Spin-Orbit-Driven Coherent Oscillations in a Few-Electron Quantum Dot

We propose an experiment to observe coherent oscillations in a single quantum dot with the oscillations driven by spin-orbit interaction. This is achieved without spin-polarized leads, and relies on changing the strength of the spin-orbit coupling via an applied gate pulse. We derive an effective model of this system which is formally equivalent to the Jaynes-Cummings model of quantum optics. For parameters relevant to an InGaAs dot, we calculate a Rabi frequency of 2 GHz.

Figure 1

Spectral features of the Rashba-coupled quantum dot as a function of magnetic field. The parameters used are typical of InGaAs: $g=-4$, $m/m_e=0.05$ with dot size $l_0=150\mathrm{nm}$. Resonance occurs at $B_0=90\mathrm{mT}$. (a) Low-lying excitation spectrum for spin-orbit coupling $\alpha =0.8\times {10}^{-12}\mathrm{eV}\mathrm{m}$. (b) Lowest-lying anticrossing. The thick line is the JC model showing anticrossing width ${\Delta }_0$ at $\delta =0$, and the thin line is the exact numerical result. (c) Plot of width ${\Delta }_n$ against central energy of anticrossing with the dot on resonance for different $\alpha$ in the range $(0.3–2.0)\times {10}^{-12}\mathrm{eV}\mathrm{m}$. The exact numerical results (circles) show excellent agreement with the square-root behavior predicted by the JC model in this $\alpha$ range.

Figure 2

Configuration of the dot in the various stages of the cycle. (a) The positions of the dot levels ${\psi }_{{\alpha }_1}^{\pm }$, chemical potentials ${\mu }_{L,R}$, and the tunneling rates ${\Gamma }_L>{\Gamma }_R$. (b) The coupling is initially ${\alpha }_1$. On average, for times $t_i>{\Gamma }_R^{-1}$ the dot will be initialized in state ${\psi }_{{\alpha }_1}^{-}$. (c) The applied voltage pulse lowers the dot levels and nonadiabatically changes the coupling to ${\alpha }_2\ne {\alpha }_1$, thus inducing Rabi oscillations. (d) Pulse is switched off after time $t_p$ and the levels return to their initial places. Tunneling to the right occurs when the electron has oscillated into the upper state. Relaxation rates ${\Gamma }_{1,2}$ are also shown.

Figure 3

Characteristics of the Rabi oscillation. (a) Probability $P(t_p)$ of finding an electron in the upper level after time $t_p$ following the nonadiabatic change ${\alpha }_1=1.5\to {\alpha }_2=0.3\times {10}^{-12}\mathrm{eV}\mathrm{m}$ as a function of magnetic field. (b) Amplitude of oscillation as a function of $B/B_0$ for changing from ${\alpha }_1=1.5$, 0.8, 0.6 to ${\alpha }_2=0.3\times {10}^{-12}\mathrm{eV}\mathrm{m}$ (top to bottom). (c) Phonon-induced relaxation rate for InAs parameters $\alpha =1.5\times {10}^{-12}\mathrm{eV}\mathrm{m}$, $P=3.0\times {10}^{-21}{\mathrm{J}}^2/{\mathrm{m}}^2$, $\rho =5.7\times {10}^3\mathrm{kg}/{\mathrm{m}}^3$, $c=3.8\times {10}^3\mathrm{m}/\mathrm{s}$. Close to $B_0$ the rate is suppressed to ${\Gamma }_{ep}<{10}^{-7}{\omega }_0$.