Abstract
The interference of quanta lies at the heart of quantum physics. The multipartite generalization of singlequanta interference creates entanglement, the coherent superposition of states shared by several quanta. Entanglement allows nonlocal correlations between many quanta and hence is a key resource for quantum information technology. Entanglement is typically considered to be essential for creating nonlocal quantum interference. Here, we show that this is not the case and demonstrate multiphoton nonlocal quantum interference that does not require entanglement of any intrinsic properties of the photons. We harness the superposition of the physical origin of a fourphoton product state, which leads to constructive and destructive interference with the photons’ mere existence. With the intrinsic indistinguishability in the generation process of photons, we realize fourphoton frustrated quantum interference. This allows us to observe the following noteworthy difference to quantum entanglement: We control the nonlocal multipartite quantum interference with a photon that we never detect, which does not require quantum entanglement. These nonlocal properties pave the way for the studies of foundations of quantum physics and potential applications in quantum technologies.
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Introduction
Quantum interference occurs only when no information to distinguish between the superposed states is knowable^{1}. Wellknown examples of quantum interference with photons include doubleslit interference of a single photon^{2} and HongOuMandel interference of two photons^{3}. A separate type of quantum interference is the interference via induced coherence, first realized by Zou, Wang, and Mandel in 1991^{4,5} in a MachZehnder interferometerlike configuration. The interference of the signal photon depends on the path identity of its twin photon, which is not even on the coherent paths of the signal photon. This mindboggling experiment “brings out that the quantum state reflects not what we know about the system, but rather what is knowable in principle”^{1}. In 1994, Herzog et al. demonstrated frustrated twophoton creation via induced coherence in a Michelson interferometerlike configuration, in which they can either enhance or suppress the generation of photon pairs in spontaneous parametric downconversion (SPDC) process by tuning the phases of various interferometers^{6}. Throughout this manuscript, we call this type of interference frustrated interference (FI).
Nonlocality is the characteristic feature of quantum correlation, such as entanglement. For instance, two spacelike separated observers—Alice (\({{{{{{{\mathcal{A}}}}}}}}\)) and Bob (\({{{{{{{\mathcal{B}}}}}}}}\)), share a pair of polarizationentangled photons and measure on specific polarization bases by adjusting the transmission angles (α/β) of their polarizers (Fig. 1a). When they compare their results, they will find that the joint probability depends on the polarizers’ angles: \({P}_{{{{{{{{\mathcal{A}}}}}}}}{{{{{{{\mathcal{B}}}}}}}}}(\alpha+\beta )={\sin }^{2}(\alpha+\beta )\), as shown in Fig. 1a. The probability here is normalized with the maximum counting rate for all the possible measurement settings of α and β. This secondorder interference of the entangled state can not be explained by local hidden variable theory and is considered nonlocal. In this setting, any mutual influence between the two observations is excluded under strict Einstein locality conditions^{7}. This phenomenon, predicted by quantum physics, cannot be accounted for by any local theory and represents one of the most profound foundational insights in physics^{8}.
In almost all scenarios in which nonlocal interference is observed, entanglement—or more generally some form of quantum correlation—is the basic ingredient. In this work, we show that this is not necessarily the case, and demonstrate multiphoton nonlocal quantum interference, which does not need entanglement. Note that nonlocality without entanglement has been discussed in the context of quantum state discrimination^{9}, which is not relevant to our work.
In this work we experimentally observe the multiphoton frustrated quantum interference (MFI)—a concept only theoretically proposed recently^{10}. Then we go beyond and demonstrate a surprising physical property: We observe nonlocal quantum interference that does not require quantum entanglement. Specifically, we tune the phase of a photon that we never detect, and observe interference with the rest photons.
Results
The property of twophoton frustrated interference
To understand the MFI, we first review the twophoton FI, of which the conceptual scheme is given in Fig. 1b. Two downconversion crystals are coherently pumped and probabilistically generate one photon pair. When we cannot distinguish which crystal the two photons come from, the coincidence of \({{{{{{{\mathcal{A}}}}}}}}\) and \({{{{{{{\mathcal{B}}}}}}}}\) oscillates as a function of phase β: \({P}_{{{{{{{{\mathcal{A}}}}}}}}{{{{{{{\mathcal{B}}}}}}}}}(\alpha+\beta )=\frac{1}{2}+\frac{1}{2}\cos (\alpha+\beta )\) (Fig. 1b). Moreover, FI even does not require the coincidence measurement as in the entanglement scenario. The single counts of A show the interference, depending on a phase β with no direct interaction: \({P}_{{{{{{{{\mathcal{A}}}}}}}}}(\alpha+\beta )=\frac{1}{2}+\frac{1}{2}\cos (\alpha+\beta )\) (Fig. 1b). This phenomenon is beyond the quantum entanglement, as a subsystem of a maximally entangled state is in a mixed state and shows no interference (Fig. 1a). Profit from this property, there has been a resurgence of interest in applying FI to quantumenhanced techniques recently, such as quantum imaging^{11}, spectroscopy^{12,13,14}, optical coherence tomography^{15}, state generation^{16,17}, microscopy^{18}, bioimaging^{19}, and quantum holography^{20}. This resurgence is fuelled by the application of nondegenerate photon pairs in FI, where one can probe objects of interest with the longerwavelength photon, and measure the result with a shorterwavelength photon that can easily be detected. For details, see the recent review on this topic^{21}. Note in the strong squeezing limit, one can use the socalled SU(1,1) interferometer for improving phase sensitivity^{22,23,24}.
However, this property of twophoton FI shows only the local interference. As shown in Fig. 1b, the twophoton case can not be nonlocal even in principle. The phase tuning event of the signals and idlers (α and β) are always in the backward light cones of the detection events. Under strict Einstein’s locality condition, a nonlocal configuration requires that the measurement result of Alice will not be influenced by the measurement setting β of Bob within the time that the light travels. As shown in Fig. 1b, the twophoton frustrated interference case^{4,5,6}, including quantum imaging with undetected light^{11}, will always be local under enforced Einstein locality conditions^{25}.
The property of fourphoton frustrated interference
Here, we extend FI to a fourphoton case to realize a nonlocal multiphoton interference. By nonlocal interference we mean an interferometer where the phase setting and port of the interferometer can be spatially separated under strict Einstein locality conditions (see Supplementary Note 7 for a spacetime diagram). We employ four photonpair sources in a configuration in which only two pairs of product states are generated from them (Fig. 1c). Alice and Bob control their phase shifters (α/β) locally and measure the fourfold coincidence counts, in which case they receive a product state. Since the settings of \({{{{{{{\mathcal{A}}}}}}}}\) (\({{{{{{{\mathcal{B}}}}}}}}\)) can be spacelike separated from detection events of \({{{{{{{\mathcal{B}}}}}}}}\)(\({{{{{{{\mathcal{A}}}}}}}}\)), they obtain the nonlocal phasedependent coincidence counts, that is, the fourfold coincidence counts oscillate as a function of α/β (Fig. 1c). Therefore, we call the fourphoton FI nonlocal quantum interference, as the photoncount dependence between \({{{{{{{\mathcal{A}}}}}}}}\) and \({{{{{{{\mathcal{B}}}}}}}}\) still remains, even if they are spacelike separated.
This measurement with the product state is very similar to the nonlocal interference with entangled states. However, here no quantum entanglement between any properties of the photons exists, but one can observe interference with the mere existence of a multiphoton state. It arrives from a coherent superposition of the origin of the multiphoton state. Moreover, when Bob varies the phase β and measures the threefold coincidence between the two detectors in \({{{{{{{\mathcal{A}}}}}}}}\) and detector 3 in \({{{{{{{\mathcal{B}}}}}}}}\) (Fig. 1c), they will observe the interference of the three photons as a function of β (Fig. 1c). We stress that the phase β, which we can tune, has no direct interaction with all the other three detected photons. This is the unique feature of MFI and in contrast to the entanglement case, where \({P}_{{{{{{{{\mathcal{A}}}}}}}}}(\alpha+\beta )\) does not depend on β (Fig. 1a). Although the visibility of P_{123}(α + β) is not 1 due to the particular construction of the setup, more complex source configurations and detection schemes may further increase the visibility. In this case, we can probe the threephoton coincidence count by tuning the phase of the fourth photon, which is undetected. We note that one cannot achieve spacelike separation between the detection on \({{{{{{{\mathcal{A}}}}}}}}\), photon 3, and the setting β. Otherwise, superluminal control would occur.
From a fundamental perspective, by extending the twophoton FI to multiphoton FI, one could separate the downconversion crystals in space and demonstrate nonlocal control of multiphoton interference that does not need entanglement. From an application perspective, one could devise more complex quantuminformation tasks, such as quantum computation^{10} and generations of complex multiphoton quantum states^{16,26,27}.
Scheme of fourphoton frustrated interference
In this work, four photons are generated in two indistinguishable generation processes and measured with four detectors, enabling the suppression and enhancement of fourphoton generation via FI^{10,28}. In twophoton FI, there is one pair of photons generated from two twophoton sources^{6}. In the fourphoton FI demonstrated here, we use four twophoton sources for generating two pairs of correlated photons, as shown in Fig. 2a. Four twophoton sources placed in sequence are pumped coherently by two laser beams. The downconverted photons from different groups (crystals I and II, and crystals III and IV) are aligned according to the geometry shown in Fig. 2a to ensure the path indistinguishability. Photons on the same path have identical properties (such as polarization, frequency, and arrival time at the detectors). We emphasize that there is no entanglement of any external or internal degrees of freedoms of photons involved, neither those created by the source (as we use singlemode fibers) nor those generated through the concept of entanglement by path identity (as we do not shift modes between pair creations, which is the key idea of entanglement by path identity^{16}).
Considering the low probability p for generating photon pairs for the SPDC process, the output state (without normalization) from modes 1 to 4 can be written as:
to secondorder approximation, where the numbers in the kets represent photon numbers in modes 1 to 4.
When we count only the event detecting the four photons simultaneously, we postselect the state \(\left1111\right\rangle\), which is a product state, as shown in the second line of Eq. (1). The probability of observing fourfold coincidence counts (P_{1,2,3,4}) varies with the phases in the system:
See Supplementary Note 1 for detailed derivation.
A more interesting phenomenon, multiphoton interference controlled by an undetected photon, is observed in our experiment. When we vary the phase of photon s1, ϕ_{s1}, the probability of observing the other three photons detected by detectors 1, 3, and 4 is:
The ideal visibility is 50% due to the multiphoton noise from \(\left1012\right\rangle\). In this case, we can detect and observe threephoton interference by tuning the phase of the fourth photon, which is undetected. This finding shows the effect that one has nonlocal quantum interference that does not require entangled states. For entangled states, such as a Greenberger–Horne–Zeilinger(GHZ) state^{29,30}, one would lose correlation when losing one particle.
Experimental setup
The scheme of our experimental setup is shown in Fig. 2b. The dimension of the optical setup is roughly 0.8 × 1.0 m^{2}. The pump is a 404nm femtosecond pulsed laser with vertical polarization. A halfwave plate (HWP1) rotates the polarization of the pump laser to 45^{∘}. One polarization beam displacer (BD1) separates the pump laser into two parallel paths with equal power of about 0.29 W, denoted as P1 (H) and P2 (V), to pump a single betabarium borate (BBO) crystal separately. The spacing between the two paths is about 4 mm. Both P1 and P2 are horizontal polarization after the semicircle HWP2 (halfHWP). The optical axis of BBO is in the horizontal plane and is aligned to be 40. 9^{∘} with respect to the two pumps to form the beamlike SPDC configuration^{31,32}.
P1 and P2 generate two pairs of photons denoted as s1, i1, and s2, i2. The photon pairs from the beamlike source are in the polarization product state \({\left{{{{{{{\rm{HV}}}}}}}}\right\rangle }_{{{{{{{{\rm{si}}}}}}}}}\), and the emission angles of signal and idler with respect to the pump are approximately 3^{∘}. The polarization of the downconverted photons is shown in Fig. 2b. The triple dots represent vertical polarization and the triple lines represent horizontal polarization. As s1/i1 is parallel with s2/i2, after the semicircle HWP, both s1 (V), s2 (H), and i1 (V), i2 (H) are combined on the BDs and are focused with lenses to improve the coupling efficiency at the couplers. Photons i1 and i2 pass through a quarter wave plate (QWP) with the angle fixed at 45^{∘}. Then they are reflected on mirror M1. Therefore, the two photons swap their path on the way back, which corresponds to the crossing between photons i1 and i2 in Fig. 2a. On the signal photons side, we separate s1 and s2 on a polarization beam splitter (PBS) to control their phase ϕ_{s1} and ϕ_{s2} independently.
After the BBO crystal, P1 and P2 are combined on BD2 and reflected by mirror M2, forming a symmetrical interferometer. The reflected pumps, denoted as P3 and P4, are used to generate photon pairs s3 and i3, and s4 and i4. By adjusting M1, M3, and M4, the paths of s1, i1, s2, and i2 overlap with s3, i4, s4, and i3, respectively, as shown in Fig. 2a, which erases the path distinguishability. Though the polarization states of signals and idlers are different, they are the same for the photons on the same path due to the symmetry of our interferometer, which is necessary for realizing the fourphoton interference. To observe the fourphoton interference successfully, we also need to erase the temporal distinguishability. We fix M2 and scan the delays of M1, M3, and M4 until the interference pattern emerges, ensuring that the reflected photons and the reflected pump laser pulses arrive at the crystal simultaneously. We note that there is a time difference between the reflected pumps P3 and P4 due to the geometrical dimension of the BDs, and so are the signals and idlers on the same side. We can still realize the fourphoton interference. We only need to ensure photons on the same path arrive simultaneously, not all the photons on different paths^{33}. This is especially important for future spacelike separated experiments of this effect. For the detailed results of path identity and analysis of timing, see Supplementary Note 4.
All four photons s1 (s3), s2 (s4), i1 (i4), and i2 (i3) are finally collected by singlemode fiber couplers. We analyze the coincidence counts while varying the phase ϕ_{s1} of s1. The result of fourfold coincidence counts is shown in Fig. 3a. The period of the interference pattern is 403.5 nm, in agreement with the 808nm central wavelength of photon s1, considering that it goes back and forth. The visibility of interference is about 75.47%. The misalignment of photons on the same path reduces the identity in spatial mode and thereby the fourphoton interference visibility. Based on the values obtained from independent experimental measurements, the estimated maximum achievable value for visibility is about 81.95% (see Supplementary Note 3), which is higher than we obtained (75.47%). This discrepancy may come from higherorder emission from SPDC, which further reduces the fourfold interference visibility.
The spatial misalignment causes experimental visibility different from identity. It can be modeled by including the transmissivity (T) in the path of photon s2 [see ref 4]. Therefore, we reduce the coupling efficiency of photon s2 (hence lower T) and measure the visibility of fourfold coincidence to verify this effect. We note that the visibility is not an exact linear correlation in T for fourfold coincidence: \(V=\frac{2\alpha T}{1+{\alpha }^{2}{T}^{2}}\) (see Supplementary Note 6), where α is the parameter used to characterize the path identity. The experimental result is shown in Fig. 3c. As the transmissivity of photon s2 decreases, the visibility of interference goes down to almost zero. That is because we know that photons on mode4 come from crystal IV when s2 is blocked, which destroys the interference. The parameter α of the fitting curve is 0.42.
We also scan phase ϕ_{i} and record the fourfold coincidence counts. The result is shown in Fig. 3b. Because both signals i1 and i2 experience phase ϕ_{i} as in Eq. (2), the interference period is approximately 200.9 nm, which is half of the period shown in Fig. 3a. The visibility of the interference is 74.26%. It is consistent with the visibility of Fig. 3a.
Multiphoton frustrated interference controlled by an undetected photon
To demonstrate the multiphoton interference controlled by an undetected photon in the frustrated interference, we change the phase of s1 and measure the threefold coincidence events on detectors 1, 3, and 4, where photon s1 is undetected (see the inset of Fig. 4a). The result is shown in Fig. 4a. As the phase ϕ_{s1} varies, the coincidence counts of the other three photons change correspondingly. Therefore, we observe multipartite frustrated interference, where multiple correlated photons are influenced by a phase that has no direct relevance. The visibility of the interference is 29.84%, which is lower than the theoretical value of 50%. The limited visibility is because of the limited path indistinguishability for idler photons. We furthermore show that even twofold coincidence, on detectors 1 and 3 (see the inset of Fig. 4b), can be controlled with the phase of undetected photon s1, ϕ_{s1}:
The result of the coincidence measurement on detectors 1 and 3 is shown in Fig. 4b. As the coincidence only occurs when more than two crystals generate photons, twin photons from the same crystal do not cover up the interference. The interference data in Figs. 3a and 4a, b are recorded simultaneously and show nearly identical interference phasedependence.
Finally, to show that our experiment is a genuine quantum mechanical effect and a consequence of induced coherence, we vary the phase ϕ_{s1} and measure the interference visibility of threephoton coincidence on detectors 1, 2, and 3 while reducing the transmissivity of photon s2, as shown in Fig. 4c. The nearly linear relation indicates that the fourphoton FI is an induced coherence rather than induced stimulation^{34,35}, which is beyond the classical optics^{4}.
Discussion
In this work, by harnessing the indistinguishability between the generation process of photons, we have shown fourphoton nonlocal quantum interference with product states. This effect occurs not because of a superposition of the photons’ external properties, such as path, polarization, and so on. Instead, it happens because of a fundamental unknowability where the photons have been generated. This underlying principle allows us to show how we can manipulate the interference of three photons by introducing a phase in the fourth photon that we never detect. Other types of interference: singlephoton interference, and twophoton interference (entanglement and HongOuMandel), all require the detection of all involved photons to observe the interference effect. If one traces out one photon, the outcomes show no interference. For instance, if one uses two identical single photons generated from two independent sources and performs the HOM interference experiment, one will not see any interference effect if one only measures the singlephoton counts. Interference only appears when the coincidence measurements of all interfering photons are performed. In this work, the situation is fundamentally different. We tune the phase of a photon that we never detect, and observe interference of the rest photons. Multiphoton interference appears when the coincidence measurements of only partial interfering photons are performed.
Also, this is a different interference phenomenon to some of the maximally entangled states, such as GHZ states. If one photon of a GHZ state remains undetected, no quantum interference will be observed in the rest photons (see Supplementary Note 9 for details). Our experiment thereby demonstrates in a direct way how the lack of knowledge about a quantum system can lead to multiphoton nonlocal quantum interference, a feature that cannot solely be described by entanglement.
Novel properties of this quantum system can be observed with improvements in our experimental setup. As we purposefully chose to build our setup with bulk optics, we can separate the distance between the crystals and measure the nonlocal interference influenced by the phase of the undetected photon, which is important for exploring fundamental questions of quantum physics and may be useful in quantum communication. This is possible because one can build an experimental setup that shows the nonlocal interference under strict Einstein locality conditions, by randomly setting the phases α or β (in Fig. 1c) after photon pairs in the lower layer (more details in Supplementary Information).
Additionally, variations of our multiphoton experiments with induced coherence can be used to explore highly diverse quantum systems. An example is a resource state for photonic quantum computers^{36}, in which one exploits the exact multiphoton frustrated generation that we experimentally showed here.
Finally, the demonstration of nonlocal interference with undetected photons relates our experiment to a very vibrant field of quantum imaging with undetected photons^{11} and its variations^{21}. Here, one striking property is the generation of multiple wavelengths of the different photons. This is interesting in the absence of suitable detectors for the wavelength of the undetected photon. Our experiment brings this applicationdriven research finally into the multiphoton regime^{11,12,13,14,15,18,19}. In this work, we have shown the multiphoton frustrated interference, which can potentially be observed even when the settings and detections of Alice and Bob are spacelike separated. Note that this unique property of multiphoton frustrated interference was not discussed in the original proposal of multiphoton frustrated interference^{10} nor was it shown in a recent related work^{37}.
Data availability
The data that support the plots within this paper and other findings of this study are available at https://github.com/NJUMalab/FrustratedInterference.
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Acknowledgements
This research was supported by the National Key Research and Development Program of China (Grants nos. 2022YFE0137000, 2019YFA0308704, and 2017YFA0303704), the National Natural Science Foundation of China (Grants nos. 11690032 and 11321063), the NSFCBRICS (Grant no. 61961146001), the LeadingEdge Technology Program of Jiangsu Natural Science Foundation (Grant no. BK20192001), the Fundamental Research Funds for the Central Universities, the Innovation Program for Quantum Science and Technology (Grant no. 2021ZD0301500), and the Jiangsu Funding Program for Excellent Postdoctoral Talent (No. 20220ZB60).
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K.Q., K.W., L.C., and Z.H. performed the experiment. K.Q., X.s.M. analysed the data. K.W., M.K., and X.s.M. designed the experiment. K.W., M.K., and X.s.M. wrote the manuscript with input from all authors. S.Z. and X.s.M. supervised and directed the project. All authors commented on the manuscript.
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Qian, K., Wang, K., Chen, L. et al. Multiphoton nonlocal quantum interference controlled by an undetected photon. Nat Commun 14, 1480 (2023). https://doi.org/10.1038/s4146702337228y
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DOI: https://doi.org/10.1038/s4146702337228y
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