Quadrature-Symmetric PulsePol for Robust Quantum Control Beyond the Ideal Pulse Approximation

Abstract

PulsePol is an elegantly designed pulse-sequence-based quantum control scheme that enables polarization transfer between electron and nuclear spins, for example, in nitrogen-vacancy (NV) centers. However, previous analyses of PulsePol assumed very strong, near-ideal, instantaneous microwave pulses, which is rarely achievable at higher magnetic fields. We revisit the PulsePol scheme under finite-pulse constraints and show that its performance significantly degrades due to finite-pulse effects. Using bimodal Floquet theory, we identify the symmetry-breaking mechanism responsible for this deterioration in fidelity. By phase adjustment, we reestablish the proper symmetry of the interaction-frame spin Hamiltonian, leading to a sequence called Q-PulsePol, where "Q" reflects the restored quadrature symmetry. Our results demonstrate robustness to finite-pulse effects and improved polarization transfer efficiency, establishing Q-PulsePol as a practical and reliable scheme for bulk hyperpolarization of nuclear spins in solids using a single-mode (zero-quantum or double-quantum) transfer. This work bridges idealized quantum control with realistic pulse engineering, establishing design rules for spin-based quantum control protocols.

Figures (7)

• Figure 1: The Problem in PulsePol and Need for Hamiltonian Engineering. Comparison of NOVEL and PulsePol pulse-sequences and their performance under finite-pulse conditions. (I) NOVEL: (a) pulse schematic and $e^-$ spin trajectory in the interaction frame, showing (b) a single dominant Fourier mode. (II) PulsePol: (c) pulse schematic and $e^-$ spin trajectory (see Eq. (5)), which is cyclic only over an even number of cycles. The free-precession delay $\tau$ is set by the DNP resonance condition $\omega_{0n} = k\omega_c$, which fixes the cycle time $T_c$ ($\omega_c = 2\pi/T_c$). (d) Fourier spectrum (see Eq. (6)) revealing multiple active resonance conditions with dominance at $k=3$. (e) $e^-$-spin inversion profile of a $\pi$-pulse along $S_y$, illustrating how increasing pulse finiteness ($T_{\pi}$ denotes the duration of the longest $\pi$ pulse) progressively distorts the ideal square-wave trajectory (red), breaking the toggling-frame symmetries that PulsePol relies upon. (f) Consequently, the nuclear polarization $\langle I_z \rangle$ (plotted for short contact time $\sim10-20 \,\mu$s for $\omega_{0n}/(2\pi) \approx 15$ MHz) degrades rapidly with increasing pulse finiteness (gradient corresponding to (e) highlights arbitrarily how pulse-finiteness decreases PulsePol efficiency).
• Figure 2: Visualization of symmetry operations in the interaction-frame spin trajectories. The central panel shows the original trajectory, or phase cycle, $X(t)$, composed of four segments (A--D) over one cycle period $T_c$. (a) Quadrature symmetry: the Y-component follows from a quarter-cycle time shift of the X-component, $Y(t) = \pm X(t \pm T_c/4)$, corresponding to a cyclic permutation of the phase segments (ABCD $\rightarrow$ DABC). (b) XY time-reversal symmetry: the Y-component is related to the X-component through time reversal, $Y(t) = X(T_c - t)$, resulting in a reversed ordering of segments (ABCD $\rightarrow$ DCBA). These symmetry constraints determine the characteristics of the Fourier coefficients and, consequently, the DNP scaling factors.
• Figure 3: Fourier analysis of PulsePol and Q-PulsePol under finite pulses. (a) Pulse-sequence schematics for standard PulsePol ($-$X central pulse, purple) and Q-PulsePol ($+$X central pulse, orange). A pulse phase of $360^\circ$ implies the completion of one $T_c$. (b,e) Interaction-frame $X(t)$ ($\langle S_x\rangle$) and $Y(t)$ ($\langle S_y\rangle$) components, respectively. Notice that the pulse-trajectories are finite, i.e., they are not delta functions and possess a slope. Critically, the central phase correction leaves $X(t)$ unchanged (black) but changes $Y(t)$ and helps in restoring the proper quadrature relationship between $X(t)$ and $Y(t)$. (c,d) Real and imaginary parts of the Fourier coefficients $a_x^{(k)}$, and (f,g) real and imaginary parts of $a_y^{(k)}$, confirming that Q-PulsePol satisfies the algebraic conditions of Eqs. (11) and (13), establishing pure DQ selectivity and recovery of high scaling factor, whereas standard PulsePol violates them both under finite pulses.
• Figure 4: DNP Scaling Factors in the Finite Pulse Limit. DQ and ZQ scaling factors $\chi_{\rm DQ}^{(k)}$ and $\chi_{\rm ZQ}^{(k)}$ as a function of Fourier index $k$ for (a) standard PulsePol and (b) Q-PulsePol, evaluated at the maximally finite pulse condition where pulse durations collectively occupy the full cycle time $T_c$ ($f = 1$, see Sec. \ref{['4']}). While standard PulsePol activates competing DQ and ZQ pathways at all harmonics, Q-PulsePol enforces quadrature symmetry and recovers pure uni-modal transfer. This demonstrates that the central phase correction restores selectivity even under the most demanding finite-pulse conditions.
• Figure 5: Effect of Pulse Finiteness on the Dominant Fourier Coefficients. Magnitude of $|a_x^{(k)}|$ (a) and $|a_y^{(k)}|$ (b) at the dominant harmonic $k=3$ as a function of the finiteness factor $f = \omega_1/(4\omega_c)$. The x-coefficients are identical for both sequences because the x-component of the trajectories is unchanged, while the y-coefficients highlight the decisive advantage of the phase adjustment in the finite-pulse regime. The original PulsePol experiments on NV centers in diamond schwartz2018robust correspond to the $f>10$ regime, whereas our experiments in Sec. \ref{['6']} probe PulsePol and Q-PulsePol in the $1<f<2$ regime.