# General classical and quantum-mechanical description of magnetic resonance: an application to electric-dipole-moment experiments

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## Abstract

A general theoretical description of a magnetic resonance is presented. This description is necessary for a detailed analysis of spin dynamics in electric-dipole-moment experiments in storage rings. General formulas describing a behavior of all components of the polarization vector at the magnetic resonance are obtained for an arbitrary initial polarization. These formulas are exact on condition that the nonresonance rotating field is neglected. The spin dynamics is also calculated at frequencies far from resonance with allowance for both rotating fields. A general quantum-mechanical analysis of the spin evolution at the magnetic resonance is fulfilled and the full agreement between the classical and quantum-mechanical approaches is shown. Quasimagnetic resonances for particles and nuclei moving in noncontinuous perturbing fields of accelerators and storage rings are considered. Distinguishing features of quasimagnetic resonances in storage ring electric-dipole-moment experiments are investigated in detail. The exact formulas for the effect caused by the electric dipole moment are derived. The difference between the resonance effects conditioned by the rf electric-field flipper and the rf Wien filter is found and is calculated for the first time. The existence of this difference is crucial for the establishment of a consent between analytical derivations and computer simulations and for checking spin tracking programs. The main systematical errors are considered.

## 1 Introduction

The magnetic resonance (MR) is a powerful tool of investigation of basic properties of particles and nuclei. The MR is also successfully used for studies of atoms in condensed matters. The theory of the MR is presented in many books (see, e.g., Refs. [1, 2]) and research articles. As a rule, the MR is a spin resonance and its classical and quantum-mechanical descriptions are equivalent. In the present work, we rigorously prove the equivalence of these descriptions in the general case and apply the general theory for an analysis of resonance effects in electric-dipole-moment experiments with polarized beams in storage rings.

The use of the MR in nuclear, particle, atomic, and condensed matter physics consists in a determination of a spin deflection from the initial vertical direction. To find the magnetic moment, one measures the dynamics of the vertical polarization. An exhaustive analysis of the spin evolution in storage ring electric-dipole-moment (EDM) experiments needs an advanced description of magnetic and quasimagnetic resonances. In this case, spin interactions of moving particles and nuclei with magnetic and electric fields are defined by the Thomas–Bargmann–Mishel–Telegdi (T-BMT) equation [3, 4, 5] or by its extension taking into account the EDM [6, 7, 8, 9]. An investigation of quasimagnetic resonances caused by the EDM is important for planned experiments with a rf electric-field flipper and a rf Wien filter (see Refs. [10, 11, 12]). A change of the spin (pseudo-)vector is orthogonal to the spin direction. One needs therefore to measure minor (horizontal) polarization components when a resonance is stimulated by a comparatively weak interaction. In particular, this situation takes place for a search for EDMs. In storage ring experiments, it can be convenient to use an initial horizontal beam polarization and to measure an evolution of the vertical spin component. A needed experimental precision is very high. In addition, the resonance fields of the rf electric-field flipper and the rf Wien filter are noncontinuous. For these reasons, general formulas describing spin dynamics at the magnetic and quasimagnetic resonances and their specific application are necessary.

In Sect. 2, we give the general description of the MR in the framework of classical spin physics and electrodynamics. The problem of the spin evolution at frequencies far from resonance is solved in Sect. 3. The general quantum-mechanical description of spin dynamics at the MR is presented in Sect. 4. Section 5 is devoted to a discussion of the magnetic and quasimagnetic resonances for moving particles and nuclei. Quasimagnetic resonances for particles and nuclei moving in noncontinuous perturbing fields of accelerators and storage rings are considered in Sect. 6. Distinguishing features of quasimagnetic resonances in storage ring electric-dipole-moment experiments are investigated in Sect. 7. Finally, we summarize the obtained results in Sect. 8.

The system of units \(\hbar =1,~c=1\) is used. We include \(\hbar \) and *c* into some equations when this inclusion clarifies the problem.

## 2 General classical description of nuclear magnetic resonance

In this section, we consider a usual design of the MR and obtain general equations describing the spin dynamics. While the results obtained are mostly known, the presented study allows us to apply a common approach for a consideration of classical and quantum-mechanical effects.

Let a spinning nucleus be placed into the magnetic field \({\varvec{B}}_0=B_0{\varvec{e}}_z\) and the angular frequency of the spin rotation is equal to \(\omega _0\). In this case, a rotating or oscillating horizontal magnetic field with a closed angular frequency \(\omega \approx \omega _0\) can significantly deflect the nucleus spin from the initial vertical direction. In particular, this effect allows one to measure magnetic moments of nuclei/particles with a high precision.

*g*-factor, and \(\mu _N\) is the nuclear magneton. For particles, \(g_N\mu _N\) should be replaced with \(eg\hbar /(2m)\), where \(g=2mc\mu /(es)\).

The direction of the (pseudo-)vector \({\varvec{\omega }}_0\) defines the orientation of the so-called stable spin axis. In the absence of oscillating fields, the spin remains stable if it is initially aligned along this direction. If the initial spin orientation is different, the spin describes a cone around the direction of \({\varvec{\omega }}_0\). The stable spin axis is a static quantity defined *before* activating the rf.

*x*axis:

*z*axis with the angular velocity \({\varvec{\omega }}\). We suppose that the direction of the frame rotation coincides with the direction of the spin rotation, \({\varvec{\omega }}_0\). The horizontal magnetic field rotating in the lab frame becomes constant in the rotating frame. In this frame, the spin rotates about the

*z*axis with the angular frequency \(\omega _0-\omega \) and the total angular velocity of the spin rotation is equal to

## 3 Spin evolution at frequencies far from resonance

It is instructive to consider the spin evolution at frequencies far from resonance. This is especially important for the storage ring electric-dipole-moment experiments, because some periodical perturbations may imitate the presence of an EDM.

In the considered case, effects of two magnetic fields rotating in opposite directions are comparable. Therefore, it is not appropriate to decompose \({\varvec{\mathcal {B}}}\) in Eq. (1) into the two rotating fields.

It is significant that the results presented in this section are also applicable to a horizontal perturbation caused by a *constant* field. In this case, \(\omega =0\).

*x*axis along the vectors \({\varvec{\mathcal {B}}}\) and \({\varvec{\mathfrak {E}}}\). When the perturbation is negligible, the spin rotates with the angular velocity \({\varvec{\omega }}_0\). Therefore, it is convenient to use the primed coordinate system rotating with this angular velocity:

An analysis of Eqs. (7), (9), and (15) displays an important property of magnetic and quasimagnetic resonances. When the perturbing field is rather weak (\(|\mathfrak {E}|\ll \Omega \)) and it needs to be determined, it is preferable to measure *minor* spin components. Specifically, the use of the initial vertical polarization requires a measurement of a horizontal polarization. It is of the order of \(|\mathfrak {E}|/\Omega \), while a change of the vertical spin component is of the order of \({\mathfrak {E}}^2/\Omega ^2\). Both of the horizontal spin components oscillate but only the component collinear to \({\varvec{\mathfrak {E}}}\) is nonzero on average. In practice, a detection of the oscillatory spin motion is easier than that of the small constant part of the horizontal spin component collinear to \({\varvec{\mathfrak {E}}}\).

When the initial polarization is horizontal, it is preferable to measure the vertical polarization which is of the order of \(|\mathfrak {E}|/\Omega \). In the equations for the horizontal spin components, terms proportional to \(\mathfrak {E}/\Omega \) vanish when the initial vertical polarization is equal to zero. Thus, monitoring of these components is not useful. This approach has been applied for a search for a muon EDM [14] in the framework of the muon *g* \(-2\) experiment at Brookhaven National Laboratory (see Ref. [15]). The search has been carried out for the initial longitudinal polarization of muons and the *constant* radial perturbing field caused by the EDM. The vertical spin component has been detected. This experiment has allowed one to obtain an upper bound on the muon EDM [14].

## 4 Quantum-mechanical description of magnetic resonance

The detailed classical description of the MR given in the previous sections exhaustively defines the spin motion caused by interactions linear in the spin. This description is well substantiated because it is based on manifestly covariant initial equations. However, spin-dependent parts of Hamiltonians contain also terms quadratic in the spin for deuteron and other nuclei with the spin \(s\ge 1\). A resonance experiment for the deuteron (\(s=1\)) is a part of the EDM program [16, 17]. The presence of the terms quadratic in the spin leads to systematical effects mimicking the EDM under the MR [18, 19, 20, 21, 22, 23, 24, 25, 26]. While the classical description of these effects is possible [18, 19, 20, 21, 22], a more general theory which has been developed in Refs. [23, 24, 25] is based on relativistic quantum-mechanical Hamiltonians in the Foldy–Wouthuysen representation (see Ref. [27, 28, 29] and the references therein). In the present study, we do not consider the effects nonlinear in the spin. Nevertheless, a need for future investigations stipulates for an advanced quantum-mechanical description of the standard MR conditioned by spin interactions *linear* in the spin. To solve this problem, we may use the Pauli spin matrices even for nuclei with the spin \(s\ge 1\). This possibility is based on universal commutation relations for spin components which are satisfied for any spins. The identity of the spin motion of particles with spins 1/2 and 1 near a resonance has been demonstrated in Ref. [30].

It is convenient to use the matrix Hamiltonian method for a quantum-mechanical description of the MR. When spin-tensor interactions are not taken into account, the spin rotation of nuclei/particles with spin 1/2 and with higher spins is very similar. Therefore, spin rotation of nuclei/particles with spin \(s\ge 1\) can also be described with the Dirac matrices acting on the two-component spin wave function. We consider the same field configuration as in Sect. 2.

*s*is the spin quantum number. Averages of the spin operators \(\sigma _i=2S_i\) are expressed by their convolutions with the wave function \(\Psi (t)\).

*C*(

*t*), respectively, has the form

*C*(

*t*).

Evidently, Eqs. (4) and (23) fully agree.

If we use Eq. (26) for a derivation of \(P'_i(t)\) in terms of \(P'_i(0)~(i=x,y,z)\), we arrive at Eq. (7). This fact clearly demonstrates the full agreement of results obtained by the classical and quantum-mechanical approaches. Similarly to the precedent section, we can use Eqs. (7), (8), and (23) for a derivation of the general equation (9). Thus, this equation is valid not only in classical spin physics but also in quantum mechanics.

The quantum-mechanical description of the spin evolution at the MR is often used in textbooks (see Ref. [2]) and research articles. In the present study, the *general* case has been considered and the full agreement between the classical and quantum-mechanical approaches has been demonstrated. This agreement seems to be very natural. However, its proof is not redundant, because we should take into account the existence of the difference between classical and quantum-mechanical descriptions of some spin effects (see Ref. [31]). In future investigations of resonance phenomena for nuclei with the spin \(s\ge 1\), taking into account spin interactions quadratic in the spin will be necessary. Such interactions are caused by the tensor electric and magnetic polarizabilities and the electric quadrupole moment. In this case, a transition to the spin-1 matrices can be necessary.

## 5 Magnetic and quasimagnetic resonances for moving particles and nuclei

*d*is the EDM.

*average*radial magnetic field is equal to zero. In a storage ring with electric focusing, it should be counterbalanced by a focusing vertical electric field. In this case, the horizontal components of the average Lorentz force vanish:

*observable*effect, the motion of the axes of the FS coordinate system should be added to the spin motion in this coordinate system. Once this circumstance is taken into account, the cylindrical and FS coordinate systems give an equivalent description of the spin motion [33].

In storage ring EDM experiments, the perturbing field should act on the EDM. For this purpose, one can use a resonance radial electric field (a rf electric-field flipper). One can also add a vertical magnetic field oscillating with the same resonance frequency. When the Lorentz force created by this device is equal to zero, one obtains a rf Wien filter [10]. The very weak perturbing field is caused by the interaction of the EDM with the main magnetic or electric field. This perturbing field is constant and rotates the spin about the radial axis. This case has been discussed at the end of Sect. 3.

We can mention that a constant perturbation rotating the spin about the radial axis is also conditioned by the Earth’s gravity [34, 35, 36]. This perturbation is very weak.

A detailed consideration of evolution of all spin components is necessary because an interaction stimulating a resonance can be very weak. Evidently, \(\mathrm{d}{\varvec{\zeta }}/(\mathrm{d}t)\bot {\varvec{\zeta }}\). When the initial beam polarization is vertical, one needs to measure horizontal spin components. When the initial beam polarization is horizontal, it is convenient to monitor the vertical spin component (see Sect. 3). These two possibilities may be realized in storage ring experiments on a search for EDMs [16, 25, 37, 38]. In these cases, the action of oscillating fields on the EDM stimulates a resonance, while their action on the magnetic moment does not bring about any resonance effects. This takes place because the quantities \({\varvec{\Omega }}_{\mathrm{T}{\text {-}}\mathrm{BMT}}\) and \({\varvec{\Omega }}_\mathrm{EDM}\) in Eq. (27) are usually orthogonal. Corresponding quantities in Eqs. (28) and (30) possess the same property. In any case, spin resonances originating from the EDM are quasimagnetic and are not magnetic. We can also mention that the case when the *x* and *z* components of the angular velocity of spin precession are constant [see Eq. (7)] corresponds to the conditions of the EDM experiment based on the frozen spin method [39] (cf. Eq. (22) in Ref. [25]).

It has been proven in Ref. [38] that the use of the initial vertical polarization cancels some systematical errors. The use of the initial horizontal polarization does not lead to such a cancellation. However, the initial vertical polarization can meet other problems [38].

We can conclude that specific conditions of the magnetic and quasimagnetic resonances for particles and nuclei moving in accelerators and storage rings influence only parameters \(\omega _0\) and \(\mathfrak {E}\) but do not change the general equations (9) and (15) defining the spin dynamics. A calculation of small corrections appearing in exact solutions needs a modification of initial equations. This problem will be considered in the next section.

It can be added that an extremely high precision of storage ring EDM experiments needs taking into account tensor electric and magnetic polarizabilities for nuclei with spin \(s\ge 1\) (e.g., deuteron) [18, 19, 20, 21]. The tensor magnetic polarizability, \(\beta _T\), produces the spin rotation with two frequencies instead of one, beating with a frequency proportional to \(\beta _T\), and causes transitions between vector and tensor polarizations [18, 19, 20, 21, 22, 24]. A beam with an initial tensor polarization acquires a final vector polarization [23, 24, 25]. Resonance effects caused by the tensor polarizabilities have been calculated in Refs. [18, 19, 20, 21, 22, 23]. A comparison of spin dynamics conditioned by the tensor polarizabilities and the EDM has been carried out in Refs. [23, 25, 26]. The corresponding spin motion without taking into account spin-tensor effects is presented by the formulas of Sect. 3 provided that \(\omega =0\).

## 6 Quasimagnetic resonance in a noncontinuous perturbing field

The next problem which should be taken into consideration in connection with the storage ring EDM experiments is a discontinuity of perturbing fields. In the planned EDM experiments with protons, deuterons, and \(^3\)He ions at COSY [16, 17], one will use resonance stimulations with a rf electric-field flipper and a rf Wien filter. The two devices create oscillatory fields. The frequencies of the perturbing fields are synchronized with that of the spin frequency. The both devices provide for standard conditions of the MR. Semertzidis [40] and Nikolaev [41] have compared the actions of the rf electric-field flipper and the rf Wien filter on the spin. Orlov has shown [42] that a part of the longitudinal spin component is frozen (constant in time) in the oscillatory radial electric field. A more advanced theoretical analysis has been fulfilled in Ref. [10]. The theoretical calculations agree with spin tracking [10, 40].

The rf Wien filter unlike the rf electric-field flipper does not affect the motion of particles and nuclei. This is a great advantage of the former device. The latter device can be used only if it does not destroy the beam stability.

JEDI collaboration plans to perform main experiments with the rf Wien filter [11, 12, 17]. The static version of this filter is frequently used to turn the spin without an effect on beam dynamics. The rf electric-field flipper may be applied in precursor experiments [16]. The initial beam polarization is planned to be vertical. The initial horizontal beam polarization can also be used.

The stimulating frequency, \(\omega '\), should either (almost) coincide with that of the spin rotation, \(\omega _0\), or differ by \(n\omega _c~(n=\pm 1,\pm 2,\ldots ,)\) where \(\omega _c\) is the cyclotron frequency. This property can be properly substantiated and a rigorous quantitative description of the spin evolution can be given.

Equation (31) shows a possibility to use resonance devices at different frequencies. In particular, an appropriate choice for the proton and the deuteron is \(K=-2,-3\) and \(K=+1,+2\), respectively.

*l*is the length of the flipper/filter,

*C*is the ring circumference. The spin-dependent part of the classical Hamiltonian is defined by Eq. (1) and the electric field \({\varvec{E}}_0\) is directed radially.

*of the EDM*with the radial electric field can be presented in the form

## 7 Distinguishing features of a quasimagnetic resonance in storage ring electric-dipole-moment experiments

Main distinguishing features of storage ring EDM experiments are a simultaneous influence of external fields on the electric and magnetic dipole moments and the existence of a resonance effect even when the stimulating torque acting *on the EDM* is equal to zero. The last situation occurs when the resonance in a EDM experiment is stimulated by the rf Wien filter with the vertical magnetic and radial electric fields [\(({\varvec{E}}_0+{\varvec{\beta }}\times {\varvec{B}}_0^{(\mathrm{osc})})_r=0\)]. The paradoxical property of the existence of the resonance effect on condition that \({\varvec{\Omega }}_\mathrm{EDM}=0\) has been first discovered by Semertzidis [40] with a computer simulation. The existence of this effect has been confirmed and has been rigorously proven by the subsequent theoretical analysis fulfilled in Refs. [10, 41, 42].

In the present work, we give a very simple explanation of the distinguishing features of a quasimagnetic resonance in storage ring EDM experiments. This explanation is valid for any initial polarization of particles or nuclei.

*z*axis. We may consider only terms linear in \(\eta \). In this approximation,

It can similarly be proven that the resonance effect does not appear in an all-electric storage ring when the main and oscillating electric fields are also radial.

As a rule, the condition \(|b_z|\ll 1\) is a satisfactory approximation because the length of the flipper is much less than the ring circumference (\(l\ll C\)). In the planned deuteron EDM precursor experiment [16], \(l/C=5\times 10^{-3}\).

The precedent studies [10, 40, 41, 42] have shown that the addition of the rf magnetic-field flipper with the vertical field to the rf electric-field flipper significantly improves beam dynamics but does not eliminate the EDM effect. This conclusion first followed from the computer simulation [40]. It is confirmed by the geometrical method.

*proportional to the EDM*is ensured by the action of the oscillating fields on the MDM. To determine this effect, it is convenient to pass to the axes \({\varvec{e}}_\vartheta \) and \({\varvec{e}}_\zeta \):

*n*is integer):

*exact*formula:

When \(\delta \rightarrow 0\), we obtain the approximate equation (51) (on condition that \(\psi =\pi /2+\chi \)).

One of the key problems in EDM experiments is the problem of systematical errors. Equations (28) and (30) show that the vertical electric field and the radial and longitudinal magnetic fields may create a resonance effect imitating the presence of the EDM. This effect can occur due to misalignments and imperfections of the oscillating fields in the rf Wien filter. Similarly directed constant imperfection fields can also exist in the storage ring. However, they do not create any *resonance* effect.

In the presence of the longitudinal magnetic field or the vertical component of the particle velocity, the spin turns around the longitudinal direction while the EDM effect consists in the spin rotation around the radial direction. As a result, the phases of rotating horizontal spin components appearing due to the above-mentioned sources of systematical errors and due to the EDM effect differ on \(\pi /2\).

When the initial beam polarization is horizontal and the rf Wien filter is well synchronized with the spin rotation, \(\omega =\omega _0\) and the phase difference in Eqs. (9) and (10) is \(\psi -\chi =\pm \pi /2\). In this case, the systematical errors caused by the longitudinal magnetic field and the vertical component of the particle velocity vanish and the only important systematical error is conditioned by the vertical electric field and the radial magnetic one. However, the former systematical errors should be taken into account when the resonance conditions are not exactly satisfied.

## 8 Summary

In the present paper, a general theoretical description of the MR is given. We have derived the general formulas describing a behavior of all components of the polarization vector at the MR and have considered the case of an arbitrary initial polarization. The equations obtained are exact on condition that the nonresonance rotating field is neglected. The spin dynamics has also been calculated at frequencies far from resonance without neglecting the above-mentioned field. A quantum-mechanical analysis of the spin evolution at the MR has been fulfilled and the full agreement between the classical and quantum-mechanical approaches has been proven.

Distinguishing features of magnetic and quasimagnetic resonances for particles and nuclei moving in accelerators and storage rings (including resonances caused by the EDM) have been investigated in detail. We have considered the quasimagnetic resonance in a noncontinuous perturbing field. We have also fulfilled a detailed description of a quasimagnetic resonance in storage ring EDM experiments. We have applied the simple geometrical method and have determined the spin dynamics in the general case. We have shown for the first time the difference between the resonance effects conditioned by the rf electric-field flipper and the rf Wien filter. The existence of this difference is crucial for the establishment of consent between analytical derivations and computer simulations and for checking spin tracking programs.

The results obtained define also the spin dynamics caused by systematical errors which appear due to misalignments and imperfections of the resonance fields in the rf Wien filter.

## Notes

### Acknowledgements

The author is grateful to N. Nikolaev for useful comments and discussions and to Y. Tsalkou for the help in graphic design. The author acknowledges the support by the Belarusian Republican Foundation for Fundamental Research (Grant No. \(\Phi \)16D-004) and by the Heisenberg–Landau program of the German Ministry for Science and Technology (Bundesministerium für Bildung und Forschung).

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