Quantum decay

Therefore, decay to a final equilibrium state should be inherent in a proper dynamical description of a system. It was indicated at the end of Section 9.10 that in classical systems this decay is well described by the formalism of the Fokker- Planck equation, which itself was shown to be an extension of the classical Liouville equation. In view of the similarities between the classical and quantum Liouville equations, it seems a natural question to ask whether it is also possible to find an extension of the quantum Liouville equation that lets any initial density operator decay to the equilibrium density operator for that system. 

We can also turn the question around. In chemical kinetics, we need a model to fit the data. This model can be simple, as in first-order reactions where the decay is exponential, or more complicated depending on a complex mechanism. If we do not have a model, our data are just that, data. We could try to fit to a variety of functions, but as there is an infinite number of different functions, that is a pointless exercise. As we have seen in the classical part of this chapter, even for a simple reaction a variety of models are possible, based on dissipative classical dynamics, and we can use these models to try to understand our data. This often involves varying the external parameters, temperature, pH, viscosity, and polarizabihty, but our model should tell us what to expect for such variations for instance, how the rate constant for a reaction depends on those parameters. If our models are quantum mechanical in nature, it is mandatory that we also provide a mechanism for decay, and show how the decay constant or constants depend on external parameters. 

One of the first fields where the quantum nature of the underlying system carmot be ignored, simply because there is no classical limit, is that of NMR. This is also a field where decay is directly observed in spin relaxation back to equilibrium. By giving a radio frequency pulse to an equilibrium system of spins, these are brought out of equilibrium, and, after the pulse, decay back to the equilibrium state. This free induction decay was first modeled phenomenologically by the Bloch equations, which pointed to the existence of two relaxation times, commonly called Tj and T2, but at a later stage Redfield used the density operator formalism to 

Interaction between quantum systems and classical flelds is not problematic. It is the basis of almost all forms of optical spectroscopy where the transition dipole operator of the system interacts with the electric and magnetic flelds of light. It is a necessary ingredient of linear response theory, and also of the Redfleld relaxation mechanism. The starting point for all these examples is the quantum Liouville equation 

In this equation, Hq is the Hamiltonian of the system of interest, and is a Hamiltonian that describes the interaction with the external fields and which 

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The positron source, 120 kBq of Na, was deposited onto a Kapton foil covered with identical foil and sealed. The foil 8 pm thick absorbed 10% of positrons in polyimides Ps does not form and annihilation in the source envelope gave one component only = 374 ps, which must be taken into account. The source was sandwiched between two samples of the material studied and placed into a container in a vacuum chamber. The source-sample sandwich was viewed by two Pilot U scintillators coupled to XP2020Q photomultipliers. The resolution of our spectrometer with a stop window broadened to 80% (in order to register the greatest number of three-quantum decays) was 300 ps FWHM. The finite resolution had no influence on the results of our experiment as FWHM was still comparable to the channel definition At = 260 ps.The positron lifetime spectra were stored in 8000 channels of the Tennelec Multiport E analyser. 

Nonetheless, with an inerease of the driving frequency, the tunneling quantum decay eventually takes over. This is predieted by Eq (35), since lim / y, ( i) = 0. However,... 

In particular, double quantum decay from 150 is allowed from ZS1 it is forbidden. From the latter state, annihilation into three quanta is the most likely process. The ratio of the decay rates by two and three quanta respectively is about 1100 [100]. The lifetimes of the singlet and triplet states are of the order 10 w and 10 7 seconds respectively. This is the basis of the experiments from which AIT is determined, which we shall now describe. It is to be noticed that the natural widths of these energy levels 104/2tt and 10/2irMc/s respectively) are much smaller than ATF( 2 x 105 Mc/s). 

Quantum transients are temporary features that appear in the time evolution of matter waves before they reach a stationary regime. They usually arise as a result of a sudden switch interaction that modifies the confinement of particles in a spatial region or after the preparation of a decaying sfafe [48]. The archetypical quantum transient phenomena is diffraction in time which consists of the sudden opening of a shutter to release a semi-infinite beam producing temporal and spatial oscillations of the time evolving wave [49]. A common feature in the mathematical description of quantum transient phenomena is the Moshinsky function, which as we have seen is closely related to the Faddeyeva function. Since in a recent review [48] there appears a discussion on transient phenomena for the dynamics of tunneling based on the present resonant state formalism [54, 66-76], here we restrict the discussion to the time evolution of quantum decay. 

The time evolution of quantum decay is a subject as old as quantum mechanics. At the end of the 1920s of the last century, the problem of a decay in radioactive atomic nuclei led to a theoretical derivation of the exponential decay law [1, 77]. In the 1950s, Khalfin [78] pointed out the approximate validity of the exponential decay law. It was argued that deviations from this law should occur both at very short and at very long times compared with the lifetime of the decaying system. These theoretical predictions have been confirmed experimentally in recent years in both the short [79] and the long time [80] regimes. 

A convenient model to study the time evolution of quantum decay is the 5-potential. This model was considered many years ago by Winter [81] and since then by many authors. In spite of its mathematical simplicity it describes correctly the main physical features of the time evolution of decay along the exponential and nonexponential long-time regimes. Here we discuss it by using the formalism of resonant states and make a comparison with the solution to the problem in terms of continuum states. 

G. Garcia-Calderon, R. Romo, J. ViUavicencio, Internal dynamics of multibarrier systems for pulsed quantum decay, Phys. Rev. A 79 (2009) 052121. 

G. Garcia-Calderon, J.L. Mateos, M. Moshinsky Resonant spectra and the time evolution of the survival and nonescape probabilities, Phy. Rev. Lett. 74 (1995) 33 G. Garcia-Calderon, 1. Maldonado, J. ViUavicencio, Resonant-state expansions and the long-time behavior of quantum decay, Phys. Rev. A 76 (2007) 012103. 

The above results, developing the classical model suggested by Newton so as to achieve an accurate match between classical and quantum decays, provide an intuitive physical picture and quantitative description of post-exponential decay of the probability density at points distant from the source. Note that quantum mechanics is required to provide the emission characteristics, but ISR plays no role in the classical, purely outgoing dynamics. We have checked with the methodology of [58] that ISR... 

This semiclassical relation is the analog of the classical relation (V.7). It is known that the function P (p) satisfies the general inequality P (1) < 2P(l/2). The equality sign holds when the repellor is periodic but not, in general, for a chaotic repellor. Consequently, we expect there will be some resonances with longer lifetimes than the classical lifetime. The difference between the classical and the quantum decay rates is a dramatic effect of underlying classical chaos on the quantum dynamics of the system. 

There are a number of situations where quantum mechanics plays a role which can be clarified using the density operator formalism. Spectroscopic problems can, in general, be solved in a systematic way by a series expansion in the interaction with classical Hght fields, with a reasonable model - the Brownian oscillator model - for the line widths and Hne shapes of the transitions. Excitation with short pulses leads to other interesting aspects, as we showed in Section 9.15, which challenge the concept of reaction rates for those processes. But also in that case the lack of a good theory for quantum decay prohibits a clear discussion of the meaning of the rate of transfer. 

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