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Advances in 3OM: Opto-Mechatronics, Opto-Mechanics, and Optical Metrology (3OM)

Quantum Kapitza pendulum

Pavel A. Golovinski, Vladimir A. Dubinkin

Published May 6, 2022• 0 citations
Publisher

Abstract

The quantum states of the Kapitza pendulum are investigated in the framework of the effective potential obtained by the method of averaging over fast oscillations. An analytical estimate of the energy spectrum of stabilized states is given using a model potential. For the lowest states of the inverted pendulum, an expression is obtained for the spectrum in the form of the energies of a harmonic oscillator, refined according to the perturbation theory. Tunneling corrections to the energies of resonance states in double-well effective potential are found. The results of calculations of the structure of vibrational and rotational spectra of the Kapitza pendulum by the semiclassical method and by the numerical Numerov’s algorithm are compared.

DOI
10.1117/12.2597725

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The quantum states of the Kapitza pendulum are investigated in the framework of the effective potential obtained by the method of averaging over fast oscillations. An analytical estimate of the energy spectrum of stabilized states is given using a model potential. For the lowest states of the inverted pendulum, an expression is obtained for the spectrum in the form of the energies of a harmonic oscillator, refined according to the perturbation theory. Tunneling corrections to the energies of resonance states in double-well effective potential are found. The results of calculations of the structure of vibrational and rotational spectra of the Kapitza pendulum by the semiclassical method and by the numerical Numerov’s algorithm are compared.
The quantum states of the Kapitza pendulum are investigated within the framework of the effective potential obtained by the method of averaging over fast oscillations. An analytical estimate of the energy spectrum of stabilized states is given using a model potential. For the lowest states of the inverted pendulum, an expression is obtained for the spectrum in the form of the energies of a harmonic oscillator, refined according to the perturbation theory. Tunneling corrections to the energies of resonance states in two wells of the effective potential are found. The results of calculations of the structure of vibrational and rotational spectra of the Kapitza pendulum by the semiclassical method and by Numerov’s numerical algorithm are compared.
We explore analytically the quantum dynamics of a point mass pendulum using the Heisenberg equation of motion. Choosing as variables the mean position of the pendulum, a suitably defined generalised variance and a generalised skewness, we set up a dynamical system which reproduces the correct limits of simple harmonic oscillator like and free rotor like behaviour. We then find the unexpected result that the quantum pendulum released from and near the inverted position executes oscillatory motion around the classically unstable position provided the initial wave packet has a variance much greater than the variance of the well known coherent state of the simple harmonic oscillator. The behaviour of the dynamical system for the quantum pendulum is a higher dimensional analogue of the behaviour of the Kapitza pendulum where the point of support is vibrated vertically with a frequency higher than the critical value needed to stabilize the inverted position. A somewhat similar phenomenon has recently been observed in the non equilibrium dynamics of a spin - 1 Bose-Einstein Condensate.
A semianalytical approach to the quantum dynamics of a plane pendulum is developed, based on Mathieu functions which appear as stationary wave functions. The time-dependent Schr\"odinger equation is solved for pendular analogs of coherent and squeezed states of a harmonic oscillator, induced by instantaneous changes of the periodic potential energy function. Coherent pendular states are discussed between the harmonic limit for small displacements and the inverted pendulum limit, while squeezed pendular states are shown to interpolate between vibrational and free rotational motion. In the latter case, full and fractional revivals as well as spatiotemporal structures in the time evolution of the probability densities (quantum carpets) are quantitatively analyzed. Corresponding expressions for the mean orientation are derived in terms of Mathieu functions in time. For periodic double well potentials, different revival schemes, and different quantum carpets are found for the even and odd initial states forming the ground tunneling doublet. Time evolution of the mean alignment allows the separation of states with different parity. Implications for external (rotational) and internal (torsional) motion of molecules induced by intense laser fields are discussed.
Comparative analysis of three stabilization mechanisms of unstable states of physical systems is presented in this review. These mechanisms are: the quantum Zeno effect, the stabilization of unstable states in an external fast oscillating field (at the example of Kapitza pendulum), and the algorithm called as the principle of whirligig. The common features of these mechanisms, as well as the differences between them, are defined in the paper. In particular, it is shown that the stabilization of quantum systems is possible without involvement of such a notion as the collapse of the wave function. For stabilization there is enough to have such a stabilizing radiation flow when the Rabi frequency of transitions does exceed some frequency. This particular frequency is a magnitude inverse-ly proportional to the lifetime of the state under stabilization. It is shown that stabilization of unstable states by impact of rapidly oscillating forces occurs by non-self-consistent exposure, i.e. the dynamics of stabilizing field is independent of the dynamics of the stabilized state. Stabilization with the principle of whirligig does occur by self-consistent forces, and thus, in many cases stabilization by the use of the principle of whirligig can be more effective.
The classical and quantum dynamics in a high frequency field are found to be described by an effective time independent Hamiltonian. It is calculated in a systematic expansion in the inverse of the frequency (omega) to order omega(-4). The work is an extension of the classical result for the Kapitza pendulum, which was calculated in the past to order omega(-2). The analysis makes use of an implementation of the method of separation of time scales and of a quantum gauge transformation in the framework of Floquet theory. The effective time independent Hamiltonian enables one to explore the dynamics in the presence of rapidly oscillating fields, in the framework of theories that were developed for systems with time independent Hamiltonians. The results are relevant, in particular, for exploring the dynamics of cold atoms.