Quantum Light: Coherent States, Thermal States, and Phase-Space Quasi-Probabilities in a Single-Mode Optical Field

In this MIT OpenCourseWare unit, the lecturer introduces the quantum nature of light within a single harmonic-oscillator mode. Starting from the light field Hamiltonian, he relates the electric field to the momentum of the oscillator, explains Heisenberg and Schrödinger pictures, and builds the state taxonomy of number states, thermal states, and coherent states. Coherent states are shown to have Poisson photon statistics and minimum uncertainty, while thermal states display broader Gaussian photon-number distributions with super-Poissonian fluctuations. A central idea is the quasi-probability representation, including the Q function, which visualizes states in phase space. The talk emphasizes state overlaps, the overcomplete nature of coherent states, and the time evolution of the field as a rotation in phase space. The framework sets the stage for open-system dynamics and cavities.

Introduction to Quantum Light in a Single Mode

The lecture begins with a compact overview of the MIT OpenCourseWare unit on the quantum nature of light, emphasizing the fundamental Hamiltonian for a single mode of the electromagnetic field. The instructor connects the electric field operator to the momentum of the harmonic oscillator and discusses the difference between the Schrödinger and Heisenberg pictures, clarifying when to apply time dependence to states versus operators. This establishes the stage for treating light as a quantum harmonic oscillator and for analyzing how light interacts with matter in controlled settings such as cavity QED.

Quantum States of Light: From Fock to Coherent and Thermal

The core of the lecture introduces the standard state bases for a single mode: photon-number (Fock) states, thermal (chaotic) states, and coherent states. The number states form the eigenbasis of the harmonic oscillator, while coherent states are defined as eigenstates of the annihilation operator with complex eigenvalues α. Thermal light is described by a Bose-Einstein distribution over number states, leading to broad photon-number fluctuations. Coherent states, in contrast, exhibit Poissonian photon statistics with a mean photon number ⟨N⟩ = |α|^2 and a variance equal to ⟨N⟩, indicating comparatively small fluctuations for a fixed mean. This dichotomy underpins much of quantum optics and its experimental signatures.

Quasi-Probabilities and Phase Space Visualization

A central concept introduced is the phase-space representation of quantum states through quasi-probability distributions. The speaker defines a quasi-probability Q(α) as a diagonal element of the density operator in the coherent-state basis, then discusses how different quantum states manifest in this representation. He illustrates that chaotic thermal states produce broad Gaussian distributions in phase space, coherent states produce narrow Gaussians with centers at α, and number states give ring-like patterns. The discussion emphasizes the overcomplete nature of coherent states, which means quasi-probabilities for coherent states are not delta functions but extended, overlapping features in the α-plane. This framework provides an intuitive bridge between abstract operator formalism and visual pictures of the electromagnetic field’s quantum state.

Time Evolution and the Electric Field Picture

The lecture explains that for a monochromatic single-mode field the coherent state remains a coherent state under time evolution, with the eigenvalue evolving as α e^{iωt}. In the visualization, this corresponds to a rotation of the quasi-probability distribution in phase space. The presenter also discusses how projecting the phase-space distribution onto the momentum quadrature (the E field’s analog) yields a time-dependent electric-field picture, where the field oscillates in a manner consistent with the underlying quantum state and its uncertainties. This leads to an intuitive picture of shot noise and the Heisenberg uncertainty limit in the electric field measurement.

Outlook: From Coherence to Open Quantum Systems

The final parts sketch how the formalism extends to more complex situations, including open systems and interactions with reservoirs. Master equations and cavity QED scenarios are mentioned as the next steps in understanding relaxation, decoherence, and the loss of entanglement in realistic settings. The lecture sets the stage for deeper treatments of non-classical light, single-photon states, and metrology applications where quantum limits such as the Heisenberg limit constrain measurement precision.