Search for Off-Diagonal Density Matrix Elements for Atoms in a Supersonic Beam

Abstract

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 energy͞momentum 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 studies [1] to modern work in atom optics and interferometry [2], invariably make assumptions about longitudinal atomic motion (e.g., classical, wave packet, plane wave). Despite some theoretical discussion [3][4][5][6][7], to date no experimental test has been performed which can discriminate 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 longitudinal quantum state of atomic beams.In this Letter, we present the first experimental limits on the size of off-diagonal elements (ODEs) in the longitudinal energy͞momentum 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 ͑hk͒ 2 ͞2m, the time dependence of any density matrix element is given by r͑n 0 , n 00 , t͒ r o ͑n 0 , n 0 1 n coh ͒e i2pn coh ͑t2t o ͒ , (1) where n coh ϵ n 00 2 n 0 , t o is an initial reference time and r o r͑t 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 emitting a stream of wave packets, each a coherent superposition of various longitudinal momenta, implying nonzero ODEs. However, the density matrix for an ensemble of such atoms contains an average over the randomly distributed 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 contain coherent, physically observable wave packets. Our beam is created via supersonic expansion of Na seeded in ϳ2 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 ϳ 50 m͞s, may contain nonstationary hydrodynamic instabilities. These could create a timedependent amplitude modulation of the beam, and, hence, nonzero ODEs, with expecte...

Single photon beat note in an acousto-optic modulator-based interferometer

“…This experiment may be considered as a photonic analogue of early days atom interferometry setups using detuned separated oscillatory fields as beam splitters. These experiments searched for off-diagonal terms of the density matrix of thermal atomic beams [19][20][21]. Non stationary beam splitters induce energy transfer and thus produce mixing of different energy components in the incoming particles' spectrum.…”

Planck-scale dissipative effects in atom interferometry

“…2) † For earlier works on the use of the formalism of density matrices in atom interferometry, see [35][36][37]33] and references therein.…”

Observation of Atom-Wave Beats Using a Kerr Modulator for Atom Waves

“…transmission by a moving or intensity-modulated grating [14][15][16][17], Zeeman interaction with a time-dependent magnetic field [18][19][20], and atom-laser emitting on two or several modes [21,22]. If the modulation frequency ω is larger than δE kin =ℏ (δE kin is the width of the kinetic energy distribution), the modulation may be detected by observing resolved sidebands in the energy spectrum, while, in the opposite case, the only possible detection is the observation of temporal beats: Our experiment belongs to this last case.…”