4. Resonance IV

The script begins with a discussion on the impact of magnetic fields on classical and quantum mechanical magnetic moments, revisiting the concept of rapid adiabatic passage. It explains how a spin system can be manipulated by sweeping the frequency of an oscillating magnetic field through resonance, creating an effective magnetic field that causes the spin to invert. The focus then shifts to defining the adiabatic condition, which requires the rotation of the effective magnetic field to be significantly slower than the Larmor precession frequency, to ensure the system remains aligned with the magnetic field. A counterexample illustrates what happens if the magnetic field changes too abruptly. The segment ends with an introduction to the Landau-Zener parameter, a key concept in understanding quantum mechanical transitions that will be discussed later.

This paragraph delves into the derivation of the adiabatic condition for spin systems. It starts by discussing the precession of a spin around a magnetic field and the critical nature of the resonance moment when the effective magnetic field changes due to the addition of a fictitious field. The paragraph then describes the calculation of the angular velocity, theta dot, during the sweep rate of the frequency and its relation to the Rabi frequency. The adiabatic condition is expressed as a requirement for the sweep rate to be smaller than the square of the Rabi frequency. This leads to a general condition for adiabatic processes, emphasizing the importance of a slow change in frequency relative to the relevant frequency squared. The paragraph concludes by connecting this classical result to the Landau-Zener parameter, which will be encountered in quantum mechanical treatments of adiabatic passage.

The script moves on to the application of Heisenberg's equation of motion to quantum spins in a magnetic field. It emphasizes that the principles learned for classical magnetic moments directly apply to quantum spins. The Hamiltonian for a two-level system is introduced, and the expectation values for the magnetic moment are derived using the equation of motion. The paragraph explains that the quantum system's dynamics are characterized by rotation and precession, which is an exact result for quantum mechanics, valid for any spin. It also touches on the special case of spin-1/2 systems, which are analogous to two-level systems, and the implications for composite angular momentum in atoms. The discussion concludes with an example of Dicke superradiance, which involves a system of N two-level systems interacting with an external field.

This section of the script discusses the dynamics of quantum systems, particularly focusing on the Landau-Zener problem. It starts by examining the Hamiltonian for a two-level system with a splitting and a drive term, which is represented as a simple complex exponential. The script then explores the possibility of realizing this Hamiltonian in nature, either through an approximation or an exact realization. The discussion leads to the conclusion that the Hamiltonian can indeed be exactly realized, as it is derived from real fields without approximation. The paragraph concludes with a clicker question to gauge the understanding of whether this Hamiltonian is an idealization or an exact representation that can be found in nature.

The script provides a detailed exploration of the quantum mechanical Hamiltonian for spin-1/2 systems. It begins by introducing the Hamiltonian for a two-level system with an energy splitting and a drive term, which is represented by a rotating magnetic field. The Hamiltonian is then expressed in terms of Pauli spin matrices, leading to a complex form involving the Rabi frequency and the detuning from resonance. The paragraph explains how the Hamiltonian can be rewritten in terms of spin operators, resulting in a form that includes a rotating field component. The discussion culminates in the derivation of the famous two-level Hamiltonian, which is a fundamental tool for analyzing the interaction of a two-level system with an electromagnetic field.

This part of the script delves into the concept of dressed states and the quantum description of electromagnetic fields. It starts by addressing a question about the distinction between dressed states in the context of a fully quantized field and the current Hamiltonian's eigenstates and eigenenergies. The explanation clarifies that both the classical picture with an analog amplitude and the fully quantized picture with photon number states lead to the same solutions. The script then discusses the dressed atom picture and the coherent state description, highlighting the correspondence between the two in the limit of a large photon number. The discussion concludes with an emphasis on the importance of understanding both descriptions and the homework assignment that will involve writing down the general solution for the Hamiltonian.

The script describes the process of solving the Hamiltonian for a two-level system. It begins by discussing the unitary transformation that simplifies the Hamiltonian into a time-independent form, which is easier to solve. The transformation involves rotating the frame of reference, which is analogous to the approach taken in classical problems. The script then explains how the diagonal elements of the transformed Hamiltonian represent the detuning from the energy splitting of the two-level system. The discussion concludes with the observation that when the system is on resonance, the diagonal elements disappear, indicating that the system's energy levels are equally spaced.

This section of the script explores the quantum mechanical aspects of rapid adiabatic passage, focusing on the transition probabilities when the system is not fully adiabatic. It starts by presenting the Hamiltonian in a rotating framework, which allows for a time-independent representation. The script then discusses the energy levels of the system as a function of detuning and the effect of the Rabi frequency on the system's resonance. The discussion leads to the formulation of the Landau-Zener problem, which describes the quantum mechanical behavior of a system swept through an avoided crossing. The script concludes with a mention of the textbook result for the non-adiabatic transition probability, expressed as an exponential function involving the Landau-Zener parameter.

The script delves deeper into the Landau-Zener parameter and the calculation of transition probabilities in the context of a diabatic sweep. It begins by explaining the concept of the Landau-Zener parameter as a ratio of the Rabi frequency squared to the sweep rate. The discussion then moves on to the perturbative approach to estimate the small probability of transition during a sweep that is not infinitely fast. The script emphasizes the importance of understanding this process, as it is commonly used in laboratory settings for transferring population between states. The paragraph concludes with a consideration of the effective time during the sweep and how it relates to the coherent build-up of amplitude in the second state.

In this final section, the script addresses the concept of effective time in the context of the Landau-Zener crossing and the nature of coherent processes. It starts by considering the source of incoherence in the system and concludes that the process is coherent due to the absence of factors that would lead to incoherent behavior. The script then discusses the importance of identifying the effective time during which the system is coherently driven, as this determines the transmission amplitude and the probability of transition. The discussion leads to a consideration of different possibilities for defining the effective time and concludes with the assertion that the effective time is related to the dephasing time, during which the system evolves coherently. The script wraps up with a brief mention of the homework and the next class meeting.