The study bases on a geometrical phase associated to the
SchrÃ¶dinger equation in a three level
system.
When two Stimulated Raman Adiabatic Passage (STIRAP)
processes are applied in a row; first the counterintuitive process and
then the intuitive order process, the population is transferred back to the initial
quantum state.
The phase of the new state can be well controlled.

As an example, let's assume we have two states; the initial state \(\psi_1(x) \),
the final state \(\psi_2(x) \) and the time shift between the laser pulses is
\( \Delta T = 1/6 \).
After the quantum transitions, the final state has obtained a phase \(\psi_2(x) = \psi_1(x)\exp(i\pi/6) \).
The probabilities of the wave functions are the same \(|\psi_1|^2 = |\psi_2|^2\),
but the quantum state is different. The situation is shown in the enclosed figure.

The applications include,
for example,
phase manipulation in quantum cryptography,
phase gates in quantum computing and
phase interactions in quantum interference.
More details are found
here.

A quantum logic gate is a basic
quantum circuit operating on a small number of qubits.
The Hadamard gate acts on a single qubit. It represents a rotation of
\(\pi \) about the axis
at the Bloch sphere.
In quantum optics, the Hadamard gate can be constructed
using 4 levels as shown in the following figure.

There are three couplings
\( \Omega_{10} \), \( \Omega_{11} \) and \( \Omega_{2} \) which connect
the levels. The initial population is either on levels
\( |10 \rangle \) or \( |11 \rangle \).
In the following
section we show some exact solutions to the
Hadamard system.

We show some visual images of the population transfer in the STIRAP system when the pulse areas are finite.
This is the case when the magnetic monopoles are present.