Vacuum Rabi oscillation

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.^[1][2][3] 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

${\displaystyle {\hat {H}}_{\text{JC}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+\hbar \omega _{0}{\frac {{\hat {\sigma }}_{z}}{2}}+\hbar g\left({\hat {a}}{\hat {\sigma }}_{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}\right)}$

where ${\displaystyle {\hat {\sigma _{z}}}}$ is the Pauli z spin operator for the two eigenstates ${\displaystyle |e\rangle }$ and ${\displaystyle |g\rangle }$ of the isolated two level system separated in energy by ${\displaystyle \hbar \omega _{0}}$; ${\displaystyle {\hat {\sigma }}_{+}=|e\rangle \langle g|}$ and ${\displaystyle {\hat {\sigma }}_{-}=|g\rangle \langle e|}$ are the raising and lowering operators of the two level system; ${\displaystyle {\hat {a}}^{\dagger }}$ and ${\displaystyle {\hat {a}}}$ are the creation and annihilation operators for photons of energy ${\displaystyle \hbar \omega }$ in the cavity mode; and

${\displaystyle g={\frac {\mathbf {d} \cdot {\hat {\mathcal {E}}}}{\hbar }}{\sqrt {\frac {\hbar \omega }{2\epsilon _{0}V}}}}$

is the strength of the coupling between the dipole moment ${\displaystyle \mathbf {d} }$ of the two level system and the cavity mode with volume ${\displaystyle V}$ and electric field polarized along ${\displaystyle {\hat {\mathcal {E}}}}$. ^[4] The energy eigenvalues and eigenstates for this model are

${\displaystyle E_{\pm }(n)=\hbar \omega \left(n+{\frac {1}{2}}\right)\pm {\frac {\hbar }{2}}{\sqrt {4g^{2}(n+1)+\delta ^{2}}}=\hbar \omega _{n}^{\pm }}$

${\displaystyle |n,+\rangle =\cos \left(\theta _{n}\right)|g,n+1\rangle +\sin \left(\theta _{n}\right)|e,n\rangle }$

${\displaystyle |n,-\rangle =\sin \left(\theta _{n}\right)|g,n+1\rangle -\cos \left(\theta _{n}\right)|e,n\rangle }$

where ${\displaystyle \delta =\omega _{0}-\omega }$ is the detuning, and the angle ${\displaystyle \theta _{n}}$ is defined as

${\displaystyle \theta _{n}=\tan ^{-1}\left({\frac {g{\sqrt {n+1}}}{\delta }}\right).}$

Given the eigenstates of the system, the time evolution operator can be written down in the form

${\displaystyle {\begin{aligned}e^{-i{\hat {H}}_{\text{JC}}t/\hbar }&=\sum _{|n,\pm \rangle }\sum _{|n',\pm \rangle }|n,\pm \rangle \langle n,\pm |e^{-i{\hat {H}}_{\text{JC}}t/\hbar }|n',\pm \rangle \langle n',\pm |\\&=~e^{i(\omega -{\frac {\omega _{0}}{2}})t}|g,0\rangle \langle g,0|\\&~~~+\sum _{n=0}^{\infty }{e^{-i\omega _{n}^{+}t}(\cos {\theta _{n}}|g,n+1\rangle +\sin {\theta _{n}}|e,n\rangle )(\cos {\theta _{n}}\langle g,n+1|+\sin {\theta _{n}}\langle e,n|)}\\&~~~+\sum _{n=0}^{\infty }{e^{-i\omega _{n}^{-}t}(-\sin {\theta _{n}}|g,n+1\rangle +\cos {\theta _{n}}|e,n\rangle )(-\sin {\theta _{n}}\langle g,n+1|+\cos {\theta _{n}}\langle e,n|)}\\\end{aligned}}.}$

If the system starts in the state ${\displaystyle |g,n+1\rangle }$, where the atom is in the ground state of the two level system and there are ${\displaystyle n+1}$ photons in the cavity mode, the application of the time evolution operator yields

${\displaystyle {\begin{aligned}e^{-i{\hat {H}}_{\text{JC}}t/\hbar }|g,n+1\rangle &=(e^{-i\omega _{n}^{+}t}(\cos ^{2}{(\theta _{n})}|g,n+1\rangle +\sin {\theta _{n}}\cos {\theta _{n}}|e,n\rangle )+e^{-i\omega _{n}^{-}t}(-\sin ^{2}{(\theta _{n})}|g,n+1\rangle -\sin {\theta _{n}}\cos {\theta _{n}}|e,n\rangle )\\&=(e^{-i\omega _{n}^{+}t}+e^{-i\omega _{n}^{-}t})\cos {(2\theta _{n})}|g,n+1\rangle +(e^{-i\omega _{n}^{+}t}-e^{-i\omega _{n}^{-}t})\sin {(2\theta _{n})}|e,n\rangle \\&=e^{-i\omega _{c}(n+{\frac {1}{2}})}{\Biggr [}\cos {{\biggr (}{\frac {t}{2}}{\sqrt {4g^{2}(n+1)+\delta ^{2}}}{\biggr )}}{\biggr [}{\frac {\delta ^{2}-4g^{2}(n+1)}{\delta ^{2}+4g^{2}(n+1)}}{\biggr ]}|g,n+1\rangle +\sin {{\biggr (}{\frac {t}{2}}{\sqrt {4g^{2}(n+1)+\delta ^{2}}}{\biggr )}}{\biggr [}{\frac {8\delta ^{2}g^{2}(n+1)}{\delta ^{2}+4g^{2}(n+1)}}{\biggr ]}|e,n\rangle {\Biggr ]}\end{aligned}}.}$

The probability that the two level system is in the excited state ${\displaystyle |e,n\rangle }$ as a function of time ${\displaystyle t}$ is then

${\displaystyle {\begin{aligned}P_{e}(t)&=|\langle e,n|e^{-i{\hat {H}}_{\text{JC}}t/\hbar }|g,n+1\rangle |^{2}\\&=\sin ^{2}{{\biggr (}{\frac {t}{2}}{\sqrt {4g^{2}(n+1)+\delta ^{2}}}{\biggr )}}{\biggr [}{\frac {8\delta ^{2}g^{2}(n+1)}{\delta ^{2}+4g^{2}(n+1)}}{\biggr ]}\\&={\frac {4g^{2}(n+1)}{\Omega _{n}^{2}}}\sin ^{2}{{\bigr (}{\frac {\Omega _{n}t}{2}}{\bigr )}}\end{aligned}}}$

where ${\displaystyle \Omega _{n}={\sqrt {4g^{2}(n+1)+\delta ^{2}}}}$ is identified as the Rabi frequency. For the case that there is no electric field in the cavity, that is, the photon number ${\displaystyle n}$ is zero, the Rabi frequency becomes ${\displaystyle \Omega _{0}={\sqrt {4g^{2}+\delta ^{2}}}}$. Then, the probability that the two level system goes from its ground state to its excited state as a function of time ${\displaystyle t}$ is

${\displaystyle P_{e}(t)={\frac {4g^{2}}{\Omega _{0}^{2}}}\sin ^{2}{{\bigr (}{\frac {\Omega _{0}t}{2}}{\bigr )}.}}$

For a cavity that admits a single mode perfectly resonant with the energy difference between the two energy levels, the detuning ${\displaystyle \delta }$ vanishes, and ${\displaystyle P_{e}(t)}$ becomes a squared sinusoid with unit amplitude and period ${\displaystyle {\frac {2\pi }{g}}.}$

Generalization to N atoms

The situation in which ${\displaystyle N}$ two level systems are present in a single-mode cavity is described by the Tavis–Cummings model ^[5] , which has Hamiltonian

${\displaystyle {\hat {H}}_{\text{JC}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+\sum _{j=1}^{N}{\hbar \omega _{0}{\frac {{\hat {\sigma }}_{j}^{z}}{2}}+\hbar g_{j}\left({\hat {a}}{\hat {\sigma }}_{j}^{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{j}^{-}\right)}.}$

Under the assumption that all two level systems have equal individual coupling strength ${\displaystyle g}$ to the field, the ensemble as a whole will have enhanced coupling strength ${\displaystyle g_{N}=g{\sqrt {N}}}$. As a result, the vacuum Rabi splitting is correspondingly enhanced by a factor of ${\displaystyle {\sqrt {N}}}$.^[6]

See also

• Jaynes–Cummings model
• Quantum fluctuation
• Rabi cycle
• Rabi frequency
• Rabi problem
• Spontaneous emission
• Isidor Isaac Rabi