Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Molecular Physics

Theory of scalar relaxation of the second kind. II

Nicholas C. Pyper

Published Jan 1, 1971• 17 citations
Publisher

Abstract

A previously presented [1] equation of motion of a reduced density matrix for a system of spins coupled to a second system of spins, the latter being strongly relaxed on account of their coupling to a thermal reservoir, is rederived without recourse to coarse-grained methods. It is shown that the procedure developed here, in contrast to the earlier approach [1], can be extended to corrections of fourth and higher orders in the strength of the coupling between the two groups of spins. The fourth-order terms are presented and applied to the 2A2B3X spin system, the expression derived being compared, for the first-order limit, with the result of an exact calculation. It is shown that the approach [1] becomes invalid in cases for which there is more than one eigenoperator of the Liouville operator appropriate to the rapidly relaxing spins whose eigenvalues are zero or very small. This is shown to occur when the rapidly relaxing group of spins possesses nuclear permutation symmetry and the relaxation of different spins in this group is strongly correlated. An equation of motion of the density matrix is derived for such cases and applied to the 2A3X3Y spin system.

DOI
10.1080/00268977100102131

Related papers

On the Theory of Relaxation Processes

Alfred G. Redfield

A general procedure is given for finding the equation of motion of the density matrix of a system in contact with a thermal bath, as for example a nuclear spin system weakly coupled to a crystal lattice. The thermal bath is treated both classically and quantum mechanically, and the theory is similar to, and a generalization of, conventional theories of time proportional transition probabilities. Relaxation of the system by the thermal bath is expressed by a linear matrix operator, and it is stressed that elements of this operator can be regarded as secular or nonsecular perturbations on the equation of motion and can be treated accordingly. When the motion of the system is slow compared to that of the thermal bath, the equation of motion can be expressed in an operator form which is independent of representation. If the system has a time-dependent Hamiltonian which varies slowly compared to the motion of the thermal bath, the same equation of motion is obeyed and the system is relaxed by the bath toward a Boltzmann distribution with respect to its instantaneous Hamiltonian. If the time variation of the Hamiltonian is more rapid, higher order corrections to the equation of motion must be applied. The theory is applied to spin-lattice relaxation of a coupled nuclear spin system in a metal, for arbitrary externally applied fixed magnetic field.

IBM Journal of Research and Development•Jan 1, 1957•1,803 citations

Systematic perturbation theory for dynamical coarse-graining

Gernot Schaller, Philipp Zedler, Tobias Brandes

We demonstrate how the dynamical coarse-graining approach can be systematically extended to higher orders in the coupling between system and reservoir. Up to second order in the coupling constant, we explicitly show that dynamical coarse-graining unconditionally preserves positivity of the density matrix---even for bath density matrices that are not in equilibrium and also for time-dependent system Hamiltonians. By construction, the approach correctly captures the short-time dynamics; i.e., it is suitable for analyzing non-Markovian effects. We compare the dynamics with the exact solution for highly non-Markovian systems and find a remarkable quality of the coarse-graining approach. The extension to higher orders is straightforward but rather tedious. The approach is especially useful for bath correlation functions of simple structure and for small system dimensions.

arXiv:0812.1507•Physical Review A•Mar 20, 2009•41 citations

Quantum relaxation with initial system-bath correlations

Klaus Kassner

A new type of projection operator is used to derive homogeneous equations of motion for the reduced density operator of a system coupled to a bath. Crucial simplifications of these equations are obtained without assuming that the projection operator and the zeroth-order Hamiltonian commute. No factorization assumption for the initial density matrix is made. The applicability of the general formalism, which is valid also for nonergodic initial conditions, to physically relevant situations is investigated. It turns out to be particularly useful for the treatment of a system initially submitted to some constraints but otherwise in equilibrium. The relaxation towards the unconstrained equilibrium of a two-level system in contact with an Ohmic bath is treated as an example. The reduced equilibrium density operator of this system is obtained analytically for the case of extremely low temperatures. The result shows and it is then proved rigorously that the zero-temperature reduced density operator does not describe a pure state in general, except in the zero-coupling limit.

Physical Review A•Dec 1, 1987•20 citations

Density Operators and Matrices

The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalisation techniques.

Royal Society of Chemistry eBooks•Dec 13, 2016•0 citations

Correlation functions in finite memory-time reservoir theory

Henk F. Arnoldus, Thomas F. George

Interaction of a small system S with a large reservoir R amounts to thermal relaxation of the reduced system density operator ρS(t). The presence of the reservoir enters the equation of motion for ρS(t) through the reservoir correlation functions fkl(τ) (defined in the text), which decay to zero for τ→∞ on a time scale τc. Commonly, this τc is much smaller than the inverse relaxation constants for the time evolution of ρS(t). Then a series of approximations can be made, which lead to a Markovian equation of motion for ρS(t). In this paper the assumption of a small reservoir correlation time is removed. The equation of motion for ρS(t) is solved, and it appears that the memory effect, due to τc≂/0, can be incorporated in a frequency dependence of the relaxation operator Γ̃(ω). Subsequently, (unequal-time) quantum correlation functions of two system operators are considered, where explicit expressions for (the Laplace transform of) the correlation functions are obtained. They involve again the relaxation operator Γ̃(ω), which accounts for the time regression. Additionally it is found that an initial-correlation operator Υ̃(ω) arises, as a consequence of the fact that the equal-time correlation functions do not factorize as ρS(t) times the reservoir density operator. It is pointed out that the frequency dependence of Γ̃(ω) and the occurrence of a nonzero T̃(ω) both arise as a result of τc≂/0, and should therefore be treated on an equal footing. Explicit evaluation of Γ̃(ω) and Υ̃(ω) shows that their matrix elements can be expressed entirely in f̃kl(ω), just as in the Markov approximation. Hence no essential complications appear if one should go beyond the limits of a small reservoir correlation time τc.

Journal of Mathematical Physics•Nov 1, 1987•3 citations

Density Operators and Matrices

The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored briefly. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalization techniques.

Royal Society of Chemistry eBooks•Sep 12, 2025•0 citations
A previously presented [1] equation of motion of a reduced density matrix for a system of spins coupled to a second system of spins, the latter being strongly relaxed on account of their coupling to a thermal reservoir, is rederived without recourse to coarse-grained methods. It is shown that the procedure developed here, in contrast to the earlier approach [1], can be extended to corrections of fourth and higher orders in the strength of the coupling between the two groups of spins. The fourth-order terms are presented and applied to the 2A2B3X spin system, the expression derived being compared, for the first-order limit, with the result of an exact calculation. It is shown that the approach [1] becomes invalid in cases for which there is more than one eigenoperator of the Liouville operator appropriate to the rapidly relaxing spins whose eigenvalues are zero or very small. This is shown to occur when the rapidly relaxing group of spins possesses nuclear permutation symmetry and the relaxation of different spins in this group is strongly correlated. An equation of motion of the density matrix is derived for such cases and applied to the 2A3X3Y spin system.
A general procedure is given for finding the equation of motion of the density matrix of a system in contact with a thermal bath, as for example a nuclear spin system weakly coupled to a crystal lattice. The thermal bath is treated both classically and quantum mechanically, and the theory is similar to, and a generalization of, conventional theories of time proportional transition probabilities. Relaxation of the system by the thermal bath is expressed by a linear matrix operator, and it is stressed that elements of this operator can be regarded as secular or nonsecular perturbations on the equation of motion and can be treated accordingly. When the motion of the system is slow compared to that of the thermal bath, the equation of motion can be expressed in an operator form which is independent of representation. If the system has a time-dependent Hamiltonian which varies slowly compared to the motion of the thermal bath, the same equation of motion is obeyed and the system is relaxed by the bath toward a Boltzmann distribution with respect to its instantaneous Hamiltonian. If the time variation of the Hamiltonian is more rapid, higher order corrections to the equation of motion must be applied. The theory is applied to spin-lattice relaxation of a coupled nuclear spin system in a metal, for arbitrary externally applied fixed magnetic field.
We demonstrate how the dynamical coarse-graining approach can be systematically extended to higher orders in the coupling between system and reservoir. Up to second order in the coupling constant, we explicitly show that dynamical coarse-graining unconditionally preserves positivity of the density matrix---even for bath density matrices that are not in equilibrium and also for time-dependent system Hamiltonians. By construction, the approach correctly captures the short-time dynamics; i.e., it is suitable for analyzing non-Markovian effects. We compare the dynamics with the exact solution for highly non-Markovian systems and find a remarkable quality of the coarse-graining approach. The extension to higher orders is straightforward but rather tedious. The approach is especially useful for bath correlation functions of simple structure and for small system dimensions.
A new type of projection operator is used to derive homogeneous equations of motion for the reduced density operator of a system coupled to a bath. Crucial simplifications of these equations are obtained without assuming that the projection operator and the zeroth-order Hamiltonian commute. No factorization assumption for the initial density matrix is made. The applicability of the general formalism, which is valid also for nonergodic initial conditions, to physically relevant situations is investigated. It turns out to be particularly useful for the treatment of a system initially submitted to some constraints but otherwise in equilibrium. The relaxation towards the unconstrained equilibrium of a two-level system in contact with an Ohmic bath is treated as an example. The reduced equilibrium density operator of this system is obtained analytically for the case of extremely low temperatures. The result shows and it is then proved rigorously that the zero-temperature reduced density operator does not describe a pure state in general, except in the zero-coupling limit.
The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalisation techniques.
Interaction of a small system S with a large reservoir R amounts to thermal relaxation of the reduced system density operator ρS(t). The presence of the reservoir enters the equation of motion for ρS(t) through the reservoir correlation functions fkl(τ) (defined in the text), which decay to zero for τ→∞ on a time scale τc. Commonly, this τc is much smaller than the inverse relaxation constants for the time evolution of ρS(t). Then a series of approximations can be made, which lead to a Markovian equation of motion for ρS(t). In this paper the assumption of a small reservoir correlation time is removed. The equation of motion for ρS(t) is solved, and it appears that the memory effect, due to τc≂/0, can be incorporated in a frequency dependence of the relaxation operator Γ̃(ω). Subsequently, (unequal-time) quantum correlation functions of two system operators are considered, where explicit expressions for (the Laplace transform of) the correlation functions are obtained. They involve again the relaxation operator Γ̃(ω), which accounts for the time regression. Additionally it is found that an initial-correlation operator Υ̃(ω) arises, as a consequence of the fact that the equal-time correlation functions do not factorize as ρS(t) times the reservoir density operator. It is pointed out that the frequency dependence of Γ̃(ω) and the occurrence of a nonzero T̃(ω) both arise as a result of τc≂/0, and should therefore be treated on an equal footing. Explicit evaluation of Γ̃(ω) and Υ̃(ω) shows that their matrix elements can be expressed entirely in f̃kl(ω), just as in the Markov approximation. Hence no essential complications appear if one should go beyond the limits of a small reservoir correlation time τc.
The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored briefly. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalization techniques.