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Physical Review A

Quantum-state measurement through ballistic expansion of matter waves

M. Jääskeläinen, Stig Stenholm

Published Oct 22, 2002• 7 citations
Publisher

Abstract

In this paper, a method for state measurement in atomic waveguides is presented. Matter waves exiting a terminated waveguide are allowed to expand ballistically. Measurement is performed by spatially resolved detection of laser-induced fluorescence. We describe in detail the motion of wave packets both in the guiding potential and in the region of free expansion. It is shown that different modes can be distinguished in a straightforward manner and that the same is true for coherent superpositions after introduction of a controlled phase shift. The sensitivity of the transition from adiabatic guiding to free expansion is investigated in detail. It is found that adiabatic readjustment, which will distort the original state, can be considerable but may be avoided by proper choice of potential parameters. The method proposed here is shown to be feasible for experiments in atom optics through utilizing numerical estimates.

DOI
10.1103/physreva.66.043612

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In this paper, a method for state measurement in atomic waveguides is presented. Matter waves exiting a terminated waveguide are allowed to expand ballistically. Measurement is performed by spatially resolved detection of laser-induced fluorescence. We describe in detail the motion of wave packets both in the guiding potential and in the region of free expansion. It is shown that different modes can be distinguished in a straightforward manner and that the same is true for coherent superpositions after introduction of a controlled phase shift. The sensitivity of the transition from adiabatic guiding to free expansion is investigated in detail. It is found that adiabatic readjustment, which will distort the original state, can be considerable but may be avoided by proper choice of potential parameters. The method proposed here is shown to be feasible for experiments in atom optics through utilizing numerical estimates.
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A simple quantum theory of recently suggested optical techniques for ultrahigh-resolution position measurement and localization of moving atoms in beams is presented. Both the internal and center-of-mass motion are treated quantum mechanically so that the limitations on the ultimate position resolution due to atomic motion and wave-mechanical diffraction are included in the analysis. The techniques utilize a miniaturized form of Raman-induced resonance imaging in which optical fields are used to make transitions from a long-lived initial state to a long-lived final state. The final state is shifted by the spatially varying potential of an applied force field in order to correlate the atomic position with its resonance frequency. Spatially varying level shifts are obtainable by using very large Zeeman field gradients or spatially varying light shifts in a small interaction volume. This results in very high spatial resolution. The atomic transit time across the optical-field region is limited by focusing to an ideal diameter that minimizes the spatial resolution length. The results of the analysis show that nanometer spatial resolution of the initial-state position distribution is attainable. Under appropriate conditions, the final-state spatial wave function can take the form of a minimum-uncertainty Gaussian wave packet obeying \ensuremath{\Delta}x\ensuremath{\Delta}p=\ensuremath{\Elzxh}/2. Such states may prove useful in studying one-dimensional wave-packet motion in applied potentials.
Summary form only given. The preparation of the quantum state of physical systems and its diagnostics by quantum-state reconstruction has been a subject of large progress in recent years. Various experiments have successfully demonstrated the feasibility of reconstructing the quantum state of optical fields, molecular vibrations, trapped ions, and matter waves. The impressive success in laser cooling of neutral atoms opens new possibilities for preparing quantum states of ensembles of atoms in optical lattices and also quantum computing. In this context, the diagnostics of the quantum state, would provide a very useful tool for studying the accuracy of prepared quantum states or exploring decoherence mechanisms. However, to our knowledge, a reconstruction method for optical lattices that accounts for typical features of optical lattice potentials such as anharmonicity and band structure, is not yet available. We propose such a method, that we think is relatively simple to implement, since only phase shifts of the lattice laser beams and measurements of the band populations in the lattice potential are needed. These techniques are feasible by use of electro/acousto-optical modulators and by measuring the momentum distribution of the atoms after adiabatically turning off the lattice lasers.
We present a general method for calculating wave-packet evolution in the context of ultracold atoms in matter-wave interferometers. This method provides an efficient tool for analyzing the performance of atomic interferometers based on atom clouds initially prepared in a trap in a Bose-Einstein condensate (BEC) state or in a thermal distribution. Without having to solve explicitely the Gross-Pitaevskii or the Schrodinger equation, this provides a good estimation of dynamic wave-packet properties such as size and phase across the wave-packet, and slows to predict the coherence of the interferometer. We develop a generalized Thomas-Fermi approximation for a BEC in a harmonic trap, which is valid for any atom-atom interaction strength. In particular, this approximation also provides a good estimation of the transition from a three-dimensional to a one-dimensional BEC in an elongated trap. The approximation is followed by a theory of dynamic wave-packet evolution when the trap is fully or partially switched off and the atoms are split and propagate, for example, in free space or in a matter waveguide. The method is applied for studying two examples of interferometric effects: reduction of coherence of two-state interferometers due to imperfect spatial recombination of the two interferometer arms (the Humpty-Dumpty effect) and phase diffusion due to number uncertainty in the two interferometer arms, which was previously studied thoroughly only for interferometric schemes where the BECs in the two arms stay trapped (for example, in a double-well potential). For both effects we extend the applicability of the theory to a wide range of interferometric scenarios that were not included in previous theories.
We present experiments investigating different aspects concerning the coherent manipulation of a Bose-Einstein condensate wavefunction with far blue detuned dipole potentials. In a first set of experiments we study a waveguide for matter waves as an important atom optical element allowing to hold and guide the atoms for long observation times. We present a flexible realization of this waveguide based on a tube shaped Laguerre Gaussian laser beam and investigate the dynamics of a Bose condensed ensemble transferred into this guide. In a second set of experiments we study the dynamics of dark solitons created by the manipulation of the phase of the condensate wave function by temporally applied dipole potentials. This method of phase imprinting represents a powerful tool for the design of matter waves and the controlled creation of various excitations. The dynamics of dark solitons in matter waves is investigated in detail by experimentally and numerically studying the temporal evolution of the condensate density as well as by mapping the velocity field of the excited condensate wavefunction. In these measurements we directly detect the flux of the wave function associated with the dark soliton using spatially resolved Bragg scattering. Experiments investigating the creation and interaction of several dark solitons conclude this chapter.