Infinitesimal holomorphic realizations for the Schrödinger-Weil representation and the discrete series representations of the Jacobi group are constructed. Explicit expressions of the basic differential operators are obtained. The squeezed states for the unitary irreducible representation of the Jacobi group are introduced. Matrix elements of the squeezed operators, expectation values of polynomial operators in infinitesimal generators of the Jacobi group, the squeezing region and a description of Mandel's parameter are presented.
The Jacobi group the semidirect product of the Heisenberg group and the symplectic group [1–3] is an important object in quantum mechanics, geometric quantization, optics [4–6]. The Jacobi group is responsible for the squeezed states [7–12] in quantum optics [5], [6], [13], [14]. Some coherent state systems based on SiegelJacobi domains [15], [16] have been investigated in the framework of quantum mechanics, geometric quantization, dequantization, quantum optics, nuclear structure, and signal processing [17], [18], [19], [20][23]. We have attached in [20] Perelomov’s coherent states [24] to the Jacobi group G1∗= H1(R)nSU(1,1), based on the Siegel-Jacobi disk D1 =C×D1, where D1 = {z ∈ C||z| . ez ∈ H denotes the Perelomov coherent state vector indexed by the points z of the homogeneous manifold M = G/H , where H is compact subgroup of G. The generalized squeezed states of the type introduced in [22], [23], [25] are not new [8, 26, 27]. The paper is organized as follows. §2 collects the main notation concerning
When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation
T of a real algebraic Lie group
G. This requires defining an inner product on the Hilbert space
mathbbH that carries the representation
T. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of
T, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet (
mathfrakFmathbbHD,
mathbbHD,
mathfrakFmathfrakHD) of Bargmann spaces. Coherent state wave functions -- the elements of
mathfrakFmathbbHD and of
mathfrakFmathbbHD -- are used to define the inner product on
mathbbHD in a way that simplifies the calculation of the overlaps. This inner product and the group action
Gamma of
G on
mathfrakFmathbbHD are used to formulate expressions for the matrix elements of
T with coefficients from the given subrepresentation.\n\nThe process of finding an explicit expression for
Gamma relies on matrix factorizations in the complexification of
G even though the representation
T does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and
Gamma action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in
mathbbH. The scalar and vector coherent state methods will both be applied to Sp(
n) and Sp(
n,mathbbR).
The relevant structure of a Bose Fock space using the notion of free commutative algebra with unit element and with a scalar product is incorporated. The squeezed states are then the exponentials of quadratic forms and can be written in the normal form of a general squeeze group. The connection of this group to polarization of the initial Hilbert space is established and its infinitesimal generator is computed. In this way, a common denominator for a series of papers is provided in the topic of light squeezing and last but not least a mathematical formalism is provided that may prove convenient in the general treatment of quantum optics.