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Constructing Dynamical Symmetries

James Russell Hamilton, F. Remacle, Raphael David Levine

Published Dec 28, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.26434/chemrxiv-2023-v5g58-v2

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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.
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.
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.
Dynamical symmetries, operators that do not quite commute with the Hamiltonian, extend the role of ordinary symmetries. They also provide an interesting insight on constants of the motion. Motivated by progress in quantum technologies we discuss a practical algebraic approach to computing such time-dependent operators. Explicitly we expand them as a linear combination with time dependent coefficients of time-dependent Schrödinger operators. We propose possible applications in determining quantum mechanical distributions of maximal entropy and to the dynamics of systems of coherently coupled coherent two state systems. We suggest that this generates an Ising-like Hamiltonian where each ’spin’ is a state and therefore of relevance to quantum computing based on qubit architecture.
The unitary operators U(t), describing the quantum time evolution of systems with a time-dependent Hamiltonian, can be constructed in an explicit manner using the method of time-dependent invariants. We clarify the role of Lie-algebraic techniques in this context and elaborate the theory for SU(2) and SU(1,1). We show that the constructions known as Magnus expansion and Wei-Norman expansion correspond with different representations of the rotation group. A simpler construction is obtained when representing rotations in terms of Euler angles. The many applications are postponed to Part II of the paper.
The matrix U' contains the full dynamics of a bilinear oscillator, because any observable of the system is an analytical function of the operators that determine the matrix representation. One cannot solve the time evolution equation in the general case, but there are many exactly solvable problems that simulate and enable one to understand the dynamics of more elaborate and realistic quantum-mechanical models. In the case of the solvable model just discussed it is easy to derive analytical expressions for operators in the Heisenberg picture as well as for invariants or constants of the motion. The mathematical treatment of the driven oscillator just discussed is trivial because the Hamiltonian operator belongs to a solvable Lie algebra. This model is a convenient starting point to illustrate the use of the Lie algebraic method, but it is also an oversimplification of most oscillatory phenomena. Nonlinear Hamiltonians are useful in many physical applications such as the simulation of frictional forces and dissipation effects.
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.