Measuring the Berry phase in a superconducting phase qubit by a shortcut to adiabaticity

With a counter-diabatic field supplemented to the reference control field, the “shortcut to adiabaticiy” (STA) protocol is implemented in a superconducting phase qubit. The Berry phase measured in a short time scale is in good agreement with the theoretical result acquired from an adiabatic loop. The trajectory of a qubit vector is extracted, verifying the Berry phase alternatively by the integrated solid angle. The classical noise is introduced to the amplitude or phase of the total control field. The mean of the Berry phase measured under either noise is almost equal to that without noise, while the variance under the amplitude noise can be described by an analytical expression.

Figure 1

(a) Schematic diagram of our experimental setup including the room-temperature control and the low-temperature phase qubit. (b) Optical micrograph of the phase qubit.

Figure 2

STA experiment of measuring the Berry phase with a ${\mathcal{C}}_{+-}$ spin-echo procedure. (a) Schematic diagram of the effective magnetic field in the $x-y$ plane. The dashed and solid lines are the reference and total control fields, while the red and blue colors refer to their $x$ and $y$ components. (b) Measurement of the $x$ projection of the final qubit state versus the solid angle $\mathcal{S}$ and the rotation time $T_{\mathrm{rot}}$. (c) With $T_{\mathrm{rot}}=30$ ns, the $x$, $y$ projections of the final qubit state (red and blue circles) are compared with the theoretical prediction (red and blue lines). (d) Measurement of the Berry phase versus $\mathcal{S}$ and $T_{\mathrm{rot}}$. (e) With $T_{\mathrm{rot}}=30$ ns, the extracted Berry phase (circles) is compared with the theoretical prediction (solid line). The comparison between the experimental measurement and the theoretical prediction in the ${\mathcal{C}}_{-+}$ procedure is also presented.

Figure 3

In the STA procedure, the evolution of the $\left|s_{\uparrow }\left(t\right)\right.〉$ state subject to a ${\mathcal{C}}_{+}$-rotating field with $T_{\mathrm{rot}}=30$ ns. (a) Trajectory of the qubit vector in the Bloch sphere. In the spherical coordinate system of the qubit vector, the time evolutions of the radius and the polar and azimuthal angles are plotted in (b), (c), and (d), respectively. In (a)–(d), the black and blue circles are the experimental results of ${\theta }_0=\pi /6$ and $\pi /4$, respectively. The associated solid lines are the results of an ideal evolution.

Figure 4

STA experiment of studying the noise influence on the measurement of the Berry phase. (a) Schematic diagram of the effective magnetic field with a noisy ${\mathcal{C}}_{+}$ rotation. The red and blue colors refer to the $x$ and $y$ components of the total control field. With $T_{\mathrm{rot}}=30$ ns, the histograms of the Berry phase subject to 300 trajectories of the amplitude ($c_{\mathrm{\Omega }}=0.1$) and phase noises ($c_{\varphi }=0.1$) are plotted in (b) and (c), respectively. From the left to right in both (b) and (c), the four distributions refer to the results of $\mathcal{S}=\pi /40,3\pi /16$, $3\pi /8$, and $\pi$ in the parameter space. In (b), the solid lines are the Gaussian fitting curves of the histograms. The mean values and standard deviations are shown in (d) and (e), where the circles and up-triangles refer to the experimental results of the amplitude and phase noises, respectively. In (d), the solid line is the result without noise. In (e), the solid line is the analytical prediction of Eq. (2).

Figure 5

Coherence parameter $\nu$ under two types of noises with the rotation time $T_{\mathrm{rot}}=30$ ns. (a) $\nu$ versus the solid angle $\mathcal{S}$ for the noise strengths of $c_{\mathrm{\Omega }}=0.1$ and $c_{\varphi }=0.1$. (b) $\nu$ versus $c_{\mathrm{\Omega }}$ and $c_{\varphi }$ for $\mathcal{S}=\pi$. In both (a) and (b), the circles and up-triangles refer to the experimental results of the amplitude and phase noises, respectively. For the amplitude noise, the two solid lines are the results of ${\nu }_{\mathrm{\Omega }}=exp(-{\sigma }_{\mathrm{\Omega }}^2/2)$ with ${\sigma }_{\mathrm{\Omega }}$ calculated by Eq. (2).