VOLUME 82, NUMBER 10 PHYSICAL REVIEW LETTERS 8MARCH 1999 Search for Off-Diagonal Density Matrix Elements for Atoms in a Supersonic Beam Richard A. Rubenstein, Al-Amin Dhirani, David A. Kokorowski, Tony D. Roberts, Edward T. Smith, Winthrop W. Smith,* Herbert J. Bernstein, ² Jana Lehner, Subhadeep Gupta, and David E. Pritchard Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received 28 October 1998; revised manuscript received 19 January 1999) We demonstrate the absence of off-diagonal elements for the density matrix of a supersonic Na atomic beam, thus showing that there are no coherent wave packets emerging from this source. We used a differentially detuned separated oscillatory field longitudinal interferometer to search for off-diagonal density matrix elements in the longitudinal energymomentum basis. Our study places a stringent lower bound on their possible size over an off-diagonal energy range from 0 to 100 kHz. [S0031-9007(99)08620-2] PACS numbers: 03.75.Dg, 39.10. + j Atomic beam experiments, from early resonance stud- ies [1] to modern work in atom optics and interferom- etry [2], invariably make assumptions about longitudinal atomic motion (e.g., classical, wave packet, plane wave). Despite some theoretical discussion [3–7], to date no ex- perimental test has been performed which can discrimi- nate between these alternative descriptions. This omission, along with the development of novel atomic sources with interesting coherence properties (e.g., cold atom beams, atom lasers [8]), demonstrates the need for experiments which probe not only the transverse [9] but also the longi- tudinal quantum state of atomic beams. In this Letter, we present the first experimental limits on the size of off-diagonal elements (ODEs) in the longi- tudinal energymomentum density matrix characterizing a supersonic atomic beam. This demonstration, together with a determination of the beam’s energy distribution, constitutes a measurement of the density matrix, the most complete possible description of the longitudinal quantum state of the ensemble of atoms in the atomic beam. In the energy basis hn   ¯ hk  2 2m, the time depen- dence of any density matrix element is given by rn 0 , n 00 , t   r o n 0 , n 0 1n coh e i 2pn coh t 2t o  , (1) where n coh  n 00 2n 0 , t o is an initial reference time and r o  rt o . Here n coh is the distance off the diagonal of the density matrix, while n 0 is the coordinate parallel to the diagonal. It is tempting to consider an atomic beam source as emit- ting a stream of wave packets, each a coherent superposi- tion of various longitudinal momenta, implying nonzero ODEs. However, the density matrix for an ensemble of such atoms contains an average over the randomly dis- tributed initial time t o of each wave packet, causing all ODEs to be zero if the source is temporally stationary (e.g., an effusive atom oven in thermal equilibrium). For a nonstationary source, the ensemble average need not make all ODEs zero because the distribution of t o ’s can be nonrandom. Thus nonstationary sources can con- tain coherent, physically observable wave packets. Our beam is created via supersonic expansion of Na seeded in 2 atm of Ar, which passes through a sharp-edged beam skimmer with a hole diameter of d  0.5 mm. This arrangement, which yields a sound velocity at the skimmer of y s  50 ms, may contain nonstationary hydrodynamic instabilities. These could create a time- dependent amplitude modulation of the beam, and, hence, nonzero ODEs, with expected frequencies n coh up to y s d  100 kHz. The time dependence of the ODEs implies that a time- dependent experiment is required to observe them, in contrast to previous measurements of angular [10] or transverse [9,11,12] correlations and contrary to assertions elsewhere [13]. In this experiment, we used the differen- tially detuned separated oscillatory field (DSOF) longi- tudinal interferometer of Ref. [14] as the time-dependent apparatus necessary to study the relative phases of differ- ent energy components and thereby to search for ODEs. We first studied intentionally applied ODEs created via amplitude modulation of the beam [14,15] for calibration purposes. Our search then employed this calibration to place an upper bound on the magnitude of the ODEs cre- ated by our source. To determine the range of our search, we measured the velocity (and hence the energy) distribution of atoms emitted from our source, using the technique of separated oscillatory fields, and found a nearly Gaussian distribution with a mean frequency of 3.4 3 10 13 Hz (1090 ms) and an rms width of 2.2 3 10 12 Hz (35 ms). As we expected off-diagonal structure on the scale of 100 kHz, we searched a narrow ribbon in the neighborhood of the diagonal of the density matrix with n coh # 100 kHz. To illustrate how DSOF can be used to detect nonzero ODEs, we consider an incident beam of ground state two-level atoms in a superposition of plane wave components whose total energies differ by hn coh [so r o n 0 , n 0 1n coh  fi 0]. This beam emerges from a source at x  0 (see Fig. 1) and propagates through two differentially detuned radio-frequency oscillatory field regions located at x 1 and x 2  x 1 1 L, where rf fields of 2018 0031-9007 99  82(10)  2018(4)$15.00 © 1999 The American Physical Society