Atoms in a BEC engineered to have spin–orbit coupling typical of electrons in a solid

Since the creation of a Bose–Einstein condensate (BEC) nearly 20 years ago, the toolbox for manipulating ultracold gases has greatly expanded. Researchers have simulated the transition from a bosonic superfluid to a Mott insulator, led fermions continuously from a molecular BEC to a Bardeen-Cooper-Schrieffer superfluid composed of widely spaced Cooper pairs, and reproduced many other phenomena seen in condensed-matter systems. And they want to do more.

Researchers envision using ultracold gases to simulate a real solid. For example, interfering laser beams can create a periodically varying potential energy in which the atoms sit, reminiscent of electrons in a crystal lattice; experimenters can adjust the depth of the potential wells or the interaction strength between the atoms. The advantage of using an ultracold atomic system is that it is much cleaner than a real solid and allows experimenters to exert much greater control over the relevant parameters.

To expand the types of simulations that can be done, however, requires the introduction of electromagnetic effects. Without such effects, there’s no way to reproduce fascinating behavior like the quantum Hall effect or to further investigate semiconductor spintronics. But how can one mimic electromagnetic effects when dealing with a gas of neutral atoms?

Engineering gauge fields

One answer is to use lasers in a creative way to produce some of the same impacts on neutral atoms that a magnetic field has on charged particles. ¹ About a year and a half ago, researchers from the Joint Quantum Institute (JQI) of NIST in Gaithersburg, Maryland, and the University of Maryland, College Park, used lasers to generate an effective magnetic field in a BEC (see PHYSICS TODAY, February 2010, page 17 ). As proof, they observed vortices that were produced while neutral atoms encircled the effective flux lines. ²

Some of the same JQI researchers have now taken an important step further and simulated the spin–orbit coupling (SOC) between an atom’s spin and its motion. ³ For electrons, SOC usually results from an electron’s motion in a charge field; there’s no such charge field for atoms moving in a BEC, so the JQI experimenters had to devise a way to simulate the same effect with lasers. They also found a quantum phase transition from a regime in which two spin states coexist in the same region of space to one in which they are spatially separate. The team consists of JQI researchers Yu-Ju Lin and Ian Spielman, as well as Karina Jiménez-García, of JQI and the National Polytechnic Institute of Mexico.

Jason Ho of the Ohio State University comments that the new work will open many doors for future studies. A natural extension will be to demonstrate SOC in fermions as well as bosons. In general, the SOC interaction stems from a nonabelian gauge field, although the kind simulated in the JQI work is abelian. Such fields are being widely explored for possible implementation of topological quantum computing ⁴ (see PHYSICS TODAY, March 2011, page 20 , and the article by Sankar Das Sarma, Michael Freedman, and Chetan Nayak, July 2006, page 32 ).

Patrik Öhberg of Heriot-Watt University in Edinburgh, UK, notes that the new SOC technique allows experimenters to adjust parameters, such as the coupling strength, that otherwise are set by nature. In addition, it has resulted in a new type of particle—a boson with SOC—that has not been studied before. In certain circumstances, he says, such a condensate “might mimic relativistic dynamics, even if it is ultracold.”

Artificial fields

In atomic physics, SOC is an interaction between an electron’s spin and its motion about the nucleus; for solids, it’s the link between the electron’s spin and its motion in the charge field of the underlying lattice. Taking a broader view of SOC, Spielman explains, his group sought a way to link the internal spin of an atom to its momentum. They did that with a pair of laser beams. He and his colleagues started with a BEC of rubidium-87 atoms. In place of an electron’s two spin states—up and down—they focused on two particular hyperfine states of the ⁸⁷Rb atom’s 5S _1/2 F = 1 electronic ground state, referring to them as pseudospin states. The hyperfine state with m_F = 0 was taken to be the spin-up state ∣↑〉, and the m_F = −1 state as the spin-down state ∣↓〉. Roughly equal populations of the two spin states were present in the BEC at the start of the experiment.

The two pseudospin states were split in energy, as shown in figure 1a, by an external magnetic field in the y-direction. A pair of lasers coupled those two states with a coupling strength Ω. The laser beams, each traveling at 45° from the x-direction in the xy-plane, intersected at right angles. By means of a two-photon process, those beams connected the spin-up state to a spin-down state whose momentum differed by 2k, where k is the component of the single-photon recoil momentum in the x-direction. Essentially, the interaction of the light with the atom creates a pair of dressed states, denoted ∣↑′〉 and ∣↓′〉. Spielman explains that the dressed spin-up state consists primarily of the bare spin-up state plus a small superposition of the bare spin-down state whose momentum is different by 2k, and likewise for the dressed spin-down state.

In its SOC paper, the JQI team showed that the Hamiltonian for the experiment corresponds to that for an electron in a spin-dependent vector potential, so that the SOC appears as a product of the momentum and the spin of the electron. For an electron in a solid, the coefficient of the SOC term is a property of the material. For a system of ultracold atoms, in which the dressed states are superpositions of states with different momenta, the Hamiltonian of those dressed states also contains an SOC term—that is, a term involving a product of the momentum and spin. In the case of the BEC, however, the coefficient of the term is controlled by the laser geometry and wavelength. It seems a bit of a cheat for the JQI experimenters to have applied an external magnetic field, but the SOC term contains no dependence on that field. The external field’s role was to split the internal spin states and thereby help create the two pseudospin states.

From the SOC Hamiltonian, the JQI team determined a dispersion relation between the energy and the quasimomentum q as a function of the laser coupling Ω. As seen in figure 1b, the dispersion curves have two branches, with the lower branch having double minima for coupling strengths up to a certain value, 4E _L, where the recoil energy E _L is taken as the natural scale of energy for the system. States near the two minima are the two dressed states, ∣↑′〉 and ∣↓′〉. At higher coupling strengths, the two minima merge into one. It was in that domain of strong coupling that the JQI group had worked a year earlier to create an effective magnetic field. The approach taken to achieve the SOC is similar to one proposed earlier as a method to produce highly entangled many-body systems. ⁵

The experimenters demonstrated that they had indeed produced dressed states in the two minima, as predicted by the calculated dispersion relation. To do that, Spielman and his collaborators started with a mixture of the two bare spin states, slowly turned on the laser beams (loading the atoms into the dressed spin states), and then suddenly turned off the laser light and let the BEC cloud expand to determine the momentum in each of the spin states. As seen in figure 1c, atoms in the bare spin-up state ∣↑〉 are found predominantly near the predicted minimum at q/k = −1, with a few spin-up atoms, arising from the dressed spin-down state, found near the degenerate minimum at q/k = +1. Likewise, the atoms with spin down were found predominantly at q/k = +1, with a small number of them at q/k = −1. At higher coupling strength, when the two minima have merged into a single minimum, both bare spin states are centered at a value of quasimomentum q = 0.

Spatially separated spins

The up and down pseudospin states of ⁸⁷Rb atoms examined in the NIST experiment are normally well mixed in a BEC, absent SOC. However, Spielman and his colleagues found that as they increased the laser coupling strength above a critical value, the two pseudospin states segregated themselves spatially. The new repulsion between the spin states results from a laser-induced interaction between the two dressed spin states. Figure 2 shows the atoms in a homogeneous phase and in spin-separated phases.

Jason Ho notes that SOC should also give rise to a periodic variation in density in the BEC that is caused by the interference between the two dressed spin states. (The spatially separated spin states constitute indirect evidence of that.) If so, he says, such a spatially varying BEC might be used to simulate the crystalline lattice in a condensed-matter system without having to create one with interfering laser beams. That’s yet another of the myriad possibilities for this new addition to the ultracold atom toolbox.