# Topological transitions of the generalized Pancharatnam-Berry phase

Manuel F. Ferrer-Garcia  Nexus for Quantum Technologies, University of Ottawa,
K1N 5N6, ON, Ottawa, Canada  Kyrylo Snizhko  Department of Condensed Matter
Physics, Weizmann Institute of Science, Rehovot, 76100 Israel  Institute for
Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021
Karlsruhe, Germany  Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS,
38000 Grenoble, France  Alessio D’Errico<EMAIL_ADDRESS>Nexus for
Quantum Technologies, University of Ottawa, K1N 5N6, ON, Ottawa, Canada
Alessandro Romito  Department of Physics, Lancaster University, Lancaster LA1
4YB, United Kingdom  Yuval Gefen  Department of Condensed Matter Physics,
Weizmann Institute of Science, Rehovot, 76100 Israel  Ebrahim Karimi  Nexus
for Quantum Technologies, University of Ottawa, K1N 5N6, ON, Ottawa, Canada
<EMAIL_ADDRESS>
(August 28, 2024)

###### Abstract

Distinct from the dynamical phase, in a cyclic evolution, a system’s state may
acquire an additional component, a.k.a. geometric phase. The latter is a
manifestation of a closed path in state space. Geometric phases underlie
various physical phenomena, notably the emergence of topological invariants of
many-body states. Recently it has been demonstrated that geometric phases can
be induced by a sequence of generalized measurements implemented on a single
qubit. Furthermore, it has been predicted that such geometric phases may
exhibit a topological transition as function of the measurement strength.
Here, we demonstrate and study this transition experimentally employing an
optical platform. We show the robustness to certain generalizations of the
original protocol, as well as to certain types of imperfections. Our protocol
can be interpreted in terms of environment-induced geometric phases.

##  I Introduction

When a quantum state undergoes a cyclic evolution, the phase acquired is given
by the well-known dynamical component plus an additional contribution,
associated with the geometrical features of the path followed by the state.
This additional contribution is known as the geometric phase. The general
framework for the emergence of a geometric phase has been pointed out first by
Berry [1] in the context of adiabatic quantum evolution. A specific
realization of this phase had been earlier considered by Pancharatnam [2] in
his study of generalized interference theory. Pancharantnam’s theory shows how
geometric phases can be acquired in a non-adiabatic cyclic evolution, noting
that these are given by the area enclosed by the respective trajectory of the
system in the state space. The Pancharantnam phase can be observed following a
sequence of running projective measurements, each of a different observable,
where the last measurement projects on the initial state [3, 4]. Geometric
phases have found applications in several fields of physics [5], in
particular, in optics [6, 7, 8, 9] and condensed matter physics [10, 11, 12,
13, 14]. Importantly, the Berry phase is a key theme for understanding
topological phases of matter [14]. For instance, the Berry phase plays the
role of a topological invariant in one-dimensional chiral symmetric systems
[15, 16] and serve as the fundamental building block in the definition of
other topological invariants, such as Chern numbers [17]. Going beyond
Hamiltonian dynamics, the emergence of geometric phases has been predicted and
observed in the context of non-Hermitian evolution [18, 19, 20]; such phases
were further shown to emerge following a sequence of weak measurements [21,
22]. Further pursuing the latter theme, a major theoretical development has
revealed that dynamics comprising multiple measurements may assign topological
features to geometric phases. In particular, the limits of weak and strong
measurement are topologically distinct [22, 23, 24]. This prediction has
recently been confirmed employing a superconducting qubit platform [25]. That
study has implemented postselection on each individual measurement. This
aligns with the original theoretical proposal [22, 23, 24], yet it leaves the
question open: to what extent is the predicted topological transition a
feature of the specific laid-down protocol?

The experiment reported here not only employs a platform different from that
of Ref. [25] (namely, an optical platform), but also introduces a protocol
which is conceptually different: rather than exercising postselection on each
individual detector’s readout, here we implement postselection on a joint
readout of all measurements of the run. We find that a topological phase
transition takes place also under such generalized conditions, with distinct
values of the topological number characterizing the respective limits of
projective and infinitely weak measurements. Our experimental procedure
consists of a sequence of measurements, each implemented by a set of optical
elements. The key optical element is a polarization-sensitive beam displacer,
which is used to execute a weak measurement on the polarization state of a
laser beam. Employing additional elements (quarter wave plates, compensating
wave plates) serves to tune the measurement to a specific observable. The
strength of the measurement is determined by the ratio of the beam width and
the difference of transverse displacements of orthogonal polarizations.
Importantly, the detector’s readout is, in fact, the polarisation degree of
freedom of the photon, which, in turn, could be viewed as the system, while
the transverse position can be viewed as the environment. Our protocol could
then be interpreted as an environment inducing a geometric phase, highlighting
the dual nature of detector/environment. Finally, we investigate the
robustness of the observed topological properties with respect to setup
imperfections.

##  II Results

Theoretical overview. We consider a class of processes where $N$ measurements
are performed on a quantum system, as shown in Fig. 1a. Each step is a post-
selected measurement associated with the polarization state
$\left|\theta,\phi\right\rangle$, where $\theta$ and $\phi$ stand for the
polar and azimuthal coordinates on the Bloch sphere, respectively. We can
define a sequence of measurements $(\theta,\phi_{n})$ for a fixed value of
$\theta\in[0,\pi]$ while the azimuth is spanned in discrete steps
$\phi_{n}=2\pi n/(N+1)$. Let us denote the acquired geometric phase
$\chi_{\eta}(\theta)$, where $\eta\in[0,\infty)$ is introduced to indicate the
strength of the measurement. It can be shown that
$\Delta\chi_{\eta}=\chi_{\eta}(\pi)-\chi_{\eta}(0)=2\pi m$, where $m$ is an
integer – see Supplementary Information (SI), Section 1.3, for more details.

As illustrated in Fig. 1b, for infinitely weak measurements, $\eta\rightarrow
0$, the effect of each measurement is vanishingly small. Therefore,
$\chi(\theta)=0$ for any value of $\theta$, implying that
$\Delta\chi_{\eta\rightarrow 0}=0$. However, in the limit of projective
measurements, strong measurement, one observes that
$\Delta\chi_{\eta\rightarrow\infty}=2\pi$, as shown in Pancharatnam’s
geometric-phase theory [2]. An example of the latter case, strong measurement
limit, is illustrated in Fig. 1c: When $\theta=0$, the measurement sequence
does not change the projected state. Thus, the trajectory on the Bloch sphere
shrinks to a single point independently of the measurement strength. In
consequence, the enclosed area – and the geometrical phase – is zero. For
$\theta\rightarrow 0$ , the state follows a loop close to the initial
projected state, acquiring a small geometric phase. As $\theta\rightarrow\pi$,
the state follows a similar loop close to the south pole of the Bloch sphere,
thus the enclosed geometric phase is close to $2\pi$. This gives
$\Delta\chi_{\eta\rightarrow\infty}=2\pi$, as stated above. The distinction
between $\Delta\chi_{\eta\rightarrow 0}=0$ and
$\Delta\chi_{\eta\rightarrow\infty}=2\pi$ suggests the existence of a
transition in the behavior of the geometric phase, as the measurement strength
$\eta$ is varied. Since $\Delta\chi=2\pi m$, the nature of the transition is
topological.

Figure 1: Measurement-induced phase and its topological transition. a. The
state trajectory of a system is determined by a series of $N$ measurements
along different directions $(\theta,\phi)$. b and c. The state trajectory on
the Bloch sphere for a sequence of three measurements with strength $\eta=0.2$
(b) and $\eta=0.7$ (c) for different values of $\theta$. The black dashed line
corresponds to $\theta$, at which the measurements are performed. The black
points connected by red arrows denote the system state trajectory, as induced
by the measurements. The colored portion of the Bloch sphere is the solid
angle subtended by the respective trajectory. Figure 2:  Optical
implementation of a sequence of weak polarization measurements. a.
Experimental setup used to detect the geometrical phase acquired due to a
sequence of polarization measurements. A 632.9 nm laser emits a vertically
polarized Gaussian beam which impinges on a spatial light modulator (SLM) to
obtain a beam with a certain width $w_{0}$. The beam is split into two paths:
in one is subjected to a sequence of transformations, while a spatially
uniform phase $\delta$ is applied to the other path. Each stage is composed of
a quarter-wave plate (QWP), whose fast axis is oriented at angle $\alpha$ with
respect to the vertical, a beam displacer (BD), and a QWP for 808 nm acting as
the compensation wave plate (CWP). Finally, the output power of the
interference is recorded after the recombined beam passes through a vertical
polarizer. b. The measurement strength $\eta=d_{x}/w_{0}$ is controlled by
varying the waist parameter $w_{0}$ of the input beam. When $w_{0}$ is much
larger than the beam displacement $d_{x}$, the displacement is ineffectual,
corresponding to a weak measurement. For $w_{0}<d_{x}$, the two polarizations
become two well-separated beams, leading to a projective measurement in the
limit $w_{0}\rightarrow 0$. c. In contrast to the examples exhibited in Fig.
1b-c, the sequence of measurements produced by this setup corresponds to a
circle of $\theta=2\alpha$, which is additionally rotated by $2\alpha$ around
the x axis of the Bloch sphere. This does not affect the subtended area and,
consequently, the accumulated geometric phase.

Experimental setup. We demonstrate the existence of this topological
transition in an optical experiment where the qubit state is associated with
the polarization of a coherent beam. As illustrated in Fig. 2a, the beam goes
through a series of $N=3$ identical optical stages which emulate the
measurement steps. Each stage is composed of a quarter-wave plate (QWP), whose
fast axis is oriented at angle $\alpha=\theta/2$ with respect to the vertical,
followed by a YVO4 beam displacer (BD) and an additional compensating wave
plate (CWP). The BD’s ordinary and extraordinary axes are aligned along
$\mathbf{\hat{y}}$ and $\mathbf{\hat{x}}$, respectively. Therefore, the BD
shifts the centroid of the horizontally polarized component by a distance
$d_{x}$, keeping the vertically polarized contribution unchanged. The BD
essentially performs a measurement in the vertical/horizontal polarization
basis, as the horizontally polarized component of the beam is spatially
displaced. If the beam waist $w_{0}$ is larger than $d_{x}$, the measurement
is weak, since there is no sharp separation between the two polarizations
components (see Fig. 2b). If the waist is much smaller than the displacement,
$w_{0}\ll d_{x}$, this implements a projective measurement, as the two
polarization components are completely separated. Therefore, we can control
the measurement strength by modifying $w_{0}$. The CWP with a vertically
aligned fast axis is used to compensate for the phase difference between the
two polarization components accumulated while propagating inside the BD.

Figure 3: Experimentally measured and theoretically simulated geometric phase.
Topological transition in the measurement-induced geometric phase
$\chi(\alpha=\theta/2)$: a. theoretical modeling, b. experimental results, c.
modeling incorporating the imperfection of the birefringent crystals. The left
column corresponds to a narrow beam (small $w_{0}$, strong measurement) and
features $\Delta\chi=2\pi$. The right column corresponds to a large beam width
(weak measurement) and exhibits $\Delta\chi=0$. The middle column represents a
point close to the transition: the phase $\chi(\alpha)$ exhibits a sharp
change near $\alpha=\pi/4$, and the interference contrast vanishes near
$\alpha=\pi/4$, enabling the change of the phase’s topological behavior.

The role of the QWPs is to implement the desired sequence of measurement
directions $(\theta,\phi_{n})$ on the Bloch sphere. The rotation by angle
$\alpha$ enables controlling the polar angle $\theta$ of the measurement axis.
The sequence of measurements induced by the setup in Fig. 2a corresponds to
the directions $(\theta,\phi_{n})$ rotated by an angle $\theta=2\alpha$ around
the y axis of the Bloch sphere, cf. Fig. 2c. The details of this
correspondence are explained in the Section 2.1. Given the geometric nature of
the induced phase, the expected topological transition remains unaffected by
this rotation, both qualitatively and quantitatively. Finally, to complete the
cyclic evolution, the polarization state is projected onto the initial state
using a polarizer.

Our aim is to investigate the geometrical phase acquired by the non-deflected
beam (corresponding to the measurement postselected to yield a null outcome).
This is done by interfering the final state with the reference beam, which
only experiences a controllable phase shift $\delta$ introduced by a variable
phase plate. The output power at the interferometer exit is recorded as a
function of $\delta$. The shift of this curve corresponds to the acquired
geometric phase. The input beam is generated by means of a spatial light
modulator (SLM) that displays a hologram allowing to tailor the beam waist
through the technique introduced in Ref. [26]. We set the input beam’s
polarization state to be vertical ($\mathbf{\hat{y}}$), corresponding to the
initial state in the direction $(\theta,\phi_{0})$ in the theoretical
protocol. Based on this setup, the measurement protocol to unveil the hidden
topological transition is given as follows. The strength of our intermediate
$N=3$ measurements is regulated by varying the waist parameter of the input
beam: the value of $w_{0}$ is inversely proportional to the measurement
strength $\eta=d_{x}/w_{0}$ (see Fig. 2c). For a fixed waist parameter, we
proceed to get power readouts as a function of the reference arm phase shift
$\delta\in[0,2\pi]$, while $\alpha=\theta/2$ is kept constant. From here, it
is possible to retrieve the accumulated geometrical phase
$\chi_{\eta}(\theta=2\alpha)$ for a given orientation of QWPs, $\alpha$, by
proper curve fitting. By varying the QWPs orientation, it is possible to
reconstruct the behavior of $\chi_{\eta}(\theta=2\alpha)$ for all $\alpha$ for
a given measurement strength.

Experimental results. Here, we discuss the experimental results and their
relation to the theoretical predictions. Since the postselection in our
experiment goes beyond the original theoretical proposal, we have modeled the
experiment to confirm the presence of a topological transition theoretically
(see Section 2 of SI). Figure 3a shows that when $w_{0}$ is sufficiently
small, i.e. strong measurement regime, the simulation predicts
$\Delta\chi=\chi(\alpha=\pi/2)-\chi(\alpha=0)=2\pi$, while for the case of
weak measurements ($w_{0}<d_{x}$), $\Delta\chi=0$. A sharp transition is
occurs at $w_{0}=0.85$ mm, where the interference contrast vanishes for
$\alpha\approx\pi/4$, enabling the abrupt change of the phase behavior. The
experiment was carried on by performing measurements for $w_{0}$ between 0.6
mm and 2.5 mm. The experimental results, shown in Fig. 3b, clearly exhibit a
similar transition between $\Delta\chi=2\pi$ for small $w_{0}$ and
$\Delta\chi=0$ for large $w_{0}$, as well as the vanishing contrast at the
transition.

The difference $\Delta\chi=\chi(\pi/2)-\chi(0)$ in the observations is not
strictly equal to $0$ or $2\pi$, but can slightly deviate from these values.
This is seen most prominently for $w_{0}=0.6$ mm. We attribute this to the
stability of the Mach-Zehnder interferometer, in particular to a small drift
in the phase between the two arms during the measurement process (which was
performed in 45 minutes). We emphasize that this does not violate the
topological quantization of $\Delta\chi$, but introduces an error in its
extraction. In all the cases, the extracted $\Delta\chi$ is close to either
$0$ or $2\pi$, making the determination of the topological index $m$
straightforward. The vanishing contrast at the transition also confirms the
expected phenomenology of the topological transition.

We note that the waist $w_{0}^{*}$ at which the transition happens clearly
deviates from the theory predictions: $w_{0}^{*}=0.85$ mm in the simulation,
while $w_{0}^{*}=1.2$ mm in the experiment. We attribute this deviation to the
fact that the surfaces of the BDs are parallel within a few tens of
arcseconds, as stated by the manufacturer and verified by us independently.
This tiny angle between the two surfaces induces a small transverse wavevector
difference between the two components. We have incorporated this effect into
our theoretical modeling, the results of which are presented in Fig. 3c. With
this, we are able to reproduce the change in the transition location. A
detailed analysis of these imperfections and the enhanced modeling can be
found in Section 2.2 of the SI.

Theoretical studies have predicted [23, 24] that the topological transition
only exists if the dynamical phases are compensated accurately enough. In our
work, this condition is satisfied. In Fig. 4 we explored theoretically the
topological phase diagram considering the additional parameter $\gamma$
corresponding to the optical retardation of the CWP. The results show that
there is a range of values of $\gamma$ where the topological transition can be
observed, both in the ideal scenario and in the case of imperfect optical
elements.

Figure 4:  Topological phase dependence on compensating waveplates. The phase
diagram (theoretical) depicting the topological properties of the measurement-
induced phase as a function of beam waist $w_{0}$ and the compensation phase
$\gamma$. We present the results of a theory simulation without (left) and
including (right) the experimental imperfections of the birefringent crystals.
Note that the trivial phase with ($\Delta\chi=0$) exists only in a narrow
interval of the phase compensation parameter The vertical line indicates the
parameters used in our experiment. The imperfections of birefringent crystals
clearly make the trivial region shrink, yet do not eliminate the topological
transition.

##  III Discussion

We have demonstrated that measurement-induced geometric phases in optical
systems exhibit a topological transition. In particular, we consider a family
of processes parameterized by a variable $\alpha$ and a measurement strength
$\eta$. We demonstrated that the geometrical phase, with respect to $\eta$ and
$\alpha$, exhibits a nontrivial topology. More precisely, the variation
$\Delta\chi$ in geometrical phase as a function of $\alpha$ undergoes a sharp
transition of $2\pi$ as $\eta$ is varied. The parameter $\eta$ can be viewed
as the coupling strength with an environment, here represented by the light’s
spatial degree of freedom. In this framework, our observations can be
interpreted as topological transitions induced by the coupling to an external
environment. The topological transition is robust to amending the protocol and
the imperfections in the measurement process. The location of the topological
transition depends on specific details of the system (quality of beam
displacers, retardation of the compensating waveplates, etc.). This
sensitivity of the transition location may be useful for characterizing
optical elements or for sensing. We leave this, however, to future
investigations.

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Supplementary Information accompanies this manuscript.

Acknowledgments This work was supported by Canada Research Chairs (CRC),
Canada First Research Excellence Fund (CFREF) Program, NRC-uOttawa Joint
Centre for Extreme Quantum Photonics (JCEP) via High Throughput and Secure
Networks Challenge Program at the National Research Council of Canada,
Deutsche Forschungsgemeinschaft (German Research Foundation) through Project
No. 277101999, TRR 183 (Project C01), and Projects No. EG 96/13-1, GO
1405/6-1, and MI 658/10-2, Helmholtz International Fellow Award, the Israeli
Science Foundation (ISF), NSF Grant No. DMR-2037654, and the U.S.–Israel
Binational Science Foundation (BSF).

Author Contributions K.S., A.R., Y.G., and E.K. conceived the idea; M.F.,
A.D., K.S., and E.K. designed the experiment; M.F. and K.S. performed the
theoretical simulations; M.F. and A.D. performed the experiment and collected
the data; M.F., A.D., and K.S. analysed the data; M.F., K.S., A.D., and A.R.
prepared the first version of the manuscript. All authors discussed the
results and contributed to the text of the manuscript.

Author Information The authors declare no competing financial interests.
Correspondence and requests for materials should be addressed to
<EMAIL_ADDRESS>
Supplementary Information for:
Topological transitions of the generalized Pancharatnam-Berry phase

##  Section 1 Theoretical protocol and its relation to the experimental setup

Here we provide a brief theoretical background on quantum measurement theory,
on measurement induced geometric phases, and the topological transition in
them, as well as connect the quantum measurement formalism to the optical
setup employed in the paper.

###  Section 1.1 Null-weak measurements of different observables in the
polarization space

The formalism applies to the transition reported in the main text when
regarding the polarization of the laser beam as a quantum polarization state
$\left|\theta,\phi\right\rangle=\cos(\theta/2)\left|\uparrow\right\rangle+e^{i\phi}\sin(\theta/2)\left|\downarrow\right\rangle$
of a photon, where $\left|\uparrow\right\rangle$ and
$\left|\downarrow\right\rangle$ label the linearly independent vertical and
horizontal polarizations respectively with $\theta\in[0,\pi]$ and
$\phi\in[0,2\pi)$.

In the most general setting, a measurement of a quantum system in a state
$\left|\psi\right\rangle$ returns an outcome $r$ with probability
$P(r)=\langle\psi|M_{r}^{\dagger}M_{r}|\psi\rangle$, while the state is
updated as
$\left|\psi\right\rangle\to\left|\psi^{\prime}\right\rangle=M_{r}\left|\psi\right\rangle/\sqrt{P(r)}$.
The process is controlled by the Kraus operators $M_{r}$, which depend on the
specific detection process and fulfil $\sum_{r}M_{r}^{\dagger}M_{r}=1$ due to
overall probability conservation [27]. For what we are concerned here, we
specialize in a measurement process, known as null weak measurement, with two
possible outcomes, $r=+,\,-$, and corresponding Kraus operators

$M_{+}=\sqrt{\zeta}\left|\downarrow\right\rangle\left\langle\downarrow\right|,\quad
M_{-}=\left|\uparrow\right\rangle\left\langle\uparrow\right|+\sqrt{1-\zeta}\left|\downarrow\right\rangle\left\langle\downarrow\right|,$
(S1)

where $0\leq\zeta\leq 1$ corresponds to the measurement strength
$\eta=\sqrt{-\ln\left(1-\zeta\right)}$. For $\zeta=1$
($\eta\rightarrow\infty$), the measurement is projective: the operation
projects the state on $\left|\downarrow\right\rangle$ if $r=+$ and
$\left|\uparrow\right\rangle$ if $r=-$. For $\zeta<1$, however, we either have
a collapse to $\left|\downarrow\right\rangle$ (“click”, described by $M_{+}$)
or no collapse (“no-click” or “null measurement”, described by $M_{-}$). In
the latter case, the system’s state is updated to a the post-measurement
state, which depends on the pre-measurement one.

The measurement procedure in Eq. (S1) corresponds to an imperfect, or weak,
measurement of $\sigma_{z}$. This can be generalized to measure arbitrary
observables, i.e. to distinguish two arbitrary basis states. A direction
$\mathbf{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ is identified
by the polar and azimuthal angles $\theta$ and $\phi$ on the Bloch sphere. The
corresponding orthogonal basis states,
$\left|\uparrow_{\mathbf{n}}\right\rangle$ and
$\left|\downarrow_{\mathbf{n}}\right\rangle$, are defined as eigenstates of
$\mathbf{n}\cdot\bm{\sigma}$ associated with the respective eigenvalue $\pm
1$. For $\phi=0$ and $\pi$ these states correspond to two mutually orthogonal
linear polarizations, whereas for all other values of $\phi$ they correspond
to general elliptic polarizations. The analog of Eq. (S1) is then given by
$M_{+}\left(\mathbf{n}\right)=\sqrt{\zeta}\left|\downarrow_{\mathbf{n}}\right\rangle\left\langle\downarrow_{\mathbf{n}}\right|$,
$M_{-}\left(\mathbf{n}\right)=\left|\uparrow_{\mathbf{n}}\right\rangle\left\langle\uparrow_{\mathbf{n}}\right|+\sqrt{1-\zeta}\left|\downarrow_{\mathbf{n}}\right\rangle\left\langle\downarrow_{\mathbf{n}}\right|$.

###  Section 1.2 Measurement-induced geometric phases

In order to induce a geometric phase, one needs to use multiple measurements.
We label them by $j=1,...,N$. Each measurement is determined by the observable
it measures, i.e., by the direction $\mathbf{n}_{j}$. The respective Kraus
operators are $M_{r_{j}}^{(j)}=M_{r_{j}}\left(\mathbf{n}_{j}\right)$, where
$r_{j}=\pm$ is the readout of measurement $j$.

For each such measurement with a given outcome, the phase of the post-
measurement state is gauge-dependent, hence non-physical. However, if a
sequence of measurements leads to a post-measurement final state
$\left|\psi_{f}\right\rangle$ which is proportional to the initial one,
$\left|\psi_{0}\right\rangle$, the phase difference between
$\left|\psi_{f}\right\rangle$ and $\left|\psi_{0}\right\rangle$ is a legit
observable given by

$\chi_{\\{r_{j}\\}}={\rm arg}\langle\psi_{0}|\psi_{f}\rangle={\rm
arg}\left\langle\psi_{0}\right|M_{r_{N}}^{(N)}\dots
M_{r_{2}}^{(2)}M_{r_{1}}^{(1)}\left|\psi_{0}\right\rangle,$ (S2)

where $r_{j}$ is the outcome (readout) of the $j$-th measurement, whose effect
is encoded in the Kraus operator $M_{r_{j}}^{(j)}$.

Note that even if the final state, $\left|\psi_{f}\right\rangle$, differs from
the initial one, the phase defined in Eq. (S2) is still well-defined: this can
be understood introducing a fake projective measurement onto the initial
state, $\left|\psi_{0}\right\rangle\left\langle\psi_{0}\right|$, after the
application of all $M_{r_{j}}^{(j)}$, in order to force
$\left|\psi_{f}\right\rangle\propto\left|\psi_{0}\right\rangle$. Therefore,
Eq. (S2) defines a legitimate observable for a general sequence of
measurements. The sequence of (normalized) post-measurement intermediate
states, $\left|\psi_{0}\right\rangle$,
$M_{r_{1}}^{(1)}\left|\psi_{0}\right\rangle$, $\dots$,
$M_{r_{N}}^{(N)}...M_{r_{1}}^{(1)}\left|\psi_{0}\right\rangle$, defines a
trajectory on the Bloch sphere. This trajectory is given by the set of
geodesics connecting the states. For Hermitian Kraus operators,
$M_{r_{j}}^{(j)\dagger}=M_{r_{j}}^{(j)}$ (which is the case in Eq. (S1)), the
measurement-induced phase, (S2), has a geometric interpretation as
$\chi_{\\{r_{j}\\}}=\Omega/2$, where $\Omega$ is the solid angle subtended by
the trajectory on the Bloch sphere [21, 22, 23, 24].

###  Section 1.3 Protocol for topological transition in measurement-induced
phases

Consider a family of measurement sequences, defined in the main text. Each
sequence consists of $N$ measurements corresponding to
$\left(\theta,\phi_{j}=2\pi j/(N+1)\right)$, where $j=1,...,N$ is the
measurement number. The family is obtained when considering such sequences at
all $\theta\in[0,\pi]$. We perform postselection, which restricts measurement
readouts to be $r_{j}=-$ for all $j$. According to Eq. (S2), this defines a
phase $\chi_{\left\\{-\right\\}}(\theta)$ for each measurement sequence in the
family. For briefness, we denote this phase $\chi(\theta)$.

The function $\chi(\theta)$ possesses a topological invariant. This follows
from the fact that at $\theta=0$ and $\theta=\pi$,
$M_{r_{j}}^{(j)}\left|\psi_{0}\right\rangle=\left|\psi_{0}\right\rangle$ for
all $j$, implying that
$\left|\psi_{f}\right\rangle=\left|\psi_{0}\right\rangle$. That is, the
measurements do not change the system state, so that the resulting phase is
trivial: $e^{i\chi(0)}=e^{i\chi(\pi)}=1$. As any phase, the measurement-
induced phase is defined up to an integer multiple of $2\pi$. Without loss of
generality, we can set $\chi(0)=0$. This, however, eliminates the freedom of
adding multiples of $2\pi$ at all other $\theta$ due to the natural demand of
continuity of $\chi(\theta)$. In particular, $\chi(\pi)$ may be non-zero; yet
$e^{i\chi(\pi)}=1$ implies that $\chi(\pi)=2\pi m$ with integer $m$. In other
words, the difference $\Delta\chi=\chi(\pi)-\chi(0)$, must be quantized in
units of $2\pi$, as stated in the main text. Further, the quantization of
$\Delta\chi$ implies that its value cannot be changed by continuous
deformations of the function $\chi(\theta)$. Therefore, $\Delta\chi$
constitutes a topological invariant.

As discussed in the main text, for infinitely weak ($\zeta\to 0$) and
projective ($\zeta\to 1$) measurements $\Delta\chi_{\zeta\rightarrow 0}=0$ and
$\Delta\chi_{\zeta\rightarrow 1}=2\pi$ respectively. This necessitates a jump
in the topological invariant, i.e., a topological transition, at some critical
measurement strength, $\zeta_{c}\in[0,1]$.

###  Section 1.4 Detection of measurement-induced geometric phases

In order to detect a measurement-induced phase, one needs to interfere the
state that underwent measurements with the initial unmeasured state, as in
Fig. 2a. Here we provide a theoretical description of this in the quantum
measurement formalism.

The incoming photon in polarization state $\left|\psi_{0}\right\rangle$
becomes, after the beam splitter, the state
$\left|\psi\right\rangle=\frac{1}{\sqrt{2}}\left|\psi_{0}\right\rangle\otimes\left(\left|0\right\rangle+\left|1\right\rangle\right)$
where $\left|0\right\rangle$ and $\left|1\right\rangle$ refer to the two arms
of the interferometer. A sequence of measurements with Kraus operators
$M_{r_{j}}\left(\mathbf{n}_{j}\right)$ is performed in the interferometer arm
denoted as $\left|0\right\rangle$. The arm denoted as $\left|1\right\rangle$
features only a phase shifter, $\delta$. Therefore, after passing through the
respective interferometer arms the photon state is

$\left|\psi\right\rangle_{int}=\frac{1}{\sqrt{2}}\left[\sum_{\\{r_{j}\\}}\left|\Psi_{0}\left(\\{r_{j}\\}\right)\right\rangle\otimes\left|0\right\rangle+\left|\Psi_{1}\right\rangle\otimes\left|1\right\rangle\right],$
(S3)

where

$\displaystyle\left|\Psi_{0}\left(\\{r_{j}\\}\right)\right\rangle$
$\displaystyle=M_{r_{N}}^{(N)}...M_{r_{1}}^{(1)}\left|\psi_{0}\right\rangle\otimes\left|\left\\{r_{j}\right\\}\right\rangle,$
(S4) $\displaystyle\left|\Psi_{1}\right\rangle$
$\displaystyle=e^{i\delta}\left|\psi_{0}\right\rangle\otimes\left|\left\\{r_{j}=-\right\\}\right\rangle.$
(S5)

Here we introduced the collective state of all the detectors,
$\left|\left\\{r_{j}\right\\}\right\rangle$. In the arm denoted by
$\left|1\right\rangle$ the detectors do not interact with the photon and thus
stay in the no-click position.

The second beam splitter converts
$\left|0\right\rangle\rightarrow\frac{1}{\sqrt{2}}\left(\left|0\right\rangle+\left|1\right\rangle\right)$,
$\left|1\right\rangle\rightarrow\frac{1}{\sqrt{2}}\left(\left|0\right\rangle-\left|1\right\rangle\right)$,
so that the output state is

$\displaystyle\left|\psi\right\rangle_{out}=\frac{1}{2}\biggl{[}\left(\left|\Psi_{1}\right\rangle+\sum_{\\{r_{j}\\}}\left|\Psi_{0}\left(\\{r_{j}\\}\right)\right\rangle\right)\otimes\left|0\right\rangle+\left(-\left|\Psi_{1}\right\rangle+\sum_{\\{r_{j}\\}}\left|\Psi_{0}\left(\\{r_{j}\\}\right)\right\rangle\right)\otimes\left|1\right\rangle\biggr{]}.$
(S6)

Therefore, the probability of the photon appearing at output ports
$\left|0\right\rangle$ or $\left|1\right\rangle$ is given by

$\displaystyle P_{0/1}$
$\displaystyle=\frac{1}{2}\left[1\pm\sum_{\\{r_{j}\\}}\mathrm{Re}\thinspace
e^{-i\delta}\langle\Psi_{1}|\Psi_{0}\left(\\{r_{j}\\}\right)\rangle\right]$
$\displaystyle=\frac{1}{2}\left[1\pm\mathrm{Re}\thinspace
e^{-i\delta}\left\langle\psi_{0}\right|M_{-}^{(N)}...M_{-}^{(1)}\left|\psi_{0}\right\rangle\right].$
(S7)

Note that only the term with no-click readouts, $\\{r_{j}=-\\}$, in the
measured interferometer arm contributes to the interference, which thus
enables the observation of the measurement-induced phase, cf. Eq. (S2) and
Section 1.2. In other words, the interference implicitly performs the
postselection to $\\{r_{j}=-\\}$ required by the definition of $\chi(\theta)$,
cf. Section 1.3.

##  Section 2 Optical implementation of the null-weak measurement

In our optical implementation of the measurements, the detectors are not two-
state systems with possible readouts $r=\pm$. We use the photon spatial degree
of freedom, i.e., its location in the $xy$ plane (transverse to the
propagation direction). The formal description of this is as follows. The
incident photon’s electric field can be described as

$\displaystyle\mathbf{E}_{0}(x,y)=\left(\begin{array}[]{c}E_{0y}(x,y)\\\
E_{0x}(x,y)\end{array}\right)=\left(\begin{array}[]{c}E_{0y}\\\
E_{0x}\end{array}\right)\sqrt{\frac{2}{\pi
w_{0}^{2}}}e^{-(x^{2}+y^{2})/w_{0}^{2}}e^{ikz}.$ (S12)

The measurement is implemented via a beam displacer (see Figure S1a) that
shifts the $x$-polarized component in space:

$\displaystyle\mathbf{E}(x,y)=\mathrm{BD}\>\mathbf{E}_{0}(x,y)=\sqrt{\frac{2}{\pi
w_{0}^{2}}}\left(\begin{array}[]{c}e^{ikn_{y}L_{y}}E_{0y}e^{-(x^{2}+y^{2})/w_{0}^{2}}\\\
e^{ikn_{x}L_{x}}E_{0x}e^{-(\left[x-d_{x}\right]^{2}+y^{2})/w_{0}^{2}}\end{array}\right)e^{ikz}.$
(S15)

Apart from the displacement, the phases associated with propagation in the
beam displacer, $kn_{x}L_{x}$ and $kn_{y}L_{y}$, are imprinted onto the
polarization components. The overall phase is not important, whereas the
difference $\gamma=kn_{x}L_{x}-kn_{y}L_{y}$ may lead to observable
consequences, cf. Fig. 4. In our protocol we compensate this phase difference,
see below. Therefore, here we put, for simplicity,
$kn_{x}L_{x}=kn_{y}L_{y}=0$.

If, after experiencing the BD, the beam were to interfere with the original
beam, the interference term would be

$\displaystyle\int
dx\,dy\>\mathbf{E}_{0}^{*}(x,y)\mathbf{E}(x,y)=\left|E_{0y}\right|^{2}+\left|E_{0x}\right|^{2}e^{-d^{2}/(2w_{0}^{2})}=\left(\begin{array}[]{c}E_{0y}\\\
E_{0x}\end{array}\right)^{\dagger}M_{-}\left(\begin{array}[]{c}E_{0y}\\\
E_{0x}\end{array}\right),$ (S20)

where $M_{-}$ is the matrix defined in Eq. (S1) if
$\left|\uparrow\right\rangle$ is interpreted as the $y$ polarization and
$\left|\downarrow\right\rangle$ as the $x$ polarization, while
$\sqrt{1-\zeta}=e^{-d_{x}^{2}/(2w_{0}^{2})}$. Therefore, a beam displacer
implements a postselected null weak measurement in the photon’s polarization
space. The measurement strength
$\eta=\sqrt{-\ln\left(1-\zeta\right)}=d_{x}/w_{0}$, as defined in the main
text. The limit of projective measurement corresponds to
$\eta\rightarrow\infty$, while the infinitely weak measurement corresponds to
$\eta\rightarrow 0$.

Note that in our actual setup, cf. Fig. 2a, the interference happens after
three beam displacements have been performed. Therefore, the postselection is
implemented not on the readout of each individual measurement, but on the
combined “readout” of all measurements. This consitutes an important
conceptual difference compared to the original definition of the measurement-
induced phase and its topological transition, cf. Section 1.2 and Section 1.3.
Observation of the topological transition in our work, thus, underlines that
the transition is not a feature of a specific narrow protocol, but a more
general phenomenon.

_Phase difference compensation._ In order to compensate the unwanted phase
difference $\gamma=kn_{x}L_{x}-kn_{y}L_{y}$, one can employ a phase plate,

$\text{P}(\varphi)=\begin{pmatrix}e^{i\varphi/2}&0\\\\[6.45831pt]
0&e^{-i\varphi/2}\end{pmatrix}.$ (S21)

Choosing $\varphi=\gamma$ and placing the phase plate after the beam displacer
leads to

$\displaystyle\text{P}(\gamma)\mathrm{\mathrm{\>BD\>}}\mathbf{E}_{0}(x,y)=\sqrt{\frac{2}{\pi
w_{0}^{2}}}e^{ik(n_{x}L_{x}+n_{y}L_{y})/2}\times\left(\begin{array}[]{c}E_{0y}e^{-(x^{2}+y^{2})/w_{0}^{2}}\\\
E_{0x}e^{-(\left[x-d_{x}\right]^{2}+y^{2})/w_{0}^{2}}\end{array}\right)e^{ikz},$
(S24)

leaving one only with an unimportant overall phase. The overall phase is
unimporant because it does not depend on the incoming polarization, and thus
can be calibrated away.

In our setup, cf. Fig. 2a, the required phase compensation is implemented with
a quarter wave plate for a wave length distinct from that of the laser we
employ. We denote it as CWP.

_Measuring different observables._ The measurement procedure described above
leads to the back action matrix $M_{-}$ as defined in Eq. (S1), i.e., to
measuring $\sigma_{z}$. In order to implement measurements of different
observables $\mathbf{n}\cdot\bm{\sigma}$, corresponding to
$\mathbf{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$, one needs to
be able to (i) discriminate different linear polarizations (not only
horizontal and veritcal) with a beam displacer and (ii) convert elliptical
polarizations to linear and back, so that they can be discriminated by the
beam displacer.

(i) can be implemented by rotating the beam displacer in the $xy$ plane:

$\mathrm{BD}(\theta/2)=R(\theta/2)\mathrm{\>BD\>}R(-\theta/2),$ (S25)

with the rotation matrix

$R(\theta/2)=\begin{pmatrix}\cos\theta/2&-\sin\theta/2\\\
\sin\theta/2&\cos\theta/2\end{pmatrix}.$ (S26)

(ii) can be implemented by placing phase plates $P(\pm\phi)$ before and after
the beam displacer.

Therefore, a measurement of $\mathbf{n}\cdot\bm{\sigma}$ as described in
Section 1.1 can be implemented via a sequence of elements that involves a
rotated BD and CWP, as well as two phase plates:

$\mathcal{M}(\theta,\phi)=\mathrm{P}(-\phi)R(\theta/2)\text{P}(\gamma)\mathrm{\>BD\>}R(-\theta/2)\mathrm{P}(\phi).$
(S27)

Note that in order to rotate the measurement axis by $\theta$, one needs to
perform real space rotations by $\alpha=\theta/2$.

This sequence uses four elements per measurement, whereas our setup in Fig. 2
features only three optical elements per measurement. We describe how this is
achieved in the next section.

###  Section 2.1 Simplifying the experimental setup

The protocol for observing the topological transition requires sending in a
laser beam with polarization

$\mathbf{E}_{in}=\left(\begin{array}[]{c}E_{0y}\\\
E_{0x}\end{array}\right)=\left(\begin{array}[]{c}\cos\theta/2\\\
\sin\theta/2\end{array}\right)=R(\theta/2)\left(\begin{array}[]{c}1\\\
0\end{array}\right).$ (S28)

and using $N$ measurements $\mathcal{M}(\theta,\phi_{j})$, where the
measurement stages are defined in Eq. (S27) and $\phi_{j}=2\pi j/(N+1)$. The
number of required optical elements can be reduced. In order to do this, one
needs two observations.

First, consider the incoming polarization and the first measurement:

$\displaystyle\mathcal{M}(\theta,\phi_{1})\left(\begin{array}[]{c}\cos\theta/2\\\
\sin\theta/2\end{array}\right)=\mathrm{P}(-\phi_{1})R(\theta/2)\mathrm{\text{P}(\gamma)}\mathrm{\>BD\>}\underbrace{R(-\theta/2)\mathrm{P}(\phi_{1})R(\theta/2)}\left(\begin{array}[]{c}1\\\
0\end{array}\right).$ (S33)

The block $R(-\theta/2)\mathrm{P}(\phi_{1})R(\theta/2)$ can be interpreted as
a phase plate rotated by the angle $\alpha=\theta/2$,
$\mathrm{P(\phi_{1},\alpha)}$.

Second, consider two sequential measurements

$\mathcal{M}(\theta,\phi_{j+1})\mathcal{M}(\theta,\phi_{j})=\mathrm{P}(-\phi_{j+1})R(\theta/2)\mathrm{\text{P}(\gamma)}\mathrm{\>BD\>}\underbrace{R(-\theta/2)\mathrm{P}(\phi_{j+1})\mathrm{P}(-\phi_{j})R(\theta/2)}\mathrm{\text{P}(\gamma)}\mathrm{\>BD\>}R(-\theta/2)\mathrm{P}(\phi_{j}).$
(S34)

The block $R(-\theta/2)\mathrm{P}(\phi_{j+1})\mathrm{P}(-\phi_{j})R(\theta/2)$
can be replaced with a single rotated phase plate
$\mathrm{P}(\phi_{j+1}-\phi_{j},\alpha)=\mathrm{P}(2\pi/(N+1),\alpha)=\mathrm{P}(\phi_{1},\alpha)$.

Therefore, instead of having a rotated incoming polarization and rotated beam
displacers, one can have vertical incoming polarization and rotated phase
plated $\mathrm{P}(\phi_{1},\alpha)$ before the beam displacers. Note that
this setup simplification involves replacing all phase plates
$\mathrm{P}(\phi_{j})$ with their rotated versions
$R(-\theta/2)\mathrm{P}(\phi_{j})R(\theta/2)$ and the input polarization
$R(\theta/2)\begin{pmatrix}1&0\end{pmatrix}^{\mathrm{T}}$ with
$\begin{pmatrix}1&0\end{pmatrix}^{\mathrm{T}}$. The simplified setup is
related to the orignial protocol by rotating all the measurement axes
$\mathbf{n}_{j}$ by angle $\theta$ around the $y$ axis of the Bloch sphere.

For our choice of $N=3$, we have $\phi_{1}=\pi/2$, making the required phase
plates $\mathrm{P}(\phi_{1},\alpha)$ quarter wave plates and leading to the
setup in Fig. 2a.

###  Section 2.2 Effect of imperfect birefringent crystals

In our previous description, we have assumed that both the extraordinary and
ordinary components of the field exit from the birefringent crystal along
parallel directions. Nevertheless, while extremely small, the beam displacer
(THORLABS BDY12) manufacturer mentions that both components are parallel to
each other within 30 arcseconds. Figure S1-b shows the intensity profile from
a YBO4 crystal when illuminated with diagonally polarized light, and a
polarizer is placed on the output. The presence of an interference pattern
corroborates the existence of a small deviation in the propagation direction
of one of the output beams.

Following these results, we modified the model for our beam displacer to
account for our crystals’ imperfections. First, we assumed that only the
extraordinary component experiences a deflection from the optical axis by an
angle $\beta$. Due to variability from one crystal to another, we allow the
deviation to occur along any direction in the transverse plane, characterized
by an angle $\nu$. Therefore, it is possible to write the effect of an
imperfect beam displacer on an impinging electric field as

$\displaystyle\mathbf{E}(x,y)=\mathrm{BD}_{\nu,\beta}\>\mathbf{E}_{0}(x,y)=\sqrt{\frac{2}{\pi
w_{0}^{2}}}\left(\begin{array}[]{c}e^{ikn_{y}L_{y}}E_{0y}e^{-(x^{2}+y^{2})/w_{0}^{2}}\\\
e^{ikn_{x}\Delta
L_{x}}E_{0x}e^{-(\left[x-d_{x}\right]^{2}+y^{2})/w_{0}^{2}}\end{array}\right)e^{ikz}.$
(S37)

where $\Delta=\sin\beta(\cos\nu x+\sin\nu y)$ implements the deflection on the
extraordinary component. Therefore, it is possible to describe the evolution
of the input beam $\mathbf{E}_{0}(x,y)$ due to a sequence of $N$ stages with
distinct imperfections as

$\mathbf{E_{u}}(x,y)=\prod_{j=1}^{N}\mathcal{M}_{\nu_{j},\beta_{j}}(\theta,\phi_{j})\mathbf{E}_{0}(x,y),$
(S38)

where $\mathcal{M}_{\nu,\beta}(\theta,\phi)$ is obtaining by substituting Eq.
(S37) into Eq. S27. At first glance at the effect of these imperfections on
the location of the transition, let us consider the case when the $N=3$
measurement stages possess identical crystals. As shown in Figure S1c, the
location of the topological transition shifts towards higher values of $w_{0}$
when the deflection angle increases.

Nevertheless, this assumption is not valid in our experiment, since each
individual crystal is different. As a consequence, the behavior of the
geometrical phase curve depends directly on three pairs $(\nu,\beta)$ which
quantify the nonparallelism between the faces of each BD. A genetic algorithm
(GA)[28] was implemented to perform the search for the set of optimal
parameters $P$ that match the experimental results.

Figure S1:  Modeling an imperfect beam displacer. a. Schematic of a beam
displacer (BD). b. Interference pattern obtained from the output beams after
projecting on circular polarization (and with a circularly polarized input
beam). c.  Geometrical phase curves for different values of the deviation
angle $\beta$.