Coherent drive Hamiltonian

Question

Sometimes in the light-matter interaction described by the Jaynes-Cummingsa model, an additional coherent drive is included which drives the light in the cavity. I've encountered this so many times in various quantum optics papers, but I still don't clearly understand what this drive is. $$ H=H_\text{r}+H_\text{a}+H_{\text{int}}+H_{\text{d}} ,$$

or $$ H=\hbar \omega_r a^+ a + \frac{1}{2}\hbar \omega _a \sigma_z + g(a^+ \sigma_-+a\sigma_+)+(\epsilon a^+ e^{-i\omega_d t} + \epsilon^* a e^{+i\omega_d t}),$$

where the first two terms, $H_\text{r}$ is the resonator's Hamiltonian, $H_\text{a}$ is the atom's Hamiltonian simplified as the two-level system. The third term $H_{\text{int}}$ is the atom-light interaction Hamiltonian, and the last term $H_{\text{d}}$ is the external drive's Hamiltonian with $\epsilon$ as the amplitude and $\omega_d$ as the drive's frequency.

The following things are not clear to me:

1. Usually, it is said in the papers that $H_{\text{d}}$ is the coupling Hamiltonian between the drive and the light in the cavity. What is the process of coupling light in the cavity with the drive's light? How do they interact with each other?

2. How is this Hamiltonian derived? Where did this relation come from? Any textbooks that you can recommend for this?

3. Since the coherent drive is the light as well, why is there no Hamiltonian like $ \hbar \omega_d a^+ a $, which accounts for drive's energy?

Answer 1

The coherent drive represents an electromagnetic field interacting with the light/atom in the cavity. Say you have a drive (electromagnetic field):

$$E\propto \epsilon b +\epsilon^*b^\dagger$$

and the light in the cavity has creation/annihilation operators $a, a^\dagger$. Then, the interaction between the drive and the light in the cavity would be represented as:

$$H_d \propto (\epsilon b +\epsilon^*b^\dagger)(a+a^\dagger)$$

The reason for such an interaction is that photons would be added (pumped into) the cavity by a laser. You can look into this more deeply by reading about cavity optomechanics. In the interaction picture, it can be seen that the operators of the light in the cavity evolve as:

$$a(t)\propto a(0) e^{-iwt}$$

where $w$ is the frequency of the light in the cavity. Also, in the interaction picture, the light of the coherent drive evolves as:

$$E(t)\propto\epsilon b e^{-iw_dt}+\epsilon^*b^\dagger e^{iw_dt}$$

where $w_d$ is the frequency of the coherent drive. Due to this, the interaction term in the interaction picture Hamiltonian looks like:

$$H_d^{I}\propto (\epsilon b e^{-iw_dt}+\epsilon^*b^\dagger e^{iw_dt})(a(0) e^{-iwt}+a^\dagger(0) e^{iwt}) \\= \epsilon b a(0) e^{-i(w_d+w)t}+\epsilon b a^\dagger(0) e^{-i(w_d-w)t}+\epsilon^*b^\dagger a(0) e^{i(w_d-w)t}+\epsilon^*b^\dagger a^\dagger(0) e^{i(w_d+w)t}$$

In the Rotating Wave Approximation (RWA), it is assumed that $|w_d-w|<<w_d+w$ is true, which is reasonable for $w_d\approx w$. So, the higher frequency terms $w_d+w$ oscillate very quickly and average out to zero at the time-scales at which the system is probed. This is a fully quantum mechanical treatment of the problem.

When the drive is introduced in the semi-classical approximation, it is a complex number. The semi-classical approximation is the case in which the light/atom in the cavity is treated quantum mechanically (hence, they are represented as operators) and the drive is treated classically (hence, it is a complex number). Then, the coherent drive is:

$$E\propto \epsilon e^{-iw_dt}+\epsilon^* e^{iw_dt} $$

Here, the coherent drive is explicitly time dependent because that's how one describes the electromagnetic wave classically, as a traveling wave. The interaction term would be represented as:

$$H_d \propto (\epsilon e^{-iw_dt}+\epsilon^*e^{iw_dt})(a+a^\dagger)$$

Under RWA, it gives the $H_d$ in the question.

Answer 2

1. The drive is just a coherent monochromatic light source with angular frequency $ \omega_{d} $. The cavity can be thought of selective amplifier which only allows discrete spectrum of frequencies to sustain, given by $ \lambda_{n} = \frac{2L}{n} $ which is nothing but the allowed modes of standing waves in cavity of length $ L $. Waves having any other frequencies will decay with time inside the cavity.

2. There are no first participles to derive the Hamiltonian of system. It is mostly through physical intuition and by trial and error.

3. This is a semi-classical model where only the energy of cavity is quantized and the drive (light) is treated classically for simplification.