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Journal of Mathematical Physics

Generalized intelligent states and squeezing

D. A. Trifonov

Published May 1, 1994• 100 citations
Publisher

Abstract

The Robertson–Schrödinger uncertainty relation for two observables A and B is shown to be minimized in the eigenstates of the operator λA+iB, λ being a complex number. Such states, called generalized intelligent states (GIS), can exhibit arbitrarily strong squeezing of A or B. The time evolution of GIS is stable for Hamiltonians which admit linear in A and B invariants. Systems of GIS for the SU(1,1) and SU(2) groups are constructed and discussed. It is shown that SU(1,1) GIS contain all the Perelomov coherent states (CS) and the Barut and Girardello CS while the spin CS are a subset of SU(2) GIS. CS for an arbitrary semisimple Lie group can be considered as a GIS for the quadratures of the Weyl generators.

DOI
10.1063/1.530553

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The Robertson–Schrödinger uncertainty relation for two observables A and B is shown to be minimized in the eigenstates of the operator λA+iB, λ being a complex number. Such states, called generalized intelligent states (GIS), can exhibit arbitrarily strong squeezing of A or B. The time evolution of GIS is stable for Hamiltonians which admit linear in A and B invariants. Systems of GIS for the SU(1,1) and SU(2) groups are constructed and discussed. It is shown that SU(1,1) GIS contain all the Perelomov coherent states (CS) and the Barut and Girardello CS while the spin CS are a subset of SU(2) GIS. CS for an arbitrary semisimple Lie group can be considered as a GIS for the quadratures of the Weyl generators.
A sufficient condition for a state |ψ> to minimize the Robertson-Schrödinger uncertainty relation for two observables A and B is obtained which for A with no discrete spectrum is also a necessary one. Such states, called generalized intelligent states (GIS), exhibit arbitrarily strong squeezing (after Eberly) of A and B. Systems of GIS for the SU(1,1) and SU(2) groups are constructed and discussed. It is shown that SU(1,1) GIS contain all the Perelomov coherent states (CS) and the Barut and Girardello CS while the Bloch CS are subset of SU(2) GIS.
The diagonalization of the uncertainty matrix and the minimization of Robertson inequality for n observables are considered. It is proved that for even n this relation is minimized in states which are eigenstates of n/2 independent complex linear combinations of the observables. In the case of canonical observables, this eigenvalue condition is also necessary. Such minimizing states are called Robertson intelligent states (RIS). The group-related coherent states (CS) with maximal symmetry (for semisimple Lie groups) are a particular case of RIS for the quadratures of Weyl generators. Explicit constructions of RIS are considered for operators of su(1,1), su(2), and sp(N,R) algebras. Unlike the group-related CS, RIS can exhibit strong squeezing of group generators. Multimode squared amplitude squeezed states are naturally introduced as sp(N,R) RIS. It is shown that the uncertainty matrices for quadratures of q-deformed boson operators (q>0) and of any k power of are positive definite and can be diagonalized by symplectic linear transformations.
Three basic properties (eigenstate, orbit and intelligence) of the canonical squeezed states (SS) are extended to the case of n arbitrary observables. The SS for n observables X i can be constructed as eigenstates of their linear complex combinations or as states which minimize the Robertson uncertainty relation. When X i close a Lie algebra L the generalized SS could also be introduced as orbit of Aut ( L C ). It is shown that for the nilpotent algebra h N the three generalizations are equivalent. For the simple su (1, 1) the family of eigenstates of uK - + vK + ( K ± being lowering and raising operators) is a family of ideal K 1 – K 2 SS, but it cannot be represented as an Aut ( su C (1, 1) orbit although the SU (1, 1) group related coherent states (CS) with symmetry are contained in it. Eigenstates | z, u, v, w ; k ⟩ of general combination of uK - + vK + + wK 3 the three generators K j of SU (1, 1) in the representations with Bargman index k = 1/2, 1, ... , and k = 1/4, 3/4 are constructed and discussed in greater detail. These are ideal SS for K 1,2,3 . In the case of the one mode realization of su (1, 1) the nonclassical properties (sub-Poissonian statistics, quadrature squeezing) of the generalized even CS | z, u, v ; + ⟩ are demonstrated. The states | z, u, v, w ; k = 1/4, 3/4 ⟩ can exhibit strong both linear and quadratic squeezing.
A complete set of solutions |z,u,v>_{sa} of the eigenvalue equation (ua^2+va^{dagger 2})|z,u,v> = z|z,u,v> ([a,a^{dagger}]=1) are constructed and discussed. These and only these states minimize the Schrödinger uncertainty inequality for the squared amplitude (s.a.) quadratures. Some general properties of Schrödinger intelligent states (SIS) |z,u,v> for any two observables X, Y are discussed, the sets of even and odd s.a. SIS |z,u,v;+,-> being studied in greater detail. The set of s.a. SIS contain all even and odd coherent states (CS) of Dodonov, Malkin and Man'ko, the Perelomov SU(1,1) CS and the squeezed Hermite polynomial states of Bergou, Hillery and Yu. The even and odd SIS can exhibit very strong both linear and quadratic squeezing (even simultaneously) and super- and subpoissonian statistics as well. A simple sufficient condition for superpoissonian statistics is obtained and the diagonalization of the amplitude and s.a. uncertainty matrices in any pure or mixed state by linear canonical transformations is proven.