Precision measurement of the ${}^{87}\text{Rb}$ tune-out wavelength in the hyperfine ground state $F=1$ at 790 nm

We report on a precision measurement of the $D$ line tune-out wavelength of ${}^{87}\text{Rb}$ in the hyperfine ground state $\left|F=1,m_F=0,\pm 1\right.〉$ manifold at $790\mathrm{nm}$, where the scalar ac Stark shifts of the $D_1$ and the $D_2$ lines cancel. This wavelength is sensitive to usually neglected contributions from vector and tensor ac Stark shifts, transitions to higher principle quantum numbers, and core electrons. The ac Stark shift is probed by Kapitza-Dirac scattering of a rubidium Bose-Einstein condensate in a one-dimensional optical lattice in free space and controlled magnetic environment. The tune-out wavelength of the magnetically insensitive $m_F=0$ state was determined to $790.01858\left(23\right)\mathrm{n}\mathrm{m}$ with sub-pm accuracy. An in situ absolute polarization, and magnetic background field measurement is performed by employing the ac vector Stark shift for the $m_F=\pm 1$ states. Comparing our findings to theory, we get quantitative insight into atomic physics beyond commonly used two-level atom approximations or the neglect of inner shell contributions.

Figure 1

The ac Stark shift $V_0$ for Rb atoms in the $m_F=0$ (solid), $m_F=+1$ (dashed), and $m_F=-1$ (dashed-dotted) ground states $\left|F=1\right.〉$, for right-handed $\left({\sigma }^{+}\right)$ polarized light $({\theta }_0=\pi /4)$. $V_0$ is given in multiples of the photon recoil $E_r$ at $790\mathrm{n}\mathrm{m}$, and a typical light intensity in our experiment of $I=136\mathrm{m}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. The inset shows a magnified view of the tune-out wavelength for the three $m_F$ states (vertical lines). The tune-out wavelengths of the $m_F=\pm 1$ states are shifted by more than $2\mathrm{n}\mathrm{m}$ with respect to the $m_F=0$ states due to the vector ac Stark shift.

Figure 2

Experimental setup. (a) Simplified artist's view on the experiment. The coil setup used for magnetic field control consists of the homogeneous field coils (1), the Stern-Gerlach coil (2), and 3D compensation coils (3). The dipole trap beam (4) and the two counterpropagating lattice beams (5) are overlapped by dichroic mirrors on the main axis through the glass cell. (b) Schematic scattering process. When applying the lattice pulse on the free falling BEC, atoms are scattered into higher momentum orders $\pm 2N\hslash k$ depending on the potential depth. (c) Schematic time line for the experimental sequence. After preparing the BEC in the dipole trap and setting the magnetic field to the desired value, the dipole trap is switched off and after a delay of $3\mu \mathrm{s}$, the lattice pulse of $\tau =12\mu \mathrm{s}$ duration is applied. Subsequently, the Stern-Gerlach field gradient is switched on and after $20\mathrm{m}\mathrm{s}$ the atoms are imaged via absorption imaging along the $y$ axis.

Figure 3

KD image for an Rb spin mixture. (a) KD scattering image for a $12\text{−}\mu \text{s}$ pulse at a lattice wavelength of $790.018\mathrm{n}\mathrm{m}$ after 20-ms TOF. The atomic density $\rho$ is given as number of atoms per image pixel (px). The $m_F$ states are distinguished by applying a magnetic Stern-Gerlach field along the gravitation axis $x$. The wavelength corresponds to the tune-out wavelength of the $m_F=0$ state. $m_F=+1$ and $m_F=-1$ show scattering into higher momentum orders due to the vector ac Stark shift. (b) The use of an improved absorption image calculation suppresses interference fringes of the imaging laser and background noise, increasing the SNR from 34 to 95. (c) The signal is vertically binned in the highlighted areas for each $m_F$ state. The improved image calculation (solid line) is compared to the standard technique (shaded). (d) A multi-Gaussian (solid line) is fitted to the line data (shaded line) from (b). The transfers are determined from the areas of the BEC peaks (shaded area). The population of the $\pm 4\hslash k$ peaks correspond to less than 1750 Rb atoms each.

Figure 4

Tune-out wavelength measurement of Rb in the $\left|F=1,m_F=0\right.〉$ state. (a) From the KD scattering images, the populations in the zero-momentum state (triangle), $\pm 2\hslash k$ (square), and $\pm 4\hslash k$ (circle) state were extracted. Insets show samples of KD images for vanishing (left) and maximal potential (right). The atomic density $\rho$ is given in atoms per pixel (px). (b) The resulting lattice potential $V_0$ (dots) was fitted with the model of Eq. (5) with the tune-out wavelength ${\lambda }_M$ (solid line). (c) For various magnetic offset fields $B_z$ the tune-out wavelength has been measured. The values scatter around the average of ${\overline{\lambda }}_M=790.01858\mathrm{n}\mathrm{m}$ (solid line). The error bars are calculated from the fitting uncertainties of (b) and the wavelength measurement calibration.

Figure 5

Lattice potential for the $m_F=\pm 1$ states. Due to the vector Stark shift, the tune-out wavelength for the $m_F=-1$ state (circles, left) is shifted to a lower, and $m_F=+1$ (triangles, right) is shifted to a higher wavelength. As a reference, the $m_F=0$ (dashed, center) potential is given. The solid line shows one single fit to both the $m_F=+1$ and the $m_F=-1$ state with our model, including a circular degree of polarization $A$ of the lattice [see Eq. (7)].

Figure 6

Measurement of the vector Stark shift. (a) An offset field $B_z$ along the $z$ axis, parallel to the lattice $\vec{k}$ vector is applied. For $B_z=-B_{0,z}$, the quantization axis along the total magnetic field ${\vec{B}}_t$ is perpendicular to $\vec{k}$, minimizing the vector light shift. The minimum at $B_{0,z}$ is fitted (solid line) according to our model [Eq. (8)]. (b) For an offset field $B_x$ along the $x$ axis, the projection of ${\vec{B}}_t$ is maximized for $B_x=-B_{0,x}$. Using $B_{0,z}$ from (a), we get $B_{0,x}$ and $B_{0,y}$ from the fit (solid line). (c) Applying an offset $B_y$ along the $y$ axis, the vector Stark shift is maximized for $B_y=-B_{0,y}$ as well. The solid line shows the model with fit results from (a) and (b). The lattice wavelength was held constant at the tune-out wavelength $790.0182\left(2\right)\mathrm{n}\mathrm{m}$ for all measurements. For each magnetic field, roughly 20 data points were taken.