On the geometric phases during radio frequency pulses with sine and cosine amplitude and frequency modulation

The concept of phase is pervasive in nuclear magnetic resonance (NMR). It has been shown that if the Hamiltonian undergoes slow changes and returns to its initial state after adiabatic evolution, the system acquires a measurable phase of purely geometric origin, in addition to the well-known dynamic phase.^1,2 The conceptualization of the geometric phase appeared initially in the works by Pancharatnam in polarization optics.³ Additionally, in his seminal contribution, Berry showed that the geometric properties of the system can define the phase that only appears during cyclic time variations of the Hamiltonian of the system and for adiabatic time traversals.^1,2 This concept was later generalized by Aharonov and Anandan to the case where the time traversal does not need to be adiabatic and the state of the system only needed to be specified as cyclic.^4,5 Observation of the geometric phase in biological systems, such as the brain and central nervous system, had been envisioned several decades ago.⁶ Recent work demonstrated the importance of the geometric phases for the description of molecular dynamics.⁷ However, direct observation of the geometric phase in vivo remains elusive because of the complexity of multiple relaxation pathways occurring in the living sample.

The geometric phase has unique features associated with the evolution of the quantum ensemble, and it is independent of the dynamic dephasing in the system. In contrast to the dynamic phase that is refocused in the Spin Echo (SE)-type experiment,^8,9 the geometric phase depends on the geometry of the environment and could be added by combining and refocusing reverse frequency sweep using radio frequency (RF) pulses.^10,11 Experimentally, several evolution circuits of magnetization had been used in MR for the detection of the geometric phases, and one of the most common is the cone circuit where in the tilted rotating frame of reference the angle varies between the axis of quantization of the laboratory and rotating frame.^12–14 

In NMR, the variations of B₀ and B₁ magnetic fields in the inhomogeneous sample cause different spin ensembles to acquire different dynamic phases, which could be refocused using spin echo and rotary echo strategies.^8,9,15 It had been understood that when spin magnetization undergoes precession in the rotating frame, the geometric phase is formed with its sign defined by the direction of the frequency sweep, and generally, the cyclic evolution of magnetization results in phase accumulation between ±π.¹¹ The geometric phase could be refocused along with the dynamic phases; however, it could be accumulated in the SE experiment by performing frequency sweep in opposite directions using frequency swept (FS) pulses.¹¹ This procedure results in the cancellation of the dynamic phase while leading to two geometric terms that add up together.¹¹ The cancellation of the dynamic phase using conventional SE approaches was utilized, such as for quantum computing by NMR¹⁶ and realization of a one-qubit quantum gates.¹⁰ For magnetic resonance applications, the refocusing could also be achieved by generating rotating frame rotary echoes when inverting the effective field halfway through the FS pulses.^17–19 Although the concept of the geometric phases is well established in different disciplines and had been evaluated previously for NMR applications,^13,20 to the best of our knowledge, the description of the geometric phase formation during FS pulses is still unavailable. Therefore, it is instrumental to evaluate how the geometric phase could be formed during the amplitude and frequency-modulated RF pulses operating in adiabatic and nonadiabatic regimes. This goal is motivated by the substantial merit of the FS pulses and their broad applicability for generating noninvasive MRI contrasts^21–25 and for protein dynamic characterization in NMR.^26,27

Recently, a rotating frame relaxation method entitled Relaxation Along a Fictitious Field (RAFF) in the rotating frame of rank n (RAFFn) has been introduced.^21–23 With RAFF2, the time-dependent Hamiltonian is transformed to a rotating frame of rank 2 resulting from nonadiabatic rotation of the effective field B_eff in the first rotating frame (FRF), which is the vector sum of B₁(t) and the fictitious component Δω/γ (Fig. 1).

This rotation produces a fictitious field component (γ⁻¹dα⁽¹⁾/dt). The orientation between B_eff(t) and the Z′ axis is described by an angle α⁽¹⁾(t). The second rotating frame (SRF) with the axis of quantization Z″ collinear to B_eff(t) evolves in the FRF. As a consequence of the time dependence of α⁽¹⁾(t), the FRF rotates around the Y′ axis leading to the SRF (n = 2). The new effective field B_E is the vector sum of two field components: one of these components B_eff(t) is the effective field in the FRF and the second is γ⁻¹dα⁽¹⁾/dt. We have shown that using different fictitious field angles α⁽²⁾ and amplitudes of the effective field, B_E, in the SRF allowed us to generate novel MRI contrasts in the human brain.²¹ The formation of the geometric phases in vivo, although predicted in prior contributions, had not been quantified so far for FS pulses. Early work by Cui³⁰ and subsequently by Lei and Zheng³¹ provided the analytical solution of the nonadiabatic geometric phases in the rotating systems. However, a detailed evaluation of the geometric phase formation for specific cases of the FS pulses operating in multiple rotating frames along with the experimental strategies for their detection in MR is not available.

In this work, we derived expressions for the geometric phases during the time-dependent RF Hamiltonian for spin ½. The RF pulses with sine amplitude modulation and cosine frequency modulation functions operating in the nonadiabatic regime were considered. We have utilized the formalism proposed by Suzuki et al.²⁸ and first explicated by Messina et al.²⁹ We elaborated on the dependence of the geometric phase on the initial conditions of the spinors $10$ and $01$ in the solution of the Schrödinger equation and evaluated geometric phases through their dependencies on the orientation of the effective field B_E in the SRF. Finally, we applied the developed formalism for the evaluation of the geometric phases during the RF pulse with sine amplitude modulation and cosine frequency modulation functions.

In a series of papers appearing soon after Berry’s work, it had been demonstrated that the geometric phases can be determined through NMR experiments.^{11–13,34,35} A detailed description of the interacting ensemble was provided by Gamliel and Freed in the context of electron spin resonance (ESR).³⁶ In this work, the conditions for experimental observations of the geometric phase by ESR were determined for the spin ensemble using the stochastic Liouville approach.³⁶ Specifically, it had been shown that the experimental observation of Berry’s phases could become plausible for the spin ensemble undergoing slow evolution as compared to the rate of change of the Hamiltonian.

Figure 1 illustrates the transformation of the spin ensemble from the FRF to the SRF. The nonadiabatic evolution of $ωeff(1)(t)$ in the FRF leads to the generation of a fictitious field component γ⁻¹dα⁽¹⁾(t)/dt.¹⁹ Thus, the effective field that is formed in the SRF, B_E, is the vector sum of two components, B_eff(t) and γ⁻¹dα⁽¹⁾(t)/dt,¹⁹ and has different α⁽²⁾ relative to Z” of the SRF depending on B_eff(t) and γ⁻¹dα⁽¹⁾(t)/dt. In Figs. 2 and 3, the calculations of sub-geometric phase components are shown for the spinor corresponding to initial conditions $10$⁠, $|Ψ11(t)〉$⁠, and $|Ψ12(t)〉$ and $01$⁠, $|Ψ21(t)〉$⁠, and $|Ψ22(t)〉$⁠, respectively, using Eq. (22), namely, $G11(t)=∫0tiΨ11(τ)|Ψ̇11(τ)dτ$ and $G12(t)=∫0tiΨ12(τ)|Ψ̇12(τ)dτ$⁠. The amplitudes of sub-geometric phases increase with the duration of the sine/cosine RF pulse and significantly depend on the angle between the effective field in the SRF and Z″, α⁽²⁾(t).

In Fig. 4, we compare analytical [Eq. (40)] and numerical solutions [Eq. (27)] for the total geometric phase, which are in remarkable agreement. It should be noted that the dissipation of the geometric phase through the imaginary part was not explicitly included in the presented treatment, implying that the relaxation phenomena were not described.

Figure 5 depicts one case of the RF pulses with sine amplitude modulation and cosine frequency modulation functions for α⁽²⁾ = 45° during P and P⁻¹ segments, along with the magnetization path during the pulses. In Figs. 5(b) and 5(c), it is shown that magnetization M undergoes a rotation from the Z″ to Y″ axis of the SRF during the period [0, T_p/2]. The rotation of M in the positive hemisphere is interrupted at the Y” axis, and M allows it to evolve in the negative hemisphere toward −Z″ during the period [T_p/2, T_p]. This rotation is achieved by instantaneously flipping B_E to π of the negative hemisphere, which is achieved by time-reversing both amplitude and phase and performing a π flip of the phase [Eq. (16)]. The pulse modulation functions are shown in Fig. 5(a). In Fig. 5(d), the numerical calculation of the geometric phase during the pulse waveforms represented in Fig. 5(a) is shown.

In this work, we solved the Schrödinger equation for spinor components to obtain sub-geometric phases in the presence of a nonadiabatic RF Hamiltonian. We considered the RF pulse with sine amplitude modulation and cosine frequency modulation functions used in the recently developed rotating frame method called RAFFn.^21–23 We provided an analytical representation of the propagators for spin ½ for the Hamiltonian that we defined in Eq. (4). This gives a closed form for the representation of the propagators in the FRF through α⁽¹⁾, ϕ_E, and ϕ(t). From two components of each spinor, we defined corresponding sub-geometric phases. For the general treatment, we solved the Schrödinger equation for different initial conditions of the spinors. Detailed analysis demonstrates that the geometric phases for initial conditions $10$ and $01$ are equal in magnitude but are opposite in their signs, which is in agreement with previously reported findings.³⁹ We evaluated geometric phases via both analytical derivation and numerical solutions of the Schrödinger equation applied to the case of the nonadiabatic RF pulses operating in the rotating frame of rank n = 2. We derived the sub-geometric phase components (Fig. 2), and we determined through the evolution operator the functional relationship between the geometric phase and the RF pulse parameters that determine the magnetization path. The analytical and numerical solutions presented in this work are in remarkable agreement (Fig. 4).

During RF irradiation with sine/cosine RF pulses, the nonadiabatic rotation of the effective field in the FRF results in a fictitious field component, which leads to a formation of the SRF (rank n = 2).^21,23 Figures 2 and 3 demonstrate the dependence of the geometric phase on α⁽²⁾ during the sine/cosine RF pulse. The calculations were performed for the initial condition of spinors $10$ and $01$⁠. It can be seen that the geometric phases are formed predominantly for the angles α⁽²⁾ ∼ π/4 and α⁽²⁾ ∼ 3/4π with the maximal values accumulated at the end of the pulse. Conversely, no geometric phase formation was obtained for α⁽²⁾ ∼ π/2. In our previous work, the detailed analysis of the magnetization trajectories during the sine/cosine RF pulse was presented [Fig. 1(a)].²¹ For these analyses, the trajectories of the magnetization M in the FRF were calculated using the Runge–Kutta algorithm.²¹ The trajectories of M for a given α⁽²⁾ demonstrate that for small [such as α⁽²⁾ = 5°] and large [such as α⁽²⁾ = 85°s] angles, M nutates only slightly from the Z′ axis in the FRF. For intermediate values of α⁽²⁾, M nutates with larger angles and reaches the Y′ axis in the FRF for α⁽²⁾ = π/4. Notably, for the large angles α⁽²⁾, despite the high amplitude of the effective field B_E in the SRF, the nutation angle of M remains small because of fast oscillations of both amplitude and frequency modulations [see Eqs. (14), (15), and (17)]. The formation of the geometric phases during the sine/cosine RF pulse is closely related to the tip angles of M for various α⁽²⁾. Specifically, our results suggest that the maximal geometric phase is obtained for α⁽²⁾ ∼ π/4 and α⁽²⁾ ∼ 3/(4π), which corresponds to a maximal nutation of M in the FRF.

Previously, Jones et al.¹¹ and subsequently Zhu and Wang³³ have suggested that the geometric phase can be added in the SE experiment when the direction of the frequency sweep is reversed after the refocusing pulse, while it can be canceled along with the dynamic phase when the same frequency sweep is utilized. Such a strategy was also used in subsequent investigations.^10,39 The geometric phase during sine/cosine RF pulses can be accumulated, such as by instantaneously reversing the effective field B_E in the SRF by π flip (Fig. 5). Magnetization evolves in the positive hemisphere during the first P segment of the RF pulses [Fig. 5(a)], but it evolves in different quadrants of the negative hemisphere during the P⁻¹ segment [Figs. 5(b) and 5(c)]. The calculations of the geometric phase for the PP⁻¹ segments shown in Fig. 5(a) are represented in Fig. 5(d). Despite M evolving in the negative hemisphere during the P⁻¹ segment of the RF pulse, which is achieved by instantaneous π flip of the effective field B_E, the accumulation of the geometric phase occurs because of the reverse frequency sweep. Such a strategy may offer an elegant solution for detecting the geometric phases in addition to the conventionally used spin echo-based approaches combined with reversed frequency sweeps.^10,11,16

In this work, the effort of evaluating geometric phases during amplitude and frequency-modulated RF pulses was motivated by a possible contribution of the geometric phase to image contrast, which could be noninvasively generated during in vivo MRI. Our analytical and numerical evaluations set the basic framework for the description of the geometric phases during a wide class of amplitude and frequency-modulated RF pulses. It is likely that the geometric phases if inaccurately taken into consideration may bias the quantification of the relaxation rate constants during MRI pulse sequences. Finally, the development of novel techniques that allow for efficient refocusing of the dynamic phase while accumulating the geometric phase can open a new horizon for investigations of tissue microstructure and function.

The formalism presented in this work can be used to describe the formation of geometric phases during RF swept pulses in MR operating in both adiabatic and nonadiabatic regimes. The formalism may be critical for accurately detailing noninvasive MRI tissue contrasts in vivo obtained during the application of RF waveforms.^22,23,40,41 Since FS pulses are frequently used for protein dynamics characterization in high-resolution NMR, the formalism presented here could also be useful for relaxation dispersion analysis when FS pulses are utilized.^26,27,42 To the best of our knowledge, a detailed evaluation of the relaxation processes with the inclusion of the effects of the geometric phases during amplitude- and frequency-modulated RF pulses has not been detailed in MR, and a limited effort had been dedicated to such a description.^36,43,44 Initially, the description of the dissipative processes with the inclusion of the geometric phases was provided using the stochastic Liouville approach by Gamliel and Freed³⁶ in Electron Paramagnetic Resonance (EPR). However, given significant differences between FS pulses operating in adiabatic and nonadiabatic regimes with substantial differences between modulation functions, a detailed description of the dissipation processes induced by geometric phases during FS pulses will require consideration in each case separately. Further investigations are thus warranted to properly consider the influence of the geometric phases on MRI contrasts generated noninvasively using FS pulses.

The authors would like to appreciate the NIH for support via core Grant Nos. P41 EB027061 and R01 NS129739. The authors thank Dr. Michael Garwood for stimulating discussions and Dr. Silvia Mangia for helpful discussions and editing the manuscript.

The authors have no conflicts to disclose.

D.J.S. participated in the design of the work, theoretical derivations and modeling, literature research, and writing the manuscript. S.M. participated in the design of the work, theoretical derivations and modeling, literature research, writing, and editing of the manuscript.

Dennis J. Sorce: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Shalom Michaeli: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

All calculations included in this work were conducted using Mathematica 12 software package, and the programs/code can be provided by the authors upon request.