Abstract
Topological insulators combine insulating properties in the bulk with scatteringfree transport along edges, supporting dissipationless unidirectional energy and information flow even in the presence of defects and disorder. The feasibility of engineering quantum Hamiltonians with photonic tools, combined with the availability of entangled photons, raises the intriguing possibility of employing topologically protected entangled states in optical quantum computing and information processing. However, while twophoton states built as a product of two topologically protected singlephoton states inherit full protection from their singlephoton “parents”, a high degree of nonseparability may lead to rapid deterioration of the twophoton states after propagation through disorder. In this work, we identify physical mechanisms which contribute to the vulnerability of entangled states in topological photonic lattices. Further, we show that in order to maximize entanglement without sacrificing topological protection, the joint spectral correlation map of twophoton states must fit inside a welldefined topological window of protection.
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Introduction
The prospect of generating topologically protected entangled states of several photons is a highly intriguing proposition^{1,2,3}. Specifically, topological protection can enable robust transport of quantum information across disordered photonic structures without degradation^{4,5}, just as efficiently as for singleparticle wavepackets^{6,7,8,9,10}.
In recent years, we have witnessed several experimental demonstrations of topological protection at the singlephoton level in integrated onedimensional lattice systems. Notably, Wang and coworkers^{11} showed that the fundamental quantum features of spatially entangled biphotonstates can be protected against disorder in the socalled SuSchriefferHeeger (SSH) topological lattice. Interestingly, SSH lattices turned out to be equally effective in protecting polarizationentangled photon pairs^{12}. Another important ingredient was provided by Tambasco et al.^{13} showing that HongOuMandel twophoton interference of topological edge–modes is feasible, by implementing a topological beamsplitter in a judiciously engineered timedependent Harpermodel.
Concurrently, on the theory front several ideas have been suggested to investigate topological twophoton effects in linear^{14,15} and nonlinear^{16} lattice systems. In this regard, an intriguing proposition was recently put forward^{17}, where the BoseHubbard model, which is topologically trivial for single particles, becomes topologically nontrivial for two interacting photons. That is, particle interactions have a dramatic impact on topological properties, not only modifying the topology of the spectra but also creating a topological order in otherwise topologically trivial systems.
In order to maximize the potential of topological photonic networks for transferring quantum information, it is indispensable to have a considerable number of edge modes at our disposal. One possibility is to use twodimensional topological systems, which intrinsically support a multitude of topological edgestates^{18,19,20}.
In twodimensional photonic topologial insulators, singleparticle edgestates reside in the gap existing between the energy bands supporting the bulk states^{21,22,23}. Thus, breaking the topological protection requires disorder with sufficient strength to close the bandgap. For states describing two indistinguishable photons, the same bandgap is fundamentally lacking. The reason is because the propagation eigenvalues \({\lambda }_{12}^{(2)}\) for twophoton eigenstates in a photonic system are given by the sum of the eigenvalues λ_{1}, λ_{2} corresponding to the constituent individual photons, \({\lambda }_{12}^{(2)}={\lambda }_{1}+{\lambda }_{2}\). This implies that we can keep \({\lambda }_{12}^{(2)}\) constant while increasing λ_{1} and simultaneously decreasing λ_{2}, or vice versa. In this way, we can combine two singlephoton bulk states, one from the lower and one from the upper band, to create a biphoton bulk–bulk state whose energy lies inside the singleparticle bandgap. This fundamental additive property of the singleparticle eigenvalues removes the bandgap and leads to massive degeneracies of the edge–edge, edge–bulk, and bulk–bulk twophoton states. Hence, considering the lack of the topological bandgap for twophoton systems, it is not clear whether topological protection will be automatically granted to twoparticle states provided single particles are topologically protected in the same system.
In solids, the degeneracies described above lead to the decay of twoelectron edge states when electron–electron correlations are substantial^{24,25,26}. This decay mechanism is reminiscent of autoionization, where electron–electron correlations lead to energy exchange between the two particles, coupling two bound electronic excitations to an energydegenerate boundcontinuum twoelectron state^{27,28}.
Still, photonic systems are fundamentally different from solids, as the two photons do not readily interact with each other^{29}. Consequently, the evolution operator for twophoton states, U^{(2)}(z), breaks down into the product of two propagators for individual singlephoton states, U^{(2)}(z) = U(z) ⊗ U(z)^{30}. Thus, a natural question to ask is whether such a factorization and the absence of bangap will prevent decoherence and dissipation of nonfactorizable twophoton edgestates into the bulk?
In this work, we analyze possible mechanisms of dissipation of twophoton edge states into the bulk of two different topological insulator system, the Haldane lattice model and an aperiodic lattice corresponding to the quantum Hall effect. Our results show that the key to topological protection is to minimize the disorderinduced overlap of the initial twophoton (joint) spectrum with the edge–bulk and bulk–bulk spectral regions.
Results
Theoretical approach
In lattice systems, static disorder can be introduced in either the site energies—termed diagonal disorder^{31}—or in the coupling coefficients—socalled offdiagonal disorder^{32}. In either case, static disorder is represented by a singleparticle operator \({\hat{V}}^{(1)}\). Since such perturbation is timeindependent, energy conserving resonant coupling into the bulk is absent within firstorder perturbation theory—the singleparticle transition induced by \({\hat{V}}^{(1)}\) does not preserve energy. The process that can resonantly couple a twophoton edge–edge state to a bulk–bulk, or to a bulk–edge, state would require a correlated change of states for both photons and it might arise within the secondorder corrections in \({\hat{V}}^{(1)}\).
To see this, we examine the secondorder transition matrix elements between an initial twophoton edge–edge state \(\left{\rm{i}}\right\rangle =\left{n}_{{\rm{i}}},{m}_{{\rm{i}}}\right\rangle\) and a final edge–bulk, or bulk–bulk, state \(\left{\rm{f}}\right\rangle =\left{n}_{{\rm{f}}},{m}_{{\rm{f}}}\right\rangle\)
where \(\leftj^{\prime} \right\rangle =\leftn^{\prime} ,m^{\prime} \right\rangle\) labels intermediate virtual states and \({\lambda }_{j^{\prime} }^{(2)}={\lambda }_{n^{\prime} }+{\lambda }_{m^{\prime} }\). The singleparticle nature of \({\hat{V}}^{(1)}\) ensures that only two terms corresponding to the two possible timeorderings of the two singleparticle transitions are left in the sum
The stationary nature of the disorder dictates that real transitions from the edge–edge states can occur only if the initial eigenvalue \({\lambda }_{{\rm{i}}}^{(2)}={\lambda }_{{n}_{{\rm{i}}}}+{\lambda }_{{m}_{{\rm{i}}}}\) is equal to the final one \({\lambda }_{{\rm{f}}}^{(2)}={\lambda }_{{n}_{{\rm{f}}}}+{\lambda }_{{m}_{{\rm{f}}}}\). Therefore, \({\lambda }_{{n}_{{\rm{f}}}}{\lambda }_{{n}_{{\rm{i}}}}=({\lambda }_{{m}_{{\rm{f}}}}{\lambda }_{{m}_{{\rm{i}}}})\), and the two terms in \({V}_{{\rm{i}},{\rm{f}}}^{(2)}\) exactly cancel each other, \({V}_{{\rm{i}},{\rm{f}}}^{(2)}=0\). That is, twoparticle dissipation from each product state is zero, and the same is true for any entangled twoparticle state ψ^{(2)} represented as a quantum superposition of two possible distinguishable configurations \({\psi }_{{\rm{i}}}^{(2)}\), e.g., \({\psi }^{(2)}={\psi }_{1}^{(2)}+{\psi }_{2}^{(2)}\). Physically, this destructive interference is a direct outcome of the indistinguishability of the photons, which ensures that the twoparticle eigenvalue is a sum of the two singleparticle eigenvalues, and that the twoparticle propagator is a product of the singleparticle counterparts.
In contrast to dissipation, the situation with dephasing can be different: while each constituent state in an entangled superposition can be protected against disorder^{1}, the overall superposition is, in general, not. To be precise, motion through different disordered regions may lead to disorderinduced random phase shifts between the states destroying the entanglement. To avoid this fate, all states in the superposition must travel across the same spatial region of the photonic structure, such that they are affected by disorder in the exact same manner^{33}. These effects have been explored for spatial pathentangled states^{1}, and for states built from an entangled superposition of an initial nonstationary state \({\psi }_{1}^{(2)}\) with its timedelayed replica \({\psi }_{1}^{(2)}(\tau )\), that is, entangled states in the timedomain^{2}. In these two cases, however, the entanglement of the states can be related to the entanglement of symmetrized wavefunction of identical particles^{34}. Consequently, the states exhibit the lowest possible amount of entanglement, as indicated by the corresponding Schmidt numbers S_{N} = 2. Throughout this work we use the Schmidt number to quantify the amount of entanglement: S_{N} = 1 denotes complete separability while S_{N} ≫ 1 corresponds to high entanglement^{35}.
A more appealing type of highly entangled twophoton states are multimode optical Gaussian states in which both photons are most likely to be found inhabiting any waveguide, within an excitation window, simultaneously^{36}. The importance of such states is based on the fact that any phase difference arising among the paths becomes enhanced by a factor of two in comparison with single photon states^{37}. Naturally, the enhanced phase sensitivity of such highly entangled twophoton states manifests as faster diffraction of the associated wavepackets propagating in any photonic system, periodic and disordered^{38}. Therefore, it is not clear to what extent topological protection will persist for these types of highly entangled states.
Propagation of entangled twophoton states in disordered topological lattices
In what follows we analyze the impact of disorder onto a continuum of twophoton states that extend from the correlated to the anticorrelated limits, passing through a completely separable state. For our analysis we consider two topological lattices, one periodic and one aperiodic. In the periodic case we consider the Haldane model^{39}, and for the aperiodic we use a square lattice whose singleparticle dynamics corresponds to the quantum Hall effect^{6,40} (information, S. Supporting material). The results for the Haldane model are presented here, while the quantum Hall effect lattice is discussed in the Supplementary Note 5.
In optics, a firstorder approximation of the Haldane model can be implemented using a honeycomb lattice composed of helical waveguides as illustrated in Fig. 1a, see pioneering work^{41}. In this system, every waveguide has a nearestneighbor coupling κ_{1} and a complex secondnearestneighbor coupling κ_{2} or \({\kappa }_{2}^{* }\), see Fig. 1b. At the singlephoton level, the Haldane lattice is governed by the Hamiltonian^{1}
where β_{i} represents the propagation constant of the ith waveguide and the corresponding optical mode is represented by the creation (annihilation) operator, \({\hat{a}}_{i}^{\dagger }\) (\({\hat{a}}_{i}\)). Notice, in a disorderfree lattice β_{i} = β. The symbols 〈〉 and 〈〈〉〉 indicate summation over nearest and nextnearestneighbor sites, respectively. The lattice used in our simulations is a ribbon with N_{y} = 90 hexagons in the ydirection and N_{x} = 10 hexagons in xdirection, Fig. 1c. We normalized the units in terms of κ_{1} throughout this work, and set κ_{2} = iκ_{1}/5.
For pure states of two indistinguishable noninteracting particles the Hamiltonian is H_{2} = H ⊗ I + I ⊗ H, where H is the singleparticle Hamiltonian and I is the identity operator^{42}. The twophoton eigenstates are given by the symmetric tensorproduct combinations of the singlephoton eigenstates
As alluded to above, the twophoton eigenvalues are the sums of the singlephoton ones, \({\lambda }_{m,n}^{(2)}={\lambda }_{m}+{\lambda }_{n}\).
In the absence of disorder, the eigenvalue spectrum for singlephoton states in a finite lattice exhibits topological edge states in the bandgap^{43}, Fig. 1d. In contrast, for two indistinguishable photons, the spectrum does not have a bandgap: the edge–edge states can have the same eigenvalues \({\lambda }_{n,m}^{(2)}={\lambda }_{n}+{\lambda }_{m}\) as those lying in the bulk–bulk region, Fig. 1e.
To include disorder, we separate the lattice shown in Fig. 1c into three regions^{1}. The left and right parts of the system are disorderfree, while its middle part exhibits diagonal disorder^{31}, that is, random modifications of the onsite refractive index taken from a normal distribution with zero mean and variance σ = 1. Importantly, taking σ = 1 ensures that the disorder strength does not destroy the topological protection for single photons, since σ = 1 corresponds to half the size of the topological bandgap. The twophoton eigenspectrum in the presence of disorder is shown in Fig. 1f.
We now send trial twophoton wavepackets into the system. They are built from singlephoton edge states and vary continuously from an unentangled product state, with Schmidt number S_{N} = 1, to highly entangled twophoton states, S_{N} ≫ 1^{44,45}, with the two photons either correlated or anticorrelated in space^{46}.
To construct these states, we begin with protected singlephoton states as a template, \({\tilde{\psi }}_{\sigma }^{(1)}\rangle =\mathop{\sum }\nolimits_{j = 1}^{{M}_{{\rm{e}}}}{(1)}^{j}{e}^{\frac{{\left({x}_{0}j\right)}^{2}}{2{\sigma }^{2}}}\leftj\right\rangle\), where \(\leftj\right\rangle\) describes a photon initialized in waveguide j, M_{e} = 20 is the selected range of waveguides in the upperleft edge of our system, and x_{0} = (M_{e} + 1)/2 = 10.5 is the center of this range. These singlephoton wavepackets travel through both clean and disordered lattice without scattering to the bulk or back scattering. Importantly, the alternating sign (−1)^{j} in the amplitude ensures that the wavepacket has proper momentum and resides in the singlephoton edge subspace.
Next, we construct our trial twophoton states as follows
Here, \(\leftj,k\right\rangle\) represents the state where a photon starts at waveguide j and its twin at k. The spatial twophoton correlations are controlled by the parameters σ_{c} and σ_{a}. For σ_{c} ≫ σ_{a} we have a spatially correlated state, in which both photons most probably enter into the same waveguide simultaneously^{38}. For σ_{a} ≫ σ_{c} we obtain a spatially anticorrelated state, in which the two photons enter at two waveguides symmetrically lying on opposite sides of the window covered by the wavefunction^{46}.
Finally, we must ensure that the initial wavepackets only include edge states. To this end, we project our state onto the twophoton eigenstates \({\phi }_{m,n}^{(2)}\rangle\) of the system and then remove the components belonging to the subspaces \({\mathcal{B}}\otimes {\mathcal{E}}\) and \({\mathcal{B}}\otimes {\mathcal{B}}\), keeping only states that belong to the edge–edge subspace
where A is the normalization constant. It is worth noting that twophoton states described by Eq. (6) are a lattice adoption of Gaussian twomode squeezed states^{47}, which are a commonplace choice in quantum optical experiments. The corresponding spatial \({P}_{j,k}= \langle j,k {\psi }_{{\sigma }_{{\rm{c}}},{\sigma }_{{\rm{a}}}}^{(2)}\rangle { }^{2}\) and spectral \({S}_{m,n}= \langle {\phi }_{m,n}^{(2)} {\psi }_{{\sigma }_{{\rm{c}}},{\sigma }_{{\rm{a}}}}^{(2)}\rangle { }^{2}\) correlation maps of our initial states, Eq. (6), are shown in Fig. 2. Tuning σ_{a} and σ_{c}, one can go from the spatially correlated state \({\psi }_{{\rm{c}}}^{(2)}\rangle\), Fig. 2a, to the product state \({\psi }_{{\rm{p}}}^{(2)}\rangle\), Fig. 2b, and to the spatially anticorrelated state \({\psi }_{{\rm{a}}}^{(2)}\rangle\), Fig. 2c. Note the relation between spatial and spectral distributions: the state \({\psi }_{{\rm{c}}}^{(2)}\rangle\), which is strongly correlated in space, Fig. 2a, is strongly anticorrelated spectrally Fig. 2d, and vice versa for \({\psi }_{{\rm{a}}}^{(2)}\rangle\). Irrespective of their correlation maps, all these states occupy the same spatial area on the upperleft edge of the lattice, see Supplementary Note 1. The Schmidt number for \({\psi }_{{\rm{c}},{\rm{a}}}^{(2)}\rangle\) is S_{N} = 13, while for \({\psi }_{{\rm{p}}}^{(2)}\rangle\) we have S_{N} = 1. We now explore the robustness of these twophoton states as they traverse the disordered lattice. We begin with the product state \({\psi }_{{\rm{p}}}^{(2)}\rangle\). To characterize the impact of disorder, we compute the fidelity^{30} that is given as the overlap of the state \({\psi }_{{\rm{p}}}^{(2)}({z}_{{\rm{f}}})\rangle\) after it has traversed the lattice with the reference state \({\psi }_{{\rm{p}}}^{(2)}({z}_{{\rm{m}}})\rangle\) obtained after propagating the same state \({\psi }_{{\rm{p}}}^{(2)}\rangle\) in a disorderfree lattice, see Supplementary Note 2. The two wavepackets are taken at slightly different propagation distances z_{f} and z_{m} to account for the somewhat different travel distance in a disordered lattice. We find the fidelity \({F}_{{\rm{p}}}= \langle {\psi }_{{\rm{p}}}^{(2)}({z}_{{\rm{f}}}) {\psi }_{{\rm{p}}}^{(2)}({z}_{{\rm{m}}})\rangle { }^{2}=0.98\), confirming that both the singlephoton states and their product are immune to disorder. The edge–mode content of the evolved state is almost 100%, \({E}_{{\rm{p}}}=\mathop{\sum }\nolimits_{n,m}^{{\mathcal{E}}\otimes {\mathcal{E}}} \langle {\phi }_{n,m}^{(2)} {\psi }_{{\rm{p}}}^{(2)}({z}_{{\rm{f}}})\rangle { }^{2}=0.9934\). The product state traverses the lattice without distortion, see Supplementary Movie M1, in spite of the degeneracy between the twophoton edge–edge and bulk–bulk states.
Figure 3a, b visualize this outcome by showing the singlephoton spatial distribution and twophoton spectral correlation maps for the twophoton product state \({\psi }_{{\rm{p}}}^{(2)}\rangle\) traversing the disordered lattice. The spatial singlephoton probability density R(n) is given by the diagonal elements \({\rho }_{nn}^{(1)}\) of the reduced singlephoton density matrix \({\hat{\rho }}^{(1)}\), \(R(n)\equiv \left\langle n\right{\hat{\rho }}^{(1)}\leftn\right\rangle \equiv {\rho }_{nn}^{(1)}\)^{48}. The reduced singlephoton density matrix \({\hat{\rho }}^{(1)}\) is obtained from the twophoton density matrix \({\hat{\rho }}^{(2)}\) in the usual way, \({\hat{\rho }}^{(1)}=\mathop{\sum }\nolimits_{m}^{M}\left\langle m\right{\hat{\rho }}^{(2)}\leftm\right\rangle\)^{30}. As expected, the spectral composition of the wavepacket remains undisturbed and the wavepacket propagates through the disordered region without leaving the edge.
We now turn our attention to entangled twophoton states. Figure 3c, d depict R(n) and the spectral correlation maps for the twophoton state \({\psi }_{{\rm{c}}}^{(2)}\rangle\) traversing the “clean” lattice, and Fig. 3e, f show the same for the disordered lattice. While in the absence of disorder, R(n) stays on the edge and the highly correlated twophoton spectral distribution is unchanged (panels c, d), the disorder strongly affects these states. Figure 3e shows strong dissipation into the bulk as soon as the entangled wavepacket encounters the disordered region. The spectral distribution spreads all over the system, with both bulk–bulk and bulk–edge states becoming occupied, Fig. 3f. A similar result is obtained for \({\psi }_{{\rm{a}}}^{(2)}\rangle\), except that the crosslike shape observed in Fig. 3f is flipped towards the opposite diagonal, Supplementary Note 3.
To quantify the probability fraction of the states scattered into the bulk we compute the edge–mode content. For \({\psi }_{{\rm{c}}}^{(2)}\rangle\) the edge–mode content after traversing the disordered lattice is E_{c} = 0.4524, while for \({\psi }_{{\rm{a}}}^{(2)}\rangle\) it gives E_{a} = 0.4453. Thus, >50% of both types of states is scattered into the bulk. The part of the states that survives the disordered region and stays on the edge remains strongly correlated in the spectral domain: the edge–edge part of its spectral content preserves the initial shape, see the right column in Fig. 3f. However, the spectral phase of the state is scrambled. To illustrate this point, we have renormalized the transmitted edge part of the twophoton wavepacket to unity and computed its fidelity F_{N} by overlapping it with the reference twophoton wavepacket from a clean system, yielding F_{N} = 0.405.
The topological window of protection
We find that the conduit for dissipation of the twophoton edge–edge states is always provided by the edge–bulk states, which are degenerate in energy with the edge–edge states. Once disorder induces transitions into the edge–bulk states, they further transfer the amplitudes into the energydegenerate bulk–bulk states, see Supplementary Movies M2 and M3. Hence, the key to topological protection is to minimize the disorderinduced overlap of the initial joint spectrum with the edge–bulk and bulk–bulk spectral regions, keeping it as close to the center as possible. That is, there is a topological protection window for twophoton states that offers the key guideline for designing robust twophoton states. To infer the protection window, we sent a probe product state with σ_{c} = σ_{a} = 0.01 through an ensemble of 1000 disordered lattices. This initial state is very well localized onto the edge region in real space, ensuring that all components within the state travel along very close paths. The spectral content of the state before and after the disorder is shown in in Fig. 4a, b. The components that have survived the impact of disorder are within the marked window—the topological window of protection. The joint spectral correlation map of any entangled state with varying σ_{a} and σ_{c} must fit inside this protection window to be robust against disorder.
In practice, to increase the amount of entanglement we need to increase σ_{a}\(\left({\sigma }_{{\rm{c}}}\right)\) while decreasing σ_{c}\(\left({\sigma }_{{\rm{a}}}\right)\), and by doing so the joint spectrum unavoidably tends to fall outside the protection window. However, we can always find twophoton states with a considerable amount of entanglement, which are protected. To elucidate this we have scanned the edgemode content of the twophoton states after propagation through the disordered region as a function of σ_{a} and σ_{c}. In Fig. 4c, d we show the contour maps of the edge–mode content as we vary σ_{a} and σ_{c}. Figure 4c shows the edge–mode content of the twophoton states after propagation through the disordered region, with the diagonal corresponding to the product states, that is, states with σ_{a} = σ_{c}. The states with the highest degree of entanglement correspond to very different σ_{a} and σ_{c} and, therefore, they are found in the top left and lower right corners in Fig. 4c. In general, highly entangled states lay on the top left \(\left({\sigma }_{{\rm{a}}}\ll {\sigma }_{{\rm{c}}}\right)\) and bottom right \(\left({\sigma }_{{\rm{a}}}\gg {\sigma }_{{\rm{c}}}\right)\) corners and the edge–mode content quickly drops below 0.5. The reason is because as one increases σ_{a}, or σ_{c}, the tails of the spectral correlation ellipse fall outside of the protection window and, as a result, the states scatter into the bulk. Similarly, uncorrelated states may experience the same fate when they are initially confined into a small spatial region, which is the case for states with \({\sigma }_{{\rm{a}}}={\sigma }_{{\rm{c}}}\in \left(0,2.5\right)\). Figure 4d shows the key figure of merit, E ⋅ S_{N}, the product of the Schmidt number S_{N} and the edge–mode content E. The bright yellow islands indicate the best twophoton states, which combine robustness against disorder with high degree of entanglement. Importantly, the spectral correlation ellipse of these states always fits into the protection window shown in Fig. 4b. It is worth mentioning the features exhibited by the contour maps are generic as similar structures are obtained for disordered Haldane lattices with different dimensions, see Supplementary Note 4. This demonstrates that, in principle, one can create states with high Schmidt number and edge–mode content close to unity.
As evidence that our results are generic, in the sense that they apply to other twodimensional topological systems, in the Supplementary Note 5 we have performed a similar analysis for an aperiodic topological lattice system^{6,40}. We have found that the contour map of the edgemode content E is not symmetric, implying that the correlated states are slightly less protected than their anticorrelated mirrorimages. Nevertheless, we obtain the same qualitative features as in the Haldane model.
Discussion
Before concluding, we would like to outline possible ways to generate the initial states and address the potential challenges for experimental observations of these effects. The initially highly correlated states can be implemented using standard spontaneousparametricdownconversion nonlinear crystals to generate photon pairs that are coupled to the edge of the lattice using a positive achromatic doublet lens as demonstrated in^{38}. Anticorrelated photon pairs can be generated by applying the fractional Fourier transform to the highly correlated states^{49}. The Haldane lattice has been previously demonstrated using femtosecond laser written waveguides as reported in^{41}. Hence, the challenges are reduced to optimizing the fabrication for minimal scattering, absorption and bending losses associated with the helical waveguides.
These results lead to the following conclusions. Two issues have to be considered when constructing twophoton entangled edge states in topological systems: their dissipation into the bulk and the relative dephasing between the different components comprising the entangled state. Regarding dissipation, the twophoton edge states can be protected just as well as the singlephoton edge states. Further, phase scrambling can also be minimized if the different components of the entangled state travel along the same path in the edge region. Both aims are achieved by keeping the spectral correlation map of the twophoton state in the center of the window of protection. Thus, attempts to increase entanglement must be balanced against keeping the spectral correlation maps of the twophoton states within the narrow spectral region at the very center of the singlephoton gap—the topological window of protection. This limits the degree of entanglement one can safely encode in practice, but presents a clear strategy for creating useful states with high degree of entanglement and robustness.
Looking forward, one could take advantage of the static nature of disorder to circumvent entanglementinduced dissipation into the bulk. While the disorderinduced relative phase between the different productstate components of the entangled wavepacket may appear random due to the random nature of disorder, for static disorder scrambling and dissipation are nevertheless fixed. This opens an opportunity to find the windows of protection as we have done in the cases considered here, and generate robust wavepackets tailored to the particular disordered system at hand. From a practical perspective, the stability of entangled states up to relatively high Schmidt numbers offers practical guidelines for generating useful entangled edge states in topological photonic systems. Finally, our work may open the door to study topological protection of highly entangled multiphoton nonGaussian states that fulfill the protection conditions.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The Matlab programs used to simulate the systems discussed in the paper are available as supplementary software.
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Acknowledgements
We acknowledge support by the Open Access Publication Fund of HumboldtUniversität zu Berlin. K.T., A.P.L., and K.B. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) within the framework of the DFG priority program 1839 Tailored Disorder (BU 1107/ 122, PE 2602/22). A.J.G. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 899794. M.I. acknowledges funding from the Deutsche Forschungsgemeinschaft under grant agreement IV 152/62. The work of DNC was partially supported by AFOSR (MURI FA95502010322).
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A.P.L., M.A.B., and D.N.C. initiated the project. A.P.L., M.A.B., D.N.C., M.I., and K.B. outlined the work. K.T., A.J.G., M.I., M.A.B., and A.P.L. developed the theory. K.T., K.B., and A.P.L. performed the simulations. All the authors discussed and analyzed the results. K.T., M.I., and A.P.L. wrote the manuscript with input from all coauthors. M.A.B. and A.P.L. coordinated the project.
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Tschernig, K., JimenezGalán, Á., Christodoulides, D.N. et al. Topological protection versus degree of entanglement of twophoton light in photonic topological insulators. Nat Commun 12, 1974 (2021). https://doi.org/10.1038/s41467021222643
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DOI: https://doi.org/10.1038/s41467021222643
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