Coherent quantum oscillations and echo measurements of a Si charge qubit

Fast quantum oscillations of a charge qubit in a double quantum dot fabricated in a Si/SiGe heterostructure are demonstrated and characterized experimentally. The measured inhomogeneous dephasing time $T_2^{*}$ ranges from 127 ps to 2.1 ns; it depends substantially on how the energy difference of the two qubit states varies with external voltages, consistent with a decoherence process that is dominated by detuning noise (charge noise that changes the asymmetry of the qubit's double-well potential). In the regime with the shortest $T_2^{*}$, applying a charge-echo pulse sequence increases the measured inhomogeneous decoherence time from 127 to 760 ps, demonstrating that low-frequency noise processes are an important dephasing mechanism.

Figure 1

Larmor oscillations. (a) SEM image of a device identical to the one used in the experiment. The current $I_{\mathrm{QPC}}$ is used for charge sensing through a measurement of its transconductance $G_{\mathrm{L}}=\partial I_{\mathrm{QPC}}/\partial V_{\mathrm{L}}$, and voltage pulses are applied to gate L. (b) (Bottom) Diagram of energy levels vs detuning $\varepsilon$, showing the anticrossing between (2,1) and (1,2) charge configurations. The red arrow represents an applied voltage pulse with the dc level and amplitude of the pulse chosen to place the peak of the pulse at $\varepsilon =0$. (Top) Example pulse showing the pulse peak position ${\varepsilon }_{\mathrm{p}}$, pulse duration $t_{\mathrm{p}}$, and the measurement phase. (c) Larmor oscillations between the (2,1) and (1,2) charge configurations, as a function of pulse duration $t_{\mathrm{p}}$ and pulse peak position ${\varepsilon }_{\mathrm{p}}$. The Larmor oscillations that reflect rotations between the states with (2,1) and (1,2) charge occupations are manifest near ${\varepsilon }_{\mathrm{p}}=0$. (d) Numerical simulation of the Larmor oscillations of (c), using an 80-ps-pulse rise time and the energy level diagram in (b) with best fit parameter $\Delta =10.8$ $\mu$eV ($\Delta /h=2.62$ GHz). (e) Solid red: a horizontal cut taken near zero detuning from the integrated data of (c), showing Larmor oscillations as a function of pulse duration. Dashed black: a corresponding cut from the simulated charge occupation data.

Figure 2

(a) Ramsey fringes: QPC transconductance $G_{\mathrm{L}}$ as a function of the base level of detuning ${\varepsilon }_{\mathrm{b}}$ and the time $\tau$ between two $5\pi /2$ pulses, as shown in the inset. The oscillations observed in the region where $\tau >280$ ps reflect the rotation of the Bloch vector around the $z$ axis in the $x-y$ plane. (b) Gray: a line cut of the integrated data of (a) after removal of a smooth background, as described in Appendix . Red: fit to the form $A\exp [-{(\tau -{\tau }_0)}^2/T_2^{*2}]\cos (\omega t+\varphi )+C$, which yields $T_2^{*}=127\pm 8$ ps. (c) Charge echo: QPC transconductance $G_{\mathrm{L}}$ as a function of ${\varepsilon }_{\mathrm{b}}$ and $\delta t$ for $t=640$ ps. Oscillations are strongest near $\delta t=0$, where equal free evolution times before and after the $3\pi$ $x$ rotation provide the best correction for slow inhomogeneous dephasing. Away from $\delta t=0$, increasingly mismatched free evolution times provide less correction, and the oscillations decay with characteristic time $T_2^{*}$. (Inset) Trace of the pulse sequence used to acquire these data. (d) Sequence of pulses used to implement charge echo: a nominal $5\pi /2$ $x$ rotation into the $x-y$ plane, free evolution for a time $t/2+\delta t$, a $3\pi$ $x$ rotation, free evolution time $t/2-\delta t$, and a $5\pi /2$ $x$ rotation out of the $x$-$y$ plane. For this experiment, $5\pi /2$ pulses have a duration of 280 ps and $3\pi$ pulses have a duration of 330 ps. (e) Dark circles: charge oscillation amplitudes $\Delta P_{(1,2)}$ as a function of the free evolution time $t$, extracted from data sets with different $t$ values. The echo amplitude decays as $t$ increases. A fit of the decay to the Gaussian form $y_0+A\exp [-{(t/T_2)}^2]$ yields $T_2=760\pm 190$ ps, significantly longer than $T_2^{*}$. Thus the echo pulse sequence corrects for slow inhomogeneous dephasing and extends the coherence time.

Figure 3

Ramsey fringe analysis. (a) Integration of the transconductance $G_{\mathrm{L}}$ from Fig. 2a of the main text yields the probability $P_{(}1,2)$ of occupying the (1,2) charge state in the regime where the (2,1) charge state is the ground state. The data are normalized by noting that the total charge transferred across the polarization line is one electron. The red dashed box indicates the location of the fringes. (b) Dark gray: line cut of the data in (a), at the detuning $\varepsilon =-120\mu$eV. Light gray: Smooth background that is subtracted from the line cut before fitting the data to a damped sinusoidal form. (c) [The same as Fig. 2b in the main text.] Gray: the data from (b) after subtraction of the smooth background. Red: fit to the form $A\exp [-{(\tau -{\tau }_0)}^2/T_2^{*2}]\cos (\omega t+\varphi )+C$, which yields $T_2^{*}=127\pm 8$ ps.

Figure 4

Analysis of echo data for extraction of the decoherence time $T_2$. (a)–(c) Transconductance $G_L$ as a function of the base level of detuning ${\varepsilon }_{\mathrm{b}}$ and $\delta t$ (defined in the main text) for total free evolution times of $t=390$, 690, and 990 ps, respectively. (d)–(f) Fourier transforms of the charge occupation $P_{(1,2)}$ as a function of detuning ${\varepsilon }_{\mathrm{b}}$ and oscillation frequency $f$ for the data in (a)–(c), respectively. We obtain $P_{(1,2)}$ (not shown here) by integrating the transconductance data in (a)–(c) and normalizing by noting that the total charge transferred across the polarization line is one electron. Fast Fourier transforming the time-domain data of $P_{(1,2)}$ allows us to quantify the amplitude of the oscillations visible near $\delta t=0$. The oscillations of interest appear as weight in the FFT that moves to higher frequency at more negative detuning (farther from the anticrossing). For an individual detuning energy, the FFT has nonzero weight for a nonzero bandwidth. (g) Echo amplitude as a function of free evolution time $t$. The data points (dark circles) are obtained at ${\varepsilon }_{\mathrm{b}}=-120$ $\mu$eV by integrating a horizontal line cut of the FFT data over a bandwidth range of 46–72 GHz, then normalizing by the echo oscillation amplitude of the first data point, as described in the text. The echo oscillation amplitudes, plotted for multiple free evolution times, decay with characteristic time $T_2$ as the free evolution time $t$ is made longer. By fitting the decay to a Gaussian, we obtain $T_2=760\pm 190$ ps. (h)–(j) Fourier transforms of the transconductance $G_L$ as a function of ${\varepsilon }_{\mathrm{b}}$ and oscillation frequency $f$ for (a)–(c), respectively. As $t$ is increased the magnitude for oscillations at a given frequency decays with characteristic time $T_2$. We take the magnitude of the FFT at the point where the central feature (black line) intersects 65 GHz. (k) Measured FFT magnitudes at 65 GHz for multiple free evolution times (dark circles) with a Gaussian fit (red line), which yields $T_2=620\pm 140$ ps, in reasonable agreement with the result shown in (g).