In cavity quantum electrodynamics, a vacuum Rabi oscillation is a damped oscillation of an initially excited atom coupled to an electromagnetic resonator or cavity in which the atom alternately emits photon(s) into a single-mode electromagnetic cavity and reabsorbs them. The atom interacts with a single-mode field confined to a limited volume V in an optical cavity. Spontaneous emission is a consequence of coupling between the atom and the vacuum fluctuations of the cavity field.
Mathematical treatment A mathematical description of vacuum Rabi oscillation begins with the Jaynes–Cummings model, which describes the interaction between a single mode of a quantized field and a two level system inside an optical cavity. The Hamiltonian for this model in the rotating wave approximation is
H ^ JC = ℏ ω a ^ † a ^ + ℏ ω 0 σ ^ z 2 + ℏ g ( a ^ σ ^ + + a ^ † σ ^ − ) {\displaystyle {\hat {H}}_{\text{JC}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+\hbar \omega _{0}{\frac {{\hat {\sigma }}_{z}}{2}}+\hbar g\left({\hat {a}}{\hat {\sigma }}_{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}\right)}
where σ z ^ {\displaystyle {\hat {\sigma _{z}}}} is the Pauli z spin operator for the two eigenstates | e ⟩ {\displaystyle |e\rangle } and | g ⟩ {\displaystyle |g\rangle } of the isolated two level system separated in energy by ℏ ω 0 {\displaystyle \hbar \omega _{0}} ; σ ^ + = | e ⟩ ⟨ g | {\displaystyle {\hat {\sigma }}_{+}=|e\rangle \langle g|} and σ ^ − = | g ⟩ ⟨ e | {\displaystyle {\hat {\sigma }}_{-}=|g\rangle \langle e|} are the raising and lowering operators of the two level system; a ^ † {\displaystyle {\hat {a}}^{\dagger }} and a ^ {\displaystyle {\hat {a}}} are the creation and annihilation operators for photons of energy ℏ ω {\displaystyle \hbar \omega } in the cavity mode; and
g = d ⋅ E ^ ℏ ℏ ω 2 ϵ 0 V {\displaystyle g={\frac {\mathbf {d} \cdot {\hat {\mathcal {E}}}}{\hbar }}{\sqrt {\frac {\hbar \omega }{2\epsilon _{0}V}}}}
is the strength of the coupling between the dipole moment d {\displaystyle \mathbf {d} } of the two level system and the cavity mode with volume V {\displaystyle V} and electric field polarized along E ^ {\displaystyle {\hat {\mathcal {E}}}} .
The energy eigenvalues and eigenstates for this model are
E ± ( n ) = ℏ ω ( n + 1 2 ) ± ℏ 2 4 g 2 ( n + 1 ) + δ 2 = ℏ ω n ± {\displaystyle E_{\pm }(n)=\hbar \omega \left(n+{\frac {1}{2}}\right)\pm {\frac {\hbar }{2}}{\sqrt {4g^{2}(n+1)+\delta ^{2}}}=\hbar \omega _{n}^{\pm }}
| n , + ⟩ = cos ( θ n ) | g , n + 1 ⟩ + sin ( θ n ) | e , n ⟩ {\displaystyle |n,+\rangle =\cos \left(\theta _{n}\right)|g,n+1\rangle +\sin \left(\theta _{n}\right)|e,n\rangle }
| n , − ⟩ = sin ( θ n ) | g , n + 1 ⟩ − cos ( θ n ) | e , n ⟩ {\displaystyle |n,-\rangle =\sin \left(\theta _{n}\right)|g,n+1\rangle -\cos \left(\theta _{n}\right)|e,n\rangle }
where δ = ω 0 − ω {\displaystyle \delta =\omega _{0}-\omega } is the detuning, and the angle θ n {\displaystyle \theta _{n}} is defined as
θ n = tan − 1 ( g n + 1 δ ) . {\displaystyle \theta _{n}=\tan ^{-1}\left({\frac {g{\sqrt {n+1}}}{\delta }}\right).}
Given the eigenstates of the system, the time evolution operator can be written down in the form
e − i H ^ JC t / ℏ = ∑ | n , ± ⟩ ∑ | n ′ , ± ⟩ | n , ± ⟩ ⟨ n , ± | e − i H ^ JC t / ℏ | n ′ , ± ⟩ ⟨ n ′ , ± | = e i ( ω − ω 0 2 ) t | g , 0 ⟩ ⟨ g , 0 | + ∑ n = 0 ∞ e − i ω n + t ( cos θ n | g , n + 1 ⟩ + sin θ n | e , n ⟩ ) ( cos θ n ⟨ g , n + 1 | + sin θ n ⟨ e , n | ) + ∑ n = 0 ∞ e − i ω n − t ( − sin θ n | g , n + 1 ⟩ + cos θ n | e , n ⟩ ) ( − sin θ n ⟨ g , n + 1 | + cos θ n ⟨ e , n | ) . {\displaystyle {\begin{aligned}e^{-i{\hat {H}}_{\text{JC}}t/\hbar }&=\sum _{|n,\pm \rangle }\sum _{|n',\pm \rangle }|n,\pm \rangle \langle n,\pm |e^{-i{\hat {H}}_{\text{JC}}t/\hbar }|n',\pm \rangle \langle n',\pm |\\&=~e^{i(\omega -{\frac {\omega _{0}}{2}})t}|g,0\rangle \langle g,0|\\&~~~+\sum _{n=0}^{\infty }{e^{-i\omega _{n}^{+}t}(\cos {\theta _{n}}|g,n+1\rangle +\sin {\theta _{n}}|e,n\rangle )(\cos {\theta _{n}}\langle g,n+1|+\sin {\theta _{n}}\langle e,n|)}\\&~~~+\sum _{n=0}^{\infty }{e^{-i\omega _{n}^{-}t}(-\sin {\theta _{n}}|g,n+1\rangle +\cos {\theta _{n}}|e,n\rangle )(-\sin {\theta _{n}}\langle g,n+1|+\cos {\theta _{n}}\langle e,n|)}\\\end{aligned}}.}
If the system starts in the state | g , n + 1 ⟩ {\displaystyle |g,n+1\rangle } , where the atom is in the ground state of the two level system and there are n + 1 {\displaystyle n+1} photons in the cavity mode, the application of the time evolution operator yields
e − i H ^ JC t / ℏ | g , n + 1 ⟩ = ( e − i ω n + t ( cos 2 ( θ n ) | g , n + 1 ⟩ + sin θ n cos θ n | e , n ⟩ ) + e − i ω n − t ( − sin 2 ( θ n ) | g , n + 1 ⟩ − sin θ n cos θ n | e , n ⟩ ) = ( e − i ω n + t + e − i ω n − t ) cos ( 2 θ n ) | g , n + 1 ⟩ + ( e − i ω n + t − e − i ω n − t ) sin ( 2 θ n ) | e , n ⟩ = e − i ω c ( n + 1 2 ) [ cos ( t 2 4 g 2 ( n + 1 ) + δ 2 ) [ δ 2 − 4 g 2 ( n + 1 ) δ 2 + 4 g 2 ( n + 1 ) ] | g , n + 1 ⟩ + sin ( t 2 4 g 2 ( n + 1 ) + δ 2 ) [ 8 δ 2 g 2 ( n + 1 ) δ 2 + 4 g 2 ( n + 1 ) ] | e , n ⟩ ] . {\displaystyle {\begin{aligned}e^{-i{\hat {H}}_{\text{JC}}t/\hbar }|g,n+1\rangle &=(e^{-i\omega _{n}^{+}t}(\cos ^{2}{(\theta _{n})}|g,n+1\rangle +\sin {\theta _{n}}\cos {\theta _{n}}|e,n\rangle )+e^{-i\omega _{n}^{-}t}(-\sin ^{2}{(\theta _{n})}|g,n+1\rangle -\sin {\theta _{n}}\cos {\theta _{n}}|e,n\rangle )\\&=(e^{-i\omega _{n}^{+}t}+e^{-i\omega _{n}^{-}t})\cos {(2\theta _{n})}|g,n+1\rangle +(e^{-i\omega _{n}^{+}t}-e^{-i\omega _{n}^{-}t})\sin {(2\theta _{n})}|e,n\rangle \\&=e^{-i\omega _{c}(n+{\frac {1}{2}})}{\Biggr [}\cos {{\biggr (}{\frac {t}{2}}{\sqrt {4g^{2}(n+1)+\delta ^{2}}}{\biggr )}}{\biggr [}{\frac {\delta ^{2}-4g^{2}(n+1)}{\delta ^{2}+4g^{2}(n+1)}}{\biggr ]}|g,n+1\rangle \\&\quad +\sin {{\biggr (}{\frac {t}{2}}{\sqrt {4g^{2}(n+1)+\delta ^{2}}}{\biggr )}}{\biggr [}{\frac {8\delta ^{2}g^{2}(n+1)}{\delta ^{2}+4g^{2}(n+1)}}{\biggr ]}|e,n\rangle {\Biggr ]}\end{aligned}}.}
The probability that the two level system is in the excited state | e , n ⟩ {\displaystyle |e,n\rangle } as a function of time t {\displaystyle t} is then
P e ( t ) = | ⟨ e , n | e − i H ^ JC t / ℏ | g , n + 1 ⟩ | 2 = sin 2 ( t 2 4 g 2 ( n + 1 ) + δ 2 ) [ 8 δ 2 g 2 ( n + 1 ) δ 2 + 4 g 2 ( n + 1 ) ] = 4 g 2 ( n + 1 ) Ω n 2 sin 2 ( Ω n t 2 ) {\displaystyle {\begin{aligned}P_{e}(t)&=|\langle e,n|e^{-i{\hat {H}}_{\text{JC}}t/\hbar }|g,n+1\rangle |^{2}\\&=\sin ^{2}{{\biggr (}{\frac {t}{2}}{\sqrt {4g^{2}(n+1)+\delta ^{2}}}{\biggr )}}{\biggr [}{\frac {8\delta ^{2}g^{2}(n+1)}{\delta ^{2}+4g^{2}(n+1)}}{\biggr ]}\\&={\frac {4g^{2}(n+1)}{\Omega _{n}^{2}}}\sin ^{2}{{\bigr (}{\frac {\Omega _{n}t}{2}}{\bigr )}}\end{aligned}}}
where Ω n = 4 g 2 ( n + 1 ) + δ 2 {\displaystyle \Omega _{n}={\sqrt {4g^{2}(n+1)+\delta ^{2}}}} is identified as the Rabi frequency. For the case that there is no electric field in the cavity, that is, the photon number n {\displaystyle n} is zero, the Rabi frequency becomes Ω 0 = 4 g 2 + δ 2 {\displaystyle \Omega _{0}={\sqrt {4g^{2}+\delta ^{2}}}} . Then, the probability that the two level system goes from its ground state to its excited state as a function of time t {\displaystyle t} is
P e ( t ) = 4 g 2 Ω 0 2 sin 2 (