Dressed photon-orbital states in a quantum dot: Intervalley spin resonance

The valley degree of freedom is intrinsic to spin qubits in Si/SiGe quantum dots. It has been viewed alternately as a hazard, especially when the lowest valley-orbit splitting is small compared to the thermal energy, or as an asset, most prominently in proposals to use the valley degree of freedom itself as a qubit. Here we present experiments in which microwave electric field driving induces transitions between both valley-orbit and spin states. We show that this system is highly nonlinear and can be understood through the use of dressed photon-orbital states, enabling a unified understanding of the six microwave resonance lines we observe. Some of these resonances are intervalley spin transitions that arise from a nonadiabatic process in which both the valley and the spin degree of freedom are excited simultaneously. For these transitions, involving a change in valley-orbit state, we find a tenfold increase in sensitivity to electric fields and electrical noise compared to pure spin transitions, strongly reducing the phase coherence when changes in valley-orbit index are incurred. In contrast to this nonadiabatic transition, the pure spin transitions, whether arising from harmonic or subharmonic generation, are shown to be adiabatic in the orbital sector. The nonlinearity of the system is most strikingly manifest in the observation of a dynamical anticrossing between a spin-flip, intervalley transition and a three-photon transition enabled by the strong nonlinearity we find in this seemly simple system.

Figure 1

(a) Multiple resonance frequencies as a function of the external magnetic field, observed for a single electron spin confined in a gate-defined Si/SiGe quantum dot, driven by low-power microwave excitation applied to one of the quantum dot gates. Resonances (1) and (2) are indistinguishable on this scale, as are resonances (3) and (4). On the horizontal axis, we plot $B_{\mathrm{tot}}=\sqrt{{(B_{\mathrm{ext}}^{\mathrm{x}}+B_{\parallel })}^2+B_{\perp }^2\phantom{{}^2}}$, with $B_{\parallel }\sim -120$ mT and $B_{\perp }\sim 50$ mT the estimated components of the stray magnetic field from the micromagnet [magnetized, in this measurement, along the $x$ direction (see Fig. S3 [26])]. (b) Zoom-in of the region indicated by a red box in panel (a). (c) Schematic energy level diagram of a generic two-level quantum system described by Eq. (1), as a function of detuning $\epsilon$ and with a harmonic driving of amplitude ${\epsilon }_1$ around the central value ${\epsilon }_0$ with energy eigenvalues $E_{\pm }$ and energy splitting $E_{01}=E_{+}-E_{-}$. (d) The four energy levels considered in this work as a function of $B_{\mathrm{tot}}$, in the absence of photonic dressing, comprised of two valley-orbit states $\left(|V_{1,2}\rangle \right)$ and two spin states $\left(|\downarrow ,\uparrow \rangle \right)$. $E_{VS}$ represents the valley-orbit energy splitting. The vertical arrows labeled 1–6 correspond to six processes observed in our simulations and to excitations observed in the experiment, with the same labeling scheme as panels (a) and (b). (1) and (2) correspond to single-photon spin flips. (3) and (4) correspond to two-photon spin flips. (5) corresponds to a combined spin flip, valley-orbit excitation, and single-photon absorption. (6) is also a combined spin-flip, valley-orbit excitation, between different states than (5), and is a three-photon process. (7) is a nonadiabatic valley-orbit excitation; it is observed in simulations but not in the experiments, because $E_{VS}\sim k_B{T}_{\text{el}}$, and therefore readout is ineffective.

Figure 2

Theoretical calculations of resonances in an ac-driven qubit with a low-lying orbital state induced by adiabatic and nonadiabatic processes. (a) Schematic illustrating that an excitation at driving frequency $\omega$ leads to several adiabatic (nonadiabatic) response frequencies of the electron dipole moment, $p\left(\mathrm{\Omega }\right)=FFT\left[p\right(t\left)\right]$, listed in green (blue). The time-dependent electron dipole moment in turn produces spin flips due to the magnetic field gradient when $\hslash \mathrm{\Omega }$ matches the Zeeman splitting. (b) Simulated resonance spectrum of the ac-driven qubit model of Eqs. (1) and (2), setting $\hslash =1$. As described in the main text, the dynamics of the dipole moment $p\left(t\right)$, defined in Eq. (3), are solved in the time domain for the driving frequency $\omega$, then Fourier transformed to obtain the response frequency $\mathrm{\Omega }$. Here we use ${\epsilon }_0={\epsilon }_1=\mathrm{\Delta }=1$. The dashed lines indicate the positions of the fundamental resonance and its first two harmonics (top to bottom). The lower inset shows the relation between the energy levels and the driving term. The upper inset shows the same results as the main panel, with the experimentally relevant resonances highlighted (compare to Fig. 1). (c) Resonance spectrum corresponding to the parameters ${\epsilon }_0=\mathrm{\Delta }=1$ and ${\epsilon }_1=0.5$. Here, the shaded region was normalized separately from the rest of the figure. The resonance features labeled 1–10 are discussed in the main text (see also Secs. I C and II B of Supplemental Material [26]). (d) A blowup of the region shown in the center box of panel (b), using the parameters $\mathrm{\Delta }={\epsilon }_0=1$ and ${\epsilon }_1=0.11$, which gives good agreement with the level repulsion observed in the experiments shown in Figs. 1 and 1.

Figure 3

Sensitivity of intravalley and intervalley spin transitions to electric fields. (a) Measured resonance frequency for the intervalley $(f_0^{\left(5\right)}$ at $B_{\mathrm{ext}}^{\mathrm{y}}=792$ mT, blue data and axes) and intravalley $(f_0^{\left(1\right)}$ at $B_{\mathrm{ext}}^{\mathrm{y}}=550$ mT, red data and axes) spin transition, as a function of gate voltage $V_2$. Inset: low-power continuous wave response of the intervalley spin transition, with $T_2^{*}$ estimated from the line width (see also Fig. S4 [26]). (b) Schematic representation of an atomic step in a Si/SiGe quantum well and of a quantum dot parabolic confinement potential (not to scale) laterally offset from the step. (c) Valley-orbit energy splitting found by a 2D tight-binding calculation using the geometry shown in panel (b), a 13-nm-wide quantum well barrier of 160 meV (corresponding to $30\%$ Ge), a parabolic confinement potential for the dot of size $\sqrt{\langle x^2\rangle }=21.1$ nm (corresponding to an orbital energy splitting of $\hslash \omega =0.45$ meV), and an electric field of $1.5\times {10}^6$ V/m (the experimental electric field is not well known). In the plots, a positive step offset corresponds to a step on the right-hand side of the dot. (d) Scatter plot showing two quantities measured every 400 s. The $x$ axis shows the average rate ${\mathrm{\Gamma }}_{IN}$ with which an electron tunnels into the quantum dot during qubit initialization (in orange in the inset); on the $y$ axis is the frequency of resonance (5) relative to its value averaged over the entire measurement (see also Fig. S6 [26]). The continuous line represents a linear fit to the data through the point (0,0). The dashed lines represent the $95\%$ confidence interval. The distribution of the points in the scatter plot indicates that the two quantities are correlated. $B_{\mathrm{ext}}^{\mathrm{x}}=590$ mT, $B_{\mathrm{ext}}^{\mathrm{y}}=598.2$ mT, $\mathrm{\Delta }{f}_0^{\left(5\right)}=f_0^{\left(5\right)}-15.6894$ GHz, $\mathrm{\Delta }{\mathrm{\Gamma }}_{IN}={\mathrm{\Gamma }}_{IN}-1.0762$ kHz.

Figure 4

(a) Measured spin-up probability as a function of microwave power at $B_{\mathrm{ext}}^{\mathrm{y}}=846$ mT for a fixed microwave burst time of $50\mu \mathrm{s}$, near the resonance condition for intervalley spin-flip transition (5). For increasing power, the line does not only become taller and wider but also moves towards higher frequencies. Panel (b) summarizes from top to bottom the center frequency, width, and height of the response. $\mathrm{\Delta }{f}_0^{\left(5\right)}=f_0^{\left(5\right)}-17.825$ GHz. (c) Top panel of (b) replotted using a linear power scale $(x$ axis $\propto {10}^{\mathbf{MW}\mathbf{power}\left(\mathbf{dBm}\right)/5})$. The blue line represents a linear fit to the black data points to the relation $\mathrm{\Delta }{E}_{\text{asymp}}\propto {\epsilon }_1^2$ [Eq. (S42)]. The points indicated by the red crosses are excluded from the fit.