A two-level system

We now add die field back into the Hamiltonian, and examine the simplest case of a two-level system coupled to coherent, monochromatic radiation. This material is included in many textbooks (e.g. [6, 7, 8, 9, 10 and 11]). The system is described by a Hamiltonian having only two eigenstates, i and with energies = and Define coq = - co. The most general wavefunction for this system may be written as... 

Equation (A1.6.64) describes the relaxation to equilibrium of a two-level system in tenns of a vector equation. It is the analogue of tire Bloch equation, originally developed for magnetic resonance, in the optical regime and hence is called the optical Bloch equation. 

In tills weakly coupled regime, ET in an encounter complex can be described approximately using a two-level system model [23]. As such, tlie time-dependent wave function is... 

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,... 

The first illustration of the concept of a partition function is that of a two-level system, e.g. an electron in a magnetic field, with its spin either up or down (parallel or anti parallel to the magnetic field) (Fig. 3.2). The ground state has energy Eq = 0 and the excited state has energy Ae. By substituting these values in Eq. (3) we find the following partition function for this two-level system ... 

Figure 3.2. Partition function and fractional occupation for a two-level system.

Magnetic resonance is actually a rate process which may be treated in terms of the usual rate equations. Consider a two-level system as shown in Fig. 7, and suppose for the moment that transitions only occur because of interactions between the unpaired electron and the oscillating (microwave) magnetic field. If this is the only interaction, it may be shown that the probability for the downward transition, Pafs, is equal to the probability for the upward transition, Pga. One can then write the rate equation... 

There may, however, be specific reasons to study a signal over an extended temperature range. For one, a linear increase in EPR amplitude with the inverse of the temperature (Curie s law) is proof that a spin system is a two-level system, i.e., an S = 1/2 or an effective S = 1/2 system. More importantly, in complex multicenter metalloproteins, overlapping spectra may be deconvoluted by virtue of their Tu value being different if two centers, a and b, have rMa < TMb then at TMb the spectrum of center a is broadened and that of center b is not. It is once more emphasized that these types of studies require determination of (anisotropic) saturation behavior at all relevant temperatures. 

Fig. 3.7. Specific heat c of a two-level system with energy separation AE.

Figure 1.10 (a) The absorption and emission energies for a two-level system (rigid lattice), (b) The absorption and emission energies showing the Stokes shift (vibrating lattice). 

For a two-level system the mapping (80b) obviously coincides with Eq. (75d) for... 

Let us start with a brief review of spin-coherent state theory. For simplicity we focus on a two-level (or spin 1/2) system. The coherent states for a two-level system with basis states /i), /2) can be written as [136, 139]... 

Within the theoretical framework of time-dependent Hartree-Fock theory, Suzuki has proposed an initial-value representation for a spin-coherent state propagator [286]. When we adopt a two-level system with quantum Hamiltonian H, this propagator reads... 

If the object is embedded into a matrix with the same average scattering properties as the considered jumping unit, then the scattering contrast from the average size of the object is matched by the matrix and the corresponding forward scattering is suppressed. It can be shown [133] that the dynamic structure factor for an object embedded in a matrix, which performs jumps in a two level system, can be obtained as ... 

The energy separation between the non-degenerate and degenerate states can be assessed by fitting the MCD intensity as a function of 1/T at a discrete wavelength to a Boltzmann population distribution for a two level system. 

The fast-forward protocol can be regarded as a prescription for finding a shortcut in state space, [50] from the initial state to the target state. There are, of course, many possible shortcuts in state space but very few proposals to find those shortcuts. In this section, we generalize the fast-forward protocol in a two-level system, developing different shortcuts in which, in contrast to fast-forward field (FFE)-driven dynamics, the amplitude and the phase of the wave function of the intermediate state are modulated [50]. 

We first examine the fast-forward protocol in a two-level system, whose Hamiltonian is represented as... 

Figure 43 (a) Schematic drawing of a two-level system with off-diagonal couphng to a continuum... 

We focus on two regimes a two-level system coupled to either an AN or PN thermal bath (Figure 4.5). The bath Hamiltonian (in either regime) will be... 

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