Quantum instanton approach to metastable collective spins

1. Problem Statement

The paper addresses the challenge of characterizing metastability in driven-dissipative collective spin systems. These systems, often modeled as an ensemble of spins coupled to a common reservoir (e.g., in cavity QED), exhibit multiple long-lived metastable states before relaxing to a unique steady state.

• The Core Issue: As the system size (extensivity parameter $V$, here the total spin $J$) increases, the system undergoes a first-order dissipative phase transition where the most probable state switches abruptly.
• Limitations of Existing Methods:
  • Quantum Master Equation (QME): Direct numerical simulation of the QME becomes computationally intractable for large $J$ due to the exponential growth of the Hilbert space ($2J+1$ levels).
  • Semiclassical Wigner (SW) Approach: Previous studies relied on semiclassical Fokker-Planck equations for the Wigner distribution. However, these methods truncate higher-order derivative terms (neglecting non-Gaussian fluctuations), leading to inaccurate predictions of activation barriers and phase transition points.
  • Keizer's Paradox: There is a known discrepancy where Mean-Field (MF) theory predicts stable fixed points, while the exact QME predicts finite lifetimes for these states. The transition rates between these states are governed by activation barriers $A_{i \to j}$, which scale as $\kappa_{i \to j} \sim \exp(-J A_{i \to j})$. Accurately calculating $A_{i \to j}$ is crucial but difficult.

2. Methodology

The authors develop a real-time instanton approach based on exact quantum quasiprobability dynamics, avoiding the truncations inherent in semiclassical approximations.

• Formalism:

  • The system is described by a Lindblad QME. The density matrix $\hat{\rho}$ is represented using Husimi ($p_H$) and Glauber-Sudarshan P ($p_P$) quasiprobability distributions in stereographic coordinates $(v, w)$.
  • The dynamics are governed by differential operators $\mathcal{L}_\alpha$ (where $\alpha \in \{H, P\}$). Unlike the SW approach, these operators are not truncated; they retain all derivative orders.
  • In the large-$J$ limit, the propagator $K_\alpha$ is analyzed using a WKB ansatz: $K_\alpha \sim \exp(-J S_\alpha)$. This leads to a Hamilton-Jacobi equation for the action $S_\alpha$, governed by an auxiliary Hamiltonian $H_\alpha(x, \pi)$.

• Instanton Trajectories:

  • The activation barrier $A_{i \to j}$ is determined by the minimum action of an "instanton" trajectory connecting two fixed points (attractors) in phase space.
  • The trajectory consists of a segment within the $\pi=0$ manifold (deterministic MF relaxation) and a segment with $\pi \neq 0$ (fluctuation-driven escape).
  • Key Technical Challenge: The auxiliary Hamiltonians $H_\alpha$ are non-convex in momentum $\pi$. This means the action can be multivalued or even negative, and the standard "minimum action" selection rule for convex systems does not apply directly.
  • Solution: The authors derive specific boundary conditions ($\pi=0$ at start and end points) and utilize the system's symmetry (rotational symmetry broken only by the drive) to reduce the problem to a 2D plane ($w$-$\pi_w$). They employ a numerical continuation method to solve the resulting boundary value problem and find the heteroclinic connections between saddle points.

3. Key Contributions

1. Generalization of Instanton Theory: The work successfully extends the real-time instanton approach (previously used for bosonic systems via Keldysh path integrals) to collective spin systems using exact quasiprobability dynamics.
2. Resolution of Non-Gaussian Fluctuations: By utilizing non-truncated equations of motion, the method captures non-Gaussian fluctuations that are neglected in the semiclassical Wigner approach. This is shown to be the critical factor in correctly predicting activation barriers.
3. Analytical and Numerical Framework: The authors provide a rigorous derivation of the auxiliary Hamiltonians $H_H$ and $H_P$ and demonstrate a robust numerical procedure (continuation method) to compute instanton trajectories in non-convex settings.
4. Proof of Concept: They demonstrate that this approach resolves the discrepancy between MF predictions and finite-size QME results (Keizer's paradox) by correctly identifying the scaling of transition rates.

4. Results

The authors apply their method to a minimal toy model of a collective spin with nonlinear dissipation and a coherent drive.

• Activation Barriers ($A_{i \to j}$):

  • The calculated barriers for transitions between the "upper" ($u$) and "lower" ($\ell$) branches of magnetization show a crossing point at $\Gamma \approx 8.9\gamma$.
  • This crossing point perfectly matches the parameter value where the finite-$J$ QME magnetization deviates from the MF upper branch and transitions to the lower branch.
  • Comparison with SW: The semiclassical Wigner (SW) approach predicts a crossing at $\Gamma \approx 7.9\gamma$ and underestimates the barrier heights. The SW approach fails because it truncates the third- and higher-order derivatives, effectively ignoring the non-Gaussian noise essential for the transition.

• Liouvillian Gap ($\lambda$):

  • The slowest relaxation timescale is characterized by the Liouvillian gap $\lambda \sim \exp(-J A_{\min})$.
  • Numerical QME simulations for large $J$ confirm that the decay rate of $\lambda$ matches the prediction of the new instanton approach.
  • The SW approach significantly underestimates the gap (predicting faster relaxation than observed), further validating the necessity of the exact quasiprobability treatment.

• Robustness: The results hold for different drive strengths ($\Omega = 0.25\gamma$ and $0.5\gamma$), confirming the generality of the method.

5. Significance

• Theoretical Advancement: This paper provides a rigorous framework for studying metastability in quantum systems beyond the semiclassical limit. It bridges the gap between exact quantum dynamics and large-deviation theory.
• Experimental Relevance: The findings are crucial for understanding and controlling dissipative phase transitions in modern quantum platforms, such as:
  • Cavity and circuit QED (atomic ensembles, superconducting circuits).
  • Schrödinger cat qubits: The method offers a way to accurately estimate bit-flip error rates in these qubits, which rely on metastability for protection.
  • Systems with nonlinear dissipation where standard Gaussian approximations fail.
• Methodological Impact: The approach avoids the complexity of Keldysh path integrals while retaining full quantum accuracy in the large-$J$ limit. It suggests that similar instanton techniques can be applied to spin-boson complexes and feedback-controlled quantum systems.

In summary, the paper demonstrates that non-Gaussian fluctuations are essential for correctly describing metastability in collective spin systems and provides a powerful, exact instanton-based tool to quantify these effects, outperforming traditional semiclassical methods.