Teachspin - Pulse Counter/Interval Timer

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Pulse Counter/Interval Timer

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"TeachSpin's Hat-Trick" - Introducing Fourier Methods, UltraSonics, Pulse Counter/Interval Timer

Introduction

Lab Topics:
Measurement
Thermodynamics & Statistical Physics

Hall Effect Probe

TeachSpin’s ‘ Two-Slit Interference, One Photon at a Time’ has become a classic demonstration of the ‘essential quantum paradox’ and an experiment in single-photon detection. Now, we’ve improved its electronics with a new Pulse Counter/Interval Timer unit which makes possible a whole new set of investigations in the statistics of random-event processes.

Our NEW Pulse Counter/Interval Timer unit will:

discriminate pulse events from a noise background, for pulses of amplitude 10 mV to > 1 Volts;
correctly count ‘events per unit time’, with 0.1, 1.0, or 10-second counting intervals;
alternatively, measure the time interval between a pulse and its successor pulse, to 1- µs resolution;
and write data files of unlimited length (via HyperTerminal, in a host computer) of successive counts per unit time, or intervals between successive counts.

An electronic discriminator, designed to detect even low-level analog pulses standing up above a noise background, is built into the counter/timer unit. The discriminator’s threshold level can be adjusted continuously and linearly over a range of about 10 mV to several Volts. A monitor output gives a view of exactly what input pulses are meeting the discriminator’s threshold criterion. Figure 1 shows the discriminator in action.

Fig. 1: Oscilloscope traces obtained at 10 ns/div
Upper trace: analog input to the discriminator;
Lower trace: pulses at monitor output of discriminator. The ‘scope is set to trigger on the lower-trace events,
so the upper trace displays all (but only) the pulses which have activated the discriminator

The discriminator’s output is a set of standardized pulses, each of which arises from a PMT pulse meeting an electronic criterion. But the time of occurrence of those events ought to obey the laws of a Poisson process – they are expected to occur uncorrelated with each other, and randomly in time, but still at some average rate. Most any counter will do for measuring just the average of the count rate in the study of single-photon events. However, for studying fluctuations in count rates, and therefore the statistical properties of random events, a long series of such counts is needed. For example, suppose we are counting the photons that arrive when the two-slit apparatus is set to monitor the intensity at the top of the ‘central fringe’ of an interference pattern. If we adjust source intensity to give an average rate of about 1000 counts per second, what will we find if we do many repeated individual measurements? The answer is shown in Fig. 2, which is a scatter plot of the pulse counts for 1200 successive 1-second trials.

Fig. 2: Scatter plot of number of photon events detected in 1-second gate times, plotted
as a function of observation number, for 1200 successive observations

The data display statistical fluctuations about the mean, of 1086 counts/second and students can process the data files from our new counter to study these fluctuations. Questions they should explore include:

What is the standard deviation of the count rate?
How does that result depend on the (variable) average count rate?
What does the chi-squared test have to say about the data in Fig. 2?
What is the histogram of the data in Fig. 2? (See Fig. 3.) Is this distribution Gaussian, and does it have the width expected from Poisson statistics?

Fig. 3: Histogram of the data of Fig. 2, showing also the Gaussian distribution expected
from the central-limit theorem distribution expected from the central-limit theorem

Back to the plot in Fig. 2: it shows counts arriving at an average rate of about 10³ /s, which means that counts are arriving at average time interval of 10^-3 s or 1 ms.

But, if that time interval between a pulse and its first successor is measured repeatedly, what individual values will result? And what distribution of values will result from repeated series of such measurements? Do you know what histogram those values will have?

Our new Counter/Timer can be configured to measure such a time interval, and to do so repeatedly, and write the results to a file, easily giving a sample of (say) 5000 time-interval measurements. In such a sample, we found (for example) nearly 1000 occurrences of time intervals in the 1-200 µs range. The histogram of all 5000 measured intervals is shown in Fig. 4.

Fig. 4: Semi-log plot of the histogram showing the distribution of 5000 measurements of time
intervals between a photon event and its first successor; horizontal axis in microseconds

The histogram is not Gaussian, but exponential, in character. The mean interval implied by this distribution is 0.910 ms, to be compared with the value 0.921 ms expected from the average count rate from Fig. 2. But the histogram does not have a peak at that value; in fact, the shortest time intervals are the most probable.

Note too, that the degree to which the distribution matches the expected exponential is a test of the random-event hypothesis: any correlation between photon-detection events, or any periodicity in their arrival, would show up directly in this plot.

There are many other sources of electronic events, some highly (but not perfectly) regular, others nearly (or wholly) random, and all of them susceptible to statistical study. You might think of a not-quite-perfect clock, or a Geiger counter, as sources of such events. The pre-requisite for such statistical studies is a large sample of data, and we’ve now made that as easy, and straightforward, as possible to obtain. We’ve paid attention to transparency of operation of our new counter/timer, ensuring (for example) that data can be taken by hand for a first mode of understanding of what’s going on. We feel sure that students would do best to take a first (small) data set manually, so that they understand the process of abstraction and aggregation that lies between a list of data and the histogram derived from it.

Returning to the Two-Slit experiment, we want students to be able to see that the photon events they’re detecting, whether at an interference maximum or a minimum, still display all the statistical properties expected for events that are random and uncorrelated, but nevertheless occurring at a well-defined mean rate. And this invites an even deeper look into the essential quantum mystery.

Finally, we expect that this counter/timer will have uses far beyond the Two-Slit experiment. We invite you to come up with some of them.

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