Abstract
Disorder and nonHermiticity dramatically impact the topological and localization properties of a quantum system, giving rise to intriguing quantum states of matter. The rich interplay of disorder, nonHermiticity, and topology is epitomized by the recently proposed nonHermitian topological Anderson insulator that hosts a plethora of exotic phenomena. Here we experimentally simulate the nonHermitian topological Anderson insulator using disordered photonic quantum walks, and characterize its localization and topological properties. In particular, we focus on the competition between Anderson localization induced by random disorder, and the nonHermitian skin effect under which all eigenstates are squeezed toward the boundary. The two distinct localization mechanisms prompt a nonmonotonous change in profile of the Lyapunov exponent, which we experimentally reveal through dynamic observables. We then probe the disorderinduced topological phase transitions, and demonstrate their biorthogonal criticality. Our experiment further advances the frontier of synthetic topology in open systems.
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Introduction
Topological edge states in topological materials are robust against weak perturbations, an ability originating from the global geometry of eigen wave functions in the Hilbert space^{1,2}. Such an intrinsic geometric feature is captured by global topological invariants that are related to edge states through the bulkboundary correspondence. However, this conventional paradigm is challenged by localization under disorder^{3,4,5,6} or nonHermiticity^{7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}, which have become the focus of study of late, particularly in light of recent experimental progress in synthetic topological systems^{25,26,27,28,29,30,31,32,33,34,35}. On one hand, despite its gapclosing tendency, the disorder can induce topology from a trivial insulator. In the resulting topological Anderson insulator, the global topology emerges in a bulk with localized states, in the absence of translational symmetry^{3,4,5,6}. On the other hand, in a broad class of nonHermitian topological systems, the nominal bulk states are exponentially localized toward boundaries under the nonHermitian skin effect^{8,9,10,11,12,13,14,15,16,17,18,19,20,21,22}. The deviation of the bulkstate wave functions from the extended Bloch waves invalidates the conventional bulkboundary correspondence, necessitating the introduction of nonBloch topological invariants^{8,9,10,11}. While the two localization mechanisms differ in origin and manifestation, the topology of the underlying system gets fundamentally modified in either case. Remarkably, in the recently proposed nonHermitian topological Anderson insulator^{36,37,38,39}, the two distinct localization mechanisms are pitted against each other, wherein the interplay of disorder, nonHermiticity, and topology leads to exotic phenomena such as the nonmonotonous localization, disorderinduced nonBloch topological phase transitions, and biorthogonal critical behaviors.
In this work, we report the experimental observation of nonHermitian topological Anderson insulators in singlephoton quantumwalk dynamics. Driven by a nonunitary topological Floquet operator, the quantum walk undergoes polarizationdependent photon loss and acquires the nonHermitian skin effect. In contrast to previously implemented quantum walks with the nonHermitian skin effect^{30,35}, our current experiment resorts to the timemultiplexed configuration, with the spatial degrees of freedom encoded in the discrete arrival time of photons at the detector^{40}. This enables us to implement quantum walks with a larger number of time steps, which is pivotal for the current experiment. We introduce static random disorder through parameters of the optical elements^{41}, which would result in a complete localization of bulk states in the largedisorder limit. In the intermediate regime with moderate loss and disorder, the competition between the nonHermitian skin effect and Anderson localization yields nonmonotonic localization features which we characterize by measuring the Lyapunov exponent^{20}. Using the biorthogonal chiral displacement, we then probe the topological phase transition, which is in qualitatively agreement with theoretical predictions. At the measured topological phase boundary, the biorthogonal localization length diverges, consistent with the biorthogonal critical nature of the phase transition^{36,37,38}. We further measure topological edge states from dynamics close to the boundary of the nonHermitian topological Anderson insulator.
Results
A timemultiplexed nonunitary quantum walk
We implement a onedimensional photonic quantum walk governed by the Floquet operator
Here the shift operator is given by \(S={\sum }_{x}\leftx1\right\rangle\left\langle x\right\otimes \leftH\right\rangle \left\langle H\right+\leftx+1\right\rangle \left\langle x\right\otimes \leftV\right\rangle \left\langle V\right\), with \(\leftH\right\rangle\) (\(\leftV\right\rangle\)) the horizontally (vertically) polarized state. The nonunitary operator \(M={\sum }_{x}\leftx\right\rangle \left\langle x\right\otimes \big({{{e}^{\gamma }}\atop {{0}}}\;{{0} \atop {{e}^{\gamma }}}\big)\) with γ the gainloss parameter. The coin operator \(R(\theta )={\sum }_{x}\leftx\right\rangle \left\langle x\right\otimes \big({{\cos \theta}\atop{\sin \theta}}\; {{\sin \theta}\atop{\cos \theta}}\big)\), where the matrix is in the basis \(\{\leftH\right\rangle ,\leftV\right\rangle \}\). For the quantumwalk dynamics, U is repeatedly acted upon the walker state, giving rise to discretetime Floquet dynamics. The quantum walk governed by U features the nonHermitian skin effect (see Supplemental Material), which originates from a nonvanishing bulk probability flow that we confirm later with dynamic measurements.
For the experimental implementation, we adopt a timemultiplexed scheme, as illustrated in Fig. 1. Photons are sent through an interferometric network consisting of optical elements for a half step of the discretetime quantum walk in Eq. (1). The shift operator is implemented by separating the two polarization components and routing them through fibers of different lengths to introduce a polarizationdependent time delay, such that the walker position is mapped to the time domain. For instance, a superposition of multiple spatial positions at a given time step is translated into the superposition of multiple wellresolved pulses within the same discretetime step. A pair of wave plates are introduced into each of the paths, to realize a polarizationdependent loss operation \({M}_{{{{{{{{\rm{E}}}}}}}}}={\sum }_{x}\leftx\right\rangle \left\langle x\right\otimes\left(\leftH\right\rangle \left\langle H\right+{e}^{2\gamma }\leftV\right\rangle\left\langle V\right\right)\), which is related to M through M = e^{γ}M_{E}. We, therefore, read out the timeevolved state driven by U by adding a timedependent factor e^{γt} to our experimental measurement. To implement the coin operator, an electrooptical modulator (EOM) is inserted into the main interferometric cycle, in combination with wave plates, to provide a carefully timesequenced control over θ. Importantly, the EOM enables an individualpulseresolved coin operation, providing the basis for the implementation of a walkerpositiondependent disorder. The disorder is introduced to the operator R(θ_{1}) in Eq. (1), where the actual rotation angle is modulated by a small positiondependent δθ(x), with δθ(x) randomly taking values within the range of \(\left[W,W\right]\). Here W indicates the disorder strength. We implement only static disorder for our experiments, such that δθ(x) does not change with time steps.
For the input and outcoupling of the interferometric network, a beam splitter (BS) with a reflectivity of 5% is introduced, corresponding to a low coupling rate of photons into the network, but also enabling the outcoupling of photons for measurement. For that purpose, two avalanche photodiodes (APDs) are employed to record the outcoupled photons’ temporal and polarization properties, yielding information regarding the number of time steps, as well as the spatial and coin states of the walker.
NonHermitian skin effect
Whereas the nonHermitian skin effect is typically associated with nonreciprocity^{8}, it can also occur in systems with onsite loss^{7,30}. Here the nonHermitian skin effect is a result of the interplay of an onsite, polarizationdependent loss (M_{E} operator) and an effective coupling between the polarization and spatial modes (S operator). While a defining signal of the nonHermitian skin effect is the accumulation of Eigen wave functions at the boundary, it also impacts dynamics in the bulk, leaving unique signatures in the Lyapunov exponent. Here the Lyapunov exponent is defined as \(\lambda (v)=\mathop{\lim }\limits_{t\to \infty }\frac{1}{t}\log \left\right.\psi (x=vt,t)\left\right.\)^{20}, where v is the shift velocity, and ψ(x, t) is the wavefunction component at position x and time step t. Remarkably, for a system with the nonHermitian skin effect, λ(v) takes a maximum value at v ≠ 0 for bulk dynamics far away from any boundary^{20}. By contrast, in the absence of the nonHermitian skin effect, λ(v) acquires a symmetric profile with respect to its peak at v = 0. Intuitively, from the definition of the Lyapunov exponent, it is understood that, if λ(v) peaks at a shift velocity v_{m} at time t, the timeevolved wave function must peak at x = v_{m}t. A finite peak shift velocity thus reflects a directional wave function propagation in the bulk (or equivalently, a persistent bulk current), which lies at the origin of the nonHermitian skin effect. Alternatively, the nonHermitian skin effect can also be confirmed by dynamics close to a boundary (see Supplemental Material).
For our experiment, we implement tenstep quantum walks without imposing any boundary or domain and measure the polarizationaveraged growth rate
Here the additional average over polarization enables us to qualitatively capture the distinctive features of the Lyapunov exponent using a relatively small number of time steps (t = 10). In Eq. (2), the polarizationresolved growth rates are defined as \({\lambda }_{i}(v)=\frac{1}{t}\log  {\psi }_{x = vt}^{(i)}\). To construct \({\psi }_{x = vt}^{(i)}=\left\langle i\right\otimes \left\langle x\right{U}^{t}\left0\right\rangle \otimes \lefti\right\rangle\) (i = H, V), we initialize the walker in the state \(\left0\right\rangle \otimes \lefti\right\rangle\), and projectively measure the probability distribution of photons in the polarization state \(\lefti\right\rangle\) of the spatial mode \(\leftx\right\rangle\), following the last time step (t = 10). Note that the average over polarization in Eq. (2) is taken for faster convergence of the growth rate at a finite evolution time to the Lyapunov exponent.
In Fig. 2, we show the measured polarizationaveraged growth rates as functions of the shift velocity, for a, c the unitary, and b, d the nonunitary cases, both without the disorder. Apparently, under the nonHermitian skin effect (γ ≠ 0), the peak of the growth rate lies with a finite v (Fig. 2b), in contrast to the more symmetric profile without skin effect (Fig. 2a). Such a growthrate profile directly originates from the directional propagation of probability in the bulk, as clearly indicated in the measured polarizationresolved probability distributions after the final time step (Fig. 2c, d). In the presence of open boundaries, the directional probability propagation naturally leads to the accumulation of population at the boundaries. Note that the ability to infer the existence of the nonHermitian skin effect from bulk dynamics confirms that the nonHermitian skin effect is not merely a finitesize effect, but has a profound impact even within the thermodynamic limit.
Competition with Anderson localization
We now switch on disorder and investigate the interplay between the nonHermitian skin effect and disorder^{36,37}. In Fig. 3, we show the measured \(\bar{\lambda }(v)\) for increasing disorder strength W, under a fixed nonHermitian parameter γ. When W is small, the asymmetric profile persists (see Fig. 3a, d), indicating the dominance of the nonHermitian skin effect. A careful comparison between Fig. 2b and Fig. 3a suggests the emergence of another peak at v = 0, though only just visible in Fig. 3a. The peak at v = 0 rapidly rises with increasing W. This leads to the twinpeak structure under an intermediate W, as shown in Fig. 3b and e. This is a direct evidence for the competition between the disorderinduced Anderson localization and the nonHermitian skin effect. Finally, for sufficiently large W, \(\bar{\lambda }(v)\) again peaks at v = 0, as Anderson localization completely suppresses probability flow in the bulk that leads to the nonHermitian skin effect. Such a competition as revealed by our experiment is consistent with the nonmonotonous localization predicted in ref. ^{36}, where the inverse participation ratio is used to characterize the competition (see Supplemental Material).
Disorderinduced topology
The Floquet operator U is topological, protected by the chiral symmetry with ΓUΓ = U^{−1}, where \({{\Gamma }}={\sum }_{x}\leftx\right\rangle \left\langle x\right\otimes {\sigma }_{x}\). While the topology of U generally persists under small random disorder, the disorder can also induce nontrivial topology from a topologically trivial state, similar to the case with the topological Anderson insulator in Hermitian systems^{3,4,5,6}. We emphasize that the topology discussed here is to be differentiated from the spectral topology of the nonHermitian skin effect, with the latter indicating closedloop structures of the eigenenergy spectra on the complex plane^{17,18}.
In Fig. 4a, we plot the theoretical phase diagram, characterized through the disorderaveraged local marker under the nonBloch band theory (see Supplemental Material). The yellow (blue) region corresponds to the topologically nontrivial (trivial) phase, thus the nonHermitian topological Anderson insulator state corresponds to the yellow region with the finite disorder (W > 0). Here the biorthogonal local marker, calculated over a unit cell deep in the bulk, plays the role of a topological invariant in the presence of disorder and converges to the nonHermitian winding number for W = 0 (see Methods). Incidentally, for our choice of U, the topological phase boundary is insensitive to γ, despite the presence of the nonHermitian skin effect and the application of the nonBloch band theory. Nevertheless, the biorthogonal localization length, rather than the conventional localization length, diverges at the topological phase boundary (solid black curve in Fig. 4a)^{37}, suggesting a unique nonHermitian criticality. From our observation in previous sections, a nonHermitian topological Anderson insulator with persistent signatures of the nonHermitian skin effect is expected in the yellow region of Fig. 4a with W ≲ 0.4, where the disorder has not become dominant.
Here we focus on the impact of the disorder on the topological phase boundary, which we experimentally probe through the time and disorderaveraged biorthogonal chiral displacement, defined for a tstep quantum walk as^{26,37}
where \(\left{\psi }_{n}(t)\right\rangle ={U}^{t}\left\psi (0)\right\rangle\) and \(\left{\chi }_{n}(t)\right\rangle ={\left[{({U}^{1})}^{{{{\dagger}}} }\right]}^{t}\left\psi (0)\right\rangle\), \(\left\psi (0)\right\rangle =\left0\right\rangle \otimes \leftV\right\rangle\), the subscript n indicates the nth disorder configuration (with a total of N configurations), and X is the position operator. Experimentally, we prepare \(\left{\psi }_{n}(t)\right\rangle\) and \(\left{\chi }_{n}(t)\right\rangle\) by separately evolving the initial state with U and \({({U}^{1})}^{{{{\dagger}}} }\), followed by state tomography to reconstruct \(\left{\psi }_{n}(t)\right\rangle\) and \(\left{\chi }_{n}(t)\right\rangle\), respectively, before calculating \(\bar{C}\) according to Eq. (3).
In Fig. 4b, we plot the measured \(\bar{C}\). Similar to ref. ^{26}, while the measured chiral displacement varies smoothly across the topological phase boundaries due to the limited number of time steps amenable to our experiment, it does show a tendency consistent with the theoretically predicted phase boundaries. Numerically, it is found that \(\bar{C}\) approaches the topological invariants given by the local marker (dashed line) at much larger time steps. The measured \(\bar{C}\) is insensitive to γ, consistent with theoretical predictions using the local marker.
To provide direct evidence for the topological nature of the nonHermitian topological Anderson insulating state, in Fig. 4c, d, we show the spatial probability distributions following tenstep quantumwalk dynamics close to a domainwall configuration, where the left (x ≤ −1) and right (x ≥ 0) regions feature different parameters (θ_{2} in our experiment). When the two regions belong to different topological phases, the timeevolved probability shows a prominent peak at the boundary, indicating the presence of topological edge states (Fig. 4c). This is in sharp contrast to Fig. 4d, where both regions are in the same topological phase. Note that to minimize the impact of the nonHermitian skin effect, we choose a parameter regime where the nonHermitian skin effect leads to a directional probability flow through the boundary (corresponding to the probability peaks in the region x ≤ −1 in Fig. 4c, d), such that the probability accumulation at the boundary in Fig. 4c is unambiguously associated with edge states.
Discussion
We report the first experimental observation of a nonHermitian topological Anderson insulator, achieved by introducing disorder to a discretetime nonunitary quantum walk with topology and nonHermitian skin effect. Using dynamic observables, we demonstrate the two competing localization mechanisms inherent in the system and reveal a disorderinduced topological phase transition. Our experiment lays the foundation for interesting theoretical questions as to the fate of localized states in a nonHermitian manybody system with skin effect, as well as the interplay of nonHermiticity, disorder, and manybody interactions therein. On the application side, disorder and nonHermiticity provide convenient control over key properties of nonHermitian Anderson insulators, opening routes toward the design of a tunable optical device for engineered quantum transport.
For future studies, it is hopeful to further increase the evolution time of the quantumwalk dynamics based on the timemultiplexed configuration, such that a more accurate determination of the Lyapunov exponent can be achieved. It would also be interesting to explore similar competitions for higher dimensional nonHermitian topological Anderson insulators.
Methods
Experimental setup
To implement quantum walks governed by the Floquet operator U in Eq. (1), we adopt a timemultiplexed configuration, encoding the internal coinstate degrees of freedom in the photonic polarization, and the external spatial modes in the discretized temporal shift within a time step^{40}. The overall experimental configuration is illustrated in Fig. 1.
The wave packets of photons are generated by a pulsed laser source with a central wavelength of 808 nm, a pulse width of 88 ps, and a repetition rate of 31.25 kHz. The pulses are attenuated to the singlephoton level using neutral density filters at the detection stage. For a unitary quantum walk, the probability that a photon undergoes a full roundtrip without getting lost or detected is about 0.59 per step and the detection efficiency is 0.03 per step (taking into account the efficiency of APDs and the reflectivity of BSs). We ensure the average photon number per pulse at the detection stage to be less than 2 × 10^{−4} so that there is a negligible probability of a multiphoton event.
To implement U with a fiber loop configuration, we rewrite the tstep timeevolution operator as \({U}^{t}={e}^{2\gamma t}{U}_{{{{{{{{\rm{E}}}}}}}}}^{t}\), where
and
Here the coin operator R(θ_{i}) and the shift operator S are the same as those in Eq. (1) and the ensuing discussions. The polarizationdependent loss operator \({M}_{{{{{{{{\rm{E}}}}}}}}}={\sum }_{x}\leftx\right\rangle \left\langle x\right\otimes \left(\leftH\right\rangle \left\langle H\right+{e}^{2\gamma }\leftV\right\rangle \left\langle V\right\right)\), which is related to M through M = e^{γ}M_{E}. For each cycle in the interferometric network, the walker state is subject to the operation M_{E}SR(θ), where θ is alternatingly modulated to be 2θ_{2} or θ_{1} for odd or even cycles. As such, one cycle in the network roughly corresponds to a half step of the quantum walk. The coin operators R(θ_{2}) and R(−θ_{2}) are implemented at the input and outcoupling stage, respectively.
More specifically, the operator R(−θ_{2}) [R(θ_{2})] in Eq. (1) is implemented using two halfwave plates (HWPs) with setting angles −θ_{2}/2 (θ_{2}/2) and 0, respectively, before (after) the photon is sent into (coupled out of) the network. For the input, photons are reflected by a lowreflectivity BS with a reflectivity of 5%, such that there is a 5% probability to couple a photon into the network. The same BS is subsequently used as the outcoupler, where photons, after completing cycles in the interferometer, have a 5% probability of being reflected out of the cycle and into the detection module.
Within each interferometer cycle, the photon is first sent through a sandwichtype, QWP(0)EOM(4θ_{2})QWP(90^{∘}) configuration^{42}, which is used to implement the coin operator R(2θ_{2}) or R(θ_{1}) in Eq. (5). Here QWP is the abbreviation for quarterwave plates. The birefringent crystal inside the EOM is set at 45^{∘} to the x/y axis so that the EOM acts on the photon polarization as \({\tilde{R}}_{{{{{{{{\rm{EOM}}}}}}}}}(\vartheta )=\big({{1}\atop{1}}\; {{1}\atop{1}}\big)\big({{{e}^{i\frac{\vartheta}{2}}}\atop{0}}\;{{0}\atop{{e}^{i\frac{\vartheta }{2}}}}\big)\big({{1}\atop{1}}\; {{1}\atop{1}}\big)=\big({{\cos \frac{\vartheta }{2}}\atop{i\sin \frac{\vartheta }{2}}}\; {{i\sin \frac{\vartheta }{2}}\atop{\cos \frac{\vartheta }{2}}}\big)\). The properties impose that ϕ_{V}(x)/ϕ_{H}(x) = −1. Thus, in combination with a pair of wave plates, an EOM can be used to modify the polarization of each pulse individually, providing the basis for realizing positiondependent coin operations \(R(\vartheta )=\big({{1}\atop{0}}\; {{0}\atop{i}}\big){\tilde{R}}_{{{{{{{{\rm{EOM}}}}}}}}}(2\vartheta )\big({{1}\atop{0}}\; {{0}\atop{i}}\big)=\big({{\cos \vartheta}\atop{\sin \vartheta}}\; {{\sin \vartheta}\atop{\cos \vartheta}}\big)\). For a disorderfree quantum walk, we sequence the EOM such that ϑ = 2θ_{2} for odd cycles and ϑ = θ_{1} for even cycles.
The shift operator S is implemented by separating different polarization components of a photon using polarizing beam splitters (PBSs) and routing them through fibers of different lengths to introduce a welldefined time delay in between. Specifically, horizontally polarized photons traverse the fiber loop in 751.680 ns, while vertical ones take 33.046 ns longer to complete the trip. The resulting temporal difference corresponds to a step in the spatial domain of x ± 1. As such, each position in each time step is represented by a unique discretetime bin, i.e., the position information is mapped into the time domain.
To implement the loss operator M_{E}, a pair of HWPs are inserted into each fiber loop, one at the entrance and one near the exit. Since the operator M_{E} induces a loss in the polarization state \(\leftV\right\rangle\) with a probability 1 − e^{−4γ}, we adjust the setting angles of the HWPs, such that only the desired components are reflected (transmitted) by the PBS at the exit of the short (long) fiber loop into the blocker, rendering the dynamic within the main cycle nonunitary. We, therefore, read out the evolved states from our experiment with M_{E} by adding a factor e^{γt}.
At the output of the shift operator, the two paths are coherently recombined, and photons are sent back to the input BS for the next splitstep. In order to realize a full time step, two cycles in the interferometer network are required, with the setting angle of the EOM alternating between 2θ_{2} (odd cycle) and θ_{1} (even cycle). We introduce static disorder to the coin operator R(θ_{1}) for odd cycles. This is achieved by modulating the setting of EOM by a small random amount \(\delta \theta \in \left[W,W\right]\) around θ_{1}. Here δθ is positiondependent but timeindependent. Such static disorder preserves the chiral symmetry of U.
Finally, after a photon has completed multiple cycles and is coupled out of the network by the BS (with a probability of 5%), the coin operator R(θ_{2}) is applied, and the photon registers a click at an APD with a time jitter 350 ps for detection.
State tomography
For the detection of the timeaveraged chiral displacement, we reconstruct the final state \(\left\psi (t)\right\rangle ={U}^{t}\left\psi (0)\right\rangle\) and its left vector \(\left\chi (t)\right\rangle ={\left[{({U}^{1})}^{{{{\dagger}}} }\right]}^{t}\left\psi (0)\right\rangle\) for each time step. Here we take the reconstruction of \(\left\psi (t)\right\rangle\) as an example. Since U and the initial state \(\left\psi (0)\right\rangle =\left0\right\rangle \otimes \leftV\right\rangle\) are purely real in the polarization basis \(\{\leftH\right\rangle ,\leftV\right\rangle \}\), we have the expansion
where the coefficients p_{μ}(t, x) (μ = H, V) are also real. Based on these, we perform three distinct measurements M_{i} (i = 1, 2, 3) to reconstruct \(\left\psi (t)\right\rangle\) in the basis \(\{\leftH\right\rangle ,\leftV\right\rangle \}\). This amounts to measuring the absolute values and the r signs of the real coefficients p_{μ}(t, x), as we detail in the following.
First, we measure the absolute values \(\left{p}_{\mu }(t,x)\right\). After the tth time step, photons in position x are sent to a detection unit M_{1}, which consists of PBS and APDs. M_{1} applies a projective measurement of the observable σ_{z} on the polarization of photons. The counts of the horizontally polarized photons N_{H}(t, x) and vertically polarized ones N_{V}(t, x) are registered by the coincidences between one of the APDs in the detection unit, and the APD for the trigger photon. The measured probability distributions are
where \(c(t)={{{{{{{\rm{Tr}}}}}}}}\left[{U}_{{{{{{{{\rm{E}}}}}}}}}^{t} \psi (0)\right\rangle \left\langle \psi (0) {({U}_{{{{{{{{\rm{E}}}}}}}}}^{{{{\dagger}}} })}^{t}\right]\). The square root of the probability distribution P_{μ}(t, x) corresponds to \(\left{p}_{\mu }(t,x)\right\).
Second, we determine the relative sign between the amplitudes p_{H}(t, x) and p_{V}(t, x) via the detection unit M_{2}, which consists of an HWP at 22.5^{∘}, a PBS, and APDs. The only difference between M_{2} and M_{1} is the HWP at 22.5^{∘}, i.e., a projective measurement of the observable σ_{x} on the polarization components of photons. The difference between the probability distributions of the horizontally and vertically polarized photons is given by
which determines the relative sign between p_{H}(t, x) and p_{V}(t, x).
Third, we probe the relative sign between the amplitudes p_{H}(t, x) and \({p}_{V}(t,x^{\prime} )\), which is necessary to calculate the summation of wave functions in different positions at each time step. We take the relative sign between the amplitudes in the positions x and x − 2 as an example. To this end, a detection unit M_{3} is introduced, consisting of an extra loop, an HWP at 22.5^{∘}, a PBS, and APDs. In the extra loop, the EOM is set to realize a rotation R(θ_{2} + 3π/4). The horizontally polarized photons at both x and x − 2 are combined at the end of the loop. The projective measurement of the observable σ_{x} is applied to the polarization components of photons via an HWP at 22.5^{∘}, a PBS, and APDs. The difference between the probability distributions of the horizontally and vertically polarized photons is given by
As we have determined the relative sign between p_{H}(t, x) and p_{V}(t, x) [between p_{H}(t, x − 2) and p_{V}(t, x − 2)] with M_{2}, we determine, using M3, the relative sign between p_{μ}(t, x) and p_{μ}(t, x − 2) for arbitrary x.
Note that, as the purpose of reconstructing the final state is to calculate the expectation value of the averaged chiral displacement, the global sign of p_{μ}(x, t) is unimportant.
Biorthogonal local marker and chiral displacement
Following refs. ^{26,37}, the biorthogonal local marker is defined as
where \(\leftm,s\right\rangle\) is the sublattice state s of the mth unit cell, and X is the unitcell position operator. The biorthogonal projection operator Q = P_{+} − P_{−}, with \({P}_{\pm }={\sum }_{n}\left{\phi }_{\pm }^{(n)}\right\rangle \left\langle {\chi }_{\pm }^{(n)}\right\). Where \(\left{\phi }_{\pm }^{(n)}\right\rangle\) is the nth right eigenstate of U, satisfying \(U\left{\phi }_{\pm }^{(n)}\right\rangle ={\lambda }_{\pm }^{(n)}\left{\phi }_{\pm }^{(n)}\right\rangle\); and \(\left\langle {\chi }_{\pm }^{(n)}\right\) is the nth left eigenstate, with \({U}^{{{{\dagger}}} }\left{\chi }_{\pm }^{(n)}\right\rangle ={\lambda }_{n,\pm }^{* }\left{\chi }_{\pm }^{(n)}\right\rangle\). Here λ_{n,+} (λ_{n,−}) lies in the lower (upper) half of the complex plane. Similar to the analysis in refs. ^{26,37}, the biorthogonal local marker serves as the topological invariant in a disordered system, and is reflected in the disorder and timeaveraged chiral displacement defined in Eq. (3).
Data availability
All other data, any related experimental background information not mentioned in the text, and other findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.
Code availability
Any simulation and computational codes for this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
This work has been supported by the National Natural Science Foundation of China (Grant Nos. 12025401, U1930402, 11974331, and 12088101). W.Y. acknowledges support from the National Key Research and Development Program of China (Grant Nos. 2016YFA0301700 and 2017YFA0304100). L.X. acknowledges support from the Project Funded by China Postdoctoral Science Foundation (Grant Nos. 2020M680006 and 2021T140045).
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Q.L. performed the experiments with contributions from K.W. and L.X. W.Y. developed the theoretical aspects and performed the theoretical analysis with contributions from T.L. and wrote part of the paper. P.X. supervised the project, designed the experiments, analyzed the results, and wrote part of the paper.
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Lin, Q., Li, T., Xiao, L. et al. Observation of nonHermitian topological Anderson insulator in quantum dynamics. Nat Commun 13, 3229 (2022). https://doi.org/10.1038/s41467022309389
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DOI: https://doi.org/10.1038/s41467022309389
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