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arXiv:2604.16985

Heteronuclear Polarization Transfers Between Spin-locked and Anti-Longitudinal Spin States in the NMR of Liquids and Spinning Solids

Sundaresan Jayanthi, Adonis Lupulescu, Julia Grinshtein, Lucio Frydman

quant-phSubmitted Apr 18, 2026• Announced Apr 21, 2026
arXivHTMLPDF

Abstract

Recently, Pang et al reported a novel polarization transfer scheme applicable to three-spin systems, whereby a rotating-frame NMR analogue of the cross effect could transfer polarization between; e.g., two 13Cs and an 15N in a single crystal. The present work furthers this scheme to the case of powder NMR under magic angle spinning (MAS) conditions, as well as to solution NMR. It is found that in all such cases a second-order average Hamiltonian can transfer polarization between non-equivalent, coupled abundant spins (e.g., two 1Hs) prepared in anti-longitudinal magnetization states, and the spin-locked magnetization of a rare spins (e.g., one 13C). The average Hamiltonian for such three-spin (S1-S2) to I transfer was derived for both liquids and solids, and found in good quantitative agreement with numerical simulations and experiments. At an optimal transfer condition whereby an I-spin RF irradiation field matches the S1-S2 chemical-shift-difference, a maximum polarization enhancement equal to the ratio of gyromagnetic ratios is achieved; as explained and demonstrated in the study, ca. half of this can be effectively obtained for I = 13C in powdered solids and in multi-spin systems in solutions. All such processes display an oscillatory nature, meaning that the transverse spin-locked polarization of a rare spin can become anti-longitudinal magnetization of abundant spins -without ever pulsing on the latter. The roles played by many-body interactions, RF inhomogeneities, and interferences of other coherences during the execution of these novel forms of cross-polarization were investigated, and are exemplified with experiments and simulations.

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In spin-related fundamental researches, there have been considerable interests in the transfer of microscopic orders between magnetic sublevels of different kinds of atoms. As for gaseous atomic systems, collisional transfer of spin polarization from optically pumped atoms A to atoms B is attained through spin exchange collisions. So far many experiments have been reported on the transfer of longitudinal spin polarization created along a static magnetic field. Transfer of coherence, on the other hand,is possible only when the energy matching condition j!A !Bj < is satisfied, where !A and !B are the Larmor frequencies of atoms A and B, and the line width. This is the reason why transfer of spin polarization precessing around the static magnetic field has not been observed except for a few experiments; The coherence transfer has been observed in the experiments where the atoms are subjected to properly chosen rf magnetic fields. Recently, it has been reported that coherence transfer between different kinds of atoms is possible by using the light field which transversely pumps only one kind of atoms under properly chosen experimetal conditions. It has also been pointed out that, as for the system of Rb and K atoms, which happen to have the same gyro-magnetic ratio, the peculier distortion of the magnetic resonance signal line shapes observed in the collisionally pumped atoms can be explained by the spin exchange coupling between the resonant Rb and K atoms. It is suggested that the precessing spin polarization in Rb can be transferred to K atoms or vice versa, since their Zeeman splittings become equal in the hyperfine multiplets jF 1⁄4 1i and jF 1⁄4 2i of the ground state 2S1=2 with the same gyromagnetic ratio 1⁄4 2 700 kHz/G. The experimental apparatus of the present experiment is shematically shown in Fig. 1. In addition to a droplet of natural rubidium and potassium, the sample cell contains 50mbar Ar as a buffer gas. It is subjected to a static magnetic field of H0 1⁄4 0:786G along the z axis and irradiated transversely to the field by a light beam from a GaAlAs laser A (output power = 10mW, beam diameter = 1mm). It is left-handed circularly polarized and tuned to the Rb absorption lines (S1=2, F 1⁄4 1 ! P1=2, F 1⁄4 1 and 2) which are not resolved owing to the Doppler broadening. The intensity of the laser A is modulated by an AOM (acousto-optic-modulator) at the Larmor frequency !A 1⁄4 2 550 (1⁄4 700 0:786) kHz of Rb in a rectangular shape pulse. As mentioned above the precessing spin polarization in Rb can be transferred to K. The precessing spin polarization of Rb and that of K are detected from the Faraday rotations of the beams of probe lasers B ( 1⁄4 794 nm, output power = 3mW, beam diameter = 2mm) and C ( 1⁄4 769 nm, output power = 3mW, beam diameter = 2mm), respectively. The frequency of laser B is chosen to be a few GHz higher than the Rb absorption lines (S1=2, F 1⁄4 1 ! P1=2, F 1⁄4 1 and 2) to prevent the detection beam from destroying the polarization. Similarly, the frequency of laser C is chosen to be a few GHz higher than the K absorption lines (S1=2, F 1⁄4 1 and 2 ! P1=2, F 1⁄4 1 and 2) which are not resolved owing to the Doppler broadening. They are linearly polarized and counter-propagated against the pumping beam,making angles of about 1 to the pumping beam. In each case the transmitted light beam of probe laser passes through the analyzer which makes an angle 45 to the polarization of the incident beam,and then it is detected by a pin-photo diode. The signals are electronically averaged over a few hundred cycles. Figure 2(a) shows the typical examples of signals on the magnetizations M x and M B x of Rb and K, respectively, Fig. 1. Schematic diagram of the experimetal apparatus. AOM, acoustooptic modulator; PD, pin-photo diode; P, polarizer; A, analyzer.

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Experiments in coherent spectroscopy correspond to control of quantum-mechanical ensembles guiding them from initial to final target states. The control inputs (pulse sequences) that accomplish these transfers should be designed to minimize the effects of relaxation and to optimize the sensitivity of the experiments. For example in Nuclear Magnetic Resonance (NMR) spectroscopy, a question of fundamental importance is what is the maximum efficiency of polarization or coherence transfer between coupled spins in the presence of relaxation. Furthermore, what is the optimal pulse sequence which achieves this efficiency? This letter is a continuation of our previous work on the above problems. Here, we attempt to answer these questions for an Ising spin chain, under the presence of transverse relaxation. The methods presented in this paper are expected to find applications not only in coherent spectroscopy but also in quantum information processing.

•May 15, 2005•0 citations
Recently, Pang et al reported a novel polarization transfer scheme applicable to three-spin systems, whereby a rotating-frame NMR analogue of the cross effect could transfer polarization between; e.g., two 13Cs and an 15N in a single crystal. The present work furthers this scheme to the case of powder NMR under magic angle spinning (MAS) conditions, as well as to solution NMR. It is found that in all such cases a second-order average Hamiltonian can transfer polarization between non-equivalent, coupled abundant spins (e.g., two 1Hs) prepared in anti-longitudinal magnetization states, and the spin-locked magnetization of a rare spins (e.g., one 13C). The average Hamiltonian for such three-spin (S1-S2) to I transfer was derived for both liquids and solids, and found in good quantitative agreement with numerical simulations and experiments. At an optimal transfer condition whereby an I-spin RF irradiation field matches the S1-S2 chemical-shift-difference, a maximum polarization enhancement equal to the ratio of gyromagnetic ratios is achieved; as explained and demonstrated in the study, ca. half of this can be effectively obtained for I = 13C in powdered solids and in multi-spin systems in solutions. All such processes display an oscillatory nature, meaning that the transverse spin-locked polarization of a rare spin can become anti-longitudinal magnetization of abundant spins -without ever pulsing on the latter. The roles played by many-body interactions, RF inhomogeneities, and interferences of other coherences during the execution of these novel forms of cross-polarization were investigated, and are exemplified with experiments and simulations.
In spin-related fundamental researches, there have been considerable interests in the transfer of microscopic orders between magnetic sublevels of different kinds of atoms. As for gaseous atomic systems, collisional transfer of spin polarization from optically pumped atoms A to atoms B is attained through spin exchange collisions. So far many experiments have been reported on the transfer of longitudinal spin polarization created along a static magnetic field. Transfer of coherence, on the other hand,is possible only when the energy matching condition j!A !Bj < is satisfied, where !A and !B are the Larmor frequencies of atoms A and B, and the line width. This is the reason why transfer of spin polarization precessing around the static magnetic field has not been observed except for a few experiments; The coherence transfer has been observed in the experiments where the atoms are subjected to properly chosen rf magnetic fields. Recently, it has been reported that coherence transfer between different kinds of atoms is possible by using the light field which transversely pumps only one kind of atoms under properly chosen experimetal conditions. It has also been pointed out that, as for the system of Rb and K atoms, which happen to have the same gyro-magnetic ratio, the peculier distortion of the magnetic resonance signal line shapes observed in the collisionally pumped atoms can be explained by the spin exchange coupling between the resonant Rb and K atoms. It is suggested that the precessing spin polarization in Rb can be transferred to K atoms or vice versa, since their Zeeman splittings become equal in the hyperfine multiplets jF 1⁄4 1i and jF 1⁄4 2i of the ground state 2S1=2 with the same gyromagnetic ratio 1⁄4 2 700 kHz/G. The experimental apparatus of the present experiment is shematically shown in Fig. 1. In addition to a droplet of natural rubidium and potassium, the sample cell contains 50mbar Ar as a buffer gas. It is subjected to a static magnetic field of H0 1⁄4 0:786G along the z axis and irradiated transversely to the field by a light beam from a GaAlAs laser A (output power = 10mW, beam diameter = 1mm). It is left-handed circularly polarized and tuned to the Rb absorption lines (S1=2, F 1⁄4 1 ! P1=2, F 1⁄4 1 and 2) which are not resolved owing to the Doppler broadening. The intensity of the laser A is modulated by an AOM (acousto-optic-modulator) at the Larmor frequency !A 1⁄4 2 550 (1⁄4 700 0:786) kHz of Rb in a rectangular shape pulse. As mentioned above the precessing spin polarization in Rb can be transferred to K. The precessing spin polarization of Rb and that of K are detected from the Faraday rotations of the beams of probe lasers B ( 1⁄4 794 nm, output power = 3mW, beam diameter = 2mm) and C ( 1⁄4 769 nm, output power = 3mW, beam diameter = 2mm), respectively. The frequency of laser B is chosen to be a few GHz higher than the Rb absorption lines (S1=2, F 1⁄4 1 ! P1=2, F 1⁄4 1 and 2) to prevent the detection beam from destroying the polarization. Similarly, the frequency of laser C is chosen to be a few GHz higher than the K absorption lines (S1=2, F 1⁄4 1 and 2 ! P1=2, F 1⁄4 1 and 2) which are not resolved owing to the Doppler broadening. They are linearly polarized and counter-propagated against the pumping beam,making angles of about 1 to the pumping beam. In each case the transmitted light beam of probe laser passes through the analyzer which makes an angle 45 to the polarization of the incident beam,and then it is detected by a pin-photo diode. The signals are electronically averaged over a few hundred cycles. Figure 2(a) shows the typical examples of signals on the magnetizations M x and M B x of Rb and K, respectively, Fig. 1. Schematic diagram of the experimetal apparatus. AOM, acoustooptic modulator; PD, pin-photo diode; P, polarizer; A, analyzer.
Taking inspiration from NMR spectroscopy, periodic irradiation schemes have recently shown remarkable performance when implemented into pulsed dynamic nuclear polarization (DNP) sequences. This has prompted considerable interest in development of broadband pulsed DNP sequences utilizing such schemes. On this background, most efforts have focused on solid-state NMR like transverse spin-locked pulse sequences whose performance in DNP applications may be compromised by the broadband capabilities of the initial excitation pulse. Leveraging the flexibility and robustness of optimal control theory combined with underlying insights from effective Hamiltonian theory, we present a new family of broadband DNP pulse sequences, termed LOOP (Longitudinally Optimized with Overarching Periodicity), that alleviates the excitation-pulse challenge by accomplishing longitudinal polarization transfer. These sequences define robust single-spin effective zzz rotations, with impressive compensation towards microwave field inhomogeneity, and are capable of delivering DNP transfer with bandwidths exceeding 100 MHz, while employing a peak microwave field amplitude of only 32 MHz, at an external magnetic field of 0.35 T.
Experiments in coherent spectroscopy correspond to control of quantum mechanical ensemble guiding them from initial to final target states. The control inputs (pulse sequences) that accomplish these transformations should be designed to minimize the effects of relaxation and to optimize the sensitivity of the experiments. For example in nuclear magnetic resonance (NMR) spectroscopy, a question of fundamental importance is what is the maximum efficiency of coherence or polarization transfer between two spins in the presence of relaxation. Furthermore what is the optimal pulse sequence which achieves this efficiency? This manuscript extends our previous work in the above problems. Here, we take into account both the longitudinal and the transverse relaxation mechanisms, thus generalizing our previous results, where the former had been neglected.
We report analytical solutions to the unitary bound problem for coherence/polarization transfer in IS two-spin-12 systems by means of unitary operations. Theoretical upper bounds for the transfer efficiency along with the associated optimum transformation operators are obtained analytically by decomposing the unitary operator as a product of exponentials in the special unitary Lie group SU(4). Addressing NMR spectroscopy as a specific example, the method is demonstrated for the non-Hermitian transfers I−→S− and 2I−Sz→S− being relevant for heteronuclear single-quantum coherence (HSQC) experiments as well as the double- to single-quantum transfer I−S−→I−Sβ+IβS− being representative for coherence-order and spin-state-selective transfer in INADEQUATE CR experiments. Furthermore, using a Lagrangian function approach it is demonstrated how the method enables analytical description of two-dimensional bounds for Iz→Sz cross polarization.
In this work, quantum correlations in two model systems which can be invesitgated by means of solid-state NMR are considered. The first is a two-spin system in NMR spin-locking experiment, where two schemes of spin-locking are considered. Concurrence is calculated for both schemes assuming the initial thermodynamic equilibrium state. Quantum discord is calculated analytically in the high-temperature approximation. The second model system is the spin-1/2 chain with the XY interaction, where the transfer of multiple quantum (MQ) coherences from one end of the chain to another is studied. The conditions under which the density matrix of the receiver is a block-scaled version of the sender’s density matrix are found.
Experiments in coherent spectroscopy correspond to control of quantum-mechanical ensembles guiding them from initial to final target states. The control inputs (pulse sequences) that accomplish these transfers should be designed to minimize the effects of relaxation and to optimize the sensitivity of the experiments. For example in Nuclear Magnetic Resonance (NMR) spectroscopy, a question of fundamental importance is what is the maximum efficiency of polarization or coherence transfer between coupled spins in the presence of relaxation. Furthermore, what is the optimal pulse sequence which achieves this efficiency? This letter is a continuation of our previous work on the above problems. Here, we attempt to answer these questions for an Ising spin chain, under the presence of transverse relaxation. The methods presented in this paper are expected to find applications not only in coherent spectroscopy but also in quantum information processing.