Time‐Dependent Schrödinger Equation

Scrinzi, Armin

doi:10.1002/9783527677689.ch8

Cited by 10 publications

(13 citation statements)

References 45 publications

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“…-Single-Active-Electron (SAE) approaches are the simplest numerical approaches to the TDSE [11], though they are only ab initio for atomic hydrogen, and require model potentials to mimic larger systems. Nevertheless, they can be used effectively to tackle problems where electron correlation effects do not play a role, and their relative simplicity has allowed multiple user-ready software packages that offer this functionality in strong-field settings [12][13][14][15].…”

Section: Ab Initio and Numerical Methodsmentioning

confidence: 99%

Dialogue on analytical and ab initio methods in attoscience

et al. 2021

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The perceived dichotomy between analytical and ab initio approaches to theory in attosecond science is often seen as a source of tension and misconceptions. This Topical Review compiles the discussions held during a round-table panel at the ‘Quantum Battles in Attoscience’ cecam virtual workshop, to explore the sources of tension and attempt to dispel them. We survey the main theoretical tools of attoscience—covering both analytical and numerical methods—and we examine common misconceptions, including the relationship between ab initio approaches and the broader numerical methods, as well as the role of numerical methods in ‘analytical’ techniques. We also evaluate the relative advantages and disadvantages of analytical as well as numerical and ab initio methods, together with their role in scientific discovery, told through the case studies of two representative attosecond processes: non-sequential double ionisation and resonant high-harmonic generation. We present the discussion in the form of a dialogue between two hypothetical theoreticians, a numericist and an analytician, who introduce and challenge the broader opinions expressed in the attoscience community.

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Section: Ab Initio and Numerical Methodsmentioning

confidence: 99%

Dialogue on analytical and ab initio methods in attoscience

et al. 2021

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“…Strictly speaking, neglecting nuclear motion, an atomic or molecular system interacting with a strong electric field pulse is described by the time-dependent Schrödinger equation (TDSE) that captures both the evolution of the (electronic) wave function and the time evolution of the physical observables. The numerical solution of the TDSE offers a full quantum mechanical description of the laser-matter interaction processes; it has been used extensively to study HHG [213][214][215][216] and ATI [142,[217][218][219][220] in atomic and molecular systems. However, the full numerical integration of the TDSE in all the degrees of freedom of the system is computationally very demanding, when it is at all possible.…”

Section: Strong Field Approximationmentioning

confidence: 99%

Symphony on strong field approximation

et al. 2019

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In the last three decades, we have witnessed incredible advances in laser technology and in the understanding of nonlinear laser-matter interactions, crowned recently by the award of the Nobel prize to Gérard Mourou and Donna Strickland [1,2]. It is now routinely possible to produce few-cycle femtosecond (1 fs = 10 −15 s) laser pulses in the visible and mid-infrared regimes [3,4]. By focusing such ultrashort laser pulses on gas or solid targets, possibly in a presence of nano-structures [5], the targets are subjected to an ultra-intense electric field, with peak field strengths approaching the binding field inside the atoms themselves. Such fields permit the exploration of the interaction between strong electromagnetic coherent radiation and an atomic or molecular system with unprecedented spatial and temporal resolution [6]. On one hand, HHG nowadays can be used to generate attosecond pulses in the extreme ultraviolet [7,8], or even in the soft X-ray regime [9]. Such pulses themselves may be used for dynamical spectroscopy of matter; despite carrying modest pulse energies, they exhibit excellent coherence properties [10,11]. Combined with femtosecond pulses they can also be used to extract information about the laser pulse electric field itself [12]. HHG sources therefore offer an important alternative to other sources of XUV and X-ray radiation: synchrotrons, free electron lasers, X-ray lasers, and laser plasma sources. Moreover, HHG pulses can provide information about the structure of the target atom, molecule or solid [13][14][15]. Of course, to decode such information from a highly nonlinear HHG signal is a challenge, and that is why a possibly perfect, and possibly "as analytical as possible" theoretical understanding of these processes is in high demand. Here is the first instance where SFA offers its basic services.Since electronic motion is governed by the waveform of the laser electric field, an important quantity to describe the electric field shape is the so-called absolute phase or carrier-envelope phase (CEP). Control over the CEP is paramount for extracting information about electron dynamics, and to retrieve structural information from atoms and molecules [13,16,17]. For instance, in HHG an electron is liberated from an atom or molecule through ionization, which occurs close to the maximum of the electric field. Within the oscillating field, the electron can thus accelerate along oscillating trajectories, which may result in recollision with the parent ion, roughly when the laser field approaches a zero value. Control over the CEP is particularly important for HHG, when targets are driven by laser pulses comprising only one or two optical cycles. In that situation the CEP determines the relevant electron trajectories, i.e. the CEP determines whether emission results in a single or in multiple attosecond bursts of radiation [16,18].The influence of the CEP on electron emission is also extremely important. It was demonstrated for instance in an anti-correlation experiment, in which the number of AT...

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“…where it does not lead to confusion, and we drop the added complex conjugate for simplicity. The resulting two-dimensional integral, (5), is now in its minimal form, and the saddle-point equations for the amended action read…”

Section: A the Strong-field Approximationmentioning

confidence: 99%

“…This is an important change, as the steepestdescent method requires integration contours to follow the contour lines of Re(S): thus, the change in the ordering of Re(S(t s , t s )) for the long-and short-trajectory saddle points means that, after the Stokes transition, the short-trajectory saddle point (in this example) can no longer form part of a suitable integration contour, and it needs to be discarded from the summation. This means, however, that the SPA harmonic yield at the cutoff is a discontinuous function of Ω, coming from the discrete jump at the points where one of the trajectories is eliminated, and this discontinuity is clearly incompatible with the initial, obviously continuous, expressions for D(Ω) in (1) and (5). This apparent paradox is resolved by noting that the SPA is not valid when saddles are close together, as the quadratic approximation to the exponent fails.…”

Section: A the Strong-field Approximationmentioning

confidence: 99%

“…This emission can be modelled using a wide range of approaches, from classical heuristics [3,4] to intensive numerical computations [5], but the quantitative models that most closely follow the overall intuition are quasiclassical methods [6,7], known as the Strong-Field Approximation (SFA), where the emission amplitude is given by a path-integral sum over discrete emission events. These are known as quantum orbits [8][9][10], i.e., quasiclassical trajectories whose start and end times are complex [11,12].…”

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The imaginary part of the high-harmonic cutoff

Pisanty,

Ciappina,

Lewenstein

2020

Preprint

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Time‐Dependent Schrödinger Equation

Cited by 10 publications

References 45 publications

Dialogue on analytical and ab initio methods in attoscience

Dialogue on analytical and ab initio methods in attoscience

Symphony on strong field approximation

The imaginary part of the high-harmonic cutoff

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