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Physical Review A

Ultrahigh-resolution two-photon optical Ramsey spectroscopy of an atomic fountain

Raymond G. Beausoleil, Theodor W. Hänsch

Published Mar 1, 1986• 55 citations
Publisher

Abstract

We present a semiclassical analysis of ultrahigh-resolution two-photon optical Ramsey spectroscopy of cold neutral atoms falling freely in a fountain. Considering atoms which interact with the same standing-wave laser field twice on their parabolic trajectories and averaging over a broad atomic velocity distribution, we predict a nearly Lorentzian line shape whose width is just the natural linewidth. We have investigated a number of systematic corrections to the atomic resonance frequency, including first-order and second-order Doppler shifts, gravitational red shifts, and ac Stark shifts. First-order Doppler shifts due to vertical motion cancel even if the counterpropagating beams are slightly misaligned, and resolutions below 1 Hz appear feasible with a fountain of modest dimensions.

DOI
10.1103/physreva.33.1661

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Using a simple semiclassical model, we explore ultrahigh-resolution optical two-photon spectroscopy of cold neutral atoms falling freely in a fountain. Considering atoms that interact with the same standing-wave laser field twice on their parabolic trajectories, and averaging over a Maxwellian velocity distribution, we predict a nearly Lorentzian line shape with just the natural linewidth. First-order Doppler shifts cancel even if the counterpropagating beams are slightly misaligned, and sub-Hertz resolution appears feasible with a fountain of modest dimensions.

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We propose an atomic fountain clock with a truncated cold atomic beam to achieve both a low collisional frequency shift and high frequency stability. In this clock, the launching velocity of a cold atomic beam can be swept to reduce the atomic density in the interrogation region for the Ramsey resonance and to increase the atomic density in the detection region. Before the top of the beam arrives at the interrogation region, the cold atomic beam is truncated by turning off the cooling laser beams to remove the unnecessary light shift. The atomic density in the interrogation region is theoretically evaluated to be 0.04 times that in an ordinary atomic fountain with optical molasses for the same number of detected atoms. The frequency stability limit due to quantum projection noise is calculated to reach $6.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}$ in 1 s from the number of detected atoms while the fractional collisional shift is estimated to be $~\ensuremath{-}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$.

Physical Review A•Jul 26, 2010•1 citation
We present a semiclassical analysis of ultrahigh-resolution two-photon optical Ramsey spectroscopy of cold neutral atoms falling freely in a fountain. Considering atoms which interact with the same standing-wave laser field twice on their parabolic trajectories and averaging over a broad atomic velocity distribution, we predict a nearly Lorentzian line shape whose width is just the natural linewidth. We have investigated a number of systematic corrections to the atomic resonance frequency, including first-order and second-order Doppler shifts, gravitational red shifts, and ac Stark shifts. First-order Doppler shifts due to vertical motion cancel even if the counterpropagating beams are slightly misaligned, and resolutions below 1 Hz appear feasible with a fountain of modest dimensions.
Using a simple semiclassical model, we explore ultrahigh-resolution optical two-photon spectroscopy of cold neutral atoms falling freely in a fountain. Considering atoms that interact with the same standing-wave laser field twice on their parabolic trajectories, and averaging over a Maxwellian velocity distribution, we predict a nearly Lorentzian line shape with just the natural linewidth. First-order Doppler shifts cancel even if the counterpropagating beams are slightly misaligned, and sub-Hertz resolution appears feasible with a fountain of modest dimensions.
Exact and asymptotic line shape expressions are derived from the semiclassical density matrix representation describing a set of closed three-level $\ensuremath{\Lambda}$ atomic or molecular states including decoherences, relaxation rates, and light shifts. An accurate analysis of the exact steady-state dark-resonance profile describing the Autler-Townes doublet, the electromagnetically induced transparency or coherent population trapping resonance, and the Fano-Feshbach line shape leads to the linewidth expression of the two-photon Raman transition and frequency shifts associated to the clock transition. From an adiabatic analysis of the dynamical optical Bloch equations in the weak field limit, a pumping time required to efficiently trap a large number of atoms into a coherent superposition of long-lived states is established. For a highly asymmetrical configuration with different decay channels, a strong two-photon resonance based on a lower states population inversion is established when the driving continuous-wave laser fields are greatly unbalanced. When time separated resonant two-photon pulses are applied in the adiabatic pulsed regime for atomic or molecular clock engineering, where the first pulse is long enough to reach a coherent steady-state preparation and the second pulse is very short to avoid repumping into a new dark state, dark-resonance fringes mixing continuous-wave line shape properties and coherent Ramsey oscillations are created. Those fringes allow interrogation schemes bypassing the power broadening effect. Frequency shifts affecting the central clock fringe computed from asymptotic profiles and related to the Raman decoherence process exhibit nonlinear shapes with the three-level observable used for quantum measurement. We point out that different observables experience different shifts on the lower-state clock transition.
We demonstrate coherent Doppler-free three-photon excitation of the $^{1}S_{0}\phantom{\rule{0.16em}{0ex}}\ensuremath{\leftrightarrow}\phantom{\rule{0.16em}{0ex}}^{3}P_{0}$ optical clock transition and the $^{1}S_{0}\phantom{\rule{0.16em}{0ex}}\ensuremath{\leftrightarrow}\phantom{\rule{0.16em}{0ex}}^{3}P_{1}$ intercombination transition in free-space thermal clouds of 88Sr^{88}\mathrm{Sr}88Sr atoms. By appropriate orientation of the wave vectors of three lasers incident on the atoms, the first-order Doppler shift can be eliminated for all velocity classes. Three-photon excitation of the $^{1}S_{0}\phantom{\rule{0.16em}{0ex}}\ensuremath{\leftrightarrow}\phantom{\rule{0.16em}{0ex}}^{3}P_{1}$ transition enables high-contrast Ramsey spectroscopy with interrogation times comparable to the $21\phantom{\rule{0.16em}{0ex}}\textmu{}\mathrm{s}$ natural lifetime using a single near-resonant laser source. Three-photon spectroscopy on the $^{1}S_{0}\phantom{\rule{0.16em}{0ex}}\ensuremath{\leftrightarrow}\phantom{\rule{0.16em}{0ex}}^{3}P_{0}$ clock transition, using only laser frequencies nearly resonant with the $^{1}S_{0}\phantom{\rule{0.16em}{0ex}}\ensuremath{\leftrightarrow}\phantom{\rule{0.16em}{0ex}}^{3}P_{0}$ and $^{1}S_{0}\phantom{\rule{0.16em}{0ex}}\ensuremath{\leftrightarrow}\phantom{\rule{0.16em}{0ex}}^{3}P_{1}$ transitions, enables a reduction in Doppler broadening by two orders of magnitude and a corresponding $\ensuremath{\sim}470\phantom{\rule{0.16em}{0ex}}\mathrm{Hz}$ linewidth without a confining potential.
We develop a field-nonlinear theory of sub-Doppler spectroscopy in a gas of two-level atoms, based on a self-consistent solution of the Maxwell-Bloch equations in the mean field and single-atom density matrix approximations. This makes it possible to correctly take into account the effects caused by the free motion of atoms in a gas, which lead to a nonlinear dependence of the spectroscopic signal on the atomic density even in the absent of a direct interatomic interaction (e.g., dipole-dipole interaction). Within the framework of this approach, analytical expressions for the light field were obtained for an arbitrary number of resonant waves and arbitrary optical thickness of a gas medium. Sub-Doppler spectroscopy in the transmission signal for two counterpropagating and co-propagating waves has been studied in detail. A previously unknown red shift of a narrow sub-Doppler resonance is predicted in a counterpropagating waves scheme, when the frequency of one wave is fixed and the frequency of the other wave is varied. The magnitude of this shift depends on the atomic density and can be more than an order of magnitude greater than the known shift from the interatomic dipole-dipole interaction (Lorentz-Lorenz shift). The found effects, caused by the free motion of atoms, require a significant revision of the existing picture of spectroscopic effects depending on the density of atoms in a gas. Apart of fundamental aspect, obtained results are important for precision laser spectroscopy and optical atomic clocks.
The paper theoretically establishes and studies the sub-Doppler linear optical Ramsey resonances that arise under certain conditions near centers of optical transitions in the absorption of a sufficiently weak monochromatic light beam during its stationary propagation in the normal direction through an ultrathin gas cell whose internal thickness is less than or of the order of the wavelength of this radiation. We consider a situation when the cross section of this beam consists of two coaxial spatially separated regions, with the absorption signal being detected in a comparatively narrow central part of the beam. The paper studies the significant dependence of the linear optical Ramsey resonances, that arise in such an absorption spectrum, on the distance between these regions, as well as on the phase difference between them. In particular, it is shown that if this phase difference is close to pi, then instead of absorption, amplification of the central part of the incident beam can occur. The Ramsey resonances under consideration are most clearly manifested when the internal thickness of the gas cell is equal to a small halfinteger number of wavelengths of the resonant radiation. However, these resonances do not arise if this thickness is equal to an integer number of such waves. The established Ramsey resonances, under certain conditions, can find application in ultrahigh resolution atomic (molecular) spectroscopy, as well as effective references in compact optical frequency standards.
We propose an atomic fountain clock with a truncated cold atomic beam to achieve both a low collisional frequency shift and high frequency stability. In this clock, the launching velocity of a cold atomic beam can be swept to reduce the atomic density in the interrogation region for the Ramsey resonance and to increase the atomic density in the detection region. Before the top of the beam arrives at the interrogation region, the cold atomic beam is truncated by turning off the cooling laser beams to remove the unnecessary light shift. The atomic density in the interrogation region is theoretically evaluated to be 0.04 times that in an ordinary atomic fountain with optical molasses for the same number of detected atoms. The frequency stability limit due to quantum projection noise is calculated to reach $6.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}$ in 1 s from the number of detected atoms while the fractional collisional shift is estimated to be $~\ensuremath{-}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$.