Time-reversal in a dipolar quantum many-body spin system

The study addresses whether and how the unitary dynamics of an interacting quantum many-body system can be reversed in time by flipping the sign of its Hamiltonian. Historically, Loschmidt argued microscopic laws are time-symmetric, creating a paradox with macroscopic irreversibility, while Boltzmann emphasized statistical irreversibility. In quantum systems with precise control, effective time reversal can be realized by changing the Hamiltonian’s sign, which has been demonstrated in spin echo (single-particle) settings and in limited many-body scenarios. This work targets a general, isolated, long-range interacting dipolar spin system (Rydberg atoms) to implement many-body time reversal, investigate its sensitivity via a Loschmidt echo, and extend it to a broad class of engineered XXZ spin models via Floquet techniques.

Previous demonstrations of time-reversal-like protocols include spin echoes (reversing single-particle dephasing by flipping effective fields) and select many-body implementations in collective systems, mixed states, and digital quantum simulators. Rydberg platforms enable isolated quantum dynamics and long-range spin models in both random and controlled geometries and have been extensively used to explore quantum magnetism and nonequilibrium dynamics. Floquet engineering has been established across platforms to transform native Hamiltonians into target forms, including tunable XXZ models. This study builds on these foundations by implementing a Hamiltonian-sign flip through state encoding changes, measuring Loschmidt echoes to probe sensitivity to perturbations, and combining with Floquet engineering to reverse general XXZ dynamics.

System: An ultracold gas of 87Rb atoms forms a spatially disordered ensemble of Rydberg spins with long-range dipolar interactions. The pseudo-spin is encoded in two Rydberg states. Initially, |↓>1 = nS and |↑>1 = nP; specifically, |↑>1 = |61S1/2, mj = 1/2> and |↓>1 coupled via microwave to |61P1/2, mj = 1/2> to prepare magnetization in the equatorial plane. Hamiltonian and sign flip: Direct dipolar exchange between S and P realizes an XX Hamiltonian Hxx = Σij Jij(Si^x Sj^x + Si^y Sj^y), with Jij = 2 C3 (1 − 3 cos^2 θij) / rij^3. A coherent two-pulse state transfer changes the spin encoding to a second pair of Rydberg states, |↑>2 = |n'P>, |↓>2 = |n'S>, chosen so that the effective dipolar coupling changes sign C3 → −κ C3 (κ = C3/C3'). This implements a time-reversal operation for the XX dynamics. Experimental sequence: After optical excitation to |61S1/2, mj = 1/2>, a microwave π/2 pulse couples to |61P1/2, mj = 1/2> to set initial transverse magnetization. The system evolves under Hxx with C3/2π = +3.2 GHz μm^3 for time t1. A pair of microwave π pulses transfers population to |↑>2 = |61P1/2, mj = −1/2> and |↓>2 = |62S1/2, mj = 1/2> with Rabi frequencies Ω/2π = 9 and 11 MHz. In this encoding C3/2π = −2.8 GHz μm^3 (κ ≈ 1.1). The system then evolves for time t2 before being transferred back for measurement. Magnetization Mx in the equatorial plane is measured via tomographic phase-contrast readout. Loschmidt echo and reversal condition: For ideal reversal, M(t1 + t2) = ⟨ψ0| e^{iHxx(t1 − κ t2)} M0 e^{−iHxx(t1 − κ t2)} |ψ0⟩ revives to M0 at t1 = κ t2. The protocol’s sensitivity to perturbations is characterized by measuring the reversed magnetization at trev = t1 + κ t1 across t1 and interaction strengths. Interaction scale and densities: Typical energy scales are set by the median nearest-neighbor interaction Jm = median{max Jij}. Densities are tuned by varying Rydberg excitation time to access Jm/2π ≈ 0.43, 0.79, 0.95 MHz. A hard-sphere superatom excitation model estimates spin distributions and Jm (less accurate at highest densities). Numerical modeling: Moving-average cluster expansion (MACE) simulations (cluster size up to n = 16) of hundreds of spins approximate dynamics to isolate perturbation effects. A simplified two-level model implements the state transfer as an ideal 2π rotation about y at the end of t1. Two key imperfections are included: (i) finite pulse width with interactions active during π pulses, and (ii) thermal atomic motion. Motion is modeled by assigning velocities from a Boltzmann distribution at T = 11 μK and updating interatomic couplings every 200 ns. Rydberg lifetimes are much longer than evolution times and are neglected. Floquet engineering for XXZ: A periodic microwave pulse sequence (π/2 pulses with tunable delays τ1, τ2 = τ3 = τ) is applied to transform the native XX Hamiltonian to an XXZ model Hxxz = Σij Jij (Si^x Sj^x + Si^y Sj^y) + J∥ Σij Si^z Sj^z, with J⊥ = J2(τ1 + 2τ2) and J∥ = J2 τ2 over total period τtot = 2(τ1 + 2τ2). The anisotropy J⊥/J∥ is tuned by τ. The same sequence applied after state transfer flips the sign to −Hxxz without additional changes. For these XXZ tests, the system is pre-evolved for tprep ≈ 100 ns to reduce contributions from the strongest pairs in the disordered sample before applying the Floquet sequence.

- Time reversal of many-body XX dynamics: Starting from a fully magnetized state, the system relaxes to Mx ≈ 0.15 at t1 = 0.4 μs. After transferring to the negative-coupling encoding and evolving for t2 = 0.41 μs (total t ≈ 0.81 μs), the magnetization revives, evidencing reversal. The observed coupling ratio is κ ≈ 1.03, slightly below the theoretical κ ≈ 1.1, attributed to interactions during finite-width microwave pulses not included in the simple estimate. - Long-time reversal efficiency and Loschmidt echo sensitivity: For Jm/2π = 0.43, 0.79, 0.95 MHz, the reversed magnetization at trev = t1 + κ t1 decreases with increasing trev, characteristic of imperfect Loschmidt echoes. For the weakest interactions (0.43 MHz), Mrev ≈ 0.2 remains at trev ≈ 6 μs, even though forward relaxation occurs by t1 ≈ 0.7 μs. Stronger interactions show faster decay of the echo amplitude. - Role of perturbations via MACE: Simulations qualitatively capture the decay trends. Atomic motion at T = 11 μK minimally changes the median interaction Jm but significantly alters the microscopic coupling matrix (Frobenius norm ||ΔJ|| grows to ≈ 20% after ~6 interaction cycles), causing Loschmidt echo decay at longer times even with perfect transfers. Finite transfer pulse width reduces the initial magnetization plateau (even at t = 0) but does not cause continued decay; it contributes a largely constant offset error. Combining both imperfections reproduces the experimental decay: short-time degradation dominated by finite transfer efficiency; long-time decay dominated by sensitivity to microscopic coupling changes from motion. - Floquet-engineered XXZ reversal across anisotropies: Using periodic driving, the team realizes tunable XXZ models with anisotropy J⊥/J∥ between ~0.14 and 1. Without reversal, magnetization at t1 = 0.5 μs increases with anisotropy and is nearly conserved at J⊥/J∥ = 1 (SU(2) symmetry). Applying the same sequence after the sign-flipping state transfer yields −Hxxz and revives the magnetization to values limited by the transfer efficiency for all probed anisotropies, demonstrating generality of the reversal protocol beyond XX. - Quantitative parameters: Native couplings C3/2π = +3.2 and −2.8 GHz μm^3 for the two encodings; Rabi frequencies for transfer Ω/2π ≈ 9 and 11 MHz; evolution times up to 6 μs; interaction scales Jm/2π ≈ 0.43, 0.79, 0.95 MHz.

The work demonstrates that reversing the sign of an interacting many-body Hamiltonian by coherently changing the spin encoding can reverse complex relaxation dynamics, causing a demagnetized state to evolve back into a magnetized state. This directly addresses the feasibility of many-body time reversal in an isolated, long-range dipolar spin system. The Loschmidt-echo analysis clarifies the protocol’s sensitivity: finite-duration transfers set a baseline reduction, while atomic motion induces microscopic Hamiltonian changes that degrade reversibility over multiple interaction cycles. Combining the state-transfer-based sign flip with Floquet engineering enables time reversal for a broad family of spin models (XXZ with tunable anisotropy), indicating the method’s versatility. These results are relevant for benchmarking quantum simulators, characterizing decoherence, studying information scrambling, and enhancing quantum metrology protocols that benefit from time-symmetric dynamics.

The study experimentally realizes time reversal in an isolated dipolar quantum many-body spin system by flipping the sign of the interaction Hamiltonian through a coherent state transfer between Rydberg state encodings. It demonstrates magnetization revival in the XX model and extends the protocol, via Floquet engineering, to arbitrary XXZ Hamiltonians with tunable anisotropy, with performance limited primarily by transfer fidelity at short times and by atomic motion at longer times. The method is broadly applicable across platforms with rich internal state structures and can be leveraged for quantum-enhanced sensing (e.g., spin squeezing and near-Heisenberg-limited phase sensitivity), validation and benchmarking of quantum devices, and studies of quantum information scrambling. Future work could optimize faster and more robust transfers, mitigate motion (e.g., via tighter confinement or lower temperatures), scale to programmable geometries (e.g., tweezers), and extend to other models such as Hubbard and t–J by appropriately engineering sign reversals of interaction and tunneling terms.

- Sensitivity to atomic motion: Even modest thermal motion at 11 μK changes microscopic coupling matrices over time, degrading Loschmidt echoes at longer durations despite a nearly constant median interaction. - Finite transfer pulse widths: Interactions during π pulses reduce transfer fidelity and cause a baseline loss in magnetization revival even at short times; faster or optimized pulses could improve this. - Modeling constraints: MACE simulations with cluster size n = 16 are not fully converged, and high-density excitation models (hard-sphere approximation) become less accurate; nevertheless, qualitative agreement is observed. - Experimental time window: Dynamics are limited to ≤6 μs to remain well within Rydberg lifetimes; longer sequences may accumulate additional errors. - Disorder and strongest pairs: In disordered samples, the strongest interacting pairs are less amenable to precise Floquet engineering, necessitating a short pre-evolution step to reduce their impact.