An example The two-level system

Show that the time evolution defined by Eqs (2.2) and (2.3) corresponds to solving Eqs (2.8) (or equivalently (2.9) with the initial conditions 

The first equality in (2.11) has the fonn of a continuity equation (see Section 1.1.4) that establishes J as a flux (henceforth referred to as a probability flux). 

Consider a two-level system whose Hamiltonian is a sum of a simple part, Hq, and a perturbation F, 

The coupling elements F,., are in principle complex, and we express them as 

Quantum dynamics using the time-dependent Schrodinger equation 

Problem 2.3. Let (r,t) be the solution of a 1 -particle Schrodinger equation with 

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As an example, consider the two-level system, with relaxation that arises from spontaneous emission. In this case there is just a single V. ... 

The frozen Rydberg gas corresponds to a quantum mesoscopic system, where the coherence of the two-level system is shared with the ensemble of the other atoms, offering an interesting example of decoherence through interaction with the environment [Joos 2003]. 

In Sections 2.2 and 2.9 we have discussed the dynamics of the two-level system and of the harmonic oscillator, respectively. These exactly soluble models are often used as prototypes of important classes of physical system. The harmonic oscillator is an exact model for a mode of the radiation field (Chapter 3) and provides good starting points for describing nuclear motions in molecules and in solid environments (Chapter 4). It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach (see Section 6.5.4). In fact, many linear response treatments in both classical and quantum dynamics lead to harmonic oscillator models Linear response implies that forces responsible for the return of a system to equilibrium depend linearly on the deviation from equilibrium—a harmonic oscillator property We will see a specific example of this phenomenology in our discussion of dielectric response in Section 16.9. 

Note that for a model characterized by an upper cutoff in the boson density of states, for example, the Debye model, these rates vanish when the level spacing of the two-level system exceeds this cutoff. Note also that the rates (12.47) and (12.48) satisfy the detailed balance relationship (12.45). 

This system in many cases can be simplified further. For example, if we have a broad spectral line excitation with a not very intense laser radiation, we have a situation for an open transition when 7 Ti, H. In practical cases this condition is often fulfilled at excitation with cw lasers operating in a multimode regime. If the homogeneous width of spectral transition usually is in the range of 10 MHz, then the laser radiation spectral width broader than 100 MHz usually can be considered as a broad line excitation. In this case we can use a procedure known as adiabatic elimination. It means that we are assuming that optical coherence pi2 decays much faster than the populations of the levels puJ = 1,2. Then we can find stationary solution for off-diagonal elements for the density matrix and afterwards find a rate equations for populations in this limit. For the two level system we will have... 

A risk assessment analyses systems at two levels. The first level defines the functions the system must perform to respond successfully to an accident. The second level identifies the hardware for the systems use. The hardware identification (in the top event statement) describes minimum system operability and system boundaries (interfaces). Experience shows that the interfaces between a frontline system and its support systems are important to the system cs aluaiion and require a formal search to document the interactions. Such is facilitated by a failure modes and effect analysis (FMEA). Table S.4.4-2 is an example of an interaction FMEA for the interlace and support requirements for system operation. 

An accurate calculation of the heat conductivity requires solving a kinetic equation for the phonons coupled with the multilevel systems, which would account for thermal saturation effects and so on. We encountered one example of such saturation in the expression (21) for the scattering strength by a two-level system, where the factor of tanh((3co/2) reflected the difference between thermal populations of the two states. Neglecting these effects should lead to an error on the order of unity for the thermal frequencies. Within this single relaxation time approximation for each phonon frequency, the Fermi golden rule yields, for the scattering rate of a phonon with Ha kgT,... 

In general, it can be very difficult to determine the nature of the boundary terms. A specific result in an exactly solvable case is discussed in Section IV.A.2. Equation (55) is the Gallavotti-Cohen FT derived in the context of deterministic Anosov systems [28]. In that case, Sp stands for the so-called phase space compression factor. It has been experimentally tested by Ciliberto and co-workers in Rayleigh-Bemard convection [52] and turbulent flows [53]. Similar relations have also been tested in athermal systems, for example, in fluidized granular media [54] or the case of two-level systems in fluorescent diamond defects excited by light [55]. 

As an example we consider a system composed of two spin-free states, [S] and [T], which are mixed by the spin interaction Q. It is assumed that the zero-order, spin-free energies °[5] and E°[T] are functions of a system parameter Q, e.g., an internal nuclear coordinate, such that t the spin-free levels cross as shown in Figure 2. The matrix element... 

The H- ll9Sn HDMR experiments may often be applied to the analysis of type A X spin systems with degenerate transitions and only two groups of magnetically equivalent nuclei. For example, the energy level diagram for an A3X system is shown in Fig. 3. 

Broad maxima in Cm versus T (the so-called Schottky anomalies) frequently indicate partially populated discrete levels that are separated by an energy difference AE in the range of knT. For a simple two-level system, the maximum occurs at B niax 0.42A/i. Phase transitions, for example, transitions between a long-range ordered ferromagnetic phase and a paramagnetic phase, produce a characteristic peak in Cm versus T graphs. 

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