Geometric phase gates with adiabatic control in electron spin resonance

High-fidelity quantum operations are a key requirement for fault-tolerant quantum information processing. Manipulation of electron spins is usually achieved with time-dependent microwave fields. In contrast to the conventional dynamic approach, adiabatic geometric phase operations are expected to be less sensitive to certain kinds of noise and field inhomogeneities. Here, we introduce an adiabatic geometric phase gate for the electron spin. Benchmarking it against existing dynamic and nonadiabatic geometric gates through simulations and experiments, we show that it is indeed inherently robust against inhomogeneity in the applied microwave field strength. While only little advantage is offered over error-correcting composite pulses for modest inhomogeneities $\lesssim 10\%$, the adiabatic approach reveals its potential for situations where field inhomogeneities are unavoidably large.

Figure 1

(a) Evolution of the spin eigenstate $|0\rangle$ (red trace ${\mathcal{C}}_1$) represented by the spin vector $S$ and effective magnetic field arising from the applied microwave field (blue trace ${\mathcal{C}}_2$). (b) The detuning $\Delta$, amplitude $\Omega$, and phase $\phi$ of the microwave field during an adiabatic $\pi$ phase gate ($\Delta$ and $\Omega$ are shown in arbitrary units). The microwave frequency is tuned from off resonance to resonance between time 0 and $t_d$ and then kept resonant for a period of $t_s$ while the phase $\varphi$ of the microwave is swept. After time $t_d+t_s$ the microwave is detuned again. The effective Hamiltonian changes sign after the $\pi$ phase shift at $t_d+t_s/2$. (c) The $x$ (blue) and $y$ (green) components of the microwave field applied to the electron spin for a $\pi$ phase gate (in arbitrary units).

Figure 2

(a) Simulated relative phase errors ${\varepsilon }_r=\gamma /\pi -1$ for $\pi$ phase gates as functions of the error in microwave amplitude $\Delta {\Omega }_0/{\Omega }_0$, where $\gamma$ is the acquired phase in the simulation. The inset shows a zoomed in view of ${\varepsilon }_r$ for a better comparison of the two geometric gates. (b) Infidelities $I$ (see text) of the operations as functions of $\Delta {\Omega }_0/{\Omega }_0$. All simulations employ a single spin $1/2$ and do not include inhomogeneous broadening of the spin packet. The parameters are ${\Omega }_0=14$ MHz, initial detuning ${\Delta }_0=6$ MHz, $t_d=2\mu$s, and $t_s=4\mu$s. Note that, since there is no microwave noise, the BB1 errors are independent of the BB1 pulse duration.

Figure 3

(a) Measured geometric phase $\gamma$ (black, dashed) and error ${\varepsilon }_{\gamma }$ (red, solid) as functions of the angle $\varphi$ that the spin rotates in the $xy$ plane during the adiabatic sequence. Inset: pulse sequence for measuring the geometric phase gate. (b) Intensities of the measured spin-echo signal normalized to corresponding Hahn echo intensities. (c) Phase error and (d) echo intensity for adiabatic phase gates of different $t_s$. In panel (c) $t_s$ decreases from bottom to top at $\varphi =5\times \pi$, whereas in (d) it increases from bottom to top at $\varphi =6\times \pi$. In these experiments $t_d=2\mu$s, maximum of the microwave amplitude ${\Omega }_0=0.64$ MHz, and initial detuning ${\Delta }_0=6$ MHz. In panel (a), $t_s=6\mu$s.

Figure 4

(a) Measured phase error and (b) echo intensity for the adiabatic geometric phase gate, and dynamic phase gates using single microwave and BB1 composite pulses. All intensities are normalized to unity to enable a more convenient comparison. The parameters used for the adiabatic sequence of the experiment are the same as in Fig. 3. (c), (d) Simulation using the parameters ${\Omega }_0=1.4$ MHz, ${\Delta }_0=6$ MHz, $t_d=2\mu$s, and $t_s=4\mu$s.