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Physical review. B, Condensed matter

Thermodynamics for many-body systems evolving under a periodic time-dependent Hamiltonian: Application to pulsed magnetic resonance

M. Matti Maricq

Published Jan 1, 1985• 28 citations
Publisher

Abstract

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 ${T}_{2}$. 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.

DOI
10.1103/physrevb.31.127

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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.
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.
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 theorem allows one to reformulate periodic time-dependent problems such as the interaction of a many-body system with a laser field in terms of time-independent, field-dressed states, also known as Floquet states. If this was possible for density-functional theory as well, one could reduce in such cases time-dependent density-functional theory to a time-independent Floquet density-functional theory. We analyze under which conditions the Floquet theorem is applicable in a density-functional framework. By employing numerical ab initio solutions of the interacting time-dependent Schr\"odinger equation with time-periodic external potentials we show that the exact effective potential in the corresponding Kohn-Sham equation is not unconditionally periodic. Whenever several Floquet states in the interacting system are involved in a physical process the corresponding Hartree-exchange-correlation potential is not periodic with the external frequency only. Using an analytically solvable example we demonstrate that, in general, the periodicity of the time-dependent Kohn-Sham Hamiltonian cannot be restored by choosing a different initial state. Only if the external periodic potential is sufficiently weak such that the initial state of the interacting system evolves adiabatically to a single, field-dressed state, the resulting Kohn-Sham system admits the application of the Floquet theorem.
How to understand the order of Floquet stationary states in the presence of external bath coupling and their statistical mechanics is challenging; the answers are important for preparation and control of those Floquet states. Here, we propose a scheme to classify the statistical distribution of Floquet states for time-periodic systems which couple to an external heat bath. If an effective Hamiltonian and a system-bath coupling operator, which are all time independent, can be simultaneously obtained via a time-periodic unitary transformation, the statistical mechanics of the Floquet states is equivalent to the equilibrium statistical mechanics of the effective Hamiltonian. In the large driving frequency cases, we also show that the conditions of this theorem can be weakened to the following: The time-period part in the system Hamiltonian commutes with the system-bath coupling operator. A Floquet-Markov approach is applied to numerically compute the Floquet state occupation distribution of a bosonic chain, and the results agree with the theoretical predictions.
This article presents the application of continuous Floquet theory in solid-state nuclear magnetic resonance (NMR). Continuous Floquet theory extends the traditional Floquet theory to non-continuous Hamiltonians, enabling the description of observable effects not fully captured by the traditional Floquet theory due to its requirement for a periodic Hamiltonian. We present closed-form expressions for computing first- and second-order effective Hamiltonians, streamlining integration with the traditional Floquet theory and facilitating application in NMR experiments featuring multiple modulation frequencies. Subsequently, we show examples of the practical application of continuous Floquet theory by investigating several solid-state NMR experiments. These examples illustrate the importance of the duration of the pulse scheme regarding the width of the resonance conditions and the near-resonance behavior.
The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in many scenarios such as driven soft matter or nuclear magnetic resonance. In this article we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly-time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.