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Journal of Chemical Theory and Computation

Compacting the Time Evolution of the Forced Morse Oscillator Using Dynamical Symmetries Derived by an Algebraic Wei-Norman Approach

James Russell Hamilton, F. Remacle, Raphael David Levine

Published Apr 29, 2025• 4 citations
Publisher

Abstract

High Resolution Image Download MS PowerPoint Slide A practical approach is put forward for a compact representation of the time evolving density matrix of the forced Morse oscillator. This approach uses the factorized product form of the unitary time evolution operator, à la Wei-Norman. This product form casts the time evolution operator in the basis of operators that form a closed Lie algebra. The further requirement that the Hamiltonian of the system be closed within this Lie algebra is satisfied by restricting the dynamics to its sudden limit. One is thereby able to propagate in time both pure and mixed quantum states. As an example, for a thermal initial state, the time-evolved density matrix of maximum entropy is derived, and it is compacted to be described by only three explicit constraints: one time-dependent constraint, which is a dynamical symmetry, and two constants of the motion, with corresponding time-independent coefficients. This representation is a significant reduction from O ( j 2 ) constraints down to just three, where j is the number of bound states of the Morse oscillator.

DOI
10.1021/acs.jctc.5c00148

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High Resolution Image Download MS PowerPoint Slide A practical approach is put forward for a compact representation of the time evolving density matrix of the forced Morse oscillator. This approach uses the factorized product form of the unitary time evolution operator, à la Wei-Norman. This product form casts the time evolution operator in the basis of operators that form a closed Lie algebra. The further requirement that the Hamiltonian of the system be closed within this Lie algebra is satisfied by restricting the dynamics to its sudden limit. One is thereby able to propagate in time both pure and mixed quantum states. As an example, for a thermal initial state, the time-evolved density matrix of maximum entropy is derived, and it is compacted to be described by only three explicit constraints: one time-dependent constraint, which is a dynamical symmetry, and two constants of the motion, with corresponding time-independent coefficients. This representation is a significant reduction from O ( j 2 ) constraints down to just three, where j is the number of bound states of the Morse oscillator.
Optical excitation in the dipole approximation and other probes couple two quantum states of an unaddressed finite quantum mechanical discrete system. Thereby the interaction of the system with the probe is bilinear in the coherence between the two states and in the time-dependent strength of the probe. The total Hamiltonian is a sum of such bilinear terms and of terms linear in the populations. The terms in the Hamiltonian form a basis for a Lie algebra that can be represented as a direct product of individual two state systems each using the population and the coherence between two states. Dynamical symmetries can be used to advantage to describe the progress of such systems in time. They also offer a compact and efficient representation for a density matrix of maximal entropy that evolves in time. Using the factorization approach of Wei and Norman, (J. Wei and E. Norman, Lie Algebraic Solution of Linear Differential Equations, J. Math. Phys. 4, 575 (1963)), we construct a unitary quantum mechanical evolution operator that is a factored contribution of individual two state systems. Thereby one can propagate, to all orders in perturbation theory, both the wave function and the density matrix with special reference to dynamical symmetries.
Many important quantum systems are characterized by Hamiltonians expressible as a linear combination of time-independent generators of a finite Lie algebra, H ̂ ( t ) = ∑ l = 1 L η l ( t ) g ̂ l . The Wei-Norman framework provides a powerful analytical method for computing the exact time-evolution operator in a factorized form, U ̂ ( t ) = ∏ l = 1 L e Λ l ( t ) g ̂ l . By exploiting the underlying algebraic structure, the method yields a system of differential equations—the Wei-Norman equations—which determine the time-evolution operator coefficients Λ l ( t ) from the Hamiltonian ones η l ( t ) . Although robust and formally extendable, applying the method by hand becomes increasingly difficult or even intractable for high-order systems ( L ≥ 6 ). In this work, we introduce , a Python library that automates the application of this method. Given the commutators between generators, the library symbolically derives the Wei-Norman equations and efficiently computes nested commutators, similarity transformations, and the differential equations that yield Baker-Campbell-Hausdorff-like relations. We validate the library by recovering known results for quantum systems associated with the Lie algebras su ( 1 , 1 ) , su ( 2 ) , sl ( 2 ) , and so ( 2 , 1 ) , and further demonstrate its capabilities by deriving the Wei-Norman equations for a system of two coupled time-dependent harmonic oscillators involving 11 generators. Additionally, we specialize the library to handle the Lie group SU ( N ) using a generic Cartan-Weyl basis for its algebra, and demonstrate its versatility by deriving the Wei-Norman equations for su ( 2 ) , su ( 3 ) , and su ( 4 ) , including the explicit construction of a set of quantum gates for universal quantum computation.
We introduce a practical method for compacting the time evolution of the quantum state of a closed physical system. The density matrix is specified as a function of a few time-independent observables where their coefficients are time-dependent. The key mathematical step is the vectorization of the surprisal, the logarithm of the density matrix, at each time point of interest. The time span used depends on the required spectral resolution. The entire course of the system evolution is represented as a matrix where each column is the vectorized surprisal at the given time point. Using singular value decomposition, SVD, of this matrix we generate realistic approximations for the time-independent observables and their respective time dependent coefficients. This allows a simplification of the algebraic procedure for determining the dominant constraints (the time-independent observables) in the sense of the maximal entropy approach. A nonstationary coherent initial state of a Morse oscillator is used to introduce the approach. We derive analytical exact expression for the surprisal as a function of time and this offers a benchmark for comparison with the accurate but approximate SVD results. We discuss two examples of a Morse potential of different anharmonicities, the H2 and I2 molecules. We further demonstrate the approach for a two coupled electronic states problem, the well studied non radiative decay of pyrazine from its bright state. Five constraints are found to be enough to capture the ultrafast electronic population exchange and to recover the dynamics of the wave packet in both electronic states.
A lossless, exact compaction of the time-evolved state of the quantum dynamical system of a perturbed anharmonic molecule is demonstrated using dynamical symmetries. The density matrix of the anharmonic molecule is a linear combination of these symmetries, and it remains so as a time-dependent perturbation is applied. Accurate, unitary-but-approximate, and thereby irreversible compaction is further shown using fewer symmetries, and the fidelity of this lossy compaction is quantified. Perturbations are typically linear in the operators of a Lie algebra. For a Hamiltonian that is also linear, one knows well how to reversibly compact the state of a dynamical system. However, anharmonic vibrations have a finite number of unequally spaced energy levels, and a good description of their spectra typically requires an algebraic-type Hamiltonian that is bilinear in the operators of a Lie algebra. For a bilinear Hamiltonian we show how a matrix-based approach allows us to compact both the populations and the coherences, either exactly reversibly or inexactly irreversibly, with fewer symmetries. A forced Morse oscillator is used as an explicit analytical and numerical example covering the entire range of dynamics from the sudden to the adiabatic limits.
Dynamical symmetries, time-dependent operators that almost commute with the Hamiltonian, extend the role of ordinary symmetries. Motivated by progress in quantum technologies, we illustrate a practical algebraic approach to computing such time-dependent operators. Explicitly we expand them as a linear combination of time-independent operators with time-dependent coefficients. There are possible applications to the dynamics of systems of coupled coherent two-state systems, such as qubits, pumped by optical excitation and other addressing inputs. Thereby, the interaction of the system with the excitation is bilinear in the coherence between the two states and in the strength of the time-dependent excitation. The total Hamiltonian is a sum of such bilinear terms and of terms linear in the populations. The terms in the Hamiltonian form a basis for Lie algebra, which can be represented as coupled individual two-state systems, each using the population and the coherence between two states. Using the factorization approach of Wei and Norman, we construct a unitary quantum mechanical evolution operator that is a factored contribution of individual two-state systems. By that one can accurately propagate both the wave function and the density matrix with special relevance to quantum computing based on qubit architecture. Explicit examples are derived for the electronic dynamics in coupled semi-conducting nanoparticles that can be used as hardware for quantum technologies.