Experimental Fock-State Bunching Capability of Non-Ideal Single-Photon States

Advanced quantum technologies, as well as fundamental tests of quantum physics, crucially require the interference of multiple single photons in linear-optics circuits. This interference can result in the bunching of photons into higher Fock states, leading to a complex bosonic behaviour. These challenging tasks timely require to develop collective criteria to benchmark many independent initial resources. Here we determine whether n independent imperfect single photons can ultimately bunch into the Fock state $|n \rangle$. We thereby introduce an experimental Fock-state bunching capability for single-photon sources, which uses phase-space interference for extreme bunching events as a quantifier. In contrast to autocorrelation functions, this operational approach takes into account not only residual multi-photon components but also vacuum admixture and the dispersion of the individual photon statistics. We apply this approach to high-purity single photons generated from an optical parametric oscillator and show that they can lead to a Fock-state capability of at least 14. Our work demonstrates a novel collective benchmark for single-photon sources and their use in subsequent stringent applications.

Multi-photon interference leads to a non-trivial redistribution of photons between optical modes. To achieve such interferences, all photons have to be indistinguishable. Several methods have been recently developed to investigate this indistinguishability using different benchmarks, e.g., fidelity [27] or specific photon correlation measures [28][29][30][31][32]. However, the joint impact of photon statistics from many imperfect single-photon states, i.e., exhibiting unwanted vacuum and residual multi-photon components, on multi-photon interference has remained elusive. The joint statistical influence of these parameters cannot be described by evaluating properties of single- ‡ Present address: Friedrich-Alexander University Erlangen-Nürnberg (FAU), Department of Physics, 91058 Erlangen, Germany. §  photon states that are averaged over many experimental runs. Hence, we need criteria, experimental data and subsequent analysis to determine whether independently generated single photons can, in principle, produce the targeted multi-photon interference effects.
Multi-photon interference effects come in a variety of flavors. An extreme event corresponds to the bunching of n single photons into the Fock state |n [33][34][35]. Such bunching can appear in a linear-optical network with inputs fed by indistinguishable single photons, as shown in Fig. 1. The elementary example is the appearance of Fock state |2 based on the HOM effect, as demonstrated in experiments with optical photons but also with microwave photons [16,36], phonons in trapped ions [37] or surface plasmons [38,39]. This extreme bunching event, i.e., the result of a clear operational procedure, enables to introduce a strong benchmark for single-photon states that evaluates their ability to undergo multi-photon interference [4]. This Fock-state bunching capability relies on negativities of the resulting Wigner function, which provide a very sensitive signature of the non-classicality of the generated higher Fock states [3,42].
In contrast to the well-known second-order autocorrelation function at zero time delay g (2) (0), which measures the suppression of the multi-photon contribution and affects the interference visibility [43], the capability is also strongly dependent on the vacuum admixture. Another crucial difference is that it collectively tests multiple photon statistics and determines the joint statistical impact of small discrepancies between them. This provides more stringent and accurate evaluation than other available characteristics.
The previous theoretical study based on Monte-Carlo simulations has only predicted that the bunching of single photons is affected by vacuum and multi-photon contri-FIG. 1: Fock-state bunching capability of non-ideal single-photon states. A single-photon source provides photons with different vacuum admixture and residual multi-photon components, as depicted by the photon-number distributions (left). These states are used as inputs of a balanced linear optics networkÛ . In an extreme case, all photons can bunch into just one output mode whereas all other modes are in the vacuum state. This stage is done computationally and provides the expected photonnumber distribution Pn for the output mode (right). The negativities of the associated Wigner function are used to determine the Fock-state capability. In contrast to other measures, this collective bechmark depends not only on the vacuum admixture and multiple-photon statistics of the imperfect input photons but also on the small discrepancies between them.
butions [4]. However, these contributions and their dispersion in non-ideal photon statistics of many independent copies are too complex to be described, specifically when the number of photons increases. Experimental data are necessary to confirm this prediction. Here, we employ the bunching capability to collectively benchmark experimental single-photon states using heralded single photons generated by parametric down-conversion from an optical parametric oscillator (OPO). By tuning the photon source properties, we address the scaling of the capability with the statistics of non-ideal single photons. We hereby provide a crucial insight into the combined effects of non-ideal photon statistics of independently generated single photons. We demonstrate that experimentally generated single photons can bunch into the Fock state |14 with high fidelity and suppressed higher Fock states contributions. We show that the Fock state capability non-linearly decreases with photon loss, providing a more stringent characterization than g (2) (0), which is independent of photon loss, and also than negative Wigner function that decreases only linearly. Our results indicate that despite the negative impact of multi-photon contributions typically reported using g (2) (0), they prevent the bunching of single-photon states into a respective Fock state less severely than optical loss.

QUANTIFIER PRINCIPLE
We first describe the quantifier principle. To collectively test the ability of the generated single photons to undergo multi-photon interference, we computationally determine the Wigner function of the higher Fock state, which can, in principle, appear from multiple copies of the single-photon state, as depicted in Fig. 1. The area in phase space, where the Wigner function of the ideal Fock state |n is negative, is composed of n/2 or (n−1)/2 concentric annuli if n is an even number or an odd number, respectively. By definition, a single-photon state has the capability of the Fock state |n if the Wigner function of the state, which can be generated from n independent copies of the single-photon state, has the same number of negative annuli as the ideal Fock state |n [4]. The negative annuli in the Wigner function witness the nonclassical nature of the multi-photon interference in phase space. The Fock-state capability, which is determined computationally, collectively tests the copies of a singlephoton state, even though any multi-copy procedure is not implemented in the laboratory.
In theory, copies of the ideal single-photon state |1 have the capability of an arbitrary Fock state |n . For states generated by single-photon sources, the negative annuli in the Wigner function are sensitive to the presence of vacuum and multi-photon contributions. Also, the exact distribution of residual multi-photon statistics in many non-ideal single-photon states is not known. As a consequence, the joint effect of small discrepancies between individual single-photon copies on multi-photon interference has to be investigated by applying the quantifier on photon statistics measured in an experiment. In this way, we can determine whether the single-photon sources have a sufficient quality for applications in quantum technology that require multi-photon interference. I: Photon-number statistics of heralded single photons. Each set is obtained by successive measurements under the same conditions (in particular pump power). The table displays the single-photon component P1, the multi-photon probability P2+, the second-order correlation function g (2) (0), and the negativity at the origin of the Wigner function. To study this benchmark, we used heralded singlephoton states generated using a two-mode squeezer, i.e., a type-II phase-matched optical parametric oscillator operated well below threshold (see Appendix). The signal and idler photons at 1064 nm are separated on a polarizing beam-splitter and the idler photon is detected via a high-efficiency superconducting nanowire single-photon detector. This detection event heralds the generation of a single photon in the signal mode. The generated state is emitted into a well-defined spatio-temporal mode [44], with a bandwidth of about 65 MHz. The state is measured via high-efficiency homodyne detection, with a visibility of the interference with the local oscillator above 99%, and reconstructed via maximum-likelihood algorithms [2]. The experimental setup has been described elsewhere [1,46]. Importantly, the OPO used in this work exhibits a close-to-unity escape efficiency, i.e., the transmission of the output coupler is much larger than the intracavity losses [48]. As a result, a large heralding efficiency can be obtained, i.e., a very low admixture of vacuum. A single-photon component up to 91% is achieved. Also, by changing the pump power the multi-photon component can be increased at will. These features enable us to explore different combinations of state imperfections. Seven sets of data were recorded, each of them being obtained by a repetitive measurement of the single-photon states generated under the same conditions. Parameters of the sets are given in Table I. They include the singlephoton component P 1 and the probability P 2+ of finding two or more photons. These measured quantities give also access to the conditional second-order autocorrelation function at zero-time delay g (2) (0) [1].

EXPERIMENTAL FOCK-STATE CAPABILITY
To test a particular data set for the Fock-state capability n, the data are randomly partitioned in n subsets from which n photon-number statistics are obtained and used as the quantifier inputs. The output-state Wigner function of the computational quantifier is averaged over 30 such random choices. From the averaged output-state Wigner function, it is determined whether the data set has the Fock-state capability n (see Appendix). The capability for all data sets is depicted in Fig. 2 as a function of P 1 and P 2+ . The quantifier is presently computationally limited by the Fock-state capability 14 (see Appendix), which is already a very large number in this operational context. All data sets for which this capability 14 is obtained may also have the capability of a higher Fock state. In the following, we describe the different measured points and typical trends.
First, single-photon states with a low purity due to a vacuum component close to 50% (brown bars in Fig. 2, sets 1 and 2 in Table I) have only the trivial capability of the Fock state |1 , despite their very low g (2) (0). This shows that the broadly used autocorrelation function does not fully characterize the ability to bunch into higher Fock states exhibiting non-classical signatures. In particular this example demonstrates that the capability is more sensitive to vacuum mixture, as a state obtained from two copies of these single photons would have a positive Wigner function. Due to their trivial capability, such states are not a useful resource for the preparation  Table I. These parameters are averaged over photon-number statistics from a given data set obtained by successive measurements under the same experimental conditions. Colors denote the Fock-state capability. The gray-shaded area excludes the unphysical probabilities P1 + P2+ > 1. The standard deviation of the probabilities are given by the thicknesses of the color bars. of large Fock states that could be used e.g. for quantum metrology [18][19][20] or error correction [24][25][26]. The necessary condition for a non-trivial capability n > 1 is to reach a single-photon component P 1 > 2/3 [4]. Above this threshold, the capability moderately grows with P 1 . As can be seen in Fig. 2, the state corresponding to the green bar (set 3 in Table I) has a multi-photon component P 2+ = 0.02 and the capability of the Fock state |3 . The state associated to the red bar (set 4) has the capability of the Fock state |4 despite having a similar single-photon component as the previous state but a larger, still low, probability P 2+ = 0.05. For a given P 1 , an increase in P 2+ may thereby lead to a larger capability. Actually, this increase in P 2+ comes in that case with a decrease in the vacuum component, indicating that the bunching is less affected by multi-photon contributions than vacuum admixture. We have shown in additional simulations that at fixed vacuum the capability decreases with the multi-photon component.
Finally, for P 1 > 0.8, the capability is expected to rapidly increase and to diverge at P (∞) 1 = 0.885, where an arbitrary capability can be reached [4]. The experimental results agree well with this prediction and highlight the nonlinearity of the quantifier. The verification of this trend is an important benchmark for the development of single-photon sources. The data sets indicated with blue bars have at least the capability 14. For the set 7, note that its g (2) (0) = 0.05 does not significantly differ from that of the states with the trivial Fock-state capability. The capability 14 is also achieved for lower single-photon fidelities P 1 and higher multi-photon contributions P 2+ , even for a state with four times larger g (2) (0) = 0.2. However, these states might have a lower capability than the set 7 due to the saturation to 14 for reason of computational power. Figure 3 presents the output of the computational quantifier with fourteen input states randomly chosen from the data set 7, i.e., the set with the highest heralding efficiency and lowest multi-photon component. Figure 3a first provides the cut through the Wigner function. The output Wigner function is fitted by the one of a lossy Fock state |14 , with a fitted attenuation parameter η = 0.9205±0.0005. The fit shows that the oscillations of the output Wigner function in phase space coincide with the ones of the attenuated Fock state |14 . The photonnumber statistics of the output state and attenuated Fock state are compared in Fig. 3b. The good cut-off of the multi-photon contributions with more than fourteen photons in the statistics of the output state is another feature that further demonstrates the high quality of the initial single-photon states. Such result was made possible only by considering single-photon states with limited multiphoton contributions and very low vacuum admixture, as provided by the OPO-based source used in this work.

DISCUSSION: EFFECT OF LOSS AND TRUNCATION
We now come to an additional characterization of the quantifier, i.e., its evolution with optical losses. This quantifier depth, in analogy to non-classicality depth [49], is tested by considering attenuation for two states randomly chosen from different data sets. Figure 4 shows the Fock-state capability as a function of the attenuation parameter η, for the state with P 1 = 0.91 and P 2 = 0.02 (blue in Fig. 2) and the state with P 1 = 0.74 and P 2 = 0.05 (red in Fig. 2). Both states exhibit a similar g (2) (0) parameter (which is preserved with attenuation), but different initial capabilities 14 and 4, respectively. As it can be seen, the capability depends nonlinearly on the attenuation η. This is in contrast to the negativity of the single-photon Wigner function which decreases linearly with the attenuation. As a result, the capability allows more sensitive benchmarking of singlephoton states than the negativity of the Wigner function.
The results in Fig. 4 are also superimposed with two plots that give the evolution of the capability with optical losses for states whose photon-number statistics are truncated, i.e., neglecting the multi-photon contribution. The discrepancy in the Fock-state capability between the experimental states and the truncated ones demonstrates that the multi-photon contributions play a significant role in such bunching experiments. The truncation of multi-photon contributions can be a limiting approximation when multi-photon interference is involved.

CONCLUSION
In conclusion, with the advance of quantum technologies, novel procedures and applications put challenging demands on resources and required benchmarking [50]. In this broad context of utmost importance, we have employed the Fock-state bunching capability to collectively benchmark experimental single-photon states for the first time. We have investigated the behavior of this test with photon statistics and loss. This quantifier, which is highly non-linear, has a clear operational meaning in terms of photon merging and moreover takes into account the unavoidable dispersion of individual copies of singlephoton states.
Thanks to high-purity states based on a state-ofthe-art OPO, this work has experimentally verified the numerically-predicted threshold, P 1 > 0.885, to observe a large Fock-state capability. Capability of at least 14 has been demonstrated thanks to the very low two-photon component and the large heralding efficiency. Importantly, we have shown that the capability is more sensitive to optical losses than the single-photon negativity of the Wigner function and fidelity. Based on our numerical data, we also deduced that a moderate increase in the ratio of the multi-photon contributions to the vacuum does not decrease the capability. This shows that despite the negative impact of multi-photon contributions, they prevent the bunching of single-photon states into a single Fock state less severely than optical losses.
In the present implementation, we have estimated photon-number distributions from homodyne detection. Multiplexed single-photon detectors [51,52] or photon-number resolving superconducting detectors [53][54][55] should enable a direct measurement of the Fock-state capability. Also, this benchmark does not depend on the nature of the source and can thereby be used to characterize microwave photons in superconducting circuits [15], plasmons at metal-dielectric surfaces [38,39], phonons in trapped-ion [56] or optomechanics experiments [57], and collective excitations in atomic ensembles [58][59][60]. Finally, the multi-photon interference quantifier can be modified to investigate the capability of other resource states, e.g. squeezed states or Schrödinger cat states [13,61], to produce different target states such as NOON states [18][19][20] or superpositions of squeezed states (GKP states) [62], opening a new avenue for testing the potential of light emitters for advanced quantum state engineering.  The computational demand is substantially reduced if one chooses a single photon-number distribution from the experimental data set and use it as an identical input, which is fed into all channels of the linear optics network. In this way, we calculate the output Wigner function for several random choices of the photon-number distribution. Then the Wigner function is averaged over these random choices. Hence the differences between the individual copies of single-photon states are not taken into account. Using this simplified method, we determine the Fock-state capabilities of the experimental data sets, which agree with the capabilities depicted in Fig. 2 obtained by the full, unsimplified multi-photon interference quantifier. However, note that the full quantifier should always be used to confirm the results of the simplified quantifier, which is not able to correctly estimate the propagation of the input state's discrepancies through the quantifier. In order to estimate the capability of a very high Fock state, the quantifier can be even further simplified by neglecting the discrepancies between photon-number distributions in the experimental data set. Working only with the average photon-number distribution, we estimate that the experimental data set 7 with P 1 = 0.91 and P 2 = 0.02 has the capability of at least the Fock state |50 . This agrees with the theoretical prediction [4] that P 1 > P (∞) 1 = 0.885 is sufficient to reach the capability of an arbitrary Fock state, if multi-photon contributions and discrepancies between photon-number distributions are neglected.