1. Particlelike Behavior in Photons
Parker Henry
April 19, 2015
Abstract
The purpose of this experiment is to demonstrate nonclassical, i.e.
particle-like behavior in light. This is achieved using two test detectors,
which will detect incoming photons after a second photon triggers a third
control detector. The classical theory of light as a pure wave demands that
both test detectors are triggered, whereas the quantum theory of light
consisting of particles called photons demands that only one or the other
is triggered. A value g is experimentally determined, which is proportional
to the ratio of instances in which both test detectors are triggered by the
light to the number of instances in which the control detector is triggered.
This paper argues that g must be greater than or equal to 1 under the
classical theory of light, but that g must be 0 or substantially less than
1 under the quantum theory. A value of g = 0.088(10) is experimentally
obtained, thus demonstrating that light behaves in a particlelike manner
in this experiment.
1 Introduction
This experiment seeks to provide experimental evidence that photons are parti-
cles. This will be achieved by shooting two beams of light, one at a light-detector
A, and the other polarized beam at a half-silvered polarizing crystal which will
split the beam to be detected at either a detector B, a detector B’, or both (see
Figure 1). The two beams will be set up such that one hits A, and the other
hits either B, B’, or both, at the same time. The event of the first light beam
triggering the first detector will trigger the detection of the light beam at B or
B’, or both. A ratio g will be measured by computer software linked to the
detectors, and g will satisfy the below proportionality relation:
g ∝
number of events where B and B’ are simultaneously triggered
number of events in which A is triggered
If the photons are wavelike, then B and B’ should be triggered simultaneously
each time A is triggered, i.e. g would be measured to be 1. If the photons are
particlelike, then either B or B’ will be triggered, but never both, so g would be
measured to be 0. As as a result of the following experiment, g will be measured
to be nearly 0, thus convincingly arguing that photons behave like particles in
this setting.
1

2. Figure 1: Schematic of the trajectory of the lasers
2 History of the Theory of the Photon
Historically, physicists have debated whether light is a wave or if light is made
of particles. The classical development of the theory of light culminated in
Maxwell’s equations, when Maxwell infered that light is an electromagnetic wave
with speed c, after having found that electromagnetic waves travel at nearly the
same measured value of the speed of light in 1862, 2.98 × 108
m/s.
However, the wave theory was not entirely satisfactory in explaining light
phenomena. For instance, in the 1880s, Heinrich Hertz demonstrated the pho-
toelectric effect, by which light waves can produce a flow of electricity. At
500nm wavelength, he found that even a faint light could eject electrons from
sodium, but no matter what intensity light was used at 600nm, no electrons
would be emitted. The wave model predicts that wave intensity should be the
only variable affecting how many electrons are emitted. Thus, the wave model
was completely unsatisfactory in explaining the photoelectric effect. [1, p. 76]
In 1905, Albert Einstein hypothesized that visible light is quantized as dis-
crete particles, and named the quantum of visible light the photon. He posited
that the energy of a photon was given by
E = ¯hf
where ¯h is Plank’s constant over 2π, and f is the frequency of the photon.
Furthermore, because the proportionality of energy to frequency implies an
inverse proportionality to wavelength, under the Einstein theory, the 500 nm
laser carried more energy than the 600 nm one. Thus, the 600 nm laser photons
did not carry enough energy to trigger the photoelectric effect, whereas the 500
2

3. nm laser did. This was a much more satisfactory explanation of the photoelectric
effect than what the wave theory of light offered.
Furthermore, with Plank’s theory of the quantization of black-body radi-
ation, Einstein showed that if this theory is true, that photons must have a
definite momentum
p = ¯h/λ
where λ is the wavelength of the photon [2]. This implies that photons are prop-
erly described as particles, rather than waves. Einstein theorized that they also
acted in a wavelike manner in the context of electromagnetic theory. This ren-
ders his theory consistent with Maxwell’s theory of light as an electromagnetic
wave.
In summary, photons exhibit wave-particle duality, in that, depending on
the situation or the experiment, they may behave as particles or as waves [1,
p. 73]. For instance, Thomas Young and Augustus Fresnel showed that light
refracts and it shows interference, in a wavelike manner. Plank’s blackbody ra-
diation and Einstein’s theory, however, assert that it can also act in a particlelike
manner.
It is the aim of this experiment to demonstrate that photons can be measured
to have definite position, and thus exhibit particlelike behavior.
3 Classical and Quantum Theories of Photon
Detection
This discussion follows that of Beck ([3]) to elucidate the theory of this experi-
ment.
If the light detected in this experiment is completely wavelike, i.e. it is
completely described by Maxwell’s equations, then the detection of the light by
the detectors is solely a function of the light intensity.
Let IB(t) denote the intensity of the B beam at time t, IB’(t) denote the
intensity of the B beam, and IA(t) denote the intensity of the A beam (refer to
Figure 1). In terms of intensities, if g is the second-order coherence of detectors
B and B’, then g is given by
g =
IB(t)IB (t)
IB(t) IB (t)
(1)
for the beams entering B and B (see Figure 1). Now let II(t) denote the
intensity of the beam entering the beam-splitting crystal. In classical wave
theory, if τ and ρ are the transmission and reflection coefficients of the crystal,
then
IB(t) = τII(t)
IB (t) = ρII(t)
3

4. Substitute these expressions to see that, classically,
g =
[II(t)]2
II(t) 2
But for any random variable X,
X2
≥ X 2
,
so classically,
g ≥ 1 (2)
Therefore, even if it is the case that by some experimental error, g manages to
be greater than 1, it is mathematically impossible that g is less than 1 under
the classical theory of light. One also notes the contrapositve of this statement,
which is that if g < 1, then the photons are not behaving classically.
More generally, in the semiclassical theory of photon detection, the number
of counts that the detector B yields in a time interval ∆t at time t is a random
variable, proportional to the average intensity of the beam striking B, IB(t) ;
the probability of obtaining one photocount in a time window ∆t at time t is
PB = ηB IB(t) ∆t
where ηB is some constant.
The joint probability PBB of detecting a photon at time t at detector B in
a time window ∆t and then detecting one at B’, again in a time window ∆t, is
PBB’ = ηBηB IB(t)IB (t) ∆t2
We combine the above equation with equation 1 to see that
gB,B’ =
PBB’
PBPB’
(3)
If the photons are behaving classically, then the reasoning of equation 2 applies,
and we see that
gB,B’ ≥ 1
In terms of the experiment, the probabilities must be expressed in terms of
the count rates of the detectors: if T is the time duration of making counts, NB
is the number of detections by detector B, and ∆t is a small time interval, then
the probability PB of detecting a count at detector B in the time window ∆t is
PB = NB
∆t
T
Similar reasoning for PB and PBB allow us to rewrite equation 3 as
gB,B’ =
NBB’
NBNB’
T
∆t
(4)
4

5. With three detectors, A, B, and B’, the detection events at B and B’ are not
reported unless detector A detects a photon. Denote by PBB |A the conditional
probability that both detectors B and B’ detect a photon, given that detector A
detects a photon, and define PB|A and PB |A similarly. Analogously to equation
3, we see that
g =
PBB |A
PB|APB |A
(5)
The conditional probabilities defined above are normalized by the total num-
ber of trials for a photon to be detected at a detector, i.e. if NA is the total
number of photons detected at A, and NAB is the total number of photons
detected at B whenever a photon was detected at A, NAB is the total number
of photons detected at B whenever a photon was detected at A, and NABB is
the total number of photons detected at both B and B whenever a photon is
detected at A, then
PB|A =
NAB
NA
, PB |A =
NAB
NA
, PBB |A =
NABB
NA
We take these and substitute them into equation 5 to finally obtain
g =
NANABB
NABNAB
(6)
Again, if the photons behave classically, then g may be rewritten in terms of
intensities, from which it is seen that g ≥ 1.
However, if a quantum model of the photon is assumed, then g must be zero,
because for the photons to be detected by the detectors, the detectors must make
a definite measurement of the position of the photon. This position must be
at one detector or the other, but never both. Even if g is not measured to be
completely zero, due to some confounding factor, a measurement of g which is
less than 1, contrary to the assertion in equation (2), will unequivocally show
that photons are not behaving in a wavelike manner at all in this experiment,
and that they are behaving in a particlelike manner.
Because the detectors are set up to count the number of photons detected
by detectors A, B, and B’, which can be conditioned on whether a photon is
detected at B or B’ within a time interval ∆t (which will be 4.76 nanoseconds
in this experiment) after detecting one at A, the quantity g defined in equa-
tion 6 is the quantity which will be experimentally determined in the following
experiment.
4 Experimental Theory of the Measurement of
g and gB,B
Having previously obtained gB,B in equation 4, it is tempting to simply ex-
perimentally measure gB,B , check if gB,B is less than 1, and thus attempt
contradict the hypothesis that the photons are behaving in a wavelike manner.
5

6. However, as following discussion shows, if this procedure is attempted, then
gB,B will remain greater than or equal to 1. This neither proves that photons
behave in a wavelike manner nor does it prove that they behave in a particle
like manner; if photons behave classically, then gB,B ≥ 1, but the converse does
not necessarily hold.
If gB,B is measured to be greater than 1, then one might think that varying
the size of ∆t would help to reduce it to a value less than 1. However, this will
not suffice; NB,B increases if detector B and detector B are triggered within
time ∆t of each other, so if ∆t is close to T in equation 4, and if T is large, then
NB,B = NBNB , so gBB will asymptotically approach 1. If ∆t is infinitesimal
in size, then NB,B may be approximated by
NB,B = NBNB
∆t
T
as may be derived by using the formulae which state that PB,B = PB ∗ PB |B,
PB,B = NB,B /T, PB = NB/T, and PB |B = NB ∗∆t/T, when ∆t is sufficiently
small. So equation (4) approaches 1 in the small-∆t limit. Thus, measuring
gB,B as such is completely useless in demonstrating particlelike behavior in
photons; a different experimental quantity is needed. As will be seen in the
experiment, that quantity is g as defined in equation 6.
It should be noted that g will not be completely zero. Indeed, PBB |A =
PBPB |A, and if ∆t is very small, then
NABB = NBNAB
∆t
T
so
g =
NANB
NAB
∆t
T
This quantity will not be zero in general (unless ∆t = 0, but then no mea-
surement would be taken at all). However, if it can be shown experimentally
that g < 1, then the photons in this experiment will be demonstrated to have
particlelike behavior.
5 Experimental Setup
Referring to the schematic of the beam trajectory in Figure 1, from the laser
source, a 405 nanometer beam of light is emitted into a reflecting mirror, which
is then sent into a crystal.
Most of the photons in the laser beam will pass through the crystal in the
same trajectory as before, with minimal distortion. However, approximately 1
in 1010
photons will undergo a process known as down-conversion. The precise
physical theory of down conversion is not yet well-understood, but the effect of
the down conversion is that one photon enters the crystal and two new photons
at a 3 degree angle to the original trajectory are emitted. These two photons will
6

7. Figure 2: Schematic of the detectors hooked up to the TAC (the time splitter).
each have approximately double the wavelength of the original 405 nanometer
photon, or 810 nanometers, as mandated by energy conservation.
It is these two trajectories of the two down-converted photons which are
aligned with detectors A and with a polarizing crystal. We refer to these beams
as the A and B beams, respectively.
The B beam is polarized with a half-wave plate, and the polarization is
rotated by various angles with a rotating filter. This beam is then put through
a polarizing beam splitting crystal, which will either transmit the beam through
or reflect it by 90 degrees, depending on the polarization. These two new beams
are sent to the detectors at B and B’.
When detector A detects a photon, the computer software checks to see if a
photon was detected at B, or B’, or both, within an interval of 7 nanoseconds.
The software then records whether both B and B’ detect photons, and repeats
this over an interval of approximately 0.1 seconds.
6 Verification that Photon Signals are Coinci-
dent
Before the detection process can be carried out, it must be verified that the
down-converting crystal is producing the two photons at the same time as one
another. Experimentally, this is equivalent to requiring that a detection event
at A and a detection event at either of the B detectors correlate in time, that
is, that if A detects a photon, then one of the B detectors detects a photon,
within a short enough time interval, and vice versa. In this experiment, this
7

8. Figure 3: Output of the Multichannel Analyzer (MCA).
verification was achieved by means of a time-to-amplitude converter and a mul-
tichannel analyzer. The basic layout is depicted in Figure 2. The TAC and
MCA apparatus may loosely be referred to as a time-splitter.
The time-to-amplitude converter (TAC) outputs a voltage proportional to
the time difference between a pair of pulses which are inputted into the START
and STOP inputs of the TAC. After detectors A and B detected photons, the
output of one detector is wired to the START input, and the output of the
other is delayed by 88 ns, then input into the STOP input. If the time delay
of 88 ns is not used, then the voltage pulses produced by the TAC are near 0V,
and cannot be distinguished from random noise.
The output voltage is sent to a multichannel analyzer (MCA), which is what
indicates the coincidence of the photon signals. The output of the MCA is
shown in Figure 3. The time range of the TAC is set to 200ns, for which the
output voltage pulse is 10V, and the MCA has a voltage range of approximately
8.2V. So the total range in time for the MCA is
8.2V ∗ 200 ns/10V ≈ 164 ns
Note the spike in the number of counts in Figure 3. The full width of the
peak, which encloses the upper half of the maximum of the peak, is approx-
imately 50 channels, plus or minus 10 channels. The MCA is set for 8192
channels, so the time which the peak indicates that the pulses are coincident
for is
50(10)/8192 ∗ 164ns ≈ (1.0 ± 0.2)ns
8

9. Thus, the time-splitter verifies that the pulses are cooincident to within ap-
proximately 1 nanosecond. This shows that the down conversion in the down-
converting crystal produces photons which are emitted within 1.0±0.2 nanosec-
onds of each other. Given that the experiment uses an interval of 7 nanoseconds
to detect photons after a photon is detected at A, the 1.0 ± 0.2 nanosecond co-
incidence of the emission of the photons via down conversion is acceptable for
this experiment.
7 Experimental Procedure
The experimental procedure consists of two phases: aligning the optical equip-
ment to properly detect the lasers, and then turning on the laser to collect
data.
To set up the experiment, an 810 nanometer red laser is inserted through the
backs of the detectors to assist in lining up the trajectories of the 405 nanometer
blue laser. Both detectors are first placed such that the red laser comes out of
the polarizing crystal at the same place. This is most easily done in a very
dimly lit room. The red laser laser is again used to place the half-wave plate,
the rotating polarizing glass, and the down-conversion crystal, such that the
blue laser, which will be used to collect the data, will indeed hit detectors A,
B, and B’.
Having properly aligned the equipment, in a completely dark room, the blue
laser is now turned on, and the detectors are wired to a computer which collects
the data on counts made by the detectors. In our setup, a filter was placed over
the detectors which blocks out green wavelengths, so a green light was on to
aid with visibility without interfering with the experimental results. However,
the windows and the door were completely covered, and the computer monitor
was shielded from the experiment by a thick black cloth, so that the room was
otherwise completely dark.
With the blue laser turned on, the computer program which monitors the
detectors is set to detect photons for a period of 1 second, and it automatically
gives the correlation coefficient g. The program does this ten times per data
file, and it also returns an average g.
In the interest of observing the effect of the polarization of the laser beam
on the value of g, this procedure was carried out with different polarizations in
the rotating polarizing filter, of 10, 20, and 30 degrees.
This experiment is carried out using not only three detectors to measure
g, as given in equation 6, but also using only detectors B and B’ to measure
gB,B , as given in equation 4, to demonstrate the discussion that measuring
gB,B yields no conclusion of interest.
9

10. Table 1: Table of Measured g
Angle of Polarization (degrees) Average g
10 0.088(10)
20 0.071(51)
30 0.069(50)
Table 2: Table of Measured gBB
Angle of Polarization (degrees) Average g
10 1.25(42)
20 1.11(03)
30 1.11(04)
Table 3: Sample counts obtained at a polarization of 10 degrees (the mean
g = 0.088(10)).
NA NB NB NAB NAB NABB g
2,468,083.000 1,230,619.000 255,573.000 121,213.000 16,351.000 76.000 0.095
2,464,934.000 1,230,038.000 255,072.000 121,549.000 15,831.000 62.000 0.079
2,455,003.000 1,229,433.000 253,870.000 121,133.000 16,041.000 79.000 0.100
2,459,668.000 1,229,690.000 254,581.000 121,351.000 16,098.000 78.000 0.098
2,468,302.000 1,231,346.000 254,529.000 121,029.000 16,120.000 69.000 0.087
2,464,590.000 1,229,359.000 254,283.000 121,119.000 16,158.000 65.000 0.082
2,468,404.000 1,230,850.000 255,158.000 120,883.000 16,076.000 66.000 0.084
2,468,365.000 1,231,082.000 255,309.000 121,757.000 16,359.000 64.000 0.079
2,472,019.000 1,229,520.000 255,650.000 121,396.000 16,033.000 81.000 0.103
2,468,130.000 1,232,493.000 255,346.000 121,484.000 16,148.000 59.000 0.074
10

11. Table 4: Sample counts obtained at a polarization of 10 degrees, in calculating
gB,B , with mean gB,B = 1.26(42). Note that this is insufficient to show either
particlelike or wavelike behavior, as discussed below.
NB NB NBB gB,B
571,940.000 117,487.000 242.000 2.437
1,228,001.000 255,508.000 550.000 1.186
1,227,122.000 255,463.000 541.000 1.168
1,229,458.000 254,767.000 525.000 1.134
1,230,179.000 256,124.000 549.000 1.179
1,230,055.000 255,809.000 546.000 1.174
1,227,588.000 255,510.000 507.000 1.094
1,228,227.000 254,064.000 506.000 1.097
1,227,778.000 255,828.000 456.000 0.982
1,230,277.000 253,744.000 524.000 1.136
8 Results
The results of measuring the values of g at various angles of polarization in the
rotating polarization filter are shown in Table 1. These values are each averaged
over 10 instances in which the values of g are determined by computer, by
counting the number of detection events, as discussed when deriving equation
6, and then using these counts to obtain a value of g. The output of such a
process is shown in Table 3, which was the result of measuring g at a polarization
of 10 degrees.
Of the three values of g for 10, 20, and 30 degrees, in Table 1, the most
precise measurement is that found for 10 degrees, which is g = 0.088(10). We
take this to be the experimentally determined value of g, because the computer
software places relatively large uncertainties on the other two measurements;
large enough such that g = 0.088(10) falls within 1 standard deviation of them;
thus, no effect of the polarization on g could be determined.
The uncertainties in the above table were also automatically found by the
detection tool, together with the average values of g in a trial. g is many
standard deviations below 1, whereas the classical wave theory of light demands
that g be 1. Therefore, this phenomenon cannot be adequately explained if the
photons behave in a wavelike manner. This is a strong argument in favor of the
quantum theory of the photon.
In Table 2, we see that each of the average values of gB,B were measured
to be greater than 1. Of these, only for a polarization of 10 degrees do values
less than 1 fall within one standard deviation of the measured gB,B . However,
given that, for this value, only one measured value of gB,B fell below 1 degrees,
at 0.982, and that a measured value was 2.437, this is most likely due to random
noise. The values corresponding to 10 degree polarization are shown in Table
4. As predicted, no conclusion can be drawn from measuring gB,B .
11

12. The value of g is not precisely zero. From the formula for g (equation 6),
this means that there were instances in which both detectors B and B’ were
triggered. We look for sources of extra photons in the experiment.
The room was kept as dark as possible for the reason of preventing additional
photons from altering the data. During data collection, the computer screen
was shielded away from the detectors, cellphones were turned off, and dark
panels were placed over the door and windows to prevent any stray light from
entering the room (as light of too much intensity would actually have damaged
the optical equipment). The only source of photons was the green light, which
aided in visibility. However, filters were placed over the detectors such that they
would not detect photons of those wavelengths associated with green light. This
light would thus have a minimal and safely ignorable effect on the data.
To understand why the value of g is not precisely measured to be 0, we look at
the exact mechanism for detecting photons. When detector A is triggered, there
is a 4.73 nanosecond interval in which the software seeks photons in detectors
B and B’. If it is the case that there is more than one photon emitted in this
time interval, and it is that at least one hits B and at least one hits B’ in this
interval, then this increases the value of g. This is especially possible if the
photons emitted from the laser, or from the down-conversion, do not follow a
uniform distribution. Indeed, it is more accurate to assume that the number of
photons emitted in a time interval follows a normal distribution.
The other possible factor is random electromagnetic noise in the detectors.
The detectors detect light by electromagnetic signals. However, because there
are other sources of electromagnetic radiation present in the room, those sources
can also trigger the detectors.
9 Conclusion
In conclusion, the value of g is much closer to 0 than to 1; in fact, g differs
from 1 by 91.2 standard deviations, and from 0 by only 8.8 standard deviations.
Because it was determined that g should be 1 if light is behaving in a wavelike
manner, the wavelike explanation of light behavior in this context is completely
inadequate to describe these results. Furthermore, the form of g in equation 6
indicates that the photons have some definite position, and so they are behaving
in a particlelike manner.
One possible improvement on the experiment would be to use a 360 nanome-
ter laser instead of a 405 nanometer laser. It should be noted that when the
photons are downconverted and two of them are produced, by energy conser-
vation, they each have approximately double the wavelength of the entering
photon. The detectors in this experiment thus are detecting photons of wave-
length 810 nanometers. Their quantum efficiency at this level is 60%, whereas
it is highest at around 700 nanometers, at 80%. This could have the effect of
improving the quality of the measurement of g. However, 360 nanometer lasers
are substantially more expensive than 405 nanometer ones, so for the purpose
of demonstrating particlelike behavior in photons, the 405 nanometer laser was
12

13. adequate.
Other potential areas of improvement involve reducing the number of pho-
tons emitted in this experiment, and thus reducing the number of stray photons
which could be present at both B and B’ in the detection time window that is
started by a detection event at detector A. This can be achieved with a lower-
intensity beam, or by using a material with a lower down conversion rate. The
lower-intensity beam does not affect the effect that the classical theory of light
detection would have on these results, which is that light intensity is the sole
factor in light detection, because this experiment is measuring a ratio of counts,
and so the change in intensity would cancel out in the ratio. Hence, in both
scenarios, the rejection of the classical wave theory of light as a descriptor of
these results would still be valid. These improvements would then reduce the
measured value of g.
Despite these possible improvements, they would only succeed at reducing
the measured value of g below what this experimental setup measures. They
would not affect the conclusion of the experiment, which is that photons are
not behaving in any sort of wavelike manner in this experiment. Any classical
explanation fails utterly to describe these results, so in this context, photons
must behave like particles.
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Leemann &], 1916. Print.
[3] Beck, Mark. Quantum Mechanics: Theory and Experiment. New York: Ox-
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[4] Jones, Eric and Travis Oliphant and Pearu Peterson and others. Scipy: Open
source scientific tools for Python. 2001–. http://www.scipy.org.
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