Floquet-Magnus Theory and Generic Transient Dynamics in Periodically Driven Many-Body Quantum Systems (1508.05797v3)

Abstract: This work explores a fundamental dynamical structure for a wide range of many-body quantum systems under periodic driving. Generically, in the thermodynamic limit, such systems are known to heat up to infinite temperature states after infinite-time evolution, irrespective of dynamical details. In the present study, instead of considering infinitely long-time scale, we aim to provide a framework to understand the long but finite time behavior, namely the transient dynamics. In the analysis, we focus on the Floquet-Magnus (FM) expansion that gives a formal expression of the effective Hamiltonian on the system. Although in general the full series expansion is not convergent in the thermodynamics limit, we give a clear relationship between the FM expansion and the transient dynamics. More precisely, we rigorously show that a truncated version of the FM expansion accurately describes the exact dynamics for a finite-time scale. Our result reveals a reliable time scale of the validity of the FM expansion, which can be comparable to the experimental time scale. Furthermore, we discuss several dynamical phenomena, such as the effect of small integrability breaking, efficient numerical simulation of periodically driven systems, dynamical localization and thermalization. Especially on thermalization, we discuss generic scenario of the prethermalization phenomenon in periodically driven systems.

• The paper demonstrates that a truncated Floquet-Magnus expansion accurately captures transient dynamics in many-body quantum systems up to timescales scaling with e^(ω).
• The methodology establishes rigorous conditions for using the truncated series, showing energy absorption is exponentially suppressed at high driving frequencies.
• The findings offer practical insights for quantum control and prethermalization strategies in systems relevant to quantum computing and materials science.

Overview of Floquet-Magnus Theory and Transient Dynamics in Periodically Driven Quantum Systems

The paper at hand presents a rigorous investigation into the generic transient dynamics of many-body quantum systems subjected to periodic driving. By leveraging the Floquet-Magnus (FM) theory, the authors develop a framework focused on understanding the dynamics over long, yet finite, timescales. In the thermodynamic limit, it is known that many-body quantum systems under periodic driving tend to heat to infinite temperature states over infinite timescales, erasing any system-specific properties. The focus here, however, is on the transient behaviors that manifest over experimentally relevant timescales.

Key Contributions

The key contributions of the paper are:

1. Floquet-Magnus Expansion and Transient Dynamics: The authors establish a formal relationship between the FM expansion and transient dynamics. Even though the full FM series is typically non-convergent in the thermodynamic limit, they demonstrate that a truncated version of this expansion can accurately depict exact dynamics for certain timescales.
2. Convergence and Utility of Truncated Series: The authors provide rigorous conditions under which the truncated FM expansion approximates the dynamics well, essentially up to a timescale proportionate to $e^{\omega}$, where $\omega$ is the driving frequency.
3. Application to Short-Range Interacting Systems: The analysis extends to systems with short-range interactions. For these systems, the correction to dynamics by the truncated FM expansion is shown to be manageable even for global driving conditions, applicable in the thermodynamic limit.
4. Energy Absorption Rates: The paper addresses energy absorption in driven systems, offering evidence that energy absorption is significantly suppressed, scaling as $e^{-\omega}$ at high frequencies. This gives insights into dynamical localization phenomena.

Implications and Future Directions

Theoretical Implications

The rigorous link between the FM expansion and transient dynamics provides a substantial theoretical advancement in understanding nonequilibrium properties of periodically driven many-body systems. This strengthens the notion of 'prethermalization', where systems relax to quasi-steady states characterized by the truncated FM Hamiltonian before eventually reaching infinite temperature.

Practical Implications

Practical implications are particularly relevant in domains like quantum computing and materials science, where controlling quantum systems dynamically is crucial. The insights into energy absorption and dynamical localization at practical timescales can influence the design and stability of quantum devices.

Speculative Future Developments

Looking forward, the strategies laid out in this paper could pave the way for new control protocols in quantum systems, especially in designing driving schemes to mitigate or exploit transient dynamics for computational gains. It provides a foundation for more detailed studies on the prethermalization phenomenon and might influence future research in dynamically driven quantum systems, including those with dissipative environments.

The paper's investigations could be extended to open quantum systems where interactions with external environments may play a critical role. Such studies would benefit from further theoretical development to encompass the full complexity of real-world scenarios where periodic driving is applied.

Through its methodical and expertly technical analysis, this paper sets a benchmark in understanding the behavior of periodically driven quantum systems, opening avenues for further exploration in the transient dynamics of many-body physics.