# Electron-electron correlation

From C. Figueira de Morisson Faria and X. Liu, J. Mod. Opt. 58, 1076 (2011)

Comparison experiment and theory, RESI with few-cycle pulses

The quintessential example of electron-electron correlation is laser-induced nonsequential double and multiple ionization (NSDI,NSMI). "Nonsequential" means that the double ionization transition amplitude cannot be described as the product of one-electron ionization processes, i.e., that, physically, both electrons interact with each other. Our work on nonsequential double ionization addresses several aspects of this phenomenon, such as electron-electron dynamics, the rescattering mechanisms of electron-impact ionization (EI) and recollision with subsequent tunneling ionization (RESI), and their potential for attosecond imaging of atoms and molecules. This work has been developed analytically to a great extent employing the strong-field approximation and saddle-point methods.

My work as a PDRA focused on the direct pathways and the work with my group at UCL on RESI. RESI is important for molecular systems and the so-called “below-threshold regime”, in which the second electron may not be dislodged by electron-impact ionization. This pathway provides many challenges, both from the physical and methodological viewpoint. Physically, a myriad of shapes has been observed experimentally for correlated electron momentum distributions, which cannot be explained using simple arguments. This raises questions about the importance of quantum interference and the intermediate state of the electron. Methodologically, Methodologically, this provides an ideal testing ground for incorporating electron-electron correlation and excitation in the SFA

## Direct impact ionization and attosecond thermalization

with Dr Wilhelm Becker (MBI Berlin), Prof Maciej Lewenstein (ICFO Barcelona), Prof Anna Sanpera (Universidad Autonoma Barcelona), Prof Henning Schomerus (Lancaster University), Prof Xiaojun Liu (CAS Wuhan), Prof Paul Corkum (NRC Ottawa)

We have performed one of the most complete analytic treatments of direct ionization pathways in NSDI using the SFA [1-4]. This work, in collaboration with world leaders in the field, broke new ground in the understanding of the influence of different types of electron-electron interaction and correlated final continuum states on the shapes of the electron- momentum distributions, resulting in signatures which were subsequently verified in experiments and ab initio computations. We also proposed NSDI as a tool to measure the carrier-envelope phase (CEP) of a few-cycle pulse [5]. This scheme has been realized in a joint experiment by the MBI- Berlin, the Technical University Vienna and the MPQ. Subsequently, we extended the classical counterpart of our model to more than two active electrons, and predicted the shortest times ever for thermalization in a multielectron atom [7]. The resulting publication was one of the 2006 highlights in J. Phys. B.

[1] C. Figueira de Morisson Faria and M. Lewenstein, J. Phys. B 38(17), 3251-3271 (2005)

[3] C. Figueira de Morisson Faria, X. Liu, A. Sanpera and M. Lewenstein,, Phys. Rev. A 70(4), 043406 (2004)

[4] C. Figueira de Morisson Faria, H. Schomerus, X. Liu, and W. Becker, Phys. Rev. A 69, 043405 (2004)

[5] C. Figueira de Morisson Faria, X. Liu, H. Schomerus and W. Becker, Phys. Rev. A 69, 021402(R) (2004)

[6] X. Liu and C. Figueira de Morisson Faria, Phys. Rev. Lett. 92, 133006 (2004),

[7] X. Liu, C. Figueira de Morisson Faria, W. Becker and P.B. Corkum, J. Phys. B 39, L305 (2006)

## RESI & Imaging

with Dr Tahir Shaaran (PhD student/PDRA), TomNygren (BSc/summer student) and Zhaohe Liang (MSc student), Dr Brad Augstein (PhDstudent/PDRA) and Prof Henning Schomerus's group (Lancaster)

RESI electron momentum distributions for different intermediate bound states and types of electron- electron interaction. From PRA 81, 063413 (2010)

In order to set up the theoretical framework for RESI, we needed to start from scratch. We first wrote down the SFA transition amplitude for the specific laser-induced process described by RESI. Subsequently, this was solved using the steepest descent method and detailed analytic studies of RESI have been performed. This included the derivation of kinematic constraints [1], a unique interpretation of RESI as two time-ordered ATI-like processes [2], imaging applications for atoms [3] and molecules [4] focused on the intermediate state of the second electron, and RESI in few- cycle pulses [J16,J18]. We have also shown how to modify steepest-descent contours and incorporate causality in the latter case (collaboration with H. Schomerus, Lancaster) [J18]. Excellent agreement between our predictions and experiments led by Prof Kling, at the MPQ, Munich, in which RESI has been isolated and controlled for the first time using carrier-envelope-phase (CEP) stabilized few cycle pulses, has been observed. We were the first group worldwide to perform analytic studies of RESI momentum distributions. The above-mentioned papers are also the most comprehensive analytic studies to date of below-threshold NSDI.

[1] T. Shaaran and C. Figueira de Morisson Faria, J. Mod. Opt. 57(11), 984- 991 (2010)

[2] T. Shaaran, M. T. Nygren and C. Figueira de Morisson Faria, Phys. Rev. A 81(6), 063413 (2010)

[3] T. Shaaran, B.B. Augstein and C. Figueira de Morisson Faria, Phys. Rev. A 84(1), 013429 (2011)

[4] T. Shaaran, C. Figueira de Morisson Faria and H. Schomerus, Phys. Rev. A 85, 023423 (2012)

[5] C. Figueira de Morisson Faria, T. Shaaran and M. T. Nygren, Phys. Rev. A 86,053405 (2012)

## Quantum Interference

with Dr Andrew Maxwell (PhD student/PDRA)

Schematic representation of the types of quantum interference that may occur in RESI, due to electron exchange and different excitation pathways leading to the same momentum. From PRL 116, 143001 (2016)

Our theoretical studies however exhibited a fourfold symmetry in the electron-momentum distributions which was absent in the experimental findings. Due to the many possible excitation channels, this lack of symmetry could be related to interference effects. In our recent research, we have shown that, in the below-threshold regime, quantum-interference does survive focal averaging and integration over several degrees of freedom [2]. We have also isolated and identified several types of quantum interference, and shown that a myriad of shapes, including those observed experimentally, could be obtained by using appropriate coherent superpositions [1,2]. Future work will include the Coulomb potential in the electron continuum propagation using the novel methods developed by us. Our work called into question 20-year old assumptions, which viewed NSDI as a purely classical phenomenon. The presence of interference in NSDI has been recently confirmed experimentally.

[1] A. S. Maxwell and C. Figueira de Morisson Faria, Phys. Rev. A 92, 023421 (2015)

[2] A. S. Maxwell and C. Figueira de Morisson Faria, Phys. Rev. Lett. 116, 143001 (2016)