Theoretical hyperfine decay functions in triple quantum dots

Coherent oscillations in multiple quantum dots decay due to hyperfine interactions with nuclear spins. The decay functions observed in several double-dot experiments [1] agree well with simple formulae derived using the group SU(2), which is defined by exchange and hyperfine interactions in the singlet-triplet system [2]. We show that in triple dots, this theory generalizes to SU(3), with convenient representation in the basis of states of the exchange-only qubit in a decoherence-free subsystem~[3]. Using some intuition from SU(3), we derive analytic formulae for the hyperfine decay functions expected in coherent oscillations in triple dots~[4]. \\{} \newcommand\mybibformat[5]{#1, \textit{#2}~\textbf{#3}, #4 (#5)} [1]~\mybibformat{B.~M.~Maune et al.}{Nature}{481}{344}{2012}; \mybibformat{E.~A.~Laird et al.}{Phys. Rev. B}{82}{075403}{2012} \\{} [2]~\mybibformat{W. A. Coish and D. Loss}{Phys. Rev.~B}{72}{125337}{2005} \\{} [3]~\mybibformat{D.~P.~DiVincenzo et al.}{Nature}{408}{339}{2000}; \mybibformat{B.~H.~Fong and S.~M.~Wandzura}{Quantum Inf. Comput.}{11}{1003}{2011} \\{} [4]~\mybibformat{T. D. Ladd}{Phys. Rev. B}{86}{125408}{2012}.