Frozen Gaussian approximation-based two-level methods for multi-frequency Schrödinger equation
Introduction
In this paper, we are interested in developing two-level methods for solving numerically the time-dependent Schrödinger equation (TDSE) modeling in particular, the interaction of intense electric fields with quantum particles in attosecond science [1]. In and under the Born–Oppenheimer and dipole approximations [2], the TDSE reads in length gauge [3] where is a space-dependent (nuclear) potential, and denotes the external electric field the quantum particle is subject to. It is well known that the interaction of short and intense electromagnetic fields with atoms or molecules leads to complex nonlinear phenomena among which, one of the most important is the generation of high frequency photons, see [3], [4], [5].
Let us start from a localized state where (resp. ) is the smallest eigenvalue (resp. associated eigenfunction) of the field-free atomic Schrödinger Hamiltonian . We consider a laser field of frequency and intensity , interacting with the atom (or the molecule), where stands for the unit direction in . Subject to such a field, the atom is first ionized, then the free electrons gain ponderomotive energy , where is the electron charge and is the nucleus mass. In a third stage, electrons are recombined with their parent ion and generate high order harmonics (HHG) by multi-photon ionization. This process is described for the hydrogen atom in the celebrated paper [6], where strong-field-approximation is used to derive an accurate 3-step HHG model, using Newton’s law coupled with a tunneling model. It is shown in [6] that after the laser–atom interaction, the time harmonic spectrum is constituted by a plateau, up to a cut-off frequency , such that , where is the cut-off frequency order, and is the ionization potential. Starting from a one-frequency problem (associated to the incoming pulse frequency), high time- and space-frequencies (wavenumbers or Fourier modes) are then generated. Appropriate numerical methods should tackle this multiscale problem, and deal simultaneously with quantum and semi-classical regimes. This paper is dedicated to the derivation of two-level methods for this multiscale problem. The principle is to rigorously couple usual real space methods for relatively low space-and-time frequency regime, with a geometric optics-type method, the frozen Gaussian approximation (FGA), for high space-and-time frequency regime. What makes this problem particularly difficult is that high frequencies are dynamically generated. Fortunately, we can benefit from the 3-step model which provides quite relevant and precise information on (i) when the multi-photon ionization occurs, as well as (ii) the recombination energy, which is related to the highest generated frequencies. This is a very relevant information which allows us to determine if and when, the FGA should be reinitialized. Between these times, which can accurately be numerically evaluated, we can solve the Schrödinger equation on two separate frequency levels: (i) one solving the Schrödinger equation in the quantum regime on a coarse grid, (ii) one solving the Schrödinger equation in the semi-classical regime, using a FGA, which provides an accurate and cheap (compared to a full quantum computation on very fine grid) computation of the TDSE solution on wide frequency range. We refer to [7], [8], [9], [10] for numerical methods for quantum and classical wave equations in the semi-classical regime.
The paper is organized as follows. Section 1.2 is dedicated to the presentation of the frozen Gaussian approximation (FGA) and the discretization for the TDSE in the semi-classical regime. Corkum’s three-step model is presented in Section 2. Section 3.2 (resp. 3.3) is devoted to the two-level method without (resp. with) dynamical generation of high frequency. Numerical experiences are presented in Section 4. We finally conclude in Section 5.
In this section, we describe the numerical evolution of the TDSE wavefunction, in the semi-classical regime using the FGA denoted by .
FGA formulation. We here recall the main steps to construct a FGA. This method was originally developed by Herman–Kluk (HK) [11] and was later mathematically analyzed in [12]. More recently, the HK-formalism was used and analyzed to derive fast numerical solvers in the semi-classical regime for different classes of partial differential equations: the Schrödinger equation [13], the classical wave equation [14], [15] and linear hyperbolic systems of conservation laws [16]. FGA was also used to accelerate the convergence of Schwarz Waveform Relaxation domain decomposition algorithms [17]. FGA allows for the computational evolution of high frequency wavefunction. More specifically, we start in -dimensions, from where . We rescale the TDSE as follows. For , we set , , and we get with and . FGA is computed at , that is with . FGA reads for with where the Hamiltonian reads with and the Hamiltonian flow satisfies The classical action function satisfies and the amplitude where and .
Discretization of FGA. The discretization of FGAs is extensively discussed in the literature [13], [16], [14], [15]. We denote by the elements of the index sets denoted by ,1 then the FGA reads as, at time and discretization nodes denoted , where (resp. ) denotes the approximation of in (3) (resp. , in (2)) at at time , and denote the elementary increment in and , and is the local truncation function with a radius [13]. In addition, for any and , Then and are updated using a fourth order Runge–Kutta (RK4) scheme, for solving at time : In 1-d, , , , , , , , , are respectively denoted by , , , , , , , , .
We recall that from [12], denoting by the exact solution to the TDSE, there exists a positive constant such that In the quantum regime, we will construct an approximate solution to (1), typically using a finite difference scheme. Denoting an approximation of on a coarse mesh at time , we have where is a discrete Laplace operator.
Section snippets
Three-step model
We recall the main features of Corkum’s three-step model [6] in one dimension, which will be useful for deriving the two-level method in the framework of field–particle interaction. Some numerical experiments will then be presented to illustrate the model. Although this model is well known by laser physicists, this summary may be useful for the non-specialists.
Two-level FGA-based method
This section is devoted to the derivation of two two-level methods for multi-frequency TDSE with or without generation of high frequency, from initial time 0 to final time . We consider the evolution of the wavefunction, where (i) we initially () separate the Cauchy data into the high spatial frequency contribution from the other spatial frequency contributions (low and moderate), and (ii) we independently make evolve in time, these two contributions, up to time . The low and moderate
Numerical simulations
Some numerical illustrations of the proposed two-level techniques are presented in this section.
Multilevel method for multi-high-frequency regime
The two-level method which was presented in this paper, can easily be extended to a multilevel method, by considering multiple FGAs. More specifically, we consider a decreasing and real finite sequence associated to increasing frequencies . As before the contribution of frequencies smaller than (which was denoted by in Section 3.2) is computed by solving (1) in the quantum regime and on a coarse mesh. By Fourier filtering, we construct a finite sequence of FGAs,
Acknowledgments
E. Lorin thanks NSERC (356075) for the financial support via the Discovery Grant program. X. Yang was partially supported by the National Science Foundation grants DMS-1418936 and DMS-1107291, and Hellman Family Foundation Faculty Fellowship, UC Santa Barbara.
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