Photon interference of second- and third-order correlation generated by two fluorescence sources

We demonstrate the second- and third-order temporal correlation of two independent pseudo-thermal fluorescence (FL) sources. The two distinguishable sources become indistinguishable following the delay in Feynman’s path. The second- and third-order correlation demonstrates the strong bunching amplitude surrounded by variable oscillations and interfering side bands (beats). Specifically, three-photon bunching is overlapped by two-photon bunching, and further decomposed into multiple lower order correlation functions. Here, the strong interference arises from source and path indistinguishable terms (varying time offset, laser frequency and the bandwidth of FL sources); The support for this idea comes from Feynman’s path integral theory and sub wavelength interference for such kind of sources with appropriate detection schemes. Such phenomena may serve as modulation and carrier for quantum communication channels.


Introduction
The correlation and interference phenomenon provide a solid foundation for the establishment and development of the coherence and the quantum theory of light [1]. Dirac considered superposition comes only from the single photon in the single photon interference [2]. But in case of generation of twin/paired photon, a similar statement can be established for interference of two photons, in which superposition comes from jointly measured pair of photon (analogous definition of Dirac) sharing the same energy level such as photons generated from multi-wave mixing (MWM) under phase matching condition [3][4][5]. However, Paul considered the Dirac's statement to be limited to first-order coherence [6]. Since then [7,8], second-order temporal and spatial coherence from two independent sources (coherent, pseudo thermal, laser-photon etc) is extensively studied using 'Hong-Ou-Mandel (HOM) dip' or 'Shih-Alley dip' interference phenomenon [9][10][11]. But up to our information till date, no one has come up with third-order temporal coherence from two independent sources of pseudo thermal fluorescence (FL) light. Continuing with the development of second-order coherence, recently, Kim et al observed cosine modulation in the spatial second-order coherence function in a HOM interferometer with entangled photon pairs [9,12] and self-coherence within the same inputs being essential for the interference. Similarly, Saleh observed beating fringes, a byproduct of interference with frequency mismatch [13,14]. Based on recent development of the second-order interference between thermal and laser light [15,16] and concept of sub wavelength interference using join detection schemes, we will study the second-and third-order temporal interference between pseudo thermal fluorescence light sources, which is employed to superposition theory in Feynman's path integral. The support for this idea comes from unified interpretation of second-order subwavelength interference based on Feynman's path integral theory from coherent and thermal sources [17][18][19][20][21] (or quantum dotted light sources [22,23]) along with their indistinguishability [21,24]. This study might help to create hope for quantum communication via classical channel.
In this paper we discuss the second-and third-order temporal correlation by treating multi-order fluorescence as two independent pseudo-thermal sources. We analyze two-and three-photon bunching and quantum beating effect at different frequency bandwidth, frequency deference and relative time delay of two sources. Our results demonstrated strong interference effect between two nondegenerate fluorescence sources based on the indistinguishability of different paths. Figure 1 illustrates the employed joint detection schemes for calculating second-and third-order correlation from two fluorescence sources. Figure 1(a) shows the two sources joint detection system, in which fluorescence source named 'S F ' produces two photons named as A and B via beam splitter. Similarly, another source named 'S f ' is employed via beam splitter and produces A and B photon. The photon A from both sources can be either detected by detector D 1 or D 2 . Similarly, the photon B from both sources can be detected by either detector D 1 or D 2 . Source and path indistinguishability for detection scheme is illustrated in figure (a1). The source and path indistinguishable terms are displayed in figures 1(a2) and (a3), respectively. Following these indistinguishable schemes, the coincidence count (CC) will produce second-order temporal correlation or 'two-photon bunching' along with interference side peaks or 'beats'. Such phenomenon mainly depends on the time offset in the detector and frequency of the sources. The third-order temporal correlation or 'three-photon bunching' can be obtained by applying detection scheme illustrated in figure 1(b1), here two independent FL sources pass through two beam splitters (BS 1 and BS 2 ). The CCC produces the center peak (three-photon bunching) along with multiple interference side peaks (beat frequency) depending on the frequency of the FL sources. The source and path indistinguishability for this detection scheme is illustrated in figures 1(b2) and (b3). We can model the intensity of second-and fourth-FL sources using diagonal density matrix elements via their perturbation chains. The fourth-order fluorescence signal can be generated using two input beams in a three level system (|0〉↔|1〉↔| 2〉). The fourth-order FL signal can be described by the perturbation chain r r r r r The transition pathways density-matrix element can be written as

Interference of second-order correlation
It is surprising that two pseudo-thermal FL sources are interfering due to Feynman's path integral theory caused by the beam splitter. Following Feynman's path integral theory, there could be four different cases to trigger twophoton coincidence count, so the second-order interference of FL sources shown in figure 1(a1) follows the path depicted in figures 1(a2) and (a3), First case happens when both photons (A and B) come from the S F source following beam splitter. Second case happens when both photons come from S f . In third case photon A comes from S F and photon B comes from S f . The fourth case is opposite of third case. In the first case, there are two different alternatives to trigger a two-photon coincidence counter, which are A→D 1 , B→D 2 and A→D 2 , B→D 1 , respectively. A→D 1 is short path for photon, photon A goes to D 1 and other symbols are defined similarly. In the second case, both photons are emitted by S f . Similarly, there should be two alternatives. However, there is only one alternative since these two alternatives are identical. In the third case, there are two alternatives, which are A→D 1 , B→D 2 and A→D 2 , B→D 1 , respectively.
Here j fjA (j FjA ) and j fjB (j FjB ) are the initial phases of photons A and B emitted by source S f (S F ) in the jth detected photon pair, respectively. There is an extra phase π/2 for the photon reflected by the beam splitter comparing to that transmitting through the same beam splitter. ab (a = f and F, b =1 and 2). The final two-photon probability distribution is the sum of all the detected two-photon probability distributions å = º á + .. is ensemble average by taking all the detected two-photon probability distributions into consideration. The four lines on the right hand side of equation (2) correspond to four different cases mentioned above, respectively. Since the two pseudo-thermal FL sources are independent, therefore á ñ all terms can be equal to 0. Equation (2) can be simplified as . 3 The first two items on the right hand side of equation (3) are two-photon bunching of the two pseudothermal FL sources, respectively. Last two items are two different situations of two-photon beating terms subjected to two photons emitted by both sources, respectively. For a point light source, Feynman's photon propagator is defined as which is the same as Green function defined for classical optics. ab k and ab r are the wave and position vectors of the photon emitted by a S and detected at b D . = ab ab | | r r is the distance between a S and b D . w a and b t are the frequency and time for the photon emitted by a S and detected at b D (a = f and F, b =1 and 2). Second order correlation between two independent thermal sources can be written as where t 12 =t 1 −t 2 . w D and w i (where i=f or F) represents the bandwidth and frequency of the pseudo-thermal FL source, respectively. The three detectors D 1 , D 2 , and D 3 are placed at equal distance r o to observe the photon counting at each individual detector, equation (5) can be simplified by assuming bandwidth of both FL sources to be equal i-e w w w D = D = D , f F therefore, equation (5) can be simplified as From above equation, one can predict that, the term 2 caused by path difference hitting coincident counter. If we remove the beam splitter, there will be no interference.

Interference of third-order correlation
To observe interference among photons in third-order correlation from two sources, we will employ the scheme illustrated in figure 1(b). Two independent FL sources are incident on the two beam splitter (BS 1 and BS 2 ) and signals are detected at three single photon detectors D 1 , D 2 and D 3 . The distance between the source and detection planes are all same. For simplicity, the polarization (linear) and intensities of these two FL sources are assumed to be the same. The possible eight combinations of jth detected three-photon probability distributions can be written as  Here j fjA j fjB and j fjC are the initial phases of photons A, B and C emitted by source S f in jth detected three photon, respectively. There is an extra phase π/2 for the photon reflected by the beam splitter comparing to that transmitting through the same beam splitter. ab K is short for a b b ( ) K t r , ,which has same meaning defined in two photon case.           where t 12 =t 1 −t 2 , t 23 =t 2 −t 3 , t 13 =t 3 −t 1 .

Results and discussions
Here, we investigated the second-order temporal intensity correlation by exciting two multi-order pseudothermal fluorescence sources on beam splitter (scheme displayed in figure 1(a1). The FL signals generated from two independent FL sources are divided into two beams using beam splitters (BS 1 ). Two beams following the BS are recorded at detectors (D 1 and D 2 ) and then sent to coincidence counter that calculates the second-order correlation ( ) ( ) G t 2 12 using equation (6). In figures 2(a1)-(a5), bandwidth of two FL sources S F (w = 1 F MHz) and S f (w = 2 f MHz) are kept equal, and t 1 time offset is fixed at 0 ms. The bandwidth w D of both source S f and S F is varied from 1 MHz (figure 2(a1)) to 10 MHz ( figure 2(a5)). By this consideration, equation (5) is simplified to equation (6). According to equation (6), as we increase the bandwidth of FL sources, the linewidth of correlation changes from broad to sharp as shown in figures 2(a1)-(a5). The transition of correlation line shape from broad to sharp can be explained from change in decoherence rate of FL sources (G = G + G FL 10 11 ). When bandwidth of FL source is w D = 1 MHz, decoherence rate G FL is low which corresponds to broad peak (figure 2(a1)) and when w D is increased to 10 MHz then decoherence rate G FL increases, which produces sharp correlation peak as shown in figure 2(a5). To study the relationship between bandwidth and oscillation, we increase the frequency w F of source S F to 10 MHz and increase the bandwidth w D of both source S f and S F from 1 MHz (figure 2(b1)) to 10 MHz (figure 2(b5)) under the condition w w w D = D = D .  G t t t , , 3 12 23 13 can be calculated using equation (8) under same condition as in figure 3(a). In figure 3(b1), multiple side peaks are observed surrounding the strong center peak. These multiple peaks result from interference between two FL sources governed by the time beat term from equation (8). Figures 3(b1)-(b5) demonstrate the strong bunching peak surrounded by variable oscillation with the shifted peak from mean position (0 ms). This time shifting of strong indistinguishable bunching amplitude with variable side bands suggests the strong interference mechanism of three photons coming from two independent FL sources following Feynman's path. Here the interference solely depends on the frequency beat (w w f F ) between two FL sources following the Feynman's path and the time offset -( ) t t 1 2 introduced in the detection of photons at D 1 .
In figure 4, we investigated the second-and third-order correlation of two-and three-photon, respectively, from two FL sources by changing the frequency w f of the once S f source from 1 MHz (low) to 62 MHz (high) and fixing frequency w F of other source S F at low that is 1 MHz. One can notice from figures 4(a1)-(a6), as the frequency w f of S f source is increased, interference generated from source and path indistinguishable terms increases, which gives rise to side peaks. Mathematically, interference side peaks and central peak amplitude arise from the term of equation (5), respectively [26,27]. However, the number of side peak precisely depends on the difference of frequencies (beat) among the The same phenomenon is observed in third-order correlation function. As w f increases to 62 MHz, oscillation factor in equation (8) cos cos cos , presenting positive correlation with interference, their amplitude and linewidth precisely follow the shape as in second-order correlation. As the frequency of the FL source is further increased, the bunching effect and strong interference are observed as shown in figures 4(b1)-(b6). From figures 4(b1)-(b3) and (a1)-(a3), the bunching dominates due to very small oscillation factor which results from small frequency difference w w -( ) f F at the condition of low frequency w .
f Whereas at high frequency w f of input FL source the interference increases dramatically as shown in figures 4(a3)-(a6) and (b3)-(b6). Based on these observations, one can conclude that beating is strongly dependent on frequency difference of FL sources.
At present, our group is conducting the experimental research on second-and third-order correlation in atomic-like ensemble. In addition to the work presented in current paper, we have investigated the third-order correlation coming from nonlinear interaction among three multi-order fluorescence signals generated from a pseudo-thermal source [28]. Also, we have demonstrated the third-order correlation (resulting from path indistinguishability) of single multi-order fluorescence signal generated from a pseudo-thermal light source [29]. Currently, we are exploring the second-and third-order correlation from two and three independent pseudo-thermal fluorescence sources experimentally in our lab but the experiments are still in its initial stages.

Conclusion
In conclusion, we demonstrated the second-and third-order correlation from two independent pseudo-thermal fluorescence (FL) sources. The correlation curves demonstrated strong bunching amplitude and beats. Threephoton bunching was overlapped by two-photon bunching, and was decomposed further into multiple lower order correlation functions. The interference generated from source and path indistinguisbality was recorded at joint detection scheme. By controlling time offset in joint triggering, photon bunching can be controlled. By increasing frequency of one S f source from 1 to 62 MHz, interference among photons also increases significantly. We believed that the results presented in our manuscript may have potential applications in quantum communication through classical channel through beating effect.