Delayed-choice quantum erasure with nonlocal temporal double-slit interference

Wave–particle duality is a counterintuitive nature of quantum physics that challenges many common-sense assumptions, and Young’s double-slit interference is a prototypical example. While most quantum erasure experiments emphasized the choice of erasing or marking the which-path information of one quantum system, we use frequency entanglement to report a nonlocal temporal double-slit interferometer such that the which-time information determines the wave-like or particle-like behaviors. Since frequency-entangled photons are created simultaneously by using spontaneous parametric down conversion, the mark of temporal distinguishability is readily prepared by delaying one of the entangled photons, and its quantum eraser is implemented by using spectrally resolved detection with a tunable delayed choice. These results may provide an alternative aspect and insight into the role of the temporal degree in quantum-light complementarity and photon interference.


Introduction
Complementarity is one of the cornerstones of quantum mechanics, which describes the world of quantum phenomena beyond the possibility of classical physics [1,2]. Heisenberg uncertainty relation is a prototypical example of such a complementary theory, that lacks any counterpart in classical optics [3][4][5]. It states that one can never expect to obtain precise knowledge of position and momentum of a quantum at the same time, and thus the quantum observables position and momentum are complementary. In 1927, Bohr illustrated another complementarity principle via the wave-like and particle-like attributes to a quantum mechanical object [6]. Therefore, wave-particle duality is a well-known manifestation of the more general complementarity principle in quantum physics. Since then, several experiments confirmed both the wave and the particle nature of light [7][8][9][10][11][12][13][14].
Over the last decades, Young's double-slit interference experiment has been emphasized as an illustrative example of quantum complementarity [15][16][17][18][19][20][21]. The broad consensus is that the position-momentum uncertainty relation makes it possible to determine which slit the photon passes through without disturbing the photon enough to destroy the interference pattern at the same time. But it can reappear when which-path information is erased by using post-selection detection. Wheeler later proposed to delay the choice whether or not to inset the beam splitter at the end of the interferometer's path until the very last moment of the photon's travel inside the interferometer [22][23][24]. This operation rules out the possibility that the photon knew the configuration beforehand and adapted its behavior according. He then pointed out the seemingly paradoxical situation that it depends on the experimenter's delayed choice whether the photon behaved as a particle or wave. In other words, one could even erase or mark the which-path information after the registration of the quantum and still determine its earlier behavior to be either wave or particle.
Since 1982, quantum eraser has been reported in a variety of experiments, most of which emphases the choice of erasing or marking the which-path information that is took place in the past or delayed-choice arrangements [9][10][11][12][13][14][25][26][27][28][29][30][31]. For instance, polarization served as the which-path markers to destroy Young's double slit interference [32] or Hong-Ou-Mandel (HOM) interference [33], and realizing the observation of quantum erasure by adjusting the mark information. As a consequence, spatial indistinguishability has long been hailed as a vital prerequisite for observing quantum interference. However, in addition to the spatial information, temporal distinguishability is an alternative route towards exploring the essence of quantum interference. Back in 1987, Hellmuth et al have performed delayed-choice experiments in time domain based on the technique of quantum beats in time-resolved atomic fluorescence [34]. Later on, the coherently diffracted atoms have been used to design a temporal Young slit interferometer [35]. Recently, temporal double-slit interference with single particles has been proposed in the elaborate experiments [16,36], wherein the active interferometric stabilization and complicated experimental configuration are essential prerequisites for observing the interference pattern. Thus, the role of the temporal degree on the investigation of quantum complementarity has been, so far, little explored.
Here, we report a nonlocal temporal double-slit experiment with the assistance of HOM inference, where two distinct temporal modes are used to mark the which-time information in addition to the conventional spatial slits. In quantum mechanics, 'nonlocal' refers to a 'spooky connection' between different parts of a system that goes beyond the local cause-and-effect relationships characteristic of classical physics. The temporal double-slit we constructed is on two distinct paths, formed by entangled instantaneous correlation, hence we refer to them as nonlocal temporal double-slit. To tackle the issue of phase stabilization, two nonlocal temporal slits are prepared by delaying one of the time-entangled photons such that the relative phase between distinct temporal modes are transformed to the global phase of two-photon state. As a direct result of the inherent stability of HOM interferometry, our scheme does not have any requirement for active interferometric stabilization [37][38][39][40][41][42]. To harness the temporal distinguishability by concise spectral modulation, we observe the interference pattern in spectral domain as the biphoton temporal and spectral wavefunctions can be connected by using a Fourier transform [43][44][45][46]. Thus, when the distinct temporal modes are measured deterministically, the photons do not show interference. If the photons are measured with narrow spectral bandpass filters such that the single-photon coherence time becomes longer and which-time information is erased, then the photons will show interference pattern. In particular, the spectrally resolved detection is delayed enough such that it rules out the possibility that photon knew the configuration beforehand. Implementing nonlocal temporal double-slit interference thus implies a significant step in the exploration of quantum eraser experiments. Recently, an approach has been theoretically proposed to provide 100% visibility of HOM interference using a time lens in one arm of the interferometer to make entangled photons indistinguishable [47]. The difference is that we use HOM interference with spectrally resolved detection, the spectral modulation of single photons enables the flexible tuning of the temporal distinguishability, and realize experimental tests of non-local temporal double-slit interference quantum complementarity.

Nonlocal temporal double-slit experiment
The schematic of the conventional nonlocal quantum eraser experiment is illustrated in figure 1(a) [25]. As a pair of entanglement can be prepared by using spontaneous parametric down conversion (SPDC) in a nonlinear crystal, the paired photons are created simultaneously such that their temporal modes τ A = τ B . Then these photons are incident on two separated spatial slits A and B respectively. The signal photon that comes from A or B passes through a lens to meet a single-photon bucket detector, while the idler photon that also comes from two slits arrives simultaneously on different input ports of a balanced beam splitter that act as an interferometer. Specifically, if the beam splitter is placed such that the which-path information is erased, the nonclassical interference can be observed in the coincidence detection as a function of the relative phase shift in the two-arm interferometer, which indicates the wave-like behavior. On the contrary, if the beam splitter is moved out of the optical path such that it is allowed to deterministically distinguish that the idler photon comes from slit A or B, the standard interference disappears that indicates the particle-like behavior. A wide range of these analogous experiments merely focus on the distinguishability of spatial modes, thus the which-path or both-path information of a quantum can be erased or marked by its entangled twin even after the registration of the quantum [10-12, 15, 18, 20, 25, 26, 29, 48-51]. Now let us consider an alternative quantum eraser experiment that focuses on the distinguishability of temporal modes. In analogy to the link between position and momentum domains as a direct result of uncertainty principle, we exploit the Fourier transform that connects the temporal and spectral domains to explore the nonlocal temporal double-slit interference [43][44][45][46]. Thus the interference pattern can be observed by scanning the spectral correlation rather than the complicated temporal measurement. Another prominent advantage is the flexible control of temporal mutual coherence function by using the concise Figure 1. Schematic of (a) nonlocal spatial double-slit interferometer [25], and (b) nonlocal temporal double-slit interferometer. (c) It is allowed to tune the level of temporal distinguishability by changing the finite bandwidths with fixed time delay between the paired photons. spectral modulation. As shown in figure 1(b), a pair of frequency entanglement is created at two separated temporal modes τ A or τ B and spatial slits A or B, which can be expressed as where ω s and ω i are the central frequencies of down-converted signal and idler photons, f(ω s , ω i ) is the Gaussian spectral amplitude function with´´dω s dω i |f(ω s , ω i )| 2 = 1, |vac⟩ is the vacuum state. Analogously, the distinct temporal modes τ A and τ B mark the which-time information in our nonlocal temporal double-slit experiment. Additionally, the difference time τ = τ A − τ B would introduce a relative wavelength-dependent phase shift as exp(i∆τ ), where ∆ = |ω s − ω i | is the difference frequency of two well-separated center frequency bins. The signal photon that created at τ A or τ B are coupled to a detector by using a lens, which acts as a bucket detection without distinguishing the distinct temporal and spatial modes.
On the other hand, the idler photon that comes from temporal modes τ A or τ B is also sent to the incident input ports of a balanced beam splitter for erasing the spatial distinguishability, whose operation is expressed as [37][38][39][40][41] where the subscripts 1/2 represent two output modes of beam splitter. This operation transforms the incident quantum state into a superposition of bunching and anti-bunching effects, and the output state is then The detection post-selection at two distinct detectors merely records the anti-bunching event such that the quantum state can be simplified to Since two photons after beam splitting cannot be distinguished, we substitute w s and w i with w 1 and w 2 .
Using e −iw i τ to cancel the global phase, we obtain the following result where f(ω 1 , ω 2 ) can be considered as a symmetric function as a direct result of the collinear SPDC process. The detection operators of two detectors in different output modes arê Thus, since the beam splitter removes the which-path information, the coincidence probability at the output of two-arm interferometer is calculated as [42,52,53] The above discussion is based on the assumption that photons are completely indistinguishable. To address more general cases, we introduce |t A ⟩ and |t B ⟩ to analyze the temporal distinguishability, and the joint spectral intensity can then be expressed as [52,54] We can use the contrast of the pattern, V = Imax−I min Imax+I min = |⟨t A | t B ⟩| 2 , as a measure of its wave properties [55].
The theoretical predictions of two-photon interference as a function of difference frequency in ideal case (V = 1) are shown in figure 2. This method has been exploited in the engineering of high-dimensional frequency entanglement [42,52]. In our work, the imperfect interference visibility can be mainly attributed to the temporal distinguishability. More specifically, a perfect visibility V = 1 indicates that the which-time information between paired photons is completely erased, which reveals the wave-like behavior. On the contrary, the interference visibility V = 0 indicates that the paired photons are deterministically distinguishable in temporal degree of freedom, which reveals the particle-like behavior. Thus, the performance of temporal double-slit interference and a measure of which-time information implies the wave-particle duality in quantum mechanics.

Quantum eraser with temporal double-slit interference
Typically, the distinguishable information in temporal degree of freedom is introduced by adding a relative time delay, whose experimental implementation has the requirement for active interferometric stabilization and complicated experimental configuration [16,36]. In this work, we use a concise yet efficient method the tune the degree of temporal distinguishability. Since the temporal and spectral domains of single photons can be linked by using a Fourier transform, harnessing spectral distribution directly leads to the modulation of temporal distribution. Namely, if the spectral bandwidth of single photons becomes narrower, its corresponding coherence time inversely becomes longer. This Fourier transform connection is sufficient for the flexible control of temporal distinguishability. As shown in figure 1(c), while the relative time delay between the central frequencies of paired photons are fixed, the overlap area of their temporal distribution is still controllable by tuning their spectral bandwidth (that is the inverse of temporal bandwidth). As a direct result, we are allowed to prepare paired photons that are completely distinguishable, partially indistinguishable and almost indistinguishable in time. We note that the paired photons that are completely indistinguishable in time can be created in the experiment if and only if the spectral distribution is expressed as a delta function (the coherence time is infinitely long), which obviously cannot be implemented in a practical experiment. That is the reason for why figure 6(a) merely demonstrates the experimental results with partial indistinguishability of which-time information.
In theory, the interference probability as depicted in equation (8) manifests itself as a perfect cosine function within a spectral Gaussian distribution. As shown in figure 2, the theoretical prediction has the prerequisite that the spectral resolution satisfy the expression of delta function, i.e. an infinitely long coherence such that the distinguishable temporal information is completely erased. However, in a practical experimental setting, the spectral resolution of detection system cannot meet the requirement of filtering out a specific wavelength with an infinitely long coherence time. Generally, if the frequency of single photons is within a spectral wavepacket around a specific central frequency that corresponds to a definitely long coherence time, all of them would be coupled into the detectors. Namely, we can harness this spectral bandwidth to determine its coherence time, which accordingly tune the temporal distinguishability as discussed above. Since the detection probability as shown in equation (8) states that the oscillation period of cosine function is determined by a specific wavelength, the spectral filtering within a bandwidth would result in the decrease of two-photon interference visibility as shown in figure 3. In summary, if the spectral bandwidth is narrowed, its corresponding coherence time becomes longer such that the which-time information of two initially well separated time-bin maybe partially erased, and consequently the two-photon interference visibility gets higher. On the contrary, if the spectral bandwidth is widened, its corresponding coherence time becomes shorter such that the which-time information of two initially well separated time-bin cannot be erased, and consequently the two-photon interference visibility is always very low. Thus, HOM interferometry provides an alternative, concise yet efficient methods for the analogous implementation of quantum eraser with temporal double-slit interference.

Complementarity relation in temporal degree of freedom
In the ideal case, the distinct temporal modes can be approximately expressed as τ A ∝ exp(−σ 2 t 2 /2) and τ B ∝ exp[−σ 2 (t + τ ) 2 /2], we use their cross-correlation functions Γ(τ ) = ⟨τ A τ * B ⟩ to define the temporal distinguishability parameter as where σ is the RMS bandwidth in temporal domain that determines the single-photon coherence time.
These results agree well with our broad consensus that the nonlocal interference can be observed if and only if the temporal delay τ is within the coherence time [37][38][39][40][41]. Additionally, this indicates that the level of temporal distinguishability can also be tuned by changing the finite bandwidth even in the case that the relative time delay between the paired photons is fixed as shown in figure 1(c). According to (8), the temporal distinguishability makes a major contribution to the imperfect interference visibility as where I min can be obtained when ∆ = |ω 1 − ω 2 | = 0, which follows the analogous theoretical calculations that are demonstrated in [54]. These predictions obviously indicate that there is a trade-off between the measure of temporal distinguishability and the double-slit interference visibility, namely the wave-particle duality. This allows us to confirm that the ideal measurement results of our nonlocal temporal double-slit experiment satisfy the equality in complementarity relation as D + V = 1 [54]. Each randomly set value of D, by tuning the spectral bandwidth of down-converted photons that is relevant to the parameter σ in (9), allows us to observe interference with visibility V and to obtain incomplete which-time information characterized by the distinguishability parameter D. The all-or-nothing cases (V = 1, D = 0) or (V = 0, D = 1) obviously fulfill the complementarity relation. Intermediate situations, corresponding to partial which-time information and reduced visibility also satisfy this inequality, which are investigated in our experimental implementation.

Experimental results
The nonlocal temporal double-slit experiment is implemented by a conventional HOM interferometer with spectrally resolved detection as shown in figure 4 [42,45,56,57]. In our experiment, a continuous-wave grating-stabilized laser diode was used to pump a periodically polarized potassium titanium phosphate crystal (5 mm in length) to generate orthogonally polarized, spectrally degenerate photon pairs via type-II collinear SPDC. We separate the two orthogonally polarized photons into distinct spatial modes deterministically by using a polarization beam splitter and introduce a relative delay τ between them. We rotate the polarization of the photons in both arms by a pair of half-wave plates so that the two photons are identically polarized and are collected independently into the single-mode 2 × 2 fiber coupler. One output port of the coupler is directly connected to a single-photon counting module (SPCM), while a Bragg grating is inserted between the other output port and SPCM to map the continuous spectrum into the spatial distribution, and a tunable slit is used to filter partial spectrum at specific wavelength. By rotating the Bragg grating, the double-slit interference pattern as expressed in equation (8) can be observed with respect to the difference frequency of paired photons.
The temporal distinguishability refers to the which-time information that one can deterministically know the precise knowledge about which temporal slit that the photon comes. As expressed in equation (6), the temporal distinguishability parameter is determined by the single-photon coherence time 1/σ and the relative time delay between paired photons τ . Thus, by varying the width of tunable slit as shown in figure 2, the spectral bandwidth becomes narrower such that the single-photon coherence time inversely becomes longer, which results in the flexible control of temporal distinguishability (see figure 1(c)). Typically, as we set τ beyond the coherence time such that the which-time information, on which of the two slits the photon takes, deterministically appears, it is not allowed to observe the HOM interference pattern. To tackle this issue, we use a narrow band spectral filter (by narrowing the width of the tunable slit in our experiment) to erase the which-time information. This method has been used in the frequency-resolved entanglement swapping and multi-photon interference such that the temporal distinguishability from independent entanglement sources can be erased by using fiber Bragg grating to achieve ultranarrow frequency bandwidth, namely ultralong coherence time [58][59][60][61][62].
Additionally, an optical delay L s − L i > 2.76 m is chosen such that any which-time information one can infer from spectrally resolved detection must be at least 9.2 ns later than the registration of bucket detection, where L s is the optical distance between the output of the beam splitter and the spectrally resolved detector, and L i is the optical distance between the output of the beam splitter and the bucket detector. Compared to the 1 ns response time of the detectors, this optical delay is sufficient for the 'delayed erasure' . Although there is an arbitrariness in the time when photons are detected, it is allowed to say that the choice in spectrally resolved detection is delayed with respect to the bucket detection because the entangled photon pairs are created simultaneously by using SPDC process in a nonlinear crystal.
In our proof-of-principle experiment, the spectral bandwidth of down-converted photons is quite narrow, and the dynamic range of relative time delay is limited. Accordingly, we note that it has the requirement for high-resolution spectrometer to observe the interference pattern. For example, the relative path (time) delay between two arms of a HOM interferometer is set as −0.36 mm (τ = −1.2 ps). Thus the  temporal distinguishability as expressed in equation (9) is determined by the single-photon coherence time, which can be controlled by using concise spectral modulation. As shown in figures 5(a)-(d), the bandwidth of HOM interference pattern becomes wider as the single-photon coherence time becomes longer. This naturally enables the tuning of temporal distinguishability within the dynamic range of [0, 1]. The experimental measurements of the corresponding interference patterns in spectral domain are shown in figures 5(e)-(h).
To explore the complementarity in this experiment, the temporal distinguishability D and its corresponding interference visibility are required. More specifically, the single-photon coherence time 1/σ is extracted by fitting the interference pattern in figures 5(a)-(d) to the conventional Gaussian HOM dip, which is the inverse of single-photon spectral bandwidth. As the relative time delay is set as τ = −1.2 ps, the temporal distinguishability D can be readily calculated by following equation (9). The experimentally measured visibility V can be extracted by the interference pattern in figures 5(e)-(h). The experimentally measured interference visibility is generally lower than the theoretical prediction, which can be mainly attributed to the imperfect experimental implementation and the deleterious noise such as the distinguishability in polarization or spatial degrees of freedom. To quantify the complementarity relation, D and V are independently measured as a function of the inverse of the characteristic bandwidth 1/σ as shown in figure 6(a). The resultant complementarity relation that is defined as V + D is shown in figure 6(b), where the average value reaches ≈ 0.94 ± 0.02 and the dashed line represent the upper bound. These experimental results agree well with the theoretical predictions.

Discussion
In summary, we report an experimental test of quantum complementarity with nonlocal temporal double-slit interference. By using HOM interference with spectrally resolved detection, the spectral modulation of single photons enables the flexible tuning of the temporal distinguishability, and confirms the wave-particle duality by investigating the complementarity relation. Our results demonstrate the viewpoint that the system photon behaves either definitely as a wave or definitely as a particle would require the erasure of which-time information in addition to the conventional which-path information. Moreover, these results also demonstrate the possibility of delayed determination of wave or particle nature via quantum entanglement, and may provide an alternative route towards fully harnessing quantum interference in various quantum application.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).