Vacuum Rabi oscillation explained

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

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

where

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

is the strength of the coupling between the dipole moment

d of the two level system and the cavity mode with volume V and electric field polarized along \hat{ l{E}} .^[4] The energy eigenvalues and eigenstates for this model are E _{\pm} (n) = \hbar\omega \left(n + (1)/(2) \right) \pm (\hbar)/(2) \sqrt{ 4 g ^{2} (n +1 ) + \delta ^{2}=\hbar} \omega ^{\pm} _{n} |n, + \rangle = \cos \left(\theta _{n\right)|g,n+1\rangle+\sin} \left(\theta _{n\right)|e,n\rangle} |n, - \rangle = \sin \left(\theta _{n\right)|g,n+1\rangle-\cos} \left(\theta _{n\right)|e,n\rangle}

where

\delta = \omega _{0} - \omega is the detuning, and the angle \theta _{n} is defined as\theta _{n} = \tan ^{-1} \left( (g \sqrt{n +1)/() }\right).

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

\begin{align} e ^{- i\hat{H} _{JC} t/\hbar} & = \sum _{|n,\pm} \sum _{|n',\pm} |n,\pm \rangle \langle n,\pm| e ^{- i\hat{H} _{JC} t/\hbar} |n',\pm \rangle \langle n',\pm|\\ & = ~e ^{i(\omega - (\omega _{0})/()t 2)} |g, 0 \rangle \langle g, 0 | \\ & ~~~ + \sum ^{inf ty{e} _{n =0} ^{- 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 ^{inf ty{e} _{n =0} ^{- 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{align}.

If the system starts in the state

|g,n +1 \rangle , where the atom is in the ground state of the two level system and there are n +1 photons in the cavity mode, the application of the time evolution operator yields\begin{align} e ^{- i\hat{H} _{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 + (1)/(2)} )} r [\cos{ r ( (t)/(2) \sqrt{ 4 g ^{2(n+1)+\delta} ^{2}} r )} r [ (\delta ^{2-4g} ^{2(n+1)})/(\delta ^{2+4g} ^{2(n+1)}) r ]|g,n +1 \rangle + \sin{ r ( (t)/(2) \sqrt{ 4 g ^{2(n+1)+\delta} ^{2 } r )} r [ (8 \delta ^{2} g ^{2(n+1)})/(\delta ^{2+4g} ^{2(n+1)})} r ]|e,n\rangle r ] \end{align}.

The probability that the two level system is in the excited state

|e,n\rangle as a function of time t is then\begin{align} P _{e(t)} & = |\langle e,n|e ^{- i\hat{H} _{JC} t/\hbar} |g,n +1 \rangle | ^{2\\ &=} \sin ^{2 { r ( (t)/(2)} \sqrt{ 4 g ^{2(n+1)+\delta} ^{2 } r )} r [ (8 \delta ^{2} g ^{2(n+1)})/(\delta ^{2+4g} ^{2(n+1)})} r ]\\ & = (4 g ^{2(n+1)})/(\Omega ^{2} _{n}) \sin ^{2 { r ( (\Omega _{n} t)/(2)} r )} \end{align}

where

\Omega ^{2 (n +1 ) + \delta} _{n = \sqrt{ 4 g} ^{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 is zero, the Rabi frequency becomes \Omega ^{2+ \delta} _{0= \sqrt{ 4 g} ^{2}} . Then, the probability that the two level system goes from its ground state to its excited state as a function of time t is P _{e(t)} = (4 g ^{2})/(\Omega ^{2} _{0}) \sin ^{2 { r ( (\Omega _{0} t)/(2)} r ).}

For a cavity that admits a single mode perfectly resonant with the energy difference between the two energy levels, the detuning

\delta vanishes, and P _{e(t)} becomes a squared sinusoid with unit amplitude and period (2 \pi)/(g) .

Generalization to N atoms

The situation in which

N two level systems are present in a single-mode cavity is described by the Tavis–Cummings model^[5], which has Hamiltonian\hat{H} _{JC} = \hbar \omega \hat{a} ^{\dagger} \hat{a} + \sum ^{N{\hbar} _{j =1} \omega _{0} (\hat{\sigma)/(_{j}) ^{z}{2} +\hbar} g _{j} \left(\hat{a}\hat{\sigma} ^{+ + \hat{a}} _{j} ^{\dagger} \hat{\sigma} ^{- \right)}.} _{j}

Under the assumption that all two level systems have equal individual coupling strength

g to the field, the ensemble as a whole will have enhanced coupling strength g _{N=g\sqrt{N}} . As a result, the vacuum Rabi splitting is correspondingly enhanced by a factor of \sqrt{N} .^[6]

See also

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

Notes and References

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