Weak-measurement-induced phases and dephasing: Broken symmetry of the geometric phase

Coherent steering of a quantum state, induced by a sequence of weak measurements, has become an active area of theoretical and experimental study. For a closed steered trajectory, the underlying phase factors involve both geometrical and dynamical terms. Furthermore, considering the reversal of the order of the measurement sequence, such a phase comprises a symmetric and an antisymmetric term. Superseding common wisdom, we show that the symmetric and the antisymmetric components do not correspond to the dynamical and geometrical parts, respectively. Addressing a broad class of measurement protocols, we further investigate the dependence of the induced phases on the measurement parameters (e.g., the measurement strength). We find transitions between different topologically distinct sectors, defined by integer-valued winding numbers, and show that the transitions are accompanied by diverging dephasing. We propose experimental protocols to observe these effects.

Figure 1

Backaction of a generalized measurement. After a projective measurement of the $S_z$ component of spin $\frac{1}{2}$ that yields a readout $S_z=+\frac{1}{2}$, the initial state (red arrow) becomes aligned with the north pole of the Bloch sphere (black arrow). A generalized measurement's backaction does not necessarily align the state with an eigenstate of the measured observable; it only pulls the state towards the north pole and may also rotate it around the $z$ axis (green arrow). These two effects of the backaction are illustrated by blue dashed lines. Such a backaction appears in the measurement protocol we consider in Sec. 3a.

Figure 2

State trajectories of the system for the direct and reversed measurement sequences may be drastically different. A spin-$\frac{1}{2}$ system, whose states are represented on the Bloch sphere, is subject to a sequence of generalized measurements, pertaining to the spin projections around a given parallel (black line), in accordance with the protocol described in Sec. 3. All measurements of either the clockwise (red) or the counterclockwise (green) sequences are read out and postselected to have $r_k=0$. The yellow lines connect $|{\psi }_N\rangle$ to $|{\psi }_0\rangle$ by the shortest geodesic on the Bloch sphere. This line represents closing the trajectories by a postselected projective measurement at the end of the measurement sequence [see the discussion between Eqs. (10) and (12)]. It is known that, similar to the Berry phase, the Pancharatnam phase (14) for a spin-$\frac{1}{2}$ system may be expressed in terms of the solid angle enclosed by the closed trajectory. The trajectories under the direct and reversed measurement sequences do not subtend the same solid angle on the Bloch sphere. Their solid angles differ by sign and in absolute value. Clearly the Pancharatnam phases [see Eq. (14)] under the two time-reversed sequences are different.

Figure 3

Emergence of asymmetry of time-reversed measurement-induced trajectories. Two measurement sequences of spin projections $({\mathbf{n}}^{\left(s\right)}·{\mathit{\sigma }}^{\left(s\right)})$ onto subsequent directions ${\mathbf{n}}^{\left(s\right)}$ located on the same parallel (black line) are performed, a clockwise and a counterclockwise one, respectively. We illustrate the effect of the first two measurements of each sequence (clockwise and counterclockwise). All measurements are assumed to yield readouts $r=0$ and are characterized by Kraus operators ${\mathcal{M}}^{\left(0\right)}$ [Eq. (28)]. The initial state is represented by the red arrow and the measurement axes are shown by black arrows. If the backaction operator ${\mathcal{M}}^{\left(0\right)}$ is Hermitian, the clockwise and the counterclockwise trajectories are mirror reflections of each other (yellow arrows). However, for non-Hermitian backaction, there is an extra rotation around the measurement axes (cf. Fig. 1) leading to the two trajectories not being simply related to each other (green arrows show the state locations after the respective measurements; these locations are connected by the blue arrows to show what the clockwise trajectory is and what the counterclockwise one is).

Figure 4

Vanishing of the probability and windings of the phase in the postselective protocol $\{r_k=0\}$. (a) Contour plot of the logarithm of the postselection probability $ln{P}_{\{r_k=0\}}^{(d=+1)}(\theta =3\pi /4)$ (value indicated by color) as a function of measurement parameters $C$ and $A$ [cf. Eq. (40)]. Note the divergence at $(C,A)\approx (1,2)$. (b) Dependence of the phase ${\chi }_{\{r_k=0\}}^{(d=+1)}(\theta =3\pi /4)$ (value indicated by color) on $C$ and $A$. The phase is ill-defined at the singularity point $(C,A)\approx (1,2)$. Following the phase value around the singular point, the phase varies continuously from $-\pi$ to $\pi$, i.e., makes a $2\pi$ winding.

Figure 5

Phase ${\chi }_{\{r_k=0\}}^{(d=+1)}\equiv \chi$ [cf. Eq. (40)] as a function of $\theta$ for $A=1$ for $C$ above and below the critical value $C_{\mathrm{crit}}\approx 1.925$. The winding number $n$ [cf. Eq. (59)] is equal to 0 for $C<C_{\mathrm{crit}}$ and to $-1$ for $C>C_{\mathrm{crit}}$. The behavior of $\chi (\theta <{\theta }_{\mathrm{crit}})$ immediately above and below the transition is identical, while the dependence of $\chi (\theta >{\theta }_{\mathrm{crit}})$ differs by a $2\pi$ shift. This leads to $\chi (\theta ={\theta }_{\mathrm{crit}},C=C_{\mathrm{crit}})$ being ill-defined. The dependence of $P_{\{r_k=0\}}^{(d=+1)}\equiv P$ on $\theta$ at $(C=C_{\mathrm{crit}},A=1)$ is shown in the inset. The indeterminacy of $\chi (\theta ={\theta }_{\mathrm{crit}},C=C_{\mathrm{crit}})$ is enabled by $P(\theta ={\theta }_{\mathrm{crit}},C=C_{\mathrm{crit}})=0$.

Figure 6

Measurement-induced system trajectories $\left\{\right|{\psi }_k\rangle \}$ on the Bloch sphere for (a) $C=2.3>C_{\mathrm{crit}}$ and (b) $C=1.5<C_{\mathrm{crit}}$, above and below the transition for $A=1$. Different trajectories correspond to the protocols executed at different $\theta$. The red segments correspond to the quasicontinuous sequences $\left\{\right|{\psi }_{k=0,...,N}\rangle \}$, while the yellow segments are the shortest geodesics on the Bloch sphere connecting $|{\psi }_N\rangle$ with $|{\psi }_{N+1}\rangle \propto |{\psi }_0\rangle$. Above the critical measurement strength, the surface wraps around the Bloch sphere, while below, it does not. Confer Fig. 5 of Ref. [20] for the special case of $A=0$.

Figure 7

Critical lines of the topological transitions in the behavior of the weak-measurement-induced phase for the (a) postselective (${\chi }_{\{r_k=0\}}^{(d=+1)}$) and (b) averaging (${\overline{\chi }}^{(d=+1)}$) protocols. The lines follow the coordinates $(C_{\mathrm{crit}},A_{\mathrm{crit}})$. The critical polar angle ${\theta }_{\mathrm{crit}}^{(d=+1)}$ is shown by the color code. For $d=-1$, the transitions take place at the same $(C_{\mathrm{crit}},A_{\mathrm{crit}})$ but at ${\theta }_{\mathrm{crit}}^{(d=-1)}=\pi -{\theta }_{\mathrm{crit}}^{(d=+1)}$, as can be inferred from the symmetries discussed in Secs. 3c and 3d. The behavior at $A<0$ can be inferred too employing those symmetries.

Figure 8

(a) Dephasing ${\alpha }^{(+1)}$ and (b) phase ${\overline{\chi }}^{(+1)}$ [cf. Eq. (21)] at $\theta =3\pi /4$ color coded as functions of the measurement strength $C$ and asymmetry $A$ parameters. Note the two singularities at $C\approx 2$, where ${\alpha }^{(+1)}$ diverges. The phase makes $\pi$ windings around the points of divergent ${\alpha }^{(+1)}$.

Figure 9

Dependence of the averaged phase ${\overline{\chi }}^{(d=+1)}\left(\theta \right)$ on $\theta$ for various $C$ at (a) $A=1$ and (b) $A=0$.

Figure 10

Projections of the critical line $(C_{\mathrm{crit}},A_{\mathrm{crit}},{\theta }_{\mathrm{crit}}^{(d=+1)})$ for the averaged phase [cf. Fig. 7] onto the (a) $(A,\theta )$ and (b) $(C,\theta )$ planes.

Figure 11

Dependence of the averaged phase ${\overline{\chi }}^{(d=+1)}\left(\theta \right)$ on $\theta$ for various $A$ at (a) $C=1$ and (b) $C=3.5$.

Figure 12

Interferometry setups for observing measurement-induced phases (a) ${\chi }_{\{r_k=0\}}$ and (b) $\overline{\chi }$ [cf. Eqs. (12) and (13)]. A particle, whose spin represents the measured system, flies through the interferometer. Weak measurements of its spin are denoted by $M_{k=1,...,N}$, while the last postselected projective measurement is denoted by $P_{N+1}$. The protocol of (b) for detecting the averaged phase involves two special features. First, the particle spin is flipped in one arm as indicated by the “FLIP” boxes. Second, the detectors interact with the two arms via different Hamiltonians (see Appendix pp3 for details).

Figure 13

Comparison of Monte Carlo simulations for (a) the averaged phase ${\overline{\chi }}^{(d=+1)}$ and (b) the dephasing parameter ${\alpha }^{(d=+1)}$ with the results obtained using the semianalytical method of Sec. 3d. The number of Monte Carlo samples (readout sequence realizations $\left\{r_k^{\left(i\right)}\right\}$) used is $N_{\mathrm{rs}}=100$. The number of measurements in the sequence is denoted by $N$. The plots correspond to $C=3$ and $A=1$.

Figure 14

Trajectories of the quantum state $|{\psi }_k\rangle$ [Eq. (10)] on the Bloch sphere in various scaling limits. Plotted are the trajectories induced by sequences of $N$ measurements around the parallel corresponding to $\theta =\pi /4$ (black) with all measurements yielding readouts $r_k=0$. The backaction matrix $M^{\left(0\right)}$ is given in Eq. (A3). The trajectories are plotted for $b=0$ (purple), 0.1 (cyan), 0.3 (green), and 0.5 (red) using $C=1$, $A=1$, and $d=+1$. All the trajectories with $b<\frac{1}{2}$ converge towards the parallel line (i.e., the trajectory of the measurement axes) as $N\to \infty$. This does not happen for $b=\frac{1}{2}$. Moreover, the trajectory seems not to change with increasing $N$, suggesting that already at $N=20$ measurements the measurement-induced trajectory has converged.

Figure 15

(a) Imaginary part $\mathrm{Im}\left(\sqrt{P_{\{r_k=0\}}^{(d=+1)}}{e}^{i{\chi }_{\{r_k=0\}}^{(d=+1)}+iA}\right)\equiv \mathrm{Im}\left(\sqrt{P}{e}^{i(A+\chi )}\right)$ as a function of $\theta$ for various $C$ at $A=1$. One can check that the probability $P_{\{r_k=0\}}^{(d=+1)}=0$ at $(C_{\mathrm{crit}}\approx 1.925,A_{\mathrm{crit}}=1,{\theta }_{\mathrm{crit}}\approx 1.894)$. The imaginary part vanishes at $\theta ={\theta }_{\mathrm{crit}}$ for any $C$. (b) Dependence of $sinb/b$ and $cosb$ on $b$. Here $b_c$ is such that $sin{b}_c/b_c=1/\pi$.