Complex-valued Trajectories Research Articles

The imaginary part of the high-harmonic cutoff

High-harmonic generation—the emission of high-frequency radiation by the ionization and subsequent recombination of an atomic electron driven by a strong laser field—is widely understood using a quasiclassical trajectory formalism, derived from a saddle-point approximation, where each saddle corresponds to a complex-valued trajectory whose recombination contributes to the harmonic emission. However, the classification of these saddle points into individual quantum orbits remains a high-friction part of the formalism. Here we present a scheme to classify these trajectories, based on a natural identification of the (complex) time that corresponds to the harmonic cutoff. This identification also provides a natural complex value for the cutoff energy, whose imaginary part controls the strength of quantum-path interference between the quantum orbits that meet at the cutoff. Our construction gives an efficient method to evaluate the location and brightness of the cutoff for a wide class of driver waveforms by solving a single saddle-point equation. It also allows us to explore the intricate topologies of the Riemann surfaces formed by the quantum orbits induced by nontrivial waveforms.

Looking inside the tunnelling barrier III: spin polarisation in strong field ionisation from orbitals with high angular momentum

Strong field ionisation is known as optical tunnelling of an electron through the barrier created by the laser field and core potential. Time-dependent analytical R-matrix takes advantage of the semiclassical nature of electron dynamics in strong laser fields and naturally incorporates complex-valued trajectories in time propagation. The language of trajectories enables straightforward ‘kinematic’ interpretation of several new phenomena such as ‘inversion’ of the conventional propensity rules in photo-ionisation (Barth and Smirnova 2011 Phys. Rev. A 84 063415) and generation of spin-polarised electrons (Barth and Smirnova 2013 Phys. Rev. A 88 013401) during ionisation in strong laser fields. Both phenomena are due to the kinematics of the electron motion under the barrier and were predicted for short-range potentials, i.e. neglecting the electron–core interaction. However, the electron–core interaction can and does affect the kinematics of tunnelling defining the ionisation propensity rules (companion paper I) and spin-polarisation. The latter is the focus of this paper. We consider orbitals with arbitrary angular momentum and extend our results to positively charged ions.

Synthesizing quantum probability by a single chaotic complex-valued trajectory

The current trajectory interpretation of quantum mechanics is based on an ensemble viewpoint that the evolution of an ensemble of Bohmian trajectories guided by the same wavefunction Ψ converges asymptotically to the quantum probability |Ψ|2. Instead of the Bohm's ensemble-trajectory interpretation, the present paper gives a single-trajectory interpretation of quantum mechanics by showing that the distribution of a single chaotic complex-valued trajectory is enough to synthesize the quantum probability. A chaotic complex-valued trajectory manifests both space-filling (ergodic) and ensemble features. The space-filling feature endows a chaotic trajectory with an invariant statistical distribution, while the ensemble feature enables a complex-valued trajectory to envelop the motion of an ensemble of real trajectories. The comparison between complex-valued and real-valued Bohmian trajectories shows that without the participation of its imaginary part, a single real-valued trajectory loses the ensemble information contained in the wavefunction Ψ, and this explains the reason why we have to employ an ensemble of real-valued Bohmian trajectories to recover the quantum probability |Ψ|2. © 2015 Wiley Periodicals, Inc.

Complex time paths for semiclassical wave packet propagation with complex trajectories.

The use of complex-valued trajectories in semiclassical wave packet methods can lead to problems that prevent calculation of the wave function in certain regions of the configuration space. We investigate this so-called bald spot problem in the context of generalized Gaussian wave packet dynamics. The analysis shows that the bald spot phenomenon is essentially due to the complex nature of the initial conditions for the trajectories. It is, therefore, expected to be a general feature of several semiclassical methods that rely on trajectories with such initial conditions. A bald region is created when a trajectory, needed to calculate the wave function at a given time, reaches a singularity of the potential energy function in the complex plane at an earlier, real time. This corresponds to passage of a branch point singularity across the real axis of the complex time plane. The missing portions of the wave function can be obtained by deforming the time path for the integration of the equations of motion into the complex plane so that the singularity is circumvented. We present examples of bald spots, singularity times, and suitable complex time paths for one-dimensional barrier transmission in the Eckart and Gaussian systems. Although the bald regions for the Eckart system are often localized, they are found to be semi-infinite for the Gaussian system. For the case of deep tunneling, the bald regions for both systems may encompass the entire portion of space occupied by the transmitted wave packet. Thus, the use of complex time paths becomes essential for a treatment of barrier tunneling.

Barrier scattering with complex-valued quantum trajectories: Taxonomy and analysis of isochrones

To facilitate the search for isochrones when using complex-valued trajectory methods for quantum barrier scattering calculations, the structure and shape of isochrones in the complex plane were studied. Isochrone segments were categorized based on their distinguishing features, which are shared by each situation studied: High and low energy wave packets, scattering from both thick and thin Gaussian and Eckart barriers of varying height. The characteristic shape of the isochrone is a trifurcated system: Trajectories that transmit the barrier are launched from the lower branch (T), while the middle and upper branches form the segments for reflected trajectories (F and B). In addition, a model is presented for the curved section of the lower branch (from which transmitted trajectories are launched), and important features of the complex extension of the initial wave packet are identified.

Complex trajectories sans isochrones: Quantum barrier scattering with rectilinear constant velocity trajectories

One of the major obstacles in employing complex-valued trajectory methods for quantum barrier scattering calculations is the search for isochrones. In this study, complex-valued derivative propagation method trajectories in the arbitrary Lagrangian-Eulerian frame are employed to solve the complex Hamilton-Jacobi equation for quantum barrier scattering problems employing constant velocity trajectories moving along rectilinear paths whose initial points can be in the complex plane or even along the real axis. It is shown that this effectively removes the need for isochrones for barrier transmission problems. Model problems tested include the Eckart, Gaussian, and metastable quadratic+cubic potentials over a variety of wave packet energies. For comparison, the "exact" solution is computed from the time-dependent Schrodinger equation via pseudospectral methods.

Semiclassical Coherent State Path Integral around a Classically Realizable Trajectory

The structure of the semiclassical Feynman kernel in the coherent state representation is studied. The analysis is focused on classically realizable (i.e. real-valued) trajectories, whose contributions to the semiclassical kernel persist in the limit → 0. A smooth deformation of the real-valued guiding trajectory that starts from the center of the “initial” coherent state and obeys the Hamilton equation induces complex-valued trajectories that contribute to the semiclassical kernel “linearized” around the guiding trajectory. The linearized semiclassical kernel describes Heller’s Gaussian wavepacket dynamics (GWD). Two mechanisms that break the GWD approximation are studied in terms of the semiclassical coherent state path integral. The rate at which a caustic approaches the guiding trajectory is estimated. In addition, the deviation of the semiclassical kernel from a Gaussian shape is estimated, with the assumption that the influence of caustics is negligible. These results provide an estimation of departure times between classical and quantum evolutions. In particular, for classically integrable systems, the present argument elucidates that the departure time due to caustics is much larger than that due to the deviation from a Gaussian shape. Furthermore, a disagreement between the estimations of the departure times obtained from the Husimi representation and the Wigner representation is pointed out.

Quantum-classical Liouville description of multidimensional nonadiabatic molecular dynamics

The quantum-classical Liouville formulation gives a quantum-mechanical density-matrix description of the “quantum” particles of a problem (e.g., the electrons) and a classical phase-space-density description of the “classical” particles (e.g., the nuclei). In order to employ this formulation to describe multidimensional nonadiabatic processes in complex molecular systems, this work is concerned with an efficient Monte Carlo implementation of the quantum-classical Liouville equation. Although an exact stochastic realization of this equation is in principle available, in practice one has to cope with two major complications: (i) The representation of nonlocal phase-space operators in terms of local classical trajectories and (ii) the convergence of the Monte Carlo sampling which is cumbersome due to complex-valued trajectories with rapidly oscillating phases. Several strategies to cope with these problems are discussed, including various approximations to determine the momentum shift associated with a nonadiabatic transition, the on-the-fly generation of new trajectories at curve-crossings, and the localization of trajectories after irreversible electronic transitions. Employing several multidimensional model systems describing ultrafast photoinduced electron transfer and internal conversion, detailed numerical studies are performed which are compared to exact quantum calculations as well as to the “fewest-switches” surface-hopping method. In all cases under consideration, the Liouville calculations are in good agreement with the quantum reference. In particular, the approach is shown to provide a correct quantum-classical description of the electronic coherence.

Complex-valued trajectories in semiclassical potential scattering

The semiclassical theory of potential scattering leads naturally to an inclusion of complex-value classical paths which describe classically forbidden processes. However, only a few of the existing complex trajectories contribute. The authors discuss which of these forbidden paths should be retained and a detailed investigation of the complex contributions is given for the special class of repulsive inverse-power potentials. The oscillatory structure of the quantum-mechanical differential cross section is resolved as an interference pattern of classically allowed and forbidden trajectories. The region of validity of the complex-trajectory approximation is investigated. For potentials with a minimum the forbidden trajectories contribute to scattering angles larger than the rainbow angle. This is demonstrated for a superposition of inverse-power potentials.

Quasiclassical trajectory calculations of He–LiF (001) diffraction scattering

A quasiclassical trajectory method for describing elastic diffraction scattering of an atom from a crystal surface is presented and applied to the He–LiF (001) system. The results for a 20 meV, oriented 4He beam are compared with the recently reported quantum ones of Wolken and the semiclassical ones of Doll. The dependence of the specular scattering intensity on the initial beam orientation is calculated and compared with the quantum results of Wolken. It is concluded that the quasiclassical results are semiquantitatively accurate, only slightly less accurate than the uniform semiclassical results (without inclusion of complex-valued trajectories), but more accurate than the classical and primitive semiclassical results.

Comparison of quasiclassical trajectory and classical S-matrix treatments of collinear collisions of F and D2

The dynamics of collinear collisions of F with D2(n 1 a=0) have been studied using standard quasiclassical trajectory and classical S-matrix (semiclassical) techniques for collision energies ranging from 0.65 to 5.50 kcal mole−1 in the center-of-mass system. The results are compared to previous quantum mechanical calculations on the same potential energy surface. At most only three transitions (NR 0→0, R 0→3, and R 0→4) are ``classically allowed'' at any energy studied. Of these, the only transition for which the ``uniform semiclassical'' and quasiclassical results are significantly different is R 0→4. The predicted energy dependence of this transition is discussed in terms of recent experimental determinations of the relative rates of R 0→4 and R 0→3. The ``classically forbidden'' R 0→2 transition is predicted (using the complex-valued trajectory method) to have a probability which exhibits distinct oscillations as a function of collision energy.

Classical S-Matrix for Vibrational Excitation of H2 by Collision with He in Three Dimensions

Complex-valued classical trajectories have been computed by direct numerical integration of the equations of motion for three-dimensional collisions of He and H2, and from such trajectories classical S-matrix elements for transitions between specific rotational-vibrational states of H2 have been constructed. At the collision energies employed (∼1–2 eV) all vibrationally inelastic transitions are classically forbidden, thus the need for analytically continued, complex-valued trajectories. Comparison with the quantum mechanical calculations of Eastes and Secrest shows excellent agreement between the quantum and uniform semiclassical transition probabilities. Since, however, for three-dimensional collision systems one is seldom concerned with individual S-matrix elements, but rather sums over many of them, an important practical feature is developed which shows how one can combine the usual Monte Carlo classical treatment of some of the internal degrees of freedom with a semiclassical state-by-state description of others; i.e., one can ``quantize'' only those degrees of freedom that are highly quantumlike (e.g., vibration). This ``partial averaging'' approach also greatly simplifies the practical aspects of applying classical S-matrix theory to systems with several internal degrees of freedom.

Theory of semiclassical transition probabilities for inelastic and reactive collisions. IV. Classically-inaccessible transitions calculated by integration along complex-valued trajectories

We calculate semiclassical S-matrix elements S mn by numerical integration along complex-valued trajectories, by avoiding (simply) previously reported trajectory divergences. Now, one can numerically calculate even ❘ S mn ❘ ≈ 10 −11 for the Secrest-Johnson system (their lowest ❘ S mn ❘ 2), instead of only ❘ S mn ❘ 2 > 10 −3. Agreement is excellent.

Analytic Continuation of Classical Mechanics for Classically Forbidden Collision Processes

Classically forbidden processes are those that cannot take place via ordinary classical dynamics. Within the framework of classical S-matrix theory, however, classical mechanics can be analytically continued and classical-limit approximations obtained for these classically forbidden, or weak transition amplitudes (i.e., S-matrix elements). The most powerful and general way of analytically continuing classical mechanics for a complex dynamical system is to integrate the equations of motion themselves through the classically inaccessible regions of phase space. Success in calculating these analytically continued trajectories is reported in this work; with certain special features of these complex-valued trajectories recognized and taken account of, it is seen that they are essentially as easy to deal with numerically as ordinary (i.e., real) classical trajectories. Application to the linear A+BC collision (vibrational excitation) gives excellent results; transition probabilities as small as 10−11 (the smallest ones available for comparison) have been obtained, agreement with the exact quantum mechanical values being within a few percent.