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Abstract
We have developed a new method to determine the photon statistics of a light source without the need for coincidence counting. Our method relies on using an asymmetric beam-splitter with intentional asymmetrical losses in a non-photon-number resolving detection system. We observe that the counts from one arm of the beam-splitter, when plotted as a function of the counts from the other arm, follow universal single-parameter curves that hint to the statistics of the photon source. We make a series of measurements on single photons detected from an attenuated laser source and show that the data clearly conforms to the universal curve established exclusively for coherent sources. We also show that the data cannot be fitted to the universal curve for thermal and photon-number sources.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As research in quantum information science and technology continues to grow, so does the demand to accurately determine the photon statistics of light sources [1–6]. In order to do so, one would generally measure the second degree of coherence ($g^{(2)}$) of the light source [7,8]. This is done by passing the light through a 50:50 (balanced) beam-splitter and recording individual counts for each path of the beam-splitter as well as counting coincidences between the two [9–11]. Measuring the coincidences is an elaborate process and requires extreme care to handle the errors and uncertainties due to the optics and electronics involved in such experiments [12–14].
In this Letter, we present a framework to determine the photon statistics using an alternative and potentially simpler method. Our framework relies on using an asymmetric beam-splitter and possibly introducing intentional asymmetrical losses in the non-photon-number resolving detection system. We observe that the counts from one arm of the beam-splitter, when plotted as a function of the counts from the other arm, follow universal single-parameter curves that hint to the statistics of the photon source.
In order to show this, consider the setup in Fig. 1, where the beam is coupled in port $\hat {a}$ of the beam-splitter $\textrm {BS}_\textrm{a}$. The reflected and transmitted photons from $\textrm {BS}_\textrm{a}$ appear in ports $\hat {a}_r$ and $\hat {a}_t$, respectively. $\hat {a}_v$ signifies the unused port (vacuum) and the associated photon operator, which is required to preserve unitarity. The intended procedure can be carried out with only $\textrm {BS}_\textrm{a}$ and counting the photons in ports $\hat {a}_r$ and $\hat {a}_t$. However, in practice, there may be losses associated with each arm, e.g. when coupling to the photon detectors. We model such losses with two additional beam-splitters [15], one on each arm: $\textrm {BS}_\textrm{b}$ and $\textrm {BS}_\textrm{c}$. Therefore, in an actual experiment $\textrm {BS}_\textrm{a}$ is the only physical beam-splitter, while $\textrm {BS}_\textrm{b}$ and $\textrm {BS}_\textrm{c}$ are only used (in theory) to model the losses of photons associated with detections from ports $\hat {a}_t$ and $\hat {a}_r$.
Fig. 1. The conceptual framework of the experiment: the light source is coupled in port $\hat {a}$ of the beam-splitter $\textrm {BS}_\textrm{a}$. The reflected and transmitted photons from $\textrm {BS}_\textrm{a}$ appear in ports $\hat {a}_r$ and $\hat {a}_t$, respectively. $\hat {a}_v$ signifies the unused port (vacuum). The losses are modeled with the hypothetical beam-splitters $\textrm {BS}_\textrm{b}$ and $\textrm {BS}_\textrm{c}$. The non-photon-number resolving detectors are coupled to $\hat {b}_t$ and $\hat {c}_t$ ports. $\hat {b}_v$ and $\hat {c}_v$ are empty vacuum ports, and $\hat {b}_r$ and $\hat {c}_r$ represent the losses in their corresponding arms.
Fig. 2. A schematic of the experiment: the input power of the laser light is controlled by using a half-wave plate, a polarizing beam-splitter, and a linear polarizer. The attenuated laser then enters a 50:50 beam-splitter ($\textrm {BS}_\textrm{a}$). One output path of the beam-splitter goes directly to detector C (Path C), while the other path passes through another LP to introduce an intentional loss and is then collected by detector B (Path B). The pulses from photodiodes are processed by the FPGA, which also synchronizes the laser and the detectors.
2. Theory
In this model, the non-photon-number resolving detectors are coupled to $\hat {b}_t$ and $\hat {c}_t$ ports. $\hat {b}_v$ and $\hat {c}_v$ are empty vacuum ports, and $\hat {b}_r$ and $\hat {c}_r$ represent the losses in their corresponding arms. Each beam-splitter is associated with a pair of amplitude transmission and reflection coefficients: $t_a$ and $r_a$ for $\textrm {BS}_\textrm{a}$, $t_b$ and $r_b$ for $\textrm {BS}_\textrm{b}$, and $t_c$ and $r_c$ for $\textrm {BS}_\textrm{c}$. All the operators indicated in Fig. 1 are proper conventional photon (annihilation) operators and satisfy the appropriate commutation relationships [11]; for example, $[ \hat {a}, \hat {a}^\dagger ]=1$ and $[ \hat {c}_v, \hat {c}_v^\dagger ]=1$.
The input-output relations for the dielectric beam-splitter $\textrm {BS}_\textrm{a}$ can be expressed as [11]
Because we assume that the detectors are non-photon-number resolving, the probability of detecting at least one photon in port $\hat {b}_t$ ($P^s_b$) can be calculated as one minus the probability that no photon is detected in port $\hat {b}_t$ ($P^v_b$). The superscripts $s$ and $v$ indicate detecting a signal or vacuum, respectively. Therefore, $P^s_b=1-P^v_b$ and $P^v_b$ can be calculated as
where the trace operation is performed on ports $\hat {b}_r$, $\hat {c}_r$, and $\hat {c}_t$ over the reduced density matrix projected onto the vacuum state in port $\hat {b}_t$. $P^v_b$ can be calculated by substituting Eqs. (2) and (3) in Eq. (5) and using a trinomial expansion. We obtainIn experiment, it is straightforward to determine $T_a$ and $R_a$, because they represent the imbalance of the typically lossless beam-splitter used in a photon-counting experiment. However, $T_b$ and $T_c$ parameterize the losses and depend on the coupling efficiencies to the detectors as well as the detection efficiencies of the detectors at the given wavelength. Fortunately, neither these parameters, nor $\beta$ are strictly required to determine whether a light source is coherent or thermal. What is important is that Eqs. (9) and (10) are universal single-parameter curves with only one free parameter $\beta$: in the following, we will measure the signal rate in port $\hat {c}_t$ versus the signal rate $\hat {b}_t$, using non-photon-counting detectors, and fit the measured data to Eqs. (9) and (10). We will see that in an unbalanced system where $\beta \neq 1$ the goodness of the fit can clearly favor one to another. It is notable that the limit of $\beta =1$ for the balanced detector results in linear curves of the form $P^s_c=P^s_b$ for both coherent and thermal sources; therefore, the imbalance is critical to distinguishing between coherent and thermal sources.
3. Experiment
In our experiment, shown in Fig. 2, we used a laser light as the coherent light source in an unbalanced beam-splitter setting. The laser was a 532 nm frequency-doubled Nd:YAG with a 0.5 ns pulse duration and a 1 kHz repetition rate. The input power was controlled by using a half-wave plate, a polarizing beam-splitter, and a linear polarizer (LP). The attenuated laser then entered a 50:50 beam-splitter ($\textrm {BS}_\textrm{a}$). One output path of the beam-splitter (in our experiments the path associated with $\hat {c}_t$) went directly to the detector (Path C), while the other path (Path B associated with $\hat {b}_t$) passed through another LP to introduce an intentional loss by creating an unbalanced setup with $\beta < 1$, and was then collected by the other detector. Each detector was a conventional non-photon-number resolving single photon counting module (SPCM). When one or more photons arrived at the SPCM, the embedded silicon avalanche photodiode sent a 25 ns electric pulse that was processed by a field-programmable gate array (FPGA) board and a LabVIEW code based on the procedure explained in Refs. [16,17].
An important characteristic of the avalanche photodiodes is the deadtime, which is the short time delay between one photon detection and the next one [18]. This delay occurs because time is needed to reset the circuit so it is ready for the next photon. For the detectors used in our experiments, the typical deadtime was about 50 ns [19]. However, prior to this experiment, we assessed the resolving time of the detection system, $\tau$, by using two independent light sources and measured the (accidental) coincidence counts [20]. Two independent continuous-wave (CW) He-Ne lasers were sent to the detectors, and the accidental coincidences $N_{acc}$ were measured as a function of single counts $N_1$ and $N_2$ at the two detectors, respectively. There is a simple relation between $N_{acc}$ and $N_2$ for a fixed $N_1$: $N_{acc} = 2\,N_1\,N_2\,\tau$. By measuring the accidental coincidence counts as a function of single counts in the second detector ($N_2$), while keeping the number of single counts in the first detector ($N_1$) constant, we found the resolving time of the detection system to be $\tau$ = 57 ns, which is consistent with the commercially specified value of 50 ns deadtime for the detectors.
Considering the deadtime of approximately 50 ns and the pulse duration of only 0.5 ns, we detected at most one signal count from each laser pulse in our experiment (ignoring after-pulsing), and the maximum count rate equaled the repetition-rate of the laser, i.e. a maximum of 1000 counts per second. We observed that ambient noise was approximately 500 counts per second, which was equal to half of the maximum number of signal counts at full saturation in our setup. In order to address this issue, we synchronized the laser and detector systems through the FPGA and implemented a selection window (SW) within the detector system to reduce the total noise. Having the laser pulses synchronized with the internal clock of the FPGA allowed us to know exactly when the laser pulses (signal) arrived at the SPCM. To remove the dark counts from our data, a SW was implemented such that, when closed, it ignored all detection signal counts coming from the SPCM that we knew to be noise. Considering the 470 ns jitter time of the laser, a 20 $\mu$s SW was implemented to ensure that every laser pulse was being captured. Adding the SW resulted in an overall 50-fold reduction of the noise in the system; the dark counts dropped from 500 counts per second to less than 10 counts per second, making our experiment feasible.
We started our measurements by setting the laser to its lowest power and adjusted the LP in Path B for minimal loss. We then collected data for photon counts from both paths for one minute. We then increased the input laser power to the beam-splitter slightly and collected new data for another minute. This procedure was repeated in incremental steps in input laser power, until we covered the full range from near-zero counts in either detectors to the fully saturated counts (1000 counts per second) in both detectors. The series of steps mentioned so far related to a single value of $\beta$. In order to change the value of $\beta$, we increased the loss to Path B by rotating the corresponding LP by 20 degrees. We then readjusted the input laser light back to the minimum power and followed the above procedure to sweep the full range of powers until reaching saturation at 1000 counts per second. This whole procedure was then repeated for different values of $\beta$, each using an increment of 20 degrees rotation in the LP in Path B. The counts collected at each power level were then divided by 60,000 (1000 counts per second over one minute) to represent $P^s_c$ (Path C) and $P^s_b$ (Path B) and then plotted in Fig. 3. Each data point Fig. 3 represents a $(P^s_b,P^s_c)$ pair, while each color represents a different value for the LP angle in Path B (different $\beta$ values). We also subtracted 20 counts per second from each arm, equivalent of subtracting 0.02 in each axis in Fig. 3, in order to subtract the background noise count, which appeared to persist in all our measurements. The measured data in each set corresponding to a fixed value of $\beta$ have been fitted to Eq. (9), and the agreement appears to be excellent in each case. The fitted curves are plotted over the data points in each case in Fig. 3.
Fig. 3. Measured data (dots) versus theoretical fits to Eq. (9) (solid lines) in an unbalanced beam-splitter using a laser beam as the coherent light source. Different colors show different values of imbalance corresponding to different values of $\beta$. From bottom to top, the fitted curves correspond to the loss induced in Path B using the LP at 0-degree angle (maximum transmission, red), 20-degree angle (green), 40-degree angle (blue), 60-degree angle (purple), and 80-degree angle (black). Excellent fits to Eq. (9) clearly show that the source is coherent.
It is now interesting to see how well the measured data can be fitted to Eq. (10) for a thermal source. In Fig. 4, we show the results for fitting to the same experimental data shown in Fig. 3 using the nonlinear fitting algorithm in Mathematica. It is clear that the more unbalanced the setup, the worse the fit is. The data clearly shows that the source is not thermal and the measurements do not follow the universal form of Eq. (10).
Fig. 4. Similar to Fig. 3 except the solid lines represent best fits using Eq. (10) for thermal light. The low quality of fits clearly indicates that the source is not thermal.
4. Discussion
The sources we have discussed so far are coherent (with Poissonian statistics) and thermal (with super-Poissonian statistics), which are both classical. It is possible to follow the same procedure for sub-Poissonian quantum sources of light. For example, for the photon-number state $|{n}\rangle$, we have $P_n=\delta _{n,\bar {n}}$, where $\bar {n}$ is equal to the number of photons (and obviously the average number of photons). This form of $P_n$ can be used following the same procedure outlined above to obtain the universal curve $P^s_c=P^s_b/\beta$, where $\beta$ is defined in Eq. (11). Therefore, for the photon-number state the $P^s_c$ versus $P^s_b$ always follows a straight line with slope $1/\beta$, regardless of the size of imbalance. Our experimental data clearly shows a significant curvature and is not a sub-Poissonian number state.
In order to decide in an experiment whether a source is thermal, coherent, or a number state using the above procedure, one needs to rely on a metric based on the goodness of the fit in an unbalanced setup. For more general sources where a closed-form expression may not be available, it will be interesting to devise a single-number metric akin to the second degree of coherence $(g^{(2)})$. A concrete connection between the results in this paper and $g^{(2)}$ is yet to be made. Such a single-number metric can possibly be constructed using the curvature of the $P^s_c-P^s_b$ curve to determine the quantumness or classicality of the source.
We point out that we conveniently avoided complications due to pulse overlaps from the detectors, because we used a low repetition-rate pulsed source with a duration substantially shorter than the deadtime of the detectors. If this technique is to be applied to CW sources, issues such as the large dark-count and pulse overlaps need to be carefully considered for the proper interpretation of the measured data. Another important point is that we have not presented any measurements for thermal sources. The main reason was our lack of access to an appropriate thermal source that could be readily used in our procedure. A rotating ground glass can be used to generate pseudo-thermal light at microsecond time scales, which is much longer than the deadtime of our detectors [21–24]. A low pressure gas discharge lamp may be a reasonable source of thermal light for our procedure [25]. As such, testing our procedure with a thermal source will be postponed to future efforts.
Finally, we would like to highlight a few other methods that have been successfully used to determine the statistics of a light source. For example, Refs. [26–28] have demonstrated a design for a photon counting detector that is capable of resolving multiphoton events using a single non-photon-counting detector and a fiber loop. Another notable work is presented in Ref. [29] that is based on on/off avalanche photo-detection assisted by maximum-likelihood estimation. This method is in a similar spirit as ours, given that it uses variable detection efficiency to determine photon statistics with photon counting.
Funding
Directorate for Engineering (ENG) (1522933, 1842712).
Acknowledgments
The authors are grateful to Francisco Elohim Becerra at the University of New Mexico for helpful consultations.
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