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Journal of Optics

Distribution of atomic momentum in resonance fluorescence

L. Mandeļ

Published Mar 1, 1979• 52 citations
Publisher

Abstract

It is shown that the distribution of the momentum transfer to an atom in resonance fluorescence may be easily derived from the angular distribution of spontaneously emitted photons. Some general properties of the momentum distribution are found, and explicit forms of the distribution are derived for several special cases. It is shown that the distribution tends to become Gaussian after a large number n of spontaneous photon emissions, with variance proportional to n, but it is already close to the asymptotic form when n=3. When the appropriate averaging is performed to allow for the velocity spread of the atoms, the derived distributions account for the results of atomic beam deflection measurements, but much of the information about the original distributions is thereby lost. In a standing wave field both stimulated and spontaneous emissions contribute to the atomic momentum, in such a way that the variance of the momentum spread is again proportional to the number n of photon emissions.

DOI
10.1088/0150-536x/10/2/001

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Physical Review A•May 1, 1983•34 citations
It is shown that the distribution of the momentum transfer to an atom in resonance fluorescence may be easily derived from the angular distribution of spontaneously emitted photons. Some general properties of the momentum distribution are found, and explicit forms of the distribution are derived for several special cases. It is shown that the distribution tends to become Gaussian after a large number n of spontaneous photon emissions, with variance proportional to n, but it is already close to the asymptotic form when n=3. When the appropriate averaging is performed to allow for the velocity spread of the atoms, the derived distributions account for the results of atomic beam deflection measurements, but much of the information about the original distributions is thereby lost. In a standing wave field both stimulated and spontaneous emissions contribute to the atomic momentum, in such a way that the variance of the momentum spread is again proportional to the number n of photon emissions.
The theory of photon-number statistics in resonance fluorescence is treated, starting with the general formula for the emission probability of nnn photons during a given time interval TTT. The results fully confirm formerly obtained results by Cook that were based on the theory of atomic motion in a traveling wave. General expressions for the factorial moments are derived and explicit results for the mean and the variance are given. It is explicitly shown that the distribution function tends to a Gaussian when TTT becomes much larger than the natural lifetime of the excited atom. The speed of convergence towards the Gaussian is found to be typically slow, that is, the third normalized central moment (or the skewness) is proportional to ${T}^{\ensuremath{-}\frac{1}{2}}$. However, numerical results illustrate that the overall features of the distribution function are already well represented by a Gaussian when TTT is larger than a few natural lifetimes only, at least if the intensity of the exciting field is not too small and its detuning is not too large.
A theory appropriate for investigating the angular spectra of atoms scattered out of an atomic beam by resonance fluorescence due to interactions with an orthogonal uniform light beam is presented. Particular emphasis is made on the scattering in the plane containing the two beams. The atomic distribution is shown to be intimately related to the angular distribution of the spontaneously emitted photons. Both distributions show marked variations with the detuning and field intensity parameters, leading to the possibility of maximum scattering of atoms to occur on either side of the atomic beam, while the accompanying photons are always emitted on one particular side of the light beam. Since the angular distributions reflect the momentum changes suffered by both the atoms and the electromagnetic field, this lends insight into the basic mechanisms involved in resonance fluorescence.
We have observed both diffractive and diffusive regimes in the transfer of momentum to an atomic beam by a standing-wave radiation field. The momentum transfer was measured by passing a well-collimated beam of sodium atoms through a focused standing-wave laser waist. A narrow velocity distribution beam was obtained by utilizing a supersonic expansion and a two-state system was realized by optically pumping the beam prior to the interaction. The transverse momentum transferred to the atomic beam by the radiation field was measured by scanning a detector downstream from the interaction. The overall momentum resolution of the apparatus is less than the momentum of a single photon. The transition from diffractive to diffusive behavior is associated with an increase in the average number of spontaneous decays occurring during the interaction. In the diffractive regime, spontaneous decay is insignificant and momentum transfer proceeds by pairs of absorption/stimulated emission processes. This produces momentum exchange quantized in even multiples of photon momentum, the so-called Kapitza-Dirac effect; the deflection patterns show oscillatory structure. In the diffusive regime, spontaneous decay is important and causes fluctuations in the radiative forces. In addition, the recoil distribution associated with spontaneous decay results in a washing out of the diffraction peaks associated with stimulated processes, resulting in a smooth deflection pattern.
A simple theoretical model is formulated which describes dynamical features of the interaction of two-level atoms with a continuum of radiation oscillators. In particular, this paper is concerned with resonance scattering by single atoms in the presence of an external classical field, and spontaneous emission by a system of identical atoms. In the case of resonance scattering, the response of an atom to rapid changes of the amplitude of the external field is studied. The model for spontaneous emission by a system of atoms is based on the assumptions that all atoms have the same resonance frequency, and that the atoms are contained in a volume whose extension is small compared with the wavelength of the emitted radiation. The complete evolution of a system of mmm atoms is determined by a set of coupled first-order linear differential equations, and explicit solutions are presented for the values m=1,2,3,4,and8m=1,2,3,4, \mathrm{and} 8m=1,2,3,4,and8. Pair correlations of the atoms and photon-number fluctuations are discussed. An approximation for resonance fluorescence by a system of atoms with a broad distribution of resonance frequencies is included in an Appendix.
Exact closed form expressions are derived for the photon number distribution functions and the factorial moments in resonance fluorescence from a two-level atom that is driven on or near resonance by a laser beam of finite bandwidth. Derivation procedure that was used by Lenstra in the theory of photon number statistics for limited cases is applied to our generalized problem with help of the theory of resonance fluorescence with excitation of finite bandwidth. All of the results are presented in a fully suitable form for practical analyses of statistical features of resonance fluorescence.
This paper discusses the connection between the atomic-momentum distribution due to photon recoil and photon statistics. A new formulation is presented based on a Fourier transform with respect to momentum. The asymptotic approximation that is equivalent to a Fokker-Planck treatment is found. Resonance fluorescence is treated in detail. The anomalous diffusion contribution recently found by Ashkin, Gordon, and Cook is separated, and its physical significance is discussed. In particular, it is pointed out that it signifies the coherence between successive photon recoil events. It follows that to obtain the correct diffusion expression one must perform the adiabatic elimination of internal atomic degrees of freedom in the frame moving with the atom, including the deflection caused by light-pressure force. Connections to earlier work are discussed.