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A squeezed state holomorphic phase space representation of equations of motion

D. Galetti, J.S. da Veiga

Published Aug 1, 1992• 0 citations
OpenAlex

Abstract

A mapping scheme is presented which takes quantum operators associated to bosonic degrees of freedom into complex phase space integral kernel representatives. The procedure consists of using the Schrödinger squeezed state as the starting point for the construction of the integral mapping kernel which, due to its inherent structure, is suited for the description of second quantized operators. Products and commutators of operators have their representatives explicitly written which reveal new details when compared to the usual q-p phase space description. The classical limit of the equations of motion for the canonical pair q-p is discussed in connection with the effect of squeezing the quantum phase space cellular structure.

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A mapping scheme is presented which takes quantum operators associated to bosonic degrees of freedom into complex phase space integral kernel representatives. The procedure consists of using the Schrödinger squeezed state as the starting point for the construction of the integral mapping kernel which, due to its inherent structure, is suited for the description of second quantized operators. Products and commutators of operators have their representatives explicitly written which reveal new details when compared to the usual q-p phase space description. The classical limit of the equations of motion for the canonical pair q-p is discussed in connection with the effect of squeezing the quantum phase space cellular structure.
Classical linear transformations in coordinate-momentum phase space are mapped to unitary quantum-mechanical operators in Hilbert space to produce a generalization of squeeze operators. The classical to quantum transition is manifestly apparent in the derivation. The unitary operators are evaluated in the coherent-state representation using the ``integration within ordered products'' technique in the one- and two-mode cases. Application of these operators results in a new generalization of coherent states.
Abstract The problem of quantizing properly the canonical pair “angle and action variables”, φ and I, is almost as old as quantum mechanics itself and since decades an intensively debated but still unresolved issue in quantum optics. The present paper proposes a new approach to the problem, namely quantization in terms of the group SO(1,2): The crucial point is that the phase space has the global structure (a simple cone) and cannot be quantized in the conventional manner. As the group SO(1,2) acts transitively, effectively and Hamilton‐like on that space its irreducible unitary representations of the positive discrete series provide the appropriate quantum theoretical framework. The phase space has the conic structure of an orbifold . That structure is closely related to a Z2 gauge symmetry which corresponds to the center of a 2‐fold covering of SO(1,2), the symplectic group . The basic variables on the phase space are the functions h0 = I, h1 = I cos φ and h2 = ‐I sin φ the Poisson brackets of which obey the Lie algebra . In the quantum theory they are represented by the self‐adjoint Lie algebra generators K0, K1, and K2 of a unitary representation, where K0 has the spectrum {k + n, n = 0, 1, …; k > 0}. A crucial prediction is that the classical Pythagorean relation h12 + h22 = h02 can be violated in the quantum theory. For each representation one can define three different types of coherent states the complex phases of which may be “measured” by means of the operators K1 and K2 alone without introducing any new phase operators! The SO(1,2) structure of optical squeezing and interference properties as well as that of the harmonic oscillator are analyzed in detail. The additional coherent states can be used for the introduction of (Husimi type) “Q” distributions and (Sudarshan‐Glauber type) “P” representations of the density operator. The three operators K0, K1, and K2 are fundamental in the sense that one can construct composite position and momentum operators out of them! The new framework poses quite a number of fascinating experimental and theoretical challenges.
The continuous-time regularization scheme for defining phase-space path integrals is briefly reviewed as a method to define a quantization procedure that is completely covariant under all smooth canonical coordinate transformations. As an illustration of this method, a limited set of transformations is discussed that have an image in the set of the usual squeezed states. It is noteworthy that even this limited set of transformations offers new possibilities for stationary phase approximations to quantum mechanical propagators.
The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born's rule and recently suggested method of dequantizer-quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical-like equations for the probability distributions determining the quantum system states. Relations to phase-space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.
Motivated by some well-known results in the phase space description of quantum optics and quantum information theory, we aim to describe the formalism of quantum field theory by explicitly considering the holomorphic representation for a scalar field within the deformation quantization program. Notably, the symbol of a symmetric ordered operator in terms of holomorphic variables may be straightforwardly obtained by the quantum field analogue of the Husimi distribution associated with a normal ordered operator. This relation also allows to establish a [Formula: see text]-equivalence between the Moyal and the normal star-products. In addition, by writing the density operator in terms of coherent states we are able to directly introduce a series representation of the Wigner functional distribution, which may be convenient in order to calculate probability distributions of quantum field observables without performing formal phase space integrals at all.