Extending the Tanguy–Reynaud–Cohen-Tannoudji unified propagator for atomic-beam deflection beyond the Raman-Nath approximation

The deflection of an atomic beam by a near-resonant laser wave spans two regimes: coherent diffraction (the resonant Kapitza-Dirac effect) for interaction times $T$ short compared with the radiative lifetime $\Gamma^{-1}$, and spontaneous-emission-induced diffusion described by a Fokker-Planck equation for $T\gg\Gamma^{-1}$. Tanguy, Reynaud and Cohen-Tannoudji (1984) unified both in a single momentum-transfer propagator — the Green's function mapping the incoming atomic Wigner function to the outgoing one for arbitrary $T$, with spontaneous emission treated exactly — but only for a two-level atom in a strictly monochromatic field under the conditions $kv_\perp\ll\Gamma$ (negligible Doppler shift) and $kv_\perp\ll T^{-1}$ (negligible transverse free flight during the interaction, the Raman-Nath approximation). The open problem stated here is whether that propagator formalism can be extended beyond these assumptions. The literature to date shows: the degenerate-level and multilevel extensions were constructed already in 1987-1992 (Hendriks and Nienhuis; Lindberg) but still within the Raman-Nath condition; the diffractive-to-diffusive transition itself was measured (Gould et al. 1991) and quantitatively reproduced by numerical solution of the 1984 equations (Wilkens et al. 1991; Tan and Walls 1991); beyond the Raman-Nath condition only the spontaneous-emission-free (far-detuned, coherent) Bragg regime has analytic treatments, while dynamics with both free flight and spontaneous emission is handled by Monte Carlo wave-function numerics. The surviving open core is therefore the analytic propagator with transverse free flight retained (and Doppler shifts admitted), with demonstrated reduction to all established limits. The practical need is partly superseded by numerical methods, so the open content is the analytic structure of the kernel itself.

Background

When an atomic beam crosses a laser wave, momentum is exchanged between light and atoms in integer multiples of the photon momentum $\hbar k$ (with $k$ the laser wavenumber and $\hbar$ the reduced Planck constant). Writing $\Gamma$ for the natural linewidth (the spontaneous decay rate of the excited state) and $\tau_R=\Gamma^{-1}$ for the radiative lifetime, two limiting regimes were understood by the early 1980s. For interaction times $T\ll\tau_R$ hardly any spontaneous photons are emitted, the interaction is coherent, and the atomic center-of-mass motion is described by a Schrödinger equation: a resonant standing light wave acts as a thin phase grating and produces the resonant Kapitza-Dirac effect — discrete outgoing momentum peaks at multiples $m\hbar k$ with weights given by squared Bessel functions $J_m^2(\Omega T/2)$, where $\Omega$ is the resonant Rabi frequency measuring the atom-field coupling (derived by Bernhardt and Shore 1981; observed for atoms by Gould, Ruff and Pritchard 1986). A wavepacket localized well below the optical wavelength is instead split into two momentum components by the intensity gradient (the optical Stern-Gerlach effect). For $T\gg\tau_R$ many spontaneous photons are emitted, each giving the atom a randomly directed recoil; the momentum distribution then obeys a Fokker-Planck equation (a drift-diffusion equation in momentum) with the mean dipole force $F(x)$ and the momentum diffusion coefficient $D(x)$ computed by Gordon and Ashkin (1980).

Tanguy, Reynaud and Cohen-Tannoudji (1984) — building on an earlier treatment of deflection profiles with spontaneous emission by the same group (Tanguy, Reynaud, Matsuoka and Cohen-Tannoudji 1983) — bridged the two regimes with a single object. They start from the generalized optical Bloch equations (GOBE): the master equation of the atom's density matrix including spontaneous emission, written in the variables $r$ (the center-of-mass position) and $u$ (the difference of the two position arguments of the density matrix). Under the condition $kv_\perp\ll T^{-1}$, where $v_\perp$ is the root-mean-square transverse atomic velocity along the laser axis, the free-flight term proportional to $(i\hbar/M)\partial^2/\partial r\,\partial u$ ($M$ the atomic mass) can be dropped: the atom does not move appreciably along the standing-wave axis while in the light. This is the Raman-Nath approximation. Together with the Doppler-negligibility condition $kv_\perp\ll\Gamma$ (so the laser detuning is not smeared by the velocity spread), the GOBE become local and act as a linear filter on the atomic correlation function, with an amplitude $L(r,u,T)$ whose Fourier transform with respect to $u$ defines the propagator $G(r,q,T)$ — the Green's function of momentum transfer, mapping the incoming Wigner function (the phase-space quasi-probability distribution of the atom) to the outgoing one through the momentum shift $q$. In the limits $T\ll\tau_R$ and $T\gg\tau_R$ this propagator reduces respectively to the coherent delta-comb (Raman-Nath diffraction) and to a Gaussian with cumulants $F(x)T$ and $2D(x)T$, i.e. exactly the Green's function of the Fokker-Planck equation; at intermediate times $T\sim\tau_R$ its Laplace transform expands in the number of spontaneously emitted photons, each term convolving stimulated momentum jumps with the spontaneous recoil distribution. The derivation is restricted to a nondegenerate two-level atom in a strictly monochromatic running or standing wave.

When $kv_\perp\ll T^{-1}$ fails, the atom traverses a significant fraction of the standing-wave period $\lambda=2\pi/k$ during the interaction. The dynamics then become Bragg scattering — resonant coherent coupling between momentum orders with Pendellösung oscillations (the coherent periodic exchange of population between the coupled orders), the matter-wave analogue of dynamical X-ray diffraction in a crystal — or channeling, guided motion along the nodes or antinodes of the light potential. These regimes lie outside the 1984 propagator, and they are analytically understood only in the far-detuned, spontaneous-emission-free limit; how to build a propagator that includes this free-flight physics while keeping spontaneous emission and all interaction times is the open question examined here.

Problem Statement

Determine whether the unified propagator of Tanguy, Reynaud and Cohen-Tannoudji (J. Phys. B 17, 4623, 1984) can be extended beyond the assumptions under which it was obtained, while preserving its defining character: a single momentum-transfer kernel that maps any incoming atomic Wigner function (the phase-space quasi-probability distribution of the atom) to the outgoing one for arbitrary interaction time $T$, with spontaneous emission fully included.

Concretely, construct a propagator $G$ for the deflection of a two-level atom by a near-resonant monochromatic running or standing laser wave that retains the transverse free-flight dynamics during the interaction — the term of the generalized optical Bloch equations (the master equation of the atom's density matrix including spontaneous emission) discarded under the condition $kv_\perp\ll T^{-1}$, where $v_\perp$ is the root-mean-square transverse atomic velocity along the laser axis and $k$ the laser wavenumber — so that the kernel is valid when the atom moves a significant fraction of the laser wavelength $\lambda=2\pi/k$ along the laser axis while in the light (the Bragg-scattering and channeling regimes), and that relaxes the Doppler-negligibility condition $kv_\perp\ll\Gamma$, where $\Gamma$ is the spontaneous emission rate of the excited state, so that velocity-dependent detuning is admitted;

and demonstrate its consistency by explicit reduction to the established limiting cases: the 1984 propagator as $kv_\perp\to 0$ (including the resonant Kapitza-Dirac comb

$\mathcal{P}_{out}(p)=\sum_m J_m^2(\Omega T/2)\,\delta(p-m\hbar k)$

at $T\ll\Gamma^{-1}$ — where $\Omega$ is the resonant Rabi frequency (the atom-field coupling strength), $J_m$ the Bessel function of the first kind of order $m$, $p$ the transverse momentum and $\delta$ the Dirac delta — and the Gaussian/Fokker-Planck limit with cumulants $F(x)T$ and $2D(x)T$ at $T\gg\Gamma^{-1}$, where $F(x)$ is the mean dipole force and $D(x)$ the momentum diffusion coefficient at transverse position $x$), and the known coherent Bragg-diffraction dynamics when spontaneous emission is suppressed ($\Gamma\to 0$) with $kv_\perp T\gtrsim 1$. If no such extension exists, characterise the obstruction that rules it out, with the same explicitness as the reductions above.

Degenerate level structure and polarization (so far treated analytically only under the Raman-Nath condition), three-dimensional deflection, and non-monochromatic fields are related directions that may be pursued alongside this kernel; they are not required by this statement.

References

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