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Novel Strong-Field Approaches

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We have developed novel strong-field approaches or brought approaches to other areas to strong-field physics. Ideally, these approaches should be orbit based, computationally inexpensive, and include the external driving fields and the residual binding
potentials on equal footing. They should also allow a clear space-time picture of the phenomena we
intend to model.
Specifically, we have employed the following theoretical methods

Bohmian trajectories

with Jie Wu (PhD student), Brad Augstein (PDRA), Angel S Sanz (long-term visitor; Institute de Fisica Fundamental, Madrid)


We were the first group worldwide to model high-order harmonic generation using Bohmian trajectories. The main questions addressed by us were:

(i) what trajectories lead to harmonic spectra with a well-defined plateau, with harmonics of comparable intensities, followed by a cutoff?; (ii) is there a relationship between a Bohmian trajectory and a classical trajectory?


We found that the innermost Bohmian trajectory, already contains the plateau and the cutoff. Once the Bohmian trajectories leave the core, these patterns are lost (Wu et al, Phys. Rev. A 88, 023415 (2013)). Time-frequency analysis has however revealed that a Bohmian trajectory corresponds to a whole ensemble of classical trajectories returning to the core, and are thus very different entities. Other Bohmian trajectories in the core region are necessary to reproduce low- and mid-plateau harmonics quantitatively (Wu et al, Phys. Rev.
A 88, 063416 (2013)
The studies also show that nonlocality plays a key role in influencing harmonic spectra.

Initial value representations

with Jie Wu (PhD student), Dr Brad Augstein (PDRA), Dr Carlos Zagoya (PDRA) and in collaboration with Prof Dmitry Shalashilin’s group at the University of Leeds

Initial value representations (IVRs) are widely used in quantum chemistry and semiclassical theory, but not in attoscience. IVRS are highly scalable and can be extended to systems with many degrees of freedom without hitting an exponential wall as they use trajectory-based grids.


We have applied two Initial Value Representations to strong-field electron dynamics: The Herman Kluk propagator and the Coupled Coherent States (CCS). Both are trajectory-guided methods that are, in principle, suitable for systems with many degrees of freedom. The main challenge encountered was that in strong-field physics the electronic wave packet is initially bound. This is in striking contrast to the initial conditions in Quantum Chemistry, where such methods are widespread. For that reason, it was necessary to perform a detailed analysis of ionization dynamics in phase space (Zagoya et al, New J. Phys. 16, 103040 (2014)) and introduce a reprojection method in the CCS to avoid degradation (Symonds et al, Phys. Rev. A 91, 023427
). Both the CCS and the HK propagator were successfully applied in the modelling of high-order harmonic generation. This was particularly challenging due to the highly coherent nature of this phenomenon and it occurring close to the core. We are pioneers in the modelling of HHG using the CCS.

The Coulomb Quantum Orbit Strong-Field Approximation (CQSFA)

with Xuanyang Lai (PDRA),
Andrew Maxwell (PhD student/PDRA), Ahmed Al-Jawahiry (MSc student) and Toni Das (PhD student); in
collaboration with Prof Henning Schomerus’s group (Lancaster University)




We developed a semi-analytic strong-field approach which accounts for the residual binding potential in the continuum: The Coulomb Quantum Orbit Strong-Field Approximation (CQSFA) (Lai et al, Phys. Rev. A 92, 043407 (2015)). This has placed us among a handful of groups across the globe who have achieved this, as it is a highly non-trivial task. The key issue is to associate trajectories with quantum mechanical transition amplitudes, in the spirit of Feynman's path integral methods, and yet  retain quantum-mechanical features such as tunneling and quantum interference.


Our method exhibits a series of advantages in comparison to other Coulomb corrected strong-field approaches. First, it has no restriction upon the scattering angle. Second, it considers the residual binding potential and the external laser field on equal footing. Third, it requires only a small number of contributed trajectories, while other Coulomb-corrected methods typically require 10  - 10   trajectories to yield converged results. For that reason, the CQSFA leads to very clear interference patterns. This makes it the ideal method to study time resolved  photoelectron holography, which we have done subsequently. The CQSFA is one of the most advanced Coulomb-distorted methods to date.

We have developed versions of the CQSFA with real and complex trajectories in the continuum. The latter require dealing with branch cuts, which are loosely associated with rescattering (Maxwell et al, Phys. Rev. A 98, 063423 (2018)). The latter work was done in collaboration with Prof Sergey Popruzhenko. 

Uniform approximation in strong-field physics

with Dr Wilhelm Becker (MBI Berlin) and  Prof Henning Schomerus

(Lancaster University)

In collaboration with H. Schomerus and W. Becker, I have employed a uniform saddle-point approximation for the first time in strong-field laser physics, which treats pairs of saddles correctly and whose only applicability requirement is that the saddles occur in pairs (C. Figueira de Morisson Faria, H. Schomerus, and W. Becker, Phys. Rev. A 66, 043413 (2002)).
Since then, this uniform approximation has become a standard approach in the modelling of strong-field phenomena. This early work has also paved the way for subsequent lines of research for my group and my collaborators'.

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