Rotating Frames, Generalized Rabi Frequency, and Rapid Adiabatic Passage in Spin Dynamics – MIT OCW Lecture

Summary

In this MIT OpenCourseWare lecture, Colin guides you through spin dynamics in a magnetic field under a rotating drive. The talk begins with the rotating frame transformation, showing how a fictitious magnetic field can cancel the real field at resonance, and how the remaining transverse field drives precession at the Rabi frequency. It then treats off resonance, introducing the generalized Rabi frequency and explaining why a spin cannot fully invert unless the drive sweeps through resonance. The discussion moves to rapid adiabatic passage, describing slow frequency ramps that robustly flip spins despite small detunings, and contrasts this with conventional pi pulses. The lecture concludes with connections to precision metrology, including cesium fountain clocks and strontium optical clocks, illustrating how controlled spin dynamics underpin high-precision timekeeping.

Overview

The lecture delves into the dynamics of a magnetic moment in a time dependent magnetic field, focusing on the rotating frame where the driving field becomes static or nearly so. This framework reveals how the combination of a static quantization axis and a transverse rotating field can be treated as a simple precession around an effective magnetic field. The key insight is that in the rotating frame, the effect of the rotating field is captured by a fictitious magnetic field whose magnitude and direction depend on the drive frequency. On resonance, this fictitious field cancels the static field along the quantization axis, leaving a single transverse field that drives precession at the Rabi frequency. This leads to intuitive pictures of spin flips and population transfer that are directly observable in experiments.

Rotating Frame and Fictitious Field

When a magnetic moment is subjected to a rotating field at the same frequency as the Larmor precession, moving to the rotating frame eliminates the explicit time dependence of the drive. In this frame, the total effective field is the vector sum of the static field along the z-axis and a fictitious field along the axis of rotation. If the rotating frame is chosen to match the drive frequency, the fictitious field cancels the static field, and the remaining field in the transverse direction drives coherent motion with a well defined frequency called the Rabi frequency. This provides a clean, exact way to describe what would be a time dependent problem in the lab frame as a time independent problem in the rotating frame.

Generalized Rabi Frequency and Off-Resonance Driving

At exact resonance, the spin undergoes a complete 180 degree flip (a pi rotation) around the transverse axis, driven by the transverse field with angular frequency equal to the resonant Rabi frequency. Moving away from resonance, the effective field in the rotating frame forms an angle with the transverse axis, and the spin precesses about this tilted effective field. The result is a generalized Rabi frequency, which combines the resonant Rabi frequency with the detuning of the drive. This produces a spiral precession and prevents full inversion unless one sweeps through resonance. The geometry can be captured with a three-dimensional rotation picture, or with explicit time dependent solutions derived from the Heisenberg or Schrödinger frameworks, showing consistency between classical and quantum pictures.

Linearly Polarized Driving and the Rotating Wave Approximation

The discussion also touches the common laboratory situation where the driving field is linearly polarized. In this case, one can view the linearly polarized field as a superposition of equal amplitude left and right circularly polarized components, each rotating at the Larmor frequency but in opposite directions. In the rotating frame, one component becomes time independent while the other rotates rapidly, which motivates the rotating wave approximation. Unlike the exact rotating frame treatment for a single rotating field, the rotating wave approximation is an approximation that neglects rapidly oscillating terms, and the speaker emphasizes that the exact rotating frame transformation already provides deep insight without relying on this approximation for rotating fields.

Rapid Adiabatic Passage and Spin Inversion

A central topic is rapid adiabatic passage, where the drive frequency is swept through resonance slowly enough that the spin follows the instantaneous eigenstate of the evolving effective field. If the sweep is sufficiently slow compared to the precession set by the effective field, the spin flips direction robustly, even in the presence of modest detunings. This contrasts with a Pi pulse, which relies on exact resonance to fully invert the population. RAP offers a robust method for state manipulation in experiments, particularly where small fluctuations in the magnetic field or drive frequency would otherwise spoil a Pi pulse. The lecture provides a geometric derivation of the generalized Rabi dynamics and shows how the tilt angle between the spin and the effective field governs the precession trajectory and the final state after the sweep.

Connections to Quantum Mechanics and Applications to Clocks

The presenter highlights the strong correspondence between the classical precession picture and quantum two-level dynamics for a spin-1/2 system. In quantum language, Rabi flopping and RAP map onto coherent population transfer between eigenstates of a two-level Hamiltonian. The discussion then turns to metrology, contrasting microwave atomic clocks based on cesium hyperfine transitions with optical clocks based on strontium. The precision achievable in these clocks illustrates the extreme sensitivity of spin dynamics to external fields and the importance of robust control methods like RAP. The lecture concludes by outlining practical considerations in real experiments, such as the need for a stable bias field, meticulous control of the transverse drive, and the role of the rotating frame in achieving intuitive understanding and exact solutions.

Atomic Clocks and Precision Metrology

The talk surveys modern atomic clocks, including cesium fountain clocks whose definition of the second rests on hyperfine transitions, and strontium optical clocks that operate in the optical domain with extremely narrow transitions. The speaker emphasizes interrogation time, Fourier-limited linewidths, and techniques for stabilizing lasers to reference cavities. These clocks push the boundaries of frequency metrology and demonstrate how precise control of spin dynamics and coherent evolution in rotating frames translates into practical, world-leading time standards. The conclusion ties the physics of rotating frames and generalized Rabi dynamics to the broader field of quantum control and high-precision measurement, illustrating how classical intuition and quantum formalism come together in cutting edge experiments.

Summary and Outlook

Overall, the lecture presents a cohesive framework for understanding spin dynamics under rotating drives through rotating frames, fictitious fields, and generalized Rabi frequencies. It demonstrates how RAP provides robust state inversion and how exact solutions in the rotating frame align with quantum mechanical predictions. The application to atomic clocks provides a compelling demonstration of how these concepts have real world impact in timekeeping and fundamental metrology. The talk also stresses the versatility of the rotating frame approach across different Hamiltonians and experimental contexts, including linear polarization and the rotating wave approximation, and hints at further explorations in quantum control and spintronics that build on these foundational ideas.