Let’s now complicate the setup just a tiny bit, as shown in Figure 129. Here, photons are split just as before by a first beam splitter into T and R photons. Each of these are then reflected 90° so they will meet at a second beam splitter. Trace the possible paths of the photons and you will see that all TT photons come out of the same exit of beam splitter 2 as all RR photons. The other exit from beam splitter 2 contains all RT and TR photons.
Figure 129 Photons sent through a Mach–Zehnder interferometer are split by a first beam splitter and then recombined by a second beam splitter to yield two outputs: one outputs all TT-and RR-photons, while the second outputs all TR- and RT-photons.
What would be your guess about the probability of a photon coming out of one or the other port of beam splitter 2? Based on the results from the prior experiment (Figure 128), you would probably expect that since RR and TT counts were 25% each, then RR + TT = 50%, and the rest would of course be TR + RT = 50%, right? WRONG!
As long as the photon paths between the beam splitters form a perfect square, ALL photons will come out of the TR + RT output, and none will come out of the TT + RR output. This is experimental fact.
Nothing is wrong with the logic. After all, we did analyze how photons interact with beam splitters:
1. The experiment of Figure 125 showed us that photons have approximately 50%/50% probability of being transmitted or reflected by a beam splitter. However, a photon is never split in half. A photon either exits at the T port or the R port of a beam splitter, but never through both at the same time.
2. The experiment of Figure 128 showed us that beam splitters can be stacked, and each simply divides the probability in half. We definitely found 25% of our photons at the TT exit of beam splitter 2, and 25% of the photons at the RR exit of beam splitter 3.
3. Photons in the experiment of Figure 129 were sent one at a time, ensuring that they wouldn’t “collide” at the second beam splitter in such a way that they would only exit through one of the ports.
As you may have figured out by now, the reason for the strange behavior is that the Mach–Zehnder interferometer behaves very similarly to a double-slit experiment. Just as in the case of the two-slit experiment with single photons (Figure 90), we are looking at the interference of a photon with itself!
If you now think about light as a wave, you will immediately realize that all that is happening is that destructive interference happens at the RR + TT port, while constructive interference happens at the RT + TR port when the legs of the interferometer are of exactly the same length. Photons would start showing up at the RR + TT port as soon as one of the path lengths changed by a fraction of a wavelength.
This last point is important. The Mach–Zehnder interferometer is sensitive to extremely small variations in path length between the beam splitters. For this reason, building a Mach-Zehnder interferometer requires that precision optical mounts and components be used. In addition, the substrate on which the interferometer is built must be very solid and free of vibration.
A very tight photon beam and small detector apertures are necessary to count photons. However, the Mach–Zehnder interferometer can be made to produce an interference pattern by using a broader beam. In essence, a wide beam spreads the lengths of the interferometer’s legs so that the RR + TT and TR + RT conditions are met at different angles at the second beam splitter’s ports. This technique is convenient if one prefers to use the presence or absence of an interference pattern as an indicator of collapse of the superposition state instead of counting photons. However, as shown in Figure 130, both methods are equivalent.
Figure 130 A wide photon beam produces interference patterns at the outputs of beam splitter 2 in a Mach–Zehnder interferometer. If the legs of the interferometer are perfectly balanced, a bright band will be seen directly around the TR + RT output, while a dark band will be observed directly around the TT + RR output. Placing detectors with a very narrow opening directly in front of the beam splitter’s outputs produces results identical to those resulting from use of a very narrow beam (Figure 129).
As shown in Figure 131, we built our interferometer on a 1/2-in.-thick aluminum optical breadboard (a surplus Edmund Optics model 56935 18-in. × 18-in. bench plate) that we set on our basement’s concrete slab atop two bubble-wrap rolls to isolate the interferometer from environmental vibrations. We used a 5-mW green laser pointer that has a permanent-on mode and good thermal dissipation. Although any visible laser will work, output stability is important to ensure the interference pattern doesn’t shift at random. In addition, lasers toward the green portion of the spectrum are best, since they produce easily visible patterns, even at low power. We mounted our laser on an adjustable V-shaped platform mount that gives us independent steering and height adjustments.
Figure 131 Our Mach–Zehnder interferometer is built on a surplus optical breadboard that sits on our basement’s slab atop two rolls of bubble-wrap. (a) Good quality optics and solid mountings are necessary to produce a usable interference pattern. (b) All of our components are surplus, which explains the unnecessarily large beam-splitter cubes.
In general, the very tight beam produced by a laser pointer is inconveniently small to produce interference fringes that can be seen with ease. For this reason, we expanded the beam to approximately 1-cm in diameter using two lenses, as shown in Figure 131. It is important to adjust the spacing between the lenses to produce a tightly collimated beam that doesn’t diverge at a distance of around 10 m. Alternatively, you can use a ready-made 5 × or 10 × beam expander, but only if you happen to have one on hand, or find it at really low cost in the surplus market.
Our beam splitters are nonpolarizing cubes. We purchased them from Surplus Shed—a great place to find high-quality surplus optical components. Preferably, you should choose beam splitters that are antireflectively coated for the laser wavelength you intend to use. You should also select mounts that allow for adjustment of the position and orientation of the beam splitters (e.g., Thorlabs model KM100B kinematic platform mount with a model PM1 clamping arm).
We use first-surface aluminized mirrors that are flat to one-tenth of a wave over an inch diameter. We found a matching pair of mirrors on eBay that came installed on a pair of Newport MM-1 kinematic mounts. Any similar kinematic mirror mount would work just as well (e.g., Thorlabs model KMM1, Edmund Optics NT58-851, etc.).
Finally, we chose to view the interference fringes produced by the system, instead of counting photons from each port. This makes it similar to the two-slit interference with which we already have some prior experience. To show the interference pattern, we simply project one of the outputs from the second beam splitter onto a piece of ground glass. If necessary, you can use a magnifying glass or any other suitable convex lens to enlarge the interference fringe pattern.
Adjusting the Mach–Zehnder interferometer takes some patience. Start without the magnifying glass or the glass screen, projecting the beam on a surface some 2 m away. You should first adjust one of the mirrors to place its beam right at the center of beam splitter 2. You should then adjust the other mirror so that its beam intersects the first beam at the center of beam splitter 2. Next, adjust the beam splitter until the spots produced by the two beams on the distant surface overlap. Move the surface closer and check again for overlap. Just a few iterations of adjusting the second mirror and beam splitter 2 should suffice. The ground glass screen can then be put in place to visualize the interference pattern.
It is interesting to press very lightly on one of the mirror mounts to change the length of one of the interferometer legs by fractions of a wavelength. This should suffice to disturb the interference pattern without permanently altering the interferometer’s setting. You can also try placing a hot soldering iron close to the path of the light on one of the legs of the interferometer. This will change the refractive index of the air, thus changing the speed of light along one of the legs. This should also disturb the interference pattern, demonstrating the extraordinary sensitivity of the interferometer.
Lastly, as shown in Figure 132, it is easy to observe the weird quantum behavior of the Mach–Zehnder interferometer when only one photon is present in the system at any one time. Just place the interferometer within a light-tight enclosure (which may be built from 1/4-in. black foam board), attenuate the intensity of the laser down to where at most one photon is within the interferometer at any one time, and use the makeshift low-light camera and software frame integrator that we built in (Figure 90).
Figure 132 The same type of setup that we built to record two-slit interference patterns one photon at a time (Figure 93) can be used to record the interference pattern produced by flying single photons through the Mach–Zehnder interferometer.
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