Cherenkov radiation from an Abrikosov-Josephson vortex

The Cherenkov radiation of generalized Swihart waves is investigated in connection with the slow motion of an Abrikosov-Josephson vortex, which corresponds to a 2 π kink in the phase difference of Cooper pairs on opposite sides of a tunnel junction. The radiative friction force acting on such a vortex is determined. An evaluation is made of the steady-state vortex velocity when the accelerating influence of an electric current through the Josephson junction is compensated by radiative slowing of the vortex due to Cherenkov radiation from the Abrikosov-Josephson vortex. Fiz. Tverd. Tela (St. Petersburg) 39, 444–448 (March 1997)     

Poisson brackets in condensed matter physics

A general method of deriving nonlinear equations of hydrodynamics for both normal liquid and superfluid ⁴He and ³He, equations of the elasticity theory, equations for spin waves in magnets and spin glasses, liquid crystals, and so on is described. The method is based on the use of the Poisson “hydrodynamic” brackets. Hydrodynamic brackets are on the one hand, a classical limit of quantum commutators, on the other hand, Poisson brackets of certain symmetry groups inherent in the given problem: groups of general coordinate transformations for hydrodynamics and elasticity theory, groups of local spin rotations for spin waves, etc. Along with well-known examples nonlinear equations of the elasticity theory for bodies with impurities, dislocations and disclinations, and equations of motion for spin glasses and multisublattice magnets are studied.     

Hamiltonian dynamics of vortex lines in hydrodynamic-type systems

It is shown that the degeneracy of the noncanonical Poisson bracket operating on the space of solenoidal vector fields that arises due to the freezing-in of the curl of the velocity [E. A. Kuznetsov and A. V. Mikhailov, Phys. Lett. A 77, 37 (1980)] is lifted when the vorticity Ω is represented in terms of vortex lines. This representation makes it possible to integrate the equation of motion of the vorticity for a system with the Hamiltonian H=∫∣Ω∣d r. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 12, 1015–1020 (25 June 1998)     

Unambiguous Quantization from the Maximum Classical Correspondence that Is Self-consistent: The Slightly Stronger Canonical Commutation Rule Dirac Missed

Dirac’s identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac’s canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables, but does imply the quantum/classical correspondence of Poisson brackets between any linear function of phase space and the sum of an arbitrary function of only configuration space with one of only momentum space. Since every linear function of phase space is itself such a sum, it is worth checking whether the assumption of quantum/classical correspondence of Poisson brackets for all such sums is still self-consistent. Not only is that so, but this slightly stronger canonical commutation rule also unambiguously determines the order of noncommuting factors within quantized dynamical variables in accord with the 1925 Born-Jordan quantization surmise, thus replicating the results of the Hamiltonian path integral, a fact first realized by E.H. Kerner. Born-Jordan quantization validates the generalized Ehrenfest theorem, but has no inverse, which disallows the disturbing features of the poorly physically motivated invertible Weyl quantization, i.e., its unique deterministic classical “shadow world” which can manifest negative densities in phase space.     

Interaction of surface acoustic waves with moving vortex structures in superconducting films

A method is proposed for describing a moving film vortex structure and its interaction with surface acoustic waves. It is shown that the moving vortex structure can amplify (generate) surface acoustic waves. In contrast to a similar effect in semiconductor films, this effect can appear when the velocity of the vortex structure is much lower than the velocity of the surface acoustic waves. A unidirectional collective mode is shown to exist in the moving vortex structure. This mode gives rise to an acoustic analogue of the diode effect that is resonant in the velocity of the vortex structure. This acoustic effect is manifested as an anomalous attenuation of the surface acoustic waves in the direction of the vortex-structure motion and as the absence of this attenuation for the propagation in the opposite direction.     

Lagrangian and Hamiltonian formalisms for relativistic dynamics of a charged particle with dipole moment

The Lagrangian and Hamiltonian formulations for the relativistic classical dynamics of a charged particle with dipole moment in the presence of an electromagnetic field are given. The differential conservation laws for the energy-momentum and angular momentum tensors of a field and particle are discussed. The Poisson brackets for basic dynamic variables, which form a closed algebra, are found. These Poisson brackets enable us to perform the canonical quantization of the Hamiltonian equations that leads to the Dirac wave equation in the case of spin 1/2. It is also shown that the classical limit of the squared Dirac equation results in equations of motion for a charged particle with dipole moment obtained from the Lagrangian formulation. The inclusion of gravitational field and non-Abelian gauge fields into the proposed formalism is discussed.Received: 4 June 2005, Published online: 27 July 2005     

Constrained dynamics of an anomalous (g≠2) relativistic spinning particle in electromagnetic background

In this paper we have considered the dynamics of an anomalous (g≠2) charged relativistic spinning particle in the presence of an external electromagnetic field. A constraint analysis is done and the complete set of Dirac brackets are provided that generate the canonical Lorentz algebra and dynamics through Hamiltonian equations of motion. The spin-induced effective curvature of spacetime and its possible connection with Analogue Gravity models are commented upon.     

Conformal Affine Toda Fields from a Gauged WZNW

The general conformal affine Toda (CAT) fields are derived as the result of imposing the constraint explicitly on the gauged WZNW action. The reduction procedure naturally indicates the integrability of the resulting system. The action, equations of motion, canonical Poisson brackets for this system are derived from WZNW model. The energy momentum tensor is derived also and is shown to be traceless after being improved.     

Decomposition of almost-Poisson structure of generalised Chaplygin’s nonholonomic systems

This paper constructs an almost-Poisson structure for the non-self-adjoint dynamical systems, which can be decomposed into a sum of a Poisson bracket and the other almost-Poisson bracket. The necessary and sufficient condition for the decomposition of the almost-Poisson bracket to be two Poisson ones is obtained. As an application, the almost-Poisson structure for generalised Chaplygin's systems is discussed in the framework of the decomposition theory. It proves that the almost-Poisson bracket for the systems can be decomposed into the sum of a canonical Poisson bracket and another two noncanonical Poisson brackets in some special cases, which is useful for integrating the equations of motion.     

Resonance attenuation of ultrasonic waves in a superconductor with a moving vortex structure

Equations describing the interaction of ultrasonic waves with a moving vortex structure are derived. The addition to attenuation and the relative change in the velocity of longitudinal ultrasonic waves due to this interaction are calculated. It is found that when a longitudinal ultrasonic wave propagates along the direction of motion of the vortex structure and the velocity V of the structure is equal to half the velocity of the wave, then anomalous acoustic attenuation occurs and the contribution from the ultrasound-vortex interaction to the velocity of the ultrasonic wave vanishes. It is shown that if the vortex structure moves at a sufficiently high velocity, then (in contrast to the case of the structure at rest) a weakly damping collective mode propagating with velocity 2V arises in the structure. It is this mode that is responsible for anomalous attenuation of longitudinal ultrasonic waves.     

Control of surface plasmon excitation via the scattering of light by a nanoparticle

We analyze the scattering of vortex pairs (the particular case of 2D dark solitons) by a single quantum vortex in a Bose–Einstein condensate with repulsive interaction between atoms. For this purpose, an asymptotic theory describing the dynamics of such 2D soliton-like formations in an arbitrary smoothly nonuniform flow of a ultracold Bose gas is developed. Disregarding the radiation loss associated with acoustic wave emission, we demonstrate that vortex–antivortex pairs can be put in correspondence with quasiparticles, and their behavior can be described by canonical Hamilton equations. For these equations, we determine the integrals of motion that can be used to classify various regimes of scattering of vortex pairs by a single quantum vortex. Theoretical constructions are confirmed by numerical calculations performed directly in terms of the Gross–Pitaevskii equation. We propose a method for estimating the radiation loss in a collision of a soliton-like formation with a phase singularity. It is shown by direct numerical simulation that under certain conditions, the interaction of vortex pairs with a core of a single quantum vortex is accompanied by quite intense acoustic wave emission; as a result, the conditions for applicability of the asymptotic theory developed here are violated. In particular, it is visually demonstrated by a specific example how radiation losses lead to a transformation of a vortex–antivortex pair into a vortex-free 2D dark soliton (i.e., to the annihilation of phase singularities).     

Steady vortices in plasmas and geophysical flows

A number of two-dimensional fluid models in geophysical fluid dynamics and plasma physics are examined to find out whether they have steady and localized monopole vortex solutions. A simple and general method that consists of two steps is used. First the dispersion relation is calculated, to find all possible values of the phase velocity of the linear waves. Then an integral relation that determines the center-of-mass velocity of localized structures must be found. The existence condition is that this velocity should be outside the region of linear phase velocities. After a presentation of the method, previous work on the plasma drift wave model and the shallow-water equations is reviewed. In both cases it is found that the center-of-mass velocity is larger than the maximum phase velocity of the linear waves if the amplitude is large enough, and steady localized vortices can therefore exist. New results are then obtained for a number of two-field models. For the coupled ion acoustic-drift modes in plasmas, it is found that the center-of-mass velocity depends on the ratio between the parallel ion velocity component and the electrostatic potential in the vortex. If this ratio is large enough, the vortex can be steady. For the drift-Alfven mode the "center-of-charge" velocity is proportional to the ratio between the parallel current and the total charge in the vortex. It can therefore be steady if this ratio satisfies the appropriate conditions. For the quasigeostrophic two-layer equations, describing stratified flow on a rotating planet, it is found that the center-of-mass velocity is determined by the ratio between the baroclinic and the barotropic components in the vortex. If a baroclinic component with an appropriate sign is added to a barotropic vortex, it propagates faster than the barotropic Rossby waves, and can be steady. Finally, the existence conditions for a vortex in an external zonal flow are examined. It is found that the center-of-mass velocity acquires an additional westward contribution in an anticyclonic shear zone in the framework of the shallow-water equations, and also that an easterly jet south of this shear zone partly shields a vortex situated in the shear zone from the dispersive influence of the fast Rossby waves on the equatorward side.     

The Lie-Poisson structure of integrable classical non-linear sigma models

The canonical structure of classical non-linear sigma models on Riemannian symmetric spaces, which constitute the most general class of classical non-linear sigma models known to be integrable, is shown to be governed by a fundamental Poisson bracket relation that fits into ther-s-matrix formalism for non-ultralocal integrable models first discussed by Maillet. The matricesr ands are computed explicitly and, being field dependent, satisfy fundamental Poisson bracket relations of their own, which can be expressed in terms of a new numerical matrixc. It is proposed that all these Poisson brackets taken together are, representation conditions for a new kind of algebra which, for this class of models, replaces the classical Yang-Baxter algebra governing the canonical structure of ultralocal models. The Poisson brackets for the transition matrices are also computed, and the notorious regularization problem associated with the definition of the Poisson brackets for the monodromy matrices is discussed.Suported by the Deutsche Forschungsgemeinschaft, Contract No. Ro 864/1-1Supported by the Studienstiftung des Deutschen Volkes     

Poisson bracket scheme for vortex dynamics in superfluids and superconductors and the effect of the band structure of the crystal

Poisson brackets for the Hamiltonian dynamics of vortices are discussed for 3 regimes, in which the dissipation can be neglected and the vortex dynamics is reversible: (i) The superclean regime, in which the spectral flow is suppressed. (ii) The regime in which the fermions are pinned by the crystal lattice. This includes the regime of extreme spectral flow of fermions in the vortex core: these fermions are effectively pinned by the normal component. (iii) The case when the vortices are strongly pinned by the normal component. All these limits are described by the single parameter C ₀, the physical meaning of which is discussed for superconductors containing several bands of electrons and holes. The effect of the topology of the Fermi surface on the vortex dynamics is also discussed. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 11, 794–800 (10 December 1996) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Classical mechanics in non-commutative phase space

In this paper the laws of motion of classical particles have been investigated in a non-commutative phase space.The corresponding non-commutative relations contain not only spatial non-commutativity but also momentum non-commutativity.First,new Poisson brackets have been defined in non-commutative phase space.They contain corrections due to the non-commutativity of coordinates and momenta.On the basis of this new Poisson brackets,a new modified second law of Newton has been obtained.For two cases,the free particle and the harmonic oscillator,the equations of motion are derived on basis of the modified second law of Newton and the linear transformation (Phys.Rev.D,2005,72:025010).The consistency between both methods is demonstrated.It is shown that a free particle in commutative space is not a free particle with zero-acceleration in the non-commutative phase space.but it remains a free particle with zero-acceleration in non-commutative space if only the coordinates are non-commutative.     

Nonlinear photonic crystals: III. Cubic nonlinearity

Abstract

Weakly nonlinear interactions between wavepackets in a lossless periodic dielectric medium are studied based on the classical Maxwell equations with a cubic nonlinearity. We consider nonlinear processes such that: (i) the amplitude of the wave component due to the nonlinearity does not exceed the amplitude of its linear component; (ii) the spatial range of a probing wavepacket is much smaller than the dimension of the medium sample, and it is not too small compared with the dimension of the primitive cell. These nonlinear processes are naturally described in terms of the cubic interaction phase function based on the dispersion relations of the underlying linear periodic medium. It turns out that only a few quadruplets of modes have significant nonlinear interactions. They are singled out by a system of selection rules including the group velocity, frequency and phase matching conditions. It turns out that the intrinsic symmetries of the cubic interaction phase stemming from assumed inversion symmetry of the dispersion relations play a significant role in the cubic nonlinear interactions. We also study canonical forms of the cubic interaction phase leading to a complete quantitative classification of all possible significant cubic interactions. The classification is ultimately based on a universal system of indices reflecting the intensity of nonlinear interactions.     

Relativistic Equations of Motion from Poisson Brackets

The inverse problem of Poisson dynamics isreviewed as well as a derivation of the Maxwellequations from a postulated set of Poisson brackets. Theformalism is extended to the relativistic case bypostulating Poisson brackets, as in the nonrelativisticcase, and using the relativistic Hamiltonian. A systemof relativistic equations of motion is obtained, and itis indicated that a system of consistency conditions remains valid in this limit.     

Hard soliton excitation regime: Stability investigation

The problem of the stability of one-dimensional solitons in the hard regime of soliton excitation, where the matrix element of the four-wave interaction has an additional smallness, is studied. It is that shown for optical solitons striction can weaken the Kerr nonlinearity. It is shown that solitons with a finite amplitude discontinuity at the critical soliton velocity, equal to the minimum phase velocity of linear waves, are unstable while solitons with a soft transition remain stable with respect to one-dimensional perurbations. Two-and three-dimensional solitons near threshold are unstable with respect to modulation perturbations. Zh. éksp. Teor. Fiz. 116, 299–317 (July 1999)