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Special issue: Robert Kaptein Festschrift

**Research article**
23 Dec 2020

**Research article** | 23 Dec 2020

# Representation of population exchange at level anti-crossings

Bogdan A. Rodin and Konstantin L. Ivanov

^{1,2}

^{1,2}

**Bogdan A. Rodin and Konstantin L. Ivanov**Bogdan A. Rodin and Konstantin L. Ivanov

^{1,2}

^{1,2}

^{1}International Tomography Center, Siberian Branch of the Russian Academy of Science, Novosibirsk, 630090, Russia^{2}Physics Department, Novosibirsk State University, Novosibirsk, 630090, Russia

^{1}International Tomography Center, Siberian Branch of the Russian Academy of Science, Novosibirsk, 630090, Russia^{2}Physics Department, Novosibirsk State University, Novosibirsk, 630090, Russia

**Correspondence**: Konstantin L. Ivanov (ivanov@tomo.nsc.ru)

**Correspondence**: Konstantin L. Ivanov (ivanov@tomo.nsc.ru)

Received: 27 Aug 2020 – Discussion started: 18 Sep 2020 – Revised: 15 Nov 2020 – Accepted: 18 Nov 2020 – Published: 23 Dec 2020

A theoretical framework is proposed to describe the spin dynamics driven by coherent spin mixing at level anti-crossings (LACs). We briefly introduce the LAC concept and propose to describe the spin dynamics using a vector of populations of the diabatic eigenstates. In this description, each LAC gives rise to a pairwise redistribution of eigenstate populations, allowing one to construct the total evolution operator of the spin system. Additionally, we take into account that in the course of spin evolution a “rotation” of the eigenstate basis case take place. The approach is illustrated by a number of examples, dealing with magnetic field inversion, cross-polarization, singlet-state nuclear magnetic resonance and parahydrogen-induced polarization.

Nuclear magnetic resonance (NMR) methods, which exploit coherent spin mixing at level anti-crossings (LACs), are widely used in various areas of research, notably, to perform broad-band excitation (Baum et al., 1985; Freeman, 1998; Tannús and Garwood, 1997) and cross-polarization (Hartmann and Hahn, 1962), to transfer spin hyperpolarization (Ivanov et al., 2014; Theis et al., 2018, 2014b; Pravdivtsev et al., 2014a, b, c; Franzoni et al., 2013), and to generate and detect long-lived nuclear singlet order (DeVience et al., 2013; Rodin et al., 2018, 2019; Pravdivtsev et al., 2016). In this work, we propose an approach aimed at a simple understanding of spin mixing at LACs and predicting the resulting spin order. The approach is applicable to spin systems with arbitrary populations of adiabatic nuclear spin states and no coherence between them; it makes use of two ingredients – permutations of the populations and rotation of the basis of spin eigenstates. In this work, we introduce the main concept and formalism and provide a number of NMR-relevant examples, showing how the approach works. These examples include a consideration of spin order transfer upon adiabatic inversion (Lukzen and Steiner, 1995; Eills et al., 2019) of the external magnetic field and, more generally, NMR experiments with field jumps (Miesel et al., 2006; Pravdivtsev et al., 2013a), as well as some pulsed NMR experiments, such as cross-polarization (Hartmann and Hahn, 1962; Pines et al., 1972). Last but not least, using the language of LACs we describe some pulse sequences, which are currently exploited in singlet-state NMR (Levitt, 2019, 2012) and parahydrogen-induced polarization (PHIP) (Natterer and Bargon, 1997; Green et al., 2012; Barskiy et al., 2019; Duckett and Mewis, 2012).

PHIP makes use of the spin order of parahydrogen, *p*H_{2}, which is the H_{2} molecule in its nuclear singlet state. It is straightforward to enrich the H_{2} gas in the *para* component to >90 %. Such a significant deviation of the singlet state population from the value expected at equilibrium conditions at high temperature, only 25 % of *p*H_{2}, provides a source of strong non-thermal polarization. In the traditional PHIP method, *p*H_{2} is attached to a substrate molecule by using a suitable catalyst. When the equivalence of the *p*H_{2}-nascent protons is broken in
the reaction product, the non-thermal spin order can be converted into
observable magnetization, giving rise to significant NMR signal enhancements (Pravica and Weitekamp, 1988; Bowers and Weitekamp, 1987). PHIP can also
be transferred from the primarily polarized protons to other nuclei in the
product molecule to enhance their NMR signals. Alternatively, one can use
the signal amplification by reversible exchange (SABRE) method (Adams et
al., 2009; Barskiy et al., 2019; Duckett and Mewis, 2012), in which no
chemical modification of the substrate occurs. Instead, *p*H_{2} and the substrate bind to an Ir-based organometallic complex, where spin order conversion gives rise to polarization of the substrate. Subsequently, the hyperpolarized substrate molecule dissociates from the complex, contributing to polarization of the free substrate pool.

A related field is singlet-state NMR (Levitt, 2012; Carravetta and Levitt, 2004; Carravetta et al., 2004), dealing with slowly relaxing symmetry-protected spin states, which can be used to probe various slow processes and to store non-equilibrium spin polarization. In many molecules (Levitt, 2012; Carravetta and Levitt, 2004; Carravetta et al., 2004; Stevanato et al., 2015; Sheberstov et al., 2019; Zhou et al., 2017; Wang et al., 2017; Buratto et al., 2014; Vasos et al., 2009; Zhang et al., 2015; Franzoni et al., 2012; Kiryutin et al., 2019; DeVience et al., 2013) singlet-order relaxes much longer than spin magnetization for the reason that it is immune to some relaxation mechanisms, for instance, in a two-spin system dipolar relaxation cannot drive singlet–triplet transitions because the dipole–dipole interaction is invariant to the exchange of the two spins (Pileio, 2010). In singlet-state NMR experiments, spin magnetization is converted into singlet order by a suitable pulse sequence; singlet-state readout is also done by singlet-to-magnetization conversion using special pulse sequences.

In the cases of PHIP and singlet-state NMR a consideration of LACs often becomes important, in particular, in molecules with pairs of nearly equivalent spins (Ivanov et al., 2014; Pravdivtsev et al., 2013b; Franzoni et al., 2013, 2012; Sheberstov et al., 2019; Stevanato et al., 2015; DeVience et al., 2013; Theis et al., 2014a), such that the symmetry breaking is due to a very small chemical shift difference of the nuclei or due to their magnetic non-equivalence, i.e., due to slightly different couplings to other spins. Such symmetry breaking is usually a minor effect, giving rise to spin mixing only under special conditions, which correspond to LACs. In this situation, the approach proposed in this work can be useful for understanding the spin dynamics.

This contribution aims at a simple description of LAC-based coherent phenomena. We illustrate the concept presented here by a number of examples, in each case showing the scheme of energy levels and discussing the type of spin mixing. For numerical calculations, we used the “SpinDynamica” software package (Bengs and Levitt, 2018). We also anticipate that the present method is easy to exploit and widely applicable to treat magnetic resonance experiments, which utilize LACs.

## 2.1 Spin mixing at LACs

Before going into detail on the method, we would like to remind the reader of the LAC concept (von Neumann and Wigner, 1929) and characterize the efficiency of spin mixing at LACs.

By a level anti-crossing, or an avoided crossing, we mean the following situation. Let us imagine a spin system described by the Hamiltonian

comprising the main term ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ and a small perturbation $\widehat{\mathcal{V}}$; we imply that the Frobenius norm of the perturbation term is much smaller: $\u2225\widehat{\mathcal{V}}\u2225\ll \u2225{\widehat{\mathcal{H}}}_{\mathrm{0}}\u2225$. The perturbation term becomes relevant only under special conditions, namely, when the difference between energies dictated by the ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ term (eigenvalues of ${\widehat{\mathcal{H}}}_{\mathrm{0}}$) is small; i.e., the energy levels tend to cross. Let us consider this situation in more detail.

Hereafter, we assume that there is a parameters *x*, which one can control
experimentally: this can be the external magnetic field strength or the
strength of an applied radiofrequency (RF) field. Upon variation in *x*, the
energies, i.e., eigenvalues of the spin Hamiltonian, change. For simplicity,
we consider what happens to a pair of levels, corresponding to the
eigenstates |*ψ*_{k}〉 and |*ψ*_{l}〉 of the “unperturbed” Hamiltonian ${\widehat{\mathcal{H}}}_{\mathrm{0}}$, with energies ${\mathcal{E}}_{k}^{\mathrm{0}}$ and ${\mathcal{E}}_{l}^{\mathrm{0}}$; i.e., we consider the solutions of the eigenproblem ${\widehat{\mathcal{H}}}_{\mathrm{0}}\mathrm{|}{\mathit{\psi}}_{k,l}\rangle ={\mathcal{E}}_{k,l}^{\mathrm{0}}\mathrm{|}{\mathit{\psi}}_{k,l}\rangle $. The next step is to figure out how
the perturbation term affects the actual energies and the corresponding
eigenstates of the full Hamiltonian. When solving this problem, we assume
that the energies ${\mathcal{E}}_{k}^{\mathrm{0}}$ and ${\mathcal{E}}_{l}^{\mathrm{0}}$ closely approach each other in a certain range of *x* values, having a crossing at *x*=*x*_{0} so that ${\mathcal{E}}_{k}^{\mathrm{0}}\left({x}_{\mathrm{0}}\right)={\mathcal{E}}_{l}^{\mathrm{0}}\left({x}_{\mathrm{0}}\right)$. We also imply that all
other states |*ψ*_{m}〉 (where $m\ne k,l)$ are remote in
energy at *x*≈*x*_{0}. Below, we discuss the reason of making such an
assumption. To solve the problem, we need to do nothing else but diagonalize
the full Hamiltonian, including the perturbation term. To determine the
actual state energies, i.e., the eigenvalues of $\widehat{\mathcal{H}}$, we solve the
following equation for ℰ and obtain the energies:

For simplicity, here we assume that the perturbation term has only
off-diagonal elements ${\mathcal{V}}_{kl}=\langle {\mathit{\psi}}_{k}\left|\widehat{\mathcal{V}}\right|{\mathit{\psi}}_{l}\rangle $ in the basis |*ψ*_{k,l}〉 (when this is not true the diagonal terms are also modified with a consequence that the actual crossing point might move from *x*_{0} to ${x}_{\mathrm{0}}^{\prime}$). One can see that when 𝒱_{kl}≠0, there are
always two different solutions for the energy, ℰ_{k}≠ℰ_{l}. Even when the unperturbed levels do cross, ${\mathcal{E}}_{k}^{\mathrm{0}}\left({x}_{\mathrm{0}}\right)={\mathcal{E}}_{l}^{\mathrm{0}}\left({x}_{\mathrm{0}}\right)$, the levels of the total Hamiltonian are always different and cannot cross: the crossing is “avoided” and we obtain an LAC instead of the level crossing (LC); see Fig. 1a. Of course, the perturbation term is inactive when $\left|{\mathcal{V}}_{kl}\right|\ll \left|{\mathcal{E}}_{k}^{\mathrm{0}}-{\mathcal{E}}_{l}^{\mathrm{0}}\right|$ (since ${\mathcal{E}}_{k,l}\approx {\mathcal{E}}_{k,l}^{\mathrm{0}})$, but it
strongly affects the energies when $\left|{\mathcal{V}}_{kl}\right|\sim \left|{\mathcal{E}}_{k}^{\mathrm{0}}-{\mathcal{E}}_{l}^{\mathrm{0}}\right|$. The range of *x* values such that $\left|{\mathcal{V}}_{kl}\right|\sim \left|{\mathcal{E}}_{k}^{\mathrm{0}}-{\mathcal{E}}_{l}^{\mathrm{0}}\right|$ determines the LAC region. The minimal splitting between ℰ_{k} and ℰ_{l} is achieved at the
LC point *x*_{0} (also giving the center of the LAC region) being equal to
2|𝒱_{kl}|. According to the widely accepted terminology, the
energy levels ${\mathcal{E}}_{k,l}^{\mathrm{0}}$, corresponding to the unperturbed Hamiltonian, are diabatic levels, whereas the levels ℰ_{k,l}, corresponding to the full Hamiltonian, are adiabatic levels.

It is important to emphasize that LACs strongly affect spin dynamics, giving
rise to coherent spin mixing. To rationalize this, we need to solve the
eigenproblem of the full Hamiltonian $\widehat{\mathcal{H}}$. The two eigenstates
corresponding to the levels ℰ_{k}(*x*) and ℰ_{l}(*x*), are superposition states of |*ψ*_{k}〉 and |*ψ*_{l}〉:

The “mixing angle” *θ*_{kl} is defined via the off-diagonal
perturbation term and the difference of the unperturbed energies:

For the sake of simplicity, we assume that 𝒱_{kl} is real. The *θ*
angle goes to zero when the unperturbed levels are very different in energy
and the |*ψ*_{k}〉 and |*ψ*_{l}〉 states are the eigenstates of the spin system. However, in the LAC region *θ*≠0 and the |*ψ*_{k}〉 and |*ψ*_{l}〉 states are superpositions of the true eigenstates |*φ*_{k}〉 and |*φ*_{l}〉. Hence, if initially the |*ψ*_{k}〉 state is populated, the spin system will not stay in this state: the population will oscillate between the states |*ψ*_{k}〉 and |*ψ*_{l}〉. From Eq. (3) we notice that this effect is particularly pronounced at ${\mathcal{E}}_{k}^{\mathrm{0}}-{\mathcal{E}}_{l}^{\mathrm{0}}=\mathrm{0}$; i.e., at the
LC, $\left|\mathit{\theta}\right|=\frac{\mathit{\pi}}{\mathrm{4}}$. In this case $\mathrm{|}{\mathit{\phi}}_{k,l}\rangle =\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\left\{\mathrm{|}{\mathit{\psi}}_{k}\rangle \pm \mathrm{|}{\mathit{\psi}}_{l}\rangle \right\}$ meaning that the population can be completely transferred between
the states |*ψ*_{k}〉 and |*ψ*_{l}〉. This is exactly the
way how LACs can be exploited: spin mixing at LACs can be utilized to
perform a complete transfer of the population from one state to another. In
Fig. 1b we demonstrate how the coefficients *c*_{kl} and ${c}_{kl}^{\prime}$,
which describe the state mixing, change upon variation in the *x* parameter:
away from the LAC one of them goes to 1 and the other one goes to 0, whereas
in the LAC region both of them are non-zero. When an LC is not turned into an LAC, mixing does not occur – for this reason, LCs are of no significance for
this work.

Here we consider two different ways of transferring population between the
diabatic states. The first method utilizes coherent spin mixing at the LAC.
The idea is that away from the LAC we prepare the spin system in an
unperturbed state, for clarity, in |*ψ*_{k}〉. A fast
(non-adiabatic) jump to *x*=*x*_{0} will keep the state the same, but |*ψ*_{k}〉 now becomes a superposition of the true eigenstates

The wave function will change in time, since the two eigenstates have
different energies (having an LAC is equivalent to having two different
energies). At time *t*, the wave function becomes (we express the energy in
units)

and the populations of the unperturbed state are (here we substitute ${\mathcal{E}}_{k}-{\mathcal{E}}_{l}=\mathrm{2}{\mathcal{V}}_{kl})$

Hence, the population oscillates between the states |*ψ*_{k}〉 and
|*ψ*_{l}〉; at $t=\mathit{\pi}/\mathrm{2}{\mathcal{V}}_{kl}$ the populations are inverted. If we
bring the system out of the LAC at this instant of time the population will
be transferred from |*ψ*_{k}〉 to |*ψ*_{l}〉. When *x*≠*x*_{0}, coherent spin mixing can still take place but the efficiency of
population exchange is reduced (e.g., population inversion is no longer
possible).

Another possibility to transfer the population is to perform a slow
(adiabatic) passage through the LAC. When the adiabaticity condition is
fulfilled, meaning that the rate of variation in |*φ*_{k,l}〉 is
much smaller than the intrinsic evolution frequency of the spin system
|ℰ_{k}−ℰ_{l}|, the populations adjust to the slow variation in the adiabatic eigenstates. As a consequence, the populations of the
adiabatic eigenstates |*φ*_{k,l}〉 do not change upon passage
through the LAC. This means that the populations of the diabatic states
|*ψ*_{k,l}〉 are swapped: *p*_{k}→*p*_{l} and *p*_{l}→*p*_{k}.
Hence, like in the previous case, a complete exchange of the populations takes
place. When complete adiabaticity is not achieved, the populations are not
swapped, but partially redistributed. This effects can be taken into account by using the Landau–Zener
approach (Zener, 1932). Specifically, assuming that initially
*p*_{k}=1 and *p*_{l}=0, after a passage through an LAC we obtain the
following state populations (Zener, 1932):

where ${\mathcal{F}}_{kl}=\frac{\mathrm{d}}{\mathrm{d}t}\left|{\mathcal{E}}_{k}-{\mathcal{E}}_{l}\right|$ gives the rate at which the splitting between the diabatic levels changes in time (in the Landau–Zener approach this speed is taken as being constant).

In many cases, adiabatic passage gives better results as compared to a coherent exchange of populations, being more robust to inaccuracies in
setting the parameters of the spin Hamiltonian. Indeed, spin mixing using
coherences requires that *x* is precisely set to satisfy the LC condition
for the Hamiltonian ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ and the timing is controlled. In the case
of adiabatic passage, it is sufficient to pass through the LAC region slowly
enough. One should note, however, that as far as the transfer time is
concerned, coherent population exchange is preferable, since it takes less
time (an adiabatic process always requires a relatively slow variation in
the control parameter).

We illustrate how population exchange can take place for spin $\frac{\mathrm{1}}{\mathrm{2}}$, i.e., in a two-level system, which is described by the following Hamiltonian:

Hence, a time-dependent field is applied along the *z* axis; additionally
there is a constant *x* field. The system has an LAC at zero magnetic field,
where the |*α*〉 and |*β*〉 eigenstates of ${\widehat{\mathcal{H}}}_{\mathrm{0}}$
have a crossing, which is avoided due to the presence of the
perturbation term. As usual, by |*α*〉 and |*β*〉 we hereafter denote the spin-$\frac{\mathrm{1}}{\mathrm{2}}$ states with the *z* projection of
$+\frac{\mathrm{1}}{\mathrm{2}}$ and $-\frac{\mathrm{1}}{\mathrm{2}}$, respectively.

If we assume that initially the system is in the |*α*〉
state, a possible way to perform the |*α*〉→|*β*〉
population transfer is to introduce a non-adiabatic jump to zero field,
where the true eigenstates, $\left(\mathrm{|}\mathit{\alpha}\rangle \pm \mathrm{|}\mathit{\beta}\rangle \right)/\sqrt{\mathrm{2}}$ are superposition states of |*α*〉 and |*β*〉. In this situation, according to Eq. (7), the population oscillates between |*α*〉 and |*β*〉, as shown in Fig. 2a.

Another possibility is to perform an adiabatic passage through the LAC, by
varying the *z* component of the field, so that *ω*_{z} goes from a
negative value −*ω*_{max} to a positive value +*ω*_{max}. Here we assume that *ω*_{max}≫*ω*_{x} and that the time dependence of *ω*_{z} is a linear dependence:

with *τ*_{sw} being the duration of the switch. The resulting state
populations would then follow from Eq. (8), with the Landau–Zener parameter
equal to

The resulting state populations are shown in Fig. 2b; for comparison we also show the result of a numerical simulation of the spin dynamics with a $\widehat{\mathcal{H}}\left(t\right)$ time-dependent Hamiltonian.

We would like to emphasize that in some cases the mixing matrix element is
zero; however, when the states |*ψ*_{k}〉 and |*ψ*_{l}〉 are both coupled to a third state |*ψ*_{m}〉 the basis wave functions also become perturbed and a mixing matrix element 𝒱_{kl} effectively becomes non-zero. In Appendix A, we explain how to calculate 𝒱_{kl} in this case, corresponding to degenerate perturbation theory. Hence, the two states |*ψ*_{k}〉 and |*ψ*_{l}〉 are never mixed (and the LC is never turned to an LAC) only when the Hamiltonian $\widehat{\mathcal{H}}$ is block-diagonal and these two states belong to different blocks.

## 2.2 Theoretical framework

The idea of this paper is to describe how spin order changes due to coherent
spin mixing at LACs. In all cases, we consider processes, in which a certain
parameter *x*(*t*) is varied so that the spin Hamiltonian ${\widehat{\mathcal{H}}}_{\mathrm{0}}\left(x\right)$ also varies with time and the system goes through LCs, which are turned into LACs by the $\widehat{\mathcal{V}}$ term. In the following, we make several assumptions.

First, we consider the initial and final spin states characterized by the
density matrices *ρ*_{i} and *ρ*_{f}, which are diagonal in the
eigenbasis of the Hamiltonian:

where $\mathrm{|}{\mathit{\psi}}_{m}^{\mathrm{i}}\rangle $ and $\mathrm{|}{\mathit{\psi}}_{m}^{\mathrm{f}}\rangle $ stand for the
diabatic eigenstates of the initial and final unperturbed Hamiltonian
${\widehat{\mathcal{H}}}_{\mathrm{0}}$. We also assume that the eigenstates of ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ can be
determined analytically at any *x* value, which is possible in many cases
when the perturbation term is dropped off. A consideration of the coherences
can be complicated, as they give rise to complex phenomena, e.g., those
described by Berry's phase (Zwanziger et al., 1990; Berry,
1984). Here we avoid such complexities assuming that the initial state is
adjusted such that the density matrix *ρ*_{i} is diagonal in the
eigenbasis of the initial Hamiltonian. This means that instead of the
density matrix we can use a vector of state populations, |*ρ*),
introduced in the following way:

Here |*ψ*_{m})=|*ψ*_{m}〉〈*ψ*_{m}| define the operator basis for the density
matrix. Note that the curly bracket introduced in this way does not
correspond to the bracket notations, used to define wave functions in quantum
mechanics (we deliberately use a different type of brackets). It is easy to
see, that this basis is orthonormal as $\left({\mathit{\psi}}_{m}\phantom{\rule{0.125em}{0ex}}\mathrm{|}\phantom{\rule{0.125em}{0ex}}{\mathit{\psi}}_{k}\right)\equiv \text{Tr}\left\{\left|{\mathit{\psi}}_{m}\rangle \langle {\mathit{\psi}}_{m}\right|{\mathit{\psi}}_{k}\rangle \langle {\mathit{\psi}}_{k}\mathrm{|}\right\}={\mathit{\delta}}_{mk}$. Since we deal with population vectors, in Eq. (10) we omit all terms |*ψ*_{m}〉〈*ψ*_{n}| when *m*≠*n*.

Second, we assume that the spin dynamics are described entirely in terms of
redistribution of the populations, occurring at LACs. The idea is that we
can determine the LC points for the levels of ${\widehat{\mathcal{H}}}_{\mathrm{0}}$, figure out
whether the LCs are turned into LACs by the $\widehat{\mathcal{V}}$ term and assume that
at each LAC redistribution of the corresponding state populations is taking
place. This means that after mixing at the LAC between |*ψ*_{k}〉
and |*ψ*_{l}〉 the populations of the diabatic eigenstates change as
follows:

Here Δ_{kl} stands for the mixing efficiency, which is
varied between zero and 1. Hence, we keep in mind that the exchange of the
populations may be incomplete, for instance, when the time of the coherent
evolution at the LAC is not optimized or when the adiabaticity condition is
not perfectly fulfilled. When Δ_{kl}=1, the populations are
swapped; when Δ_{kl}=0, there is no population exchange
taking place. The precise Δ_{kl} value can be determined by
simulating the spin dynamics at the LAC. For coherent spin mixing and
adiabatic passage, Δ_{kl} can be determined from Eqs. (7) and (8), respectively.

Third, we assume that LACs are isolated from each other, meaning that the
spin mixing is occurring independently at different LACs. For instance, the
region of LAC occurring between the states |*ψ*_{k}〉 and |*ψ*_{l}〉 should not overlap with that of the LAC between the states
|*ψ*_{k}〉 and |*ψ*_{m}〉. LACs between different pairs of
states are allowed to occur at similar values of *x*. Under such assumptions
we can describe the spin dynamics in terms of a pairwise redistribution of
populations at isolated LACs.

Finally, we need to consider that the eigenstates of the Hamiltonian
${\widehat{\mathcal{H}}}_{\mathrm{0}}$ can differ when the *x* parameter is varied: a “rotation” of
the eigenbasis can take place. The state basis $\mathrm{|}{\mathit{\psi}}_{m}^{\mathrm{f}}\rangle $ is
then “tilted” with respect to the basis $\mathrm{|}{\mathit{\psi}}_{m}^{\mathrm{i}}\rangle $. Hence,
when we compute an expectation value of a certain spin operator in the basis
of $\mathrm{|}{\mathit{\psi}}_{m}^{\mathrm{f}}\rangle $ states, it might correspond to a different
operator in the $\mathrm{|}{\mathit{\psi}}_{m}^{\mathrm{i}}\rangle $ basis. This happens, for instance,
when the direction of a quantization axis changes upon variation in *x*. We
will discuss such examples separately.

Using these assumptions, we can formulate the theory for evaluating the spin
evolution driven by LACs. Redistribution of the diabatic state populations
given by Eq. (12) can be described by an operator ${\widehat{\mathrm{\Pi}}}^{\left(kl\right)}\left({\mathrm{\Delta}}_{kl}\right)$, hereafter, termed “population
redistribution operator”, which is a square matrix with the following
non-zero elements (here *δ*_{mn} is the Kronecker delta):

This operator can be explicitly written as

One can see that this operator does not change the populations of states $m\ne k,l$. When acting on a certain state, which gets mixed with another state, for example, $\mathrm{|}{\mathit{\psi}}_{k}^{\mathrm{i}})$, we obtain

This expression agrees with Eq. (13). Acting on the vector of populations by ${\widehat{\mathrm{\Pi}}}^{\left(kl\right)}$ we get the result

The elements of the new population vector
|*ρ*^{′}) are ${p}_{j}^{\prime}={p}_{j}$ for $j\ne k,l$; the ${p}_{k}^{\prime}$ and ${p}_{l}^{\prime}$ populations are given by Eq. (12). To be more precise, one should term $\widehat{\mathrm{\Pi}}$ “super operator” (as it is an operator acting in the operator space); however, we do not use double “hats” and omit this complexity for the sake of brevity.

If the system passes through a sequence of LACs (occurring in pairs of state $kl,\mathrm{\dots},pq,rs$), the resulting redistribution operator is

The operators, describing population redistribution at subsequent LACs, are multiplied one after another from right to left to obtain the resulting operator $\widehat{\mathrm{\Pi}}$.

In some cases, the actual permutation of the state populations is performed
via several consecutive permutations, for example, $i\to p\to f$. Such a
sequence of simple permutations gives rise to a more complex permutation.
When Δ=1 for each permutation, the actual form of the
$\widehat{\mathrm{\Pi}}$ operator is simplified, corresponding to cyclic permutation.
For instance, for permutations $i\to p\to f$ we obtain $\widehat{\mathrm{\Pi}}={\widehat{\mathrm{\Pi}}}^{\left(pf\right)}\left(\mathrm{1}\right)\cdot {\widehat{\mathrm{\Pi}}}^{\left(ip\right)}\left(\mathrm{1}\right)$. This is
equivalent to the following permutations: *i*→*f*, *f*→*p*, *p*→*i*. In
this work we will mostly consider spin order transfer pathways with a single
permutation. Nevertheless, we also discuss cases where more complex
permutations come into play (Rodin et al., 2020).

Knowing the final vector or state populations, we are able to evaluate the final density matrix from Eq. (15) and to compute the expectation values of a spin operator ${\widehat{Q}}_{A}$ of interest:

It is important to note that for many operators the *Q*_{A} expectation
value will be zero because all off-diagonal elements of the density matrix
are zero. In some cases, it is desirable to express the resulting spin order
in the eigenbasis $\mathrm{|}{\mathit{\psi}}_{k}^{\mathrm{i}}\rangle $ of the initial Hamiltonian: an
additional transformation is then required described by a basis rotation
operator ${\widehat{\mathrm{\Psi}}}_{i\to f}$ (whereas ${\widehat{\mathrm{\Psi}}}_{f\to i}$ gives the
inverse transformation). If we then express the final density matrix in the
initial $\mathrm{|}{\mathit{\psi}}_{k}^{\mathrm{i}}\rangle $ basis, it becomes the following:

In some cases, the basis rotation is equivalent to a physical rotation of
spins in the three-dimensional space. In this situation, we can introduce
the rotation axis **n** and the rotation angle *ϑ*,
so that ${\widehat{\mathrm{\Psi}}}_{i\to f}={\widehat{\mathrm{\Psi}}}_{\mathbf{n}}\left(\mathit{\vartheta}\right)$, where ${\widehat{\mathrm{\Psi}}}_{\mathbf{n}}\left(\mathit{\vartheta}\right)$ is the super operator describing the actual rotation (all super operators here are denoted by capital Greek letters). If the operator generating the basis rotation is given by $(\mathbf{n}\cdot \widehat{\mathbf{I}})$, the basis rotation corresponds to the physical spin rotation. In the general
case, rotation in the Hilbert space does not necessarily correspond to
rotation in the physical 3D space. Using Eq. (18), we can evaluate the expectation value of any operator of interest.

Basis rotation becomes an important concern in some NMR experiments: an example is given by our recent work (Rodin et al., 2020) on “algorithmic cooling” of a spin system exploiting long-lived singlet order. The protocol for algorithmic cooling requires specific permutations of state populations in a four-level system, which are carried out by using NMR pulses with adiabatically increased or decreased field strength (which make use of adiabatic passage through LACs). Such pulses not only swap state populations but also rotate the basis of spin eigenstates. Consequently, additional pulses are required to compensate for this effect (Rodin et al., 2020). Examples, in which basis rotation is taking place, are discussed below in Sect. 3.4.

The conversion of spin order can be illustrated by a diagram, as the one
depicted in Fig. 3. In the diagram above, we plot the *x*(*t*)
trajectory in a schematic way, showing only the passages through LACs or
jumps to LACs. In the diagram below, we show the energy levels as functions
of *x* and indicate the pathway for redistribution of the state populations.
The resulting spin order can be represented by the populations of the
eigenstates $\mathrm{|}{\mathit{\psi}}_{n}^{\mathrm{f}}\rangle $ in the cases of either complete
population exchange or partial redistribution of the populations.

In this section, we consider a number of examples of LC- or LAC-based analysis of the spin dynamics. In each case, we start from introducing the ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ Hamiltonian (along with its eigenvalues) and the perturbation term $\widehat{\mathcal{V}}$. After that, we explain how the spin order of the system is modified due to the evolution at LACs.

## 3.1 Adiabatic zero-field passage

The first example we consider here is given by adiabatic inversion of the
external magnetic field **B**||*z*. The simplest
example is given by a two-spin system with spins *I* and *S* of different
kind, i.e., two heteronuclei with the gyromagnetic ratios *γ*_{I}≠*γ*_{S}.

The Hamiltonian of the spin system is given by expression (in
units; here *J*_{IS} is the coupling strength, given in Hz)

Here we assume that the first two terms and the secular part of the coupling term give the main Hamiltonian,

while the non-secular coupling term is a perturbation:

The unperturbed states of the spin system are the Zeeman states |1〉=|*α**α*〉, |2〉=|*α**β*〉, |3〉=|*β**α*〉 and |4〉=|*β**β*〉.

When *B*=0, the unperturbed energy levels cross: |*α**α*〉
crosses with |*β**β*〉 and |*α**β*〉 crosses with
|*β**α*〉. The first LC cannot be turned into an LAC, since
$\langle \mathit{\alpha}\mathit{\alpha}\left|\widehat{\mathcal{V}}\right|\mathit{\beta}\mathit{\beta}\rangle =\mathrm{0}$, but the second LC is turned into an LAC by the perturbation term, since $\langle \mathit{\alpha}\mathit{\beta}\left|\widehat{\mathcal{V}}\right|\mathit{\beta}\mathit{\alpha}\rangle =\mathit{\pi}{J}_{IS}$. The true LCs are completely irrelevant for spin mixing, but at the LAC the populations of the states |2〉 and |3〉 can be exchanged. The energy levels are schematically shown in Fig. 4. One can see that there are two more LCs at *B*≠0 (an LC at a positive field and an LC at a negative field), which are never turned to LACs when the Hamiltonian has the form given by Eq. (19) because the corresponding states are characterized by different values of the *z* projection of the total spin $\widehat{\mathbf{F}}=\widehat{\mathbf{I}}+\widehat{\mathbf{S}}$ and are
not mixed by the perturbation term. However, mixing at this LC may become a
concern (Lukzen and Steiner, 1995) in the presence of an additional transverse field. Discussing such effects is beyond the scope of this work.

If we assume that the two spins have different polarizations, the initial density matrix is given by expression

The coefficients ${M}_{I}=\text{Tr}\left\{{\widehat{I}}_{z}\mathit{\rho}\right\}$ and
${M}_{S}=\text{Tr}\left\{{\widehat{S}}_{z}\mathit{\rho}\right\}$ (here we omit division
by $\text{Tr}\left\{{\widehat{I}}_{z}^{\mathrm{2}}\right\}$ and $\text{Tr}\left\{{\widehat{S}}_{z}^{\mathrm{2}}\right\}$, which are equal to 1) give the polarizations of
the two nuclei, which are taken to be different; *M*_{I}≠*M*_{S}. They can also
be determined directly from the populations as ${M}_{I}={p}_{\mathit{\alpha}\mathit{\alpha}}+{p}_{\mathit{\alpha}\mathit{\beta}}-{p}_{\mathit{\beta}\mathit{\alpha}}-{p}_{\mathit{\beta}\mathit{\beta}}$ and ${M}_{S}={p}_{\mathit{\alpha}\mathit{\alpha}}-{p}_{\mathit{\alpha}\mathit{\beta}}+{p}_{\mathit{\beta}\mathit{\alpha}}-{p}_{\mathit{\beta}\mathit{\beta}}$, ranging from 1 to −1. The population vector in the basis of Zeeman states, $\mathbb{Z}=\left\{\mathit{\alpha}\mathit{\alpha},\phantom{\rule{0.125em}{0ex}}\mathit{\alpha}\mathit{\beta},\phantom{\rule{0.125em}{0ex}}\mathit{\beta}\mathit{\alpha},\phantom{\rule{0.125em}{0ex}}\mathit{\beta}\mathit{\beta}\right\}$,
is as follows:

Now we consider a passage through zero field from −*B*_{0} to +*B*_{0},
assuming that $\left|{\mathit{\gamma}}_{I}-{\mathit{\gamma}}_{S}\right|{B}_{\mathrm{0}}\gg \mathrm{2}\mathit{\pi}\mathrm{|}{J}_{IS}\mathrm{|}$ (this condition simply means that at $B=\pm {B}_{\mathrm{0}}$ the spin
system is away from the LAC region). If we redistribute the populations of
the states |2〉 and |3〉 by an adiabatic passage through the LAC, we arrive at the following expression for the populations:

Rewriting the ${\widehat{I}}_{z}$ and ${\widehat{S}}_{z}$ operators in their vector form (i.e., omitting zero off-diagonal elements),

we can determine the polarization values:

Hence, redistribution of polarizations occurs. When the efficiency
Δ=1, we obtain the spins exchange polarizations,
${M}_{I}^{\prime}={M}_{S}$ and ${M}_{S}^{\prime}={M}_{I}$, in accordance with an earlier
result on polarization transfer in electron–nuclear systems
(Lukzen and Steiner, 1995). The actual efficiency can be
estimated from Eq. (8), by evaluating the parameter $\frac{\mathrm{2}\mathit{\pi}{\left|{\mathcal{V}}_{kl}\right|}^{\mathrm{2}}}{{\mathcal{F}}_{kl}}$. In the present case, 𝒱_{kl}=*π**J*_{IS} and
${\mathcal{F}}_{kl}=\frac{\mathrm{1}}{\mathrm{2}}\left({\mathit{\gamma}}_{I}-{\mathit{\gamma}}_{S}\right)\frac{\mathrm{d}B}{\mathrm{d}t}$.

Polarization transfer can be carried out in other ways. For instance, one
can perform a non-adiabatic jump ${B}_{\mathrm{0}}\to B=\mathrm{0}$, i.e., to the LAC, to
convert the population difference (*p*_{αβ}−*p*_{βα}) into the coherences between the new eigenstates $\mathrm{|}\mathrm{2},\mathrm{3}\rangle =\left\{\mathrm{|}\mathit{\alpha}\mathit{\beta}\rangle \pm \mathrm{|}\mathit{\beta}\mathit{\alpha}\rangle \right\}/\sqrt{\mathrm{2}}$. As explained above, by controlling the evolution time *t*_{mix} at zero field, one can change the sign of the coherence. After that, a non-adiabatic field jump to *B*_{0} will swap the populations of the states
|2〉 and |3〉. If we assume that the mixing efficiency Δ is less than 1, we get the general result given by Eq. (24). As follows from Eq. (7), the optimal mixing time, which guarantees
Δ→1, is achieved when 2𝒱_{kl}*t*_{mix}=*π*; i.e., ${t}_{\mathrm{mix}}=\mathrm{1}/{J}_{IS}$.

In this context, it is useful to consider a more complex problem of
enhancing NMR signals of “insensitive” nuclei, such as ^{13}C or
^{15}N, by transferring PHIP upon adiabatic passage through zero field. This method has been successfully implemented (Eills et al., 2019) to polarize ^{13}C nuclei in a system of two protons prepared in the
singlet spin state and a carbon nucleus. In this case of two protons (spins
*I*_{a} and *I*_{b}) coupled to a ^{13}C nucleus (spin *S*), the spin
Hamiltonian takes the form

The proton–proton coupling is *J*_{HH}; the coupling on the first proton and
second proton to the carbon nucleus are denoted as *J*_{aS} and *J*_{bS}.
The key issue is how to separate the Hamiltonian into two parts. Hereafter,
we follow the results of Eills et al. (2019) introducing the main Hamiltonian as (keeping Zeeman interactions, proton–proton coupling and the secular part of the heteronuclear couplings)

and the perturbation as

For ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ the eigenbasis of states is the
“singlet–triplet–Zeeman” basis. In Eills et al. (2019) such a basis is introduced in two ways. The obvious one is to use the basis $\mathbb{S}\mathbb{T}\mathbb{Z}={\left\{\mathrm{|}S\rangle ,\phantom{\rule{0.125em}{0ex}}\mathrm{|}{T}_{+}\rangle ,\mathrm{|}{T}_{\mathrm{0}}\rangle ,\phantom{\rule{0.125em}{0ex}}\mathrm{|}{T}_{-}\rangle \right\}}_{\mathrm{12}}\otimes {\left\{\mathrm{|}\mathit{\alpha}\rangle ,\mathrm{|}\mathit{\beta}\rangle \right\}}_{S}$.
This is the singlet–triplet basis of the *I* spins and Zeeman basis of the
*S* spin. As usual, the singlet–triplet states are

However, one should note that the true eigenbasis of ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ is given by $\mathbb{S}\mathbb{T}{\mathbb{Z}}^{\prime}\ne \mathbb{S}\mathbb{T}\mathbb{Z}$, which takes into account
that the only the states |*T*_{±}*α*〉 and |*T*_{±}*β*〉 are true eigenstates of ${\widehat{\mathcal{H}}}_{\mathrm{0}}$, while the other four states are superposition states of |*S**α*〉, |*S**β*〉, |*T*_{0}*α*〉 and |*T*_{0}*β*〉. However, when *J*_{HH} is significantly larger than the other two couplings in the spin system, the following expressions hold approximately: $\mathrm{|}S\mathit{\alpha}{\rangle}^{\prime}\approx \mathrm{|}S\mathit{\alpha}\rangle $, $\mathrm{|}S\mathit{\beta}{\rangle}^{\prime}\approx \mathrm{|}S\mathit{\beta}\rangle $, $\mathrm{|}{T}_{\mathrm{0}}\mathit{\alpha}{\rangle}^{\prime}\approx \mathrm{|}{T}_{\mathrm{0}}\mathit{\alpha}\rangle $ and $\mathrm{|}{T}_{\mathrm{0}}\mathit{\beta}{\rangle}^{\prime}\approx \mathrm{|}{T}_{\mathrm{0}}\mathit{\beta}\rangle $. In this situation, assuming a special case of the spin system prepared in the |*S*〉 state of the *I* spins, we can approximately set only four populations to a non-zero value, namely, the populations of the |*S**α*〉^{′}, |*S**β*〉^{′}, |*T*_{0}*α*〉^{′} and
|*T*_{0}*β*〉^{′}.

In the spin system, there is a number of LCs and LACs; see Fig. 5. At
zero field, in any multi-spin system there are always several LCs present
(for symmetry reasons, groups of spin states become degenerate): in the
present case six levels with a proton triplet character are degenerate, as
are the two states having a singlet character. There is also a number of
LCs at non-zero fields; however, not all of them are turned into LACs. The
reason is the same as in the case of an *I**S* two-spin system: all terms in
$\widehat{\mathcal{H}}$ do not alter the *z* projection of all three spins;
$\widehat{\mathbf{F}}={\widehat{\mathbf{I}}}_{a}+{\widehat{\mathbf{I}}}_{b}+\widehat{\mathbf{S}}$. For this reason, we need to consider only four LCs, which turn into LACs. The LC positions have been determined in the previous work (Eills et al., 2019); they are as follows (${B}_{\mathrm{LC}}^{\left(\mathrm{1}\right)}$ and ${B}_{\mathrm{LC}}^{\left(\mathrm{2}\right)}$):

where ${J}_{\mathrm{\Sigma}}={J}_{\mathrm{1}S}+{J}_{\mathrm{2}S}$; this expression is valid when $\left|{J}_{HH}\right|\gg \left|{J}_{\mathrm{1}S}-{J}_{\mathrm{2}S}\right|$. Upon adiabatic passage $-{B}_{\mathrm{0}}\to +{B}_{\mathrm{0}}$ (where ${B}_{\mathrm{0}}\gg {B}_{\mathrm{LC}}^{\left(\mathrm{1}\right)},{B}_{\mathrm{LC}}^{\left(\mathrm{2}\right)})$, the following population swapping occurs:

Strictly speaking, upon the field inversion two more population swaps occur: additionally there are population swaps of the kind $\mathrm{|}{T}_{\mathrm{0}}\mathit{\beta}{\rangle}^{\prime}\u27f7\mathrm{|}{T}_{-}\mathit{\alpha}{\rangle}^{\prime}$ and $\mathrm{|}{T}_{+}\mathit{\beta}{\rangle}^{\prime}\u27f7\mathrm{|}{T}_{\mathrm{0}}\mathit{\alpha}{\rangle}^{\prime}$. Hence, in both state manifolds with ${F}_{z}=\pm \frac{\mathrm{1}}{\mathrm{2}}$, we have cyclic permutations of the populations of three states. However, initially only one of the three states of each manifold (the one with singlet character of the protons) is populated, which simplifies the description. Specifically, in the ${F}_{z}=+\frac{\mathrm{1}}{\mathrm{2}}$ manifold it is sufficient to consider a single population swap, whereas in the ${F}_{z}=-\frac{\mathrm{1}}{\mathrm{2}}$ manifold two swaps should be taken into account.

The initial density matrix in the case under study can be written as

After the adiabatic swap, the final density matrix becomes (when Δ=1 for the relevant LACs)

As a result, the singlet order is converted into *z* polarization of protons
and *S* spins. The polarizations of the *I* spins and *S* spins becomes (if
we assume that only two states are populated at *B*=*B*_{0})

Hence, the singlet order is converted into the polarization of the *I* spins
and *S* spins; *M*_{I} and *M*_{S} are the same in size but have opposite
signs, since the *F*_{z} value is conserved. To optimize the conversion
efficiency, so that Δ→1, one can use Eq. (8).

Spin order transfer in this system can be carried out in a different
(perhaps, simpler) way. For instance, one can perform a sweep from *B*=0 to
+*B*_{0}: the populations are swapped between the states $\mathrm{|}S\mathit{\alpha}{\rangle}^{\prime}\leftrightarrow \mathrm{|}{T}_{+}\mathit{\beta}{\rangle}^{\prime}$, whereas the population of the |*S**β*〉^{′} state remains the same. One more possibility is to perform a non-adiabatic field jump ${B}_{\mathrm{0}}\to {B}_{\mathrm{LC}}^{\left(\mathrm{1}\right)}$ to generate the coherence between the states |*S**α*〉^{′}
and $\mathrm{|}{T}_{+}\mathit{\beta}{\rangle}^{\prime}$, let it evolve for half a period and
perform a field jump ${B}_{\mathrm{LC}}^{\left(\mathrm{1}\right)}\to {B}_{\mathrm{0}}$. If the timing is properly set, the states |*S**α*〉^{′} and $\mathrm{|}{T}_{+}\mathit{\beta}{\rangle}^{\prime}$ exchange populations. In both cases, there is a single step of redistributing the populations. The resulting spin order is the same as in the case of the adiabatic field inversion. The experiments exploiting adiabatic passage are, most likely, easier to implement as they do not require precise control of the timing. The optimal mixing time can be evaluated using Eq. (7).

## 3.2 Cross-polarization

Cross-polarization (CP) is a widely used method (Hartmann and Hahn, 1962; Pines et al., 1972; Hediger et al., 1994) to enhance NMR signals of
“rare” nuclei in high-field NMR experiments, in particular, in solid-state
NMR. The idea of CP is to transfer polarization from protons, hereafter
denoted as *I* spins, to insensitive nuclei, hereafter *S* spins. Here
we consider polarization transfer in a two-spin *I**S* system with *γ*_{I}>*γ*_{S}.

In the CP experiment (Hartmann and Hahn, 1962), see Fig. 6a, the
*I* spins are first flipped by a 90^{∘} pulse, here a 90_{y} pulse, and then the transverse magnetization is locked by a continuous-wave
(CW) pulse. After that, an RF pulse is applied at the frequency of the *S*
spins. When the amplitudes of the two RF fields are set in a proper way (see
explanation below), the transverse polarization is transferred from the *I*
spins to *S* spins. To detect this polarization, the RF field applied to the
*S* spins is instantaneously turned off. Polarization transfer enables
the enhancement of the NMR signals of the *S* spins due to the transfer of the
higher polarization of the *I* spins.

To describe this experiment, we write down the Hamiltonian in the doubly rotating frame:

In such a frame the Zeeman interactions of the two spins are
time-independent; for simplicity we assume that they are applied exactly on
resonance so that the spins interact only with the RF fields; here ${\mathit{\omega}}_{\mathrm{1}I}=-{\mathit{\gamma}}_{I}{B}_{\mathrm{1}I}$ and ${\mathit{\omega}}_{\mathrm{1}S}=-{\mathit{\gamma}}_{S}{B}_{\mathrm{1}S}$. The
coupling term ${\widehat{\mathcal{H}}}_{C}$ is time-dependent and contains contributions which oscillate at the frequencies *ω*_{I}, *ω*_{S}, (*ω*_{I}+*ω*_{S}) and (*ω*_{I}−*ω*_{S}). Such terms rapidly average out to zero; the only exception is
given by the *z**z* term, ${\widehat{\mathcal{H}}}_{zz}={H}_{zz}{\widehat{I}}_{z}{\widehat{S}}_{z}$, which commutes with ${e}^{i{\mathit{\omega}}_{S}t{\widehat{S}}_{z}}$ and ${e}^{i{\mathit{\omega}}_{I}t{\widehat{I}}_{z}}$ and remains time-independent in the doubly rotating frame.

In the following, it is convenient to go to the doubly tilted frame, in
which the quantization axes are parallel to the effective fields, i.e., to
the *x* axes of the doubly rotating frame. In the new frame, the Hamiltonian
takes the form

These expressions are obtained from Eq. (32) by making a substitution of
spin operators: ${\widehat{I}}_{x}$, ${\widehat{S}}_{x}\to {\widehat{I}}_{z}$, ${\widehat{S}}_{z}$
and ${\widehat{I}}_{z}{\widehat{S}}_{z}\to {\widehat{I}}_{x}{\widehat{S}}_{x}$. This can also be
achieved by using the operator of frame rotation ${\widehat{U}}_{IS}=\mathrm{exp}\left[i\frac{\mathit{\pi}}{\mathrm{2}}{\widehat{I}}_{y}\right]\mathrm{exp}\left[i\frac{\mathit{\pi}}{\mathrm{2}}{\widehat{S}}_{y}\right]$ and “sandwiching” the Hamiltonian ${\widehat{\mathcal{H}}}_{\mathrm{drf}}$ between ${\widehat{U}}_{IS}$ and ${\widehat{U}}_{IS}^{-\mathrm{1}}$. The initial state of the spin system in the doubly tilted frame can be described by the following density matrix (*I* spins are polarized along the corresponding RF field):

The eigenstates of ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ are obviously the Zeeman states |1〉=|*α**α*〉, |2〉=|*α**β*〉, |3〉=|*β**α*〉 and |4〉=|*β**β*〉. In this basis the density matrix in Eq. (34) can be written as

The perturbation term, which contains the raising and lowering spin
operators, can mix the states |1〉 and |4〉 as well as |2〉 and |3〉. When *ω*_{1I} and *ω*_{1S} are of the same sign, the states |2〉 and |3〉 have a crossing which can be turned into LAC by the $\widehat{\mathcal{V}}$ term; see Fig. 6b. The LC condition

is known as the Hartmann–Hahn condition (Hartmann and Hahn, 1962). In accordance with this condition, the fields *B*_{1I} and *B*_{1S} should be set inversely proportional to the corresponding gyromagnetic ratios; i.e., $\frac{{B}_{\mathrm{1}S}}{{B}_{\mathrm{1}I}}=\frac{{\mathit{\gamma}}_{I}}{{\mathit{\gamma}}_{S}}$. By virtue of the
perturbation term, the populations of the states |2〉 and |3〉 are redistributed and polarization transfer takes place. As a result, the density matrix takes the form

Hence, in the ideal case ${M}_{S}^{\prime}=\left({M}_{S}\mathrm{|}{\mathit{\rho}}^{\prime}\right)\stackrel{\mathrm{\Delta}\to \mathrm{1}}{\u27f6}{M}_{I}$ and *z* polarization is completely transferred to the *S* spin. In
the non-tilted rotating frame, this would correspond to the transfer of
transverse polarization among the spins of the heteronuclei.

The CP experiment can be done in a different way (Metz et al., 1994). The RF field *B*_{1S} can be increased in an adiabatic fashion from a
value below $\frac{{\mathit{\gamma}}_{I}}{{\mathit{\gamma}}_{S}}{B}_{\mathrm{1}I}$ (corresponding to the
LC) to a value above this field, in order to enable passage through the LAC.
The result of such an experiment, ramped CP, will be the same as for
conventional CP: passage through the LAC will enable population swapping
between the same states: |2〉 and |3〉. Such a technique is often more robust, as explained above.

Like in the cases described above, one can use Eqs. (7) and (8) for quantitative analysis of the Δ value and for optimization of the polarization transfer.

## 3.3 Singlet order

Experiments with long-lived singlet order are attracting increased attention, as they allow one to investigate various slow processes and to preserve non-thermal spin order from relaxation losses (Levitt, 2012; Carravetta and Levitt, 2004; Carravetta et al., 2004). Presently, there is a number of NMR methods, reviewed in detail by Pileio (2017), known to convert magnetization into singlet order and to perform backward conversion of such a long-lived order into detectable magnetization. In strict terms, the long-lived order is given by the expectation value of the singlet-order operator 〈SO〉. The singlet-order operator is written as

In the present work, we only focus on LAC-based methods, which can be
applied to pairs of nearly equivalent spins $\frac{\mathrm{1}}{\mathrm{2}}$, meaning that the difference {*ω*_{a}−*ω*_{b}} in their Zeeman interaction with the external field is much smaller than the spin–spin coupling strength *J*. In the weak coupling regime LAC-based consideration is typically not applicable, whereas in strongly coupled spin pairs the
magnetization-to-singlet conversion commonly occurs at LACs in the RF-rotating frame, carried out in the manner of SLIC (spin-locking-induced
crossing) (DeVience et al., 2013)

The Hamiltonian of a homonuclear two-spin system, comprising spins *I*_{a}
and *I*_{b}, in the presence of an RF field can be written as follows in the
rotating frame:

Here $\mathit{\delta}{\mathit{\omega}}_{a,b}={\mathit{\omega}}_{a,b}-{\mathit{\omega}}_{\mathrm{rf}}$, where *ω*_{a,b} stands for the NMR frequency of the corresponding spin and *ω*_{rf} is the RF-frequency. The definition of the main term and the
perturbation is then as follows:

where $\langle \mathit{\delta}\mathit{\omega}\rangle =\frac{\mathrm{1}}{\mathrm{2}}\left\{\mathit{\delta}{\mathit{\omega}}_{a}+\mathit{\delta}{\mathit{\omega}}_{b}\right\}$ and ${\mathit{\omega}}_{\mathrm{\Delta}}=\frac{\mathrm{1}}{\mathrm{2}}\left\{\mathit{\delta}{\mathit{\omega}}_{a}-\mathit{\delta}{\mathit{\omega}}_{b}\right\}=\frac{\mathrm{1}}{\mathrm{2}}\left\{{\mathit{\omega}}_{a}-{\mathit{\omega}}_{b}\right\}$. Hence, the perturbation is given by the small difference in the resonance frequencies of the two spins. To determine the eigenvalues and eigenstates of the main Hamiltonian it is convenient to tilt the reference frame such that the new *z* axis is parallel to the effective field vector ${\mathit{\omega}}_{\mathrm{eff}}=\left({\mathit{\omega}}_{\mathrm{1}},\mathrm{0},\langle \mathit{\delta}\mathit{\omega}\rangle \right)$. In this frame the ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ term takes the form

where ${\mathit{\omega}}_{\mathrm{eff}}=\sqrt{{\mathit{\omega}}_{\mathrm{1}}^{\mathrm{2}}+\langle \mathit{\delta}\mathit{\omega}{\rangle}^{\mathrm{2}}}$. The scalar coupling term remains unchanged, since the operator $\left({\widehat{\mathbf{I}}}_{a}\cdot {\widehat{\mathbf{I}}}_{b}\right)$ is invariant to spatial rotations. The eigenstates of ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ correspond to the singlet–triplet basis of states in the tilted frame:

where $\mathrm{tan}{\mathit{\theta}}_{t}={\mathit{\omega}}_{\mathrm{1}}/\mathit{\delta}\mathit{\omega}$. The primes in
the notations of the triplet states indicate that they are defined in the
tilted reference frame with *z*||*ω*_{eff} (we do not use the prime for the |*S*〉 state, which is the same in any
frame). The energies of these states of ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ are

The ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ Hamiltonian has a single LC occurring when *ω*_{eff}=2*π*|J|, which is an *S*-*T*_{+} or *S*-*T*_{−} crossing (depending on the sign of *J*). The coupling term gives rise to mixing of the crossing states; hence, the LC is turned into an LAC. Let us now consider how spin mixing at this LAC can be exploited to perform spin order conversion.

The simplest way to convert spin order is given by the SLIC (DeVience et al., 2013) method, which utilizes a
resonant RF pulse; i.e., 〈*δ**ω*〉=0, with ${\mathit{\omega}}_{\mathrm{1}}=\mathrm{2}\mathit{\pi}\left|J\right|$. The application of such a pulse brings the spin system to the LC, where the perturbation term becomes active. Hence, *S*-*T*_{±} mixing
takes place; in the ideal case it swaps the populations of the two states; see Fig. 7.

Efficient conversion of magnetization into singlet state requires that first
the magnetization vector is set parallel to the effective field; in the case
〈*δ**ω*〉=0, the *ω*_{eff} vector is
parallel to the *x* axis of the tilted frame. Hence, starting with
*z* polarization one should first apply a 90_{y} pulse and then apply a
SLIC pulse with the *x* phase. Under such conditions the initial density
matrix in the tilted frame takes the form

Represented as a state population vector, it is as follows:

Hence, the longitudinal magnetization in the tilted frame (corresponding to
the transverse magnetization in the original frame) is non-zero, while the
singlet order is zero; 〈SO〉=0. By applying a SLIC pulse, however, one can swap the populations of the states |*S*〉 and $\mathrm{|}{T}_{+}{\rangle}^{\prime}$. When the RF field is resonant, i.e.,
*δ**ω*=0, and the tilt angle is ${\mathit{\theta}}_{t}=\mathit{\pi}/\mathrm{2}$, we obtain permutation occurring between the states |*S*〉 and $\mathrm{|}{T}_{+}{\rangle}^{\prime}$, which is obtained from |*T*_{+}〉 after a 90_{x} rotation. Consequently, after spin mixing at the LAC the state populations become

According to the definition given by Eq. (37), the singlet-order operator is as follows:

Hence, we obtain that $\langle \text{SO}\rangle =\left(\text{SO}\mathrm{|}{\mathit{\rho}}^{\prime}\right)=\frac{\mathrm{4}}{\mathrm{3}}{M}_{I}$ and the polarization is reduced. The same kind of pulse can be used to convert the singlet order back into transverse polarization.

A possible way (Theis et al., 2014a) of implementing SLIC is to apply a pulse
with time-dependent amplitude *ω*_{1}(*t*), which is varied in an
adiabatic way such that the minimal *ω*_{1} is smaller than 2*π*|J| and the maximal *ω*_{1} is greater than 2*π*|J|. In this particular case, it does not matter if *ω*_{1} is
increased or decreased: the permutation of the populations is the same;
namely, the |*S*〉 and $\mathrm{|}{T}_{+}{\rangle}^{\prime}$ populations are swapped. As in
the previous example, spin order conversion by an adiabatic pulse is usually
more robust, although a pulse with *ω*_{1}=2*π*|J|
provides faster conversion.

Spin order conversion by SLIC pulses is not the unique method of driving
singlet–triplet transitions. It is also possible to apply off-resonant
pulses to perform the desired conversion. At a first glance, by using an RF
pulse with 〈*δ**ω*〉≠0 and with a ramped amplitude *ω*_{1}(*t*), designed such that the LC at *ω*_{eff}=2*π*|J|
is passed, one can perform the same kind of transformation as in the SLIC
case. However, this is not true because the direction of
*ω*_{eff} changes upon variation in RF-field
amplitude. Indeed, when *ω*_{1}=0, the effective field is directed
along the *z* axis (for any small, but non-zero, value of *ω*_{1}),
since there is only the 〈*δ**ω*〉 term in the Hamiltonian
${\widehat{\mathcal{H}}}_{\mathrm{0}}$, whereas at *ω*_{1}≫〈*δ**ω*〉 the effective field is parallel to the *x* axis. As a consequence, a pulse with an adiabatically increased *ω*_{1}(*t*) converts the *z* magnetization of spins into singlet order. A pulse with adiabatically decreased *ω*_{1}(*t*) converts the singlet order into *z* magnetization. This type of conversion is exploited in the APSOC (Adiabatic Passage Spin Order Conversion) method (Pravdivtsev et al., 2016), which has an advantage that additional pulses are not required for locking
spin magnetization; furthermore, there is no need to control the phase of
the pulses. In order to estimate the efficiency of spin order conversion in
all outlined cases, Eqs. (7) and (8) should be used.

## 3.4 Parahydrogen-induced polarization

PHIP also frequently relying (Franzoni et al., 2013, 2012; Pravdivtsev et al., 2014b; Theis et al., 2014b; Pravdivtsev et al., 2013b) on spin mixing occurring at LACs. In this section, we discuss possible methods for transferring PHIP to polarize rare spins, such as ^{13}C or ^{15}N. We consider here a three-spin system, comprising two
*I* spins (protons), *I*_{a} and *I*_{b}, prepared in the singlet state and
an *S* spin. Such a consideration is relevant in the context of transferring
SABRE-derived polarization to rare spins, such as ^{15}N. A number of
methods has been suggested to solve this problem (Theis et al., 2014b, 2018; Knecht et al., 2018); here we provide a unified view on such methods. For simplicity, we assume that the *I* spins are chemically equivalent, but not magnetically equivalent nuclei: the unequal *J*_{IS} couplings lift the magnetic equivalence. We also consider a particular method of spin order transfer, assuming that it is performed at a high magnetic field by applying RF excitation solely to the *S* channel. In the rotating frame (with the frame rotation done only for the *I* spins) the Hamiltonian of the spin system is as follows:

Here *ω*_{H} is the proton NMR frequency, *δ**ω*_{S} is the
offset of the RF field from the frequency of the *S* spins, *ω*_{1} is
the RF-field strength expressed in the frequency units and *J*_{II} is the
couplings of the *I* spins. For the *I**S* couplings we assume that there is
interaction only for the *I*_{a}−*S* spin pair and that *J*_{IS}≪*J*_{II}
(so that perturbation theory treatment is applicable). For the same reason
as explained above, in the *I**S* coupling term we keep only the products of
*z* operators. Hence, we set the main part of the Hamiltonian as

and the perturbation as

Now we again go to the tilted frame and modify the Hamiltonian in the following way for the main term,

and for the perturbation term,

Here ${\mathit{\omega}}_{S,\mathrm{eff}}=\sqrt{{\mathit{\omega}}_{\mathrm{1}}^{\mathrm{2}}+\mathit{\delta}{\mathit{\omega}}_{S}^{\mathrm{2}}}$ and *θ*_{eff} is the tilt angle; $\mathrm{tan}{\mathit{\theta}}_{\mathrm{eff}}=\frac{{\mathit{\omega}}_{\mathrm{1}}}{\mathit{\delta}{\mathit{\omega}}_{S}}$. Here the frame tilt is introduced only for the *S* spin, which is subject to RF excitation.

The next step is solving the eigenproblem of the unperturbed Hamiltonian.
To do so, we introduce a suitable basis, which is given by the direct
product of the singlet–triplet bases of each spin pair: ${\left\{\mathrm{|}S\rangle ,\mathrm{|}{T}_{+}\rangle ,\mathrm{|}{T}_{\mathrm{0}}\rangle ,\mathrm{|}{T}_{-}\rangle \right\}}_{II}\otimes {\left\{\mathrm{|}{\mathit{\alpha}}^{\prime}\rangle ,\mathrm{|}{\mathit{\beta}}^{\prime}\rangle \right\}}_{S}$; in the basis of the *S*
spin the primes indicate that the Zeeman states are written in the tilted
frame. In this basis, the ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ Hamiltonian is diagonal. It is then straightforward to evaluate the diabatic energy levels. One can determine that two LCs emerge, when the following matching conditions are fulfilled:

Here we consider only LCs in the manifold of the |*S*〉 and |*T*_{0}〉 states. The reason is that the |*S*〉 states and |*T*_{±}〉 are split by the large proton Zeeman interaction, *ω*_{H}, and the corresponding crossings cannot occur in high magnetic fields; furthermore, there is no perturbation term, which would mix these states. Therefore, in the present case of single-frequency excitation, it is sufficient to consider only *S*−*T*_{0} mixing of the *I* spins.

At each of the two LCs, the perturbation terms become active: the
${\widehat{I}}_{az}$ operator can mix the |*S*〉 and |*T*_{0}〉 states, while the ${\widehat{S}}_{x}$ operator can mix the |*α*〉 and |*β*〉 states. One should only be careful that when the matching condition

is fulfilled, the *θ*_{eff} should not be approaching zero (which is the case when $\mathit{\delta}\mathit{\omega}\approx \mathrm{2}\mathit{\pi}{J}_{II}\gg {\mathit{\omega}}_{\mathrm{1}})$: under
such conditions the coupling term becomes too small to provide a fast and
efficient exchange of the state populations at the LAC. Of course, both
conditions ${\mathit{\omega}}_{S,\phantom{\rule{0.125em}{0ex}}\mathrm{eff}}=\pm \mathrm{2}\mathit{\pi}{J}_{II}$ cannot be fulfilled
simultaneously. Therefore, for the sake of clarity, we assume that ${\mathit{\omega}}_{S,\phantom{\rule{0.125em}{0ex}}\mathrm{eff}}=\mathrm{2}\mathit{\pi}{J}_{II}$. The relevant energy levels are shown in Fig. 8.

Now, let us consider the LAC-driven spin dynamics of the process. In the
case *J*_{II}≫*J*_{IS}, the initial density matrix can be written as

After population swapping the density matrix becomes

That is, spin mixing gives rise to population exchange between the states
|*S**α*^{′}〉 and |*T*_{0}*β*^{′}〉. As a consequence,
singlet order is converted into magnetization of the *S* spin. The resulting
polarization of the *S* spin is then as follows:

Here ${M}_{S}^{\prime}$ is the magnetization value in the tilted frame. The resulting spin order of the *S* spins depends on how the experiment is carried out. In the simplest case, where a single pulse with ${\mathit{\omega}}_{S,\mathrm{eff}}=\mathrm{2}\mathit{\pi}{J}_{II}$ is applied for a sufficiently long time (so that spin mixing can occur), the *S* spin is polarized along the
*ω*_{S, eff} vector. In the situation *δ**ω*=0 and *ω*_{1}=2*π**J*_{II} (resonant pulse), magnetization of the *S* spin is the purely transverse magnetization: the ${\widehat{S}}_{z}$ spin order in the tilted frame corresponds to ${\widehat{S}}_{x}$ in the non-tilted frame. If it is necessary to generate longitudinal magnetization, an additional RF pulse
should be applied (Theis et al., 2014b). By applying a pulse with ${\mathit{\omega}}_{S,\mathrm{eff}}=\mathrm{2}\mathit{\pi}{J}_{II}$ and *δ**ω*≠0, one can again generate the ${\widehat{S}}_{z}^{\prime}$ order (Knecht et al., 2018) in the tilted
frame, which corresponds to spin order

in the non-tilted frame. Hence, the magnetization vector has transverse as
well as longitudinal components. If the RF pulse is applied such (Theis et al., 2018) that *ω*_{1}(*t*) is adiabatically reduced to zero in such a way that the LC ${\mathit{\omega}}_{S,\phantom{\rule{0.125em}{0ex}}\mathrm{eff}}=\mathrm{2}\mathit{\pi}{J}_{II}$ is passed and *δ**ω*≠0, the *S* spin is polarized along the *ω*_{S, eff} vector, which
becomes parallel to *z* when *ω*_{1} becomes zero. It means that
longitudinal magnetization of the *S* spins is generated. It is important
that necessarily *δ**ω*≠0 in this case: when the offset from
the resonance frequency is zero, the effective field does not have any
preferred direction and the *S* spins cannot be preferentially polarized
parallel or anti-parallel to the external magnetic field.

A similar situation arises upon transfer of the singlet order into the
magnetization of heteronuclei in a four-spin system of the AA^{′}XX^{′} type.
In this situation, the Hamiltonian is written as follows (in the RF-rotating
frame for the *S* spins):

The perturbation term is given by the expression

Now it is convenient to rewrite the Hamiltonians in the tilted frame. The main Hamiltonian becomes

and the perturbation is

Here ${\mathit{\omega}}_{S,\mathrm{eff}}=\sqrt{{\mathit{\omega}}_{\mathrm{1}}^{\mathrm{2}}+\mathit{\delta}{\mathit{\omega}}_{S}^{\mathrm{2}}}$ and *θ*_{eff} is the tilt angle; $\mathrm{tan}{\mathit{\theta}}_{\mathrm{eff}}=\frac{{\mathit{\omega}}_{\mathrm{1}}}{\mathit{\delta}{\mathit{\omega}}_{S}}$. The eigenbasis for the ${\widehat{\mathcal{H}}}_{\mathrm{0}}$ Hamiltonian for now is given by the direct product of the singlet–triplet bases in each spin pair: ${\left\{\mathrm{|}S\rangle ,\phantom{\rule{0.125em}{0ex}}\mathrm{|}{T}_{+}\rangle ,\mathrm{|}{T}_{\mathrm{0}}\rangle ,\phantom{\rule{0.125em}{0ex}}\mathrm{|}{T}_{-}\rangle \right\}}_{II}\otimes {\left\{\mathrm{|}S\rangle ,\phantom{\rule{0.125em}{0ex}}\mathrm{|}{T}_{+}{\rangle}^{\prime},\mathrm{|}{T}_{\mathrm{0}}{\rangle}^{\prime},\mathrm{|}{T}_{-}{\rangle}^{\prime}\right\}}_{SS}$; in the basis of the *S* spins the
primes indicate that the singlet–triplet states are written in the tilted
frame (there is no frame tilt introduced for the *I* spins).

The perturbation term (62) can drive *S*→*T*_{0} transitions for the *I*
spins accompanied by $S\to {T}_{\pm}^{\prime}$ transitions for the *S* spins
resulting in increasing the $\mathrm{|}{T}_{\pm}{\rangle}^{\prime}$ populations and,
consequently, causing the enhanced heteronuclei magnetization along the
RF-field directions. The LCs of the system are the following (LC conditions
are also specified):

The generalized LAC condition is then ${\mathit{\omega}}_{S,\mathrm{eff}}=\pm \mathrm{2}\mathit{\pi}({J}_{II}+{J}_{SS})$. The spin dynamics and polarization behavior is similar to the three-spin case described above; hence, we do not consider further detail here. In order to learn more about this subject, the reader is advised to read previous publications (Knecht et al., 2018; Theis et al., 2014b).

Finally in this section, we would like to note that similar LAC-driven spin
dynamics have been reported for homonuclear systems of the AA^{′}MM^{′} type,
where AA^{′} and MM^{′} stand for the two groups of chemically equivalent but
magnetically non-equivalent spins, with the AA^{′} spins prepared in the
singlet state. Discussion of this case is beyond the scope of the present
work. We only mention that spin order transfer is based on the same
principles as those described above: upon RF excitation, polarization
transfer occurs at LACs (in the rotating frame) and gives rise to
polarization of the AA^{′} and MM^{′} spins along their respective effective
fields. One can also vary the actual spin magnetization by introducing a
single RF pulse, which brings the spin system to an LAC, or by passing
through LACs using adiabatically ramped RF-field amplitudes. Further
information can be found in the original publications (Pravdivtsev et
al., 2014b; Franzoni et al., 2013).

In this work, we present a general approach to treat spin mixing occurring at LACs. The approach is formulated assuming that the spin system has a set of LACs, which do not overlap with each other, for the state described in terms of the populations of diabatic states; i.e., we ignore the possible presence of spin coherences in the initial and final state. Upon variation in a control parameter (magnetic field strength, RF frequency, RF-field strength), the spin system passes through LACs and permutations of the state populations occur. Introducing the operators of permutations, we can compute the final spin order. We also take into account that upon variation in the control parameter the basis of the diabatic eigenstates may be altered. This consideration of the spin dynamics proposed here is summarized by a flowchart diagram, shown in Fig. 9.

The treatment presented here is supported by a number of examples. These examples deal with spin order conversion via adiabatic passage through zero field, with cross-polarization, with singlet-state NMR and with PHIP. To conclude, utilizing LACs provides powerful methods to manipulate spin order and to design experimental protocols for robust and efficient spin order conversion. LAC-based methods have proven to be a useful tool. For instance, in our lab we have developed several methods based on harnessing LACs, such as the APSOC method and techniques for manipulating PHIP. Further applications of this method can be found in solid-state NMR using magic angle spinning, which is a commonly used way to improve resolution and sensitivity. Notably, LAC-based descriptions can be utilized to describe spin-locking experiments with quadrupolar nuclei (Vega, 1992; Ashbrook and Wimperis, 2009) and dynamic nuclear polarization (Thurber and Tycko, 2014, 2012; Mentink-Vigier et al., 2015) in rotating solids.

A possible extension of the theory presented here (which goes beyond the scope of the present work) is given by a consideration of relaxation effects, which also give rise to population exchange between spin eigenstates. To treat relaxation, one should introduce a relaxation super operator, which acts in between passages through individual LACs. Hence, population swaps would be accompanied by the relaxation of populations between subsequent swaps. Of course, such a treatment would be limited to the relaxation of populations only, whereas the relaxation of coherence would be beyond its reach.

Here we present calculations of the effective coupling element 𝒱_{kl} in the situation where |*ψ*_{k}〉 and |*ψ*_{l}〉 are not mixed
by the perturbation but are mixed with other states |*ψ*_{m}〉. If
we assume that there is only one such state, we can evaluate the
second-order correction to the wave functions:

The new wave functions can be mixed because 𝒱_{km} and 𝒱_{lm} are both non-zero. Assuming ${\mathcal{E}}_{k}^{\mathrm{0}}\approx {\mathcal{E}}_{l}^{\mathrm{0}}$ we can estimate the matrix element, which mixes them, as follows:

If there are multiple states |*ψ*_{m}〉 through which the |*ψ*_{k}〉 and |*ψ*_{l}〉 are coupled, we generalize this expression as

The effective coupling element vanished only when for any *m*, both 𝒱_{km}
and 𝒱_{lm} are zero, meaning that |*ψ*_{k}〉 and |*ψ*_{l}〉 must belong to different blocks of a block-diagonal Hamiltonian. In
this situation, a true LC between these states is possible; otherwise, it is
turned into an LAC.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Both authors planned the research, conducted the research and wrote the paper.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Robert Kaptein Festschrift”. It is not associated with a conference.

We acknowledge Jorg Matysik for inspiring us to describe the representation of LAC-driven spin mixing, Geoffrey Bodenhausen for valuable comments on the text of the paper and Malcolm H. Levitt for drawing our attention to the discussion of the “basis rotation” concept.

This research has been supported by the Russian Foundation for Basic Research (grant no. 20-53-15004) and the Ministry of Science and Education of the Russian Federation.

This paper was edited by Perunthiruthy Madhu and reviewed by two anonymous referees.

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