One of the primary questions when characterizing Earth-sized and super-Earth-sized exoplanets is whether they have a substantial atmosphere like Earth and Venus or a bare-rock surface like Mercury. Phase curves of the planets in thermal emission provide clues to this question, because a substantial atmosphere would transport heat more efficiently than a bare-rock surface. Analyzing phase-curve photometric data around secondary eclipses has previously been used to study energy transport in the atmospheres of hot Jupiters. Here we use phase curve, Spitzer time-series photometry to study the thermal emission properties of the super-Earth exoplanet 55 Cancri e. We utilize a semianalytical framework to fit a physical model to the infrared photometric data at 4.5 µm. The model uses parameters of planetary properties including Bond albedo, heat redistribution efficiency (i.e., ratio between radiative timescale and advective timescale of the atmosphere), and the atmospheric greenhouse factor. The phase curve of 55 Cancri e is dominated by thermal emission with an eastward-shifted hotspot. We determine the heat redistribution efficiency to be 1.47–0.25+0.30, which implies that the advective timescale is on the same order as the radiative timescale. This requirement cannot be met by the bare-rock planet scenario because heat transport by currents of molten lava would be too slow. The phase curve thus favors the scenario with a substantial atmosphere. Our constraints on the heat redistribution efficiency translate to an atmospheric pressure of ∼1.4 bar. The Spitzer 4.5 µm band is thus a window into the deep atmosphere of the planet 55 Cancri e.