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Adiabatic guiding of matter waves

M. Jääskeläinen

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

Recent efforts in the field of ultracold atoms have goneinto creating wave guides of sub-micron sizes. In this thesis,the quantum dynamics of matter waves in such confiningpotential structures with minima extended in space isinvestigated. A general framework based upon the separation ofthe wave function for the quantum particle using a discrete setof mode functions is introduced to reduce the dimensionality ofthe problem. Conditions for propagation in the form ofindependent modes, i.e. adiabatic propagation, are determinedfor the case of matter waves with spatially varying width andfound to be connected to the diffraction of matter wavesthrough a dimensionless parameter, the Fresnel parameter. Further, the analysis is extended to include situationswhere a transition to completely non-adiabatic dynamics takesplace. Here it is found that focusing of matter waves due toenergy redistribution at the end of adiabatic guiding isdetermined by the Fresnel parameter found earlier. In theadiabatic regime, the essential dynamics in the directiontransverse to the minimal valley of the guiding potentialstructure occurs at a time-scale much shorter than that ofchanges in the propagating direction. As a result of this,reection of matter waves is likely to occur unless the changesare made over very long distances. The formalism of adiabatic propagation is also applied thesituation of splitting of matter waves in potential structures.It is found that the adiabaticity criteria are identical tothose for a single guide of spatially varying width. Aformulation of adiabatic splitting in terms of states localizedclose to either of the two minima is developed. The inuence oflongitudinal localization on the splitting of coherentsuperposition states is examined and found to be described by asimple analytical expression.

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Recent efforts in the field of ultracold atoms have goneinto creating wave guides of sub-micron sizes. In this thesis,the quantum dynamics of matter waves in such confiningpotential structures with minima extended in space isinvestigated. A general framework based upon the separation ofthe wave function for the quantum particle using a discrete setof mode functions is introduced to reduce the dimensionality ofthe problem. Conditions for propagation in the form ofindependent modes, i.e. adiabatic propagation, are determinedfor the case of matter waves with spatially varying width andfound to be connected to the diffraction of matter wavesthrough a dimensionless parameter, the Fresnel parameter. Further, the analysis is extended to include situationswhere a transition to completely non-adiabatic dynamics takesplace. Here it is found that focusing of matter waves due toenergy redistribution at the end of adiabatic guiding isdetermined by the Fresnel parameter found earlier. In theadiabatic regime, the essential dynamics in the directiontransverse to the minimal valley of the guiding potentialstructure occurs at a time-scale much shorter than that ofchanges in the propagating direction. As a result of this,reection of matter waves is likely to occur unless the changesare made over very long distances. The formalism of adiabatic propagation is also applied thesituation of splitting of matter waves in potential structures.It is found that the adiabaticity criteria are identical tothose for a single guide of spatially varying width. Aformulation of adiabatic splitting in terms of states localizedclose to either of the two minima is developed. The inuence oflongitudinal localization on the splitting of coherentsuperposition states is examined and found to be described by asimple analytical expression.
In this paper we present an analysis of how matter waves, guided as propagating modes in potential structures, are split under adiabatic conditions. The description is formulated in terms of localized states obtained through a unitary transformation acting on the mode functions. The mathematical framework results in coupled propagation equations that are decoupled in the asymptotic regions as well before as after the split. The resulting states have the advantage of describing propagation in situations, for instance matter-wave interferometers, where local perturbations make the transverse modes of the guiding potential unsuitable as a basis. The different regimes of validity of adiabatic propagation schemes based on localized versus delocalized basis states are also outlined. Nontrivial dynamics for superposition states propagating through split potential structures is investigated through numerical simulations. For superposition states the influence of longitudinal wave-packet extension on the localization is investigated and shown to be accurately described in quantitative terms using the adiabatic formulations presented here.
In this work the adiabatic approximation is applied to the propagation of matter waves in confined geometries like those experimentally realized in recent atom optical experiments. Adiabatic propagation along a channel is assumed not to mix the various transverse modes. Nonadiabatic corrections arise from the potential squeezing and bending. Here we investigate the effect of the former. Detailed calculations of two-dimensional propagation are carried out both exactly and in an adiabatic approximation. This offers the possibility to analyze the validity of adiabaticity criteria. A semiclassical (sc) approach, based on the sc Massey parameter is shown to be inadequate, and the diffraction due to wave effects must be included separately. This brings in the Fresnel parameter well known from optical systems. Using these two parameters, we have an adequate understanding of adiabaticity on the system analyzed. Thus quantum adiabaticity must also take cognizance of the intrinsic diffraction of matter waves.
This dissertation explores the spatial patterns that evolve dynamically in atomic clouds and beams after they interact with one or more standing wave fields, typically formed by laser beams. The main systems under study are ensembles of free atoms, but a model Bose-Einstein condensate and the harmonic oscillator are considered as well. Focused atomic patterns with features much smaller than the wave length of the light fields are possible owing to the nonlinear atom-field interaction and are emphasized. The quantitative results combine numerical simulations and analytical expressions. Computer algorithms were developed from known methods in partial differential equations to perform the numerical simulations. The analytical results relied on standard and newly contrived techniques in quantum physics. The analytical expressions were evaluated by computer in order to extract valuable quantitative information and to compare different analytical methods with one another and with the numerical simulations. The thesis is written as a self-contained piece, and as such, develops specific examples of atom-field configurations and atomic patterns of interest from a general theory of the atoms interacting with a spatially varying light field that influences the subsequent center-of-mass motion. Beyond those already published by our group and expanded upon in the thesis, significant advances are made in: the link between transient interactions in the time- and spatial-domains; an understanding of the Raman-Nath approximation, its range of validity, and its corrections; atomic focusing by pulsed adiabatic interactions between an atom and a single standing wave light field; the differences between classical and quantum free motion after periodic interactions; the distinctions between free motion after amplitude and phase gratings; harmonic oscillator ground states kicked by spatially varying pulses; and quasiperiodic echo effects. The general theoretical structure that was detailed can be applied effectively, as intended from the outset, to other problems in atom optics.
Ultracold atoms whose de Broglie wavelength is of the same order as an extended confining potential can experience waveguiding along the potential. When the transverse kinetic energy of the atoms is sufficiently low, they can be guided in the lowest order mode of the confining potential by analogy with light guided by a single mode optical fiber. We have obtained the first images of the transverse mode structure of guided matter waves in a confining potential with up to 6565%65 of atoms in the lowest order mode. The coherence of the guided atomic de Broglie waves is demonstrated by the diffraction pattern produced when incident upon a two dimensional periodic structure. Such coherent waveguides will be important atom optic components in devices with applications such as atom holography and atom interferometry.
In this thesis we relate the formal description of various cold atomic systems in the\nenergy eigenbasis, to the observable spatial mode dynamics. Herein the `spatial mode\ndynamics' refers to the direction of photon emission following the spontaneous emission\nof an excited fermion in the presence of a same species and spin ideal anisotropic Fermi\nsea in its internal ground state. Due to the Pauli principle, the presence of the ground\nstate Fermi sea renders the phase space, anisotropic and only partially accessible, thereby\na ecting the direction of photon emission following spontaneous emission.\nThe spatial and energetic mode dynamics also refers to the quantum `tunneling' interaction\nbetween localised spatial modes, synonymous with double well type potentials.\nHere we relate the dynamics of the wavefunction in both the energetic and spatial representations.\nUsing this approach we approximate the relationship between the spatial\nand energetic representations of a wavefunction spanning three spatial and energetic\nmodes. This is extended to a process known as Spatial Adiabatic Passage, which is a\ntechnique to transport matter waves between localised spatial modes. This approach\nallows us to interpret the transport of matter waves as a signature of a geometric phase\nacquired by the one of the internal energy eigenstates of the system during the cyclical\nevolution. We further show that this geometric phase may be used to create spatial\nmode qubit and qutrit states.
We study experimentally and theoretically a beam splitter setup for guided atomic matter waves. The matter wave is a guided atom laser that can be tuned from quasimonomode to a regime where many transverse modes are populated, and propagates in a horizontal dipole beam until it crosses another horizontal beam at 45°. We show that depending on the parameters of this X configuration, the atoms can all end up in one of the two beams (the system behaves as a perfect guide switch), or be split between the four available channels (the system behaves as a beam splitter). The splitting regime results from a chaotic scattering dynamics. The existence of these different regimes turns out to be robust against small variations of the parameters of the system. From numerical studies, we also propose a scheme that provides a robust and controlled beam splitter in two channels only.