Continuously tracked, stable, large excursion trajectories of dipolar coupled nuclear spins Ozgur Sahin, 1, ∗ Hawraa Al Asadi, 1, ∗ Paul M. Schindler, 2, ∗ Arjun Pillai, 1 Erica Sanchez, 1 Matthew Markham, 3 Mark Elo, 4 Maxwell McAllister, 1 Emanuel Druga, 1 Christoph Fleckenstein, 5 Marin Bukov, 2, 6 and Ashok Ajoy 1, 7 1 Department of Chemistry, University of California, Berkeley, Berkeley, CA 94720, USA. 2 Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany. 3 Element Six Innovation, Fermi Avenue, Harwell Oxford, Didcot, Oxfordshire OX11 0QR, UK. 4 Tabor Electronics, Inc., Hatasia 9, Nesher 3660301, Israel. 5 Department of Physics, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden. 6 Department of Physics, St. Kliment Ohridski University of Sofia, 5 James Bourchier Blvd, 1164 Sofia, Bulgaria. 7 Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. We report an experimental approach to excite, stabilize, and continuously track Bloch sphere orbits of dipolar- coupled nuclear spins in a solid. We demonstrate these results on a model system of hyperpolarized 13 C nuclear spins in diamond. Without quantum control, inter-spin coupling leads to rapid spin decay in  ∗ 2 ≈1.5ms. We elucidate a method to preserve trajectories for over  ′ 2 >27s at excursion solid angles up to 16 ◦ , even in the presence of strong inter-spin coupling. This exploits a novel spin driving strategy that thermalizes the spins to a long-lived dipolar many-body state, while driving them in highly stable orbits. We show that motion of the spins can be quasi-continuously tracked for over 35s in three dimensions on the Bloch sphere. In this time the spins complete >68,000 closed precession orbits, demonstrating high stability and robustness against error. We experimentally probe the transient approach to such rigid motion, and thereby show the ability to engineer highly stable “designer” spin trajectories. Our results suggest new ways to stabilize and interrogate strongly-coupled quantum systems through periodic driving and portend powerful applications of rigid spin orbits in quantum sensing. I. INTRODUCTION Periodically driven quantum systems have attracted enormous interest for their ability to host novel far-from-equilibrium phases of matter [1–5], and for sustaining long-lived, highly stable states wherein absorption of energy can be controllably suppressed [6– 8]. Consider, for instance, a network of dipolar coupled solid- state spins; their interaction drives rapid free induction decay of prepared transverse states in a very short time  ∗ 2 . However, via Floquet prethermalization [8–11], periodically driving the spins can greatly extend these state lifetimes from  ∗ 2 to  ′ 2 in a manner that parametrically increases with the driving frequency. However, this approach remains restricted to specific initial states, typically those aligned parallel to the driving field. More generally applicable schemes for stabilization, especially along arbitrary axes, remain elusive. In this paper, we propose and demonstrate a strategy to excite and stabilize closed spin orbits on the Bloch sphere, including those that span large angular ex- cursions away from the driving axis. Our approach exploits novel applications of the eigenstate thermalization hypothesis (ETH) and quantum thermalization [12, 13], combined with the engi- neering of a family of effective Hamiltonians [14] under the si- multaneous action of two orthogonal and frequency-separated driving fields. Stable orbits are then excited within the micro- motion dynamics between these Hamiltonians [15]. In a model system of strongly interacting 13 C nuclear spins in diamond with an intrinsic  ∗ 2 ≈1.5ms [16], we demonstrate the ability to stabi- lize highly tunable orbits with excursion angles >16 ◦ for a lifetime  ′ 2 >27s. This corresponds to an increase in the spin lifetimes of over 18, 000-fold, even in the presence of strong inter-spin dipolar couplings. This same method allows us to quasi-continuously track the resulting spin motion in three-dimensions on a Bloch sphere over long periods; here we demonstrate the quasi-continuous track- ing of spins as they traverse >68,000 cycles of the engineered ∗ Equal contribution Laser 13 C 13 C 13 C 13 C 13 C 13 C 13 C 13 C 13 C V- N 0.9 -0.9 0 0 0 0.9 0.9 -0.9 A . . . Control π 2 y B AC time time θ x θ x θ x θ x τ 1/f AC t acq tpulse B C |Z| Y X Φ Fig. 1. Concept. (A) System: ensemble of dipolar-coupled 13 C nuclear spins hyperpolarized by NV centers at 38mT. (B) Protocol:  -pulses spin-lock the 13 C nuclei at 7T, which are simultaneously subject to a time- varying field with amplitude  AC and angular frequency  AC =( 2)  AC respectively. Pulse spacing is , pulse width is  pulse , and spin precession amplitude and phase are interrogated in  acq windows between pulses. (C) Experimentally tracked “8-point star” spin trajectory on a Bloch sphere during a Δ =4ms window (zoomed in inset). Trajectory remains rigid (stable) for  ′ 2 ≈26.4s, and is continuously tracked for the entire period (see Fig. 3). Extent of the spin excursion on the Bloch sphere is denoted by angle Φ (here Φ≈11.5 ◦ ) . stabilized orbits. While such state tracking is difficult in many experimental scenarios [17,18], our method makes it easily at- tainable by (1) exploiting weak coupling of the nuclear spins to a readout cavity in a manner that imposes no back-action upon them [19], (2) employing “hyperpolarization”, yielding consider- able gains in measured signal to noise [20,21], and (3) arranging a hierarchical, >10 5 -fold separation of time scales between the rates of cavity-induced spin interrogation (signal sampling) and that of the spin orbiting. Ultimately, this rapid spin tracking un- ravels the emergence of stable spin orbits, including insights into the underlying dynamical thermalization processes that are key to their rigidity. Leveraging these special features, we demonstrate the ability to excite and track “designer” spin trajectories created arXiv:2206.14945v2 [quant-ph] 13 Jul 2022