A note on squeezed states

Vasileios Bartzis, E. Vlahos, A. D. Jannussis

Alexander N. Korotkov

In these notes, we discuss squeezed states using the elementary quantum language based on one-dimensional Schrödinger equation. No operators are used. The language of quantum optics is mentioned only for a hint to solve a differential equation. Sections II and III (squeezed vacuum and pure squeezed states) can be used in a standard undergraduate course on quantum mechanics after discussion of a harmonic oscillator. Section IV discusses density matrix of mixed squeezed states in $x$-representation. These notes present explicit (therefore somewhat lengthy) step-by-step derivations. These notes are not intended for publication as a journal paper.

Arash Azizi

Abstract These lecture notes provide a comprehensive exploration of coherent and squeezed states, which are central to quantum optics and continuous-variable quantum information. We begin with the traditional formalism, where displacement and squeezing operators are defined via the creation and annihilation operators. Coherent states are presented as quantum analogues of classical fields with minimal uncertainty, while squeezed states are introduced as nonclassical resources that reduce quantum noise below the vacuum limit, enabling next-generation quantum technologies. A central feature of these notes is a parallel development from a group-theoretical standpoint. Here, these operators are defined abstractly by their algebraic properties, revealing their structure as bosonic realizations of the Weyl–Heisenberg and $$SU(1,1)$$ S U ( 1 , 1 ) structures, respectively. This advanced perspective not only provides a more rigorous mathematical foundation but also leads to new insights, including a proof of the uniqueness of the single-mode squeezing operator and a no-go theorem for certain multi-mode squeezing configurations. Designed for both pedagogical clarity and advanced study, these notes bridge the foundational operator algebra of quantum optics with the modern group-theoretical structures essential for research in continuous-variable quantum systems.

John R. Klauder

Gerard J Milburn

The dynamic response of the inversion in a two-level atom to single-mode squeezed states of the radiation field is examined. Depending on the direction of squeezing, an increase or decrease in the collapse time is demonstrated. A number of additional departures from the response of the atom to a coherent field are noted. In particular, for certain squeezed states the response of the atom is similar to that expected for chaotic radiation.

L. Mandeļ