Abstract
The ability of indistinguishable particles to interfere with one another is a core principle of quantum mechanics. The interplay of interference and particles exchange statistics^{1,2,3,4} gives rise to the Hong–Ou–Mandel (HOM) effect^{5}, where the bunching of bosons suppresses twoparticle coincidences between the output ports of a balanced beamsplitter. Conversely, fermionic antibunching can yield up to a twofold enhancement of coincidences compared to the baseline of distinguishable particles. As such, the emergence of dips or peaks in the HOM effect may appear indicative of the particles’ bosonic/fermionic nature. Here, we demonstrate experimentally that the coincidence statistics of boson pairs can be seamlessly tuned from full suppression to enhancement by an appropriate choice of the observation basis. Our photonic setting leverages birefringent couplers^{6} to introduce differential dissipation in the photons’ polarization. In contrast to previous work^{7,8,9}, the mechanism underpinning this unusual behaviour does not act on individual phases accumulated by pairs of particles along specific paths, but instead allows them to jointly evade losses as indistinguishable photons are prevented from inhabiting orthogonal modes. Our findings reveal a new approach to harnessing nonHermitian settings for the manipulation of multiparticle quantum states and as functional elements in quantum simulation.
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, nonHermitian 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 groundbreaking work of Bender and Boettcher on paritytime (PT) symmetry^{10} and its subsequent adaptation to optical settings^{11}, a slew of works^{12} have revealed fascinating features arising from the complex interplay of gain and loss, for example, in nonorthogonal eigenmodes^{13}, peculiar transport properties^{14}, lossinduced transparency^{15} and unidirectional invisibility^{16} as well as exceptional points^{16,17} with enhanced response^{18}, PTsymmetric lasers^{19,20} and even lightfunnelling^{21}. However, although gain and loss are readily incorporated into classical optics as a complexvalued 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 light^{22}. Fortunately, inadvertently introducing additional quantum noise can be avoided by entirely passive configurations. For example, quasiPTsymmetric loss distributions in directional couplers have been shown to systematically accelerate the onset of fully destructive twophoton quantum interference^{23}. Along similar lines, lossy beamsplitters^{7}, opaque scattering media^{8}, nonunitary metasurfaces^{24} or photonically implemented quantum decay processes^{9} may enhance twophoton 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 antibunching behaviour.
In this Letter we experimentally demonstrate that, even if lossinduced antibunching 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 nonHermitian twoport couplers based on femtosecond laserwritten birefringent waveguides^{25}, 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 loss^{26,27} through polarizationsensitive coupling^{28}. Notably, indistinguishable photons are known for being able to partially evade lossy domains in unison^{29}, 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 C_{ind} compared to the baseline C_{dis} 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 twophoton interference on a balanced beamsplitter remains strictly basisindependent (Fig. 1e). To gain more detailed insights into these dynamics, we study the HOM patterns for representative basis orientations associated with visibilities of v = C_{ind}/C_{dis} − 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 transmission^{24}.
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 space^{6}. 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 antidiagonal 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 polarizationdependent coupling strengths^{28} C_{H} and C_{V}. 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 diffraction^{27}. Although the resulting polarizationsensitive loss rates γ_{H} and γ_{V} 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 singlephoton state
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).
As shown in Fig. 3a, horizontally and vertically polarized excitations of the target waveguide are indeed attenuated at markedly different rates, as confirmed by tightbinding simulations. Notably, the Zeno dynamics^{26} in tightbinding lattices do not yield a purely exponential decay of quantum states, and the resulting deviations at short propagation distances^{30} can be accounted for by defining effective loss rates ɣ_{H/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 γ_{V} = 0.02433 cm^{−1}), while the substantially larger coupling in the horizontal polarization component was able to reduce the respective transmission to 4.5% (ɣ_{H} = 0.1035 cm^{−1}).
To study the impact on HOM dynamics, a polarizationduplexed twophoton state, with one photon each in the diagonal and antidiagonal states, was synthesized from a spontaneous parametric downconversion (SPDC) photonpair 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 singlephoton counting modules (SPCMs) to register coincidences between the two output ports. Crucially, a halfwave 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/antidiagonal (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.
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 ɣ_{H/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 v_{HV} = −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 v_{AD} = +54.1 ± 0.9% in the A/D basis (θ = 45°). Note that an ideal HOM peak visibility of v_{AD} = 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 highcontrast 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 nontrivial 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 basisdependent 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 twophoton coincidences (v_{HV} ≈ −1). This prevalence of the conventional HOM dip is inextricably linked to the structure of the twophoton state
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 twophoton wavefunction of distinguishable particles injected as \({{{{\mathrm{A}}}},{{{\mathrm{D}}}}\rangle}\) evolves as
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 nonHermiticity 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 polarizationindependent, 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 (v_{AD} = 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 Vpolarized 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 v_{AD} = 1, whereas finite loss contrasts yield maximal visibilities in the range of 0 < v_{AD} < 1 (cf. Fig. 4b, where a maximal visibility of v_{AD} = 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 v_{AD}: 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, twophoton interference for a balanced splitting ratio \({({{\Delta}{\beta } {z}} = {\uppi }/{4})}\) exhibits basisindependent visibilities v ≈ −1, which also holds for fermionic twophoton states (as calculated in Supplementary Section 3) with visibility v ≈ 1. By contrast, a nontrivial 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 nontrivial loss contrasts. This confirms that a new mechanism for tailoring twophoton coincidences has indeed been identified.
In conclusion, our findings outline a new approach to manipulate the coincidence dynamics of biphotons through the interplay of nonHermitian systems and multipleparticle 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 nonHermitian photon dynamics and the basisindependent HOM visibilities for Hermitian quantum interference. Notably, the techniques described here can be readily combined with the versatility afforded by the threedimensional spatial arrangement of birefringent waveguides^{25}, specific exchange symmetries of the input states^{31} and waveguide structures in higher synthetic dimensions^{6}, 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 quantumoptical transformations such as the singular value decomposition^{32} can be emulated in photonic systems, and even experiments on the evolution of multipleparticle quantum states in complex nonHermitian systems become experimentally accessible. Taking a broader perspective, the reported basissensitive nonclassical 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 nonHermitian HOM interferometry to explore photonpair evolution in topological edge states^{33} and PTsymmetric quantum systems^{23}, as well as quantum interference phenomena that arise from the Harper–Hofstadter model^{34}, all of which are brought into experimental reach by our platform^{6}.
Methods
Waveguide fabrication and birefringence characterization
We used the femtosecondlaser directwriting technique^{25} to inscribe systems of evanescently coupled singlemode waveguides. To this end, ultrashort laser pulses with 270fs 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 150mmlong fusedsilica 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 laserwritten waveguides with an inherent birefringence, the strength of which can be tuned via the writing speed^{35}. We characterized the waveguides using classical light and crossed polarizers^{36}. 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 (I_{T}) and total intensities (I_{total}) was measured with two photodiodes and evaluated according to the theoretical model
allowing for the relevant values of 0 and π/4 of the cumulative phase difference \({{{{\mathrm{{\Delta}}}}}{\beta } {z}}\) to be identified by max(I_{T}/I_{total}) = 0.5 and max(I_{T}/I_{total}) = 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 C_{H} = 0.154 cm^{−1} and C_{V} = 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 intraarray 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 100mW continuouswave laser diode (Coherent OBIS) at λ = 407 nm, producing wavelengthdegenerate 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 fibrebased 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).
Data availability
All experimental data that have been used to produce the results reported in this manuscript are available in an openaccess data repository^{37}.
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Acknowledgements
We thank C. Otto for preparing the highquality fusedsilica samples used in this work. We acknowledge funding from the European Research Council (grant no. 899368 ‘EPIQUS’), Deutsche Forschungsgemeinschaft (grants nos. SCHE 612/61, SZ 276/121, BL 574/131, SZ 276/151 and SZ 276/201) and the Alfried Krupp von Bohlen and Halbach Foundation.
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M.E. fabricated the samples and carried out the measurements. All authors jointly interpreted the measured data and cowrote the manuscript.
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Extended data
Extended Data Fig. 1 Dependence of the coupling strength on distance and polarization.
Horizontally/vertically polarized classical light at a wavelength of λ = 814 nm excites appropriately spaced waveguide pairs in order to calibrate the coupling strength from the waveguides’ output intensities.
Extended Data Fig. 2 HOM traces in the strongly nonHermitian limit.
These HOM traces are observed by replacing the birefringent photonic chip with a standard polarizing beam splitter cube with an intensity extinction ration of 10^{3} between H and V polarization, corresponding to the limit of \({{{\mathrm{{\Delta}}}}}\gamma \to \infty\). In accordance with the calculated behavior, shown in Supplementary Information, the peak visibility remains near unity irrespective of the basis orientation, while the overall count rate is maximal in the AD basis (θ=±45°) and approaches zero in the HV configuration (θ=0°).
Extended Data Fig. 3 Full set of measured HOM traces.
These HOM pattern are corresponding to the visibilities in Fig. 4c: Hermitian polarization coupler a without (\({{{\mathrm{{\Delta}}}}}\beta z = 0\)) and b with interference (\({{{\mathrm{{\Delta}}}}}\beta z = \pi /4\)), and lossy coupler (\({{{\mathrm{{\Delta}}}}}\gamma = 0.0396\,{{{\mathrm{cm}}}}^{  1}\)) c without (\({{{\mathrm{{\Delta}}}}}\beta z = 0\)) and d with interference (\({{{\mathrm{{\Delta}}}}}\beta z = \pi /4\)), respectively. Note that the data shown in Fig. 4b of the main manuscript is reproduced here in subfigure b for ease of comparison with the other configurations.
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Ehrhardt, M., Heinrich, M. & Szameit, A. Observationdependent suppression and enhancement of twophoton coincidences by tailored losses. Nat. Photon. 16, 191–195 (2022). https://doi.org/10.1038/s41566021009433
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DOI: https://doi.org/10.1038/s41566021009433
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