The Transition from Coherent to Incoherent Tunneling

A rigorous quantum mechanical theory of the transition from coherent to incoherent tunneling was developed by Szymanski [84] and Scheurer [85]. The results of this elaborate theory can be reproduced by a combination of density matrix theory and formal kinetics as was shown in Ref [19]. 

The transition from coherent to incoherent tunneling is most easily understood by considering the ground state of the system and the first excited state. In the systems under consideration the typical energy difference between the ground state and the first excited state is in the range 40-50 meV. At temperatures below 50 K practically only the ground state is populated (nj, 10-5 d only a small 

As a starting point let us see what happens, if we consider not only the ground state, but also the first excited state of the tunneling pairs. Both states are connected via thermal excitation with rates k j, and kjj. Since the transition between even and odd states is both symmetry and spin forbidden, these rates connect only states of the same symmetry and the spin is conserved in the transition. 

From the principle of detailed balance we find for the ratio of the rates 

The coherent evolution of the density matrix is described by the corresponding Liouville operators  

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Figure 21.3 Sketch of the transition from coherent to incoherent tunneling ...

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