Single Photon Interference

## Introduction

With the previous experiments we have clear evidence to verify the existence of photons. We now set out on a new experiment, reminiscent of the famous double slit experiment.

The probability amplitude is a key concept in the study of quantum mechanics. Interference arises from the square of the sum of the probability amplitudes for the alternative ways that an event can be observed .

## Experimental Setup

We allow single photons to pass through and Mach-Zehnder interferometer, as diagrammed in Figure 1. Figure 1: Diagram of Mach-Zender Interferometer. Green line represents the laser coming in, as the light passes through the first beamsplitter two seperate paths are avalible for the light to travel, this is represented by the blue and yellow lines.

The photons enter through a half-silvered mirror, then instead of proceeding directly to a pair of detectors, the two beams reflect off of mirrors and then recombine on a second beam-splitter. There are now two distinct paths from input to output allowing for interference. If light were purely a wave it would split at the first beam splitter and then recombine at the second one with some phase shift between them that would determine the amplitude. The phase difference can easily be adjusted by adjusting the position of the mirror. But the question is, would this interference occur for individual photons? Using our same down converted light and triggering scheme we were able to generate a single photon as it entered the apparatus. If interference is observed then we would have to think of each photon particle splitting at the first beam splitter and then recombining at the second after traveling through both arms. But from our previous experiment proving the existence of the single photon, we know that our down converted light source produces something that never splits, but always goes one way or the other.

## What Do we Expect?

We know from our previous experiment that the down converted light that we have entering the interferometer is made up of single photons. We also know that the photon never splits at a beam splitter. Therefore the photon can only go through one path of the interferometer. If the photon only goes through one path then we would not expect any interference, constructive or deconstructive

## Prediction

Feynman had an approach that can predict the interference pattern produced by an interferometer with nearly equal arm lengths . This does not work for our case because we can assume that the amplitudes of the plane waves are modified by reflection and transmission at the beam splitter. At beam splitter one in the figure above the photon has two possible options: either transmit through the beam splitter with amplitude A into arm one or reflect into arm 2 with amplitude B. The plane wave associated with path 1 or 2 assumes the phase ${\delta} = k_sl_j$ where $l_j$ is the length of the arm j=1 or j=2 and $k_s$ is the wave number of the photon and is defined by 

(1)
\begin{align} k_s={2{\pi} \over {\lambda}_s} \end{align}

The probability of the photon emerging from path 1 is $tre^{i{\delta}_1$. Likewise, the probability of the photon emerging from path 2 is $tre^{i{\delta}_2$.

The probability that the photon will be detected at one port of the second beam splitter is then 

(2)
\begin{align} P = {|{tre^{i{\delta}_1} + {tre^{i{\delta}_1}|}^2 \end{align}
(3)
\begin{align} P = rr*tt*[2 + e^{i({\delta}_1 - {\delta}_2})} + e^{-i({\delta}_1 - {\delta}_2})}] \end{align}
(4)
\begin{align} P = 2RT[1 + cos{\delta}] \end{align}

where R=rr* is the reflection probability and T=tt* is the transmission probability. ${\delta} = {\delta}_1 - {\delta}_2$ is the phase difference caused by the difference in the lengths of the two arms of the interferometer. For our case we assume that our beam-splitters are identical. Our beam-splitters are 50-50 beam-splitters which reflect half of the incoming light, $R = {1 \over 2}$ and transmit half of the incoming light, $T = {1 \over 2}$. Eq.(4) becomes

(5)
\begin{align} P = {1 \over 2}(1 + cos{\delta}) \end{align}

This shows that the varying path length will produce a varying number of photons, otherwise known as an interference pattern.

Greenberger, Horne, and Zeilinger  offered a slightly more complex explanation that introduces bra and ket notation, which is the language of quantum mechanics. A photon is in a state $|s>$ when it enters the interferometer. After interacting with the first beam-splitter the photon is transformed to the state 

(6)
\begin{equation} |s> = A|l_1> + B|l_2> \end{equation}

where $|l_1>$ and $|l_2>$ are the states of the photons in arm 1 an arm 2 of the interferometer respectively. If we let $|a>$ and $|b>$ represent the state of a photon exiting in arm 1 an arm 2 of the interferometer respectively then 

(7)
\begin{align} |l_1> = e^{i{\delta}_1}(A|a> + B|b>) \end{align}
(8)
\begin{align} |l_2> = e^{i{\delta}_2}(A|a> + B|b>) \end{align}

The overall state of the photon after the transformation is

(9)
\begin{align} |s> = AB(e^{i{\delta}_1} + e^{i{\delta}_2})|a> + (AAe^{i{\delta}_1} + BBe^{i{\delta}_2})|b> \end{align}

The interference pattern arises from the probability of detecting a photon either in state $|a>$ or detecting a photon in state $|b>$. These probabilities should be the same and should be defined as 

(10)
\begin{align} P({\delta}) = {|<a|s>|}^2 = 2RT(1 + cos{\delta}) \end{align}

This results is essentially the same as Eq.(4).

## Results

After taking data for this experiment we see exactly what we would not expect if we treat the photon as a single indivisible particle. However, the data does agree with the quantum mechanical prediction. Our data clearly shows an interference pattern as the path length of one arm is changed relative to the path length of the other arm. The photon, which was clearly shown to take only one path, now takes both. The data fits perfectly to a sine curve defined by Eq.(11)

(11)
\begin{align} N_c = N_0((1-Vcos({\delta})) \end{align}

where N and V (defined by Eq.(12)) are fitting parameters, and ${\delta}$ is the phase difference between the two paths of the interferometer. V is an important parameter called the visibility and is defined by

(12)
\begin{align} V = {{{max}-{min}} \over {{max}+{min}}} \end{align}

where max is the maximum of the oscillations (also called fringes) and min the minimum of the oscillations. The quality of our data was good as is shown by the visibility of our interference pattern was 85%.

So far we have done an experiment that proves that the photon behaves as a particle and this experiment which proves that the photon interferes with itself. If we look at the results of the two experiments individually neither is a problem. They can both be explained with relatively simple physics. But it is when we look at the results together that our understanding is shattered. This brings back the problem of wave-particle duality. Is light a wave or a particle? This also brings back the great debate caused by this mystery. There were many experiments that demonstrated light’s wave nature and many others that demonstrated its particle properties. The work of this experiment combined with our previous experiments can be summed up in saying that light appears as a wave in some circumstances and as a particle in others. This brings up the point that light is truly neither wave nor particle, but something completely different.

The quantum-mechanical calculation of the interference pattern always makes use of the superposition principle. This always involves the sum of two or more states. With this even a single, indivisible photon has no location . This is what accounts for the ability of the photon to interfere with itself.

References
1. Greenstein and Zajonc. The Quantum Challenge. Jones and Bartlett Publishers: Massachusetts. (2006)
2. E. J. Galvez, C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courtemanche, L. Heilig, and J. Spencer, "Interference with correlated photons: Five quantum mechanics experiments for undergraduates," Am. J. of Phys. 73, 127-140 (2005)
3. R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, 1965), Vol. 3 p. 1-1
4. D.M. Greenberger, M.A. Horne, A. Zeilinger, "Multiparticle interferometry an the superposition principle," Phys. Today 46(8), 22-29 (1993).
page revision: 35, last edited: 30 Apr 2009 19:03