the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 04 Nov 2024
Analytical expressions for time evolution of spin systems affected by two or more interactions
Abstract. Analytical expressions for the description of the time evolution of spin systems beyond the product-operator formalism (POF) can be obtained if a low-dimensional subspace of the Liouville space has been found. The latter can be established by a procedure that consists of a repeated application of the commutator of the Hamiltonian with the density operator. This iteration continues as long as the result of such a commutator operation contains a term that is linearly independent of all the operators appearing in the previous commutator operations. The coefficients of the resulting system of commutator relations can be immediately inserted into the generic propagation formulae given in this article if the system contains two, three or four equations. In cases not satisfying the validity conditions of any of these propagation formulae, the coefficients are used as intermediate steps to obtain both the Liouvillian and propagator matrices of the system. Several application examples are given where an analytical equation can be obtained for the description of the time evolution of small spin systems under the influence of two or more interactions.
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CC1: 'Comment on mr-2024-18', Tom Barbara, 11 Nov 2024
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The author gives some nice examples of propagation for the Liouville operator space for special cases. It may be of some interest to readers to mention that if the Hamiltonian eigenvalues can be found, then these can be used to get the eigenvalues for the Liouville operator as well, as they are just the differences between them. In other words the |a><b| projectors are eigen- matrices of the Liouvillian in the diagonal basis of the Hamiltonian. This provides a very direct path to providing some of results mentioned in this work. There is a large body of work on this topic that unfortunately is not mentioned in the references, especially so with two coupled spins and spin one dynamics, such as the work of Vega and Pines, Levitt, Suter and Ernst, etc. Since the Liouvillian is a real orthogonal matrix, there are also useful occasions when the matrix can be blocked into groups of four dimensions, leading to the interesting relationship between three and four dimensional rotations as illustrated in JMR 67,491 (1986).
Citation: https://doi.org/10.5194/mr-2024-18-CC1 -
AC1: 'Reply on CC1', Günter Hempel, 13 Nov 2024
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Thank you very much for the comment. It will help me to improve some parts of the manuscript and to change some paragraphs to avoid misunderstandings.
The application examples should serve as a kind of confirmation of the procedure proposed in this manuscript. I.e. they should illustrate that in some cases the procedure given here leads to known results which have been obtained in other ways, perhaps very directly in some cases. (Nevertheless, some of these propagation formulae also seem to be new, for example some of the CP-related ones.) But I agree, that in a revised version more papers should be cited which present other ways of calculating. Especially the paper you mentioned, where a matrix of two 4x4 blocks represents the Liouvillian for the case of quadrupolar interaction + rf_x irradiation.
The connection between the Liouvillian and Hamiltonian eigenvalues was not mentioned in the original version of the manuscript, because eigenvalues do not immediately play a role here. (In fact, they do in some methods of matrix exponentialisation, but this was not considered in detail here.)
Do you mean "the propagator matrix is a real orthogonal matrix"? The matrix associated with the Liouville operator is skew symmetric and pure imaginary if a Hermitian operator basis is chosen. Therefore, it cannot be orthogonal (or even unitary) in general. This can be seen by inspecting Eqs. (35) and (54) of the main part, or the matrices A and B in the paper mentioned in the comment. At least for odd dimensions, the determinant of the Liouvillian matrix will be zero. An orthogonal matrix, on the other hand, has a determinant = 1. However, the matrix of the propagator is orthogonal in an Hermitian basis, otherwise "only" unitary, because it is the exponential of a skew-symmetric matrix.
The occurrence of a block structure of a matrix means that it can be written as direct product of the corresponding matrices related to subspaces. However, using the original procedure from the manuscript of finding an appropriate subspace (Section 3), no block structure will appear for the Liouvillian matrix. Instead, the procedure ends as soon as the dimension of a block is reached; the commutator of the last equation can be expanded without remainder in the operators already present. This result corresponds to a subspace whose basis includes the operator of the initial state. We could say that the procedure finds the block of the total matrix corresponding to a subspace containing the operator of the initial state.
However, a block structure may occur if we use the modified procedure (last paragraph of subsection 3.4).Citation: https://doi.org/10.5194/mr-2024-18-AC1 -
CC2: 'Reply on AC1', Tom Barbara, 13 Nov 2024
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Thanks for you kind reply Gunter, I apologize for my confusing language as to what the "Liouvillian" exactly is. I somewhat sloppily refer to it in the case when the imaginary unit is not factored out, such that (i*L) is still a Liouvillian. Maybe (i*L) should be the "Barbarian"! (Not!)
I confess that I have not collated the equations to see exactly what the Hamiltonian is for each case explored. I only ventured a guess that since you have successfully propagated the solutions for these cases, then surely the Hamiltonian eigenvalues are relatively easy to find, because those for the superoperator dynamics are just + or -i*(wa-wb) or zero when a=b. Of course one still needs to construct the eigenvectors, which can be rather tedious to do. Furthermore, the zero eigenvalues for the superoperator dynamics are usually degenerate, so you have to be careful there as well. If my guess is incorrect, then that is great news!
A few years back I wrote up some notes when working out aspects of relaxation theory and the problem of vectorizing a matrix related to my own submission to MR. I put these up on the now official sister site "ResearchGate" to MR and the thought occurred to me that perhaps it is useful to have them placed here on the MR site proper, so I have done so as a supplement to our discussion. Beware they are hand written, but my writing style is pretty good....
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AC2: 'Reply on CC2', Günter Hempel, 14 Nov 2024
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Thanks again, Tom. After a first look at your handwritten text, I realize that I had also tried something similar - treating matrices of tensors as vectors to perform coordinate transformations. I look forward to seeing more details in your file in the next few days. I would like to discuss this with you afterwards, perhaps via our actual e-mail addresses? Greetings, Günter.
Citation: https://doi.org/10.5194/mr-2024-18-AC2
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AC2: 'Reply on CC2', Günter Hempel, 14 Nov 2024
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CC2: 'Reply on AC1', Tom Barbara, 13 Nov 2024
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AC1: 'Reply on CC1', Günter Hempel, 13 Nov 2024
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RC1: 'Comment on mr-2024-18', Anonymous Referee #1, 19 Nov 2024
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The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr-2024-18/mr-2024-18-RC1-supplement.pdf
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