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2.3 Feynman's Path Integral Matter waves from localized sources in homogeneous force fields Dissertation von Tobias Kramer 30. Januar 2003 Physik-Department der Technischen Universit¨at Munchen¨ Institut fur¨ theoretische Physik T30 Matter waves from localized sources in homogeneous force fields Tobias Kramer Vollst¨andiger Abdruck der von der Fakult¨at fur¨ Physik der Technischen Universit¨at Munchen¨ zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr. Gerhard Abstreiter Prufer¨ der Dissertation: 1. Univ.-Prof. Dr. Manfred Kleber 2. Hon.-Prof. Dr. Ignacio Cirac Die Dissertation wurde am 30.1.2003 bei der Technischen Universit¨at Munchen¨ eingereicht und durch die Fakult¨at fur¨ Physik am 3.4.2003 angenommen. Contents Contents i List of Figures iii 1 Introduction 1 2 Sources in quantum mechanics 3 2.1Interferenceofmatterwaves.......................... 3 2.2Currentsgeneratedbyquantumsources.................... 11 2.3Feynman’spathintegral............................ 14 3 Properties of Green functions 17 3.1Theenergy-dependentGreenfunction..................... 17 3.2ConnectiontoFermi’sgoldenrule....................... 21 3.3ThegeneralquadraticHamiltonian...................... 23 3.4Multipolesources................................ 25 3.5Greenfunctionsfromcomplexintegration.................. 28 4 Uniform magnetic and electric fields 35 4.1GaugepropertiesoftheGreenfunction.................... 35 4.2 Propagator in crossed magnetic and electric fields . 37 4.3Purelyelectricfield............................... 38 4.4Parallelmagneticandelectricfields...................... 39 4.5Purelymagneticfield.............................. 47 4.6Crossedmagneticandelectricfields...................... 49 5 Currents in 2D electric and magnetic fields 53 5.1 Two-dimensional Green function for crossed fields . 53 5.2 Properties of the two-dimensional density of states . 59 5.3Spinandeffectivemass............................. 62 5.4 Conductivity in the quantum Hall regime . 63 i ii CONTENTS 6 Source model in strong fields 69 6.1Connectiontoexperiments........................... 69 6.2Predictionsforthephotodetachmentrate................... 75 7 Spatially extended quantum sources 77 7.1 Gaussian source distribution in a linear force field . 78 7.2Gaussiansourcesinparallelfields....................... 82 7.3Generalsourcedistribution........................... 83 7.4TheFranck-Condonprinciple......................... 86 8 Atom laser 91 8.1Bose-Einsteincondensation........................... 91 8.2RadiooutputcouplingfromaBEC...................... 93 8.3 Transition from a BEC state to the gravitational field . 94 8.4 Energy range for the operation of an atom laser . 95 8.5Vorticesandatomlasers............................ 96 9 Summary 107 A Linear canonical transformation 111 A.1Evaluationofthematrixrepresentation.................... 111 A.2Homogeneousfieldsatarbitraryangles.................... 112 B Integrals involving Airy functions 115 C Useful relations involving Hermite polynomials 119 D The reflection approximation 121 References 123 List of publications 131 Danksagung 133 Lebenslauf 135 List of Figures 2.1Twoslitexperiments............................... 4 2.2 Probability density for a two-slit configuration . 6 2.3 Probability density for a uniform field . 8 2.4Trajectoriesintheuniformfield........................ 9 2.5 Pulsed emission from a quantum source in a linear field . 10 2.6 Potential scattering and long range forces . 11 3.1Saddlepointsinthecomplexplane...................... 29 3.2UniformAiryandsaddlepointapproximation................ 34 4.1 Total current from multipole sources in parallel fields . 41 4.2Currentdensityinparallelfields:overview.................. 43 4.3 Current density in parallel fields: four path interference . 45 4.4Lateraluncertaintyproductinparallelfields................. 46 4.5 Density of states in a 3D electron gas with magnetic field . 49 4.6 Propagator for crossed fields in three dimensions. 50 4.7Totalcurrentincrossedfieldsinthreedimensions.............. 51 5.1 2D electronic density of states in crossed fields . 60 5.2 Hall resistance ρxy and longitudinal resistance ρxx as a function of the mag- neticfield..................................... 65 5.3BreakdownofthequantizedHallresistance.................. 66 6.1 Photodetachment in crossed fields at different temperatures. 73 6.2Zeemansplittedenergylevelsinsulfur..................... 74 6.3 Photodetachment rate of S− incrossedfields................. 75 7.1 Virtual point source for a Gaussian distribution . 79 7.2 Transition from a point-like to an extended source in a linear potential. 81 7.3 Transition from a point-like to an extended source in a quadratic potential. 84 7.4Franck-Condonfactorinthequantumsourcemodel............. 87 8.1 Ground state of a Bose-Einstein condensate . 93 8.2 Total current of an atom-laser: theory vs. experiment . 96 iii iv LIST OF FIGURES 8.3Currentdensityofanatom-laserfordifferentBECsizes........... 97 8.4Totalcurrentoriginatingfromasinglevortex................ 100 8.5Atomlaserdensityprofilesforasinglevortex................ 101 8.6BECwavefunctionwithanembeddedvortexlattice............. 103 8.7 Total current originating from a vortex lattice . 105 8.8 Atom laser beam profile originating from a vortex lattice . 106 Chapter 1 Introduction The motion of particles in a uniform force field is a standard subject of classical mechanics. For given initial momentum and position, the equations of motion contain all information about the further path of the particle. In quantum mechanics the simultaneous specifica- tion of position and momentum is not possible. However, the question of how a particle evolves, that starts at a given time, or, alternatively, with fixed energy from a well-defined place is meaningful. A convenient way to deal with these initial conditions is the use of the time-evolution operator in its coordinate representation (the propagator) and its Laplace transform, the so-called energy-dependent Green function. The systematic study of propa- gators is progressing well, as recent compilations of known solutions show (i.e. about 1000 references in [GS98]). Unfortunately, the situation for three-dimensional energy-dependent Green functions is not keeping pace, and “simple” analytic solutions for only six(!) cases are known: the free particle; a particle subject to uniform electric or magnetic fields; the isotropic harmonic oscillator; the Coulomb potential; and the combination of parallel electric and magnetic fields. More analytic solutions are urgently needed, since perturbation theory for constant or linear (and therefore extremely long range) potentials is not applicable. The availability of the Green function offers the possibility to treat stationary scattering problems by obtaining the total scattering current and the spatial density distribution. Combined with quantum- source formalism, more complicated emission characteristics and distributions of particle emitting sources can be incorporated in the calculations. Due to the known differentiation properties of an analytically given Green function, most of the results for the currents can again be evaluated as closed expressions. The density of states in electronic systems is also directly related to the energy-dependent Green function and is highly relevant for the calculation of electric, magnetic and thermal properties of materials. The six already available Green functions have profound implications for many different physical systems: the angular momentum dependence of continuum cross sections was studied by Wigner; photodetachment experiments into external fields probe the binding energy of electrons with unprecedented accuracy; and the interference of atomic matter waves in the gravitational field has become a standard topic in quantum optics. Likewise many effects in solid state physics depend on the density of states in the presence of external 1 2 CHAPTER 1. INTRODUCTION fields. In all these cases a formulation in terms of Green functions is possible and this, in turn, leads to a coherent theoretical description. How can we find new energy-dependent Green functions? A thoroughly mathematically understanding of the propagator is needed, since it provides one possibility to construct the corresponding Green functions by a Laplace transform. The theory of residues is one main ingredient for the calculations. However, the singularities of the propagator often show a complicated structure and render this method unusable. A numerical integration is frequently difficult, since the strong oscillations of the propagator lead to the cancellation of terms that affect the accuracy critically. The analytic construction of the Green function is a main topic of this thesis. The second one is the application and demonstration of the Green function in a variety of systems and processes in quantum physics. In the next chapter we will show how in the quantum realm interference phenomena in uniform fields emerge and establish a theoretical description that is suitable for scattering from localized sources into external fields. The quantum-source approach provides an excellent theoretical basis for the embedding of localized sources into external potentials. As a first example we review two-slit experiments and show how a uniform field provides a virtual double-slit for emitted waves from a quantum source. In Chapter 3 we review different mathematical concepts and apply them to the calcu- lation of Green functions. The presence of magnetic and electric fields leads to quadratic
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