In atomic physics, a dark state refers to a state of an atom or molecule that cannot absorb (or emit) photons. All atoms and molecules are described by quantum states; different states can have different energies and a system can make a transition from one energy level to another by emitting or absorbing one or more photons. However, not all transitions between arbitrary states are allowed. A state that cannot absorb an incident photon is called a dark state. This can occur in experiments using laser light to induce transitions between energy levels, when atoms can spontaneously decay into a state that is not coupled to any other level by the laser light, preventing the atom from absorbing or emitting light from that state.

A dark state can also be the result of quantum interference in a three-level system, when an atom is in a coherent superposition of two states, both of which are coupled by lasers at the right frequency to a third state. With the system in a particular superposition of the two states, the system can be made dark to both lasers as the probability of absorbing a photon goes to 0.

Two-level systems

Three-level systems

A three-state Λ-type system

We start with a three-state Λ-type system, where ${\displaystyle |1\rangle \leftrightarrow |3\rangle }$ and ${\displaystyle |2\rangle \leftrightarrow |3\rangle }$ are dipole-allowed transitions and ${\displaystyle |1\rangle \leftrightarrow |2\rangle }$ is forbidden. In the rotating wave approximation, the semi-classical Hamiltonian is given by

${\displaystyle H=H_{0}+H_{1}}$

with

${\displaystyle H_{0}=\hbar \omega _{1}|1\rangle \langle 1|+\hbar \omega _{2}|2\rangle \langle 2|+\hbar \omega _{3}|3\rangle \langle 3|,}$

${\displaystyle H_{1}=-{\frac {\hbar }{2}}\left(\Omega _{p}e^{+i\omega _{p}t}|1\rangle \langle 3|+\Omega _{c}e^{+i\omega _{c}t}|2\rangle \langle 3|\right)+{\mbox{H.c.}},}$

where ${\displaystyle \Omega _{p}}$ and ${\displaystyle \Omega _{c}}$ are the Rabi frequencies of the probe field (of frequency ${\displaystyle \omega _{p}}$) and the coupling field (of frequency ${\displaystyle \omega _{c}}$) in resonance with the transition frequencies ${\displaystyle \omega _{1}-\omega _{3}}$ and ${\displaystyle \omega _{2}-\omega _{3}}$, respectively, and H.c. stands for the Hermitian conjugate of the entire expression. We will write the atomic wave function as

${\displaystyle |\psi (t)\rangle =c_{1}(t)e^{-i\omega _{1}t}|1\rangle +c_{2}(t)e^{-i\omega _{2}t}|2\rangle +c_{3}(t)e^{-i\omega _{3}t}|3\rangle .}$

Solving the Schrödinger equation ${\displaystyle i\hbar |{\dot {\psi }}\rangle =H|\psi \rangle }$, we obtain the solutions

${\displaystyle {\dot {c}}_{1}={\frac {i}{2}}\Omega _{p}c_{3}}$

${\displaystyle {\dot {c}}_{2}={\frac {i}{2}}\Omega _{c}c_{3}}$

${\displaystyle {\dot {c}}_{3}={\frac {i}{2}}(\Omega _{p}c_{1}+\Omega _{c}c_{2}).}$

Using the initial condition

${\displaystyle |\psi (0)\rangle =c_{1}(0)|1\rangle +c_{2}(0)|2\rangle +c_{3}(0)|3\rangle ,}$

we can solve these equations to obtain

${\displaystyle c_{1}(t)=c_{1}(0)\left[{\frac {\Omega _{c}^{2}}{\Omega ^{2}}}+{\frac {\Omega _{p}^{2}}{\Omega ^{2}}}\cos {\frac {\Omega t}{2}}\right]+c_{2}(0)\left[-{\frac {\Omega _{p}\Omega _{c}}{\Omega ^{2}}}+{\frac {\Omega _{p}\Omega _{c}}{\Omega ^{2}}}\cos {\frac {\Omega t}{2}}\right]\quad -ic_{3}(0){\frac {\Omega _{p}}{\Omega }}\sin {\frac {\Omega t}{2}}}$

${\displaystyle c_{2}(t)=c_{1}(0)\left[-{\frac {\Omega _{p}\Omega _{c}}{\Omega ^{2}}}+{\frac {\Omega _{p}\Omega _{c}}{\Omega ^{2}}}\cos {\frac {\Omega t}{2}}\right]+c_{2}(0)\left[{\frac {\Omega _{p}^{2}}{\Omega ^{2}}}+{\frac {\Omega _{c}^{2}}{\Omega ^{2}}}\cos {\frac {\Omega t}{2}}\right]\quad -ic_{3}(0){\frac {\Omega _{c}}{\Omega }}\sin {\frac {\Omega t}{2}}}$

${\displaystyle c_{3}(t)=-ic_{1}(0){\frac {\Omega _{p}}{\Omega }}\sin {\frac {\Omega t}{2}}-ic_{2}(0){\frac {\Omega _{c}}{\Omega }}\sin {\frac {\Omega t}{2}}+c_{3}(0)\cos {\frac {\Omega t}{2}}}$

with ${\displaystyle \Omega ={\sqrt {\Omega _{c}^{2}+\Omega _{p}^{2}}}}$. We observe that we can choose the initial conditions

${\displaystyle c_{1}(0)={\frac {\Omega _{c}}{\Omega }},\qquad c_{2}(0)=-{\frac {\Omega _{p}}{\Omega }},\qquad c_{3}(0)=0,}$

which gives a time-independent solution to these equations with no probability of the system being in state ${\displaystyle |3\rangle }$.^[1] This state can also be expressed in terms of a mixing angle ${\displaystyle \theta }$ as

${\displaystyle |D\rangle =\cos \theta |1\rangle -\sin \theta |2\rangle }$

with

${\displaystyle \cos \theta ={\frac {\Omega _{c}}{\sqrt {\Omega _{p}^{2}+\Omega _{c}^{2}}}},\qquad \sin \theta ={\frac {\Omega _{p}}{\sqrt {\Omega _{p}^{2}+\Omega _{c}^{2}}}}.}$

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 ${\displaystyle |3\rangle }$ 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.

See also

• Electromagnetically induced transparency
• List of laser articles

References