Publications

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.

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.

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 timeindependent and timedependent 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 timeindependent Schrödinger equation (TISE) and the application to the timedependent Schrödinger equation.

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.

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 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 ₂O 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.

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.

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 Li₂ molecule.

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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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 Γⁿ₁(1/Tⁿ₁) and Γ^nm₂(1/T^nm₂), i.e., population and phase relaxation rates for individual matrix elements in the energy representation. Our results show that Γⁿ₁(t) = Γ₁(t) is independent of n; this is contrary to the widely held belief that Γⁿ₁ is proportional to n. We also derive the simple result that Γ^nm₂(t) = |n-m|Γ₁(t)/2, a generalization of the two-level system result Γ₂ = Γ₁/2. We show that Γ₁(t) is the classical rate of energy relaxation, which has periodic modulations characteristic of the classical damped oscillator; averaged over a period Γ₁(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 Γ₁(t) and Γ₂(t) are derived, and shown to satisfy the relationship Γ₂ = Γ₁/2. This familiar result, when applied to these collective rate constants, is seen to have a simple geometrical interpretation in phase space.

Scattering matrix elements and symmetric transition-state resonances for the collinear H₂ + H → H + H₂ 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{ρ̂²}. 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.