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arXiv:2004.08829 · International Journal of Modern Physics A

A pedestrian introduction to coherent and squeezed states

Bijan Bagchi, Rupamanjari Ghosh, Avinash Khare

Published Jul 8, 2020• 15 citations
arXivHTMLPDF

Abstract

This review is intended for readers who want to have a quick understanding on the theoretical underpinnings of coherent states and squeezed states which are conventionally generated from the prototype harmonic oscillator but not always restricting to it. Noting that the treatments of building up such states have a long history, we collected the important ingredients and reproduced them from a fresh perspective but refrained from delving into detailed derivation of each topic. By no means we claim a comprehensive presentation of the subject but have only tried to recapture some of the essential results and pointed out their interconnectivity.

DOI
10.1142/s0217751x20300112

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This review is intended for readers who want to have a quick understanding on the theoretical underpinnings of coherent states and squeezed states which are conventionally generated from the prototype harmonic oscillator but not always restricting to it. Noting that the treatments of building up such states have a long history, we collected the important ingredients and reproduced them from a fresh perspective but refrained from delving into detailed derivation of each topic. By no means we claim a comprehensive presentation of the subject but have only tried to recapture some of the essential results and pointed out their interconnectivity.
Both the coherent states and also the squeezed states of the harmonic oscillator have long been understood from the three classical points of view: the 1) displacement operator, 2) annihilation- (or ladder-) operator, and minimum-uncertainty methods. For general systems, there is the same understanding except for ladder-operator and displacement-operator squeezed states. After reviewing the known concepts, I propose a method for obtaining generalized minimum-uncertainty squeezed states, give examples, and relate it to known concepts. I comment on the remaining concept, that of general displacement-operator squeezed states.
The Quantum Chemist’s Companion: Background Classical and Quantum Mechanics is the first part of a comprehensive two volume reference designed for students and researchers seeking a deeper, more intuitive understanding of quantum chemistry. Filling a long standing gap in the traditional chemistry curriculum, this volume builds the essential bridge between classical mechanics, quantum mechanics and the mathematical tools that underpin them. Ideal as both a course companion and a long-term desk reference, this book brings clarity to concepts that often challenge chemists especially those rarely explored in standard chemistry training. Detailed treatments of Lagrangian and Hamiltonian mechanics, the structure of vector spaces and the role of linear algebra in quantum theory make this volume an indispensable resource. Every derivation is presented step by step, ensuring that even the most mathematically inexperienced reader can follow the logic with confidence. Inside this volume: • Part 1 provides a guided introduction to the central ideas of classical and quantum mechanics, emphasizing how the two theories connect; • Part 2 dives deeper into classical mechanics, exploring key systems, such as the harmonic oscillator, that form the backbone of later quantum discussions; • Part 3 covers systematic development of quantum mechanics built directly on top of the foundations established in Parts 1 and 2; and • Part 4 consolidates the tools and techniques needed for the mathematical material in this volume and beyond. Whether you are navigating quantum mechanics for the first time or revisiting the fundamentals with a fresh perspective, this book supports learning at every stage. Use it cover to cover or consult it as a precise, cross referenced guide to specific topics. Either way, it provides the clarity and depth needed to transform quantum theory from an abstract challenge into a coherent, intuitive framework.
In this work, we introduce a generalized harmonic oscillator (GHO) model, which is constructed based on normal harmonic oscillators (HO) and inverted harmonic oscillators (IHO). Employing theoretical frameworks, we meticulously derive the normal ordering form of the time evolution operator and its corresponding quantum transformation for the GHO. Our proposed approach enables a comprehensive comparative analysis of the time evolution of the input coherent states (CSs) in the HO and the IHO. By adjusting the frequency value within the GHO, we conduct a detailed study through the evaluation of key parameters such as fidelity, squeezing effect, and photon distribution characteristics. This novel method provides a fresh perspective on understanding the quantum state evolution under different quantum systems, opening up new research directions in the field of quantum dynamics.