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

References and notes

1. Hiroyuki Yokoyama & Ujihara K (1995). Spontaneous emission and laser oscillation in microcavities. Boca Raton: CRC Press. p. 6. ISBN   0-8493-3786-0.
2. Kerry Vahala (2004). Optical microcavities. Singapore: World Scientific. p. 368. ISBN   981-238-775-7.
3. Rodney Loudon (2000). The quantum theory of light. Oxford UK: Oxford University Press. p. 172. ISBN   0-19-850177-3.
4. Marlan O. Scully, M. Suhail Zubairy (1997). Quantum Optics. Cambridge University Press. p. 5. ISBN   0521435951.
5. Schine, Nathan (2019). Quantum Hall Physics with Photons (PhD). University of Chicago.
6. Mark Fox (2006). Quantum Optics: An Introduction. Boca Raton: OUP Oxford. p. 208. ISBN   0198566735.