Two neutrino double beta decay of $^{76}$Ge to excited states of $^{76}$Se has been studied using data from Phase I of the GERDA experiment. An array composed of up to 14 germanium detectors including detectors that have been isotopically enriched in $^{76}$Ge was deployed in liquid argon. The analysis of various possible transitions to excited final states is based on coincidence events between pairs of detectors where a de-excitation $gamma$ ray is detected in one detector and the two electrons in the other. No signal has been observed and an event counting profile likelihood analysis has been used to determine Frequentist 90,% C.L. bounds for three transitions: ${0^+_{rm g.s.}-2^+_1}$: $T^{2 u}_{1/2}>$1.6$cdot10^{23}$ yr, ${0^+_{rm g.s.}-0^+_1}$: $T^{2 u}_{1/2}>$3.7$cdot10^{23}$ yr and ${0^+_{rm g.s.}-2^+_2}$: $T^{2 u}_{1/2}>$2.3$cdot10^{23}$ yr. These bounds are more than two orders of magnitude larger than those reported previously. Bayesian 90,% credibility bounds were extracted and used to exclude several models for the ${0^+_{rm g.s.}-0^+_1}$ transition.
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