Exploring the effect of noise on the Berry phase

We experimentally investigate the effects of noise on the adiabatic and cyclic geometric phase, also termed the Berry phase. By introducing artificial fluctuations in the path of the control field, we measure the geometric contribution to dephasing of an effective two-level system for a variety of noise powers and different paths. Our results, measured using a microwave-driven superconducting qubit, clearly show that only fluctuations which distort the path lead to geometric dephasing. In a direct comparison with the dynamic phase, which is path independent, we observe that the Berry phase is less affected by noise-induced dephasing. This observation directly points towards the potential of geometric phases for quantum gates or metrological applications.

Figure 1

(a) The path of the effective field describes a circle in the $B_x$-$B_y$ plane at constant $B_z$. Noise in $x$ and $y$ directions with noise powers $P_x$ and $P_y$ can be decomposed into noise in $\rho$ and $\phi$ directions with noise powers $P_{\rho }$ and $P_{\phi }$. (b) and (c) The path of the effective field without noise (dashed lines lying in the plane with constant $B_z$) is drawn alongside the same path exposed to two kinds of noise (solid lines): angular noise in panel (b), where the velocity of precession is proportional to line thickness, and radial noise in panel (c). The projection of the paths on the unit sphere $\left|\mathbf{B}\right|=1$ is also shown. In panel (b), the difference in solid angle due to noncyclic evolution is highlighted by a stripe pattern.

Figure 2

(a) Histograms of Berry phases and (b) measured expectation values $\langle X\rangle$, $\langle Y\rangle$ of 600 realizations of radial noise for each solid angle $A=\pi /16,3\pi /16,8\pi /16$, and $15\pi /16$ (indicated in red, orange, purple, and black online). Fits of a Gaussian to the measured histograms are also shown in panel (a). The circle in panel (b) indicates unit coherence. (c) and (d) Measurements analogous to panels (a) and (b) for angular noise. (e) and (f) Coherence $\nu$ and phase difference $\Delta \gamma$ as a function of solid angle $A$ for radial noise (filled circles) and angular noise (open circles). The experimental data points are shown alongside the theory curve (solid lines) and the results from numerical simulations (the shaded area indicates the standard deviation about the mean). Data in panels (a)–(f) are recorded at fixed noise bandwidths ${\Gamma }_i=10\mathrm{MHz}$, normalized noise amplitudes $s_i=1/15$, and evolution time $\tau =100\mathrm{ns}$.

Figure 3

(a) and (b) Sketches of the pulse schemes used to measure (a) Berry and (b) dynamic phases with radial noise. Pulses applied along the $x$ and $y$ quadratures are shown as solid and dashed lines, respectively. The readout pulse (see text) concludes the sequence after $t\approx 400$ ns. (c) and (d) Experimentally measured coherence $\nu$ of the Berry phase and phase difference $\Delta \gamma =\gamma -{\gamma }_0$ as a function of normalized noise amplitude $s$ for radial noise (filled circles) and angular noise (open circles), plotted on a logarithmic scale. For every value of $s$, 300 noise realizations were measured with noise bandwidth $\Gamma =10\mathrm{MHz}$ at solid angle $A=7\pi /16$ and evolution time $\tau =100\mathrm{ns}$. The continuous line is computed from Eq. (3). The dashed line is a fit to the function $\exp (-{\left(4as\right)}^2/2)$ with fitting parameter $a=0.25\pm 0.01$. (e) and (f) Quantities analogous to panels (c) and (d) but for the dynamic phase, with $\Delta \delta =\delta -{\delta }_0$ and fitting parameter $a=0.60\pm 0.03$.

Figure 4

(a) Standard deviation ${\sigma }_{\gamma }$ of the Berry phase (dots) and ${\sigma }_{\delta }$ of the dynamic phase (squares) as a function of evolution time $\tau$, based on 300 noise realizations with $\Gamma =10\mathrm{MHz}$ and $s_{\rho }=1/15$. The solid lines result from calculations based on Eqs. (3) and (5). The vertical dashed line approximately separates the nonadiabatic from the adiabatic regime. (b) Coherence $\nu$ vs evolution time $\tau$ of the Berry phase (dots) and the dynamic phase (squares).