Physics:Vacuum Rabi oscillation

Short description: Damped oscillation in quantum optics

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

[math]\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) }[/math]

where [math]\displaystyle{ \hat{\sigma_z} }[/math] is the Pauli z spin operator for the two eigenstates [math]\displaystyle{ |e \rangle }[/math] and [math]\displaystyle{ |g\rangle }[/math] of the isolated two level system separated in energy by [math]\displaystyle{ \hbar \omega_0 }[/math]; [math]\displaystyle{ \hat{\sigma}_+ = |e \rangle \langle g | }[/math] and [math]\displaystyle{ \hat{\sigma}_- = |g \rangle \langle e | }[/math] are the raising and lowering operators of the two level system; [math]\displaystyle{ \hat{a}^{\dagger} }[/math] and [math]\displaystyle{ \hat{a} }[/math] are the creation and annihilation operators for photons of energy [math]\displaystyle{ \hbar \omega }[/math] in the cavity mode; and

[math]\displaystyle{ g=\frac{\mathbf{d}\cdot\hat{\mathcal{E}}}{\hbar}\sqrt{\frac{\hbar \omega}{2 \epsilon_0 V}} }[/math]

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

[math]\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 }[/math]

[math]\displaystyle{ |n,+\rangle= \cos \left(\theta_n\right)|g,n+1\rangle+\sin \left(\theta_n\right)|e,n\rangle }[/math]

[math]\displaystyle{ |n,-\rangle= \sin \left(\theta_n\right)|g,n+1\rangle-\cos \left(\theta_n\right)|e,n\rangle }[/math]

where [math]\displaystyle{ \delta = \omega_0 - \omega }[/math] is the detuning, and the angle [math]\displaystyle{ \theta_n }[/math] is defined as

[math]\displaystyle{ \theta_n = \tan^{-1}\left(\frac{g \sqrt{n+1}}{\delta}\right). }[/math]

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

[math]\displaystyle{ \begin{align} 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{align}. }[/math]

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

[math]\displaystyle{ \begin{align} 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{align}. }[/math]

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

[math]\displaystyle{ \begin{align} 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{align} }[/math]

where [math]\displaystyle{ \Omega_n=\sqrt{4g^2(n+1)+\delta^2} }[/math] is identified as the Rabi frequency. For the case that there is no electric field in the cavity, that is, the photon number [math]\displaystyle{ n }[/math] is zero, the Rabi frequency becomes [math]\displaystyle{ \Omega_0=\sqrt{4g^2+\delta^2} }[/math]. Then, the probability that the two level system goes from its ground state to its excited state as a function of time [math]\displaystyle{ t }[/math] is

[math]\displaystyle{ P_e(t) =\frac{4g^2}{\Omega_0^2} \sin^2{\bigr(\frac{\Omega_0 t}{2}\bigr).} }[/math]

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

Generalization to N atoms

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

[math]\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)}. }[/math]

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

See also

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