Decaying state

The resolvent can be continued analytically from above through the cut into a second Riemann sheet. The decaying state is now associated with a pole on the second sheet. Let us write the coordinates of the pole as... 

Fig. 20. Bound states, approximately bound states and decaying states.

A theory of photochemical processes which relates macroscopic observables to molecular properties should have great appeal to the physical chemist, since in these cases, unlike the case of thermal reactions, the nature of the metastable decaying states can be unambiguously defined. Furthermore, since these states are routinely prepared in photochemical experimentation, they are worthy of extensive study. Perhaps a more complete understanding of the properties of isolated molecule metastable states can play the role of precursor to the understanding of the states involved in thermal reactions. 

The Siegert resonance state provides a method of calculating the lifetime and position of the decaying state without the need to solve the TDSE. Aside from the practical advantage, they also greatly facilitate the understanding... 

In the previous sections, we have seen that a resonance can be described through both a time-dependent approach and a time-independent approach. The goal of this section is to create a unified picture that joins both methodologies and explain how general wave packets evolve into pseudostationary decaying states. 

Electronic configurations and terms of the decaying states are specified in the first row. The literature values in parentheses are taken from Ref. [55]. See Ref. [44] for the details of the Fano-ADC computation. 

For the metastable states in which the decaying electron spends some time in the vicinity of the target, Gamow /52/ and Dirac /53/ suggested that the width of these resonances T is related to the lifetime by the uncertainty relation T = h/r where r is the lifetime of the metastable state and that the time development of these decaying states be described by... 

Development in Time of the Probability Amplitude for a Decaying State... 

In Chapter 3 we investigated the development in time of a decaying state, expressed in terms of the time-independent eigenfunctions satisfying a system of two coupled differential equations, resulting from the separation of the Schrodinger equation in parabolic coordinates. In this analysis we obtained general expressions for the time-dependent wave function and the probability amplitude. 

To sum up, our treatment has elucidated the short-time dynamics of low-temperature MQT through time-modulated barriers. Current-bias modulation has been shown to imitate either frequent measurements or correlated perturbations of a decaying state, between successive impulses (shocks) [Fan-dau 1977 Ivlev 2002], Such modulation has been demonstrated to either enhance or suppress the MQT rate (causing the AZE or QZE, respectively). Remarkably, quantum gates based on JJ qubits [Averin 2000] or their atomic-condensate counterparts [Smerzi 1997 Anderson 1998 (a)] may benefit from the ability to suppress the decoherence due to MQT to the continuum. 

It is evident that the antisymmetric state is populated by the coherent coupling to the symmetric state. Since the decay rate of the antisymmetric state, T(1 — p), is very small for p 1, the population stays in this state for a long time. If A = 0 the state is decoupled from the symmetric state and is zero if its initial value is zero. In the latter case the system reduces to a two-level atom. In the former case the transfer of the population to a slowly decaying state leaves the symmetric state almost unpopulated even if the driving field is strong. This is shown in Fig. 17, where we plot the steady-state populations pss as a... 

For intermediate systems, such as the HF dimer, mode specificity can be attributed to the differences in bonding of the two hydrogens.85 Likewise, in the NO dimer the two neighboring states with drastically different lifetimes correspond to two types of stretch vibrations. In the larger systems, however, such direct identification of the regular states is not always possible. Moreover, it is clear83,86 that the slowly decaying states are superimposed on other states with much faster decay, a behavior that can be anticipated theoretically. 

After the coordinates entering the Hamiltonian have been transformed, the Hamiltonian is no longer hermitian and thus can support complex eigenenergies associated with decaying states. Hence the complex energy in spherical coordinates is given by... 

Multiphoton processes taking place in atoms in strong laser fields can be investigated by the non-Hermitian Floquet formalism (69-71,12). This time-independent theory is based on the equivalence of the time-dependent Schrodin-ger description to a time-independent field-dressed-atom picture, under assumption of monochromaticity, periodicity and adiabaticity (69,72). Implementation of complex coordinates within the Floquet formalism allows direct determination of the complex energy associated with the decaying state. The... 

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