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Physical Review A

Measurement of the electric polarizability of sodium with an atom interferometer

Christopher R. Ekstrom, Jörg Schmiedmayer, Michael Chapman, Troy D. Hammond, David E. Pritchard

Published May 1, 1995• 261 citations
Publisher

Abstract

We have constructed an atom interferometer with interfering beams that are physically isolated by a metal foil. By applying an interaction to one of the two interfering beams, we can measure ground-state energy shifts with a spectroscopic precision of 6.6\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}14}$ eV/\ensuremath{\surd}min , or 16 Hz/\ensuremath{\surd}min . Applying an electric field to one beam of the interferometer, we have measured the phase shift induced from the quadratic Stark effect. By analyzing these phase shifts, we have determined the ground-state polarizability of sodium, with much improved accuracy, to be 24.11(6${)}_{\mathrm{statistical}}$(6${)}_{\mathrm{systematic}}$\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}24}$ ${\mathrm{cm}}^{3}$.

DOI
10.1103/physreva.51.3883

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We have constructed an atom interferometer with interfering beams that are physically isolated by a metal foil. By applying an interaction to one of the two interfering beams, we can measure ground-state energy shifts with a spectroscopic precision of 6.6\ifmmode\times\else\texttimes\fi{}10\ensuremath−14{10}^{\mathrm{\ensuremath{-}}14}10\ensuremath−14 eV/\ensuremath{\surd}min , or 16 Hz/\ensuremath{\surd}min . Applying an electric field to one beam of the interferometer, we have measured the phase shift induced from the quadratic Stark effect. By analyzing these phase shifts, we have determined the ground-state polarizability of sodium, with much improved accuracy, to be 24.11(6)statistical{)}_{\mathrm{statistical}})statistical​(6)systematic{)}_{\mathrm{systematic}})systematic​\ifmmode\times\else\texttimes\fi{}10\ensuremath−24{10}^{\mathrm{\ensuremath{-}}24}10\ensuremath−24 cm3{\mathrm{cm}}^{3}cm3.
We have constructed an atom interferometer using three 200 nm period transmission gratings. The two interfering portions of each atom’s wave pass on opposite sides of a stretched metal foil. A fringe amplitude of up to 900 cps is observed, which allows us to determine the phase to 15 milliradians in 1 minute. By applying different interactions to the two sides of the interferometer, we have determined the electric polarizability of the ground state of sodium, the coherence length of our beam, and the differential phase shifts and coherences of the different Zeeman ground states in a magnetic field. Some future prospects of separated beam experiments are discussed.
We have built an atom interferometer and, by applying an electric field on one of the two interfering beams, we have measured the static electric polarizability of lithium $\ensuremath{\alpha}=(24.33\ifmmode\pm\else\textpm\fi{}0.16)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}30}\phantom{\rule{0.3em}{0ex}}{\mathrm{m}}^{3}$ with a 0.66% uncertainty. Our experiment is similar to an experiment done on sodium in 1995 by Pritchard and co-workers, with several improvements: the electric field can be calculated analytically and the interference signals have a large intensity and a high visibility, resulting in accurate phase measurements. This experiment illustrates the extreme sensitivity of atom interferometry: when the atom enters the electric field, its velocity increases and the fractional change, equal to $4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$ for our largest field, is measured with a ${10}^{\ensuremath{-}3}$ accuracy.
We measured the ground-state electric-dipole polarizability of sodium, potassium, and rubidium using a Mach-Zehnder atom interferometer with an electric-field gradient. We find ${\ensuremath{\alpha}}_{\mathrm{Na}}=24.11(2){}_{\mathrm{stat}}(18){}_{\mathrm{sys}}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}24}{\mathrm{cm}}^{3}$, ${\ensuremath{\alpha}}_{\mathrm{K}}=43.06(14)(33)$, and ${\ensuremath{\alpha}}_{\mathrm{Rb}}=47.24(12)(42)$. Since these measurements were all performed in the same apparatus and subject to the same systematic errors, we can present polarizability ratios with 0.30.3%0.3 uncertainty. We find ${\ensuremath{\alpha}}_{\mathrm{Rb}}/{\ensuremath{\alpha}}_{\mathrm{Na}}=1.959(5)$, ${\ensuremath{\alpha}}_{\mathrm{K}}/{\ensuremath{\alpha}}_{\mathrm{Na}}=1.786(6)$, and ${\ensuremath{\alpha}}_{\mathrm{Rb}}/{\ensuremath{\alpha}}_{\mathrm{K}}=1.097(5)$. We combine our ratio measurements with the higher-precision measurement of sodium polarizability by Ekstrom et al. [Phys. Rev. A 51, 3883 (1995)] to find ${\ensuremath{\alpha}}_{\mathrm{K}}=43.06(21)$ and ${\ensuremath{\alpha}}_{\mathrm{Rb}}=47.24(21)$.
We observed the phase shift of the atomic wave function induced from the dc Stark effect using an atom interferometer comprised of four copropagating traveling laser beams. We calculated the interference signal for the interferometer in the case of a thermal atomic beam. By comparing the observed interference fringes with the calculated ones, the difference between the polarizabilities of the 3P1^{3}P_{1}3P1​ state and the 1S0^{1}S_{0}1S0​ state of calcium was obtained to be $\ensuremath{\alpha}(^{3}P_{1})\ensuremath{-}\ensuremath{\alpha}(^{1}S_{0})=(13.4\ifmmode\pm\else\textpm\fi{}2.0)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}24}$ cm3{\mathrm{cm}}^{3}cm3. This value was consistent with that deduced from the Stark frequency shift.
We measured the ground-state static electric-dipole polarizabilities of Cs, Rb, and K atoms using a three-nanograting Mach-Zehnder atom beam interferometer. Our measurements provide benchmark tests for atomic structure calculations and thus test the underlying theory used to interpret atomic parity-nonconservation experiments. We measured ${\ensuremath{\alpha}}_{\mathrm{Cs}}=4\ensuremath{\pi}{\ensuremath{\epsilon}}_{0}\ifmmode\times\else\texttimes\fi{}59.39(9)\phantom{\rule{0.16em}{0ex}}{\AA{}}^{3},{\ensuremath{\alpha}}_{\mathrm{Rb}}=4\ensuremath{\pi}{\ensuremath{\epsilon}}_{0}\ifmmode\times\else\texttimes\fi{}47.39(8)\phantom{\rule{0.16em}{0ex}}{\AA{}}^{3}$, and ${\ensuremath{\alpha}}_{\mathrm{K}}=4\ensuremath{\pi}{\ensuremath{\epsilon}}_{0}\ifmmode\times\else\texttimes\fi{}42.93(7)\phantom{\rule{0.16em}{0ex}}{\AA{}}^{3}$. In atomic units, these measurements are ${\ensuremath{\alpha}}_{\mathrm{Cs}}=401.2(7),{\ensuremath{\alpha}}_{\mathrm{Rb}}=320.1(6)$, and ${\ensuremath{\alpha}}_{\mathrm{K}}=290.0(5)$. We report ratios of polarizabilities ${\ensuremath{\alpha}}_{\mathrm{Cs}}/{\ensuremath{\alpha}}_{\mathrm{Rb}}=1.2532(10),{\ensuremath{\alpha}}_{\mathrm{Cs}}/{\ensuremath{\alpha}}_{\mathrm{K}}=1.3834(9)$, and ${\ensuremath{\alpha}}_{\mathrm{Rb}}/{\ensuremath{\alpha}}_{\mathrm{K}}=1.1040(9)$ with smaller fractional uncertainty because the systematic errors for individual measurements are largely correlated. Since Cs atom beams have short de Broglie wavelengths, we developed measurement methods that do not require resolved atom diffraction. Specifically, we used phase choppers to measure atomic beam velocity distributions, and we used electric field gradients to give the atom interference pattern a phase shift that depends on atomic polarizability.
We have demonstrated an interferometer for atoms. A three-grating geometry is used, in which the interfering beams are distinctly separated in both position and momentum. We used a highly collimated beam of sodium atoms with a de Broglie wavelength of 16 pm and high-quality 0.4-\ensuremath{\mu}m-period free-standing gratings which we fabricated using a novel method. The interference signal is 70 counts/s, which allows us to determine the phase to 0.1 rad in 1 min. Applications of atom interferometers are briefly discussed.