We investigate the prospects of the diffractive production of $J/psi$ mesons at large momentum transfer $|t|$ at the future Electron Ion Collider in electron-proton collisions. In particular, we focus on the measurements of the rapidity gap size. The model used for the calculations is based on the diffractive exchange of the Balitsky-Fadin-Kuraev-Lipatov perturbative Pomeron. Calculations for the cross section and the estimates for the rates assuming integrated luminosity of $10 , rm fb^{-1}$ are provided. Two experimental strategies were considered. First, measuring the rapidity gap size directly, by observing the activity in the forward part of the central detector, and second by putting a lower limit on the rapidity gap size in the case when the detector cannot measure forward activity. We find that, it is possible to measure at the EIC the dependence of the cross section on rapidity gap interval up to four units in rapidity. This should allow to measure the change of the cross section by a factor 1.6 expected due to the BFKL exchange. This is possible with the present setup of the detector which projects the coverage up to 3.5 units of rapidity. We conclude however, that the extension of the detector up to higher rapidity, for example to 4.5 would be desirable and provide even better lever arm for testing rapidity gap physics at the EIC.
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