Defining coherent states: why must they be eigenstates of the annihilation operator?

TL;DR

This pedagogical paper explores why coherent states of a harmonic oscillator are eigenstates of the annihilation operator, linking classicality, phase space proximity, and energy expectation properties through rigorous theorems.

Contribution

It provides a physically motivated derivation of coherent states' mathematical definition and proves their unique classicality properties via two key theorems.

Findings

States closest to phase space points have minimum uncertainty product.

Energy expectation deviations are minimized in eigenstates of the annihilation operator.

Coherent states are the most classical states due to these properties and decoherence processes.

Abstract

This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.