# given an AC field, how many photons are there?

Say you are applying a time varying potential across double quantum dots in the form of $V_{ac}$cos($2\pi f t$). We know that each photon has an energy $E_{photon}= hf$.

Is it correct to say that the number of photons from the oscillating field is $eV_{ac}/hf$ ?

I suppose the question above is ill-defined, in the sense that one should ask "the number of photons per unit area per second," which is given by $\frac{P}{Ahf}$, where P is power and A is the area over which the field is being applied.

I ended up asking this question because I couldn't understand the plot below. (originally from enter link description here)

Without the quantum dot context, in the caption, a positive n corresponds to the absorption of n photons. I found it odd that $J_1^2$ peaks at $\alpha = eV_{ac}/hf = 2$ as opposed to 1. It seems odd that all $J_n^2$ is shifted to the right by 1 for $n>0$.

• Maybe I don't get your question, but the plot shows a splitting of an energy level due to an external driving field. (compare ac Stark effect and Franz–Keldysh effect). The hf, 2hf is not the numer of photons in the microwave field but the energy of the forming sidebands. Does this go in the direction of your question? – user_na Jul 24 '18 at 16:41
• @user_na I thought given some energy difference between left and right dots, when $hf$ or some integer multiple of it matches the energy difference, then you get photon-assisted tunneling. So I thought the plot of the peaks of the Bessel functions squared basically tells you at what $V_{ac}$ does the energy difference match some integer times the energy of a single photon. For example, the peak position of $J_1^2$ tells me that the probability of the energy transition from E to E+hf is maximized when it absorbs 2 photons. (because $J_1^2$ hits its maximum at $α=2$. – Blackwidow Jul 24 '18 at 17:38
• So what does the second maximum of $J_1^2$ at $\alpha=5.5$ tells you then? – user_na Jul 24 '18 at 18:17
• @user_na the 2nd peak of $J_1^2$ tells me that at the $V_{ac}$ value where $\alpha =$ e$V_{ac}$/hf = 5.5, it's 2nd most likely to absorb 1 photon and achieve photon-assisted tunneling. – Blackwidow Jul 24 '18 at 20:57