Main

Energy exchange with the environment is an inescapable feature for any physical system. As such, the fundamental assumption of Hermiticity, as convenient as it may be, for example, in the theoretical description of quantum systems, necessarily constitutes an approximation. Historically, non-Hermitian characteristics tended to be neglected as a matter of course, while the utility of their twofold appearance as gain and loss was primarily seen in their capacity for mutual cancellation. Yet, sparked by the ground-breaking work of Bender and Boettcher on parity-time (PT) symmetry10 and its subsequent adaptation to optical settings11, a slew of works12 have revealed fascinating features arising from the complex interplay of gain and loss, for example, in non-orthogonal eigenmodes13, peculiar transport properties14, loss-induced transparency15 and unidirectional invisibility16 as well as exceptional points16,17 with enhanced response18, PT-symmetric lasers19,20 and even light-funnelling21. However, although gain and loss are readily incorporated into classical optics as a complex-valued refractive index profile, the consequences of changing the number of particles pose fundamental constraints on how these concepts can be brought to bear on genuine quantum states of light22. Fortunately, inadvertently introducing additional quantum noise can be avoided by entirely passive configurations. For example, quasi-PT-symmetric loss distributions in directional couplers have been shown to systematically accelerate the onset of fully destructive two-photon quantum interference23. Along similar lines, lossy beamsplitters7, opaque scattering media8, non-unitary metasurfaces24 or photonically implemented quantum decay processes9 may enhance two-photon coincidence rates for indistinguishable photon pairs, yielding a peak instead of the conventional dip in the Hong–Ou–Mandel (HOM) experiment, which in Hermitian systems would be indicative of fermionic anti-bunching behaviour.

In this Letter we experimentally demonstrate that, even if loss-induced anti-bunching in the form of a HOM peak is observed, the photons’ bosonic characteristics of destructive quantum interference—a HOM dip—can nevertheless be retained by an appropriate choice of the observation basis. To this end, we implement non-Hermitian two-port couplers based on femtosecond laser-written birefringent waveguides25, where polarization oscillations of photons play the role of continuous coupling between orthogonal polarization bases,6 and extended ancillary arrays provide the desired amounts of loss26,27 through polarization-sensitive coupling28. Notably, indistinguishable photons are known for being able to partially evade lossy domains in unison29, which affects the transition probability of photon pairs depending on their degree of indistinguishability, resulting in an apparent enhancement of the contribution of indistinguishable photon pairs (Fig. 1a). Regardless of the number of transmitted photon pairs however, bunching can be enforced for indistinguishable photons through the choice of the observation basis. As outlined in Fig. 1b, bunched photon pairs cannot be detected via coincidence measurement in specific bases, allowing for a suppression of their registered coincidences Cind compared to the baseline Cdis of distinguishable photon pairs that remain unaffected (Fig. 1c). We harness a combination of these two effects to induce a continuous transition from enhancement (that is, a HOM peak) in one polarization basis set to full suppression (a HOM dip) in its counterpart that has been rotated by 45° (Fig. 1d), whereas the conventional HOM dip as a sign of bosonic two-photon interference on a balanced beamsplitter remains strictly basis-independent (Fig. 1e). To gain more detailed insights into these dynamics, we study the HOM patterns for representative basis orientations associated with visibilities of v = Cind/Cdis − 1 < 0 (that is, the conventional HOM dip) to peaks (ν > 0). Furthermore, by experimentally probing the influence of the paths taken by photon pairs through the sample, we show that HOM interference actually remains unaffected by the specific phases accumulated by their constituent particles, revealing an entirely new approach to systematically tailor photon coincidence statistics beyond techniques based on selective transmission24.

Fig. 1: Propagation and coincidence detection of photon pairs in lossy systems.
figure 1

a, When pairs of distinguishable (red/blue) and indistinguishable (red/red) photons are launched in identical quantities, lossy regions (here represented by the coupling to ancillary arrays) affect the correlations of distinguishable photon pairs more strongly. In turn, the transmission of indistinguishable photon pairs is effectively enhanced. b, Illustration of observation-dependent coincidence detection for indistinguishable photon pairs. In the observation in basis 1, the photons inhabit different states, allowing them to jointly register in a coincidence measurement. By contrast, when the photons occur in the same state (bunching), they are rendered invisible to coincidence measurement between the states spanning basis 2. c, Distinguishable photons do not occur in bunched states, allowing them to be measured as coincidences independently of the measurement basis, as long as neither of them has been annihilated. d, Schematic geometric interpretation of the basis-dependent outcome of the HOM experiment in a non-Hermitian polarization coupler. Depending on the perspective, the structure may be perceived either as a dip or a peak, respectively. e, By contrast, for the conventional Hermitian structure, the appearance remains invariant against rotations of the observation basis.

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Our setting is based on birefringent waveguides, which can be described with a pair of propagation constants, \(\beta_{\rm{H}}\) and \(\beta_{\rm{V}}\), associated with photons polarized horizontally and vertically, in line with the structure’s principal axes (Fig. 2a). As demonstrated recently, such waveguides can be employed as a directional coupler in polarization space6. Viewed through this lens, they promote a periodic, coherent transfer of excitation between the diagonal state \({|{{{\mathrm{D}}}}\rangle }={ (|{{{\mathrm{V}}}}\rangle + |{{{\mathrm{H}}}}\rangle )}/{\sqrt 2}\) and the anti-diagonal one \({|{{{\mathrm{A}}}}\rangle }= {(|{{{\mathrm{V}}}}\rangle - |{{{\mathrm{H}}}}\rangle )}/{\sqrt 2}\) upon propagation along the longitudinal coordinate z. Although both \({|{{{\mathrm{D}}}}\rangle}\) and \({|{{{\mathrm{A}}}}\rangle}\) evolve with the mean propagation constant \({\bar \beta }= {{(\beta _{{{\mathrm{V}}}}} + {\beta _{{{\mathrm{H}}}})}/{2}}\), the rate of coupling between them is governed by the mismatch \(\Delta \beta = (\beta_{\rm{V}} - \beta_{\rm{H}})/2,\) that is the strength of birefringence. To impose dissipation to the system, we employ (spatial) coupling to ancillary waveguide arrays placed in its vicinity (Fig. 2b). Although no actual absorption is introduced, the presence of these extended arrangements of similarly birefringent waveguides provides a flexible means to drain light from the states \({|{{{\mathrm{H}}}}\rangle}\) and \({|{{{\mathrm{V}}}}\rangle}\) with polarization-dependent coupling strengths28 CH and CV. Once photons have tunnelled out of the target waveguide, the comparatively stronger coupling within the arrays immediately conveys the photons to the sides by means of ballistic discrete diffraction27. Although the resulting polarization-sensitive loss rates γH and γV indeed selectively affect the evolution of the initial amplitudes \({\alpha }_{{{{\mathrm{H}}}}/{{{\mathrm{V}}}}}\) (\({\alpha }_{{{\mathrm{H}}}}^{2} + {\alpha }_{{{\mathrm{V}}}}^{2} = {1}\)) of a single-photon state

$${|\psi (z)\rangle} = {\alpha }_{{{\mathrm{H}}}}{\rm{e}}^{ - {\gamma }_{{{\mathrm{H}}}}z} {\rm{e}}^{ - {{{\mathrm{i}}}}{\beta }_{{{\mathrm{H}}}}z}{|{{{\mathrm{H}}}}\rangle} + {\alpha }_{{{\mathrm{V}}}} {\rm{e}}^{ - {\gamma }_{{{\mathrm{V}}}}z} {\rm{e}}^{ - {{{\mathrm{i}}}}{\beta }_{{{\mathrm{V}}}}z}{|{{{\mathrm{V}}}}\rangle}$$
(1)

when described as a superposition of the polarization eigenstates \({|{{{\mathrm{H}}}}\rangle}\) and \({|{{{\mathrm{V}}}}\rangle}\), the mean loss rate \({\bar \gamma} = {{({\gamma }_{{{\mathrm{H}}}} + {\gamma }_{{{\mathrm{V}}}})}/{2}}\) represents the balanced global loss rate imposed on the states \({|{{{\mathrm{D}}}}\rangle}\) and \({|{{{\mathrm{A}}}}\rangle}\). In turn, the difference in the individual loss rates \({\Delta \gamma} = {{({\gamma }_{{{\mathrm{H}}}} - {\gamma }_{{{\mathrm{V}}}})}/{2}}\) appears as the imaginary part in their coupling rate (Fig. 2c).

Fig. 2: Loss implementation in a polarization coupler.
figure 2

a,b, When the birefringent target waveguide with propagation constants βH/V along its horizontal/vertical principal axes (a) is brought into contact with ancillary arrays comprised of tightly coupled waveguides (b), the resulting polarization-sensitive coupling strengths CH and CV establish polarization-dependent loss rates \(\gamma_{\rm{H}}\) and \(\gamma_{\rm{V}}\) for the target (central) waveguide. Details on the polarization dependence of coupling are provided in Extended Data Fig. 1. c, In the AD-polarization basis, the mean propagation constant \({\bar \beta }= {{({\beta }_{{{\mathrm{V}}}} + {\beta }_{{{\mathrm{H}}}})}/{2}}\) and the real part \({\Delta \beta }= {{({\beta }_{{{\mathrm{V}}}} - {\beta }_{{{\mathrm{H}}}})}/{2}}\) of the coupling rate directly result from the waveguide’s birefringence, whereas the global loss \(\bar \gamma = (\gamma _{{{\mathrm{H}}}} + \gamma _{{{\mathrm{V}}}})/2\) as well as the imaginary part \(\Delta \gamma = (\gamma _{{{\mathrm{H}}}} - \gamma _{{{\mathrm{V}}}})/2\) of the coupling arise from the differential losses of the H and V components, respectively. Together, these four quantities make up the non-Hermitian single-particle Hamiltonian of the polarization coupler.

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As shown in Fig. 3a, horizontally and vertically polarized excitations of the target waveguide are indeed attenuated at markedly different rates, as confirmed by tight-binding simulations. Notably, the Zeno dynamics26 in tight-binding lattices do not yield a purely exponential decay of quantum states, and the resulting deviations at short propagation distances30 can be accounted for by defining effective loss rates ɣH/V and their respective mean \(\bar \gamma\) and mismatch \(\Delta \gamma\) (see Supplementary Information for details). Figure 3b shows the observed classical intensity distribution excitations after a propagation length of z = 15 cm, where a fraction of 48.3% of vertically polarized light has remained in the target waveguide (corresponding to a loss rate of γV = 0.02433 cm−1), while the substantially larger coupling in the horizontal polarization component was able to reduce the respective transmission to 4.5% (ɣH = 0.1035 cm−1).

Fig. 3: Light dynamics under the influence of differential losses.
figure 3

a, Calculated evolution of the probability of photons in H (red) and V (black) polarization remaining in the target waveguide as determined by tight-binding dynamics (solid lines). Provided the intra-array couplings are sufficiently large to efficiently transport the photons away once they have been extracted, the initial Zeno/anti-Zeno dynamics quickly transition into the exponential decay behaviour associated with constant losses. For ease of handling, we employ effective loss rates that yield the same net decays after the sample length of 15 cm (dashed lines). b, Normalized output intensity distribution of the system observed with a charge-coupled device camera when the target waveguide is illuminated with appropriate polarized classical light at λ = 814 nm. Whereas only 4.5% of H-polarized light emerges from the target waveguide at z = 15 cm, 48.3% of V-polarized light polarization remains in the target guide. The corresponding effective loss rates are \(\gamma_{\rm{H}} = 0.1035\, {\rm{cm}}^{-1}\) and \(\gamma_{\rm{V}}=0.02433\, {\rm{cm}}^{-1}\), respectively.

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To study the impact on HOM dynamics, a polarization-duplexed two-photon state, with one photon each in the diagonal and anti-diagonal states, was synthesized from a spontaneous parametric downconversion (SPDC) photon-pair source and injected into the target waveguide. At the other end of the sample, a polarization beamsplitter (PBS) served to separate the photons remaining in the target waveguide according to their polarization, and route them to dedicated single-photon counting modules (SPCMs) to register coincidences between the two output ports. Crucially, a half-wave plate (HWP) placed in front of the PBS allowed us to freely choose the orientation θ of the observation polarization between the horizontal/vertical (H/V, θ = 0°) and diagonal/anti-diagonal (D/A, 45°) cases by an appropriate orientation θ/2 of its fast axis (Fig. 4a). In turn, the characteristic HOM patterns were recorded by varying the photons’ relative delay τ at the injection facet between τ = 0 for the indistinguishable configuration and \({\left| \tau \right| > {1}\,{{{\mathrm{ps}}}}}\) for the fully distinguishable case.

Fig. 4: Switching between dip and peak in HOM interference.
figure 4

a, Schematic of the experimental set-up. Photon pairs from a type I SPDC source are collected with polarizing fibres and transformed into the desired input state with one photon each in the A and D polarizations by means of a fibre-based polarization combiner. After propagation through the sample, the remaining photons in the target waveguide are analysed with a PBS. Crucially, the orientation angle θ of the basis in which the coincidence counts are detected with SPCMs can be seamlessly adjusted via a HWP. C.w., continuous wave. b, HOM patterns recorded in the lossy coupler with \({{{{\mathrm{{\Delta}}}}}{\beta } {z}} = {0}\) for different detection basis orientations ranging from the HV (θ = 0°) to the AD orientation (θ = 45°). As a guide to the eye, the basis orientations with respect to the HV frame (dashed lines) are show as red and black arrows. c, Calculated (lines) and observed (circles) basis-dependent HOM visibilities in the Hermitian system (left) and under the influence of differential losses (right) for polarization couplers with accumulated phases of \({{{{\mathrm{{\Delta}}}}}{\beta } {z} = {0}}\) modulo π (green) and \({{{{\mathrm{{\Delta}}}}}{\beta } {z}} = {\uppi }/{4}\) modulo π/2 (magenta). The full set of corresponding experimental HOM traces is shown in Extended Data Fig. 3. The polar plots illustrate the underlying physics by means of colour-encoded HOM visibilities for arbitrary basis orientations \({\theta \in [{0},\,{90}^\circ ]}\) and accumulated phases \({{{\mathrm{{\Delta}}}}}{\beta } {z} \in [{0},\,{\uppi }/{2}]\). The green and magenta dashed traces indicate the theory datasets in the adjacent plots.

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In a first set of experiments we set the birefringence of the target waveguide such that the cumulative phase difference along the sample yielded \({{{{\mathrm{{\Delta}}}}}{\beta } {z}} = {0}\) modulo π, meaning that any observed change to the photons’ polarization state at the output is directly associated with the respective loss rates ɣH/V instantiated by the ancillary arrays. Notably, this configuration removes any impact of the polarization coupling dynamics in the target waveguide on the HOM interference, allowing us to directly characterize the influence of observation basis choice. As shown in Fig. 4b for six representative orientation angles θ of the HWP, the obtained HOM pattern can be continually tuned between the conventional HOM dip regime in the H/V basis (θ = 0°) with a visibility of vHV = −94.6 ± 1.0%, via the case of HOM suppression (v = 2.9 ± 0.9% at θ = 18°), to a pronounced HOM peak displaying a visibility of vAD = +54.1 ± 0.9% in the A/D basis (θ = 45°). Note that an ideal HOM peak visibility of vAD = 1 would be reached in the limiting case of zero transmission in one of the polarizations H or V (Extended Data Fig. 2 presents an equivalent realization of this configuration with a high-contrast polarizer).

To gain a more detailed understanding of the interplay between the choice of basis and the polarization coupling dynamics, we implemented three additional settings: a system with identical losses (\({\bar \gamma }= {0.0639}\,{{{\mathrm{cm}}}}^{ - 1}\), \({\Delta \gamma }= {0.0396}\,{{{\mathrm{cm}}}}^{ - 1}\)) but a non-trivial cumulative phase of \({{{{\mathrm{{\Delta}}}}}{\beta }{z}} = {\uppi }/{4}\) modulo π/2, representing a balanced A/D polarization coupler, and the ‘lossless’ counterparts (\({\bar \gamma }= {0}\); that is, in the absence of ancillary arrays) for the cases of \({{{{\mathrm{{\Delta}}}}}{\beta } {z}} = {0}\) and \({{{{\mathrm{{\Delta}}}}}{\beta } {z}} = {\uppi }/{4}\). Figure 4c summarizes the dependence of the visibility on basis orientation and polarization coupling (Extended Data Fig. 3 presents individual basis-dependent HOM traces). Although the A/D basis clearly allows for both HOM dips and HOM peaks to be observed depending on the specific combination of cumulative phase and basis orientation, the H/V basis universally enforces a suppression of the two-photon coincidences (vHV ≈ −1). This prevalence of the conventional HOM dip is inextricably linked to the structure of the two-photon state

$${\varPsi}_{{{{\mathrm{ind}}}}}{\left( {z} \right)} = {\rm{e}}^{ - 2{{{\mathrm{i}}}}{\bar \beta }{z}}{\left( {\frac{{{\rm{e}}^{ - {2}{{{\mathrm{i}}}}{\Delta}\beta z}{\rm{e}}^{ - {2}{\gamma }_{{{\mathrm{V}}}}z}}}{{\sqrt 2 }}{|{{{\mathrm{VV}}}}\rangle }- {\frac{{{\rm{e}}^{ - {2}{{{\mathrm{i}}}}{\Delta}\beta z}{\rm{e}}^{ - {2}{\gamma }_{{{\mathrm{H}}}}z}}}{{\sqrt 2 }}}{|{{{\mathrm{HH}}}}\rangle }} \right)}.$$
(2)

As it evolves from the injected state \({\varPsi}{\left( {0} \right)} = {\left( {{|{{{\mathrm{AD}}}}\rangle } + {|{{{\mathrm{DA}}}}\rangle }} \right)}/{\sqrt 2} = {\left( {{|{{{\mathrm{VV}}}}\rangle } - {|{{{\mathrm{HH}}}}\rangle }} \right)}/{\sqrt 2}\) by propagating though the system, the bunching of indistinguishable photons in the H and V polarizations is retained, regardless of the propagation phasors \({\rm{e}}^{ - 2{{{\mathrm{i}}}}{\left( {{\bar \beta } \pm {{{\mathrm{{\Delta}}}}}\beta } \right)}z}\) or the specific values of the loss rates \(\gamma_{\rm{H/V}}\). It follows that the indistinguishable photons never coincide in the H/V basis (as indicated in Fig. 1b). By contrast, the two-photon wavefunction of distinguishable particles injected as \({|{{{\mathrm{A}}}},{{{\mathrm{D}}}}\rangle}\) evolves as

$$\begin{array}{ll}{{{\varPsi}_{{\rm{dis}}}}\left( z \right)}=&{ {{\rm{e}}^{ - 2{\rm{i}}\bar \beta z}}\left( {\frac{{{{\rm{e}}^{ - 2{\rm{i}}\Delta \beta z}}{{\rm{e}}^{ - 2{\gamma _{\rm{V}}}z}}}}{2}|{\rm{V}},{\rm{V}}\rangle + \frac{{{{\rm{e}}^{ - \left( {{\gamma _{\rm{V}}} + {\gamma _{\rm{H}}}} \right)z}}}}{2}\left( {|{\rm{V}},{\rm{H}}\rangle } \right.} \right.}\\{}&\quad \quad \quad \quad {\left. {\left. { - |{\rm{H}},{\rm{V}}\rangle } \right) - \frac{{{{\rm{e}}^{2{\rm{i}}\Delta \beta z}}{{\rm{e}}^{ - 2{\gamma _{\rm{H}}}z}}}}{2}|{\rm{H}},{\rm{H}}\rangle } \right)}\end{array}$$
(3)

and, therefore, yields detectable photon pairs in the states \({|{{{\mathrm{V}}}},{{{\mathrm{H}}}}\rangle}\) and \({|{{{\mathrm{H}}}},{{{\mathrm{V}}}}\rangle}\) for any finite z. In this vein, the impact of non-Hermiticity on the HOM trace is most apparent in the absence of interference terms (\({{{{\mathrm{{\Delta}}}}}{\beta } {z}} = {0}\) modulo π). If, in addition, the losses are polarization-independent, that is for a trivial loss mismatch \(\left(\Delta \gamma = 0\right)\), the coincidence rates of distinguishable and indistinguishable particles necessarily coincide and the HOM visibility is suppressed (vAD = 0). By contrast, in the limiting case of \({{\gamma }_{\rm{H}} \to \infty}\), only the first terms of equations (2) and (3) remain: both photons are then V-polarized and register as pairs in the AD basis with a probability of 1/4. Comparing the transmitted amplitudes of the \({|{{{\mathrm{VV}}}}\rangle}\) and \({|{{{\mathrm{V}}}},{{{\mathrm{V}}}}\rangle}\) terms, respectively, one notices that they differ by a factor of \({\sqrt 2}\): compared to the baseline of the fully distinguishable case, twice as many indistinguishable photon pairs can, in principle, be observed, corresponding to a HOM peak with ideal maximal visibility of vAD = 1, whereas finite loss contrasts yield maximal visibilities in the range of 0 < vAD < 1 (cf. Fig. 4b, where a maximal visibility of vAD = 54.1 ± 0.9% was observed for a loss contrast of \(\Delta \gamma = 0.0396\, {\rm{cm}}^{-1}\)).

As shown in Fig. 4c, the loss contrast is the crucial variable in determining the specific impact that the HOM interference within the sample has on vAD: the polar plot representation of the visibility landscape depending on θ and \({{{{\mathrm{{\Delta}}}}}{\beta }{z}}\) illustrates that the case of vanishing loss contrast (\(\Delta \gamma = 0\), left box) only allows for deviations (v ≈ 0) from strong suppression (v ≈ −1) in a narrow region surrounding the AD basis (θ = 45°) for trivial cumulative phases (\({{\Delta}{\beta } {z}} = {0}\)). On the other hand, two-photon interference for a balanced splitting ratio \({({{\Delta}{\beta } {z}} = {\uppi }/{4})}\) exhibits basis-independent visibilities v ≈ −1, which also holds for fermionic two-photon states (as calculated in Supplementary Section 3) with visibility v ≈ 1. By contrast, a non-trivial loss mismatch (\(\Delta \gamma = 0.0396\, {\rm{cm}}^{-1}\), right box) leads to nearly \({{{{\mathrm{{\Delta}}}}}{\beta } {z}}\)-independent visibilities for any given basis orientation θ. This calculated behaviour is confirmed by the observed visibilities. These measurements clearly show that the requirement of \({{{{\mathrm{{\Delta}}}}}{\beta } {z} \approx {\uppi }/{4}}\) for quantum interference in the lossless case can be overcome by introducing non-trivial loss contrasts. This confirms that a new mechanism for tailoring two-photon coincidences has indeed been identified.

In conclusion, our findings outline a new approach to manipulate the coincidence dynamics of biphotons through the interplay of non-Hermitian systems and multiple-particle quantum mechanics. To this end, the polarization degree of freedom allows both for a seamless adjustment of the observation basis and for the implementation of complex loss profiles, facilitating a clear distinction between non-Hermitian photon dynamics and the basis-independent HOM visibilities for Hermitian quantum interference. Notably, the techniques described here can be readily combined with the versatility afforded by the three-dimensional spatial arrangement of birefringent waveguides25, specific exchange symmetries of the input states31 and waveguide structures in higher synthetic dimensions6, and open up a number of fascinating opportunities. For example, probabilistic quantum gates clearly stand to benefit greatly from the capability to introduce sophisticated loss realization into integrated optical platforms. Arbitrary quantum-optical transformations such as the singular value decomposition32 can be emulated in photonic systems, and even experiments on the evolution of multiple-particle quantum states in complex non-Hermitian systems become experimentally accessible. Taking a broader perspective, the reported basis-sensitive non-classical interference during photon evolution in open systems provides a first glance at previously unexplored physics behind the HOM dip. Along these lines, it will be of particular interest to leverage non-Hermitian HOM interferometry to explore photon-pair evolution in topological edge states33 and PT-symmetric quantum systems23, as well as quantum interference phenomena that arise from the Harper–Hofstadter model34, all of which are brought into experimental reach by our platform6.

Methods

Waveguide fabrication and birefringence characterization

We used the femtosecond-laser direct-writing technique25 to inscribe systems of evanescently coupled single-mode waveguides. To this end, ultrashort laser pulses with 270-fs duration at a carrier wavelength of λ = 517 nm and a repetition rate of 333 kHz from a fibre amplifier system (Coherent Monaco) were focused into the bulk of 150-mm-long fused-silica samples (Corning 7980) through a ×50 objective (NA = 0.60), forming waveguides along desired trajectories by translating the sample with a precision positioning system (Aerotech ALS180).

The combination of elliptical material modifications with residual stress fields from the rapid quenching immediately following the inscription pass imbues the laser-written waveguides with an inherent birefringence, the strength of which can be tuned via the writing speed35. We characterized the waveguides using classical light and crossed polarizers36. Having placed the sample between two PBSs and simultaneously rotating them by an angle φ (while maintaining their respective crossed orientation), the ratio of the transmitted (IT) and total intensities (Itotal) was measured with two photodiodes and evaluated according to the theoretical model

$${\frac{{I_{{{\mathrm{T}}}}}}{{I_{{{{\mathrm{total}}}}}}}} = {\sin ^2}{\left( {2\varphi } \right)} {\sin ^2}{\left( {{{{\mathrm{{\Delta}}}}}{\beta } {z}} \right)},$$

allowing for the relevant values of 0 and π/4 of the cumulative phase difference \({{{{\mathrm{{\Delta}}}}}{\beta } {z}}\) to be identified by max(IT/Itotal) = 0.5 and max(IT/Itotal) = 0 up to multiples of π/2 and π, respectively.

For our experiments we chose a separation of 27.5 μm between the target waveguide and the ancillary arrays, corresponding to coupling strengths of CH = 0.154 cm−1 and CV = 0.065 cm−1 of the polarization eigenstates to their respective ‘sinks’. To ensure that photons, once extracted from the target guide, were swiftly transported away, an array pitch of 20 μm was used to set the intra-array coupling strengths to 0.551 cm−1 and 0.335 cm−1 for the horizontal and vertical polarizations, respectively (see Extended Data Fig. 1 for more details on the dependence of the coupling strength on polarization and waveguide separation).

Biphoton creation and detection

For the generation of photon pairs, we used a type I SPDC source (Fig. 4a). A bismuth borate (BiBO) crystal was pumped by a 100-mW continuous-wave laser diode (Coherent OBIS) at λ = 407 nm, producing wavelength-degenerate horizontally polarized photon pairs at λ = 814 nm, which were subsequently collected by polarizing fibres and routed to the sample via a polarization combiner with adjustable principal axes of the output fibre. A variable delay τ between the arrival times of the two photons at the sample was implemented by means of a motorized translation stage (PI). The degree of indistinguishability of the photons was characterized by recording the HOM dip with a fibre-based beamsplitter, yielding a visibility of up to v = −97.2 ± 1.1%. To record coincidences, the single photons at the output of the PBS were detected with SPCMs (Excelitas Technologies, detection efficiency of >50%, dark counts of <50 s−1, dead time of 20 ns), the signals of which were processed with a correlation card (Becker & Hickl).