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Physical review. B, Condensed matter

Spin thermodynamics of periodically time-dependent systems: The quasistationary state and its decay

M. Matti Maricq

Published Jul 1, 1987• 14 citations
Publisher

Abstract

The effective, or Floquet, Hamiltonian for a periodically time-dependent many-body system plays a role analogous to the energy of a conservative system and can therefore be used to describe the equilibrium state. Restrictions on this analogy and the necessity of a series approximation of the effective Hamiltonian complicate the interpretation of the long-time evolution. Analysis of the effective Hamiltonian for the examples of a dipolar coupled spin system in a weak nonresonant field and for the pulsed spin-locking problem shows it to be characteristically the sum of two (or more) commuting observables plus small off-diagonal terms. This implies that the system evolves to a quasistationary state comprised of two or more independent thermodynamic baths. The properties of this state and its subsequent decay to equilibrium are discussed in detail and compared to the results of magnetic resonance experiments. This analysis overcomes the difficulties, noted in the past as the ``Magnus paradox,'' with the use of time-independent model Hamiltonians to describe approximately the true evolution of the system.

DOI
10.1103/physrevb.36.516

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On the equivalence between different averaging schemes in magnetic resonance

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Evolution of quantum mechanical systems under time-dependent Hamiltonians has remained a challenging problem of interest across all disciplines. Through suitable approximations, different averaging methods have emerged in the past for modeling the time-evolution under time-dependent Hamiltonians. To this end, the development of analytic methods in the form of time-averaged effective Hamiltonians has gained prominence over other methods. In particular, the advancement of spectroscopic methods for probing molecular structures has benefited enormously from such theoretical pursuits. Nonetheless, the validity of the approximations and the exactness of the proposed effective Hamiltonians have always remained a contentious issue. Here, in this report, we reexamine the equivalence between the effective Hamiltonians derived from the Magnus formula and Floquet theory through suitable examples in magnetic resonance.

The Journal of Chemical Physics•Sep 1, 2020•11 citations
The effective, or Floquet, Hamiltonian for a periodically time-dependent many-body system plays a role analogous to the energy of a conservative system and can therefore be used to describe the equilibrium state. Restrictions on this analogy and the necessity of a series approximation of the effective Hamiltonian complicate the interpretation of the long-time evolution. Analysis of the effective Hamiltonian for the examples of a dipolar coupled spin system in a weak nonresonant field and for the pulsed spin-locking problem shows it to be characteristically the sum of two (or more) commuting observables plus small off-diagonal terms. This implies that the system evolves to a quasistationary state comprised of two or more independent thermodynamic baths. The properties of this state and its subsequent decay to equilibrium are discussed in detail and compared to the results of magnetic resonance experiments. This analysis overcomes the difficulties, noted in the past as the ``Magnus paradox,'' with the use of time-independent model Hamiltonians to describe approximately the true evolution of the system.
A discussion is presented on the use of statistical thermodynamics to describe the long-time behavior of a many-body Hamiltonian that also depends periodically on time. Floquet's theorem is applied to the dynamics of the system in order to exploit the time symmetry. This approach produces an effective Hamiltonian that propagates the system over one time cycle. This is taken as the fundamental constant of the motion for the system and forms the basis for the thermodynamic description. Under suitable conditions, the resulting formulas for the equilibrium values of observables have a strong analogy to the usual thermodynamic expressions for time independent systems. These ideas are applied to describe the equilibrium magnetization for multiply pulsed spin systems, in particular to the Ostroff-Waugh and Waugh-Huber-Haeberlen pulse sequences. For these systems a quasistationary state develops after a few times T2{T}_{2}T2​. This is followed by a slow decay to an equilibrium state. The dynamics of the decay is discussed in terms of an application of the Provotorov theory of saturation to the effective Hamiltonian.
Two difficulties arise in the Magnus expansion for the evolution operator of a periodic time dependent Hamiltonian when the eigenvalues of the Hamiltonian are separated by more than 2π/τ, where τ is the period: (1) The Magnus expansion generally does not converge, and (2) the equilibrium properties of the system cannot be determined by statistical mechanical techniques analogous to those for time independent systems. A transformation is introduced that folds the eigenvalue spectrum into the interval −π/τ<λi<π/τ, yet retains the periodicity of the Hamiltonian, thereby alleviating the above difficulties. The method is compared to the approach taken by Shirley [Phys. Rev. B 138, 979 (1965)] for the Floquet solution to the evolution operator. It is applied to a dipole coupled spin system in an oscillating transverse field and the implications of the folding transformation on the spin thermodynamics of this system and the ‘‘Magnus paradox’’ are discussed.
The Floquet-Magnus expansion is a widely used tool to derive effective descriptions of time-periodic quantum systems by approximating their dynamics with a time-independent Hamiltonian. However, its standard formulation is, strictly speaking, restricted to bounded Hamiltonians. In this work, we extend its definition and analysis to a broad class of time-periodic unbounded Hamiltonians. Our approach is based on an a priori distinct nonperturbative framework for the construction of effective Hamiltonians, which we show to reproduce the Floquet-Magnus expansion. A particular strength of our framework is that it allows us to prove that the resulting effective dynamics approximates the original time evolution propagators to arbitrary order in the high-frequency limit without requiring convergence of the Floquet-Magnus expansion, a condition that is already highly restrictive even in the bounded setting. We illustrate the scope of the method on representative models: the quantum Rabi Hamiltonian in the interaction picture, and the periodically driven quantum harmonic oscillator.
A periodically driven quantum system, when coupled to a heat bath, relaxes to a non-equilibrium asymptotic state. In the general situation, the retrieval of this asymptotic state presents a rather non-trivial task. It was recently shown that in the limit of an infinitesimal coupling, using the so-called rotating wave approximation (RWA), and under strict conditions imposed on the time-dependent system Hamiltonian, the asymptotic state can attain the Gibbs form. A Floquet–Gibbs state is characterized by a density matrix which is diagonal in the Floquet basis of the system Hamiltonian with the diagonal elements obeying a Gibbs distribution, being parametrized by the corresponding Floquet quasi-energies. Addressing the non-adiabatic driving regime, upon using the Magnus expansion, we employ the concept of a corresponding effective Floquet Hamiltonian. In doing so we go beyond the conventionally used RWA and demonstrate that the idea of Floquet–Gibbs states can be extended to the realistic case of a weak, although finite system-bath coupling, herein termed effective Floquet–Gibbs states.
An accurate theoretical treatment of periodically driven quantum systems is crucial for various fields in the exact sciences, for instance Nuclear Magnetic Resonance (NMR) spectroscopy. Conventionally, either average Hamiltonian theory or Floquet theory is used to predict or describe experimental outcomes, such as the time evolution or the spectra yielding the information of the sample under study. A detailed analysis of the equivalence of these two approaches with an emphasis on applications in NMR will help to improve the theoretical understanding of NMR experiments. In this work, we identify the Floquet-Magnus expansion as essential to prove the mathematical equivalence of Floquet theory and average Hamiltonian theory. We advocate a calculation scheme which is less prone to algebraic mistakes because explicit integration is avoided. On this basis, we provide the first four orders of both theories. We further examine their applicability to some experiments in NMR. As examples, we investigate the Bloch-Siegert shift and dipolar coupled spin systems under magic-angle spinning. Based on our analysis, we recommend the use of the Floquet-Van Vleck approach including both the effective Hamiltonian and the kick operator. The consistent separation of secular and non-secular contributions appears to be especially advantageous for numerical robustness. Its accuracy is about three times better than the one provided by average Hamiltonian theory despite their formal equivalence. Our findings provide important insights into the theoretical background of Floquet theory and average Hamiltonian theory. This includes the extent of their algebraic and perturbative equivalence, with an emphasis on how these findings are of relevance in the analysis of solid-state NMR experiments.
Evolution of quantum mechanical systems under time-dependent Hamiltonians has remained a challenging problem of interest across all disciplines. Through suitable approximations, different averaging methods have emerged in the past for modeling the time-evolution under time-dependent Hamiltonians. To this end, the development of analytic methods in the form of time-averaged effective Hamiltonians has gained prominence over other methods. In particular, the advancement of spectroscopic methods for probing molecular structures has benefited enormously from such theoretical pursuits. Nonetheless, the validity of the approximations and the exactness of the proposed effective Hamiltonians have always remained a contentious issue. Here, in this report, we reexamine the equivalence between the effective Hamiltonians derived from the Magnus formula and Floquet theory through suitable examples in magnetic resonance.