Improving frequency selection of driven pulses using derivative-based transition suppression

Many techniques in quantum control rely on frequency separation as a means for suppressing unwanted couplings. In its simplest form, the mechanism relies on the low bandwidth of control pulses of long duration. Here we perform a higher-order quantum-mechanical treatment that allows for higher precision and shorter times. In particular, we identify three kinds of off-resonant effects: (i) simultaneous unwanted driven couplings (e.g., due to drive crosstalk), (ii) additional (initially undriven) transitions such as those in an infinite ladder system, and (iii) sideband frequencies of the driving wave form such as we find in corrections to the rotating-wave approximation. With a framework that is applicable to all three cases, in addition to the known adiabatic error responsible for a shift of the energy levels we typically see in the spectroscopy of such systems, we derive error terms in a controlled expansion corresponding to higher-order adiabatic effects and diabatic excitations. We show, by also expanding the driving wave form in a basis of different order derivatives of a trial function (typically a Gaussian) these different error terms can be corrected for in a systematic way, hence strongly improving quantum control of systems with dense spectra.

Figure 1

Absolute Fourier transform of a Gaussian pulse (solid blue line), its derivative (dotted yellow line), and their sum (dashed red line). Panel (a) is the first derivative (with $\sigma =3$); panel (b) is the second derivative (with $\sigma =2$).

Figure 2

Population inversion error [Eq. (5.7)] for two uncoupled qubits of energy difference $\Delta$. The dotted blue line shows the error using standard Gaussian shaping [Eq. (3.3)], while the solid red line uses the pulse shape given by DRAG Eqs. (5.3)–(5.6), which is an exact (infinite-order) solution for $T>2.5\pi /\Delta$. The dot-dashed orange line is the error when using the first-order shape given by the second-derivative solution [Eq. (5.12)]; the dashed green line corresponds to the first-order shape using the third derivative [Eq. (5.13)]; and the dot-dot-dashed black line is for the first-order, fourth-derivative solution [Eq. (5.14)].

Figure 3

Selection error [Eq. (5.7)] for three uncoupled qubits for energy differences from the first qubit of ${\Delta }_2$ and ${\Delta }_3=1.7{\Delta }_2$. The dotted blue line shows the combined inversion error using standard Gaussian shaping [Eq. (3.3)]. The solid red line gives the error for the solution using the first and second derivatives [Eq. (5.15)]; the dot-dashed orange line is for the first and third derivatives [Eq. (5.16)]; the dashed green line is for the second and third derivatives [Eq. (5.17)]; and the dot-dot-dashed black line is for the second and fourth derivatives [Eq. (5.18)].

Figure 4

Selection error [Eq. (5.7)] for four uncoupled qubits for energy differences ${\Delta }_2$, ${\Delta }_3=1.7{\Delta }_2$, and ${\Delta }_4=-1.3{\Delta }_2$ from the driven qubit. The dotted blue line shows the error using standard Gaussian shaping [Eq. (3.3)], while the solid red line is for the pulse shape given by the first-order DRAG solution [Eq. (5.19)].

Figure 5

Selection error as a function of frequency offset when using a Gaussian (dotted blue line) or Gaussian and its second derivative (solid red line).

Figure 6

For a three-level system driven by a $T=4\pi /\Delta$ with $\pi$ pulse, gate error is plotted vs leakage transition strength $\lambda$ for solutions to the adiabatic expansion to different orders $H^{D\left(h\right)}$, correcting for errors to order $h$. The solid blue line shows the error for a standard Gaussian [Eq. (3.3)]. Each line under that gives the error when correcting for the next order in the adiabatic expansion of the control operators. Panel (a) gives the solutions to different orders when no derivative controls are used. Panel (b) shows when the 0–1 transition is removed using a first derivative. Panel (c) is the error plotted when an additional perturbative control is used that includes the second derivative, which enables the removal of the 0–2 transition. How the different lines for each order are calculated is discussed in the text.

Figure 7

Error from counter-rotating terms in qubit frame. Gate error for Gaussian (dotted blue line), Gaussian with derivative (dot-dashed yellow line), and Gaussian with derivative and detuning (solid red line) are shown.