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Register a new userA time-dependent harmonic oscillator with two frequency jumps: an exact algebraic solution
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We consider a harmonic oscillator (HO) with a time dependent frequency which undergoes two successive abrupt changes. By assumption, the HO starts in its fundamental state with frequency omega_{0}, then, at t = 0, its frequency suddenly increases to omega_{1} and, after a finite time interval tau, it comes back to its original value omega_{0}. Contrary to what one could naively think, this problem is a quite non-trivial one. Using algebraic methods we obtain its exact analytical solution and show that at any time t > 0 the HO is in a squeezed state. We compute explicitly the corresponding squeezing parameter (SP) relative to the initial state at an arbitrary instant and show that, surprisingly, it exhibits oscillations after the first frequency jump (from omega_{0} to omega_{1}), remaining constant after the second jump (from omega_{1} back to omega_{0}). We also compute the time evolution of the variance of a quadrature. Last, but not least, we calculate the vacuum (fundamental state) persistence probability amplitude of the HO, as well as its transition probability amplitude for any excited state.
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Using operator ordering techniques based on BCH-like relations of the su(1,1) Lie algebra and a time-splitting approach,we present an alternative method of solving the dynamics of a time-dependent quantum harmonic oscillator for any initial state. We find an iterative analytical solution given by simple recurrence relations that are very well suited for numerical calculations. We use our solution to reproduce and analyse some results from literature in order to prove the usefulness of the method and, based on these references, we discuss efficiency in squeezing, when comparing the parametric resonance modulation and the Janszky-Adam scheme.
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