Flight Crew Instrument; Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP)
Flight Crew Last 90 Days 57; Flight Crew Total 9094; Flight Crew Type 698
Situational Awareness; Workload; Human-Machine Interface
Aircraft Equipment Problem Less Severe; Conflict Airborne Conflict; Deviation - Altitude Overshoot; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy
Air Traffic Control Provided Assistance; Flight Crew FLC complied w / Automation / Advisory; Flight Crew Took Evasive Action
Aircraft; Software and Automation
Briefly; as I can write and remember...TCAS advisory of traffic 12:00 closing +3;900 [ft.] descending and less than 15 miles.We were estimated FL210 for FL230 on the SID; the process of switching from ZZZ to ZZZ1 ATC.Traffic continued to descend and closing now 'amber' and 'Traffic Traffic.'Keying the mic to say Aircraft X climbing from FL222 to FL230 when in the midst receive 'Monitor Vertical Speed' 'Climb Climb. Announcing to ATC we were on an RA climb for traffic (I think the conflict was being announced during the transmission).The target was now 'red' and near. (We never saw any traffic outside.)Controller returned with 'We have no traffic in your vicinity climb maintain FL230.'We complied and began our level off as TCAS instructed and 'Clear of Conflict' at FL230.Again ATC advised NO traffic in our vicinity (and apologized at least 2x.)ATC cleared us to climb from FL230 to FL430.The target was still red and directly below us (estimated as I don't remember 3;000 feet; but immediately below our aircraft.As I keyed the mic we heard and saw 'Monitor Vertical Speed' and 'Climb Climb' perhaps we heard 'Traffic Traffic' too... I don't remember.Again as keying the mic; I replied (while RA was announced over the radio) we are on an RA emergency climb from [Flight Level] 230 to 430.Again ATC replied we have no traffic in your vicinity climb to FL430.Once we had 'Clear of Conflict' and climbing quickly I took 2 pictures to show our displays. The target (never seen by us visually nor ATC on their radar) was depicted immediately below our aircraft in 'red' and at that time of the picture 6;100 feet and descending and then disappeared."
C680 Captain reported getting two RAs during climbout; however; they were unable to visually locate the traffic and ATC advised there was no other traffic in the vicinity.