Periodically-driven quantum systems: Effective Hamiltonians and engineered gauge fields (1404.4373v3)

Abstract: Driving a quantum system periodically in time can profoundly alter its long-time dynamics and trigger topological order. Such schemes are particularly promising for generating non-trivial energy bands and gauge structures in quantum-matter systems. Here, we develop a general formalism that captures the essential features ruling the dynamics: the effective Hamiltonian, but also the effects related to the initial phase of the modulation and the micro-motion. This framework allows for the identification of driving schemes, based on general N-step modulations, which lead to configurations relevant for quantum simulation. In particular, we explore methods to generate synthetic spin-orbit couplings and magnetic fields in cold-atom setups.

• The paper derives effective Hamiltonians for periodically-driven quantum systems that facilitate the simulation of gauge fields and topological phases.
• It employs a perturbative 1/ω expansion to systematically capture higher-order dynamic effects crucial for realizing synthetic magnetic fields.
• Numerical simulations confirm that the effective Hamiltonian approach accurately reproduces the full time-dependent behavior in optical lattice and cold atom setups.

Periodically-Driven Quantum Systems: Effective Hamiltonians and Engineered Gauge Fields

The paper by Nathan Goldman and Jean Dalibard provides a comprehensive examination of periodically-driven quantum systems, focusing particularly on the development of effective Hamiltonians and the engineering of synthetic gauge fields within such systems. This investigation is grounded in the broader context of quantum simulation and the quest for novel states of matter in condensed-matter physics. The authors approach this by analyzing the effects of periodic driving on quantum systems, extending the considerations to both free space and optical lattice configurations.

Key Contributions and Methodological Approach

The authors present a robust formalism that decomposes the time evolution of periodically-driven quantum systems into three crucial components: the effective Hamiltonian that governs long-time dynamics, the initial phase of the driving, and the micro-motion. This formalism builds on earlier approaches by providing a clear separation of these dynamic aspects, enabling a thorough understanding of how each contributes to the evolution of the system. Furthermore, the formalism is expressed perturbatively, expanding in powers of $1/\omega$, where $\omega$ is the driving frequency. This allows for a systematic approach to capturing higher-order effects.

Critical to the discussion is the distinction between the roles played by different terms in the system's Hamiltonian: the time-independent Hamiltonian $\hat{H}_0$ and the time-dependent perturbation $\hat{V}(t)$. By carefully engineering these terms through a combination of static and time-varying potentials, the authors demonstrate how synthetic magnetic fields and spin-orbit couplings can be realized in cold atom systems.

Significant Results and Implications

1. Effective Hamiltonians and Gauge Potentials: The paper derives expressions for effective Hamiltonians in periodically-driven systems, demonstrating how non-trivial topological and gauge-like properties can emerge. This is particularly illustrated through their derivation of Hofstadter-like Hamiltonians in optical lattices, which are pertinent for the quantum simulation of magnetic fields and exploring topological phases.
2. Spin-Orbit Coupling in Cold Atoms: Through a sequence of tailored driving pulses, the authors show how Rashba spin-orbit coupling can be synthetically induced in neutral atoms, providing a platform for studying quantum phases analogous to those in topological insulators and superconductors.
3. Numerical Simulations and Theoretical Predictions: The paper is supported by numerical simulations that confirm the theoretical predictions of the effective Hamiltonian approach. Simulations show the agreement between full time-dependent dynamics and those predicted by the effective Hamiltonians, reinforcing the utility of the perturbative method and the relevance of higher-order corrections.

Future Directions and Open Questions

The implications of this research extend to various domains within quantum technologies, offering potential pathways for simulating complex quantum materials using cold atom systems. Future research could explore the interplay of interactions within such driven systems, potentially leading to the discovery of new quantum phases. Additionally, understanding how dissipation and noise in real-world implementations affect the stability and lifetime of these states remains an open question.

Moreover, the scalability of these approaches to engineer synthetic gauge fields in more complex systems, such as those involving multi-component or higher-dimensional lattice structures, could further bridge the gap between theoretical predictions and experimental realizations. As the exploration of non-equilibrium quantum systems progresses, the methods and insights from this paper will likely play a crucial role in shaping our understanding of dynamical gauge fields and topological matter.