# Distinguishability and many-particle interference

###### Abstract

Quantum interference of two independent particles in pure quantum states is fully described by the particles’ distinguishability: the closer the particles are to being identical, the higher the degree of quantum interference. When more than two particles are involved, the situation becomes more complex and interference capability extends beyond pairwise distinguishability, taking on a surprisingly rich character. Here, we study many-particle interference using three photons. We show that the distinguishability between pairs of photons is not sufficient to fully describe the photons’ behaviour in a scattering process, but that a collective phase, the triad phase, plays a role. We are able to explore the full parameter space of three-photon interference by generating heralded single photons and interfering them in a fibre tritter. Using multiple degrees of freedom—temporal delays and polarisation—we isolate three-photon interference from two-photon interference. Our experiment disproves the view that pairwise two-photon distinguishability uniquely determines the degree of non-classical many-particle interference.

The famous Hong-Ou-Mandel (HOM) experiment in 1987 provided the first important example of non-classical two-photon interference Hong et al. (1987). Two independent photons impinging on a beam splitter exhibit bunching behaviour at the output ports that cannot be explained by a classical field model. The degree of bunching depends on how similar the two photons are in all degrees of freedom, for example time, frequency, polarisation, and spatial mode. This interference effect lies at the heart of photonic quantum information Knill et al. (2001) and has also become the standard tool for testing photon sources. Extending the study of interference to many particles is of interest from a fundamental as well as from a technological viewpoint Spagnolo et al. (2013); Tillmann et al. (2015); Mährlein et al. (2015). The scattering of multiple photons in linear networks is related to solving problems in quantum information processing, metrology, and quantum state engineering Aaronson and Arkhipov (2011); Spagnolo et al. (2012); Broome et al. (2013); Spring et al. (2013); Tillmann et al. (2013); Crespi et al. (2013); Gräfe et al. (2014); Motes et al. (2015); Carolan et al. (2015). Thus, understanding multiphoton interference is also of great relevance for practical applications.

Here, we demonstrate how many-particle interference is fundamentally richer than two-particle interference Tichy (2014). Two situations with the same pairwise distinguishability can lead to a different output distribution. This is due to a phase, the triad phase, that occurs only when more than two photons interfere.

We use independent photons and a tritter, a three-port symmetric beam splitter, as our tools for investigating multi-particle interference. We isolate the triad phase for the first time by interfering three photons in a tritter and exploiting multiple degrees of freedom, here time and polarisation. We show that interfering three identical photons and varying time delays between them, as demonstrated in previous experiments Spagnolo et al. (2013); Spring et al. (2016), is not sufficient to study three-photon interference in full generality Tichy et al. (2015). Further, we demonstrate that pairwise distinguishability between photons alone is not sufficient to fully describe the tritter’s output statistics Weihs et al. (1996). Our experiment allows us to isolate and tune the three-photon interference term as distinct from two-photon interference. Our work thus challenges the usual view that a general theory of photon (in)distinguishability can be reduced to time-delays de Guise et al. (2014); Tillmann et al. (2015).

### .1 Theory

The inner scalar product of two pure states and is:

(1) |

where is the real modulus and is the argument. The modulus can be interpreted as a measure of the distinguishability of two photons in states and , and equals zero (one) for two orthogonal (identical) states Jozsa and Schlienz (2000). The argument has, so far, received little attention due to its irrelevance in two-photon interference.

We consider two examples of devices that can be used to probe interference: a beam splitter and a tritter. The simplest device to probe interference is a balanced two-port beam splitter (see Fig. 1a). When two photons and are injected into the beam splitter, the output statistics depend on the pairwise distinguishability of the incident photons:

(2) |

where is the probability for detecting one photon in each of the output ports. If the photons are completely indistinguishable they always exit the same output port, in contrast to the classical behaviour.

A tritter maps three spatial input modes onto three spatial output modes (see Fig. 1b); a linear transformation corresponding to a balanced tritter is given by the unitary matrix:

(3) |

where each output is equally likely and .

If we inject three photons into the tritter—a single photon in state into each mode for each —the probability of having one photon in each of the output modes of the tritter is (see Appendix) Tichy (2015); Shchesnovich (2015a):

(4) |

where we define the collective triad phase as the sum of the three arguments. The dependence on appears only if the photons are partially distinguishable. If the states are orthogonal, the three moduli are zero; if they are identical, their scalar product will be equal to one and vanishes. Similar expressions can also be derived for the probabilities of having two or three photons in one of the output modes of the tritter (see Appendix).

Note that a global phase applied onto one of the input states does not lead to any change in the triad phase . Each phase is only defined up to a global arbitrary phase. The sum of the phases, the triad phase, has physical meaning and is a measurable quantity. It remains unaffected by any global phase transformation and is crucial for the description of partially distinguishable photons Bergou et al. (2012); Sugimoto et al. (2010).

However, dependence on the triad phase only occurs in measurements with more than two photons. The two-photon output coincidence probabilities (one photon in outputs 2 and 3), , when sending two photons into different input ports of the tritter (as in Fig. 1e) are:

(5) |

and depend only on the mutual distinguishability of the incident photons and .

### .2 Probing the triad phase and genuine three-photon interference

We introduce a convenient implementation that allows us to control the moduli and the arguments independently. We use two degrees of freedom for each spatial mode—time and polarisation—to show that the addition of non-identical polarisation states can be used to create a non-zero . We consider the following input states to the tritter (see Fig. 2):

(6) |

where is a temporal mode delayed by time , and denote horizontal and vertical polarisation, respectively, and denotes the spatial mode. Using only temporal modes, , and otherwise identical photons with symmetric spectral intensities, the triad phase would always vanish, since is real and non-negative (see Appendix for more information on temporal modes).

In a first experiment, we aim to probe the triad phase directly. As a first step, we prepare the photons with the same polarisation in states

(7) |

for , which sets .

In the next step, we prepare photons in states (as depicted in the inset in Fig. 3b):

(8) | |||||

Here the scalar products and , but , setting . These two configurations demonstrate that using polarisation as an additional degree of freedom allows us to vary the triad phase (see Appendix for more details).

In a second experiment, we isolate three-photon interference from two-photon interference. We explicitly show that control of allows manipulation of the three-photon term whilst leaving the two-photon interference terms constant. To do so, we prepare the following as input states to the tritter:

(9) | ||||

where the state depends on a polarisation rotation with angle and the polarisations of and are kept constant. With these states, we obtain the following moduli

(10) | ||||

(11) | ||||

(12) |

and the triad phase

(13) |

The angle affects both the triad phase and the moduli , ; the temporal state only affects and , but not . Combining control of both and allows us to manipulate whilst and remain unchanged. For example, to keep , must be chosen such that

(14) |

with and being the variance in time of the Gaussian wave packet (see Appendix).

### .3 Experiment and Results

To study the triad phase experimentally, we generate three heralded photons using spontaneous four-wave mixing (SFWM) in silica-on-silicon waveguides Spring et al. (2016). Using wave plates and delay stages, we prepare the polarisation and temporal state of each input photon before coupling into the fibres to the tritter (see Fig. 2 and Appendix for technical details).

We first probe the triad phase directly by choosing the input polarisations of the photons as given in Eqns. (7) and (8). By setting , and , and varying smoothly over the range shown in Fig. 3c and d, we tune the degree of two- and three-photon interference. The results are shown in Fig. 3; we see a clear qualitative difference in behaviour for the two cases of and . In the former case we observe a W-like shape, whereas for the other case we observe a dip; deviations from the ideal curves are discussed below (see caption of Fig. 3).

We then demonstrate genuine three-photon interference by choosing the input states as given in Eqn. (9), but now setting the time delay differences as in Eqn. 14. We determined from a set of two-photon HOM dips with polarisations chosen as in Fig. 4a (first and third panel). The results are shown in Fig. 4; we observe good agreement of the measured curves with the theoretical prediction. The three-photon data follow a cosine shape as predicted by Eqn. 4. The two-photon contributions , , (see Eqn. 5) are nearly constant and show fluctuations of only on average 6%. and the single photon detections at the tritter outputs vary only by a maximum of 3% due to polarisation dependence. This verifies that these two-photon contributions are independent of the arguments. Detailed analysis suggests that polarization dependence of the tritter contributes to these fluctuations (see Appendix).

Our experimental data, both in Fig. 3 and Fig. 4, show the expected behaviour, but there are some deviations from the probabilities given by Eqns. (7), (8), and (9).
These are primarily due to an imperfect tritter operation, imperfections in the photon preparation (polarisation, purity (), distinguishability), and higher-order photon emission (squeezing parameter , see Appendix). Further, along with photons that are produced by the SFWM-process, uncorrelated photons are created in other processes such as Raman scattering and fluorescence Spring et al. (2016). To understand the influence of all these effects on the measured visibilities, we performed a simulation of our experiment.
Our model includes terms corresponding to up to photons in total (signals and heralds) and up to 3 uncorrelated noise photons. This provides sufficient accuracy as terms corresponding to higher photon numbers are negligibly small.
Based on our model, we calculated theory curves including realistic experimental parameters. These curves are shown in Fig. 3 and Fig. 4 as dashed lines and agree well with our measured data (see Appendix for a more detailed analysis) (see also ^{1}^{1}1We became aware that similar results to those in Fig. 4 have been obtained using a three-photon entangled state Agne et al. ()).

### .4 Conclusion

In this work, we identify and describe a new phase that arises at the level of three photons: the triad phase. This new phase manifests itself in quantum interference and therefore has implications for the scattering of many particles. In particular, the outcome of scattering events of more than two particles is determined not only by pairwise distinguishabilities of the particles’ wavefunctions, but also on the collective properties of the particles. In this context, the triad phase initially emerges as a formal artifact Tichy et al. (2011); Tan et al. (2013); Ra et al. (2013); de Guise et al. (2014); Shchesnovich (2014); Tamma and Laibacher (2014); Tichy (2014); Shchesnovich (2015b, a); Tamma and Laibacher (2015); we show here that it is of physical relevance. Two situations involving the scattering of multiple particles with the same pairwise distinguishability can nevertheless exhibit a different outcome for scattering depending on the triad phase.

There is a formal similarity between the triad phase and the geometric phases that can be acquired by single photons, for instance in the Pancharatnam-Berry phase Berry (1987); Jozsa and Schlienz (2000); Hartley and Vedral (2004); Kobayashi et al. (2010). Scaling up our study to more than three photons is ongoing work, but for example four interfering photons can be described by six two-photon measurements and three three-photon measurements.

Our work has implications for both linear-optical quantum computing and general multiparticle scattering. It shows having truly indistinguishable particles is a crucial ingredient for all types of scattering experiments. However, our work also opens up new opportunities as the triad phase can be seen as a tool to engineer the output state of a scattering process. Furthermore, extending applications such as boson sampling to partial distinguishabilities and using multiple degrees of freedom will be an interesting avenue to explore.

## References

- Hong et al. (1987) C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).
- Knill et al. (2001) E. Knill, R. Laflamme, and G. J. Milburn, Nature 409, 46 (2001).
- Spagnolo et al. (2013) N. Spagnolo, C. Vitelli, L. Aparo, P. Mataloni, F. Sciarrino, A. Crespi, R. Ramponi, and R. Osellame, Nat. Commun. 4, 1606 (2013).
- Tillmann et al. (2015) M. Tillmann, S.-H. Tan, S. E. Stoeckl, B. C. Sanders, H. de Guise, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, Phys. Rev. X 5, 041015 (2015).
- Mährlein et al. (2015) S. Mährlein, J. Von Zanthier, and G. S. Agarwal, Opt. Express 23, 15833 (2015).
- Aaronson and Arkhipov (2011) S. Aaronson and A. Arkhipov, in Proceedings of the 43rd annual ACM symposium on Theory of computing (2011) pp. 333–342.
- Spagnolo et al. (2012) N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, and F. Sciarrino, Sci. Rep. 2 (2012).
- Broome et al. (2013) M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, Science 339, 794 (2013).
- Spring et al. (2013) J. Spring et al., Science 339, 798 (2013).
- Tillmann et al. (2013) M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, Nat. Photon. 7, 540 (2013).
- Crespi et al. (2013) A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, Nat. Photon. 7, 545 (2013).
- Gräfe et al. (2014) M. Gräfe, R. Heilmann, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, Nat. Photon. 8, 791 (2014).
- Motes et al. (2015) K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson, and P. P. Rohde, Phys. Rev. Lett. 114, 170802 (2015).
- Carolan et al. (2015) J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, Science 349, 711 (2015).
- Tichy (2014) M. C. Tichy, J. Phys. B 47, 103001 (2014).
- Spring et al. (2016) J. B. Spring, P. L. Mennea, B. J. Metcalf, P. C. Humphreys, J. C. Gates, H. L. Rogers, C. Soeller, B. J. Smith, W. S. Kolthammer, P. G. Smith, et al., arXiv preprint arXiv:1603.06984 (2016).
- Tichy et al. (2015) M. C. Tichy, Y.-S. Ra, H.-T. Lim, C. Gneiting, Y.-H. Kim, and K. Mølmer, New J. Phys. 17, 023008 (2015).
- Weihs et al. (1996) G. Weihs, M. Reck, H. Weinfurter, and A. Zeilinger, Phys. Rev. A 54, 893 (1996).
- de Guise et al. (2014) H. de Guise, S.-H. Tan, I. P. Poulin, and B. C. Sanders, Phys. Rev. A 89, 063819 (2014).
- Jozsa and Schlienz (2000) R. Jozsa and J. Schlienz, Phys. Rev. A 62, 012301 (2000).
- Tichy (2015) M. C. Tichy, Phys. Rev. A 91, 022316 (2015).
- Shchesnovich (2015a) V. S. Shchesnovich, Phys. Rev. A 91, 013844 (2015a).
- Bergou et al. (2012) J. A. Bergou, U. Futschik, and E. Feldman, Phys. Rev. Lett. 108, 250502 (2012).
- Sugimoto et al. (2010) H. Sugimoto, T. Hashimoto, M. Horibe, and A. Hayashi, Phys. Rev. A 82, 032338 (2010).
- (25) We became aware that similar results to those in Fig. 4 have been obtained using a three-photon entangled state Agne et al. ().
- Tichy et al. (2011) M. C. Tichy, H.-T. Lim, Y.-S. Ra, F. Mintert, Y.-H. Kim, and A. Buchleitner, Phys. Rev. A 83, 062111 (2011).
- Tan et al. (2013) S.-H. Tan, Y. Y. Gao, H. de Guise, and B. C. Sanders, Phys. Rev. Lett. 110, 113603 (2013).
- Ra et al. (2013) Y.-S. Ra, M. C. Tichy, H.-T. Lim, O. Kwon, F. Mintert, A. Buchleitner, and Y.-H. Kim, Proc. Natl. Acad. Sci. USA 110, 1227 (2013).
- Shchesnovich (2014) V. Shchesnovich, Phys. Rev. A 89, 022333 (2014).
- Tamma and Laibacher (2014) V. Tamma and S. Laibacher, Phys. Rev. A 90, 063836 (2014).
- Shchesnovich (2015b) V. S. Shchesnovich, Phys. Rev. A 91, 063842 (2015b).
- Tamma and Laibacher (2015) V. Tamma and S. Laibacher, Phys. Rev. Lett. 114, 243601 (2015).
- Berry (1987) M. V. Berry, Journal of Modern Optics 34, 1401 (1987).
- Hartley and Vedral (2004) J. Hartley and V. Vedral, J. Phys. A 37, 11259 (2004).
- Kobayashi et al. (2010) H. Kobayashi, S. Tamate, T. Nakanishi, K. Sugiyama, and M. Kitano, Phys. Rev. A 81, 012104 (2010).
- (36) S. Agne, T. Kauten, J. Jin, E. Meyer-Scott, J. Z. Salvail, D. R. Hamel, K. J. Resch, G. Weihs, and T. Jennewein, arXiv:1609.07508 .
- Tichy et al. (2010) M. C. Tichy, M. Tiersch, F. de Melo, F. Mintert, and A. Buchleitner, Phys. Rev. Lett. 104, 220405 (2010).

## Appendix A Acknowledgements

We thank Carlo Di Franco, Luca Rigovacca, Myungshik Kim, and Jan Sperling for helpful discussions. A.J.M is supported by the James Buckee scholarship from Merton College. A.E.J. is supported by the EPSRC Controlled Quantum Dynamics CDT. M.C.T. acknowledges support from Danish Council for Independent Research and the Villum Foundation. S.B. acknowledges support from the Marie Curie Actions within the Seventh Framework Programme for Research of the European Commission, under the Initial Training Network PICQUE (Photonic Integrated Compound Quantum Encoding, grant agreement no. 608062) and from the European Union’s Horizon 2020 Research and Innovation program under Marie Sklodowska-Curie Grant Agreement No. 658073. I.A.W. acknowledges an ERC Advanced Grant (MOQUACINO) and the UK EPSRC project EP/K034480/1.

## Appendix B Author contributions

A.J.M. and A.E.J. performed the experiments, theoretical modeling, and data analysis. B.J.M., S.B., and W.S.K assisted with data-taking and data analysis. All authors discussed the results. M.C.T. developed the theory. B.J.M., S.B., W.S.K. and I.A.W conceived the project. S.B., W.S.K. and I.A.W. supervised the project. All authors wrote the manuscript.

## Appendix C Appendix: Additional theory

### c.1 Transition probability for three partially distinguishable bosons in a three mode-setup

We inject three partially distinguishable bosons into the three input modes of a scattering setup described by a unitary matrix . The distinguishing degrees of freedom of the bosons – in our case the photon polarisation and the time-of-arrival – are described by the states . The mutual pairwise distinguishabilities are then encoded in the positive semi-definite hermitian matrix , which accommodates both the scalar product moduli as well as the relative phases . The probability to find one particle in each output mode is obtained as a sum over all possible double-sided Feynman diagrams Tichy (2015), giving a multidimensional permanent

(15) |

comprising fully distinguishable particles (, ) and perfectly identical bosons (, ) as extremal cases.

Our aim is to understand the dependence of three-photon interference on the distinguishability parameters and in detail. For this purpose, we write out the sums over the permutation group explicitly,

(16) | |||||

where is the element-wise complex conjugated matrix with the rows (corresponding to the input modes) permuted as , i.e. for , we leave the matrix unchanged, while for , we exchange the first two rows. The product is meant as Hadamard element-wise multiplication, not the usual matrix-product.

For final states with particles in the th output mode, we adapt Eqn. (16) formally by replacing by a matrix of the same dimensions that repeats the th column of times , i.e. the column multiplicity reflects the final mode population. To ensure the proper normalization of the final result, a factor needs to be included, where is the mode occupation list of the final state.

We see clearly how the dependence on the scattering matrix is separated from the dependence on the scalar products . For a fixed scattering matrix , the output signals depend on precisely six parameters, , of which only four have physical significance: the three scalar product moduli and the collective triad phase . Whereas each relative phase can be transformed away by a global phase transformation, the sum of the three relative phases – the collective triad phase – remains independent of any choice of basis or global phase. The dependence of scattering probabilities on a phase that describes the particles’ collective indistinguishability has no precedent in single- or two-particle scattering. The triad phase only carries physical meaning in the context of the full three-particle state, and is thus a purely collective quantity.

### c.2 Event probabilities in the symmetric tritter

For a symmetric tritter,

(17) |

the output event probabilities take particularly simple forms:

(18) | |||||

(19) | |||||

(20) | |||||

(21) |

Here, denotes the probability of measuring photons in output mode 1, photons in output mode 2, and photons in output mode 3. The probability to find two particles in one output mode vanishes for indistinguishable photons, a result of the suppression law for Fourier matrices Tichy et al. (2010).

## Appendix D Internal space dimensionality and triad phase

The triad phase encodes a purely collective property, which can only take non-trivial values for partially distinguishable particles: If all pairs of particles are mutually perfectly indistinguishable, we have , such that . The three states then span a trivial one-dimensional Hilbert-space, and . On the other hand, when the particles are fully distinguishable, all scalar products , , vanish, and the value of the phase is neither defined, nor does it have any impact on any observable, since it comes only in conjunction with the product . In this case, the three states span a three-dimensional Hilbert-space.

In the intermediate case in which all particles are neither mutually distinguishable nor indistinguishable ( for all ), the question arises whether the three internal states span a three- or merely a two-dimensional Hilbert-space. This question arises, e.g. when three photons are deliberately prepared in different polarisation states, but are supposed to be indistinguishable in all other degrees of freedom.

In order to see how the measurement of the triad phase can resolve this question, let us first consider three states living in a qubit-like two-dimensional Hilbert-space. Without restrictions to generality, we can then find two states and , such that

(22) | |||||

(23) | |||||

(24) |

where , . We note that the three states are described by three parameters – precisely those required to describe the relative positions of three points on a Bloch-sphere describing a qubit. In Eqn. 4 (main paper), however, four parameters – – dictate the degree of three-particle interference.

By evaluating the relevant scalar products

(25) | |||||

(26) | |||||

(27) |

we can express as a function of , to see that is fixed by the three scalar product moduli , i.e.

(28) |

In that sense, when restricted to a qubit-like Hilbert-space, the triad phase is not an independent degree of freedom, but it is fully fixed by the geometry of the three vectors on the Bloch sphere, or, equivalently, by .

When we lift the restriction to a qubit-like Hilbert-space and admit states of the more general form

(29) | |||||

(30) | |||||

(31) |

the new parameter describes to what extent the third state lives outside the Hilbert-space spanned by and . Hence, we now need four parameters to describe the three states, and even when are fixed, now remains an independently tuneable parameter.

As a consequence, the three-photon measurements yielding and reveal whether or not the three states span a three-dimensional Hilbert-space () or merely a two-dimensional one (). The latter is clearly a collective property of all three states, and invisible to any combination of two-photon measurement data, which only yield , but not the triad phase . In an experiment, a measurement of that is compatible to implies that the three photons live in a two-dimensional space, while a measurement that is incompatible with shows that the three photons are distinguishable in more than a qubit-like degree of freedom.

## Appendix E Dependence of triad phase on delay in three-photon interference

A single photon in spectral mode is denoted

(32) |

where the state of a single photon with angular frequency is given by and the mode is described by the complex-valued spectral amplitude . The same state can be described in the temporal domain with the mode transformation , where describes a single photon with arrival time . With this substitution, we find where the temporal amplitude is the inverse Fourier transform of .

If we delay a single photon initially in mode by a time , the resulting state is

(33) |

In the following, we write this as and forgo the subscript since only one initial mode will be considered. The inner product of two single-photons in initial mode delayed by times and is

(34) |

where the function is defined as the Fourier transform of the spectral intensity and .

Now consider the triad phase of three photons each in initial mode with a distinct delay

(35) |

We are interested in the conditions for which is independent of the delays. Accordingly we require that derivative with respect to delay vanishes

(36) | ||||

For this to be true for all values of and , it must be that is independent of . Therefore is linear

(37) |

The inverse Fourier transform of , the spectral intensity, is real. It follows that . Therefore, is odd and .

If is independent of the delays, we can therefore write

(38) |

Consider spectral intensity functions for which , which we denote as . For this case, since and its Fourier transform are real valued, is an even function. Now consider the general case for non-zero . The shift property of the Fourier transform along with Eqn. (38) tells us that is an even function. Therefore we conclude that the triad phase is independent of delays if the three photons start in identical spectral modes for which the spectral intensity is symmetric about its mean value.

## Appendix F Inner products of Gaussian wavepackets with relative delays

The state of a single photon in the time-frequency modes , delayed by time is given by:

(39) |

For a Gaussian wave-packet delayed by time t, central frequency and variance in time , takes the form:

(40) |

We can express the overlap of the temporal modes of two photons with identical Gaussian spectra at times and as:

(41) |

Next, we show that for time-delays the products of overlaps that appear in the expressions for the multi-photon coincidence probabilities are always real and positive and hence do not give rise to a triad phase. In the case of the two photon interference terms, which contain expressions of the form

(42) |

this is easy to see, as the expression is purely real. For the three photon interference term:

(43) |

As we can see this expression is also real. This also holds for any number of photons.

## Appendix G Experimental details

We pump three separate waveguides in a silica-on-silicon chip Spring et al. (2016) with a Ti-Saph femtosecond pulsed laser running at MHz and nm (fs pulses). In each guide, we generate one signal and one idler photon at nm and nm, respectively. The pump has an orthogonal polarisation to the daughter photons and so can be separated using a polarising beam splitter after the chip. The signal and idler are spatially separated using a dichroic mirror before final filtering to remove residual pump and to factor out their spectral components, removing spectral correlations to give pure single photons. Pumping three of these guides yields three signal photons and three idler photons. By heralding on the former using three silicon APDs, we are left with three heralded, highly pure identical single photons with central wavelength nm at a rate of about Hz when pumping with mW per guide. The setup for measuring different output configurations after the interference tritter is shown in Fig. 5.

## Appendix H Raw experimental data

### h.1 Polarisations set for (cf. Equation 7 in main paper)

#### h.1.1 HOM dips for temporal alignment of photons

In order to align the generated photons temporally and verify their indistinguishability, we perform heralded HOM measurements for the three possible pairs injected into the tritter. We also use these to verify our polarisation state preparations. The results are shown in the Figures below.

For the case where the photons have identical polarisation, we expect a theoretical visibility of 50% (since the two-photon coincidence probability is ) for a tritter, and so the visibility should be half the scalar product magnitude), we record closer to 40% due to all effects mentioned in the main paper. The dip in Figure 6 is twice as narrow as the others, corresponding to the dip between two photons which are both being translated in time on injection. The other two dips are from when only one of the photons injected into the tritter is translated in time (see Figure 3 in main text). The dips are all centred such that the three photons overlap in time when the stages are at their zero positions.

#### h.1.2 Additional output event plots

Here we present plots for count rates corresponding to in the case where all photons are injected into the tritter with the same polarisation. In the ideal case when all photons are completely indistinguishable in time and polarisation, and these outputs are completely suppressed Tichy et al. (2010). Our simulations demonstrate this is not the case when taking into account experimental imperfections, and the visibility is reduced from 100% to around 57%. The theory and simulation curves have been rescaled for comparison with experimental count rates.

### h.2 Polarisations set for (cf. Equation 8 in main paper)

#### h.2.1 HOM dips for temporal alignment of photons

Again to align the three photons temporally before injection into the tritter, we perform HOM measurements for the three pairs of photons. We expect 12.5% visibility but record closer to 10%, again due to the effects mentioned in the main paper. The dip in Figure 12 is twice as narrow as the others, corresponding to the dip between two photons which are both being translated in time on injection. The other two dips are from when only one of the photons injected into the tritter is translated in time (see Figure 3 in main text). The dips are all centred such that the three photons overlap in time when the stages are at their zero positions.

We also recorded coincidences for the outputs corresponding to but these statistics are all predicted to have lower visibilities for this case of compared to . Our recorded statistics are not sufficient to resolve these features.

### h.3 Probing the triad phase (cf. Equation 9 in main paper)

#### h.3.1 Polarisation dependence of the tritter

For isolating three-photon interference, we scan the triad phase by varying the polarisation of one of the photons. In order to study the polarisation-dependence of the tritter, we send heralded single photons into different tritter inputs and record the output counts (see Figures 15 and 16).

The total number of counts is relatively constant (see Fig. 16), whilst some individual heralded singles events in the bottom row of Figure 15 vary as the triad phase (and thus polarisation of the photon injected into the first input) changes. This suggests that the couplings of the tritter have a slight polarisation dependence.

#### h.3.2 Heralded two-photon coincidences

We monitored the heralded two-fold coincidences to verify that we have as little variation as possible as a function of the triad phase. In Figure 17 all possible combinations of heralded two-photon events are displayed. The largest variation in counts is observed for channels containing the first input channel, arising, as discussed in the previous section, from the tritter’s polarisation dependence.

#### h.3.3 Additional output event plots

## Appendix I Simulation of the experiment

In order to provide a simulation of the experiment, we used the formalism developed in Tichy et al. (2015); Shchesnovich (2015b, a) to simulate general mixed, squeezed states, contaminated with distinguishable noise photons, that are input into a lossy unitary.

### i.1 Impure input states

It was noted previously in Tichy et al. (2015) that the counting statistics for a mixed state input can be expressed as a function of the density matrices for each photon in input mode . For three photons input to an interferometer described by the unitary this leads to the following expression for the coincidence probability :

(44) |

For simplicity in the simulation we make the assumption that we can decompose the density matrix into a mixed and a pure subspace, where the full density matrix for each photon is given by their tensor product:

(45) |

may be represented as the tensor product of a density matrix which contains the temporal modes and another containing the polarisation degree of freedom.

(46) |

For general temporal modes , we find a representation of the states in terms of orthonormal modes using the Gram-Schmidt decomposition:

(47) | |||||

(48) | |||||

(49) |

Where

and are a set of orthonormal vectors. We can then construct the density matrices in mode basis:

(50) |

The polarisation density matrix is constructed from basis states and . Mixedness is modelled on a two dimensional Hilbert-space which is chosen to be orthogonal to time-frequency and polarisation modes.

### i.2 Higher order photon contributions

The state of a single ideal two-mode-squeezer is given by:

(51) |

Furthermore, we assume that in each source uncorrelated photons are created with probabilities for the idlers and for the signals. In particular is the probability of producing no uncorrelated noise photons. is the probability of creating exactly one uncorrelated photon pair.

We can then construct the density matrix for one source’s emission:

(52) |

where for each total number of photons , we include cases where they come from four-wave mixing or noise processes. The indices and label the number of signal and idler noise photons which are assumed to be completely distinguishable from all other photons.

### i.3 Parameter values

In the following table we give the parameter values that were used for the simulation:

Name | Symbol | Value |
---|---|---|

Squeezing-parameter | 0.16 | |

Purity | 0.9 | |

Fluorescence probability idler | 0.035 | |

Fluorescence probability signal | 0.009 |

The squeezing parameter was taken to be the same as in Spring et al. (2016); the experiment reported in Spring et al. (2016) was performed with the same power of the pump beam). The purity is a lower bound estimate and primarily affected by our ability to filter out non-factorable components in the (signal/idler) joint spectral distribution. We were limited in the signal/idler filtering bandwidth as we used a single pair of angle tuned bandpass filters in the beam path of signal and idler photons, immediately after a dichroic mirror. Since the three beams pass through the filters at slightly different angles the filters’ spectral edges are slightly shifted with respect to each other, effectively limiting our tuning range. We calculate the degree of spectral purity for the given filter bandwidth of nm and obtain a value of approximately purity. The uncorrelated noise probability is obtained from a measurement of the heralded in Spring et al. (2016) (supplementary). We perform a fit of the to our model and use as a free parameter. is chosen to be of as the background noise for the signals is significantly smaller. The ratio of was obtained by comparing background noise levels of signal and idler photons with a single photon spectrometer. When the pump polarisation is rotated by 90 degree we lose phase-matching, allowing us to observe the background noise only at the given input power.