Quantitative multilevel theory-experiment comparison for probe-transmission spectroscopy of the one-dimensional lin-perp-lin cesium optical lattice (Verkerk et al., 1992)

In the 1992 experiment of Verkerk, Lounis, Salomon, Cohen-Tannoudji, Courtois and Grynberg (Phys. Rev. Lett. 68, 3861), cold cesium atoms in a one-dimensional lin-perp-lin optical molasses spontaneously organized into a periodic spatial arrangement, and a weak probe beam revealed transmission spectra containing stimulated Raman resonances between vibrational bands and a narrow central stimulated Rayleigh resonance. Three quantitative puzzles were left open: the vibrational frequencies grow with pump intensity as an apparent power law $I^{0.4}$ instead of the predicted $I^{1/2}$ ; the Raman linewidths (35-50 kHz) lie roughly an order of magnitude below the optical pumping rate while the Rayleigh linewidth (about 8 kHz) is narrower still, a Lamb-Dicke coherence-preservation effect the paper argued only qualitatively; and the complete theory of the spectra, promised for a forthcoming publication, was deferred. The promised theory appeared as Courtois and Grynberg, Phys. Rev. A 46, 7060 (1992), but was worked out in detail only for a simplified $J_g=1/2 \to J_e=3/2$ level structure, with a single illustrative computation for the cesium-like $J_g=4 \to J_e=5$ transition; its authors explicitly left a quantitative comparison between experiment and theory open. Later work extended the program to steady two- and three-dimensional lattices and reinterpreted the Rayleigh resonance, but the quantitative multilevel comparison for the 1992 one-dimensional cesium spectra, and a quantitative explanation of the sub- $\sqrt{I}$ intensity scaling, remain open. The problem is stated below in that narrowed, precise form.

Background

An optical lattice is created when counterpropagating laser beams interfere to give a spatially periodic light shift (AC Stark shift) for atoms. In the one-dimensional lin-perp-lin configuration, two counterpropagating beams with orthogonal linear polarizations produce a light field whose polarization cycles through pure $\sigma^+$ , linear, and $\sigma^-$ (right-circular, linear, and left-circular) over each quarter wavelength, so that the light-shift potential $U(z)$ for each ground-state Zeeman sublevel is periodic with $\lambda/2$ , the potential minima of the two sublattices being interleaved at a spacing of $\lambda/4$ ; its depth $U_0$ is conventionally quoted in units of the recoil energy $E_R = \hbar^2 k^2/(2m)$ , the kinetic energy an atom of mass $m$ gains by absorbing a photon of wave number $k$ . Because the light shift of each ground-state Zeeman sublevel depends on the local polarization, Sisyphus cooling (a polarization-gradient mechanism) arises: optical pumping cycles, occurring at rate $\Gamma'$ (absorption followed by spontaneous emission), preferentially transfer atoms near polarization-dependent potential maxima into sublevels that are low-energy at the neighboring minima. The dissipation localizes atoms near the bottoms of the wells, where their quantized motion forms vibrational bands. In the two-level, unsaturated limit the vibrational frequency at a well bottom is proportional to the atomic Rabi frequency (the atom-light coupling strength, proportional to the light-field amplitude and hence to $\sqrt{I}$ ), so the band spacing scales as $\Omega_v \propto \sqrt{U_0} \propto \sqrt{I}$ with the laser intensity $I$ . Castin and Dalibard (Europhys. Lett. 14, 761 (1991)) developed the dressed-atom band formalism for such molasses, treating the atom dressed by the laser field so that its quantized center-of-mass motion organizes into vibrational bands; for a ground-state angular momentum $J_g = 1/2$ the eigenstates factorize into an internal (Zeeman) part and an external (motional) part, and this factorization fails for larger $J_g$ . In the unsaturated two-level picture the band spacing therefore grows as $\sqrt{I}$ , but the 1992 measurement instead shows a weaker power law (approximately $I^{0.4}$ ); candidate explanations for this sub- $\sqrt{I}$ scaling include weak saturation of the transition (the mechanism conjectured in that paper), multilevel light shifts, and anharmonicity of the vibrational bands.

The experiment concerned cesium atoms cooled in such a one-dimensional lin-perp-lin molasses on the cyclic (closed) hyperfine transition $6S_{1/2}, F=4 \to 6P_{3/2}, F'=5$ , i.e. between the ground-state manifold of total angular momentum $F=4$ and the excited-state manifold of total angular momentum $F'=5$ , treated theoretically as $J_g = 4 \to J_e = 5$ . A weak probe beam was sent through the molasses and its transmission recorded versus probe-pump detuning. The spectra showed (i) lateral resonances interpreted as stimulated Raman transitions between adjacent vibrational bands, whose positions track the band spacing, and (ii) a narrow central structure interpreted as a stimulated Rayleigh resonance arising from two-wave mixing between pump and probe (coherent scattering of one beam off the grating the two beams jointly write in the atoms): forward scattering on the orientation (magnetization) grating that the Sisyphus mechanism imprints on the atoms, and backscattering on the density and orientation (magnetization) gratings; for ground states with $J_g > 1/2$ a modulated alignment (rank-2 moment) can also contribute.

Two further concepts are essential. First, the Lamb-Dicke regime: when an atom is spatially localized to much less than the optical wavelength ( $\lambda/25$ in the experiment), an absorption-spontaneous-emission cycle returns it to the same vibrational state with probability close to one, so Raman coherences between bands survive many pumping cycles and the observed resonances are far narrower than the optical pumping rate $\Gamma'$ (the 1992 paper reports Raman widths of 35-50 kHz, about an order of magnitude below $\Gamma'$ , and a Rayleigh width of about 8 kHz). Second, the spatial order itself: Sisyphus cooling sorts atoms into planes of alternating local magnetization spaced $\lambda/4$ apart, an antiferromagnetic-like arrangement of magnetization that constitutes large-scale order of cold atoms driven by light. The 1992 one-dimensional molasses was transient, with the atomic density decaying on a time constant of 5-50 ms, which limited steady-state characterization.

The 1992 paper left open the quantitative theory of these probe-transmission spectra for the real multilevel cesium system: the line positions as functions of lattice depth (including the observed weaker-than- $\sqrt{I}$ intensity scaling), the linewidths set by optical-pumping-induced dephasing with Lamb-Dicke suppression, and the decomposition and amplitudes of the central Rayleigh structure.

Problem Statement

Consider the experiment of Verkerk et al. (Phys. Rev. Lett. 68, 3861 (1992)): cesium atoms cooled in a one-dimensional lin-perp-lin optical molasses (lattice period $\lambda/4$ ) on the cyclic transition $6S_{1/2}, F=4 \to 6P_{3/2}, F'=5$ , i.e. a $J_g=4 \to J_e=5$ system with all ground-state Zeeman sublevels, probed by a weak beam whose transmission spectrum versus probe-pump detuning exhibits stimulated Raman resonances between vibrational bands and a narrow central stimulated Rayleigh resonance. The measured facts to be explained quantitatively are: (i) the Raman resonance positions follow an apparent power law in the pump intensity, $I^{x}$ with $x \approx 0.4$ , visibly weaker than the $I^{1/2}$ law expected for an unsaturated two-level atom; (ii) the Raman linewidths are 35-50 kHz, roughly an order of magnitude below the optical pumping rate $\Gamma'$ , while the Rayleigh linewidth (about 8 kHz) is narrower still; and (iii) the central structure has a definite shape and amplitude reflecting scattering on two distinct atomic gratings, the rank-0 density and rank-1 orientation (magnetization) gratings (plus possible rank-2 alignment for $J_g > 1/2$ ). The quantitative comparison is to be made against the measured line positions, widths, and relative amplitudes of Verkerk et al. (reference 1).

Deliver a quantitative theory of this probe-transmission spectrum for the real multilevel cesium transition, without reducing the internal structure to an effective $J_g=1/2 \to J_e=3/2$ system, and confront it with the measured spectra. Concretely, the solution must provide: (a) the vibrational band structure and Raman line positions as functions of lattice depth $U_0/E_R$ and pump intensity, including anharmonicity, together with a quantitative account of the sub- $\sqrt{I}$ scaling that determines what governs the observed exponent and demonstrates it quantitatively over the full multilevel dynamics; (b) Raman and Rayleigh linewidths computed from optical-pumping-induced dephasing with Lamb-Dicke spatial averaging, compared with the measured 35-50 kHz and about-8-kHz values; (c) the predicted relative amplitudes of the central Rayleigh structure, computed from the multilevel dynamics; and (d) a comparison of the computed spectra with the measured spectra of the 1992 experiment (line positions, widths, and relative amplitudes) at the experimental parameters of reference 1, stating the experimental reference uncertainties (those on the quoted lattice depth $U_0/E_R$ , detuning, and intensity) and the theoretical uncertainties (the numerical and truncation error on the computed line positions and widths). The target is a multilevel theory whose computed spectra, confronted with the 1992 measurements, account for the observed line positions, widths, and relative amplitudes and for the sub- $\sqrt{I}$ intensity scaling of the Raman positions over the measured intensity range.

References

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