Quantum Dynamical Problems Research Articles

A multilayer multiconfiguration time-dependent Hartree simulation of the reaction-coordinate spin-boson model employing an interaction picture

The multilayer multiconfiguration time-dependent Hartree method is applied in an interaction picture to simulate dynamics of the spin-boson model in the reaction-coordinate representation. The use of the interaction picture allows a more effective description of correlation effects, especially when the coupling strength between the reaction coordinate and the bath is very strong. Examples show that in most physical regimes the efficiency is improved significantly, in some cases up to several orders of magnitude. This opens up new avenues for studying quantum dynamical problems.

A MATRIX-FREE LEGENDRE SPECTRAL METHOD FOR INITIAL-BOUNDARY VALUE PROBLEMS

We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrodinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable.The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space.

An introduction to the problem of bridging quantum and classical dynamics

Simulating the exact quantum dynamics of realistic interacting systems is presently a task beyond reach but for the smallest of them, as the numerical cost for solving the time-dependent Schrodinger equation scales exponentially with the number of degrees of freedom. Mixed quantum-classical methods attempt to solve this problem by starting from a full quantum description of the system and subsequently partitioning the degrees of freedom in two subsets: the quantum subsystem and the bath. A classical limit is then taken for the bath while preserving, at least approximately, the quantum evolution of the subsystem. A key, as yet not fully resolved, theoretical question is how to do so by constructing a consistent description of the overall dynamics. An exhaustive review of this class of methods is beyond the scope of this paper and we shall limit ourselves to present, as an example, a specific approach, known as the LANDM-Map method. The method stems from an attempt at taking a rigorous limit for the classical degrees of freedom starting from a path integral formulation of the full quantum problem. The results that we discuss are not new, but our intent here is to present them as an introduction to the problem of mixed quantum classical dynamics. We shall also indicate a broad classification of the available approaches, their limitations, and some open questions in this field.

Basis Set Generation for Quantum Dynamics Simulations Using Simple Trajectory-Based Methods

Methods for solving the time-dependent Schrödinger equation generally employ either a global static basis set, which is fixed at the outset, or a dynamic basis set, which evolves according to classical-like or variational equations of motion; the former approach results in the well-known exponential scaling with system size, while the latter can suffer from challenging numerical problems, such as singular matrices, as well as violation of energy conservation. Here, we suggest a middle road: building a basis set using trajectories to place time-independent basis functions in the regions of phase space relevant to wave function propagation. This simple approach, which potentially circumvents many of the problems traditionally associated with global or dynamic basis sets, is successfully demonstrated for two challenging benchmark problems in quantum dynamics, namely, relaxation dynamics following photoexcitation in pyrazine, and the spin Boson model.

Feynman’s clock, a new variational principle, and parallel-in-time quantum dynamics

We introduce a discrete-time variational principle inspired by the quantum clock originally proposed by Feynman and use it to write down quantum evolution as a ground-state eigenvalue problem. The construction allows one to apply ground-state quantum many-body theory to quantum dynamics, extending the reach of many highly developed tools from this fertile research area. Moreover, this formalism naturally leads to an algorithm to parallelize quantum simulation over time. We draw an explicit connection between previously known time-dependent variational principles and the time-embedded variational principle presented. Sample calculations are presented, applying the idea to a hydrogen molecule and the spin degrees of freedom of a model inorganic compound, demonstrating the parallel speedup of our method as well as its flexibility in applying ground-state methodologies. Finally, we take advantage of the unique perspective of this variational principle to examine the error of basis approximations in quantum dynamics.

Propagation of excitation in long 1D chains: Transition from regular quantum dynamics to stochastic dynamics

The quantum dynamics problem for a 1D chain consisting of 2N + 1 sites (N ≫ 1) with the interaction of nearest neighbors and an impurity site at the middle differing in energy and in coupling constant from the sites of the remaining chain is solved analytically. The initial excitation of the impurity is accompanied by the propagation of excitation over the chain sites and with the emergence of Loschmidt echo (partial restoration of the impurity site population) in the recurrence cycles with a period proportional to N. The echo consists of the main (most intense) component modulated by damped oscillations. The intensity of oscillations increases with increasing cycle number and matrix element C of the interaction of the impurity site n = 0 with sites n = ±1 (0 < C ≤ 1; for the remaining neighboring sites, the matrix element is equal to unity). Mixing of the components of echo from neighboring cycles induces a transition from the regular to stochastic evolution. In the regular evolution region, the wave packet propagates over the chain at a nearly constant group velocity, embracing a number of sites varying periodically with time. In the stochastic regime, the excitation is distributed over a number of sites close to 2N, with the populations varying irregularly with time. The model explains qualitatively the experimental data on ballistic propagation of the vibrational energy in linear chains of CH2 fragments and predicts the possibility of a nondissipative energy transfer between reaction centers associated with such chains.

Compact MCTDH Wave Functions for High-Dimensional System-Bath Quantum Dynamics

We employ a simple multiconfiguration time-dependent Hartree (MCTDH) ansatz tailored to an effective-mode transformation of environmental variables that brings the bath into a linear chain form. In this form, important (primary) degrees of freedom can be easily identified and treated at a high correlation level, whereas secondary modes are left uncorrelated. The resulting approach scales linearly with the bath dimensions and allows us to easily access recurrence times much longer than usually possible, at a very small computational cost. Test calculations for model atom-surface problems show that the system dynamics is correctly reproduced in the relevant time window, and quantitative agreement is attained for energy relaxation and sticking, particularly in non-Markovian environments. These results pave the way for tackling realistic system-bath quantum dynamical problems on the picosecond scale.

Time‐dependent schrödinger equation with Markovian outgoing wave boundary conditions: Applications to quantum tunneling dynamics and photoionization

AbstractMarkovian outgoing wave boundary conditions are introduced as an approximate method to reduce the size of the computational grid for time integration of the time‐dependent Schrödinger equation. The ratio and polynomial methods developed as open boundary conditions are applied to the wave function at the boundaries of the computational grid. This computational method is used to study the wave packet dynamics for a metastable well, a double well, and strong‐field ionization of a model atom. Accurate results demonstrate that this method can significantly reduce the number of grid points required in a dynamical calculation for quantum dynamical problems. © 2012 Wiley Periodicals, Inc.

Dissipative tunneling in nanosystems

A quantum dynamical problem has been analytically solved for a two-level system where localized states L0 and R0 are strongly coupled with reservoirs of local oscillations {Ln} and {Rn}. It is additionally assumed that the spectra of reservoirs are equidistant and the coupling constants are the same. It has been shown that the evolution of states L0 and R0 in recurrence cycles depends on three independent factors, which characterize exchange with the two-level system, exchange of L0 with {Ln} (R0 with {Rn}) and the phonon-induced decay of {Ln} and {Rn}. In addition to coherent oscillations with the frequency of the two-level system, Δ, and dissipative tunneling with a rate Δ2/πC2 (where C is the matrix element of the coupling of L0 and R0 with Ln and Rn), a new regime appears where L-R transitions are induced by the partial recovery of the populations of L0 and R0 in each recurrence cycle due to synchronous transitions from reservoirs. These transitions induce repeating changes in the populations of the states of the two-level system (Loschmidt echo). The number and width of the echo components increase with the cycle number. Evolution becomes irregular because of the mixing of the contributions from pulses of the neighboring cycles, when the cycle number k exceeds the critical value kc = π2C2. Unlike the populations, their cycle-average values remain regular at k ≫ kc. When Δ ≪ πC2, the cycle-average populations oscillate with a frequency of ΔΩ/πC2 irrespective of mixing. The frequency of oscillations of the populations of the states {Ln} and {Rn} is approximately nΩ(Δ/2πC2)2, where Ω is the spacing between the neighboring levels of the reservoir and nΩ is the difference between the energies of the states L0 and Ln. The appearance of the mentioned low-frequency oscillations is due to the formation of collective states of the two-level system that are “dressed” by the reservoir. The predicted oscillations can be detected by femtosecond spectroscopy methods.

Generalized CC-TDSCF and LCSA: The system-energy representation

Typical (sub)system-bath quantum dynamical problems are often investigated by means of (approximate) reduced equations of motion. Wavepacket approaches to the dynamics of the whole system have gained momentum in recent years and there is hope that properly designed approximations to the wavefunction will allow one to correctly describe the subsystem evolution. The continuous-configuration time-dependent self-consistent field (CC-TDSCF) and local coherent-state approximation (LCSA) methods, for instance, use a simple Hartree product of bath single-particle-functions for each discrete variable representation (DVR) state introduced in the Hilbert space of the subsystem. Here we focus on the above two methods and replace the DVR states with the eigenstates of the subsystem Hamiltonian, i.e., we adopt an energy-local representation for the subsystem. We find that stable and semiquantitative results are obtained for a number of dissipative problems, at the same (small) computational cost of the original methods. Furthermore, we find that both methods give very similar results, thus suggesting that coherent-states are well suited to describe (local) bath states. As a whole, present results highlight the importance of the system basis-set in the selected-multiconfiguration expansion of the wavefunction. They suggest that accurate and yet computationally cheap methods may be simply obtained from CC-TDSCF/LCSA by letting the subsystem states be variationally optimized.

Full-dimensional quantum dynamics of vibrationally highly excited NHD2

We report on full-dimensional vibrational quantum dynamics of the highly excited ammonia isotopologue NHD(2) using a newly developed potential energy surface and the MCTDH program package. The calculations allow to realistically simulate an infrared laser induced stereomutation reaction at the pyramidal nitrogen atom in the femtosecond time domain. Our results allow for a thorough qualitative and quantitative understanding of infrared photoinduced stereomutation kinetics, the underlying quantum dynamics, and the reaction mechanisms. Comparison is made with a previous, reduced dimensionality study of the same reaction [R. Marquardt, M. Quack, I. Thanopulos, and D. Luckhaus, J. Chem. Phys. 118, 643 (2003)], and it is shown that slight variances of reduced spaces lead to significantly different kinetics. Because the quantum dynamics depends subtly on variances of reduced spaces, reduced dimensionality treatments are not reliable even for qualitative predictions of the stereomutation kinetics. The first direct comparison between the Multiconfigurational Time Dependent Hartree [M. H. Beck, A. Jäckle, G. A. Worth et al., Phys. Rep. 324, 1 (2000)] and Unimolecular Reactions Induced by Monochromatic Infrared Radiation [M. Quack and E. Sutcliffe, QCPE Bulletin 6, 98 (1986)] program packages on a specific, four dimensional quantum dynamical problem allows for their full validation in the present work.

Nonexponential decay in the quantum dynamics of nanosystems

The quantum dynamical problem is solved for a system coupled to an equidistant-spectrum bath with the energy difference Ω between the neighboring levels n and n + 1 and the coupling matrix elements Cn2 = C2(1 + Δ−2n2)−1 constraining the energy interval comprising the bath states interacting with the system. The evolution in the strong-coupling limit is determined by two parameters, Γ = πC2/Ω ≫ 1 and α = Γ/Δ. If α ≠ 0, then the decrease in the population in the initial cycle with a period of 2π/Ω is not exponential and the effective rate constant increases with time. The results qualitatively explain the appearance of nonexponential relaxation regimes for a dense-spectrum nanosystem and predict the possibility of the multiple recovery of the initial-state population.

Conjugate and cut loci of a two-sphere of revolution with application to optimal control

The objective of this article is to present a sharp result to determine when the cut locus for a class of metrics on a two-sphere of revolution is reduced to a single branch. This work is motivated by optimal control problems in space and quantum dynamics and gives global optimal results in orbital transfer and for Lindblad equations in quantum control.

The photodissociation of ozone: A quasi-classical approach to a quantum dynamics problem

It is extremely difficult to execute full quantum dynamics calculations on complex systems (more then three degrees of freedom) due to the exponential increase in computer resources required by these methods. Classical mechanics simulations do not suffer from this problem, but are unable to treat quantum mechanical phenomena such as non-adiabatic effects, which often play a vital role in photochemical processes. A method has been implemented for carrying out dynamical calculations using quasi-classical theory. The time dependent Schrödinger equation is solved using a swarm of trajectories treated under Newtonian laws and Tully's fewest switches trajectory surface hopping is applied to implement the surface switches. The method was applied to ozone, looking at the photodissociation that takes place after excitation into the Chappuis band of the absorption spectrum. While the goal is to treat larger systems, comparison can be made for ozone with numerically exact wavepacket calculations. The method proved successful at calculating quantities such as the rate of population transfer, but there were discrepancies in the details, especially when surface switching occurred from the lower state.

Continuous configuration time-dependent self-consistent field method for polyatomic quantum dynamical problems

A new continuous configuration time-dependent self-consistent field method has been developed to study polyatomic dynamical problems by using the discrete variable representation for the reaction system, and applied to a reaction system coupled to a bath. The method is very efficient because the equations involved are as simple as those in the traditional single configuration approach, and can account for the correlations between the reaction system and bath modes rather well.

The irreducible unitary representations of the extended Poincaré group in (1+1) dimensions

We prove that the extended Poincaré group in (1+1) dimensions P̄ is non-nilpotent solvable exponential, and therefore that it belongs to type I. We determine its first and second cohomology groups in order to work out a classification of the two-dimensional relativistic elementary systems. Moreover, all irreducible unitary representations of P̄ are constructed by the orbit method. The most physically interesting class of irreducible representations corresponds to the anomaly-free relativistic particle in (1+1) dimensions, which cannot be fully quantized. However, we show that the corresponding coadjoint orbit of P̄ determines a covariant maximal polynomial quantization by unbounded operators, which is enough to ensure that the associated quantum dynamical problem can be consistently solved, thus providing a physical interpretation for this particular class of representations.

Using quantum trajectories and adaptive grids to solve quantum dynamical problems

Techniques using quantum trajectories and dynamic adaptive grids offer possible solutions to quantum mechanics' hydrodynamical equations, facilitating the design of new computational methods to predict quantum system evolution.

Analytic dynamics of coupled two-level systems

We solve a problem of quantum dynamics of two coupled Schr\"odinger equations. Utilizing a matrix transformation analogous to the Darboux method we find analytic solutions valid in special cases, ``quasiexact solutions.'' The general conditions for matrix Darboux transformations are formulated. The method is applied to a special case of wave-packet dynamics.

Applications of the Karhunen-Loève Decomposition to Quantum Dynamical Problems

The Karhunen-Loève (KL)-decomposition is a standard method to analyze spatiotemporal patterns which are characteristic for non-linear chemical dynamics on surfaces. We apply the KL-decomposition to quantum dynamical problems. Using a variational principle it is shown how to arrive at the KL-expansion of an arbitrary complex valued function. In the case of an one-dimensional bound state motion, the KL-decomposition yields the eigenstates of the system in a straightforward manner. The time-resolved spectroscopy of an atom-molecule collision serves as another application. We demonstrate that the orthonormal decomposition into KL-modes provides an efficient tool to calculate long-time pump-probe signals.

Path Integral Quantization of the Aharonov–Bohm–Coulomb System in Momentum Space

The Coulomb system with a charge moving in the fields of Ahanorov and Bohm is quantized via path integral in momentum space. Due to the dynamics of the system in momentum space being in curve space, our result not only gives the Green function of this interesting system in momentum space but provides the second example to answer an open problem of quantum dynamics in curved spaces posed by DeWitt in 1957: We find that the physical Hamiltonian in curved spaces does not contain the Riemannian scalar curvature R.