Generalized Multiphoton Quantum Interference

Nonclassical interference of photons lies at the heart of optical quantum information processing. Here, we exploit tunable distinguishability to reveal the full spectrum of multiphoton nonclassical interference. We investigate this in theory and experiment by controlling the delay times of three photons injected into an integrated interferometric network. We derive the entire coincidence landscape and identify transition matrix immanants as ideally suited functions to describe the generalized case of input photons with arbitrary distinguishability. We introduce a compact description by utilizing a natural basis that decouples the input state from the interferometric network, thereby providing a useful tool for even larger photon numbers.

Popular Summary

The nonclassical interference of particles is a genuine quantum phenomenon. This effect is both of fundamental scientific interest and essential for optical quantum technologies such as quantum computing, quantum communication, and quantum metrology. Surprisingly, nonclassical interference is not only tied to the coherence of the particles but also to a second property—the particles’ symmetry under permutation. A first experiment was realized nearly 30 years ago using two single photons propagating through a balanced beam splitter. However, increasing the number of photons only slightly and simultaneously allowing for arbitrary interferometric networks makes the problem significantly more complex. Here, we present a novel description of complex multiphoton interference covering the full range of possible distinguishability of photons. We consider all cases ranging from maximal interference where photons are rendered totally indistinguishable, intermediate cases where photons are partially indistinguishable, and minimal interference where photons are completely distinguishable (i.e., the classical case).

Using integrated photonic quantum technology and a linear optical quantum network with three photons spread over five interferometric modes, we demonstrate theoretically and experimentally that control over deviations from perfect distinguishability is not only of scientific interest but also of immediate importance for applications. By tuning the temporal delay of the photons, we find that the degree of interference is modulated by the properties of the photons themselves. Our method works for arbitrary particle distinguishability and any interferometric architecture, and it therefore can be applied to a large class of quantum optical scenarios. As an example, we show how different instances of a recent model of quantum computing harnessing multiphoton interference are affected.

We expect that our results will motivate future studies with a larger numbers of photons.

Figure 1

The coincidence landscape and its substructure. (a) Coincidence landscape which is the visual representation of the nonclassical interference of three photons of tunable distinguishability. (b) Cut along the diagonal and antidiagonal lines of pairwise delays $\mathrm{\Delta }{\tau }_1=\mathrm{\Delta }{\tau }_2$ and $\mathrm{\Delta }{\tau }_1=-\mathrm{\Delta }{\tau }_2$, also indicated as a purple trajectory in (a). The height of the landscape corresponds to the coincidence output probability of a three-photon scattering event. The substructure, expressed in terms of contributions from the permanent “per”, the determinant “det”, and immanants “imm”, reveals the nature of the scattering event. Beside the plateaux (color coded in red), which can be explained classically, the whole landscape is governed by a genuine quantum interference of the photons.

Figure 2

Two-photon nonclassical interference. Two photons of temporal coherence ${\tau }_c$ enter a beam splitter through different input ports. (a) The coincidence output probability $P_c$ that they leave in two different output ports is plotted with respect to a relative temporal delay $\mathrm{\Delta }\tau$. This delay is used to tune the distinguishability of the otherwise identical photons. The blue curve shows $P_c$ for a $50/50$ beam splitter, and the green one for a $67/33$ beam splitter. (b) Contribution of the permanent (per) and determinant (det) to the output probability for a coincident detection event $P_c$. It is the same for both beam splitters because this description is independent of the interferometer. In the case for zero delay ($\mathrm{\Delta }\tau =0$), only the permanent contributes. By explicitly calculating the permanent, which is zero for a $50/50$ beam splitter, the vanishing coincidence probability $P_c$ (a) for zero delay is obtained.

Figure 3

Three-photon coincidence landscapes. Three photons enter the interferometric network, one each in modes 1, 2, and 4 (highlighted in yellow), and exit the network in modes (a) 3, 4, and 5 and (d) 1, 3, and 4, respectively (highlighted in blue). The intersections of the inputs’ columns and the outputs’ rows uniquely select matrix entries that constitute $3\times 3$ submatrices. Tuning the temporal delay of the three photons with respect to each other ($\mathrm{\Delta }{\tau }_1$ and $\mathrm{\Delta }{\tau }_2$) gives rise to coincidence landscapes [(b) and (e)]. Their temporal distinguishability determines the degree of nonclassical interference and therefore the probability to detect such an event. Six characteristic points ($P1–P6$) of each landscape are experimentally sampled. Theoretical prediction (left bars, shaded) and experimentally obtained output probabilities (right bars) for the six points and both output combinations are shown in (c) and (f). The reduced ${\chi }^2$ is 1.38 and 1.10, respectively, and the experimental errors are calculated as standard deviations.

Figure 4

Experimental output distribution for indistinguishable, semidistinguishable, and distinguishable photons. Various temporal delays for three photons lead to different contributions of permanents, immanants, and determinants. The normalized output probability distributions for the three photons is measured as coincidences from different spatial modes, resulting in ten elements. For all temporal delays the three photons exhibit slight spectral mismatch. Panel (a) depicts the case for a small temporal offset $\mathrm{\Delta }{\tau }_1\ll {\tau }_c\gg \mathrm{\Delta }{\tau }_2$ ($P1$ of Fig. 3), whereas for (b) and (c) this delay is increased along a diagonal axis $\mathrm{\Delta }{\tau }_1\approx \mathrm{\Delta }{\tau }_2$ ($P3$ and $P4$ of Fig. 3). The extreme case of complete distinguishability and therefore classical behavior is shown in (d) ($P6$ of Fig. 3). As a reference, the gray bars illustrate the case for perfect indistinguishability and therefore only contribution from the permanent of the scattering submatrix. The interferometer independent contribution $F_{\text{per}}$, $F_{\det }$, and $F_{\text{imm}}$ is shown in the figure legend. The error bars of the experimental data are standard deviations over 19 independent runs.

Figure 5

Five-photon BosonSampling including distinguishability. Simulation of a BosonSampling instance of five photons propagating through an interferometric network of nine modes. The output probability distribution of five photons exiting the interferometer in five different modes is normalized and contains 126 elements. Owing to the large size of this probability distribution, we choose not to list them individually, and we label the horizontal axes by “elements.” Panel (a) depicts the close-to-the-ideal case where realistic errors such as slight spectral mismatch and temporal delay ($\mathrm{\Delta }{\tau }_i\le \frac{1}{20}{\tau }_c$) of the photons lead to a small degree of partial distinguishability. Panel (b) shows a case where the interfering photons exhibit increased partial distinguishability ($\mathrm{\Delta }{\tau }_i\le \frac{1}{5}{\tau }_c$). In these exemplary output probability distributions [(a) and (b)], contributions from permanents (per) and immanants (, , ) arise. The immanant contributions cover physical scenarios with different symmetries under exchange of five photons. The labels of the immanants describe the number of photons that are distinguishable. , for instance, is the contribution from the case when four photons are indistinguishable from one another but distinguishable from the fifth photon.

Figure 6

Experimental setup. Four photons are generated via spontaneous parametric down-conversion and distributed to four spatial modes with two PBSs. A fourfold coincidence event consisting of three photons exiting the network and a trigger event postselects the desired input state. The delay lines allow us to tune the distinguishability and therefore the quantum interference of the three photons propagating through the waveguide. The integrated circuit is shown in a Mach-Zehnder decomposition and consists of eight beam splitters and 11 phase shifters.

Figure 7

Integrated photonic network. Schematic drawing of the optical network. The circuit consists of eight directional couplers (${\eta }_1,\dots ,{\eta }_8$), 11 phase shifters (${\varphi }_1,\dots ,{\varphi }_{11}$), five input modes $(1,\dots ,5)$, as well as five output modes ($1^{\prime },\dots ,5^{\prime }$). To allow coupling to the waveguide with standard fiber arrays, the input and output modes are separated $127\mu \mathrm{m}$ and the total length of the chip is 10 cm.

Figure 8

Example for one data set used for the reconstruction of $U_5$. The best fit of Eq. (a27) to the data set is shown in blue. Here, the visibility is calculated from the best fit parameters $Y_0$ and $A$. The reduced ${\chi }^2$ resulting from the fit shown in blue is ${\chi }_{\text{blue}}^2=2.02$. Fluctuations in the count rate for values of $|\mathrm{\Delta }\tau |>500\mathrm{fs}$ drive the reduced ${\chi }^2$ away from 1. These fluctuations can be interpreted as the random noise background in the lab and the increased reduced ${\chi }^2$ reflects that. However, the precision of the fitted parameters $Y_0$ and $A$ and ultimately the extracted visibility $V$ is only marginally affected by these fluctuations. The $5\times 5$ unitary description of the interferometric network is reconstructed from 84 of these visibilities. The curve in green, $N_c^{(\mathrm{th})}(t)$ [see Eq. (a33)] is calculated from four matrix entries of the reconstructed unitary and results in an overlap with the data of ${\chi }_{\text{green}}^2=2.70$. The agreement of these two curves and corresponding reduced ${\chi }^2$’s is a qualitative measure for the precision of the reconstruction.

Figure 9

State generation. A pump beam is focused into a 2-mm $\beta \text{−}{\text{BaB}}_2{\mathrm{O}}_2$ crystal cut for noncollinear, degenerate, type-II down-conversion. The generated state is emitted into the spatial modes $a$ and $b$. A compensation scheme consisting of half-wave plates and 1-mm-thick BBO crystals is applied for countering temporal and spatial walk-off. Narrow band interference filters (${\lambda }_{\mathrm{FWHM}}=3\mathrm{nm}$) are applied to increase the temporal coherence of the photons and render them close to spectral indistinguishability. The modes $a$ and $b$ are subsequently split by polarizing beam splitter cubes and two half-wave plates in their reflected ports are set to 45° to ensure the same polarization in all four output modes ($a^{\prime \prime }$, $a^{\prime }$, $b^{\prime }$, and $b^{\prime \prime }$). With this scheme, three indistinguishable photons in modes $a^{\prime }$, $b^{\prime }$, and $b^{\prime \prime }$ each can be heralded from a fourfold emission by a successful trigger event in mode $a^{\prime \prime }$.