A time-dependent harmonic oscillator with two frequency jumps: an exact algebraic solution

Understand

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.