Phase-Sensitive Nuclear Spectroscopy

• Phase-Sensitive Nuclear Spectroscopy is a set of advanced techniques that measure both amplitude and phase information to overcome limitations of conventional amplitude-only methods.
• It utilizes controlled perturbations, such as pulsed-strain NMR and quantum sensing protocols, to encode subtle frequency shifts as measurable phase changes.
• Techniques like maximum likelihood phase retrieval and ptychographic reconstruction enable quantitative analysis of nuclear responses, improving sensitivity and resolution in complex systems.

Phase-sensitive nuclear spectroscopy refers to a class of experimental and analytical techniques designed to extract not only the amplitude (or intensity) but also the phase information of the nuclear response (e.g., resonance frequency shifts, coherence, and quantum correlations) under various probing conditions. Such methodologies have become central in NMR, Mössbauer, and x-ray nuclear resonance experiments for advancing sensitivity, resolution, and the full reconstruction of underlying physical phenomena, especially in regimes challenged by inhomogeneous broadening, quantum noise, or complex many-body environments.

1. Principles of Phase-Sensitive Detection in Nuclear Magnetic Resonance

Traditional NMR detection relies on measuring the voltage induced in a receiver coil from precessing nuclear magnetization, providing access primarily to amplitude (i.e., spectral intensity) of the nuclear response. However, amplitude-based detection is fundamentally limited in scenarios where resonance shifts ($\Delta f$) are smaller than the inhomogeneous linewidth ($T_2^*{}^{-1}$), as line broadening from sample inhomogeneity or experimental artifacts can mask subtle features such as symmetry-breaking susceptibilities or fine energy shifts.

Phase-sensitive detection, by contrast, encodes frequency shifts as phase accumulations of the transverse magnetization in the rotating frame. If a brief, controllable perturbation (such as pulsed strain or magnetic field) is applied during free precession, the nuclear magnetization acquires an extra phase,

$$\varphi = \int_0^{t_p} 2\pi\,\Delta f\,dt = 2\pi\,\Delta f\,t_p$$

For a Hahn echo sequence with perturbing pulses before and after the $\pi$-pulse, the net phase shift becomes $\varphi = 2\pi\,\Delta f\,2t_p$, as the phase acquired prior to the $\pi$-pulse is retraced in the inverted frame. In a pulsed-strain NMR context, where the perturbation is uniaxial strain $\epsilon(t)$, this phase shift reads

$$\varphi = 4\pi\,t_p\,\frac{\partial f}{\partial\epsilon}\,\epsilon$$

allowing direct measurement of frequency derivatives with respect to strain, and thus of fundamental susceptibilities.

Notably, in the phase-sensitive scheme, the minimum resolvable frequency shift is set by phase precision and homogeneous $T_2$, not by inhomogeneous $T_2^*$. This results in dramatically increased detection sensitivity, overcoming the typical penalty from sample broadening (e.g., an improvement by up to three orders of magnitude in the case of BaFe$_2$As$_2$ (Chaffey et al., 2023)).

2. Experimental Implementations: Pulsed-Strain NMR and Quantum Sensing

A "standard" implementation of phase-sensitive detection in NMR employs a Hahn echo ($\pi/2$–$\tau$–$\pi$–$\tau$–echo) with the addition of pulsed uniaxial strain fields precisely interleaved with RF pulses. Strain is applied using a piezoelectric device, where synchronized voltage pulses alter lattice symmetry and modulate local electric field gradients (EFG) or Knight-shift tensors coupling to the nuclear spin Hamiltonian. The timings of strain pulses are matched with transverse evolution windows ($\tau$) such that the post-echo phase directly encodes the strain-induced frequency shift. Using this approach, the echo phase

$$\frac{\varphi}{2\pi} = 2 \left(\frac{\partial f}{\partial\epsilon}\right) \epsilon t_p$$

is extracted from a standard quadrature measurement of the echo signal.

In quantum sensing modalities, e.g., using shallow NV centers in diamond, phase-sensitive nuclear spectroscopy is implemented by stroboscopically applying weak measurement interactions that entangle a single nuclear spin to an NV electronic spin meter via Hamiltonians of the form $H_{\text{int}} = (g/2)\,\sigma_z\,I_x$, where $g$ is tuned via dynamical decoupling. Each NV readout imprints partial information about the nuclear phase and imparts back-action, which is minimized by controlling the measurement strength. Autocorrelation and Fourier analysis of the weak measurement outcome trains reconstructs both amplitude and phase, effectively realizing a single-spin analog of classical lock-in detection with spectral resolutions reaching a few Hz (Pfender et al., 2018, Cujia et al., 2018).

Mössbauer and x-ray nuclear phase-sensitive experiments employ 2D time- and energy-resolved detection with either phase retrieval algorithms (e.g., NPRS (Yuan et al., 2022)) or advanced ptychographic reconstruction (nuclear ptychoscopy (Yuan et al., 7 Nov 2025)), which enable recovery of the full complex response function $R(\Delta)$ (both modulus and argument) from intensity-only measurements by exploiting redundancy in overlapped "views" (Doppler shifts or driving phases).

3. Relation to Response Functions and Material Susceptibility

Phase-sensitive nuclear spectroscopy directly accesses response functions that encode fundamental susceptibilities, including but not limited to quadrupolar, Knight-shift, or nematic susceptibilities: $$\frac{\partial f_i}{\partial \epsilon} = \sum_{j} \frac{\partial f_i}{\partial x_j} \chi_j$$ where $x_j$ denotes EFG asymmetry, the principal EFG eigenvalue, or the Knight-shift tensor component, and $\chi_j$ are the susceptibilities. By carefully measuring $\partial f/\partial \epsilon$ via the echo phase, one can extract specific components (e.g., the nematic susceptibility $\chi_\eta$), as demonstrated in BaFe$_2$As$_2$, yielding quantitative, temperature-dependent susceptibilities that obey Curie–Weiss laws: $\chi_\eta(T) = \frac{C}{T - T_0}$

In other systems, e.g., high-pressure hydrogen, the chemical shielding tensor $\sigma$ probed by phase-sensitive NMR provides spectral fingerprints that distinguish dynamically competing quantum crystal phases, as the phase-sensitive NMR spectrum (number, split, and position of resonance lines) is intimately linked to the underlying structure and nuclear dynamics, including quantum zero-point fluctuations (Monserrat et al., 2019).

4. Data Analysis Methods: Phase Retrieval and Ptychographic Reconstruction

Recovering the full complex response from intensity-only data in nuclear and Mössbauer spectroscopy requires advanced analysis methods:

• Maximum Likelihood and Gradient Descent: For 2D data (time, detuning), the nuclear phase retrieval problem is cast as an optimization of a non-convex cost function (Poisson or least squares) over the trial response vector $\mathbf{R}$: $$\ell(\mathbf{R}) = \sum_{k,l} \left[ |\mathbf{F}_{:,k}^H(\mathbf{R} \odot \mathbf{T}_l)|^2 - I(k,l)\ln |\mathbf{F}_{:,k}^H(\mathbf{R} \odot \mathbf{T}_l)|^2 \right]$$ Optimization is accelerated by momentum terms (Nesterov), adaptive reweighting, and amplitude-projection updates.
• Ptychographic Algorithms: The redundancy arising from overlapped energy/time binning (e.g., successive Doppler detunings) is exploited via alternating projection methods (AP), Douglas–Rachford (DR), or RAAR, alternating between modulus enforcement and product-structure enforcement for sample and analyzer responses. Blind and semi-blind variants handle unknown analyzer profiles (Yuan et al., 7 Nov 2025).
• Fourier and Multidimensional Analysis: Sequential/weak measurement protocols generate time-ordered records whose autocorrelations and higher-order moments (second, fourth, etc.) are Fourier transformed to yield the full spectrum and higher-order phase correlations (Meinel et al., 2021, Herb et al., 4 Mar 2024). Multidimensional (2D/3D) spectra disentangle hyperfine, dipolar, and J-couplings, allowing assignment of nuclear spins in complex systems.
• Sparse Signal Recovery: For phase-incremented SSFP (steady-state free precession) experiments, spectra are reconstructed via LASSO-regularized inversion of the linear forward model, allowing use of large flip angles (for SNR) without sacrificing spectral resolution (Shif et al., 29 Jan 2025).

5. Sensitivity, Resolution, and Overcoming Experimental Limitations

A core advantage of phase-sensitive techniques is that their sensitivity is set by $T_2$ (homogeneous dephasing time), not $T_2^*$ (inhomogeneous linewidth), enabling detection of smaller frequency shifts and thus smaller susceptibilities or couplings. The fundamental measurement limit becomes: $$\delta f_\mathrm{min} = \frac{\delta\varphi}{4\pi\,T_2}$$ with the minimum detectable strain or field scaling inversely with both $T_2$ and the respective coupling derivative.

Practically, this enables:

• Detection of frequency shifts an order of magnitude or more below the inhomogeneous linewidth (Chaffey et al., 2023).
• Sub-10 Hz frequency resolution on single nuclear spins, unattainable by population-only NMR (Pfender et al., 2018).
• Disentangling true quantum spin dynamics from classical or random-phase backgrounds via fourth-order (and higher) correlation functions—possible only in a phase-sensitive, quantum nonlinear measurement regime (Meinel et al., 2021).
• Operation in challenging environments, such as in the superconducting state (where resistivity-based probes fail) or in materials with strong inhomogeneous broadening.

Remaining limitations include the need for precise calibration of pulsed perturbations (e.g., strain-piezo responses), suppression of mechanical or electrical artifacts (e.g., avoiding piezo resonances), and managing photon or detector noise in high-resolution x-ray regimes.

6. Applications and Broader Impact

Phase-sensitive nuclear spectroscopy is impactful across a range of condensed-matter, quantum sensing, and metrology problems:

• Nematic and Electronic Susceptibility Probing: Pulsed-strain NMR measures nematic susceptibilities in pnictides, heavy-fermion, and oxide superconductors for which static-shift detection is broadened beyond feasibility (Chaffey et al., 2023).
• Single-Spin NMR and Quantum Sensing: Weak-measurement protocols with NV centers reach single-nuclear-spin sensitivity, enabling atomic-scale chemical structure mapping and single-molecule NMR (Pfender et al., 2018, Cujia et al., 2018, Herb et al., 4 Mar 2024).
• Metrological Quantum Enhancement: Global-phase spectroscopy using collective atomic ensembles on clock transitions achieves sub-SQL (standard quantum limit) phase estimation by leveraging geometric phase accumulation and reversibility, providing enhanced stability for optical lattice clocks (Zaporski et al., 2 Apr 2025).
• Mössbauer and x-ray Spectroscopy: Nuclear phase-retrieval and ptychography yield both phase and amplitude response functions, enabling ultrahigh energy precision, quantum optics with nuclei, and eventual implementation in XFELs and nuclear clocks (Yuan et al., 7 Nov 2025, Yuan et al., 2022, Herkommer et al., 2020).
• Quantum Foundations and Many-body Physics: Extraction of higher-order quantum correlations (e.g., Leggett–Garg inequality tests), investigation of quantum phase transitions in measurement backaction, and spectral editing in nuclear clusters (Meinel et al., 2021, Herb et al., 4 Mar 2024).

A plausible implication is that phase-sensitive nuclear methods will become central for next-generation quantum sensors, precision metrology, and spectroscopic probes in systems where conventional amplitude-only detection can no longer resolve key phenomena due to noise, decoherence, or overlapping signals.