Unitary propagators

Here tp denotes the conjugate transpose of ip. Another conserved quantity is the norm of the vector ip, i.e., ip ip = const, due to the unitary propagation of the quantum part. 

In its most general form, a quantum graph is defined in terms of a (finite) graph G together with a unitary propagator U it describes the dynamics of wavefunctions

[Pg.79]

The classical dynamics corresponding to a quantum graph defined by a unitary propagator SG is given by a stochastic process with transition matrix T defined by... 

After the preparation of the initial wave packet, it is then propagated under the operation of system Hamiltonian by the unitary propagator U(/, to) =... 

Each of these operators is unitary U —t) = U t). Updating a time step with the propagator Uf( At)U At)Uf At) yields the velocity-Verlet algorithm. Concatenating the force operator for successive steps yields the leapfrog algorithm ... 

An alternative relax-and-drive procedure can be based on a strictly unitary treatment where the advance from Iq to t is done with a norm-conserving propagation such as provided by the split-operator propagation technique.(49, 50) This however is more laborious, and although it conserves the norm of the density matrix, it is not necessarily more accurate because of possible inaccuracies in the individual (complex) density matrix elements. It can however be used to advantage when the dimension of the density matrix is small and exponentiation of matrices can be easily done.(51, 52)... 

Our wish to study how the quantum mechanical subsystem evolves in time requires that we select appropriate operators, that will provide time propagation of the wave functions. These unitary transformation operators are... 

This demonstrates again the special form of the propagator, a doubly periodic operator times an unitary operator defined by a time independent effective Hamiltonian. 

In this section we have studied the cascaded quadratic processes with an input two-mode coherent state in order to characterize the quantum phase shift. We have assumed the steady-state fields and illustrated this situation by the Deutsch-Garrison technique. To fit in the framework of such a technique, we perform a linearization around a classical solution. Further we have adopted the traditional approach to the propagation. We have determined a -dependent unitary progression operator of the two-mode system in the Schrbdinger picture by direct integration. We have compared the results in the large-mismatch limit with a model of an ideal Kerr-like medium, whose properties are effectively those of the cascaded quadratic nonlinearities. 

By analogy with quantum mechanics, C(t) [c.f. Eq. (11.2.7)] can be regarded as an expectation value of eiLt in the state Wa — Poll2A. Let us recall that in quantum mechanics the Hamiltonian operator H is Hermitian and the operator exp [IHtjh] is unitary. Correspondingly here the Liouvillian L is Hermitian and the propagator... 

As we have already seen, the Liouvillian L is an Hermitian operator and the propagator eiLt is unitary. Likewise since Q is Hermitian, QLQ is Hermitian (QLQ)+ = Q+L+Q+ = (QLQ). It follows that e QLQt is a unitary operator. These properties allow us to prove the following theorems. 

The above propagation of P(Z) is explicitly unitary, which is a main factor contributing to the numerical stability of the SP method. 

Oscillations in time of quantal states are usually much faster than those of the quasiclassical variables. Since both degrees of freedom are coupled, it is not efficient to solve their coupled differential equations by straightforward time step methods. Instead it is necessary to introduce propagation procedures suitable for coupled equations with very different time scales short for quantal states and long for quasiclassical motions. This situation is very similar to the one that arises when electronic and nuclear motions are coupled, in which case the nuclear positions and momenta are the quasiclassical variables, and quantal transitions lead to electronic rearrangement. The following treatment parallels the formulation introduced in our previous review on this subject [13]. Our procedure introduces a unitary transformation at every interval of a time sequence, to create a local interaction picture for propagation over time. 

G. Bom and I. Shavitt, A Unitary Group Formulation of Open-Shell Electron Propagator Theory, J. Chem. Phys. 76, 558-567 (1982). 

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