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

${\hat{H}}_{\text{JC}}=\hslash \omega {\hat{a}}^{†}\hat{a}+\hslash {\omega }_0\frac{{\hat{\sigma }}_z}{2}+\hslash g(\hat{a}{\hat{\sigma }}_{+}+{\hat{a}}^{†}{\hat{\sigma }}_{-})$

where $\widehat{{\sigma }_z}$ is the Pauli z spin operator for the two eigenstates $|e⟩$ and $|g⟩$ of the isolated two level system separated in energy by $\hslash {\omega }_0$; ${\hat{\sigma }}_{+}=|e⟩⟨g|$ and ${\hat{\sigma }}_{-}=|g⟩⟨e|$ are the raising and lowering operators of the two level system; ${\hat{a}}^{†}$ and $\hat{a}$ are the creation and annihilation operators for photons of energy $\hslash \omega$ in the cavity mode; and

$g=\frac{\mathbf{𝐝}\cdot \widehat{{ℰ}}}{\hslash }\sqrt{\frac{\hslash \omega }{2{ϵ}_{0}V}}$

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

$E_{\pm }(n)=\hslash \omega (n+\frac{1}{2})\pm \frac{\hslash }{2}\sqrt{4g^2(n+1)+{\delta }^2}=\hslash {\omega }_n^{\pm }$

$|n,+⟩=\cos \left({\theta }_n\right)|g,n+1⟩+\sin \left({\theta }_n\right)|e,n⟩$

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

where $\delta ={\omega }_0-\omega$ is the detuning, and the angle ${\theta }_n$ is defined as

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

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

$\begin{array}{rl}{e}^{-i{\hat{H}}_{\text{JC}}t/\hslash }& =\sum _{|n,\pm ⟩}\sum _{|n^{\prime },\pm ⟩}|n,\pm ⟩⟨n,\pm |e^{-i{\hat{H}}_{\text{JC}}t/\hslash }|n^{\prime },\pm ⟩⟨n^{\prime },\pm |\\ & =e^{i(\omega -\frac{{\omega }_0}{2})t}|g,0⟩⟨g,0|\\ & +\sum _{n=0}^{\mathrm{\infty }}{e}^{-i{\omega }_n^{+}t}(\cos {\theta }_n|g,n+1⟩+\sin {\theta }_n|e,n⟩)(\cos {\theta }_n⟨g,n+1|+\sin {\theta }_n⟨e,n|)\\ & +\sum _{n=0}^{\mathrm{\infty }}{e}^{-i{\omega }_n^{-}t}(-\sin {\theta }_n|g,n+1⟩+\cos {\theta }_n|e,n⟩)(-\sin {\theta }_n⟨g,n+1|+\cos {\theta }_n⟨e,n|)\\ \end{array}.$

If the system starts in the state $|g,n+1⟩$, 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

$\begin{array}{rl}{e}^{-i{\hat{H}}_{\text{JC}}t/\hslash }|g,n+1⟩& =(e^{-i{\omega }_n^{+}t}({\cos }^2({\theta }_n)|g,n+1⟩+\sin {\theta }_n\cos {\theta }_n|e,n⟩)+e^{-i{\omega }_n^{-}t}(-{\sin }^2({\theta }_n)|g,n+1⟩-\sin {\theta }_n\cos {\theta }_n|e,n⟩)\\ & =(e^{-i{\omega }_n^{+}t}+e^{-i{\omega }_n^{-}t})\cos (2{\theta }_n)|g,n+1⟩+(e^{-i{\omega }_n^{+}t}-e^{-i{\omega }_n^{-}t})\sin (2{\theta }_n)|e,n⟩\\ & =e^{-i{\omega }_c(n+\frac{1}{2})}[\cos \left(\frac{t}{2}\sqrt{4g^2(n+1)+{\delta }^2}\right)[\frac{{\delta }^2-4g^2(n+1)}{{\delta }^2+4g^2(n+1)}]|g,n+1⟩+\sin \left(\frac{t}{2}\sqrt{4g^2(n+1)+{\delta }^2}\right)\left[\frac{8{\delta }^2{g}^2(n+1)}{{\delta }^2+4g^2(n+1)}\right]|e,n⟩]\end{array}.$

The probability that the two level system is in the excited state $|e,n⟩$ as a function of time $t$ is then

$\begin{array}{rl}{P}_e(t)& =|⟨e,n|e^{-i{\hat{H}}_{\text{JC}}t/\hslash }|g,n+1⟩{|}^2\\ & ={\sin }^2\left(\frac{t}{2}\sqrt{4g^2(n+1)+{\delta }^2}\right)\left[\frac{8{\delta }^2{g}^2(n+1)}{{\delta }^2+4g^2(n+1)}\right]\\ & =\frac{4g^2(n+1)}{{\mathrm{\Omega }}_n^2}{\sin }^2\left(\frac{{\mathrm{\Omega }}_{n}t}{2}\right)\end{array}$

where ${\mathrm{\Omega }}_n=\sqrt{4g^2(n+1)+{\delta }^2}$ is identified as the Rabi frequency. For the case that there is no electric field in the cavity, that is, the photon number $n$ is zero, the Rabi frequency becomes ${\mathrm{\Omega }}_0=\sqrt{4g^2+{\delta }^2}$. Then, the probability that the two level system goes from its ground state to its excited state as a function of time $t$ is

$P_e(t)=\frac{4g^2}{{\mathrm{\Omega }}_0^2}{\sin }^2\left(\frac{{\mathrm{\Omega }}_{0}t}{2}\right).$

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 $\frac{2\pi }{g}.$

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 , which has Hamiltonian

${\hat{H}}_{\text{JC}}=\hslash \omega {\hat{a}}^{†}\hat{a}+\sum _{j=1}^N\hslash {\omega }_0\frac{{\hat{\sigma }}_j^z}{2}+\hslash g_j(\hat{a}{\hat{\sigma }}_j^{+}+{\hat{a}}^{†}{\hat{\sigma }}_j^{-}).$

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 $\sqrt{N}$.

See also

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