Small Transport; Low Wing; 2 Turbojet Eng
Flight Crew Instrument; Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP)
Communication Breakdown
ATC Issue All Types; Conflict NMAC; Deviation / Discrepancy - Procedural Clearance
Flight Crew Took Evasive Action
Airspace Structure; Procedure
On takeoff we were climbing out on our initial climb following our climb via clearance. I was flying and saw Aircraft Y cross the windscreen above us from left to right less than 500 feet. I reacted by pulling back power as we received a Resolution Advisory to descend which I started to follow. That Resolution Advisory went away after a couple seconds. CA informed Tower that we responded to a Resolution Advisory and asked where the traffic was since the Resolution Advisory disappeared so fast. I had told him I saw Aircraft Y pass above and ATC confirmed they had cleared traffic to cross overhead. We continued the flight as normal. I had heard ATC clear traffic to cross overhead thinking they would be over the approach end and we had a slightly steeper climb since we were light. After landing we called the ZZZ Tower to ask what happened and they informed us that the Aircraft Y had crossed further west than they anticipated. After the fact; pulling power and lowering the nose after traffic passed and following the Resolution Advisory for a couple seconds.
Corporate jet First Officer reported ATC failed to provide separation between traffic resulting in an NMAC.