reduced finite elements have all afore-mentioned qualities. Formulas (20) and (21) are derived expressing the middle node displacements by the final node displacements. These formulas allow to compute the middle section displacements of the bar already after the solution of equation system. The proposed reduced elements can be introduced and applied in engineering practice very easily, because their stiffness matrix coincide with the stiffness matrix of a usual bar finite element. The created elements with internal freedom degrees are very important for the problems of structures optimization with displacement constraints, because the constraint of bar middle section displacement can form just in case, when this displacement is one of the problem's unknown. Also it is very important to decrease the number of unknowns of optimization problem.

Anticommuting variables, internal degrees of freedom, and the Wilson loop

To the quantum mechanics specialists community it is a well-known fact that the famous original Stern–Gerlach experiment (SGE) produces entanglement between the external degrees of freedom (position) and the internal degree of freedom (spin) of silver atoms. Despite this fact, almost all textbooks on quantum mechanics explain this experiment using a semiclassical approach, where the external degrees of freedom are considered classical variables, the internal degree is treated as a quantum variable, and Newton's second law is used to describe the dynamics. In the literature there are some works that analyze this experiment in its full quantum mechanical form. However, astonishingly, to the best of our knowledge the original experiment, where the initial states of the spin degree of freedom are randomly oriented coming from the oven, has not been analyzed yet in the available textbooks using the Schrödinger equation (to the best of our knowledge there is only one paper that treats this case: Hsu et al (2011 Phys. Rev. A 83 012109)). Therefore, in this contribution we use the time-evolution operator to give a full quantum mechanics analysis of the SGE when the initial state of the internal degree of freedom is completely random, i.e. when it is a statistical mixture. Additionally, as the SGE and the development of quantum mechanics are heavily intermingled, we analyze some features and drawbacks in the current teaching of quantum mechanics. We focus on textbooks that use the SGE as a starting point, based on the fact that most physicist do not use results from physics education research, and comment on traditional pedagogical attitudes in the physics community.

Lipid bilayers exhibit a phase behavior that involves two distinct, but coupled, order-disorder processes, one in terms of lipid-chain crystalline packing (translational degrees of freedom) and the other in terms of lipid-chain conformational ordering (internal degrees of freedom). Experiments and previous approximate theories have suggested that cholesterol incorporated into lipid bilayers has different microscopic effects on lipid-chain packing and conformations and that cholesterol thereby leads to decoupling of the two ordering processes, manifested by a special equilibrium phase, "liquid-ordered phase," where bilayers are liquid (with translational disorder) but lipid chains are conformationally ordered. We present in this paper a microscopic model that describes this decoupling phenomena and which yields a phase diagram consistent with experimental observations. The model is an off-lattice model based on a two-dimensional random triangulation algorithm and represents lipid and cholesterol molecules by hard-core particles with internal (spin-type) degrees of freedom that have nearest-neighbor interactions. The phase equilibria described by the model, specifically in terms of phase diagrams and structure factors characterizing different phases, are calculated by using several Monte Carlo simulation techniques, including histogram and thermodynamic reweighting techniques, finite-size scaling as well as non-Boltzmann sampling techniques (in order to overcome severe hysteresis effects associated with strongly first-order phase transitions). The results provide a consistent interpretation of the various phases of phospholipid-cholesterol binary mixtures based on the microscopic dual action of cholesterol on the lipid-chain degrees of freedom. In particular, a distinct small-scale structure of the liquid-ordered phase has been identified and characterized. The generic nature of the model proposed holds a promise for a unifying description for a whole series of different lipid-sterol mixtures.

The coefficient of frictional torque acting on a rough hard sphere (Brownian particle) which rotates about a fixed axis in a fluid composed of finite-sized particles with internal rotational degrees of freedom is computed. Inertial effects are included. The long-time limit of the angular-velocity-autocorrelation function is obtained. We find that angular-velocity-autocorrelation function decays as ${t}^{\frac{\ensuremath{-}5}{2}}$, but with a coefficient that is altered from the usual Navier-Stokes result to include contributions from transport processes involving internal rotational degrees of freedom of the fluid.

The friction coefficient for a smooth hard sphere (Brownian particle) moving through a fluid composed of finite-sized particles with internal rotational degrees of freedom is computed. Inertial effects are included. The long-time limit of the velocity-autocorrelation function is obtained. We find that the velocity-autocorrelation function decays as ${t}^{\ensuremath{-}\frac{3}{2}}$ but with a coefficient that is altered from the usual Navier-Stokes results to include contributions from transport processes involving internal rotational degrees of freedom of the fluid.

This paper reports on a proposal for cooling internal and external degrees of freedom of molecules. Cooling is achieved by suitably tailored Raman processes, where absorption of photons from a laser beams is followed by emission into the cavity resonances, thereby removing in average rovibrational and translational excitations, such that at the end of the process the internal degrees of freedom are cooled into the ground state and the external degrees of freedom are cooled to the cavity linewidth. The method relies on the enhancement of emission along certain resonances of a suitably designed resonator, while the laser, driving the molecules, is sequentially set to different frequencies in order to empty the higher excited ro-vibrational states of the molecules. The cooling efficiency is investigated numerically for the case of the OH radical, using ab-initio data and taking into account the rovibrational dependence of the Raman scattering into the cavity modes.

This paper presents a brief overview including recent results obtained from simulation studies of models of pseudo-two-dimensional systems of molecules with translational (crystalline) as well as internal (conformational) degrees of freedom. The models, which include both lattice-gas Potts models and models of hard discs with varying sizes, have a general sphere of applicability. The models are here being used to describe the phase transitions between the condensed phases in lipid monolayers or lipid bilayers. The simulation results reveal an intricate interplay between ordering processes governed by the two different degrees of freedom.

Physics and Chemistry with Cold Molecules.

Generalized Taylor dispersion theory is extended so as to enable the analysis of the transport in unbounded homogeneous shear flows of Brownian particles possessing internal degrees of freedom (e.g. rigid non-spherical particles possessing orientational degrees of freedom, flexible particles possessing conformational degrees of freedom, etc.). Taylor dispersion phenomena originate from the coupling between the dependence of the translational velocity of such particles in physical space upon the internal variables and the stochastic sampling of the internal space resulting from the internal diffusion process.Employing a codeformational reference frame (i.e. one deforming with the sheared fluid) and assuming that the eigenvalues of the (constant) velocity gradient are purely imaginary, we establish the existence of a coarse-grained, purely physical-space description of the more detailed physical-internal space (microscale) transport process. This macroscale description takes the form of a convective–diffusive ‘model’ problem occurring exclusively in physical space, one whose formulation and solution are independent of the internal (‘local’-space) degrees of freedom.An Einstein-type diffusion relation is obtained for the long-time limit of the temporal rate of change of the mean-square particle displacement in physical space. Despite the nonlinear (in time) asymptotic behaviour of this displacement, its Oldroyd time derivative (which is the appropriate one in the codeformational view adopted) tends to a constant, time-independent limit which is independent of the initial internal coordinates of the Brownian particle at zero time.The dyadic dispersion-like coefficient representing this asymptotic limit is, in general, not a positive-definite quantity. This apparently paradoxical behaviour arises due to the failure of the growth in particle spread to be monotonic with time as a consequence of the coupling between the Taylor dispersion mechanism and the shear field. As such, a redefinition of the solute's dispersivity dyadic (appearing as a phenomenological coefficient in the coarse-grained model constitutive equation) is proposed. This definition provides additional insight into its physical (Lagrangian) significance as well as rendering this dyadic coefficient positive-definite, thus ensuring that solutions of the convective–diffusive model problem are well behaved. No restrictions are imposed upon the magnitude of the rotary Péclet number, which represents the relative intensities of the respective shear and diffusive effects upon which the solute dispersivity and mean particle sedimentation velocity both depend.The results of the general theory are illustrated by the (relatively) elementary problem of the sedimentation in a homogeneous unbounded shear field of a size-fluctuating porous Brownian sphere (which body serves to model the behaviour of a macromolecular coil). It is demonstrated that the well-known case of the translational diffusion in a homogeneous shear flow of a rigid, non- fluctuating sphere (for which the Taylor mechanism is absent) is a particular case thereof.

Simulations of ferrofluid dynamics: Rigid dipoles model versus particles with internal degrees of freedom