Observations of Deterministic Quantum Correlation Using Coherent Light

: Complementarity theory is the essence of the Copenhagen interpretation in quantum mechanics. Since the Hanbury Brown and Twiss experiments, the particle nature of photons has been intensively studied for various quantum phenomena such as anticorrelation and Bell inequality violation over the last several decades. Regarding the quantum features based on the particle nature of photons, however, no clear answer exists for how to generate such an entangled photon pair or what causes the maximum correlation between them. Here, we experimentally demonstrate the physics of quantumness on anticorrelation using well defined and nearly sub Poisson distributed coherent photons, where a particular photon number is post - selected using a photon resolving coincidence measurement technique. As a results, unprecedented wavelength dependent first - order intensity correlation has been observed in the two - photon second - order intensity correlation with 99.9 % visibility, where this result demonstrates the anticorrelation theory in Scientific Reports 10 , 7309 (2020) and opens the door to the on - demand quantum correlation control.


Introduction
Nonclassical quantum features such as Bell inequality violation 1-6 and anticorrelation 7-10 between two-mode quantum particles or engineered individual photon pairs have been intensively studied over the last few decades to understand the quantum nature of entanglement beyond the classical limits of individuality and separability 11 . The physical understanding of quantum entanglement based on the particle nature of photons governed by the energy-time relation via the uncertainty principle has been applied for various quantum information technologies. Although the complementarity theory of quantum mechanics is normally limited to the microscopic world of conjugate entities satisfying the uncertainty principle 11,12 , the Schrodinger cat [13][14][15] as a macroscopic quantum feature should also be confined by the same physics. Here, we report the first observation of on-demand quantum correlation using sub-Poisson distributed coherent photons obtained from an attenuated continuous wave (cw) laser. Since the wave characteristics of such sub-Poisson distributed photons relies on the intrinsic properties of the coherent light source, the degree of quantum correlation among photon pairs can also be deterministically manipulated not only for microscopic entities, but also for a macroscopic Schrodinger cat. This paper opens the door to on-demand quantum entanglement generation from coherent light source if a relative phase is controllable between paired entities.
Since the seminal paper of EPR 16 , entanglement representing the maximum correlation between two individual photons or atoms has been demonstrated via the second-order intensity correlation (2) ( ), where τ is the time delay between two particles for coincidence detection in an interference scheme [7][8][9][10][11] . Bell inequality violation is for a nonlocal quantum feature in a non-interferometric scheme [1][2][3][4][5][6] , while anticorrelation, the socalled Hong-Ou-Mandel (HOM) dip, is for the same quantum feature in an interferometric scheme [7][8][9][10] . If a single photon as a Fock state is considered 17 , the phase of a Fock state must be random due to the energy-time relation of the uncertainty principle. Thus, all quantum features denoted by quantum operators in a space-time domain have no definite phase relation between them. This is the fundamental limitation of conventional approach for quantum information. On the contrary, a definite phase relation between paired photons is an essential requirement to understand the fundamental physics of quantum entanglement in a deterministic way. In that sense, the terminology of quantum coherence has been used differently in terms of higher-order correlation 18 ever since the temporal correlation observation between two individual photons was demonstrated by Hanbury Brown and Twiss 19 .
Recently, a completely different approach has been pursued for the study of anticorrelation to unveil the quantum nature of photon bunching on a beam splitter (BS) 20,21 . Such a pure coherence optics-based interpretation for a quantum feature has been successful to describe the nonclassical phenomenon of a HOM dip.
As already known, experimental results of Young's double slits or a Mach-Zehnder interferometer (MZI) show the same interference fringe whether the input entities are single photons or coherence lights [22][23][24] . This is based on Born rule of quantum mechanics regarding superposition of a single photon, where a single photon do not interfere or interact with others, but with itself 25 . The wave-particle duality has been demonstrated for the interference scheme, where the definition of particle-like or wave-like photons is based on coherence [26][27][28] . As a result, there is no fundamental difference between the wave and particle approaches for a single photon in an interferometric scheme 29 . In the wave approach, self-interference in Young's double-slit experiments or an MZI is the bedrock of all quantum phenomena in an interferometric system regardless of photon numbers or photon characteristics, whether it is interpreted as a particle or a wave. Here, we experimentally demonstrate nearperfect nonclassical second-order intensity correlation using nearly sub-Poisson distributed coherent photons obtained from an attenuated laser to prove recently proposed theory 20 . This is the first step toward on-demand quantum entanglement generation based on the wave nature of photons under the Copenhagen interpretation of quantum mechanics.  of each photon from a 532 nm laser (Coherent, Verdi-V10) is quite long to be 60 m. The 532 nm laser is continuous wave (cw), whose power stability error is ~1 %. With proper attenuation using neural density filters (OD~10), a nearly sub-Poisson distributed coherent photon stream has been obtained, where the mean photon number is measured at 〈 〉~0.04 (see Sections A and B of the supplementary Information). The coincidence measurements in Fig. 1a, however, are not for single photons but for doubly-bunched photons. Because the used single photon counting modules (SPCMs: Excelitas AQRH-SPCM-15) cannot resolve photon numbers, all bunched photons must contribute to the measured data. However, the higher photon number-caused measurement error with three or more bunched photons is negligibly small to be ~1 % according to Poisson statistics as well as measurement results (see Fig. S2 of the Supplementary information). The concept of conventional coincidence detection measurements in Fig. 1a is unrealistic due to the limited scan range (~10 μm) of a Piezo-electric transducer (PZT; Thorlabs, POLARIS-K2S3P) compared with and even unnecessary under the present wave nature approach because the λ-dependet MZI fringe of (1) correlation is much more sensitive than the τ-dependnet coincidence detection of (2) (τ) or a HOM dip 21 .

An experimental design for on-demand quantum correlation
The ultimate purpose of this paper is to investigate the missing (1) fringe in a HOM dip of (2) ( ) measurements to prove the validity of ref. 20. Because Fig. 1a is for a pure coherence optics scheme which is limited to ≪ in a conventional HOM dip, we need to intentionally induce coherence walk-off via a spatial decoherence technique to set a reference of an incoherence (classical) feature of (2) ( ≫ ) = 0.5 (see Section C of the Supplementary Information). Figure 1b is for an intentional incoherence induction scheme via asymmetric control of two-beam overlap on BS2 (see the Inset and Fig. 2). Regarding the asymmetric spatial overlap control, only one axis out of two-axis PZT control unit is scanned by a PZT controller (Thorlabs, MDT693B) to induce intentional decoherence between coincident photon pairs. For this, the PZT at 75 V is set for a good overlap between both beams on BS2 (see Fig. 2). Thus, the present on-demand decoherence control technique gives us a clear understanding of the wave nature of photons via comparison with a conventional τbased HOM dip scheme [7][8][9][10] . Until recently 21 , the SPDC-based entangled photon characteristics has been vailed for the anticorrelation as to why the coincidence measurements are λ independent. In fact, the missing (1) feature in a typical SPDC-based HOM dip is not due to a mysterious quantum phenomenon but because of the entangled photons' characteristics generated from SPDC processes 21 .
Very recently, we have experimentally demonstrated that an attenuated coherent single photon has the same bandwidth as the original cw laser light 30 . This is the essence of the self-interference governed by Born rule 29 , where the two-photon correlation of a HOM dip is not due to photon interactions but due to superposition of each photon's self-interference on a BS. Regarding the MZI in Fig. 1a, the first BS (BS1) plays a role of a phase shift of π/2 between the reflected and transmitted photon pairs 31 . This is the first necessary condition for the anticorrelation 20 , where the second requirement is randomness or superposition between paired photons 21 . To be clear, the second condition can be satisfied even classically 21 and has not been involved in the present manuscript for simplicity. Under the first condition, thus, the paired photons in Fig. 1a can experience a destructive quantum interference in the second BS (BS2), resulting in photon bunching to either A or B port depending on the PZT-induced phase difference either π or 0, respectively. Unlike general understanding of nondeterministic anticorrelation in most HOM dips, the quantum feature measurements here in this manuscript must be phase dependent, resulting in a deterministic process of quantum correlation. This is the novelty of the present paper to expand and clarify our understanding on quantumness in an interferometric system.
Unlike conventional anticorrelation experiments based on SPDC processes 7 , whose generated paired photons have a pre-determined π/2 phase relation between them 20,21 , the attenuated photons in Fig. 1 have the same phase within the narrow bandwidth of 5 MHz. Each MZI output signals (A; ) are detected by a pair of SPCMs ( 1 ; 2 ). The time resolving of the SPCMs is 350 ps at λ = 825 nm with a dead time of 22 ns, and its dark count rate is measured at 27 ± 5 counts per second (cps). Compared with the present coherence time of a single photon at 0.2 μs, the SPCM's photon resolving time is much shorter, resulting in no bottleneck in (2) correlation. Both electrically converted single photon signals from the SPCMs are fed into a coincidence counting unit (CCU; Altera DE2) for two-photon coincidence detection measurements.

Preparation of a single photon stream
To provide nearly sub-Poisson distributed single photons, we have analyzed the attenuated photon statistics from the narrow 532 nm laser using two different methods based on SPCMs and a high-speed digital oscilloscope (Yokogawa DL9040; 500 MHz). For these measurements, the photon statistics is pursued in a noninterference regime by simply removing BS2 in Fig. 1. First, we perform photon counting measurements, analyze the obtained data, and compare them with the Poisson statistics for dozens of different OD values from the ND filter combinations. Second, the same measurements are conducted using the oscilloscope to visualize single photon streams, to obtain separate photon statistics, and to compare with the CCU-based ones (see Sections A and B of the Supplementary Information). With these two different measurement techniques, we confirm that the attenuated photon statistics satisfies nearly perfect sub-Poisson distribution for the rests of experiments.   Fig. 1 without BS2. b, Expansion of the box in a. c, Expansion of the box in b for a coincident (doubly bunched) photon pair.

Observations of nonclassical features using coherent photons
For Fig. 4, the PZT in Fig. 1 is continuously and repeatedly scanned for the same voltage region via a series of saw tooth ramps generated by a function generator (Tecktronics AFG3102; 0~150V), where the center position is fixed for a nearly perfect overlap on BS2, resulting in a symmetric decoherence feature (discussed in Fig. 2) as shown in Fig. 4a. Figure 4a shows only the half of the symmetric PZT scan range originally set for 0~150 V (see Fig. S7 of the Supplementary Information). For the optimum counting rate with respect to the PZT scanning, accumulation time of 0.1 s is set for CCU. Considering ~10 fringes in Fig. 4a, ~200 data points are allocated to each fringe. This corresponds to ~0.01 π radians (1.8 degrees) per data point, which is good enough for high resolution measurements of λ dependency in (2) correlation.   Fig. 1 with BS2. As shown in Fig. 4a, the one-axis PZT scan-induced symmetric decoherence technique is demonstrated successfully, where both ends of the fringes represent incoherence, i.e., the classical bound. Each fringe pattern (black or green dots) from each SPCM shows a typical single photon-based self-interference 30 . The red curve is a Gaussian fit for the measured fringes. Once again, this decoherence feature in Fig. 4a is not from the single photons' characteristics but from the spatial decoherence discussed in Fig. 2. According to the equality between visibility and coherence 32 , the asymmetric PZT scan technique in Fig. 1b is confirmed for the intentional decoherence induction as a function of a scan position (time) of PZT for the coincidence measurements. Unlike SPDC generated entangled photons, each MZI output intensity shows a typical MZI fringe governed by (1) correlation. As mentioned above for the second requirement of anticorrelation, we have not intentionally included randomness of photon superposition, otherwise results in the Gaussian-fit curve without the observed by (1) correlation 21 . Reminding of sub-Poisson photon statistics with 〈 〉~0.04, the deterministic (1) fringe patterns in Fig. 4a result from self-interference of each photon. The measured single photon's fringe visibility at near center is nearly perfect with 99.9 % from the minimum counts of 105 with respect to the maximum counts of ~200,000. Figure 4b shows the coincidence detection counts for the doubly bunched photon pairs of Fig. 3c for Fig. 1. The blue dots are the experimental data measured by CCU via two SPCMs. The red fringe curve is for a weighted best-fit curve based on the product of each corresponding data set in Fig. 4a, which is nearly perfectly coincides with the coincidence measurements. This is also a direct proof of (2) definition of (1) product average: (2) Here, this observed fringe pattern of (2) in Fig. 4b is independent of the randomness via photon superposition required for the second condition of anticorrelation mentioned above. Unlike conventional HOM dip measurements with broadband SPDC generated photons whose SPCM's resolving time is much longer than the photon's coherence time, the present coherent photons have an opposite situation, resulting in no side effect on the detectors. This functional backgrounds with photon characteristics provide near perfect measurement probability except for the quantum efficiency. Thus, the incoherence feature of side bands in Fig. 4 can be treated as an ideal reference of (2) = 0.5 regardless of the actual counting rate. Due to the fact that MZI fringe results from self-interference according to Born rule, there is no discrepancy between single photons and bunched photons in the interferometric fringe in Fig. 4 (see Fig. 4e as an extreme case of cw light). The black Gaussian curve in Fig. 4b is the best-fit curve obtained from the blue dots, where the measured visibility is near perfect and unprecedented, demonstrating the coherent photon-based anticorrelation with a (1) correlation fringe.
The minimum photon counting rate in Fig. 4b at center is 2 counts per 0.1 s, which is within the dark count rate of 5 per 0.1 s of the SPCM. Due to the intentional setup of decoherence with the asymmetric PZT scanning method, all maxima in Fig. 4b indicate the classical lower bound of (2) = 0.5. According to the weighted product of individual data in Fig. 4b (see the red curve), all crossing points between the black and green dots in Fig. 4a should coincide with the maxima in Fig. 4b, indicating a phase dependent incoherence feature of the coherent photons (discussed in Fig. 5). Thus, all measurement data points below the maxima at ~1,300 counts per 0.1 s represent a quantum feature. Thus, Fig. 4b clearly demonstrates new physics of the wave nature-based anticorrelation whose characteristics is phase dependent. This new physics of anticorrelation with sub-Poisson distributed coherent photons expands our understanding on quantum mechanics and opens the door to ondemand quantum information processing.
On the contrary of conventional HOM dips based on SPDC-generated entangled photon pairs, the coincidence measurements in Fig. 4b results in λ dependency as proposed in ref. 20. The λ dependency implies deterministic manipulation of quantum correlation. This striking observation confirms the existence of a specific phase relation φ between the paired entangled photons in SPDC, where the anticorrelation occurs at φ = ±π/2 as theorized in ref. 20. The mismatching between Fig. 4b and conventional HOM dips has been fully analyzed recently under the same physics via coherence washout among symmetrically detuned paired photons generated from SPDC 21 .The matched intensity product-based best fit curve with the coincidence measurements fully supports the present photon number-independent wave approach for quantum mechanics. Figures 4c and 4d are expansions of Figs. 4a and 4b, respectively. As shown in Fig. 4c, both fringes of single photon measurements are not perfectly aligned (see the horizontal dotted line). Such misalignment is due to the imperfect overlap on BS2, intensity fluctuation of the laser, or MZI phase fluctuation by air turbulence. As a result, the maxima of intensity correlation in Fig. 4d, i.e., the crossing point products in Figs. 4c fluctuate, too. As analyzed in Fig. S8 of the Supplementary Information, this fluctuation, however, is within 1 % misalignment error (discussed in Fig. 5). Figures 4e and 4f are for the corresponding cw cases of Figs. 4a and 4b, respectively. For this, the SPCMs in Fig. 1 is replaced by avalanche photon diodes (Thorlabs, APD 110A) and connected directly to the oscilloscope (YOKOGAWA DL9040) used in Fig. 3. The plots of Figs. 4e and 4f are from the stored data in the oscilloscope. Both black and green fringes in Fig. 4e are from both APDs corresponding to the outputs A and B in Fig. 1, respectively. Figure 4f is generated using the product function of the oscilloscope for the data in Fig.  4e, which corresponds to the red curve in Fig. 4b. Because the physics of the present anticorrelation is based on the wave nature of photons, there should be no difference between coincidence measurements of single photons in Fig. 4b and conventional wave optics of MZI outputs in Fig. 4f. According to Born rule of quantum mechanics regarding self-interference, the MZI in Fig. 1 limits the path superposition to the case of a single photon-two slit system, where the coincidence measurement should be the same as the weighted product of each single photon measurement [22][23][24] . This rule is also applied to the conventional SPDC-based anticorrelation, where the HOM dip is caused by photon bunching due to destructive interference in an MZI between two selfinterferences in Figs. 4b and 4f 20,21 . As already demonstrated by Born rule tests 29 , Fig. 4 also demonstrates Born rule of quantum mechanics that there is no difference between the particle (single) and wave (cw) natures of photons in an interferometric scheme. This experimental demonstration for two-photon coincidence measurements in Fig. 4b is unprecedented and opens the door to new physics of deterministic quantum correlation. Figure 5 provides reconfigured data from Fig. 4a for the perfect overlap of the observed nonclassical features. For this, all data from D1 in Fig. 4a are moved by two pixel points out of 2000 points in a row, resulting in nearly equal maxima as shown in Fig. 5b, resulting from the nearly perfect crossing points in Fig. 5a. As discussed by the logic gate operation and coherence optics in Fig. 4b, Fig. 5b shows a perfect coincidence detection rates for anticorrealtion with ~100% visibility, whose nonclassicality is φ dependent. Figures 5c and  5d are numerical simulations corresponding to Figs. 5a and 5b, respectively, with sine and cosine functions multiplied by a Gaussian function, representing the MZI outputs (see Fig. S8 of the Supplementary Information). The coincidence rate in Fig. 5d is normalized for the product of D1 and D1 in Fig. 5a. Here, the meaning of the same maxima is self-evident by the definition of product between D1 and D2 for AND gate operation in CCU as well as (2) correlation. The maxima (minima) in Fig. 5b reveals the perfect incoherent (coherent) characteristics as a classical (quantum) bound. Except for maxima in Figs. 5b, all data points are nonclassical by definition of (2) correlation in quantum mechanics due to the preset reference of maxima. Thus, the observed quantum correlation in Fig. 4b is strongly related with a relative phase between paired photons, resulting in ondemand control of quantum features. The maximum anticorrelation with zero coincidence count rate occurs at the MZI phase basis, φ ∈ {0, }. This phase basis relation is the essence for the fundamental physics of quantum correlation.

Conclusion
We experimentally demonstrated λ-dependent two-photon correlation using nearly sub-Poisson distributed coherent photons and compared the results with conventional SPDC-based HOM dips. The λ dependency in coincidence measurements reveals new physics of quantum features interpreted by the wave nature of photons without violating quantum mechanics. Unlike conventional nondeterministic quantum features based on the particle nature of photons, the observed (1) fringe in the two-photon coincidence measurements directly proves the phase basis-caused interpretation of quantum mechanics, resulting in deterministic manipulations of quantum correlations. For the demonstrations of anticorrelation using coherent photons, a simple MZI scheme was used, where the first BS plays a role of a π/2 phase shift between coincident photon pairs demonstrating the phase relation between entangled photon pairs interacting on a BS. To induce an artificial incoherence effect for a classical bound, an asymmetric PZT scanning method was developed, otherwise a long coherence time makes meaningless coincidence detection because all photons are coincident at any time of the MZI scanning regime. For coincidence detection measurements between two output ports of the MZI, we confirmed the new physics of anticorrelation proposed in ref. 20 by demonstrating near perfect visibilities (99.9%) in the coincidence measurements, where phase dependent interference fringe appeared. To support the experimental data, a simple Gaussian adjusted MZI simulations were compared, resulting in a near perfect coincidence with the experimental data. Thus, the observed coincidence measurements based on phase controlled coherent photons opens the door to the deterministic control of quantum features without violating quantum mechanics. Moreover, the present demonstration witnesses the new interpretation of quantum mechanics, satisfying Copenhagen interpretation of the wave-particle duality as well as self-interference of Born rule. By replacing the single photons with a cw laser, the thumb rule of no distinction between quantum particles and classical waves in an interferometric scheme by the Born rule tests was experimentally demonstrated via the concept of selfinterference. This result opens the door to deterministic quantum information science compatible with coherence optics.
The attenuated light source is Coherent cw laser (Verdi V10). The output intensity is stabilized at ~1 % variation for the fixed output power of 10 mW. For the cw MZI experiments in Fig. 4, OD 3 of ND filters are placed before MZI, resulting in 10 μW. For the single photon MZI experiments in Figs. 2~5, another OD 10 is added, resulting in 1 fW power and a mean photon number of 〈 〉~0.04.

Counting method for single and bunched photons by a coincidence counting module
The SPCM-generated electrical pulses are sent to the coincidence counting module (CCU: DE2 FPGA, Altera). The pulse duration of each electrical signal from SPCM for single photon detection is ~10 ns. Both single and coincidence counting numbers are counted in parallel by DE2 for 100 ms acquisition time for each data point, otherwise specified. The measured data is transferred to Labview via RS232 cable for a coincidence_rs232(4_5).vi application program in real time. For single and coincidence measurements in Fig. 4, the measured photon number is compared with that by an oscilloscope-based visualization technique below. The acquisition time of 100 ms for the CCU was set for an optimum number between the MZI fringe duration and the counted coincidence detections. Additionally, photon statistics for single photons, doubly bunched, and triply bunched photons, a conventional photon resolving measurement technique are used. The results are shown in Fig. S2 of the Supplementary Information. In this data, higher-order bunched photon ratio to the doubly bunched photons is less than 1 %. The mean photon number is measured at 〈 〉 = 0.04.

Counting method for single and bunched photons by a fast digital oscilloscope
For this, SPCMs were directly connected to the oscilloscope (YOKOGAWA; DL9040). Channel 1 of the oscilloscope is for D1, while Channel 2 is for D2. Due to the storage limitations of the oscilloscope for a displaced wave form on a screen, a stored 1 ms data length is analyzed for both sub-Poissonian photon distribution and coincidence detections. The results are also compared with the SPCM-based photon statistics. To count single photons in each data set, a homemade MATLAB program is used (see code in Fig. S6 of the Supplementary Information). For 10 measured data samples (not shown), the counted numbers for single photons are tabulated in Table S3 of the Supplementary Information.

Asymmetric PZT scanning
The coherence-length modification technique in Fig. 1b is based on the linearly increased phase shift via walkoff cross section between two lights on BS2 in Fig. 1, where such misalignment is conducted by an asymmetric control of a PZT kinematic mirror (Thorlabs, POLARIS-K2S3P) via a PZT controller (Thorlabs, MDT693B). For this, a PZT scanning applies to one of two mirror control knobs, where the midpoint of the scanning range is set to be maximally coherent at 75 V, whose PZT voltage resolution is 1.5 mV.

Data availability
Data for figures are available upon reasonable requests.    Experimental results for Fig. 1. a, Single photon counting rates measured by CCU via D1 (black dots) and D2 (green dots) in Fig. 1. b, Measured two-photon coincidence counting rates for D1 and D2. c,d, Respective expansions of a and b. e, Conventional cw MZIs for D1 (black) and D2 (green), where SPCMs are replaced by avalanche photon diodes. f, Products of two data points in e (performed on the oscilloscope). The cw laser power used for e and f is 10 μW.

Supplementary Files
This is a list of supplementary les associated with this preprint. Click to download. SupplementaryInformation0529r.pdf