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International Journal of Theoretical Physics

Parametric Amplifier Studied as Time Evolution of Angular Momentum System in Entangled State Representation

Xing-Lei Xu, Shi-Min Xu, Hongqi Li, Hong-Yi Fan

Published Feb 13, 2015• 2 citations
Publisher
DOI
10.1007/s10773-015-2566-6

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We study how can an angular momentum coherent state ∣ τ 〉 keeps its form-invariant during time evolution governed by the Hamiltonian H = f ( t ) J + + f * ( t ) J − + g ( t ) J z . We discuss this topic in the context of boson realization of ∣ τ 〉 . By employing the entangled state representation ∣ ζ 〉 and deriving a new binomial theorem involving two-subscript Hermite polynomials, we derive the wave function 〈 ζ ∣ τ 〉 , which turns out to be a single-subscript Hermite polynomial. Based on this result the maintenance of angular momentum coherent state during time evolution is examined, and the value of τ ( t ) is totally determined by the parameters involved in the Hamiltonian.
The probability representation of angular momentum states and the connection of the representation with the formalism of the star-product quantization procedure are reviewed. The Schrödinger equation of a general time-dependent Hamiltonian, linear in angular momentum operators, and the evolution equation for the density operator in the probability representation are solved analytically by means of the formalism of linear time-dependent constants of motion. These analytical solutions define wavefunctions and tomograms of states (generic Dicke states), which contain the atomic coherent states as a particular case. The statistical properties of these new states are also evaluated. General forms of analytically solvable Hamiltonians are established in terms of the Euler angle parametrization of the three-dimensional rotations.