Dark State - Three-level Systems

Three-level Systems

We start with a three-state Λ-type system, where and are dipole-allowed transitions and is forbidden. In the rotating wave approximation, the semi-classical Hamiltonian is given by

with

where and are the Rabi frequencies of the probe field (of frequency ) and the coupling field (of frequency ) in resonance with the transition frequencies and, respectively, and H.c. stands for the Hermitian conjugate of the entire expression. We will write the atomic wave function as

Solving the Schrodinger equation, we obtain the solutions

\begin{align} \dot c_1&=\frac i2\Omega_p c_3\\ \dot c_2&=\frac i2\Omega_c c_3\\ \dot c_3&=\frac i2(\Omega_p c_1+\Omega_c c_2).\end{align}

Using the initial condition

we can solve these equations to obtain

\begin{align} c_1(t)&=c_1(0)\left+c_2(0)\left\\ &\quad-ic_3(0)\frac{\Omega_p }{\Omega}\sin\frac{\Omega t}{2}\\ c_2(t)&=c_1(0)\left+c_2(0)\left\\ &\quad-ic_3(0)\frac{\Omega_c }{\Omega}\sin\frac{\Omega t}{2}\\ c_3(t)&=-ic_1(0)\frac{\Omega_p }{\Omega}\sin\frac{\Omega t}{2}-ic_2(0)\frac{\Omega_c }{\Omega}\sin\frac{\Omega t}{2}+c_3(0)\cos\frac{\Omega t}{2}\end{align}

with . We observe that we can choose the initial conditions

which gives a time-independent solution to these equations with no probability of the system being in state . This state can also be expressed in terms of a mixing angle as

with

This means that when the atoms are in this state, they will stay in this state indefinitely. This is a dark state, because it can not absorb or emit any photons from the applied fields. It is, therefore, effectively transparent to the probe laser, even when the laser is exactly resonant with the transition. Spontaneous emission from can result in an atom being in this dark state or another coherent state, known as a bright state. Therefore, in a collection of atoms, over time, decay into the dark state will inevitably result in the system being "trapped" coherently in that state, a phenomenon known as coherent population trapping.

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