Single qubit operations using microwave hyperbolic secant pulses

It has been known since the early days of quantum mechanics that hyperbolic secant pulses possess the unique property that they can perform full-cycle Rabi oscillations on two-level quantum systems independently of the pulse detuning. More recently, it was realized that they induce detuning-controlled phases without changing state populations. Here, we experimentally demonstrate the properties of hyperbolic secant pulses on superconducting transmon qubits and contrast them with the more commonly used Gaussian and square waves. We further show that these properties can be exploited to implement phase gates, nominally without exiting the computational subspace. This enables us to demonstrate a microwave-driven $Z$ rotation with a single control parameter, the detuning.

Figure 1

Three different excitation profiles for sech (solid red), Gaussian (dotted green), and square (dashed blue) pulse shapes. The sech and Gaussian pulses shown here are truncated at $t=\pm 4\sigma$, where $\sigma$ is the standard deviation of the pulse. For the sech pulse, the standard deviation $\sigma$ is related to its bandwidth $\rho$ by $\sigma =\pi /\left(2\rho \right)$.

Figure 2

The excited-state probability as a function of pulse amplitude (vertical axis) and detuning (horizontal) for (a) sech, (b) Gaussian, and (c) square pulses. The left and right panels compare experimental and theoretical simulations. The simulations include the four lowest energy levels of the transmon and decoherence process $T_1=T_2=10\mu \mathrm{s}$.

Figure 3

Line cuts at various detunings vs pulse amplitude from Fig. 2.

Figure 4

The pulse amplitude for the first full Rabi cycle plotted vs the drive detuning for sech (solid red), Gaussian (dotted green), and square (dashed blue) pulse shapes.

Figure 5

Tomography of a single-qubit phase gate, $Z_{\mathrm{sech}}(\mathrm{\Delta }$), for a state prepared by a $\pi /2$ rotation. Panel (a) shows the pulse sequence and panel (b) shows the theory predicted $\theta$ (dashed cyan line) and $\varphi$ (solid magenta line) and the measured $\theta$ (blue square) and $\varphi$ (red circle), where $\theta$ and $\varphi$ are the polar and azimuthal angles of the resulting Bloch vector of the qubit state.

Figure 6

The average fidelity for the sequence in Fig. 5 averaged over the six different initial states prepared as described in the text.

Figure 7

Optical microscope image of the device.

Figure 8

Color scale of readout resonator response [(a) magnitude (b) phase] vs frequency ($x$ axis) and power ($y$ axis).

Figure 9

Readout resonator response [(a) magnitude (b) phase] vs qubit drive frequency showing the qubit energy spectra for high (red, lower line) and low (blue, upper line) driver powers.

Figure 10

(a) Energy decay, $T_1$ of qubit. (b) Histogram of multiple $T_1$ measurements over 1 h.

Figure 11

(a) Phase decoherence, $T_2^{*}$ of qubit. (b) Histogram of multiple $T_2^{*}$ measurements over 1 h.

Figure 12

(a) Schematic for the generation of the pulse-shaped microwave drive. Drive 1 is the $Z_{\mathrm{sech}}\left(\mathrm{\Delta }\right)$ gate and drive 2 is for the state preparation and tomography rotations. (b) Schematic for the generation of the pulse-shaped microwave drive. Drive 1 is the $Z_{\mathrm{sech}}\left(\mathrm{\Delta }\right)$ gate and drive 2 is for the state preparation and tomography rotations.

Figure 13

Tomographies of the $Z_{\mathrm{sech}}\left(\mathrm{\Delta }\right)$ gate for states initially prepared by a (a) $Y_{-\pi /2}$, (b) $Y_{\pi /2}$, (c) $X_{\pi /2}$, (d) $X_{-\pi /2}$, (e) $X_{\pi }$, (f) $I$ rotation.