Multi-directional Periodic Driving of a Two-Level System Yields Exact Transition Probabilities Beyond Floquet Formalism

Understanding how quantum systems respond to external forces is fundamental to developing advanced technologies, and Michael Warnock from Brown University and the Naval Undersea Warfare Center, along with David A. Hague from the Naval Undersea Warfare Center and Vesna F. Mitrovic, now present a significant advance in this field. They have developed an exact method for calculating the behaviour of a simple quantum system, a two-level system, when subjected to complex, oscillating forces. Current techniques, such as Floquet theory, often rely on approximations that can obscure crucial details and limit control precision, but this new approach utilises a mathematical technique to derive a precise solution to Schrödinger’s equation. The resulting formulation provides a complete description of the system’s response, offering a powerful tool for designing and controlling quantum devices with unprecedented accuracy.

Beyond Floquet Theory, Multi-Directional System Driving

Researchers investigate the dynamics of a two-level quantum system driven by multiple, precisely controlled fields, extending beyond conventional analytical methods. This approach allows exploration of complex scenarios inaccessible through standard techniques, particularly when the driving forces are strong and interact in intricate ways. The research focuses on understanding how the frequency, strength, and spatial arrangement of these driving fields influence the system’s behaviour, specifically examining changes in population and the maintenance of quantum coherence. The method involves solving the time-dependent Schrödinger equation under the influence of these multiple, coherently driven fields, and using numerical simulations to map the system’s response.

A key achievement is the development of a framework to characterise the system’s response using effective Hamiltonians and quasi-energy levels, providing a more intuitive understanding of the underlying physics. The results demonstrate that driving the system with fields from multiple directions significantly alters its response compared to single-direction driving, leading to novel phenomena such as enhanced oscillations and the emergence of multiple quasi-energy levels. By carefully controlling the spatial orientation of the driving fields, the team finds it is possible to manipulate the system’s coherence and achieve high-fidelity control over its quantum state, with potential applications in quantum sensing, information processing, and precision measurement.

Periodic Driving, Exact Solutions, and Kernel Formulation

This work presents a new analytical approach for determining the response of a two-level quantum system subjected to periodic driving, offering an exact solution to Schrödinger’s equation. Researchers developed a method using a resolvent formalism and path-sum theorem, resulting in a series solution expressed through a compact kernel containing all information about the periodic drive. This kernel, expanded as a non-harmonic Fourier series using generalized Bessel functions, provides a precise formulation for both analysis and control applications. The team demonstrated the accuracy of this approach by comparing analytical predictions with numerical computations of transition probabilities, achieving strong agreement in resonance positions and magnitudes.

They highlight the potential for this kernel expression to optimise signal-to-noise ratios in quantum sensing and to create novel control algorithms for quantum gates, potentially leading to higher fidelity operations. The study acknowledges that the analytical description deviates from numerical results when the strength of the driving force increases or a large number of harmonics are present, and that long-time dynamics can fall outside the approximation regime. Future work may focus on refining the model to better capture these effects and extending the method to more complex quantum systems.

Floquet Theory and Time-Dependent Perturbations

This body of work encompasses a broad range of theoretical foundations, mathematical tools, and computational techniques relevant to understanding and simulating the dynamics of quantum systems and signal processing. Researchers have extensively explored Volterra equations and integral equations, mathematical tools used to model systems with memory or delayed responses, and divided differences and polynomial expansions, techniques for approximating functions and improving the efficiency of numerical simulations. Furthermore, significant attention has been given to generalized Bessel functions, important in signal processing and solving certain differential equations, and the omega matrix calculus, a basis-and integral-free representation of time-dependent perturbation theory.

This theoretical foundation supports a wide range of applications in quantum dynamics and simulation, including the study of driven quantum systems, ultrafast control, and geometric quantum computation. Researchers have also explored open quantum systems, quantum simulation algorithms, and approximations such as the counter-rotating wave approximation, aiming to improve the accuracy and efficiency of simulations. Surprisingly, signal processing and waveform design play a prominent role, with significant work dedicated to multi-tone sinusoidal frequency modulation and constant envelope OFDM waveforms. This connection highlights the shared mathematical tools and techniques used in both quantum physics and signal processing.

Numerical methods and computational techniques are also central, with researchers developing exact formulations of the time-ordered exponential, novel frame changes for quantum physics, and open-source software frameworks for simulating the dynamics of open quantum systems. This research extends to specific physical systems and applications, including solid-state artificial atoms, nuclear magnetic resonance, and circuit quantum electrodynamics. Key themes and connections emerge, including the importance of resonances, time-periodic driving, and efficient simulation techniques. The shared mathematical tools and techniques used in both quantum physics and signal processing underscore the interdisciplinary nature of this research.