Theoretical approach

In lattice systems, static disorder can be introduced in either the site energies—termed diagonal disorder³¹—or in the coupling coefficients—so-called off-diagonal disorder³². In either case, static disorder is represented by a single-particle operator $$ {\widehat{V}}^{\left(1\right)}$$. Since such perturbation is time-independent, energy conserving resonant coupling into the bulk is absent within first-order perturbation theory—the single-particle transition induced by $$ {\widehat{V}}^{\left(1\right)}$$ does not preserve energy. The process that can resonantly couple a two-photon 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 second-order corrections in $$ {\widehat{V}}^{\left(1\right)}$$.

To see this, we examine the second-order transition matrix elements between an initial two-photon edge–edge state $$ ∣\mathrm{i}⟩=∣n_{\mathrm{i}},m_{\mathrm{i}}⟩$$ and a final edge–bulk, or bulk–bulk, state $$ ∣\mathrm{f}⟩=∣n_{\mathrm{f}},m_{\mathrm{f}}⟩$$

In contrast to dissipation, the situation with dephasing can be different: while each constituent state in an entangled superposition can be protected against disorder¹, the overall superposition is, in general, not. To be precise, motion through different disordered regions may lead to disorder-induced 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 manner³³. These effects have been explored for spatial path-entangled states¹, and for states built from an entangled superposition of an initial non-stationary state $$ {\psi }_1^{\left(2\right)}$$ with its time-delayed replica $$ {\psi }_1^{\left(2\right)}\left(\tau \right)$$, that is, entangled states in the time-domain². In these two cases, however, the entanglement of the states can be related to the entanglement of symmetrized wavefunction of identical particles³⁴. 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 entanglement³⁵.

A more appealing type of highly entangled two-photon states are multimode optical Gaussian states in which both photons are most likely to be found inhabiting any waveguide, within an excitation window, simultaneously³⁶. 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 states³⁷. Naturally, the enhanced phase sensitivity of such highly entangled two-photon states manifests as faster diffraction of the associated wavepackets propagating in any photonic system, periodic and disordered³⁸. Therefore, it is not clear to what extent topological protection will persist for these types of highly entangled states.

Propagation of entangled two-photon states in disordered topological lattices

In what follows we analyze the impact of disorder onto a continuum of two-photon states that extend from the correlated to the anti-correlated 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 model³⁹, and for the aperiodic we use a square lattice whose single-particle 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 first-order approximation of the Haldane model can be implemented using a honeycomb lattice composed of helical waveguides as illustrated in Fig. 1a, see pioneering work⁴¹. In this system, every waveguide has a nearest-neighbor coupling κ₁ and a complex second-nearest-neighbor coupling κ₂ or $$ {\kappa }_2^{*}$$, see Fig. 1b. At the single-photon level, the Haldane lattice is governed by the Hamiltonian¹

Fig. 1

The Haldane photonic lattice.

a Photonic implementation of the Haldane system using a honeycomb lattice of helical waveguides. b Elementary hexagonal cell of the Haldane system, with real-valued nearest-neighbor coupling (blue arrows) κ₁ = 1 and imaginary next-nearest-neighbor coupling (red arrows); κ₂ = iκ₁/5 along the arrow and − iκ₁/5 in the opposite direction. c Pictorial view of the finite lattice used in our numerical analysis. d Single-photon spectrum formed by eigenvalues λ_n. In e and f we show the two-photon eigenspectra without and with disorder, respectively. Colors encode the two-photon eigenvalue $$ {\lambda }_{n,m}^{\left(2\right)}={\lambda }_n+{\lambda }_m$$. For better appreciation we have separated the spectrum into the subspaces edge–edge $$ \left(\mathcal{E}\otimes \mathcal{E}\right)$$, bulk–edge $$ \left(\mathcal{B}\otimes \mathcal{E}\right)$$, and bulk–bulk $$ \left(\mathcal{B}\otimes \mathcal{B}\right)$$.

For pure states of two indistinguishable noninteracting particles the Hamiltonian is H₂ = H ⊗ I + I ⊗ H, where H is the single-particle Hamiltonian and I is the identity operator⁴². The two-photon eigenstates are given by the symmetric tensor-product combinations of the single-photon eigenstates

In the absence of disorder, the eigenvalue spectrum for single-photon states in a finite lattice exhibits topological edge states in the bandgap⁴³, Fig. 1d. In contrast, for two indistinguishable photons, the spectrum does not have a bandgap: the edge–edge states can have the same eigenvalues $$ {\lambda }_{n,m}^{\left(2\right)}={\lambda }_n+{\lambda }_m$$ as those lying in the bulk–bulk region, Fig. 1e.

To include disorder, we separate the lattice shown in Fig. 1c into three regions¹. The left and right parts of the system are disorder-free, while its middle part exhibits diagonal disorder³¹, that is, random modifications of the on-site 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 two-photon eigenspectrum in the presence of disorder is shown in Fig. 1f.

We now send trial two-photon wavepackets into the system. They are built from single-photon edge states and vary continuously from an unentangled product state, with Schmidt number S_N = 1, to highly entangled two-photon states, S_N ≫ 1^44,45, with the two photons either correlated or anti-correlated in space⁴⁶.

To construct these states, we begin with protected single-photon states as a template, $$ \mid {\tilde{\psi }}_{\sigma }^{\left(1\right)}⟩={\sum }_{j=1}^{M_{\mathrm{e}}}{\left(-1\right)}^j{e}^{-\frac{{\left(x_0-j\right)}^2}{2{\sigma }^2}}∣j⟩$$, where $$ ∣j⟩$$ describes a photon initialized in waveguide j, M_e = 20 is the selected range of waveguides in the upper-left edge of our system, and x₀ = (M_e + 1)/2 = 10.5 is the center of this range. These single-photon 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 single-photon edge subspace.

Next, we construct our trial two-photon states as follows

Finally, we must ensure that the initial wavepackets only include edge states. To this end, we project our state onto the two-photon eigenstates $$ \mid {\varphi }_{m,n}^{\left(2\right)}⟩$$ 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

Fig. 2

Initial two-photon states.

a, b, c Spatial correlation maps P_n,m over the 91 sites of the upper edge of the lattice. d, e, f Spectral correlation maps S_n,m in the $$ \mathcal{E}\otimes \mathcal{E}$$-subspace. a, d correspond to the strongly spatially correlated state $$ ∣{\psi }_{\mathrm{c}}^{\left(2\right)}⟩$$, $$ \left({\sigma }_{\mathrm{c}},{\sigma }_{\mathrm{a}}\right)=\left(\sqrt{40},0.01\right)$$, b, e stand for the product state $$ \mid {\psi }_{\mathrm{p}}^{\left(2\right)}⟩$$, $$ \left({\sigma }_{\mathrm{c}},{\sigma }_{\mathrm{a}}\right)=\left(\sqrt{40},\sqrt{40}\right)$$, and c, f for the strongly spatially anti-correlated state $$ ∣{\psi }_{\mathrm{a}}^{\left(2\right)}⟩$$, $$ \left({\sigma }_{\mathrm{c}},{\sigma }_{\mathrm{a}}\right)=\left(0.01,\sqrt{40}\right)$$.

Figure 3a, b visualize this outcome by showing the single-photon spatial distribution and two-photon spectral correlation maps for the two-photon product state $$ \mid {\psi }_{\mathrm{p}}^{\left(2\right)}⟩$$ traversing the disordered lattice. The spatial single-photon probability density R(n) is given by the diagonal elements $$ {\rho }_{nn}^{\left(1\right)}$$ of the reduced single-photon density matrix $$ {\widehat{\rho }}^{\left(1\right)}$$, $$ R\left(n\right)\equiv ⟨n∣{\widehat{\rho }}^{\left(1\right)}∣n⟩\equiv {\rho }_{nn}^{\left(1\right)}$$⁴⁸. The reduced single-photon density matrix $$ {\widehat{\rho }}^{\left(1\right)}$$ is obtained from the two-photon density matrix $$ {\widehat{\rho }}^{\left(2\right)}$$ in the usual way, $$ {\widehat{\rho }}^{\left(1\right)}={\sum }_m^M⟨m∣{\widehat{\rho }}^{\left(2\right)}∣m⟩$$³⁰. As expected, the spectral composition of the wavepacket remains undisturbed and the wavepacket propagates through the disordered region without leaving the edge.

Fig. 3

Propagation of two-photon edge states.

a Probability density distribution for the reduced single-photon state and b the spectral correlation map for the product state $$ \mid {\psi }_{\mathrm{p}}^{\left(2\right)}⟩$$, which survives disorder. c, d The same for the spatially correlated entangled state $$ ∣{\psi }_{\mathrm{c}}^{\left(2\right)}⟩$$ propagating in the clean lattice, while e, f show the impact of disorder on this highly entangled state. In all cases the rightmost panels show a magnification of the edge–edge subspace. Dashed lines in a, c, e indicate the disordered region.

We now turn our attention to entangled two-photon states. Figure 3c, d depict R(n) and the spectral correlation maps for the two-photon state $$ \mid {\psi }_{\mathrm{c}}^{\left(2\right)}⟩$$ 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 two-photon 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 $$ \mid {\psi }_{\mathrm{a}}^{\left(2\right)}⟩$$, except that the cross-like 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 $$ \mid {\psi }_{\mathrm{c}}^{\left(2\right)}⟩$$ the edge–mode content after traversing the disordered lattice is E_c = 0.4524, while for $$ \mid {\psi }_{\mathrm{a}}^{\left(2\right)}⟩$$ 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 two-photon wavepacket to unity and computed its fidelity F_N by overlapping it with the reference two-photon wavepacket from a clean system, yielding F_N = 0.405.

The topological window of protection

We find that the conduit for dissipation of the two-photon 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 energy-degenerate bulk–bulk states, see Supplementary Movies M2 and M3. Hence, the key to topological protection is to minimize the disorder-induced 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 two-photon states that offers the key guideline for designing robust two-photon 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.

Fig. 4

Topological window of protection.

In order to identify the topological window of protection, we considered a spectrally broad product state with (σ_a = σ_c = 0.01) as initial state and propagate it through an ensemble of 1000 random Haldane lattices. In a we depict the spectral correlation map for the initial state. b depicts the ensemble-average of the spectral correlation maps inside the edge–edge subspace after the propagation through the ensemble of disordered lattices. We find that the only two-photon amplitudes that survive the disorder lie in the region indicated by the black square, which is the protection window. c shows the edge–mode content E and d the product of the edge–mode content and the Schmidt-number E ⋅ S_N as a function of the parameters σ_a, σ_c of the initial states in the range σ_a, σ_c ∈ [0.01, 10].

In practice, to increase the amount of entanglement we need to increase σ_a$$ \left({\sigma }_{\mathrm{c}}\right)$$ while decreasing σ_c$$ \left({\sigma }_{\mathrm{a}}\right)$$, and by doing so the joint spectrum unavoidably tends to fall outside the protection window. However, we can always find two-photon states with a considerable amount of entanglement, which are protected. To elucidate this we have scanned the edge-mode content of the two-photon 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 two-photon 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 $$ \left({\sigma }_{\mathrm{a}}\ll {\sigma }_{\mathrm{c}}\right)$$ and bottom right $$ \left({\sigma }_{\mathrm{a}}\gg {\sigma }_{\mathrm{c}}\right)$$ 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 $$ {\sigma }_{\mathrm{a}}={\sigma }_{\mathrm{c}}\in \left(0,2.5\right)$$. 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 two-photon 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 two-dimensional 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 edge-mode content E is not symmetric, implying that the correlated states are slightly less protected than their anti-correlated mirror-images. Nevertheless, we obtain the same qualitative features as in the Haldane model.