Cavity QED

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Cavity QED, or Cavity Quantum Electrodynamics, is a central paradigm for the study of open quantum systems. The "Cavity" in "Cavity QED" refers (typically) to the optical or microwave resonator being employed, while "QED" refers to the quantum nature of the coherent interactions between the material system (i.e. atoms) and the electro-magnetic field (i.e. photons) inside the cavity. In addition to the coherent atom-field interactions, the system possesses two prominent decay channels: an excited atom may spontaneously emit light out of the cavity mode, and light may leak out through the cavity mirrors. However, the great advantage of employing a cavity is that these decoherence rates can be made small relative to the cavity mediated atom-field interaction.

Most experimental and theoretical investigations in Cavity QED study the interactions of a single mode of the electro-magnetic field with a two-level atom. In this case, the dynamics may be modeled by the unconditional master equation ( $\hbar=1$ )

$\dot{\rho}=-i [\mathcal{H},\rho]+ 2\kappa\mathcal{D}[a]\rho + \gamma_{||}\mathcal{D}[\sigma_{-}]\rho$

where $\mathcal{H}$ is the Hamiltonian in a frame rotating at the frequency of the driving field

$\mathcal{H}=\Delta_c a^{\dagger}a+\Delta_a \sigma_{+} \sigma_{-} +i g_0 (a^{\dagger}\sigma_{-} -a\sigma_{+} )+i\mathcal{E}(a^\dagger-a)$

The first two parts of the Hamiltonian involve detunings of the atom and cavity mode from the driving laser and result in a modulation of the dynamics. The third piece in the Hamiltonian is the Jaynes-Cummings Hamiltonian (which utilizes the electric dipole approximation) and the fourth piece allows for driving of the intercavity mode (all in the rotating wave approximation). Also, $κ$ is the cavity field decay rate, $γ | |$ is the atomic spontaneous emission rate, and the Lindblad superoperator $\mathcal{D}$ is defined by

$\mathcal{D}[O]\rho = O\rho O^{\dagger} - \frac{1}{2}\left(O^{\dagger} O \rho + \rho O^\dagger O\right)$

for an arbitrary system operator $O$ . The atom-field coupling $g 0$ measures the coherent coupling rate between the atom and the cavity, the rates $γ | |$ , and $κ$ characterize processes which tend to inhibit the build up of coherence. This master equation for the system dynamics (involving a partial density matrix for the system, where the baths responsible for atomic spontaneous emission and cavity decay have been traced over) is obtained in the Born-Markov approximation.
The qualitative nature of the dynamics may be determined by two dimensionless parameters which measure the relative strengths of the coherent and incoherent processes: the critical photon number

$n_{0} = \frac{\gamma_{||}\gamma_\perp}{4 g_0^2} \;,$

and the critical atom number

$N_0 = \frac{2\gamma_\perp\kappa}{g_0^2} \;,$

where $\gamma_\perp$ is the transverse relaxation rate, and $\gamma_\perp = \gamma_{||}/2$ for purely radiative relaxation. Qualitatively, the critical photon number measures the number fo photons required in the cavity mode in order to saturate the atomic response. Similarly, the critical atom number roughly measures the number of resonantly coupled atoms needed to dramatically change the response of the cavity. The strong-coupling regime of Cavity QED is typically refers to the parameter regime $(n 0, N 0) < 1$ , although some authors choose the less stringent requirement $n 0 < 1$ or $N 0 < 1$ . If $n 0$ is much larger than one (and surprisingly this can be as small as 10!), the field can often be treated classically. When both $N 0$ and $n 0$ are much larger than one, the dynamics of the system can often be well approximated by Maxwell-Bloch equations, where one assumes that atom-field expectation values factorize, i.e. $\langle a \sigma_-\rangle=\langle a\rangle\langle \sigma_-\rangle$ . The resulting nonlinear dynamics can show interesting biufurcations and chaos. Quantum noise resulting in squeezing and antibunching can be treated in the small noise limit.

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Categories: Concept | Quantum Optics

Cavity QED

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