Exact Nonadiabatic Holonomic Transformations of Spin-Orbit Qubits

An exact analytical solution is derived for the wave function of an electron in a one-dimensional moving quantum dot in a nanowire, in the presence of time-dependent spin-orbit coupling. For cyclic evolutions we show that the spin of the electron is rotated by an angle proportional to the area of a closed loop in the parameter space of the time-dependent quantum dot position and the amplitude of a fictitious classical oscillator driven by time-dependent spin-orbit coupling. By appropriate choice of parameters, we show that the spin may be rotated by an arbitrary angle on the Bloch sphere. Exact expressions for dynamical and geometrical phases are also derived.

Figure 1

(a) Displacement $\xi (t)$ and spin-orbit coupling $\alpha (t)$, in scaled units as (sinusoidal) functions of driving time and (b) scaled responses $x_c(t)$, $a_c(t)$, and ${\dot{a}}_c(t)$ with $T=12\pi /\omega$. Thin dashed lines indicate discontinuous changes in displacement with $T=4\pi /\omega$ and the corresponding smooth response. Contours $[\xi (t),a_c(t)]$ for square and circular loops are shown in (c) and (d). Note that the acquired phase in (c) is always unity in scaled units—the adiabatic result, as indicated. By contrast the harmonically driven case in (d) is a circle only in the adiabatic limit, with oscillatory contour otherwise.