Detection and manipulation of nuclear spin states in fermionic strontium

Fermionic ${}^{87}$Sr has a nuclear spin of $I=9/2$, higher than any other element with a similar electronic structure. This large nuclear spin has many applications in quantum simulation and computation, for which preparation and detection of the spin state are requirements. For an ultracold ${}^{87}$Sr cloud, we show two complementary methods to characterize the spin-state mixture: optical Stern-Gerlach state separation and state-selective absorption imaging. We use these methods to optimize the preparation of a variety of spin-state mixtures by optical pumping and to measure an upper bound of the ${}^{87}$Sr spin-relaxation rate.

Figure 1

Principle of OSG separation. (a) ${\sigma }^{+}$- and ${\sigma }^{-}$-polarized laser beams propagating in the $y$ direction create dipole forces on an atomic cloud that is located on the slopes of the Gaussian beams. (b) The laser beams are tuned close to the ${}^1{S}_0$($F=9/2$)–${}^3{P}_1$($F^{\prime }=11/2$) intercombination line, creating attractive (${\sigma }^{-}$ beam) or repulsive (${\sigma }^{+}$ beam) dipole potentials. Each $m_F$ state experiences a different potential because of the varying line strength of the respective transition. (c) Potentials resulting from dipole potentials and the gravitational potential. The dashed line marks the initial position of the atoms. Inset: The relevant region of the potentials, offset shifted to coincide at the position of the atoms, which clearly shows the different gradients on each $m_F$ state.

Figure 2

OSG separation of the 10 ${}^{87}$Sr nuclear spin states. (a) Atomic density distribution after OSG separation integrated over the $(\widehat{\mathbf{x}}+\widehat{\mathbf{y}})$ direction obtained in experiment and simulation. At the right, the density distribution of the experiment integrated along the $x$ and $y$ directions is shown together with a fit consisting of 10 Gaussian distributions. (b) Atom number distribution determined from the area of the Gaussian distributions [(red) squares] and the fit of the simulation to the experiment [(black) circles].

Figure 3

Simulated density distributions of atoms after OSG separation. Shown are distributions integrated along $x$ (aI), along the $(\widehat{\mathbf{x}}+\widehat{\mathbf{y}})$ direction (aII, b), and along $y$ (aIII, c). Angle symbols indicate the angle between the direction of integration and the direction of the OSG beams, the $y$ direction. (a) Density distribution obtained from a simulation with parameters fitted to the outcome of the experiment, integrated along different directions. The experimental situation corresponds to (aII). (b) Influence of the two OSG beams and the magnetic field on OSG separation. (bI) As (aII), but no ${\sigma }^{-}$ beam. (bII) As (aII), but no ${\sigma }^{+}$ beam. (bIII) As (aII), but no magnetic field and reduced OSG beam detuning. (c) Alternative scheme of OSG separation using two blue detuned OSG beams and variations of that scheme; see the text.

Figure 4

$m_F$-state-resolved absorption imaging on the ${}^1{S}_0$($F=9/2$)–${}^3{P}_1$($F^{\prime }=11/2$) intercombination line. (a) Spectrum of an ${}^{87}$Sr sample with a nearly homogeneous $m_F$-state distribution. The spectrum was obtained using ${\sigma }^{-}$-polarized light and shifting transitions corresponding to different $m_F$ states in frequency by applying a magnetic field of 0.5 G. Circles show the line strengths of the transitions. (b) Absorption images taken on the maxima of absorption of each $m_F$ state using ${\sigma }^{+}$ or ${\sigma }^{-}$ polarized light.

Figure 5

Examples of state mixtures prepared by optical pumping and analyzed using OSG separation. (a) The $m_F=1/2$ and $5/2$ states were pumped to the $3/2$ state using ${\sigma }^{+}$ and ${\sigma }^{-}$ light, respectively. (b) Negative $m_F$ states were pumped to positive $m_F$ states. (c) A two-state mixture obtained by pumping the lower $m_F$ states to the $m_F=5/2$ and $7/2$ states and subsequently pumping the $m_F=7/2$ state to the $m_F=9/2$ state.

Figure 6

Absence of spin relaxation in ${}^{87}$Sr. Shown are absorption images of the $m_F=5/2$ and $7/2$ states averaged over 25 runs of the experiment. Atoms were initially removed from the $m_F=5/2$ state by optical pumping, whereas all other $m_F$ states remained populated (a). After a 10-s hold at a magnetic field of 5 G (b) or 500 G (c), no $m_F=5/2$ atoms are detectable, showing the low rate of spin relaxation.