Harmonic Oscillators, Frequency Measurements, and Quantum Limits: Insights from an MIT OpenCourseWare Lecture

The lecture investigates how precisely one can measure frequency across classical and quantum systems, starting from the Fourier limit and Heisenberg uncertainty and moving through classical harmonic oscillators, laser pulses, and quantum two level systems. It explains how signal to noise, line shape, and beat notes with stable local oscillators enable improved frequency measurements, and it contrasts classical and quantum limits using optics, cavity QED, and rotating frame methods. The talk also unpacks the subtle differences between harmonic oscillators and two level systems, saturation, and nonlinearities, and introduces how rotating frames and Lamor frequency underpin magnetic resonance concepts.

Introduction to Frequency Measurements and Fundamental Limits

The lecture opens with a discussion of the Fourier limit and the Heisenberg uncertainty relation as fundamental constraints on frequency determination. It emphasizes that frequency measurements are among the most precise in physics and that classical and quantum measurements alike are affected by the observation time and spectral width of the signal.

Using a classical harmonic oscillator as a baseline, the instructor explains that pure classical measurements can beat naive limits if the signal to noise ratio is high and the spectral line is well understood. The discussion then moves to quantum systems, asking whether a quantum oscillator can be measured more precisely than the Heisenberg limit would suggest, depending on the measurement context and the use of nonlinear interactions.

The lecture then examines a laser pulse as a practical measurement probe. A laser is described as a quantum optical field in a coherent state with many photons, whose classical limit emerges at large photon number. Beat notes against a stable local oscillator demonstrate how arbitrarily high signal to noise can be achieved in principle, thus mapping the classical and quantum pictures.

For a single quantum system such as a two level atom driven by a single photon, the Heisenberg limit applies to single measurements, but there are ways to improve precision by using multiple photons, correlations, and nonlinear processes which enable what is sometimes referred to as the Heisenberg limit or beyond in certain regimes.

The instructor draws a careful analogy and distinction between two level systems and harmonic oscillators. In the weak excitation limit, a two level system behaves like a harmonic oscillator, but saturation prevents the oscillator from behaving identically in all regimes. Cavity QED and the normal mode splitting are used to illustrate how highly nonclassical states can arise when a two level system interacts with a single mode of the electromagnetic field.

Rotating frame techniques are introduced as a powerful method to simplify dynamics in driven systems. The Lamor frequency and gyromagnetic ratio provide the link between magnetic moments and precessional motion, with a careful discussion of factors of two that often appear in magnetic resonance experiments. The lecture concludes by contrasting the quantum and classical pictures of a rotating magnetic moment and outlining when a truly quantum measurement remains indispensable.