# Extension of the input-output relation for a Michelson interferometer to arbitrary coherent-state light sources: --- Gravitational-wave detector and weak-value amplification ---

## TL;DR

This paper extends the input-output relation for Michelson interferometers to include arbitrary coherent light sources, revealing that conventional gravitational-wave detectors inherently incorporate weak-value amplification effects.

## Contribution

It provides a generalized input-output relation for Michelson interferometers with arbitrary coherent states, linking gravitational-wave detection to weak-measurement concepts.

## Key findings

- The extended relation applies to both standard and weak-measurement scenarios.
- Conventional detectors inherently include weak-value amplification effects.
- Quantum noise reduction is connected to measurement at the dark port.

## Abstract

An extension of the input-output relation for a conventional Michelson interferometric gravitational-wave detector is carried out to treat an arbitrary coherent state for the injected optical beam. This extension is one of necessary researches toward the clarification of the relation between conventional gravitational-wave detectors and a simple model of a gravitational-wave detector inspired by weak-measurements in [A.~Nishizawa, Phys. Rev. A {\bf 92} (2015), 032123.]. The derived input-output relation describes not only a conventional Michelson-interferometric gravitational-wave detector but also the situation of weak measurements. As a result, we may say that a conventional Michelson gravitational-wave detector already includes the essence of the weak-value amplification as the reduction of the quantum noise from the light source through the measurement at the dark port.