Photodriven germanium hole qubit

Hole qubits in germanium quantum dots are promising candidates for coherent control and manipulation of the spin degree of freedom through electric dipole spin resonance. We theoretically study the time dynamics of a single heavy-hole qubit in a laser-driven planar germanium quantum dot confined laterally by a harmonic potential in the presence of linear and cubic Rashba spin-orbit couplings and an out-of-plane magnetic field. We obtain an approximate analytical formula of the Rabi frequency using a Schrieffer-Wolff transformation and establish a connection of our model with the ESDR results obtained for this system. For stronger beams, we employ different methods such as unitary transformation and Floquet theory to study the time evolution numerically. We observe that high radiation intensity is not suitable for the qubit rotation due to the presence of high-frequency noise superimposed on the Rabi oscillations. We display the Floquet spectrum and highlight the quasienergy levels responsible for the Rabi oscillations in the Floquet picture. We study the interplay of both the types of Rashba couplings and show that the Rabi oscillations, which are brought about by the linear Rashba coupling, vanish for typical values of the cubic Rashba coupling in this system.

Figure 1

Schematic diagram of the model.

Figure 2

Bloch sphere dynamics of the qubit on application of the laser beam for half Rabi cycle, i.e., $T_{\pi }=\pi /{\omega }_{\text{res}}$. We use the following system parameters: ${\tilde{F}}_0=0.02\times 300$, $B=0.5$ T, ${\tilde{\alpha }}_l=0.0047,{\tilde{\omega }}_c=0.303,{\tilde{\omega }}_z=0.214,\omega ={\omega }_{\text{21}}$ which are defined in Sec. 5. The spin-up state spirals down to the spin-down state on the Bloch sphere because the driving frequency is much larger than the Rabi frequency. Here, ${\tilde{F}}_0$ has been magnified 300 times the actual value for better visualization of the rotations (although a stronger drive leads to other effects which is discussed later). As a result, the spin vector rotates $\sim 9$ times about the $z$ axis during the process of spin-flip.

Figure 3

Comparison between analytical and numerical values of resonant Rabi frequencies. (a) Variation of ${\omega }_{\text{res}}$ with the radiation amplitude $E_0$ for ${\alpha }_l=2.01$ meV $\AA$/$\hslash$ and $B=0.5$ T. (b) Variation of ${\omega }_{\text{res}}$ with ${\alpha }_l$ for $E_0=2.172$ kV/m and $B=0.5$ T.

Figure 4

Plot of maximum transition probability as a function of $\tilde{\omega }$ and ${\tilde{\alpha }}_l$ for a fixed radiation amplitude ${\tilde{F}}_0=0.08$ and different values of magnetic field: (a) $B=0.5$ T, (b) $B=1.0$ T, and (c) $B=1.5$ T. The dark curves indicate the resonances. The width of the resonances increase with $B$ and ${\alpha }_l$.

Figure 5

Plots of occupation probability as a function of time for realistic system parameters (i.e., $B=0.5$ T, ${\tilde{\alpha }}_l=0.0047,{\tilde{\omega }}_c=0.303,{\tilde{\omega }}_z=0.214,\omega ={\omega }_{\text{21}}$) and different values of driving strength: (a) ${\tilde{F}}_0=0.02$, (b) ${\tilde{F}}_0=0.04$, (c) ${\tilde{F}}_0=0.06$, and (d) ${\tilde{F}}_0=0.08$. The resonant Rabi frequency is an increasing function of the driving strength.

Figure 6

Plots of occupation probability as a function of time for realistic system parameters (i.e., $B=0.5$ T, ${\tilde{\alpha }}_l=0.0047,{\tilde{\omega }}_c=0.303,{\tilde{\omega }}_z=0.214,\omega ={\omega }_{\text{21}}$) and larger values of radiation amplitude: (a) ${\tilde{F}}_0=0.1$, (b) ${\tilde{F}}_0=0.2$, (c) ${\tilde{F}}_0=0.4$, and (d) ${\tilde{F}}_0=0.6$.

Figure 7

Variation of quasienergies of certain Floquet levels with radiation amplitude for realistic system parameters (i.e., $B=0.5$ T, ${\tilde{\alpha }}_l=0.0047,{\tilde{\omega }}_c=0.303,{\tilde{\omega }}_z=0.214,\omega ={\omega }_{21}$). The quasienergies of the levels represented by the red dotted curves decrease with $F_0$ and also do not contribute to the Rabi oscillations for the given initial and final states. The levels represented by the blue solid curves participate in the Rabi rotations and the Rabi frequency, equal to the gap between them, increases (linearly) with $F_0$.

Figure 8

Plots of occupation probability as a function of time for realistic system parameters (i.e., $B=0.5$ T, ${\tilde{\alpha }}_l=0.0047,{\tilde{\omega }}_c=0.303,{\tilde{\omega }}_z=0.214,{\tilde{F}}_0=0.02,\omega ={\omega }_{21}$) and different values of cubic Rashba coupling: (a) ${\tilde{\alpha }}_c=0.002$, (b) ${\tilde{\alpha }}_c=0.003$, (c) ${\tilde{\alpha }}_c=0.0047$ (equal to ${\alpha }_l$), and (d) ${\tilde{\alpha }}_c=0.013$ (typical value). The Rabi oscillations become increasingly nonresonant with increase in ${\alpha }_c$ and nearly vanish for typical values for ${\alpha }_c$ for this system.