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Revisiting self-interference in Young’s double-slit experiments S. Kim1 and Byoung S. Ham1,* 1Center for Information Processing, School of Electrical Engineering and Computer Science, Gwangju Institute of Science and Technology 123 Chumdangwagi-ro, Buk-gu, Gwangju 61005, South Korea (Submitted on April. 16, 2021 *[email protected]) Abstract: Over the past several decades, single photon-based path interference in Young’s double-slit experiment has been well demonstrated for the particle nature of satisfying complementarity theory, where the mechanical interpretation of the single photon interference fringe is given by Born’s rule for complex in measurements. Unlike most conventional methods using entangled photon pairs, here, a classical coherent source is directly applied for the proof of the same self-interference phenomenon. Instead of Young’s double slits, we use a Mach-Zehnder interferometer as usual, where the resulting self-interference fringe of coherent single photons is exactly the same as that of the coherent photon ensemble of the laser, demonstrating that the classical based on the nature is rooted in the single photon self- interference. This unexpected result seems to contradict our common understanding of decoherence phenomena caused by multi- among bandwidth distributed photons in coherence .

Introduction Young’s double-slit experiments [1] have been studied over the last century using not only the wave nature of coherent light but also the particle nature of [2-4], atoms [5,6], and photons [7-10] for quantum mechanical understanding of self-interference (SI) [11,12]. According to the Copenhagen interpretation, a single photon can be viewed as either a particle nature or a wave nature, so that the Young’s double-slit interference fringe based on single photons has been interpreted as self-interference based on the wave nature, satisfying the complementarity theory [13]. In other words, the single photon-based fringe cannot be compatible with the particle nature simply because a single photon as a minimum energy unit cannot be split into two parts. Thus, of a single photon itself between two paths of Young’s double slits needs to be interpreted in terms of the wave nature via indistinguishable entities or quantum superposition, otherwise the fringe disappears [7,14,15]. This is the essence of the quantum interference in Young’s double-slit experiments for SI satisfying the complementarity in quantum . Without violating generality, a Mach-Zehnder interferometer (MZI) has replaced the Young’s double slits due to its experimental feasibilities [7,12,14-20]. The SI of single photons in an MZI has also been applied for the proof of quantum features such as quantum erasers [7,15], Franson-type nonlocal correlation [16], and N00N state generations for higher order entangled photon pairs [17-21]. Based on the quantum features of SI in an MZI, indistinguishability between two entities in a paired path has been demonstrated with a basis, where a selection of perpendicularly polarized bases negates the interference fringe due simply to distinguishable entities regardless of mutual coherence [7,14]. So far, those experiments of SI have been performed using quantum particles of single photons generated mostly from spontaneous parametric down conversion (SPDC) processes [15,16,18-20]. Keeping in mind that multi- interference among many results in decoherence, each individual photon’s coherence time is supposed to be longer than that of an ensemble of photons. According to the Copenhagen interpretation regarding wave- particle duality or complementarity theory, the particle and wave natures are incompatible with each other. Thus, SI should be related with the wave nature of photons, otherwise the physical reality of a single photon in the paired path would be contradictory. In that sense, the first question for SI in an MZI is whether the coherence time of each single photon is longer than that of an ensemble of photons, resulting in a classical bound. According to the given (measured) bandwidth of a laser, each individual photon’s or is supposed to be slightly different from the others, resulting in the ensemble as a classical bound. Thus, it can be assumed that a single photon should have a longer coherence length than . Here, in this 𝑙𝑙𝑐𝑐 paper, we experimentally demonstrate that it is not true; thus, a fundamental question is followed𝑐𝑐 by what the difference between coherent light as a classical entity and a single photon as a quantum entity𝑙𝑙 should be. As

1 long as a is defined as a linear superposition of Fock states, a laser cannot be treated as a quantum entity [22-24]. By answering these simple questions, we confirm quantum characteristics of MZI via SI. In other words, the Fock state representation of a coherent state may not be applied at least for the MZI.

Fig. 1. Schematic of quantum superposition for self-interference in an MZI. L: 405 nm laser, A: attenuator, BS: , G1/G2: 500 m glass plate, M: , D1/D2: single photon detector, CCU: coincidence counting unit. μ Figure 1 shows the schematic of coherent photon-based MZI for the test of SI. For this, we set a reference as a classical bound with a typical MZI fringe using a commercial 405 nm laser (Omicron, PhoxX405-120), whose coherence time (length) is = 0.5 ( = 150 ). For a single photon MZI, the 405 nm laser light is attenuated with neutral density filters𝑐𝑐 (OD 10)𝑐𝑐 until a single photon stream is achieved with ~2 Mega counts per a second (cps). To confirm SI, 𝑡𝑡a heralded𝑝𝑝𝑝𝑝 detection𝑙𝑙 technique𝜇𝜇𝜇𝜇 is applied, where doubly bunched photons are split into two parts by the first beam splitter (BS) in Fig. 1. Bunched photons grouped into three or more are simply neglected, because their occurrence rates are less than 1% compared with the doubly bunched photons (see Fig. S1 of the Supplementary Information). Thus, the measured single photon-based SI in the present experiments has an error of < 1%.

Fig. 2. Attenuated laser light for a single and bunched photons. (a) and (b) From SPCM output D2. (c) On the oscilloscope (some examples). Expansion from the top to the bottom. Yellow/green: D1/D2, where D2 is relocated between the first and second BSs in Fig. 1. In (a), the red (black) dots are from D1 (D2).

Figure 2 shows experimental data for both single photons and doubly bunched photons detected by a coincidence counting unit (Altera, DE2) fed by a pair of single photon counting modules (Excelitas SPCM- AQRH 15), whose resolving time and dark count rate are 350 ps and 50 cps, respectively. For Fig. 2, the detector D2 in Fig. 1 is relocated to a position between the first and second BSs before the MZI. Both SPCMs show a ~1 Mcps detection rate as shown by red and black dots in Fig. 2(a), while Fig. 2(b) shows the coincidence detection rate for the doubly bunched photon pairs. The detected coincidence counting rate for the doubly bunched photons is ~12 kcps. As mentioned above, this coincidence counting rate includes three or more bunched photons, resulting in a < 1% error experimentally. With this error, Fig. 2(b) confirms stable coherent 2 photon pair generation by the coherent light source of 405 nm laser, where one photon in each pair is used for the SI experiment, while the other photon is used for the heralded detection to confirm the validity of SI. To visualize the doubly bunched photon pairs, both detector outputs are fed into a fast (500 MHz) digital oscilloscope (Yokogawa, DL9040). As shown in Fig. 2(c), a pair of single photon streams detected by D1 and D2 are observed in both top and middle panels, respectively. Of them are doubly bunched photon pairs occasionally detected as shown in the yellow box of the middle panel (see Figs. S2 and S3 of the Supplementary Information for details). The bottom panel of Fig. 2(c) shows some examples of the doubly bunched photon pairs. The detected number of photon pairs are 11 for 1 ms in the top panel. Figure 3 shows both coherent light-based MZI at 200 W (cw; 405 nm) for the classical bound (upper panels) and the single photon-based SI for the quantum bound (lower panels) with attenuation of the cw light. For the coherent light (cw) MZI, the single photon detector D1μ is blocked, and the SPCM of D2 is replaced by an avalanche silicon photodiode (Thorlabs, APD-110A) connected to the fast digital oscilloscope. For both cases, the fringe is expanded to ~2 cm in diameter using a (see Fig. 1). For the measurement of the fringe, a homemade single slit of width < 1 mm is inserted right after the collimated fringe and is linearly scanned across the fringe cross-section as shown in Fig. 1. The scanning slit-filtered light (photon) is focused onto the detector D2 by a lens, so that the slit scanning position does not affect the measurement efficiency. The number of MZI fringes is preset to be about six, resulting in each fringe width being much wider than the slit width to neglect potential effects. For the intentional path-length difference of the MZI, identical glass plates (G1 and G2) are inserted in the MZI paths, where one (G1) is changed, while the other (G2) is fixed. For G1 control, one end of the plate is linearly pushed using a picometer (Thorlabs, Z825B) with respect to the other hinged end, resulting in a rotation. The path-length difference of the MZI corresponds to the rotation angle of G1 (see Fig. S4 of the Supplementary Information). The reference position of G1 for = 0 is confirmed by the maxima of the MZI fringe. Four other different Ls are fixed at 100, 200, 300, and 350 m. The spatial resolution of the picometer is 29 nm. To minimize air turbulence-caused path length 𝛥𝛥𝛥𝛥fluctuations, the MZI is covered by a box, where the fringe maxima continueΔ for several tens of minutes with less thanμ a few per center fringe fluctuations. The upper panels of Fig. 3 are for the coherent light-based MZI outputs as a function of the slit scanning, while the lower ones are for the single photons. For the coherent light, a fast digital oscilloscope is used to record the captured data. The slit scanning time for the upper panels is 500 s, while it is 2,500 s for the lower panels. For the lower panels, the acquisition time of each data point is 1 s as set by CCU under the continuous scanning mode of the slit, where total data points are 2,500. All data in Fig. 3 are raw without any treatments. For each panel, the visibility is measured from Origin software generated best-fit Gaussian curves (see Fig. S5 of the Supplementary Information). Table 1 shows the calculated visibilities for Fig. 3 (see also Fig. 4).

Fig. 3. Observations of self-interference in an MZI using (Upper) cw coherent light at 200 W and (Lower) single photons with OD 10. μ

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Table 1. Calculated visibility for Fig. 3. L unit: m.

0 100 200∆ μ300 350 Source ∆𝐿𝐿 CW 1 0.801 0.539 0.3 0.153 SP 1 0.8 0.525 0.326 0.171 n n 0 0.037 0.01 0.007 -0.022 (CW) 𝑖𝑖 − � n n 0 0.031 -0.012 0.02 -0.019 (SP) 𝑖𝑖 − �

The measured visibilities for Fig. 3 in Table 1 are plotted in Fig. 4. Figures 4(a) and (b) show examples of how the Gaussian function fitting is applied for the visibility calculations in Table 1. Figure 4(c) shows two best- fit lines for the visibility sets in Table 1 for both coherent light (CW) and single photon (SP) cases. From Fig. 4(c), the measured coherence lengths for both coherent light and single photon cases are 268 m and 273 m, respectively, where the cw coherence length is well satisfied by the factory specification of the 405 nm laser. To our surprise, both visibility decay lines nearly perfectly coincide each other within the standardμ deviationμ = 0.0214 and = 0.0212, demonstrating that both cases result from the same , i.e., single photon𝐶𝐶𝐶𝐶 SI. In other words,𝑆𝑆𝑆𝑆 the conventional MZI fringe of coherent light originates from the single photon SI. Thisσ result is unexpectedσ because our common understanding of MZI with coherent light is classical. On the contrary, Fig. 4(c) demonstrates that there is no mutual interference among different photons. Figures 3 and 4 are the direct proofs of single photon SI mentioned by Dirac [11], such that photons do not interfere with each other. Thus, MZI cannot be discriminated depending on the input source whether the light is a coherent ensemble, thermal, or quantum particles of single photons. This result is not obvious in the community because coherent light has been excluded from quantum features. In a brief summary, the quantum superposition between two entities passing through different paths of an MZI is only caused by single photon SI regardless of the photon characteristics. This means that a common prejudice on the coherent light as a form of Fock state superposition needs to be reinterpreted at least for an interferometric scheme such as Hong-Ou- Mandel dip and Franson-type nonlocal correlation.

Fig. 4. Best-fit Gaussian curves for (a) the top and (b) bottom panels of Fig. 3 at L = 200 m. Blue: maximum, Green: minimum, Red: average. (c) Best-fit lines for visibilities in Table 1. Red/black: single photon/coherent light. Standard deviation : 0.0214 (cw), 0.0212 (single photon). The bestΔ-fit Gaussianμ curves and error bars are automatically generated from Origin. CW: continuous wave, SP: single photon. σ Discussion One point worth discussing regarding SI and the Copenhagen interpretation of complementarity theory is as follows. In generalized multi-photon multi-slit experiments for the Sorkin parameter satisfying Born’s rule in 4 [25], it is already known that there is no fundamental difference between the classical (coherent states) and quantum interferences [26]. In fact, even the observed nonzero Sorkin parameter indicating violation of Born’s rule does not necessarily contradict quantum mechanics [27,28]. As shown in Fig. 4(c), the matched best-fit lines from visibility calculations in Fig. 3 and Table 1 for both coherent light and single photon cases strongly support the fundamental rule of quantum mechanics for quantum superposition in terms of SI, where the SI is the origin of classical interference in an MZI system. Then the following two questions are raised: 1. Is there no mutual interference among different photons? 2. Does the two-photon interaction (such as the Hong-Ou-Mandel dip) for coincidence measurements result from SI? The answer to the first question must be yes according to Fig. 4(c), otherwise there should be accelerated decoherence. The second question response must be also yes because of the same role of BS in the MZI of Fig. 1. This is very intriguing and will be discussed elsewhere for coherent light-based photonic de Broglie waves and compared with the Sorkin experiments.

Conclusion Self-interference (SI) was experimentally demonstrated for coherent single photons in an MZI and compared with a classical bound of a typical laser light-based interference. Regarding the same coherence length observed in both cases, it was concluded that SI is the origin for classical coherence. This result initially appears contradictory to our common understanding of quantum mechanics because a coherent state indicates the laser light is a linear superposition of Fock states and cannot be considered as quantum in nature. Thus, MZI for arguments of quantumness based on the particle nature of photons may be reignited for the fundamental discussions of complementarity theory. In other words, common understanding of the classical interpretation for an MZI with coherent states of a laser needs to be corrected as the sum of individual quantum effects of single photon-based SI. According to the Copenhagen interpretation regarding the wave-particle duality, the wave nature of photons must not be mixed with the particle nature of Fock states. With the wave nature-based characteristics of single photons, photon number counting and MZI interference must not be dealt with simultaneously. The observation of SI with coherent light-attenuated single photons sheds light on a new vision of quantum mechanics for complementarity theory and may lead us toward better understanding of quantumness in topics such as anticorrelation, photonic de Broglie waves, and Bell inequality violations within an interferometric scheme.

Acknowledgment This works was supported by GIST via GIR 2021.

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