Xing-Chu Zhang(張興初) and Wei-Long She(佘衛(wèi)龍)

1Department of Physics and Information Engineering,Guangdong University of Education,Guangzhou 510303,China

2School of Physics,Sun Yat-Sen University,Guangzhou 510275,China

3Sino-French Institute of Nuclear Engineering and Technology,Sun Yat-Sen University,Zhuhai 519082,China

Keywords: classic plane light wave,discrete wavelet structure,discrete energy

1. Introduction

One of the apparitions in physics is the wave–particle duality of particles.[1,2]The property of light quantum (photon)is a typical example.[1–4]Although there have been Dirac,[5]Gupta,[6]and Fermi’s[7]approaches towards quantizing the radiation field and various models presented for photon,[8–13]it is still a challenge to understand the property of photon thoroughly due to its multi-faceted and elusive nature.[14,15]The difficulty comes mainly from the depiction of its wave feature. Unlike an electron, whose wave state can be described by a coordinates function (the probability amplitude of spatial localization), the single photon with energy hω has no analogous probability amplitude available[16]despite a light wave involving the photon. It is known that the interaction between a light wave and a matter will result in discrete energy exchange in the basic unit of hω. However, it is still a question on what change will happen about the wave state of light due to the absorption of photons. The following thought experiment may be enlightened for giving an insight into the property of photon(s). In the thought experiment, a coherent plane light wave with frequency ω is incident onto an absolute blackbody, and then the state change of the light wave is monitored during the process of light absorption. The question then arises: what happens to the plane light wave when the photons are absorbed by the absolute blackbody? Does the amplitude of the whole plane wave change constantly,or does the wave disappear segment by segment? In our experiment,a chopped beam of He–Ne laser is incident onto a photomultiplier tube in a black cavity,which shows that the absorption of photons in the front of the light wave is impossible to affect the amplitude of the succeeding part of the light wave.This suggests that the plane light wave would disappear segment by segment during the process of light absorption.And it also indicates that,by the time inversion,the plane light wave could be reconstructed by the translation and superposition of a segment by segment wave packet or wave train.This leads to another question: what method can realize the reconstruction?The wavelet transform[17]should be a good candidate. In fact,the wavelet transform has been adopted to investigate physical problems for many years.[17–19]And Morlet wavelet function has been widely used owing to its advantage in simplicity and time-frequency analysis.[20]It has been found that the wavelet is more suitable for present physics. In this paper,the wavelet transform is used to decompose the classical linearly polarized plane light wave into a series of discrete Morlet wavelets (or basic wavelets). And we find that the energy of the light wave can be related to its discrete wavelet structure and can be discrete as well. Finally,the random light wave packets are used to simulate the Mach–Zehnder interference of single photons,showing the wave-particle duality of light.

2. Discrete wavelet structure and discrete energy of classical plane light waves

Here, we start to investigate the discrete wavelet structure and discrete energy of classical plane light waves. We first consider the radiation field in free space. As well known,the classical radiation fields in free space are governed by Maxwell’s equations. The light wave equations for the electric field can be derived from Maxwell’s equations as follows:

where K is the central wave number of the Morlet wavelet,and s is a parameter.For the function Ek(z)with k=K,the wavelet transform can be expressed as[17]

We notice that the inverse wavelet transform[17]can be performed only by translation of wavelets, i.e., with a=1, the function Ek(z)can be reconstructed as

where n is the discrete wavelet structure parameter. Equation(7)shows the discrete wavelet structure of the plane light wave, which is the translation and superposition of infinite wavelets. In order to see clearly the way to reconstruct the plane light wave,we consider the cases of a finite n. Figure 1 shows the results of wavelet reconstruction,where Figs.1(a)–1(d) show the cases for n=0, 1, 2, and 6, respectively. The dotted line in the figure is the waveform of the plane light wave,while the solid lines denote the reconstructed light wave packets or wave trains with discrete wavelet structure.One can see that there is not a complete period for the case of n<2.But when n=2, there are two complete periods. When n increases by 1,two complete periods will be added.When n=6,there are 10 complete periods. The larger n is,the more complete periods appear. When n →∞, the whole plane wave is reconstructed finally. These results will be used below. It is easy to check that,if n is finite in Eq.(7),by making the transformation z →z?ct,then

Fig.1. Morlet wavelet reconstruction of the classical plane wave with Ek0 =1 V/m. The dotted line in the figure is the waveform of the plane light wave. Solid lines are the reconstructed light wave packets or wave trains with discrete wavelet structure. (a)–(d)show the cases for n=0,1,2,and 6 in Eq.(7),respectively.

is also the solution to Eqs. (1) and (2), representing a plane light wave packet or wave train with λ =2π/k (the idler frequency is kc=ω). Figure 1 can be taken as the waveforms of light wave packets or wave trains reconstructed by wavelets at t=0.

By far, we have obtained the discrete wavelet structure of plane light waves. Now we study the discrete energy of these light waves associated with their discrete wavelet structure.

The result is valid for any value of m. In other words,the energy of a segment of an m-period light wave in an infinitely long one is proportional to the number of periods. This result is consistent with the energy division of Planck radiation theory.[23]

i.e.,

A similar discussion can be made on a general cross-section for a plane light wave packet or wave train. With n →∞, the result turns to that for the infinite plane wave of Eq.(7).

Let us continue our discussion on the light wave described by Eq.(8). For a general value of n,if we regard it as a state of an electromagnetic wave mode(degree of freedom)with idler frequency ω =kc rather than simply take it as a wave packet or wave train not indispensable,then Ek0in Eq.(8)could not be zero. Since it is well known that electromagnetic vibration in free space does not depend on a medium. And the electromagnetic wave modes only depend on the electromagnetic vibrations. If we assume that each admissive electromagnetic wave mode is a physical existence and always nonempty,then the electromagnetic vibration for its mode should always exist. If Ek0=0, the mode with it will disappear. Therefore,Ek0would have a minimum value of nonzero, Ek0min, which cannot be further divided. In other words, pkin Eq.(15)has a minimum value of nonzero,denoted by p0k,being indivisible either,which suggests that a general pkwould be composed of some p0k. For the case of pk=p0k,Eq.(15)reads

Therefore,the plane light wave packet or wave train of Eq.(8)with Ek0minis the basic one. It can be seen that,for such kind of basic light wave packet with n=1, the minimum changeable energy is nearly a portion of p0kω in an absorption process. The remaining part of 0.59p0kω could not be further absorbed for ensuring the existence of its electromagnetic wave vibration mode. Therefore,for a basic plane light wave packet or wave train described by Eq.(8),practically,its changeable energy can only take the form of H0k=np0kω(n=1,2,3,...).Now there is a question:how to determine the minimum value of p0k?The answer is experiment,for example,the experiment of photoelectric effect.[3,24]And the value of p0kis expected to be h/2π. One would also ask whether the wave packets of Eq. (8) can show the wavelength λ (=2π/kλ) in an experiment? We will investigate this by the simulation on Mach–Zehnder interference(MZI)of single photons.[4]

3. Simulation on Mach–Zehnder interference of single photons

We use the interference field of random wave packets with minimum amplitude Ek0minto perform the simulation. The interference field of each pair of wave packets can be expressed as

where T =λ/c with λ =500 nm; x is the optical path difference between two arms of Mach–Zehnder interferometer,ranging from ?3λ to 3λ; and n is set to 10. Two terms on the right side of Eq.(17)represent two wave packets passing through two arms of the interferometer,respectively,simulating single photons passing through two different arms of the interferometer. For simplicity, we take Ek0minas two units.And the intensity distribution of the interference after counting the pairs of wave packets m times per round of measurement is

where uris a random number taken from the sequence[0,1,2,3, 4, 5], meaning that the pairs of wave packets in each time of count would be zero,one,or more than one; and e?uris a weight factor, meaning that the probability for more pairs of wave packets is less than that for fewer pairs,in each time of count.Figure 2 shows the numerical results for m=3,30,300,respectively. Figure 2(a)is the case for m=3,where none of the interference fringes can be observed. When m=30, see Fig.2(b),the interference fringes appear but not so clear.For a large enough number of m,for example,m=300,the interference fringes become very distinct, as shown in Fig.2(c). For comparison,a classical interference figure of MZI is presented in Fig.2(d), whose amplitude and period are like those in Fig.2(c).One can see that the space of the interference fringes(corresponding to the optical path difference) in Fig.2(c) is 500 nm, being the wavelength λ (=2π/kλ). The result here looks like the experiment of MZI of single photons,[4]and like that of the electron two-hole interference[25]as well. The light wave packets here behave like photons with wave–particle duality.In the simulation,the wavelength of light can be revealed by numerous wave packets through interference but cannot be identified by a few wave packets,which show no wave information about light; instead, they behave as if they are “particles”like electrons.[25]

Fig.2. Simulation on Mach–Zehnder interference of single photons by using the random light wave packets with discrete wavelet structure of n=10, where m is the number of wave packet pairs for each optical path difference x; (a)m=3, (b)m=30, (c)m=300, and(d)a classical interference figure of Mach–Zehnder interference for comparison.

4. Conclusion

By wavelet transform,a classical plane light wave of linear polarization can be viewed as a string of discrete Morlet wavelets and is found to be with discrete energies. And the changeable energy of basic plane light wave packet or wave train with discrete wavelet structure is shown to be that of H0k=np0kω (n=1,2,3,...), practically. Finally, the waveparticle duality from the Mach–Zehnder interference of single photons is simulated by using random basic plane light wave packets.