## 3.2 Quantum interference, revisited (still about beam-splitters)

One of the simplest quantum devices in which quantum interference can be controlled is a **Mach–Zehnder interferometer** — see Figure 3.5.^{45}

It consists of two beam-splitters (the square boxes, bottom left and top right) and two slivers of glass of different thickness which are inserted into each of the optical paths connecting the two beam-splitters.
The slivers are usually referred to as “phase shifters” and their thicknesses, ^{46}

A photon (the coloured dot in the figure) impinges on the first beam-splitter from one of the two input ports (here input 1) and begins its journey towards one of the two photodetectors.
Let^{47}

For example, let us calculate

The “classical” part of this expression, *relative* phase

If we do not care about the experimental details, we can represent the action of the Mach–Zehnder interferometer in terms of a diagram: see 3.6.

Here we can follow, from left to right, the multiple different paths that a photon can take in between specific input and output ports.
The amplitude along any given path is just the product of the amplitudes pertaining to the path segments (Rule 1), while the overall amplitude is the sum of the amplitudes for the many different paths (Rule 2). You can, for example, see that the probability amplitude ^{48} by the four probability amplitudes

The most popular instance of a Mach–Zehnder interferometer involves only symmetric beam-splitters ^{49} you obtain

To better understand why we don’t worry about global phase factors, think about the eigenvalues of a matrix

You can play around with a virtual Mach–Zehnder interferometer at Virtual Lab by Quantum Flytrap. (There are also lots of other things you can do in this virtual lab: go have a look!).↩︎

The two diagonal objects in the top-left and bottom-right corners of 3.5 are simply mirrors to make the two possible paths meet at the second beam-splitter.↩︎

We will often use

i as an index even though it is also used for the imaginary unit. Hopefully, no confusion will arise for it should be clear from the context which one is which.↩︎In general, any isolated quantum device, including a quantum computer, can be described by a matrix of probability amplitudes

U_{ij} that inputj generates outputi . Watch the order of indices.↩︎That is, when you write down the matrices describing the action of the symmetric beam-splitters and the phase gates, and then multiply them all together (which is an exercise worth doing).↩︎