Quantum dynamics of the damped harmonic oscillator

TL;DR

This paper presents a canonical quantization method for the damped harmonic oscillator using a continuum reservoir, enabling accurate modeling of Ohmic damping and general damping behaviors in quantum systems.

Contribution

It introduces a novel approach to quantize damped oscillators with a continuum reservoir, overcoming limitations of discrete models and capturing a wide range of damping effects.

Findings

Successfully quantized the oscillator with Ohmic damping.

General damping behaviors are incorporated via an effective susceptibility.

Provides a framework applicable to nano-mechanical and opto-mechanical systems.

Abstract

The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a damped oscillator is determined by an arbitrary effective susceptibility that obeys Kramers-Kronig relations. This approach offers an alternative description of nano-mechanical oscillators and opto-mechanical systems.