To lift a weight requires force, and not energy. To lift 1kg weight on the Earth requires around 9.81N force (I think we can round it to 10N).
It counterweights the gravitation of the Earth. Thus, with it, the 1kg mass levitates. If you want it to leave the Earth, you have to accelerate it (upwardly). As many g you want it to accelerate, as many times 10N you have to give it (over the initial 10N which compensates the Earth's gravitation).
Most human-carrying rockets start with around 2-5g. To start as they do, you have to give 30-60N after every kg of mass your rocket has. The mass of the ion engine is of course in it.
Now to produce a force continuously, what you require is still not energy. To do that, you require power. Power means, how many energy do you produce in a second.
From that point is the situation a little bit more complex.
To produce 1N force (upwardly) for a second, you have to shot out (downwardly) 1 kg mass (typically, some gaseous material) with 1m/s^2 speed during this second.
Or you can shot out 0.1kg mass with 10m/s^2 as well. Or you can 0.001 kg (=1g) with 1 km/s.
If the outgoing speed of the gaseous material leaving the ion drive with 100km/s (=100000m/s), it only requires 0.00001 kg of fuel after every kg it every second, to produce this 1N thrust.
Note, the maximal outgoing speed with chemical engines is around 4-5km/s. The best actually used fuel (LOX, fluid oxygen and hydrogen) has 3.6km/s. Current ion engines can go until some tens of km/s, experimental ones can go until even 50. But it is a nanotechnologic thing, so we can calculate with 100km/s.
Now the problem is the following: energy increases quadratically with the speed, so to accelerate 1kg fuel to 100km/s requires 100 times more energy as to accelerate it to 10km/s. But it gives only 10 times more force, because it depends only linearly.
To levitate an 1kg thing 1s long with a drive producing 100km/s outgoing speed, you need to:
- use 0.0001g fuel
- produce 0.5*0.0001*100000^2 = 500000J energy (energy can be calculated by 1/2mv^2, where m is the mass of the fuel (in kg) and v is its speed (in m/s))
Now the problem is that this 500kJ energy will actually mean a gas hitting the ground and the air with cosmical speed. Yes you lose this energy to drive the engine, but you get it back on the spot. The drive will be in a plasma cloud and everything will evaporate around it. I wouldn't be in the near, it will be like an explosion.
It the space it could work.
In the atmosphere I would suggest to elevate like an airplane or like an air-breathing rocket until the atmosphere is too dense.