Near-100% two-photon-like coincidence-visibility dip
with classical light and the role of complementarity
The famous Hong-Ou-Mandel two-photon coincidence-visibility dip (TPCVD), which accepts one photon into each port of a balanced beam splitter and yields an equal superposition of a biphoton from one output port and vacuum from the other port, has numerous applications in photon-source characterization and to quantum metrology and quantum computing. Exceeding 50% two-photon-coincidence visibility is widely believed to signify quantumness. Here we show theoretically that classical light can yield a 100% TPCVD for controlled randomly chosen relative phase between the two beam-splitter input beams and experimentally demonstrate a 99.635 0.002% TPCVD with classical microwave fields. We show quantumness emerges via complementarity for the biphoton by adding a second beam splitter to complete an interferometer thereby testing whether the biphoton interferes with itself: Our quantum case shows the proper complementarity trade-off whereas classical microwaves fail.
The Hong-Ou-Mandel (HOM) TPCVD[1, 2], is a salient fourth-order interference effect, that serves as one of the most important effects and tools in quantum optics. Myriad applications include measuring photon purity and distinguishability, heralding in optical quantum computing gates, conceptually underpinning the complexity of Boson Sampling, and realizing a N00N state for quantum metrology.
TPCVD is manifested by injecting a single photon into each input port of a balanced beam splitter and observing anti-correlated output in the form of bunching, i.e., signature of the biphoton , which is a pair of identical photons, emerging from one port and vacuum from the other; mathematically, the input state is the pure product state for one photon in each input port and the output state is the symmetric/anti-symmetric superposition
The coincidence rate is defined as being the ratio of joint-detection intensity integrated from detector turn-on time to detector turn-off time to the denominator corresponding to the detector on-time . Given some delay time between photon arrival times at the two input ports, the TPCVD is [9, 10] i.e., the ratio of the maximum-minimum coincidence difference to the maximum coincidence and is unity for in the ideal case. The TPCVD temporal width corresponds to temporal wave-packet width for the single-photon source.
Despite the immense importance, applicability and success of TPCVD, a widespread misconception accompanies this phenomenon, namely the myth that exceeding a 50% TPCVD falsifies classical electromagnetic field theory. Example quotations include “fourth-order interference of classical fields can not yield visibility larger than 50%”, “as long as the visibility of the coincidence dip is greater than 50%, no semiclassical field theory can account for the observed interference”, “visibility, being greater than 50%, is clear evidence of non-classical interference”, and “classical theory of the coherent superposition of electromagnetic waves, however, can only explain a HOM dip with V0.5”. This myth matters as the 50% dip threshold is widely accepted as proving that two-photon interferometry has entered the quantum domain.
The reason for the common belief in the 50% TPCVD threshold is that independent classical input fields injected into each balanced-beam-splitter input port yields a maximum of , if the phases of the two input beams are randomized over relative phase.[14, 15] However, this proof leaves open the possibility that classical fields without noisy phases could yield unity . Here we prove that can indeed approach unity for classical fields with definite relative phase, and we show that the real quantum signature lies in completing the interferometer as a two-photon analogue of the experimental proof of complementarity for a single photon.
Now we prove the 50% and 100% dips for classical input fields with controlled relative phases (chosen randomly from defined sets) between the two input fields. We consider both inputs as pulses with amplitude and slowly varying envelope function for input 1 and amplitude and slow varying envelope function for input 2, with the relative time delay between pulses 1 and 2. Thus, the complex electric fields for the two pulses are and , respectively, for , and the corresponding output fields are from the two output ports.The normalized coincidence rate is[17, 18, 19]
which involves averaging over some continuous or discrete (sum over Dirac functions) probability distribution with . For phase-insensitive square-law photodetection, which holds in the infrared (IR) and optical regimes, the coincidence rate (13) is modified by considering only the numerator as the denominator is dominated by pump- and thermal-photon counts.
For integration time much larger than pulse width and delay time ,
which yields for chosen uniformly randomly from the pair , for and for chosen uniformly randomly from the quadruplet or chosen uniformly randomly from the continuous interval (see Appendices). The last case of is the 50% dip generally accepted as the quantum-classical boundary, but our formula (3) shows the 50% dip to be a consequence of ignorance, or lack of control, of the phase.
If exceeding the 50% dip threshold does not signify quantumness, where does quantumness appear? Contraria sunt complementa, or opposites are complementary, as Niels Bohr’s coat of arms says: a particle is opposite to a wave. In the context of complementarity, particle-like behaviour is characterized by detection that reveals indivisibility of the object. Wave-like behaviour, on the other hand, is characterized by interference fringes that signify the object is delocalized and coherent across some region. For instance, single-photon complementarity is demonstrated by a particle-like test that shows that a photon passing through a beam splitter is indivisible. The wave-like property is demonstrated by allowing this indivisible entity to take two different paths and observing interference fringes as a function of path-length difference through a difference measurement of the photo detectors of the two output ports.
We extend the above argument to the TPCVD biphoton case in which we first have one photon incident on each input port of a beam splitter and then using the HOM dip as a signature of the particle-like behaviour, manifested as a biphoton in a superposition of two different paths (1). To realize experimentally the classical limit, we produce one microwave pulse incident at each input port of a power splitter with fixed phase difference between the inputs chosen randomly from a chosen distribution; thus, we can control whether the pulses constructively or destructively interfere at the output, leading, as we show, either to the biphoton appearing to be an indivisible particle through a cross-correlation measurement or appearing to be a wave by introducing a second beam splitter to show interference fringes due to the two biphoton paths interfering with themselves. Thus, the quantumness of the HOM dip is in complementarity of the biphoton, not the more than 50% dip.
We confirm this theory and these concepts with two experiments whose schematics are shown in Fig. 1 and Fig. 2. To fix the relative phase between the two input fields to a beam splitter, we employ sine-modulated Gaussian pulses which are 1.162 GHz microwave pulses injected into a 180 power splitter (should be a 50:50 beam splitter but measured to be 49.3:50.7) and show a near-100% dip despite the fields being effectively classical; in the wavelike experiment we add a second 180 power splitter (also 50.3:49.7) to investigate possible biphoton like interference. In our second experiment, we inject two IR quantum fields and show a near-100% dip at the beam-splitter output and then show a output for the Mach Zehnder Interferometer (MZI) output field, thus empirically confirming complementarity. The microwave case represents classical optics, and the source relative-phase choice can give either a near-100% dip or near-perfect interference but not both. On the other hand, the IR (quantum) case yields both near-perfect HOM dip and near-perfect interference depending on the measurement but not depending on the source setting, thereby demonstrating complementarity for the IR case and failing complementarity, but achieving a near-perfect HOM dip in the classical microwave case.
For the optical photon case, we employ a Type II heralded photon source  in the collinear configuration and the resultant degenerate twin photons are incident on a 50:50 polarizing beam splitter. One of the photons (the V-polarized one) is incident on a half-wave plate that rotates the polarization (to H), thus making the two photons identical in polarization. These two now indistinguishable photons are incident on a fibre beam splitter (with a verified splitting ratio of 52:48) and the two outputs are then coupled to two single-photon Avalanche Photo Detectors for coincidence measurement using a Universal Quantum Device (UQD) time tagging unit. Further details concerning choice of parameters for both experimental set-ups are in the Appendices.
Our results are presented as plots of the cross-correlation function as a function of time delay between input pulses/photons in the classical and quantum case, respectively. As shown in Figure 3, we obtain a TPCVD of 99.635 0.002% when the phase between the two input pulses is averaged over the set with the two values of phase difference being equally likely. A TPCVD of 48.03 3.50% is obtained when the phase between the two input pulses is averaged over the set with all four values of phase difference being equally likely. In the supplementary section, we show that we get a similar TPCVD (48.13 2.47%) even when we randomly chose the phase difference between the two pulses to be any phase value in the continuous interval .
Figure 4 shows IR coincidence counts as a function of time delay between the two input photons. In this case, we obtain a TPCVD of 96.06 0.16%. Note that, in the photon case, the phase is uncontrollable and unobservable. In the classical case, instrumental errors are negligible (relative error of the order of ), and, in the IR case, we have accounted for the systematic errors in the theory estimate graph. Error bars for both have been plotted from the 95 % confidence intervals for the mean data at each value of time delay between the pulses/photons, respectively (see supplementary material).
Classical pulses indeed demonstrate a high degree of anti-correlation indicated by this HOM dip-like behaviour by applying appropriate relative-phase control between the two inputs. As the question of classical-vs-quantum nature of the microwave field is established by trying to realize particle-wave complementarity, we recombine outputs from the power splitter on a second power splitter to complete the MZI to observe interference fringes. Whether particle-like or wave-like behaviour can be observed depends crucially on the relative-phase choice for the inputs. Figures 5 and 6 show the schematics for the classical pulse and IR versions of this experiment.
In the IR case, we observe both HOM dip and interference successively without any change required in the source setting which demonstrates that the same source manifests both particle-like and wave-like behaviour i.e., complementarity. As explained in supplementary material, the ratio between the Gaussian-envelope amplitude for the interference pattern and the zero-interference amplitude when one path is blocked is expected to be 0.25 in the IR case. Our experiment successfully yields this value, namely .
In contrast, for the classical microwave case, a given relative-phase source setting either results in a near-perfect HOM dip or near-perfect interference but not both. In the near-perfect HOM-dip case, the ratio between the Gaussian-envelope amplitude with both paths open vs one path blocked is half (as the intensity drops to half), which is what we measure experimentally (with negligible instrumental error). Our experiment yields the value to be . If the source is arranged to observe interference, we do not see more than a 50% HOM dip. Thus, we observe either particle-like or wave-like behaviour but never both with the same source settings which means we do not demonstrate complementarity in the classical case.
In summary, we have shown that the famous 50% Hong-Ou-Mandel dip threshold is not a quantum threshold but rather due to a lack of controllability of the relative phase of the two interferometer inputs, and we show that quantumness is manifested as particle-wave complementarity by showing that particle-like and wavelike features are observable without changing the source setting. We show that an IR source manifests complementarity whereas the classical microwave source, with exquisite relative-phase control, yields either particle-like or wave-like behaviour depending on the choice of relative phase of the two inputs. Furthermore we have explained how particle-like vs wave-like behaviour can be understood in terms of the biphoton apparently being indivisible in an anticorrelation experiment vs the biphoton taking a superposition of both paths when interference is measured. Our theory and experiment elucidates the quantumness of one of the most famous, ubiquitous and useful quantum optics experiments.
Appendix A Theory calculation details for the classical case
For localized pulse shape and for the detector turned on before the first pulse arrives and turned off well after the second pulse passes into the detector, we take the limits and . We care about two limits: zero time delay corresponding to simultaneously arrival of the two pulses
and long time delay between pulses , with the assumption that the pulse is negligible in the large- limit, thus proving Eq. (3).
Now we evaluate in Eq. (13) for and so that we can subsequently determine the visibility. If is fixed to any value , then so , with identically being unity due to normalization of each field forcing the correlation product always to be one. Now consider averaging over two phases and :
which is not unity. For the pair corresponding to , implying the visibility ; for the pair corresponding to , so . Averaging over the four phases and averaging uniformly over yields so , which is the famous 50% Hong-Ou-Mandel dip.
Appendix B Theory calculation details for the quantum case
The coincidence probability () at the output of a beam-splitter, as a function of the time delay () between two input photons, is given by the well known result by Hong-Ou-Mandel,
with and the transmission and reflection probabilities of the beamsplitter Here is the Fourier transform of , which is the normalized joint spectral amplitude (JSA) of the two input photons. Equation (6) is equivalent to
after accounting for instrumental errors, and is the probability that the input photons maintain indistinguishability. The quantity is the probability that both photons fall on the same input arm of the beam splitter.
Appendix C Experimental details for the photonic experiment
A diode laser (Toptica-iwave) at 405 nm with 50 mW power pumps a 5 mm5 mm10 mm Type II BBO crystal. A pair of lenses of focal lengths 22.5 cm and 25 cm are used to focus the pump beam at the crystal and to collimate the down-converted photons, respectively. Pairs of orthogonally polarized frequency degenerate photons with central wavelength of 810 nm are split in two directions by a polarizing beam splitter (PBS). A half-wave plate in one of the arms of the PBS is used to make the two photons have identical polarization. An arrangement of long pass filters allows wavelengths above 405 nm to be transmitted. A band-pass filter with a 3.1 nm bandwidth centred at 810 nm restricts the bandwidth of the transmitted photons also blocking any residual pump. Two fibre couplers collect photons from the same pair and inject them in a 22 polarization-maintaining fused fibre splitter (FBS).
One of the couplers is mounted on a motorized stage which translates along the direction of the beam to set a variable delay between the two photons. We measure the coincidence count rate as a function of relative beam displacement between the signal and idler photons. A 10 m step size for the stage provides necessary precision to resolve the HOM-dip width. The data acquisition time is 5 s at each position and we measure at around 160 positions, resulting in one complete experimental run requiring close to eight minutes. We repeat the measurement 100 times for better averaging and to estimate an error bar.
Illustration of complementarity requires injecting the superposition state (1), which is the output state of the HOM dip position, to a second beam splitter to complete the MZI. Thus, photons from the two output fibres of the FBS are incident on a 50:50 beam splitter through two collimators. One of the collimators is placed on a motorized translational stage which gives a variable path difference () in the interferometer. Photons from the two output terminals of the beam splitter are coupled to single-mode fibres that are connected to two single-photon counting modules (-SPAD from Pico Quant) for coincidence measurement using a Universal Quantum Device (UQD) time tagging unit.
We measure the coincidence counts as a function of path difference. As path difference approaches zero, we observe an interference zone which is due to second order correlation also known as fourth order interference. We compare the coincidence counts at the maximum of the interference pattern with the counts when one of the arms of the MZI is blocked. This ratio is expected to be 1/4 (details in Supplementary material).
We did not stablilize the phase of our interferometer as we are only interested in the above comparison so phase noise is present in the interferometer. Moreover the efficiency of the two collimators as well as the fibre coupling efficiency are not identical. Thus, the single-photon rates in the output arms of the interferometer are not identical, making the coincidence rate also a fluctuating quantity, which leads to a coincidence-count difference if the individual arms are blocked. We circumvent this problem by blocking the arms individually and then averaging of the two coincidence counts as a representative value for the blocked MZ case.
To determine the position of the motorized translational stage such that the path difference () in the interferometer is zero, we ensure non-zero time delay between the input photons to the FBS (non-HOM condition). MZI interference is then completely second-order, i.e., first-order coherence. The position of the motorized translational stage, at the point that interference in the singles count appears, determines the value equal to zero position. After finding the stage’s zero position, we go back to the HOM condition and vary by 0.5 mm on each side with a 0.01 mm step size.
We accumulate the coincidence counts for 5 sec at each position and repeat the profile measurement 50 times. The rapid fluctuation in the data points arises due to instability of our MZI. Phase instability shifts the position of the maximum of the interference pattern between runs. As we are interested in the ratio between the coincidence counts at the maximum of the interference pattern and the counts when one of the arms is blocked, we need to know the value of the coincidence counts at the maximum accurately. We shift the interference patterns such that all maxima coincide at one point and then take the average of all the counts at that position to determine the number of coincidences representing the maximum of the interference.
Now we discuss sources of systematic and random errors in our experimental setup. The major contribution to systematic error comes from the PBS. Ideally the PBS is supposed to transmit and reflect horizontally and vertically polarized photons, respectively, but, in practice, transmission for vertical polarization and reflection for horizontal polarization occurs with nonzero probability. To elucidate, let us say that P1 and P2 are the measured powers in the transmitted arm when H polarized and V polarized light are incident on a PBS, respectively. We assume that the input power is the same for both cases: according to this specification, the ratio of P1 to P2 is 1000:1. One of these systematic errors also arises due to the fact that the minimum resolvable angle for our half-wave plate rotational mount is 2. These two errors are accounted for in the theoretical estimation of the HOM correlation profile.
Systematic error due to mode mismatch also appears for the coupling of the photons in a single-mode fibre. Equality of fibre length is limited to a 0.1 m mismatch. A systematic error can also appear due to backlash by the motorized actuator, which is less than 15 m. The random-error source is mainly from pump-laser intensity fluctuations, which are quantified by statistical averaging.
We move our stage for a total traverse of 1 mm in 0.01 mm step size. Thus, there are 160 positions and, at each position, we accumulate the photons for 5 s. We use bootstrapping with the number of photons accumulated at each position as the initial population to decide the number of repetitions of the profile measurement. This leads to a choice of 100 repetitions to claim 95% confidence for the error in the mean to be within 2%.
Appendix D Experimental details for the classical experiment
The two power splitters for the experiment were broadband power splitters ( ET industries model: J-076-180) which work from 770 MHz to 6 GHz. Though the power splitters are from the same company and same part number, frequency response of both may not be the same due to variation in accuracy level at manufacturing end. Therefore, both power splitters are characterized and frequency of the experiment is concluded based on the ratio of the power splitting. From the S-parameter analysis, the operation frequency is chosen to be 1162 MHz, where the power splitting ratios of PS1 and PS2 are closest to each other as well as the ideal expectation of 50:50 ( 49.3:50.7 for PS1 and 50.3:49.7 for PS2 respectively) and phase error is around 2-3 (ideal phase error would have been 0).
An arbitrary waveform generator (Keysight Technologies-33622A) with dual-channel is programmed using the LabVIEW to generate a sine signal modulated with Gaussian amplitude. The two output channels of the waveform generator are set to frequency 1 KHz and amplitude 100 mV. These two channels are synced using the LabVIEW ARB:SYNC command.
At the receiver end, an oscilloscope (Agilent Technologies DSO60 14A) is used as a detector with frequency detection up to 100 MHz. As the desired operation frequency is at 1162 MHz, signals from AWG should be up converted at transmitter end and down converted at receiver end which requires a mixer and a local oscillator. The local oscillator is a signal generator (Keysight Analog Signal generator N5173B), which is tuned at 1161.999 MHz and is then connected to a one-input four-output power divider (Mini circuits part number: ZA4PD-2). Four mixers (Mini circuits part number: ZFM-2000, level 7 mixer), which operate in the range of 100-2000 MHz, are driven from four output ports of the power divider. The power level from local oscillator is set to +14 dBm to take care of power loss in the cable losses and loss in power divider.
At the transmitter end, the AWG dual-channel signal is connected to intermediate frequency ports of mixers 1 and 2 and the LO is driven by a signal generator. The resulting RF is connected to 0 and 180 ports of PS1 and the output from the and ports of PS1 is given to RF port of mixers 3 and 4. The resulting IF is connected to oscilloscope channel 1 and channel 2. The output shown on the oscilloscope screen is saved as a MATLAB® data file onto a PC.
The random phase choice between different phase values as required to generate Figure 3 is coded and given as the instruction to one of the AWG channels without disturbing the other channel. At the same time the signal from one of the channels is distinguished from another from another in milliseconds time scale, where represents 1 ms. The maximum distinguishable case observed is at time delay of and . Cross-correlation between the data recorded from two channels of oscilloscope is then calculated in the post-processing stage which leads to the graphs for different phase choices. The choice of statistics is explained in the supplementary material.
In order to perform the complementarity experiment, we need to complete the MZI after PS1 with another power splitter PS2. The time delay between the two input pulses to PS1 is kept “zero” i.e. . The two outputs of PS1 are given as inputs to 0 and 180 ports of PS2, respectively. Now the output signal from the two output ports of PS2 are given to Radio Frequency ports of mixers M3 and M4, respectively. The output signal from Intermediate Frequency (IF) port of mixers M3 and M4 is given to channels 1 and 2 of a Digital Storage Oscilloscope (DSO). Output data from the DSO are recorded with the help of a LabVIEW interface to be saved for post-processing. Cross-correlation between the two output signals is calculated.
Next we block one arms of the MZI. The connection between the port of PS1 and 180 port of PS2 is removed, and both ends are terminated with 50 terminators. Subsequently, the output signal after down-conversion process is connected to the two channels of the DSO. The data of output signals from DSO are recorded with the help of LabVIEW interface to be saved for post-processing. The ratio between the cross correlation values (MZI with both arms open and one arm blocked respectively) is computed and found to be 0.4919 0.0242.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
BCS and US conceived the experiment; SS, DG, KJ, ANL and US performed the experiments; SS and KJ performed the theory calculations under BCS’s supervision; US devised and supervised the experiments and analysis. All authors contributed to writing of the manuscript.
We thank Sai Dheeraj Nadella, Gareeyasee Saha and Prathwiraj Umesh for their initial technical assistance.
-  C. K. Hong, Z. Y. Ou, and L. Mandel. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett., 59:2044–2046, Nov 1987.
-  H. Fearn and R. Loudon. Quantum theory of the lossless beam splitter. Optics Communications, 64(6):485 – 490, 1987.
-  P. Hariharan and B.C. Sanders. II Quantum phenomena in optical interferometry. volume 36 of Progress in Optics, pages 49 – 128. Elsevier, 1996.
-  N. C. KatiÃºscia, L. Kaisa, and C. Silberhorn. Accessing the purity of a single photon by the width of the hongâouâmandel interference. New Journal of Physics, 12(11):113052, 2010.
-  Z. Y. Ou. Multi-photon interference and temporal distinguishability of photons. Int. J. Mod. Phys. B, 21(30):5033–5058, Dec 2007.
-  E. Knill, R. Laflamme, and G. J. Milburn. A scheme for efficient quantum computation with linear optics. Nature, 409:46–52, Jan 2001.
-  S. Aaronson and A. Arkhipov. The Computational Complexity of Linear Optics. Theory Comput., 9(4):143–252, 2013.
-  J. P. Dowling. Quantum optical metrology—the lowdown on high-N00N states. Cont. Phys., 49(2):125–143, 2008.
-  P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao. Observation of a “quantum eraser”: A revival of coherence in a two-photon interference experiment. Phys. Rev. A, 45:7729–7739, Jun 1992.
-  I. Dhand, A. Khalid, H. Lu, and B. C. Sanders. Accurate and precise characterization of linear optical interferometers. Journal of Optics, 18(3):035204, 2016.
-  R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel. Interference of two photons in parametric down conversion. Phys. Rev. A, 34:3962–3968, Nov 1986.
-  Z. Y. Ou. Investigation of Nonclassical and Nonlocal Effects in Fourth-Order Interference. PhD thesis, University of Rochester, Rochester NY, 1990.
-  J G Rarity, P R Tapster, and R Loudon. Non-classical interference between independent sources. Journal of Optics B: Quantum and Semiclassical Optics, 7(7):S171, 2005.
-  Y. S. Kim, O. Slattery, P. S. Kuo, and X. Tang. Conditions for two-photon interference with coherent pulses. Phys. Rev. A, 87:063843, Jun 2013.
-  H. J. Carmichael. Private Communication.
-  P. Grangier, G. Roger, and A. Aspect. Experimental evidence for a photon anticorrelation effect on a beam splitter: A new light on single-photon interferences. Europhys. Lett., 1(4):173–179, 1986.
-  L Mandel, E C G Sudarshan, and E Wolf. Theory of photoelectric detection of light fluctuations. Proceedings of the Physical Society, 84(3):435, 1964.
-  L. Mandel and E. Wolf. Optical Coherence and quantum optics. Cambridge University Press, first edition, 1995.
-  L Mandel. Fluctuations of photon beams and their correlations. Proceedings of the Physical Society, 72(6):1037, 1958.
-  M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko. Theory of two-photon entanglement in type-ii optical parametric down-conversion. Phys. Rev. A, 50:5122–5133, Dec 1994.
-  F. M. Dekking, C. Kraaikamp, H. P. Lopuhaä, and L. E. Meester. A Modern Introduction to Probability and Statistics: Understanding why and how. Springer Science & Business Media, 2005.
-  D. M Pozar. Microwave engineering. John Wiley & Sons, 2009.
Appendix E HOM effect using semi-classical theory
e.1 Probability of coincidence using semi-classical theory
In classical electromagnetism, light is represented by classical fields. Consider two pulses of linearly polarized light, traveling in perpendicular directions, i.e.,
where, is the electric field of the pulse. In general the field is vectorial, but if both signals have the same polarization, the vector notation can be dropped. is the amplitude modulation of the signals (such as Gaussian), is the propagation constant, is the frequency, is the relative phase between the two signals and is the time-delay between the two pulses. The perpendicularity of the travelling directions of the two pulses necessarily and sufficiently implies that . Furthermore, if the setup is such that the path-lengths of both beams are equal, then we can ignore the dependence as well. With a simplified notation, if the two perpendicular pulses are input to a balanced beam-splitter, the outputs will be
We now introduce fluctuation in the signals by making the phase a random variable with a fixed probability distribution. The reason for doing this is the following. The interference of light in classical description is, in quantum mechanics, explained as a photon interfering with itself and not with another photon (second-order interference). Hong-Ou-Mandel effect, on the other hand, is described as occurring because of two-photon interactions (fourth-order interference), which has no counterpart in the classical description. When we randomize the relative-phase over a distribution such that
and hence there is no second-order interference between and .
In the semi-classical theory of photodetection[17, 19], the probability of coincidence at the two detectors is proportional to the cross-correlation of the integrated intensities at the detectors. The correlation function here is defined by
where the phase has been averaged over the probability distribution . is the time interval over which each signal is measured. Ideally, should be infinite, but in practice, it is several times the width of the pulse. Using Eqs. (8), (9) and (10), with the simplified notation we get,
In the special case in which the amplitude profiles of both the input signals are identical, the first two terms of (14) are equal, if is large compared to the width of the input pulses. The third term, which captures the overlap of the two pulses (in time) is a function of the time-delay and the phase.
The visibility of the plot of is then
e.2 Signal processing in the experiment
In the experiment, sinusoidally varying voltages play the role of the EM radiation. and are variable parameters. For each value of an ensemble of output signals is generated with values of chosen from a chosen probability distribution. The ensemble is used to calculate cross-correlation as defined in Eq. (13), but with normalization,
Normalization makes the cross-correlation independent of the amplitudes of the signals and sensitive only to their ratio. Moreover, due to finite acquisition time, owing to the fixed window over which the data is recorded by the oscilloscope, the output pulses are truncated. The normalization takes into account the effect of this truncation. The normalized cross-correlation is expressed as
Mathematical expressions of the signals generated using an Arbitrary Waveform Generator (AWG) are
where are the peaks of the Gaussian envelopes of the two signals. Both signals have Gaussian envelopes with , and the sinusoidal wave has a frequency ().
The frequency range in which the power-splitter has a splitting ratio of 50:50 is well beyond the maximum frequency that the AWG can produce. So, the effective input is generated by up-converting the frequency of the input signal by using a frequency mixer. The frequency of the local oscillator (LO) signal used for the mixing is MHz. A 4-port power-divider is used to branch the LO signal into 4 channels, two of which are used to up-convert the input signals, and the other two are used to down-convert the output of the power splitter. After the power-splitter splits the inputs, the outputs need to be down-converted for measurement.
The measurement device used is an oscilloscope, two of whose channels are used to measure the two outputs of the power-splitter. As the oscilloscope has an upper limit to the frequencies that it can measure, the high frequency output of the power-splitter is down-converted using the LO. The result of down-conversion is a mixture of low and high frequencies. The oscilloscope acts as a low-pass filter and measures only the low frequency components, making the effective output
e.3 Data acquisition and post-processing
The digital oscilloscope samples the voltage signal at a sampling rate of two Giga-samples per second and consequently the recorded signal is a time-series of voltages and . Each time-series contains a thousand points which implies that the time interval between successive points is s and the total acquisition time is s.
To calculate we take the ensemble average over the fluctuating . We focus our attention to fluctuation that is ergodic, described by a time-independent probability distribution . The fluctuation can then be simulated by a discrete random process in which a pair of signal is generated with phase-difference , chosen from the said distribution. If we generate such samples then for large the average is approximated by the sample means, i.e.,
where we calculate the integral using the trapezoidal rule,
where this integration is carried out for each value of in the sample.
e.4 Minimum for good statistics
The relative phase () of the input signals is averaged over so that there is no second-order interference as we are interested in studying only the fourth-order interference between the inputs. Consequently, the average integrated intensities of the output in both arms of the beam-splitter (the normalization factor for the cross-correlation) are independent of . Therefore, to ensure absence of the second-order interference, we determine the minimum number of samples (of ) required to estimate the constant average-integrated-intensity within 5% error with a confidence level of 95% using the two-tailed test.
be the true average-integrated-intensity of the output signals and
be the average calculated from the sample outputs, where is the number of samples recorded. Let
where is the true standard deviation of the integrated intensities for each . From the central limit theorem we know that for large enough , the distribution of will be closely approximated by the Normal distribution. As the 95% confidence interval for a normal distribution is , the 95% confidence interval (C.I.) for is
If we want the error in the estimation of to be with 95% confidence, the width of the C.I. must be 10% of ; i.e.,
Equation (29) is then solved to get the minimum number of samples required,
In the 100% TPCVD case, the phase between the input signals is chosen uniformly from the set , so the probability distribution of is
Figure 7 shows as a function of .
The 50% TPCVD experiment is done in two versions. First the phase is chosen uniformly from the continuous interval for which
In the other version, the phase is chosen uniformly from the set with probability distribution
In either case, the means and standard deviations for all values of are identical and hence is given by the graph in Fig. 8.
e.5 Sources of error
The dominant source of error was the amplitude mismatch of the outputs, which is a result of the each signal undergoing unequal attenuation as they pass through different paths of the circuit. The remaining sources of error have negligible contribution, however, for completeness, they must be discussed.
Every instrument used in the circuit has an uncertainty in the measurement of the respective quantity. From the data-sheets of each instrument provided by the manufacturer the instrumental errors in frequency, time-delay, voltage, frequency of the local oscillator and amplitude of the local oscillator were used to find the maximum error due to instrumentation. The error in the beam-splitting ratio of the power-splitters was found from their characterization. The relative error is plotted in figure 9.
Other sources of errors include finite sampling rates of the AWG and the oscilloscope. However the rates are sufficiently high for near perfect sampling of the signals.
e.6 The cross-correlation graph for 50% with continuous variation of phase
The main text presents the cross-correlation graph for the case in which the relative phase between the input signals is varied over the discrete set . The same 50% dip in visibility is also seen when the phase is varied continuously over the interval as shown in figure 10.
Appendix F Classical description fails complementarity
A distinction can be made between the classical and the quantum description with a test for complementarity. For this a second beam-splitter can be added to the setup to make a Mach-Zehnder interferometer with a phase-shifter (creating a phase-shift of ) in one of the arms.
In the biphoton case, the input state to the second beam-splitter becomes The output of the second beam-splitter is then and therefore, an interference pattern is observed as the phase-shifter is rotated. For the trivial case of (no phase-shift), the output is as expected.
On the other hand, in the case with classical pulses, the phase-shifter doesn’t give rise to such an interference at the output, as the output of the first beam-splitter always travels entirely through one of the arms of the MZI. The outputs fields of the second beam-splitter are
where is the phase introduced in one arm, and therefore
Note that when one of is always zero and consequently, the terms involving always vanish, making independent of the phase that is introduced by the phase-shifter. In terms of coincidence probability, the phase-shifter affects the the coincidence rate in the quantum case but leaves it untouched in the classical case.
Alternatively, if one of the arms is blocked, the coincidence count drops. Say is blocked, then a projection operator on the state yields
Further, applying the second beam-splitter on this state makes the output
This means that the coincidence probability drops from one down to a quarter. This drop in probability is a combined effect of the loss of half of the biphotons due to blocking and the absence of interference between the two arms.
In the classical case the drop in probability of coincidence is different. If the arm with field is blocked
On the other hand, if is blocked,
In either case, the un-normalized cross-correlation function is
and . Specifically, compared to when none of the arms is blocked. However, in either case and hence perfectly correlated, which implies that classical theory predicts that there will never be a click in only one detector. Therefore the classical description predicts that the coincidence probability drops down by a factor of 2 which is different from the prediction of the quantum theory according to which this factor is 4. This result is equivalent to the complementarity test, and separates the classical theory from the quantum one.
Appendix G Quantum description of the HOM effect
Using the quantum description, the coincidence probability at the output of a (lossless) beam-splitter is given by the well known result by Hong-Ou-Mandel,
Where, is the fourier transform of ; is the normalized joint spectral amplitude (JSA) of the two input photons, being the difference in the frequency components of the two photons (). is the time delay between the two input photons to the beam-splitter. If the two photons are completely indistinguishable, then a balanced beam-splitter causes the coincidence probability to drop to zero, i.e, . for , where is the coherence time of the single photons.
In practice, however, different components may introduce some distinguishability in different degrees of freedom (frequency, polarization, spatial mode, etc.) of the two input photons. Let be the probability that the input photons retain indistinguishability, then the coincidence probability has correction terms, i.e.,
The second term in the equation (43) is the contribution from two perfectly distinguishable photons. Another case may appear when both photons fall on the same input arm of the beam splitter. If we assume that is the probability that this case might happen, can be written as:
g.1 Theoretical estimate for HOM profile
We have plotted coincidence counts vs time delay, taking into account various instrumental errors (Eq. (45)). The dominant contributions to these errors are from the non-ideal extinction ratio of the polarizing beam-splitter (PBS), imperfect rotation of the half-waveplates and the effect of the interference filter. Ideal PBS transmits only horizontally () polarized light and reflects vertically () polarized light. But in practice this may not happen owing to the imperfect extinction ratios. Similarly, errors in the half-waveplate rotation may introduce distinguishability in polarization. The transmittance of the band-pass filter affects the JSA of the input photons. These error parameters were measured based on the datasheets of the respective instruments. The PBS has extinction ratios 1000:1 in transmission arm and, 52:1 in the reflection arm. So, and , where represents transmission and reflection probabilities of polarized light through the PBS. represent transmission and reflection probabilities of polarized light. Probability of both and photons being in any one of the output arm of the PBS ( in Eq. (45)) becomes: . The transmission and reflection coefficients of the fiber-beam-splitter (FBS) are . The Half-waveplate (HWP) rotation should be such that the angle () of the HWP axis w.r.t. the horizontal axis is ; in order to make both the photons possess the same polarization. But the HWP rotation has an error of , leading to . The transmission-vs-frequency data for the filter is provided by the manufacturer. With the filter, the joint-spectral-amplitude of the two input photons becomes , where is the joint-spectral-amplitude of the photons without the filter. We have assumed to be a Gaussian distribution with standard deviation .
In order to compare with the experimental data, we have multiplied a scaling factor () to the coincidence probability [coincidence counts=]. We put values to all relevant parameters, i.e. and fit the function with the experimental data, while taking only as fit parameters.
g.2 Finding a fit to the experimental result
Although the major systematic errors have been taken into account above, there are additional errors that are untraceable, like dispersion and rotation of polarization as the photon passes through different fibres and components, spatial mode mismatch in the two fiber inputs of the FBS, etc. Consequently, the experimental result has a slight deviation from the theoretical estimate. To get a better fit to the data, we use the expression (see Eq. (45)) to find a fit with , , , as fit-parameters.
g.3 Confidence intervals for photon counts using bootstrap
The result of the experiment is a graph of coincidence-counts vs actuator-position (which is proportional to the time-delay between the input photons). 100 iterations of the experiment were done, which resulted in a sample of 100 values of the coincidence counts for each actuator-position. Consider the sample of coincidence counts at a particular actuator position, say, . The estimate of the counts is then the mean over which is the sample mean, i.e., . The confidence interval for was then found by employing a statistical bootstrap. The method involves creating a large number of sets by resampling-with-replacement from the original set. The size of the resampled set is the same as that of the original. Let one such resampled set be and its mean . We define a quantity
Repeating the above step a large number of times, in our case 10,000, we get a sample for . To find a 95 percentile confidence interval, we pick the and the percentile of this sample, and . The confidence interval for is then simply, . We then repeat the bootstrap method for all values of the actuator-positions. These confidence intervals were plotted as the error bars for the mean value at each actuator position.