2017“北京东盟文化之旅”交流团访问越南

Question

I was following lecture notes on light-matter interactions by Claudiu Genes (pdf link). Refer to chapter 1 : we are dealing with a two level atomic system governed by the ladder operators $\sigma = |g\rangle\langle e|$ and $\sigma^{\dagger}$. The energy gap between the ground and excited state corresponds to photon mode $\textbf{k}_0$. There is an electromagnetic field which interacts with the two level system, and it's $H_{int}$ is given by

$$H_{int}=\sum_{\textbf{k}}g_{\textbf{k}}(a_{\textbf{k}}\sigma^{\dagger}+\sigma a^{\dagger}_{\textbf{k}})$$ This two level system is coupled to it's environment, where it leaks photon of mode $\textbf{k}_0$ (see page 10 of the link). Now, jump to section 1.5 : If we start with an initial state $|i\rangle = |e\rangle \otimes |0,0,\cdots , 1_{\textbf{k}_0}, 0,0\cdots \rangle$ , then we get

$$H_{int}|i\rangle = \sqrt{2}g_{\textbf{k}_0}|g\rangle \otimes |0,0,\cdots , 2_{\textbf{k}_0}, 0,0\cdots \rangle+\sum_{k\neq k_0}g_{\textbf{k}}|g\rangle \otimes |0,0,\cdots , 1_{\textbf{k}_0}, 0,0\cdots , 1_{\textbf{k}},\cdots \rangle$$

This second term is then attributed to spontaneous emission.

The math is okay. What I do not understand conceptually is that electron transition from $|e\rangle \to |g\rangle $ should only produce photon mode $\textbf{k}_0$ and not any arbitrary $\textbf{k}$. It seems to be loosing or gaining energy of the amount proportional to $|\textbf{k}-\textbf{k}_0|$. Where is this extra energy coming from or going to? I tried to motivate myself that, there are probably other kinds of interactions with the environment going on under the hood ? However, the Hamiltonian is concerned with EM interactions only. Also, the system seems to be only leaking mode $\textbf{k}_0$ to the environment.

Note : similar question has been asked before, for eg here. Here OP doesn't mention about the environmental conditions. But the answer to the question :" In particular, are several modes of the field populated with photons at the various frequencies?" seems affirmative. My question is where are these extra modes $\textbf{k}$ coming from , in the setup mentioned in the lecture note

Answer 1

What I do not understand conceptually is that electron transition from $|e\rangle \to |g\rangle $ should only produce photon mode $\textbf{k}_0$ and not any arbitrary $\textbf{k}$.

But the calculation does not show what gets produced after some time, it shows the result of action of the interaction Hamiltonian (with off-diagonal terms) to that initial state; it's just a result of a mathematical operation. This result gives rate of change of quantum state, not the state itself.

To get state at a later time, we have to integrate the time-dependent evolution equation over time, and then the state is sum of terms like above, but weighed by appropriate time-dependent amplitudes.

It seems to be loosing or gaining energy of the amount proportional to $|\textbf{k}-\textbf{k}_0|$. Where is this extra energy coming from or going to?

The initial state isn't an eigenstate of the total Hamiltonian, so energy in the usual sense in quantum theory (value of total Hamiltonian) does not have definite value. We should be able to calculate expectation value of energy of that state and its variance, and those should remain constant.

My question is where are these extra modes k k coming from , in the setup mentioned in the lecture note

They are always there, they just get "populated" due to interaction Hamiltonian. It's spontaneous emission - radiation coming off the atom can go in all possible directions, thus having different wavenumbers, of different direction and magnitude. Directions are weighed as dipole radiation pattern (in this case, assuming a system with a non-zero off-diagonal term of the dipole moment matrix), magnitudes are weighed so that most of the weight is near the frequency $k_0 c$ (but not all of it). Again, this does not violate any conservation, because we don't have sharp energy value in the first place.