What will happen if a ground state hydrogen atom is placed in a low frequency, but high intensity laser field? Similar questions have been asked before, but this one extends the scope of   interpretation and applications. 
Let us ask the question: What will happen if we have a laser gun that produces a high intensity laser beam, of the order $10^{15}\:\mathrm{ W \: m}^{-2}$ say, and frequency  about $4\times 10^{14}\:\mathrm{Hz}$ (so a wavelength of $750\:\mathrm{nm}$), and shoot a hydrogen atom in its ground state? The duration of the laser pulse is sufficiently long. Answers in terms of physical arguments will be appreciated.
 A: For low-frequency, high-intensity laser fields, atomic and molecular targets can be ionized by multi-photon ionization (which is a nonlinear process and thus requires high intensity, but is nevertheless possible).
If you turn on the intensity further, the mechanism shifts over to tunnel ionization (which can also happen in a sufficiently strong DC, or quasi-DC, field), in which the atomic potential barrier is pushed low enough that the ground state can escape. By this time the laser electric field comparable or stronger than the atomic one, which means that perturbation-theoretic concepts like photon absorption no longer mean much.
These are rough terms, though, and they are mostly used as tools to understand the physics rather than strict definitions. In particular, there is a sliding scale between quasistatic optical tunnelling and perturbation-theory-like multiphoton ionization, which follows the so-called Keldysh adiabaticity parameter,
$$
\gamma = \sqrt{\frac{2I_p}{U_p}} = \frac{\kappa\omega}{F}
$$
(where $I_p$ is the ionization potential of the target, $U_p$ is the ponderomotive energy, and in atomic units $\kappa=\sqrt{2I_p}$, $\omega = 45.6\:\mathrm{nm}/\lambda$ and $F = 0.053\sqrt{I/10^{14}\:\rm W/cm^2}$ are the characteristic momentum of the ground state, the angular frequency in a.u. in terms of the wavelength, and the peak electric field strength in a.u. in terms of the intensity $I$.) For $\gamma<1$ the ionization has optical-tunnelling characteristics (and the smaller the $\gamma$ the more quasi-static the regime), whereas for large $\gamma$ the more perturbative the configuration becomes.
It's also worth emphasizing that the rest of the configuration also matters, including things like how long the pulse is and how sharp of an on-ramp it has, since e.g. if the pulse is too long then it is perfectly possible for the on-ramp to completely deplete the system via perturbative behaviours before the tunnel-ionization regime is achieved.
And a couple of quick notes: 


*

*For whatever reason, intensities are never reported in $\rm W/m^2$, but in $\rm W/cm^2$ instead. This makes no objective sense (neither unit is objectively better than the other, other than $\rm W/m^2$ sticking to the base units) but it's the standard in the field, so beware of unit conversions if you want to use $\rm W/m^2$.

*The intensity you mention, $10^{11}\:\rm W/cm^2$, is not usually considered to be a "high intensity laser beam", and as far as hydrogen is concerned it is more of an insistent tickling than a high-intensity driver. It can still ionize if it's on for long enough, but the rate will be rather low.
If, instead, you meant $10^{15}\:\rm W/cm^2$, then the story is wildly different ─ by the time you get to $I = 1.5\times 10^{14}\:\rm W/cm^2$ the tunnelling barrier will be so low that the peak of the barrier goes below the level of $I_p$, a condition known as over-the-barrier ionization, and the system will be blasted to bits. Under those conditions, the only chance you have of keeping any population in bound states is via ionization stabilization into Kramers-Henneberger states (for which this recent paper is a good starting point).

Apart from that, one very interesting phenomenon that can happen is the generation of very high order harmonics of the laser field. This is caused by the ionized electron being reaccelerated by the laser field towards the parent ion and colliding with it. The electron can then recombine and emit all of its considerable energy as a remarkably short (sub-femtosecond) pulse of light. This means that harmonics of order up to ~130 have been detected. A good reference for this is

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, Anne L’Huillier, and P. B. Corkum. Theory of high-harmonic generation by low-frequency laser fields. Phys. Rev. A 49 no.3, pp. 2117–2132 (1994).


A: Even in an "insufficient frequency" wave an atom can get excited and ionized due to multi-photon absorption. Many years ago I read an article of Keldysh about the corresponding cross sections.
A: \begin{align}
\gamma 
& = \sqrt{\frac{E_i}{1.87\cdot10^{-13} I \lambda^2}}
 = \sqrt{\frac{13.6\:\mathrm{eV}}{1.87\cdot10^{-13} \cdot 10^{11}\mathrm{W/cm}^2\cdot (0.750\:\mu\mathrm m)^2}}
\\ & = 35.95 > 1
\end{align}
As mentioned in the comments the Keldysh parameter helps understanding if you have a multiphoton absorption or a tunneling process. If the Keldysh parameter is above $1$ then you have multiphoton absorption (in the picture below). Tunneling ionization if $<1$. (See ref. 1 for more details.)
Using your experimental parameters, it turns out that the Keldysh parameter is equal to 36 so you are definitely not in a tunneling regime:
You can try to test different experimental parameters with a calculator I wrote, available at https://flyhigher.000webhostapp.com/



*

*Intensity dependence of the multielectron dissociative ionization of N$_2$ at 305 and 610 nm. C. Cornaggia et al. Phys. Rev. A 42, 5464 (1990).

