Dynamical Spin Properties of Confined Fermi and Bose Systems in the Presence of Spin–Orbit Coupling

Abstract

Due to the recent experimental progress, tunable spin–orbit (SO) interactions represent ideal candidates for the control of polarization and dynamical spin properties in both quantum wells and cold atomic systems. A detailed understanding of spin properties in SO-coupled systems is thus a compelling prerequisite for possible novel applications or improvements in the context of spintronics and quantum computers. Here, we analyze the case of equal Rashba and Dresselhaus couplings in both homogeneous and laterally confined two-dimensional systems. Starting from the single-particle picture and subsequently introducing two-body interactions we observe that periodic spin fluctuations can be induced and maintained in the system. Through an analytical derivation, we show that the two-body interaction does not involve decoherence effects in the bosonic dimer, and, in the repulsive homogeneous Fermi gas, it may be even exploited in combination with the SO coupling to induce and tune standing currents. By further studying the effects of a harmonic lateral confinement—a particularly interesting case for Bose condensates—we evidence the possible appearance of nontrivial spin textures, whereas the further application of a small Zeeman-type interaction can be exploited to fine-tune the system’s polarizability.

Appendix: Alternative Approach to Harmonically Confined 2D Particles

Due to the dependence of $\hat{L}'_z$ on $\hat{\mathbf {p}}'$ (which satisfies $[\hat{p}'_x,x]=-i$ and analogously for the y components), this operator commutes with the Hamiltonian (46), so that a common basis set can be defined which contemporarily diagonalizes $\hat{L}'_z$ and $\hat{h}_\mathrm{{sp}}$. Using polar coordinates ($x=r\cos \varphi$ and $y=r\sin \varphi$), the ground-state solutions given in the previous section read as

$$\begin{aligned} \psi _0^+(r,\varphi ) =\left( \frac{m\omega }{4\pi }\right) ^{1/4} e^{-iK r(\cos \varphi +\sin \varphi )} e^{-\frac{m\omega }{2}r^2} \left( \begin{array}{c} \frac{1+i}{\sqrt{2}} \\ 1 \\ \end{array} \right) \nonumber \\ \psi _0^-(r,\varphi ) =\left( \frac{m\omega }{4\pi }\right) ^{1/4} e^{iK r(\cos \varphi +\sin \varphi )} e^{-\frac{m\omega }{2}r^2} \left( \begin{array}{c} -\frac{1+i}{\sqrt{2}} \\ 1 \\ \end{array} \right) \end{aligned}$$

(67)

It is easily shown that these are at the same time eigenstates of $\hat{h}_\mathrm{{sp}}$ with energy $\omega -2m\alpha ^2$, and of $\tilde{L}_z$ with eigenvalue 0. Other eigenstates of $\tilde{L}_z$ relative to the eigenvalue l can be obtained by defining the following creation and destruction operators:

$$\begin{aligned} \hat{a}_{\pm }^{'\dagger }=\frac{1}{\sqrt{2}}(\hat{a}_x^{'\dagger }\pm i\hat{a}_y^{'\dagger }) \nonumber \\ \hat{a}'_{\pm }=\frac{1}{\sqrt{2}}(\hat{a}'_x \mp i\hat{a}'_y). \end{aligned}$$

(68)

These operators satisfy the usual commutation properties, in analogy with $\hat{a}^{'\dagger }$, $\hat{a}'$.

Moreover, from the application of $\hat{a}_{\pm }^{'\dagger }$, it becomes evident how $\hat{L}'_z$ and $\hat{L}_z$ eigenstates only differ by the phase factor $\pm Kr(\cos \varphi +\sin \varphi )$, induced by the SO coupling.