Properties of the Geometric Phase in Electromechanical Oscillations of Carbon-Nanotube-Based Nanowire Resonators

Abstract

The geometric phase is an extra phase evolution in the wave function of vibrations that is potentially applicable in a broad range of science and technology. The characteristics of the geometric phase in the squeezed state for a carbon-nanotube-based nanowire resonator have been investigated by means of the invariant operator method. The introduction of a linear invariant operator, which is useful for treating a complicated time-dependent Hamiltonian system, enabled us to derive the analytical formula of the geometric phase. By making use of this, we have analyzed the time behavior of the geometric phase based on relevant illustrations. The influence of squeezing parameters on the evolution of the geometric phase has been investigated. The geometric phase, in large, oscillates, and the envelope of such oscillation increases over time. The rate of the increase of the geometric phase is large when the parameters, such as the classical amplitude of the oscillation, the damping factor, and the amplitude of the driving force, are large. We have confirmed a very sharp increase of the geometric phase over time in the case that the angular frequency of the system reaches near the resonance angular frequency. Our development regarding the characteristics of the geometric phase is crucial for understanding the topological features in nanowire oscillations.

Analysis of the geometric phase for a nanowire-bridged superconducting Fabry-Perot resonator

“…(4). Using the Liouville-von Neumann equation for an invariant , which is given by , we derive a linear invariant operator as 36 where is an annihilation operator of the system and : The exact formula of is given bywhere C = (2 ℏL T Ω) −1/2 , and Q p ( t ) and P p ( t ) are particular solutions of the classical equation of motion of the system in q and p spaces, respectively. From Eq.…”

Classical Limit of Quantum Mechanics for Damped Driven Oscillatory Systems: Quantum–Classical Correspondence

“…To investigate quantum-classical correspondence, I consider a damped driven harmonic oscillator of mass m and frequency ω 0 , whose Hamiltonian is given by [13][14][15][16]]…”

“…4 in the text (see Ref. 13). Notice that the Hermitian adjoint of this operator, I † , is also an invariant operator.…”

“…To deduce Newtonian mechanics from quantum one along this line, canonical quantum mechanics with fundamental Hamiltonian dynamics will be used. My theory is based on an invariant operator method [10][11][12][13] which is generally employed for mathematically treating quantum mechanical systems. This method enables us to derive exact quantum mechanical solutions for time-varying Hamiltonian systems.…”

Classical limit of quantum mechanics for damped driven oscillatory systems: Quantum-classical correspondence

“…The purpose of this research is to establish a theoretical formalism concerning the classical limit of quantum mechanics for damped driven oscillatory systems, which reveals quantum and classical correspondence, without any approximation or assumption except for the fundamental limitation h → 0. Our theory is based on an invariant operator method [10][11][12][13] which is generally used for mathematically treating quantum mechanical systems. This method enables us to derive exact quantum mechanical solutions for time-varying Hamiltonian systems.…”