Stern-Gerlach splitters for lattice quasispin

We design a Stern-Gerlach apparatus that separates quasispin components on the lattice, without the use of external fields. The effect is engineered using intrinsic parameters, such as hopping amplitudes and on-site potentials. A theoretical description of the apparatus relying on a generalized Foldy-Wouthuysen transformation beyond Dirac points is given. Our results are verified numerically by means of wave-packet evolution, including an analysis of Zitterbewegung on the lattice. The necessary tools for microwave realizations, such as complex hopping amplitudes and chiral effects, are simulated.

Figure 1

A visualization of the FW transformation. Using a rotation around the $z$ axis by an angle $\varphi$, followed by a rotation around $y$ by an angle $\theta$, would rotate the eigenstates of ${\sigma }_z$ to the eigenstates of the Hamiltonian given by Eq. (1). In this visualization, the angles are considered to be scalars since they are operators that commute with the Hamiltonian.

Figure 2

Lattice topologies corresponding to the polarizer, up to second neighbors (top) and third neighbors (bottom). Thick lines correspond to the (strongest) nearest-neighbor interaction, thin lines are the next-to-nearest-neighbor interactions, and, finally, dashed lines are the weakest (and in the top model neglected) third-nearest-neighbor interactions.

Figure 3

Top: Matrix form of the nonlocal complete polarizer potential. The interaction zone contains different on-site energies indicated by the alternating pixel intensities in the diagonal. Bottom: Matrix form of a geometrical polarizer potential with range $\rho =10$ and no on-site potential. Only couplings to first and second order have been included. Both potentials are given in units of $\mathrm{\Delta }$.

Figure 4

Zitterbewegung of the wave packet. On the left panel, we see the oscillations of $x_{\mathrm{zitt}}$ without ballistic motion, as well as the decay of amplitude predicted by stationary phase approximations. On the right panel, we see the same rate of amplitude decay for three different effective masses, $\mu$. $\overline{\mathrm{\Delta }x}$ is the averaged maximum amplitude of the oscillations, while $\tau =t/T_{\chi }$ and $T_{\chi }$ is the characteristic time of the simulation given by $T_{\chi }=\hslash /\mathrm{\Delta }$.

Figure 5

Evolution of a wave packet going through the lattice polarizer. The first picture shows the initial condition of the complete wave packet, whereas the second and third pictures portray the dynamics of the upper and lower band components of the wave packet. The collision time with the polarizer is $T_c=\frac{\hslash N}{2\mathrm{\Delta }\kappa }$, where $N$ is the number of sites on the lattice.

Figure 6

The dotted line represents the reflection coefficient for the upper band component of the wave packet as a function of the variables $\kappa$ and the polarizer size $\rho$. The continuous line represents the transmission coefficient for the lower band component of the wave packet as a function of the same variables.

Figure 7

Mean reflection (red) and transmission (blue) coefficients as functions of the standard deviation of the coupling ${\sigma }_{\delta }$ (see text) for a $\rho =600$ splitter. The error bars represent the fluctuations obtained from multiple realizations. The value $\kappa =0.5$ has been chosen for optimality.

Figure 8

Configuration of disks (indicated as circles, with a number) giving rise to the coupling structure specified by Eq. (43). The six disks are organized in pairs $(1,2),(3,4)$, and $(5,6)$, each of which interacts strongly via the coupling constant $d$. The inner disks interact via the coupling constant $f$. The outer disks interact with just one of the other four disks (for example, 2 with 3), as the others remain screened geometrically. The $C_{3v}$ symmetry is broken by tilting the outer disks with an angle $\theta$.

Figure 9

Two particular realizations that keep the full $C_{3v}$ symmetry are illustrated, one in which $\theta ={\theta }_c$, and the other for $\theta =0$ in which screening sets $g=0$.

Figure 10

Spectrum of the configuration shown in Fig. 8; dots correspond to a full 3D simulation of a microwave cavity using comsol 5.2 and continuous lines correspond to tight-binding calculations. The lower band shows the desired inversion level due to effective negative coupling.

Figure 11

Upper row: simulated 3D system; dielectric disks depicted in light brown, nonreflective walls in cyan, and perfect conductors in yellow. Lower row: lowest modes for $\theta =60$ (before level inversion) and $\theta =85$ (after level inversion). Two different scales have been used inside and outside the cylinders for better visibility. The wave functions outside the cylinders exhibit the nature of couplings.