Dispersively Probed Microwave Spectroscopy of a Silicon Hole Double Quantum Dot

Owing to ever increasing gate fidelities and to a potential transferability to industrial CMOS technology, silicon spin qubits have become a compelling option in the strive for quantum computation. In a scalable architecture, each spin qubit will have to be finely tuned and its operating conditions accurately determined. In view of this, spectroscopic tools compatible with a scalable device layout are of primary importance. Here we report a two-tone spectroscopy technique providing access to the spin-dependent energy-level spectrum of a hole double quantum dot defined in a split-gate silicon device. A first gigahertz-frequency tone drives electric dipole spin resonance enabled by the valence-band spin-orbit coupling. A second lower-frequency tone (approximately $500\mathrm{MHz}$) allows for dispersive readout via rf-gate reflectometry. We compare the measured dispersive response to the linear response calculated in an extended Jaynes-Cummings model and we obtain characteristic parameters such as $g$ factors and tunnel and spin-orbit couplings for both even and odd occupation.

Figure 1

Double quantum dot device and working points. (a) Simplified 3D schematic of a split-gate, silicon-on-insulator field-effect transistor. An $LC$ resonator wired to gate 1 is used for reflectometry readout. Static voltages $V_{G1}$ and $V_{G2}$ are applied to gates 1 and 2. Using bias tees, they are combined with about a $500\mathrm{MHz}$ reflectometry tone ($f_r$) and a 1–20 GHz microwave spectroscopy tone ($f_{\mathrm{exc}}$), respectively. The inset shows a false color scanning electron micrograph of the device (scale bar is 100 nm). (b),(d) Phase response of the $LC$ resonator as a function of $V_{G1}$ and $V_{G2}$ showing interdot charge transitions ICT 2 and ICT 3 for even and odd parities, respectively. The insets show the equivalent one- and two-electron charge configurations just above and below ICT 3 and ICT 2, respectively. The first (second) number represents the equivalent hole occupation in the dot under gate 1 (gate 2). (c),(e) Phase response as a function of $V_{G2}$ and $B_z$ at fixed $V_{G1}$ [see the dashed lines in (b) and (d)], revealing the ground-state evolution in the magnetic field.

Figure 2

Photon-assisted spectroscopy at zero magnetic field. (a) Phase response of the resonator as a function of gate voltage, $V_{G2}$, and microwave frequency, $f_{\mathrm{exc}}$, at zero magnetic field. The output power of the microwave generator is adjusted for each $f_{\mathrm{exc}}$ in order to deliver a constant power at the device [27]. We estimate the power at the device level to be around $-70\mathrm{dBm}$. In addition to the central interdot transition signal vanishing at $5.72\pm 0.04$ GHz, two side branches mark photon-assisted charge transitions between the quantum dots. (b) Theoretical simulation of the driven DQD phase response. The central dip at $\epsilon =0$ vanishes when the excitation energy matches $\mathrm{\Delta }$. (c) Multiphoton processes arising at higher driving power. The delivered microwave power is increased by 10 dBm compared to (a). Additional side branches appear at one-half and one-third of the side-branch frequency in (a), indicating two-photon and three-photon processes, respectively. (d) Energy diagram of a DQD near the “(1,1)” $\leftrightarrow$ “(2,0)” transition at zero magnetic field. The double arrows mark the processes giving rise to the branches observed in (a).

Figure 3

Photon-assisted spectroscopy at finite magnetic field for an even-parity interdot charge transition. (a) Phase response of the $LC$ resonator around the spin-orbit anticrossing of $|\Downarrow \Downarrow ⟩$ with S(2,0) as a function of $V_{G2}$ and $f_{\mathrm{exc}}$ at $B_z=600\mathrm{mT}$. The dashed horizontal lines delimit regions in which the spectroscopy tone power is held constant at room temperature. The dash-dot vertical line marks the position of the ICT at zero magnetic field, corresponding to $\epsilon =0$. The strong vertical structure is associated with the spin-orbit anticrossing at $\epsilon ={\epsilon }_{\mathrm{SO}}$. It vanishes at $f_{\mathrm{exc}}={\mathrm{\Delta }}_{\mathrm{SO}}/h=4.6\pm 0.1$ GHz. The three side branches around the central signal correspond to photon-induced charge transitions between the quantum dots, as indicated in (c). (b) Corresponding simulated phase response of the driven DQD. (c) Energy diagram of DQD of around a “(1,1)” $\leftrightarrow$ “(2,0)” transition at finite magnetic field and with $g_1\ne g_2$. The photon-induced charge transitions responsible for the side branches in (a) and (b) are indicated by arrows and corresponding symbols.

Figure 4

Photon-assisted spectroscopy at finite magnetic field for an odd-parity interdot charge transition. (a) Phase response of the $LC$ resonator around zero detuning as a function of $V_{G2}$ and $f_{\mathrm{exc}}$ at $B_z=$ 1.3 T. The dispersive response due to the charge qubit is visible as the vertical feature at $V_{G2}\approx -737.3\mathrm{mV}$. At resonance, $2h{f}_{\mathrm{exc}}=\mathrm{\Omega }$, the resonator also undergoes a phase shift (white dashed line). Around 12 GHz, spin-orbit anticrossings between states with opposite spin localized in different dots are detectable in the driven response. (b) Energy diagram of a single hole in a DQD [“(0,1)” $\leftrightarrow$ “(1,0)” transition] at finite magnetic field with $g_1\simeq g_2$. At zero detuning, the down spin states of the left and right dots undergo an anticrossing due to tunneling $t$. A small anticrossing due to the spin-orbit interaction appears between the down spin states of one dot and the up spin states of the other dot. Microwave-induced transitions that give rise to the two branches are highlighted with arrows.

Figure 5

Phase response of the driven double quantum dot as given by Eqs. (A15) and (A16) for the one- and two-photon processes at zero magnetic field. The parameters are $Q=24$, $\hslash {\omega }_r=2\mu \mathrm{eV}$, $\hslash g_c=0.15\mu \mathrm{eV}$. The double quantum dot is modeled by the charge tunneling $t=11.8\mu \mathrm{eV}$, tunneling with spin flip $t_{\mathrm{SO}}=8.3\mu \mathrm{eV}$, and $g$ factors $g_1=4$, $g_2=1.91$. Detuning energy $\epsilon$ is counted from the point of charge degeneracy at zero magnetic field. The amplitude of the resonant drive is $A_{\mathrm{exc}}=15\mu \mathrm{eV}$. The phonon equilibrium temperature is significantly smaller than all energy separations between the levels, which corresponds to the limit ${\mathrm{\Gamma }}_{\uparrow }\ll {\mathrm{\Gamma }}_{\downarrow }$. With our environment model we set $S_{AA}(E_{ϵ}-E_{\gamma })/\hslash =S_{AA}(0)/\hslash =S_{AA}({\omega }_R)/\hslash =0.5\mu \mathrm{eV}$ for simplicity. To account for quasistatic noise due to the electrical environment, we furthermore convolute the computed signal with a Gaussian function of width $\sigma =5\mu \mathrm{eV}$.

Figure 6

Tunnel coupling extraction for ICT 2. Phase response of the $LC$ resonator as a function of $V_{G2}$ and $f_{\mathrm{exc}}$ is shown on the left, and a line cut (black dashed line) is shown on the right. The line cut is fitted with Eq. (B6) to extract the framed parameters.

Figure 7

Spin-orbit interaction extraction for ICT 1. Phase response of the $LC$ resonator as a function of $V_{G2}$ and $f_{\mathrm{exc}}$ is shown on the left, and a line cut (black dashed line) is shown on the right. The line cut is fitted with Eq. (B6) to extract the framed parameters.

Figure 8

Alpha factor extraction for ICT 2. Phase response of the $LC$ resonator as a function of $V_{G2}$ and $f_{\mathrm{exc}}$ (with power calibration) is shown on the left. The scatter plot on the right retraces the positions of the central line as well as the right dip-peak structure from the left graph. The orange curve is a fit using Eq. (B7) that yields the framed results.

Figure 9

Charge photon coupling extraction for ICT 1. The experimental data (blue) are fitted (orange) with Eqs. (1) and (2) to extract the relevant parameters.

Figure 10

Charge photon coupling extraction for ICT 2. The experimental data (blue) are fitted (orange) with Eqs. (1) and (2) to extract the relevant parameters.

Figure 11

The $g$-factor extraction. Simulated response of the driven DQD system at finite magnetic field in the case of an even charge parity (top panel) and the corresponding schematic of the energy diagram (bottom panel) highlighting the transition energies $E_{f1}$ and $E_{f2}$ necessary for $g$-factor extraction.