Detection of Spin-Spatial-Coupling-Induced Dynamical Phase Transitions in Real Time

Abstract: We demonstrate the real-time detection of dynamical phase transitions (DPTs) in lattice-confined spinor gases subject to a priori unknown time-variant interactions, via the temporal behaviors of both the system energy and spinor phases extracted from the observed spin dynamics. Using this technique, we describe the observed nonequilibrium spin dynamics, governed by intricate spin-spatial couplings, across a range of conditions. This work also introduces an observable that can quickly identify DPTs at holding times when commonly-used order parameters still exhibit transient, nonuniversal behavior. Our approach can naturally extend to other complex systems subject to time-dependent parameters, such as Floquet systems under driven magnetic fields, driven interactions, or spin-flopping fields, with potential applications in the study of DPTs in nonintegrable models.

• The paper introduces a cutoff time (tc) observable that rapidly signals dynamical phase transitions in spinor Bose gases.
• It employs a robust SMA-based method to iteratively extract time-dependent interaction strengths from spin-resolved data.
• The technique proves effective in both free-space and lattice-confined systems, enhancing studies of nonintegrable dynamics.

Real-Time Detection of Dynamical Phase Transitions in Spinor Gases with Spin-Spatial Coupling

Overview

The paper "Detection of Spin-Spatial-Coupling-Induced Dynamical Phase Transitions in Real Time" (2604.03521) presents an in-depth experimental and theoretical investigation of dynamical phase transitions (DPTs) in ultracold spinor Bose gases when subjected to time-variant interactions. It introduces a methodology for real-time detection of DPTs based on temporal spinor phase and energy observables, particularly under conditions where system parameters governing the quantum dynamics are not known a priori. The approach is demonstrated in both well-characterized free-space spinor BECs and highly nonequilibrium, lattice-confined spinor gases experiencing spin-spatial coupling. The work establishes the utility of a new observable—the cutoff time $t_c$—for prompt identification of DPTs and outlines a robust technique for extracting time-dependent interaction strengths from observed spin populations, applicable even in nonintegrable scenarios.

Experimental Techniques and System Dynamics

The experimental platform is an $F=1$ sodium spinor BEC held in an optical dipole trap, initialized with controlled spin populations and magnetization. Two major experimental protocols manipulate the ratio $c_2/q$:

• Quench-Q Sequence: Abrupt changes in the quadratic Zeeman energy $q$ via magnetic field quenches drive transitions across the dynamical phase diagram.
• Moving-Lattice Sequence: A moving optical lattice induces rapid, a priori unknown variation in the spin-dependent interaction parameter $c_2$ via intricate spin-spatial coupling mechanisms.

After these protocols, spin populations and phases are extracted via spin-resolved imaging. The moving-lattice sequence, in particular, produces significant atom loss and spatial redistribution, rendering traditional density-based calculations of $c_2$ unreliable. The authors solve this by iterative extraction of $c_2$ and $\theta$ using the dynamical single-mode approximation (SMA), which remains valid when all spin states share a common, time-dependent spatial mode.

Theoretical Modeling and Dynamical Regimes

The SMA-based Hamiltonian and ensuing equations of motion for $\rho_0$ (population) and $\theta$ (relative phase) generalize previously established models to accommodate time-dependent $F=1$0 and $F=1$1. Dynamical regimes are classified as:

• Interaction-Dominated Regime: $F=1$2 remains bounded, order parameter oscillations are minimal.
• Zeeman-Dominated Regime: $F=1$3 becomes unbounded, order parameter oscillates between extremal values.

Notably, $F=1$4 serves as an effective phase-based order parameter, with abrupt changes signaling DPTs and delineating dynamical regimes.

Real-Time Identification of DPTs

A central claim is the introduction of the cutoff time $F=1$5, defined as the instant when $F=1$6, corresponding to $F=1$7. $F=1$8 demonstrates superior performance relative to typical order parameters (e.g., oscillation amplitude $F=1$9) by enabling DPT detection with minimal observations, even when transient dynamics preclude direct measurement of steady-state order parameters.

Empirical results show that after magnetic or lattice-induced quenches, phase-based observables ($c_2/q$0, $c_2/q$1) promptly signal DPTs, while population-based observables require comparison to theory or prolonged monitoring. The extracted $c_2/q$2 curves exhibit rapid convergence regardless of initial guesses, evidencing robustness of the iterative extraction scheme for unknown and rapidly changing interaction strengths.

Spin-Spatial Coupling and Nonintegrable Dynamics

For the moving-lattice system with unknown time-dependent $c_2/q$3, the methodology elucidates complex DPTs driven by spin-spatial coupling. The system can cross from interaction-dominated to Zeeman-dominated regimes as $c_2/q$4 decays, and the energetic landscape transitions accordingly. When $c_2/q$5 (with $c_2/q$6 for $c_2/q$7), the system undergoes a DPT; this matches the observed phase dynamics and order parameter traces.

The successful identification of DPTs in nonintegrable, highly nonequilibrium spinor gases underscores the technique’s broad applicability for systems with time-dependent or unknown parameters, where conventional theoretical modeling is impracticable.

Implications and Future Directions

The paper’s methodology extends real-time experimental detection of DPTs beyond integrable models. By enabling prompt and reliable characterization of phase transitions via phase-based observables and energy dynamics, it facilitates studies of universality, crossover phenomena, quantum scars, and entanglement generation in complex spinor systems. The robust extraction of microscopic Hamiltonian parameters from dynamics empowers quantum simulation efforts and paves the way for advanced sensing applications in driven quantum systems, including Floquet engineering and magnetically modulated spinor gases.

Further developments could focus on leveraging these tools in strongly driven, chaotic, and nonintegrable regimes, enhancing control over quantum phases and exploring the emergence of nonthermal states, ergodicity breaking, and prethermalization in larger systems with tunable interactions.

Conclusion

The work establishes real-time, phase-based detection of dynamical phase transitions in spin-spatially coupled ultracold spinor gases, introducing the cutoff time $c_2/q$8 as a rapid and reliable DPT indicator. Iterative extraction of time-dependent interaction strengths from spin dynamics provides a powerful diagnostic tool for complex and time-dependent many-body systems. These advances enable deepened exploration of nonequilibrium quantum criticality and broaden the experimental accessibility of dynamical phase diagrams, with practical applications for quantum simulation, sensing, and study of nonintegrable dynamics in driven spinor platforms (2604.03521).