Beyond attoscience

My main area of research is strong-field and attosecond physics (theory). However, I have also been active in other areas such as mathematical physics (non-Hermitian Hamiltonian systems) and short- pulse propagation in nonlinear media. Below are a few examples.

Non-Hermitian Hamiltonians

In collaboration with Prof Andreas Fring (City, University of London)

Contrary to the widespread, but mistaken belief that, in order to possess a real and positive spectrum, a Hamiltonian system should be Hermitian, a Hamiltonian system should be Hermitian, hermiticity is only a sufficient, but not necessary condition for this to hold. Traditionally, non-Hermitian Hamiltonians in quantum physics have been associated with decay in open systems, and hence complex eigenvalues. However, since the late 1990s and early 2000s, non-Hermitian Hamiltonian systems with real eigenvalues have been under intense investigation. This requires the Hamiltonian be pseudo-Hermitian, which means that it is related to a Hermitian counterpart by a similarity transformation.

Most studies, however, are restricted to Hamiltonians which are not explicitly time dependent, and deal with eigenvalue problems. In our work, we follow a different direction and address explicitly time-dependent Hamiltonians. In particular, we provide time evolution operators, gauge transformations and a perturbative treatment for such Hamiltonians. As a direct application, we compute transition probabilities for a harmonic oscillator perturbed by a cubic non-Hermitian term, in an external linearly polarized laser field. We have also developed a systematic procedure for finding new equivalence pairs, and have employed it to obtain generalizations of, among others, the Swanson Hamiltonian, anharmonic and harmonic oscillators with complex perturbations. The fact that such systems may have real spectra brought a new perspective to the foundations of quantum mechanics and has provided a myriad of tools which can be used in many areas of knowledge. Further work involves the use of Moyal Brackets in this context J37]. These studies are part of an interdisciplinary effort between mathematical and optical physics.

[1] C. Figueira de Morisson Faria and A. Fring, J. Phys. A 39, 9269-9289 (2006)

[2] C. Figueira de Morisson Faria and A. Fring, Laser Phys. 17(4), 424-437 (2007)

[3] C. Figueira de Morisson Faria and A. Fring, Czech. J. Phys. 56(9), 899-908 (2006), doi:10.1007/s10582-006-0386-x.

Short pulse propagation

In collaboration with H. Steudel, M.G.A. Paris, A. Kamchatnov and O. Steuernagel

Second harmonic generation with strictly monochromatic light is a well-known problem of physics. However, nowadays, in several situations short-pulsed laser radiation is required. In this regime, even for fields which are relatively weak, second harmonic generation is not a completely understood process, since the propagation equations of the fundamental and harmonic waves are no longer analytically solvable. Physically, this implies an incomplete understanding of effects which start to play a role in this pulse-length region, such as the influence of the group velocity mismatch between fundamental and harmonic waves, the energy transfer between both waves and time-dependent effects which may influence the conversion efficiency and the pulse shapes of fundamental and harmonic radiation. We show that, under the assumption that both waves are amplitude-modulated, and that the slowly-varying-envelope approximation is valid, the propagation equations of fundamental and harmonic waves can be reduced to an intial value problem. Within this context, we derive a general analytical solution for an initial pulse of arbitrary shape, using the connection between second harmonic generation and the Liouville equation. The inverse problem, i.e., to determine the input pulse shape, given an asymptotic second harmonic pulse, is also solvable.

H. Steudel, C. Figueira de Morisson Faria, M.G.A. Paris, A.M. Kamchatnov and O. Steuernagel, Opt. Comm. 150, 363 (1998)
H. Steudel, C. Figueira de Morisson Faria, A. M. Kamchatnov and M.G.A. Paris, Phys. Lett. A 276, 267 (2000)