Second-Harmonic Coherent Driving of a Spin Qubit in a Si/SiGe Quantum Dot

P. Scarlino, E. Kawakami, D. R. Ward, D. E. Savage, M. G. Lagally, Mark Friesen, S. N. Coppersmith, M. A. Eriksson, and L. M. K. Vandersypen

Phys. Rev. Lett. 115, 106802 – Published 1 September 2015

Abstract

We demonstrate coherent driving of a single electron spin using second-harmonic excitation in a Si/SiGe quantum dot. Our estimates suggest that the anharmonic dot confining potential combined with a gradient in the transverse magnetic field dominates the second-harmonic response. As expected, the Rabi frequency depends quadratically on the driving amplitude, and the periodicity with respect to the phase of the drive is twice that of the fundamental harmonic. The maximum Rabi frequency observed for the second harmonic is just a factor of 2 lower than that achieved for the first harmonic when driving at the same power. Combined with the lower demands on microwave circuitry when operating at half the qubit frequency, these observations indicate that second-harmonic driving can be a useful technique for future quantum computation architectures.

Received 24 April 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.106802

Authors & Affiliations

P. Scarlino¹, E. Kawakami¹, D. R. Ward², D. E. Savage², M. G. Lagally², Mark Friesen², S. N. Coppersmith², M. A. Eriksson², and L. M. K. Vandersypen^1,*

¹Kavli Institute of Nanoscience, TU Delft, Lorentzweg 1, 2628 CJ Delft, Netherlands
²University of Wisconsin-Madison, Madison, Wisconsin 53706, USA

^*Corresponding author. L.M.K.Vandersypen@tudelft.nl

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 115, Iss. 10 — 4 September 2015

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Measured resonance frequencies as a function of externally applied magnetic field $B_{ext}$ . The long microwave burst time $t_{p} = 700 μ s ≫ T_{2}^{*}$ means that the applied excitation is effectively continuous wave. The microwave source output power was $P = - 33 dBm$ to $- 10 dBm$ ( $- 20 dBm$ to $- 5 dBm$ ) for the case of fundamental (second) harmonic excitation, decreasing for lower microwave frequency in order to avoid power broadening. The red and green lines represent fits with the relation $h f = g μ_{B} \sqrt{(B_{ext} - B_{∥})^{2} + B_{⊥}^{2}}$ , respectively, to the resonance data labeled (2) and (3) (we excluded points with $B_{ext} < 700 mT$ from the fit because the micromagnet apparently begins to demagnetize there) [30]. (b) Schematic of the energy levels involved in the excitation process, as a function of the total magnetic field at the electron location. The dashed arrows correspond to the four transitions in panel (a), using the same color code. (c) Schematic of an anharmonic confinement potential, leading to higher harmonics in the electron oscillatory motion in response to a sinusoidally varying excitation. (d) Measured spin-up probability, $P_{↑}$ , as a function of applied microwave frequency, $f_{MW}$ , for $B_{ext} = 560.783 mT$ ( $P = - 30 dBm$ for the fundamental response, $P = - 12 dBm$ for the second harmonics), averaged over 150 repetitions per point times 80 repeated frequency sweeps (160 mins in total). The frequency axis (in red on top) has been stretched by a factor of 2 for the second-harmonic spin response (red data points). From the linewidths, we extract a lower bound for the dephasing time $T_{2}^{* (1)} = 760 \pm 100 ns$ , $T_{2}^{* (2)} = 810 \pm 50 ns$ , $T_{2}^{* (3)} = 750 \pm 40 ns$ and $T_{2}^{* (4)} = 910 \pm 80 ns$ . The Gaussian fits through the four peaks use the same color code as in panels (a) and (b).
Reuse & Permissions
Figure 2
Rabi oscillations. (a) Measured spin-up probability, $P_{↑}$ , as a function of microwave burst time ( $B_{ext} = 560.783 mT$ , $f_{MW} = 6.4455 GHz$ ) at four different microwave powers, corresponding to a rms voltage at the source of 998.8 mV, 1257.4 mV, 1410.9 mV, 1583.0 mV. (b) Rabi frequencies recorded at the fundamental harmonic, $f_{0}^{(1)}$ (blue triangles, adapted from [30]), and at the second harmonic, $f_{0}^{(3)}$ (green squares), as a function of the microwave amplitude emitted from the source (top axis shows the corresponding power). For the second harmonic, the amplitude shown corresponds to a 5 dB higher power than the actual output power, to compensate for the 5 dB lower attenuation of the transmission line at 6 GHz versus 12 GHz (estimated by measuring the coax transmission at room temperature). The green solid (dashed black) line is a fit of the second-harmonic data with the relation $\log (f^{R}) \propto 2 \log (E_{ac})$ [ $\log (f^{R}) \propto \log (E_{ac})$ ]. The large error bars in the FFT of the data in Fig. 2 arise because we perform the FFT on only a few oscillations. $B_{ext} = 560.783 mT$ .
Reuse & Permissions
Figure 3
Phase control of oscillations. (a) Probability $P_{↑}$ measured after applying two $π / 2$ rotations via second-harmonic excitation, as a function of the relative phase between the two microwave bursts, $Δ ϕ$ . The two rotations are separated by $τ = 100 ns$ (black) and $τ = 2 μ s$ (red). ( $P = 16.0 dBm$ , $B_{ext} = 560.783 mT$ , $f_{MW} = f_{0}^{(3)} = 6.44289 GHz$ ). (b) Similar to panel (a), but now driving the fundamental harmonic for $τ = 20 ns$ (black) and $τ = 2 μ s$ (red). ( $P = 12.0 dBm$ , $B_{ext} = 560.783 mT$ , $f_{MW} = f_{0}^{(2)} = 12.885 77 GHz$ ). Inset: Microwave pulse scheme used for this measurement. (c) Measured spin-up probability, $P_{↑}$ (1000 repetitions for each point), as a function of $f_{MW}$ and the relative phase $Δ ϕ$ between two $π / 2$ microwave bursts (130 ns, $P = 16.0 dBm$ ) for second-harmonic excitation, with $τ = 50 ns$ . The measurement extends over more than 15 hours.
Reuse & Permissions
Figure 4
Ramsey fringes. (a) Measured spin-up probability, $P_{↑}$ , as a function of $f_{MW}$ and waiting time $τ$ ( $B_{ext} = 560.783 mT$ , $P = 13.0 dBm$ ) between two $π / 2$ pulses (130 ns) with equal phase, showing Ramsey interference. Each data point is averaged over 300 cycles. Inset: Microwave pulse scheme used for this measurement. (b) Fourier transform over the waiting time, $τ$ , of the data in panel (a), showing a linear dependence on the microwave frequency, with vertex at $f_{MW} = f_{0}^{(3)}$ and slope $f_{Ramsey} = 2 Δ f_{MW}$ (black dashed lines). The expected position of the FFT of the signal arising from resonance $f_{0}^{(4)}$ is indicated by the dotted black line. For comparison, the white dashed line represents the relation $f_{Ramsey} = Δ f_{MW}$ . (c)–(e) Sections of the Ramsey interference pattern in (a) along the three white dashed lines; the respective waiting times are indicated also in the inset of each panel. (f) Measured spin-up probability as a function of the total free evolution time, $τ$ , in a Hahn echo experiment (pulse scheme shown in inset). The decay curve is fit well to a single exponential (blue). Here, $f_{MW} = f_{0}^{(3)}$ , $B_{ext} = 560.783 mT$ .
Reuse & Permissions

Physical Review Letters