Nuclear spin effects in optical lattice clocks

We present a detailed experimental and theoretical study of the effect of nuclear spin on the performance of optical lattice clocks. With a state-mixing theory including spin-orbit and hyperfine interactions, we describe the origin of the ${}^{1}S_0\text{−}{}^{3}P_0$ clock transition and the differential $g$ factor between the two clock states for alkaline-earth-metal(-like) atoms, using ${}^{87}\mathrm{Sr}$ as an example. Clock frequency shifts due to magnetic and optical fields are discussed with an emphasis on those relating to nuclear structure. An experimental determination of the differential $g$ factor in ${}^{87}\mathrm{Sr}$ is performed and is in good agreement with theory. The magnitude of the tensor light shift on the clock states is also explored experimentally. State specific measurements with controlled nuclear spin polarization are discussed as a method to reduce the nuclear spin-related systematic effects to below ${10}^{-17}$ in lattice clocks.

Figure 1

(Color online) Simplified ${}^{87}\mathrm{Sr}$ energy level diagram (not to scale). Relevant optical transitions discussed in the text are shown as solid arrows, with corresponding wavelengths given in nanometers. Hyperfine structure sublevels are labeled by total angular momentum $F$, and the magnetic dipole $\left(A\right)$ and electric quadrupole ($Q$, equivalent to the hyperfine $B$ coefficient) coupling constants are listed in the inset. State mixing of the ${}^{1}P_1$ and ${}^{3}P_1$ states due to the spin-orbit interaction is shown as a dashed arrow. Dotted arrows represent the hyperfine induced state mixing of the ${}^{3}P_0$ state with the other $F=9∕2$ states in the $5s5p$ manifold.

Figure 2

(Color online) A Breit-Rabi diagram for the ${}^{1}S_0\text{−}{}^{3}P_0$ clock transition using Eq. (A8) with $\delta g{\mu }_0=-109\mathrm{Hz}∕\mathrm{G}$. Inset shows the linear nature of the clock shifts at the fields relevant for the measurement described in the text.

Figure 3

(Color online) (a) Schematic of the experimental apparatus used here. Atoms are confined in a nearly vertical optical lattice formed by a retroreflected $813\mathrm{nm}$ laser. A $698\mathrm{nm}$ probe laser is coaligned with the lattice. The probe polarization $E_P$ can be varied by an angle $\theta$ relative to that of the linear lattice polarization $E_L$. A pair of Helmholtz coils (blue) is used to apply a magnetic field along the lattice polarization axis. (b) Nuclear structure of the ${}^{1}S_0$ and ${}^{3}P_0$ clock states. The large nuclear spin $(I=9∕2)$ results in 28 total transitions, and the labels $\pi$, ${\sigma }^{+}$, and ${\sigma }^{-}$ represent transitions where $m_F$ changes by 0, $+1$, and $-1$, respectively. (c) Observation of the clock transition without a bias magnetic field. The ${}^{3}P_0$ population (in arbitrary units) is plotted (blue dots) versus the probe laser frequency for $\theta =0$, and a fit to a sinc-squared lineshape yields a Fourier-limited linewidth of $10.7\left(3\right)\mathrm{Hz}$. Linewidths of $5\mathrm{Hz}$ have been observed under similar conditions and when the probe time is extended beyond $200\mathrm{ms}$.

Figure 4

(Color online) Observation of the ${}^{1}S_0\text{−}{}^{3}P_0$ $\pi$ transitions $(\theta =0)$ in the presence of a $0.58\mathrm{G}$ magnetic field. Data is shown in gray and a fit to the eight observable line shapes is shown as a blue curve. The peaks are labeled by the ground state $m_F$ sublevel of the transition. The relative transition amplitudes for the different sublevels are strongly influenced by the Clebsch-Gordan coefficients. Here, transition linewidths of $10\mathrm{Hz}$ are used. Spectra as narrow as $1.8\mathrm{Hz}$ have been achieved under similar conditions if the probe time is extended to $500\mathrm{ms}$.

Figure 5

(Color online) Observation of the 18 $\sigma$ transitions when the probe laser polarization is orthogonal to that of the lattice $(\theta =\frac{\pi }{2})$. Here, a field of $0.69\mathrm{G}$ is used. The spectroscopic data is shown in gray and a fit to the data is shown as a blue curve. Peak labels give the ground state sublevel of the transition, as well as the excitation polarization.

Figure 6

(Color online) Calculation of the 18 $\sigma$ transition frequencies in the presence of a $1\mathrm{G}$ bias field, including the influence of Clebsch-Gordan coefficients. The green (red) curves show the ${\sigma }^{+}$ $\left({\sigma }^{-}\right)$ transitions. (a) Spectral pattern for $g$ factors $g_I{\mu }_0=-185\mathrm{Hz}∕\mathrm{G}$ and $\delta g{\mu }_0=-109\mathrm{Hz}∕\mathrm{G}$. (b) Same pattern as in (a) but with $\delta g{\mu }_0=+109\mathrm{Hz}∕\mathrm{G}$. The qualitative difference in the relative positions of the transitions allows determination of the sign of $\delta g$ compared to that of $g_I$.

Figure 7

(Color online) Summary of $\delta g$ measurements for different lattice intensities. Each data point (and uncertainty) represents the $\delta g$ value extracted from a full ${\sigma }^{\pm }$ spectrum such as in Fig. 5. Linear extrapolation (red line) to zero lattice intensity yields a value $-108.4\left(1\right)\mathrm{Hz}∕\mathrm{G}$.

Figure 8

(Color online) Measurement of the tensor shift coefficients $\mathrm{\Delta }{\kappa }^T$ (blue triangles), and ${\kappa }_e^T$ (green circles), using $\sigma$ spectra and Eq. (14). The measured coefficients show no statistically significant trap depth dependence while varying the depth from $0.85–1.7$ $U_0$.

Figure 9

(Color online) The effect of optical pumping via the ${}^{3}P_1$ $(F=7∕2)$ state is shown via direct spectroscopy with $\theta =0$. The red data shows the spectrum without the polarizing light for a field of $0.27\mathrm{G}$. With the polarizing step added to the spectroscopy sequence the blue spectrum is observed. Even with the loss of $\sim 15\%$ of the total atom number due to the polarizing laser, the signal size of the $m_F=\pm 9∕2$ states is increased by more than a factor of 4.

Figure 10

(Color online) Magnetic sensitivity of the ${}^{1}P_1$ state calculated with the expression in Eq. (A10) using $A=-3.4\mathrm{MHz}$ and $Q=39\mathrm{MHz}$ [55]. Note the inverted level structure.

Figure 11

(Color online) Magnetic sensitivity of the ${}^{3}P_1$ state calculated with the expression in Eq. (A10) using $A=-260\mathrm{MHz}$ and $Q=-35\mathrm{MHz}$ [61].

Figure 12

(Color online) Magnetic sensitivity of the ${}^{3}P_1$ state calculated numerically with Eq. (A9) using $A=-212\mathrm{MHz}$ and $Q=67\mathrm{MHz}$ [62].