Quantum instanton approach to metastable collective spin systems: Numerical implementation and applications

The gist Driven-dissipative collective spin systems can host several long-lived metastable states; rare switching between them governs slow relaxation and first-order dissipative phase transitions associated with changes in the dominant stationary state

Instanton Framework and Numerical Implementation

The quantum instanton framework is constructed from the dynamics of an appropriate quasiprobability distribution pα, where the differential operator Lα(x,∇x) generates the dynamics The activation barriers Ai→j governing the stochastic switching between them can be calculated as the action along an instanton trajectory satisfying Hamilton’s equations The numerical implementation combines collocation and shooting methods to approximate the π ≠ 0 instanton segment by a finite-time trajectory with boundary conditions approximating the exact asymptotic ones For effectively two-dimensional (d = 2) instanton problems studied here, collocation was preferable for trajectories escaping nodal attractors, while shooting was more effective for trajectories escaping spiral attractors

Applications to Physical Models

The framework is applied to two microscopically motivated systems: the driven-dissipative Lipkin–Meshkov–Glick (LMG) model with strong permutation symmetry and cooperative resonance fluorescence (CRF) with weak permutation symmetry Cooperative resonance fluorescence (CRF), relevant to a driven atomic ensemble in an optical cavity, combines coherent driving and collective emission with local decay and dephasing The computed barriers are used to predict steady-state magnetization, first-order transition boundaries, and the exponential scaling of relaxation rates For the CRF model without dephasing, an exact solution supplies a further benchmark The instanton approach provides a direct prediction for the location of the first-order dissipative phase transition from the crossing of the activation barriers

Analysis of Model Behavior and Scaling

For both models, mean-field dynamics in the limit J → ∞ yields long-lived metastable states corresponding to fixed points (FPs) The relative weight of attractors is set to exponential accuracy by the difference Ai→j − Aj→i The Liouvillian gap sets the slowest relaxation timescale of the system and is controlled by escape from the u± attractor The Liouvillian gap scaling follows λ ≍ e −JAu±→l

Technical Considerations and Comparisons

The auxiliary Hamiltonians derived from quantum quasiprobability dynamics are nonconvex in momentum, which precludes the direct use of efficient minimum-action algorithms developed for classical systems with convex Hamiltonians The choice of quasiprobability representation affects the numerical construction of instantons even when physical activation barriers agree The semiclassical Wigner (SW) approach yields effectively classical Fokker–Planck problems amenable to minimum-action methods but omits contributions that can change activation barriers at leading exponential order

Conclusion and Validation

The instanton approach successfully predicts the location of the first-order dissipative phase transition from the crossing of activation barriers in both models The results confirm that this framework is applicable to experimentally relevant collective spin systems with both collective and local dissipation Systematic comparisons show that while the SW approach works for some models, it cannot be assumed accurate in general because it misses significant nonGaussian effects The framework is validated by its ability to predict the asymptotic exponential scaling of the Liouvillian gap beyond exactly solvable regimes

Appendix Details

Appendix A sketches the derivation of mean-field equations and shows how they are obtained by replacing operators with classical variables<ref:2610.216843,Here, we sketch the derivation of the mean-field equations (11) and (21), which is rigorously proven in Refs. [54–57] The starting point is the Heisenberg evolution of spin operators dtJˆk = L† Jˆk, where L† is the superoperator adjoint to L<ref:2610.216843,Appendix A: Mean-field dynamics Here, we sketch the derivation of the mean-field equations (11) and (21), which is rigorously proven in Refs. [54–57] The starting point is the Heisenberg evolution of spin operators dtJˆk = L† Jˆk, where L† is the superoperator adjoint to L> Appendix A: Mean-field dynamics Here, we sketch the derivation of the mean-field equations (11) and (21), which is rigorously proven in Refs.

1. The AI system can accurately predict the stationary magnetization and the first-order dissipative phase-transition point of driven-dissipative collective spin systems by determining the leading exponential scaling of switching rates between metastable attractors, as these predictions are derived from the instanton approach.

2. The system can predict the asymptotic behavior of the Liouvillian gap, which sets the slowest relaxation timescales of the system, by calculating exponential scalings such as λ ≍ e − J A u±→l.

3. The AI can analyze and characterize metastable states in systems with both collective and local dissipation, such as those modeled by the Lipkin–Meshkov–Glick (LMG) model and cooperative resonance fluorescence (CRF), by comparing instanton results with quantum master equation calculations.

4. The system can determine the relevant stationary state occupations of metastable states by using the relationship derived from barrier differences: Pi/Pj = κj→i/κi→j; ∗ krzysztof.ptaszynski@ifmpan.edu.pl hence their relative weight is set, to exponential accuracy, by the difference Ai→j − Aj→i.

5. The system can perform robust numerical simulations of rare events in complex quantum systems by employing a combined collocation and shooting methods, with performance optimized based on the attractor type (e.g., collocation was preferable for trajectories escaping nodal attractors).

6. The AI can identify potential inaccuracies in simplified models by comparing predictions from the instanton framework against the semiclassical Wigner approach, noting that the SW approach omits contributions that can change activation barriers at leading exponential order.

Driven-dissipative collective spin systems can host several long-lived metastable states; rare switching between them governs slow relaxation and first-order dissipative phase transitions associated with changes in the dominant stationary state. In the companion Letter [arXiv:2604.15091], we introduced a quantum instanton framework based on exact quasiprobability dynamics to describe this switching and illustrated it with a minimal model possessing a simplifying symmetry. Here we develop its numerical implementation, combining collocation and shooting methods, and apply it to two microscopically motivated systems: the Lipkin-Meshkov-Glick model with purely collective dissipation and the cooperative resonance fluorescence with collective and local dissipation. By determining the leading exponential scaling of switching rates between metastable attractors, we predict the stationary magnetization, the first-order dissipative phase-transition point, and the exponential scaling of slow relaxation rates. These predictions agree with quantum master-equation calculations and an exact analytical result for the second model without dephasing. We also examine how the choice of quasiprobability representation affects numerical convergence and find that the alternative semiclassical Wigner approach, based on Gaussian truncation, has varying accuracy across models and parameter regimes. These results substantiate the framework's applicability to experimentally relevant collective spin systems with both collective and local dissipation.