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

Technical Summary: Quantum Instanton Approach to Metastable Collective Spin Systems

Problem Statement
Driven-dissipative collective spin systems often exhibit multiple long-lived metastable states, even when their finite-size dynamics possesses a unique stationary state. These metastable states arise from mean-field dynamics developing multiple attractors in the large-system limit. At finite sizes, fluctuations induce rare switching events between these attractors, governing slow relaxation dynamics and first-order dissipative phase transitions. The primary challenge is to accurately determine the activation barriers ($A_{i \to j}$) that dictate the exponential scaling of switching rates ($\kappa_{i \to j} \asymp \exp(-V A_{i \to j})$) and to identify the transition points where the dominant stationary state changes. While a companion letter introduced a theoretical framework based on exact quantum quasiprobability dynamics, it relied on a minimal model with simplifying symmetries. The current work addresses the need for a robust numerical implementation applicable to more complex, microscopically motivated systems lacking such symmetries, particularly those involving both collective and local dissipation.

Methodology
The authors develop a numerical implementation of the quantum instanton framework, which calculates activation barriers as the action of rare switching trajectories (instantons) in an auxiliary classical phase space derived from exact quantum quasiprobability dynamics.

• Theoretical Framework: The approach utilizes the dynamics of a quasiprobability distribution $p_\alpha(x,t)$ governed by a differential operator $L_\alpha$. In the large-system limit ($V \to \infty$), this yields an auxiliary Hamiltonian $H_\alpha(x, \pi)$. The activation barrier is the action along a trajectory satisfying Hamilton's equations with boundary conditions connecting stable fixed points (attractors) via unstable fixed points (saddles).
• Numerical Implementation: Unlike classical stochastic systems where auxiliary Hamiltonians are convex in momentum, the quantum-derived Hamiltonians here are nonconvex in momentum. This precludes the use of standard minimum-action algorithms. The authors employ two complementary boundary-value methods:
  1. Collocation Method: Solves the two-point boundary-value problem by approximating the trajectory in a finite-dimensional space. It is most effective for trajectories escaping "nodal" attractors (real eigenvalues).
  2. Shooting Method: Integrates Hamilton's equations from a point near the unstable manifold of the initial attractor and varies initial conditions to minimize the distance to the target saddle. It is more effective for trajectories escaping "spiral" attractors (complex-conjugate eigenvalues).
• Quasiprobability Representations: The study investigates the impact of different representations ($\alpha$), specifically the Husimi ($H$), $P$, Normal-Order ($NO$), and Generalized Coherent-state ($GC$) representations, on numerical convergence and barrier accuracy.
• Comparison Benchmarks: The results are compared against:
  • Quantum Master Equation (QME) calculations for finite system sizes.
  • Exact analytical results for the Cooperative Resonance Fluorescence (CRF) model in the zero-dephasing limit.
  • The Semiclassical Wigner (SW) approach, which truncates the quasiprobability dynamics at second-order derivatives to yield a classical Fokker-Planck equation.

Applications and Systems Studied
The framework is applied to two distinct models:

1. Lipkin–Meshkov–Glick (LMG) Model: A system with strong permutation symmetry (total spin conserved). Metastability arises from the competition between Hamiltonian spin interactions and collective decay.
2. Cooperative Resonance Fluorescence (CRF) Model: A system with weak permutation symmetry (local and collective dissipation). It models an atomic ensemble in an optical cavity with local decay and dephasing. This model tests the framework in the presence of local dissipative channels that mix different angular momentum sectors.

Key Results

• Prediction of Phase Transitions: The calculated activation barriers successfully predict the location of first-order dissipative phase transitions. The transition point is identified by the crossing of barriers ($A_{i \to j} = A_{j \to i}$), where the stationary magnetization jumps discontinuously in the thermodynamic limit.
• Relaxation Rate Scaling: The framework correctly predicts the exponential scaling of the Liouvillian gap (the slowest relaxation rate), $\lambda \asymp \exp(-V A_{\min})$, where $A_{\min}$ is the minimum activation barrier. This holds for both symmetric and antisymmetric relaxation modes in the LMG model.
• Validation against Exact Solutions: For the CRF model without dephasing ($\kappa=0$), the instanton barriers agree with exact analytical results derived from an effective potential.
• Finite Dephasing: The approach successfully predicts metastable behavior and transition points for the CRF model with finite dephasing ($\kappa \neq 0$), a regime where exact analytical solutions are unavailable.
• Nonconvexity Manifestation: The study confirms that the nonconvexity of the auxiliary Hamiltonians leads to transient decreases in the action along instanton trajectories, a behavior distinct from classical convex systems.
• Representation Dependence:
  • In the LMG model, the $H$ representation showed robust convergence across the parameter space, while the $P$ representation failed to converge in some regions.
  • In the CRF model, the $NO$ representation provided consistent results, whereas the $GC$ representation converged only near specific bifurcation points.
• Semiclassical Wigner (SW) Limitations: The SW approach, while often yielding quantitatively accurate predictions for observables like stationary magnetization in the LMG model, fails to reproduce activation barriers exactly. In the CRF model (zero dephasing), the SW approach produced a discernible shift in the predicted phase transition point compared to exact results. The authors note that the SW approach relies on uncontrolled truncation and can yield incorrect barriers in other regimes (as seen in the companion letter).

Significance and Claims
The paper establishes the applicability of the quantum instanton framework beyond minimal, highly symmetric models to experimentally relevant collective spin systems involving both collective and local dissipation. The authors claim that their numerical implementation, combining collocation and shooting methods, effectively handles the nonconvexity of quantum auxiliary Hamiltonians.

The work substantiates that:

1. Exact quasiprobability dynamics can be used to calculate activation barriers that accurately predict first-order dissipative phase transitions and relaxation scaling.
2. The framework remains valid in the presence of local dissipation and dephasing, extending beyond exactly solvable limits.
3. The choice of quasiprobability representation significantly impacts numerical convergence, necessitating careful selection (e.g., $H$ or $NO$) for robust calculations.
4. The semiclassical Wigner approach, while useful in specific regimes, cannot be generally assumed to provide accurate activation barriers or transition points due to its uncontrolled truncation of higher-order derivative terms.

The authors conclude that while the SW approach works well for some models and parameter ranges, the full quantum instanton framework is necessary for a rigorous description of metastability in systems where non-Gaussian effects are significant.