Measurements of the ground-state polarizabilities of Cs, Rb, and K using atom interferometry

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 ${\alpha }_{\mathrm{Cs}}=4\pi {\epsilon }_0\times 59.39\left(9\right){\AA }^3,{\alpha }_{\mathrm{Rb}}=4\pi {\epsilon }_0\times 47.39\left(8\right){\AA }^3$, and ${\alpha }_{\mathrm{K}}=4\pi {\epsilon }_0\times 42.93\left(7\right){\AA }^3$. In atomic units, these measurements are ${\alpha }_{\mathrm{Cs}}=401.2\left(7\right),{\alpha }_{\mathrm{Rb}}=320.1\left(6\right)$, and ${\alpha }_{\mathrm{K}}=290.0\left(5\right)$. We report ratios of polarizabilities ${\alpha }_{\mathrm{Cs}}/{\alpha }_{\mathrm{Rb}}=1.2532\left(10\right),{\alpha }_{\mathrm{Cs}}/{\alpha }_{\mathrm{K}}=1.3834\left(9\right)$, and ${\alpha }_{\mathrm{Rb}}/{\alpha }_{\mathrm{K}}=1.1040\left(9\right)$ 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.

Figure 1

Diagram of the Mach-Zehnder atom interferometry apparatus with phase choppers (green) and electrodes that produce an electric field gradient (blue). Dimensions are shown in red and tabulated in Table 1. The supersonic atom beam, shown in gray, is collimated by two slits, s1 and s2, with widths $w_1$ and $w_2$ before entering the first grating. The nanogratings, g1, g2, and g3, are spaced longitudinally such that $L_1=L_2$, which causes an interference pattern to form at the position of g3. We consider the four separate interferometers in our data analysis, labeled with $k=+2,+1,-1,-2$, that form via zeroth-, $\pm \text{first}$-, and $\pm \text{second}$-order diffraction from g1. The atoms are detected by a platinum Langmuir-Taylor detector [21], indicated in orange. The pair of blue circles represents oppositely charged cylindrical electrodes (extending perpendicular to the page) that form a virtual ground plane between them. The electric field from these electrodes polarizes the atoms and thereby shifts the interference pattern's phase. The phase choppers are shown in green; each phase chopper is a charged wire next to a grounded plane. The geometry terms relevant to the pillars and phase choppers are displayed in Fig. 2 and discussed in Sec. 2a. Due to the rotation of the Earth, the laboratory has a rotation rate about the vertical axis of ${\mathrm{\Omega }}_{\mathrm{lab},y}=38.88\mu \mathrm{rad}$/s that is also relevant to our analysis.

Figure 2

Diagram showing the dimensions, shown in red, that describe the static electric fields created by the pillars and by the phase choppers. The circle represents the cross section of a metal pillar or charged wire with radius $R$ held at voltage $V$. The GND line represents the ground plane, which may by physical (in the case of the choppers) or virtual (in the case of the pillars). $a$ is the closest distance between the pillar edge and the ground plane. The parameter $b$ is the distance between the ground plane and the effective line charge within the pillar. The atom beam center is a distance $x_b$ away from the ground plane. The different interferometer arms are separated from their neighbors by multiples of ${\theta }_d{z}_0$, where ${\theta }_d$ is the diffraction angle and $z_0$ is the longitudinal distance to grating g1 (in the case of the pillars and chopper c1) or grating g3 (in the case of chopper c2). $a$ and $R$ dimensions for the pillars and phase choppers are given in Table 1.

Figure 3

An example of a measurement of contrast $C$ vs phase chopper frequency $f_c$ for a Cs beam. We fit a model to these data that has $v_0$ and $v_r$ as fit parameters in order to measure the velocity distribution.

Figure 4

Two data plots showing how the phase $\mathrm{\Phi }$ of the interference fringes changes linearly as a function of $\mathrm{\Delta }{x}_{\det }$ and $\mathrm{\Delta }L$. In plot (a), the black dots represent coordinates at which data were acquired, and the colors represent contours inferred from the data. These figures show how $\mathrm{\Phi }$ vs $\mathrm{\Delta }{x}_{\det }$ is a line with slope proportional to $\mathrm{\Delta }L$.

Figure 5

Systematic uncertainty budget for measurements of $v_0$ and $v_r$ for our Cs, Rb, and K beams. The total systematic error in $v_0$ and $v_r$ in turn contributes toward the total systematic uncertainty in $\alpha$ measurements (shown in Fig. 7). A nominal value for ${\theta }_g$ is not listed because ${\theta }_g$ changed from $-2.37\pm 1.39$ to $1.73\pm 0.59$ mrad toward the end of the experiment.

Figure 6

An example of a measurement of phase shift vs $x$ position of the pillars for a Rb beam. The two fit parameters used to fit the model to these data are polarizability ${\alpha }_{\mathrm{Rb}}$ and the pillars' position at which the phase shift is null $x_{b0}$.

Figure 7

Systematic uncertainty budget for polarizability measurements for our Cs, Rb, and K beams. The uncertainties in knowledge of $v_0$ and $v_r$ are propagated forward from Fig. 5. Values of $V_{\mathrm{pillars}}$ ranged from 5 to 7 kV, always with 0.05% uncertainty.

Figure 8

Our (a) direct measurements and (b) measured ratios compared with other measurements, ab initio calculations, and semiempirical calculations [4, 7, 8, 11, 17, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 60, 61, 62, 63]. The references are represented on the $x$ axis by the first three letters of the first author's last name, followed by the year of publication. For the semiempirical calculations: Reference Fue82 used semiempirical pseudopotentials [42], Pat97 used experimentally determined energy levels [48], Der99 used experimentally determined electric-dipole transition matrix elements [50], and Mit03 and Yon14 used semiempirical core polarization potentials [52, 63]. Values in this plot from 1999 and later are also reported in Table 8 at the end of this paper.

Figure 9

Comparisons of our laboratory's polarizability measurements (this work as well as [11]) and Amini and Gould's ${\alpha }_{\mathrm{Cs}}$ measurement [8] to polarizabilities derived from measured lifetimes and lifetime ratios [64, 65, 66, 67, 68, 69, 70, 71, 72, 73], lifetimes inferred from photoassociation data [74, 75], theoretical $D^2$ values [76], and van der Waals $C_6$ measurements [17, 77, 78]. Values in this plot from 1999 and later are also reported in Table 8 at the end of this paper.