Forward propagation

It can be seen from the algorithm model stated above that in the Krotov method the electric field obtained in the feth iteration is used immediately to propagate /(f), which has a direct contribution to the new electric field in the next time step. In one iteration, the Krotov method involves three wave packet propagations, that is, the forward propagations of and x (t) in Steps 8.3... 

The Ss take care of the Boltzmann factor in first order, i.e. 8 = (exp (ss/lkBT) - l) /6 es/l2kBT. The off-diagonal elements indicate the values for forward propagation, whereas the diagonal stands for the back step probabilities. The zeros indicate transitions that are not possible for geometrical reasons (the ice-bias of all four bonds would have to be inverted), i.e. successive visits of two T3 or two Tl are needed to perform such a transition. It is... 

This is an exact system of equations that describes the evolution of modal amplitudes along the z-axis for the forward propagating field. A similar equation holds for the backward propagating component, of course. 

In other words, to obtain a closed system to solve numerically, we must require that the nonlinear polarization is well approximated by the nonlinear polarization calculated only from the forward propagating field. This means that the equation is only applicable when the back-reflected portion of the field is so small that its contribution to the nonlinearity can be neglected. 

We next introduce a discretized path integral representation for the nuclear part of the propagators, and choose to do so in a hybrid momentum-coordinate representation [18,41]. This can be accomplished, for example for the forward propagator, by first using the identity... 

Other than the Ornstein-Uhlenbeck process, the remaining piece of the splitting corresponds to Newtonian constant-energy (microcanonical) Hamiltonian dynamics. Since Hamiltonian dynamics leaves invariant any function of the energy, its corresponding Fokker-Planck operator (in this case the Liouvillian, = — //) will preserve distributions that are functions of the Hamiltonian H. This implies in particular that it preserves pp, which is proportional to exp(— 6//). Thus the forward propagator associated to the Hamiltonian system automatically preserves the Gibbs distribution, and we have... 

Solving this equation for 100°C (373 K) predicts that log k is -16.1 k = 7.94 x 10sec ). The uncertainty in the log k value is found by forward propagation of the standard errors associated with the fitting parameters in Eq. (2.44). 

P + o) (curve 4). The dashed (1,2) and solid (3, 4) lines correspond to back and forward propagating waves, respectively, because their slopes corresponding to group velocities c/co/c/p have different signs. The crossover of lines 2 and 3 at P = 0 and k = qo determines the Bragg frequency cdb < 2d> = qo = 2ti/Po-... 

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