Abstract
Bohr’s complementarity is one central tenet of quantum physics. The paradoxical waveparticle duality of quantum matters and photons has been tested in Young’s doubleslit (doublepath) interferometers. The object exclusively exhibits wave and particle nature, depending measurement apparatus that can be delayed chosen to rule out toonaive interpretations of quantum complementarity. All experiments to date have been implemented in the doublepath framework, while it is of fundamental interest to study complementarity in multipath interferometric systems. Here, we demonstrate generalized multipath waveparticle duality in a quantum delayedchoice experiment, implemented by largescale siliconintegrated multipath interferometers. Singlephoton displays sophisticated transitions between wave and particle characters, determined by the choice of quantumcontrolled generalized Hadamard operations. We characterise particlenature by multimode whichpath information and wavenature by multipath coherence of interference, and demonstrate the generalisation of Bohr’s multipath duality relation. Our work provides deep insights into multidimensional quantum physics and benchmarks controllability of integrated photonic quantum technology.
Introduction
Famous doubleslit or doublepath experiments, implemented in a Young’s or Mach–Zehnder interferometer, have confirmed the dual nature of quantum matters^{1}. When a stream of photons^{2}, neutrons^{3}, atoms^{4}, or molecules^{5}, passes through two narrow slits, either wavelike interference fringes build up on a screen, or particlelike whichpath distribution can be ascertained. These quantum objects exhibit both wave and particle properties but exclusively, depending on the way they are measured^{1}. In the equivalent Mach–Zehnder configuration, quantum objects display either wave or particle nature in the presence or absence of a beamsplitter, respectively, where the beamsplitter represents the choice of measurement apparatus^{6}. Wheeler further proposed a Gedanken experiment^{7}, in which the choice of particle or interference measurement is made after the object has already entered the interferometer, so as to exclude the possibility of predicting with which measurement it will be confronted. Delayedchoice experiments have enabled significant demonstrations of the genuine twopath duality of different quantum objects^{8,9,10,11,12,13,14,15}. Moreover, a quantitative description of twoslit duality relation was initialized in Wootters and Zurek’s seminal work^{6} and then formalized by Greenberger, Yasin, Jaeger, and Englert^{16,17,18} as \({{\mathcal{D}}}^{2}+{{\mathcal{V}}}^{2}\le 1\), where \({\mathcal{D}}\) is the distinguishability of whichpath information (a measure of particleproperty), and \({\mathcal{V}}\) is the contrast visibility of interference (a measure of waveproperty). This doublepath duality relation has been tested in pioneer experiments^{19} and in delayedchoice measurements^{11,13}.
Since the birth of quantum mechanics, it has long been of fundamental interests to understand multipath interference of quantum mechanical wavefunction in complex quantum systems^{20,21,22,23,24,25}. Quantum nature represented as the principles of complementarity and superposition however remains ambiguous in multipath interferometric quantum systems^{17,26,27,28,29,30}. Figure 1a sketches a general multipath Mach–Zehnder interferometric delayedchoice experiment with a single photon. Whether the photons take one or multiple paths to a given detector depends on the absence or presence of the second dmode beamsplitter. In a delayedchoice experiment, the photons can either take all d paths simultaneously (multipath wave character), or one of the d paths (multimode particle character), or everything in between, determined in a delayed manner by the choice of the dmode particle measurement or interference measurement. In contrast to the doubleslit implementation, the duality relation well describes the waveparticle complementarity, however, it cannot be simply generalized in the multipath experiment^{26,27,28,29,30}. There are several major open questions remaining: Can Bohr’s duality relation still hold in the multipath interferometric experiment? Are there any good measures of multipath wave and multimode particle properties that are accessible in experiment? Does single photon preserve the inherent dual nature in the multipath delayedchoice scenario? Revealing these unknowns are essential to understand multimode quantum superposition and quantization in complex quantum systems. Apart from fundamental interests, the characterization of multimode quantum properties in controllable systems may provide the ground of developing multidimensional quantum technology^{31}. For example, quantifying multimode coherence from sophisticated multipath interference patterns is of practical significance^{32}, in the light of recent reassessment of coherence as a key resource in quantum information^{33,34}, while it has always been a core concept underlying the theory of quantum mechanics. In general, when single photons pass through multiple paths, a superposition of multiple modes naturally forms a qudit state. Promising prospects of quditbased quantum applications have been well acknowledged, such as noiserobust multidimensional entanglement^{35,36}, resourceefficient quantum computations, and simulations^{37,38}, and highcapacity quantum communications^{39,40}; however, the deep understanding of the most elementary physics of multidimensional quantum systems is highly demanded. Any explorations of multidimensional quantum science and technology strongly rely on the quantum platform that can be operated with highlevel controllability, efficiency, and versatility^{31}. It is here the integratedoptics implementation provides one of the most competitive multidimensional quantum platforms^{41}.
Here, we report a quantum delayedchoice multipath experiment and demonstrate a generalization of waveparticle duality relation. The waveparticle nature of single photons propagating in a dpath (d up to 8) interferometer is observed, determined in a delayed manner by the state of a dmode quantumcontrolled beamsplitter. Qualitative waveparticle transitions having a 0.99 fidelity of theoretical and experimental results, and quantitative multipath duality relation are both confirmed in the context of delayedchoice. We show that quantum coherence is a good measure of the waveproperty in dpath interference, and the amount of coherence can be directly probed from interference patterns, without accessing the density matrix. The dmode whichpath information is identified, and quantum randomness is efficiently generated. All demonstrations are enabled by realizing a multipath delayedchoice interferometric system on a largescale silicon nanophotonic quantum chip that monolithically integrates 355 optical components and 95 phaseshifters.
Result
Scheme of the generalized multipath delayedchoice experiment
Figure 1a shows a diagram of general dpath Mach–Zehnder interferometer (dMZI) consisting of d arms (paths) and two dmode beamsplitters (dBSs). An array of individually reconfigurable phases of \({\{{\theta }_{k}\}}_{k = 0}^{d1}\) are applied on the dpath between the two beamsplitters of dBS1 and dBS2. To implement the dBSs, a general scheme is to nest (d^{2} − d)/2 standard 2BSs (see a bulkoptic example in Fig. 1b). For simplicity, we consider the case that dBSs are well balanced, thus the state emerging from the dBS1 is a maximally coherent state as \({\rho }_{0}={\left{{\Psi }}\right\rangle }_{00}\left\langle {{\Psi }}\right\) and \({\left{{\Psi }}\right\rangle }_{0}=\frac{1}{\sqrt{d}}\mathop{\sum }\nolimits_{k = 0}^{d1}\leftk\right\rangle\), where \({\{\leftk\right\rangle \}}_{k = 0}^{d1}\) is the logical basis that defines the reference frame. The state of dBS2 represents the measurement apparatus photons will confront. When the dBS2 is inserted (removed), the dMZI is closed (open), the detected probabilities at {D_{0}, … D_{d−1}} are dependent (independent) on the \({\{{\theta }_{k}\}}_{k = 0}^{d1}\) configuration, thus revealing the dpath wavelike interference (dmode particlelike quantization).
In order to probe the genuine waveparticle duality, the state of dBS2 has to be determined after the photon has entered the dMZI. In Fig. 1c, we adopt a modified version of the quantumcontrolled delayedchoice experiment, recently proposed by Ionicioiu and Terno^{42} and implemented in several doublepath experiments^{12,13,14,15}. By introducing a quantumcontrolled BS that is in a coherent superposition of presence and absence, it represents a controllable experiment platform that can reveal wave or particle character, and their intermediate character^{42}. In our multipath quantum delayedchoice scheme, the state of a general ddimensional Hadamard operator \({\hat{H}}_{d}\) is coherently entangled with the control qubit of \({\left\psi \right\rangle }_{C}\). The Hadamard \({\hat{H}}_{d}\) is implemented by a balanced dBS, with elements \({h}_{i,j}^{(d)}=\frac{1}{\sqrt{d}}{(1)}^{i\odot j}\), where i ⊙ j denotes the bitwise dot product of the binary representations of i and j. Note that in general the implementation of ddimensional controllable Hadamard operation however is highly challenging in the delayedchoice experiment^{10,11}. This is because of the difficulty of actively and rapidly operating the dBS2—simultaneously operating the entire (d^{2} − d)/2 array of 2BSs, and thus quickly reconfiguring the whole dMZI to an either open or closed state. We adopted a similar scheme as the doublepath quantum delayedchoice experiments^{12,13}. The two multimode complementary measurements performed on a target photon are determined by the state of another control photon, in which all operations only require passive optical components without any active operation of the whole dBS2 and dMZI. The key idea is to create a coherent entanglement between a control qubit and the state of dBS2.
We devise a largescale siliconintegrated quantum nanophotonic device for the implementation of the delayedchoice dpath interferometric experiment, that features high phase stability and scalability^{41}. Figure 1e illustrates a simplified diagram of the device to implement the circuit in Fig. 1c. Our task is to test the waveparticle dual nature of the target photon ρ_{0} by choosing measurement apparatus \({\hat{\text{M}}}_{m}\) (Fig. 1d). The device includes four parts: an entangled photonpair source, a dpath MZI, a quantumcontrolled dBS, and a dmode eraser. As shown by Fig. 1e, all these parts are monolithically integrated on a single silicon chip. We prepare a maximally entangled Bell state \(({\left0\right\rangle }_{C}{\left0\right\rangle }_{T}+{\left1\right\rangle }_{C}{\left1\right\rangle }_{T})/\sqrt{2}\) in two integrated spontaneous fourwave mixing (SFWM) sources^{35}, where \({\left0\right\rangle }_{C,T}\) and \({\left1\right\rangle }_{C,T}\) are pathencoded logical states of the control and target photons. The target photon undergoes a \({\hat{H}}_{d}\) transformation by the dBS1, and a phase operator \({\hat{\theta }}_{d}\) by the phase array. Depending on the control state of \({\left0\right\rangle }_{C}\) or \({\left1\right\rangle }_{C}\), the target photon coherently evolves either as a wave or particle, resulting in a stateprocess entanglement:
where \(\left\,\text{P}\,\right\rangle\), \(\left\,\text{W}\,\right\rangle\) represent the states taking the particle and wave processes, respectively. The two processes are implemented by two distinct physical waveguide circuits, with dBS2 (red circuits) and without dBS2 (green circuits) (Fig. 1e). Which process the target photon experiences is coherently entangled with the state of control photon. If the control takes the state \(\left0\right\rangle\), the target undergoes the dmode particleprocess, revealing particle nature; if the control takes state \(\left1\right\rangle\), the target undergoes the dmode wave process, exhibiting wave character; if the control photon is in a superposition state, it can reveal intermediate particlewave characters. Importantly, the whichprocess information is erased at a dmode quantum eraser (Fig. 1e), ensuring quantum mechanical indistinguishability between the wave and particle processes. These result in the realization of generalized quantumcontrolled \({\hat{H}}_{d}\) operation or dBS2. This stateprocess entanglement approach has been adopted for implementing doublepath delayedchoice experiments^{13} and for quantum simulations^{43}. See Supplementary Note 1 and 3 for details.
We operate the control photon state as \(\sin \frac{\alpha }{2}{\left0\right\rangle }_{C}+{e}^{i\delta }\cos \frac{\alpha }{2}{\left1\right\rangle }_{C}\), where {α, δ} represents the angles of {σ_{y}, σ_{z}} Pauli rotations, and selected the events when the detector \({\,\text{D}\,}_{8}^{\prime}\) is clicked (Fig. 1e). Owing to the presence of entanglement between the control photon and the quantum state of dBS2, the dBS2 is thus in a superposition of presence and absence as (\(\cos \frac{\alpha }{2}\hat{I}i{e}^{i\delta }\sin \frac{\alpha }{2}{\hat{H}}_{d}\)). Note the first term represents the measurement of particle character, while the second term represents the measurement of wave character. Figure 1d represents the framework of delayed choice of measurement \({\hat{\text{M}}}_{m}=\leftm\right\rangle \left\langle m\right\) performing on the target photon ρ_{0}, where \(\leftm\right\rangle\) forms the waveparticle measurement basis:
where m = 0 …, d − 1 is measurement settings; Δ_{x} is the Kronecker delta function; N_{d} is a normalization coefficient. See details in Supplementary Note 4. The probability of obtaining the mth measurement outcome is quantified by \(\,{\text{Tr}}\,[{\hat{\text{M}}}_{m}\rho ^{\prime} ]\), where \(\rho ^{\prime}\) is the state after the \({\hat{\theta }}_{d}\). In the experiment, we measured twophoton coincidences between a control port and any one of the d target ports. The probabilities were then calculated by normalizing over the d target ports.
The choice of measurement apparatus \({\hat{\text{M}}}_{m}\)—determined by the {α, δ} state of the control photon in a delayed manner, allows us to observe the dpath waveparticle transition and to test the dpath duality relation of the target photon. In Eq. (3), α refers to the amplitude of wave and particle properties, and the inherent δ phase identifies the genuine quantum particlewave superposition. If α = 0, dBS2 is in the offstate and the \({\hat{\text{M}}}_{m}\) discloses the particlenature. Hence, the photon registers each of the detectors (Fig. 1e) with a probability of 1/d, leading to the observation of dmode quantized distributions. If α = π, dBS2 is in the onstate and \({\hat{\text{M}}}_{m}\) reveals the wave nature. In this regard, the probability of detecting the photon relies on {θ_{d}}, building up dpath wave interference patterns. If 0 < α < π, dBS2 is in a superposition of the on and offstate and it thus allows the observation of waveparticle nature simultaneously. We remark that the choice of \({\hat{\text{M}}}_{m}\) and the dual property of the target photon remain undetermined, until the control photon has been detected. This is because of the essence of entanglement that information is nonlocally shared between the two photons.
Our quantum chip is designed for dpath (d ≤ 8) experiments, in which the number of paths and mode number of dBSs can be reconfigured. The dBSs are formed by a squared mesh of 2BSs (each is a 2path MZI for full reconfigurability). The chip integrates 95 phaseshifters that are individually addressed and electronically driven. A telecomband laser was used to pump two integrated SFWM sources and generate a pair of entangled photons. The signal photon is regarded as the target photon, while the idler photon is regarded as the control. The two photons were ultimately routed off the chip for detection by superconducting nanowire singlephoton detectors {\({\text{D}}_{i},{\text{D}\,}_{i}^{\prime}\)}, i = 0, ... 8 (Fig. 1e). The fabricated devices and waveguides are shown in Fig. 1f–h. See Methods and Supplementary Note 1 for more details of device fabrication and setup.
Ruling out highorder interference
Born’s rule implies that multipath interference consists of all possible combinations of mutual interference. The possible presence of highorder interference could mask the test of waveparticle duality in the dpath interferometric experiment. Prior to testing the multipath waveparticle duality, we first rule out the presence of highorder interference. As an example, we implemented a fourpath interference experiment. We measured the normalized Sorkin parameter κ, a ratio of highorder interference to secondorder interference^{23,24,25}, and obtained the tight bound of −0.0031 ± 0.0047 for the fourthorder interference (Fig. 2). Our experimental results confirm the absence of highorder interference, within an accuracy of 10^{−3} of the bound, which is comparable to the most precise result obtained so far in ref. ^{44}. Main experimental errors come from the nonperfect opening/closure of paths and thermal crosstalk between paths, as well as background photon noises. See Supplementary Note 2 for more measurement details.
Multipath waveparticle transitions
Figure 3 reports experimental results for dpath waveparticle transitions. Here, probability distributions for the \({\hat{\text{M}}}_{0}\)measurement are plotted, corresponding to the detection of the target photon at the ports of \(\{{\text{D}}_{0},{\text{D}\,}_{0}^{\prime}\}\). Measurement results for other ports \({\{{\text{D}}_{i}\}}_{i = 1}^{d1}\) are provided in Supplementary Fig. 4. In our measurement, the phase θ_{k} was chosen as k(θ − π), where θ ∈ [0, 2π], but an arbitrary θ_{k} phase can be set. We obtained the continuous transition of dpath duality, from the fullparticle nature at α = 0 to the fullwave nature at α = π, in two different scenarios: classical mixture (Fig. 3a–e) and quantum superposition (Fig. 3f–j). Classical waveparticle mixture represents the state \(({\cos }^{2}\frac{\alpha }{2}\left\,{\text{P}}\,\right\rangle \left\langle \,{\text{P}}\,\right+ {\sin }^{2}\frac{\alpha }{2}\left\,{\text{W}}\,\right\rangle \left\langle \,{\text{W}}\,\right)\), while quantum waveparticle superposition represents the state \((\cos \frac{\alpha }{2}\left\,\text{P}\,\right\rangle i{e}^{i\delta }\sin \frac{\alpha }{2}\left\,\text{W}\,\right\rangle )/\sqrt{N}\), where N is a normalization coefficient, denoting the ultimate states after the quantum erasure. Details are given in Supplementary Note 3.
In the case of 2path classical mixture, Fig. 3a shows a sinusoidal interference fringe at α = π representing the fullwave nature, and the detection probability approaches nearly 1/2 at α = 0 representing the fullparticle nature. The observation of 2path waveparticle transition is consistent with the results in refs. ^{13,14,15}. In contrast, in the dpath experiments (Fig. 3b, c), at α = π we observed interference patterns that feature sharper distributions with an increment of d, confirming the dpath wave nature; at α = 0 we observed a 1/4 (1/8) probability for d = 4 (8), confirming the dmode particle nature; for 0 < α < π, intermediate particlewave behaviors were revealed. The results for α = {0, π} are replotted in Fig. 3d, e, which are expected in classical optical multislit interference.
We now report the unique feature of quantum waveparticle superposition in dpath experiments. In Fig. 3f–h, the probability distributions represent asymmetry with respect to θ = π, while the distributions for classical mixture in Fig. 3a–c remain symmetric. The extraordinary asymmetry comes from quantum interference between the wave and particle properties (Supplementary Eq. 14). Only when choosing the fullparticle (α = 0) or fullwave (α = π) point, the distributions for classical (Fig. 3d, e) and quantum cases (Fig. 3i, j) are in agreement. When α ≠ {0, π}, the quantum distributions are remarkably distinct from the classical ones. Quantum distributions however tend to be less asymmetric for high dpath interference, becoming more classical (see analysis in Supplementary Note 5). Figure 3k–m shows quantum interference of multipath wave and multimode particle properties regarding the inherent phase δ. We set α = 3π/2 as an example (it works as well for π/2) that corresponds to the maximal waveparticle superposition. The δdependence of distributions confirms the genuine quantum waveparticle superposition, while δvariation is absent in the case of classical mixture (see the explicit forms in Supplementary Eqs. 14 and 17). It is notable that by controlling the δ phase, the quantum interference of the wave and particle properties can be steered. For example, in the 2path case, Fig. 3n shows constructive interference for δ = 0 and destructive interference for \(\delta =\frac{\pi }{2}\). In the case of 4path interference (Fig. 3o), more wavelike characters appear when δ is set as \(\frac{\pi }{2}\).
All measurements in Fig. 3 were performed in the computational basis {\(\left0\right\rangle\), \(\left1\right\rangle\)}, which are well in agreement with theoretical predictions. We also estimated highlevel classical fidelities (see definition in Fig. 3’s caption) of 0.998 ± 0.001 for 2dpath, 0.991 ± 0.003 for 4dpath, and 0.980 ± 0.007 for 8dpath experiments, respectively. Since entanglement is playing an enabling role in the delayedchoice measurement of dpath waveparticle duality, we repeated the experiments in the complementary basis {\(\left+\right\rangle\), \(\left\right\rangle\)} to verify the presence of coherent entanglement, where \(\left\pm \right\rangle\) denotes \((\left0\right\rangle \pm \left1\right\rangle )/\sqrt{2}\). Our circuit allows any local qubit rotation by the MZIs together with posterior phaseshifters (Fig. 1e). In particular, local Pauli σ_{x} ⊗ σ_{x} operations on the Bell state were implemented to perform measurements in the {\(\left+\right\rangle\), \(\left\right\rangle\)} basis. We again obtained coherent waveparticle transitions with high fidelities in the complementary basis, as shown in Supplementary Fig. 5. Moreover, to exclude the presence of local hidden variables that may contribute the delayedchoice measurement, we verified entanglement by both performing quantum state tomography of the entangled state, and demonstrating the violation of the BellCHSH (Clauser–Horne–Shimony–Holt) type inequality^{45}. The quantum state fidelity of 0.962 ± 0.002 was obtained (Supplementary Fig. 3). And the Bell value of 2.75 ± 0.04 was measured, which violates the classical bound by 18.8σ, confirming the existence of strong entanglement through the device.
Generalized multipath waveparticle duality relation
We next report experimental results of a generalized multipath waveparticle duality relation in the delayedchoice interferometer. It is of fundamental interest to develop a general framework to describe the multipath duality and to quantify wave and particle properties^{17,27,28,29,30}. The conventional visibility defined as the contrast of 2path interference fringe, fails to be a good wave measure of dpath (d > 2) interference^{26}; however, quantum coherence is believed to be a good quantifier^{27,28,29,30}. We adopt the l_{1}norm coherence (\({\widetilde{{\mathcal{C}}}}_{l1}={\sum }_{i\ne j} {\rho }_{ij}\)) proposed by Streltsov et al. as a wave measure^{34}. Moreover, the capability of distinguishing which path the photon taken represents the dpath distinguishability. The formation of pathdistinguishability for the 2path case can be generalized to the dpath case^{27,28,46}. The normalized coherence \({{\mathcal{C}}}_{d}\) and pathdistinguishability \({{\mathcal{D}}}_{d}\) that we have used are given as:
where ρ represents the state for entire system having the target photon and measurement, as photon displays particle or wave nature is dependent on the measurement apparatus. The offdiagonal elements ρ_{ij} determine the dpath wave interference, while the diagonal elements ρ_{ii} determine the distinguishability of dmode pathinformation. The explicit forms of ρ for dpath classical and quantum experiments are given in Supplementary Note 4. Bohr’s multipath duality rule is thus quantitatively generalized as^{46}:
which saturates for the pure state, i.e., waveparticle superposition state. Proof is provided in Supplementary Note 4. The definitions of \({{\mathcal{C}}}_{d}\) and \({{\mathcal{D}}}_{d}\) well meet Dürr’s criteria^{27}, and importantly, they represent macrovariables that capture the global features of the dpath interferometric system. Note when d = 2, the generalized duality relation in Eq. (6) reduces to the fundamental statement by Jaeger et al.^{17}.
The \({{\mathcal{C}}}_{d}\) and \({{\mathcal{D}}}_{d}\) were measured in the context of delayed choice of measurement \({\hat{\text{M}}}_{0}\) under the control of {α, δ}. It is notable that the measurement of whichpath information \({{\mathcal{D}}}_{d}\) in our experiment is a posteriori one^{11,13}. This is because of the fact that \({\hat{\text{M}}}_{0}\) enables the projection into the particlewave superposition basis, that is delayed controlled from the fullwave to fullparticle measurements. In order to measure the \({{\mathcal{C}}}_{d}\), we here adopt a newly derived general visibility \({{\mathcal{V}}}_{d}\) proposed by Qureshi et al.^{32,46},
Importantly, \({{\mathcal{V}}}_{d}\) is exactly equivalent to the normalized coherence, i.e., \({{\mathcal{C}}}_{d}={{\mathcal{V}}}_{d}\), and it can be directly read out from interference patterns characterized by {\({I}_{\max }\), I_{inc}} (Fig. 4a, b), without knowledge of the density matrix of the system. We obtained the primary maxima \({I}_{\max }\) of the interference fringes, and the incoherence contribution \({I}_{{\rm{inc}}}=\frac{1}{d}\mathop{\sum }\nolimits_{i = 0}^{d1}{\rho }_{ii}\). Take the case for α = π as examples, Fig. 4a, b shows the interference patterns for the 2path and 4path cases, from which the {\({I}_{\max }\), I_{inc}} values and thus the \({{\mathcal{V}}}_{d}\) values can be directly obtained (results for d ∈ [2, 8] are reported in Supplementary Fig. 6). To get whichpath information \({{\mathcal{D}}}_{d}\), we measured the diagonal elements ρ_{ii}, by recording the probability of ioutcomes only having the ipath open. We measured \({{\mathcal{V}}}_{d}\) and \({{\mathcal{D}}}_{d}\) for different {α, δ, d}. Figure 4c, e (d, f) reports the experimental results for the 2path (4path) case, when δ is set as 0. The results for δ = \(\frac{\pi }{2}\) are shown in Fig. 4g, h.
Figure 4c–h reports experimental results of the generalized multipath duality relation for both waveparticle quantumsuperposition and classicalmixture cases. In the genuine waveparticle quantumsuperposition case (Fig. 4e–h), we demonstrate that the generalized duality relation holds the tight bound of unity as \({{\mathcal{C}}}_{d}^{2}+{{\mathcal{D}}}_{d}^{2}=1\) in the different setting of {d, α, δ}. This is however not true for the classicalmixture case, where one cannot reach the upper bound and the equality does not saturate at α ≠ {0, π}, resulting in \({{\mathcal{C}}}_{d}^{2}+{{\mathcal{D}}}_{d}^{2}\, <\, 1\). The bound can only be approximated at α = {0, π} for fullparticle or fullwave states, as shown in Fig. 4c, d. The quantitative results in Fig. 4 are consistent with the qualitative observations in Fig. 3a–j. The quantity of \({{\mathcal{L}}}_{d}=1{{\mathcal{C}}}_{d}^{2}{{\mathcal{D}}}_{d}^{2}\) represents the missing information, which is due to a classical lack of knowledge about the quantum system. The \({{\mathcal{L}}}_{d}\) is directly related to the Tsallis 1/2entropy as \({{\mathcal{L}}}_{d}=\frac{1}{d1}\left(\frac{1}{4}{({S}_{1/2}(\rho )+2)}^{2}1\right)\), where S_{1/2}(ρ) = 2(Tr(ρ^{1/2}) − 1) represents the Tsallis entropy. Note \({{\mathcal{L}}}_{d}\) is exactly the linear entropy if the l_{2}norm coherence is used^{28}. In Supplementary Note 5, we show how \({{\mathcal{L}}}_{d}\) scales for a large value of d.
Discussion
Waveparticle duality is a fundamental feature of quantum physics. The sophisticated dpath interference patterns in fact contain rich information. For example, a direct quantification of the amount of coherence embedded in the dpath interference patterns is allowed by measuring the visibility \({{\mathcal{V}}}_{d}\) (Fig. 4a, b and Supplementary Fig. 6), without the need for explicitly referring to the ρ of the system. We measured the l_{1}norm coherence (see refs. ^{33,34}) \({\widetilde{{\mathcal{C}}}}_{l1}\) for d ∈ [2, 8], directly from the dpath fullwave interference fringes (α = π). Figure 5a shows the results that identify a linear scaleup of \({\widetilde{{\mathcal{C}}}}_{l1}\) information. In addition, the whichpath information of doutcomes is fundamentally nondeterministic and provides a way for quantum randomness generation. Based on the dmode quantizations, it is possible to generate more than one bit of randomness^{35,47,48}. The randomness is quantified by minentropy \({H}_{\min }={{\rm{log}}}_{2}{p}_{d}\), where p_{d} is the probability of correctly guessing which path the photons take. Figure 5b shows the measured \({H}_{\min }\) values, in the fullparticle case (α = 0), generating more than one bit of randomness.
Imperfect device fabrication and noisy operation ultimately contribute to the degradation of our experiment results, such as an accuracy of 10^{−3} of the bound for the highorder interference, and noises in the measurements of generalized waveparticle transition and duality relation. For example, photon leakages between neighboring paths in the dMZI, owning to the presence of thermal crosstalks and imperfect extinction ratio of 2MZIs (~30 dB), bring in noises in the measurement of highorder interference. Further optimizations of device fabrication and alleviations of crosstalk can improve the performance. Moreover, accidental counts may be induced from the undesirable SFWM process when the residual bright light propagates through the whole chip, i.e., the parts after the sources, though they are negligible in the measurement of coincidences. Such background noises can be further suppressed by the adoptions of pump rejection filter^{49} and optical microresonator photonpair source with a high coincidencetoaccidental ratio^{50}. In our experiment, we relied on the assumptions that photons were faithfully sampled, and the choicemaker and the observer were independent from each other; these assumptions could be further relaxed in future^{51}.
In conclusion, we have reported an experimental generalization of Bohr’s duality relation in a delayedchoice experiment, on a largescale siliconintegrated quantum optical chip. The multipath waveparticle transition and generalized duality relation have now been confirmed by the delayed choice of measurement performed on single photons. Such a multimode quantum system provides a versatile platform to study multimode quantum superposition and coherence—the most fundamental quantum properties and resources^{33}. The direct probing of quantum coherence from interference distributions may allow the study of quantum processes and dynamics in complex quantum physical systems^{52} and biological systems^{53}. Going beyond the qubitbased quantum systems, highly controllable multidimensional quantum devices and systems that bases on the largescale integrated quantum photonics platform are expected to continuously advance quantum information science and technologies^{41,54,55}.
Methods
Device fabrication
The largescale integrated quantum photonic device for the implementation of the multipath delayedchoice experiment was designed and fabricated on the silicon nanophotonics platform, a versatile system for photonic quantum information technologies. The quantum device was fabricated by the standard CMOS (complementary metaloxidesemiconductor) processes. A layer of photoresist was the first spin on an 8inch SOI (silicononinsulator) wafer with 220nmthick top silicon and 3 μmthick buried oxide. The 248 nm DUV (deep ultraviolet) lithography was adopted to define the circuit patterns on the photoresist. Double inductively coupled plasma (ICP) etching processes were applied to transfer the patterns from the photoresist layer to the silicon layer, forming waveguides and circuits. Deep etching waveguides (see SEM image in Fig. 1g) with an etched depth of 220 nm were used for the SFWM photon sources, beamsplitters, and phaseshifters. Shallow etching waveguides (see SEM image in Fig. 1h) with an etched depth of 70 nm were used for the waveguide crossers and grating couplers. A SiO_{2} layer of 1μm thickness was deposited on top of the SOI wafer by plasmaenhanced chemical vapor deposition (PECVD), working as an isolation layer between the waveguides and metal heaters to avoid potential optical losses. Then, a 10nmthick Ti glue layer, a 20nmthick TiN barrier layer, an 800nmthick AlCu layer, and a 20nmthick TiN antireflective layer, were consequently deposited by physical vapor deposition (PVD) and patterned by DUV lithography and etching process to form the electrode. A 50nmthick TiN layer for thermaloptical phaseshifters was deposited and also patterned by DUV lithography and etching process. Finally, another 1μmthick SiO_{2} was deposited as the top cladding layer, and followed by the bonding pad opening process. More SEM images of the fabricated optical components and their characterizations are provided in Supplementary Note 1.
Experimental setup
Our experimental setup bases on offtheshelf telecommunication instruments. A tunable continuouswave laser (EXFO) central at a wavelength of 1550.11 nm was amplified to 40 mW by an erbiumdoped fiber amplifier (EDFA, Pritel), and then fed in to the siliconphotonics quantum chip as a pump source for the SFWM nonlinear process. Before the chip, the polarization state of the pump light was optimized by a fiber polatisation controller in order to excite the transverse electric (TE) mode of silicon waveguides. A pair of pathcoded entangled photons was onchip generated by the two SFMW sources (Fig. 1), and further adopted for the implementation of multipath quantum delayedchoice experiment. We selected the signal photon at 1545.31 nm and idler photon at 1554.91 nm. After the chip, wavelengthdivision multiplexing (WDM) filters with a bandwidth of 1.1 nm and 200 GHz channel spacing were used to remove the residual pump light. Single photons were detected by an array of superconducting nanowire singlephoton detectors (SNSPDs, Photonspot). A multichannel time interval analyzer (Swabian) was used to record twofold photon coincidences. All of the 95 thermaloptical phaseshifters were accessed and controlled individually by multichannel electronics (Qontrol) with a 16bits precision and KHz speed. The quantum chip was glued on a printed circuit board (PCB), and packaged by wire bonding (Fig. 1f). To achieve the full 2π operation of each phaseshifter, it requires about 40 mW power consumption. When all phaseshifters were on, total power consumption was about several watts. To ensure the stability of quantum operations and suppress thermal noises and thermal crosstalk, the chip was mounted on a Peltiercell and a thermosink with liquid cooling in order to dissipate power rapidly and efficiently. A thermistor mounted on the chip together with a proportional integrative derivative controller was used to monitor and stabilize the temperature of the photonic device. See Supplementary Fig. 2 for more experimental details.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The codes that support the findings of this study are available from the corresponding author upon reasonable request.
References
Bohr, N. The quantum postulate and the recent development of atomic theory. Nature 121, 580–590 (1928).
Grangier, P., Roger, G. & Aspect, A. Experimental evidence for a photon anticorrelation effect on a beam splitter: a new light on singlephoton interferences. Europhys. Lett. 1, 173–179 (1986).
Zeilinger, A., Gähler, R., Shull, C. G., Treimer, W. & Mampe, W. Single and doubleslit diffraction of neutrons. Rev. Mod. Phys. 60, 1067–1073 (1988).
Dürr, S., Nonn, T. & Rempe, G. Origin of quantummechanical complementarity probed by a ‘whichway’ experiment in an atom interferometer. Nature 395, 33–37 (1998).
Gerlich, S. et al. Quantum interference of large organic molecules. Nat. Commun. 2, 263 (2011).
Wootters, W. K. & Zurek, W. H. Complementarity in the doubleslit experiment: Quantum nonseparability and a quantitative statement of Bohr’s principle. Phys. Rev. D. 19, 473–484 (1979).
Wheeler, J. A. Mathematical Foundations of Quantum Mechanics 9–48 (1978).
Ma, X.S., Kofler, J. & Zeilinger, A. Delayedchoice gedanken experiments and their realizations. Rev. Mod. Phys. 88, 015005 (2016).
Manning, A. G., Khakimov, R. I., Dall, R. G. & Truscott, A. G. Wheeleras delayedchoice gedanken experiment with a single atom. Nat. Phys. 11, 539–542 (2015).
Jacques, V. et al. Experimental realization of wheeler’s delayedchoice gedanken experiment. Science 315, 966–968 (2007).
Jacques, V. et al. Delayedchoice test of quantum complementarity with interfering single photons. Phys. Rev. Lett. 100, 220402 (2008).
Tang, J.S. et al. Realization of quantum wheeleras delayedchoice experiment. Nat. Photonics 6, 600–604 (2012).
Kaiser, F., Coudreau, T., Milman, P., Ostrowsky, D. B. & Tanzilli, S. Entanglementenabled delayedchoice experiment. Science 338, 637–640 (2012).
Peruzzo, A., Shadbolt, P., Brunner, N., Popescu, S. & O’Brien, J. L. A quantum delayedchoice experiment. Science 338, 634–637 (2012).
Wang, K., Xu, Q., Zhu, S. & Ma, X.S. Quantum waveparticle superposition in a delayedchoice experiment. Nat. Photonics 13, 872–877 (2019).
Greenberger, D. M. & Yasin, A. Simultaneous wave and particle knowledge in a neutron interferometer. Phys. Lett. A 128, 391–394 (1988).
Jaeger, G., Shimony, A. & Vaidman, L. Two interferometric complementarities. Phys. Rev. A 51, 54–67 (1995).
Englert, B.G. Fringe visibility and whichway information: an inequality. Phys. Rev. Lett. 77, 2154–2157 (1996).
Dürr, S., Nonn, T. & Rempe, G. Fringe visibility and whichway information in an atom interferometer. Phys. Rev. Lett. 81, 5705–5709 (1998).
Einstein, A., Podolsky, B. & Rosen, N. Can quantummechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935).
Schrödinger, E. Die gegenwärtige situation in der quantenmechanik. Naturwissenschaften 23, 823–828 (1935).
Lapkiewicz, R. et al. Experimental nonclassicality of an indivisible quantum system. Nature 474, 490–493 (2011).
Sorkin, R. Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A 09, 3119–3127 (1994).
Sinha, U., Couteau, C., Jennewein, T., Laflamme, R. & Weihs, G. Ruling out multiorder interference in quantum mechanics. Science 329, 418–421 (2010).
Cotter, J. P. et al. In search of multipath interference using large molecules. Sci. Adv. 3, e1602478 (2017).
Mei, M. & Weitz, M. Controlled decoherence in multiple beam ramsey interference. Phys. Rev. Lett. 86, 559–563 (2001).
Dürr, S. Quantitative waveparticle duality in multibeam interferometers. Phys. Rev. A 64, 042113 (2001).
Hiesmayr, B. C. & Huber, M. Multipartite entanglement measure for all discrete systems. Phys. Rev. A 78, 012342 (2008).
Bera, M. N., Qureshi, T., Siddiqui, M. A. & Pati, A. K. Duality of quantum coherence and path distinguishability. Phys. Rev. A 92, 012118 (2015).
Bagan, E., Bergou, J. A., Cottrell, S. S. & Hillery, M. Relations between coherence and path information. Phys. Rev. Lett. 116, 160406 (2016).
Erhard, M., Krenn, M. & Zeilinger, A. Advances in highdimensional quantum entanglement. Nat. Rev. Phys. 2, 365–381 (2020).
Paul, T. & Qureshi, T. Measuring quantum coherence in multislit interference. Phys. Rev. A 95, 042110 (2017).
Streltsov, A., Adesso, G. & Plenio, M. B. Colloquium: quantum coherence as a resource. Rev. Mod. Phys. 89, 041003 (2017).
Baumgratz, T., Cramer, M. & Plenio, M. B. Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014).
Wang, J. et al. Multidimensional quantum entanglement with largescale integrated optics. Science 360, 285–291 (2018).
Friis, N., Vitagliano, G., Malik, M. & Huber, M. Entanglement certification from theory to experiment. Nat. Rev. Phys. 1, 72–87 (2019).
Reimer, C. et al. Highdimensional oneway quantum processing implemented on dlevel cluster states. Nat. Phys. 15, 148–153 (2019).
Neeley, M. et al. Emulation of a quantum spin with a superconducting phase qudit. Science 325, 722–725 (2009).
Cerf, N. J., Bourennane, M., Karlsson, A. & Gisin, N. Security of quantum key distribution using dlevel systems. Phys. Rev. Lett. 88, 127902 (2002).
Ecker, S. et al. Overcoming noise in entanglement distribution. Phys. Rev. X 9, 041042 (2019).
Wang, J., Sciarrino, F., Laing, A. & Thompson, M. G. Integrated photonic quantum technologies. Nat. Photonics 14, 273–284 (2020).
Ionicioiu, R. & Terno, D. R. Proposal for a quantum delayedchoice experiment. Phys. Rev. Lett. 107, 230406 (2011).
Wang, J. et al. Experimental quantum Hamiltonian learning. Nat. Phys. 13, 551–555 (2017).
Kauten, T. et al. Obtaining tight bounds on higherorder interferences with a 5path interferometer. New. J. Phys. 19, 033017 (2017).
Clauser, J. F., Horne, M. A., Shimony, A. & Holt, R. A. Proposed experiment to test local hiddenvariable theories. Phys. Rev. Lett. 23, 880–884 (1969).
Roy, P. & Qureshi, T. Path predictability and quantum coherence in multislit interference. Phys. Scr. 94, 095004 (2019).
Gräfe, M. et al. Onchip generation of highorder singlephoton wstates. Nat. Photonics 8, 791–795 (2014).
Skrzypczyk, P. & Cavalcanti, D. Maximal randomness generation from steering inequality violations using qudits. Phys. Rev. Lett. 120, 260401 (2018).
Paesani, S. et al. Generation and sampling of quantum states of light in a silicon chip. Nat. Phys. 15, 925–929 (2019).
Llewellyn, D. et al. Chiptochip quantum teleportation and multiphoton entanglement in silicon. Nat. Phys. 16, 148–153 (2020).
Giustina, M. et al. Significantloopholefree test of Bell’s theorem with entangled photons. Phys. Rev. Lett. 115, 250401 (2015).
Leggett, A. J. Testing the limits of quantum mechanics: motivation, state of play, prospects. J. Condens. Matter Phys. 14, R415–R451 (2002).
Lambert, N. et al. Quantum biology. Nat. Phys. 9, 10–18 (2013).
Kim, J.H., Aghaeimeibodi, S., Carolan, J., Englund, D. & Waks, E. Hybrid integration methods for onchip quantum photonics. Optica 7, 291–308 (2020).
Elshaari, A. W., Pernice, W., Srinivasan, K., Benson, O. & Zwiller, V. Hybrid integrated quantum photonic circuits. Nat. Photonics 14, 285–298 (2020).
Acknowledgements
We thank T. Qureshi, P. Skrzypczyk, Y. Ding, and X. Yuan for useful discussions and comments. We acknowledge support from the National Key Research and Development (R&D) Program of China (nos 2019YFA0308702, 2018YFB1107205, 2016YFA0301302), the Natural Science Foundation of China (nos 61975001, 61590933, 61904196, 61675007, 11975026, 12075159), Beijing Natural Science Foundation (Z190005), and Key R&D Program of Guangdong Province (2018B030329001). S.F. acknowledges support from Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology (grant no. SIQSE202005), the Key Project of Beijing Municipal Commission of Education (grant no. KZ201810028042), and Academy for Multidisciplinary Studies, Capital Normal University. M.H. acknowledges support from the Austrian Science Fund (FWF) through the START project Y789N27.
Author information
Authors and Affiliations
Contributions
J.W. conceived the project. X.C., Y.D., T.P., J.M., J.B, C.Z., T. D., and H.Y. built the setup and carried out the experiment. Y.Y., B.T., and Z.L. fabricated the device. X.C., Y.D., S.L., T.P., J.G., S.M.F., M.H., and Q.H. performed the theoretical analysis. Q.H., Q.G., and J.W. managed the project. X.C., Y.D., S.L., and J.W. wrote the manuscript. All authors discussed the results and contributed to the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Nature Communications thanks Robert Keil and the other anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chen, X., Deng, Y., Liu, S. et al. A generalized multipath delayedchoice experiment on a largescale quantum nanophotonic chip. Nat Commun 12, 2712 (2021). https://doi.org/10.1038/s41467021228876
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467021228876
This article is cited by

The potential and global outlook of integrated photonics for quantum technologies
Nature Reviews Physics (2022)

A programmable quditbased quantum processor
Nature Communications (2022)

Notes on quantum coherence with $$l_1$$norm and convexroof $$l_1$$norm
Quantum Information Processing (2021)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.