Kuntal Mukherjee (IISER Mohali, India)
LinkedIn: @Kuntal Mukherjee; X: @kuntalsherlock
Abstract: Development of analytical methods for studying periodically driven quantum systems has been key for gaining insights into the physical phenomena in spectroscopy. The success of analytic methods relies on its operational aspects and exactness in replicating (known) experimental results. The analytical methods based on the Magnus expansion (ME) scheme have been preferred in time-evolution studies, though recently, the splitting of the time-propagator into a product of exponential operators in the Fer expansion (FE) scheme has gained wider attention. Hence, the operational advantages between the two has always remained contentious and is discussed herein with a two-spin model system supported by the numerical simulations with a heteronuclear spin system based on CP (Cross-Polarization) and a homonuclear spin system based on DQ-HORROR (Double Quantum Homonuclear Rotary Resonance) experiments. Here, we highlight the serious discrepancies observed in time-evolution studies based on time-propagators derived from both the FE and ME schemes. The exactness of the FE scheme is problem specific and highly dependent on the commutator relations among propagator operator. Only in certain cases, it results in agreement to those obtained from exact numerical methods. By contrast, the ME scheme in an appropriate interaction frame presents a reliable framework for evaluating the observables at stroboscopic time-intervals.
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Thank you for the presentation. Is there a scenario where you would recommend using the Fer expansion?
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Hi Jonas,
Unfortunately, there are very limited opportunities to use Fer expansion to obtain exact results. In general, It would work when following conditions are satisfied.
[Fn(t),ρ(0)] ≠ 0 & [Fn(t), D] ≠ 0
There must be non-commutating operators present in the F-operators with the initial density matrix operator and detection operator simultaneously, since, the terms showing the effect of anisotropic interactions must be reflected in the final signal expression. In case of CP and DQ-HORROR experiments, one of the F-operator commutes with the density operator, the anisotropic terms do not participate in the final signal expression. Hence, we have to choose such experiments where the above relation holds and the choice of experiment is totally contextual and need to verify the applicability of the Fer expansion.
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Hi Jonas,
Unfortunately, there are very limited opportunities to use Fer expansion to obtain exact results. In general, It would work when following conditions are satisfied.
[Fn(t),ρ(0)] ≠ 0 & [Fn(t), D] ≠ 0
There must be non-commutating operators present in the F-operators with the initial density matrix operator and detection operator simultaneously, since, the terms showing the effect of anisotropic interactions must be reflected in the final signal expression. In case of CP and DQ-HORROR experiments, one of the F-operator commutes with the density operator, the anisotropic terms do not participate in the final signal expression. Hence, we have to choose such experiments where the above relation holds and the choice of experiment is totally contextual and need to verify the applicability of the Fer expansion. -
Thank you for the effort in presenting such a topic!
It seems that the Fer expansion cannot work in the presence of CSA, as you discuss. So how and why was this method introduced in the first place?
Based on your conclusions, would there be any reason moving forward to use the Fer expansion over the Magnus one?-
Hi Nicolas, thank you for watching my presentation.
The original Fer expansion was applied to the classical systems. But for quantum mechanical systems, the commutation relations play a big role. In this presentation as well as the publication, to keep things simple, we have shown the applicability of Fer expansion over cycle time detection or stroboscopic detection. The conclusion is straightforward in this case which is presented. But for continuous detection or non-stroboscopic detection, the form of Fn-operators become very complex upon going to higher order. Hence, to work with Magnus expansion, you need to add all the Fn-operators in a single exponent and operate it on density operator to evaluate signal (through BCH expansion). It is highly probable that in such scenario, deducing closed form expression is quite cumbersome though possible and in worse cases, the presence of off-diagonal terms will not give any closed form solution. On the other hand, Fer expansion would allow to operate the Fn-operators individually and obtain a product of simpler expressions, although, it still needs to satisfy the following two conditions, i.e. [Fn(t),ρ(0)] ≠ 0 & [Fn(t), D] ≠ 0 for which the applicability of Fer expansion becomes limited. To sum up, in stroboscopic detection, Magnus expansion’s efficiency is greater or equal to Fer expansion’s efficiency. For non-stroboscopic detection, Fer expansion is convenient to use if the above two conditions satisfy. There is a paper by Shreyan et. al. (https://doi.org/10.1080/00268976.2023.2231107)on decoupling that discuss on the non-stroboscopic detection also.
I hope this explanation helps and answers your query.
Thank you again.
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Hi Kuntal,
Great to see your work here!
Have you ever explored using the Fer expansion approach for systems involving quadrupolar nuclei?
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Hi Sajith,
Thank you very much for watching my presentation.
I have been exploring the Fer expansion scheme involving spin-1/2 nuclei only. Although, for systems involving quadrupolar nuclei, the method would be same, like evaluating the time-propagator by writing down the F1 and F2 operators. Due to quadrupolar nuclei, there will be addition of Quadrupolar interaction. So, starting with a single spin quadrupolar nuclei under single pulse followed by stroboscopic detection of signal and assuming no anisotropic interaction present with offset=0 kHz, we can have simplest problem to start with. As a time-independent Hamiltonian, we would have external RF-Hamiltonian and as a time-dependent Hamiltonian, we would have internal Quadrupolar Hamiltonian. Then, we can write down the F1 and F2-operators. During detection, we would know how much Fer expansion is exact depending upon the condition it needs to satisfy, i.e. [Fn(t),ρ(0)] ≠ 0 & [Fn(t), D] ≠ 0. Further addition of anisotropies like CSA would only bring new set of challenges that Fer expansion has to deal with. As of now, I can comment on a primitive level about it. For better assessment, we have to calculate it thoroughly.
I hope this satisfies your query.
Thank you again for showing the interest.-
Yes. Thank you.
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