Editing Dark state (section)

==Three-level systems==
[[File:Lambda-type system.pdf|thumb|A three-state Λ-type system]]
We start with a three-state Λ-type system, where <math>|1\rangle\leftrightarrow|3\rangle</math> and <math>|2\rangle\leftrightarrow|3\rangle</math> are dipole-allowed transitions and <math>|1\rangle\leftrightarrow|2\rangle</math> is forbidden. In the [[rotating wave approximation]], the semi-classical [[Hamiltonian (quantum mechanics)|Hamiltonian]] is given by
:<math>H=H_0+H_1</math>
with
:<math>H_0=\hbar\omega_1|1\rangle\langle 1|+\hbar\omega_2|2\rangle\langle 2|+\hbar\omega_3|3\rangle\langle 3|,</math>
:<math>H_1=-\frac \hbar 2\left(\Omega_p e^{+i\omega_p t}|1\rangle\langle 3|+\Omega_c e^{+i\omega_c t}|2\rangle\langle 3|\right)+\mbox{H.c.},</math>
where <math>\Omega_p </math> and <math>\Omega_c </math> are the [[Rabi frequency|Rabi frequencies]] of the probe field (of frequency <math>\omega_p</math>) and the coupling field (of frequency <math>\omega_c</math>) in resonance with the transition frequencies <math>\omega_3-\omega_1</math> and <math>\omega_3-\omega_2</math>, respectively, and H.c. stands for the [[Hermitian conjugate]] of the entire expression. We will write the atomic wave function as
:<math>|\psi(t)\rangle=c_1(t)e^{-i\omega_1 t}|1\rangle+c_2(t)e^{-i\omega_2 t}|2\rangle+c_3(t)e^{-i\omega_3 t}|3\rangle.</math>
Solving the [[Schrödinger equation]] <math>i\hbar|\dot\psi\rangle=H|\psi\rangle</math>, we obtain the solutions
:
<math>\dot c_1=\frac i2\Omega_p c_3</math>

<math>\dot c_2=\frac i2\Omega_c c_3 </math>

<math>\dot c_3=\frac i2(\Omega_p c_1+\Omega_c c_2).</math>

Using the initial condition
:<math>|\psi(0)\rangle=c_1(0)|1\rangle+c_2(0)|2\rangle+c_3(0)|3\rangle,</math>
we can solve these equations to obtain
:<math>
c_1(t)=c_1(0)\left[\frac{\Omega_c ^2}{\Omega^2}+\frac{\Omega_p ^2}{\Omega^2}\cos\frac{\Omega t}{2}\right]+c_2(0)\left[-\frac{\Omega_p \Omega_c }{\Omega^2}+\frac{\Omega_p \Omega_c }{\Omega^2}\cos\frac{\Omega t}{2}\right]
\quad-ic_3(0)\frac{\Omega_p }{\Omega}\sin\frac{\Omega t}{2}</math>

:<math>
c_2(t)=c_1(0)\left[-\frac{\Omega_p \Omega_c }{\Omega^2}+\frac{\Omega_p \Omega_c }{\Omega^2}\cos\frac{\Omega t}{2}\right]+c_2(0)\left[\frac{\Omega_p ^2}{\Omega^2}+\frac{\Omega_c^2}{\Omega^2}\cos\frac{\Omega t}{2}\right]
\quad-ic_3(0)\frac{\Omega_c }{\Omega}\sin\frac{\Omega t}{2}</math>

:<math>
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}</math>

with <math>\Omega=\sqrt{\Omega_c ^2+\Omega_p ^2}</math>. We observe that we can choose the initial conditions
:<math>c_1(0)=\frac{\Omega_c }{\Omega},\qquad c_2(0)=-\frac{\Omega_p }{\Omega},\qquad c_3(0)=0,</math>
which gives a time-independent solution to these equations with no probability of the system being in state <math>|3\rangle</math>.<ref>{{cite book|author=P. Lambropoulos|author2=D. Petrosyan|name-list-style=amp|title=Fundamentals of Quantum Optics and Quantum Information|publisher=Springer|location=Berlin; New York|year=2007|bibcode=2007fqoq.book.....L }}</ref> This state can also be expressed in terms of a mixing angle <math>\theta</math> as
:<math>|D\rangle=\cos\theta|1\rangle-\sin\theta|2\rangle</math>
with
:<math>\cos\theta=\frac{\Omega_{c}}{\sqrt{\Omega_{p}^2+\Omega_{c}^2}},\qquad \sin\theta=\frac{\Omega_{p}}{\sqrt{\Omega_{p}^2+\Omega_{c}^2}}.</math>

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 <math>|3\rangle</math> 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''.