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Journal of Physics A Mathematical and General

Eigenvalue equation of squeezed coherent states and time evolution of squeezed states

G. M. Saxena

Published Oct 8, 2002• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/35/42/306

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Inverses of the harmonic oscillator creation and annihilation operators by their actions on the number states are introduced. Three of the two photon annihilation operators, viz., a(sup +/-1)a, aa(sup +/-1), and a(sup 2), have normalizable right eigenstates with nonvanishing eigenvalues. The eigenvalue equation of these operators are discussed and their normalized eigenstates are obtained. The Fock state representation in each case separates into two sets of states, one involving only the even number states while the other involving only the odd number states. It is shown that the even set of eigenstates of the operator a(sup +/-1)a is the customary squeezed vacuum S(sigma) O greater than.
Abstract We introduce a pseudo-squeezed bosonic ladder operator defined as a time-dependent non-Hermitian linear invariant and related to their adjoint operators via the bounded Hermitian invertible operator or metric operator. In fact, they are obtained from the squeezed transformation of the pseudo-bosons annihilation and creation operators. Thus, the pseudo-bosonic squeezed coherent states are just obtained as pseudo-displacement operator method acting on the ground pseudo-squeezed state. To our knowledge, the time-dependent pseudo-squeezed coherent states have not been constructed until now. As an illustration, we study the time-dependent non-Hermitian displaced harmonic oscillator, and the properties of these states are analysed with respect to the localization in position and to uncertainty principle.
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.
Making use of the recently introduced inverses of boson creation and annihilation operators, the authors show that the 'photon-added coherent states' are eigenstates of the operator (a Dagger m aa Dagger -m ) and also of the operator (a-ma Dagger -1 ) with eigenvalue alpha . The authors discuss completeness of these states. They obtain another family of photon-added coherent states which are proportional to a -m mod alpha ), where m is a positive integer and mod alpha ) is a coherent state. They show that these are the eigenstates of the operator (a -m a m+1 ) with eigenvalue alpha . Analogously, the authors introduce 'photon-depleted coherent states', a Dagger -m mod alpha ), and show that these are the eigenstates of the operator (a+ma Dagger -1 ) with eigenvalue alpha . The conditions for squeezing and antibunching of photons have also been discussed.