Coherence and Decoherence in Tunnelling between Quan- tum Dots

Coupled quantum dots are an example of the ubiquitous quantum double potential well. In a typical transport experiment, each quantum dot is also coupled to a continuum of states. Our approach takes this into account by using a Green’s function formalism to solve the full system. The time-dependant solution is then explored in different limiting cases. In general, a combination of coherent and incoherent behavior is observed. In the case that the coupling of each dot to the macroscopic world is equal, however, the time evolution is purely coherent. The double-well potential is one of the simplest and best understood problems in modern quantum mechanics. Its utility is likewise unparalleled. Potential applications of double-well devices have been noted in Refs. [1-5]. For such devices to be useful, an understanding of the processes which couple the microscopic device to the macroscopic environment is paramount. That is to say, the decoherence processes of such systems must be well understood. To this end, we consider a simple exactly-solvable model of two coupled quantum dots. An excellent review of related systems and some approximate solutions is given in Ref. [6]. Each dot is coupled to an environment (a triple-barrier system) as in Fig. 1. The environment consists of a continuum of states, as would be appropriate for a macroscopic lead. The initial condition is a single electron localized in dot 1 at time t = 0. Only one state in each quantum dot is considered, which amounts to the assumption that the tunnelling parameters connecting our two states to any neglected state are much less than the energy differences with that state. The Hamiltonian of the system can be written as the sum of two terms H = H0 + Hc. The first term, H0, represents the dynamics of the simple two-state system H0 = ε1d † 1d1 + ε2d † 2d2 + V (d † 1d2 + d † 2d1) . = ε1 V V ε2 , (1) where di creates an electron in dot i, and the other symbols are defined in Fig. 1. V may be chosen real and positive without loss of generality. The second term, Hc, reflects the coupling of each dot to its respective lead