text: 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 Schrödinger equation , we obtain the solutions
  Using the initial condition
  we can solve these equations to obtain
  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.