Tunable Mpemba Effect in a Prethermal Many-Body Spin Network

AI summary

Overview

Research area: Quantum many-body dynamics and nonequilibrium statistical physics — specifically anomalous relaxation (the Mpemba effect) in a disordered, interacting nuclear-spin network, observed under Floquet driving that stabilizes a prethermal regime.

Technical level: Intermediate. The core physics question is stated in accessible terms, but full engagement requires familiarity with NMR-style spin control, dipolar interactions, hyperpolarization, Floquet driving, and eigenmode decompositions of relaxation.

Scope: An experimental and semiclassical-theoretical study showing that the Mpemba effect — where a state initially farther from equilibrium relaxes faster than one closer to it — can be observed, explained, and continuously tuned in a many-body ¹³C spin network in diamond, with the crossing time shifted over several orders of magnitude into a prethermal plateau.

What This Paper Is About

When an interacting quantum system relaxes back to equilibrium, how fast it gets there depends not only on how far from equilibrium it starts, but on which collective relaxation modes its initial state happens to populate. This paper tests that idea in a real material: a disordered network of ¹³C nuclear spins in diamond, where randomly placed paramagnetic defects create fast-relaxing regions and slow-relaxing regions. The goal is to prepare states with different spatial polarization profiles, watch them relax, and demonstrate that the state starting farther from equilibrium can overtake the one starting closer — then show that the time at which they cross can be deliberately tuned.

Key Contributions

1. First experimental observation of the Mpemba effect in an extended, interacting many-body spin network. Prior experimental demonstrations (trapped ions, a superconducting processor, few-spin NMR systems, a colloidal particle in an optical trap, a levitated nanoparticle) involved one or a few controlled degrees of freedom, an engineered potential, or a phase transition. Here the effect emerges collectively from native spatial disorder in a network of roughly 2350 active ¹³C spins per simulated disorder realization.

2. First experimental demonstration of Mpemba dynamics inside a prethermal state. A previous theoretical study predicted an inverse Mpemba effect during relaxation toward a prethermal state in a periodically driven isolated system; this work observes crossings not just en route, but within the long-lived prethermal plateau itself, and tunes the crossing between the prethermal and late-time thermalizing regimes.

3. A tunable state-preparation knob for engineering the crossing time. By independently varying the hyperpolarization duration t_h and the low-field wait duration t_w in a field-cycling protocol, the authors shift the crossing time t_× over several orders of magnitude without changing the drive or the final equilibrium state.

4. A mode-overlap explanation backed by semiclassical simulation. A polarization-transport model over many random disorder realizations identifies the slowest relaxation eigenmode as localized in defect-poor regions, and shows that the Mpemba crossing is governed by each initial state's overlap a_0 with that mode — with a phase diagram in (t_h, t_w) space predicting where crossings occur.

Main Findings

• Reproducible Mpemba crossings are observed directly. Comparing states prepared with (t_h, t_w) = (60, 0) s and (150, 150) s, the (60, 0) s state starts with larger total polarization (farther from equilibrium) yet relaxes more rapidly and is overtaken, producing a clear crossing at t_× ≈ 8 s.

• Hyperpolarization time controls both polarization and lifetime. With t_w = 0 s fixed, increasing t_h yields a larger initial signal (quantified in the upper-right inset of Fig. 2A) and progressively slower normalized relaxation, because polarization injected near NV centers spreads farther from electrons via dipolar spin diffusion.

• Wait time selectively depletes the fast-relaxing regions. With t_h = 60 s fixed, increasing t_w decreases the initial signal while slowing relaxation, because fluctuating NV and P1 electron spins preferentially destroy polarization stored near paramagnetic defects.

• The crossing time is broadly tunable. Varying the preparation protocol systematically shifts t_× over several orders of magnitude, from the late-time thermalizing regime into the prethermal plateau (Fig. 2C).

• The slowest relaxation mode lives in defect-poor regions. The eigenmode decomposition p(t) = Σ_j a_j v_j e^{-λ_j t} gives a slowest rate averaging λ_0 = (7.75 ± 0.16) × 10⁻³ s⁻¹ over 100 disorder realizations; the corresponding mode |v_0| is localized away from the randomly positioned paramagnetic defects.

• Overlap with the slow mode, not total polarization, sets the crossing. For the two states above, the initially less polarized (150, 150) s state has a larger slow-mode overlap (a_0 = 1.71 × 10⁻³) than the more polarized (60, 0) s state (a_0 = 1.47 × 10⁻³), so it retains more polarization at long times.

• The effect survives a rigorous distance metric. Recomputing distance from equilibrium with the Kullback–Leibler divergence (under a product-state approximation, which neglects interspin correlations) produces curves that cross at approximately the same time as the polarization curves, confirming the effect is not an artifact of using total polarization as the measure.

• A predicted phase diagram delimits the effect. Mapping the overlap structure across (t_h, t_w) space yields regions (orange) where a (60, 0) s reference state relaxes faster than the comparison state, and regions (green) where no crossing occurs.

• Driving monitors, rather than creates, the crossover. The authors state explicitly that the Floquet sequence is used to resolve each trajectory quasi-continuously and to extend the nuclear-spin lifetime; it does not itself produce the crossover.

Methodology in Plain English

The experiment uses a single-crystal diamond (type-Ib, 3.4 × 3.2 × 2.1 mm) containing naturally abundant ¹³C nuclei at 1.1%, NV centers at roughly 1 ppm, and P1 (substitutional nitrogen) centers at roughly 30 ppm. The ¹³C nuclei form an interacting network coupled by dipolar interactions, with an average nearest-neighbor coupling of about 60 Hz. Because the paramagnetic defects are sparse and randomly placed, nuclei sitting close to a defect relax much faster (rates scaling as r⁻⁶) than nuclei far from any defect, creating a naturally heterogeneous relaxation landscape.

State preparation happens at low field. The sample sits at 36 mT, where thirty 1-W, 532-nm fiber-coupled lasers optically pump the NV centers and chirped microwaves at roughly 4 GHz (about 15 W) transfer polarization into nearby ¹³C nuclei over a duration t_h. Dipolar spin transport simultaneously spreads that polarization outward. The sample then waits at low field for a duration t_w, during which fluctuating electron spins selectively erase polarization near NV and P1 centers. These two knobs together set both the total polarization and its spatial profile.

Readout happens at high field. A belt-driven actuator shuttles the sample into a 7.3 T superconducting magnet in about 1 s. There, a home-built NMR probe with a copper saddle coil interrogates the ¹³C ensemble at roughly 75 MHz. A (π/2)_y pulse rotates the spins transverse, then a train of (π/2)_x pulses (56.32 μs pulse duration, 64.92 μs inter-pulse delay, typically about 1 million pulses per experiment) engineers an effective dipolar Hamiltonian that produces a long-lived prethermal state. Between pulses, 3 μs of precession signal is digitized at 0.74 ns intervals by a Proteus P9484M transceiver (16 GB memory, up to 2.7 GS/s), mixed down with an on-board numerically controlled oscillator, and Fourier transformed so the spectral peak amplitude can be tracked as a function of time. This yields quasi-continuous relaxation trajectories over several minutes.

To explain the results, the authors build a semiclassical polarization-transport model. They construct a 30 × 30 × 30 diamond-cubic lattice of conventional unit cells (about 2.16 × 10⁵ sites), randomly populate it at the experimental concentrations, exclude ¹³C spins within 7 Å of any paramagnetic center (the strong Fermi-contact region), and impose periodic boundary conditions. A typical realization contains 1 NV center, 5 P1 centers, and about 2350 active ¹³C spins. Polarization evolves as ṗ(t) = M(t)p(t) + I(t), with hopping rates built from dipolar couplings and bath correlation times (dimensionless prefactor κ = 0.13), low-field electron-induced relaxation (η_LF = 0.0022), and an injection term derived from Landau–Zener anticrossing probabilities during the microwave chirp (sweep rate 750 Hz, bandwidth 10 MHz, Rabi frequency about 2.36 MHz). Simulated polarization is summed only over spins whose hyperfine coupling falls below the readout bandwidth of about 17.8 kHz. Ensemble averages are taken over roughly 100 disorder realizations.

Why This Matters

This work reframes disorder as a design resource rather than a nuisance: random defect placement shapes the eigenmode structure of relaxation, spin transport sets how spatially extended those modes are, and state preparation sets how much weight each mode carries. It gives experimental grounding to the mode-overlap picture of anomalous relaxation in a genuinely many-body, interacting, disordered setting, and it opens the prethermal regime as a controllable venue for studying nonequilibrium dynamics — a regime where the relaxation-mode hierarchy can be resolved before final thermalization.

Real-world applications:

• Quantum sensing: preserving initialized spin states for longer coherence or signal-averaging windows, since the paper frames state preservation for sensing as a motivating application.
• Rapid qubit reset: accelerating equilibration on demand, the opposite face of the same mode-control problem, cited as motivation in the introduction.
• Hyperpolarized magnetic resonance: the diamond platform itself is a hyperpolarization technology, and understanding which spatial polarization profiles survive longest informs how to store and transport enhanced nuclear polarization.
• Materials and device design: the proposed design rules (defect density and geometry shape slow modes; transport sets their extent) extend to disordered magnetic materials, molecular spin networks, semiconductor spin ensembles, and wide-bandgap materials used for hyperpolarization and quantum sensing.

Industry relevance: The work speaks to quantum-hardware developers working on spin qubits and reset protocols, to the magnetic-resonance and hyperpolarization community (NMR/MRI signal enhancement), and to groups engineering defect densities in diamond and related wide-bandgap materials for sensing. The instrumentation — a field-cycling shuttle, high-power chirped microwave hyperpolarization, and a high-speed arbitrary-waveform transceiver for continuous spin tracking — is also relevant to commercial magnetic-resonance instrumentation.

Future Directions

• Extending mode-selective preparation beyond (t_h, t_w) knobs. The current protocol controls overlap indirectly through diffusion and defection-mediated depletion; explicitly designed pulse sequences that target the slow eigenmode could sharpen or accelerate crossings.

• Testing the generality of the mechanism in other disordered spin hosts. The authors argue the required ingredients — spatially heterogeneous relaxation plus control over the initial spatial distribution — arise naturally in molecular spin networks, semiconductor spin ensembles, and other wide-bandgap materials, but these are not tested here.

• Connecting the semiclassical treatment to a fully quantum description. The model neglects interspin correlations and evaluates the Kullback–Leibler divergence only under a product-state approximation; a treatment retaining quantum correlations, and the role of entanglement, is left open.

• Pushing crossings deeper into or beyond the prethermal plateau. The paper demonstrates tuning from late-time thermalization into the prethermal regime but does not report how far the crossing can be pushed, how long the prethermal state persists in absolute terms, or whether driving parameters can be used to reshape the mode structure itself rather than only the mode populations.

Target Audience

Physicists and chemists working on nonequilibrium quantum many-body dynamics, spin-based quantum information, and magnetic resonance. It will be most useful to researchers already comfortable with NMR-style ensemble control and eigenmode relaxation analysis, and to experimentalists in hyperpolarization and defect-based quantum sensing who want concrete guidance on how defect disorder and state preparation jointly determine spin lifetime. Readers seeking an introduction to the Mpemba effect will find the mechanism clearly stated, but should expect the experimental detail to assume familiarity with Floquet engineering and dipolar spin transport.

Authors’ abstract

Relaxation in an interacting system is determined not only by its initial distance from equilibrium, but also by the relaxation modes populated by the initial state. Here we experimentally observe and control the Mpemba effect, in which a state farther from equilibrium overtakes one initially closer, in an extended, disordered $^{13}$C nuclear-spin network in diamond. Field cycling allows us to prepare distinct spatial polarization profiles by independently controlling hyperpolarization and defect-mediated relaxation. We then track their evolution under Floquet driving, which stabilizes a long-lived prethermal regime. We observe reproducible Mpemba crossings and tune the crossing time over several orders of magnitude, from late-time thermalization into the prethermal plateau. Semiclassical simulations show that randomly positioned paramagnetic defects create fast-relaxing regions and defect-poor regions that support the slowest collective relaxation mode. The Mpemba crossings are set by the initial state overlap with this mode. Our results demonstrate anomalous relaxation within a prethermal many-body regime and identify disorder, transport, and mode-selective state preparation as resources for controlling relaxation in extended spin networks.