Fractional Model of Driven Harmonic Oscillator Outside Rotating Wave Approximation
H. A. Batarfi
The fractional derivative form of the model equations of a dissiptive, quantised harmonic oscillator (HO) driven by a laser field are examined within and outside the rotating wave approximation (RWA). Interplay of the two parameters, fraction derivative order, 0 < α < 1 and the fast oscillatory terms outside the RWA, exp(±2iωLt), where ωL is the laser frequency, affect significantly the dynamical behaviour of the average photon number, n(t), of the HO and the second order auto-correlation function g(2)(t). Of particular, in the fractional derivative case (0 < α < 1) as compared with the ordinary derivative case (α = 1) our results show that:
- n(t) shows faster increasing (with initial vacuum state of the HO) and decreasing (with initial number state of the HO) evolution and much reduced oscillator behaviour outside the RWA,
- For initial number state of the HO, and non-resonant weak laser field, anti-bunching (g(2)(t) < 1) shows over longer time period. Super-chaotic behaviour (g(2)(t) > 2) with relatively reduced oscillations exhibited with non-resonant strong laser field, within and outside RWA. For initial coherent state of the HO, g(2)(t) < 1 in the weak field case over longer time period, within and outside the RWA.
Keywords: Harmonic oscillator, fractional derivative, auto-correlation function, rotating wave approximation