A dressed singlet-triplet qubit in germanium

Abstract

In semiconductor hole spin qubits, low magnetic field ($B$) operation extends the coherence time ($T_\mathrm{2}^*$) but proportionally reduces the gate speed. In contrast, singlet-triplet (ST) qubits are primarily controlled by the exchange interaction ($J$) and can thus maintain high gate speeds even at low $B$. However, a large $J$ introduces a significant charge component to the qubit, rendering ST qubits more vulnerable to charge noise when driven. Here, we demonstrate a highly coherent ST hole spin qubit in germanium, operating at both low $B$ and low $J$. By modulating $J$, we achieve resonant driving of the ST qubit, obtaining an average gate fidelity of $99.68\%$ and a coherence time of $T_\mathrm{2}^*=1.9\,μ$s. Moreover, by applying the resonant drive continuously, we realize a dressed ST qubit with a tenfold increase in coherence time ($T_\mathrm{2ρ}^*=20.3\,μ$s). Frequency modulation of the driving signal enables universal control, with an average gate fidelity of $99.63\%$. Our results demonstrate the potential for extending coherence times while preserving high-fidelity control of germanium-based ST qubits, paving the way for more efficient operations in semiconductor-based quantum processors.

Figures (9)

• Figure 1: Germanium hole ST qubit device.a. A false-coloured tilted scanning electron microscope image of a nominally identical six QD device to the one used in the experiments. Inset: Schematic of the device cross section along the dotted black line in (a). b. Double QD charge stability diagram as a function of vP1 and vP2 with the different charge configurations in QD1 and QD2 indicated by ($N_1, N_2$). The points and arrows describe the pulse sequence used in the experiment. L: Reset with one hole loaded in QD1. I: Loading of second hole and initialization of the two-spin system in the S(2,0) state. T: Turning point before and after crossing the interdot transition. M: Manipulation point where all control pulses are applied. With $\varepsilon$ we denote the detuning from the interdot in units of its vP1 coordinate. R: The latched readout point, resulting in the $|{\downarrow\downarrow}\rangle$ state being distinguished from all other spin states $|\widetilde{\downarrow\uparrow}\rangle$, $|\widetilde{\uparrow\downarrow}\rangle$, $|{\uparrow\uparrow}\rangle$. c. Spectroscopy of transitions from the $|{\downarrow\downarrow}\rangle$ state inside the (1,1) region as a function of $\varepsilon$, measured with a chirped pulse applied to vP2 around the frequency ($f_\mathrm{chirp}$). Two transition lines are visible that are interpreted as the $|{\downarrow\downarrow}\rangle$$\leftrightarrow$$|\widetilde{\downarrow\uparrow}\rangle$ and $|{\downarrow\downarrow}\rangle$$\leftrightarrow$$|\widetilde{\uparrow\downarrow}\rangle$ transitions. d. Rabi chevron patterns of Q1 and Q2 with the drive signals applied to vP1 and vP2, respectively. $\tilde{P}$ is calibrated using the average current values in the (2,1) and (1,1) regions. e. Step-by-step illustration of the dressing procedure for the ST qubit, depicted through Bloch spheres at each stage. (i) Resonantly driven ST qubit in the lab frame, with a frequency $f_\mathrm{ST}$ and a drive term $J_\mathrm{AC}\,(t)$. (ii) Applying a rotating wave approximation (RWA) and transitioning to the rotating frame removes the time dependence of $J_\mathrm{AC}$ along with $f_\mathrm{ST}$. (iii) Transformed basis states where $\Omega_\mathrm{ST}$ points along the poles. The driving term $\Delta\nu$ emerges from either a second tone or a detuning in $\Omega_\mathrm{ST}$. (iv) The application of a second RWA results in the dressed ST frame where the time dependence of the new drive along with $\Omega_\mathrm{ST}$ have been removed.
• Figure 2: Resonantly-driven singlet-triplet qubit.a. Rabi chevron pattern with a drive signal on vB12 of duration $t_\mathrm{d}$ and frequency $f_\mathrm{d}$. Initialization and readout of the $|\widetilde{\uparrow\downarrow}\rangle$ state is described in the main text. $\widetilde{P}$ is normalized with the application of either an $\mathrm{I^{Q1}}$ or an $\mathrm{X^{Q1}_\pi}$ before the measurement sequence. Above each plot, the corresponding pulse sequence or circuit diagram is illustrated. b. Dependence of the Rabi frequency, $\Omega_\mathrm{ST}$, on the drive amplitude $A$. The divergence from linearity (white dashed line) seen in the fast Fourier transform (inset) stems from the exponential dependence of the exchange to vB12. c. Demonstration of rotation axis control via the addition of a phase $\phi$. An $\mathrm{X^{ST}_{\pi/2}}$ pulse initializes the system in a state pointing along the Y axis of the Bloch sphere. The Rabi oscillations disappear for $\phi=\pm \pi/2$, indicating that the drive axis and state align. d. Randomized benchmarking of the single-qubit gates in the singlet-triplet subspace. We extract an average gate fidelity of $\mathcal{F} = 99.68(2)\%$. e. Ramsey (above) and Hahn echo (below) experiments fitted to a decaying exponential, with $T_2^\mathrm{*}=1.9\,\mathrm{\mu{s}}$ and $T_2^\mathrm{H}=4.2\,\mathrm{\mu{s}}$ respectively. f. Exponential decay fit of the Rabi oscillation with $T_2^\mathrm{R}=20.3\,\mathrm{\mu{s}}$.
• Figure 3: Dressed singlet-triplet qubit via resonant exchange.a. Two-tone spectroscopy of the dressed ST qubit with a pump and a probe signal applied for a duration of $3\,\mu$s. Varying the pump signal amplitude ($A_\textrm{pump}$) results in Rabi oscillations in the {$|\widetilde{\uparrow\downarrow}\rangle$, $|\widetilde{\downarrow\uparrow}\rangle$} subspace. When the frequency of the probe signal $f_\textrm{probe}$ is resonant with $f_\textrm{ST}$ or $f_\textrm{ST}\pm\Omega_\mathrm{ST}$, the measured Rabi oscillation pattern is disrupted. The two signals have a phase difference of $\pi/4$. This choice was made to achieve maximum visibility of all three branches. Above each plot, the corresponding pulse sequence or circuit diagram is illustrated. b. A frequency modulated driving signal gives rise to an oscillating (with frequency $f_\mathrm{FM}$) driving term that enables rotations in the dressed ST subspace. The state $|\widetilde{\mathrm{S}}_\mathrm{\rho}\rangle$ is initialized, then rotated by FM modulation of the drive signal before being projected to $|\widetilde{\mathrm{T}}_\mathrm{0\rho}\rangle$ for readout. (i) FM driven Rabi chevron pattern. (ii) Fourier transform of the amplitude ($\mathrm{\Delta}\nu_\mathrm{FM}$) dependence of the dressed Rabi frequency $\Omega^\mathrm{FM}_\mathrm{ST}$ (inset), plotted in arbitrary units (a.u.). c. (Top) Measurement of the dressed qubit decay time $T_\mathrm{1}{}_\rho=0.8$ ms. (Bottom) Hahn echo experiment to estimate the dressed qubit echo coherence time $T_{\mathrm{2}\mathrm{\rho}}^\mathrm{H}=56\,\mu$s, where $t_\mathrm{w,\,FM}$ is the free evolution time not including the duration of $\mathrm{X^{FM}_{\pi}}$. d. A fidelity of $\mathcal{F}=99.63(7)\%$ is extracted for the FM dressed qubit gates via randomized benchmarking performed for both $|\widetilde{\mathrm{S}}_\mathrm{\rho}\rangle$ and $|\widetilde{\mathrm{T}}_\mathrm{0\rho}\rangle$ initial states. e. Comparison between the performance metrics of the resonantly-driven and dressed ST qubits.
• Figure S1: Initialization and readout.a. The $|{\downarrow\downarrow}\rangle$ state is initialized at point 1. Next we apply an identity I (do nothing for no time) (left panel) or an $\mathrm{X}^\mathrm{Q1}_\mathrm{\pi}$ pulse (right panel) to Q1, then cross to point 2 (point T in the main text), then sweep the readout point R. We observe a difference in $I_\mathrm{sensor}$ in a region above the (2,1) charge configuration, in the middle of which we place the R point. The ramp time between points 1$\leftrightarrow$2 was set at $t_\mathrm{ramp}=1\,\mu\text{s}$. Current is plotted in arbitrary units (a.u.). b. Energy diagram of the DQD spin system. The red arrow indicates the desired initialization (and readout) process, with a super slow adiabatic passage through the $\mathrm{S(2,0)}$-$\mathrm{T_-(1,1)}$ ensuring initialization (and readout) of the $|{\downarrow\downarrow}\rangle$ state. c. Spin oscillations originating from a square detuning pulse, mapping out the $\mathrm{S(2,0)}$-$\mathrm{T_-(1,1)}$ energy difference. d. By introducing a ramp at the beginning of the (2,0)-(1,1) detuning pulse, we observe that the oscillations disappear, indicating successful slow adiabatic passage through the $\mathrm{S(2,0)}$-$\mathrm{T_-(1,1)}$ anticrossing. e. By applying a chirped drive pulse, and adding a ramp in the return pulse to the (2,0), we confirm the selection rules of the spin-to-charge conversion process. By operating with a symmetric (in and out) ramp time of $1\,\mu$s, we ensure initialization and readout of the $|{\downarrow\downarrow}\rangle$ state. f. Monitoring the spin transition frequency (via a chirped pulse) while varying $B$ allows us to extract the g-factor of each spin.
• Figure S2: Single spin initialization and readout fidelity.a. Initialization and readout sequences of the $|\widetilde{\uparrow\downarrow}\rangle$ state. b. Calibration of the timing of single qubit gates. (Upper) Drive pulse duration versus the number of consecutive $\pi/2$ pulses applied. (Lower) Fourier transform revealing the oscillation speed, in units of number of $\pi$/2 pulses, from where the duration of a full and half rotation can be extracted. Current is plotted in arbitrary units (a.u.). c. Initialization and readout (SPAM) fidelity estimation after performing and I and X pulse on Q1.