Spin-orbit (SO) coupled cold atomic systems have becoming very active researching field in the latest decades. It is not only because that SO coupled neutral gases provide a promising platform for quantum simulating such novel condense matter physics like quantum spin Hall effect, super-conductors and -insulators, but also present high experimental controbility. The first experimental realization of SO coupled BEC system was at 2011 by Speilman's group through optical Raman process, and then various theoretical models as well as experimental researches have been reported giving rise to a variety of rich many-body quantum phaes and novel topological structures. However, one big challenge of such optical manupulating senarios is optical heating problem. More specifically, in a Raman process, the spontaneous emission rate and Raman transitional strength are on the same order relying on the ratio between light power and single-photon detuning, and as a result the atomic system can hardly reach to equlibrium under a strong Raman dressing. On the other hand, recently we notice from many experimental works of dual-species mixtures that interspecies interactions are also able to transfer the momentum from one to another species, and hence pave a good path to transfer the SO coupling. This SO coupling transfer provides an alternative way to induce the SO coupling for those troubled by the optical heating problem mentioned above, and will greatly broaden the study of SO coupling into more different kinds of atoms.

Fig. 1 (a) Schemetic model setup. (b) Atomic levels.

In our work, we consider a two-species spinor BEC, where one of the species is subjected to a pair of Raman laser beams that induces spin-orbit (SO) coupling, whereas the other species is not coupled to the Raman laser, as schematically shown in Fig.1. In the rotating frame, one can easily find that species A with a nonvanishing Raman strength $Omega$ possesses a Raman-induced SO coupling along direction of x, while species B with a vanishing Raman coupling does not essentially has a SO coupling. However, the interspecies spin-exchange interaction will play a Raman-beam role coupling the two spin components of species B. In such a case, species B will also possess a SO coupling.

To clearly demonstrate how the how the spin-exchange interaction induces SO coupling in species, we first consider a situation with the atomic number of species is far lager than that of species B, such that the back influced from B to A can be neglected. Now, species A is equally to be an isolated system which can be analytically solved by a variational method. Then, we turn to species B which is now strongly influced by A. We find that the interspecies spin-exchange interaction term can be rewritten as a effective Raman coupling term. More specifically, if A is in homogenous phases such as plane-wave or zero-momentum phase, the effective Raman coupling is a constant for given parametric condition, and then the ground-state properties of species B is also analytically soluble by the afformentioned variationla method. Moreover, if A is in striped phase, even though this effective Raman coupling term is much more complicated, we can still get an analytical phase boundary between in-phase and out-of-phase phase regions. In the in-phase parameter space, total density stripes of species A and B are oscillating in an in-phase way, while vise verse in the out-of-phase region. According to the above analyses, we obtain the analytical phase diagram in Fig.2(a).

Fig. 2 (a) Analytical phase diagram. (b) Numerical phase diagram. (c) Typical density profiles in different phases. In the calculation, we take NA=25NB.

Furthermore, to confirm the analytical results, we directly solve the coupled Gross-Pitaevskii equation derived from the total energy functional. In the numerical calculation, we include a box potential with hard walls for both species. We present the numerically obtained phase diagram in Fig.2(b), with typical density profiles for different phases plotted in Fig.2(c). One can see that, the two phase diagrams are in general in good qualitative agreement. The main difference is that the analytical phase diagram does not produce the two phases labeled as PS1 and PS2 in the middle of the numerical phase diagram. These two phases correspond to phase separation. The reason why PS phases are not present in Fig.1(a) can be mainly attributed to the breakdown of the assumption that species A is not affected by B, which underlies the analytical calculation. This assumption generally holds when species A is in either ST or ZM modes. However, when species A is in the PW phase, it can possess a large spin polarization. Hence even though the total atom number NA is much larger than NB, the number in the minority spin component of A may be comparable to NB. As a result, species B may have a significant influence on A.

(a) Numerical phase diagram. (b) Typical density profiles in different phases. In the calculation, we take NA=NB.

Finally, we also study the situation where two species are equally populated. Now the mutual influence between the two species is important, and we have to resort to numerical calculations to investigate this system. The phase diagram and several representative density profiles are presented in Fig.3. One striking feature one can immediately notice from the phase diagram is that the stripe phase dominates the parameter space. Through examination of the ground-state wave function, we discuss the key features of each phase in detail.

This work has been published as a Rapid Communition in Physical Review A. In detail please see https://journals.aps.org/pra/abstract/10.1103/PhysRevA.97.031601 .