Quantum Origin of Limit Cycles, Fixed Points, and Critical Slowing Down (2405.08866v2)

Abstract: Among the most iconic features of classical dissipative dynamics are persistent limit-cycle oscillations and critical slowing down at the onset of such oscillations, where the system relaxes purely algebraically in time. On the other hand, quantum systems subject to generic Markovian dissipation decohere exponentially in time, approaching a unique steady state. Here we show how coherent limit-cycle oscillations and algebraic decay can emerge in a quantum system governed by a Markovian master equation as one approaches the classical limit, illustrating general mechanisms using a single-spin model and a two-site lossy Bose-Hubbard model. In particular, we demonstrate that the fingerprint of a limit cycle is a slow-decaying branch with vanishing decoherence rates in the Liouville spectrum, while a power-law decay is realized by a spectral collapse at the bifurcation point. We also show how these are distinct from the case of a classical fixed point, for which the quantum spectrum is gapped and can be generated from the linearized classical dynamics.

• The paper explores how classical behaviors like limit cycles, fixed points, and algebraic decay emerge from quantum dissipative systems by analyzing the spectral properties of the Liouvillian operator.
• It identifies distinct Liouvillian spectral signatures for quantum limit cycles (parabolic branch) and fixed points (wedge-shaped pattern) as systems approach the classical limit.
• The study shows that critical slowing down and algebraic decay at classical critical points manifest quantum mechanically as a collapse of the Liouvillian spectrum onto the imaginary axis with infinite degenerate eigenmodes.

Quantum Origin of Limit Cycles, Fixed Points, and Critical Slowing Down

The paper "On the quantum origin of limit cycles, fixed points, and critical slowing down" by Shovan Dutta, Shu Zhang, and Masudul Haque explores the emergence of classical behaviors such as limit cycles, fixed points, and algebraic decay in quantum systems subjected to Markovian dissipation. This investigation is set against the backdrop of the well-established dilemma of reconciling classical phenomena with quantum mechanics, a topic that has been extensively analyzed in Hamiltonian systems but less so in dissipative ones.

Main Contributions and Results

The authors focus primarily on two quantum models: a single-spin model and a two-site lossy Bose-Hubbard model, demonstrating the emergence of classical behavior through the spectral properties of their Liouvillian operators. The paper establishes that the transition to classical dynamics can be inferred from specific patterns in the Liouvillian spectrum as the classical limit is approached.

1. Limit Cycles:
   • The paper identifies a signature of quantum limit cycles as a parabolically structured branch in the Liouvillian spectrum, which becomes purely imaginary and equally spaced in the classical limit. The parabolic distortion at finite system sizes results in dephasing along the limit cycle trajectory, represented by $\Delta_c \sim S^{-1}$, where $S$ is the spin length or an equivalent parameter in other systems.
2. Fixed Points:
   • Around classical fixed points, the quantum spectrum shows a wedge-shaped pattern where eigenvalues are linear combinations of the classical Jacobian's eigenvalues. This pattern emerges from a linear drift and diffusion in the Fokker-Planck equation, yielding a universal spectral form independent of specific model details.
3. Algebraic Decay at Hopf Bifurcations:
   • At critical points such as Hopf bifurcations, where classical systems exhibit algebraic relaxation, the quantum spectrum collapses onto the imaginary axis. This collapse involves an infinite number of degenerate eigenmodes, each contributing to the long-time algebraic relaxation characteristically absent in purely quantum, exponential decay processes.

Theoretical and Practical Implications

This work provides a crucial step in understanding how classical nonlinear dynamics can be explained and predicted from the quantum Liouvillian framework. It suggests a pathway for describing quantum-to-classical transitions in general dissipative systems, highlighting the importance of the spectral properties of the Liouvillian superoperator.

From a theoretical perspective, this approach offers tools for analyzing quantum systems that approximate classical phenomena, pointing to potential applications in fields such as quantum control and quantum thermodynamics. It also raises intriguing questions about other nonlinear dynamics, such as period doubling and the onset of chaos, and how these might manifest in a quantum setting. Furthermore, the findings encourage investigations into whether statistical characteristics of Liouvillian spectra could conjecturally resemble those of Hamiltonian systems, as seen in established quantum chaos theories.

Broader Context and Future Directions

In the broader context of quantum mechanics, this paper aligns with contemporary efforts to bridge classical and quantum domains—an endeavor critical for the development of technologies like quantum computing and quantum simulations. Future work might explore more complex systems beyond the two models examined, and establish connections between Liouvillian statistics and emergent macroscopic phenomena, potentially offering new insights into the foundational aspects of quantum mechanics and the dynamics of open quantum systems.

This paper does not just add to the ongoing discussion on quantum-classical correspondence but also sets a solid foundation for addressing how classical dynamical behavior, including chaos and decoherence-free subspaces, emerges in the classical limit of quantum systems with gain and loss mechanisms.