Adiabatic spin-dependent momentum transfer in an SU($N$) degenerate Fermi gas

We introduce a spin-orbit coupling scheme, where a retroreflected laser beam selectively diffracts two spin components in opposite directions. Spin sensitivity is provided by sweeping through a magnetic-field sensitive transition while dark states ensure that spontaneous emission remains low. The scheme is adiabatic and thus inherently robust. This tailored spin-orbit coupling allows simultaneous measurements of the spin and momentum distributions of a strontium degenerate Fermi gas, and thus opens the path to momentum-resolved spin-correlation measurements on $\text{SU}\left(N\right)$ quantum magnets.

Figure 1

Principle of the experiment. (a) Geometry: the ${}^{87}\mathrm{Sr}$ cloud is exposed to a retroreflected laser in ${\sigma }^{+}/{\sigma }^{-}$ configuration. (b) Level scheme: a linear Zeeman effect in the excited state ${}^{3}P_1$ is used to selectively couple two spin states in the ground manifold, $|{}^{1}S_0,m_F\rangle$ and $|{}^{1}S_0,m_F+2\rangle$, quasiresonantly via $|{}^{3}P_1,m_F+1\rangle$. The process, from either of the ground states, is associated with a momentum transfer $\pm 2\hslash k$. (c) Dressed state energies in the rotating wave picture as a function of the laser frequency, with (solid lines) and without (dashed lines) the couplings with the light. A frequency sweep of the laser adiabatically connects one spin state at rest to the second spin state with recoil. (d) Absorption image of the atomic cloud following the adiabatic momentum transfer and a time of flight, and corresponding one-dimensional integrated profile. Two selected spin states, here $m_F=-3/2$ and $-7/2$ (initially), are extracted from the SU(10) Fermi sea and diffracted in opposite directions while the other eight spin states stay at rest.

Figure 2

Fraction of the entire cloud diffracted in either direction, as a function of the frequency center of the chirp. At each diffraction peak, the two specific spin states selectively diffracted towards opposite momenta $\pm 2\hslash k$ are labeled. Inset: momentum transfer at $-3\mathrm{MHz}$ from the spin states $-3/2$ and $+1/2$, where a prior ${\sigma }^{-}$ optical pumping sequence targeted $+1/2$ and $-1/2$ consecutively (see Appendix pp3). The state $+1/2$, diffracted to $-2\hslash k$, is therefore empty here.

Figure 3

Adiabatic transfer dynamics, (a) from $|-3/2,0\hslash k\rangle$ to $|+1/2,2\hslash k\rangle$ at $-3\mathrm{MHz}$, and (b) from $|-7/2,0\hslash k\rangle$ to $|-3/2,2\hslash k\rangle$ at $-15\mathrm{MHz}$. The points are measured transfer efficiencies as a function of ramp duration, assuming a $10\%$ occupation of the spin states. The black lines result from numerical simulations without adjustable parameters: the black solid line accounts for the atoms collecting two recoils or having undergone spontaneous emission, and the dashed line is the fraction that underwent precisely two recoils. The data are finally compared with the LZ scaling $P_{\mathrm{adiab}}^{LZ}\left({\mathrm{\Omega }}_1\right)\times P_{\mathrm{adiab}}^{LZ}\left({\mathrm{\Omega }}_2\right)$ (solid green line).

Figure 4

Three-pulse measurement of the five spin states in an SU(5) Fermi gas prepared by optical pumping, providing a complete single-shot spin-resolved momentum distribution. Each cloud is assigned to its initial spin state. The central cloud, which in principle should be mostly $m_F=-9/2$, is admixed with residuals from the other states given our moderate transfer efficiency.

Figure 5

Momentum distribution of a degenerate SU(10) Fermi gas, with ${10}^4$ atoms. (left) Time-of-flight picture. (right) Two-dimensional fit residuals, from left to right, from a classical Boltzmann distribution and from a noninteracting Fermi distribution [47]. From the latter fit we estimate $T=31$ nK and $T/T_F=0.2$.