## 3.2 Beam-splitters: quantum interference, revisited

One of the simplest quantum devices in which quantum interference can be controlled is a **Mach–Zehnder interferometer** — see Figure 3.5.^{67}

This is a slightly modified version of the construction shown in Figure 3.2, where we have added two slivers of glass of different thickness into each of the optical paths connecting the two beam-splitters.
The slivers are usually referred to as “phase shifters”, and their thicknesses *symmetric* beam-splitters with **non-symmetric** ones, i.e. they no longer transmit or reflect with equal probability, but instead reflect with some (fixed) probability amplitude

A photon (the coloured dot in the figure) impinges on the first beam-splitter from one of the two input ports (here input ^{68}

In quantum theory we almost always work with probability *amplitudes*, and only once we have a full expression for the amplitude of a given outcome do we square its absolute value to get the corresponding probability.

For example, let us calculate

The “classical” part of this expression, *not* correct — it doesn’t agree with physical experiments! — whence the interference term *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, as in Figure 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, Section 1.1), while the overall amplitude is the sum of the amplitudes for the many different paths (Rule 2, Section 1.1). You can, for example, see that the probability amplitude ^{69} by the four probability amplitudes

The most popular instance of a Mach–Zehnder interferometer involves only *symmetric* beam-splitters (i.e. ^{70}

You can play around with a virtual Mach–Zehnder interferometer at Quantum Flytrap’s

*Virtual Lab*. (There are also lots of other things you can do in this virtual lab — go have a look!).↩︎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.↩︎*Any*isolated quantum device can fully be described by the matrix of probability amplitudesU_{ij} that inputj generates outputi .↩︎Really, 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!), you actually obtain

ie^{i\frac{\varphi_0+\varphi_1}{2}}U rather thanU , but as we have already said, we can ignore global phase factors.↩︎