Weak-Measurement-Induced Asymmetric Dephasing: Manifestation of Intrinsic Measurement Chirality

Geometrical dephasing is distinct from dynamical dephasing in that it depends on the trajectory traversed, hence it reverses its sign upon flipping the direction in which the path is traced. Here we study sequences of generalized (weak) measurements that steer a system in a closed trajectory. The readout process is marked by fluctuations, giving rise to dephasing. Rather than classifying the latter as “dynamical” and “geometrical,” we identify a contribution which is invariant under reversing the sequence ordering and, in analogy with geometrical dephasing, one which flips its sign upon the reversal of the winding direction, possibly resulting in partial suppression of dephasing (i.e., “coherency enhancement”). This dephasing asymmetry (under winding reversal) is a manifestation of intrinsic chirality, which weak measurements can (and generically do) possess. Furthermore, the dephasing diverges at certain protocol parameters, marking topological transitions in the measurement-induced phase factor.

Figure 1

Measurement-induced trajectories. (a) The backaction of a generalized measurement. Under a projective measurement yielding readout $r=0$, the system’s initial state (red arrow) would become $|\uparrow ⟩$ (black arrow). For finite measurement strength, the state is only pulled towards $|\uparrow ⟩$ and also rotated around the $z$ axis (green arrow). These two backaction aspects (illustrated by dashed blue lines) are quantified by parameters $S$ and $R$ in Eq. (2). The direction of the rotation (the sign of $R$) sets the measurement’s chirality. (b) Schematic illustration of trajectories induced by a sequence of measurements along some parallel (black line). Depending on the sequence of readouts $\{r_k\}$, different trajectories (red, cyan, purple) and different phases ${\chi }_{\{r_k\}}^{(d)}$ are induced.

Figure 2

Dephasing ${\alpha }^{(+1)}$ (a) and phase ${\overline{\chi }}^{(+1)}$ (b), cf. Eq. (3), at $\theta =3\pi /4$ color-coded as functions of the measurement strength ($C$) and asymmetry ($A$) parameters. Note the difference in behavior for $A>0$ and $A<0$, showing the effect of the measurement chirality. Most prominent are the two singularities at $C\approx 2$, where ${\alpha }^{(+1)}$ diverges. The phase makes $\pi$ windings around the points of divergent ${\alpha }^{(+1)}$.

Figure 3

Topological transition in the measurement-induced phase ${\overline{\chi }}^{(+1)}$. Main panel—The critical line of points $(C_{\mathrm{crit}},A_{\mathrm{crit}})$ for which there exists ${\theta }_{\mathrm{crit}}$ such that the dephasing ${\alpha }^{(+1)}(C_{\mathrm{crit}},A_{\mathrm{crit}},{\theta }_{\mathrm{crit}})$ diverges. The values of ${\theta }_{\mathrm{crit}}\in [0;\pi ]$ are shown with color. The averaged phase ${\overline{\chi }}^{(+1)}$ exhibits three distinctly different topological behaviors, corresponding to regions I, II, and III. Inset—Dependence of ${\overline{\chi }}^{(+1)}$ on $\theta$ for the measurement parameters $(C,A)$ marked with squares in the main plot. As $\theta$ varies from 0 to $\pi$, ${\overline{\chi }}^{(+1)}(\theta )$ varies from 0 to 0 (region I), 0 to $-\pi$ (reigon II), or 0 to $-2\pi$ (region III). The values of ${\theta }_{\mathrm{crit}}$ corresponding to the two transitions at $A=1$ are marked as ${\theta }_{\mathrm{c}1}$ and ${\theta }_{\mathrm{c}2}$.