diff options
Diffstat (limited to 'single-photon-interference.tex')
| -rw-r--r-- | single-photon-interference.tex | 537 |
1 files changed, 288 insertions, 249 deletions
diff --git a/single-photon-interference.tex b/single-photon-interference.tex index 1b20598..350d5aa 100644 --- a/single-photon-interference.tex +++ b/single-photon-interference.tex @@ -5,23 +5,43 @@ \noindent \textbf{Experiment objectives}: Study wave-particle duality for photons by measuring - interference patterns in the Young double-slit experiment using a conventional light source (laser) and a + interference patterns in the double-slit experiment using a conventional light source (laser) and a single-photon source (strongly attenuated lamp). \section*{History} -There is a rich historical background behind the experiment you are about to perform. Isaac Newton first -separated white light into its colors, and, in the 1680's, hypothesized that light was composed of `corpuscles', -supposed to possess some properties of particles. This view reigned until the 1800's, when Thomas Young first -performed the two-slit experiment now known by his name. In this experiment he discovered a property of -destructive interference, which seemed impossible to explain in terms of -corpuscles, but was very naturally -explained in terms of waves. His experiment not only suggested that such `light waves' existed; it also -provided a result that could be used to determine the wavelength of light, measured in familiar units. Light -waves became even more acceptable with dynamical theories of light, such as Fresnel's and Maxwell's, in the 19th -century, until it seemed that the wave theory of light was incontrovertible. +There is a rich historical background behind the experiment you are about to perform. Isaac Newton first separated white light into its colors, and, in the 1680's, hypothesized that light was composed of `corpuscles', supposed to possess some properties of particles. This view reigned until the 1800's, when Thomas Young first performed the two-slit experiment now known by his name. Young directed light through a single aperture (slit) followed by a pair of double apertures, and then observed the result on a viewing screen (see Fig.~\ref{young.fig}). + + +\begin{figure}[h!] +\centering +\includegraphics[width=0.8\linewidth]{./pdf_figs/young2.png} \\ +\includegraphics[width=0.8\linewidth]{./pdf_figs/young1.png} + + \caption{\label{young.fig} Young's double slit experiment. Top: The experimental setup. Bottom: Young's interpretation of the result as interference, analogous to the corresponding phenomenon in water waves. The figure was drawn by Young himself. {\it Images from Wikimedia.} } +\end{figure} + +Young discovered that the light appeared as light and dark fringes on the screen, a fact that seemed impossible to explain in terms of corpuscles, but was very naturally explained in terms of waves, much like the interference observed in water and sound waves. In particular, wave theory predicts that the intensity of light $I(x)$ from double slit interference should be distributed as: + +\begin{equation} \label{2slit_wDif} +I(x)= 4 I_0 \cos^2\left(\frac{\pi d}{\lambda}\frac{x}{\ell} \right)\left[\frac{\sin (\frac{\pi +a}{\lambda}\frac{x}{\ell})}{\frac{\pi a}{\lambda}\frac{x}{\ell}} \right]^2 +\end{equation} + +Here, $x$ is the position on the viewing screen, with $x=0$ corresponding to the center of the interference pattern. The light wavelength is $\lambda$, the slits have a width of $a$, and they are separated by a distance $d$. The distance between the double slit and the viewing screen is $\ell$. + +Interference can also occur if there is only one slit. In this case the intensity distribution is described by Fraunhofer diffraction (see Fig.~\ref{interference.fig} and the derivations in the Appendix): +\begin{equation} \label{1slit} +I(x) = I_0 \frac{(\sin (\frac{\pi a}{\lambda}\frac{x}{\ell}))^2}{(\frac{\pi +a}{\lambda}\frac{x}{\ell})^2} +\end{equation} + + +Young's experiment not only suggested that such `light waves' existed but was also able to determine the wavelength of light. You should spend some time thinking about the latter point. How would the interference pattern change if we change $\lambda$? Given a particular interference pattern how might we determine $\lambda$? % refer to the calculation later in the lab + + \begin{figure}[h] @@ -29,13 +49,7 @@ century, until it seemed that the wave theory of light was incontrovertible. \includegraphics[width=0.4\linewidth]{./pdf_figs/ambigram} \caption{\label{ambigram.fig} ``Light is a Particle / Light is a Wave'' oscillation ambigram (from \emph{For the Love of Line and Pattern}, p. 30).} \end{figure} -And yet the discovery of the photoelectric effect, and its explanation in terms of light quanta by Einstein, -threw the matter into dispute again. The explanations of blackbody radiation, of the photoelectric effect, and -of the Compton effect seemed to point to the existence of `photons', quanta of light that possessed definite and -indivisible amounts of energy and momentum. These are very satisfactory explanations so far as they go, but -they throw into question the destructive-interference explanation of Young's experiment. Does light have a dual -nature, of waves and of particles? And if experiments force us to suppose that it does, how does the light know -when to behave according to each of its natures? + Light waves became even more acceptable with dynamical theories of light, such as Fresnel's and Maxwell's, in the 19th century, until it seemed that the wave theory of light was incontrovertible. And yet the discovery of the photoelectric effect, and its explanation in terms of light quanta by Einstein, threw the matter into dispute again. The explanations of blackbody radiation, of the photoelectric effect, and of the Compton effect seemed to point to the existence of `photons', quanta of light that possessed definite and indivisible amounts of energy and momentum. These are very satisfactory explanations so far as they go, but they throw into question the destructive-interference explanation of Young's experiment. Does light have a dual nature, of waves and of particles? And if experiments force us to suppose that it does, how does the light know when to behave according to each of its natures? It is the purpose of this experimental apparatus to make the phenomenon of light interference as concrete as possible, and to give you the hands-on familiarity that will allow you to confront wave-particle duality in a @@ -44,270 +58,295 @@ Feynman asserts that nobody really does -- but you will certainly have direct experience with the actual phenomena that motivates all this discussion. -\section*{Experimental setup} \textbf{Equipment needed}: Teachspin ``Two-slit interference'' apparatus, -oscilloscope, digital multimeter, counter. +\newpage + +\section*{Experimental setup} + +\subsection*{Equipment needed} + +You will need the Teachspin ``Two-slit interference'' apparatus, an oscilloscope, a digital multimeter, two $\sim$1m RG58 coaxial cables with BNC connectors, a BNC to banana plug adapter, and a frequency counter (such as the BK Precision 1823A). + +\subsection*{Inspecting the apparatus} + +\begin{framed} +{\center{\large \bf \textcolor{red}{Important equipment safety information}\\} } + The double slit apparatus has a very sensitive and somewhat fragile sensor called a photomultiplier tube (PMT) that you will use to observe individual photons. The tube is inside the detector box at the right end of the apparatus as shown in Fig~\ref{tsifig1.fig}. The PMT works by being supplied a DC high voltage (500-800V) by circuitry in the same box. The PMT should never be exposed to room light, especially with the HV on. Before taking the top off of the long and narrow interference channel you need to verify that: +\begin{enumerate} +\renewcommand{\theenumi}{\Alph{enumi}} +\item the shutter rod is down +\item the high voltage switch on the detector box is also off +\item the high voltage dial on the detector box is turned all the way to zero. +\end{enumerate} + As long as the top is off the rod must remain down and the HV must remain off. Finally, when the HV is on the laser/off/bulb switch at the far left end of the interference channel must be on bulb mode, or off. The laser is far too intense for the PMT. Make sure you have read this, understood, and got sign-off from the instructor before proceeding. +\end{framed} \begin{figure} \centering \includegraphics[width=0.8\linewidth]{./pdf_figs/tsisetup} \caption{\label{tsifig1.fig}The double-slit interference apparatus.} \end{figure} -\textbf{Important}: before plugging anything in or turning anything on, confirm that the shutter (which -protects the amazingly sensitive single-photon detector) is \textbf{closed}. Locate the detector box at the -right end of the apparatus, and find the rod which projects out of the top of its interface with the long -assembly. Be sure that this rod is pushed all the way down: take this opportunity to try pulling it vertically -upward by about $2$~cm, but then ensure that it's returned to its fully down position. Also take this occasion -to confirm, on the detector box, that the toggle switch in the HIGH-VOLTAGE section is turned off, and that the -10-turn dial near it is set to $0.00$, fully counter-clockwise. - -To inspect the inside of the apparatus, open the cover by turning the four latches that hold it closed. The details -of the experimental apparatus are shown in Fig.~\ref{tsifig1.fig}. Take time to locate all the important -components of the experiment: +\begin{figure} +\centering +\includegraphics[width=0.8\linewidth]{./pdf_figs/bnc_and_banana_annotated} \caption{\label{bnc.fig} RG58 coaxial cable with a BNC connector and a BNC$\leftrightarrow$banana connector.} +\end{figure} + +\begin{figure} +\centering +\includegraphics[width=0.8\linewidth]{./pdf_figs/shutter_up_down} \caption{\label{shutter.fig} The apparatus's shutter. When down it protects the PMT and exposes the photodiode to light in the interference channel. When up, the PMT is exposed to the light. The shutter must be kept down when the box is open.} +\end{figure} + + +To inspect the inside of the apparatus, open the cover by turning the four latches that hold it closed. The details of the experimental apparatus are shown in Fig.~\ref{tsifig1.fig}. Take time to locate all the important components of the experiment: + \begin{itemize} -\item Two distinct light sources at the left end: one a red \emph{laser} and the other a green-filtered \emph{light -bulb}. A toggle switch on the front panel of the light source control box switches power from one source to the -other. - -\item Various \emph{slit holders} along the length of the long box: one to - hold a two-slit mask, one for the slit blocker, -and one for a detector slit. Make sure you locate the \emph{slits} (they may be installed already) and two -\emph{micrometer drives}, which allow you to make mechanical adjustments to the two-slit apparatus. \textbf{Make -sure you figure out how to read the micrometer dials!} On the barrel there are two scales with division of -$1$~mm, shifted with respect to each other by 0.5~mm; every fifth mark is labeled with an integer 0, 5, 10 and -so on: these are at 5-mm spacing. The complete revolution of the drum is $0.5$~mm, and the smallest division on -the rotary scale is $0.01$~mm. - -\item Two distinct light detectors at the right-hand end of the apparatus: a \emph{photodiode} -and a \emph{photomultiplier tube} (PMT for short). The photodiode is used with the much brighter laser light; -it's mounted on the light shutter in such a way that it's in position to use when the shutter is closed (pushed -down). The photomultiplier tube is an extremely sensitive detector able to detect individual photons (with energy -of the order of $10^{-19}$~J), and it is used with the much dimmer light-bulb source. Too much light can easily -damage it, so \textbf{the PMT is safe to use only when the cover of the apparatus is in place, and only when the -light bulb is in use}. It is exposed to light only when the shutter is in its up position. +\item There are two distinct light sources at the left end: one a red \emph{laser} and the other a green-filtered \emph{light bulb}. A toggle switch on the front panel of the light source control box switches power from one source to the other. + +\item There are two distinct light detectors at the right-hand end of the apparatus: a {\em photodiode} and the aforementioned {\em PMT}. The photodiode is used with the much brighter laser light; it's mounted on the bottom of the light shutter rod so that it's in position to use when the shutter is closed (pushed down, See Fig.~\ref{shutter.fig}). The photomultiplier tube is an extremely sensitive detector able to detect individual photons (with energy of the order of $10^{-19}$~J); it is used with the much dimmer light-bulb source. + +\item There are various \emph{slit holders} along the length of the long box. From left to right in Fig.~\ref{tsifig1.fig} these are: a single slit to establish the point source of light diagrammed in Fig.~\ref{young.fig}; a double slit to create the interference pattern; a slit blocker just downstream of the double slit; and finally the detector slit, which sits in front of the photodetectors, allowing only a portion of the interference pattern to reach them. The distance $\ell$ between the double slit and detector slit is important. Be sure to measure it! + +\item There are two \emph{micrometer drives}, which allow you to make mechanical adjustments to the two-slit apparatus. The right-hand dial controls the position of the detector slit. The dial near the center of the apparatus controls the position of the slit blocker. \textbf{Make sure you figure out how to read these micrometer dials!} One unit on the dial corresponds to $10~\mu$m\footnote{Yes, this is confusing! We have an unfortunate language overlap, where ``micrometer'' refers to both the device and the unit of distance -- $\mu$m. There is not much we can do about it. If it helps, you can refer to $1~\mu$m as a ``micron''.} so turning from 100 to 200 on the dial moves the detector slit by $100\times 10~\mu\mathrm{m} = 1000~\mu\mathrm{m} = 1~\mathrm{mm}$. + + \end{itemize} \section*{Experimental procedure} The experiment consists of three steps: +\begin{description} +\item[Observing interference and finding dial settings] You will first identify single and double slit interference patterns directly with the top off of the apparatus. You'll do this by looking at interference fringes on a bit of paper you'll insert into the long box, to the right of the double slit in Fig.~\ref{tsifig1.fig}. You will record the slit blocker position corresponding to single and double slit interference in your lab book so you can switch between the two when the top is on. + +\item[Measuring interference with an intense source] Using the photodiode and laser you will measure the intensity of the single- and double-slit interference patterns as a function of $x$, the position transverse to the laser beam. Your analysis will consist of fitting the data with equations \ref{1slit} and \ref{2slit_wDif}. A good fit demonstrates that the wave theory of light is correct and allows you to determine the slit width and slit separation. This is a recreation of Young's original experiment. + +\item[Measuring interference with a weak source] Then, using a very weak light source, you will record the intensity of the single- and double-slit interference pattern by counting the number of photons as a function of $x$. This measurement will introduce you to the technology we use to observe single-photons detection technology and will also demonstrate that interference occurs even if we only have one photon going through the apparatus at a time. This is the very essence of wave/particle duality! + +\end{description} + + +\subsection*{Observing interference and finding dial settings} +\begin{framed} +{\center{\large \bf \textcolor{red}{Important personal safety information}\\} } +The apparatus has a 5-mW diode laser with an output wavelength of $670 \pm 5$~nm. It could potentially harm your eyes if you are not careful. Don't observe the beam directly (i.e., by staring into it) and be careful of reflections off of reflective surfaces. +\end{framed} + + + +For this mode of operation, you will be working with the cover of the apparatus open. + \begin{enumerate} -\item You will first observe two-slit interference directly by observing the intensity -distribution of a laser beam on a viewing screen. -\item Using the photodiode you will accurately measure the intensity distribution after single- and two-slit interference patterns, -which can be compared to predictions of wave theories of light. \\ -These two steps recreate Young's original experiment. -\item Then, using a very weak light source, you will record the two-slit interference pattern one photon at a time. -While this measurement will introduce you to single-photon detection technology, it will also show you that -however two-slit interference is to be explained, it must be explained in terms that can apply to single -photons. +\item Switch on the laser. + +\item Make a viewing screen by cutting a bit of white paper to the dimensions of the interference channel. Insert your screen just to the right of the double slit and slit blocker. + +\item Adjust the position and angle of laser so that it impinges on the single slit, then the double slit, and travels all the way to the detector slit. You may need to make sure, by visual inspection, that the slit blocker is not blocking one or both of the double slits. + +\item Move your screen along the beam path toward the detector slit to see the interference pattern forming. By the time your viewing card reaches the right-hand end of the apparatus, you'll see that the two overlapping ribbons of light combine to form a pattern of illumination displaying the celebrated ``fringes'' named after Thomas Young. + \end{enumerate} -\subsection*{Visual observation of a single- and two-slit interference} - -For this mode of operation, you will be working with the cover of the apparatus open. Switch the red diode laser -on using the switch in the light source control panel, and move the laser -to the center of its magnetic pedestal -so that the red beam goes all the way to the detector slit. The diode laser manufacturer asserts that its output -wavelength is $670 \pm 5$~nm, and its output power is about 5~mW. \emph{\textbf{As long as you don't allow the -full beam to fall directly into your eye, it presents no safety hazard.}} Place a double-slit mask on the holder -in the center of the apparatus, and then put your viewing card just after the mask to observe the two ribbons of -light, just a third of a millimeter apart, which emerge from the two slits. -Move your viewing card along the beam -path to see the interference pattern forming. By the time your viewing card reaches the right-hand end of the -apparatus, you'll see that the two overlapping ribbons of light combine to form a pattern of illumination -displaying the celebrated ``fringes'' named after Thomas Young. - -Position a viewing card at the far-right end of the apparatus so you can refer to it for a view of the fringes. -Now it is the time to master the control of the slit-blocker. By adjusting the multi-turn micrometer screw, make -sure you find and record the ranges of micrometer reading where you observe the following five situations: +You are going to take data with the box closed and will want to be able to move the blocking slit to go from double slit mode to single slit mode without opening it. We'll also want to be able to completely block the light from both slits in order to measure the background signal from the photodiode and the PMT. Adjust the position of the blocking slit using the micrometer to find the following situations: + \begin{enumerate} -\item both slits are blocked; -\item light emerges only from one of the two slits; -\item both slits are open; -\item light emerges only from the other slit; +\item both slits are open +\item light emerges only from a single slit \item the light from both slits is blocked. \end{enumerate} -It is essential that you are confident enough in your ability to read, and -to set, these five positions and that -you are able to do so even when the box cover is closed. In your lab book describe what you see at the viewing -card at the far-right end of the apparatus for each of the five settings. - -\textbf{One slit is open:} According to the wave theory of light, the intensity distribution of light on the -screen after passing a single-slit is described by Fraunhofer diffraction (see Fig.~\ref{interference.fig} and -the derivations in the Appendix): -\begin{equation} \label{1slit} -I(x) = I_0 \frac{(\sin (\frac{\pi a}{\lambda}\frac{x}{\ell}))^2}{(\frac{\pi -a}{\lambda}\frac{x}{\ell})^2}, -\end{equation} -where $I$ is the measured intensity in the point $x$ in the screen, $I_0$ is the intensity in the brightest -maximum, $a$ is the width of the slit, and $\ell$ is the distance between the slit and the screen (\emph{don't -forget to measure and record this distance in the lab journal!}) - -In your apparatus move the slit blocker to let the light go through only one slit and inspect the -light pattern in the viewing screen. Does it look like the intensity distribution you expect from -the wave theory? Take a minute to discuss how this picture would change if the slit was much wider or -much narrower. - -\textbf{Two slits are open:} Now move the slit blocker to the position that opens both slits to observe Young's -two-slit interference fringes. Again, compare what you see on the screen with the interference picture predicted -by wave theory: -\begin{equation} \label{2slit_wDif} -I(x)= 4 I_0 \cos^2\left(\frac{\pi d}{\lambda}\frac{x}{\ell} \right)\left[\frac{\sin (\frac{\pi -a}{\lambda}\frac{x}{\ell})}{\frac{\pi a}{\lambda}\frac{x}{\ell}} \right]^2, -\end{equation} -where an additional parameter $d$ is the distance between the centers of the two slits. Discuss how this picture -would change if you were to vary the width or the separation of the two -slits or the wavelength of the laser. Make a -note of your predictions in the lab book. - -\subsection*{Quantitative characterization of interference patterns using laser light} - -At this stage you will use a photodiode to measure the intensity distribution of the interference pattern by -varying the position of the detector slit. You will continue using the red laser. While you may conduct these -measurements with the box cover open, room light will inevitably add some varying background to your signals, so -it is a good idea to dim the room lights or (even better!) to close up the cover of the apparatus. For -convenience, have the slit-blocker set to that previously determined setting which allows light from both slits -to emerge and interfere. - -The shutter of the detector box will still be in its closed, or down, position: this blocks any light from -reaching the PMT, and correctly positions a 1-cm$^{2}$ photodiode, which acts just like a solar cell in actively -generating electric current when it's illuminated. The output current is proportional to total power -illuminating the detector area, so it is important to use a narrow slit to allow only a selected part of the -interference pattern to be measured. Make sure that a detector slit mask (with a single narrow slit) on a -movable slit holder at the right-hand side of the apparatus is in place. By adjusting the micrometer screw of the -detector slit, you can move the slit over the interference pattern, eventually mapping out its intensity -distribution quantitatively. For now, ensure that the detector slit is located somewhere near the middle of the -two-slit interference pattern, and have the slit-blocker set to the setting which allows light from both slits -to emerge and interfere. - -The electric \emph{current} from the photodiode, proportional to the \emph{light intensity}, is conducted by a -thin coaxial cable to the INPUT BNC connector of the photodiode-amplifier section of the detector box, and -converted to \emph{voltage} signal at the OUTPUT BNC connector adjacent to it. Connect to this output a digital -multimeter set to 2 or 20-Volt sensitivity; you should see a stable positive reading. Turn off the laser first -to record the ``zero offset'' - the reading of the multimeter with no light. You will need to subtract this reading -from all of the other readings you make with this photodiode, amplifier, and voltmeter -combination. - -Turn your laser source back on, and watch the photodiode's voltage-output signal as you vary the setting of the -detector-slit micrometer. If all is well, you will see a systematic variation of the signal as you scan over the -interference pattern. Check that the maximum signal you see is about 3-8 Volts; if it is much less than this, -the apparatus is out of alignment, and insufficient light is reaching the detector. - -\textbf{Initial tests of the wave theory of light:} If we assume that the light -beam is a stream of particles, we -would naively expect that closing one of two identical slits should reduce the measured intensity of light at -any point on the screen by half, while the wave theory predicts much more dramatic variations in the different -points in the screen. Which theory provides a more accurate description of what you see? -\begin{itemize} -\item Find the highest of the maxima --- this is the ``central fringe'' or - the ``zeroth-order fringe'' that the theory -predicts --- and record the photodiode reading. Then adjust the position of the slit-blocker to let the light -to pass through only one of the slits, and measure the change in the photodiode signal. +Record the dial setting for each. Make sure that you are fully in the single and double-slit conditions as it is possible to be halfway, with the top of the pattern different than the bottom due. It is essential that you are confident enough in your ability to read, and to set, these three positions and that you are able to do so even when the box cover is closed. In your lab book describe and sketch what you see on the viewing card at the far-right end of the apparatus for each of the three settings. -\item To see another and even more dramatic manifestation of the wave nature of light, set the slit-blocker again -to permit light from both slits to pass along the apparatus, and now place the detector slit at either of the -minima immediately adjacent to the central maximum; take some care to find the very bottom of this minimum. -Record what happens when you use the slit-blocker to block the light from one, or the other, of the two slits. -\item Check your experimental results against the theoretical predictions using Eqs.~(\ref{1slit}) and (\ref{2slit_wDif}). -Do your observations confirm or contradict wave theory? -\end{itemize} +\subsection*{Measuring interference with an intense source} -Once you have performed these spot-checks, and have understood the -motivation for them and the obtained results, you are ready to conduct -systematic measurements of intensity distribution (the photodiode -voltage-output signal) as a function of detector slit position. You will -make such {\bf measurements in two slit-blocker positions: when both slits -are open, and when only one slit is open}. You will need to take enough data points to reproduce the intensity distribution in each case. Taking points systematically every 0.05 or \unit[0.1]{mm} on the tick lines will produce a very high quality dataset. One person should turn the dial and the other should record readings directly to paper or a spreadsheet (if you do this, print it out and tape into your logbook). Estimate your uncertainties from the dial and the voltmeter. Cycle through multiple maxima and minima on both sides of the central maximum. It is a good idea to plot the data points immediately along with the data taking -- nothing beats an emerging graph for teaching you what is going on, and your graph will be pretty impressive. \emph{Note: due to large number of points you don't need to include the tables with these measurements in the lab report -- the plotted distributions should be sufficient. Be clear on your uncertainties though.} - -\textbf{Slit separation calculations}: Once you have enough data points for each graph to clearly see the -interference pattern, use your data to extract the information about the -distance between the two slits $d$. To do -that, find the positions of consecutive interference maxima or minima, and calculate average $d$ using -Eq.~\ref{2slit_wDif}. Estimate the uncertainty in these parameters due to laser wavelength uncertainty. Check if -your measured values are within experimental uncertainty from the manufacturer's specs: the center-to-center -slit separation is 0.353 mm (or 0.406 or 0.457 mm, depending on which two-slit mask you have installed). - -Fit your data with Eqs.~(\ref{1slit}) and (\ref{2slit_wDif}). You will need to add these functions using ``Add new function'' option. Note that in this case you will have to provide a list of initial guesses for all the fitting parameters. A few tips: -\begin{itemize} -\item Make sure that units of all your measured values are self-consistent - the program will go crazy trying to combine measurements in meters and micrometers together! -\item Try to plot your function for guesstimated values before doing a fit with it. This catches many silly errors. -\item You will have to fit for a term to account for the overall normalization and also for the fact that the maximum is not set at $x=0$. In other words, substitute $x \rightarrow x-x_0$. Estimate both of these (for the normalization, look at what happens when $x-x_0 = 0$) but include them as free parameters in the fit. -\item You need to add a parameter to account for the non-zero background you observed when the laser was off. -\item Include the minimal number of parameters. If I have parameters $a$ and $b$ but they always appear as $ab$ in my function, then I am much better off including a term $c=ab$ when doing the fit. Otherwise the fit is underconstrained. Adjusting $a$ up has the same effect as adjusting $b$ down so it's impossible to converge on unique values for $a$ and $b$. Fitting algorithms really dislike this situation. -\item If the program has problems fitting all the parameters, first hold the values of the parameters you know fairly well (such as the light wavelength, maximum peak intensity, background, etc.). Once you determine the approximate values for all other parameters, you can release the fixed ones, and let the program adjust everything to make the fit better. +In this part of the experiment you will use a photodiode to measure the intensity distribution of the interference pattern by varying the position of the detector slit. You will do this using the red laser with the shutter down. -\end{itemize} +\subsubsection*{\textcolor{blue}{Orienting yourself with the laser and photodiode}} -\subsection*{Single-photon interference} - -Before you start the measurements you have to convince yourself that the -rate of photons emitted by the weak filtered light bulb is low enough to -have, on average, less than one photon detected in the apparatus at any -time. Roughly estimate the number of photons per second arriving to the -detector. First, calculate the number of photons emitted by the light bulb -in a 10~nm spectral window of the green filter (between $541$ and -$551$~nm), if it runs at 6V and 0.2A. Note that only 5\% of its electric -energy turns into light, and this optical energy is evenly distributed in -the spectral range between 500~nm and 1500~nm. These photons are emitted in -all directions, but all of them are absorbed inside the box except for -those passing through two slits with area approximately $0.1\times -10~\mathrm{mm}^2$. Next, assuming that the beam of photons passing through -the slits diffracts over a $1~\mathrm{cm}^2$ area by the time it reaches -the detector slit, estimate the rate of photons reaching the detector. -Finally, adjust the detected photon rate by taking into account that for -PMT only 4\% of photons produce output electric pulse at the output. That's -the rate of events you expect. Now estimate the time it takes a photon to -travel through the apparatus, and estimate the average number of detectable -photons inside at a given moment of time. \emph{You may do these -calculations before or after the lab period, but make sure to include them in the lab report.} - -Now you need to change the apparatus to use the light bulb. Open the cover and slide the laser source to the side (do not remove the laser from the stand). Now set the 3-position toggle switch to the BULB position and dial the bulb adjustment up from 0 until you see the bulb light up. (\emph{The flashlight bulb you're using will live longest if you minimize the time you spend with it dialed above 6 on its scale, and if you toggle its power switch only when the dial is set to low values}). If the apparatus has been aligned, the bulb should now be in position to send light through the apparatus. Check that the green filter-holding structure is in place: the light-bulb should look green, since the green filter blocks nearly all the light emerging from the bulb, passing only wavelengths in the range 541 to 551 nm. The filtered light bulb is very dim, and you probably will not be able to see much light at the double-slit position even with room light turned off completely. No matter; plenty of green-light photons will still be reaching the double-slit structure -- in fact, you should now dim the bulb even more, by setting its intensity control down to about 3 on its dial. - -Now close and lock the cover - you are ready to start counting photons. But first a WARNING: a photomultiplier tube is so sensitive a device that it should not be exposed even to moderate levels of light when turned off, and must not be exposed to anything but the dimmest of lights when turned on. In this context, ordinary room light is intolerably bright even to a PMT turned off, and light as dim as moonlight is much too bright for a PMT turned on. - -\textbf{Direct observation of photomultiplier pulses} You will use a digital oscilloscope for first examination of the PMT output pulses, and a digital counter for counting the photon events. Set the oscilloscope level to about 50~mV/division vertically, and 250 - 500~ns/division horizontally, and set it to trigger on positive-going pulses or edges of perhaps $>20$~mV height. Now find the PHOTOMULTIPLIER OUTPUT of the detector box, and connect it via a BNC cable to the vertical input of the oscilloscope. Keeping the shutter closed, set the HIGH-VOLTAGE 10-turn dial to 0.00, and turn on the HIGH-VOLTAGE toggle switch. Start to increase the voltage while watching the scope display. \emph{If you see some sinusoidal modulation of a few mV amplitude, and of about 200 kHz frequency, in the baseline of the PMT signal, this is normal. If you see a continuing high rate ($>10$~kHz) of pulses from the PMT, this is not normal, and you should turn down, or off, the bias level and start fresh -- you may have a malfunction, or a light leak.} Somewhere around a setting of 4 or 5 turns of the dial, you should get occasional positive-going pulses on the scope, occurring at a modest rate of $1-10$ per second. If you see this low rate of pulses, you have discovered the ``dark rate'' of the PMT, its output pulse rate even in the total absence of light. You also now have the PMT ready to look at photons from your two-slit apparatus, so finally you may open the shutter. The oscilloscope should now show a much greater rate of pulses, perhaps of order $10^3$ per second, and that rate should vary systematically with the setting of the bulb intensity. \emph{You may find a small device called Cricket in your table. It allows you to ``hear'' the individual photon arrivals - ask your instructor to show you how it works.} - -To count the pulses using a counter you will use another PMT output -- the OUTPUT TTL -- that generates a single pulse, of fixed height and duration, each time the analog pulse exceeds an adjustable threshold. To adjust the TTL settings display the OUTPUT TTL on the second oscilloscope channel and set it for 2 V/div vertically. By simultaneously watching both analog and TTL-level pulses on the display, you should be able to find a discriminator setting, low on the dial, for which the scope shows one TTL pulse for each of, and for only, those analog pulses which reach (say) a $+50$~-mV level. If your analog pulses are mostly not this high, you can raise the PMT bias by half a turn (50 Volts) to gain more electron multiplication. If your TTL pulses come much more frequently than the analog pulses, set the discriminator dial lower on its scale. - -Now send the TTL pulses to a counter, arranged to display successive readings of the number of TTL pulses that occur in successive 1-second time intervals. To confirm that this is true, record a series of ``dark counts'' obtained with the light bulb dialed all the way down to 0 on its scale. Now choose a setting that gives an adequate photon count rate (about $10^3$/second) and use the slit-blocker, according to your previously obtained settings, to block the light from both slits. This should reduce the count rate to a background rate, probably somewhat higher than the dark rate. Next, open up both slits, and try moving the detector slit to see if you can see interference fringes in the photon count rate. You will need to pick a detector-slit location, wait for a second or more, then read the photon count in one or more 1-second intervals before trying a new detector-slit location. If you can see maxima and minima, you are ready to take data. Finally, park the slit near the central maximum and choose the PMT bias at around $5$ turns of the dial and the bulb intensity setting to yield some -convenient count rate ($10^3 - 10^4$ events/second) at the central maximum. - -%Before you begin the data collection you need to set the PMT bias voltage to a suitable range. \emph{This -%procedure is \textbf{optional}, and necessary if the event rate you observe is too low and/or the dark rate -%count is too high.} To do that you'll need to measure the dependence of dark count rate (PMT shutter closed) and -%the count rate at the central maximum of the interference pattern (both slits open) on the PMT bias voltage over -%the range 300 to 650 V. When you plot the two count rates on a semi-logarithmic graph, you should see the -%``light rate'' reach a plateau, with the interpretation that you have reached a PMT bias which allow each -%photoelectron to trigger the whole chain of electronics all the way to the TTL counter; you should also see the -%(much lower) `dark rate' also rising with PMT bias. Based on your graph choose the PMT bias setting at which -%you are counting substantially all true photon events, but minimizing the number of ``dark events''. - -\textbf{Single-photon detection of the interference pattern}. Most likely the experimental results in the -previous section have demonstrated good agreement with the wave description of light. However, the PMT detects -individual photons, so one can expect that now one has to describe the light beam as a stream of particles, and -the wave theory is not valid anymore. To check this assumption, you will repeat the measurements and take the -same sort of data as in the previous section, except now characterizing the light intensity as photon count -rate. -\begin{itemize} -\item Like previously, slowly change the position of the detection slit and record the average count rate in each -point. Start with the two-slit interference. Plot the data and confirm that you see interference fringes. +{\em The apparatus should be open for this part.} + +\begin{description} +\item[The photodiode] When down, the shutter positions the $\sim 1 \times \unit[1]{cm^2}$ photodiode in the path of the interference pattern. The output voltage from the photodiode is proportional to the total amount of light hitting it. + +\item[Connecting to the photodiode] You need to connect a multimeter in DCV mode to the photodiode output on the detector box. To do this you will need a coaxial cable with BNC connectors on each end and a BNC $\leftrightarrow$ banana plug converter. These items are shown in Fig.~\ref{bnc.fig}. + +\item[The detector slit] The detector slit masks most of the photodiode, only allowing light through in a narrow window range along the horizontal ($x$) axis. By adjusting the micrometer screw of the detector slit, you can move the slit over the interference pattern, eventually mapping out its intensity distribution quantitatively. + +\item[Getting in position] For now, ensure that the detector slit is located somewhere near the edge middle of the two-slit interference pattern, and have the slit-blocker set to give you double-slit interference. + +\item[Quick scan of the pattern] Now, turn the micrometer dial to move the detector slit. Observe how the voltage output of the photodiode changes as you translate the slit across light and dark fringes. To take good data you will do the same thing with the box closed, stopping at even intervals to record the photodiode output. + +\end{description} -\item Repeat the measurement with one slit blocked and make the plot. +\subsubsection*{\textcolor{blue}{Taking photodiode data}} -\item Use the spacing of the interference maxima to check that the light source has a different wavelength than -the red laser light you used previously. Using the previously determined value of the slit separation $d$, -calculate the wavelength of the light, and check that it is consistent with the green filter specs -($541-551$~nm). +{\em Now, close the apparatus.} Be gentle when putting the top on! \textcolor{red}{Note: you will have to follow the procedure below twice, once for double slit, and then again for single slit.} +\begin{description} +\item[Record the pedestal] Turn off the laser and observe that the photodiode produces an output even without light. This is the ``zero offset'' or ``pedestal'' reading. Record it in your lab book, along with an uncertainty. You will need this for your analysis. + +\item[Take a datapoint] Now you are ready to take data. Move the detector slit to one end of the interference pattern. Collect one datapoint by recording the detector slit position (by reading the micrometer) and the voltage from the photodiode. You will be taking a lot of datapoints and it is wise to plot them as you collect them. Therefore you should record the readings in a spreadsheet and do not need to put them in your lab book. + +\item[Uncertainty -- Part 1] Estimate your uncertainty from the micrometer and multi-meter precision and stability. Record your uncertainty and any comments you have about it in your lab book. We will come back to this later. + +\item[Collect more points] Move the detector slit by some small interval, collecting additional datapoints. You want to map out the interference pattern with sufficient details that you can easily resolve the light and dark fringes. For double-slit interference, taking data every $\unit[100]{\mu m}$ should be OK. The single slit pattern is less detailed and you need only take data every $\unit[200]{\mu m}$. + +\item[Plot the data] Plot your data as you take it to verify that you are seeing an interference pattern. Remake the plot every 20 steps or so. (This is easiest to do as a scatterplot in a spreadsheet). + +\item[The goal] The goal is to take about 70 points in double slit mode and 35 in single slit mode. This should allow you to see multiple maxima and minima in double slit, and the central maximum in single slit mode, along with (hopefully) a secondary maximum on at least one side. \textcolor{red}{Note: before moving from double to single-slit mode you should estimate your uncertainties by following the procedure described below.} +\end{description} + +\subsubsection*{\textcolor{blue}{Uncertainties}} + +Are your uncertainties really just due to the measurement precision of the micrometer dial and DVM? It is wise to test this by repeating some measurements in both single and double slit modes.. To do so: + +\begin{description} +\item[Back to the beginning] + Move the detector slit all the way back to the starting point. Go a little past the point as you do so, then come back to it going in the direction that you took all the data. Record the old and new reading of that point in your lab book. + +\item[Repeat points] Now, translate the detector slit and remeasure 8 representative points in your dataset. Recording the old and new values. + +\item[And again] Do the same thing again, so that you have three repeated readings at each of the 8 points. + +\item[Uncertainty -- Part 2] Use the standard deviation in each of the 8 measurements to estimate your uncertainty due to repeatability. Is it larger than that due to the precision? Talk this over with the instructor and how to generalize what you found to all the points. +\end{description} + +\subsection*{Measuring interference with a weak source} + + +\subsubsection*{\textcolor{blue}{Orienting yourself with the bulb and PMT}} + +Now you need to change the apparatus to use the light bulb and PMT. Keep the shutter down for now. + +\begin{description} +\item[Open up] Open the cover and slide the laser source to the side (do not remove the laser from the stand). + +\item[Bulb settings] Now set the 3-position toggle switch to the ``bulb'' position and dial its intensity to 8. As long as you haven't touched anything else the apparatus is still aligned and light from the bulb will form interference patters that you can measure with the PMT. + +\item[Double slit mode] Use the dial to move the slit blocker into double slit mode. + +\item[Close it up] Now close and latch the cover. + +\item[High voltage and PMT pulses] Have an instructor show you how to use the DVM to measure the voltage supplied to the PMT and how to observe the PMT output on an oscilloscope. With the instructor, gradually turn on the HV to get up to a setting of around \unit[700]{V}. You should see pulses on the oscilloscope screen. + +% instructor: going for a darkrate of ~100Hz and a signal rate of at least 1kHz at the center of the pattern. May need to play with HV and discriminator settings. + +\item[Counting Pulses] The instructor will also show you how to look at the discriminator output on the scope and how to count pulses with a frequency counter. The counter should be set to count for 1 second. {\em Pause to document the setup procedure in your lab book. Sketch connections, record the HV rate, etc. You should document this well enough to repeat it in a year. } + +\item[Unstable Counter!] You've noticed that the counting rate isn't stable. This is because you are observing a random process. So, how do we assign an uncertainty? Begin by recording 10 different readings $N_i$ from the frequency counter in your lab notebook. + +\item[Counting uncertainty] Compute the mean of the readings and their standard deviation and record those too. Compare the standard deviation to $\sqrt{N_i}$ for a couple of the points. Are the values similar? It turns out that for a random counting process, if you count $N_i$ counts then the associated uncertainty is $\delta N_i = \sqrt{N_i}$. This is what you will use in your analysis\footnote{So, if you count $N=9$ counts in some period of time the uncertainty is apparently $\delta N = \sqrt{9} = 3$. But, what if you are sure it was 9 counts, not 8, not 10, etc? You are really positive you didn't miss any. What then? You should discuss this with your instructor.}. + +% The issue is that the number of counts N is an exact number which itself has no uncertainty. However, we are trying to apply it to estimate the true (asymptotic) counting rate, which is just the underlining probability multiplied by some unimportant scale. In that case, the appropriate uncertainty is sqrt(N). In fact, we should be doing a real Pearson-Chi2 fit in which the denominator in the Chi-2 is the value of the function (e.g. (delta fit-funct)^2) not a number from the data. But I think it's too far in the weeds and I don't know offhand how to do it in matlab. + +\item[The dark rate] By the way, you have been observing counts even without light shining on the PMT, since the shutter is down. This is the so-called ``dark rate'' of the PMT. It is similar to the ``zero offset'' or ``pedestal'' that you saw for the photodiode. You should record it in your lab book. Ask your instructor what causes it and to explain how the PMT works. {\bf Then, finally, go ahead and raise the shutter before moving on to the next section.} + +\end{description} + +\subsubsection*{\textcolor{blue}{Taking PMT data}} + +{\em The data taking must be done with the box closed and high voltage on. If for any reason you have to open the box you need to make sure the high voltage is off and the shutter is down.} + +You are now going to measure single and double slit interference just like you did with the laser and photodiode. In particular you should take about the same number of points with the same step size. Now, however you will be using the PMT counting rate as a measure of the light intensity, rather than the photodiode output, and you are observing a very dim light source. In fact the source is so dim that you can convince yourself with a ``back of the envelope'' computation that there is usually only one photon in the region downstream of the double slits at any given time. That calculation is described in the data analysis section and it should appear in your report. The ramification is that the interference pattern must be due to photons interfering with themselves! This experimentally establishes wave-particle duality. + +You don't need to repeat datapoints to establish an uncertainty. The dominant contribution is just due to the statistical noise in the counting rate ($\delta N_i = \sqrt{N_i}$). In Matlab, if you have your $N_i$ in an array {\tt N} then the $\delta N_i$ can just be computed as {\tt dN=sqrt(N)}, or similar in Python {\tt dN=np.sqrt(N)}. You can then plot the data and uncertainties with the {\tt errorbar} function in either Matlab or Python + +You should be careful not to wait to record ``round numbers'' or other aesthetically pleasing readings from the counter. The best way of doing this is to establish a routine. For example, turn the micrometer dial and then look at the counter. Record the second reading that you see since the first might be corrupted by the move. You need only record the rate as an integer (i.e., you can neglect any numbers after the decimal place). + +\subsubsection*{Qualitative observations} + +In both the one and two slit cases make some qualitative observations. Do the patterns appear to agree with Equations.~\ref{2slit_wDif}-\ref{1slit}? What would happen if we increased or decreased $d$, $a$, $\lambda$ and $\ell$. Record your thoughts in your labbook, and use these insights in setting the initial parameters when fitting your data. + +%The plots of your experimental data are clear evidence of particle-wave duality for photons. You've made contact with the central question of quantum mechanics: how can light, which so clearly propagates as a wave that we can measure its wavelength, also be detected as individual photon events? Or alternatively, how can individual photons in flight through this apparatus nevertheless `know' whether one, or both, slits are open, in the sense of giving photon arrival rates which decrease when a second slit is opened? Discuss these issues in your lab report. + +\vspace{1cm} +\hrule + +{\huge Your instructor may have told you to do a ``brief writeup'' of this experiment. If so, please see the instructions starting on the next page (page~\pageref{pag:briefwriteup}).} + +\hrule + +\newpage + +\section*{Instructions for a brief writeup}\label{pag:briefwriteup} + +In this lab you have collected 4 independent datasets (red laser and green bulb with single or double slit) and the analysis will require fits to these data. Therefore, a formal lab report with abstract, introduction, conclusion, etc. is \textbf{not} required. For this brief writeup: you only need to include the tables and figures requested below, as well as answering the questions listed in this section. {\bf Please restate each one so it's easy for us to follow along.} {\it Refer to the relevant section earlier in the manual or later in Appendices for additional discussion of the analysis procedure. This is just a bullet point summary of the deliverables.} + +%\subsection*{How does the interference patterns depend on $d$, $a$, and $\lambda$?} +% +%What would happen if we increased or decreased $d$, $a$, and $\lambda$? Explore this by finding the locations of the minima of Eqs.~(\ref{1slit}) and %(\ref{2slit_wDif}). Then look at what happens to those locations as you change $d$, $a$, and $\lambda$. Record the results in your lab book and report. Knowing how the pattern changes as you change the variables is useful as you pick starting values for those parameters to use in your fits. + +\subsection*{Data Tables and Uncertainties} + +\begin{itemize} +\item Tabulate your data for each of the four interference patterns and include them (with labels and captions). +\item Describe how uncertainties on position, photodiode voltage, and photon counting rate were determined, and report your estimated uncertainty values. \end{itemize} -The plots of your experimental data are clear evidence of particle-wave duality for photons. You've made contact with the central question of quantum mechanics: how can light, which so clearly propagates as a wave that we can measure its wavelength, also be detected as individual photon events? Or alternatively, how can individual photons in flight through this apparatus nevertheless `know' whether one, or both, slits are open, in the sense of giving photon arrival rates which decrease when a second slit is opened? Discuss these issues in your lab report. +\subsection*{Data Analysis and Fitting} + +For each of the four interference patterns, address the following points: + +\begin{enumerate} +\item Plot the measurements for the photodiode voltage or photon counting rate (y-axis) vs position (x-axis) with uncertainties and fit to Eqs.~(\ref{2slit_wDif}) and (\ref{1slit}) for Double and Single Slit, respectively. Include a figure with the plot of your data and the resulting fit in your report (with labels and captions). +\item Is it a good fit: in terms of the observed $\chi^2$ and number of degrees of freedom? +\item What are the best fit parameters and uncertainties? +\item Is B consistent with what you expect from the background (\textit{i.e.} pedestal or dark rate) measurements you collected during lab? Comment on any discrepancies. +\end{enumerate} + +\noindent Finally, quantitatively compare your measurements of $a$ and $d$ between the four datasets. Are they consistent? + +\subsection*{Did we really see single photon interference?} + +To conclude that photons interfere with themselves, rather than with other nearby photons, we have to be sure that the rate of photons is low enough that only one of them is in the vicinity of the double slit at any given time. You need to work through the steps in the Appendix ``Computing the photon rate" below, writing numbers in your lab book. In this brief writeup summarize the results in a paragraph or two, including the numbers you calculate at each stage. Write a nice bit of narrative text, not just a bunch of bullets with numbers. + +\newpage +\section*{Appendix: Computing the photon rate} +{\em This is a so-called back of the envelope calculation.} + +\begin{enumerate} + +%\item Begin by roughly estimating the number of photons per second arriving to the detector. +\item Begin by computing the amount of electrical power that is converted to light by the light bulb. Assume we operate the bulb at \unit[6]{VDC} and it draws \unit[0.2]{A} of current. From this you can get the power drawn by the bulb. % Ans: 1.2W + +\item Most of the power goes into heating the bulb. Assume 5\% of it is converted to light evenly distributed from 500-1500~nm. + +\item The bulb has a green filter that blocks the light outside the range 541-551~nm. From this you can get the amount of power radiated as green light. %1.2W*0.05*10nm/1500nm = 6e-4W + +\item You can convert the power (energy/time) into a rate of photons using the fact that $E = \frac{hc}{\lambda}$. From this you can find the rate of green photons emitted from the bulb+filter. % One 546nm photon has E=6.626e-34 J sec * 3e8 m/sec / 546e-9m = 3.64e-19 J +% (6e-4 J/sec) / (3.64e-19 J/photon) = 1.65e15 photons/sec + +\item The photons emitted by the bulb get diluted by the apertures in the interference channel. The first is a single slit about $R=\unit[10]{cm}$ from the bulb with a width of \unit[0.1]{mm} and a height of \unit[1]{cm}. Assuming that the bulb radiates evenly onto a spherical surface of radius $R$, compute the fraction of the light that would pass through the slit. % slit area/4 pi R^2=7.96e-6 +% rate passing slit is 7.96e-6 * 1.65e15 photons/sec = 1.31e10 photons/sec + +\item After the single slit the beam is diffracted into a single slit interference pattern and impinges on the double slit. If you had the apparatus in front of you, you could estimate the area that the interference pattern covers. A good guess is $\unit[1]{cm^2}$. Assume this and that the double slits have an area twice that of the single slit. You can compute the rate of photons passing through the double slits. % 1.31e10 photons/sec * 2 * 7.96e-6 = 2.1e5 photons/sec + +\item We know that a photon travels at a speed $c=\unit[3\times 10^8]{m/sec}$. Divide that by the number of photons per second that pass through the double slits. This gives you the typical space between photons in meters/photon. % 1428 m/photon + +\item Invert the number from the previous bullet point and multiply by the 0.5~m between the double slit and detector slit to determine how many photons are in the setup on average. Is it a number much less than one? If so, there is a good reason to believe that the photons must be interfering with themselves to form the light and dark fringes you see. % 3.5e-4 photons in the half meter downstream of the double slit + +\end{enumerate} + +\newpage +\section*{Appendix: Fitting the interference data} + +You need to fit your data with Eqs.~(\ref{2slit_wDif}) and (\ref{1slit}). A detailed example is provided in ``Help on the double slit analysis'' in the Matlab folder on Blackboard or ``Fitting Help: Single and Double Slit" in the Python folder on Blackboard. {\em Please utilize this, it will save you a lot of time!} + +A few fitting tips and notes: + +\begin{itemize} +\item Make sure that units of all your measured values are self-consistent - the program will go crazy if you mix meters and micrometers together! +\item Try to plot your function using estimated values of the parameters before doing a fit with it. This catches many silly errors. +\item The fitting function has a term to account for the overall normalization and also for the fact that the maximum is not set at $x=0$. In other words, we substitute $x \rightarrow x-x_0$ when coding the fit function. Estimate both of these (for the normalization, look at what happens when $x-x_0 = 0$) but include them as free parameters in the fit. +\item The fitting function has a term to account for the ``pedestal'' (for the photodiode) or ``dark rate'' (for the PMT) you observed when the lights were off. The estimated value should be changed to agree with what you observed. +\item Just a note: fitting functions should include the minimal number of parameters. If I have parameters $a$ and $b$ but they always appear as $ab$ in my function, then I am much better off including a term $c=ab$ when doing the fit. Otherwise the fit is underconstrained. Adjusting $a$ up has the same effect as adjusting $b$ down so it's impossible to converge on unique values for $a$ and $b$. Fitting algorithms really dislike this situation. +\item If the program has trouble fitting all the parameters, first hold the values of the parameters you know fairly well (such as the light wavelength, maximum peak intensity, background, etc.). Once you determine the approximate values for all other parameters, you can release the fixed ones, and let the program adjust everything to make the fit better. +\end{itemize} -\section*{\emph{Two-Slit interference with atoms}\footnote{Special thanks to Prof. Seth Aubin for providing the materials for this section}} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Appendices %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newpage +\section*{Appendix: Two-Slit interference with atoms\footnote{Special thanks to Prof. Seth Aubin for providing the materials for this section}} \emph{According to quantum mechanics, the wave-particle duality must be applied not only to light, but to any ``real'' particles as well. That means that under the right circumstance, atoms should behave as waves with |