[2] A.C. Melissinos, J. Napolitano. Experiments in Modern Physics 2nd Ed., Academic Press (2003).

There are only two known physical phenomena for which classical physics fails to provide an explanation of the experimental results, Wave Particle Duality and Quantum Entanglement.  For both of these phenomena, attempts to explain the results of experiment using only classical physics concepts leads to nonsensical contradictions.  Quantum mechanics however, correctly predicts the outcomes of these experiments.  In this experiment you will investigate the behavior of individual photons interacting with beam splitters and passing through an interferometer.  The results of the measurements illustrate the nature of wave particle duality as well as quantum mechanical concepts such as the behavior of wave packets, distinguishability of paths, and the effects of observation on experimental outcomes.

The concept of wave particle duality is perhaps best expressed by the following quote, 

“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.

Experimentally light, as well as matter at the subatomic scale, can be shown to have the properties of both waves and particles.  Furthermore, whether light behaves as a wave or a particle depends on the nature of the measurement being made and the information available to the observer.  For this lab you are provided with:

• A light source which produces photon pairs with well defined trajectories, wavelengths and polarizations.
• Optical components called half-waveplates which allow you to change the polarization state of the light.
• An optical setup consisting of a beamsplitter with a pair of photodetectors, sensitive enough to detect individual photons, positioned to detect light which has passed through or been reflected by the beamsplitter.
• An optical interferometer.  The length of one arm of the interferometer can be varied by sub-micron amounts.  A second pair of photodetectors are positioned to record photons exiting the interferometer.
• A computer based data acquisition system which allows you to count individual hits on the photodetectors as well as simultaneous hits on more than one detector at a time.

For purposes of this discussion we will define something as being a particle or a wave based on the following criteria.

1. Localization:  Particles have well defined discrete locations in space at any particular time, whereas waves extend continuously throughout space.   - Interference:  Waves can combine at a given location and exhibit interference effects, whereas particles do not.  

Your task in this lab is to devise and perform a series of measurements from which you can make the case that:

• Contrary to the dictates of classical physics, photons are observed to behave as both particles and waves.
• That whether the photon reveals itself as being wavelike or particle like changes based on what information is available to the observer.

The interferometer is a very useful device for investigating the wave nature of light.  As part of this lab you will conduct experiments using a type of interferometer known as a Mach-Zhender interferometer.  The geometry of a Mach-Zhender interferometer is shown in Fig. 1.

{ ${/download/attachments/162727370/Mach_zehnder.png?version=1&modificationDate=1490895430000&api=v2}$ Figure 1: The Mach-Zehnder interferometer.

Here we present a quantum mechanical treatment of a single photon passing through a Mach-Zhender interferometer.  The analysis will illustrate how quantum mechanics allows a single photon to interfere with itself under certain conditions related to what information about the photon is available to the observer.

Quantum mechanics assumes that light is composed of indivisible quanta called photons. (This assumption can be demonstrated experimentally, for example via the 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,

{ ${/download/attachments/162727370/image2015-7-24%208%3A53%3A34.png?version=1&modificationDate=1490895432000&api=v2}$.

(1)

Following the approach of Feynman [3], if there is more than one path to reach a particular outcome, the probability amplitudes for each path have to have to be added. If the different paths to the same outcome are indistinguishable, the probability amplitude is the sum of the probability amplitudes for each path 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 amplitudes are summed as,

{ ${/download/attachments/162727370/image2015-7-24%208%3A55%3A14.png?version=1&modificationDate=1490895432000&api=v2}$

(2)

such that the probability of an outcome is,

{ ${/download/attachments/162727370/image2015-7-24%208%3A56%3A45.png?version=1&modificationDate=1490895432000&api=v2}$.

(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:

{ ${/download/attachments/162727370/image2015-7-24%208%3A58%3A41.png?version=1&modificationDate=1490895432000&api=v2}$

(4)

and

{ ${/download/attachments/162727370/image2015-7-24%208%3A58%3A0.png?version=1&modificationDate=1490895432000&api=v2}$.

(5)

Using the above rules, we can now calculate, for example, the probability that a photon which enters the interferometer exits from port a. Let us indicate each path by the index j. The probability amplitudes for path j are plane waves { ${/download/attachments/162727370/image2015-7-24%209%3A3%3A51.png?version=1&modificationDate=1490895432000&api=v2}$, 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 beam splitter resulting in the probability amplitude

{ ${/download/attachments/162727370/eqn_6.png?version=1&modificationDate=1490895430000&api=v2}$

(6)

A photon which takes path 2 and exits port a likewise has the probability amplitude

{ ${/download/attachments/162727370/image2017-1-5%2012%3A13%3A32.png?version=1&modificationDate=1490895432000&api=v2}$

(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

{ ${/download/attachments/162727370/image2015-7-24%209%3A17%3A38.png?version=1&modificationDate=1490895431000&api=v2}$,

(8)

which is simplified as

{ ${/download/attachments/162727370/image2015-7-24%209%3A20%3A29.png?version=1&modificationDate=1490895432000&api=v2}$,

(9)

and finally

{ ${/download/attachments/162727370/image2015-7-24%209%3A22%3A24.png?version=1&modificationDate=1490895431000&api=v2}$,

(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 .   In this case interference effects are observed due to the _cosδ _term .

If we now arrange the experiment so that it is possible to know which path the photon took the probability of detecting the photon emerging from port a would be calculated as

{ ${/download/attachments/162727370/image2015-7-24%209%3A44%3A11.png?version=1&modificationDate=1490895431000&api=v2}$,

(11)

or

{ ${/download/attachments/162727370/image2015-7-24%209%3A45%3A15.png?version=1&modificationDate=1490895431000&api=v2}$.

(12)

Note that in this case there is no cosine dependence on the path length difference and thus no interference effects are observed.  It is not necessary for the experiment to measure which path the photon took 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.

Figure 3 shows the layout of the components on the optical table. Functionally, the components can be broken down into the following parts:

• A 405nm pump laser and BBO crystal for producing vertically polarized 810nm photon pairs.
• A Mach-Zehnder interferometer.
• A beamsplitter apparatus.

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.

The photodetectors are avalanche photodiodes which are sensitive enough to detect individual photons.  Signals from the photodiodes are sent to a computer data acquisition system which has the ability to count the number of detections on each of the photodetectors, as well as to count the number of simultaneous detections on pairs of detectors.

In the following sections we explain how these different components function.  It will be up to you to decide what measurements to make with the equipment provided.

IMPORTANT: Accidentally knocking a single optical component out of alignment by the slightest amount could result in a full day's delay. Think about where you are putting your hands. Take care not to bump into things. Don't adjust the optics yourself except where explicitly instructed.

{ ${/download/attachments/162727370/QE_layout_v5.png?version=1&modificationDate=1490895432000&api=v2}$\\

Figure 3: Beam paths and layout of components on the optical bread board.

Our source of correlated photon pairs is a Beta-Barium-Borate (BBO) crystal. Approximately one out of every 106 photons from the pump laser 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 with by the incident pump photon trajectory. (See Fig. 4.)

{ ${/download/attachments/162727370/worddav9f52c63beeca65de042fd8918d9533e8.png?version=1&modificationDate=1490895431000&api=v2}$ 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.  

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 pump laser 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.

{ ${/download/attachments/162727370/ILX.png?version=1&modificationDate=1490895430000&api=v2}$ 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. Always check your goggles to verify that they are the correct ones for the wavelength of the laser. The lenses of the goggles will indicate the wavelength ranges and their optical density (OD) ratings as indicated in the image below. If the goggles are not correct for the wavelength of the laser, or if they appear damaged in any manner, bring them to the attention of an instructor before turning on the laser.

{ ${/download/attachments/162727370/Goggles.png?version=1&modificationDate=1490895433000&api=v2}$

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.

The details of our Mach-Zehnder interferometer are shown in Fig. 3. Signal photons from the BBO crystal are reflected into the interferometer.

There is an 808nm half wave plate in each arm of 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. If a waveplate is rotated by 45º, the vertical polarized photons entering them will be rotated by 90º into the horizontal plane.

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.  According to the manufacturers specifications the piezo expansion is 61±15nm/V. 

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. 

As shown in Fig. 3, the idler photon goes into a setup involving a 1/2 waveplate, a polarizing beam splitter and two APDs.  The APD's are positioned such that any photons which pass through the beam splitter will be detected by APD #1, those which are reflected by the beam splitter are detected by APD #4.  Unlike the other beam splitters used in this experiment, here we use a polarizing beam splitter.  In a polarizing beam splitter whether or not a photon is transmitted or reflected depends on its polarization with respect to the orientation of the beam splitter.  Our polarizing beam splitter is setup so that vertically polarized photons are reflected into APD #4 and horizontally polarized photons are transmitted straight into APD #1.  Down converted photons from the BBO crystal are vertically polarized and would all be reflected by the beam splitter.  The 1/2 waveplate allows us to rotate the plane of polarization of the down converted photons before they reach the beam splitter.  We could use the 1/2 waveplate to rotate the DC photon polarizations into the horizontal plane, in which case they would all be transmitted and detected by APD #1.  We could also use the 1/2 waveplate to rotate the DC photon polarizations to 45 degrees ( halfway between vertical and horizontal ).  In this case each photon would have a 50/50 chance to be reflected or transmitted so that half of the photons would be detected by APD #1 and half by APD #4.  

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.

{ ${/download/attachments/162727370/worddav032d73a252dceb03ec496ae072c2e2e0.png?version=1&modificationDate=1490895431000&api=v2}$ 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.

{ ${/download/attachments/162727370/worddavd8469e0eca1f7f39b6cabe8e0b2812cd.png?version=1&modificationDate=1490895431000&api=v2}$ Figure 7a: Layout of interface connections.

{ ${/download/attachments/162727370/Wiring%20diag_v2.png?version=1&modificationDate=1490895430000&api=v2}$ 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.

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.

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.

{ ${/download/attachments/162727370/PCIe-6341_coinc_v2.png?version=1&modificationDate=1490895430000&api=v2}$ 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.

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.

Your task is to use the apparatus provided to perform measurements which will allow you to show that:

1. Photons behave as both particles and as waves, in a manner which is inconsistent with a classical interpretation. - Whether a photon behaves as a particle or a wave changes based on the information available to the 'observer'.

Some notes to consider.

• We will assume knowledge of the photoelectric effect which is the basic principle of operation of our photodetectors.  Experimentally the photoelectric effect showed that the ejection of a photoelectron from a metal exposed to light cannot be explained by the classical wave model of light, but can be explained by assuming that light is composed of discrete particles each carrying a quantized chunk of (wavelength dependent) energy.  Thus we can interpret a single pulse from one of our photodetectors as the detection of a single photon.
• The interferometer can be used to test for wave-like behavior based on the presence or absence of interference effects.  
• When you observe interference effects, try to think of scenarios in which particle(s) passing through the interferometer might interfere.  Then make measurements to either confirm or rule out these scenarios.
• You may find that the uncertainty principle can be useful in strengthening your argument for the first task. { ${/download/attachments/162727370/image2017-1-4%2011%3A11%3A38.png?version=1&modificationDate=1490895432000&api=v2}$. * Demonstrating how the behavior of the photon changes according to the information available to the observer can be accomplished by measurements made using only the interferometer apparatus.
• For the purposes of our experiment all of the information carried by a photon ( and hence available to an observer ) is contained in; x - the location of the photon in space, t - the time at which the photon is detected, E - the energy of the photon, and ε - the photons polarization. 

| * You may also find it useful to calculate the expected number of accidental coincidences between two detectors, Racc=R1R2Δt.  For more details on this calculation see this paper on accidental coincidences. |

Sometime near the end of the first day you should go over your plan for taking measurements with an instructor.  The concepts involved are not complicated but they are subtle and in cases anti-intuitive, so we don't expect you to work them all out on your own.  Our goal is to help guide you through the process of grappling with the logical disconnect between the classical and quantum mechanical interpretations of how nature works at the sub-atomic scale of photons, electrons, etc.  

When writing your report, consult the rubric and notes below for the appropriate quarter.