Control of concerted back-to-back double ionization dynamics in helium.

Double ionization (DI) is a fundamental process that despite its apparent simplicity provides rich opportunities for probing and controlling the electronic motion. Even for the simplest multielectron atom, helium, new DI mechanisms are still being found. To first order in the field strength, a strong external field doubly ionizes the electrons in helium such that they are ejected into the same direction (front-to-back motion). The ejection into opposite directions (back-to-back motion) cannot be described to first order, making it a challenging target for control. Here, we address this challenge and optimize the field with the objective of back-to-back double ionization using a (1 + 1)-dimensional model. The optimization is performed using four different control procedures: (1) short-time control, (2) derivative-free optimization of basis expansions of the field, (3) the Krotov method, and (4) control of the classical equations of motion. All four procedures lead to fields with dominant back-to-back motion. All the fields obtained exploit essentially the same two-step mechanism leading to back-to-back motion: first, the electrons are displaced by the field into the same direction. Second, after the field turns off, the nuclear attraction and the electron-electron repulsion combine to generate the final motion into opposite directions for each electron. By performing quasi-classical calculations, we confirm that this mechanism is essentially classical.

Communication: Systematic elimination of Stokes divergences emanating from complex phase space caustics.

Stokes phenomenon refers to the fact that an asymptotic expansion of complex functions can differ in different regions of the complex plane, and that beyond the so-called Stokes lines the expansion has an unphysical divergence. An important special case is when the Stokes lines emanate from phase space caustics of a complex trajectory manifold. In this case, symmetry determines that to second order there is a double coverage of the space, one portion of which is unphysical. Building on the seminal but laconic findings of Adachi, we show that the deviation from second order can be used to rigorously determine the Stokes lines and therefore the region of the space that should be removed. The method has applications to wavepacket reconstruction from complex valued classical trajectories. With a rigorous method in hand for removing unphysical divergences, we demonstrate excellent wavepacket reconstruction for the Morse, Quartic, Coulomb, and Eckart systems.

Multivalued classical mechanics arising from singularity loops in complex time.

Complex-valued classical trajectories in complex time encounter singular times at which the momentum diverges. A closed time contour around such a singular time may result in final values for q and p that differ from their initial values. In this work, we develop a calculus for determining the exponent and prefactor of the asymptotic time dependence of p from the singularities of the potential as the singularity time is approached. We identify this exponent with the number of singularity loops giving distinct solutions to Hamilton's equations of motion. The theory is illustrated for the Eckart, Coulomb, Morse, and quartic potentials. Collectively, these potentials illustrate a wide variety of situations: poles and essential singularities at finite and infinite coordinate values. We demonstrate quantitative agreement between analytical and numerical exponents and prefactors, as well as the connection between the exponent and the time circuit count. This work provides the theoretical underpinnings for the choice of time contours described in the studies of Doll et al. [J. Chem. Phys. 58(4), 1343-1351 (1973)] and Petersen and Kay [J. Chem. Phys. 141(5), 054114 (2014)]. It also has implications for wavepacket reconstruction from complex classical trajectories when multiple branches of trajectories are involved.

Quantum Dynamics in Phase Space using Projected von Neumann Bases.

We describe the mathematical underpinnings of the biorthogonal von Neumann method for quantum mechanical simulations (PvB). In particular, we present a detailed discussion of the important issue of nonorthogonal projection onto subspaces of biorthogonal bases, and how this differs from orthogonal projection. We present various representations of the Schrödinger equation in the reduced basis and discuss their relative merits. We conclude with illustrative examples and a discussion of the outlook and challenges ahead for the PvB representation.

Excited-state wavepacket and potential reconstruction by coherent anti-Stokes Raman scattering.

Among the major challenges in the chemical sciences is controlling chemical reactions and deciphering their mechanisms. Since much of chemistry occurs in excited electronic states, in the last three decades scientists have employed a wide variety of experimental techniques and theoretical methods to recover excited-state potential energy surfaces and the wavepackets that evolve on them. These methods have been partially successful but generally do not provide a complete reconstruction of either the excited state wavepacket or potential. We have recently proposed a methodology for reconstructing excited-state molecular wavepackets and the corresponding potential energy surface [Avisar and Tannor, Phys. Rev. Lett., 2011, 106, 170405]. In our approach, the wavepacket is represented as a superposition of the set of vibrational eigenfunctions of the molecular ground-state Hamiltonian. We assume that the multidimensional ground-state potential surface is known, and therefore these vibrational eigenfunctions are known as well. The time-dependent coefficients of the basis functions are obtained by experimental measurement of the resonant coherent anti-Stokes Raman scattering (CARS) signal. Our reconstruction strategy has several significant advantages: (1) the methodology requires no a priori knowledge of any excited-state potential. (2) It applies to dissociative as well as to bound excited-state potentials. (3) It is general for polyatomics. (4) The excited-state potential surface is reconstructed simultaneously with the wavepacket. Apart from making a general contribution to the field of excited-state spectroscopy, our method provides the information on the excited-state wavepacket and potential necessary to design laser pulse sequences to control photochemical reactions.

The von Neumann basis in non-Cartesian coordinates: application to floppy triatomic molecules.

We extend the periodic von Neumann basis to non-Cartesian coordinates. The bound states of two isomerizing triatomic molecules, LiCN/LiNC and HCN/HNC, are calculated using the vibrational Hamiltonian in Jacobi coordinates. The phase space localization of the basis functions leads to a flexible and accurate representation of the Hamiltonian. This results in significant savings compared to a basis localized just in coordinate space. The favorable scaling of the method with dimensionality makes it promising for applications to larger systems.

Communication: overcoming the root search problem in complex quantum trajectory calculations.

Three new developments are presented regarding the semiclassical coherent state propagator. First, we present a conceptually different derivation of Huber and Heller's method for identifying complex root trajectories and their equations of motion [D. Huber and E. J. Heller, J. Chem. Phys. 87, 5302 (1987)]. Our method proceeds directly from the time-dependent Schrödinger equation and therefore allows various generalizations of the formalism. Second, we obtain an analytic expression for the semiclassical coherent state propagator. We show that the prefactor can be expressed in a form that requires solving significantly fewer equations of motion than in alternative expressions. Third, the semiclassical coherent state propagator is used to formulate a final value representation of the time-dependent wavefunction that avoids the root search, eliminates problems with caustics and automatically includes interference. We present numerical results for the 1D Morse oscillator showing that the method may become an attractive alternative to existing semiclassical approaches.

Non-adiabatic molecular dynamics with complex quantum trajectories. I. The diabatic representation.

We extend a recently developed quantum trajectory method [Y. Goldfarb, I. Degani, and D. J. Tannor, J. Chem. Phys. 125, 231103 (2006)] to treat non-adiabatic transitions. Each trajectory evolves on a single surface according to Newton's laws with complex positions and momenta. The transfer of amplitude between surfaces stems naturally from the equations of motion, without the need for surface hopping. In this paper we derive the equations of motion and show results in the diabatic representation, which is rarely used in trajectory methods for calculating non-adiabatic dynamics. We apply our method to the first two benchmark models introduced by Tully [J. Chem. Phys. 93, 1061 (1990)]. Besides giving the probability branching ratios between the surfaces, the method also allows the reconstruction of the time-dependent wavepacket. Our results are in quantitative agreement with converged quantum mechanical calculations.

Non-adiabatic molecular dynamics with complex quantum trajectories. II. The adiabatic representation.

We present a complex quantum trajectory method for treating non-adiabatic dynamics. Each trajectory evolves classically on a single electronic surface but with complex position and momentum. The equations of motion are derived directly from the time-dependent Schrödinger equation, and the population exchange arises naturally from amplitude-transfer terms. In this paper the equations of motion are derived in the adiabatic representation to complement our work in the diabatic representation [N. Zamstein and D. J. Tannor, J. Chem. Phys. 137, 22A517 (2012)]. We apply our method to two benchmark models introduced by John Tully [J. Chem. Phys. 93, 1061 (1990)], and get very good agreement with converged quantum-mechanical calculations. Specifically, we show that decoherence (spatial separation of wavepackets on different surfaces) is already contained in the equations of motion and does not require ad hoc augmentation.

Communication: Phase space wavelets for solving Coulomb problems.

Recently we introduced a phase space approach for solving the time-independent Schrödinger equation using a periodic von Neumann basis with bi-orthogonal exchange (pvb) [A. Shimshovitz and D. J. Tannor, Phys. Rev. Lett. 109, 070402 (2012)]. Here we extend the approach to allow a wavelet scaling of the phase space Gaussians. The new basis set, which we call the wavelet pvb basis, is simple to implement and provides an appealing alternative to other wavelet approaches. For the 1D Coulomb problems tested in this paper, the method reduces the size of the basis relative to the Fourier grid method by a factor of 13-60. The savings in basis set size is predicted to grow steeply as the dimensionality increases.

Phase-space approach to solving the time-independent Schrödinger equation.

We propose a method for solving the time-independent Schrödinger equation based on the von Neumann (vN) lattice of phase space Gaussians. By incorporating periodic boundary conditions into the vN lattice [F. Dimler et al., New J. Phys. 11, 105052 (2009)], we solve a longstanding problem of convergence of the vN method. This opens the door to tailoring quantum calculations to the underlying classical phase space structure while retaining the accuracy of the Fourier grid basis. The method has the potential to provide enormous numerical savings as the dimensionality increases. In the classical limit, the method reaches the remarkable efficiency of one basis function per one eigenstate. We illustrate the method for a challenging two-dimensional potential where the Fourier grid method breaks down.

Communication: phase space approach to laser-driven electronic wavepacket propagation.

We propose a phase space method to propagate a quantum wavepacket driven by a strong external field. The method employs the periodic von Neumann basis with biorthogonal exchange recently introduced for the calculation of the energy eigenstates of time-independent quantum systems [A. Shimshovitz and D. J. Tannor, Phys. Rev. Lett. (in press) [e-print arXiv:1201.2299v1]]. While the individual elements in this basis set are time-independent, a small subset is chosen in a time-dependent manner to adapt to the evolution of the wavepacket in phase space. We demonstrate the accuracy and efficiency of the present propagation method by calculating the electronic wavepacket in a one-dimensional soft-core atom interacting with a superposition of an intense, few-cycle, near-infrared laser pulse and an attosecond extreme-ultraviolet laser pulse.

Multi-dimensional wavepacket and potential reconstruction by resonant coherent anti-Stokes Raman scattering: application to H2O and HOD.

We have recently proposed a methodology for reconstructing excited-state (ExS) molecular wavepackets, and the corresponding potential energy surface, from three-pulse resonant coherent anti-Stokes Raman scattering and knowledge of the ground-state potential [Avisar and Tannor, Phys. Rev. Lett. 106, 170405 (2011)]. The methodology is general for polyatomics and applies to any form of ExS potential--bound or dissociative. In our previous work we demonstrated the method on diatomics. Here, we demonstrate the method on the triatomics H(2)O and HOD, reconstructing the ExS wavepacket and potential in the two bond-stretching coordinates.

Complete reconstruction of the wave function of a reacting molecule by four-wave mixing spectroscopy.

Probing the real time dynamics of a reacting molecule remains one of the central challenges in chemistry. Here we show how the time-dependent wave function of an excited-state reacting molecule can be completely reconstructed from resonant coherent anti-Stokes Raman spectroscopy. The method assumes knowledge of the ground potential but not of any excited potential. The excited-state potential can in turn be constructed from the wave function. The formulation is general for polyatomics and applies to bound as well as dissociative excited potentials. We demonstrate the method on the Li(2) molecule.

Are there traps in quantum control landscapes?

There has been great interest in recent years in quantum control landscapes. Given an objective J that depends on a control field ε the dynamical landscape is defined by the properties of the Hessian δ²J/δε² at the critical points δJ/δε=0. We show that contrary to recent claims in the literature the dynamical control landscape can exhibit trapping behavior due to the existence of special critical points and illustrate this finding with an example of a 3-level Λ system. This observation can have profound implications for both theoretical and experimental quantum control studies.

Adaptive coherent control using the von Neumann basis.

Recently we introduced the von Neumann representation as a joint time-frequency description for femtosecond laser pulses. Here we show that the von Neumann basis can be implemented into an evolutionary algorithm for adaptive optimization in coherent control. We perform simulations that demonstrate the efficiency compared to other parametrizations in the frequency domain. We also illustrate pulse-shape simplification by basis-function reduction. Essential structures using the von Neumann basis are retained without losing control performance significantly. In an optical demonstration experiment we show the practicality by producing double pulses with a given time separation. Adaptive control in time-frequency space will be especially valuable for quantum systems requiring specific transition frequencies at definite times.

Wavepacket and potential reconstruction by four-wave mixing spectroscopy: preliminary application to polyatomic molecules.

We have recently shown how the excited-state wavepacket of a polyatomic molecule can be completely reconstructed from resonant coherent anti-Stokes Raman spectroscopy [Avisar and Tannor, Phys. Rev. Lett., 2011, 106, 170405]. The method assumes knowledge of the ground-state potential but not of any excited-state potential, however the latter can be computed once the excited-state wavepacket is known. The formulation applies to dissociative as well as bound excited potentials. We demonstrate the method on the Li2 molecule with its bound first excited-state as well as with a model dissociative excited state potential. Preliminary results are shown for a model two-dimensional molecular system. The calculations assume constant transition dipole moment (Condon approximation), delta-pulse excitation and a single excited-state potential, but we discuss the implications of removing these assumptions.

Molecular quantum control landscapes in von Neumann time-frequency phase space.

Recently we introduced the von Neumann representation as a joint time-frequency description for femtosecond laser pulses and suggested its use as a basis for pulse shaping experiments. Here we use the von Neumann basis to represent multidimensional molecular control landscapes, providing insight into the molecular dynamics. We present three kinds of time-frequency phase space scanning procedures based on the von Neumann formalism: variation of intensity, time-frequency phase space position, and/or the relative phase of single subpulses. The shaped pulses produced are characterized via Fourier-transform spectral interferometry. Quantum control is demonstrated on the laser dye IR140 elucidating a time-frequency pump-dump mechanism.

Trajectory based non-markovian dissipative tunneling.

The influence of a dissipative environment on scattering of a particle by a barrier is investigated by using the recently introduced bohmian mechanics with complex action [J. Chem. Phys. 125, 231103 (2006)]. An extension of this complex trajectory based formalism to include the interaction of the tunneling particle with an environment of harmonic oscillators with a continuous spectral density and at a certain finite temperature allows us to calculate transmission probabilities beyond the weak system bath coupling regime. The results display an increasing tunneling probability for energies below the barrier and a decreased transmission above the barrier due to the coupling. Furthermore, we demonstrate that solutions of a markovian master equation fail to do so in general.

Complex trajectory method in time-dependent WKB.

We present a significant improvement to a complex time-dependent WKB (CWKB) formulation developed by Boiron and Lombardi [J. Chem. Phys. 108, 3431 (1998)] in which the time-dependent WKB equations are solved along classical trajectories that propagate in complex space. Boiron and Lombardi showed that the method gives very good agreement with the exact quantum mechanical result as long as the wavefunction does not exhibit interference effects such as oscillations and nodes. In this paper, we show that this limitation can be overcome by superposing the contributions of crossing trajectories. Secondly, we demonstrate that the approximation improves when incorporating higher order terms in the expansion. Thirdly, equations of motion for caustics and Stokes lines are implemented to help overcome Stokes discontinuities. These improvements could make the CWKB formulation a competitive alternative to current time-dependent semiclassical methods.