Quantum Interferometry with a $g$-Factor-Tunable Spin Qubit

We study quantum interference effects of a qubit whose energy levels are continuously modulated. The qubit is formed by an impurity electron spin in a silicon tunneling field-effect transistor, and it is read out by spin blockade in a double-dot configuration. The qubit energy levels are modulated via its gate-voltage-dependent $g$ factors, with either rectangular, sinusoidal, or ramp radio frequency waves. The energy-modulated qubit is probed by the electron spin resonance. Our results demonstrate the potential of spin qubit interferometry implemented in a silicon device and operated at a relatively high temperature.

Figure 1

High-temperature TFET-based single-spin qubit. (a) Schematic of the device and measurement set up. The transistor is defined on a SOI structure with $n$-type source and $p$-type drain electrodes. The channel length and width are 80 nm and $10\mu \mathrm{m}$, respectively. The source-drain current $I_{SD}$ of the device is measured for the source-drain voltage $V_{SD}$; the gate voltage $V_G$ at temperature $T=1.6\mathrm{K}$ is achieved with a pumped ${}^4\mathrm{He}$ cryostat. A magnetic field $B$ is directed along the source-drain current. A microwave (MW) signal is applied on the substrate. A gate-voltage modulation in the MHz regime is applied through a high-pass block capacitor with cutoff frequency ($<10\mathrm{kHz}$) much smaller than the modulation frequency ($\sim \mathrm{MHz}$). (b) Schematic of the potential landscape of the device. (c) Schematic of the single-electron tunneling cycle in the spin-blockade regime; see the Supplemental Material [41] for details.

Figure 2

Square wave modulation of the spin qubit. (a) Shape of the rf signal. (b),(c), and (d) The source-drain current $I_{SD}$ of the device with a square modulation of its full amplitude 32 mV added to the gate at $V_G=-0.36\mathrm{V}$ through the high-pass block capacitor. Other conditions are the same as in Fig. 1. In panel (c) we present the intensity plot of $d{I}_{SD}/df$ vs the frequency $f$ and the square-wave modulation frequency $\mathrm{\Omega }$ (log scale from 0.5 to 50 MHz), showing the evolution from the two ESR peaks into the strong main ESR peak and weak sideband peaks. Note that the distance between the main and the sideband peaks (seen in the upper area $>10\mathrm{MHz}$) is linear in the modulation frequency $\mathrm{\Omega }$. Panels (b) and (d) present the source drain current $I_{SD}$ at modulation frequencies of 50 and 0.5 MHz, respectively. (e), (f), and (g) Calculation of the upper-state population of the qubit under square-wave modulation of its energy. For calculations, the following parameters were used for all the graphs: $G/2\pi =1.1\mathrm{MHz}$, ${\mathrm{\Gamma }}_1/2\pi =0.2\mathrm{MHz}$, ${\mathrm{\Gamma }}_2/2\pi =1\mathrm{MHz}$, and also for the right panels (e)–(g) here: $\delta /2\pi =24\mathrm{MHz}$.

Figure 3

Weighted averaging for the spin qubit vs the duty ratio $d$. (a) Shape of a square wave with $d=20\%$. (b)–(d) Similar plot of $I_{SD}$ as in Figs. 2 but with a square wave with a 20% duty ratio, which corresponds to $d=0.2$. The modulation frequency is changed from 0.25 to 25 MHz in a log scale. (e)–(g) Calculation of the qubit upper-state population under square-wave modulation ($d=0.2$) of its energy. Besides the duty ratio, other parameters are the same as in Figs. 2. Note that the main weighted averaged peak appears again at $f=f_0$ (here $f_0=9.02\mathrm{GHz}$), independent of the duty ratio. (h) Duty ratio dependence of the ESR peak heights at $\mathrm{\Omega }/2\pi =0.25\mathrm{MHz}$ for lower frequency, $\mathrm{\Delta }{I}_L$, and higher frequency, $\mathrm{\Delta }{I}_H$. The $\mathrm{\Delta }{I}_L$ and $\mathrm{\Delta }{I}_H$ for $d=0.2$ are indicated in (d). (i) The duty-ratio dependence among three ESR peaks, i.e., the motional averaged main peak at $\mathrm{\Omega }/2\pi =25\mathrm{MHz}$ ($f=9.021\text{GHz}$), to the ESR peaks of lower or higher frequency at $\mathrm{\Omega }/2\pi =0.25\mathrm{MHz}$. The distances between the peaks, $\mathrm{\Delta }{f}_L$ and $\mathrm{\Delta }{f}_H$, are indicated in (e) for $d=0.2$. (j) Duty-ratio dependence of the contrasts for the peak heights and distances.

Figure 4

Sinusoidal and ramp modulation. (a) Shape of the rf sinusoidal signal. (b) Source-drain current $I_{SD}$ under sinusoidal modulation with amplitude 24 mV. Intensity plot of $d{I}_{SD}/df$ vs the frequency $f$ and the modulation frequency $\mathrm{\Omega }$ (linear scale from 0 to 80 MHz). (c) Calculation of the upper-state population of the qubit under the sinusoidal modulation of its energy. The parameters are the same as for Fig. 2 besides the amplitude, which here is $\delta /2\pi =30\mathrm{MHz}$. (d) Shape of the ramp wave signal. (e) Source-drain current $I_{SD}$ under the ramp modulation with amplitude 36 mV. Intensity plot of $d{I}_{SD}/df$ as a function of the frequency $f$ and the modulation frequency $\mathrm{\Omega }$ (log scale from 0.25 to 25 MHz). (f) Calculation of the upper-state population of the qubit under ramp modulation of its energy. The parameters are the same as for Fig. 2 besides the amplitude, which here is $\delta /2\pi =27\mathrm{MHz}$.