Squeezing Classical Oscillations into Quantum Light: Squeezed States, Operators, and Detection

Overview

This lecture tracks the journey from classical squeezing of a harmonic oscillator to quantum squeezing of light. It explains how a parametric drive at twice the resonance frequency can amplify one quadrature while dampening the other, forming an ellipse in the phase space of the field. The talk then translates these ideas into the quantum domain, introducing the squeezing operator, squeezed vacuum, and the displacement operator, and discusses how nonlinear optics in an optical parametric oscillator generates squeezed light. The session also covers measurement through homodyne detection and touches on quantum state teleportation as a broader application context.

Introduction to Squeezing

The speaker begins with a classical harmonic oscillator, describing how a parametric drive at twice the resonance frequency causes the two quadrature components, cosine and sine, to evolve differently. Depending on the drive phase, one quadrature is exponentially amplified while the other is exponentially damped, transforming a circular trajectory in phase space into an ellipse. This is the classical squeezing picture, illustrating how fluctuations in one quadrature can be reduced at the cost of increased fluctuations in the conjugate quadrature.

Quantum Extension: Squeezed States

Moving to the quantum regime, the discussion connects a single mode of the electromagnetic field to a quantum harmonic oscillator. To achieve squeezing in the quantum domain, a nonlinear interaction is required, provided by an optical parametric oscillator (OPO) driven at two Omega. The Hamiltonian is simplified by treating the strong pump mode as a classical field, so the focus becomes the quantum mode E that will be squeezed. The down-conversion process creates photon pairs at the signal frequency, and the squeezing operator S emerges from this time evolution, describing a unitary transformation that reshapes the quadrature variances in phase space.

Key States and Operators

The squeezed vacuum is introduced as the vacuum state acted on by the squeezing operator, producing an elliptical uncertainty distribution in the X and P quadratures. In the limit of large squeezing, the E- and P-like quadratures become highly anisotropic in their fluctuations, with energy considerations explained by the drive illuminating how the process injects energy into the field. The displacement operator is then defined and shown to translate states in phase space, converting the vacuum into a coherent state when applied to the vacuum and displacing more general states accordingly through commutation relations.

Physical Realization and Detection

The lecture discusses practical realizations, notably how a beam splitter with a strong input field can implement displacement, effectively realizing a coherent state. It also emphasizes the role of homodyne detection as a phase-sensitive measurement technique that can capture the reduced noise in one quadrature, a necessity for exploiting squeezed states in metrology and sensing applications. The quasi-probability pictures (Wigner, Q, and P representations) are connected to the intuitive ellipse in phase space and to the time evolution of the quadratures.

Broader Context and Teleportation

Towards the end, the talk links squeezing and displacement to a broader quantum information framework by illustrating how these tools underpin schemes such as quantum teleportation. The essential message is that the language of squeezing and displacement provides a clear, scalable way to discuss quantum state manipulation in laboratory settings and in information-processing tasks.