On quantum analogue of dynamical stabilisation of inverted harmonic oscillator by time periodical uniform field

Abstract

Quantum analogue of stabilised forced oscillations around an unstable equilibrium position is explored by solving the non-stationary Schrödinger equation (NSE) of the inverted harmonic oscillator (IHO) driven periodically by spatial uniform field of frequency $\Omega$, amplitude $F_{0}$ and phase $\phi$, i.e. the system with the Hamiltonian of $\hat{{H}}=(\hat{{p}}^{2}/2m)-(m\omega ^{2}x^{2}/2)-F_0 x\sin$ $\left( {\Omega t+\phi } \right)$. The NSE has been solved both analytically and numerically by Maple 15 in dimensionless variables $\xi = x\sqrt{m\omega /\hbar }\hbox {, }f_0 =F_0 /\omega \sqrt{\hbar m\omega }$ and $\tau =\omega t$. The initial condition (IC) has been specified by the wave function (w.f.) of a generalised Gaussian type which suits well the corresponding quantum IC operator. The solution obtained demonstrates the non-monotonous behaviour of the coordinate spreading $\sigma \left( \tau \right) \hbox { =}\sqrt{\big ( {\overline{\Delta \xi ^{2}\big ( \tau \big )} } \big )}$ which decreases first from quite macroscopic values of $\sigma _{0} =2^{12,\ldots ,25}$ to minimal one of $\sim \!(1/\sqrt{2})$ at times $\tau <\tau _0 =0.125\ln \!\left( {16\sigma _0^4 +1} \right)$ and then grows back unlimitedly. For certain phases $\phi$ depending on the $\Omega /\omega$ ratio and $n=\log _2\!\sigma _0$, the mass centre of the packet $\xi _{\mathrm {av}}( \tau )= \overline{\hat{{x}}(\tau )} \cdot \sqrt{m\omega /\hbar }$ delays approximately two natural ‘periods’ $\sim \!(4\pi /\omega )$ in the area of the stationary point and then escapes to ‘$+$’ or ‘−’ infinity in a bifurcating way.  For ‘resonant’ $\Omega =\omega$, the bifurcation phases $\phi$ fit well with the regression formula of Fermi–Dirac type of argument n with their asymptotic $\phi ( {\Omega ,n\rightarrow \infty } )$ obeying the classical formula $\phi _{\mathrm {cl}} ( \Omega )=-\hbox {arctg} \, \Omega$ for initial energy $E = 0$ in the wide range of $\Omega =2^{-4},...,2^{7}$.