We predict the preservation of temporal indistinguishability of photons propagating through helical coupled-resonator optical waveguides (H-CROWs). H-CROWs exhibit a pseudospin-momentum locked dispersion, which we show suppresses on-site disorder-induced backscattering and group velocity fluctuations. We simulate numerically the propagation of two-photon wave packets, demonstrating that they exhibit almost perfect Hong–Ou–Mandel dip visibility and then can preserve their quantum coherence even in the presence of moderate disorder, in contrast with regular CROWs, which are highly sensitive to disorder. As indistinguishability is the most fundamental resource of quantum information processing, H-CROWs may find applications for the implementation of robust optical links and delay lines in the emerging quantum photonic communication and computational platforms.
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Topological photonics is emerging as a way to create disorder-immune waveguides for light using edge modes of media with nontrivial topological properties [1–3]. Since the first proof-of-concept experiments using gyro-magnetic microwave photonic crystals [4,5], various approaches have emerged to demonstrate topological transport at different length and energy scales [2,3]. Moreover, this field is developing by harmonizing with existing subfields of photonics, i.e., exploring the role of topology in nonlinear optical effects , dynamically modulated systems , lasers [8–10], and other non-Hermitian systems with structured gain and loss .
One promising application of topological photonics is the robust generation and transport of quantum state of light [12–15]. It is a more challenging problem, as the preservation of quantum properties, such as indistinguishability, depends on the phase information, which is often not robust against disorder even for topological edge states . Protecting indistinguishability has turned out to be important, as it provides a desirable resource in quantum technologies to generate entanglement . In optical delay lines, indistinguishability is determined by the degree of temporal overlap, as depicted in Fig. 1. An obstacle to preservation of temporal overlap through standard delay lines is disorder, which can not only induce backscattering and Anderson localization of propagating waves but also destroy information related to the relative phase, which occurs due to the dephasing of the ensemble-averaged state [18,19].
A potential solution is to use the robustness of topological edge states to protect quantum states of light [13,20–23]. For example, photonic quantum spin-Hall phases host bi-directional edge states, which are protected against backscattering by time-reversal and internal (e.g., crystalline) symmetries. Quantum spin-Hall edge states were demonstrated in 2D silicon ring resonator arrays, where the circulation direction within the rings plays the role of spin [24–27]. Other approaches introduce sublattices as a pseudospin degree of freedom emulating real spins [28,29]. The quantum spin-Hall phase supports transport and is robust against certain types of disorder, as backscattering requires a spin flip.
Thus far, topologically protected waveguiding has been largely demonstrated in two or higher dimensional photonic systems, while 1D topologically protected transport has required synthetic dimensions  or time modulation such as adiabatic pumping . There is another approach to achieve disorder-resistant waveguiding in 1D without a topological bandgap, based on directly implementing a spin-momentum locked dispersion by breaking time-reversal symmetry, which has been demonstrated using 1D electronic quantum wires . Recently, we proposed a model of a quasi-1D coupled-resonator optical waveguide (CROW) exhibiting a similar helical spin-momentum locked dispersion (H-CROW), by combining circulation direction and sublattice spin-like degrees of freedom. The former effectively breaks time-reversal symmetry, while the coupling between different sublattices can be tuned to create a sublattice-momentum locked dispersion relation . This results in a suppression of backscattering and enhancement of localization length compared with the regular CROW model as well as preservation of phase information. It is noteworthy, as it provides the way to miniaturize a disorder-resistant waveguide by inducing helical transport.
In this paper, we study the propagation of quantum states of light through H-CROWs and demonstrate that temporal indistinguishability can be robust against moderate disorder, thereby enabling the protection of entangled states. First, in Section 2, we compute the delay time distribution of single-photon states propagating through disordered H-CROWs and find that, for a given disorder strength, H-CROWs yield a narrower distribution of delays compared with regular CROWs. Thus, the temporal overlap of photons travelling along different paths can be preserved. In Section 3, we compute coincidence probability as a witness of indistinguishability and show the Hong–Ou–Mandel (HOM) dip . We then study in Section 4 the propagation of a path-entangled photonic state (N00N state) and show that H-CROWs protect their entanglement. We quantify their purity, associated with an inverse HOM dip, and obtain larger entanglement entropy compared with regular CROWs. We present conclusions and an outlook in Section 5.
2. MODEL OF H-CROWS WITH DISORDER-ROBUST TRANSPORT
The helical-CROW (H-CROW) consists of two legs of resonant ring cavities coupled via off-resonant link rings, as illustrated in Fig. 2. We assume decoupled circulation modes of each cavity, which allows one to effectively break time-reversal symmetry by making the link rings asymmetric. We are interested in the propagation of wave packets close to the band center, where the tight-binding approximation is valid [24,35]. Under this approximation, each unit cell hosts two sites, with coupling between nearest and next-nearest neighbors.33]. Note that we measure frequencies with respect to a resonance of a single isolated ring, such that the eigenvalues of are modal frequency detunings with respect to this resonance.
Let us now consider a periodic lattice, which enables us to compactly write the Hamiltonian in -space as33], is analogous to a Zeeman field, and and resemble intrinsic and Rashba-like spin-orbit couplings, respectively.
For the opposite excitation (clockwise), propagation is governed by the time-reversed Hamiltonian , which exhibits opposite hopping phases due to time-reversal symmetry . We introduce a total Hamiltonian composed of both circulations given in the direct product form, . We obtain pseudospin-momentum locking in the center of the passband (), where , , and the wave group velocity becomes . It supports the most resistant light propagation against disorder, since for the small momentum deviations has the form3) is written in the basis ; the first index labels the circulation direction, and the second indicates the pseudospins. Note that the off-diagonal component describing pseudospin-flipping vanishes in the first (linear) order of deviation.
At the band center, H-CROWs show the maximum Anderson localization length and most resistant temporal pulse propagation, since the most significant disorder is misalignment of the rings’ resonant frequencies, which is diagonal in the sublattice basis and does not flip the pseudospin [12,16,25,26,33,38].
Importantly, we can employ the circulation degree of freedom to use H-CROWs as two-mode delay lines (see Fig. 2). By exciting both circulations through different sublattices, the simultaneous pseudospin-momentum locking phases (red and blue) can be obtained. We now consider the propagation of light in the presence of disorder and losses. For the sake of simplicity, we only take the dominant on-site disorder into account, which has the form 40]. The derivation is given in Appendix A. We note the following relation for the output field operators : 26].
We present in Fig. 3(a) intensity profiles in the first half of an H-CROW for a monochromatic input at (the middle of the transmission band), averaged over an ensemble of 500 disorder realizations. For comparison, Fig. 3(b) shows the intensity profile of a regular CROW. The numerical calculations are performed using parameters similar to the experiments in Refs. [25,26]: , , and disorder standard deviation . In physical units, this corresponds to an operating wavelength , with , , and hopping strength [25,26]. In the H-CROW, the intensity remains confined to the upper () sublattice, signifying pseudospin-momentum locking, with the attenuation of the intensity along the array occurring only due to the internal losses . The regular CROW also exhibits Anderson localization, resulting in more rapid attenuation of the intensity in Fig. 3(b).
We plot in Fig. 3(c) the transmission spectra in Eq. (6), where the shaded regions indicate the 65% confidence intervals . H-CROWs achieve higher transmission than regular CROWs through the entire passband, with almost five times higher transmission at and smaller deviations between different disorder realizations as a result of the disorder resistant unidirectional propagation. Figure 3(d) demonstrates that H-CROWs maintain the amplitude of the output field for a broad range of disorder strengths , while CROWs show increasingly poor performance as increases. Based on these results, we use as a disorder strength in the following, as this regime clearly illustrates the advantage of H-CROWs versus regular CROWs in the presence of the moderate disorder typically present in experiments . We note that the comparatively lower transmission of the H-CROW for weak disorder is due to nonoptimal input coupling, which can be improved by, e.g., coupling the input waveguide to the second unit cell of the array.
The effects of disorder on the group delay time versus frequency detuning are presented in Fig. 3(e). We see that H-CROWs exhibit a slightly smaller average delay but much lower fluctuations. Furthermore, the statistics of delay times in the vicinity of presented in Fig. 3(f) show that H-CROWs exhibit more ballistic transport than regular CROWs; as for H-CROWs, the delay time distribution around the root mean square average has a Gaussian shape with smaller variance.
3. PRESERVATION OF PHOTON INDISTINGUISHABILITY
In this section, we consider the transmission of two identical photons forming a separable quantum state at the input and analyze the degree of temporal photon indistinguishability at the output of the H-CROW and CROW. Specifically, we consider an input state , with one photon in the clockwise mode and a temporally identical photon in the counterclockwise mode. The output state in the frequency domain has the form
We compare a degree of the temporal overlap of the two photons after each one propagates through a different part of the device by calculating the coincidence probability. It was the first experimental witness of quantum property, as Hong et al. showed quantumness by generating entangled photons and measuring their coincidence counts versus the controlled delay to one of the paths . When the total time delay is zero, coincidence rates after a 50:50 beam splitter reach a minimum and vanish due to the quantum interference when photons are temporally indistinguishable. Accordingly, we analyze the photon interference at the output with a tunable temporal delay, as illustrated in Figs. 4(a) and 4(b). When a two-photon output state passes a beam splitter with ratio , where and represent reflection and transmission, respectively, the field operators obey the unitary relation 42]. Thus, , where the density matrix is . Now, we introduce a tunable delay parameter between two outputs , which controls the temporal overlap between the photons before their interference on the beam splitter. Without loss of generality, we apply this delay to the lower output port and obtain a final expression for a balanced beam splitter () : 41], which quantifies the degree of interference. In quantum mechanics, it is also referred to as indistinguishability of photons . The meaning of coincidence probability is thus the resultant distribution of indistinguishability . Note that coincidence probability is zero for an ideal case of identical single photons, indicating indistinguishability preservation, since the transformation for the indistinguishable photon input state reads [46,47] B.
We perform a comparison against regular CROWs with the same input and same propagation length, as illustrated in Figs. 4(a) and 4(b). We consider wave packets with a Gaussian envelope at the input, i.e., , where is the envelope width. We plot in Fig. 4(c) the calculated coincidence probability versus controlled time delay after propagation through the 20-ring CROW, as discussed above. We note that the separable two-photon state is insensitive to the relative phase delay accumulated along the different paths, as sketched in Fig. 1(a), but can be affected by disorder-induced variations in the group delay and wave packet distortions, as indicated in Figs. 1(b) and 1(c). We observe that the minimum coincidence probability for H-CROWs remains close to zero (), indicating photon indistinguishability is well-preserved. On the contrary, for regular CROWs, the mean minimum coincidence probability is 0.22, which means that photons become partially distinguishable due to disorder. Even though the two photons with opposite spins travel along different disordered paths, the temporal shape of photons at the H-CROW outputs remains almost identical or indistinguishable, as a result of disorder-resistant transport. This advantage becomes more pronounced for longer CROWs, as shown in Fig. 4(d). The minimum coincidence for H-CROWs remains small (), whereas the minimum coincidence probability for the regular CROWs keeps increasing, indicating reduced indistinguishability as the length of the delay line increases.
4. PROTECTION OF PHOTON ENTANGLEMENT
We now aim to show that H-CROWs can preserve a peculiar quantum property of transmitted photons, entanglement inherently originating from the quantum coherence. Let us consider an N00N state as an input, . Hereafter we omit the ‘ab’ notation. Such states are strongly sensitive to all effects of the disorder, as sketched in Fig. 1, including phase fluctuations , in contrast with the separable states we analyzed in the previous section. The corresponding output state in the frequency domain is49], which we express as follows: 50].
Next, we determine the coincidence probability in the general case, considering all the effects due to disorder. We compute the density matrix of the output state using the projection operators16), here phase fluctuations arise from phase mismatches between the fields . Note that the coincidence probability is , which we derive in Appendix C.
To quantify the mixture of the output state induced by disorder, we also analyze another quantity, the purity , which is bounded by , where is the dimension of Hilbert space, i.e., for the two-photon case. The maximum value corresponds to pure states and the minimum to fully mixed states. The state purity after passing the controlled delay is
The form of Eq. (19) reveals that the output state remains pure when indistinguishability is preserved. Namely, since we consider noninteracting particles, we can relate the N00N state purity to the coincidence probability of the separable input state , according to Eq. (11). Since , we have , indicating that the two output fields have identical intensities and group delays. The expression in Eq. (19) thereby approaches 1, even though coincidence probability for may fluctuate due to the phase sensitivity of N00N state.
To verify this reasoning, we plot in Figs. 5(a) and 5(b) the distributions of coincidence probability for N00N states transmitted through H-CROWs and regular CROWs, respectively, with a zero-controlled time delay (). We observe that, for H-CROWs, the oscillation of probability occurs in the full range of [0,1], which is evidence that entanglement is preserved. In contrast, the coincidences from the regular CROWs show a peak at 0.5, indicating the output state is mixed and the entanglement is lost due to disorder.
We additionally show in Fig. 5(c) the average purity and the 65% confidence interval versus the delay time. We see that purity stays at 1 for H-CROWs at zero delay, while CROWs exhibit loss of coherence with huge fluctuations due to disorder. As the delay is increased, entangled photons in H-CROW become a completely mixed state with purity 0.5, as their coherence is washed out. On the other hand, in the regular CROW the average purity remains higher but with large fluctuations. This is because the transmission amplitude is sensitive to the disorder. Suppose we obtain the much higher transmission through one leg compared with the other. Then, according to Eq. (19), the temporal purity will be unity regardless of controlled delay. In addition, large group delay fluctuation in CROWs also affects the distribution of purity. For instance, even though we apply time delay to one of the waveguides, packets can be detected at the same moment due to the group delay induced by disorder.
To further quantify the effect of disorder, we consider the entanglement entropy , which indicates the capacity for encoding quantum information ,C,
We show in Fig. 5(d) the scaling of the entanglement entropy with the number of photons, revealing huge fluctuations and loss of entanglement for the regular CROW, while the H-CROW can preserve some amount of entanglement. As we increase , we observe larger fluctuations of the entanglement entropy. The entangled photons in the regular CROW become a mixed state due to the large transmission fluctuations, i.e., appreciable transmission only occurs along one leg. Thus, entanglement entropy becomes 0 and the error bar of entanglement entropy becomes smaller. We emphasize that, for a given delay or , the H-CROW exhibits better performance of protecting entanglement than the regular CROW. One caveat on entanglement entropy is that it is not sensitive to the controlled delay time, as it only depends on the intensity at each output port. To obtain the maximum entanglement entropy, the intensity at each output port should be identical, which can be seen from Eq. (22).
In this paper, we have studied the propagation of quantum states of light through helical coupled-resonator waveguides (H-CROWs). Regular CROWs can serve as delay lines in integrated photonic circuits; however, they exhibit strong sensitivity to fabrication disorder preventing reliable transmission of wave packets. H-CROWs exploit an additional sublattice degree of freedom to achieve disorder-resistant transport, which arises due to one-way modes at the center of their transmission band, whose propagation direction is fixed by the excited sublattice (known as pseudospin-momentum locking). Using numerical solutions of tight-binding models describing H-CROWs and regular CROWs, we have shown that the former can be used to more reliably transport quantum states of light in the presence of disorder.
We first showed that transmission probability and wave packet delay times have narrower fluctuations and provide more ballistic-like transport compared with regular CROWs. Next, we showed that two identical photons transmitted through an H-CROW can preserve the indistinguishability of their temporal wave packets and, accordingly, demonstrate the quantum Hong–Ou–Mandel interference. Finally, we showed that path-entangled two-photon N00N states are preserved as pure entangled states, while the effect of disorder is only expressed through the accumulation of a relative phase between the photon pairs. We note that this relative phase fluctuation can be compensated by simply placing a single tunable phase shifter at one of the output ports . The H-CROWs perform better at reliably transporting both types of quantum states, while quantum features are strongly suppressed by disorder in regular CROWs.
In the future, it will be interesting to generalize our findings to multimode entangled states and multimode H-CROWs. Moreover, in this work we considered weakly coupled resonators described by the tight binding approximation, similar to the experiments in Refs. [25,26]. In order to increase the operating bandwidth, it would be essential to consider more strongly coupled lattices, which must be modeled using the transfer matrix method . We expect that our results can provide a practical way to create robust integrated photonic delay lines, which can serve as essential components facilitating reliable generation and guiding of the quantum state of light for multiple applications, including scalable quantum information processing.
APPENDIX A. INPUT–OUTPUT RELATION
When we induce the coupling between an external probe waveguide characterized by field operator and the system, the dynamics are governed by total Hamiltonian,39]. Here, we assume that the probe waveguide has an almost continuous spectrum. The Heisenberg equation of the bath operator reads  39] A4) can be written differently: 39]: 6).
APPENDIX B. COINCIDENCE PROBABILITY FOR GENERAL BEAM SPLITTER
Suppose identical photons come toward a beam splitter (BS) from the opposite ports as depicted in Figs. 4(a) and 4(b). The BS exhibits the ratio , where , () are the reflectivity and transmissivity, respectively, . The transformation for indistinguishable photons reads [46,47]B1) is then 41,47,52].
Similar to our analysis for state, we calculate the transformation of the state after a beam splitter,4.
APPENDIX C. DENSITY MATRIX OF 2002 STATE
To compute coincidence probability and entanglement entropy, one has to introduce a density matrix defined as an outer product of output states. Density matrix of output state for a input in terms of Fock basis has the form with normalization,18). This expression can be obtained equivalently using the unitary operator in the operator basis we defined in Eq. (10). The unitary matrix for two particles in the tensor product form reads C3) is equivalent to . The purity thus has the form 53].
Entanglement entropy can be obtained from the above ingredients. Tracing out the lower port degree of freedom () yields the expression about the reduced density matrix ,
Institute for Basic Science (IBS-R024-Y1, IBS-R024-D1); Australian Research Council (DP190100277).
The authors declare no conflicts of interest.
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