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Our Day return guarantee still applies. Advanced Book Search Browse by Subject. Make an Offer. 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.
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 Lambda 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 h over bar 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. 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 , delta-pulse excitation and a single excited-state potential, but we discuss the implications of removing these assumptions. 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.
Collecting P. T. Barnum
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. 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. The shaped pulses produced are characterized via Fourier-transform spectral interferometry.
Quantum control is demonstrated on the laser dye IR 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.
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 1D optical lattice. Comparisons with optimization results based on the exact solution of the Schrodinger 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.
Nonadiabatic dynamics: The SHARC approach
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. 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. 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 root N points in time and root 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 N-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. The new derivation is used for two purposes. 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. 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. 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. 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 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. 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. Littlejohn, M. Cargo, T. Carrington, Jr. Mitchell, and B. Poirier J. 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 I 10 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 L Degani, J. Schiff, and D. Tannor Num. 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.
Sklarz, D. 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. Sklarz et al. 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 Lionville space.
As shown in [S.
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. 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.
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.
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. 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. 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 non-commuting matrices A 1 , We prove a correspondence between cubature formulae and "commuting extensions" of A 1 , 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 J.
B: At. 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 Phys. A 65 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. We present examples of qualitatively different entropic correlations. In particular, we explore the regime of entropy exchange between light and matter, i. 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. We study the problem of optimal control of dissipative quantum dynamics. Although under most circumstances dissipation leads to an increase in entropy or a decrease in purity of the system, there is an important class of problems for which dissipation with external control can decrease the entropy or increase the purity of the system.
An important example is laser cooling. In such systems, there is an interplay of the Hamiltonian part of the dynamics, which is controllable, and the dissipative part of the dynamics, which is uncontrollable. The strategy is to control the Hamiltonian portion of the evolution in such a way that the dissipation causes the purity of the system to increase rather than decrease. The goal of this paper is to find the strategy that leads to maximal purity at the final time.
Under the assumption that Hamiltonian control is complete and arbitrarily fast, we provide a general framework by which to calculate optimal cooling strategies. These assumptions lead to a great simplification, in which the control problem can be reformulated in terms of the spectrum of eigenvalues of rho, rather than rho itself.
By combining this formulation with the Hamilton-Jacobi-Bellman theorem we are able to obtain an equation for the globally optimal cooling strategy in terms of the spectrum of the density matrix. For the three-level Lambda system, we provide a complete analytic solution for the optimal cooling strategy. For this system it is found that the optimal strategy does not exploit system coherences and is a "greedy" strategy, in which the purity is 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 Phys. 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.
We demonstrate that the synchronization effect observed [Pedersen et al. 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. 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. 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. Using a set of general methods developed by Krotov [A. Konnov and V. Krotov, Automation and Remote Control 60, ], we extend the capabilities of optimal control theory to the nonlinear Schrodinger equation NLSE. The paper begins with a general review of the Krotov approach to optimization. Although the linearized version of the method is sufficient for the linear Schrodinger equation, the full flexibility of the general method is required for treatment of the nonlinear Schrodinger equation.
Formal equations for the optimization of the NLSE, as well as a concrete algorithm are presented. A phase develops across the BEC when an optical lattice potential is turned on. The goal is to counter this effect and keep the phase flat by adjusting the trap strength. The problem is formulated in the language of optimal control theory OCT and solved using the above methodology. To our knowledge, this is the first rigorous application of OCT to the nonlinear Schrodinger equation, a capability that is bound to have numerous other applications. We examine, both numerically and analytically, the limits of controllability and the dynamics of population transfer in systems with degenerate target states embedded in a finite manifold of states.
The limits of the controllability of these systems are linked to the symmetry of the transition dipole moments connecting the states. Numerical simulations of the optimized population transfer to one of a pair of degenerate states in the system agree quantitatively with the analytically predicted limits of population transfer in different regimes in the joint material-field characteristics of the system.
The applicability of the four-state model to selective population transfer in vibrationally mediated photodissociation experiments is discussed. Knopp [J. Raman Spectrosc. The experiment uses a sequence of three resonant femtosecond pulses with two independently variable time delays.
The first two pulses act as a pump and dump sequence to create a predefined, highly excited wave packet on the ground electronic state, whose amplitude is optimized by selecting the proper pump-dump Raman frequency difference and varying the time delay. The third pulse promotes the pump-dump wave packet to an excited electronic state, resulting in subsequent coherent emission of light at the anti-Stokes frequency. This fully-resonant CARS signal, measured as a function of time delay between the second and third pulses, oscillates at a frequency characteristic of the pump-dump wave packet.
Due to anharmonicity, this frequency is a sensitive measure of the amount of vibrational excitation. Knopp observed that under certain conditions the signal exhibits pronounced beating between the pump-dump wave packet frequency and the frequency characteristic of the bottom of the ground state well.
In this paper we show that these beats arise only when the final pump-dump-pump wave packet is above the excited state dissociation threshold of the molecule. We derive analytical expressions showing that under these conditions, where the polarization is short-lived, there may be strong interferences between the contributions from molecules originally in different vibrational states of the thermal ensemble. In contrast, the CARS polarization in the below threshold case is long-lived, and these interferences cancel.
Numerical evaluation of the CARS signal through vibrational wave packet propagation confirms the predictions of the analytical theory and reproduces the distinctive beating pattern observed in the experiments. 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 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.
C Elsevier Science B. We consider the equilibrium state of a quantum system weakly coupled to a quantum bath within second order perturbation theory. We reproduce this result via a different derivation, starting from the non-Markovian, rather than the Markovian, quantum Master equation. Our derivation sheds new light on the mechanism that stabilizes the deviation from the canonical form and shows that it involves an interplay between a static distortion to the equilibrium state and dynamical system-bath correlations.
We show that this deviation is a necessary consequence of translational invariance and vanishes when the rotating-wave-approximation is applied. The deviation is also shown to vanish for a two-level system off-diagonally coupled to a heat bath or when the Lamb shifts are neglected. Two ways for numerically evaluating the second order deviations are described. Finally, the deviations from canonical equilibrium are given an illuminating geometrical interpretation in terms of the phase space Wigner distribution.
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 s 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. 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 rho 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.
Advances in Chemical Physics, The Role of Degenerate States in Chemistry
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 rho 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.
A numerical method is described for integration of the time-dependent Schrodinger equation within the presence of a Coulomb held.
The sampling points are chosen, following E. Fattal, R. Baer, and R. Kosloff [Phys. E 53, ], 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 N ln N, 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. 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 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. We present an accurate, efficient, and flexible method for propagating spatially distributed density matrices in anharmonic potentials interacting with solvent and strong fields.
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.
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 stud. 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 NO 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.
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. 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 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.
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. 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 Il, although the accuracy degrades for higher frictions. A correlation function formulation for the state-selected total reaction probability, N-alpha E , is Suggested.
A wave packet, correlating with a specific set of internal reactant quantum numbers, alpha, 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 fetal reaction probability from the internal state, alpha. 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. Miller, S. Schwartz, J. Tromp, J. 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 ID 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, white 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. 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 N-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 N-2 intermediate states with intensities about an order of magnitude greater than those of the Stokes and pump pulses.
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.
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 lone times. It follows that the YM ansatz is also a solution to the Redfield master equation. 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. We show that Gamma 1 t is the classical rate of energy relaxation, which has periodic modulations characteristic of the classical damped oscillator; averaged over a period Gamma t is directly proportional to the classical friction, gamma.
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, omega nm , a quantity which has received little attention previously. This familiar result, when applied to these collective rate constants, is se. 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. 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. The scheme is demonstrated by solving the time-dependent Schrodinger 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.