Spectral Flow, the Magnus Force, and the. Josephson-Anderson Relation
|
|
- Abigayle Ferguson
- 5 years ago
- Views:
Transcription
1 Spectral Flow, the Magnus Force, an the arxiv:con-mat/ v1 16 Feb 1996 Josephson-Anerson Relation P. Ao Department of Theoretical Physics Umeå University, S , Umeå, SWEDEN October 18, 2018 Abstract We show that the spectral flow ue to a moving vortex is ientical to the phase slippage process, an conclue that its evaluation confirms the results by the Berry s phase calculation of the Magnus force. 1
2 Consier a rectilinear vortex moving in a superconuctor with a small velocity v v relative to the backgroun of the crystal lattice, both the Berry s phase[1] an the irect total transverse force[2] calculations lea to the Magnus force: F M = L q v h ρ s 2 v v ẑ. (1) Here ρ s is the superflui electron number ensity, q v = ±1 is the vorticity, an the vortex line is along the z-irection with the length L. The crystal lattice backgroun of the superconuctor is hel at rest in the laboratory frame to avoi further complications. Looking from the point of view of the electron flui, a Fermi liqui, moving vortices cause phase slippages, generating a potential ifference. Therefore the electron flui also feels a force, corresponing to the measure electric fiel. This process has been analyze by Josephson[3] an Anerson[4]. Recently, it has been reanalyze by the calculation of the spectral or momentum flow in the electron flui[5, 6]. Since this spectral flow has a sign opposite to that of the Magnus force, one might be tempte to conclue that it cancels the Magnus force. The purpose of the present letter is to show that those two views of looking at the consequences of a moving vortex are equivalent as the action-reaction forces. This equivalence must have been implicitly containe in the literature, but it has not been explicitly state yet. Instea, some confusion still exists.[7, 8] We start by stuying the momentum flow in the electron flui ue to a moving vortex trappe by a potential locate at r 0. The trapping potential can be arbitrarily weak, an just efines the vortex position. The electron flui system is homogeneous other than this trapping potential, an the system of vortex-electron flui is translational invariant. The momentum flow in the electron flui ue to the moving of the trapping potential, the moving of the vortex, is t P(t) = Tr[ ˆρ(t)ˆP]. (2) 2
3 Here the ensity matrix for the whole electron flui is governe by the equation i h ˆρ(t) = [Ĥ, ˆρ(t)], with ˆP the total momentum operator an Ĥ the truncate Hamiltonian of the system: the lattice phonon inuce effective electron-electron interaction is treate as an attractive one, which gives arise to the superconuctivity. One may inclue the lattice ynamics into the formulation. In this case, because the electron-lattice system consists of a clean superconuctor, our following conclusion will remain unmoifie. We first evaluate the momentum flow from a safe istance far away from the vortex core. Later we point out that it can be evaluate near the vortex core by the spectral flow metho an the results of the two calculations are equal as guarantee by the momentum conservation law. We note that for a slow moving vortex, the aiabatic conition hols: At a given time the ensity matrix ˆρ of the electron flui system can be approximate by its instantaneous equilibrium ensity matrix ˆρ 0 an the eviation ˆρ 1 : ˆρ = ˆρ 0 + ˆρ 1. Here ˆρ 1 is etermine by the equation [Ĥ, ˆρ 1] = i h ˆρ 0, an ˆρ 0 (t) = n f n Ψ n >< Ψ n, (3) with f n = e En/k BT / ne En/k BT the normalize Boltzmann factor an Ψ n the n-th eigenstate of the Hamiltonian. The eigenenergies {E n } are inepenent of time. The time epenence of the ensity matrix is through the vortex position epenence of the wavefunction. Then we have, to the lowest orer in the vortex velocity, { } P(t) = Tr f n v v [ r0 Ψ n >< Ψ n + Ψ n >< r0 Ψ n ]ˆP t n. (4) Here the operator r0 is the graient with respect to the vortex position, an v v = ṙ 0. Since the vortex position, or the position of the trapping potential, is the only reference point in the electron flui system, we have the ientity r0 Ψ n ({r j },r 0 ) = j rj Ψ n ({r j },r 0 ), (5) 3
4 with r j the position of the j-th electron in the system. Note that the total momentum operator is ˆP = i h j rj i h R, Eq.(4) gives us { } P(t) = i h f n [ < R Ψ n v v R Ψ n > + < v v R Ψ n R Ψ n >] t n = i hv v n f n j 3 r j [ R R Ψ n ({r j})ψ n({r j })] {r j }={r j }. (6) Following the same proceure as in Ref.[2], we first reuce the N-boy ensity to the oneboy ensity matrix ρ 1 : t P(t) = i hv v 3 r 1 [ r1 r 1 ρ 1 (r 1 ;r 1)] r 1 =r 1, (7) with ρ 1 (r 1;r 1 ) N n f n j 1 3 r j Ψ n (r 1,{r j })Ψ n(r 1,{r j }), (8) an N the total electron number. Then we use the Stokes theorem to evaluate the integral far away from the vortex core, an obtain t P(t) = L hv v ẑ r i 2 [( r r )ρ 1 (r ;r)] r =r = L q v h ρ s 2 v v ẑ, (9) which shows that the force felt by the electron flui has the same magnitue as the Magnus force, but with the opposite sign. We will return to this point below. In the integration leaing to Eq.(9) we have use the fact that the integran is the momentum ensity. The two-flui moel has been employe to account for the fact that the momentum generate by the vortex correspons to a supercurrent. The momentum flow calculation can also be performe near the vortex core by counting the flow of the energy spectrum, the so-calle spectral flow metho, anhave been oneinref.[5]. The mainiea isthat, by linearizing the time-epenent Bogoliubov-e Gennes equation near the Fermi surface, a pseuorelativistic Dirac equation will be obtaine. This linearization reuces the original 3+1 imensional problem to an effective 1+1 imensional one. As a vortex moves, there is a continuous flow 4
5 of the energy spectrum, emerging from(or sinking into) the Fermi sea. The spectral flow rate is proportional to the vortex velocity. By counting their contributions to the momentum, the same result as Eq.(9) has been arrive at in Ref.[5]. This proceure has been further verifie by a moel calculation.[8] The agreement between two seemingly totally ifferent ways, lookingfromasafeistance away fromthecoreanwatching thespectral flownear the core, of calculating the momentum flow may first appear surprising. It has been, however, explicitly emonstrate in Ref.[9] in a slightly ifferent context in the 3 He-A phase that they are inee the complementary two ways of keeping track of the same physics, an they are equal as a result of the momentum conservation in the electron flui system. In the present situation the link between those two ways has been iscusse in Ref.[6]. It is the ifferential form of the momentum flow: t (m v s )+ µ = q v hv v ẑ δ 2 (r r 0 ), (10) with the electrochemical potential µ as the sum of the chemical potential µ 0, the flui kinetic energyantheelectricpotential µ = µ 0 + m 2 v2 s+e (φ+ 1 r r A(r,t)), anthesuperflui c t velocity istribution m v s = h θ e c A. Here m an e are effective mass an charge of a Cooper pair, respectively, an θ is the phase of the superconucting conensate wavefunction. Eq.(10) is gauge invariant, an is a precise statement about the phase slippage process ue to a moving vortex. Its various consequences have been explore by Josephson[3] an Anerson[4]. An ientical equation for the special case of v v = v s in the neutral superflui has been stuie by Anerson[4]. We may call Eq.(10) the Josephson-Anerson relation. It isclear now that a moving vortex feels a transverse force, the Magnus force; if one looks from the point of view of the electron flui, the flui also feels a force with the magnitue equal to but thesign oppositeto themagnus force. The two forces areacting ontwo ifferent objects, the vortex an the electron flui, an are the action-reaction forces. In the following 5
6 we strengthen this point by a straightforwar proof. We write Eq.(2) in its equivalent form of the Ehrenfest theorem: Using R Ĥ = r0 Ĥ, we obtain P(t) = Tr[ˆρ(t) RĤ]. (11) t t P(t) = Tr[ˆρ(t) r 0 Ĥ] = F M. (12) This shows that the two forces uner iscussion are inee the action-reaction forces. In the last equality we have use the formal efinition of the Magnus force[2]. The present result, the equivalence between the Magnus force an the spectral flow as action-reaction forces, may seem obvious. Nevertheless it has never been explicitly spelle out in the literature. Instea, some recent work have treate the spectral flow as a way to cancel the Magnus force[5, 7, 8], which is incorrect accoring to the present emonstration. It shoul be pointe out that the spectral flow is a counting of contributions from extene states. There is no involvement of the localize core states. This can be explicitly checke by expressing the one-boy ensity matrix in Eq.(9) in terms of extene an localize states, an latter gives zero contribution, as having been note in Ref.[2] in the evaluation of the Magnus force. Incientally, in Ref.[5] the Wess-Zumino term for a moving vortex has been ientifie as the same force as the one ue to the spectral flow. Recent as well as earlier work have shown that the Wess-Zumino term gives exactly the Magnus force[10, 11, 12, 6], not the spectral flow. Comparing the Berry s phase calculation away from the vortex core with the spectral or momentum flow counting near the vortex core, we fin that the former only epens on a few global properties of a superconuctor, namely the topology of a vortex, an the latter is a rather etaile calculation. The topological constraints behave like conservations laws. 6
7 Results obtaine uner them shoul, an have to, be borne out by etaile calculations, which are concrete realizations. For the Magnus force, it is inee the case. Acknowlegements: The author thanks Davi Thouless an Qian Niu for numerous iscussions, an Mike Stone an Frank Gaitan for informative corresponences. The paper was initiate at the Institute of Scientific Information at Turin in the fall of 1993, an was shape into the present form at the Aspen Center for Physics in the summer of Their hospitalities are gratefully acknowlege. The work was supporte in part by Sweish Natural Science Research Council an by US NSF Grant No. DMR Present aress: Department of Physics, Box , University of Washington, Seattle, WA 98195, USA. References [1] P. Ao an D.J. Thouless, Phys. Rev. Lett. 70, 2158 (1993); P. Ao, Q. Niu, an D.J. Thouless, Physica B , 1453 (1994). [2] D.J. Thouless, P. Ao, an Q. Niu, Transverse force on a quantize vortex in a superflui, preprint, High-Tc Upate, Nov. 15, [3] B.D. Josephson, Phys. Lett. 1, 251 (1962); ibi. 16, 242 (1965). [4] P.W. Anerson, Rev. Mo. Phys. 38, 298 (1966). [5] G.E. Volovik, JETP Lett. 57, 244 (1993); JETP 77, 435 (1993). [6] F. Gaitan, J. Phys. Con. Matt. 7, L165 (1995); Phys. Rev. B51, 9061 (1995). [7] G.E. Volovik, JETP Lett. 62, 65 (1995). 7
8 [8] Y.G. Makhlin an T.S. Misirpashaev, JETP Lett. 62, 83 (1995). [9] M. Stone an F. Gaitan, Ann. Phys.(N.Y.) 178, 89 (1987). [10] M. Hatsua, S. Yahikozawa, P. Ao, an D.J. Thouless, Phys. Rev. B49, (1994); an references therein. [11] P. Ao, D.J. Thouless, an X.-M. Zhu, Mo. Phys. Lett. B9, 755 (1995); I.J.R. Aitchison, P. Ao, D.J. Thouless, an X.-M. Zhu, Phys, Rev, B51, 6531 (1995). [12] M. Stone, Int. J. Mo. Phys. B9, 1359 (1995). 8
Lecture 2 Lagrangian formulation of classical mechanics Mechanics
Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,
More informationensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y
Ph195a lecture notes, 1/3/01 Density operators for spin- 1 ensembles So far in our iscussion of spin- 1 systems, we have restricte our attention to the case of pure states an Hamiltonian evolution. Toay
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite
More informationAPPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France
APPROXIMAE SOLUION FOR RANSIEN HEA RANSFER IN SAIC URBULEN HE II B. Bauouy CEA/Saclay, DSM/DAPNIA/SCM 91191 Gif-sur-Yvette Ceex, France ABSRAC Analytical solution in one imension of the heat iffusion equation
More informationHow the potentials in different gauges yield the same retarded electric and magnetic fields
How the potentials in ifferent gauges yiel the same retare electric an magnetic fiels José A. Heras a Departamento e Física, E. S. F. M., Instituto Politécnico Nacional, México D. F. México an Department
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationDelocalization of boundary states in disordered topological insulators
Journal of Physics A: Mathematical an Theoretical J. Phys. A: Math. Theor. 48 (05) FT0 (pp) oi:0.088/75-83/48//ft0 Fast Track Communication Delocalization of bounary states in isorere topological insulators
More informationPHYS 414 Problem Set 2: Turtles all the way down
PHYS 414 Problem Set 2: Turtles all the way own This problem set explores the common structure of ynamical theories in statistical physics as you pass from one length an time scale to another. Brownian
More informationOn the number of isolated eigenvalues of a pair of particles in a quantum wire
On the number of isolate eigenvalues of a pair of particles in a quantum wire arxiv:1812.11804v1 [math-ph] 31 Dec 2018 Joachim Kerner 1 Department of Mathematics an Computer Science FernUniversität in
More information4. Important theorems in quantum mechanics
TFY4215 Kjemisk fysikk og kvantemekanikk - Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking three-imensional potentials in the next chapter, we shall in chapter 4 of this
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More informationarxiv: v1 [physics.class-ph] 20 Dec 2017
arxiv:1712.07328v1 [physics.class-ph] 20 Dec 2017 Demystifying the constancy of the Ermakov-Lewis invariant for a time epenent oscillator T. Pamanabhan IUCAA, Post Bag 4, Ganeshkhin, Pune - 411 007, Inia.
More informationThe total derivative. Chapter Lagrangian and Eulerian approaches
Chapter 5 The total erivative 51 Lagrangian an Eulerian approaches The representation of a flui through scalar or vector fiels means that each physical quantity uner consieration is escribe as a function
More informationCONSERVATION PROPERTIES OF SMOOTHED PARTICLE HYDRODYNAMICS APPLIED TO THE SHALLOW WATER EQUATIONS
BIT 0006-3835/00/4004-0001 $15.00 200?, Vol.??, No.??, pp.?????? c Swets & Zeitlinger CONSERVATION PROPERTIES OF SMOOTHE PARTICLE HYROYNAMICS APPLIE TO THE SHALLOW WATER EQUATIONS JASON FRANK 1 an SEBASTIAN
More informationTime-of-Arrival Estimation in Non-Line-Of-Sight Environments
2 Conference on Information Sciences an Systems, The Johns Hopkins University, March 2, 2 Time-of-Arrival Estimation in Non-Line-Of-Sight Environments Sinan Gezici, Hisashi Kobayashi an H. Vincent Poor
More informationChapter 4. Electrostatics of Macroscopic Media
Chapter 4. Electrostatics of Macroscopic Meia 4.1 Multipole Expansion Approximate potentials at large istances 3 x' x' (x') x x' x x Fig 4.1 We consier the potential in the far-fiel region (see Fig. 4.1
More informationθ x = f ( x,t) could be written as
9. Higher orer PDEs as systems of first-orer PDEs. Hyperbolic systems. For PDEs, as for ODEs, we may reuce the orer by efining new epenent variables. For example, in the case of the wave equation, (1)
More informationarxiv: v1 [physics.flu-dyn] 8 May 2014
Energetics of a flui uner the Boussinesq approximation arxiv:1405.1921v1 [physics.flu-yn] 8 May 2014 Kiyoshi Maruyama Department of Earth an Ocean Sciences, National Defense Acaemy, Yokosuka, Kanagawa
More informationOn Characterizing the Delay-Performance of Wireless Scheduling Algorithms
On Characterizing the Delay-Performance of Wireless Scheuling Algorithms Xiaojun Lin Center for Wireless Systems an Applications School of Electrical an Computer Engineering, Purue University West Lafayette,
More informationChapter 6: Energy-Momentum Tensors
49 Chapter 6: Energy-Momentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energy-momentum tensors, then applies these ieas to the case of Bohm's moel.
More informationThermal conductivity of graded composites: Numerical simulations and an effective medium approximation
JOURNAL OF MATERIALS SCIENCE 34 (999)5497 5503 Thermal conuctivity of grae composites: Numerical simulations an an effective meium approximation P. M. HUI Department of Physics, The Chinese University
More informationarxiv:hep-th/ v1 3 Feb 1993
NBI-HE-9-89 PAR LPTHE 9-49 FTUAM 9-44 November 99 Matrix moel calculations beyon the spherical limit arxiv:hep-th/93004v 3 Feb 993 J. Ambjørn The Niels Bohr Institute Blegamsvej 7, DK-00 Copenhagen Ø,
More informationStable and compact finite difference schemes
Center for Turbulence Research Annual Research Briefs 2006 2 Stable an compact finite ifference schemes By K. Mattsson, M. Svär AND M. Shoeybi. Motivation an objectives Compact secon erivatives have long
More informationarxiv: v1 [cond-mat.stat-mech] 9 Jan 2012
arxiv:1201.1836v1 [con-mat.stat-mech] 9 Jan 2012 Externally riven one-imensional Ising moel Amir Aghamohammai a 1, Cina Aghamohammai b 2, & Mohamma Khorrami a 3 a Department of Physics, Alzahra University,
More informationSwitching Time Optimization in Discretized Hybrid Dynamical Systems
Switching Time Optimization in Discretize Hybri Dynamical Systems Kathrin Flaßkamp, To Murphey, an Sina Ober-Blöbaum Abstract Switching time optimization (STO) arises in systems that have a finite set
More informationON THE OPTIMALITY SYSTEM FOR A 1 D EULER FLOW PROBLEM
ON THE OPTIMALITY SYSTEM FOR A D EULER FLOW PROBLEM Eugene M. Cliff Matthias Heinkenschloss y Ajit R. Shenoy z Interisciplinary Center for Applie Mathematics Virginia Tech Blacksburg, Virginia 46 Abstract
More informationPhysics 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel. Problem Set 3. 2 (x x ) 2 + (y y ) 2 + (z + z ) 2
Physics 505 Electricity an Magnetism Fall 003 Prof. G. Raithel Problem Set 3 Problem.7 5 Points a): Green s function: Using cartesian coorinates x = (x, y, z), it is G(x, x ) = 1 (x x ) + (y y ) + (z z
More informationDust Acoustic Compressive Waves in a Warm Dusty Plasma Having Non-Thermal Ions and Non-Isothermal Electrons
Plasma Science an Technology, Vol.17, No.9, Sep. 015 Dust Acoustic Compressive Waves in a Warm Dusty Plasma Having Non-Thermal Ions an Non-Isothermal Electrons Apul N. DEV 1, Manoj K. DEKA, Rajesh SUBEDI
More informationG4003 Advanced Mechanics 1. We already saw that if q is a cyclic variable, the associated conjugate momentum is conserved, L = const.
G4003 Avance Mechanics 1 The Noether theorem We alreay saw that if q is a cyclic variable, the associate conjugate momentum is conserve, q = 0 p q = const. (1) This is the simplest incarnation of Noether
More informationCentrum voor Wiskunde en Informatica
Centrum voor Wiskune en Informatica Moelling, Analysis an Simulation Moelling, Analysis an Simulation Conservation properties of smoothe particle hyroynamics applie to the shallow water equations J.E.
More informationProblem Set 6: Workbook on Operators, and Dirac Notation Solution
Moern Physics: Home work 5 Due ate: 0 March. 014 Problem Set 6: Workbook on Operators, an Dirac Notation Solution 1. nswer 1: a The cat is being escribe by the state, ψ >= ea > If we try to observe it
More informationQubit channels that achieve capacity with two states
Qubit channels that achieve capacity with two states Dominic W. Berry Department of Physics, The University of Queenslan, Brisbane, Queenslan 4072, Australia Receive 22 December 2004; publishe 22 March
More informationA Review of Multiple Try MCMC algorithms for Signal Processing
A Review of Multiple Try MCMC algorithms for Signal Processing Luca Martino Image Processing Lab., Universitat e València (Spain) Universia Carlos III e Mari, Leganes (Spain) Abstract Many applications
More informationLie symmetry and Mei conservation law of continuum system
Chin. Phys. B Vol. 20, No. 2 20 020 Lie symmetry an Mei conservation law of continuum system Shi Shen-Yang an Fu Jing-Li Department of Physics, Zhejiang Sci-Tech University, Hangzhou 3008, China Receive
More information1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7.
Lectures Nine an Ten The WKB Approximation The WKB metho is a powerful tool to obtain solutions for many physical problems It is generally applicable to problems of wave propagation in which the frequency
More informationarxiv:nlin/ v1 [nlin.cd] 21 Mar 2002
Entropy prouction of iffusion in spatially perioic eterministic systems arxiv:nlin/0203046v [nlin.cd] 2 Mar 2002 J. R. Dorfman, P. Gaspar, 2 an T. Gilbert 3 Department of Physics an Institute for Physical
More informationShort Intro to Coordinate Transformation
Short Intro to Coorinate Transformation 1 A Vector A vector can basically be seen as an arrow in space pointing in a specific irection with a specific length. The following problem arises: How o we represent
More informationSemiclassical analysis of long-wavelength multiphoton processes: The Rydberg atom
PHYSICAL REVIEW A 69, 063409 (2004) Semiclassical analysis of long-wavelength multiphoton processes: The Ryberg atom Luz V. Vela-Arevalo* an Ronal F. Fox Center for Nonlinear Sciences an School of Physics,
More informationSurvey Sampling. 1 Design-based Inference. Kosuke Imai Department of Politics, Princeton University. February 19, 2013
Survey Sampling Kosuke Imai Department of Politics, Princeton University February 19, 2013 Survey sampling is one of the most commonly use ata collection methos for social scientists. We begin by escribing
More informationIntroduction to the Vlasov-Poisson system
Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its
More informationThe Ehrenfest Theorems
The Ehrenfest Theorems Robert Gilmore Classical Preliminaries A classical system with n egrees of freeom is escribe by n secon orer orinary ifferential equations on the configuration space (n inepenent
More informationIntroduction to variational calculus: Lecture notes 1
October 10, 2006 Introuction to variational calculus: Lecture notes 1 Ewin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE-106 91 Stockholm, Sween Abstract I give an informal summary of variational
More informationEntanglement is not very useful for estimating multiple phases
PHYSICAL REVIEW A 70, 032310 (2004) Entanglement is not very useful for estimating multiple phases Manuel A. Ballester* Department of Mathematics, University of Utrecht, Box 80010, 3508 TA Utrecht, The
More information05 The Continuum Limit and the Wave Equation
Utah State University DigitalCommons@USU Founations of Wave Phenomena Physics, Department of 1-1-2004 05 The Continuum Limit an the Wave Equation Charles G. Torre Department of Physics, Utah State University,
More informationQuantum optics of a Bose-Einstein condensate coupled to a quantized light field
PHYSICAL REVIEW A VOLUME 60, NUMBER 2 AUGUST 1999 Quantum optics of a Bose-Einstein conensate couple to a quantize light fiel M. G. Moore, O. Zobay, an P. Meystre Optical Sciences Center an Department
More informationStatics. There are four fundamental quantities which occur in mechanics:
Statics Mechanics isabranchofphysicsinwhichwestuythestate of rest or motion of boies subject to the action of forces. It can be ivie into two logical parts: statics, where we investigate the equilibrium
More informationconrm that at least the chiral eterminant can be ene on the lattice using the overlap formalism. The overlap formalism has been applie by a number of
The Chiral Dirac Determinant Accoring to the Overlap Formalism Per Ernstrom an Ansar Fayyazuin NORDITA, Blegamsvej 7, DK-00 Copenhagen, Denmark Abstract The chiral Dirac eterminant is calculate using the
More informationarxiv: v1 [hep-ex] 4 Sep 2018 Simone Ragoni, for the ALICE Collaboration
Prouction of pions, kaons an protons in Xe Xe collisions at s =. ev arxiv:09.0v [hep-ex] Sep 0, for the ALICE Collaboration Università i Bologna an INFN (Bologna) E-mail: simone.ragoni@cern.ch In late
More informationVertical shear plus horizontal stretching as a route to mixing
Vertical shear plus horizontal stretching as a route to mixing Peter H. Haynes Department of Applie Mathematics an Theoretical Physics, University of Cambrige, UK Abstract. The combine effect of vertical
More informationFormulation of statistical mechanics for chaotic systems
PRAMANA c Inian Acaemy of Sciences Vol. 72, No. 2 journal of February 29 physics pp. 315 323 Formulation of statistical mechanics for chaotic systems VISHNU M BANNUR 1, an RAMESH BABU THAYYULLATHIL 2 1
More informationAgmon Kolmogorov Inequalities on l 2 (Z d )
Journal of Mathematics Research; Vol. 6, No. ; 04 ISSN 96-9795 E-ISSN 96-9809 Publishe by Canaian Center of Science an Eucation Agmon Kolmogorov Inequalities on l (Z ) Arman Sahovic Mathematics Department,
More informationPhysics 2212 K Quiz #2 Solutions Summer 2016
Physics 1 K Quiz # Solutions Summer 016 I. (18 points) A positron has the same mass as an electron, but has opposite charge. Consier a positron an an electron at rest, separate by a istance = 1.0 nm. What
More informationOptimal Control of Spatially Distributed Systems
Optimal Control of Spatially Distribute Systems Naer Motee an Ali Jababaie Abstract In this paper, we stuy the structural properties of optimal control of spatially istribute systems. Such systems consist
More information1. At time t = 0, the wave function of a free particle moving in a one-dimension is given by, ψ(x,0) = N
Physics 15 Solution Set Winter 018 1. At time t = 0, the wave function of a free particle moving in a one-imension is given by, ψ(x,0) = N where N an k 0 are real positive constants. + e k /k 0 e ikx k,
More informationApplication of the homotopy perturbation method to a magneto-elastico-viscous fluid along a semi-infinite plate
Freun Publishing House Lt., International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 Application of the homotopy perturbation metho to a magneto-elastico-viscous flui along a semi-infinite
More informationNonlinear Schrödinger equation with a white-noise potential: Phase-space approach to spread and singularity
Physica D 212 (2005) 195 204 www.elsevier.com/locate/phys Nonlinear Schröinger equation with a white-noise potential: Phase-space approach to sprea an singularity Albert C. Fannjiang Department of Mathematics,
More information1 Heisenberg Representation
1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.
More informationConservation laws a simple application to the telegraph equation
J Comput Electron 2008 7: 47 51 DOI 10.1007/s10825-008-0250-2 Conservation laws a simple application to the telegraph equation Uwe Norbrock Reinhol Kienzler Publishe online: 1 May 2008 Springer Scienceusiness
More informationThe effect of nonvertical shear on turbulence in a stably stratified medium
The effect of nonvertical shear on turbulence in a stably stratifie meium Frank G. Jacobitz an Sutanu Sarkar Citation: Physics of Fluis (1994-present) 10, 1158 (1998); oi: 10.1063/1.869640 View online:
More informationProof of SPNs as Mixture of Trees
A Proof of SPNs as Mixture of Trees Theorem 1. If T is an inuce SPN from a complete an ecomposable SPN S, then T is a tree that is complete an ecomposable. Proof. Argue by contraiction that T is not a
More informationChapter 2 Lagrangian Modeling
Chapter 2 Lagrangian Moeling The basic laws of physics are use to moel every system whether it is electrical, mechanical, hyraulic, or any other energy omain. In mechanics, Newton s laws of motion provie
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More informationHyperbolic Moment Equations Using Quadrature-Based Projection Methods
Hyperbolic Moment Equations Using Quarature-Base Projection Methos J. Koellermeier an M. Torrilhon Department of Mathematics, RWTH Aachen University, Aachen, Germany Abstract. Kinetic equations like the
More informationWUCHEN LI AND STANLEY OSHER
CONSTRAINED DYNAMICAL OPTIMAL TRANSPORT AND ITS LAGRANGIAN FORMULATION WUCHEN LI AND STANLEY OSHER Abstract. We propose ynamical optimal transport (OT) problems constraine in a parameterize probability
More informationA Simple Model for the Calculation of Plasma Impedance in Atmospheric Radio Frequency Discharges
Plasma Science an Technology, Vol.16, No.1, Oct. 214 A Simple Moel for the Calculation of Plasma Impeance in Atmospheric Raio Frequency Discharges GE Lei ( ) an ZHANG Yuantao ( ) Shanong Provincial Key
More informationChapter 2 Governing Equations
Chapter 2 Governing Equations In the present an the subsequent chapters, we shall, either irectly or inirectly, be concerne with the bounary-layer flow of an incompressible viscous flui without any involvement
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More informationMaking a Wavefunctional representation of physical states congruent with the false vacuum hypothesis of Sidney Coleman
Making a Wavefunctional representation of physical states congruent with the false vacuum hypothesis of Siney Coleman A. W. Beckwith Department of Physics an Texas Center for Superconuctivity an Avance
More informationEvaporating droplets tracking by holographic high speed video in turbulent flow
Evaporating roplets tracking by holographic high spee vieo in turbulent flow Loïc Méès 1*, Thibaut Tronchin 1, Nathalie Grosjean 1, Jean-Louis Marié 1 an Corinne Fournier 1: Laboratoire e Mécanique es
More informationThe maximum sustainable yield of Allee dynamic system
Ecological Moelling 154 (2002) 1 7 www.elsevier.com/locate/ecolmoel The maximum sustainable yiel of Allee ynamic system Zhen-Shan Lin a, *, Bai-Lian Li b a Department of Geography, Nanjing Normal Uni ersity,
More informationSlide10 Haykin Chapter 14: Neurodynamics (3rd Ed. Chapter 13)
Slie10 Haykin Chapter 14: Neuroynamics (3r E. Chapter 13) CPSC 636-600 Instructor: Yoonsuck Choe Spring 2012 Neural Networks with Temporal Behavior Inclusion of feeback gives temporal characteristics to
More informationOptimal Control of Spatially Distributed Systems
Optimal Control of Spatially Distribute Systems Naer Motee an Ali Jababaie Abstract In this paper, we stuy the structural properties of optimal control of spatially istribute systems. Such systems consist
More informationarxiv:cond-mat/ v2 [cond-mat.mes-hall] 20 Feb 2006
Pair tunneling through single molecules arxiv:con-mat/5249v2 con-mat.mes-hall] 2 Feb 26 Jens Koch, M.E. Raikh, 2 an Felix von Oppen Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee
More informationExperimental demonstration of metamaterial multiverse in a ferrofluid
Experimental emonstration of metamaterial multiverse in a ferroflui Igor I. Smolyaninov, 1,* Braley Yost, Evan Bates, an Vera N. Smolyaninova 1 Department of Electrical an Computer Engineering, University
More informationImpurities in inelastic Maxwell models
Impurities in inelastic Maxwell moels Vicente Garzó Departamento e Física, Universia e Extremaura, E-671-Baajoz, Spain Abstract. Transport properties of impurities immerse in a granular gas unergoing homogenous
More informationTHE VAN KAMPEN EXPANSION FOR LINKED DUFFING LINEAR OSCILLATORS EXCITED BY COLORED NOISE
Journal of Soun an Vibration (1996) 191(3), 397 414 THE VAN KAMPEN EXPANSION FOR LINKED DUFFING LINEAR OSCILLATORS EXCITED BY COLORED NOISE E. M. WEINSTEIN Galaxy Scientific Corporation, 2500 English Creek
More informationSelf-focusing and soliton formation in media with anisotropic nonlocal material response
EUROPHYSICS LETTERS 20 November 1996 Europhys. Lett., 36 (6), pp. 419-424 (1996) Self-focusing an soliton formation in meia with anisotropic nonlocal material response A. A. Zoulya 1, D. Z. Anerson 1,
More informationinvolve: 1. Treatment of a decaying particle. 2. Superposition of states with different masses.
Physics 195a Course Notes The K 0 : An Interesting Example of a Two-State System 021029 F. Porter 1 Introuction An example of a two-state system is consiere. involve: 1. Treatment of a ecaying particle.
More informationNonlinear Dielectric Response of Periodic Composite Materials
onlinear Dielectric Response of Perioic Composite aterials A.G. KOLPAKOV 3, Bl.95, 9 th ovember str., ovosibirsk, 639 Russia the corresponing author e-mail: agk@neic.nsk.su, algk@ngs.ru A. K.TAGATSEV Ceramics
More informationSOLUTIONS for Homework #3
SOLUTIONS for Hoework #3 1. In the potential of given for there is no unboun states. Boun states have positive energies E n labele by an integer n. For each energy level E, two syetrically locate classical
More informationPhysics 5153 Classical Mechanics. The Virial Theorem and The Poisson Bracket-1
Physics 5153 Classical Mechanics The Virial Theorem an The Poisson Bracket 1 Introuction In this lecture we will consier two applications of the Hamiltonian. The first, the Virial Theorem, applies to systems
More informationThe Hamiltonian structure of a 2-D rigid cylinder interacting dynamically with N point vortices
The Hamiltonian structure of a -D rigi cyliner interacting ynamically with N point vortices Banavara Shashikanth, Jerrol Marsen, Joel Burick, Scott Kelly Control an Dynamical Systems, 07-8, Mechanical
More informationarxiv:physics/ v2 [physics.ed-ph] 23 Sep 2003
Mass reistribution in variable mass systems Célia A. e Sousa an Vítor H. Rorigues Departamento e Física a Universiae e Coimbra, P-3004-516 Coimbra, Portugal arxiv:physics/0211075v2 [physics.e-ph] 23 Sep
More informationCalculus of Variations
Calculus of Variations Lagrangian formalism is the main tool of theoretical classical mechanics. Calculus of Variations is a part of Mathematics which Lagrangian formalism is base on. In this section,
More informationTransmission Line Matrix (TLM) network analogues of reversible trapping processes Part B: scaling and consistency
Transmission Line Matrix (TLM network analogues of reversible trapping processes Part B: scaling an consistency Donar e Cogan * ANC Eucation, 308-310.A. De Mel Mawatha, Colombo 3, Sri Lanka * onarecogan@gmail.com
More informationImplicit Lyapunov control of closed quantum systems
Joint 48th IEEE Conference on Decision an Control an 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 29 ThAIn1.4 Implicit Lyapunov control of close quantum systems Shouwei Zhao, Hai
More informationEmergence of quantization: the spin of the electron
Journal of Physics: Conference Series OPEN ACCESS Emergence of quantization: the spin of the electron To cite this article: A M Cetto et al 2014 J. Phys.: Conf. Ser. 504 012007 View the article online
More informationGeneralization of the persistent random walk to dimensions greater than 1
PHYSICAL REVIEW E VOLUME 58, NUMBER 6 DECEMBER 1998 Generalization of the persistent ranom walk to imensions greater than 1 Marián Boguñá, Josep M. Porrà, an Jaume Masoliver Departament e Física Fonamental,
More informationEffective Rheological Properties in Semidilute Bacterial Suspensions
Noname manuscript No. (will be inserte by the eitor) Effective Rheological Properties in Semiilute Bacterial Suspensions Mykhailo Potomkin Shawn D. Ryan Leoni Berlyan Receive: ate / Accepte: ate Abstract
More informationDiagonalization of Matrices Dr. E. Jacobs
Diagonalization of Matrices Dr. E. Jacobs One of the very interesting lessons in this course is how certain algebraic techniques can be use to solve ifferential equations. The purpose of these notes is
More informationOptimized Schwarz Methods with the Yin-Yang Grid for Shallow Water Equations
Optimize Schwarz Methos with the Yin-Yang Gri for Shallow Water Equations Abessama Qaouri Recherche en prévision numérique, Atmospheric Science an Technology Directorate, Environment Canaa, Dorval, Québec,
More informationConvective heat transfer
CHAPTER VIII Convective heat transfer The previous two chapters on issipative fluis were evote to flows ominate either by viscous effects (Chap. VI) or by convective motion (Chap. VII). In either case,
More information6. Friction and viscosity in gasses
IR2 6. Friction an viscosity in gasses 6.1 Introuction Similar to fluis, also for laminar flowing gases Newtons s friction law hols true (see experiment IR1). Using Newton s law the viscosity of air uner
More informationA coupled surface-cahn-hilliard bulk-diffusion system modeling lipid raft formation in cell membranes
A couple surface-cahn-hilliar bulk-iffusion system moeling lipi raft formation in cell membranes Haral Garcke, Johannes Kampmann, Anreas Rätz, Matthias Röger Preprint 2015-09 September 2015 Fakultät für
More informationThe proper definition of the added mass for the water entry problem
The proper efinition of the ae mass for the water entry problem Leonaro Casetta lecasetta@ig.com.br Celso P. Pesce ceppesce@usp.br LIE&MO lui-structure Interaction an Offshore Mechanics Laboratory Mechanical
More informationarxiv: v2 [quant-ph] 27 Apr 2016
Optimal state iscrimination an unstructure search in nonlinear quantum mechanics Anrew M. Chils 1,,3 an Joshua Young 3,4 arxiv:1507.06334v [quant-ph] 7 Apr 016 1 Department of Computer Science, Institute
More informationNOTES ON EULER-BOOLE SUMMATION (1) f (l 1) (n) f (l 1) (m) + ( 1)k 1 k! B k (y) f (k) (y) dy,
NOTES ON EULER-BOOLE SUMMATION JONATHAN M BORWEIN, NEIL J CALKIN, AND DANTE MANNA Abstract We stuy a connection between Euler-MacLaurin Summation an Boole Summation suggeste in an AMM note from 196, which
More informationA note on asymptotic formulae for one-dimensional network flow problems Carlos F. Daganzo and Karen R. Smilowitz
A note on asymptotic formulae for one-imensional network flow problems Carlos F. Daganzo an Karen R. Smilowitz (to appear in Annals of Operations Research) Abstract This note evelops asymptotic formulae
More informationThe Kepler Problem. 1 Features of the Ellipse: Geometry and Analysis
The Kepler Problem For the Newtonian 1/r force law, a miracle occurs all of the solutions are perioic instea of just quasiperioic. To put it another way, the two-imensional tori are further ecompose into
More information