Optimal speed-up of multi-step Pontus-Mpemba protocols

The classical Mpemba effect is the counterintuitive phenomenon where hotter water freezes faster than colder water due to the breakdown of Newton’s law of cooling after a sudden temperature quench. The genuine nonequilibrium post-quench dynamics allows the system to evolve along effective shortcuts absent in the quasi-static regime. When the time needed for preparing the (classical or quantum) system in the hotter initial state is included, we encounter so-called Pontus-Mpemba effects. We here investigate multi-step Pontus-Mpemba protocols for open quantum systems whose dynamics is governed by time-inhomogeneous Lindblad master equations. In the limit of infinitely many steps, one arrives at continuous Pontus-Mpemba protocols. We study the crossover between the quasi-static and the sudden-quench regime, showing the presence of dynamically generated shortcuts achieved for time-dependent dissipation rates. Time-dependent rates can also cause non-Markovian behavior, highlighting the existence of rich dynamical regimes accessible beyond the Markovian framework.

💡 Research Summary

The manuscript investigates how to accelerate relaxation in open quantum systems by extending the classical Mpemba effect—where a hotter sample can freeze faster than a colder one—into a multi‑step “Pontus‑Mpemba” protocol. The authors focus on a two‑level (spin‑½) system coupled to a thermal environment and described by a time‑inhomogeneous Lindblad master equation. The Hamiltonian (H(t)=\mathbf{h}(t)\cdot\boldsymbol{\sigma}) provides coherent precession, while three dissipative channels (excitation, relaxation, pure dephasing) are represented by jump operators (\sigma_{+},\sigma_{-},\sigma_{z}) with time‑dependent rates (\gamma_{+}(t),\gamma_{-}(t),\gamma_{z}(t)).

A key conceptual step is to treat the preparation of the “hot” initial state as part of the protocol. In a standard Mpemba experiment only the initial condition is varied; in a Pontus‑Mpemba experiment the relaxation pathway itself is engineered. The simplest two‑step protocol first quenches the system to an auxiliary state (effectively heating it) and then performs a second quench to the final target parameters. By including the time required for the auxiliary heating, the total duration can be shorter than a direct sudden quench, provided the intermediate state lies in a region of state space where subsequent relaxation is faster.

The authors generalize this idea to an arbitrary number (N) of steps. In the limit (N\to\infty) the protocol becomes continuous: the control parameters (\mathbf{h}(t)) and the rates (\gamma_{\lambda}(t)) are varied smoothly in time. This continuous Pontus‑Mpemba protocol interpolates between the quasi‑static limit (infinitely slow parameter change, no speed‑up) and the sudden‑quench limit (instantaneous change, the usual Mpemba scenario).

Mathematically, the dynamics of the Bloch vector (\mathbf{r}(t)) obeys a linear affine differential equation (\dot{\mathbf{r}}=\Lambda(t)\mathbf{r}+ \mathbf{b}(t)), where (\Lambda(t)) and (\mathbf{b}(t)) are explicit functions of the time‑dependent rates and fields. An analytical formal solution exists in terms of the fundamental matrix (\Xi(t)=\mathcal{T}\exp!\int_{0}^{t}!\Lambda(s)ds), but for generic control schedules the authors resort to numerical integration.

A central novelty is the deliberate use of “pseudo‑Lindblad” dynamics: when any rate (\gamma_{\lambda}(t)) becomes negative, the generator no longer guarantees complete‑positive divisibility, signalling non‑Markovian behavior. The authors quantify non‑Markovianity by (\mathcal{F}(t)=\sum_{\lambda}\int_{0}^{t}!ds,\min