Beyond-Born-Oppenheimer effects in sub-kHz-precision photoassociation spectroscopy of ytterbium atoms

We present high-resolution two-color photoassociation spectroscopy of Bose-Einstein condensates of ytterbium atoms. The use of narrow Raman resonances and careful examination of systematic shifts enabled us to measure 13 bound-state energies for three isotopologues of the ground-state ytterbium molecule with standard uncertainties of the order of 500 Hz. The atomic interactions are modeled using an ab initio based mass-scaled Born-Oppenheimer potential whose long-range van der Waals parameters and total WKB phase are fitted to experimental data. We find that the quality of the fit of this model, of about 112.9 kHz (rms) can be significantly improved by adding the recently calculated beyond-Born-Oppenheimer (BBO) adiabatic corrections [J. J. Lutz and J. M. Hutson, J. Mol. Spectrosc. 330, 43 (2016)] and by partially treating the nonadiabatic effects using distance-dependent reduced masses. Our BBO interaction model represents the experimental data to within about 30.2 kHz on average, which is 3.7 times better than the “reference” Born-Oppenheimer model. We calculate the $s$-wave scattering lengths for bosonic isotopic pairs of ytterbium atoms with error bars over two orders of magnitude smaller than previous determinations. For example, the $s$-wave scattering length for ${}^{174}\mathrm{Yb}$ is $+5.55812\left(50\right)$ nm.

Figure 1

(a) Main Yb transitions used in the experiment. (b) Two-color PA spectroscopy in a Raman configuration: a pair of 556 nm lasers transfer a colliding ground-state atomic pair to a ground molecular bound state through a virtual excited state. The difference between the laser frequencies $f_1-f_2$ yields the bound-state energy $E_b$. (c) Schematic setup for PA lasers. The two laser beams for the Raman transition were split by a polarized beam splitter (PBS). The frequency difference between the two beams was controlled by acousto-optic modulators (AOMs). Two beams are combined by a PBS and then the polarization of both lights is set to be parallel to each other. In order to remove PA light-intensity fluctuations after optical fibers, we utilized an intensity-feedback system by use of an AOM and photodiode (PD). HWP: half-wave plate; M: mirror.

Figure 2

Extrapolation of photoassociation line positions shown on the example of the ${}^{170}\mathrm{Yb}v=2$, $J=2$ line. (a) An example two-color PA spectrum fitted with a Lorentz function; the FWHM is about 1 kHz. (b) The optical light shift by the PA lasers as a function of laser intensity. (c) FORT light shift as measured by an additionally applied 532 nm laser as a function of its power. (d) Mean-field shift by the BEC as a function of atomic density. See Sec. 3 for details. The final ${\mathrm{Yb}}_2$ binding energies are shown in Table 1.

Figure 3

The mass-independent Born-Oppenheimer potential curve $V\left(R\right)$ used to describe the ground $0_g^{+}$ [or ${}^1{\mathrm{\Sigma }}_g^{+}$ in Hund's case (a) notation] state of ${\mathrm{Yb}}_2$. The short-range ab initio potential [58] (green dots) is interpolated using cubic splines, scaled by a fitted parameter $s$, and smoothly connected to an analytic $-C^6/R^6-C^8/R^8$ long-range potential (dashed line) using a transition function described by Eq. (2) over the distances between $a=10a_0$ and $b=19a_0$. The depth of the fitted potential $D_e=739.7{\mathrm{cm}}^{-1}$.

Figure 4

Mass (solid lines) and volume (dashed lines) corrections to the interaction potential $V^{\prime }\left(R\right)$ for bosonic isotopologues ${}^{168}\mathrm{Yb}_2$–${}^{174}\mathrm{Yb}_2$ with respect to ${}^{176}\mathrm{Yb}_2$, where they are defined as uniformly zero. The short-range parts of the curves are based on raw data published by the authors of Ref. [44] and, like the main potential $V\left(R\right)$, are smoothly connected to an analytic $A_6{R}^{-6}+A_8{R}^{-8}$ asymptote using Eq. (2). The mass shift depends on the nuclear reduced mass ${\mu }_n$ [Eq. (3)], while the volume shift is defined by the dimensions of the appropriate nuclei [Eq. (4)].

Figure 5

(a) The effective, distance-dependent reduced mass as defined by Eq. (5) shown for the ${}^{174}\mathrm{Yb}_2$ isotopomer. The $R_m$ parameter defines the distance where the effective mass $\mu \left(R\right)$ is exactly halfway between the nuclear ${\mu }_n$ and atomic ${\mu }_a$ reduced masses. At shorter distances, $\mu$ is closer to the nuclear mass limit, while as $R\to \infty$, $\mu$ approaches the atomic reduced mass as $R^{-6}$. The case of $R_m=18a_0$ shown in the figure is used in our BBO model. (b) The dependence of fit quality factor ${\chi }^2$ for the BBO interaction model as a function of the effective mass range parameter $R_m$. The lowest ${\chi }^2=9555$ (${\chi }^2/\mathrm{dof}=1062$) is reached for $R_m=18a_0$.