> //"But what is light really? Is it a wave or a shower of photons? There seems no likelihood for forming a consistent description of the phenomena of light by a choice of only one of the two languages. It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do."// > > //--// A.Einstein and I. Leopold. //The evolution of physics: the growth of ideas from early concepts to relativity and quanta//. CUP Archive, 1961. In this experiment you will investigate the behavior of individual photons passing through an interferometer. A quantum mechanical treatment of this system yields surprising, and non-intuitive, results. These results focus attention on some unique aspects of quantum mechanics such as the behavior of wave packets, distinguishability of paths, wave particle duality, and the effects of observation on experimental outcomes. **WARNING:** The pump laser beam **WILL** cause permanent eye damage, including possible blindness, if used without protection. Laser safety goggles **MUST** be worn at all times and the door **MUST** be closed when the laser is on, even when the table is covered. Wait for instructions before turning on the pump laser. ====== References ====== ---- |[1] [[download/attachments/130155799/interference-with-correlated-photons.pdf?version=1&modificationDate=1438176442000&api=v2 |E.J. Galvez et. al. "Interference with Correlated photons: Five quantum mechanics experiments for undergraduates", //Am. J. Phys.// **73**, 127-140 (2005)]]. | [2] A.C. Melissinos, J. Napolitano. //Experiments in Modern Physics 2nd Ed//., Academic Press (2003). [3] R. P. Feynman, R. B. Leighton, M. Sands. //The Feynman Lectures on Physics, Vol III,// (Ch. 1.), Addison-Wesley Publishing Company, 1971. 1 Theory =========== ---- ===== 1.1 Classical behavior of an interferometer ===== Classically, we model light as a continuous electromagnetic wave. When such a wave passes through a 50/50 beamsplitter, half the intensity passes through (leading it down one path), while half is reflected (leading it down another). These two paths later are brought back together and the two waves sum; this recombination leads to //interference//. If the path lengths are different, a phase difference is introduced between the two waves before they are recombined. The amplitude of the recombined wave is proportional to //cos//(//δ//), where //δ// = 2_πd_///λ// is the phase difference, //d// is the path length difference ,and //λ// is the wavelength of the light. ===== 1.2 Mach-Zhender interferometer ===== In this experiment we will be using a Mach-Zhender interferometer, rather than the more familiar Michelson interferometer. The geometry of a Mach-Zhender interferometer is shown in Fig. 1. {FIXME ${/download/attachments/130155799/Mach_zehnder.png?version=1&modificationDate=1443721883000&api=v2}$ **Figure 1**: The Mach-Zehnder interferometer. |A Mach-Zehnder interferometer consists of two 50/50 beam splitters and two mirrors. A continuous wave entering the interferometer is split at the first beam splitter. Half of the light intensity travels "Path 1" and the other half travels "Path 2". After reflecting from a mirror, the waves from each path come together at the second beam splitter and sum to form two output waves exiting at ports //a// and //b//. If the light is monochromatic and the lengths of Path 1 and Path 2 are _l_1, and _l_2 respectively, there will be a phase difference of _δ = _2π |_l_1 - _l_2 |///λ// between the two beams when they recombine at the second beam splitter. The resulting amplitude of the sum wave will be proportional to cos(//δ//). | ===== 1.3 Quantum mechanical treatment of a single photon passing through a Mach-Zhender interferometer ===== Quantum mechanics assumes that light is composed of indivisible quanta called photons. (This assumption can be demonstrated experimentally, for example via the [[https://en.wikipedia.org/wiki/Photoelectric_effect|photoelectric effect]].) Consider the case of a single photon entering an interferometer. If the photon is indivisible, it must travel along one path or the other, but not both. Thus, it would seem that there could not be any interference when a single photon passes through the interferometer. However, quantum mechanics predicts that under certain circumstances a single photon //will// interfere with itself as if it simultaneously took both paths though the interferometer. Let us look at this more carefully. In quantum mechanics, one calculates probability amplitudes for different possible outcomes of an experiment. The probability //p// of a particular outcome being observed is the square of the absolute value of the probability amplitude //Φ,// | {FIXME ${/download/attachments/130155799/image2015-7-24%208%3A53%3A34.png?version=1&modificationDate=1438176442000&api=v2}$. | (1) | Following the approach of Feynman [3], if there is more than one way to reach a particular outcome, the probability amplitudes for each way have to have to be added. If the different ways to reach the same outcome are indistinguishable, the probability amplitude is the sum of the probability amplitudes for each way considered separately. Indistinguishable in this context means that there is no property of the particle which can be measured, which would indicate which way it went. So, for our case of a photon which can travel two possible paths through an interferometer the probability is calculated as, | {FIXME ${/download/attachments/130155799/image2015-7-24%208%3A55%3A14.png?version=1&modificationDate=1438176442000&api=v2}$ | (2) | such that | {FIXME ${/download/attachments/130155799/image2015-7-24%208%3A56%3A45.png?version=1&modificationDate=1438176442000&api=v2}$. | (3) | However, if the experiment is arranged so that it is possible to determine which path the photon took through the interferometer, then the probability is the sum of the probabilities for each path considered separately: | {FIXME ${/download/attachments/130155799/image2015-7-24%208%3A58%3A41.png?version=1&modificationDate=1438176442000&api=v2}$ | (4) | and | {FIXME ${/download/attachments/130155799/image2015-7-24%208%3A58%3A0.png?version=1&modificationDate=1438176442000&api=v2}$. | (5) | ==== 1.3.1 Indistinguishable paths ==== Using the above rules, we can now calculate, for example, the probability that a photon enters the interferometer, and exits from //port a//. Let us indicate each path by the index //j.// The probability amplitudes for //path j// are plane waves {FIXME ${/download/attachments/130155799/image2015-7-24%209%3A3%3A51.png?version=1&modificationDate=1438176442000&api=v2}$, where //δ=klj// , //lj// is the length of arm //j//, and _k = 2π/λ _is the wave number of the photon. At each beam splitter, the photon is either reflected with amplitude //r,// or transmitted with amplitude //t//. A photon which takes path 1 through the interferometer and exits //port a// will be transmitted at the first beam splitter and reflected at the second resulting in the probability amplitude | {FIXME ${/download/attachments/130155799/eqn_6.png?version=1&modificationDate=1475853108000&api=v2}$ | (6) | A photon which takes path 2 and exits //port a// likewise has the probability amplitude | {FIXME ${/download/attachments/130155799/eqn_7.png?version=1&modificationDate=1475853140000&api=v2}$ | (7) | Therefore, in the case when paths 1 and 2 are indistinguishable, as depicted in Fig. 2(A), the probability for a photon to exit _port _//a// becomes | {FIXME ${/download/attachments/130155799/image2015-7-24%209%3A17%3A38.png?version=1&modificationDate=1438176442000&api=v2}$, | (8) | which is simplified as | {FIXME ${/download/attachments/130155799/image2015-7-24%209%3A20%3A29.png?version=1&modificationDate=1438176442000&api=v2}$, | (9) | and finally | {FIXME ${/download/attachments/130155799/image2015-7-24%209%3A22%3A24.png?version=1&modificationDate=1438176442000&api=v2}$, | (10) | where //T = tt*// = 1/2 and //R = rr*// = 1/2 are the reflection and transmission probabilities and _δ = δ_1 _- δ_2 is the phase difference acquired from the difference between the two paths** .** {FIXME ${/download/attachments/130155799/worddav98ec1cb559ddffba5377830a7ba48351.png?version=1&modificationDate=1438176443000&api=v2}$ **Figure 2**: Using photon polarizations to illustrate distinguishability of paths. ==== 1.3.2 Distinguishable paths ==== We can arrange the experiment so that photons taking path 1 emerge with orthogonal planes of polarization relative to those taking path 2, as in Fig 2 (B). In this case the two paths are distinguishable because a measurement of the photon's polarization state after passing through the interferometer would allow us to determine which path was taken. Now, the probability of detecting the photon emerging from //port a// can be calculated as | {FIXME ${/download/attachments/130155799/image2015-7-24%209%3A44%3A11.png?version=1&modificationDate=1438176442000&api=v2}$, | (11) | or | {FIXME ${/download/attachments/130155799/image2015-7-24%209%3A45%3A15.png?version=1&modificationDate=1438176442000&api=v2}$. | (12) | It is not necessary for the experiment to //measure// the polarization state of the photon in order to eliminate the interference. Simply performing the experiment in such a way that the path taken by the photon can //in principle// be determined will cause the interference to disappear. ==== 1.3.3 Quantum eraser ==== Consider again the case where the experiment is performed so that a photon which takes path 1 emerges vertically polarized while one which takes path 2 is horizontally polarized. In this case, the two paths are distinguishable and there should be no interference. Suppose that we now place a linear polarizer outside of the interferometer and in front of detector //a,// such that the pass axis of the polarizer is at a 45º angle from horizontal. (See Fig. 2(C).) Now, horizontally and vertically polarized photons both have a 50% probability of passing through the polarizer. Additionally, any photon which passes through the polarizer will emerge plane polarized along the pass axis of the polarizer. Thus the detector at //port a// will no longer be able to distinguish which path the photon took. This means that the paths are again indistinguishable so quantum mechanics predicts that the interference pattern will re-appear, even though the polarizations of photons traveling the two arms remain orthogonal. The calculation of the probability of observing a photon in a detector at //port a// is somewhat more complicated because now the polarization state has to be included; details of the calculation can be found in [1]. Here, we simply state the final result which is | {FIXME ${/download/attachments/130155799/image2015-7-24%2012%3A34%3A19.png?version=1&modificationDate=1438176442000&api=v2}$. | (13) | Note that the total amplitude of the restored interference pattern is reduced by a factor of 2 (where //R// = //T// = 1/2) relative to the calculation done in Sec. 1.3.1. This reduction is due to the linear polarizer's filtering out half of the photons. ===== 1.4 Classical versus quantum mechanical interpretation ===== In the preceding section we showed how a quantum mechanical analysis of a single photon passing through a Mach-Zender interferometer leads to interference effects under certain circumstances. The conditions for interference to occur depend on whether or not the photon carries with it information about which path it took through the interferometer. The concept of a single particle simultaneously taking multiple paths through an experiment appears to go against common sense. However, this is exactly what quantum mechanics predicts. The question is not whether or not the prediction is sensible, but rather whether or not it is supported by observations which also //rule out// a classical interpretation. Classically, the interference effects, or lack thereof, described in Sec. 1.3 can all be accounted for if one assumes that rather than single photons we have a very weak electromagnetic wave entering the interferometer. By analyzing how the electric fields of the waves traveling both paths would recombine at the output of the interferometer, one can derive classical predictions for interference for light exiting the interferometer that match those provided by quantum mechanics. Where the two models differ in an irreconcilable manner, however, is when we take into account the particle-like nature of the experiment. If one can demonstrate that the experiment is detecting discrete, indivisible particles, then the classical models have to be abandoned in favor of the non-intuitive quantum mechanical interpretation. ====== 2 Apparatus ====== ---- Figure 3 shows the layout of the components on the optical table. Functionally, the components can be broken down into the following four parts: * the correlated photon source; * the Mach-Zehnder interferometer; * a HeNe laser for aligning the interferometer; and * mirrors for directing light into the interferometer. Positioning and alignment of the various components must be done with great precision. Due to time constraints, the optical components have been aligned on the table for you. You will perform only a limited number of manipulations during the experiment. **IMPORTANT**: Accidentally knocking a single optical component out of alignment by the slightest amount could result in a full day's delay. Keep the following points in mind whenever you work on the optical table: * Think about where you are putting your hands. Take care not to bump into things. * When rotating the half wave plates, do not twist the filter holder in the post holder. The face of the wave plate must remain normal to incident light. * When inserting and adjusting the beam splitter during the "indivisibility" portion, do not adjust or bump the APD filter/lens holders as their alignment is critical. {FIXME ${/download/attachments/130155799/worddav45d537e7e1094ddd809bee749bad5b10.png?version=1&modificationDate=1438176443000&api=v2}$ **Figure 3:** Beam paths and layout of components on the optical bread board. ===== 2.1 BBO crystal ===== Our source of correlated photon pairs is a Beta-Barium-Borate (BBO) crystal. Approximately one out of every 106 photons which enter the crystal will undergo a process called //spontaneous parametric down conversion//. In this process the incident (//pump//) photon is down converted (DC) into a pair of photons which we will refer to as DC photons. Some of the DC photon pairs will have equal energy and, by conservation of momentum, will leave the crystal with trajectories lying on a cone with opening angle θ with respect to the axis defined by the incident pump photon. (See Fig. 4.) {FIXME ${/download/attachments/130155799/worddav9f52c63beeca65de042fd8918d9533e8.png?version=1&modificationDate=1438176443000&api=v2}$ **Figure 4**: Parametric photon down conversion (DC) in the BBO crystal. The DC photons are produced with vertical polarizations. By convention, one of the DC photons in a pair is designated as the //idler// photon, the other is the //signal// photon. In our experiment we will refer to the photon which enters the interferometer as the signal, and the other half of the pair as the idler. ===== 2.2 405nm pump laser ===== Our source of pump photons is a 20 mW, 405 nm diode laser. The collimated beam from this laser reflects off of two mirrors and is directed onto the BBO crystal. The DC photons lying on the cone described above will have wavelengths of 810 nm.   The pump laser diode is controlled by the ILX Lightwave laser diode controller. (See Fig. 5.) The controller has two functions – it maintains the diode at a constant temperature (TEC Mode) and it controls the current which operates the laser (Laser Mode). Both the operating temperature (24 ºC) and operating current (50 mA) have been preset. The controller should already be on and the TEC should be running when you enter the lab. If not, turn on the ILX using the key switch on the lower left corner of the operating panel and then press the button indicated by the GREEN dot in the //TEC Mode// section of the control panel. The temperature of the diode is indicated by the TEC LED display and should reach 24 ºC in a few minutes. When it is time to turn on the pump laser, press the button indicated by the //Orange// dot on the //Laser Mode// section of the control panel. The diode current, as indicated on the LED display, should read 50mA when the laser is on. {FIXME ${/download/attachments/130155799/ILX.png?version=1&modificationDate=1443797370000&api=v2}$ **Figure 5**: Laser diode controller. **WARNING:** The pump laser beam **WILL** cause permanent eye damage, including possible blindness, if used without protection. Laser safety goggles **MUST** be worn at all times and the door **MUST** be closed when the laser is on, even when the table is covered. ===== 2.3 Avalanche photodiodes ===== The photon detectors we use are avalanche photodiodes (APDs). Ordinary photodiodes are semiconductor devices which generate an electric current via the photoelectric effect when struck by visible light photons. The magnitude of the current is proportional to the intensity of the light striking the diode. APDs operate analogously to a photomultliplier tube in that they use an avalanche process to produce a large number of liberated electrons for each photon captured. The APD modules we use are designed for single photon counting and produce a TTL output pulse for each photon detected. APDs can be damaged if directly exposed to too much light, such as room lights. Our APDs are protected from the room lights by coupling them to a fiber optic cable with a collimating lens which only allows photons coming straight into the lens to reach the APD. Further, we place a narrow bandpass interference filter in front of the fiber optic collimating lens which passes only photons with wavelength 810 ± 5 nm. The bandpass data for the filters is posted in the lab. **CAUTION**: Power to the APDs should never be turned on if the APDs are not connected to the fiber optic cables and with the narrow bandpass filters in place. Check that this is the case before turning on the APDs. In this experiment the DC photons produced in the crystal propagate outward along the surface of a cone as described earlier. One of the APDs (the idler APD, labeled APD #1) is positioned to detect DC idler photons directly. Two other APDs (Port A APD and Port B APD, labeled APD #2 and #3) are positioned to detect DC signal photons emerging from the two output ports of the Mach-Zehnder interferometer. ===== 2.4 Mach-Zehnder interferometer ===== The details of our Mach-Zehnder interferometer are shown in Fig. 3. DC photons from the BBO crystal are reflected into the interferometer by //Flip Mirror 2//. We will use the half wave plates to manipulate the degree of distinguishability of the two paths through the interferometer. A waveplate (//or retardation plate//) is a material with two optical axes, called the fast and slow axes. The waveplate has different indicies of refraction along these two axes such that light whose polarization is parallel to the fast axis propagates through the material faster than light whose polarization is parallel to the slow axis. When the polarization of the light lies between the fast and slow axes of the wave plate, the components of the light which project onto the fast and slow axes will propagate at different speeds and will emerge from the wave plate with a phase difference. A half waveplate has been designed so that for a specific wavelength of light the components along the two axes will be out of phase by one half a wavelength after passing through the waveplate. As a result, depending on the relative orientation of the half wave plate axes and the polarization of the incident light, the polarization of the light leaving the half waveplate can be rotated. When the half waveplate filter holders are rotated to read 0º the slow axis of the wave plate will be vertical. Since the DC photons are vertically polarized their polarization axis will not be rotated, and the two paths through the interferometer are indistinguishable. When one of the wave plates is rotated by 45º, the polarization of photons passing through them will be rotated by 90º into the horizontal plane. If the other wave plate is left at the 0º mark we have a configuration where photons traveling through one arm of the interferometer are horizontally polarized and photons traveling the other arm are vertically polarized. The two paths can now be distinguished by measuring the polarization of the photon after passing through the interferometer. One of the corner mirrors is mounted to a //piezoelectric crystal//. A //piezoelectric crystal// (piezo) has the property that it expands when a voltage is applied to it. This expansion is highly repeatable and can be controlled at the level of fractions of a micron. By applying a voltage to the piezo we can displace the attached mirror, and hence change the path length of this arm of the interferometer, by a fraction of the wavelength of a DC photon. The piezo voltage is controlled by the y-axis of the //3-Axis Piezo Controller//. The other corner mirror is mounted on a //Micrometer Stage// which allows the length of the two arms of the interferometer to be precisely matched. The //Micrometer Stage// has already been adjusted so that when the piezo voltage on the opposite mirror mount is set to 50V, both arms of the interferometer are the same length. Photons leaving the second //Beam Splitter// are detected by Port A APD or Port B APD depending on which output path they took. In front of Port A APD is a post holder which can hold either a //Linear Polarizer// or an //Empty Filter Holder//. The linear polarizer is used for the quantum eraser part of the experiment. Otherwise an empty filter holder should be placed here so that the effective aperture of the APD remains constant for all measurements. ===== 2.5 Computer data acquisition ===== The APD signals are collected and the piezo voltage is controlled by a computer data acquisition (DAQ) system. A National Instruments PCIe-6341 multifunction DAQ card (NI-DAQ), installed in the computer tower, connects to an SCB-68A interface as shown in Fig. 6. {FIXME ${/download/attachments/130155799/worddav032d73a252dceb03ec496ae072c2e2e0.png?version=1&modificationDate=1438176442000&api=v2}$ **Figure 6**: Computer data acquisition system. Signals from the APDs and to the ThorLabs piezo driver connect to the SCB-68A through a BNC connector plate as shown schematically in Fig. 7. {FIXME ${/download/attachments/130155799/worddavd8469e0eca1f7f39b6cabe8e0b2812cd.png?version=1&modificationDate=1438176443000&api=v2}$ **Figure 7a**: Layout of interface connections. {FIXME ${/download/attachments/130155799/Wiring%20diag_v2.png?version=1&modificationDate=1443721145000&api=v2}$ **Figure 7b**: Wiring diagram for making connections to the NI interface. The DAQ card is capable of both generating and collecting a variety of analog and digital signals. For this experiment we use one analog voltage output to control the piezo voltage, and four digital counter inputs to record pulses from the APDs. The card is controlled by a LabView program called //SP_DAQ//. ===== 2.6 DAQ software ===== //SP_DAQ// has three operating modes selectable by tabs in the interface window; //Singles Rates//, //Coincidence Rates// and //Coincidence Scan//. In //Singles Rates// mode the interface displays an analog and digital rate meter for each counter on the DAQ card. There is an input box for the user to set the //Count Time// which is the time, in seconds, for which the DAQ will count pulses from the APDs. Clicking the //Start// button initiates a measurement. While running the analog meters display the pulse rate for each counter. At the end of the measurement the average event rate and the total number of counts recorded are displayed along with the actual elapsed run time. The //Abort// button stops a measurement in progress. In _Coincidence Rates _mode the interface displays analog and digital rate meters for Idler singles rates, Idler/Port A coincidence rate, and Idler/Port B coincidence rate.  This mode operates just like the singles rates mode. In //Coincidence Scan// mode the interface displays controls and readings for two photon coincidence counting while scaning over a range of piezo voltages. The user can set a //Starting Piezo Voltage//, an //Ending Piezo Voltage//, and the //Voltage Step// size for the piezo scan using the sliders. The //Run Length// can be set which determines how long the software spends counting at each voltage step. Clicking the //Start// button initiates a scan. The //Abort// button stops a scan in progress. A //Save// button allows the user to save the measurements from the most recent scan to a text file. ===== 2.7 Coincidence counting ===== In order to count coincident hits on two different APD's we utilize the counters on the NI-DAQ card in a start/stop mode as shown in Fig. 8. In this mode each pulse from the Idler APD is split into two signals, one (prompt) going to the start input of a counter, the other (delayed) going through an additional 100 ns of delay before arriving at the stop input of the same counter. The signal from a second APD runs through a 50 ns delay before connecting to the counter input. Thus a photon detected by the Idler APD will cause the counter to run for a 100 ns interval of time. If a photon strikes the APD connected to the input of that counter within this 100 ns interval it will be recorded. If both APDs detect photons simultaneously, the signal from the second APD will arrive at the counter input 50 ns after the counter has been started by the signal from the Idler APD and will thus be counted. {FIXME ${/download/attachments/130155799/PCIe-6341_coinc_v2.png?version=1&modificationDate=1443721083000&api=v2}$ **Figure 8**: NI-DAQ card counter connections. Used this way, the counter increments only when the two APD's are hit within ±50 ns of one another. ===== 2.8 DAQ interface connections ===== The BNC side of the interface connector panel is labeled as shown in Fig. 7a. Verify that the connections are correct. The four APD's should all have 3 ft long BNC cables coming from the outputs on the APD modules. The APDs from Ports A and B each connect to an additional 50 ns long BNC cable before connecting to the interface connector panel. The 3 ft long BNC from the idler APD connects to the Idler Prompt input on the interface connector panel with a "T" adapter. The other end of the "T" adapter connects to a 100 ns long BNC cable the other end of which connects to the Idler Delay input. The Piezo voltage output on the interface connector panel should be connected to the voltage input for the //y//-axis of the ThorLabs 3-Axis Piezo Controller. Use the manual knob on the Piezo controller to set the y-axis voltage to 50 V. ====== 3 Procedure ====== ---- ===== 3.1 APD pulses and background noise rates ===== APDs, like photomultiplier tubes, will produce pulses at some non-zero average rate even in the absence of light. This //dark rate// is an intrinsic characteristic of the detectors and is a source of background noise in the measurements. Another source of background is stray room light. We would like rate of detected DC photons to be significantly greater than the rate due to noise. There is a maximum rate of pulses above which it is possible to cause damage to the detectors. Although the APD modules are designed to protect themselves in the even that count rates go too high, it is best to operate them under conditions that do not approach the maximum allowable count rates. With the pump laser off and the optical table covered, turn on the APDs and measure the rate of pulses coming from each of the detectors, also known as the //singles rates//. Under these conditions the singles rate of each APD should be less than 2000 counts/sec. **WARNING**: If you ever measure APD count rates over 1 MHz (106 counts/sec) immediately turn the detectors off and get the TA or lab staff to assist you in determining the cause of the excessive rates. Look at the output from one of the APDs on the scope. Think about what the signal from the APDs tells you about the nature of light and whether or not it arrives as a continuous electromagnetic wave or in discrete chunks (i.e. photons). > **NOTEBOOK**: Sketch a typical APD output pulse, include details of the pulse width and height. Leaving the table covered, put on safety goggles and turn on the 405 nm pump laser to produce a flux of DC photons from the BBO crystal. Measure the APD count rates.  > ** NOTEBOOK: ** Record the singles rates for all four detectors in your lab notebook. Include sufficient of the conditions under which the measurements were made that you could duplicate them.   ===== 3.2 Accidental coincidence rates ===== Verify that the components on the optical table are laid out as shown in Fig. 3. All APDs should have their narrow bandpass filters in place. - Both half waveplates should be rotated to the 0º mark. - The piezo voltage should be set to 50 V. By coincidence counting, we mean we wish to count the number of times a given pair of APDs (for example APD #1 and Port B APD) simultaneously record a photon hit. For our purposes photon detections within ±50 ns of each other are considered simultaneous. With the pump laser off, room lights on and table covered, measure the coincidence rates between the Port A and Idler APDs (A/I), and the Port B and Idler APDs (B/I). With the laser off you are measuring accidental coincidences due to uncorrelated photons which happen to arrive at the detectors within 50 ns of each other due to random chance. Make sure you count for a long enough time to accumulate a statistically significant number of coincidences. Now measure the singles rates for these three APDs under the same conditions as you just measured the coincidence rates. We can also compute the expected accidental rate from what we know about the singles rate on each detector individually. In this type of coinicidence counting experiment, the accidental coincidences are caused by uncorrelated events at the two detectors which just happen to occur at the same time due to random chance. If the rate of events at each of the detectors (_R_1 and //R_2) is not large in comparison to the resolving time (Δ_t//) of the experiment, then the rate of accidental coincidences (_R_acc) can be calculated according to | _R_acc = _R_1_R_2Δt. | (14) | For a derivation of this relation see page 412 of reference [2]. > **NOTEBOOK:** From the singles rates calculate the expected number of coincidences for A/I and B/I. Are the measured coincidence rates consistent with the expected rate of accidental coincidences? Turn on the pump laser to produce DC photons from the BBO crystal. The APDs should now be recording simultaneous hits from correlated signal and idler DC photon pairs in addition to any accidental coincidences. Repeat the measurements of the singles rates and coincidence rates. > **NOTEBOOK:** From the singles rates calculate the expected number of coincidences for A/I and B/I. Are the measured coincidence rates consistent with the expected rate of accidental coincidences? ===== 3.3 Divisibility of photons ===== One of the central ideas in interpreting single photon interference results is that light is made up of //indivisible// bundles of energy (photons). The optical setup shown in Figure 9 can be used to demonstrate whether or not individual photons are observed to split. If a photon were to split at the beam splitter, you would observe coincident detections at the Idler APD and APD #4. (See Fig. 9.) {FIXME ${/download/attachments/130155799/worddava796465c4a09936dfeb6d0147a73da0d.png?version=1&modificationDate=1438176443000&api=v2}$ Figure 9: Optical setup for testing whether or not photons split. In order to measure coincidences between the Idler APD and APD #4 swap the cable from APD #4 with the cable from one of the interferometer output ports. Measure the singles rates of both APDs and their coincidence rate both with and without the beam splitter in place. Note, the TA or lab staff can assist you with placing and aligning the beam splitter. > ** NOTEBOOK: ** Sketch the arrangement of the beamsplitter and detectors.  Record the number of counts, singles and coincidences, for both sets of measurements. Estimate the number of accidental coincidences you would expect from random counting statistics. ===== 3.4 Single photon interference (indistinguishable paths) ===== Return the piezo voltage to 35 V on the controller. Measure coincidence rates, for A/I and B/I, and singles rate for the Idler APD for piezo voltages from 35 V to 65 V in 1 V steps. > **NOTEBOOK:** Record the settings you use in the DAQ as well as the name(s) of any saved data file(s). > > **NOTEBOOK:** Sketch the interferometer, identifying the different components.  Record the length of the two arms of the interferometer. ===== 3.5 Single photon interference (distinguishable paths) ===== Rotate one of the half wave plates by 45º while leaving the other waveplate at 0º. The two paths through the interferometer are now distinguishable. Measure the coincidence and singles rates again over the same voltage range. The degree to which the two paths are distinguishable is determined by the relative alignment of the two half wave plates. If they are exactly 45º apart, the photon polarizations for the two paths through the interferometer will be exactly orthogonal. As a result quantum mechanics predicts that there will be no single photon interference. However, if the two half wave plates are not 45º apart the photon polarizations will not be completely orthogonal and you still will observe a single photon interference pattern, but at a reduced amplitude. Plot your measured coincidence rates against piezo voltage. If your two half wave plates are set correctly your data should be statistically consistent with a straight line and there should be no apparent sinusoidal variation of the rates with piezo voltage. If your data still show some interference effect, you will need to fine tune the angle between the half wave plates until the interference pattern disappears. To do this change the angle of the 0º wave plate by a small amount, in either direction, and repeat the measurement of coincidence rates vs. piezo voltage. Plot this data along with the first data set. If the amplitude of the interference effect has decreased, continue to change the angle of the waveplate in the same direction. If the amplitude of the interference effect has increased, you need to rotate the waveplate in the other direction. By repeating this process, you will eventually find the angle at which the interference pattern vanishes. It should take you no more than 4 or 5 iterations to make the two paths through the interferometer completely distinguishable. > **NOTEBOOK:** Record the settings you use in the DAQ as well as the name(s) of any saved data file(s). ===== 3.6 Single photon interference (quantum eraser) ===== In order to erase the information about which path the photon took, we will place a linear polarizer in front of the Port B APD. The linear polarizer needs to be adjusted so that its pass axis makes a 45º angle with respect to the polarization axes of the two paths through the interferometer. Use the following procedure to set the pass axis of the linear polarizer. - Use a piece of black foam board to block one arm of the interferometer. - Place the linear polarizer in front of the Port B APD . - Rotate the polarizer while observing the singles count rate for the Port B APD. Find the two angles on the polarizer filter holder which correspond to the cases when the pass axis of the polarizer is parallel to the polarization of the photons. Make note of these angles. - Now move the light blocking foam board to the other arm of the interferometer. Determine the polarizer angles associated with the polarization axis light passing through this arm of the interferometer. - Use this information to set the polarizer to an angle which is as close as possible to being 45º with respect to the two possible photon polarization axis. Now that you have set the polarizer to the proper angle, make sure that both arms of the interferometer are unblocked. Cover the table. Measure the coincidence rates as a function of piezo voltage. > ** NOTEBOOK: ** Record the settings you use in the DAQ as well as the name(s) of any saved data file(s). ====== 4 Analysis ====== ---- Unlike some of the other experiments in this course, the analysis for this lab is primarily //qualitative//. In addition to the data collected in this lab, you may also draw on your knowledge of general wave and particle phenomena or experiments you have done previously – e.g. wave experiments from PHYS 133/143 or the photoelectric effect experiment from PHYS 154 – to support your arguments. To help you understand the logic needed to explain your observations, consider the following. === Interferometer – classical waves === Present your observations from the three interferometer experiments – plot coincident count rate as a function of piezoelectric voltage on the two interferometer outputs, and discuss. Are your observations here consistent with a classical wave model or with a classical indivisible particle model? Be detailed and handle each of the three cases individually. Consider concepts such as superposition and polarization. It may help to consider specifically the orientation of the electric field vector of a classical electromagnetic wave in each case.  === Single photons – classical particles === At what points in this experiment do you observe single photons? Consider how the avalanche photodiode works and consider your observations from the divisibility experiment. Present your measurements of the singles and coincident count rates for APDs 1 and 4 both with and without the beam splitter and discuss. Are your observations here consistent with a classical wave model or with a classical indivisible particle model? === Rectifying the two sets of observations === You now have some observations consistent with classical waves and some consistent with classical particles. Can we rectify the two sets of observations classically? Given your singles rates measurements, can you show that there is (on average) no more than one particle in the interferometer at a time? If this is true, how can quantum mechanics explain all your observed measurements? Where necessary, detail how the classical explanation fails in the face of single photon evidence, but can be replaced by a quantum mechanical one which is consitent with all observations. Rubric ---- When writing your report, consult the rubric and notes below **for the appropriate quarter**. ===== Autumn Quarter ===== | **Description** | **Percent** | | Classical Mechanical Treatment | 30 | | Use the classical wave model of light to explain the interferometer data for the three parts of the interferometer experiment: (1) both wave plates aligned to 0°, (2) one wave plate rotated by 45º, (3) adding the polarizer set to 45º in front of one of the detectors. Include a plot of the coincidence rates vs. piezo voltage for each case. **HINT:**//Using the principle of superposition and considering what is going on with the electric field vector of a classical electromagnetic wave, you should be able to completely explain your data in a couple short paragraphs with no equations. Close examination of Fig. 2 may be useful in aiding your understanding//. | | Single Photons? | 30 | | Do you have evidence that the light emitted from the BBO crystal comes in discrete chunks (i.e. quanta)? **HINT:** //The answer is actually so simple it may not be obvious, but can be given in a single sentence with no math//. Show that statistically it is highly improbable for there to be more than one photon passing through the interferometer at a time. **HINT**** :** //Using the 10 nm bandwidth of the interference filters at the entrance of the APDs as the uncertainty of the detected photon wavelength, calculate the length of the photon wave packet from the uncertainty principle. Then, make use of the dimensions of the interferometer on the optical table and the average time interval between successive photons striking the output port detectors.// Comment on whether or not single photons are ever observed to split at a beamsplitter. | | Quantum Mechanical Treatment | 40 | | You now have the following six separate observations of the behavior of light emitted from the BBO crystal: the three different interferometric measurements and the three points discussed in the //Single Photons?// box above. In the theory section of this experiment we show how a quantum mechanical treatment of the system accounts for all of the observed behaviors. Can you come up with a classical interpretation of all six measurements using the wave model of light? If not, explain how classical physics fails, but how the quantum mechanical explanation prevails. **HINT**: //If you can explain all six measurements classically, submit your work to a journal and await your Nobel Prize!// | ===== Winter Quarter ===== | **Description** | **Percent** | | Experimental Method | 15 | | Give an overview of the optical setup. Describe how the photon divisibility measurement is performed. Describe how the Mach-Zhender interferometer is used to manipulate the distinguishability of the two paths, and how the eraser works. | | Classical Mechanical Treatment | 25 | | Use the classical wave model of light to explain the interferometer data for the three parts of the interferometer experiment: (1) both wave plates aligned to 0°, (2) one wave plate rotated by 45º, (3) adding the polarizer set to 45º in front of one of the detectors. Include a plot of the coincidence rates vs. piezo voltage for each case. **HINT:**//Using the principle of superposition and considering what is going on with the electric field vector of a classical electromagnetic wave, you should be able to completely explain your data in a couple short paragraphs with no equations. Close examination of Fig. 2 may be useful in aiding your understanding//. | | Single Photons? | 25 | | Do you have evidence that the light emitted from the BBO crystal comes in discrete chunks (i.e. quanta)? **HINT:** //The answer is actually so simple it may not be obvious, but can be given in a single sentence with no math//. Show that statistically it is highly improbable for there to be more than one photon passing through the interferometer at a time. **HINT**** :** //Using the 10 nm band width of the interference filters at the entrance of the APDs as the uncertainty of the detected photon wavelength, calculate the length of the photon wave packet from the uncertainty principle. Then, make use of the dimensions of the interferometer on the optical table and the average time interval between successive photons striking the output port detectors.// Comment on whether or not single photons are ever observed to split at a beamsplitter. | | Quantum Mechanical Treatment | 35 | | You now have the following six separate observations of the behavior of light emitted from the BBO crystal: the three different interferometric measurements and the three points discussed in the //Single Photons?// box above. In the theory section of this experiment we show how a quantum mechanical treatment of the system accounts for all of the observed behaviors. Can you come up with a classical interpretation of all six measurements using the wave model of light? If not, explain how classical physics fails, but how the quantum mechanical explanation prevails. **HINT**: //If you can explain all six measurements classically, submit your work to a journal and await your Nobel Prize!// | ===== Spring Quarter ===== | **Description** | **Percent** | | Abstract | 5 | | Summarize the motivation, methods, results, and conclusions of your experiment. Include numerical values (with uncertainties) where appropriate. | | Theory, Introduction | 5 | | Introduce the idea of interference and discuss (in general terms) how it might appear in a Mach-Zender interferometer. **NOTE**: you will be explaining interference more carefully below for both the classical and quantum cases, so you need not repeat yourself here. | | Experimental Method | 10 | | Give an overview of the optical setup. Describe how the photon divisibility measurement is performed. Describe how the Mach-Zhender interferometer is used to manipulate the distinguishability of the two paths, and how the eraser works. | | Classical Mechanical Treatment | 25 | | Use the classical wave model of light to explain the interferometer data for the three parts of the interferometer experiment: (1) both wave plates aligned to 0°, (2) one wave plate rotated by 45º, (3) adding the polarizer set to 45º in front of one of the detectors. Include a plot of the coincidence rates vs. piezo voltage for each case. **HINT:**//Using the principle of superposition and considering what is going on with the electric field vector of a classical electromagnetic wave, you should be able to completely explain your data in a couple short paragraphs with no equations. Close examination of Fig. 2 may be useful in aiding your understanding//. | | Single Photons? | 20 | | Do you have evidence that the light emitted from the BBO crystal comes in discrete chunks (i.e. quanta)? **HINT:** //The answer is actually so simple it may not be obvious, but can be given in a single sentence with no math//. Show that statistically it is highly improbable for there to be more than one photon passing through the interferometer at a time. **HINT**** :** //Using the 10 nm band width of the interference filters at the entrance of the APDs as the uncertainty of the detected photon wavelength, calculate the length of the photon wave packet from the uncertainty principle. Then, make use of the dimensions of the interferometer on the optical table and the average time interval between successive photons striking the output port detectors.// Comment on whether or not single photons are ever observed to split at a beamsplitter. | | Quantum Mechanical Treatment | 35 | | You now have the following six separate observations of the behavior of light emitted from the BBO crystal: the three different interferometric measurements and the three points discussed in the //Single Photons?// box above. In the theory section of this experiment we show how a quantum mechanical treatment of the system accounts for all of the observed behaviors. Can you come up with a classical interpretation of all six measurements using the wave model of light? If not, explain how classical physics fails, but how the quantum mechanical explanation prevails. **HINT**: //If you can explain all six measurements classically, submit your work to a journal and await your Nobel Prize!// |