Measuring the Berry Phase in a Superconducting Phase Qubit by a Shortcut to Adiabaticity

TL;DR

This paper demonstrates a fast method to measure the Berry phase in a superconducting qubit using a shortcut to adiabaticity, verifying the phase through trajectory analysis and noise robustness.

Contribution

It introduces a shortcut to adiabaticity protocol for efficient Berry phase measurement in superconducting qubits, with experimental validation and noise analysis.

Findings

Berry phase measured accurately in short time

Trajectory verification via solid angle

Robustness of phase under amplitude noise

Abstract

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. In the statistics of the Berry phase, the mean with either noise is almost equal to that without noise, while the variation with the amplitude noise can be described by an analytical expression.

Equations

$$\displaystyle H_{\mathrm{cd}}(t)=i\hbar\sum_{n}\big{[}\left\lvert{\partial_{t}n(t)}\middle\rangle\!\middle\langle{n(t)}\right\rvert-\left\langle{n(t)}\middle|{\partial_{t}n(t)}\right\rangle\left\lvert{n(t)}\middle\rangle\!\middle\langle{n(t)}\right\rvert\big{]},$$

$${\sigma }_{\Omega }=2\sqrt{2}{c}_{\Omega }\pi {\sin }^2{\theta }_0\cos {\theta }_0\frac{\sqrt{\Gamma T_{\mathrm{rot}}-1+e^{-\Gamma T_{\mathrm{rot}}}}}{\Gamma T_{\mathrm{rot}}}.$$

Full text

In quantum mechanics, a geometric phase is acquired as a quantum state propagates on a curved surface and this interesting phenomenon is observed in many quantum systems ranging from microscopic particles to condensed matter materials BerryPhase ; Chiao ; Bitter ; WilczekBook ; BerrySolidState . The precise control and measurement of the geometric phase lead to geometric phase gates in quantum computation SjoqvistPhysics ; NMRgate ; Iongate ; nonAbelianGate ; SpinGate . A typical example to understand the geometric phase is a spin-$1/2$ particle subject to a slowly-varying magnetic field which undergoes a closed path in the parameter space. In response to the changing field, the spin remains in an instantaneous eigenstate and follows a cyclic path in the Bloch sphere. The geometric phase, termed the Berry phase BerryPhase , is acquired by the spin state when the spin vector returns to its initial position. The Berry phase in such an adiabatic process is independent of the speed of the field’s evolution if the adiabatic condition is satisfied.

In practice, a quantum manipulation is often performed in a short time scale to avoid dissipation induced errors, incompatible with the presumption of the adiabatic process. The advance of large-scale quantum devices requires fast operations to improve efficiency of information processing XMon2 . Various procedures have thus been proposed for the realization of a fast ‘adiabatic’ process DemirplakRice ; Berry09 ; ChenXi ; Masuda ; Shortcuts ; CompositPulse ; fastAdiabaticgate . One general strategy is to apply a ‘shortcut to adiabaticity’ (STA) where an additional Hamiltonian is employed to cancel the non-adiabatic contribution in a fast evolution DemirplakRice ; Berry09 ; ChenXi ; Masuda ; Shortcuts . For a given reference Hamiltonian $H_{0}(t)$, the counter-diabatic Hamiltonian $H_{\mathrm{cd}}(t)$ is formally written as Berry09 ; SI

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where $|n(t)\rangle$ is an instantaneous eigenstate of $H_{0}(t)$. For each eigenstate $|n(t)\rangle$, $H_{\mathrm{cd}}(t)$ suppresses its non-adiabatic transition to other eigenstates. The quantum system driven by the total Hamiltonian $H_{\mathrm{tot}}(t)=H_{0}(t)+H_{\mathrm{cd}}(t)$ can evolve fast but remain in the state $\left\lvert n(t)\right\rangle$. Regardless of the evolution of $H_{\mathrm{tot}}(t)$, the system state evolves along a closed path with respect to $H_{0}(t)$ and acquires the associated Berry phase AAphase ; ZhuSL2002 .

The STA protocol has been implemented in atomic, molecular and optical systems for the state preparation, population transfer and optimal control Shortcuts ; Bason ; Suter . Compared with these microscopic systems, a superconducting circuit is fabricated on chips with lithographic scalability. The superconducting qubit is realized based on nonlinear quantized energy levels of the circuit. Sophisticated microwave techniques allow a reliable generation of the counter-diabatic Hamiltonian in superconducting qubits. In this paper, we focus on the realization of the STA protocol in a superconducting phase qubit MartinisReview ; phaseQubit . We achieve the Berry phase measurement with the STA protocol and study the influence of external field fluctuations, which extends previous studies of the Berry phase in a Cooper pair pump MottononPhase and in an adiabatically-steered superconducting qubit LeekPhase ; NoisePhase .

The circuit of a superconducting phase qubit is a nonlinear resonator comprised of a Josephson junction, a loop inductance and a capacitor SI ; MartinisReview ; phaseQubit . A flux current biases this resonator in an anharmonic cubic potential, and the lowest two energy levels of the nonlinear resonator are used as the ground ($\left\lvert 0\right\rangle$) and excited ($\left\lvert 1\right\rangle$) states of a qubit. Through the same bias line, a microwave drive signal is coupled to the qubit, providing a fast and reliable control of the qubit state. The phase qubit is mounted in a sample box and cooled in a dilution refrigerator whose base temperature is $\sim 10$ mK. In our experiment, the resonance frequency of the phase qubit is set at $\omega_{10}/2\pi=5.7\,\mathrm{GHz}$, and the qubit dissipation is characterized by a relaxation time, $T_{1}=270\,\mathrm{ns}$, and a spin-echo decoherence time, $T_{2}^{\mathrm{echo}}=450\,\mathrm{ns}$. For our phase qubit, it is difficult to perform an adiabatic operation LeekPhase ; NoisePhase , but feasible to implement a fast STA protocol. With the qubit modelled as a spin-$1/2$ particle, we also treat the microwave signal as an effective external magnetic field. In the rotating frame of the external field, the Hamiltonian is expressed as $H(t)=\hbar\bm{B}(t)\cdot\bm{\sigma}/2$ after the rotating wave approximation SI . Here $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the vector of Pauli operators and $\bm{B}(t)$ is the effective magnetic field expressed in the unit of angular frequency. Throughout this paper, our experiment will be described and discussed in the rotating frame.

In an adiabatic process, the Berry phase can be measured with a spin-echo scheme LeekPhase ; NoisePhase ; FilippPhase . In our STA experiment, the reference Hamiltonian, $H_{0}(t)=\hbar\bm{B}_{0}(t)\cdot\bm{\sigma}/2$, is used to construct the spin-echo trajectory in a short time scale. As shown in Fig. 1(a), the reference magnetic field $\bm{B}_{0}(t)$ evolves as follows. A $\pi/2$-pulse is applied to the ground-state qubit, preparing an initial superposition state at $(\left\lvert 0\right\rangle+\left\lvert 1\right\rangle)/\sqrt{2}$. After the initialization, the first sequence of the field, $\bm{B}_{\mathrm{ramp},0}(t_{1})=(\Delta_{0}\tan\theta(t_{1}),0,\Delta_{0})$ with $0<t_{1}<T_{\mathrm{ramp}}$, ramps up its $x$-component by $\theta(t_{1})=\theta_{0}t_{1}/T_{\mathrm{ramp}}$. The ramping time is fixed at $T_{\mathrm{ramp}}=10$ ns, while $\Delta_{0}=\omega_{\mathrm{d}}-\omega_{10}$ is the detuning between the microwave drive frequency $\omega_{\rm d}$ and the qubit resonance frequency $\omega_{10}$. We set $\Delta_{0}/2\pi=7$ MHz to reduce the influence of higher excited states. The second sequence builds a rotating field, $\bm{B}_{\mathrm{rot},0}(t_{2})=(\Omega_{0}\cos\phi(t_{2}),\Omega_{0}\sin\phi(t_{2}),\Delta_{0})$ with $0<t_{2}<T_{\mathrm{rot}}$. The drive amplitude is $\Omega_{0}=\Delta_{0}\tan\theta_{0}$, while the rotation is operated under a constant speed $\omega_{0}=2\pi/T_{\mathrm{rot}}$ along either the counterclockwise ($\mathcal{C}_{+}$) or clockwise ($\mathcal{C}_{-}$) direction. The time-dependent phase of the effective field is given by $\phi(t_{2})=\pm\omega_{0}t_{2}$. In our experiment, the pre-designed polar angle $\theta_{0}$ and the rotation time $T_{\mathrm{rot}}$ are varied as control parameters. To further reduce the influence of higher excited states, a reversed ramping-down field, $\bm{B}_{\mathrm{ramp},0}(T_{\mathrm{ramp}}-t_{3})$ with $0<t_{3}<T_{\mathrm{ramp}}$, is subsequently used to finish the first (dephasing) part of the spin-echo scheme 3level . A refocusing $\pi$-pulse is then applied to invert the qubit states. During the second (rephasing) part of our spin-echo scheme, the three sequences in the first part are reversed, as shown in Fig. 1(a). Based on the cyclic rotations in the dephasing and rephasing parts, the symbols of $\mathcal{C}_{+-}$ and $\mathcal{C}_{-+}$ represent two different types of spin-echo procedures.

To fulfill the STA protocol, the counter-diabatic Hamiltonian, $H_{\mathrm{cd}}(t)=\hbar\bm{B}_{\mathrm{cd}}(t)\cdot\bm{\sigma}/2$, is calculated by Eq. (1). For each ramping or rotating step, the counter-diabatic magnetic field is given by $\bm{B}_{\mathrm{cd}}(t)=\bm{B}_{0}(t)\times\dot{\bm{B}}_{0}(t)/|\bm{B}_{0}(t)|^{2}$, which is perpendicular to the reference magnetic field SI . In particular, the counter-diabatic ramping field is written as $\bm{B}_{\mathrm{ramp},\mathrm{cd}}(t)=(0,\pm\theta_{0}/T_{\mathrm{ramp}},0)$, where the $\pm$ signs correspond to the ramping-up and ramping-down steps, respectively. The counter-diabatic rotating field is written as $\bm{B}_{\mathrm{rot,cd}}(t)=\left(\Omega_{\mathrm{cd}}\cos\phi(t),\Omega_{\mathrm{cd}}\sin\phi(t),\Delta_{\mathrm{cd}}\right)$, where the signs of $\Omega_{\mathrm{cd}}=\mp\omega_{0}\sin\theta_{0}\cos\theta_{0}$ and $\Delta_{\mathrm{cd}}=\pm\omega_{0}\sin^{2}\theta_{0}$ refer to the $\mathcal{C}_{+}$ and $\mathcal{C}_{-}$ rotations, respectively. The total magnetic field is obtained as $\bm{B}_{\mathrm{tot}}(t)=\bm{B}_{0}(t)+\bm{B}_{\mathrm{cd}}(t)$, an example of which is shown in Fig. 1(a).

In the STA protocol, the qubit is designed to follow the path of $\bm{B}_{0}(t)$ when driven by the total Hamiltonian $H_{\mathrm{tot}}(t)=\hbar\bm{B}_{\mathrm{tot}}(t)\cdot\bm{\sigma}/2$. Initially, the $|0\rangle$ and $|1\rangle$ states are the instantaneous spin-up ($\left\lvert s_{\uparrow}(t)\right\rangle$) and spin-down ($\left\lvert s_{\downarrow}(t)\right\rangle$) eigenstates of the reference Hamiltonian, respectively. In an ideal adiabatic scenario, the Bloch vector of the $\left\lvert s_{\uparrow}(t)\right\rangle$ state always points to the same direction of $\bm{B}_{0}(t)$, while the opposite occurs for the $\left\lvert s_{\downarrow}(t)\right\rangle$ state. The ramping sequences only produce a dynamic phase, while both the dynamic and Berry phases are acquired during circular rotations. For the $\left\lvert s_{\uparrow}(t)\right\rangle$ state, the Berry phase accumulated in one cycle is given by $\gamma_{\uparrow}=\mp\mathcal{S}/2$, where $\mathcal{S}=2\pi(1-\cos\theta_{0})$ is the solid angle of the cone subtended by the cyclic path at the origin and the $\mp$ signs refer to the $\mathcal{C}_{+}$ or $\mathcal{C}_{-}$ paths, respectively. Since the $\left\lvert s_{\downarrow}(t)\right\rangle$ state follows the opposite path of the $\left\lvert s_{\uparrow}(t)\right\rangle$ state, its accumulated Berry phase is opposite, i.e., $\gamma_{\downarrow}=\pm\mathcal{S}/2$. After the refocusing $\pi$ pulse, the phases associated with the $\left\lvert s_{\uparrow}(t)\right\rangle$ and $\left\lvert s_{\downarrow}(t)\right\rangle$ states are swapped. For each instantaneous eigenstate, the dynamic phase is cancelled in the rephasing part, while the Berry phase is doubled due to a reversed rotating direction. At the echo time, the $\left\lvert s_{\uparrow}(t)\right\rangle$ and $\left\lvert s_{\downarrow}(t)\right\rangle$ states return to the two poles of the Bloch sphere, and the final qubit state is written as $(e^{-i\gamma/2}\left\lvert 0\right\rangle+e^{i\gamma/2}|1\rangle)/\sqrt{2}$ with a global phase excluded SI . The total relative Berry phase after two circular rotations is theoretically given by $\gamma=\mp 2\mathcal{S}$ for the $\mathcal{C}_{+-}$ or $\mathcal{C}_{-+}$ procedures, respectively.

In our experiment, the total field $\bm{B}_{\mathrm{tot}}(t)$ is realized by the microwave control signal, and the qubit is driven in the spin-echo scheme as above. To measure the Berry phase $\gamma$, we apply the quantum state tomography (QST) to extract the density matrix $\rho$ of the final state phaseQubit . The $x$- and $y$-projections of the qubit vector are given by $\langle x\rangle=\mathrm{Tr}\{\sigma_{x}\rho\}$ and $\langle y\rangle=\mathrm{Tr}\{\sigma_{y}\rho\}$. The two-dimensional (2D) diagram in Fig. 1(b) presents the experimental measurement of $\langle x\rangle$ with the change of the rotation time $T_{\mathrm{rot}}$ and the pre-designed solid angle $\mathcal{S}$ (through the change of $\theta_{0}$) in the $\mathcal{C}_{+-}$ procedure. The rotation time, $20~{}\mathrm{ns}\leq T_{\mathrm{rot}}\leq 60~{}\mathrm{ns}$, in our experiment is much shorter than the necessary time ($T_{\mathrm{rot}}\gtrsim 1~{}\mu$s) of an adiabatic operation LeekPhase . We observe that $\langle x\rangle$ oscillates with $\mathcal{S}$ and varies slowly with $T_{\mathrm{rot}}$; the same conclusion is applied to $\langle y\rangle$ (not shown). From the results of $T_{\mathrm{rot}}=30$ ns in Fig. 1(c), these two components are consistent with the theoretical prediction of $\langle x\rangle=r\cos\gamma$ and $\langle y\rangle=r\sin\gamma$ with $\gamma=-2\mathcal{S}$. Here $r=0.72$ is an adjusted fitting parameter due to dissipation. On the other hand, these two oscillations are nearly undamped with $\mathcal{S}$, indicating a weak geometric dephasing LeekPhase . For each final density matrix, the Berry phase $\gamma$ is estimated using $\arctan[\langle y\rangle/\langle x\rangle]$. In the $\mathcal{C}_{+-}$ procedure, the Berry phase with $\theta_{0}<\pi/3$ (or $\mathcal{S}<\pi$) is assigned in the range of $(-2\pi,0)$ after considering the signs of $\langle x\rangle$ and $\langle y\rangle$. With $\pi/3<\theta_{0}<\pi/2$ (or $\pi<\mathcal{S}<2\pi$), an extra $-2\pi$ is included to assign $\gamma$ in the range of $(-4\pi,-2\pi)$. An opposite range is specified for $\gamma$ in the $\mathcal{C}_{-+}$ procedure. Figure 1(d) presents the measurement of $\gamma$ for the $\mathcal{C}_{+-}$ procedure. From the result of $T_{\mathrm{rot}}=30$ ns in Fig. 1(e), the linear relation, $\gamma=-k\mathcal{S}$ ($k=2.04\pm 0.02$), is extracted, in good agreement with the theoretical prediction of $\gamma=-2\mathcal{S}$. The same relation with an opposite sign, $\gamma=k^{\prime}\mathcal{S}$ ($k^{\prime}=2.06\pm 0.02$), is extracted for the $\mathcal{C}_{-+}$ procedure. The two slopes, $k$ and $k^{\prime}$, are almost unchanged with the increase of the rotation time $T_{\mathrm{rot}}$. Our experiment thus demonstrates that the Berry phase of a cyclic adiabatic path can be successfully measured in the superconducting phase qubit following the fast STA protocol.

To further illustrate the accumulation of the Berry phase, we modify the external microwave signal and inspect the trajectory of the $\left\lvert s_{\uparrow}(t)\right\rangle$ state in the STA protocol. Without the excitation of the $\pi/2$-pulse, the ground-state qubit (or the $\left\lvert s_{\uparrow}(t)\right\rangle$ state) is driven by the ramping-up field $\bm{B}_{\mathrm{ramp,tot}}(t_{1})$ and the subsequent $\mathcal{C}_{+}$-rotating field $\bm{B}_{\mathrm{rot,tot}}(t_{2})$ with $T_{\mathrm{rot}}=30$ ns. We measure the density matrix of the qubit by interrupting the rotation and performing the QST every 0.5 ns. Figure 2 presents the time evolution of the qubit vector during the rotation period for two pre-designed polar angles, $\theta_{0}=\pi/6$ and $\pi/4$, in the parameter space. As shown in Figs. 2(c) and 2(d), the polar angles of both qubit vectors vary weakly with time (around the given value of $\theta_{0}$), and their azimuthal angles increase almost linearly with the time as $\phi(t_{2})=\omega_{0}t_{2}$. The time evolution of the radii in Fig. 2(b) shows that the two qubit vectors are initially on the surface of the Bloch sphere and slightly shrink due to dissipation. Our experiment confirms that the $\left\lvert s_{\uparrow}(t)\right\rangle$ state follows the same direction of $\bm{B}_{\mathrm{rot,0}}(t_{2})$ and takes a circular path along the latitude of $\pi/2-\theta_{0}$ in the northern hemisphere. An integration, $\mathcal{S}=\int[1-\cos\theta(t_{2})]d\phi(t_{2})$, further allows us to estimate the actual solid angles enclosed by the circular path. The approximate integration results, $\mathcal{S}(\theta_{0}=\pi/6)=0.788$ and $\mathcal{S}(\theta_{0}=\pi/4)=1.72$, are roughly close to their theoretical values of $\mathcal{S}=2\pi(1-\cos\theta_{0})$ in an ideal adiabatic scenario, also consistent with the measurement of the Berry phase in Fig. 1.

In a previous experiment of a superconducting transmon qubit with the adiabatic protocol, the stability of the Berry phase was studied under the influence of slowly-varying noises NoisePhase ; NoiseTheory . Similarly, we introduce an artificial fluctuation $\delta\bm{B}_{\mathrm{rot}}(t)$ to the rotation field $\bm{B}_{\mathrm{rot,tot}}(t)$. Two independent stochastic noises, $\delta\Omega(t)$ and $\delta\phi(t)$, are assigned to the drive amplitude and phase of the effective magnetic field in the $x$-$y$ plane, respectively. Both noises are assumed to follow the Ornstein-Uhlenbeck (O-U) process. The amplitude noise satisfies $\langle\delta\Omega(t)\rangle=0$ and $\langle\delta\Omega(t)\delta\Omega(0)\rangle=c^{2}_{\Omega}\Omega^{2}_{\mathrm{tot}}\exp(-\Gamma t)$ with $\Omega_{\mathrm{tot}}=\Omega_{0}+\Omega_{\mathrm{cd}}$, while the phase noise satisfies $\langle\delta\phi(t)\rangle=0$ and $\langle\delta\phi(t)\delta\phi(0)\rangle=c_{\phi}^{2}\exp(-\Gamma t)$. The dimensionless coefficients, $c_{\Omega}$ and $c_{\phi}$, represent the reduced noise strengths. A small noise bandwidth, $\Gamma=10$ MHz, is chosen so that noises are correlated through the operation time. In our experiment, we investigate the influence of the amplitude noise $\delta\Omega(t)$ and the phase noise $\delta\phi(t)$ separately. With respect to each noise parameter ($c_{\Omega}$ and $c_{\phi}$), the total 300 stochastic trajectories of $\delta\bm{B}_{\mathrm{rot}}(t)$ are generated. Since the relative dynamic phase between the $\left\lvert s_{\uparrow}(t)\right\rangle$ and $\left\lvert s_{\downarrow}(t)\right\rangle$ states cannot be cancelled by a noisy spin-echo sequence, we only assign a single rotating field, $\bm{B}_{\mathrm{rot,tot}}(t)+\delta\bm{B}_{\mathrm{rot}}(t)$, to measure the Berry phase accumulated in a single cycle. To reduce the influence of the intrinsic noise, the spin-echo scheme is still used, while the effective magnetic field is only applied in the second part following the $\pi$-pulse. An example of the noisy magnetic field with the $\mathcal{C}_{+}$ rotation is shown in Fig. 3(a). For each noisy sequence, we measure the final density matrix by the QST and unwrap the total accumulated phase, similar to the method in Fig. 1. The final relative Berry phase $\gamma$ ($|s_{\downarrow}(t)\rangle$ relative to $|s_{\uparrow}(t)\rangle$) from one circular loop is then estimated from the total phase subtracted by a theoretical calculation of the dynamic phase SI . This indirect approach could introduce a small error of the dynamic phase into the estimation of $\gamma$, which however does not affect the main conclusion of our experiment. After collecting data over 300 trajectories, we obtain the statistics of $\gamma$ subject to the amplitude noise $\delta\Omega(t)$ or the phase noise $\delta\phi(t)$.

With a fixed rotation time, $T_{\mathrm{rot}}=30$ ns, we first study the variation of the Berry phase under a given noise strength, $c_{\Omega}=0.1$ or $c_{\phi}=0.1$. For four pre-designed solid angles, $\mathcal{S}=\pi/40,3\pi/16,3\pi/8$, and $\pi$ in the parameter space, the histograms of $\gamma$ subject to the amplitude and phase noises are presented in Figs. 3(b) and 3(c), respectively. For the amplitude noise, each histogram is well fitted by a Gaussian distribution, $\exp[-(\gamma-\bar{\gamma}_{\Omega})^{2}/2\sigma^{2}_{\Omega}]$, predicted by a perturbation theory SI ; NoiseTheory . As verified in Fig. 3(d), the mean value of the Berry phase over various noise trajectories, $\bar{\gamma}_{\Omega}$, is very close to the result of $\gamma=\mathcal{S}$ from a single circular loop without noise. In the leading order of $c_{\Omega}$, the standard deviation $\sigma_{\Omega}$ is analytically expressed as SI

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Figure 3(e) shows that the experimental measurement of $\sigma_{\Omega}$ agrees well with Eq. (2) over a broad range of $\mathcal{S}$. At small solid angles, a residue $\sim 0.02\pi$ is observed for $\sigma_{\Omega}$, which may be attributed to the intrinsic noise of the qubit. Equation (2) indicates an upper deviation limit, $\sigma_{\Omega}\sim 2c_{\Omega}\pi\sin^{2}\theta_{0}\cos\theta_{0}$, for the STA process. For the parameters $\Gamma T_{\mathrm{rot}}=0.3$ and $c_{\Omega}=0.1$ in our experiment, Fig. 3(e) demonstrates that the fluctuation of the Berry phase is still tolerable with $\sigma_{\Omega}<0.1\pi$. As a comparison, the influence of the phase noise is much weaker than that of the amplitude noise, which can be identified by the narrow distributions of $\gamma$ in Fig. 3(c). This behavior is due to the fact that the Berry phase depends on the geometry of the circular path instead of the rotating speed NoiseTheory . Accordingly, the mean $\bar{\gamma}_{\phi}$ is close to the result without noise [Fig. 3(d)] and the standard deviation $\sigma_{\phi}$ is consistently small, $\sigma_{\phi}\sim 0.02\pi$, over the whole range of the solid angle [Fig. 3(e)].

A coherence parameter, $\nu=|\left\langle\exp(i\gamma)\right\rangle|$, can alternatively quantify the fluctuation of the Berry phase NoisePhase . For the amplitude noise ($c_{\Omega}=0.1$) and phase noise ($c_{\phi}=0.1$), the $\mathcal{S}$-dependencies of $\nu_{\Omega}$ and $\nu_{\phi}$ in Fig. 4(a) are consistent with the results of $\sigma_{\Omega}$ and $\sigma_{\phi}$ in Fig. 3(e). The effect of the phase noise is essentially negligible whereas the influence of the amplitude noise is well described by an analytical expression. Furthermore, we explore the change of the coherence parameter as the noise strength ($c_{\Omega}$ or $c_{\phi}$) varies. For a fixed polar angle, $\theta_{0}=\pi/3$ (or $\mathcal{S}=\pi$), Fig. 4(b) confirms the weak influence of the phase noise. As a comparison, $\nu_{\Omega}$ can maintain a large value for $c_{\Omega}<0.2$ while a major drop of $\nu_{\Omega}$ occurs for $c_{\Omega}>0.5$. In both Figs. 4(a) and 4(b), the results of $\nu_{\Omega}$ can be well described by $\nu_{\Omega}=\exp(-\sigma^{2}_{\Omega}/2)$, where the analytical values of $\sigma_{\Omega}$ from Eq. (2) are used.

In summary, we have implemented the STA protocol in a superconducting phase qubit. In good agreement with the theoretical prediction, the Berry phase is successfully measured in a time ($20~{}\mathrm{ns}\leq T_{\mathrm{rot}}\leq 60$ ns) much shorter than that required by the adiabatic theorem ($T_{\mathrm{rot}}\gtrsim 1~{}\mu$s). The measurement of the Berry phase is almost independent of the operation time, which is a characteristic property of the STA protocol. The trajectory of a qubit state is verified, from which the solid angle enclosed by the path is calculated to understand the accumulation of the Berry phase. Classical fluctuations of the drive amplitude or phase are artificially introduced to the total control field. Our experiment shows that the mean value of the Berry phase is unchanged, while the standard deviation with the amplitude noise can be described by an analytical expression. To further understand the stability of the Berry phase, an extended study of quantum noise will be necessary in the future DMTong04 ; JFDu08 . The fast measurement of the Berry phase in this paper will hopefully encourage more applications of the STA protocol in superconducting qubit systems.