We investigate many characteristic features of revival and fractional revival phenomena via derived analytic expressions for an autocorrelation function of a one-dimensional Rydberg atom with weighting probabilities modelled by a Gaussian or a Lorentzian distribution. The fractional revival phenomenon in the ionization probabilities of a one-dimensional Rydberg atom irradiated by two short half-cycle pulses is also studied. When many states are involved in the formation of the wave packet, the revival is lower and broader than the initial wave packet and the fractional revivals overlap and disappear with time.
After a Rydberg wave packet forms, it is known to undergo a series of collapses and revivals within a time period called the revival time
trev, at the end of which it resembles its original shape. We study the behavior of Rydberg wave packets on time scales much greater than
trev. We find that after a few revival cycles the wave packet ceases to reform at multiples of the revival time. Instead, a new series of collapses and revivals commences, culminating after a time period
tsr≫trev with the formation of a wave packet that more closely resembles the initial packet than does the full revival at time
trev. Furthermore, at times that are rational fractions of
tsr, we show that the motion of the wave packet is periodic with periodicities that can be expressed as fractions of the revival time
trev. These periodicities indicate a new type of fractional revival, occurring for times much greater than
trev. We also examine the effects of quantum defects and laser detunings on the revival structure of Rydberg wave packets for alkali-metal atoms.
We study the dynamics of superposed wave packets in a specific nonlinear Hamiltonian which models the wave packet propagation in Kerr-like media and the dynamics of Bose–Einstein condensates. We show the dependence of fractional revival times on the initial superposed wave packets using analysis based on the expectation values, Rényi entropy and Wigner function. We also show how the selective identification of fractional revivals using moments of appropriate observables depends on the number of wave packets present in the initial state.
Gazeau-Klauder coherent states of the extended trigonometric Scarf potential, underlying the quadratic energy spectrum and associated with Jacobi-type Xm exceptional orthogonal polynomials P(a,b,m) n (x), are constructed. The temporal evolution of wave-packet coherent states are performed by means of an autocorrelation function and the full revival properties are investigated in the usual time-domain analysis. This latter seems to be less useful for describing the fractional revivals due to the complicated nature of coherent wave-packet. Fortunately the autocorrelation function revels a little signature of fractional revivals at the vicinity of quarters of the revival time Trev due to the quadratic energy spectrum and the use of the wavelet-based time-frequency analysis of the autocorrelation function provides an analytical and numerical observation of the fractional revivals at different orders of the system.
We study revival and fractional revival phenomena of wave packets in a one-dimensional Rydberg atom irradiated by two time-delayed half-cycle pulses using an autocorrelation function characterized by electronic transition probabilities as weighting factors rather than modeling them by a Gaussian or Lorentzian distribution.If the momentum (q2) delivered to the atom by the second kick is much smaller than that (q1) imparted by the first one, the times of revival and fractional revival coincide with those of the single kicked atom.For q2 ≥ q 1 4 appearance of revival and fractional revival depends on both the values of q2 and time delay t1 between the pulses but more sensitively on t1.The number of fractional revivals tends to become numerous as the value of t1 increases.
The general properties of the time-dependent characteristics of one-dimensional vibrational wave packets in an anharmonic system with the quadratic spectrum are discussed. The local behavior of the autocorrelation function and its square modulus (the survival function) is studied near the moments of fractional revival t=(p/q)Trev, where Trev is the period of the complete revival, p/q is an irreducible fraction, and q is the order of the revival. It is shown that all the wave packets have the same values of the survival function at the moments of fractional revival provided that the number of states in the packet exceeds the order of the revival q. It is also found that for the given packet, all the fractional revivals are identical to each other in the sense that their lifetimes do not depend on the order of the revival. The lifetimes of all fractional revivals are expressed in terms of the parameters of the initial packet (the effective number of states and the period of classical motion of the packet center) and the anharmonicity of the molecular system.