Tunable atomic spin-orbit coupling synthesized with a modulating gradient magnetic field

We report the observation of synthesized spin-orbit coupling (SOC) for ultracold spin-1 87Rb atoms. Different from earlier experiments where a one dimensional (1D) atomic SOC of pseudo-spin-1/2 is synthesized with Raman laser fields, the scheme we demonstrate employs a gradient magnetic field (GMF) and ground-state atoms, thus is immune to atomic spontaneous emission. The strength of SOC we realize can be tuned by changing the modulation amplitude of the GMF, and the effect of the SOC is confirmed through the studies of: 1) the collective dipole oscillation of an atomic condensate in a harmonic trap after the synthesized SOC is abruptly turned on; and 2) the minimum energy state at a finite adiabatically adjusted momentum when SOC strength is slowly ramped up. The condensate coherence is found to remain very good after driven by modulating GMFs. Our scheme presents an alternative means for studying interacting many-body systems with synthesized SOC.

This supplemental material addresses a few points of the main text in more detail.

THE EFFECTIVE HAMILTONIAN WITH A MODULATING GRADIENT MAGNETIC FIELD
We first discuss how a one dimensional (1D) gradient magnetic field (GMF) is created. The effective Hamiltonian with synthesized spin-orbit coupling (SOC) is then derived for atoms moving in a modulating GMF [1].
As demonstrated in the main text, the key element for synthesizing SOC is the spin rotation transformation U z = exp{−ik so zF z }, which can be realized by a 1D GMF and the corresponding rotation angle scales linearly with z-coordinate. According to Maxwell's equations, both the divergence and curl of a static magnetic field vanish in free space. Hence a 1D GMF B ′ zẑ cannot exist alone. However, as we show below, this can be approximately produced by a combination of a pulsed 3D quadruple field B ′ (t)(−xx/2 − yŷ/2 + zẑ) with a constant strong bias field B 0ẑ . Neglecting quadratic Zeeman shift (QZS), the interaction between atomic spin and magnetic field is described by x/2, the magnetic field direction is approximately slaved to the z-axis or n ≈ẑ.

The magnetic field strength
then gains a term linearly proportional to z-coordinate, or approximately H (0) The strong bias field induces spin precession at the Larmor frequency ω 0 = g F µ B B 0 / , which can be transformed away in a frame rotating around z-axis at a frequency ω = ω 0 . The transformed Hamiltonian H B in the rotating frame corresponds to where R t = exp (iωtF z ). Thus we generate an effective 1D GMF B ′ (t)zẑ.
Now consider an atom moving in a periodically modulating 1D GMF B(t) = β(t)( k so /g F µ B )zẑ where β(t + T ) = β(t), ∫ T momentum during modulation. The dynamics for an atom is described by the Hamiltonian After a unitary transformation R z (t) = exp , the above Hamiltonian is changed where with an initial phase φ, one easily arrives at c 1 = (1/2) cos φ and c 2 = 3/8. In our experiments φ = 0 is used, thus c 1 = 1/2 and c 2 = 3/8. The above derivation neglects QZS from the bias field qF 2 z , because it commutes with Eq. (5), thus can be directly absorbed into the effective Hamiltonian

ES
In our experiment, a modulating GMF generates an effective 1D SOC which does not contain any explicit spin flip interaction. If wanted, spin flip interaction can be introduced from additional RF pulses, following an earlier suggestion in Ref.
[2]. Fig. S1(a) illustrates schematically the new scheme. Each period is now composed of two parts with durations T 1 and T 2 respectively for pulsed interactions with GMF and RF magnetic fields. The interaction between an atom and the 3D quadrupole magnetic field, the bias field, and the RF magnetic field b(t) cos(ω RF t)x is governed by the Hamiltonian In the frame rotating around z-axis at a frequency ω = ω RF , it is transformed to where |q| = ω 2 0 /∆ HFS is the bias field QZS, and ∆ HFS denotes the ground state hyperfine splitting. If ω RF ≫ g F µ B B ′ y/(2 ) and g F µ B B ′ x/(2 ), which is equivalent to B 0 ≫ B ′ x/2 and B ′ y/2 as ω RF ≈ ω 0 , all fast rotating terms at frequencies ω RF and 2ω RF can be neglected, thus the Hamiltonian reduces to where ∆ = ω RF − ω 0 is the detuning of the RF field frequency from the linear Zeeman shift, Ω(t) = g F µ B b(t)/2 is the corresponding resonant Rabi frequency of the RF field.
Including the kinetic term, the evolution operator for one period is given by where U so (T 1 ) and U RF (T 2 ) are the evolution operators for the two parts respectively. In the first part a periodically modulating GMF is applied with Ω = 0. Following the same steps outlined in the previous section, we obtain U so (T 1 ) = exp{−iH so T 1 / }, where with E so = 2 k 2 so /2m. In the second part, the quadruple field is turned off with B ′ = 0. According to Eq. (9), we obtain U RF (T 2 ) = exp{−iH RF T 2 / }, where When the pulses are sufficiently short, we can simply add the noncommuting exponents to arrive at wherec 1 = c 1 T 1 /T , q = q + c 2 E so T 1 /T , andΩ = ΩT 2 /T . A gap now opens up between different eigenstate dispersions as is shown in Fig. S1(c). This fine tuning is essential to the rich physics already probed in previous SOC experiments [3][4][5][6][7][8][9], and constitutes one of the most important future directions for the present experiment.
In this generalized scheme, the SOC term ∝ k z F z and the spin flip term ∝ F x do not commute with each other, thus the net gauge field in Eq. (13) cannot be transformed away. Our protocol augmented with RF pulses can be understood intuitively. An atom moves a distance depending on its spin state after one period of gradient magnetic field pulses. It acquires a spin dependent group velocity (dz/dt) due to the modulating GMF while its canonical momentum (p z ) remains unchanged. It is this step that couples atomic spin and velocity. The additional RF pulses then couples different spin states. This coupling is also immune to atomic spontaneous emission, like GMF induced coupling. The two steps together give a velocity changing spin coupling, which can be contrasted to the Raman scheme, where SOC is accomplished by a momentum sensitive twophoton Raman transition. Thus our scheme (with additional RF pulses) and the Raman scheme are equivalent in their physical effects.

ESTIMATION OF ENERGY SCALE
The characteristic energy scales for the SOC and the effective QZS are both of the order of atomic recoil energy E so = 2 k 2 so /2m, which is typical for most ultracold atomic quantum gas systems. For the particular parameters adopted in our study, this is about 1 kHz for k so ∼ 0.5k L (one photon recoil momentum). The bias field QZS is about 1 kHz. The coherent spin exchanging collision interaction at atomic densities of our experiment is about 10 Hz for 87 Rb atoms in F = 1 states. The relaxation or dissipation is mainly caused by parametric heating from the time modulating GMF, which we estimate based on the inverse of the observed life time ( 300 ms for k so ∼ 0.5k L ), to be about 3Hz, much smaller than the characteristic energy of SOC. Three-body loss is negligible under our experimental conditions. This is affirmed by our observed life time, which is long enough to observe interesting phenomena associated with SOC. In our experiments, the linear Zeeman splitting or the distance between the states |1⟩, |0⟩, |− 1⟩ is about 4 MHz, which is much larger than other energy scale, due to the strong bias magnetic field used to select out the one dimensional gradient. The largest energy scale of our system as implemented is the bias field Zeeman energy, which is much larger than the coherent spin coupling, thus prevented us from directly observing spin flip oscillations. In section II we present a generalized scheme to resolve this difficulty by introducing a near resonant coherent drive coupling different atomic spin states, which could nullify the off scale Zeeman shift, as employed in the Raman scheme. Besides, even with a large Zeeman shift, we can still observe interesting many-body phenomena. For example, coherent spin dynamics where two atoms in the |0⟩ state oscillate into a pair of atoms in the |1⟩ and |− 1⟩ states was observed experimentally with linear Zeeman shift much larger than coherent spin exchange, due to conservation of the spin magnetization [10].