## Interferometers

Many modern interference experiments are performed using internal degrees of freedom of atoms and ions.
For example, **Ramsey interferometry**, named after American physicist Norman Ramsey, is a generic name for an interference experiment in which atoms are sent through two separate resonant interaction zones, known as **Ramsey zones**, separated by an intermediate dispersive interaction zone.

Many beautiful experiments of this type were carried out in the 1990s in Serge Haroche’s lab at the Ecole Normale Supérieure in Paris.
Rubidium atoms were sent through two separate interaction zones (resonant interaction in the first and the third cavity) separated by a phase inducing dispersive interaction zone (the central cavity).
The atoms were subsequently measured, via a selective ionisation, and found to be in one of the two preselected energy states, here labeled as |0\rangle and |1\rangle.
The fraction of atoms found in states |0\rangle or |1\rangle showed a clear dependence on the phase shifts induced by the dispersive interaction in the central cavity.
In 2012, Serge Haroche and Dave Wineland shared the Nobel Prize in physics for “ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems.”

The three rectangular boxes in Figure 1.1 represent three cavities, each cavity being an arrangement of mirrors which traps electromagnetic field (think about standing waves in between two mirrors).
The oval shapes represent rubidium atoms with two preselected energy states labelled as |0\rangle and |1\rangle.
Each atom is initially prepared in a highly excited internal energy state |0\rangle and zips through the three cavities, from the left to the right.
In each cavity the atom interacts with the cavity field.
The first and the third cavities are, for all theoretical purposes, identical: their frequencies are tuned to the resonant frequency of the atom, and the atom exchanges energy with the cavity, going back and forth between its energy states |0\rangle and |1\rangle.
In contrast, in the second (central) cavity, the atom undergoes the so-called dispersive interaction: it is too off-resonance to exchange energy with the field but its energy states “feel” the field and acquire phase shifts.
After experiencing this well timed sequence of resonant–dispersive–resonant interactions, the energy of the atom is measured and the atom is found to be either in state |0\rangle or state |1\rangle.
The fraction of atoms found in state |0\rangle or |1\rangle shows a clear dependence on the phase shifts induced by the dispersive interaction in the central cavity.

We can understand this interference better if we follow the two internal states of the atom as it moves through the three cavities.

Suppose we are interested in the probability that the atom, initially in state |0\rangle, will be found, after completing its journey through the three cavities, in state |1\rangle.
As you can see in Figure 1.2, this can happen in two ways, as indicated by the two red paths connecting the input state |0\rangle on the left with the output state |1\rangle on the right.
Again, let U_{ij} denote the probability amplitude that input |j\rangle generates output |i\rangle (for i,j=0,1).
We can see from the diagram that
\begin{aligned}
U_{10}
&= \frac{1}{\sqrt2} e^{i\varphi_0}\frac{1}{\sqrt2} + \frac{1}{\sqrt2} e^{i\varphi_1}\frac{-1}{\sqrt2}
\\&= \frac12 e^{i\varphi_0} - \frac12 e^{i\varphi_1}
\\&= -ie^{i\frac{\varphi_0+\varphi_1}{2}}\sin\frac{\varphi}{2},
\end{aligned}
where \varphi = \varphi_1-\varphi_0 is the relative phase.
The corresponding probability reads
\begin{aligned}
P_{10}
&= \vert U_{10}\vert^2
\\&= \left\vert \frac12 e^{i\varphi_0} - \frac12 e^{i\varphi_1}\right\vert^2
\\&= \frac12 - \frac12\cos\varphi.
\end{aligned}
You should recognise the first term, \frac12, as the “classical” probability and the second one, -\frac12\cos\varphi, as the interference term.
We can repeat such calculations for any other pair of input–output states.
This approach works fine here but, in general, tracking all possible paths in evolving quantum systems can become messy when the number of input and output states increases.
There is, however, a neat way of doing it via matrix multiplication.

The effect of each interaction on atomic states can be described by a matrix of transition amplitudes, as illustrated in Figure 1.3.
Then a sequence of independent interactions is described by the product of these matrices.
\begin{aligned}
U &=
\begin{bmatrix}
\frac{1}{\sqrt2} & \frac{1}{\sqrt2}
\\\frac{1}{\sqrt2} & \frac{-1}{\sqrt2}
\end{bmatrix}
\begin{bmatrix}
e^{i\varphi_0} & 0
\\0 & e^{i\varphi_1}
\end{bmatrix}
\begin{bmatrix}
\frac{1}{\sqrt2} & \frac{1}{\sqrt2}
\\\frac{1}{\sqrt2} & \frac{-1}{\sqrt2}
\end{bmatrix}
\\&= e^{i\frac{\varphi_0+\varphi_1}{2}}
\begin{bmatrix}
\cos\frac{\varphi}{2} & -i\sin\frac{\varphi}{2}
\\\ -i\sin\frac{\varphi}{2}& \cos\frac{\varphi}{2}
\end{bmatrix},
\end{aligned}
where \varphi = \varphi_1-\varphi_0, as before.

In general, quantum operation A followed by another quantum operation B is a quantum operation described by the matrix product BA (watch the order of matrices).
Indeed, the expression (BA)_{ij}=\sum_k B_{ik}A_{kj} is the sum over amplitudes that input |j\rangle generates output |i\rangle via a specific intermediate state |k\rangle.
As you can see, the matrix approach is a wonderful bookkeeping tool for in one swap it takes care of both multiplying and adding probability amplitudes corresponding to all the contributing paths.