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Conference on Lasers and Electro-Optics

A coherence length for matter waves

Sonja Franke‐Arnold, Stephen Mark Barnett, G. Huyet

Published Aug 29, 2005• 0 citations
Publisher

Abstract

Summary form only given. We introduce the concept and mathematical description of a coherence length and a coherence momentum for matter states, and explain their significance via the example of the motional states of a single cold trapped ion. These parameters describe the coherence properties of matter waves analogously to the way in which optical coherence determines the coherence time and spectral width. They become apparent in the interference pattern of a matter or light wave interferometer, respectively. While the coherence length is a characteristic length by which two parts of a matter wave packet can be separated so that they still interfere, the coherence momentum describes the time interval of this overlap.

DOI
10.1109/iqec.2000.907899

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Summary form only given. We introduce the concept and mathematical description of a coherence length and a coherence momentum for matter states, and explain their significance via the example of the motional states of a single cold trapped ion. These parameters describe the coherence properties of matter waves analogously to the way in which optical coherence determines the coherence time and spectral width. They become apparent in the interference pattern of a matter or light wave interferometer, respectively. While the coherence length is a characteristic length by which two parts of a matter wave packet can be separated so that they still interfere, the coherence momentum describes the time interval of this overlap.
We define and discuss measures of coherence for matter waves. These characterize the interference pattern arising from the overlap of one point of a matter wave with another. We identify two coherence quantities, one related to the distance between the interfering parts of a matter wave, and one describing the influence of the position within the wave. We show that our coherence properties are suitable for describing both single particles such as cold trapped ions, and many-particle systems such as Bose-Einstein condensates and Fermi gases.
The concept of coherence has been applied in the quantum theory of radiation and matter as well as in classical optics. We review here the meaning of the term in each case and show how these apparently different uses of the word coherence can be understood in terms of a single concept involving interference (nonzero cross terms), minimum uncertainty, and cohering (uniform variance) of the wave packet that describes the system. Self-induced transparency, superradiance, and superconductivity are discussed as dramatic examples of coherence.
The goal of this project was to develop an apparatus to measure the coherence length of the light waves produced by different lasers.The coherence length is the distance over which two interacting waves maintain a constant relative phase.The method selected to achieve the goal of the project was to introduce the laser light into a modified Michelson interferometer arrangement while observing and measuring the resulting interference pattern. This pattern can be interpreted in terms of the properties of the laser light from which the pattern was produced, including, it was predicted, its coherence length. In order to characterize the apparatus and develop a means of interpreting the interference patterns, two different lasers were used: a red diode laser taken from an inexpensive laser pointer, and a red He-Ne laser. As part of this project the quantum eraser phenomenon and bandwidth of the diode and He-Ne lasers were also investigated. The project is continuing an investigation of the utility of the apparatus for coherence length measurements.
Coherence lengths of one particle states described by quantum wave functions are studied. We show that one particle states in various situations are not described by simple plane waves but are described by wave packets that are superpositions of plane waves. Wave packet is an approximate eigenstate of the free Hamiltonian and has a finite spatial size which we call the coherence length. The coherence lengths in the coordinate space and in the momentum space are studied in the present paper. We investigate several mechanisms of forming wave packets, stabilities of wave packets, and transformations of wave packets. 1 1
Ultra-cold atoms provide ideal platforms for interferometry. The macroscopic matter-wave property of ultra-cold atoms leads to large coherent length and long coherent time, which enable high accuracy and sensitivity to measurement. Here, we review our efforts to improve the performance of the interferometer. We demonstrate a shortcut method for manipulating ultra-cold atoms in an optical lattice. Compared with traditional ones, this shortcut method can reduce the manipulation time by up to three orders of magnitude. We construct a matter-wave Ramsey interferometer for trapped motional quantum states and significantly increase its coherence time by one order of magnitude with an echo technique based on this method. Efforts have also been made to enhance the resolution by multimode scheme. Application of a noise-resilient multi-component interferometer shows that increasing the number of paths could sharpen the peaks in the time-domain interference fringes, which leads to a resolution nearly twice compared with that of a conventional double-path two-mode interferometer. With the shortcut method mentioned above, improvement of the momentum resolution could also be fulfilled, which leads to atomic momentum patterns less than 0.6 ℏkL . To identify and remove systematic noises, we introduce the methods based on the principal component analysis (PCA) that reduce the noise in detection close to the 1 / 2 of the photon-shot noise and separate and identify or even eliminate noises. Furthermore, we give a proposal to measure precisely the local gravity acceleration within a few centimeters based on our study of ultracold atoms in precision measurements.
We describe a method to create superposition states from a mixed state of a harmonic oscillator. If the initial state is described by a thermal state, then the resulting superposition state will be a ``hot'' superposition state. Such a state can be distinguished from a statistical mixture by its coherence properties. Here we suggest how to demonstrate the coherence of the superposition state by observing interference fringes when the two parts of the wave packet are overlapped. In the case of the mixed superposition state, a partial overlap may not be sufficient to observe the presence of fringes. We introduce therefore the idea of a coherence length for the wave packet and demonstrate its relevance for interferometry of mixed states. We illustrate our ideas with the example of superposition states for the motion of a single trapped ion.