Survival amplitudes

As we discussed in Section II in relation to (2.41), a survival amplitude has a semiclassical behavior that is directly related to the periodic orbits by the Gutzwiller or the Berry-Tabor trace formulas, in contrast to the quasi-classical quantities (2.42) or (3.3). Therefore, we may expect the function (3.7) to present peaks on the intermediate time scale that are related to the classical periodic orbits. For such peaks to be located at the periodic orbits periods, we have to assume that die level density is well approximated as a sum over periodic orbits whose periods Tp = 3eSp and amplitudes vary slowly over the energy window [ - e, E + e]. A further assumption is that the energy window contains a sufficient number of energy levels. At short times, the semiclassical theory allows us to obtain... 

The survival probability S(t) associated with the initial state is the absolute square of the survival amplitude, S(t) = A(/) 2. The survival amplitude A(t) is the overlap between the state that evolves in time from this initial state and the initial state... 

As time advances from t = 0, the survival probability starts at the value 1, then (at least in many cases) gradually decreases. There are frequently a number of oscillations superimposed upon the declining function S(t) these are due to partial recurrences of the time evolving wave packet upon the starting wave packet. The relationship between the survival amplitude and the lineshape function will be described in Section II. Examples of survival probability plots will be shown later in Sections III and IV. 

It is significant that all spectral intensity information is contained within the first row of the eigenvector matrix associated with the tridiagonal matrix T. We will now show that it is possible to evaluate the survival amplitude in terms of the same quantities. 

Survival amplitude and the chain propagator. The time-development operator, U(t), also referred to as the propagator, associated with the Hamiltonian operator is given by... 

The survival amplitude for the initial state (), which is represented by the column vector C(0) = U0, is then... 

Recall that the survival probability is the absolute square of the survival amplitude, S(t)... 

Equation (75) for the survival amplitude may also be expressed in terms of the chain propagator. From Eq. (67), we may express H in terms of the chain Hamiltonian matrix T, H = QTQ. Using this result in Eq. (75) then gives... 

This equation states that the survival amplitude is the 1,1 upper-left element of the matrix representation of the chain propagator. This result is analogous to Eq. (68). which expresses the Green function in terms of the 1,1 element of the inverse of the matrix (T - ElM). 

Computation of transition probabilities. In addition to survival amplitudes, the RRGM can also be used to compute state-to-state time-dependent transition amplitudes. If we denote the initial state at t = 0 as i), then the state that evolves from this initial state is i(t)) = U(t) i), where U t) is again the evolution operator. At time t, the amplitude for finding state /) in this evolved state is given by A,f i) = (f U(t) i). If we know the eigenvectors, i / ), and eigenvalues, Ea, for this Hamiltonian, then the transition amplitude can be written... 

We will now recast this amplitude into a form that is more convenient for RRGM calculations. It may not be obvious, but this transition amplitude can be written in terms of the difference between two survival amplitudes. This result can be shown as follows. If we define two orthonormal transition vectors, w ) = [ i) + /)]/ /2, and v ) = [ i) - /)]/V/2, then the transition amplitude can be expresed as... 

The Fourier transform of the frequency spectrum yields the survival amplitude. We define the limited Fourier transform in the frequency window [w, w ] as... 

It is quite remarkable that all information needed to compute the survival amplitude (autocorrelation function) is contained in E(z) as given by... 

The survival amplitude yields the probability amplitude that at time t a particle remains in its initial state. The survival amplitude is defined as... 

It follows from the results presented in the Section 5.2 that the survival amplitude may be expanded in terms of resonant states as... 

In general, at asymptotically long times, the behavior of the survival amplitude may depend on many resonance terms. However, in view of Eq. (110), it may happen that a single coefficient dominates over the others, i.e., CmCm 1, which means that the overlap of the initial state with that that resonant state is large. In that case it is justified to make use of a single resonant term for the long-time behavior. This has been a usual assumption in the literature [78,90-93]. 

Khalfin discovered a very general result [55] that the long-time decay for Hamiltonians with spectra bounded from below is slower than exponential. The argument is this consider a system described by a time-independent Hamiltonian, H, initially in a normalized nonstationary state ho). The survival amplitude of fhat stafe is defined as the overlap of the initial state with the state at time t,... 

In potential scattering models the basic "mathematical" reason for exponential decay is a complex pole in the fourth quadrant of the momentum complex plane (second Riemann sheet of the energy plane), which, through its exponentially decaying residue, dominates the dynamics for some time. A simple analytical example of the deviation from exponentiality follows from the integral expression for the survival amplitude. 

It turns out to be an excellent approximation to write Ci(t) as the sum of the exponential term plus the asymptotic approximation, for all buf fhe very shortest times t. The survival amplitude then becomes... 

Figure 9.6 Log modulus of survival amplitude for = 0.5 versus t in dimensionless units, in Longhi s discrete model. Solid lines exact analytical solution dashed line approximation of Eq. (100) (upper curve) and Eq. (99) (lower curve, offset by two decades).

Prom (41) we obtain the survival amplitude of the initial state (j) at time t (autocorrelation function)... 

---