Compacting the Time Evolution of the Forced Morse Oscillator Using Dynamical Symmetries Derived by an Algebraic Wei-Norman Approach

A practical approach is put forward for a compact representation of the time evolving density matrix of the forced Morse oscillator. This approach uses the factorized product form of the unitary time evolution operator, à la Wei-Norman. This product form casts the time evolution operator in the basis of operators that form a closed Lie algebra. The further requirement that the Hamiltonian of the system be closed within this Lie algebra is satisfied by restricting the dynamics to its sudden limit. One is thereby able to propagate in time both pure and mixed quantum states. As an example, for a thermal initial state, the time-evolved density matrix of maximum entropy is derived, and it is compacted to be described by only three explicit constraints: one time-dependent constraint, which is a dynamical symmetry, and two constants of the motion, with corresponding time-independent coefficients. This representation is a significant reduction from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>j</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math> constraints down to just three, where <i>j</i> is the number of bound states of the Morse oscillator.