  We present a multi-wavelength analysis of the galaxy cluster SPT-CL
J0607-4448 (SPT0607), which is one of the most distant clusters discovered by
the South Pole Telescope (SPT) at $z=1.4010\pm0.0028$. The high-redshift
cluster shows clear signs of being relaxed with well-regulated feedback from
the active galactic nucleus (AGN) in the brightest cluster galaxy (BCG). Using
Chandra X-ray data, we construct thermodynamic profiles and determine the
properties of the intracluster medium. The cool core nature of the cluster is
supported by a centrally-peaked density profile and low central entropy
($K_0=18_{-9}^{+11}$ keV cm$^2$), which we estimate assuming an isothermal
temperature profile due to the limited spectral information given the distance
to the cluster. Using the density profile and gas cooling time inferred from
the X-ray data, we find a mass cooling rate of
$\dot{M}_\mathrm{cool}=100_{-60}^{+90}~M_\odot$ yr$^{-1}$. From optical
spectroscopy and photometry around the [O II] emission line, we estimate that
the BCG star formation rate is SFR$_\mathrm{[O~II]}=1.7_{-0.6}^{+1.0}~M_\odot$
yr$^{-1}$, roughly two orders of magnitude lower than the predicted mass
cooling rate. In addition, using ATCA radio data at 2.1 GHz, we measure a radio
jet power of $P_\mathrm{cav}=3.2_{-1.3}^{+2.1}\times10^{44}$ erg s$^{-1}$,
which is consistent with the X-ray cooling luminosity
($L_\mathrm{cool}=1.9_{-0.5}^{+0.2}\times10^{44}$ erg s$^{-1}$ within
$r_\mathrm{cool}=43$ kpc). These findings suggest that SPT0607 is a relaxed,
cool core cluster with AGN-regulated cooling at an epoch shortly after cluster
formation, implying that the balance between cooling and feedback can be
reached quickly. We discuss implications for these findings on the evolution of
AGN feedback in galaxy clusters.

Astrophysics of Galaxies;Cosmology and Nongalactic Astrophysics;High Energy Astrophysical Phenomena