# Quantum tunneling versus over-the-barrier ionization

I was doing a quantum physics past paper and there was this question about ionizing atoms with laser. My answers made reference to the photons within the laser knocking off or transferring their energy to the electrons and allowing them to escape if this energy exceeds their biding energy. I then checked on the markscheme and all it mentioned was the laser modifying the nuclear potential to allow the electron to quantum-mechanically escape. I am confused.

I know quantum tunneling happens when a wave function meets a potential barrier, but I don't understand how the laser provides this barrier. Isn't a laser just a stream of photons basically? Also, the markscheme made reference to OTBI ionization for hydrogen (the second part of the question was specifically about hydrogen).

I did some reading and I get the idea it is the same thing as quantum tunneling, so I am not sure why in the markscheme the two seem like different ideas altogether. Unfortunately, the markscheme is not a full answer but just a string of key words to award marks so that doesn't help.

• Related: Is quantum tunneling related to imaginary time?, not quite at the level you need but with information and references that might help sharpen your question. Similarly, try reading the introduction of my PhD thesis, which can hopefully clarify some of the issues. – Emilio Pisanty Jun 3 '18 at 22:22
• To provide a useful answer, it'll help a good deal to have the exact text of the exam question. The difference between single-photon, multiphoton, tunnel and over-the-barrier ionization is partly in how we conceptualise the process but also in the parameter regime of the laser source and the target, which are likely to be specified in the text, but it's hard to know what they meant without the specific wording. – Emilio Pisanty Jun 3 '18 at 22:25
• It's easier for other to read questions if the text follows some basic English punctuation conventions: 1) Capitalize the first word of each sentence, 2) Capitalize the word "I", 3) Use an apostrophe in contractions, e.g. "can't" instead of "cant", and 4) Spell words correctly. – DanielSank Jun 4 '18 at 3:18
• Maybe links in my answer here physics.stackexchange.com/questions/409959/… will help clarify that the word "tunneling " in quantum effects has more than one use. – anna v Jun 4 '18 at 7:16

I suspect that the core of your question is this:

Isn't a laser just a stream of photons basically?

The answer to that core problem is essentially "yes, but there are situations where that isn't a useful description".

In particular, tunnel ionization and over-the-barrier ionization are phenomena that take place when the laser's frequency (and thus its photon energy) is very small, so that you need many photons to make it all the way up to the ionization threshold. Since you need many photons, the process basically won't happen unless you have a very high field intensity, but this can be problematic in its own way, because the perturbation-theory expansion that allows you to talk about 'absorption of photons' loses its usefulness at those high intensities:

• For one, we don't really know that the expansion converges at all intensities, and indeed it could just outright break.
• More practically, if the intensity is strong enough to allow for six-photon ionization, then absorbing a seventh photon isn't all that crazy. (This is generally known as Above-Threshold Ionization.)
• And, to make things worse, it's also possible (in the perturbation-theory sense) to re-emit that seventh photon into the field, giving an eighth-order process that interferes with the six-photon process. Furthermore, if the intensity is high enough, those two processes can be comparable, and their interference will be very difficult to calculate. As the intensity increases, more and more of these sub-leading-order processes need to be accounted for, until the expansion loses all of its usefulness.

This is where tunnel ionization comes in. Supposing that you have a field of angular frequency $\omega$, intensity $I$ and peak field strength $F=\sqrt{2I/\varepsilon_0c}$ ionizing a target of ionization potential $I_p=\frac12 \kappa^2$, then you're in the multi-quantum regime whenever $I_p/\hbar\omega$ is greater than one, but the real discriminant is known as the Keldysh adiabaticity parameter, $$\gamma = \sqrt{\frac{I_p}{2U_p}} = \sqrt{\frac{I_p/2}{e^2F^2/4m_e\omega^2}} = \frac{\kappa\omega}{F}$$ (with that last equation on atomic units; $U_p$ is the ponderomotive energy). If the field is strong enough and slow enough that $\gamma < 1$, then the perturbative expansion that explains ionization as the absorption of individual photons loses its usefulness, and you need to move to a different picture.

That picture is tunnel ionization. Since we're in a roughly adiabatic regime, where the suitable measure for $\omega$ (namely, $\gamma$) is small, then we can basically think of the interaction with the laser, the dipole coupling $$\hat H_\mathrm{interaction} = e \hat{\mathbf r} \cdot \mathbf F(t)$$ as a time-dependent linear potential that gets added to the initial atomic potential well. And, if that field is strong enough, then it can bend the potential low enough to make a potential barrier that the electron can then tunnel through:

Image source

If you do the calculation for the rate of (quasi-static) tunnelling through that barrier, the core result is that the rate goes as $\omega_t = F/\kappa$ (or $\omega_t = eF/\sqrt{2I_pm_e}$ with all the constants put in), and this mechanism will only produce a sizeable tunnelled population if the field peak is sustained longer than that, i.e. if the Keldysh parameter $\gamma = \omega/\omega_t$ is small enough.

Thus far for tunnel ionization, which is relatively 'clean' as far as things go. Now, it should be clear from my diagram above that if you turn up the intensity even further (thereby taking you even further away from the validity of the photon picture) then at some point the peak of the tunnelling barrier will be so low that it falls below the energy level the electron was initially sitting on: this is the situation for Over-the-Barrier ionization, and it should be clear that it is swift and effective at removing the population from the bound state and onto the continuum.

• There is a separate, deleted answer in this thread by @pietro, which is full of very valuable content ─ in my opinion, it should be restored. – Emilio Pisanty Jun 13 '18 at 13:01
• you can vote to undelete pietro's answer, no? – anna v Jun 13 '18 at 13:20

The laser doesn’t provide the barrier, on the contrary, the laser helps to overcome the barrier. That means that when you give visible light, you achieve just an excited state of the atoms, and it’s impossible to get them ionized. But these excited electrons can easily escape through quantum tunneling, something that is not possible in classic physics. Using a laser you can give a pure wavelenght of visible light, exactly towards a target.