Two-level states

Specific heat measurements40 indicate that the assumption of static frozen glass disorder at low temperatures may be too restrictive and that the intra-H-bond hydrogen motion may still persist in the form of quantum tunnelling. To check this hypothesis, 2D deuteron NMR and 87Rb and 2H SLR measurements were carried out at low temperatures. With site-specific NMR measurements, it was also hoped to identify the microscopic nature of the "two-level" states which determine the low T glassy properties of these systems. 

A special feature of amorphous materials is the anomalous behaviour of several properties at very low temperatures, such as in the low temperature specific heats. Most of these anomalies are attributed to the presence of two-level states (TLS) separated by small barriers, which gives rise to tunneling excitations. These excitations are characterized by wide distribution of relaxation times and energies. Several ultrasonic and low temperature specific heat measurements have been performed to characterize the TLS but their physical nature such as their structures, etc. is far from having been understood. These are the ADWP states discussed earlier in Chapter 7 in some detail. 

The tunneling in two-level states takes place when both the barrier heights and the widths are very small and thus the barrier can be breached by heavy atoms, unlike in the case of electrons or protons, which involves only tunneling. It is not clear whether TLS is a distinct class of objects involved in relaxation or whether they are just the tail-end states of secondary relaxations with very broad distributions, which continue to be active down to cryogenic temperatures. Another remarkable feature of the low temperature specific heat is that its magnitude depends only weakly on chemical composition, implying that the TLS present in glasses is almost a universal feature. 

In the previous section we discussed light and matter at equilibrium in a two-level quantum system. For the remainder of this section we will be interested in light and matter which are not at equilibrium. In particular, laser light is completely different from the thennal radiation described at the end of the previous section. In the first place, only one, or a small number of states of the field are occupied, in contrast with the Planck distribution of occupation numbers in thennal radiation. Second, the field state can have a precise phase-, in thennal radiation this phase is assumed to be random. If multiple field states are occupied in a laser they can have a precise phase relationship, something which is achieved in lasers by a teclmique called mode-locking Multiple frequencies with a precise phase relation give rise to laser pulses in time. Nanosecond experiments... 

It follows that there are two kinds of processes required for an arbitrary initial state to relax to an equilibrium state the diagonal elements must redistribute to a Boltzmaim distribution and the off-diagonal elements must decay to zero. The first of these processes is called population decay in two-level systems this time scale is called Ty The second of these processes is called dephasmg, or coherence decay in two-level systems there is a single time scale for this process called T. There is a well-known relationship in two level systems, valid for weak system-bath coupling, that... 

Much of the previous section dealt with two-level systems. Real molecules, however, are not two-level systems for many purposes there are only two electronic states that participate, but each of these electronic states has many states corresponding to different quantum levels for vibration and rotation. A coherent femtosecond pulse has a bandwidth which may span many vibrational levels when the pulse impinges on the molecule it excites a coherent superposition of all tliese vibrational states—a vibrational wavepacket. In this section we deal with excitation by one or two femtosecond optical pulses, as well as continuous wave excitation in section A 1.6.4 we will use the concepts developed here to understand nonlinear molecular electronic spectroscopy. 

Figure Al.6.13. (a) Potential energy curves for two electronic states. The vibrational wavefunctions of the excited electronic state and for the lowest level of the ground electronic state are shown superimposed, (b) Stick spectrum representing the Franck-Condon factors (the square of overlap integral) between the vibrational wavefiinction of the ground electronic state and the vibrational wavefiinctions of the excited electronic state (adapted from [3]).

Figure A3.13.14. Illustration of the quantum evolution (pomts) and Pauli master equation evolution (lines) in quantum level structures with two levels (and 59 states each, left-hand side) and tln-ee levels (and 39 states each, right-hand side) corresponding to a model of the energy shell IVR (liorizontal transition in figure...

Application of an oscillating magnetic field at the resonance frequency induces transitions in both directions between the two levels of the spin system. The rate of the induced transitions depends on the MW power which is proportional to the square of oi = (the amplitude of the oscillating magnetic field) (see equation (bl.15.7)) and also depends on the number of spins in each level. Since the probabilities of upward ( P) a)) and downward ( a) p)) transitions are equal, resonance absorption can only be detected when there is a population difference between the two spin levels. This is the case at thennal equilibrium where there is a slight excess of spins in the energetically lower p)-state. The relative population of the two-level system in thennal equilibrium is given by the Boltzmaim distribution... 

Here all couplings are ignored except the direct couplings between the initial and final states as in a two-level atom. The coupled equations to be solved are... 

The situation in singlet A electronic states of triatomic molecules with linear equilibrium geometry is presented in Figme 2. This vibronic structure can be interpreted in a completely analogous way as above for n species. Note that in A electronic states there is a single unique level for K =, but for each other K 0 series there are two levels with a unique character. 

At this point, we make two comments (a) Conditions (1) and (2) lead to a well-defined sub-Hilbert space that for any further treatments (in spectroscopy or scattering processes) has to be treated as a whole (and not on a state by state level), (b) Since all states in a given sub-Hilbert space are adiabatic states, stiong interactions of the Landau-Zener type can occur between two consecutive states only. However, Demkov-type interactions may exist between any two states. 

To extend the above kinetie model to this more general ease in whieh degenerate levels oeeur, one uses the number of moleeules in eaeh level (N and Nf for the two levels in the above example) as the time dependent variables. The kinetie equations then governing their time evolution ean be obtained by summing the state-to-state equations over all states in eaeh level... 

Here, gi and gf are the degeneraeies of the two levels (i.e., the number of states in eaeh level) and the Ri f and Rf i, whieh are equal as deseribed above, are the state-to-state rate eoeffieients introdueed earlier. 

At steady state, the populations of these two levels are given by setting dNf... 

There will be incidences when the foregoing assumptions for a two-tailed test will not be true. Perhaps some physical situation prevents p from ever being less than the hypothesized value it can only be equal or greater. No results would ever fall below the low end of the confidence interval only the upper end of the distribution is operative. Now random samples will exceed the upper bound only 2.5% of the time, not the 5% specified in two-tail testing. Thus, where the possible values are restricted, what was supposed to be a hypothesis test at the 95% confidence level is actually being performed at a 97.5% confidence level. Stated in another way, 95% of the population data lie within the interval below p + 1.65cr and 5% lie above. Of course, the opposite situation might also occur and only the lower end of the distribution is operative. 

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