Spin coherences

The low MW power levels conuuonly employed in TREPR spectroscopy do not require any precautions to avoid detector overload and, therefore, the fiill time development of the transient magnetization is obtained undiminished by any MW detection deadtime. (3) Standard CW EPR equipment can be used for TREPR requiring only moderate efforts to adapt the MW detection part of the spectrometer for the observation of the transient response to a pulsed light excitation with high time resolution. (4) TREPR spectroscopy proved to be a suitable teclmique for observing a variety of spin coherence phenomena, such as transient nutations [16], quantum beats [17] and nuclear modulations [18], that have been usefi.il to interpret EPR data on light-mduced spm-correlated radical pairs. 

Fig. 2.9.7 Hahn spin-echo rf pulse sequence combined with bipolar magnetic field gradient pulses for hydrodynamic-dispersion mapping experiments. The lower left box indicates field-gradient pulses for the attenuation of spin coherences by incoherent displacements while phase shifts due to coherent displacements on the time scale of the experiment are compensated. The box on the right-hand side represents the usual gradient pulses for ordinary two-dimensional imaging. The latter is equivalent to the sequence shown in Figure 2.9.2(a).

P, Pohl, H.-J., Schenkel, T., I hewall, M.L.W., Itoh, K.M. and Lyon, S.A. (2012) Electron spin coherence exceeding seconds in high-purity silicon. Nat. Mater., 11, 143-147. 

As an alternative to QDs, silicon can be doped with single atom impurities, in particular phosphorus, which acts as an electron donor. Donors can be implanted individually with a precision of about 10 nm. Either the 31P nuclear spin or the unpaired electron can be used as qubits [63, 64]. An advantage of silicon is its widespread use in current electronics, meaning that QC might profit from methods and technologies already developed for their classical cousins . Also, spins in silicon can attain extremely high coherence times experiments on 28 Si-enriched silicon show spin coherence times T2 exceeding 10 s [65]. The read-out and coherent manipulation of individual spin qubits in silicon have been recently achieved [66]. 

Spin-Hamiltonians that are linear in the spin operators, HSpm propagate spin-coherent states exactly,... 

Second, the mapping approach to nonadiabatic quantum dynamics is reviewed in Sections VI-VII. Based on an exact quantum-mechanical formulation, this approach allows us in several aspects to go beyond the scope of standard mixed quantum-classical methods. In particular, we study the classical phase space of a nonadiabatic system (including the discussion of vibronic periodic orbits) and the semiclassical description of nonadiabatic quantum mechanics via initial-value representations of the semiclassical propagator. The semiclassical spin-coherent state method and its close relation to the mapping approach is discussed in Section IX. Section X summarizes our results and concludes with some general remarks. 

Finally, it is noted that there exist alternative mappings of spin to continuous DoF. For example. Ref. 219 discusses various mappings that (like the Holstein-Primakoff transformation) represent a spin system by a single-boson DoF. The possibility of utilizing spin coherent states for this purpose is discussed in Section IX. 

A further simplification of the semiclassical mapping approach can be obtained by introducing electronic action-angle variables and performing the integration over the initial conditions of the electronic DoF within the stationary-phase approximation [120]. Thereby the number of trajectories required to obtain convergence is reduced significantly [120]. A related approach is discussed below within the spin-coherent state representation. 

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]... 

To discuss the semiclassical spin-coherent state propagator, we consider a general transition amplitude ( / e which can be expressed as an integral... 

The path integral representation of the spin-coherent state propagator is formally given by [139]... 

Employing the stationary-phase approximation to the path integral, the semiclassical spin-coherent state propagator is obtained [139, 140, 143, 281, 282] ... 

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... 

Another possibility to introduce a semiclasscial initial value representation for the spin-coherent state propagator is to exploit the close relation between Schwinger s representation of a spin system and the spin-coherent state theory [100, 133-135]. To illustrate this approach, we consider an electronic two-level system coupled to Vvib nuclear DoF. Within the mapping approach the semiclassical propagator for this system is given by... 

In order to express the propagator (134) in terms of spin-coherent states, we introduce the following parameterization of the complex electronic variables [133] ... 

Figure 44. Diabatic population (a) and modulus of the autocorrelation function (b) of the initially prepared state for Model IVa. The full line is the quantum result, the dashed-dotted line is the result of the semiclassical spin-coherent state propagator, and the dashed line depicts the result of Suzuki s propagator. The semiclassical data have been normalized. Panel (c) shows the norm of the semiclassical wave functions.

The results obtained for the three-mode Model IVb are depicted in Fig. 45. As was found for the semiclassical mapping approach, the spin-coherent state propagators can only reproduce the short-time dynamics for the electronic population. The autocorrelation function, on the other hand, is reproduced at least qualitatively correctly by the semiclassical spin-coherent state propagator. 

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