Observing a topological transition in weak-measurement-induced geometric phases

Measurement plays a quintessential role in the control of quantum systems. Beyond initialization and readout which pertain to projective measurements, weak measurements, in particular through their back action on the system, may enable various levels of coherent control. The latter ranges from observing quantum trajectories to state dragging and steering. Furthermore, just like the adiabatic evolution of quantum states that is known to induce the Berry phase, sequential weak measurements may lead to path-dependent geometric phases. Here we measure the geometric phases induced by sequences of weak measurements and demonstrate a topological transition in the geometric phase controlled by measurement strength. This connection between weak measurement-induced quantum dynamics and topological transitions reveals subtle topological features in measurement-based manipulation of quantum systems. Our protocol could be implemented for classes of operations (e.g., braiding) that are topological in nature. Furthermore, our results open new horizons for measurement-enabled quantum control of many-body topological states.

Figure 1

Measurement-induced topological transition. (a) A sequence of measurements along a fixed latitude drags the state on a trajectory displayed on the surface of the Bloch sphere [arrows indicate the back action of the measurements for the first two and last (of six) measurements for one latitude]. When an additional, final projective measurement closes the path (green arrow), the state acquires a geometric phase ($\chi$). Considering all latitudes, these trajectories form a closed surface winding around the Bloch sphere. (b) Weaker measurements result in smaller back action on the state; as a result, the trajectories form a closed surface that does not wrap around the Bloch sphere. (c) Dependence of the geometric phase on the polar angle $\theta$ for the measurement sequences with measurement strengths slightly below (blue dashed line) and above (black dashed line) the critical value. The black solid line shows the case of infinitely strong measurements, the blue solid line represents zero measurement strength, and faint lines indicate intermediate measurement strengths. The values of $\chi$ for $\theta =0$ and $\theta =\pi$ must differ by a multiple of $2\pi$. This difference cannot be changed by continuous deformation of the dependence of $\chi$ on $\theta$. Thus the behaviors above and below the transition are topologically distinct. The insets illustrate the origin of the transition. For sufficiently strong measurements the equatorial trajectory circumnavigates the Bloch sphere while for weak measurements it does not.

Figure 2

Experiment setup. (a) A superconducting Transmon qutrit is dispersively coupled to a high quality factor microwave cavity. (b) A coherent state probe acquires a qutrit-state ($|g\rangle ,|e\rangle ,$ or $|f\rangle$)-dependent displacement on the $I/Q$ plane. (c) The dispersive interaction results in a state-dependent cavity transmission, allowing for measurements that resolve one state while leaving the other two unresolved. (d) The measurement is quantified via the frequency-dependent dephasing rates of each two-level subspace of Transmon states. The green arrow indicates the operating probe frequency, which corresponds to a selective measurement of the $|f\rangle$ state, which preserves the coherence in the $\{|g\rangle ,|e\rangle \}$ subspace. A typical Ramsey measurement is shown in the inset.

Figure 3

Experimental sequence (a) Measurements along an arbitrary axis (solid arrows) are composed of rotations before and after measurement along the $z$ axis (dashed arrows). Partial (projective) measurements are indicated in blue (red). (b) The full experiment consists of a sequence of six measurements in the qubit manifold at decreasing azimuthal angles $\varphi$. (c) The geometric phase, $\chi$, and interference contrast, $c$, are determined from the final interference pattern between $|g\rangle$ and $|e\rangle$, where $P\left(g\right)$ is the probability of obtaining outcome $g$ from the final projective measurement, ${\mathrm{\Pi }}_g$.

Figure 4

(a) The geometric phase under varying measurement strength (${\gamma }_{ef}\tau$) gained with sequential measurements along different latitudes of the Bloch sphere for polar angle $0\to \pi$. The black (blue) arrows mark measurement strengths above (below) the transition. (b) The corresponding contrast of the interference vanishes at the transition. (c) The geometric phase along fixed measurement strengths before the transition (blue lines), where the set of trajectories does not wrap the Bloch sphere, and after the transition (black lines) where the set of trajectories wraps the Bloch sphere.

Figure 5

Cancellation of relative dynamical phases. The observed interference phase (a) and contrast (b) for a reference experiment with fixed $\varphi =0$, showing that the dynamical phase is smooth and effectively canceled through appropriate rotating frames.