We show an example of the optical phase manipulation.

Stimulated Raman adiabatic passage (STIRAP) is a process that
transfers of a population between two quantum
states via two coherent electromagnetic pulses. Here we consider an example
of two concequtive STIRAP processes.
The first
counterintuitive STIRAP process
transfers the population from level 1 to 3, and
the other STIRAP is applied in the reverse order (intuitive order),
in which the population is transferred back from state 3 to 1. The
couplings between the levels are \(\Omega_1\) and \(\Omega_2\).

We consider the SchrÃ¶dinger equation
\begin{equation}
i\hbar\frac{\partial}{\partial t} \Psi(R,t) =
H_{3D}\Psi(R,t),
\end{equation}
where the Hamiltonian is
\begin{equation}
H_{3D}
=
\left[
\matrix{
T_1(p)+V_1(x) & \Omega_1(t) & 0 \cr
\Omega_1(t) & T_2(p)+V_2(x) & \Omega_2(t)
\cr
0 & \Omega_2(t) & T_3(p)+V_3(x) \cr
} \right].
\end{equation}
Here \(T_i\) is a kinetic energy,
\(V_i\) is a potential energy and \(\Omega_i\) is a coupling between the states.
We choose a normalized minimum wave function which propagates in an harmonic oscillator potential
\begin{equation}
V \sim \frac{1}{2}m\omega^2x^2.
\end{equation}
The couplings \(\Omega_1\) and \(\Omega_2\) are applied in a
counterintuitive and reverse (intuitive) order.

\( A \) is the pulse height, \( T \) is
the pulse width and \(\Delta T\) is a time between
the pulses. The final state on the level 1 is then
\begin{equation}
\tilde{\psi}_1(x) = \psi_1(x) e^{i\pi\Delta T}.
\end{equation}
The population is the same as in the initial quantum state on the level 1, but
the wave function has obtained an additional phase \(\Delta T\).
When the pulse areas are finite, the populations on the levels oscillate due to the Rabi cycle.
One can tune the
parameters \( AT \) and \(\Delta T\) in order to get a maximum transition between the states.

Generally, the measurement destroys the quantum
state and the wave function cannot be changed. However, in our example
we are able to manipulate the phase of the wave function and the state is not destroyed.
This has applications, for example, in quantum cryptography
and in quantum computing .
More details and discussions can be found from the
reference paper.