Damped vacuum Rabi oscillations

The optical Bloch equations can be generalized to describe not just an atom interacting with the vacuum, but also an atom and a single cavity mode, each interacting with its own reservoir. This is the master equation for cavity QED, and using such a master equation we can revisit the phenomenon of vacuum Rabi oscillations and see what happens in the presence of damping.

Generalization of the optical Bloch equations

The starting point for generalizing the optical Bloch equations is the Lindblad form we previously saw in the full-derivation walkthrough,

${\dot {\rho }}_{A}=-{\frac {i}{\hbar }}[H,\rho _{A}]-{\frac {\Gamma }{2}}\left[{\sigma _{+}\sigma _{-}\rho _{A}-2\sigma _{-}\rho _{A}\sigma _{+}+\rho _{A}\sigma _{+}\sigma _{-}}\right]\,.$

In this expression, $\sigma _{+}$ and $\sigma _{-}$ are "jump operators," and represent changes that occur to the atom when distinct dissipative events happen ("distinct" meaning that the environment changes between orthogonal states).

We may write the master equation for our more general scenario by replacing the atomic density matrix $\rho _{A}$ by a general density matrix $\rho$ representing the atom and cavity field, and by replacing the atomic jump operators with general jump operators $L_{k}$,

${\dot {\rho }}=-{\frac {i}{\hbar }}[H,\rho ]+\sum _{k}\left[{L_{k}\rho L_{k}^{\dagger }-{\frac {1}{2}}(L_{k}^{\dagger }L_{k}\rho +\rho L_{k}^{\dagger }L_{k})}\right]\,.$

Note that the $L_{k}$ include normalization factors which reflect their probabilities of occurrence. In other words, for the atom + vacuum model, $L={\sqrt {\Gamma }}\sigma _{-}$.

Master equation for cavity QED system

For the cavity QED model, the atom and cavity field each have possible jump operators. In general, the atom and cavity may both couple to a thermal field with $n_{th}$ average photons. In such a case, the jump operators are $L_{1}={\sqrt {\Gamma (n_{th}+1)}}\sigma _{-}$, $L_{3}={\sqrt {\Gamma n_{th}}}\sigma _{+}$, $L_{2}={\sqrt {\kappa (n_{th}+1)}}a$, and $L_{4}={\sqrt {\kappa n_{th}}}a^{\dagger }$, where $\Gamma$ parameterizes the spontaneous emission rate of the atom in free space, and $\kappa$ is parameterizes the cavity $Q$ factor.

Experimentally, typically the environment, the vacuum, is essentially at zero temperature, so $n_{th}=0$, in which case the only two relevant jump operators are $L_{1}={\sqrt {\Gamma }}\sigma _{-}$ and $L_{2}={\sqrt {\kappa }}a$.

Damped vacuum Rabi oscillations

Vacuum Rabi oscillations, in the absence of damping, involve only two states of the atom and cavity: $|e,0{\rangle }$ and $|g,1{\rangle }$. When damping is added, the $|g,0{\rangle }$ state must be included, since both the atom and cavity states can decay and loose their quanta of energy. Moreover, because only one quantum of excitation is involved in this system, we can observe the essential physics by considering the case when $\Gamma$ is zero (no spontaneous emission), but $\kappa$ is nonzero (the cavity is leaky). Let $\Omega _{0}$ be the vacuum Rabi frequency, and denote this three state space by $|1{\rangle }=|e,0{\rangle }$, $|2{\rangle }=|g,1{\rangle }$, and $|3{\rangle }=|g,0{\rangle }$. Written out explicitly in terms of the $3\times 3$ density matrix elements $\rho _{jk}$, the master equation is (for the case $\delta =0$ with the atom starting in the ground state)

${\begin{array}{rcl}{\dot {\rho }}_{11}&=&i{\frac {\Omega _{0}}{2}}(\rho _{12}-\rho _{21})\\{\dot {\rho }}_{22}&=&-i{\frac {\Omega _{0}}{2}}(\rho _{12}-\rho _{21})-\kappa \rho _{22}\\{\dot {\rho }}_{12}&=&i{\frac {\Omega _{0}}{2}}(\rho _{11}-\rho _{22})-{\frac {\kappa }{2}}\rho _{12}\\{\dot {\rho }}_{21}&=&-i{\frac {\Omega _{0}}{2}}(\rho _{11}-\rho _{22})-{\frac {\kappa }{2}}\rho _{21}\\{\dot {\rho }}_{33}&=&\kappa \rho _{22}\,.\end{array}}$

When the cavity damping rate is small, $\kappa \ll \Omega _{0}$, then the vacuum Rabi oscillations are damped, with average damping rate $\kappa /2$ since the atoms spend roughly half their time in the excited state.

When the cavity damping rate is large, $\kappa \gg \Omega _{0}$, then the atomic excitation is irreversibly damped, and no oscillations occur. Let $P_{e}$ be the probability of being in the $|1\rangle =|e,0{\rangle }$ state. We can combine two of the above equations to write

${\begin{array}{rcl}{\dot {\rho }}_{12}-{\dot {\rho }}_{21}&=&i\Omega _{0}(\rho _{11}-\rho _{22})-{\frac {\kappa }{2}}(\rho _{12}-\rho _{21})\,.\end{array}}$

Since in this case ${\dot {\rho }}_{12}+{\dot {\rho }}_{21}\approx 0$, and $\rho _{22}\approx 0$, it follows that

$(\rho _{12}-\rho _{21})\approx {\frac {2i\Omega _{0}}{\kappa }}(\rho _{11}-\rho _{22})\,,$

Since in this regime, we slowly try to populate state $|2{\rangle }$ while it quickly decays away, we can approximate $\rho _{22}\ll \rho _{11}$ giving

$(\rho _{12}-\rho _{21})\approx {\frac {2i\Omega _{0}}{\kappa }}\rho _{11}\,,$

Finally plugging this back into the optical bloch equation for $\rho _{11}$ yields

${\dot {\rho }}_{11}=-{\frac {\Omega _{0}^{2}}{\kappa }}\rho _{11}\,,$

so $P_{e}$ decays exponentially, with rate $\Gamma _{c}=\Omega _{0}^{2}/\kappa$. Note that a large cavity damping rate slows down the decay of the atom's excited state. This is due to the fact that the stimulated transition (Rabi oscillation) to the lower state is slowed down by the cavity damping (which is a manifestation of the quantum Zeno effect).

Purcell factor: cavity enhanced spontaneous emission

How does this compare with the free space spontaneous emission rate $\Gamma$? Recall that the vacuum Rabi frequency is

$\Omega _{0}=2{\frac {dE_{0}}{\hbar }}\,,$

where $d$ is the atomic dipole moment and $E_{0}$ is the electric field amplitude of a single photon at the atomic transition frequency $2\pi \omega _{0}$,

$E_{0}={\sqrt {\frac {\hbar \omega _{0}}{2\epsilon _{0}V}}}\,,$

and $V$ is the cavity volume. This gives

$\Omega _{0}=d{\sqrt {\frac {2\omega _{0}}{\epsilon _{0}\hbar V}}}\,.$

Letting $Q=\omega _{0}/\kappa$, we thus find that

$\Gamma _{c}={\frac {2d^{2}}{\epsilon _{0}\hbar }}{\frac {Q}{V}}\,$

as the decay rate of the atom in the cavity.

Recall that the spontaneous emission rate of an atom in free space, as determined by Fermi's golden rule, is

$\Gamma ={\frac {d^{2}\omega _{0}^{3}}{3\pi \epsilon _{0}\hbar c^{3}}}$

The ratio of this rate to the decay rate in the cavity is

$\eta ={\frac {\Gamma _{c}}{\Gamma }}={\frac {3}{4\pi ^{2}}}{\frac {Q\lambda ^{3}}{V}}\,,$

where we take $\lambda =2\pi c/\omega _{0}$ as being the wavelength of the cavity field, which is assumed to be resonant with the atomic transition frequency $2\pi \omega _{0}$. Note that $\eta$ is independent of the atomic dipole strength, and determined solely by cavity parameters. Moreover, note that for small, high-$Q$ cavities, the decay rate of the atom in the cavity can be much larger than the free space spontaneous emission rate. This "cavity enhanced" spontaneous emission rate was predicted by Purcell (1946), an observation credited as being the starting point of cavity QED.