Quantum Many-Body Mpemba Effect through Ruelle-Pollicott Resonances

The gist: The authors provide a general framework for understanding the quantum Mpemba effect (QME) in closed quantum many-body chaotic systems by reformulating local subsystem equilibration in terms of Ruelle-Pollicott (RP) resonances, showing that suppressing the initial-state overlap with dominant RP resonant modes accelerates subsystem equilibration and yields the QME The gist.

The Mpemba Effect and Quantum Analogue

The Mpemba effect is an anomalous relaxation phenomenon where an initial state farther from equilibrium can approach it faster than another initial state prepared closer to equilibrium in quantum systems, termed the quantum Mpemba effect (QME) This QME provides a microscopic perspective on the anomalous relaxation and has practical relevance for accelerated state preparation and cooling protocols in quantum platforms The universal origin of the Mpemba effect remains under debate due to its very sensitivity to experimental setup

Mechanism via Ruelle-Pollicott Resonances

The authors characterize local equilibration of the reduced density matrix in terms of Ruelle-Pollicott (RP) resonances, originally developed for classical chaos and extended to quantum many-body systems The QME occurs when the initial state farther from equilibrium has a smaller overlap for slowly decaying RP resonant modes In the thermodynamic limit L → ∞, the reduced density matrix is asymptotically expanded in terms of RP resonances as ρs(t) ≃ I/2l + Xα Z π−πdk2πλtα,k Tr[ρ(0)V Rα,k]ϱα,k The QME occurs when the global initial state of the system that is initially closer to equilibrium has a larger weight c1,0 of the dominant RP resonance at k = 0

Strong QME via Translational Symmetry Breaking

A novel type of strong QME can occur via complete translation-symmetry breaking of initial states When the initial state completely breaks translational symmetry, the late-time relaxation law is qualitatively modified, so which state relaxes faster is solely governed by this modified relaxation law This complete translation-symmetry breaking emerges for novel yet experimentally relevant initial states constructed from “Legendre sequence”, well-known concept in number theory When translational symmetry is absent, the overlap function Tr[ρ(0)V Rα,k] exhibits a continuum of momenta instead of discrete delta-function peaks The large-time behaviour is obtained by evaluating the integral over k in equation (2) via the method of steepest descent This leads to an algebraic prefactor t−1/2 compared to the purely exponential form in equation (3)

Application to Kicked Ising Chain

The framework is applied to the chaotic kicked Ising chain, where RP resonances are studied, and it is shown that the QME occurs when a pair of initial states with angles θi (i = I,II) satisfy two conditions: (i) θI − π/2 < θII − π/2 and (ii) θI < θII Numerical simulations demonstrate that these trends imply the occurrence of the QME for a pair of initial states with angles θi (i = I,II) These numerical results agree well with the asymptotic form predicted by equation (10) at late times

Methodology and Extensions

The analysis relies on decomposing the whole system into a local subsystem S of finite length l and its complement E, leading to the reduced density matrix ρs(t) = TrE[ρ(t)] = Utρs(0) The RP resonances are obtained by introducing a cutoff r on operator size by defining a truncated propagator Ek,r = Πk,rEkΠk,r The method extends to Markovian open quantum systems where the mechanism is understood from the spectral decomposition of the corresponding Liouvillian The authors also provide an asymptotic expansion of the reduced density matrix in terms of RP resonances for both translationally invariant and completely broken symmetry cases

Conclusion and Outlook

The QME in closed quantum many-body chaotic systems can be understood within the framework of RP resonances, where the QME occurs when the state initially farther from equilibrium carries a smaller weight of the dominant (slowest) resonant mode Complete breaking of translational symmetry can induce a strong QME, where which initial state equilibrates faster is dictated by this change in the asymptotic relaxation behavior rather than by a mere difference in their overlaps with the same slowest mode The predictions agree with numerical simulations of the kicked Ising chain at experimentally accessible sizes This formalism provides a route to analyse late-time local relaxation beyond the QME by giving access to the reduced density matrix rather than a particular observable It enables principled predictions for the late-time behaviour of nonlinear functionals of density matrices, such as quantum Fisher information, R´enyi entanglement entropies, and other information-theoretic quantities The authors suggest extending this framework to continuous-time Hamiltonian dynamics and long-range interacting systems

Methods Details

The asymptotic analysis shows that when Tr[ρ(0)V Rα,q] is non-zero, the equation reduces to equation (9) in the main text For the kicked Ising chain, this leads to an enhanced power ∼ t−(1+γ)/2 with γ > 0 when the initial state satisfies Tr[ρ(0)V Rα,k] = 0 for k ∈ Σα Such an initial state can be constructed from a de Bruijn sequence [2] This construction suppresses Tr[ρ(0)V Rα,k] and thereby realizes γ ≥ 1 approximately The time evolution of the Bures distance D(t) for the dynamics starting from the initial state (sm-12) exhibits a decay consistent with D(t) = O(λ1,0 t−1) This corresponds to γ = 1, i.e., a faster approach to equilibrium than in the γ = 0 cases discussed in the main text The data show that curves corresponding to different values of θ intersect at certain times, signaling the occurrence of the QME

Data Availability and Acknowledgements

The data that support the plots within this paper are provided in the Source Data file The computer codes used to generate the results that are reported in this paper are available from the authors upon reasonable request S. Y. was supported by Grant-in-Aid for Young Scientists (Start-up) No. 25K23355 and Institute for Advanced Science, University of Electro-Communications R. H. was supported by JSPS KAKENHI Grant No. JP24K16982 and by JST ERATO Grant Number JPMJER2302, Japan S.Y and R.H contributed to developing the theory, interpreting the results, and writing the manuscript The authors declare no competing interests

References

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1. Accelerated State Preparation in Chaotic Systems The improved AI system can achieve accelerated state preparation protocols by leveraging initial states that completely break translational symmetry, as this leads to a relaxation law where which state relaxes faster is solely governed by this modified relaxation law.

2. Universal Framework for Quantum Relaxation Analysis The system can be equipped with a universal mechanism to understand the QME in closed quantum many-body chaotic systems by characterizing local equilibration in terms of Ruelle-Pollicott (RP) resonances, which provides a microscopic perspective on the anomalous relaxation.

3. Predictive Modeling of Non-Markovian Dynamics The AI system can predict the late-time behavior of nonlinear functionals like quantum Fisher information, R´enyi entanglement entropies, and other information-theoretic quantities, enabling principled predictions beyond tracking a single observable.

4. Design of Number-Theoretic Initial States The system can generate experimentally relevant initial states using concepts from number theory, such as those inspired by number theory or constructed from Legendre sequence, to specifically probe the strong QME and realize faster equilibration rates.