# Publications

### 2023

Reducing decoherence is an essential step toward realizing general-purpose quantum computers beyond the present noisy intermediate-scale quantum (NISQ) computers. To this end, dynamical decoupling (DD) approaches in which external fields are applied to qubits are often adopted. We numerically study DD using a two-level model system (qubit) under the influence of Markovian decoherence by using quantum optimal control theory with slightly modified settings, in which the physical objective is to maximally create and maintain a specified superposition state in a specified control period. An optimal pulse is numerically designed while systematically varying the values of dephasing, population decay, pulse fluence, and control period as well as using two kinds of objective functionals. The decrease in purity due to the decoherence limits the ability to maintain a coherent superposition state; we refer to the state of maximal purity that can be maintained as the saturated value. The optimally shaped pulse minimizes the negative effect of decoherence by gradually populating and continuously replenishing the state of saturated purity.

*Encyclopedia of Chemical Physics and Physical Chemistry*. p. 187-244 Abstract

### 2021

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. .

A dual formalism for Lagrange multipliers is developed. The formalism is used to minimize an action function $S(q_2,q_1,T)$ without any dynamical input other than that $S$ is convex. All the key equations of analytical mechanics -- the Hamilton-Jacobi equation, the generating functions for canonical transformations, Hamilton's equations of motion and $S$ as the time integral of the Lagrangian -- emerge as simple consequences. It appears that to a large extent, analytical mechanics is simply a footnote to the most basic problem in the calculus of variations: that the shortest distance between two points is a straight line.

We present a new trajectory formulation of high harmonic generation that treats classically allowed and classically forbidden processes within a single dynamical framework. Complex trajectories orbit the nucleus, producing the stationary Coulomb ground state. When the field is turned on, these complex trajectories continue their motion in the field-dressed Coulomb potential and therefore tunnel ionization, unbound evolution and recollision are described within a single, seamless framework. The new formulation can bring mechanistic understanding to a broad range of strong field physics effects.

### 2019

The absorption spectrum of the vibronically allowed S1(1A2) ← S0(1A1) transition of formaldehyde is computed by combining multiplicative neural network (NN) potential surface fits, based on multireference electronic structure data, with the two-layer Gaussian-based multiconfiguration time-dependent Hartree (2L-GMCTDH) method. The NN potential surface fit avoids the local harmonic approximation for the evaluation of the potential energy matrix elements. Importantly, the NN surface can be constructed so as to be physically well-behaved outside the domain spanned by the ab initio data points. A comparison with experimental results shows spectroscopic accuracy of the converged surface and 2L-GMCTDH quantum dynamics.

We propose and demonstrate, numerically and experimentally, use of sparsity as prior information for extending the capabilities and performance of techniques and devices for laser pulse diagnostics. We apply the concept of sparsity in three different applications. First, we improve a photodiode-oscilloscope system’s resolution for measuring the intensity structure of laser pulses. Second, we demonstrate the intensity profile reconstruction of ultrashort laser pulses from intensity autocorrelation measurements. Finally, we use a sparse representation of pulses (amplitudes and phases) to retrieve measured pulses from incomplete spectrograms of cross-correlation frequency-resolved optical gating traces.

### 2018

*Advances in Chemical Physics, volume 163*. Whaley KB.(eds.). p. 273-323 (trueAdvances in Chemical Physics). Abstract

This chapter provides a simple pedagogical presentation of the discrete variable representation (DVR). It reviews the von Neumann (vN) basis of phase‐space Gaussians which include the Projected von Neumann Basis (PvN) and the Biorthogonal von Neumann Basis (PvB). The chapter includes a variety of interesting formal properties of nonorthogonal bases that are an extension of the DVR presentation and provides insight into the method. It presents an analysis of multidimensional considerations, including details of a highly efficient tensor formulation for performing pruned multidimensional DVR calculations for sparse but unstructured grids. The chapter contains illustrative applications. Pruned phase‐space methods have been successfully used for computing eigenenergies of (ro‐) vibrational systems. The chapter focuses on the applications in the context of solving the time‐dependent Schrodinger equation (TDSE). The efficiency of phase‐space versus coordinate‐space methods will certainly depend on the particular system studied and the strength of coupling between degrees of freedom.

Quantum computation places very stringent demands on gate fidelities, and experimental implementations require both the controls and the resultant dynamics to conform to hardware-specific constraints. Superconducting qubits present the additional requirement that pulses must have simple parameterizations, so they can be further calibrated in the experiment, to compensate for uncertainties in system parameters. Other quantum technologies, such as sensing, require extremely high fidelities. We present a novel, conceptually simple and easy-to-implement gradient-based optimal control technique named gradient optimization of analytic controls (GOAT), which satisfies all the above requirements, unlike previous approaches. To demonstrate GOAT's capabilities, with emphasis on flexibility and ease of subsequent calibration, we optimize fast coherence-limited pulses for two leading superconducting qubits architectures-flux-tunable transmons and fixed-frequency transmons with tunable couplers.

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.

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.

### 2017

Complex-valued semiclassical methods hold out the promise of treating classically allowed and classically forbidden processes on the same footing. In addition, they provide a natural way to describe optical excitation with complex fields within the trajectory framework. Despite their promise, these methods have until now been limited to short time propagation, due to the numerical difficulties introduced by the complexification. Using a new Final Value Representation of the Coherent State Propagator (FINCO), combined with an analysis of the complex classical phase space, we achieve accurate wavepacket propagation all the way to the revival time of a strongly anharmonic system.

### 2016

We present an efficient implementation of dynamically pruned quantum dynamics, both in coordinate space and in phase space. We combine the ideas behind the biorthogonal von Neumann basis (PvB) with the orthogonalized momentum-symmetrized Gaussians (Weylets) to create a newbasis, projected Weylets, that takes the best from both methods. We benchmark pruned time-dependent dynamics using phase-space-localized PvB, projectedWeylets, and coordinate-space-localized DVR bases, with real-world examples in up to six dimensions. For the examples studied, coordinate-space localization is the most important factor for efficient pruning and the pruned dynamics is much faster than the unpruned, exact dynamics. Phase-space localization is useful for more demanding dynamics where many basis functions are required. There, projected Weylets offer a more compact representation than pruned DVR bases. Published by AIP Publishing.

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.

### 2015

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.

### 2014

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.

*Advances In Chemical Physics*. Vol. 156. p. 1-34 Abstract

One of the striking parallels in the development of the method in quantum mechanics and signal processing is that the method never became mainstream in either community. A key reason is undoubtedly the problems encountered in converging the method, problems reported independently in both fields. This chapter presents a theory that optimizes the phase space localization to obtain the most compact possible representation. It shows applications to shaped femtosecond pulses, to the solution of time‐independent and time‐dependent quantum mechanical problems, and to audio and image compression. The chapter then discusses the application of the periodic von neumann basis with biorthogonal exchange (pvb) method to quantum mechanics. Finally, it discusses the application to the time‐independent Schrödinger equation (TISE) and the application to the time‐dependent Schrödinger equation.

We consider manipulation of the transmission coefficient for a quantum particle moving in one dimension where the shape of the potential is taken as the control. We show that the control landscape, the transmission as a functional of the potential, has no traps, i.e., any maxima correspond to full transmission.

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.

### 2012

We extend a recently developed quantum trajectory method [Y. Goldfarb, I. Degani, and D. J. Tannor, J. Chem. Phys. 125, 231103 (2006)]10.1063/1.2400851 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)]10.1063/1.459170. 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.

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)]10.1063/1.4739845. We apply our method to two benchmark models introduced by John Tully [J. Chem. Phys. 93, 1061 (1990)]10.1063/1. 459170, 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.

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.

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, 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.

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.

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)10.1103/PhysRevLett.106.170405]. 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.

We show that the second-order traps in the control landscape for a three-level Λ-system found in our previous work (Phys. Rev. Lett. 2011, 106, 120402) are not local maxima: there exist directions in the space of controls in which the objective grows. The growth of the objective is slow - at best 4th order for weak variations of the control. This implies that simple gradient methods would be problematic in the vicinity of second-order traps, where more sophisticated algorithms that exploit the higher order derivative information are necessary to climb up the control landscape efficiently. The theory is supported by a numerical investigation of the landscape in the vicinity of the ε(t)=0 second-order trap, performed using the GRAPE and BFGS algorithms.

In a recent paper [B. Poirier, Chem. Phys. 370, 4 (2010)], a formulation of quantum mechanics was presented for which the usual wavefunction and Schrdinger equation are replaced with an ensemble of real-valued trajectories satisfying a principle of least action. It was found that the resultant quantum trajectories are those of Bohmian mechanics. In this paper, analogous ideas are applied to Bohmian Mechanics with Complex Action (BOMCA). The standard BOMCA trajectories as previously defined are found not to satisfy an action principle. However, an alternate set of complex equations of motion is derived that does exhibit this desirable property, and an approximate numerical implementation is presented. Exact analytical results are also presented, for Gaussian wavepacket propagation under quadratic potentials.

We consider the optimal control problem of transferring population between states of a quantum system where the coupling proceeds only via intermediate states that are subject to decay. We pose the question whether it is generally possible to carry out this transfer. For a single intermediate decaying state, we recover the stimulated Raman adiabatic passage process, which we identify as the global optimum in the limit of infinite control time. We also present analytical solutions for the case of transfer that has to proceed via two consecutive intermediate decaying states. We show that in this case, for finite power the optimal control does not approach perfect state transfer even in the infinite time limit. We generalize our findings to characterize the topologies of paths that can be achieved by coherent control under the assumption of finite power. If two or more consecutive states in an N-level chain are subject to decay, complete population transfer with finite-power controls is not possible.

### 2011

One of the most popular methods for solving numerical optimal control problems is the Krotov method, adapted for quantum control by Tannor and coworkers. The Krotov method has the following three appealing properties: (1) monotonic increase of the objective with iteration number, (2) no requirement for a line search, leading to a significant savings over gradient (first-order) methods, and (3) macrosteps at each iteration, resulting in significantly faster growth of the objective at early iterations than in gradient methods where small steps are required. The principal drawback of the Krotov method is slow convergence at later iterations, which is particularly problematic when high fidelity is desired. We show here that, near convergence, the Krotov method degenerates to a first-order gradient method. We then present a variation on the Krotov method that has all the advantages of the original Krotov method but with significantly enhanced convergence (second-order or quasi-Newton) as the optimal solution is approached. We illustrate the method by controlling the three-dimensional dynamics of the valence electron in the Na atom.

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.

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.

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.

Path-integral derivations are presented for two recently developed complex trajectory techniques for the propagation of wave packets: complex WKB and Bohmian mechanics with complex action (BOMCA). The complex WKB technique is derived using a standard saddle-point approximation of the path integral, but taking into account the dependence of both the amplitude and the phase of the initial wave function, thus giving rise to the need for complex classical trajectories. The BOMCA technique is derived using a modification of the saddle-point technique, in which the path integral is approximated by expanding around a near-classical path, chosen so that up to some predetermined order there is no need to add any correction terms to the leading-order approximation. Both complex WKB and BOMCA techniques give the same leading-order approximation; in the complex WKB technique higher accuracy is achieved by adding correction terms, while in the BOMCA technique no additional terms are ever added: higher accuracy is achieved by changing the path along which the original approximation is computed. The path-integral derivation of the methods explains the need to incorporate contributions from more than one trajectory, as observed in previous numerical work. On the other hand, it emerges that the methods provide efficient schemes for computing the higher-order terms in the asymptotic evaluation of path integrals. The understanding we develop of the BOMCA technique suggests that there should exist near-classical trajectories that give exact quantum dynamical results when used in the computation of the path integral keeping just the leading-order term. We also apply our path-integral techniques to give a compact derivation of the semiclassical approximation to the coherent-state propagator.

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 Li _{2} 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), δ-pulse excitation and a single excited-state potential, but we discuss the implications of removing these assumptions.

### 2010

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.

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.

We analyze in detail the so-called pushing gate for trapped ions, introducing a time-dependent harmonic approximation for the external motion. We show how to extract the average fidelity for the gate from the resulting semiclassical simulations. We characterize and quantify precisely all types of errors coming from the quantum dynamics and reveal that slight nonlinearities in the ion-pushing force can have a dramatic effect on the adiabaticity of gate operation. By means of quantum optimal control techniques, we show how to suppress each of the resulting gate errors in order to reach a high fidelity compatible with scalable fault-tolerant quantum computing.

The non-Markovian master equation is applied to the calculation of reaction rates. Starting from the flux-side correlation function form, we treat both the thermal and real time evolution consistently within second order perturbation theory in the system-bath coupling. It is shown that the non-Markovian dynamics enter formally not only in the time propagation but also in the expressions for the initial system-bath correlations. We show that these initial correlations can have a significant effect on the reaction rate. The method presented, although approximate, is an effective way to calculate reaction rates for weakly coupled systems over a wide range of temperatures. As such it provides a complementary approach to the exact treatment based on the ML-MCTDH method of Craig et al. [1], which serves as reference in this work.

### 2009

We combine optimal control theory with the multi-configuration time-dependent Hartree-Fock method to control the dynamics of interacting particles. We use the resulting scheme to optimize state-to-state transitions in a one-dimensional (1D) model of helium and to entangle the external degrees of-freedom of two rubidium atoms in a ID optical lattice. Comparisons with optimization results based on the exact solution of the Schrödinger equation show that the scheme can be used to optimize even involved processes in systems consisting of interacting particles in a reliable and efficient way.

We recently introduced the von Neumann picture, a joint time-frequency representation, for describing ultrashort laser pulses. The method exploits a discrete phase-space lattice of nonorthogonal Gaussians to represent the pulses; an arbitrary pulse shape can be represented on this lattice in a one-to-one manner. Although the representation was originally defined for signals with an infinite continuous spectrum, it can be adapted to signals with discrete and finite spectrum with great computational savings, provided that discretization and truncation effects are handled with care. In this paper, we present three methods that avoid loss of accuracy due to these effects. The approach has immediate application to the representation and manipulation of femtosecond laser pulses produced by a liquid-crystal mask with a discrete and finite number of pixels.

It is well known that a finite level quantum system is controllable if and only if the Lie algebra of its generators has full rank. When the rank of the Lie algebra is not full, there is a rich mathematical and physical structure to the subalgebra that to date has been analyzed only in special cases. We show that uncontrollable systems can be classified into reducible and irreducible ones. The irreducible class is the more subtle and can be related to a notion of generalized entanglement. We give a general prescription for revealing irreducible uncontrollable systems: the fundamental representation of su (N), where N is the number of levels, must remain irreducible in the subalgebra of su (N). We illustrate the concepts with a variety of physical examples.

In this chapter we have reviewed selected applications of Local Control Theory (LCT) to the control of processes taking place in molecules. The scheme rests on the construction of electric fields taking the instantaneous response of a perturbed system into account. In the simplest version, this is done by calculating the rate of an observable and the fields are adjusted to either increase or decrease this rate, depending on which objective is chosen. From the given examples, it is revealed that the construction scheme is successful and easy to implement in a calculation. We have emphasized that the emerging fields can be interpreted in a straightforward way, often leading to a simple classical understanding of the underlying physics which determines the properties of the field and thus the outcome of the control process. These obvious advantages suggest that other applications will be presented in the future. For example, preliminary work has shown that LCT can be applied to problems in quantum computing [51]. Here, the present approach could open new perspectives. Also, it is interesting to address the question of how the scheme can be realized experimentally. We are optimistic that there are many more interesting problems that can be treated using the local control scheme, and we hope that the present compilation of work will stimulate research in that direction.

### 2008

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.

### 2007

In recent years, the use of joint time-frequency representations to characterize and interpret shaped femtosecond laser pulses has proven to be very useful. However, the number of points in a joint time-frequency representation is daunting as compared with those in either the frequency or time representation. In this article we introduce the use of the von Neumann representation, in which a femtosecond pulse is represented on a discrete lattice of evenly spaced time-frequency points using a non-orthogonal Gaussian basis. We show that the information content in the von Neumann representation using a lattice of √N points in time and √N points in frequency is exactly the same as in a frequency (or time) array of N points. Explicit formulas are given for the forward and reverse transformation between an iV-point frequency signal and the von Neumann representation. We provide numerical examples of the forward and reverse transformation between the two representations for a variety of different pulse shapes; in all cases the original pulse is reconstructed with excellent precision. The von Neumann representation has the interpretational advantages of the Husimi representation but requires a bare minimum number of points and is stably and conveniently inverted; moreover, it avoids the periodic boundary conditions of the Fourier representation.

We present a unified derivation of Bohmian methods that serves as a common starting point for the derivative propagation method (DPM), Bohmian mechanics with complex action (BOMCA), and the zero-velocity complex action method (ZEVCA). The unified derivation begins with the ansatz Ψ = e^{iSlh} where the action (S) is taken to be complex, and the quantum force is obtained by writing a hierarchy of equations of motion for the phase partial derivatives. We demonstrate how different choices of the trajectory velocity field yield different formulations such as DPM, BOMCA, and ZEVCA. The new derivation is used for two purposes. First, it serves as a common basis for comparing the role of the quantum force in the DPM and BOMCA formulations. Second, we use the new derivation to show that superposing the contributions of real, crossing trajectories yields a nodal pattern essentially identical to that of the exact quantum wavefunction. The latter result suggests a promising new approach to deal with the challenging problem of nodes in Bohmian mechanics.

We present a new semiclassical method that yields an approximation to the quantum mechanical wavefunction at a fixed, predetermined position. In the approach, a hierarchy of ODEs are solved along a trajectory with zero velocity. The new approximation is local, both literally and from a quantum mechanical point of view, in the sense that neighboring trajectories do not communicate with each other. The approach is readily extended to imaginary time propagation and is particularly useful for the calculation of quantities where only local information is required. We present two applications: the calculation of tunneling probabilities and the calculation of low energy eigenvalues. In both applications we obtain excellent agrement with the exact quantum mechanics, with a single trajectory propagation.

In recent years, intensive effort has gone into developing numerical tools for exact quantum mechanical calculations that are based on Bohmian mechanics. As part of this effort we have recently developed as alternative formulation of Bohmian mechanics in which the quantum action S is taken to be complex [Y. Goldfarb et al., J. Chem. Phys. 125, 231103 (2006)]. In the alternative formulation there is a significant reduction in the magnitude of the quantum force as compared with the conventional Bohmian formulation, at the price of propagating complex trajectories. In this paper we show that Bohmian mechanics with complex action is able to overcome the main computational limitation of conventional Bohmian methods - the propagation of wave functions once nodes set in. In the vicinity of nodes, the quantum force in conventional Bohmian formulations exhibits rapid oscillations that present a severe numerical challenge. We show that within complex Bohmian mechanics, multiple complex initial conditions can lead to the same real final position, allowing for the accurate description of nodes as a sum of the contribution from two or more crossing trajectories. The idea is illustrated on the reflection amplitude from a one-dimensional Eckart barrier. We believe that trajectory crossing, although in contradiction to the conventional Bohmian trajectory interpretation, provides an important new tool for dealing with the nodal problem in Bohmian methods.

### 2006

Finding multidimensional nondirect product discrete variable representations (DVRs) of Hamiltonian operators is one of the long standing challenges in computational quantum mechanics. The concept of a "DVR set" was introduced as a general framework for treating this problem by R. G. Littlejohn, M. Cargo, T. Carrington, Jr., K. A. Mitchell, and B. Poirier (J. Chem. Phys. 2002, 116, 8691). We present a general solution of the problem of calculating multidimensional DVR sets whose points are those of a known cubature formula. As an illustration, we calculate several new nondirect product cubature DVRs on the plane and on the sphere with up to 110 points. We also discuss simple and potentially very useful finite basis representations (FBRs), based on general (nonproduct) cubatures. Connections are drawn to a novel view on cubature presented by I. Degani, J. Schiff, and D. J. Tannor (Num. Math. 2005, 101, 479), in which commuting extensions of coordinate matrices play a central role. Our construction of DVR sets answers a problem left unresolved in the latter paper, namely, the problem of interpreting as function spaces the vector spaces on which commuting extensions act.

The central objective in any quantum computation is the creation of a desired unitary transformation; the mapping that this unitary transformation produces between the input and output states is identified with the computation. In [S.E. Sklarz, D.J. Tannor, arXiv:quant-ph/0404081 (submitted to PRA) (2004)] it was shown that local control theory can be used to calculate fields that will produce such a desired unitary transformation. In contrast with previous strategies for quantum computing based on optimal control theory, the local control scheme maintains the system within the computational subspace at intermediate times, thereby avoiding unwanted decay processes. In [S.E. Sklarz et al.], the structure of the Hilbert space had a direct sum structure with respect to the computational register and the mediating states. In this paper, we extend the formalism to the important case of a direct product Hilbert space. The final equations for the control algorithm for the two cases are remarkably similar in structure, despite the fact that the derivations are completely different and that in one case the dynamics is in a Hilbert space and in the other case the dynamics is in a Liouville space. As shown in [S.E. Sklarz et al.], the direct sum implementation leads to a computational mechanism based on virtual transitions, and can be viewed as an extension of the principles of Stimulated Raman Adiabatic Passage from state manipulation to evolution operator manipulation. The direct product implementation developed here leads to the intriguing concept of virtual entanglement - computation that exploits second-order transitions that pass through entangled states but that leaves the subsystems nearly separable at all intermediate times. Finally, we speculate on a connection between the algorithm developed here and the concept of decoherence free subspaces.

In recent years there has been a resurgence of interest in Bohmian mechanics as a numerical tool because of its local dynamics, which suggest the possibility of significant computational advantages for the simulation of large quantum systems. However, closer inspection of the Bohmian formulation reveals that the nonlocality of quantum mechanics has not disappeared-it has simply been swept under the rug into the quantum force. In this paper we present a new formulation of Bohmian mechanics in which the quantum action, S, is taken to be complex. This leads to a single equation for complex S, and ultimately complex x and p but there is a reward for this complexification-a significantly higher degree of localization. The quantum force in the new approach vanishes for Gaussian wave packet dynamics, and its effect on barrier tunneling processes is orders of magnitude lower than that of the classical force. In fact, the current method is shown to be a rigorous extension of generalized Gaussian wave packet dynamics to give exact quantum mechanics. We demonstrate tunneling probabilities that are in virtually perfect agreement with the exact quantum mechanics down to 10-7 calculated from strictly localized quantum trajectories that do not communicate with their neighbors. The new formulation may have significant implications for fundamental quantum mechanics, ranging from the interpretation of non-locality to measures of quantum complexity.

### 2005

Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are obtained by extending (adding rows and columns to) certain noncommuting matrices A _{1},...,A _{d} , related to the coordinate operators x _{1},...,x _{d} , in R ^{d} . We prove a correspondence between cubature formulae and "commuting extensions" of A _{1},...,A _{d} , satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices to be zero. Thus, the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions and briefly describe our attempts at computing them.

We extend a recently introduced mapping model, which explains the bunching phenomenon in an ion beam resonator for two ions (Geyer and Tannor 2004 J. Phys. B: At. Mol. Opt. Phys. 37 73), to describe the dynamics of the whole ion bunch. We calculate the time delay of the ions from a model of the bunch geometry and find that the bunch takes on a spherical form at the turning points in the electrostatic mirrors. From this condition we derive how the observed bunch length depends on the experimental parameters. We give an interpretation of the criteria for the existence of the bunch, which were derived from the experimental observations by Pedersen et al (2002 Phys. Rev. A 65 042704).

The Jaynes-Cummings model (JCM) is the simplest fully quantum model that describes the interaction between light and matter. We extend a previous analysis by Phoenix and Knight [Ann. Phys. 186, 381 (1988)] of the JCM by considering mixed states of both the light and matter. We present examples of qualitatively different entropic correlations. In particular, we explore the regime of entropy exchange between light and matter, i.e., where the rate of change of the two are anticorrelated. This behavior contrasts with the case of pure light-matter states in which the rate of change of the two entropies are positively correlated and in fact identical. We give an analytical derivation of the anticorrelation phenomenon and discuss the regime of its validity. Finally, we show a strong correlation between the region of the Bloch sphere characterized by entropy exchange and that characterized by minimal entanglement as measured by the negative eigenvalues of the partially transposed density matrix.

The calculation of chemical reaction rates in the condensed phase is a central preoccupation of theoretical chemistry. At low temperatures, quantum-mechanical effects can be significant and even dominant; yet quantum calculations of rate constants are extremely challenging, requiring theories and methods capable of describing quantum evolution in the presence of dissipation. In this paper we present a new approach based on the use of a non-Markovian quantum master equation (NM-QME). As opposed to other approximate quantum methods, the quantum dynamics of the system coordinate is treated exactly; hence there is no loss of accuracy at low temperatures. However, because of the perturbative nature of the NM-QME it breaks down for dimensionless frictions larger than about 0.1. We show that by augmenting the system coordinate with a collective mode of the bath, the regime of validity of the non-Markovian master equation can be extended significantly, up to dimensionless frictions of 0.5 over the entire temperature range. In the energy representation, the scaling goes as the number of levels in the relevant energy range to the third power. This scaling is not prohibitive even for chemical systems with many levels; hence we believe that the current method will find a useful place alongside the existing techniques for calculating quantum condensed-phase rate constants.

### 2004

The problem of optimal control of dissipative quantum dynamics was investigated. It was observed under most circumstances dissipation leads to an increase in entropy of the system. The hamiltonian portion was controlled in a way that the dissipation causes the maximal purity at the final time. The results show that the optimal strategy does not exploit a system coherences and was a 'greedy' strategy, in which the purity was increased maximally at each instant.

We present a two particle model to explain the mechanism that stabilizes a bunch of positively charged ions in an 'ion trap resonator' (Pedersen et al 2001 Phys. Rev. Lett. 87 055001). The model decomposes the motion of the two ions into two mappings for the free motion in different parts of the trap and one for a compressing momentum kick. The ions' interaction is modelled by a time delay, which then changes the balance between adjacent momentum kicks. Through these mappings we identify the microscopic process that is responsible for synchronization and give the conditions for that regime.

### 2002

The application of optimal control theory (OCT) is extended in a systematic way to systems governed by the nonlinear Schrodinger equation (NLSE), such as solitons in fiber optics and Bose-Einstein condensates (BEC's) in atomic physics. It begins with a general description of the Krotov iterative method. It describes first its application to quantum systems governed by the linear Schrodinger equation, and then shows how a generalized version of this method can be used to treat nonlinear problems.

Cooling of internal atomic and molecular states via optical pumping and laser cooling of the atomic velocity distribution, rely on spontaneous emission. The outstanding success of such examples, taken together with general arguments, has led to the widely held notion that radiative cooling requires spontaneous emission. We here show by specific examples and direct calculation, based primarily on breaking emission-absorption symmetry as in lasing without inversion, that cooling of internal states by external coherent control fields is possible. We also show that such coherent schemes allow us to practically reach absolute zero in a finite number of steps, in contrast to some statements of the third law of thermodynamics.

It has been proposed that the adiabatic loading of a Bose-Einstein condensate (BEC) into an optical lattice via the Mott-insulator transition can be used to initialize a quantum computer [D. Jaksch , Phys. Rev. Lett. 81, 3108 (1998)]. The loading of a BEC into the lattice without causing band excitation is readily achievable; however, unless one switches on an optical lattice very slowly, the optical lattice causes a phase to accumulate across the condensate. We show analytically and numerically that a cancellation of this effect is possible by adjusting the harmonic trap force constant of the magnetic trap appropriately, thereby facilitating quick loading of an optical lattice for quantum computing purposes. A simple analytical theory is developed for a nonstationary BEC in a harmonic trap.

The limits of controllability and the dynamics of population transfer in systems with degenerate target states embedded in a finite manifold of states were examined. A pair of shaped laser pulses acting upon a four-state system with intermediate degenerate states and a diamond-shaped array of transition moments were used to show the existence of a control scheme. Quantitative agreement was found between the numerical simulations of the optimized population transfer to one of a pair of degenerate states in the system and the analytically predicted limits of population transfer in different regimes. The use of the four-state model to selective population transfer in vibrationally mediated photodissociation experiments were also discussed.

We demonstrate that the synchronization effect observed [Pedersen et al., Phys. Rev. Lett. 87, 055001 (2001)], when a bunch of ions oscillates between two mirrors in an electrostatic ion beam trap, can be explained as a negative mass instability. We derive simple necessary conditions for the existence of a regime in which this dispersionless behavior occurs and demonstrate that in this regime, the ion trap can be used as a high resolution mass spectrometer.

### 2001

The vibrational polarization beats in femtosecond coherent anti-Stokes Raman spectroscopy (CARS) were studied. The experiment used a sequence of three femtosecond pulses with two variable time delays. The first two pulses act as a pump and dump sequence to create highly excited wave packet and the third pulse promotes the pump-dump wave packet to an excited electron state. It was shown that the betas arise when the final pump-dump-pump wave is above the excited state dissociation threshold of the molecule. The predictions of analytical theory were confirmed through the numerical evaluation of CARS signal through vibrational wave packet propagation.

Optimal control theory (OCT) is applied to laser cooling of molecules. The objective is to cool vibrations, using shaped pulses synchronized with the spontaneous emission. An instantaneous in time optimal approach is compared to solution based on OCT. In both cases the optimal mechanism is found to operate by a "vibrationally selective coherent population trapping". The trapping condition is that the instantaneous phase of the laser is locked to the phase of the transition dipole moment of v = 0 with the excited population. The molecules that reach v = 0 by spontaneous emission are then trapped, while the others are continually repumped. For vibrational cooling to v = 2 and rotational cooling, a different mechanism operates. The field completely changes the transient eigenstates of the Hamiltonian creating a superposition composed of many states. Finally this superposition is transformed by the field to the target energy eigenstate.

### 2000

Romero-Rochin and Oppenheim claimed that deviations from the equilibrium state are inevitable. In this study, an alternative derivation of their result was made, starting from the non-Markovian quantum Master equation (QME). It was demonstrated that the equilibrium state corresponding to the second order non-Markovian QME deviation from the canonical Boltmann form.

Calculation of chemical reaction dynamics is central to theoretical chemistry. The majority of calculations use either classical mechanics, which is computationally inexpensive but misses quantum effects, such as tunneling and interference, or quantum mechanics, which is computationally expensive and often conceptually opaque. An appealing middle ground is the use of semiclassical mechanics. Indeed, since the early 1970s there has been great interest in using semiclassical methods to calculate reaction probabilities. However, despite the elegance of classical S-matrix theory, numerical results on even the simplest reactive systems remained out of reach. Recently, with advances both in correlation function formulations of reactive scattering as well as in semiclassical methods, it has become possible for the first time to calculate reaction probabilities semiclassically. The correlation function methods are contrasted with recent flux-based methods, which, although providing somewhat more compact expressions for the cumulative reactive probability, are less compatible with semiclassical implementation.

### 1999

Optimal control theory (OCT) is applied to the problem of cooling molecular rotations. The optimal field gives rise to a striking behavior, in which there is no noticeable increase in the lowest rotational state population until the last percent or so of the control interval, at which point the population jumps to 1. Further analysis of the intermediate time interval reveals that cooling is taking place all along, in the sense that the purity of the system, as measured by Tr(ρ^{2}), is increasing monotonically in time. Once the system becomes almost completely pure, the external control field can transfer the amplitude to the lowest rotational state by a completely Hamiltonian manipulation. This mechanism is interesting because it suggests a possible way of accelerating cooling, by exploiting the cooling induced by spontaneous emission to all the ground electronic state levels, not just the lowest rotational level. However, it also raises a major paradox: it may. be shown that external control fields, no matter how complicated, cannot change the value of Tr(ρ^{2}); changing this quantity requires spontaneous emission which is inherently uncontrollable. What place is there then for control, let alone optimal control, using external fields? We discuss the resolution to this paradox with a detailed analysis of cooling in a two-level system.

We present an accurate, efficient, and flexible method for propagating spatially distributed density matrices in anharmonic potentials interacting with solvent and strong fields. The method is based on the Nakajima-Zwanzig projection operator formalism with a correlated reference state of the bath that takes memory effects and initial/final correlations to second order in the system-bath interaction into account. A key feature of the method proposed is a special parametrization of the bath spectral density leading to a set of coupled equations for primary and N auxiliary density matrices. These coupled master equations can be solved numerically by representing the density operator in eigenrepresentation or on a coordinate space grid, using the Fourier method to calculate the action of the kinetic and potential energy operators, and a combination of split operator and Cayley implicit method to compute the time evolution. The key advantages of the method are: (1) The system potential may consist of any number of electronic states, either bound or dissociative. (2) The cost for arbitrarily long solvent memories is equal to only N+1 times that of propagating a Markovian density matrix. (3) The method can treat explicitly time-dependent system Hamiltonians nonperturbatively, making the method applicable to strong field spectroscopy, photodissociation, and coherent control in a solvent surrounding. (4) The method is not restricted to special forms of system-bath interactions. Choosing as an illustrative example the asymmetric two-level system, we compare our numerical results with full path-integral results and we show the importance of initial correlations and the effects of strong fields onto the relaxation. Contrary to a Markovian theory, our method incorporates memory effects, correlations in the initial and final state, and effects of strong fields onto the relaxation; and is yet much more effective than path integral calculations. It is thus well-suited to study chemical systems interacting with femtosecond short laser pulses, where the conditions for a Markovian theory are often violated.

At the heart of the search for a quantum transition state theory is the partitioning of dynamic from thermal factors in the quantum rate expression. We explore the possibility of achieving an approximate partitioning by using the coherent state basis. The coherent states provide a tetradic representation of both the dynamic and thermal factors; the degree to which these factors partition is tied to the degree to which one or both of the tetradics is diagonal, and hence phase space localized. We find that for the dynamical factor the off-diagonal contributions are small, except for matrix elements between coherent states positioned anywhere along the stable branch of the classical separatrix. The thermal factor is nearly diagonal at high temperatures, but has significant off-diagonal contributions at low temperatures. As a result, at high temperatures the thermal factor cuts off long range correlation, leading to the classical limit. At low temperatures, there is a subtle interplay of the thermal and dynamical factors, with the long range off-diagonal portions of the thermal factor combining with the long range off-diagonal portions of the dynamical factor. This phase space picture sheds light on the physical assumptions underlying several commonly applied approximations for calculating thermal reaction rates. In particular, by elucidating the subtlety of the contributions to the low temperature rate it becomes clear why a simple, yet accurate, estimate of the rate in this regime is elusive, if not impossible.

Calculation of chemical reaction rates lies at the very core of theoretical chemistry. The essential dynamical quantity which determines the reaction rate is the energy-dependent cumulative reaction probability, N(E), whose Boltzmann average gives the thermal rate constant, k(T). Converged quantum mechanical calculations of N(E) remain a challenge even for three- and four-atom systems, and a longstanding goal of theoreticians has been to calculate N(E) accurately and efficiently using semiclassical methods. In this article we present a variety of methods for achieving this goal, by combining semiclassical initial value propagation methods with a reactant- product wavepacket correlation function approach to reactive scattering. The correlation function approach, originally developed for transitions between asymptotic internal states of reactants and products, is here reformulated using wavepackets in an arbitrary basis, so that N(E) can be calculated entirely from trajectory dynamics in the vicinity of the transition state. This is analogous to the approaches pioneered by Miller for the quantum calculation of N(E), and leads to a reduction in the number of trajectories and the propagation time. Numerical examples are presented for both one- dimensional test problems and for the collinear hydrogen exchange reaction.

We present new expressions for the cumulative reaction probability (N(E)), cast in terms of time-correlation functions of reactant and product wave packets. The derivation begins with a standard trace expression for the cumulative reaction probability, expressed in terms of the reactive scattering matrix elements in an asymptotic internal basis. By combining the property of invariance of the trace with a wave packet correlation function formulation of reactive scattering, we obtain an expression for N(E) in terms of the correlation matrices of incoming and outgoing wave packets which are arbitrary in the internal coordinates. This formulation, like other recent formulations of N(E), allows calculation of the quantum dynamics just in the interaction region of the potential, and removes the need for knowledge of the asymptotic eigenstates. However, unlike earlier formulations, the present formulation is fully compatible with both exact and approximate methods of wave packet propagation. We illustrate this by calculating N(E) for the collinear hydrogen exchange reaction, both quantally and semiclassically. These results indicate that the use of wave packet cross-correlation functions, as opposed to a coordinate basis and flux operators, regularizes the semiclassical calculation, suggesting that the semiclassical implementation described here may be applied fruitfully to systems with more degrees of freedom.

We study different mechanisms of adiabatic population transfer in N-level systems by means of optimal control algorithms. Using two-dimensional topographic maps of the yield of population transfer as a function of time delay and intensity of the pulses we analyze the global properties of the schemes and the conditions that lead to optimization. For three-level systems it is shown that the optimal pulse sequence is the well-known STIRAP (stimulated Raman adiabatic passage) scheme. For five-level systems a family of solutions ranging from the alternating STIRAP scheme to the new straddling STIRAP (S-STIRAP) scheme is obtained and the behavior of the solutions is compared. For four-level systems we obtain as optimal a S-STIRAP type sequence that behaves as an effective two-level system. For both odd and even numbers of N-level systems, the crucial role of the straddling pulse in reducing the population of all intermediate levels is demonstrated.

A numerical method is described for integration of the time-dependent Schrödinger equation within the presence of a Coulomb field. Because of the singularity at [Formula Presented] the wave packet has to be represented on a grid with a high density of points near the origin; at the same time, because of the long-range character of the Coulomb potential, the grid must extend to large values of r. The sampling points are chosen, following E. Fattal, R. Baer, and R. Kosloff [Phys. Rev. E 53, 1217 (1996)], using a classical phase space criterion. Following those workers, the unequally spaced grid points are mapped to an equally spaced grid, allowing use of fast Fourier transform propagation methods that scale as [Formula Presented] where N is the number of grid points. As a first test, eigenenergies for the hydrogen atom are extracted from short-time segments of the electronic wave-packet autocorrelation function; high accuracy is obtained by using the filter-diagonalization method. As a second test, the ionization rate of the hydrogen atom resulting from a half-cycle pulse is calculated. These results are in excellent agreement with earlier calculations.

In the last several years we have discovered a variety of remarkable pulse strategies for manipulating molecular motion by employing a design strategy we call "local optimization." Here we review the concept of local optimization and contrast it with optimal control theory. By way of background, we give highlights from two recent examples of the method: (1) a strategy for eliminating population transfer to one or many excited electronic states during strong field excitation, an effect we call 'optical paralysis'; (2) a generalization of the counterintuitive STIRAP (stimulated Raman adiabatic passage) pulse sequence from three levels to N levels, a strategy we call 'straddling STIRAP.' We then turn to a third example, which is the main subject of this paper: laser cooling of molecular internal degrees of freedom. We study a model that includes both coherent interaction with the radiation field and spontaneous emission; the latter is necessary to carry away the entropy from the molecule. An optimal control calculation was performed first and succeeded in producing vibrational cooling, but the resulting pulse sequence was difficult to interpret. Local optimization subsequently revealed the cooling mechanism: the instantaneous phase of the laser is locked to the phase of the transition dipole moment between the excited state amplitude and v = 0 of the ground state. Thus, the molecules that reach v = 0 by spontaneous emission become decoupled from the field, and no longer absorb, while molecules in all other states are continually repumped. The mechanism could be called "vibrationally selective coherent population trapping," in analogy to the corresponding mechanism of velocity selective coherent population trapping in atoms for sub-Doppler cooling of translations.

### 1998

In a recent series of papers we showed how a phase space distribution function approach could be combined with the method of reactive flux to obtain the rate of barrier crossing as a function of solvent friction in the intermediate to high friction limit. Those studies dealt with both the Markovian and non-Markovian cases, but were restricted to analytic results for parabolic barriers. Here we extend the approach to anharmonic barriers. The guiding approximation is to assume that the phase space distribution for each initial velocity, starting at the barrier top, remains Gaussian for all time, with Gaussian parameters given by time-dependent mean field equations. We expect this approximation to be accurate for short times, up to the "Ehrenfest" time; if this time exceeds the "plateau" time - the time for the distribution to reach its asymptotic partitioning - the quality of the results should be high. There are no adjustable parameters, although some reasonable criterion is needed for ending the integration of the mean field equations to prevent divergence. Numerical results for the linear cusp and the quartic potential show that the method is quite accurate for dimensionless frictions ≤1, although the accuracy degrades for higher frictions.

Optimal control theory (OCT) applied to driving molecular systems by means of femtosecond pulses is now a mature area, but many of its intricacies are as yet unexplored. As a numerical tool, the many variations on the basic method differ not only in computer efficiency but in the type of solutions obtained. In this paper we survey this diversity, focusing on the use of multiphoton IR laser excitation to control either (1) the state selectivity or (2) the photodissociation in a 1D Morse potential. We compare two distinct algorithms, the Krotov method and the gradient method. The former method generates large changes in the field at each iteration, while the latter does not. As a result, the Krotov method virtually always leads to pulses that are very different from the initial guess, while with the gradient method this is not always the case. We then analyze the effect of changing the final time, T, and find that it also can have a profound effect on the nature of the optimal solutions. Finally, we compare the solutions obtained using two different projectors to describe the bond-breaking process: a coordinate projector and a projector over scattering states. Again we observe that the optimal pulses and the dynamics they generate are markedly different in the two cases. This ambiguity in the definition of the optimal pulses may be viewed as a shortcoming of the approach, or alternatively it may be viewed as giving the method extra flexibility.

A correlation function formulation for the state-selected total reaction probability, N_{α}(E), is suggested. A wave packet, correlating with a specific set of internal reactant quantum numbers, α, is propagated forward in time until bifurcation is complete at which time the nonreactive portion of the amplitude is discarded. The autocorrelation function of the remaining amplitude is then computed and Fourier transformed to obtain a reactivity spectrum. Dividing by the corresponding spectrum of the original, unfiltered, wave packet normalizes the reactivity spectrum, yielding the total reaction probability from the internal state, α. The procedure requires negligible storage and just one time-energy Fourier transform for each initial reactant state, independent of the number of open channels of products. The method is illustrated numerically for the one-dimensional Eckart barrier, using both quantum-mechanical and semiclassical propagation methods. Summing over internal states of reactants gives the cumulative reaction probability, N(E). The relation to the trace formula [W. H. Miller, S. D. Schwartz, J. W. Tromp, J. Chem. Phys. 79, 4889 (1983)], N(E)= 1 2 (2π)^{2}tr(F̄δ(H-E)F̄δ(H-E)), is established, and a new variant of the trace formula is presented.

### 1997

Six major theories of quantum dissipative dynamics are compared: Redfield theory, the Gaussian phase space ansatz of Yan and Mukamel, the master equations of Agarwal, Caldeira-Leggett/ Oppenheim-Romero-Rochin, and Louisell/Lax, and the semigroup theory of Lindblad. The time evolving density operator from each theory is transformed into a Wigner phase space distribution, and classical-quantum correspondence is investigated via comparison with the phase space distribution of the classical Fokker-Planck (FP) equation. Although the comparison is for the specific case of Markovian dynamics of the damped harmonic oscillator with no pure dephasing, certain inferences can be drawn about general systems. The following are our major conclusions: (1) The harmonic oscillator master equation derived from Redfield theory, in the limit of a classical bath, is identical to the Agarwal master equation. (2) Following Agarwal, the Agarwal master equation can be transformed to phase space, and differs from the classical FP equation only by a zero point energy in the diffusion coefficient. This analytic solution supports Gaussian solutions with the following properties: the differential equations for the first moments in p and q and all but one of the second moments (q^{2} and pq but not p^{2}) are identical to the classical equations. Moreover, the distribution evolves to the thermal state of the bare quantum system at long times. (3) The Gaussian phase space ansatz of Yan and Mukamel (YM), applied to single surface oscillator dynamics, reduces to the analytical Gaussian solutions of the Agarwal phase space master equation. It follows that the YM ansatz is also a solution to the Redfield master equation. (4) The Agarwal/ Redfield master equation has a structure identical to that of the master equation of Caldeira-Leggett/ Oppenheim-Romero-Rochin, but the two are equivalent only in the high temperature limit. (5) The Louisell/Lax HO master equation differs from the Agarwal/Redfield form by making a rotating wave approximation (RWA), i.e., keeping terms of the form ââ†,â†â and neglecting terms of the form â†â†,ââ. When transformed into phase space, the neglect of these terms eliminates the modulation in time of the energy dissipation, modulation which is present in the classical solution. This neglect leads to a position-dependent frictional force which violates the principle of translational invariance. (6) The Agarwal/Redfield (AR) equations of motion are shown to violate the semigroup form of Lindblad required for complete positivity. Considering the triad of properties: complete positivity, translational invariance and asymptotic approach to thermal equilibrium, AR sacrifices the first while Lindblad's form must sacrifice either the second or the third. This implies that for certain initial states Redfield theory can violate simple positivity; however, for a wide range of initial Gaussians, the solution of the AR equations does maintain simple positivity, and thus for these states appears to be distinctly more physical than the solution of the semigroup equations.

In a recent paper we showed the equivalence, under certain well-characterized assumptions, of Redfield's equations for the density operator in the energy representation with the Gaussian phase space ansatz for the Wigner function of Yan and Mukamel. The equivalence shows that the solutions of Redfield's equations respect a striking degree of classical-quantum correspondence. Here we use this equivalence to derive analytic expressions for the density matrix of the harmonic oscillator in the energy representation without making the almost ubiquitous secular approximation. From the elements of the density matrix in the energy representation we derive analytic expressions for Γ^{n}_{1}(1/T^{n}_{1}) and Γ^{nm}_{2}(1/T^{nm}_{2}), i.e., population and phase relaxation rates for individual matrix elements in the energy representation. Our results show that Γ^{n}_{1}(t) = Γ_{1}(t) is independent of n; this is contrary to the widely held belief that Γ^{n}_{1} is proportional to n. We also derive the simple result that Γ^{nm}_{2}(t) = |n-m|Γ_{1}(t)/2, a generalization of the two-level system result Γ_{2} = Γ_{1}/2. We show that Γ_{1}(t) is the classical rate of energy relaxation, which has periodic modulations characteristic of the classical damped oscillator; averaged over a period Γ_{1}(t) is directly proportional to the classical friction, γ. An additional element of classical-quantum correspondence concerns the time rate of change of the phase of the off diagonal elements of the density matrix, ω^{nm}, a quantity which has received little attention previously. We find that ω^{nm} is time-dependent, and equal to |n -m|Ω(t), where Ω(t) is the rate of change of phase space angle in the classical damped harmonic oscillator. Finally, expressions for a collective Γ_{1}(t) and Γ_{2}(t) are derived, and shown to satisfy the relationship Γ_{2} = Γ_{1}/2. This familiar result, when applied to these collective rate constants, is seen to have a simple geometrical interpretation in phase space.

The possibility of using phase coherent optical pulse sequences to generate large-amplitude vibrational motion while locking excited state population has been demonstrated by us previously [Kosloff et al., Phys. Rev. Lett. 69 (1992) 2172]. While the approach is compatible in principle with strong fields, in practice strong fields excite many higher electronic states and may produce multiphoton ionization, processes that are frequently neglected in model calculations. Here we demonstrate that it is possible to lock any number of unwanted electronic excitations by a single condition on the instantaneous phase of the pulse sequence. We call this scheme "optical paralysis". Since only the phase of the field is determined by this condition, the amplitude of the field is still unspecified and can be chosen to achieve some desired objective, e.g. monotonic increase of the ground-state vibrational energy. The scheme is demonstrated by solving the time-dependent Schrödinger equation for nine coupled electronic states of Na_{2}, with energy deposited in the ground state and a single excited state while the population in all other excited states is kept locked.

Scattering matrix elements and symmetric transition-state resonances for the collinear H_{2} + H → H + H_{2} reaction are obtained using a time-dependent approach. The correlation function between reactant channel wavepackets and product channel wavepackets is used to determine the S-matrix elements. In a similar fashion, autocorrelation functions are used to extract the positions and widths of transition-state resonances. The time propagation of the wavepackets is performed by the improved semiclassical frozen Gaussian method of Herman and Kluk, which is an initial value, uniformly converged method. The agreement between the quantum and semiclassical results is far better than that obtained previously for this system by other semiclassical methods.

Theoretical progress in the cooling of internal degrees of freedom of molecules using shaped laser pulses is reported. The emphasis is on general concepts and universal constraints. Several alternative definitions of cooling are considered, including reduction of the von Neumann entropy, -tr{ρ̂logρ̂} and increase of the Renyi entropy, tr{ρ̂^{2}}. A distinction between intensive and extensive considerations is used to analyse the cooling process in open systems. It is shown that the Renyi entropy increase is consistent with an increase in the system phase space density and an increase in the absolute population in the ground state. The limitations on cooling processes imposed by Hamiltonian generated unitary transformations are analyzed. For a single mode system with a ground and excited electronic surfaces driven by an external field it is shown that it is impossible to increase the ground state population beyond its initial value. A numerical example based on optimal control theory demonstrates this result. For this model only intensive cooling is possible which can be classified as evaporative cooling. To overcome this constraint, a single bath degree of freedom is added to the model. This allows a heat pump mechanism in which entropy is pumped by the radiation from the primary degree of freedom to the bath mode, resulting in extensive cooling.

STIRAP (stimulated Raman adiabatic passage) has proven to be an efficient and robust technique for transferring population in a three-level system without populating the intermediate state. Here we show that the counterintuitive pulse sequence in STIRAP, in which the Stokes pulse precedes the pump, emerges automatically from a variant of optimal control theory we have previously called “local” optimization. Since local optimization is a well-defined, automated computational procedure, this opens the door to automated computation of generalized STIRAP schemes in arbitrarily complicated [Formula Presented]-level coupling situations. If the coupling is sequential, a simple qualitative extension of STIRAP emerges: the Stokes pulse precedes the pump as in the three-level system. But, in addition, spanning both the Stokes and pump pulses are pulses corresponding to the transitions between the [Formula Presented] intermediate states with intensities about an order of magnitude greater than those of the Stokes and pump pulses. This scheme is amazingly robust, leading to almost 100% population transfer with significantly less population transfer to the [Formula Presented] intermediate states than in previously proposed extensions of STIRAP.

### 1996

A procedure is described for calculating both the magnitude and phase of an autocorrelation function from the Wigner distribution. The method is fully compatible with both quantum and classical-based procedures for propagating the Wigner distribution. Numerical results are presented for an initial Gaussian in a considerably anharmonic Morse potential, showing that the method is accurate for several vibrational periods.

Scattering matrix elements for the collinear H_{2}(v)+H→H+H_{2}(v′) reaction are obtained using a recently developed time-dependent approach to scattering. The correlation function between reactant channel wave packet and product channel wave packet is used to determine the S-matrix elements. The time propagation of the reactant wave packet is performed by the semiclassical method of Herman and Kluk, which is an initial value, uniformly converged method. The agreement between the quantum and semiclassical results is far better than that obtained previously for this system by other semiclassical methods.