Rigorous Bound on Energy Absorption and Generic Relaxation in Periodically Driven Quantum Systems

Abstract: We discuss the universal nature of relaxation in isolated many-body quantum systems subjected to global and strong periodic driving. Our rigorous Floquet analysis shows that the energy of the system remains almost constant up to an exponentially long time in frequency for arbitrary initial states and that an effective Hamiltonian obtained by a truncation of the Floquet-Magnus expansion is a quasi-conserved quantity for long time scale. These two general properties lead to intriguing classification on the initial stage of relaxation, one of which is similar to the prethermalization phenomenon in quasi-integrable systems.

• The paper rigorously demonstrates that energy absorption is suppressed for exponentially long times relative to the driving frequency.
• It exploits the Floquet-Magnus expansion to establish a quasi-conserved truncated Hamiltonian until higher-order interactions prevail.
• The derived bound on the site-specific energy absorption rate decays exponentially with frequency, guiding experimental control of heating.

Analysis of Periodic Driving in Many-Body Quantum Systems

The paper "Rigorous Bound on Energy Absorption and Generic Relaxation in Periodically Driven Quantum Systems" explores the dynamic behavior of many-body quantum systems under the influence of periodic driving. This study utilizes the Floquet theoretical framework to analyze how the energy absorption and relaxation processes evolve over time for quantum systems subjected to global and intensive periodic driving.

Key Findings

1. Energy Conservation Over Extended Timescales: The study rigorously shows that the energy of a periodically driven quantum system remains nearly invariant for exponentially long periods relative to the driving frequency. This behavior holds for any initial quantum state, illustrating that the energy absorption rate is remarkably low, thus delaying the onset of heating in the system.
2. Existence of a Quasi-Conserved Quantity: The research leverages the Floquet-Magnus expansion to demonstrate that the truncated Floquet Hamiltonian serves as a quasi-conserved quantity over long timescales. This truncated Hamiltonian retains its validity up until a certain period, beyond which the higher-order terms in the expansion, which account for collective spin interactions, begin to dominate. The truncation approach helps circumvent convergence issues often associated with the full Floquet-Magnus series.
3. Bound on Energy Absorption Rate: It is established that the site-specific energy absorption rate is bounded and diminishes exponentially with the driving frequency. The formulation derived shows that for high-frequency drives, the system's propensity to absorb energy is significantly curtailed, providing a quantitative measure of transient stability in experiments.

Implications and Speculations

• Prethermalization in Quantum Systems:

The paper’s findings indicate a classification of relaxation regimes similar to the prethermalization phenomenon observed in undriven nearly-integrable systems. Specifically, the system may reach a quasi-stationary state characterized by a truncated Floquet Hamiltonian before eventual thermalization to an infinite-temperature state, assuming the Floquet ETH holds.

• Experimental Relevance:

Understanding the timescale and dynamics of heating in periodically driven systems is vital for experimental investigations involving ultrafast or high-frequency drivings, such as those seen in cold atom experiments or laser-driven quantum systems. The theoretical results provide a framework to predict the limits of transient stability and energy retention in such setups.

• Future Directions:

The exploration of quantum dynamics in driven systems further opens questions on identifying phases with localized energy states that resist heating and understanding the conditions under which the Floquet ETH may or may not apply. The quantitative estimates provided by the study on energy retention timescales could be refined through extended numerical simulations or by considering broader classes of quantum systems including bosonic models.

In summary, this work advances the understanding of periodically driven quantum systems, offering rigorous insights into their dynamic relaxation processes and the effective conservation of energy on extended timescales. The theoretical underpinnings laid out in this research are poised to guide future experimental and theoretical explorations in the domain of nonequilibrium quantum physics.