Observation of vector and tensor light shifts in ${}^{87}\mathrm{Rb}$ using near-resonant, stimulated Raman spectroscopy

We present the derivation of the frequency-dependent scalar, vector, and tensor dynamical polarizabilities for the two hyperfine levels of the ${}^{87}\mathrm{Rb}$ atom $5s$ ground state. Based on the characterization of the dynamical polarizabilities, we analyze and measure the differential vector and tensor light shift between the $5s$ ground-state sublevels with near-resonant, stimulated Raman transitions. These results clarify that the tensor polarizabilities for the ground states of alkali atoms are absent when the light field is far detuned from the atomic resonance and the total electronic angular momentum $J$ is a good quantum number. In the near-resonant case, the light shifts are nontrivial and the determination of the frequency-dependent vector and tensor dynamic polarizabilities will help to achieve higher fidelities for applications of neutral atoms in quantum information and precision measurements.

Figure 1

${}^{87}\mathrm{Rb}$ atom $D2$ line hyperfine levels and its Raman transition schemes in circularly polarized laser configurations in the presence of Zeeman and light shifts.

Figure 2

(a) Scalar, (b) vector, (c) tensor, and (d) zoomed-in tensor polarizabilities of the ${}^{87}\mathrm{Rb}$ atom $F=1$ (red) and $F=2$ (blue) ground states. For each subgraph, the polarizabilities induced by the near-resonant lasers have an apparent difference, and are much larger than the polarizabilities induced by the far-detuned Raman lasers. Specifically, the polarizabilities ${\alpha }_1^l\left({\omega }_2\right)$ (red solid lines) and ${\alpha }_2^l\left({\omega }_1\right)$ (blue dashed lines) with single-photon detuning $|{\mathrm{\Delta }}_{\mathrm{s}}|\le 2\mathrm{GHz}$ are much larger than the polarizabilities ${\alpha }_1^l\left({\omega }_1\right)$ (red dash-dotted lines) and ${\alpha }_2^l\left({\omega }_2\right)$ (blue dotted lines) with single-photon detuning equal to ${\mathrm{\Delta }}_{\mathrm{s}}+6.8\mathrm{GHz}$.

Figure 3

Schematic diagram of the experimental setup.

Figure 4

Raman spectra obtained with different Raman π pulse intensities for the (a) ${\sigma }^{+}{\sigma }^{+}$ and (b) ${\sigma }^{-}{\sigma }^{-}$ polarization configurations. The laser intensities $I_1,I_2,I_3,I_4,I_5$ in the plots are $0.241,0.480,1.196,2.493,4.784\mathrm{mW}/\phantom{\mathrm{mW}\mathrm{c}{\mathrm{m}}^2}\mathrm{c}{\mathrm{m}}^2$, respectively, and $P_{F=1}$ are given with an offset for better readability.

Figure 5

The dependence of the transition peak frequencies ${\omega }_{p_{m_F}}$ and ${\omega }_{p_{m_F}}^{\prime }$ ($m_F=0,\pm 1$) on Raman laser intensity in ${\sigma }^{+}{\sigma }^{+}$ (dashed lines) and ${\sigma }^{-}{\sigma }^{-}$ (solid lines) polarization configurations. The experimentally measured results (points) are in good agreement with the theoretically calculated results (lines). The horizontal uncertainties come from the power fluctuations of the laser in 1 min measured with a power meter and the vertical uncertainties (too small to be visible) come from the fitting uncertainties.

Figure 6

The differential vector and tensor light shifts for different Raman laser intensities. (a) Differential vector light shifts $\mathrm{\Delta }{\omega }_{m_F}^T$, where the measured results for $m_F=-1,0,1$ are denoted as magenta rhombuses, red squares, and blue circles, respectively; the theoretically inferred results are shown as magenta dash-dotted line, red dashed line, and blue solid line, respectively. (b) Difference between the differential tensor light shifts, where the measured results $\mathrm{\Delta }{\omega }_{1(-1)}^T-\mathrm{\Delta }{\omega }_0^T$ for ${\sigma }^{+}{\sigma }^{+}$ and ${\sigma }^{-}{\sigma }^{-}$ polarization configurations are denoted as blue circles and red squares, respectively; the theoretically inferred results are shown as a magenta solid line. The horizontal uncertainties come from the power fluctuations of the laser in 1 min measured with a power meter and the vertical uncertainties come from the fitting uncertainties; see Ref. [35] for detail.