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Mach–Zehnder interferometer

The Mach–Zehnder interferometer (MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the delayed choice quantum eraser, the Elitzur–Vaidman bomb tester, and in studies of quantum entanglement.

We can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the "lower" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the "upper" path which starts from the bottom, goes straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector ψ ∈ C 2 {\displaystyle \psi \in \mathbb {C} ^{2}} that is a superposition of the "lower" path ψ l = (1 0) {\displaystyle \psi _{l}={\begin{pmatrix}1\\0\end{pmatrix}}} and the "upper" path ψ u = (0 1) {\displaystyle \psi _{u}={\begin{pmatrix}0\\1\end{pmatrix}}}, that is, ψ = α ψ l + β ψ u {\displaystyle \psi =\alpha \psi _{l}+\beta \psi _{u}} for complex α, β {\displaystyle \alpha,\beta }. In order to respect the postulate that ⟨ ψ, ψ ⟩ = 1 {\displaystyle \langle \psi,\psi \rangle =1} we require that | α | 2 + | β | 2 = 1 {\displaystyle |\alpha |^{2}+|\beta |^{2}=1}.

Both beam splitters are modelled as the unitary matrix B = 1 2 (1 i i 1) {\displaystyle B={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&i\\i&1\end{pmatrix}}}, which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of 1 / 2 {\displaystyle 1/{\sqrt {2}}}, or be reflected to the other path with a probability amplitude of i / 2 {\displaystyle i/{\sqrt {2}}}. The phase shifter on the upper arm is modelled as the unitary matrix P = (1 0 0 e i Δ Φ) {\displaystyle P={\begin{pmatrix}1&0\\0&e^{i\Delta \Phi }\end{pmatrix}}}, which means that if the photon is on the "upper" path it will gain a relative phase of Δ Φ {\displaystyle \Delta \Phi }, and it will stay unchanged if it is in the lower path.

A photon that enters the interferometer from the left will then be acted upon with a beam splitter B {\displaystyle B}, a phase shifter P {\displaystyle P}, and another beam splitter B {\displaystyle B}, and so end up in the state

and the probabilities that it will be detected at the right or at the top are given respectively by

One can therefore use the Mach–Zehnder interferometer to estimate the phase shift by estimating these probabilities.

It is interesting to consider what would happen if the photon were definitely in either the "lower" or "upper" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired). In both cases, there will be no interference between the paths anymore, and the probabilities are given by p (u) = p (l) = 1 / 2 {\displaystyle p(u)=p(l)=1/2}, independently of the phase Δ Φ {\displaystyle \Delta \Phi }. From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.