Quantum Metrology with Entangled States and Squeezed Light: From NOON States to LIGO

In this MIT OpenCourseWare lecture, the instructor surveys quantum metrology, showing how entangled states and squeezed light can enhance phase sensitivity in interferometers. Beginning with the idea of using two qubits and Hadamard and CNOT gates to create Bell states, the talk then scales to multi-qubit entanglement, NOON states, and the Heisenberg limit. The discussion covers optical and atomic implementations, including nitrogen vacancy centers in diamond and atom interferometers, and extends to practical considerations like photon indistinguishability, photon-number squeezing, and loss. A substantial portion ties these concepts to real-world precision measurements such as gravitational wave detection with LIGO, where squeezed vacuum input can improve sensitivity near the quantum limit. The lecture also contrasts classical and quantum views of interference and the two-photon G2 function.

Overview

This MIT OpenCourseWare lecture on Quantum Metrology does not introduce new concepts but demonstrates how existing ideas about quantum states, entanglement, and interferometry can push measurement precision beyond classical limits. The instructor frames precision measurements as a resource-driven problem, focusing on phase estimation in a Mach-Zehnder type interferometer and the transition from standard shot-noise limits to Heisenberg-limited scaling.

From Coherent States to the Heisenberg Limit

The session begins by reviewing coherent light input, which yields standard shot-noise limited phase sensitivity. It then explores single-photon inputs and the 1/sqrt(N) scaling, before introducing the concept of entangling N photons to reach the Heisenberg limit with a phase sensitivity that scales as 1/N. The intuitive picture is presented by invoking the idea of combining N photons into an effective frequency N times higher, reducing phase uncertainty accordingly.

Bell Gates, NOON States, and Multi-Qubit Entanglement

Using two-qubit gates, specifically the Hadamard and controlled-NOT, the lecture shows how to create zero-one superpositions that evolve into a Bell state, and then, with a phase shift, into a state with enhanced phase sensitivity. By repeatedly adding qubits, the speaker describes the construction of highly entangled states whose phase sensitivity scales as 1/N. The discussion includes the Greenberger–Horne–Zeilinger (GHZ) family of states and the Jerky state for Bose-Einstein condensates, illustrating pathways to Heisenberg-limited interferometry.

Alternative Routes to Heisenberg-Limited Sensing

The talk surveys three additional methods to achieve Heisenberg-limited precision: (1) Super Beams Splitter Method using a highly nonclassical beam splitter, (2) attractive Bose-Einstein condensates in a symmetric double-well potential that produce macroscopic superpositions, and (3) squeezed light interferometry where squeezed vacuum input reduces measurement noise. Each approach highlights the trade-offs between enhanced phase sensitivity and practical issues such as particle loss and experimental imperfections.

Squeezed Light and Practical Quantum Enhancement

The instructor derives how squeezing changes the noise properties of the interferometer by modifying quadrature variances. In the strong local oscillator limit, the phase uncertainty appears reduced by squeezing, but an important caveat is discussed: increasing squeezing shifts photons into the squeezed vacuum, creating a nonzero average photon number there and necessitating corrections to the simple 1/N result. An optimal squeezing level emerges, balancing improvements in one quadrature with possible degradations in the other.

Quantum Metrology and Gravitational Wave Detection

Applied examples connect theory to practice via gravitational wave detectors like LIGO. The lecturer explains how squeezing is integrated into Advanced LIGO, typically by injecting squeezed vacuum into an open port to reduce quantum noise without needing to tailor the laser itself. The discussion covers fundamental quantum limits on displacement measurements, including the balance between photon counting noise and radiation-pressure noise, and how squeezing shifts this balance to approach the standard quantum limit more closely under high-power operation.

G2: Classical and Quantum Perspectives

Concluding the module is a discussion of the G2 function, with four distinct derivations: random intensity fluctuations producing chaotic light, wave interference from two modes, quantum counting statistics, and a quantum-indistinguishability viewpoint. The goal is to illuminate why G2 can take values 1 or 2 in different scenarios and to distinguish genuine quantum effects from classical correlations arising from indistinguishable bosons and interference phenomena.