Quantum Interferometry Explained: Beam Splitters, Phase Shifts, and State Vectors

Overview

In this video a physicist walks through a two-beam interferometer with beam splitters and mirrors, describing how a single photon is represented by a two-component quantum state. The key ideas are normalization of probabilities, superposition of the photon across the two paths, and a pure phase shifter that changes the phase without absorbing energy.

The beam splitter is treated as a linear, unitary operator that maps input path states to output path states, represented by a 2x2 matrix acting on the state vector. The talk emphasizes that different, balanced beam splitters may differ by phase, yet preserve overall probability and the interference phenomena characteristic of interferometry.

Introduction to the Interferometer and State Representation

The video begins with a Mach-Zehnder style interferometer consisting of two beam splitters and mirrors, with detectors D0 and D1 at the outputs. A photon entering the device must be described in quantum mechanics by a state that places amplitude on either the upper or the lower path, or a superposition of both. This is formalized by a two-component state vector (alpha, beta), where alpha is the amplitude for the photon to be in the upper beam and beta for the lower beam. Normalization requires |alpha|^2 + |beta|^2 = 1, which ensures a total probability of 1 for finding the photon somewhere in the interferometer.

Phase Shifter and Phase Evolution

Between components of the interferometer, a device such as a phase shifter can be inserted. It multiplies the left amplitude by e^{i delta} and leaves the right side unchanged, thereby adding a controllable phase without absorbing the photon. This preserves the norm due to the unit modulus of the phase factor.

Beam Splitter as a Unitary Operator

The beam splitter is the first nontrivial element that mixes the two paths. A photon input in the upper path (state 10) or the lower path (state 01) is transformed into a superposition of the two outputs. The device is described by 4 complex numbers S, T, U, V that encode reflection and transmission amplitudes. Linearity implies that the effect on any input state is given by a matrix multiplication, so the beam splitter acts on the general input state alpha beta as a 2x2 matrix S U; T V acting on the vector (alpha, beta).

Balanced Beamsplitter and Probability Conservation

A balanced 50-50 beam splitter is defined by equal intensities in each output, which imposes S^2 = T^2 = U^2 = V^2 = 1/2. Yet there remains freedom in the relative phases, so the actual values can differ by signs or complex phases while still conserving probability. The video illustrates how to check this by applying the matrix to normalized input states and verifying that the output remains normalized for all possible alpha and beta.

Concrete Beam Splitter Matrices

Two unitary, balanced beam splitters are discussed. One is the standard symmetric form U1 = (1/√2) [[1, 1], [1, -1]], which preserves normalization and yields familiar interference patterns. A second beam splitter, also 50-50, exists with a different phase structure, illustrating that multiple unitary representations can model the same physical beam-splitter behavior up to phase conventions. The key takeaway is that the specific matrix is a matter of design choice, as long as it remains unitary and balanced.

Putting It Together

With phase shifters between components and a unitary beam splitter acting on the two-path state, the entire interferometer can be described by simple linear algebra. The math predicts how probabilities at detectors D0 and D1 depend on phases and beam-splitter parameters, and shows why interference arises from the coherent superposition of the two paths. The lecture closes by stressing the linearity of quantum mechanics and the practical utility of the matrix formalism for analyzing interferometric experiments.