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diff --git a/single-photon-interference.tex b/single-photon-interference.tex index 7680c6a..f98f9ae 100644 --- a/single-photon-interference.tex +++ b/single-photon-interference.tex @@ -38,9 +38,10 @@ nature, of waves and of particles? And if experiments force us to suppose that 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 which will allow you to confront wave-particle duality in a +possible, and to give you the hands-on familiarity that will allow you to confront wave-particle duality in a precise and definite way. When you have finished, you might not fully understand the mechanism of duality -- -Feynman asserts that nobody really does -- but you will certainly have direct experience of the actual phenomena +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, @@ -48,13 +49,13 @@ oscilloscope, digital multimeter, counter. \begin{figure} \centering -\includegraphics[width=0.8\linewidth]{./pdf_figs/tsisetup} \caption{\label{tsifig1.fig}The double slit interference apparatus.} +\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 +\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 +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. @@ -79,7 +80,7 @@ the rotary scale is $0.01$~mm. and a \emph{photomultiplier tube} (PMT for short). The photodiode is used with the much brighter laser light; it's mounted on light the 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 +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{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. \end{itemize} @@ -106,7 +107,7 @@ For this mode of operation, you will be working with the cover of the apparatus on using the switch in the light source control panel, and move the laser in 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 +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 @@ -129,7 +130,7 @@ you'll be able to do so even when the box cover is closed. In your lab book desc 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 +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 @@ -152,7 +153,8 @@ I(x)= 4 I_0 \cos^2\left(\frac{\pi d}{\lambda}\frac{x}{\ell} \right)\left[\frac{\ 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 centers of the two slits. Discuss how this picture -would change if you vary the width and the separation of the two slits, and the wavelength of the laser. Make a +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} @@ -187,18 +189,18 @@ detector-slit micrometer. If all is well, you will see a systematic variation of 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 wave theory of light:} If we assume that the light +\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'' which theory -predicts, -- and record the photodiode reading. Then adjust the position of the slit-blocker to let the light +\item Find the highest of the maxima --- this is the ``central fringe'' or the ``zeroth-order fringe'' that 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. -\item To see another and even more dramatic manifestation of the wave nature of light, set the slit blocker again +\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. @@ -220,7 +222,7 @@ interference pattern, use your data to extract the information about the distanc 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 what two-slit mask you have installed). +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} @@ -237,9 +239,9 @@ Fit your data with Eqs.~(\ref{1slit}) and (\ref{2slit_wDif}). You will need to a 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, 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 then are absorbed inside the box except for those passing through two slits with area approximately $0.1\times 10~\mathrm{mm}^2$. Next, if we assume that the beam of photons passing through the slits diffract over a $1~\mathrm{cm}^2$ area by the time they reach the detector slit, estimate the rate of photons reaching the detector. Finally, we have to 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 event 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 this calculations before or after the lab period, but make sure to include them in the lab report.} +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 this 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 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. @@ -290,23 +292,23 @@ wavelength $\lambda_{\mathrm{atom}}=h/\sqrt{2mE}=h/p$ (often called de Broglie w constant, $m$ is the mass of the particle, and $E$ and $p$ are respectively the kinetic energy and the momentum of the particle. In general, wave effects with ``massive'' particles are much harder to observe compare to massless photons, since their wavelengths are much shorter. Nevertheless, it is possible, especially now when -scientists has mastered the tools to produce ultra-cold atomic samples at nanoKelvin temperatures. As the energy +scientists have mastered the tools to produce ultra-cold atomic samples at nanoKelvin temperatures. As the energy of a cooled atom decreases, its de Broglie wavelength becomes larger, and the atom behaves more and more like -waves. For example, in several experiments, researches used a Bose-Einstein condensate (BEC) -- the atomic -equivalent of a laser -- to demonstrate the atomic equivalent of the Young's double slit experiment. As shown in +waves. For example, in several experiments, researchers used a Bose-Einstein condensate (BEC) -- the atomic +equivalent of a laser -- to demonstrate the atomic equivalent of the Young's double-slit experiment. As shown in Fig.~\ref{BECinterferfometer.fig}(a), an original BEC sits in single-well trapping potential, which is slowly deformed into a double-well trapping potential thus producing two phase-coherent atom wave sources. When the trapping potential is turned off, the two BECs expand and interfere where they overlap, just as in the original -Young's double slit experiment.} +Young's double-slit experiment.} % \begin{figure}[h] \centering \includegraphics[width=0.8\linewidth]{./pdf_figs/BECinterferfometer} \caption{\label{BECinterferfometer.fig} -Atom interferometry version of Young's double slit experiment: \emph{(a)} schematic and \emph{(b)} +Atom interferometry version of Young's double-slit experiment: \emph{(a)} schematic and \emph{(b)} experimentally measured interference pattern in an ${}^{87}$Rb Bose-Einstein condensate.} \end{figure} -\emph{Fig.~\ref{BECinterferfometer.fig}(b), shows the resulting interference pattern for a ${}^{87}$Rb BEC. Atom +\emph{Fig.~\ref{BECinterferfometer.fig}(b) shows the resulting interference pattern for a ${}^{87}$Rb BEC. Atom interferometry is an area of active research, since atoms hold promise to significantly improve interferometric resolution due their much shorter de Broglie wavelength compared to optical photons. In fact, the present most accurate measurements of accelerations, rotations, and gravity gradients are based on atomic interference. } @@ -314,7 +316,7 @@ accurate measurements of accelerations, rotations, and gravity gradients are bas \section*{Appendix: Fraunhofer Diffraction at a Single Slit and Two-Slit interference} \textit{Diffraction at a Single Slit} We will use a \emph{Fraunhofer} diffraction model to calculate the -intensity distribution resulting from light passing a single slit of width $a$, as shown in +intensity distribution resulting from light passing a single-slit of width $a$, as shown in Fig.~\ref{interference.fig}(a). We will assume that the screen is far away from the slit, so that the light beams passed through different parts of the slit are nearly parallel. \begin{figure}[h] @@ -365,7 +367,7 @@ $\frac{\pi D}{\lambda}\sin\theta=(m+\frac{1}{2})\pi$ for $m = 0, \pm1, \pm2, \c Let us now consider the case of interference pattern from two identical slits separated by the distance $d$, as shown in Fig.~\ref{interference.fig}(b). We will assume that the size of the slits is much smaller than the distance between them, so that the effect of Fraunhofer diffraction on each individual slit is negligible. Then -going through the similar steps the resulting intensity distribution on the screen is given my familiar Young +going through the similar steps the resulting intensity distribution on the screen is given by familiar Young formula: \begin{equation} %\label{2slit_noDif} I(\theta)=\left|E_0e^{ikd/2\sin\theta} +E_0e^{-ikd/2\sin\theta} \right|^2 = 4I_0\cos^2\left(\frac{\pi |