## Abstract

The first- and second-order temporal interference between two independent thermal and laser light beams is discussed by employing the superposition principle in Feynman’s path integral theory. It is concluded that the first-order temporal interference pattern can not be observed by superposing two independent thermal and laser light beams, while the second-order temporal interference pattern can be observed in the same condition. These predictions are experimentally verified by employing pseudothermal light to simulate thermal light. The relationship between the indistinguishability of alternatives and photons is analyzed. The conclusions are helpful to understand the interference of different kinds of light and the difference between the coherence properties of thermal and laser light.

© 2015 Optical Society of America

## 1. Introduction

Interference is essential to understand coherence [1–3] and superposition principle in quantum physics [4, 5]. Based on the conservation of energy, Dirac concluded that *“Each photon then interferes only with itself. Interference between two different photons never occurs.”* This simple and famous statement is the key to understand the first-order interference of light in quantum physics, which triggers lots of discussions [3,6]. Mandel *et al.* experimentally verified that there is transient first-order interference pattern by superposing two independent lasers [7], which seems contradicting with Dirac’s statement. However, they concluded that their experiments does not contradict Dirac’s statement after further experiments [8]. The observed interference pattern is due to the detection of a photon *“forces the photon into a superposition state in which it is partly in each beam. It is the two components of the state of one photon which interference, rather than two separate photons* [3]”. Paul pointed out that the second part of Dirac’s statement is not always correct by taking the second- and higher-order interference of light into consideration [6]. However, Dirac’s statement is for the first-order interference of light. When considering the second- and higher-order interference of light, Dirac’s statement should be generalized to that a multi-photon state of independent photons only interferes with itself and interference between two different multi-photon states never occurs [9]. From this point of view, Dirac’s statement has not been disproved when considering the second- and higher-order interference.

There is an alternative interpretation for the interference of light to avoid the discussions about Dirac’s statement, which is the superposition principle in Feynman’s path integral theory [5]. For instance, there are two different alternatives to trigger a photon detection event in the first-order interference of two independent lasers [7], which are emitted by the superposed two lasers, respectively. If these two different alternatives are indistinguishable, the probability distribution for the *j*th detected photon is given by [5]

The first- and second-order interference of two independent light beams has been studied extensively by employing different kinds of light sources, such as lasers [7, 8, 10–15], thermal light sources [16–22], and nonclassical light sources [3, 6, 23–25]. The interference of photons in different kinds of light seems more interesting and important to understand the coherence properties of light, such as the interference of photons emitted by laser and single-photon sources [26], laser and quantum down-converted light [27], and laser and thermal light beams [28, 29]. Based on our recent studies about the second-order interference between thermal and laser light [28, 29], we will study the first- and second-order temporal interference between thermal and laser light by employing the superposition principle in Feynman’s path integral theory, hoping to understand coherence properties of thermal and laser light better. We also studied the second-order temporal interference of thermal and laser light when the polarization of the superposed light varies.

The following parts are organized as follows. In Sect. 2, we will theoretically study the first-and second-order temporal interference of thermal and laser light based on the superposition principle in Feynman’s path integral theory. The experiments are presented in Sect. 3 by employing pseudothermal light to simulate thermal light. The discussions and conclusions are in Sects. 4 and 5, respectively.

## 2. Theory

We will employ the scheme in Fig. 1 for the calculations of the first- and second-order interference between thermal and laser light. Two independent thermal and laser light beams are incident to the two adjacent input ports of a 1:1 non-polarizing beam splitter (BS), respectively. S* _{T}* and S

*are point thermal and laser light sources, respectively. D*

_{L}_{1}and D

_{2}are two single photon detectors. The distance between the source and detection planes are all equal. C and CC are single-photon count and two-photon coincidence count detection systems, respectively. For simplicity, the polarizations and intensities of these two light beams are assumed to be the same. The mean frequencies of light emitted by S

*and S*

_{L}*are*

_{T}*ω*and

_{L}*ω*, respectively.

_{T}In the first-order interference of thermal and laser light beams shown in Fig. 1(a), there are two different alternatives to trigger a photon detection event at D_{1}. One is the detected photon is emitted by S* _{L}*. The other one is the detected photon is emitted by S

*. Although the frequencies of the photons emitted by S*

_{T}*and S*

_{L}*are different, these two different alternatives are indistinguishable if the time measurement uncertainty of the detection system is less than 1*

_{T}*/|ω*| [29] (and references therein). The probability distribution for the

_{L}− ω_{T}*j*th detected photon is given by [5]

*φ*and

_{Lj}*φ*are the initial phases of the

_{Tj}*j*th detected photon emitted by S

*and S*

_{L}*, respectively. ${K}_{L}{}_{1}(\overrightarrow{r},t)$ and ${K}_{T}{}_{1}(\overrightarrow{r},t)$ are the Feynman’s photon propagators from the S*

_{T}*and S*

_{L}*to D*

_{T}_{1}at $(\overrightarrow{r},t)$, respectively. The extra phase

*π/*2 is due to the photon reflected by the beam splitter will gain an extra phase comparing to the transmitted one [30]. The magnitudes of these two amplitudes in Eq. (2) are equal, since the intensities of these two light beams are assumed to be identical.

The first-order interference pattern is proportional to the final probability distribution, which is the sum of all the detected single-photon probability distributions

In the second-order interference of thermal and laser light shown in Fig. 1(b), there are four different cases to trigger a two-photon coincidence count. The first one is both photons are emitted by S* _{T}*. The second one is both photons are emitted by S

*. The third one is photon A is emitted by S*

_{L}*and photon B is emitted by S*

_{T}*. The fourth one is photon A is emitted by S*

_{L}*and photon B is emitted by S*

_{L}*. In the first case, there are two different alternatives to trigger a two-photon coincidence count, which are*

_{T}*A → D*

_{1}

*,B → D*

_{2}and

*A→ D*

_{2}

*, B →D*

_{1}, respectively.

*A → D*

_{1}is short for photon A goes to D

_{1}and other symbols are defined similarly. In the second case, both photons are emitted by laser source, S

*. There should be two alternatives, too. However, there is only one alternative since these two alternatives are identical [29]. In the third case, there are two alternatives, which are*

_{L}*A → D*

_{1}

*, B→D*

_{2}and

*A→D*

_{2}

*, B→D*

_{1}, respectively. The fourth case is similar as the third one. If all the seven different alternatives are indistinguishable, the

*j*th detected two-photon probability distribution is [5, 29]

Where *φ _{T jA}* and

*φ*are the initial phases of photons A and B emitted by thermal source in the

_{T jB}*j*th detected photon pair, respectively.

*φ*is the initial phase of photon emitted by laser in the

_{Lj}*j*th detected photon pair.

*K*is short for ${K}_{\alpha}({\overrightarrow{r}}_{\beta},{t}_{\beta})$, which means the Feynman’s photon propagator from the light source S

_{αβ}*to the detector*

_{α}*β*at $({\overrightarrow{r}}_{\beta},{t}_{\beta})$ (

*α*=

*L*and T,

*β*= 1 and 2). The final two-photon probability distribution is the sum of all the detected two-photon probability distributions,

The first line on the righthand side of Eq. (7) is two-photon bunching of thermal light [32]. The second line corresponds to two-photon probability distribution of single-mode continuous-wave laser light, which is a constant [1, 2]. The third and fourth lines are two-photon beating terms when two photons are emitted by two light sources, respectively.

For a point light source, Feynman’s photon propagator is [33]

*and detected at D*

_{α}*, respectively.*

_{β}*r*= |

_{αβ}**r**

*| is the distance between S*

_{αβ}*and D*

_{α}*·*

_{β}*ω*and

_{α}*t*are the frequency and time for the photon that is emitted by S

_{β}*and detected at D*

_{α}*, respectively (*

_{β}*α*=

*L*and

*T*,

*β*= 1 and 2). We will concentrate on the first- and second-order temporal interference between thermal and laser light. Generalizing the discussions to the spatial part is straight forward. Substituting Eq. (8) into Eq. (6) and with similar calculations as the ones in Refs. [9, 13, 22, 28, 35], it is straight forward to have one-dimension temporal two-photon probability distribution as

Where paraxial and quasi-monochromatic approximations have been employed to simplify the calculations. The positions of D_{1} and D_{2} are assumed to be the same in order to concentrate on the temporal part. *sinc*(*x*) equals sin*x/x*. Δ*ω _{T}* is the frequency bandwidth of thermal light. Δ

*ω*is the difference between the mean frequencies of thermal and laser light. When the mean frequencies of thermal and laser light are different, the second-order temporal beating can be observed as shown by the last term of Eq. (9).

_{TL}When a fiber beam splitter is put after the BS in Fig. 1(b) and two single-photon detectors are connected to the fiber beam splitter to measure the second-order temporal interference pattern as in Fig. 2, Eq. (9) should be changed into

It is easy to find that Eqs. (9) and (10) are identical except the minus sign in Eq. (9) is changed into plus sign in Eq. (10). The visibility and period are the same for these two schemes.

## 3. Experiments

In last section, we have calculated the first- and second-order temporal interference of thermal and laser light shown in Fig. 1(a) and 1(b), respectively. It is concluded that the first-order interference pattern can not be observed, while the second-order interference pattern can be observed. In this section, we will employ experimental scheme in Fig. 2 to verify the predictions. The laser is a single-mode continuous wave laser with 780 nm central wavelength and 200 kHz frequency bandwidth. P is a polarizer. BS_{1} and BS_{2} are 1:1 non-polarizing beam splitters. W is a half-wave plate to control the polarization. RG is Rotating ground glass to randomize the phases of photons passing through it. M_{1} and M_{2} are mirrors. L_{1} and L_{2} are two identical lens with focus length of 100 mm and the distance between them are 200 mm. Acoustooptic modulator (AOM) is at the confocal point of L_{1} and L_{2} to change the frequency of laser light. H is a pinhole to block the laser light that does not change frequency after passing through AOM. L_{3} and L_{4} are two identical lens with focus length of 50 mm. S* _{T}* and S

*are point pseudothermal and laser light sources, respectively. The coherence time of true thermal light is usually very short and the degeneracy factor of true thermal light is much less than 1 [3], which makes it is difficult to observe the interference pattern between thermal and laser light. Hence we will employ pseudothermal light [16] to simulate true thermal light in our experiments. FBS is a 1:1 non-polarizing fiber beam splitter, which is employed to ensure that the positions of these two detectors are identical. The distance between L*

_{L}_{3}and the collector of FBS is equal to the distance between L

_{4}and the collector of FBS via BS

_{2}. The optical length between the laser and detector via M

_{1}is 4.24 m. The single-photon counting rates of D

_{1}and D

_{2}are both about 50000 c/s, which means on average there is only 1.41

*×*10

^{−3}photon in the experimental setup at one time. Our experiments are done at single photon’s level.

We first measure the first- and second-order temporal interference patterns when the half-wave plate, W, is removed. The measured single-photon counting rates and two-photon coincidence counts are shown in Fig. 3(a) and 3(b), respectively. The dark counts of both detectors are less than 100 c/s. The single-photon counting rates of D_{1} and D_{2} are shown by the squares and circles in Fig. 3(a), respectively. t is collection time. The single-photon counting rates of both detectors are constant, which is consistent with the prediction of Eq. (4). In the same condition, the second-order temporal interference pattern is observed in Fig. 3(b), which is consistent with the prediction of Eq. (9). The red lines in Figs. 3(b) and 4(a)–4(c) are sine function fitting with minimum sum of squares of error. The reason why the background of the observed second-order temporal beating is flat is the second-order coherence time of pseudothermal light is much longer than the beating period. The second-order coherence time of pseudothermal light is measured to be 51 *μ*s in our experiment. The beating period in Fig. 3(b) is 4.85 ns. Figures 4(a), 4(b), and 4(c) correspond to the second-order temporal beatings when the frequency shifts of AOM are 212.51 MHz, 200.66 MHz, and 195.28 MHz, respectively. The calculated beating frequencies are 212.04 MHz, 199.92 MHz, 194.86 MHz, respectively, which are consistent with the frequency shifts of AOM.

We also measured the second-order temporal beating when the polarizations of thermal and laser light beams are different. The visibility of the second-order temporal beating is shown in Fig. 5 when the angle of the half-wave plate is varied. The error bars of the visibility in Fig. 4 are less than the size of the squares. The observed maximal visibility is 35.12(*±*0.63)% when the polarizations of these two light beams are parallel. The visibility is defined as *v* = (*I _{max} − I_{min}*)/(

*I*+

_{max}*I*), where

_{min}*I*(

_{max}*I*) is the maximal (minimal) value of the measured interference pattern. The theoretical limit for the second-order interference pattern with two independent thermal and laser light beams is 44.4% (4/9), which is consistent with the conclusion that the maximum visibility of the second-order interference pattern with two independent classical light beams is 50% [36]. The reasons why the visibility can not reach 44.4% in our experiments as as follows. (I) The intensities of thermal and laser light beams are not strictly equal. (II) The alignment of our experimental setup is not perfect. (III) The polarization of light beam is not 100% linearly polarized. We have tried corresponding method to minimize these effects. For instance, a fiber beam splitter is employed to ensure that the positions of these two detectors are identical. A linear polarizer is put after the laser to ensure the light is linearly polarized. The observed minimal visibility is 1.88(

_{min}*±*0.46)% when the polarizations of these two light beams are orthogonal, which approaches 0. The reasons why the visibility can not reach zero may be the polarization of the light beam is not 100% polarized in one direction or the polarizations of these two light beams are not strictly orthogonal in the measurement.

## 4. Discussions

Although the first- and second-order interference of classical light can be interpreted by both quantum and classical theories [1,2,37], we will employe one-photon and two-photon interference based on the superposition principle in Feynman’s path integral theory to discuss the first- and second-order interference of thermal and laser light. Not only because it is simple, but also it will give a unified interpretation for all order interference of classical and nonclassical light. For instance, the first-order interference of light is interpreted by the superposition principle of electromagnetic fields in classical theory [34]. The second- and higher-order interference of light is interpreted by the intensity fluctuation correlations based on the first-order interference of light [38,39]. The first-order interference of light is the foundation of the second- and higher-order interference of light in classical theory. In quantum theory, the first-order interference of light is interpreted by one-photon interference based on the superposition principle in quantum physics. The second- and higher-order interference is interpreted by multi-photon interference based on the same superposition principle in quantum physics. The first-, second-, and higher-order interference of light is interpreted by the same theory in a unified way [5]. Further more, the superposition principle in Feynman’s path integral theory can be generalized to interpret the interference of massive particles, such as electrons, neutrons, and atoms. The classical theory of interference of light, on the other hand, can not be generalized to the interference of massive particles or nonclassical light.

The superposition principle in Feynman’s path integral theory is based on the indistinguishability of different alternatives [5]. The indistinguishability of alternatives is related, but not equivalent, to the indistinguishability of particles. For instance, there are two different situations for two photons in a Hong-Ou-Mandel (HOM) interferometer as shown in Fig. 6. I_{1} and I_{2} are two input ports. F* _{j}* is frequency filter that only let photon with frequency

*ω*passes (

_{j}*j*= 1 and 2). In Fig. 6(a), D

_{1}and D

_{2}can only be triggered by photons A and B, respectively. For simplicity, we only consider the case that one photon comes from one input port, respectively. There are two different alternatives to trigger a two-photon coincidence count in Fig. 6(a). The first one is photon A coming from I

_{1}is detected by D

_{1}and photon B coming from I

_{2}is detected by D

_{2}. The second one is photon A coming from I

_{2}is detected by D

_{1}and photon B coming from I

_{1}is detected by D

_{2}. Although photons A and B are distinguishable, these two different alternatives are indistinguishable if it is impossible to tell which photon comes from which input port. It is the reason why the beating between photons of different colors can be observed with ordinary detectors [40, 41]. In the scheme shown in Fig. 6(a), the indistinguishability of alternatives is independent to the indistinguishability of photons.

In the scheme shown in Fig. 6(b), photons A and B come from I_{1} and I_{2}, respectively. There are no filters before detectors. There are two different ways to trigger a two-photon coincidence count, which are *A → D*_{1}*,B → D*_{2} and *A → D*_{2}*,B → D*_{1}, respectively. If these two photons are distinguishable for the detection system, these two alternatives are distinguishable. If these two photons are indistinguishable, these two different ways are indistinguishable, too. The indistinguishability of alternatives is equivalent to the indistinguishability of photons in the scheme shown in Fig. 6(b).

Our experimental setup is similar as the one in Fig. 6(b) except there are more than two alternatives to trigger a two-photon coincidence count. The second-order temporal beating between photons of different frequencies is observed as the ones in Figs. 3 and 4. The reason why there are two-photon interference for photons of different frequencies is photons with different frequencies can be indistinguishable [29] (and references therein). The different alternatives to trigger a two-photon coincidence count are indistinguishable and there is two-photon interference [5]. When the polarization of pseudothermal light is varied by rotating the half-wave plate, the photons in these two light beams gradually become distinguishable. The visibility of the second-order temporal beating drops from 35.12(*±*0.63)% to nearly zero when the polarizations of pseudothermal and laser light change from parallel to orthogonal. When the polarizations of the photons in these two light beams are orthogonal, these photons are distinguishable by polarizations. These two different alternatives to trigger a two-photon coincidence count are distinguishable. There is no two-photon interference and no second-order interference pattern can be observed in this condition.

## 5. Conclusions

In conclusions, we have discussed the first- and second-order temporal interference of thermal and laser light based on the superposition principle in Feynman’s path integral theory. It is concluded that the first-order interference pattern can not be observed by superposing thermal and laser light, while the second-order interference pattern can be observed in the same condition. These predictions are experimentally verified by employing pseudothermal light to simulate thermal light. The relationship between the indistinguishability of alternatives and photons is analyzed. These conclusions can be generalized to the third- and higher-order interference of light. By changing Feynman’s propagators and superposition principle for fermions, the same method can be employed to calculate the first-, second- and higher-order interference of massive particles.

## Acknowledgments

The authors wish to thank D. Wei for the help on the AOM. This project is supported by National Science Foundation of China (No. 11404255), Doctoral Fund of Ministry of Education of China (No. 20130201120013), the 111 Project of China (No. B14040) and the Fundamental Research Funds for the Central Universities.

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