Quantum states of the Kapitza pendulum

Oscillatory states of quantum Kapitza pendulum

“…Near the bottom, the potential u(φ) is approximately a harmonic potential, the discrepancy is represented by higher order corrections. The eigenvalue can be computed by the stationary perturbation method [21]. Among others, a quantum number n which takes value in natural numbers, is introduced by matching the leading order eigenvalue with harmonic oscillator eigenvalue.…”

“…The relation (25) leads to the same eigenvalue (21); in fact, the expression ( 21) is invariant under the simultaneous inversions B 1/2 → −B 1/2 and μ → −μ, i.e. µ → −(µ + 1).…”

“…The dimensionless angular frequency is Ω 0 = 2(B + A/2) 1/2 ; the factor 1/2 is related to the zero point energy. When the depth of the well satisfies B ≫ B 1/2 μ, that is μ ≪ B 1/2 , we can treat states with small μ as confined in the well, and the expression (21) gives a good approximation for the eigenvalue.…”

“…In section 3, we compute the eigensolution for the oscillatory states spatially localise around a stable saddle position, by extending the coordinate to the complex plane to utilize a complexified Floquet theorem. The series expansion of eigenvalue is given in (21), and approximate wavefunctions are given in (27) and (36). In particular, the factor 1/2 related to the zero point energy arises naturally.…”

Kapitsa pendulum effects in Josephson junction + nanomagnet under external periodic drive

“…Moreover, it was analytically and experimentally demonstrated that if the oscillation direction of the pendulum suspension point change over time, so does the pendulum equilibrium point and active damping control can take place. The Kapitsa quantum pendulum can be stabilized in the form of quantum states near a local minimum of the effective potential energy [17].…”