Simulation of temporal second-order correlation function based on Monte Carlo algorithm

Abstract

The second-order correlation function of photons is the primary means to quantitatively describe the second-order coherence of a light field. In contrast to the stationary second-order correlation function, the temporal second-order correlation function can be used to study the second-order coherence of a transient light field. Based on the Monte Carlo algorithm, we carried out theoretical simulation on the temporal second-order correlation function from the perspective of photon statistics. By introducing experimental factors into the simulation, such as intensity jitter of the light field and time resolution of the instruments, the effects of imperfect experimental conditions on the measurement of second-order correlation function have also been elucidated. Our results provide theoretical guidance and analysis methods for experimental measurements on the second-order coherence of light fields.

Graphical abstract

Keywords

temporal second-order correlation function / Monte Carlo algorithm / autocorrelation / cross-correlation

1 Introduction

As an important concept in quantum optics, the second-order coherence of a light field is the primary means to describe the correlation between photons [1]. In practice, such second-order coherence is described quantitatively by the second-order correlation function of a light field. To date, the second-order correlation function has found application in a vast variety of fields, such as quantum information and computation [2-5], quantum imaging [6, 7], and materials science [8]. It also provides theoretical basis and experimental guidance for optical technology and quantum information processing [9-17]. Experimentally, the Hanbury−Brown−Twiss (HBT) experiment is well known as a classical, and stationary, second-order correlation measurement technique [18]. Thanks to the rapid progresses of optical instrumentation, temporal second-order correlation function measurements have become possible in recent years using photon counting technique of streak cameras [19-21]. Compared with the steady state second-order correlation function, the temporal second-order correlation function can accurately give the photon statistical information of a light field at every moment. This excellent feature can be used to analyze the characteristics of the light source in the time domain, which provides an efficient approach to study the ultrafast dynamics of the optical field. With this new technique, interesting experiments have been successfully performed for various physics, such as superradiance [22], polariton lasers [23], and exciton−polariton vortices [24]. Theoretically, Monte Carlo algorithm is widely accepted as an efficient method for simulating second-order correlation functions, and has been applied to a variety of scientific frontiers, such as open quantum systems [25], optical coherence propagation [26], and polariton lasers [27]. Although remarkable progresses have been made, both experimentally and theoretically, it is worth pointing out that much remains to be done to promote further the application of the second order correlation function. Especially, measurement of the second order correlation function is rather sensitive to experimental environment. Elucidating the effect of experimental environment, such as limited instrument resolution, intensity jitter and pseudo photon counting, on the measurement of the second order correlation function is an objective that has been pursued by the scientific community for a long time. Monte Carlo simulation of the temporal second-order correlation function from the perspective of photon statistics, a method that provides fast and qualitative analyses for experiments, remains unsolved either.

In this paper, we carried out theoretical studies on the temporal second-order correlation function using Monte Carlo algorithm. We performed simulations for the autocorrelation function of typical light fields and the cross-correlation function between different light fields. Our simulation results show satisfactory agreement with the expected temporal second-order correlation function. Moreover, effects of imperfect experimental conditions on measurement results can also be simulated by introducing experimental factors. Using our method, no additional complex modeling is required when simulating time-dependent second-order coherence. Therefore, it is easy to introduce experimental factors and then qualitatively analyze the influence of experimental factors on measurement results. The uncertainty in experimental measurements can be eliminated reliably.

2 Theoretical simulations

2.1 Stationary and temporal second-order correlation function for photons

The classical method for the measurement of the second-order correlation function is usually carried out through HBT experiments, as shown in Fig.1(a). In general, the distance between the two detectors and the beam splitter is equal. Such experiment essentially measures the probability that one detector has a count at time t and the other has a count at $t+\tau$. Therefore, the expression of the second-order correlation function (intensity autocorrelation function) can be written as

Fig.1 Experimental setup for the measurement of second-order correlation function. (a) Schematic diagram for HBT experiments. TCSPC refers to time-correlated single photon counting. (b) Schematic diagram for the measurement of second-order correlation function using streak cameras. The streak camera has a picosecond time resolution and can record the time of each photon arriving at the streak camera. Based on the transverse scanning capability, photon emission events for a series of excitation pulses can be recorded within one streak image, thus improving the detection and calculation efficiency.

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(1)$\begin{array}{rl}{g}^2\left(t,\tau \right)=& {\displaystyle \frac{⟨n\left(t\right)n\left(t+\tau \right)⟩}{⟨n\left(t\right)⟩⟨n\left(t+\tau \right)⟩}}\\ =& {\displaystyle \frac{⟨a^{†}\left(t\right)a^{†}\left(t+\tau \right)a\left(t\right)a\left(t+\tau \right)⟩}{⟨a^{†}\left(t\right)a\left(t\right)⟩⟨a^{†}\left(t+\tau \right)a\left(t+\tau \right)⟩}},\end{array}$

where $n$ is the photon number operator, $a^{†}$ and $a$ are the creation and annihilation operators, respectively. Accordingly, the cross-correlation function $g_{XY}^2\left(t,\tau \right)$ between two different light fields can be expressed as

(2)$\begin{array}{rl}{g}_{XY}^2\left(t,\tau \right)=& \frac{⟨n_X\left(t\right)n_Y\left(t+\tau \right)⟩}{⟨n_X\left(t\right)⟩⟨n_Y\left(t+\tau \right)⟩}\\ =& \frac{⟨a_X^{†}\left(t\right)a_Y^{†}\left(t+\tau \right)a_X\left(t\right)a_Y\left(t+\tau \right)⟩}{⟨a_X^{†}\left(t\right)a_X\left(t\right)⟩⟨a_Y^{†}\left(t+\tau \right)a_Y\left(t+\tau \right)⟩}.\end{array}$

Based on the value of the second-order correlation function, the population distribution of photons can be classified into three types: bunching [$g^2\left(t,0\right)>1$], coherent [$g^2\left(t,0\right)=1$] and anti-bunching [$g^2\left(t,0\right)<1$]. Here, it is worth pointing out that HBT experiments usually measure the second order correlation function under different delay time τ, generating a function $g^2\left(\tau \right)$. The evolution of the second-order correlation function, i.e., $g^2\left(t,\tau \right)$, cannot be obtained.

In contrast, by means of streak cameras, time-dependent second-order correlation function $g^2\left(t,\tau \right)$ can be obtained readily. As shown in Fig.1(b), the streak camera firstly turns the incident photons into electrons using a photocathode. The electrons are then tilted in the vertical direction by an applied time-dependent voltage V(t) which translates the time information of the incident photons into the position information [19-21]. The tilted electrons are turned into photons by a phosphor screen and finally recorded by a detector. In this way, evolution of the incident light field can be measured, which usually manifests itself as a longitudinal bright stripe on the screen. Employing the transverse scanning capability of the streak camera, evolution of the light field under a series of excitation pulses can be recorded in one streak image, as demonstrated in Fig.1(b). It is based on the inspiration of streak cameras that we simulate a series of photon detection events using Monte Carlo algorithm.

2.2 Simulation of autocorrelation photon detection events using Monte Carlo algorithm

The second-order correlation function of photons describes the ability of the light source to emit photon pairs. It can also be considered as the probability of detecting a second photon with delay $\tau$ after one photon is detected [1]. As an example, the second-order correlation function of coherent light is always $g^2\left(\tau \right)=1$. This fact is quite straightforward. After one photon is detected in the coherent light field, the detection of a second photon is still completely random. Thus the probability of detecting a second photon is the same as that of detecting the previous one. For bunching light, $g^2\left(0\right)=2$. This means that the probability of detecting a second photon immediately after the detection of one photon in thermal light field is twice the probability of the occurrence of a completely random photon. According to this physical meaning, we can generate a series of photon detection events can be simulated using Monte Carlo algorithm.

For given measurement time $\mathrm{\Delta }t$, it can be considered that the average light intensity detected in $\mathrm{\Delta }t$ is $\overline{n}\ll 1$ when $\mathrm{\Delta }t\to 0$. The value of $\overline{n}$ is related to the emission intensity of the light field to be measured, and the expression can be written as $\overline{n}=\eta I\left(t\right)$. Here, $I\left(t\right)$ is the time-dependent intensity of the normalized light field and $\eta$ is the photon count rate. To make sure that $\overline{n}\ll 1$, $\eta$ should be small enough (this will be discussed later). $\mathrm{\Delta }t$ is taken as the time step in our simulation. Its magnitude must be far smaller than the characteristic time for the change of the intensity of the light field. The following steps can be used to simulate photon detection events with different distribution patterns:

(i) Choose an appropriate time step $\mathrm{\Delta }t$.

(ii) Time ($t=0$) and delay ($\tau =0$) initialization.

(iii) Generate a random number $x$ belonging to $\left(0,1\right)$.

(iv) Compare the value of $x$ and $\overline{n}\cdot g^2\left(\tau \right)$.

(v) If $x\le \overline{n}\cdot g^2\left(\tau \right)$, a photon has been detected. Record the corresponding time $t$, set $\tau =0$.

(vi) If $x>\overline{n}\cdot g^2\left(\tau \right)$, no photon has been detected. Set $\tau =\tau +1$.

(vii) After performing the above steps, one cycle ends and the next cycle starts (set $t=t+1$). Return to (iii) and repeat the preceding steps until $t$ reaches the specified value.

Here, $g^2\left(\tau \right)$ is the second-order correlation function that needs to be generated. For simplicity, the expression used in this paper is $g^2\left(\tau \right)=1$ (Coherent), $g^2\left(\tau \right)=1+\exp (-2\left|\tau \right|/{\tau }_c)$ (Bunching) and $g^2\left(\tau \right)=1-\exp (-\pi \tau /{\tau }_c)$ (Anti-bunching) [21], with ${\tau }_c$ being the coherence time of the light field. It is also worth explaining why $x\le \overline{n}\cdot g^2\left(\tau \right)$ is used as the criterion to judge whether a photon has been detected. As mentioned earlier, assume that light field intensity in the whole evolution process remains the same, the second-order correlation function $g^2\left(\tau \right)$ can be understood as the probability that the detection of one photon is followed by the detection of another (interval of time $\tau$). The variation of light field intensity is reflected in $\overline{n}=\eta I\left(t\right)$. Since the value range of $x$ is constant, $x\le \overline{n}\cdot g^2\left(\tau \right)$ is a simple criterion for photon detection events. At the same time, it also provides convenience for the subsequent simulation experiment conditions.

According to the above process, a series of photon detection data can be obtained. The corresponding second-order correlation function can be calculated through these data and Eq. (1). Typical results are shown in Fig.2. Fig.2(a) shows the zero-delay autocorrelation function of an anti-bunching light field. The red curve shows the evolution of the normalized mean photon count, which manifests the evolution of the light field in the time domain. The blue curve shows the evolution of the second-order correlation function with delay time τ = 0. Obviously, $g^2\left(t,0\right)$ is always less than 1. This tells that the light field exhibits anti-bunching effect at any time. Another feature is that the error bar becomes larger with time. This is caused by the decrease in the number of photon meters due to the decrease of the light intensity, as shown by the red line. Fig.2(b) shows the second order correlation function under different delay time. When the delay time is shorter than the coherence time ${\tau }_c$, $g^2\left(\tau {<\tau }_c\right)<1$. When the delay time exceeds the coherence time, $g^2\left(\tau {>\tau }_c\right)=1$. Fig.2(c, d) and Fig.2(e, f) show the second-order correlation functions of coherent and bunched light fields, respectively. Here, it is worth noting that $g^2\left(\tau \right)$ in Fig.2(f) does not reach 2 at τ = 0. This is caused by the value of $\eta$, which will be discussed in detail later.

Fig.2 Autocorrelation photon detection events simulated by Monte Carlo algorithm. (a, b) Simulation results for anti-bunching light field. (c, d) Results for coherent light field. (e, f) Results for bunching light field. (a), (c) and (e) show the zero-delay autocorrelation function $g^2\left(t,0\right)$ as a function of time. The red curve shows the evolution of the normalized mean photon count over time. The blue curve shows the evolution of the second-order correlation function over time with delay time τ = 0. (b), (d) and (f) show the second order correlation function under different delay time, i.e., $g^2\left(\tau \right)$. Here, $\mathrm{\Delta }t=0.08\mathrm{p}\mathrm{s}$, ${\tau }_c=12\mathrm{p}\mathrm{s}$, and $\eta ={10}^{-3}$. Lifetime of the light field is 32 ps. The number of simulations is ${10}^7$. The time resolution of the simulation was $t_r=2\mathrm{p}\mathrm{s}$. These parameters are used in subsequent simulations unless otherwise specified.

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3 Cross-correlation photon detection events simulated by Monte Carlo algorithm

Like the autocorrelation function, cross-correlation photon detection events can also be simulated using Monte Carlo algorithm. The corresponding simulation steps are as following:

(i) Choose an appropriate time step $\mathrm{\Delta }t$.

(ii) Time ($t=0$) and delay ($\tau =0$) initialization.

(iii) Generate four random numbers $x_1$, $x_2$, $x_3$ and $x_4$ which belong to $\left(0,1\right)$.

(iv) If the previous photon only appears in light field 1, compare the value of $x_1$ with ${\overline{n}}_1\cdot g_{11}^2\left(\tau \right)$ and $x_2$ with ${\overline{n}}_2\cdot g_{12}^2\left(\tau \right)$. If the previous photon only appears in light field 2, compare the value of $x_3$ with ${\overline{n}}_2\cdot g_{22}^2\left(\tau \right)$ and $x_4$ with ${\overline{n}}_1\cdot g_{12}^2\left(\tau \right)$. If the preceding photon appears in both light field 1 and light field 2, all four formulas are compared. In particular, at the initial moment (no photon appears in two light field), all four equations are compared.

(v) If $x_1\le {\overline{n}}_1\cdot g_{11}^2\left(\tau \right)$ or $x_3\le {\overline{n}}_1\cdot g_{12}^2\left(\tau \right)$, a photon has been detected in light field 1. If $x_2\le {\overline{n}}_2\cdot g_{22}^2\left(\tau \right)$ or $x_4\le {\overline{n}}_2\cdot g_{12}^2\left(\tau \right)$, a photon has been detected in light field 2. In case that both conditions are met, photons are recorded simultaneously in the corresponding light field. Record the corresponding time $t$, set $\tau =0$. In addition, it is necessary to record which light field the photon appears in, so that we can continue to make judgements in following moments.

(vi) If no conditions are met, no photon has been detected, set $\tau =\tau +1$.

(vii) After performing the above steps, one cycle ends and the next cycle starts (set $t=t+1$). Return to (iii) and repeat the preceding steps, until $t$ reaches the specified value.

With the above steps, correlation between any two light fields can be simulated. Fig.3(a, b) show the autocorrelation properties of coherent light field 1. The second-order correlation function of light field 2 with bunching effect is shown in Fig.3(c, d). The intensity of light field 2 is set to be 0.6 times that of light field 1 (${\overline{n}}_2$ = $0.6{\overline{n}}_1$). Fig.3(e, f) show the simulation results with an anti-bunching effect between the two light fields. The above results show that we have successfully simulated the autocorrelation and cross-correlation characteristics of typical light fields based on Monte Carlo algorithm.

Fig.3 Cross-correlation photon detection events simulated by Monte Carlo algorithm. (a, b) Intensity autocorrelation function $g_{11}^2\left(t,0\right)$ with zero delay (a) and autocorrelation function $g_{11}^2\left(\tau \right)$ under different delay (b) for light field 1. (c, d) Intensity autocorrelation function $g_{22}^2\left(t,0\right)$ with zero delay (c) and autocorrelation function $g_{22}^2\left(\tau \right)$ under different delay (d) for light field 2. (e, f) Corresponding intensity cross-correlation function $g_{12}^2$ between light field 1 and light field 2.

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4 Discussion

Based on the above simulation, some problems encountered in experiments can be analyzed. But before that, we need to answer another question from the previous simulation: why $g^2\left(\tau \right)$ of the bunching light field does not decrease strictly from 2. As mentioned earlier, the reason why this happens is the chosen value of the photon count rate $\eta$. Fig.4(a) shows the zero-delay intensity autocorrelation function $g^2\left(t,0\right)$ of a bunching light field with $\eta$ set to be ${10}^{-4}$, ${10}^{-3}$ and ${10}^{-2}$, respectively. When $\eta ={10}^{-4}$, as shown by the green line, the $g^2\left(t,0\right)$ value oscillates around 2 with large amplitude. This is because the low photon count rate results in a low average number of detected photons, thus causing oscillations. To reduce the oscillations, one can increase the number of simulations and the photon count rate $\eta$. However, although the oscillation becomes significantly smaller when $\eta$ was increased to ${10}^{-3}$, $g^2\left(t,0\right)$ decreases meanwhile from 2 to ~1.82, as shown by the blue curve. The reason for such drop is the way we make judgments for the occurrence of photons [$x$ and $\overline{n}\cdot {\mathrm{g}}^2\left(\tau \right)$]. It should be remembered that we are comparing the size of a random number $x$ (value ranges from 0 to 1) with that of $\overline{n}\cdot g^2\left(\tau \right)$. If the value of $\eta$ is too large, the proportion of $\overline{n}=\eta I\left(t\right)$ will be large in our judgment as well. This also means the influence of $g^2\left(\tau \right)$ becomes small. For example, if $\eta$ = 1, the condition $x<\overline{n}\cdot g^2\left(\tau \right)$ is always met regardless of $g^2\left(\tau \right)$. In other words, it can always be regarded as $g^2\left(\tau \right)=1$ in this case. The larger $\eta$, the more the simulation results tend to coherent light. If $\eta$ is further increased to $\eta ={10}^{-2}$, as shown by the black curve, the value of $g^2\left(t,0\right)$ continues to drop, thus justifying our claim. Particularly, it gets smaller when the light intensity $I\left(t\right)$ is higher, as shown by the red line. In order to show more clearly the influence of photon count rate on the simulation of second-order correlation function, Fig.4(b) shows $g^2\left(\tau =0\right)$ as a function of $\eta$. Obviously, larger $\eta$ leads to smaller $g^2\left(\tau =0\right)$. Therefore, to obtain more accurate results, one needs to reduce the photon count rate and increase the number of simulations.

Fig.4 Effect of photon count rate on second-order correlation function simulation. (a) The zero-delay intensity autocorrelation function of the bunching light field with different $\eta$. (b) $g^2\left(\tau =0\right)$ as a function of photon count rate $\eta$.

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In actual experiments, measurement results are often affected by the system and the environment. Therefore, factors causing deviations should be found out and corrected. These could be done by taking experimental conditions into simulation. Here, we take the instrumental resolution as an example. To measure the second-order correlation function of a light field, time resolution of the instrument should be much smaller than the coherence time of the light field. Fig.5(a) shows the second order correlation function $g^2\left(\tau \right)$ of a bunching light field when the ratio between coherence time $t_c$ and system time resolution $t_r$ is changed. As one can see from the red curve, $g^2\left(\tau \right)$ can be accurately measured when the coherence time is much larger than the time resolution. However, when the time resolution is at the same order of magnitude as the coherence time, the value of $g^2\left(\tau \right)$ becomes significantly smaller, as shown by the blue and green curves. The sky-blue curve shows the second-order correlation function at $t_c/t_r=0.1$. Clearly, the value of $g^2\left(\tau =0\right)$ now approaches 1. The reason for these lies in the fact that correlation between photons is only present within their coherence time. If the coherence time is smaller than the resolution, some uncorrelated photons will be counted. The smaller the coherence time is, more irrelevant photons are counted. This effect can be shown even more clearly in Fig.5(b), where the correlation function $g^2\left(\tau =0\right)$ for a bunching light field is plotted as a function of the ratio between the coherence time and the resolution of the detector. As one can see, the correlation function $g^2\left(\tau =0\right)$ versus $t_c/t_r$ shows a “S” shape, in agreement with previous theoretical reports [21].

Fig.5 Effects of coherence time of the bunching light field and time resolution on the measurement of the second-order correlation function. (a) Second order correlation function $g^2\left(\tau \right)$ when the ratio between coherence time $t_c$ and time resolution $t_r$ is changed. (b) $g^2\left(\tau =0\right)$ as a function of $t_c/t_r$. The time resolution of the simulation was $t_r=2\mathrm{p}\mathrm{s}$. Red asterisks represent results of the actual fitting. The blue curve is a guide line for the trend.

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In addition to the system performance, effects of the experimental environment (such as background light, time jitter of the detector and the intensity jitter of the light field) on the measurement results can also be simulated. Here, we take the intensity jitter of the light field to be measured as an example. Fig.6(a) simulates the effect of intensity jitter on the second-order correlation function of an anti-bunching light field. Based on the discussion in previous text, we know that we make judgements on the photon detection events by comparing the value of $x$ to $\overline{n}\cdot g^2\left(\tau \right)$. Light intensity jitter will affect $\overline{n}$. Therefore, a coefficient $\left({\mathrm{\Delta }}_{\mathrm{I}}\cdot c+1\right)$ can be added to $\overline{n}$ to simulate the light intensity jitter. Specifically, the condition $x<\left({\mathrm{\Delta }}_{\mathrm{I}}\cdot c+1\right)\cdot \overline{n}\cdot g^2\left(\tau \right)$ indicates that a photon has been detected. ${\mathrm{\Delta }}_{\mathrm{I}}$ is the jiitter intensity. $c$ is a random number that conforms to a Gaussian distribution. As can be seen from Fig.6(a), simulation results without intensity jitter (${\mathrm{\Delta }}_{\mathrm{I}}=0$) perfectly reproduce the second-order correlation function image of the anti-bunching light field. When the intensity jitter is small (${\mathrm{\Delta }}_{\mathrm{I}}=0.1$), the simulation results do not change much. However, when the jitter was further increased, the curve of the correlation function was vertically shifted obviously. The larger the jitter is, the more the curve is shifted. Fig.6(b) shows the $g^2\left(\tau =0\right)$ and $g^2\left(\tau >{\tau }_c\right)$ as a function of the light intensity jitter ${\mathrm{\Delta }}_{\mathrm{I}}$. It is obvious that the intensity jitter has little effect on $g^2\left(\tau =0\right)$, but has a significant effect on $g^2\left(\tau >{\tau }_c\right)$. Therefore, in experiments where measurement results are not normalized at $\tau >{\tau }_c$, deviations caused by light intensity jitter should be corrected before any data analyses.

Fig.6 Effects of intensity jitter on second-order correlation function measurement. (a) The change of $g^2\left(\tau \right)$ under different light intensity jitter ${\mathrm{\Delta }}_{\mathrm{I}}$. The light source chosen here is an anti-bunching field. (b) $g^2\left(\tau =0\right)$ (blue curve) and $g^2\left(\tau >{\tau }_c\right)$(red curve) as a function of light intensity jitter ${\mathrm{\Delta }}_{\mathrm{I}}$.

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In summary, based on the idea of photon statistics, we successfully simulate the autocorrelation and cross-correlation functions of light fields using Monte Carlo algorithm. This temporal second-order correlation function can be used to analyze the time-resolved properties of light fields. Our work shows that more accurate results can be obtained by increasing the number of simulations and by decreasing the photon count rate. Moreover, time resolution of the instrument and jitter of the laser intensity were found to have significant effects on the measurement results. Results in this work provides a simplified simulation method for the theoretical study of the second-order correlation function of photons in complex optical fields. Our work also provides theoretical support and analytical method for the experimental measurement of second-order correlation function.

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Competing interests

The authors declare no competing interests.

Data availability

All data in the manuscript are available from the corresponding author upon reasonable request.

Acknowledgements

The work was supported by the National Natural Science Foundation of China (Grant Nos. 12474070, 12174112, and 12274159) and the Natural Science Foundation of Chongqing Province and Shanghai (Grant Nos. 2023NSCQ-MSX1489 and 23ZR1419800).

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