Synthetic spin-orbit coupling mediated by a bosonic environment

We study a mobile quantum impurity, possessing internal rotational degrees of freedom, confined to a ring in the presence of a many-particle bosonic bath. By considering the recently introduced rotating polaron problem, we define the Hamiltonian and examine the energy spectrum. The weak-coupling regime is studied by means of a variational ansatz in the truncated Fock space. The corresponding spectrum indicates that there emerges a coupling between the internal and orbital angular momenta of the impurity as a consequence of the phonon exchange. We interpret the coupling as a phonon-mediated spin-orbit coupling and quantify it by using a correlation function between the internal and the orbital angular momentum operators. The strong-coupling regime is investigated within the Pekar approach, and it is shown that the correlation function of the ground state shows a kink at a critical coupling, that is explained by a sharp transition from the noninteracting state to the states that exhibit strong interaction with the surroundings. The results might find applications in such fields as spintronics or topological insulators where spin-orbit coupling is of crucial importance.

Figure 1

Schematic of a rotating impurity immersed into the bosonic bath. The COM transverse motion is confined to a ring of radius $r_0$ in the $xy$ plane (dashed line). Here, $(x,y,z)$ and $(x^{\prime },y^{\prime },z^{\prime })$ are the laboratory and circulation frames, respectively.

Figure 2

The spectral function of the system ${\mathcal{A}}_{M_J,L{M}_L}$ as a function of the normalized energy and bath density for different configurations of the free parameters $\{L,M_L,M_J\}$. State $|2,1,2\rangle$ exhibits an additional transition from the $|2,0,1\rangle$ state (green circles), whereas the same state with the inverted orientation of internal rotation $|2,-1,2\rangle$ does not reveal such a transition (red circle).

Figure 3

The enlarged regions of Fig. 2 in the vicinity of the crossings between the metastable state of the phonon continuum associated with impurity state $|2,0,1\rangle$ and stable quasiparticle branches of states (a) $|2,-1,2\rangle$ and (b) $|2,1,2\rangle$. From the behavior of the spectral function, one can deduce the information about the suppressed (a) and allowed (b) transitions.

Figure 4

The dependence of angular momentum correlation on the density of the bosonic bath. In the ground-state $|0,0,0\rangle$, the ${\widehat{J}}_z$ and ${\widehat{L}}_z$ operators exhibit infinitesimally small correlation (the inset), whereas, in the excited states, the two are coupled to each other in the instability regime with the maximum amplitude dependent on the relative projections of momenta.

Figure 5

(a) Ground-state energy dependence on the impurity-bath interaction strength. The values are obtained through the minimization of Pekar functional (25). (b) Angular momentum correlation dependence on the interaction strength. At the critical value of $a_c\approx 8.3$, the system exhibits transition to the spin-orbit coupled regime. (c) Dependence of the variational coefficient amplitudes on the interaction amplitude. The transition to the correlated regime is explained by the redistribution of coefficients' amplitudes at values higher than the critical strength $a_c$.