Real-Time Feedback Control of Charge Sensing for Quantum Dot Qubits

Measurement of charge configurations in few-electron quantum dots is a vital technique for spin-based quantum information processing. While fast and high-fidelity measurement is possible by using proximal quantum dot charge sensors, their operating range is limited and prone to electrical disturbances. Here we demonstrate real-time operation of a charge sensor in a feedback loop to maintain its sensitivity suitable for fast charge sensing in a $\mathrm{Si}$/${\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}$ double quantum dot. Disturbances to the charge sensitivity, due to variation of gate voltages for operating the quantum dot and $1/f$ charge fluctuation, are compensated by a digital proportional-integral-differential controller with the bandwidth of approximately equal to $100\mathrm{kHz}$. The rapid automated tuning of a charge sensor enables unobstructed charge-stability-diagram measurement facilitating real-time quantum dot tuning and submicrosecond single-shot spin readout without compromising the performance of a charge sensor in time-consuming experiments for quantum information processing.

Figure 1

Schematic of the experimental setup. The $\mathrm{Si}$/${\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}$ QD sample is mounted on the mixing chamber plate of the dilution refrigerator, where the measured electron temperature is 50 mK (false-colored scanning electron microscope image). The charge arrangement of the electron spin qubits confined in the DQD (white circles) is controlled by the voltages $V_{L,R}$ applied to the plunger gates (blue), and it is detected by the QD charge sensor (yellow circle). The rf carrier reflected from the $LC$ resonant circuit is sampled by the M3300A digitizer and processed by the digital PID controller. The control voltage $u(t-\tau )$ is combined with the dc voltage $V_S$ and applied to the sensor plunger gate (yellow).

Figure 2

Stability diagrams measured with the feedback control turned off (a)–(c) and on (d)–(f). The values of $V_L$ and $V_R$ are offset from their values at the center of each diagram. Signal is integrated for $14\mu \mathrm{s}$ at each data point and the whole diagram is captured at a refresh rate of about $4$ frames per second. (a),(b) Reflectometry signal $v_m$ as a function of $V_L$ and $V_R$ (a) and its derivative (b). (c) Illustration of the typical sensor signal trace during the stability-diagram measurement. The sensor is sensitive to the charge arrangement only near the Coulomb peaks while it is mostly blind in the Coulomb blockade regime (red shaded region). (d),(e) Control voltage $u$ as a function of $V_L$ and $V_R$ (d) and its derivative (e), taken while the feedback control is active. The PID control is reset ($t=0$) at the lower left corner and $V_{L,R}$ are swept upward. Red arrows in (d) denote jumps from one Coulomb peak to another in the charge sensor. (f) Illustration of the sensor signal traces as functions of $u$ for different values of $V_R$. Yellow circles show the stable points toward which $u$ is controlled to satisfy $v_m=v_s$. As $V_R$ is increased, the signal trace shifts to the left (from black to green) and the value of $u$ at the stable point decreases. The Coulomb peak height reduces as $V_R$ is increased further (from green to brown), leading to the jumps of the stable point as indicated by the red arrows in (d).

Figure 3

Performance characterization of the feedback control. (a) Tuned-up step response of the charge sensor. A step waveform is applied to $V_R$ (upper panel) and the responses of $v_m$ (middle panel) and $u$ (lower panel) are recorded. The traces are averaged for 1000 trials. (b) Noise PSDs of the charge-sensor signal $v_m$. When the feedback is off (blue), we observe the typical $1/f$ tail due to the conductance fluctuation of the charge sensor along with a 50-Hz peak of the power-line cycles. These noise components are suppressed by using the feedback control (orange) up to frequencies above 100 kHz.

Figure 4

Single-shot spin measurement with the feedback-controlled charge sensor. (a) Pulse cycle for the spin measurement using PSB [24]. Solid black lines show the DQD charge transitions. The electron in the left QD is emptied ($E$), reloaded ($R$), measured ($M$), and parked ($P$) at the points marked by yellow circles. The dwell times at these points are $10\mu \mathrm{s}$ ($E$), $10\mu \mathrm{s}$ ($R$), $445\mu \mathrm{s}$ ($M$), and $35\mu \mathrm{s}$ ($P$), respectively. PSB is lifted outside the gray shaded region. The PID control is enabled or disabled at point $P$, while it is disabled elsewhere. (b) Histograms of two sets of the single-shot measurement carried out in succession without the feedback control (the PID control is disabled at point $P$). Each histogram is constructed from 100 000 measurement outcomes taken by integrating $v_m$ for $t_{\mathrm{int}}=5\mu \mathrm{s}$. The left (right) peak shows the outcomes indicating the $(1,1)$ [$(0,2)$] charge arrangement that corresponds to spin triplet (singlet). The peak positions $v_{11}$ and $v_{02}$ fluctuate with time. (c) Time traces of $v_{02}$ taken by turning on (orange) or off (blue) the feedback control. The values of $v_{02}$ are extracted by fitting the histograms similar to those in (b) [25]. (d) SNRs of the single-shot measurement as functions of the integration time $t_{\mathrm{int}}$ with the feedback control on (orange circles) or off (blue squares). Solid curves are the fits with ${\sigma }_0=1.81\pm 0.01{\mathrm{Vs}}^{-1/2}$ and $t_0=0.03\pm 0.01\mu \mathrm{s}$ (see the main text).