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The European Physical Journal Plus

From operators to groups: a dual perspective on coherent and squeezed states

Arash Azizi

Published Jun 9, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1140/epjp/s13360-026-07777-4

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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)SU(1,1)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.
Nonclassical states of light and their correlations lie at the heart of quantum optics, serving as fundamental resources that underpin both the exploration of quantum phenomena and the realisation of quantum information protocols. These lecture notes provide an accessible yet rigorous introduction to the foundations of quantum optics, emphasising their relevance to quantum information science and technology. Starting from the quantisation of the electromagnetic field and the bosonic formalism of Fock space, the notes develop a unified framework for describing and analysing quantum states of light. Key families of states -- thermal, coherent, and squeezed -- are introduced as paradigmatic examples illustrating the transition from classical to nonclassical behaviour. The concepts of convexity, classicality, and quasiprobability representations are presented as complementary tools for characterising quantumness and defining operational notions such as P-nonclassicality. The discussion extends naturally to Gaussian states, composite systems, and continuous-variable entanglement, highlighting how nonclassicality serves as a resource for generating and quantifying quantum correlations. Theoretical developments are complemented by computational and experimental perspectives, including simulations of optical states using the Python library Strawberry Fields and data analysis from simulated data. Together, these notes aim to bridge the foundational concepts of quantum optics and modern quantum information, offering both conceptual insight and practical tools for students and researchers entering the field.
The concept of coherent states for a quantum system has been generalized in many different ways. One elegant way is the dynamical group approach. The subject of this thesis is the physical application of some dynamical group methods in quantum optics and Bose-Einstein Condensation(BEC) and their use in generalizing some quantum optical states and BEC states. We start by generalizing squeezed coherent states to the displaced squeezed phase number states and studying the signal-to-quantum noise ratio for these states. Following a review of the properties of Kerr states and the basic theory of the deformation of the boson algebra, we present an algebraic approach to Kerr states and generalize them to the squeezed states of the q-parametrized harmonic oscillator. Using the eigenstates of a nonlinear density-dependent annihilation operator of the deformed boson algebra, we propose general time covariant coherent states for any time-independent quantum system. Using the ladder operator approach similar to that of binomial states, we construct interpolating number-coherent states, intermediate states which are generalizations of some fundamental states in quantum optics. Salient statistical properties and non-classical features of these interpolating numbercoherent states are investigated and the interaction with an atomic system in the framework of the Jaynes-Cummings model and the scheme to produce these states are also studied in detail. After briefly reviewing the realization of Bose-Einstein Condensates and relevant theoretical research using mean-field theory, we present a dynamical group approach to Bose-Einstein condensation and the atomic tunnelling between two condensates which interact via a minimal coupling term. First we consider the spectrum of one Bose-Einstein condensate and show that the mean-field dynamics is characterised by the semi-direct product of the 8U(1,1) and Heisenberg-Weyl groups.We then construct a generalized version of the BEC ground states and weakly excited states. It is shown that our states for BEC provide better fits to the experimental results. Then we investigate the tunnelling between the excitations in two condensates which interact via a minimal coupling term. The dynamics of the two interacting Bose systems is characterised by the 80(3,2) group, which leads to an exactly solvable model. Further we describe the dynamics of the tunnelling of the two coupled condensates in terms of the semi-direct product of 80(3,2) and two independent Heisenberg-Weyl groups. From this we obtain the energy spectrum and eigenstates for the two interacting Bose-Einstein condensates, as well as the Josephson current between the two coupled condensates.
A quick review of some Lie algebras related to well-known groups is given. We start with the Heisenberg-Weyl algebra and after the definitions of the Fock states we give the definition of the coherent state of this group. This is followed by the exposition of the SU(2) and SU(1,1) algebras and their coherent states. From there we go on to describe the binomial states and their extensions as the finite dimensional pair coherent states and their nonlinear versions as realizations of the SU(2) group. This is followed by considering the negative binomial states, the single mode and two-mode squeezed states and their variants as realizations of the SU(1,1) group. Generation schemes based on physical systems are considered for some of these states.
In this paper, we define the eigenvalue equation of the squeezed coherent state using recently introduced inverse annihilation and creation operators. The salient feature of the eigenvalue equation of the squeezed coherent state is that it consistently yields the eigenvalue equation of the squeezed vacuum when the coherent amplitude is reduced to zero. The squeezed coherent state defined by others, without using the inverse operators, cannot be reduced to the squeezed vacuum state. The time evolution of the squeezed states is also discussed. The squeezed states have phase-dependent quantum noise. The phase operators have an important role in the squeezing experiments. We show that the inverse operators are also useful in defining the phase operators. We also discuss the eigenvalue equation of the squeezed coherent state defined by Yuen vis-à-vis that is introduced in this paper using inverse annihilation and creation operators for bosons.
We start with the Heisenberg–Weyl algebra and after the definitions of the Fock states we give the definition of the coherent state of this group. This is followed by the exposition of the SU(2) and SU(1, 1) algebras and their coherent states. From there we go on describing the binomial state and its extensions as realizations of the SU(2) group. This is followed by considering the negative binomial states, and some squeezed states as realizations of the SU(1, 1) group. Generation schemes based on physical systems are mentioned for some of these states.
Using Lie algebra methods, we find a new class of Bogoliubov transformations which generalize the notion of squeezed states. The Hamiltonians for the simple harmonic and anharmonic oscillators, turn out to be the generators of a Lie group, whose other generators may be found exactly, or up to any desired order of the perturbation parameter. An element of this Lie group, which is realized as the multi-photon operator, transforms the anharmonic Hamiltonian to the harmonic one. The transformation of the ordinary annihilation and creation operators under this unitary transformation leads to the introduction of multi-photon coherent states. We specifically consider four-photon coherent states in detail and study the time dependent position and momentum uncertainties in these states.