Polarizabilities of the ${}^{87}\mathrm{Sr}$ clock transition

In this paper, we propose an in-depth review of the vector and tensor polarizabilities of the two energy levels of the ${}^{87}\mathrm{Sr}$ clock transition whose measurement was reported in P. G. Westergaard et al. [Phys. Rev. Lett. 106, 210801 (2011)]. We conduct a theoretical calculation that reproduces the measured coefficients. In addition, we detail the experimental conditions used for their measurement in two Sr optical lattice clocks and exhibit the quadratic behavior of the vector and tensor shifts with the depth of the trapping potential and evaluate their impact on the accuracy of the clock.

Figure 1

Energy levels for Sr. The two clock states are represented in red. The $F=9/2$ hyperfine component of the three green (light gray) states $(5s5p{}^{3}P_1,5s5p{}^{3}P_2$, and $5s5p{}^{1}P_1)$ is involved in the vector and tensor polarizabilities of the excited clock state $5s5p{}^{3}P_0$, as explained in Sec. 3. The other states are considered to calculate the polarizability coefficients of the clock transition.

Figure 2

Spectroscopy of the saturated $5s^2{}^{1}S_0\to 5s5p{}^{3}P_1(F=9/2)$ transition by depletion of ${}^{87}\mathrm{Sr}$ atoms in an optical lattice for a linearly polarized trapping light, no magnetic field, and a trap depth of $1520E_R$, where $E_R=h\times 3.47$ kHz is the recoil energy associated with the absorption of a lattice photon with frequency ${\omega }_l$. In this configuration, the polarizability operator (5) is already diagonal when choosing the quantization axis along the lattice polarization. Its eigenvalues are $\alpha \left(m\right)={\alpha }_s+{\alpha }_t\frac{3m^2-F(F+1)}{F(2F-1)}$. The graph is centered on the light-shift-free frequency of the transition and displays the splitting between magnetic sublevels induced by the scalar and tensor polarizabilities of both states. The arrows indicate the theoretical light shift $\delta \nu =-\frac{1}{4}\alpha \left(m\right){\left|\mathcal{E}\right|}^2$ for each value of $\left|m\right|$, as calculated from the numerical values of Tables 1 and 2.

Figure 3

(left) Parametrization of the light polarization $\vec{\varepsilon }$ and the static bias magnetic field ${\vec{B}}_s$. Here ${\vec{e}}_x$ $\left({\vec{e}}_y\right)$ is a unit vector along the major (minor) axis of the trapping light polarization $\vec{\varepsilon }$, such that the polarization ellipsis [red (medium)] is included in the $({\vec{e}}_x,{\vec{e}}_y)$ plane. (right) Projection in the plane defined by the trapping light wave vector $\vec{k}$ and ${\vec{B}}_s$. The fictitious magnetic field representing the vector light shift adds to ${\vec{B}}_s$ to form an effective magnetic field ${\vec{b}}_{\text{eff}}$. If the vector component of the Hamiltonian dominates its tensor component, the Hamiltonian is diagonal for a quantization axis ${\vec{e}}_Z$ defined along ${\vec{b}}_{\text{eff}}$. When the polarization $\vec{\varepsilon }$ is elliptical, the quantization axis therefore depends on the trap depth $U_0$, introducing nonlinear effects in the Zeeman shift and average clock frequency as a function of $U_0$.

Figure 4

Differential Zeeman shift $\mathrm{\Delta }{Z}_s$ of the ${}^{87}\mathrm{Sr}$ clock resonance as a function of the trap depth $U_0$ for an elliptical polarization $\left|\xi \right|\simeq 0.9$ and $|m=9/2|$. The experimental data points are fit with Eq. (22), assuming ${\kappa }^v\left({}^{1}S_0\right)=0$. These four data points are sufficient to determine the three free parameters of this equation that describe the geometry of the experimental setup, here $B_s=355.0\pm 0.2\mu \text{T},\xi {\kappa }^v\left({}^{3}P_0\right)=212\pm 2\text{mHz}/E_R$, and $\theta =\text{6}3.9\pm 0.5\text{mrad}$.

Figure 5

Ratio $R$ between the differential Zeeman shifts $\mathrm{\Delta }{Z}_s$ (for linear polarization, so that no dependence on $U_0$ is observed) for the $\sigma \left(\right|m|=9/2\to |m|=7/2)$ and $\pi \left(\right|m|=9/2\to |m|=9/2)$ transitions. This ratio is related to the ratio between Landé factors by $\tilde{g}\left({}^{3}P_0\right)/\tilde{g}\left({}^{1}S_0\right)=9(R-1)/(9R-7)$.