Collective dipole oscillations of a spin-orbit coupled Fermi gas

The collective dipole mode is induced and measured in a spin-orbit (SO) coupled degenerate Fermi gas of 173Yb atoms. Using a differential optical Stark shift, we split the degeneracy of three hyperfine states in the ground manifold, and independently couple consecutive spin states with the equal Raman transitions. A relatively long-lived spin-orbit-coupled Fermi gas, readily being realized with a narrow optical transition, allows to explore a single-minimum dispersion where three minima of spin-1 system merge into and to monitor collective dipole modes of fermions in the strong coupling regime. The measured oscillation frequency of the dipole mode is compared with the semi-classical calculation in the single-particle regime. Our work should pave the way towards the characterization of spin-orbit-coupled fermions with large spin s >\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2}$$\end{document}12 in the strong coupling regime.

, multiple Raman transitions need to be set up, which in particular causes severe light-induced heating for alkali fermions.
The recent implementation of SO coupling in non-alkali fermions [14][15][16] allows for flexible configuration with SO coupling owing to a narrow optical transition with significantly reduced light-induced heating. So far various schemes have been demonstrated to implement spin-1 2 SO coupling with non-alkali fermions in a homogeneous gas 15 , in an optical Raman lattice 19,20 and in an optical lattice clock 21,22 . Here, we extend our capability to engage three hyperfine states that are consecutively coupled through Raman transitions 23 . A narrow optical transition of 173 Yb atoms is used to induce Raman coupling with minimal heating. Consequently, we monitor a collective dipole mode in the strong coupling regime where multiple minima merge into the single minimum with long-lived spin-orbit-coupled fermions.
The effect of SO coupling in atomic gases has been extensively investigated. The phase digram is experimentally measured to uncover the (non-)magnetic and stripe phases 24 , followed by a recent observation of a stripe superfluid phase in a superlattice with SO coupling 25 . Furthermore, the dynamic properties such as the collective mode 6 , the excitation spectrum 24 , the Zitterbewegung 9 and the negative mass 26 have been studied with spin-orbit-coupled bosons. Nevertheless, the property of the spin-orbit-coupled fermions still remains to be explored in experiments. Here, we, for the first time, investigate the collective mode of the spin-orbit-coupled degenerate fermions. Although the collective mode measurement has been widely used to understand the properties of atomic gases in various contexts including spin-orbit-coupled bosons 6 , its application to spin-orbit-coupled fermions has remained a challenge due to the limited lifetime of fermions in the presence of SO coupling. In this work, we implement relatively long-lived spin-orbit-coupled fermions, and experimentally measure a collective dipole mode of a spin-orbit-coupled Fermi gas in the collisionless regime. We observe the minimum dipole oscillation frequency around the two-photon detuning δ ∼− 2.7 E r at which three spin states are symmetrically coupled in good agreement with the semi-classical theory. Our observation of the collective dipole mode will pave the way towards not only the spin-orbit-coupled fermions in hydrodynamic regime 27 but also quantum dynamics with large spin 23

Experiments and Results
We begin experiments with a degenerate Fermi gas of 173 Yb atoms by loading cold atoms pre-cooled in the intercombination magneto-optical trap into a crossed optical dipole trap (ODT). The crossed ODT, generated by 1064 nm lights, consists of a tight ODT (tODT) in the horizontal plane and a shallow ODT (sODT), the latter of which is tilted by 15 degree from the horizontal plane 15,28 . In the middle of the forced evaporative cooling, we optically pump more than 70% atoms into the = m F  from other hyperfine levels of the ground manifold. During the experiment, the magnetic field of 10 G is applied along the − +x z ( ) direction. A pair of beams, blue-detuned by ∼1 GHz from the intercombination λ 0 = 556 nm transition 1 S 0 = ↔ ( )  , intersect at the atomic cloud with an angle of θ = 90°, which imparts the momentum onto atoms and changes the energy dispersion as described in Fig. 1 . After the evaporative cooling in the crossed ODT, the two Raman coupling beams are linearly ramped up from zero intensity to the final value within 1 ms. The two-photon detuning δ is simultaneously ramped from ∼50 E r to the final value where E r is the recoil energy. We induce the collective dipole mode by briefly switching off the tODT beam for 1 ms, during which the atomic cloud is accelerated down by the gravity as described in Fig. 1(c). Subsequently the tODT is switch back on while the Raman beam being kept on, followed by the variable hold time. With a weak probe light along the ŷ direction, a time-of-flight (TOF) image is obtained by abruptly switching off all the ODT and Raman lights at the end of the hold time. After a 4 ms TOF ballistic expansion, we record the momentum distribution of the atomic cloud and monitor the center of mass oscillation.
Raman transitions successively couple three hyperfine levels in the 1 S 0 ground manifold as described in Fig. 1(b). Each Raman beam induces the spin-dependent ac Stark shift. With an additional light lifting the degenacy, we separate out = = F m , , , states from the remaining hyperfine levels, independently controlling both the detuning δ from the Raman resonance and the quadratic Zeeman shift ε. We thus achieve the effective Hamiltonian in the quasi-momentum basis that describes spin-1 SO coupling: where q x is the quasi-momentum of atoms along x-direction, k r is the recoil momentum k r = k 0 sin(θ/2) for . With a given single-photon detuning of ∼1 GHz, we can approximate Ω R = Ω ∼ R with current experimental setting. All values of detuning can be tuned by using spin-dependent light shifts induced by Raman beams and far-detuned beams that co-propagate with the beam R2 (see Fig. 1(a)). Spin-dependent level shifts from different lights are independently calibrated by monitoring the two-photon Rabi oscillation between = = F m , , induced by a pair of co-propagating Raman beams 15 .
Let us note that a relatively long-lived spin-orbit-coupled Fermi gas is a critical ingredient to monitor a collective dipole mode in the strong coupling regime around Ω R ∼ 4 E r . A narrow optical transition, readily available in ytterbium atoms 15 or other lanthanides 14 , allows to create SO coupling with minimal light-induced heating. In contrast to fermionic alkali atoms, alkaline-earth-like atoms have effective large fine-structure splitting for the triplet 3 P 1 excited state with a narrow line width (e.g. the natural linewidth of 182 kHz for 173 Yb atoms). In our experimental setting, the 1 GHz detuning of the Raman beam still limits the lifetime upto ∼50 m in the strong coupling regime but much longer lifetime is expected with larger detuning in the red-detuned regime.
The collective dipole mode is a simple center-of-mass motion which usually exhibits the mode frequency same as the harmonic trap frequency according to the Kohn theorem 29 . With SO coupling that breaks the Galilean invariance, however, a wave function of the particle becomes sensitive to the velocity of particles through spin-momentum locking, which affects the frequency of the dipole mode. In the experiment, we start with a degenerate Fermi gas prepared at rest in the crossed ODT. The perturbation induced by the tODT generates a irregular elliptical motion in the −x z plane after a TOF expansion (see Fig. 2(b)). The oscillation along the ẑ and x are obtained by projecting the oscillation trace onto the corresponding axes, which are also two of three eigenaxes of the crossed ODT. The eigenaxes of the system are further confirmed by the principle component analysis so that a single frequency fit works for each axis. The collective mode frequency is then obtained by fitting the oscillation with a single frequency sinusoidal function with a exponential decay. All parameters are left as free parameters in the fitting process. To determine the reduced oscillation frequency with SO coupling, we independently measure the trap frequency of the crossed ODT with Raman beams at large two-photon detuning δ > 50 Er. In this regime, the dispersion relation is almost the same as the one without SO coupling, but the lift beam or the Raman beam may slightly affect the bare trap frequency. The collective dipole mode is significantly affected along the x direction with SO coupling while the dipole oscillation frequency does not change along the ẑ direction. Figure 2(c) shows a typical dipole oscillation with and without SO coupling measured in the ODT.
The aforementioned oscillation measured so far has shown the capability of probing the collective dipole mode in the strong coupling regime. Here we set the Raman coupling strength Ω R = 4 E r and scan the two-photon detuning δ from −7 E r to 8 E r . In the schematic phase diagram of the δ − Ω R plane (Fig. 3(b)), the white dashed line shows the δ − Ω R parameters scanned in the experiment. For each point in the δ − Ω R plane, we obtain the normalized oscillation frequency as ω ω / i SOC i where ω i SOC (ω i ) is the measured frequency with (without) SO coupling for i = {x, z}. The central result of our work is to identify the single minima dispersion relation that involves three spin states. Intuitively, the spin configuration adiabatically follows the dipole motion as long as atomic gases stay in the lowest energy band. When only two consecutive spin states are resonantly coupled (e.g. δ ∼− 6 E r or 0 E r ), the bottom of the dispersion curve is dominated by a spin-1 2 system as shown in Fig. 3(a). Around δ ∼− 2.7 E r , three band minima corresponding to each spin state , , merge into the single minimum at the bottom of the dispersion. Therefore, a sufficiently large initial momentum kick of ∼3 k r (indicated by dashed arrow in Fig. 3(a)) couples spin configuration with momentum during the dipole mode at δ = −2.7 E r while only two spin states are coupled for δ = −6 E r or 0 E r .
In the experiment, we monitor collective dipole oscillations along two perpendicular axes with and without SO coupling after imparting an initial momentum kick. The dipole oscillation along the z axis is not affected by SO coupling, which allows us to monitor and compensate systematic fluctuations of the trap frequency. Along the x direction, however, we observe the reduction of the dipole oscillation due to the presence of SO coupling. Around the detuning δ ∼− 2.7 E r where three spin states are symmetrically coupled, we observe the smallest value of the dipole oscillation frequency which is in good agreement with the prediction as follows.
For a quantitative comparison of the dipole oscillation to the measurement, we use a variational wave-function calculation assuming that the fermions stay in the lowest energy during the oscillation 30 . We note that the higher bands may be populated during the initial momentum transfer when the Raman coupling is detuned by |δ| > 10 E r in our system. In the experiment, however, the detuning δ is restricted within ±8 E r . Furthermore, the collisional rate τ −1 of the current system satisfies τ −1 < ω x,y,z and therefore we ignore collisional effects including spin-dependent interactions. This allows us to consider a spin-orbit-coupled Fermi gas in a semi-classical way as follows.
Our theoretical simulation is conducted based on the single particle Hamiltonian H SOC for the spin-1 manifold. With the collisionless approximation of the atomic gas, the oscillation of atomic could in momentum space can be numerically calculated from the dispersion relation of the lowest band E(k x ) with the following set of equations: where ω x is the trap frequency along the SO coupling direction. Thus, it is straigthforward to have the equation . By setting the intial value of k x as the momentum kick transferred to the atomic cloud, we thus simulate the evolution of the atomic cloud in momentum space. The momentum and time steps to run the numerical simulation are set to around 0.005 k r and 0.5 ms. To extract the frequency of numerically calculated oscillation, we conduct the fitting of the numerical results with a sinusoidal function with proper amplitude and initial fitting parameters. In this way, we calculate the reduced dipole oscillation frequency in the presence of SO coupling as for the initial momentum kick of 3 k r (see solid curve in Fig. 3(c)). In the experiment, an atomic cloud has a finite momentum distribution due to the Fermi sea formed at the finite temperature.This momentum distribution not only causes the fast decay of the collective dipole oscillation but also enhance anharmonic behaviour associated. We attribute the deviation of the observed frequency from the prediction to the the anharmonicity as observed in the previous work 6 .

Discussion
We investigate the collective dipole mode in a spin-orbit-coupled fermions for the first time. Using the Raman transition consecutively coupling three internal hyperfine states, the Raman-dressed spin-1 spin-orbit-coupled Fermi gas is experimentally realized. Owing to a narrow optical transition of 173 Yb atoms, the collective dipole mode can be monitored in the strong coupling regime with a long-lived sample. The measured oscillation frequency of the dipole mode is compared with the semi-classical calculation around a single-minimum dispersion, at which the minimum oscillation frequency is observed as expected for the spin-1 case. Our observation of the dipole mode will pave the way towards the study of spin-orbit-coupled fermions in hydrodynamic regime 23 . Furthermore, this work highlights the remarkable capabilities to study the effect of SO coupling with large spin > s 1 2 in a degenerate Fermi gas. The large spin system would allow us to explore unprecedented quantum phenomena including SU(N) Fermi liquids 31,32 and SU(N) Hubbard model 33,34 , which can be further enriched by spin-momentum locking.
In near-term experiments, the measurement of spin susceptibility is conceivable by monitoring the spin oscillation over the dipole oscillation 35 . In a lanthanide cold atom system where inter-particle interaction strength can be easily tuned 14 , our present work shall push forward the future studies of interacting spin-orbit-coupled states, which are broadly discussed in theory but very hard to investigate in solid-state materials. Figure 3. Measurement of the dipole oscillation frequency of spin-1 SO coupled fermions in the strong coupling regime. The schematic phase diagram in the δ − Ω R plane is shown in (b). In the strong coupling regime at Ω R = 4 E r , multiple band minima, corresponding to three spin states, merge into the single minimum when the consecutive Raman coupling is identical around δ ∼− 2.7 E r as described in ②. When only two consecutive spins, 5 2 and 3 2 , are resonantly coupled, the third spin 1 2 does not play the role at the bottom of the dispersion (see ① in (a)). In (a), the dashed arrow indicates the initial momentum kick of ∼3 k r applied to atoms. The dipole oscillation frequency is monitored along the x and ẑ axes, the former of which has SO coupling. The oscillation is not affected along the ẑ direction (open circle) while the oscillation becomes slower in the presence of SO coupling (solid circle). For this measurement, the Raman coupling is set to Ω R = 4 E r . The solid curve is the numerical prediction based on the semi-classical calculation.