1. Introduction

In a duality experiment, the photon’s wavelike behavior is responsible for the interference fringe, and its particlelike behavior is represented by the photons’ trajectory. Although it is almost impossible to exactly predict what trajectory a particular photon took in a duality experiment, the corresponding probability is usually obtainable. Historically, many schemes have been proposed to simultaneously record the photons’ trajectory and the interference pattern, e.g. Feynman’s light microscope [1], Scully’s quantum eraser [2], and Englert’s interferometer [3], and so on. However, most of them are very hard to realize in experiment due to the limitation of current technology. The widely-used way to obtain the photons’ trajectory information is to measure the probability of detecting a photon in every path [4, 5], and the detection destroys the interference pattern, in accord with the mutual exclusiveness between the wavelike and particlelike properties of a photon [6]. In other duality experiments with electron [7] and atom [8, 9], the particle information is obtained in a similar way.

Recently, a new method is introduced to reveal the trajectory of the photons in an interferometer, where some mirrors are placed in the paths of the photons, and vibrate at different frequencies [10]. In order not to disturb the interference obviously, the vibration amplitude of the mirrors is set very small so that the change in the optical length due to the mirrors’ rotation is much smaller than the wavelength. At one output port of the interferometer, the photons are detected by a quad-cell photodetector, and a Fourier analysis is made on the recorded data. It is demonstrated that every vibrating mirror, so long as it ever interacted with the photons, contributes a peak in the frequency spectrum of the output beam. As is well known that Fourier transformation is widely used in many subjects [11], such as signal processing, analysis of partial differential equations, and so on. Even at present, it still be a very important and basic tool in many theoretical and experimental works [12, 13]. It is a novel ideal to use the Fourier method for revealing the trajectory of photons. At the same time, the interference in the system is only affected slightly, due to the substantially small vibration amplitude. So it provides us another potential way to simultaneously measure the wave and particle information in a duality experiment, and some more details will be introduced in the next section.

2. Fourier analysis based on differential detection

This interesting method for tracing the trajectory of a photon [10] can be illustrated through the a simple example. In a Mach-Zehnder interferometer, see Fig. 1, two mirrors A and B, placed in the upper and lower arms, vibrate periodically around their horizontal axes at the frequency f_A and f_B, respectively. Here we assume that the reflectivity of the both beam splitters, BS1 and BS2, is 0.5. A Gaussian light beam with width Δ, described by $\mathrm{\Psi }(x,y)=𝒜e^{-\frac{x^2+y^2}{2{\mathrm{\Delta }}^2}}$ in the cross section, is injected into the interferometer, which is split equally by BS1, and then deflected in the y-direction by the two mirrors A and B due to their vibration. A quad-cell photodetector, D, placed in the output port of the interferometer, is used to measure the difference between the optical intensities detected in the upper and lower cells. After leaving the interferometer, the output beam in the horizontal direction can be described by,

 

Fig. 1 A Mach-Zehnder interferometer. Two mirrors A and B vibrate at the frequency f_A = 282Hz and f_B = 296Hz, and the two beam splitters, BS1 and BS2, equally split the input light.

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Here “+” (“−”) in Eq. (1) corresponds to the constructive (destructive) interference in the output beam D. δ_A and δ_B stand for the y-direction shift of the light beam, due to the vibration of the mirrors A and B, respectively.

The intensity difference of the optical fields measured in the upper and lower cells is calculated by,

 

Fig. 2 Fourier spectrum of the output beam based on the intensity difference (3). (a) In the case of constructive interference, two peaks f_A and f_B appear in the Fourier spectrum as the signature of the two vibrating mirrors A and B in the interferometer; (b) In the case of destructive interference, some “noisy” peaks appear in the Fourier spectrum, besides the two peaks f_A and f_B.

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However, we point out here that the peaks in the frequency spectrum do not always have a one-to-one correspondence with the vibrating mirrors based on this differential detection technique. If we slightly modify the experiment by setting the output beam D from constructive interference (“+” in Eq. (1)) to destructive interference (“−” in Eq. (1)), and remaining other parameters unchanged, four peaks will appear in the Fourier spectrum in the range [260Hz,340Hz] with almost equal height, see Fig. 2(b). In this case, although a destructive interference is set for the output beam D, the vibrating mirrors destroy the complete destructive interference, and results in many leaky photons. Among the four peaks, only the two middle ones are directly associated with the vibrating mirrors A and B. The other two “noisy” peaks, f₁ and f₂, origin from the combined contribution of the two mirrors in the Fourier analysis, which is negligibly small when the output beam D is far from destructive interference. In fact, the two “noisy” peaks are linearly dependent on the frequencies of the two mirrors in a simple way, i.e. f₁ = 2f_A − f_B and f₂ = 2f_B − f_A. Please note that we here only plot part of the Fourier spectrum in the frequency ranging from 260Hz to 340Hz. In fact, some other peaks exist outside this range, and they are also linearly dependent on f_A and f_B. In addition, this differential detection technique for revealing the trajectory of a photon, based on the Fourier analysis on the intensity difference in Eq. (3), is valid only in the case that the vibration amplitude of the mirrors is substantially small, so that there is an interference between the zeroth order and the first order terms in the signal [14], and higher order small terms are negligible. Based on this technique, our simulation shows that the larger vibration amplitude, the more “noisy” peaks in the Fourier spectrum, and the smaller signal-noise ratio.

3. Fourier analysis based on CLSID

The quantify in Eq. (3), i.e. the intensity difference between the photons detected in the upper and lower cell of the quad-cell photodetector is not the only one we can use in the Fourier analysis for revealing the trajectory of photons. Since the vibrating mirrors deflect the light beam in the y-direction, the intensity distribution of the output beam should also carry the vibration information of the mirrors, and the photons’ trajectory can thus be deduced. Based on this idea, we use the centroid location of the symmetric intensity distribution (CLSID) of the output beam to represent its spatial coordinate, and record the time evolution of the CLSID. Since it is very hard to define the centroid of an irregular pattern, we only concern the symmetric intensity distribution of the output beam. It will be shown that every vibrating mirror contributes a peak in the Fourier spectrum of CLSID, so long as it ever interacted with the detected photons.

We briefly introduce a possible way to collect the CLSID data in experiment. Firstly, we use a charge-coupled device (CCD) to record real-time intensity distribution of the output beam, i.e. |Ψ(y, t)|² (The integral over x-axis contributes a constant to this intensity distribution), which is supposed to be with high precision and resolution. For a particular intensity distribution at time t_j, we secondly find out its maximum intensity points, |Ψ(y₁, t_j)|² = |Ψ(y₂, t_j)|² = ··· = |Ψ(y_N, t_j)|². When an odd number of maximum intensity points are found out, the location of the center one is the potential centroid of the output beam. On the other hand, if an even number of maximum intensity points are found out, the middle of the two center ones is the potential centroid of the output beam. In the following, we check whether the potential centroid is a true CLSID or not. The judgement criterion is just the symmetry condition of an intensity distribution. For the intensity distribution |Ψ(y, t_j)|² at time t_j, if its potential centroid y_j satisfies the condition, |Ψ(y_j + d)|² = |Ψ(y_j − d)|², for any distance d, it is a true CLSID. Otherwise, it is a fake CLSID, and should be discarded. Finally, we collect all the true CLSID data (y_j, t_j), and substitute them into the discrete Fourier transformation,

In the experiment described in the above section, since the output beam expressed in Eq. (1) is composed of two Gaussian light beams with equal width, its intensity always has a symmetric distribution in the y-direction, whose centroid is located at the middle of the two Gaussian lights,

 

Fig. 3 Fourier spectrum of the output beam based on CLSID. The peaks in the Fourier spectrum have a one-to-one correspondence with the vibrating mirrors, no matter which type of interference, constructive or destructive, is set for the output beam.

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In the following, we use a more complicated example to verify the validity of this CLSID technique. In a modified Mach-Zehnder interferometer system, a small Mach-Zehnder interferometer is nested in a larger Mach-Zehnder interferometer in its upper arm, see Fig. 4. The four beam splitters, BS1, BS2, BS3 and BS4, with reflectivity $\frac{1}{3}$, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{2}{3}$, respectively, are suitably arranged so that the nested Mach-Zehnder interferometer leads to a destructive interference in the beam F. Five mirrors, denoted as A, B, C, E and F, are placed in the paths of the photons inside the interferometer, and vibrate at the frequency f_A = 282Hz, f_B = 296Hz, f_C = 309Hz, f_E = 318Hz, and f_F = 337Hz, respectively, around their horizontal axes with a small amplitude δ. In this special case, the output beam D can be described as,

 

Fig. 4 A modified Mach-Zehnder interferometer. A small Mach-Zehnder interferometer is nested in the upper arm of the larger one, and a destructive interference is set for the beam F initially. Five mirrors slightly vibrate in the experiment at the frequency f_A = 282Hz, f_B = 296Hz, f_C = 309Hz, f_E = 318Hz and f_F = 337Hz, respectively.

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Based on Eq. (6), the output beam D has an asymmetric intensity distribution at most time. The symmetric intensity distribution appears when anyone of the following conditions is satisfied, (i) δ_A = δ_B; (ii) δ_C = δ_B + δ_E + δ_F ; (iii) δ_A + δ_C = 2δ_B + δ_E + δ_F. As a nonlinear function of time t, each one of the three equations has a series of nonuniform roots. In our simulation, we randomly choose N = 3k points among about 108k effective CLSID within 60 seconds. The Fourier spectrum based on these nonuniform CLSID data [15, 16] is plotted in Fig. 5. Here we see that all five frequencies appear in the Fourier spectrum of the CLSID, accompanied by a few “noisy” peaks. Each “noisy” peak in the spectrum in Fig. 5 can be expressed as a linear combination of the five frequencies in a simple way, e.g. f₁ = f_B + f_F − f_C and f₂ = f_B + f_E − f_A. To verify whether a peak is a true signature of a particular mirror, a two-step test can be applied, (i) checking whether the peak in the spectrum is located at the same frequency as the mirrors vibration frequency; (ii) modifying the vibration frequency of this mirror, and checking whether the corresponding peak moves to the new location, and whether the other peaks, one-to-one corresponding to the other mirrors without any modification, keep their locations.

 

Fig. 5 Fourier spectrum of the output beam in the experiment with nested interferometer based on CLSID. Every mirror in the interferometer contributes a peak in the Fourier spectrum, see the five red peaks, denoted as f_A, f_B, f_C, f_E and f_F. In addition, many “noisy” peaks appear in the spectrum as a result of the combined contribution of two or more vibrating mirrors, see the lower blue peaks, e.g. f₁ and f₂.

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Compared with the quantity in Eq. (3), the CLSID of output beam is more difficult to measure, where strict spatial resolution, fast response and high precision is required for the CCD. However, the method of CLSID for deducing the trajectory of photons is valid in a more general situation. For example, it is independent on the vibration amplitudes of the mirrors, and still works when the vibration amplitude of the mirrors is not very small, which is not allowed in the Fourier analysis based on the quantity (3).

4. Conclusion

To summarize, the vibration of the mirrors inside an interferometer causes the motion of output beam’s spatial intensity distribution, thus the trajectory of photons can be revealed from the Fourier spectrum of the output beam. Since no photon is blocked inside the interferometer or branched out for other uses, this technique can be potentially used in a duality experiment to measure the wave and particle information simultaneously. Different from the differential detection technique based on the intensity difference in Eq. (3), here we focus on the centroid location of the symmetric intensity distribution (CLSID) of the output beam, and deduce the photons’ trajectory based on its Fourier analysis. It has been shown that the Fourier spectrum based on CLSID provides us reliable trajectory information of photons, and each vibrating mirror in the interferometer, if only it ever interacted with the photons in the output beam, contributes a peak in the spectrum. Furthermore, the Fourier analysis based on CLSID is not limited by the vibration amplitude of the mirrors, and can be used widely.