Quantum Entanglement, Hong-Ou-Mandel Interference, and Bell Inequalities in MIT OpenCourseWare

Overview

MIT OpenCourseWare presents a lecture on quantum entanglement, measurement, and how optical experiments reveal non classical correlations. The instructor compares two harmonic oscillators to real implementations with ions and photons, highlighting the ambiguity between a single particle in two modes and two distinct systems. The talk emphasizes how density matrices relate to pure states via partial traces, setting up deeper questions about what constitutes entanglement in different physical contexts.

Key Points

The talk covers entanglement as a resource, probabilistic generation of entangled states using beam splitters, and the role of measurement in projecting atomic states. It also touches on experimental challenges such as detector efficiency and mode matching, and ends with a look ahead to larger-scale entanglement and metrological applications.

Introduction and Core Concepts

The lecture from MIT OpenCourseWare delves into quantum entanglement, starting with two harmonic oscillators in different states and asking whether their joint state is entangled. The instructor then translates these ideas into photonic systems, where a single mode of the electromagnetic field in two optical fibers can be treated as a two-mode oscillator. A central theme is the ambiguity between describing the system as one photon distributed across two fibers versus two fibers each in a different excitation state. This ambiguity motivates a discussion of what is meant by a particle and what is meant by a state, especially when considering how a density matrix can be obtained as a partial trace of a larger pure state.

Hong-Ou-Mandel Interference and Entanglement Creation

The Hong-Ou-Mandel (HOM) effect is introduced as a key tool for entangling remote atoms. Two identical photons enter a 50:50 beam splitter from different input ports, and due to bosonic statistics they exit together in the same output mode. By coupling each photon to a distant atom and performing coincidence detection on the output photons, one can project the two atoms into a Bell state. The talk emphasizes that this entanglement is probabilistic and relies on high detector efficiency and indistinguishability of the photons in polarization, timing, and spatial mode. Losses and post-selection limits are discussed as major practical hurdles in scaling to more atoms.

Bell Inequalities and Local Realism

The lecturer then introduces Bell's inequality in the CHSH form, describing a scenario with two distant observers (Bob and Alice) choosing measurement bases and obtaining outcomes ±1. He explains how classical local realism implies a bound on the CHSH expression, while quantum mechanics predicts violations up to 2√2. Experiments with photons and with atoms are discussed, including loopholes such as detector efficiency and potential hidden communications. The conclusion is that violations of Bell inequalities challenge the notions of local realism and provide evidence for non-classical correlations in quantum systems.

Entanglement as a Resource and Quantum Metrology

A major theme is treating entanglement as a resource that can enhance measurement precision beyond the standard quantum limit. The discussion covers beam splitter interferometers, coherent states, vacuum noise, and squeezing as a means to reduce quantum noise. A thought experiment shows that in an idealized lossless setting, phase resolution could reach the Heisenberg limit scaling as 1/n with the number of photons, suggesting nonlinear or entangled strategies might surpass the standard 1/√n shot-noise limit in practice. The lecturer notes that real experiments must balance entanglement, loss, and detection efficiency when pursuing metrological advantages, with applications to high-precision clocks and spectroscopy.

Quantum Circuits and Advanced Entanglers

Towards the end, the talk sketches a quantum gate perspective where a Hadamard-like operation implemented by a beam splitter with a phase shift serves as a single-qubit rotation, and a controlled NOT gate couples two qubits to generate entanglement. The demonstration shows how a simple input state can be transformed into an entangled two-qubit state across spatial modes, with further hints at even more powerful schemes to extend entanglement to larger networks. The lecturer teases a future class where larger scale entanglement could yield additional metrological gains.

Experimental Challenges and Philosophical Notes

Throughout the lecture there is an emphasis on practical challenges, including optical mode matching, magnetic field sensitivity, and the distinction between particle versus excitation pictures. The speaker also notes ongoing debates about the proper definitions of entanglement and the frontier status of foundational questions in quantum physics, highlighting the ongoing evolution of the field as research pushes into new regimes and experimental capabilities.

Closing Reference

The talk references experiments that realize atom-photon entanglement over macroscopic distances and notes the role of HOM interference in enabling probabilistic entanglement generation. The session closes with a prompt for feedback and a preview of future topics, including higher-order entanglement schemes and more advanced metrological techniques.