Rapid Adiabatic Passage in Spin-1/2 Systems: From Classical Precession to Landau-Zener Transitions

In this MIT OpenCourseWare lecture, the instructor develops rapid adiabatic passage for both classical and quantum spins, deriving the adiabaticity condition and showing how a sweep through resonance creates an effective rotating magnetic field. The discussion covers the quantum two-level system, the spin-1/2 Hamiltonian driven by a rotating field, rotating-frame transformation, and the emergence of Rabi oscillations. The talk then connects these ideas to Landau-Zener transitions in an avoided crossing, contrasts adiabatic and diabatic limits, and discusses coherence versus decoherence in driven spin dynamics, including applications to atomic traps and evaporative cooling experiments.

Overview of Rapid Adiabatic Passage in a Two-Level Spin System

The lecture begins by revisiting RAP as a robust method to invert populations in spins when an oscillating field sweeps through resonance. The key adiabaticity condition is introduced: the rotation rate of the effective magnetic field must be much slower than the Larmor precession frequency, formalized as Omega-dot much less than Omega_Rabi squared. The instructor emphasizes the roles of the real magnetic field, the fictitious field arising from the rotating frame, and how the effective field traces a path in parameter space as the drive sweeps. This sets up the classical intuition behind RAP before transitioning to the quantum description.

From Classical to Quantum Spin Dynamics

The transition to quantum mechanics is made via Heisenberg’s equations of motion for angular momentum and the magnetic moment. The precession equation emerges exactly as in the classical picture, and the two-level system limit (spin-1/2) is highlighted as the central example. The electron’s negative gyromagnetic ratio is discussed, explaining why spin aligns differently from the magnetic moment depending on the sign of gamma. The lecturer then introduces the standard two-level Hamiltonian for a spin in a magnetic field, including the diagonal splitting and the drive term that couples the levels through the Rabi frequency.

The Rotating Frame and the Rabi Model

Transforming to the rotating frame eliminates explicit time dependence from the coupling in many cases, yielding a time-independent problem in the appropriate frame. The detuning, Delta, arises as the difference between the drive frequency and the intrinsic level splitting, and the dressed-state picture emerges naturally. The Rabi frequency, together with detuning, governs the coherent oscillations between the two levels, or Rabi oscillations, with the rotating frame making the dynamics particularly transparent. The connection to the Landau-Zener problem is then made explicit by considering a sweep of the drive through resonance, producing an avoided crossing whose properties determine adiabatic versus diabatic transitions.

Landau-Zener Transitions and Coherence

The Landau-Zener framework is invoked to describe transitions as a system passes through an avoided crossing. In the adiabatic limit, the system follows the adiabatic eigenstate, while in the diabatic limit it effectively remains in its initial instantaneous basis. The non-adiabatic transition probability depends on the Landau-Zener parameter, which combines the squared Rabi frequency and the sweep rate of the detuning. The lecture emphasizes that for a slow sweep, the population tracks the adiabatic path; for a rapid sweep, the probability of remaining in the initial state grows, illustrating the continuum between the two limits. A perturbative and intuitive perspective is offered, linking the quantum Landau-Zener result to the classical adiabatic criterion discussed earlier.

Coherent Versus Incoherent Dynamics and Experimental Contexts

The speaker contrasts coherent evolution, where amplitudes accumulate with fixed phase relationships, with incoherent, rate-driven processes governed by Fermi’s golden rule. In a clean, closed two-level system with no reservoir, the evolution is fully coherent, and transition probabilities scale with the square of the drive time in the short-time limit. In realistic experiments, residual fluctuations can induce apparent incoherence, but the idealized model remains a powerful guide to understanding how RAP and Landau-Zener sweeps manipulate quantum states. The discussion then touches on laboratory practice, including how adiabatic sweeps appear in atom-trap and evaporative cooling experiments, and how the Landau-Zener parameter sets the efficiency of population transfer in near-resonant driving.

Summary and Outlook

The lecture closes by reinforcing the core idea that RAP is a fundamental tool for controlling spin dynamics, with a clear quantum mechanical description that naturally connects to the rotating frame, Rabi physics, and Landau-Zener theory. The coherence window, the effective time during which population transfer accumulates coherently, emerges as a central concept, tying together the semiclassical intuition and the exact quantum solution. The material lays the groundwork for further exploration of Dicke superradiance, multi-atom dynamics, and more complex driving fields in subsequent lectures.