# Validity of flat coupling approximation in input-output theory + why is it called Markov approximation?

In standard input-output theory, we model the interaction between light and a two level system by the following hamiltonian which assumes rotating wave approximation:

The interaction between a classical electric field, polarized along a single axis: $$x$$, and a dipole is the following:

$$H=-\frac{\hbar \omega_0}{2} \sigma_z + \epsilon(e^{j \phi} \sigma + e^{-j\phi} \sigma^{\dagger})E_x(t)$$

When ones apply the rotating wave approximation we have:

$$H=-\frac{\hbar \omega_0}{2} \sigma_z + \sum_{n=-\infty}^{+\infty} \epsilon (E^*(k_n) e^{j(\phi+\omega_n t)} \sigma + E(k_n)e^{-j(\phi+\omega_n t)} \sigma^{\dagger})$$

And, to make easy "comparisons" with the quantized version of the field, we can rewrite:

$$E(k_n)=\sqrt{\frac{\hbar \omega_n}{2 \epsilon_0 V}} \alpha(k_n)$$

$$H=-\frac{\hbar \omega_0}{2} \sigma_z +\sum_n \epsilon \sqrt{\frac{\hbar \omega_n}{2 \epsilon_0 V}} (e^{j \phi} \alpha^{*}(k_n) \sigma + e^{-j \phi} \alpha(k_n) \sigma^{\dagger})$$

In the quantized version (E.M field quantized), we just have to replace $$\alpha \rightarrow a^{\dagger}$$ which leads to:

$$H=-\frac{\hbar \omega_0}{2} \sigma_z + \sum_{n=-\infty}^{+\infty} \hbar \omega_n \widehat{a}^{\dagger}(k_n)\widehat{a}(k_n)+\sum_n \hbar g(\omega_n) (e^{j \phi} \widehat{a}^{\dagger}(k_n) \sigma + e^{-j \phi} \widehat{a}(k_n)\sigma^{\dagger})$$

$$\hbar g(\omega_n)=\epsilon \sqrt{\frac{\hbar \omega_n}{2 \epsilon_0 V}}$$

At this point, if we look at the classical Hamiltonian, because the coupling is flat in frequency between the field and the dipole, there are no memory effect. It can be seen on the first equation: if I replace $$E_x(t)$$ by its Fourier serie each fourier coefficient of the field will be coupled by a constant term to the TLS operators (the interaction is independant of $$\omega$$).

More directly, if I cut the electric field at time $$t_0$$, the interaction is instantaneously cut. There is no latence (it is another way of saying the coupling is flat in frequency).

In the quantum regime it is thus the case as well. The fact the constant $$g(\omega_n)$$ depends on $$\omega_n$$ is simply a matter of rewriting that can be seen on the classical level in the derivations done here. (It is basically when the change of variable $$E \rightarrow \alpha$$ is done).

Now, in input output theory, we usually approximate: $$g(\omega_n) \approx g$$, i.e the coupling constant becomes frequency-independant. This is called Markov approximation.

## My questions are the following.

1. Why is the approximation $$g(\omega_n) \approx g$$ valid ? The context of what I want to understand is basically how to drive a two level system to induce single qubit gates under the R.W.A. Here, Electric dipole interaction: flat coupling semi-classically, non flat with E.M field quantized: why ? What does that mean? it is said the it is implied by the R.W.A but I am not sure to exactly see why.
2. Why is it called Markov approximation ? Indeed on the classical level we can directly see that there is no memory effect seen by the system, the coupling being flat in frequency. For me, stating $$g(\omega_n) \approx g$$, if we go back to the classical Hamiltonian it will induce a frequency dependent coupling between the field and the dipole and will thus induce memory effect. It confuses me.

: I think I understand somehow the first point, but I would like to check it.

In the context of performing Rabi oscillations with this field, we will apply a pulse of duration $$\Delta T$$, where $$\Delta T \sim \frac{1}{\Omega}$$ ($$\Omega$$ being the classical Rabi frequency induced). For example to perform a $$\pi$$ pulse, it requires a pulse length = time of interaction being $$\Delta T = \frac{\pi}{\Omega}$$.

The width of the modes composing the pulse is $$\Delta \omega \sim \frac{1}{\Delta T} = \Omega$$.

Finally, as the RWA is valid if $$\Omega \ll \omega_0$$, we have: $$RWA \Rightarrow \frac{\Delta \omega}{\omega_0} \ll 1$$: the range of populated frequencies of the drive is a narrow band around $$\omega_0$$. Thus, we can say $$g(\omega_n) \approx g(\omega_0)$$.

Does that make sense for you ?

This will not qualify as a full answer, but it is too long for comments.

The approximation of frequency-independent coupling is widely used in tunneling problems, where it is called broad band limit. Typically one deals with a tunneling rate like $$\Gamma = 2\pi|V(E)|^2\rho(E),$$ where the matrix element $$V(E)$$ and the density-of-states $$\rho(E)$$ are energy dependent. One usually assumes that the energies involved are very close to the Fermi surface, so that one can neglect the slow variation of these quantities (in addition the density-of-states is exactly constant in 2D case). I suspect the same reasons are valid here

The use of Markov terminology in optics also always struck me as rather stretched. I suspect that it has to do with the influence of Phil Anderson and his influential co-workers, see here and here. (Their very use of the Germanized spelling Markoff testifies to the ancient roots of this term.)

Remark
Markov approximation often appears as Born-Markov approximation, as in Born approximation in scattering problems. The claim to Markovian is probably the strongest when truncating the expansion of the density matrix, as, e.g., discussed here.