Jaynes-Cummings Hamiltonian

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The Jaynes-Cummings Hamiltonian models the interaction of a single quantized mode of the electro-magnetic field with a two-level atom and is commonly used in Cavity QED. It takes the form

$\mathcal{H}_{JC}=i\hbar g_0\left( a^{\dagger}\sigma_{-} -a\sigma_{+} \right)$

The operator $a$ is the boson (here photon) annihilation operator and $a^{\dagger}$ is the boson creation operator. $σ +$ and $σ -$ are atomic (raising) lowering operators. Treating the atom as a two level system, these reduce to the Pauli the spin matrices.

This Hamiltonian has its origin in an electric dipole coupling $\mathcal{H}=-\vec{\mu}\cdot \vec{E}$ where one takes the applied electric field and atomic dipole to be parallel, the applied field to be resonant with an atomic transition, and the quantized field and dipole operators

$E=i\sqrt{\hbar\omega/2\epsilon_o V}f(z)\left(a^{\dagger}-a\right)$
$~\mu=|\mu_{eg}|(\sigma_++\sigma_-)~$

where, $~f(z)~$ is the classical electric field mode evaluated at the location of the atom, ${\vec{\mu}}_{eg}=\langle e|\vec{r}|g\rangle$ , and $V$ is the volume of the electromagnetic mode. Note that is not literally the volume of the cavity but the effective volume of the electromagnetic mode,

$V=\int TBA .$

The J-C Hamiltonian can be interpreted as follows: the first term describes the process where a photon is created in the cavity and the atom relaxes to the ground state. The second term describes the opposite process; the atom absorbs a photon and is excited. Energy non-conserving terms such as $~a^{\dagger}\sigma_+~$ and $~a\sigma_-~$ are dropped in the rotating wave approximation.

The Jaynes-Cummings model (that is, the Hamiltonian shown above, possibly generalized to include detuning between the electomagnetic mode and the atomic transition, but without the counter-rotating terms) is arguably the simplest exactly solvable, nontrivial system in quantum optics. When the counter-rotating terms are included, however, the reslting spin-boson Hamiltonian has no known exact solution. While at optical frequencies, the counter-rotating (energy non-conserving) terms are negligible to an excellent approximation, in many solid-state systems (where the bosons are, typically, phonons) the full spin-boson Hamiltonian needs to be considered.

The atom-field coupling described $g_0=\sqrt{\omega/2\hbar\epsilon_o V}f(z)|\mu_{eg}|$ . (In a more general treatment, $g$ depends on the location of the atom in the spatial distribution of the (gaussian) mode profile.) It is generally increased by making the field mode volume smaller. The reduction in mode volume is accomplished in practice by constructing small volume optical resonators such as Fabry-Perot cavity, photonic crystal resonator or micro-disc resonator. For a Fabry-Perot type cavity (standing waves) we have $g = T.B.A.$ while for a traveling wave cavity one has $g = T.B.A.$

In the Jaynes-Cummings model the cavity field is assumed to be initially prepared in some state, and thereafter it evolves by interacting with the atom without any external influence. Closer to experimental reality is the case of an externally driven optical cavity with resonant frequency $~\omega_c~$ , a two-level atom, with ground and excited states separated by a frequency $~\omega_a~$ , and an external (classical) field with drive amplitude $~\mathcal{E}~$ and frequency $~\omega_l~$ . For an atom-field coupling constant $~g_0~$ the Hamiltonian written in a frame rotating at the drive frequency is given by ( $~\hbar=1~$ ):

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

where $~\Delta_c = \omega_c - \omega_l~$ and $~\Delta_a = \omega_a - \omega_l~$ are, respectively, the generally nonzero cavity-field and atom-field detunings.