Modeling quantum interference

Diagram 1: Mach-Zehnder interferometer

In this post, I'm going to model a device that exhibits quantum behavior in a simple but striking way.

The device pictured at the left is called a Mach-Zehnder interferometer. The beam splitter splits a beam of light into two paths. 50% of the beam is reflected towards mirror 1 and 50% of the beam is transmitted towards mirror 2. When the beams reach the second beam splitter, each beam is split again and is reflected or transmitted towards the detectors.

Intuitively, it would seem that half the light should end up at detector 1 and half at detector 2.^[1] However, assuming the two paths are the same length, all the light actually ends up at detector 1 at the right and none at the top detector.

This result is due to quantum interference at the second beam splitter where light heading towards the top detector destructively interferes and light heading towards the right detector constructively interferes. In true quantum style, this result always occurs even if only a single photon of light is emitted towards the first beam splitter.

In quantum mechanical terms, the photon is in a superposition of travelling along both paths simultaneously. At the second beam splitter, each path forms a further superposition (again with one path reflecting and one transmitting - see the four arrows heading towards the detectors in the diagram). The two paths heading towards detector 2 destructively interfere (i.e., they are 180° out of phase) and thus cancel each other out. Whereas the two paths heading toward detector 1 constructively interfere and so the photon is always detected there.

So how does the device actually work? The mathematics is actually fairly straightforward. The basic strategy is to model each path that the photon can take and combine identical paths at the end. Each path segment has a complex value associated with it called an amplitude which can be visualized as an arrow that can rotate around a center point (like a clock hand).^[2]

The initial (blue) path amplitude is 1 (see Diagram 2 below which specifies the calculated amplitudes for each path segment). The basic rule at the beam splitter is that the path splits into two paths and each path takes the amplitude of the source path value and multiplies it by 1/√2 (this is the normalization condition - the squares of the amplitudes in a superposition of paths must always sum to 1, i.e., 1/2 + 1/2 = 1).^[3] Also, the path of the reflected beam additionally multiplies the amplitude by -1 which represents a phase change of 180°. So the upper (green) path has an amplitude of -1/√2 (-0.707) and the lower (red) path has an amplitude of 1/√2 (0.707).

(Note: If the photon passes through the rear of the beam splitter, the result is the same except that the phase change rule does not apply.^[4] This is the case for the upper beam path when it reaches the second beam splitter. The front of each beam splitter is indicated by the dot.)

At each mirror, the amplitude for each path is multiplied by -1 (i.e., a phase change of 180°). So the upper path now has a value of 1/√2. The lower path now has a value of -1/√2. At the second beam splitter, the upper (reflection) path itself splits into reflection and transmission paths toward the two detectors. The upper beam reflection path value is 0.5 (1/√2 * 1/√2) and the upper beam transmission path value is also 0.5 (1/√2 * 1/√2). The lower beam reflection path value is 0.5 (-1/√2 * 1/√2 * -1) and the lower beam transmission path value is -0.5 (-1/√2 * 1/√2).

Diagram 2: Path amplitudes

This is where the quantum magic happens. The upper beam reflection path and the lower beam transmission path coincide. They are both directed towards detector 2. So the paths merge and the amplitudes are added to give a value of 0 (0.5 + -0.5) which is destructive interference. Similarly, the upper beam transmission path and the lower beam reflection path also coincide. They are both directed towards detector 1. So the paths merge and the amplitudes are added to give a value of 1 (0.5 + 0.5) which is constructive interference.

The probability of finding the photon at a particular detector is given by the amplitude squared, which is 100% for detector 1. Thus the photon always ends up at detector 1.

Note that this result depends on the physical configuration of the interferometer. In this case, the two paths between the beam splitters are the same length. However if the length of one of the paths is changed, the results also can change such that the photon is instead always found at detector 2 (i.e., change a path phase by 180° by multiplying by -1 and recalculate the subsequent path values), or found at either detector with equal probability (i.e., change a path phase by 90° by multiplying by the imaginary number i and recalculate), or any other probabilistic combination.

For further interferometer fun, see Part 2.

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[1] If the second beam splitter were removed, the light would be distributed between both detectors. In the case of one emitted photon, the photon would be observed at one detector or the other with 50% probability. No interference between the photon paths would occur since the paths are different (one is directed towards detector 1 and one is directed towards detector 2 when they cross).

[2] The amplitude actually continually changes as the photon travels (i.e., the arrow rotates). To simplify the example, the path segments are of lengths that are multiples of the wavelength. So a photon that leaves the beam splitter with a particular phase angle will have the same phase angle when it arrives at the mirror. Also, the top path and lower path are the same length.

[3] This is in accordance with the Born rule. The probability that the photon will be observed on a particular path is given by the square of the amplitude.

[4] There is a phase change for a reflection at a surface with a higher refractive index which is true at the front of the beam splitter (the glass refracts more than the air the photon is travelling in) but not at the rear (where the photon is already travelling in the glass before it reflects).