Reconstruction of the phase of matter-wave fields using a momentum-resolved cross-correlation technique

Abstract

We investigate the potential of the so-called XFROG cross-correlation technique originally developed for ultrashort laser pulses for the recovery of the amplitude and phase of the condensate wave function of a Bose-Einstein condensate. Key features of the XFROG method are its high resolution, versatility and stability against noise and some sources of systematic errors. After showing how an analogue of XFROG can be realized for Bose-Einstein condensates, we illustrate its effectiveness in determining the amplitude and phase of the wave function of a vortex state. The impact of a reduction of the number of measurements and of typical sources of noise on the field reconstruction are also analyzed. PACS numbers: 39.20.+q,06.90.+v Many of the remarkable properties of atomic Bose-Einstein condensates (BECs) originate from the fact that in those systems a single wave-function is occupied by a macroscopic number of particles [1,2,3,4,5]. The amplitude of this complex wave function corresponds to the atomic density and can comparatively easily be measured using for example absorption or phase contrast imaging [6,7,8,9,10]. The phase of the wave function, on the other hand, is relatively hard to measure directly. Still, its measurement is essential for the characterization of many quantum-degenerate atomic systems, such as e.g. rotating BECs with vortices [11].A situation in many ways similar is encountered in the context of ultrashort laser pulses of a few optical cycles duration. The characterization of these pulses, which are of considerable interest in both fundamental science and applications, requires likewise the knowledge of both amplitude and phase.The measurement of the time-dependent phase of ultrashort laser pulses has found a solution that is in many respects optimal in the so-called Frequency-Resolved Optical Gating (FROG) methods [12,13,14,15]. These methods, which can be adapted to many different situations, offer very high resolution and precision, are stable against noise, and can even compensate for or detect some sources of systematic errors. Single-shot measurements are straightforward [16], and measurements of fields with less than one photon per pulse on average have also been successfully demonstrated. Almost every aspect of the FROG-methods has been studied in depth. Good starting points for accessing the wealth of research literature on that subject are the review article [17] and Ref. [18].The similarity of the phase-retrieval problems for ultrashort pulses and atomic condensates naturally leads one to ask whether the optical FROG techniques can be extended to the atom-optical domain. This paper answers this question in the affirmative by demonstrating how a specific version of the general FROG method, called cross-correlation FROG (XFROG), can easily be translated to the atom-optical case [19,20,21]. Section I briefly reviews the fundamentals of XFROG phase retrieval and discusses how it can be adapted to cold-atom scenarios. As an illustration, section II shows the reconstruction of the spa...

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