Coincidences

# Coincidence Counting Project

## Purpose

An important part of understanding the larger quantum picture is to understand the coincidences that are an integral part of the quantum experiments[1]. Primarily, the purpose of this coincidence project is to provide students preparing to study the interferometer with a better understanding of the procedure and expected results of coincidence counting experiments. Also, with the coincidence counting data, it is possible to comment on the validity of the coincidence equations previously formulated.

## Introduction

In order to understand the experiments that rely on coincidence counting, first it is important to understand what a coincidence actually is. A coincidence is defined to be n events that occur nearly or completely simultaneously. Coincidence counting involves two or more detectors connected to an electronic coincidence circuit. The detectors send out a signal in the form of pulses. These pulses have a uniform pulse width, which is referred to by the greek letter tau, and is not measure in units of length as you might imagine, but rather in units of time. In a situation where two-fold coincidences are being counted, the pulses can overlap normally like so:

or the leading edge can coincide directly with the trailing edge,

Having a good understanding of the ways pulses can align will help when we consider the coincidence formulas for 2-fold and especially 3-fold coincidences. A coincidence is when two or more events happen simultaneously, but an accidental coincidence, which is the point of interest for this project, is when two events occur simultaneously but are causally unrelated (in other words, completely independent) of each other. So, sources that rely on each other to produce counts are not suitable for the experiment detailed on this page; radioactive decay, on the other hand, is perfect since the sources decay completely randomly, and it is in fact what we used.
Pulse shortening is another important idea in coincidence counting. The pulse width is directly related to the number of accidental coincidences we see, so it needs careful consideration. The DE2 is equipped with switches that can shorten the pulse width from 132 x 10-4 seconds to 4 x 10-9 seconds, which is a large amount. This can greatly reduce the number of accidental coincidences collected.
The phenomenon of dead time is the last concept that really begs an introduction. When an electrical system is running, there is a small window of time- between the instant when the system accepts incoming data and passes it on and the instant that it begins accepting more incoming data- when the system will not detect incoming information. This small window is called the dead time. However, simply because the system stops detecting pulses during this time, it does not mean that the pulses stop coming. And if pulses are coming through the system that are not being detected, we are missing data! If we are missing data, we’re not getting an accurate picture of the actual total number of counts.

## Accidental Coincidence Formula and Related Equations

There are a few important formulas needed to calculate and describe coincidences. First, the Anti- Correlation Parameter (as discussed previously) is useful when determining whether an event is purely classical, or something else entirely. If the Anti-Correlation Parameter is calculated to be 1, then the behavior we are seeking to define can be described completely classically. However, if this equation returns anything less than 1, classical descriptions fail. As the Anti-Correlation Parameter goes to and reaches zero, quantum descriptions are needed[1]. The formula for the Anti-Correlation Parameter is

(1)
\begin{align} A = {{N_c} \over {N_1N_2}}{T \over {\Delta}t} \end{align}

where T is the time or duration of the experiment itself and ${\Delta}t$ is the time resolution. Essentially, the time resolution is a property of the mechanical system we use, and it tells us how far apart incidents (events that trigger the detector) must be for the system to register them. Thus, the term $T \over {\Delta}t$ tells us how many experiments (and coincidences) could possibly take place. For further discussion of the use and meaning of the Anti-Correlation Parameter, see the single photon detection experiment page. Next, we have several formulas to express coincidences specifically. First, we have the formula for accidental coincidences,

(2)
\begin{align} N_{accidental} = 2{\tau}N_1N_2 \end{align}

This equation shows the relation between the number of counts from a first detector ($N_1$), the number of counts from a second detector ($N_2$), and the pulse width, or resolving time of the coincidence counting circuit tau. Since it is possible to have two pulses of width tau line up right at the edges to produce a coincidence, we take that into account by doubling the pulse width represented in the formula. Equation 2 can also be written as

(3)
\begin{align} R_{accidental} = 2{\tau}R_1R_2 = 2{N_1 \over t}{N_2 \over t}{\tau}<<N_{coincidences} \end{align}

This version of the formula simply relates the rate of accidental coincidences to the rates at which the first and second detector are getting counts. One important thing to remember is that these formulas only hold for counts (from two different detectors) that are completely independent of each other. For my first experiments, the 2-fold coincidence formula (shown as Eqs. 2 and 3 above) was important. However, the next experiments I performed involved 3-fold coincidences. Thus, the 2-fold coincidence formula needed to be revised. From working with the 2-fold formula, we knew that we needed to incorporate the pulse width into the equation. However, for 3- fold coincidences, we do not have a simple two pulse overlap to consider. The three pulses entering the DE2 can align in many ways; thus, we incorporate this by using a coefficient of 3${\tau}$2. Knowing this, we can modify the 2-fold coincidence formula to apply to 3-fold coincidences. The equation for this is

(4)
\begin{align} R_{real} = 3{\tau}^{2}R_{1}R_{2}R_{3} \end{align}

These equations are correct, but with one small oversight. With the full coincidence counting circuit, there will always be some form of noise in the system that can give false counts and thus false coincidences. These false counts are referred to as dark counts, and we need to take them into account, and thus adjust our previous formula. Thus, for 2-fold coincidences we have

(5)
\begin{align} N_{real} = N_{coincidences} - 2{\tau}(N_{1dark}N_{2dark} + N_{1dark} N_{2} + N_1N_{2dark}) \end{align}

Here, $N_{real}$ will be the number of real coincidences (i.e. not due to dark counts or noise within the system), $N_{1dark}$ will be the number of dark counts coming through the first detector and $N_{2dark}$ will be the number of dark counts coming through the second detector. This equation is not appropriate for the radiation coincidence experiments discussed in this section because for these experiments, there is no simulation of dark counts. However, this equation becomes very important when working with the Single Photon Detection Module.

In addition to these coincidence formulas, I also conducted an experiment to determine the dead time of the system I used. The equation that I used to find the dead time is as follows

(6)
\begin{align} R_{reall} = {R_{measured}\over(1- R_{measured}{\tau})}} \end{align}

where Rreal is the actual rate of counts detected, Rmeasured is the measured rate of counts detected and ${\tau}$ is the dead time[3].

## Experimental Set-Up

There was a lot of set up involved before actually beginning the data collection process. First, I had to obtain a working computer. I used a Dell PC to do all the coincidence counting experiments. I found that it is important to be able to install hardware onto the computer and thus, having administrator privileges on the computer used is extremely helpful. Once I had the computer up and running, I needed to install a few different programs onto the computer. I needed two programs to run the coincidence counting unit (CCU), LabView and Quartus II. The DE2 is a preassembled programmable development board that is very convenient, since we simply download the programs or files to it that we need, and no further programming is required. With this in mind, I installed the Quartus and LabView design files needed to run the CCU through the Altera DE2. These files allowed me to program the DE2 to detect coincidences, and they were developed by Jessie Lord, of Whitman College. Additionally, Marc Beck is responsible for the overall coincidence counting unit. I followed several manuals (written by Whitman College) to enact the steps above[2]. For more information on the CCU, see the detection system page in the development section.

I experienced some trouble once I had programmed the Altera DE2. When I ran the newly installed coincidence counting circuit, the LabView program failed. Eventually, we narrowed the problem down to the fact that the computer was not detecting the connection between the COM port of the Altera DE2 and the computer system. We tried to solve this problem in many ways. First, I switched out the RS232 cable to make sure that the wiring was not the problem. It wasn’t. The next step would be to take the Altera DE2 itself to another computer to ensure that the board was programmed correctly. However, before resorting to this, we found that another application on the computer system was using the COM port and once we disabled that application, the circuit ran properly.

The next important piece of the coincidence counting circuit is the voltage converter box. When we performed similar coincidence counting experiments last semester, we ran into trouble converting the 5.0 V pulses output by the Single Photon Counting Modulus into the 3.3V logic required by the Altera DE2, and originally needed to run the 5.0V pulses through a separate circuit, using several resistors to divide the voltage to 3.3V. However, this semester, and towards the end of the last, we were able to use a premade adaptor box that performed the same function, without the need for a separate circuit. This box has 4 input channels (A, B, A’, B’), which gather data from up to four detectors. With those, the coincidence counting circuit is able to detect 2-fold (i.e. AB, A’B, etc), 3-fold (i.e. ABB’), and 4-fold (i.e. ABA’B’) coincidences; switches on the DE2 board determine which channels are being "watched" for coincidences.

Now, we need to add the source of the coincidences. For my purposes this semester, I used radioactive decay- specifically multiple Radium samples undergoing β-decay- instead of using single photons to create coincidences and interference (as seen in the single photon interference section). Digital radiation monitors plug directly into the adaptor box. The radioactive source is placed in a position so that the radiation monitor will detect decay, but the specific location and distance of the source from the monitor can vary. I began by using only two digital radiation monitors at first, since I felt it best to start with 2-fold coincidences. However, I then added a third detector to measure the effect that 3-fold coincidences have on the validity of the coincidence equations.

One important feature of my model of the premade adaptor box are the switches which control the pulse length of the signal from the box sent to the DE2. Each channel has a switch located on top of the box, which allows the pulse to be read either at the tip of the (plug) or across the ring. When the signal is read at the tip, the pulse length is shorter than when the signal is read across the ring. This difference in pulse length will be important when detecting coincidences, since it will be more unusual to see a coincidence when the pulses are shorter. This can also be instrumental in helping establish a good idea of whether the coincidence counting formulas hold for different pulse lengths.

However, once data collection began, it became clear very quickly that there was a problem. The digital radiation monitors and the DE2 were not in agreement on the number of counts. A small difference can be expected, primarily due to the efficiency of the system. However, the difference was incredibly large, and definitely too large to be explained away. Upon preliminary evaluation, the immediate cause of the problem could not be determined.

In light of the problems we now have with the new version of the adaptor box, we needed to implement a new experimental set up. Still using the same DE2, we switched out the new adaptor box for an older box, which we know from previous experiments to be functioning correctly. This older adaptor box is not compatible with the digital radiation monitors we had been using, so we subsituted RadAlert radiation detectors for the digital detectors. The RadAlert detectors themselves are only able to detect counts by the minute and so I had to set the LabView software to detect the number of counts per 60 seconds. This way, the count read off the detector should be very close or equal to the counts detected by the DE2. Running several tests, I found that this new set up was working correctly. Below is an image of this set up. In it, you can see the DE2, the older adaptor box, and three RadAlert detectors.

#### Dead Time

The next experiment I ran was a dead time experiment. In order to find the dead time I used multiple sources with only one detector. The main idea of the experiment is to measure counts entering a detector from one source at a time, for each of three different sources, and then measuring the counts entering the detector from all three sources at once. For this experiment, it was important to be able to reproduce everything. The placement of each source needed to be the same when taking counts from that source alone and also when taking counts from all three. We found that in order to be able to see a noticeable effect, the detector had to be getting atleast 300 counts per second from each source[3]. First, we tried to use 3 ${\beta}$-sources placed at 3 different locations, but we were not getting enough counts. So, we substituted a ${\gamma}$-source for the third ${\beta}$-source. The two ${\beta}$-sources were placed directly against the detector, one over the sensor and one at the side. The ${\gamma}$-source was placed a small distance from the detector, since the radiation is much more penetrating. All positions were clearly marked so that the experiment could be completely recreated. In theory, if the system did not experience an effect due to dead time, we could expect the sum of number of counts recorded for each individual source to be equal to the number of counts recorded for all three sources together. However, if we do not see this, it means that there is an effect due to the dead time, which we can find using our results and Eq. 6 above[3].

## Data and Results

Once we implemented the experimental set up that functioned properly, we were able to collect 2- and 3- fold coincidence data. For each setup, I ran 10 tests, one test being 60 minutes long, taking one data point per minute. I did this for the two versions of the 2-fold coincidence counting set up (with and without pulse shortening) and also for the 3-fold set up.

One true disadvantage of not being able to use the new adaptor box is that now, the pulse shortening switches built directly into the adaptor box are not accessible. The new box allowed the pulse width of each channel to be changed individually, so that we could test the rate of 2-fold coincidences between two long pulses, two short pulses, and a short pulse and a long pulse, as well as the applications for 3- and 4-fold coincidences. The number of coincidences between two long pulses should be higher than the number of coincidences between two short pulses, but now we are not as easily able to determine the exact relationship that exists between them. This is not lost to us, since we do have an alternative method of pulse shortening. The DE2 comes complete with 2 switches that effectively change the pulse width. It is not as extensive as the adaptor box was built to be, but it offers us another opportunity to determine the relationship between a changing pulse width and the rate of accidental coincidences.

So, as I said, first I looked at 2-fold coincidences, with no pulse shortening; thus, the pulse width was 132 microseconds. The results that I obtained were

After that, I worked on 2-fold coincidences with pulse shortening. The results are below.

Finally, I worked with 3-fold coincidences, and found

For all three experiments, the standard deviation of the mean was quite small, which showed us that the experimental results verify the accidental coincidence equations.

For the dead time results, I found that there was a very noticeable difference between the sum of the number of counts taken from each source individually and the number of counts taken when all three sources were placed together. From the data I collected, I found the dead time to be 7.5 x 10-4 seconds.

References
1. Greenstein and Zajonc. The Quantum Challenge. Jones and Bartlett Publishers: Massachusetts. (2006)
3. Jackson, D. Lectures given during 2 meetings of PHYS 492. (2008, November)
page revision: 119, last edited: 26 May 2009 07:49
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