Search for an exosphere in sodium and calcium in the transmission spectrum of exoplanet 55 Cancri e

[Abridged] The aim of this work is to search for an absorption signal from exospheric sodium (Na) and singly ionized calcium (Ca$^+$) in the optical transmission spectrum of the hot rocky super-Earth 55 Cancri e. Although the current best-fitting models to the planet mass and radius require a possible atmospheric component, uncertainties in the radius exist, making it possible that 55 Cancri e could be a hot rocky planet without an atmosphere. High resolution (R$\sim$110000) time-series spectra of five transits of 55 Cancri e, obtained with three different telescopes (UVES/VLT, HARPS/ESO 3.6m & HARPS-N/TNG) were analysed. Targeting the sodium D lines and the calcium H and K lines, the potential planet exospheric signal was filtered out from the much stronger stellar and telluric signals, making use of the change of the radial component of the orbital velocity of the planet over the transit from -57 to +57 km/sec. Combining all five transit data sets, we detect a signal potentially associated with sodium in the planet exosphere at a statistical significance level of 3$\sigma$. Combining the four HARPS transits that cover the calcium H and K lines, we also find a potential signal from ionized calcium (4.1 $\sigma$). Interestingly, this latter signal originates from just one of the transit measurements - with a 4.9$\sigma$ detection at this epoch. Unfortunately, due to the low significance of the measured sodium signal and the potentially variable Ca$^+$ signal, we estimate the p-values of these signals to be too high (corresponding to <4$\sigma$) to claim unambiguous exospheric detections. By comparing the observed signals with artificial signals injected early in the analysis, the absorption by Na and Ca$^+$ are estimated to be at a level of approximately 2.3$\times 10^{-3}$ and 7.0$\times 10^{-2}$ respectively, relative to the stellar spectrum.