Main

There have been many theoretical proposals for creating multi-dimensional SOC in ultracold atoms9,10,11,12,13,14, so as to access novel macroscopic quantum phenomena and quantum topological states15,16,17,18,19,20,21,22,23,24. Whereas these proposals have not been realized in laboratories, physicists have also just begun to explore topological phenomena in optical lattices25,26,27,28. Here, we use the Raman scheme to produce a highly controllable 2D synthetic SOC for an ultracold Fermi gas of 40K. Such SOC allows us to create and manipulate a single stable Dirac point on a 2D plane, which is detected by spin-injection rf spectroscopy4.

We apply three far-detuned lasers propagating on the x–y plane to couple three ground hyperfine spin states, within the 42S1/2 ground electronic manifold, |1〉 = |F = 9/2, mF = 3/2〉, |2〉 = |F = 9/2, mF = 1/2〉 and |3〉 = |F = 7/2, mF = 1/2〉, where (F, mF) are the quantum numbers for hyperfine spin states as shown in Fig. 1a, to the electronically excited states. Unlike the tripod scheme, where a single excited state is considered9,10,11,15,16,17,18, in the 40K used here the excited states include a fine-structure doublet 42P1/2 (D1 line) and 42P3/2 (D2 line) with a fine-structure splitting of ∼3.4 nm. Each of two D-line components also has hyperfine structures. After adiabatically eliminating excited states, the ring scheme proposed in ref. 12 is realized for three cyclically coupled states, with a generalization to arbitrary laser configurations.

Figure 1: Two-dimensional synthetic spin–orbit coupling.
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a, Schematic of energy levels of 40K for creating 2D SOC. Each of the three Raman lasers dresses one hyperfine spin state from |9/2, 3/2〉 (|1〉), |9/2, 1/2〉 (|2〉) and |7/2, 1/2〉 (|3〉). Atoms are initially prepared in the free reservoir spin state |9/2, 5/2〉. b, Experimental geometry and laser configuration. Raman lasers 1 and 2 counter-propagate along the y axis and Raman laser 3 propagates along the x axis, and they are linearly polarized along the z, x and y directions respectively. c, Energy–momentum dispersions in the absence of SOC (Ω12 = Ω13 = Ω23 = 0). The three parabolae are displaced from the origin by ki, respectively, in the (px, py) plane. e, Energy–momentum dispersions of three dressed states after turning on SOC (−Ω12 = 2Ω13 = 2Ω23 = Er), showing that the dispersions of two dressed states touch at a Dirac point. Here, δ2 = δ3 = 0. d,f, Plots enlarging the low-energy parts of c and e, respectively. In e,f, the weights of different hyperfine spin states in the dressed states—the eigenstates of the Hamiltonian with SOC—are represented by assigning different colours to the hyperfine spin states. Red, blue and green represent |1〉, |2〉 and |3〉 respectively. The depth of each colour indicates the percentage of the corresponding spin state in a dressed state.

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The Hamiltonian is written as

where p denotes the momentum of atoms, ki′ (|ki′| = 2π/λi) and ωi are the wavevectors and frequencies of the three lasers, Ωi are the Rabi frequencies, i, j are the indices for the three ground hyperfine spin and the excited states respectively, ɛi and Ej are the ground and excited state energies, n is the total number of the excited states and Mij is the matrix element of the dipole transition. Different from refs 9,10,15, each hyperfine ground spin state here is dressed by only one laser field, regardless of the excited states it is coupled to. A gauge transformation, |i〉→ e − i ( k i ' . r+0 i ) |i〉, is applied to eliminate the phase θi. All results discussed here are therefore insensitive to the phase difference, and the sophisticated and challenging phase lockings are no longer necessary.

Applying the rotating wave approximation and the adiabatic elimination of excited states, the Hamiltonian is written as Ha = pz2/(2m) + Hxy,

The light-induced energy shifts in the adiabatic elimination have been recast into the diagonal terms, δ1 is set as zero (energy reference) for simplification, δ2 (δ3) corresponds to the two-photon Raman detuning between Raman laser 1 and 2 (1 and 3), and ki = ℏ ki′. All three Ωii′ = Ωi′i are real Raman coupling strengths regarding |i〉 ↔ |i′〉. Their values are derived from calculations (see Supplementary Information) and measured in experiments3. The single-photon recoil momentum kr = 2πℏ/λ and recoil energy Er = kr2/2m are taken as natural momentum and energy units. Since the dispersion along the z direction is not affected by the lasers, we will focus on the 2D Hamiltonian Hxy.

Using the two lemmas in Supplementary Information, we find that a doubly degenerate point p0 exists in the momentum space, where p0 satisfies two independent equations

Projecting the Hamiltonian in equation (2) to the pseudo-spin-1/2 formed by the two dressed states near the degenerate point p0, an effective Hamiltonian at low energies is obtained,

where pi=x, y has been redefined as pi − p0, i for simplicity. By rotating the momentum and the spin (see Supplementary Methods), the Hamiltonian can be simplified as

where both λx and λy are finite. Equation (5) describes a 2D SOC, which is equivalent to the Dresselhaus coupling if a simple transformation σx′ → σy′, σz′ → σx′ is applied. Such a 2D SOC directly tells one that the doubly degenerate point at p0 corresponds to a Dirac point with a linear dispersion at low energies. In particular, both the amplitude and anisotropy λx/λy can be largely tuned (see Supplementary Information). As a demonstration, Figs 1e (1f) and 1c (1d) show a comparison of the energy–momentum dispersion with and without SOC. The latter shows that energy dispersions of two dressed states touch at a Dirac point.

In this experiment, a homogeneous bias magnetic field B0 = 121.4 G along the z axis (the gravity direction shown in Fig. 1b) produces a Zeeman shift to isolate these three hyperfine spin states from other ones in the Raman transitions, as shown in Fig. 1a. When the three Raman lasers are nearly resonant with the three ground states (δi=1,2,3 ≈ 0), the nearest Raman transitions with other hyperfine states are |7/2, 3/2〉 ↔ |2〉 and |7/2, 3/2〉 ↔ |3〉, which have large two-photon Raman detunings about h × 60 kHz. A three-level system is thus a good description of our experiment.

The Raman coupling strength Ωii′ includes contributions from all excited states of two D-line components, as shown in Fig. 2a (see Supplementary Information). When the wavelength of the Raman lasers is longer than the D1 line (770.1 nm) or shorter than the D2 line (766.7 nm), Ω12Ω13Ω23 > 0 (see the eigenvalues of dressed states in Supplementary Information), the Dirac point emerges in the highest two dressed states, as shown in Fig. 2b, d. In contrast, when the wavelength of the Raman lasers is tuned between the D1 line and D2 line, Ω12Ω13Ω23 < 0, the Dirac point emerges when the lowest two dressed states become degenerate, as shown in Fig. 2c. This is different from the work in ref. 19, where only one excited state is considered and far-blue-detuned lasers are used to make the degenerate dark states the low-lying ones in the manifold of ground electronic states.

Figure 2: Controlling the Dirac point by changing the wavelength of the Raman lasers.
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a, Red, green and blue curves represent the Raman coupling strength Ω12, Ω13 and Ω23 as functions of the wavelength λ of the Raman lasers. Depending on λ, Ω12Ω13Ω23 can be either positive or negative. Ii is the intensity of the ith laser. b–d, Energy bands for wavelengths of 763.5 nm (blue-detuning D2 line), 768.85 nm (between D1 and D2 line) and 772.4 nm (red-detuning D1 line), respectively. The two higher dispersions touch at the Dirac point in b and d, where Ω12Ω13Ω23 is positive. The two lower dispersions touch at the Dirac point in c, where Ω12Ω13Ω23 is negative.

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We investigate the energy–momentum dispersions of three dressed states by spin-injection rf spectroscopy, which drives the atoms from a free spin-polarized state into SOC dressed states. We start with a degenerate Fermi gas of 2 × 106 40K atoms at the free reservoir spin state |9/2, 5/2〉 in a crossed optical dipole trap. A homogeneous bias magnetic field is ramped to B0 = 121.4 G. Then the three Raman lasers are ramped up in 60 ms from zero to their final value. Subsequently, a Gaussian shape pulse of the rf field is applied for 450 μs to drive atoms from |9/2, 5/2〉 to the final empty state with 2D SOC. The rf field does not transfer momentum to the atoms, and spin injection occurs when its frequency matches the energy difference between the initial and final states. Since the spin state |9/2, 5/2〉 is coupled via rf to the state |1〉, rf spectroscopy also measures the weight of the |1〉 state, in addition to the energy dispersions with 2D SOC. Following the spin-injection process, the Raman lasers, the optical trap and the magnetic field are switched off abruptly, and atoms freely expand for 12 ms in a magnetic field gradient applied along the x axis. Absorption images are taken along the z direction. By counting the number of atoms in state |1〉 as a function of the momentum and the rf frequency from the absorption image, we determine the energy band structure and locate the Dirac point.

Figure 3a2 shows the momentum-resolved spin-injection spectrum when the one-photon detuning of the Raman lasers is set at a wavelength of 772.4 nm (red-detuning D1); the corresponding theoretical plot is shown in Fig. 3a1. The two higher-energy dispersions touch at a Dirac point, as shown in Fig. 3a2. To further visualize the Dirac point, we plot the energy as a function of py for various px, as shown in Fig. 3a3–a5. When the wavelength of the Raman lasers is tuned to 768.85 nm, between the D1 line and D2 line, two lower-energy dispersions touch at a Dirac point, as shown in Fig. 3b1–b5. These two wavelengths were used to investigate the 1D SOC (refs 3,5). For the wavelength of the Raman lasers at the 763.5 nm blue-detuning D2 line, we have observed that two higher-energy dispersions touch at a Dirac point, similar to the 772.4 nm case. We also perform numerical calculations for the eigenvalues of the Hamiltonian in equations (2) and (5) according to experimental parameters and have found a reasonably good agreement between theory and experiments. In practice, several mechanisms, such as the fluctuation of the magnetic field and the finite momentum resolution, lead to the differences in rf spectrum between the theoretical and experimental results, as well as the error bars presented in Fig. 3 (see Supplementary Information).

Figure 3: Energy dispersions of dressed atoms measured by rf spin-injection spectroscopy.
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a1–a5, The one-photon detuning of the Raman lasers is set at a wavelength of 772.4 nm (red-detuning D1). Ω12 = 3.58Er, Ω13 = −3.94Er, Ω23 = −4.66Er, δ2 = −5.14Er, δ3 = −3.23Er. b1–b5, The wavelength is set at 768.85 nm, between the D1 line and D2 line. Ω12 = −4.96Er, Ω13 = 5.46Er, Ω23 = 6.46Er, δ2 = −5.2Er, δ3 = −2.13Er. a1,b1, Theoretical results for the |1〉 component using the realistic experimental parameters. a2,b2, Experimental results measured by rf spin-injection spectroscopy. Each line on the surfaces of the dispersions is constructed from time-of-flight images at the given rf frequency (see Supplementary Information). a3–a5, Cross-section drawings of a1 and a2 in energy-py coordinates for different quasimomenta px. Red dots with error bars and solid curves are experimental and theoretical results, respectively. The inset in a4 is a plot (without error bars) enlarging the region near the Dirac point. Here, λx1 = 0, λx2 = −0.42kr/m, λy1 = 1.5kr/m, and λy2 = −0.07kr/m are obtained by the experimental parameters. b3–b5, Cross-section drawings of b1 and b2. The inset in b4 is a plot enlarging the region near the Dirac point. λx1, λx2λy1 and λy2 are the same as in a, since Ω12/Ω13 and Ω12/Ω23 remain the same.

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Although a pair of Dirac points has been produced in optical lattices25, to our best knowledge, we have created a single Dirac point for the first time in the continuum. Moreover, the Dirac point created in this scheme is robust, in the sense that it moves in momentum space without opening a gap when experimental parameters change. This is because the Hamiltonian in equation (2) is real—consequently, the SOC in equation (5) cannot contain the σy term. By modulating δi, Ωi and ki, an arbitrary trajectory of the Dirac point in the momentum space can in principle be designed. As a demonstration, we fix δ2 and measure the positions of the Dirac point on the (px, py) plane as a function of δ3, as shown in Fig. 4a (blue dots). When fixing δ3 and changing δ2, the Dirac point moves along a different line (red dots). We also measure the energy at the Dirac point, as well as the energy separation ΔE from highest dressed state, as a function of the Raman coupling strength. Zero energy in Fig. 4b corresponds to the Zeeman energy splitting between |9/2, 5/2〉 and |9/2, 3/2〉. The energy of the Dirac point decreases with increasing Raman coupling strength (red diamonds). ΔE increases linearly with increasing Raman coupling strength (right inset of Fig. 4b). Since the diagonal energy shift induced by the Raman field has no influence on the effective SOC, we further plot the energy difference between the Dirac point and the highest dressed state, as highlighted by the green vertical arrow in the left inset of Fig. 4b. We find that ΔE increases linearly with increasing Raman coupling strength (right inset of Fig. 4b), and observe a good agreement between experimental and theoretical results.

Figure 4: Moving the Dirac point.
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a, Controlling the Raman detuning δ2 and δ3, the Dirac point moves along certain trajectories in momentum space. Blue dots: δ2 is fixed at −5.45Er and δ3 is tuned in the range [−2Er, 0.45Er]. Red dots: δ3 is fixed at −0.2Er and δ2 is tuned in the range [−7Er, −2.5Er]. The other parameters are Ω12 = −4.96Er, Ω13 = 5.46Er, Ω23 = 6.46Er. b, Energy at the Dirac point (red diamonds) and corresponding energy of the highest dressed state with the same momentum p0 (blue boxes) as a function of the Raman coupling strength. −Ω12 = 0.91Ω13 = 0.77Ω23 = Ω, δ2 = −5.2Er and δ3 = −2.13Er. The left inset in b shows the energy dispersions. The right inset in b represents the energy separation ΔE between the Dirac point and the highest dressed state as a function of the Raman coupling strength. Here, the one-photon detuning of three Raman lasers is at a wavelength of 768.85 nm. The error bars represent the standard deviation of repeated measurements.

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We hope that a highly controllable 2D SOC will help physicists to explore both gapless topological matter in the continuum, analogous to graphene and Weyl semimetals in lattices, and interesting topological or other exotic superfluids when the gap at the Dirac point is opened20,21,22,23,24. Whereas our scheme is very general and can be applied to any atomic species, we hope that the specific heating effect in alkali atoms due to the fine-structure splitting may be overcome by novel cooling schemes or the quantum quenching approach, which may allow one to extract topological properties from quantum dynamics in a relatively short timescale29.

Methods

Experimental set-up.

After sympathetic cooling with bosonic 87Rb in a quadrupole-Ioffe configuration magnetic trap, 40K atoms in the |9/2, 9/2〉 state and 87Rb atoms in the |2,2〉 are transferred into an optical dipole trap formed by two orthogonal 1,064 nm laser beams. The Fermi gas is further evaporatively cooled to T/TF ≈ 0.3, giving about 2 × 106 40K atoms with 87Rb atoms in the optical trap30, where TF is the Fermi temperature defined by and the geometric mean of trapping frequencies  Hz in our system, N is the number of fermions. After removal of the 87Rb atoms, the fermionic atoms are transferred into the state |9/2, 5/2〉 via a rapid adiabatic passage induced by a rf field of 80 ms at 19.6 G. Here, the transition is addressed by a rf ramp that starts at 6.56 MHz to the end of 6.28 MHz. Then a homogeneous bias magnetic field along the z axis (gravity direction) is ramped to B0 = 121.4 G by a pair of coils operating in the Helmholtz configuration.

Raman lasers.

Three Raman lasers are derived from a CW Ti–sapphire single frequency laser. Two Raman beams 1 and 2 are frequency-shifted around +201.144 × 2 MHz and +220.531 × 2 MHz by two double-pass acousto-optic modulators (AOM), respectively. Raman laser 3 is sent through two AOMs in double-pass and frequency-shifted −212.975 × 4 MHz. Then the three Raman beams are coupled into three polarization-maintaining single-mode fibres, respectively, in order to improve stability of the beam pointing and achieve a better beam-profile quality. After the fibres, these three Raman beams have maximum powers of 80 mW for each beam and overlap in the atomic cloud with 1/e2 radii of 200 μm. Raman lasers 1 and 2 counter-propagate along the y axis and Raman laser 3 propagates along the x axis, and they are linearly polarized along the z, x and y directions, respectively, corresponding to π, σ and σ polarizations relative to the quantization axis z shown in Fig. 1b.