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,

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.

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

Hereafter, we assume that there is a parameters

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

For the sake of simplicity, we assume that

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

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

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

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

We illustrate how population exchange can take place for spin

If we assume that initially the system is in the

Another possibility is to perform an adiabatic passage through the LAC, by
varying the

We would like to emphasize that in some cases the mixing matrix element is
zero; however, when the states

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

First, we consider the initial and final spin states characterized by the
density matrices

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

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

Finally, we need to consider that the eigenstates of the Hamiltonian

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

If the system passes through a sequence of LACs (occurring in pairs of state

In some cases, the actual permutation of the state populations is performed
via several consecutive permutations, for example,

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

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

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

The first example we consider here is given by adiabatic inversion of the
external magnetic field

The Hamiltonian of the spin system is given by expression (in

When

Correlation diagram for an adiabatic inversion. Simulation
parameters:

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

Now we consider a passage through zero field from

Polarization transfer can be carried out in other ways. For instance, one
can perform a non-adiabatic jump

In this context, it is useful to consider a more complex problem of
enhancing NMR signals of “insensitive” nuclei, such as

However, one should note that the true eigenbasis of

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

The two state manifolds of a three-spin

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

Spin order transfer in this system can be carried out in a different
(perhaps, simpler) way. For instance, one can perform a sweep from

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

In the CP experiment (Hartmann and Hahn, 1962), see Fig. 6a, the

To describe this experiment, we write down the Hamiltonian 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

The CP experiment can be done in a different way (Metz et al., 1994). The RF field

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

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

The Hamiltonian of a homonuclear two-spin system, comprising spins

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.,

Efficient conversion of magnetization into singlet state requires that first
the magnetization vector is set parallel to the effective field; in the case

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

A possible way (Theis et al., 2014a) of implementing SLIC is to apply a pulse
with time-dependent amplitude

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

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

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:

At each of the two LCs, the perturbation terms become active: the

Population swapping upon increase in the amplitude of the
RF field applied to the

Now, let us consider the LAC-driven spin dynamics of the process. In the
case

A similar situation arises upon transfer of the singlet order into the
magnetization of heteronuclei in a four-spin system of the AA

The perturbation term (62) can drive

Finally in this section, we would like to note that similar LAC-driven spin
dynamics have been reported for homonuclear systems of the AA

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.

Flowchart diagram indicating

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

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.