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CRC Press eBooks

Parametric excitation and generation of nonclassical states in linear media V V Dodonov

V.V. Dodonov, V.I. Man'ko

Published Mar 13, 2003• 1 citation
Publisher

Abstract

The single-mode case is equivalent to the model of a one-dimensional generalized oscillator with the time-dependent Hamiltonian H(t) = i [fi(t)p2 + v(t)x2 + p(t) [xp + px)] + ci(t)p + C2(t)x, (1) where the quadrature operators x and p satisfy the canonical commutation relation [x,p] = ih. The same Hamiltonian can also be written in terms of the bosonic annihilation and creation operators as H = \h [ D 0 ( a f a + aat) + DYa2 + D*tf2] +h(ga + tfg*) , (2) 154 PARAMETRIC EXCITATION AND GENERATION OF NONCLASSICAL STATES 2N 2N •qB(t)q + C(t)q. (3) (5) PROPAGATORS AND INTEGRALS OF MOTION 155 156 PARAMETRIC EXCITATION AND GENERATION OF NONCLASSICAL STATES SINGLE MODE: QUADRATURE REPRESENTATION 157 Ai = Mp-A2/*, Ai(0) A 2 = A i i / — A2P, A 2(0) A 3 - A 3 p - A4/1, A 3(0) A 4 = A 3z/ - A 4 p , A 4(0) Putting expressions (21) into equation (20) we arrive at the set of first-order ordinary differential equations and initial conditions 1 (22) 0 (23) 0 (24) 1 (25) : 0 (26) : 0 (27) (the dot means the time derivative). An immediate consequence of equations (22)- (25) and the initial conditions is the identity Its physical meaning is very simple: the commutation relations between the operators £ and p are the same as between X and P, [x , p] = [X , P] = ih, i.e., transformation (21) is canonical (and the evolution operator is unitary).

DOI
10.1201/9781482288223-8

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The single-mode case is equivalent to the model of a one-dimensional generalized oscillator with the time-dependent Hamiltonian H(t) = i [fi(t)p2 + v(t)x2 + p(t) [xp + px)] + ci(t)p + C2(t)x, (1) where the quadrature operators x and p satisfy the canonical commutation relation [x,p] = ih. The same Hamiltonian can also be written in terms of the bosonic annihilation and creation operators as H = \h [ D 0 ( a f a + aat) + DYa2 + Dtf2] +h(ga + tfg) , (2) 154 PARAMETRIC EXCITATION AND GENERATION OF NONCLASSICAL STATES 2N 2N •qB(t)q + C(t)q. (3) (5) PROPAGATORS AND INTEGRALS OF MOTION 155 156 PARAMETRIC EXCITATION AND GENERATION OF NONCLASSICAL STATES SINGLE MODE: QUADRATURE REPRESENTATION 157 Ai = Mp-A2/*, Ai(0) A 2 = A i i / — A2P, A 2(0) A 3 - A 3 p - A4/1, A 3(0) A 4 = A 3z/ - A 4 p , A 4(0) Putting expressions (21) into equation (20) we arrive at the set of first-order ordinary differential equations and initial conditions 1 (22) 0 (23) 0 (24) 1 (25) : 0 (26) : 0 (27) (the dot means the time derivative). An immediate consequence of equations (22)- (25) and the initial conditions is the identity Its physical meaning is very simple: the commutation relations between the operators £ and p are the same as between X and P, [x , p] = [X , P] = ih, i.e., transformation (21) is canonical (and the evolution operator is unitary).
The statistical properties of systems of coupled quantum-mechanical harmonic oscillators are analyzed. The Hamiltonian for the system is assumed to be an inhomogeneous quadratic form in the creation and annihilation operators, and is allowed to have an explicit time dependence. The relationship to classical theory is emphasized by expressing pure states in terms of the coherent-state vectors, and density operators by means of the PPP representation and an analogous representation involving the Wigner function. The state which evolves from an initially coherent state of the system is found, and equations governing the time evolution of the Wigner function and the weight function for the PPP representation are derived, in differential and integral form, for arbitrary initial states of the system. The results remain valid for couplings which do not preserve the vacuum state, and for cases in which the time dependence of the coupling parameters gives rise to large-scale amplification of the initial field intensities. The analysis is performed by first treating general linear inhomogeneous canonical transformations on the oscillator variables, and then specializing to the case in which these transformations represent the solutions for the Heisenberg operators in terms of their initial values. The results are illustrated within the context of a model of parametric amplification.
This thesis is concerned with the properties of dynamical systems with Hamiltonians which are general quadratic combinations of classical or quantum canonical variables or of Boson or Fermion annihilation and creation operators.Chapter I, consisting of six sections, deals mainly with sets of canonical forms for quadratic Hamiltonians under the action of the group of linear canonical transformations.By viewing the group of C.C.R.Bogoliubov transformations as Sp(2tf, i?) , a full set of canonical Boson Hamiltonians, labelled by the invariants for symplectic conjugacy classes, is obtained.Since the conjugacy class of a real orthogonal matrix is determined by its eigenvalues, there is only one type of canonical form for quadratic Fermion Hamiltonians.Unlike the Fermion case, indefinite Boson Hamiltonians can not in general be reduced to a sum of quasi-particle number operators.However, it is shown how spectral theory in Krein space rigorizes the full reduction of a strictly positive quadratic Boson Hamiltonian for which the single particle space is infinite dimensional.In the case of infinite degrees of freedom, there are various existence criteria for a unitary operator which implements a given Bogoliubov transformation and the equivalence of these is discussed in Section 6.Cahpter II, consisting of seven sections, applies the techniques of Chapter I to algebraic quantization.The Boson commutation relations , among formal mode operators obtained in heuristic quantization, are shown to be incompatible with the canonical commutation relations among conjugate variables unless the motion is stable.Segal quantization of any real
A simple prescription for quantizing fields is developed and applied to the sytsem of interacting spins in an electromagnetic field. The classical equations of motion are written in the standard form $\ensuremath{-}i(\frac{d}{\mathrm{dt}})|V〉+\mathfrak{M}|V〉=0$, where the state vector ∣V〉|V〉∣V〉 has the state variables, such as the generalized coordinates and momenta, as its components, and the time development operator M\mathfrak{M}M contains derivative, integral, and matrix operators in general. By diagonalizing M\mathfrak{M}M and replacing the resulting classical variables with operators obeying boson commutation relations, the system is quantized, and the Hamiltonian and Heisenberg equations of motion are automatically cast into the harmonic-oscillator form. Thus the quantization is reduced to the simple problem of diagonalizing a matrix. This avoids the laborious process of finding a Lagrangian density (by trial and error in general), determining the canonical variables, forming the Hamiltonian density, and inventing transformations to creation and annihilation operators. Problems with zero-frequency modes, which can be annoying in the standard treatment, are automatically eliminated by the prescription. The bookkeeping of forward and backward waves, different branches of dispersion curves, etc., is simplified. In addition to affording a straightforward and simple method of quantizing fields, the method also gives the necessary and sufficient conditions for a Hamiltonian to be transformable into harmonic-oscillator form, gives a direct relation between the classical equations of motion and the Heisenberg quantum-mechanical equations, and gives a simple method of determining the constants of the motion.