[Paper Review] Exponentially accelerated approach to stationarity in Markovian open quantum systems through the Mpemba effect

Federico Carollo, Antonio Lasanta+1 more|arXiv (Cornell University)|Mar 8, 2021

Quantum many-body systemsPhysics and Astronomy61 references123 citations

TL;DR

This paper demonstrates an exponentially accelerated approach to stationarity in Markovian open quantum systems by applying an optimal unitary transformation to the initial state, rendering it orthogonal to the slowest-decaying dynamical mode. By eliminating excitation of this mode, the system bypasses long-lived metastable regimes and converges to the stationary state orders of magnitude faster, inspired by the classical Mpemba effect.

ABSTRACT

Ergodicity-breaking and slow relaxation are intriguing aspects of nonequilibrium dynamics both in classical and in quantum settings. These phenomena are typically associated with phase transitions, e.g. the emergence of metastable regimes near a first-order transition or scaling dynamics in the vicinity of critical points. Despite being of fundamental interest the associated divergent time scales are a hindrance when trying to explore steady-state properties. Here we show that the relaxation dynamics of Markovian open quantum systems can be accelerated exponentially by devising an optimal unitary transformation that is applied to the quantum system immediately before the actual dynamics. This initial "rotation" is engineered in such a way that the state of the quantum system becomes orthogonal to the slowest decaying dynamical mode. We illustrate our idea -- which is inspired by the so-called Mpemba effect, i.e., water freezing faster when initially heated up -- by showing how to achieve an exponential speed-up in the convergence to stationarity in Dicke models, and how to avoid metastable regimes in an all-to-all interacting spin system.

Motivation & Objective

• To address the challenge of slow relaxation in open quantum systems due to long-lived dynamical modes.
• To overcome the practical hindrance of divergent time scales in studying steady-state properties.
• To develop a method that enables fast, exponential convergence to stationarity in Markovian open quantum systems.
• To demonstrate the feasibility of the Mpemba effect in quantum systems through unitary engineering.
• To avoid metastable regimes in strongly correlated quantum systems like Dicke models and all-to-all spin systems.

Proposed method

• Apply a unitary transformation U to the initial state |ψ⟩ such that Tr(ℓ₂Uρ₀U†) = 0, where ℓ₂ is the left eigenmatrix associated with the slowest decaying mode.
• Leverage the spectral decomposition of the Lindblad generator to isolate the slowest decaying mode λ₂ with Re(λ₂) = −1/τ.
• Ensure the transformed state has no overlap with the slowest mode, thereby accelerating relaxation to the stationary state with time scale 1/|Re(λ₃)|.
• Use the adjoint Lindblad map L⁺ to identify left eigenmodes ℓₖ and compute the overlap Tr(ℓ₂ρ₀) to determine the required unitary.
• Construct the unitary operation U such that Uρ₀U† is orthogonal to the slowest mode in the Hilbert-Schmidt inner product.
• Illustrate the method in two paradigmatic models: the Dicke model (via adiabatic elimination to a spin-only description) and an all-to-all interacting spin system.

Experimental results

Research questions

• RQ1 Can the relaxation to stationarity in Markovian open quantum systems be exponentially accelerated by pre-processing the initial state?
• RQ2 Is the Mpemba effect—faster cooling from a hotter initial state—realizable in quantum open systems through unitary control?
• RQ3 Can metastable regimes arising from slow decay modes be avoided via engineered initial states?
• RQ4 What is the role of the slowest decaying mode in determining the relaxation time scale in open quantum systems?
• RQ5 How can the unitary transformation be constructed to ensure orthogonality to the slowest mode without altering the system's dynamics?

Key findings

• The relaxation time scale is reduced from τ = 1/|Re(λ₂)| to 1/|Re(λ₃)|, resulting in an exponential speed-up when |Re(λ₃)| ≫ |Re(λ₂)|.
• The method successfully avoids long-lived metastable regimes in the all-to-all interacting spin system, enabling rapid convergence to stationarity.
• In the Dicke model, the spin-only effective dynamics confirms that the unitary pre-processing leads to faster relaxation, as verified via adiabatic elimination of the bosonic mode.
• The unitary transformation is constructed such that Tr(ℓ₂Uρ₀U†) = 0, ensuring no excitation of the slowest mode.
• The approach is general and applicable to any Markovian open quantum system with a unique, real, and non-degenerate slowest-decaying mode.
• The Mpemba effect in quantum systems is realized not by temperature differences, but by state engineering via unitary operations that suppress slow modes.