### What is quantum tunneling?

When a classical particle, having a pre-defined and well-known energy, encounters a potential energy barrier higher than the energy possessed by the particle itself, the particle is bound to bounce back the obstacle and go back from where it came from.

If, for example, we roll a marble on a hypothetical frictionless plane and up a hill, the marble will roll until its kinetic energy is fully converted into potential energy. At that point, let’s call it the “classical turning point”, the ball stops and, by experiencing the gravitational pull, will roll back downhill reverting the direction of its motion. Only if the kinetic energy of the particle is greater than the potential energy assumed by the particle on top of the hill, the particle will be able to overcome the barrier and roll down the other side of the hill.

This is not true for a quantum particle. It turns out that quantum particles can enter “classically forbidden regions” in which the repelling potential is greater than the energy possessed by the particle itself and in which the linear momentum of the particle assumes purely imaginary values. Being able to enter “classically forbidden regions” of the barrier also means that a particle can cross it reaching the “classically allowed region” on the other side. In summary, a quantum particle has a non-zero probability of “tunneling through” a potential energy barrier, crossing it, even if its energy is smaller than the barrier itself.

If we think about it, admittingly loosening our formal tone, the concept of a “classical turning point” is, by itself, ill-defined. Having a “classical turning point” for a quantum particle would mean, in a broad sense, being able to define a position in space where the particle is located and has a precisely known zero linear momentum. This is incompatible with the constraints imposed by the Heisenberg uncertainty principle for which conjugate variables, such as the position of a particle and its momentum, cannot be determined simultaneously with arbitrary precision.

### Why is quantum tunneling interesting?

Quantum tunneling is omnipresent, it directly impacts the way many physical phenomena happen and often enables otherwise forbidden processes to take place. Quantum tunneling is fundamental in the understanding of many physical events and plays a substantial role in many modern technologies.

Quantum tunneling is fundamental in electronics in which the tunneling of electrons through isolation layers has, especially on microelectronic devices, a great effect on leakage currents. Quantum tunneling is also fundamental in understanding the working of semiconductor and superconductor junctions in which the tunneling of electrons or Cooper pairs plays often a crucial role.

Quantum tunneling is also a fundamental aspect of many nuclear processes such as nuclear fusion, in which tunneling is fundamental in the overcoming of the Coulomb barrier preventing the reaction from happening, and the radioactive decay enabling particles to cross in and out of potential wells of stability.

Quantum tunneling is fundamental also in many aspects of chemistry in which electron tunneling has a great effect on electron and excitation transfer while light atom tunneling has noticeable effects on the kinetic of many reactive and isomerization processes and in the spectroscopic response of many systems. In this context, tunneling splittings play a significant role and represent the focus of my research.

### What are tunneling splittings?

In general, tunneling splittings occur when a system, characterized by some sort of symmetry, has two or more equivalent configurations separated by a potential energy barrier. Quantum tunneling allows the interaction of states located at energies lower than the barrier top resulting in the splitting of otherwise degenerate localized energy levels.

To clarify what I mean by that, let me consider a double well potential in which two sites, one on the right ($R$) and one on the left ($L$), are separated by a somewhat relevant potential energy barrier. If we observe the site on the right ignoring the one on the left, working under the hypothesis that the barrier is large enough to keep the two configurations apart, we could study the system localized in the right well and compute a good approximation of the lowest energy states $\lvert \varphi_n^{(R)}\rangle$ localized within it. The same can be done for the left well obtaining the states $\lvert \varphi_n^{(L)}\rangle$. Given that the two wells are symmetrical, the two sets of states will have the same energy progression, or, in technical terms, they will be degenerate.

This however is a good depiction of the system only in the classical limit in which the barrier is sufficient to keep apart the two configurations. In reality, the two configurations can interact by tunneling through the barrier. The interaction determines the symmetric and anti-symmetric mixing of the site states that, in turn, results in the splitting of each pair of degenerate states in a doublet of closely spaced energy levels $\lvert \psi_{n\pm}\rangle$.

A textbook example of this is observed for the ammonia molecule in which the umbrella inversion of the pyramidal structure of the molecule generates a structure equivalent to the starting one. If a vibrational spectrum of the molecule is recorded in the gas phase, a clear splitting of energy levels, with the emergence of two Q-branches, can be observed. The observed transitions are associated with the excitations connecting, according to the selection rules, states of different symmetries of the ground and first excited vibrational states namely $\lvert \psi_{0+}\rangle$ to $\lvert \psi_{1-}\rangle$ and $\lvert \psi_{0-}\rangle$ to $\lvert \psi_{1+}\rangle$.

### What did I work on?

The aim of my research work has been the development of new theoretical tools to estimate tunneling splitting in systems subject to the phenomenon of nuclear quantum tunneling.

The starting point of the work has been the well-known isomorphic relation existing between the Born-Oppenheimer nuclear Hamiltonian and the diffusion operator. This similarity, already used for example in the derivation of the Diffusion Monte-Carlo method, is a powerful tool that allows the connection between the formal structure of two different fields of physics such as quantum mechanics, and the description of stochastic processes.

The starting point of our work was recognizing that, if the ground state of a system is known or can be properly modeled to produce a desired potential profile, the Hamiltonian operator can be rewritten, apart from an additive constant equal to the ground state energy, in a form completely isomorphic to the Fokker-Planck-Smoluchowski operator.

This observation opened an interesting parallelism between the two problems allowing us to show how the computation of tunneling splittings and localized states from the ground state of a quantum system is equivalent, in formal terms, to the problem of computing site populations and transition rates of a thermally activated stochastic process in the diffusive regime. This allowed us to adapt the so-called localization function theory, developed exactly to that purpose in the stochastic context, to the computation of tunneling splittings in simple model potentials.

The advantage of this treatment is that, due to its nature, it introduces the concept of asymptotic limit in the treatment of quantum tunneling. This is interesting both because the computed tunneling splitting estimate is generally increasingly accurate with the increasing of the barrier, a trend that is opposite to that observed for many numerical methods, but it is also attractive because it suggests that in the limit of high barriers, an analytical asymptotic limit can be recovered for the tunneling splittings. This, once again, is in perfect analogy with the stochastic case in which, in the asymptotic limit of high barriers, the kinetic constants associated with the activated process can be computed with Kramers’ theory.

The basic theory of how the localization function and its asymptotic limit can be adapted to the study of quantum tunneling has been described in two peer-reviewed papers an is also available on ArXiv as a pre-print:

- P.Pravatto, B.Fresch, G.J.Moro,
*Chem Phys*, 561, 111608, (**2022**) [Paper link] [ArXiv link] - P.Pravatto, B.Fresch, G.J.Moro,
*J. Chem. Phys*., 158, 144110, (**2023**) [Paper link] [ArXiv link]