The time evolution of the rotational distribution of diatomic molecules induced by a train of broadband optical pulses can be modeled as a class of continuous-time quantum walk (CTQW) on a half line. As the transition moments asymptotically decrease to zero like Gaussian distribution, the rotational distribution cannot reach infinity far from the initial state and exhibits a localization-like behavior where the upper limit of the distribution is not trivial. We investigated a time evolution of the density distribution with the transition moments of the Gaussian distribution peaked at the boundary of the half line for the initial state of J =0. Even though the time evolution exhibited oscillatory motion inside a certain region repeatedly, we observed a clear time-averaged distribution that is well fitted to the stretched exponential distribution with parameters close to the Gaussian distribution by numerical simulation. By varying the coefficients of the Gaussian distribution of the transition moments, the coefficients of the stretched exponential distribution can be determined by regression.
We present a detailed analysis of continuous time quantum walks (CTQW) with both position and transition defects defined at a single point in the line. Analytical solutions of both traveling waves and bound states are obtained, which provide valuable insight into the dynamics of CTQW. The number of bound states is found to be critically dependent on the defect parameters, and the localized probability peaks can be readily obtained by projecting the state vector of CTQW on to these bound states. The interference between two bound states are also observed in the case of a transition defect. The spreading of CTQW probability over the line can be finely tuned by varying the position and transition defect parameters, offering the possibility of precision quantum control of the system.
We consider a discrete-time quantum walk, called the Grover walk, on a distance regular graph
X. Given that
X has diameter
d and invertible adjacency matrix, we show that the square of the transition matrix of the Grover walk on
X is a product of at most
d commuting transition matrices of continuous-time quantum walks, each on some distance digraph of the line digraph of
X. We also obtain a similar factorization for any graph
X in a Bose Mesner algebra.
Continuous-time quantum walks on a lattice spread ballistically and converge to a limiting distribution for the rescaled position. On the hard-wall half line the boundary reflects the walker but does not change the bulk dispersion, so the ballistic front remains set by the maximal group velocity. We ask how close one can get to this front when the initial state is constrained to occupy only the first
M sites. We show that optimizing the moment-generating function of the limiting-velocity distribution over this finite-support class reduces to the principal eigenvalue of an explicit
M×M Hermitian matrix, which yields the optimal state by direct diagonalization. For large
M, a near-front scaling limit produces a continuum description that controls the entire peak region. In particular, the maximal mean drift approaches the front with an inverse-square deficit in
M and a constant prefactor
π2/4. Numerical results verify both the finite-
M spectral formulation and the predicted scaling.