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S U(3) Squeezing: Quantum and Semiclassical Evolutions

Hossein T. Dinani

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

In this thesis we describe a squeezing for SU(3) systems and compare squeezing in the phase space in the semiclassical approximation with squeezing obtained by the full quantum mechanical calculation. We show that the equations of motion in phase space for SU(3) Wigner function are given in terms of the Poisson bracket plus quantum correction terms which depend on the inverse dimension of the system. In the semiclassical approximation, for large values of the SU(3) representation label , we can ignore the quantum correction terms and use the truncated Liouville equation; squeezing in SU(3) systems is well described by this truncated Liouville equation. Finally, we find some scaling behaviors associated with squeezing in SU(3) and compare these with the corresponding SU(2) calculations.

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In this thesis we describe a squeezing for SU(3) systems and compare squeezing in the phase space in the semiclassical approximation with squeezing obtained by the full quantum mechanical calculation. We show that the equations of motion in phase space for SU(3) Wigner function are given in terms of the Poisson bracket plus quantum correction terms which depend on the inverse dimension of the system. In the semiclassical approximation, for large values of the SU(3) representation label , we can ignore the quantum correction terms and use the truncated Liouville equation; squeezing in SU(3) systems is well described by this truncated Liouville equation. Finally, we find some scaling behaviors associated with squeezing in SU(3) and compare these with the corresponding SU(2) calculations.
Abstract. We establish the difference between the propagation of semiclassical Wigner functions and classical Liouville propagation. First we re-discuss the semiclassical limit for the propagator of Wigner functions, which on its own leads to their classical propagation. Then, via stationary phase evaluation of the full integral evolution equation, using the semiclassical expressions of Wigner functions, we provide the correct geometrical prescription for their semiclassical propagation. This is determined by the classical trajectories of the tips of the chords defined by the initial semiclassical Wigner function and centered on their arguments, in contrast to the Liouville propagation which is determined by the classical trajectories of the arguments themselves. 1. introduction The Wigner [11] function W is a representation of the quantum density operator ˆρ as a real function on phase space, for a non-relativistic dynamical system with classical-quantum correspondence and n degrees of freedom. Here, all the underlying geometry is euclidean and we can include the Wigner function within the general framework of Weyl representation of operators: Â → A ∈ Ck IR (IR2n), so W(x) = (2π�) −n ρ(x) , where x ≡ (p1,...pn, q1,...qn) denotes a point in phase space. In this context, the quantum evolution of the density operator ˆρt effected by an autonomous external hamiltonian
The semiclassical evolution of the generalized SU (2) Wigner function for quantum systems with a variable number of excitations is discussed in detail. It is shown that in the framework of the Liouvillian approach quantum dynamics can be approximately described in terms of classical trajectories on an appropriate four-dimensional symplectic manifold. The analysis of this manifold allows the introduction of a phase-like observable that can be interpreted as a relative phase ‘between’ invariant subspaces.
A general approximation scheme for solving the quantum Liouville equation has been developed and is presented here specifically within the Hartree approximation. This method uses the connection between the density matrix and the Wigner phase-space representations. By use of the Wigner representation the off-diagonal elements of the density matrix are expanded in powers of deviations from the mean momentum value. The time evolution of these deviations is governed by a closed self-consistent set of equations.
We show unexpected behaviour in simulations of generalized squeezing performed with finite-dimensional truncations of the Fock space: even for extremely large dimension of the state space, the results depend on whether the truncation dimension is even or odd. This situation raises the question whether the simulation results are physically meaningful. We demonstrate that, in fact, the two truncation schemes correspond to two well-defined, distinct unitary evolutions whose generators are defined on different subsets of the infinite-dimensional Fock space. This is a consequence of the fact that the generalized squeezing Hamiltonian is not self-adjoint on states with finite excitations, but possesses multiple self-adjoint extensions. Furthermore, we present results on the spectrum of the squeezing Hamiltonians corresponding to even and odd truncation size that elucidate the properties of the two different self-adjoint extensions corresponding to the even and odd truncation scheme. To make the squeezing operator applicable to a physical system, we must regularize it by other terms that depend on the specifics of the experimental implementation. We show that the addition of a Kerr interaction term in the Hamiltonian leads to uniquely converging simulations, with no dependence on the parity of the truncation size, and demonstrate that the Kerr term indeed renders the Hamiltonian self-adjoint and thus physically interpretable.
We generalized the squeeze and displacement operators of the one-dimensional harmonic oscillator to the three-dimensional case and based on these operators we construct the corresponding coherent and squeezed states. We have also calculated the Wigner function for the three-dimensional harmonic oscillator and from the analysis of time evolution of this function, the quantum Liouville equation is also presented. Further properties of the quantum states including Mandel's Q and quadrature squeezing parameters are discussed as well.
Quantum mechanics has been formulated in phase space, with the Wigner function as the representative of the quantum density operator, and classical mechanics has been formulated in Hilbert space, with the Groenewold operator as the representative of the classical Liouville density function. Semiclassical approximations to the quantum evolution of the Wigner function have been defined, enabling the quantum evolution to be approached from a classical starting point. Now analogous semiquantum approximations to the classical evolution of the Groenewold operator are defined, enabling the classical evolution to be approached from a quantum starting point. Simple nonlinear systems with one degree of freedom are considered, whose Hamiltonians are polynomials in the Hamiltonian of the simple harmonic oscillator. The behavior of expectation values of simple observables and of eigenvalues of the Groenewold operator are calculated numerically and compared for the various semiclassical and semiquantum approximations.