Geometric phases in semiconductor spin qubits: Manipulations and decoherence

We describe the effect of geometric phases induced by either classical or quantum electric fields acting on single electron spins in quantum dots in the presence of spin-orbit coupling. On one hand, applied electric fields can be used to control the geometric phases, which allows performing quantum coherent spin manipulations without using high-frequency magnetic fields. On the other hand, fluctuating fields induce random geometric phases that lead to spin relaxation and dephasing, thus limiting the use of such spins as qubits. We estimate the decay rates due to piezoelectric phonons and conduction electrons in the circuit, both representing dominant electric noise sources with characteristically differing power spectra.

Figure 1

Spin relaxation and decoherence rates in semiconducting quantum dots. Results of the present work (solid lines) are compared to those of Ref. 21 (dashed lines), where the effect of piezoelectric phonons is analyzed to second order in the coupling to piezoelectric phonons. We find that geometric dephasing (evaluated to fourth order) leads to a saturation at low magnetic fields. Deviations between both results at intermediate field values are mainly due to the Nyquist noise of the conduction electrons. At higher magnetic fields (not shown), the results coincide. Parameters correspond to a parabolic quantum dot in GaAs with ${\omega }_0=5\mathrm{K}$ at $T=100\mathrm{mK}$.

Figure 2

(Color online) (a) The geometric spin precession due to SO interaction for an electron adiabatically moving within a 2DEG (left) is equivalent to the changing orientation of a sphere rolling on a plane (right). Both paths are related in a one-to-one fashion to each other. (b) The length of a straight path required to perform a spin flip versus the angle of the path relative to a crystal axis for several ratios $\beta ∕\alpha$ with a constant $l_{\mathrm{SO}}$. The area of the ellipses remains constant and equal to $\pi l_{\mathrm{SO}}^2$.

Figure 3

(Color online) The dressed spin-orbit length ${\tilde{l}}_{\mathrm{SO}}$ as a function of the typical orbital size $x_0=\sqrt{1∕m{\omega }_0}$ for different ratios $\beta ∕\alpha$. Exchanging $\alpha$ and $\beta$ yields identical curves

Figure 4

Relaxation rates due to piezoelectric phonons and Ohmic fluctuations in two scenarios, both as a function of the magnetic field $\mathit{B}$ applied along the [100] direction. For the strengths of Rashba and Dresselhaus spin-orbit coupling, a ratio $\alpha ∕\beta =4$ is assumed (Ref. 41). In (a), we plot the unfavorable scenario for quantum information precessing, with ${\lambda }_{\Omega }=5\times {10}^{-3}$, $l_{\mathrm{SO}}=1500\mathrm{nm}$, and a large dot $x_0=115\mathrm{nm}$ $({\omega }_0\approx 1\mathrm{K})$. In (b), we plot the rates for favorable conditions ${\lambda }_{\Omega }={10}^{-4}$, $l_{\mathrm{SO}}=3000\mathrm{nm}$, and $x_0=36\mathrm{nm}$ $({\omega }_0=10\mathrm{K})$. Lines range from a bath temperature of $100\mathrm{mK}$ (lower rates) to $900\mathrm{mK}$ (higher rates) in steps of $200\mathrm{mK}$. Shaded in gray is the region $T_1^{-1}>{\omega }_B=-g{\mu }_B\mid \mathit{B}\mid$ for which the rotating wave approximation fails and the Zeno regime sets in (see Sec. 2C). Note the extreme dependence of relaxation rates on the specific conditions.

Figure 5

Total relaxation and decoherence rates for a $51\mathrm{nm}$ dot $({\omega }_0=5\mathrm{K})$ at $T=100\mathrm{mK}$. Other parameters are $l_{\mathrm{SO}}=3\mu \mathrm{m}$, $\alpha ∕\beta =4$, $\theta =0$, and ${\lambda }_{\Omega }={10}^{-3}$. Three regimes are clearly visible. In the inset we give a polar plot of the dependence of the rates with magnetic field angle $\theta$. This angular dependence is negligible in the saturated regime, and of the form $T^{-1}\propto {\alpha }^2+{\beta }^2-2\alpha \beta \sin 2\theta$ at higher fields.

Figure 6

(Color online) Phase diagram of the three relaxation regimes at ${\lambda }_{\Omega }={10}^{-3}$, $\alpha ∕\beta =4$, $l_{\mathrm{SO}}=3\mu \mathrm{m}$, and $\theta =0$. A window of Ohmic fluctuations—dominated relaxation and dephasing opens up at lower temperatures. The saturation regime in this plot is dominated by phonon fluctuations, although regions exist (not shown) at lower fields and temperatures where the saturation is mainly due to Ohmic fluctuations.

Figure 7

Diagrams involved in the evaluation of the relaxation times. The $4\times 4$ self-energy matrix $\Sigma$ that dresses the propagator $\Pi$ is calculated to fourth order in the dot-bath coupling (blue dots). Contractions of bath fields in ${\Sigma }^{\left(4\right)}$ can be classified in two distinct types (upper and lower rows of diagrams). Times $\{0,{\tau }_1,{\tau }_2,t\}$ are ordered, and ${\tau }_1$, ${\tau }_2$ must be integrated.

Figure 8

The kernels $K_1$ and $K_2$ involved in the result for $T_{1,2}^{\left(4\right)}$ [Eq. (C10)] for generic values of the coupling constants ${\mathbf{\gamma }}^{\perp ,\parallel }$. A resolving imaginary part $ϵ=0.01{\omega }_B$ was used to make the delta functions visible.