The Kubo-Thermalization Correspondence

Abstract: Quantum thermalization describes how interacting quantum systems relax toward thermal equilibrium, a central problem in modern physics. Yet most experimental information on many-body systems comes from short-time transition spectroscopy, typically interpreted within Kubo's linear-response framework. These perspectives - long-time equilibration versus short-time response - seem fundamentally disconnected. Here we establish an exact link between them: the Kubo-Thermalization correspondence, which connects long-time thermalized magnetization under weak driving to short-time linear-response spectra for a spin coupled to a thermal bath. The correspondence holds even when the steady state differs substantially from the initial state and when each regime is individually difficult to describe theoretically. We experimentally confirm the correspondence using effective spin-1/2 impurities realized with ultracold fermions in two internal states coupled to a Fermi sea. Our results provide a rare exact statement about quantum thermalization and offer a novel route to infer thermalization dynamics from equilibrium response measurements in strongly interacting quantum systems, independent of microscopic details of the system-bath coupling.

• The paper introduces an exact correspondence that connects short-time linear-response observables to long-time quantum thermalization in driven spin systems.
• Experimental validation with ultracold 6Li atoms confirms the relation across weak and strong coupling regimes and reveals systematic scaling of resonance deviations.
• The study establishes a model-independent framework enabling inference of non-equilibrium thermal properties from readily accessible spectroscopic measurements.

The Kubo-Thermalization Correspondence: Bridging Linear Response and Long-Time Quantum Thermalization

Introduction

This work introduces and experimentally substantiates the Kubo-Thermalization correspondence, an exact theoretical link between short-time linear-response observables and the asymptotic properties of quantum thermalization in driven spin systems weakly coupled to a thermal bath (2605.06666). The correspondence provides a formal bridge connecting two regimes in quantum many-body physics that are typically analyzed using fundamentally different frameworks: linear response theory (Kubo) for near-equilibrium and short times, and long-time, out-of-equilibrium thermalization—both especially difficult to analyze in the presence of strong system-bath interactions.

Theoretical Framework

The canonical setup considered involves a spin-1/2 particle embedded in a thermal bath at temperature $T$, with driving provided by a near-resonant oscillatory field. The system's Hamiltonian in the rotating frame is given by

$H = H_s + H_B + H_{\text{int}},$

where $H_s$ encodes the detuning, Rabi frequency, and Zeeman splitting; $H_B$ is an unspecified bath Hamiltonian, and $H_{\text{int}}$ represents a generic system-bath coupling not involving spin flips. The system is initialized in one spin state and its magnetization $M = \langle \sigma_z \rangle$ is monitored as a function of time and detuning.

In the weak drive regime:

• Short times: The transition rate $R_I(\Delta)$ between spin states is governed by Fermi's Golden Rule, and spectra are interpreted within linear-response (Kubo) theory.
• Long times: The spin magnetization approaches a steady-state value $M_0(\Delta)$, characterizing the thermalized driven system.

The main theoretical result is an exact, model-independent functional relation (the Kubo-Thermalization correspondence) between the asymptotic resonance $\Delta_0$ of the thermalized system and the short-time transition spectrum:

$$h\Delta_0 = -\frac{1}{\beta} \ln \int d\Delta\, R_I(\Delta) e^{-\beta h \Delta},$$

where $H = H_s + H_B + H_{\text{int}},$0, and $H = H_s + H_B + H_{\text{int}},$1 is the normalized linear response transition rate.

Remarkably, this relation is fully independent of the detailed microscopic properties of the bath or system-bath coupling and is readily generalized to $H = H_s + H_B + H_{\text{int}},$2-level systems.

Experimental Implementation

The correspondence was tested using ultracold $H = H_s + H_B + H_{\text{int}},$3Li atoms in a box potential as an isolated, tunable quantum simulator. The spin-impurity was realized in two hyperfine states, with the bath formed by a third state. Interactions were tuned across the BCS-BEC crossover regime via a magnetic Feshbach resonance, providing access to both weakly and strongly correlated limits.

Key experimental protocols:

• Short-time (Kubo) regime: Weak radio-frequency drive was applied for a few milliseconds, yielding the transition spectrum $H = H_s + H_B + H_{\text{int}},$4.
• Long-time (Thermalization) regime: Longer and/or stronger drives allowed the system to reach the steady-state magnetization $H = H_s + H_B + H_{\text{int}},$5, from which the thermalization resonance $H = H_s + H_B + H_{\text{int}},$6 was extracted.

Numerous quantitative checks were performed:

• For narrow, symmetric response spectra (weak coupling), the resonance positions extracted from both short-time and long-time measurements coincide, as predicted by the correspondence.
• For broader, asymmetric spectra (strong coupling/BEC side), systematic deviations between peak position ($H = H_s + H_B + H_{\text{int}},$7) and zero-crossing ($H = H_s + H_B + H_{\text{int}},$8) were observed and found to scale with the spectral width, consistent with the functional dependence in the correspondence.
• A symmetrized version of the functional, using both forward and reverse spectrum measurements, provided robust agreement between experimental and theoretically expected values for $H = H_s + H_B + H_{\text{int}},$9, even in strongly correlated regimes where no independent theoretical predictions exist for either side of the correspondence.

The correspondence was further validated in the metastable, repulsive polaron regime ($H_s$0), where thermalization is limited to a long-lived upper branch, demonstrating its applicability when equilibration is sector-dependent on relevant timescales.

Numerical Results and Experimental Validation

Strong numerical results are reported:

• Across the entire BCS-BEC crossover, the resonance positions computed using the Kubo-Thermalization correspondence match experimental steady-state magnetization spectra to within experimental uncertainty.
• Systematic scaling of the peak-to-zero-crossing deviation with spectral width is quantitatively consistent with theoretical predictions, including a power-law exponent close to that expected for Gaussian lineshapes.
• In the repulsive polaron regime, agreement is maintained only when the metastable state's lifetime substantially exceeds the thermalization timescale, and deviations are well accounted for by relaxation dynamics.

A noteworthy claim is the correspondence's exactness under minimal and physically reasonable assumptions (weak but nonzero drive, system-bath coupling that preserves spin projection). The principle is demonstrated both for specific spin-bath models and in experimentally challenging regimes where alternative theoretical approaches are intractable.

Implications and Future Directions

Practically, the Kubo-Thermalization correspondence provides a general, model-independent route to infer non-equilibrium steady-state properties from equilibrium linear-response measurements. This result is potentially transformative for experimental quantum many-body physics, where long-time dynamics are often inaccessible but short-time spectroscopy is routine (e.g., ARPES, Raman, and neutron scattering).

Theoretically, this correspondence challenges the presumed independence between the regimes of Kubo (short-time, equilibrium) and long-time thermalization (non-equilibrium, arbitrary bath). The universal functional nature of the relation also suggests possible applications in diverse platforms—such as NMR, trapped ions, and Rydberg arrays—where similar driven quantum spin dynamics are encountered.

Open research avenues include:

• Extension to more complex observables and multi-level systems.
• Exploration of the correspondence in presence of strong non-Markovian or nonthermalizing baths.
• Application to condensed matter systems where direct access to non-equilibrium steady states is challenging.

Conclusion

The Kubo-Thermalization correspondence establishes a rigorous, model-independent bridge linking equilibrium linear-response spectra and long-time thermalization properties in driven quantum impurity systems. This connection is experimentally validated across a range of interaction regimes using ultracold fermions. The result provides a new pathway to quantify thermalization dynamics from readily accessible spectroscopic measurements, with broad implications for both theoretical understanding and experimental investigation of non-equilibrium quantum many-body systems (2605.06666).