Phil, for helicopters the problem boils down to three areas:
1. Fuel source.
2. Engine.
3. Aerodynamics.
For 1, i am not as convinced as others about hydrogen. The main reason is that it has lousy energy volume density (even as a liquid), although good energy mass density. Consider that the mighty Saturn V used liq H2/O2 combustion in it's 2nd and 3rd stages (J2 engines), but good old kerosine for the heavy lift 1st stage (F1 engine). CO2 + H2O + Energy can be converted into liquid hydrocarbons, but it will take some investment to develope the chemical engineering required. It is doable.
Where the energy comes from is the key. We get 1.4kW of power for every square meter from sunlight, so there is a start. Water turbines in the gulfstream look a more practical way for UK to capture this energy. Ultimately our power hungry civilisation will use fusion, which will be with us in 3 or 4 decades. Fusion is already here, for ~1min burst, but it takes time to commercialise these things.
For 2, the ultimate solution will be solid state energy conversion. There are some very interesting technologies using quantum mechanics to convert combustion directly into electrical power. They will probably work their way through the space industry first while they mature. Timescale probably similar to fusion, mostly due to market resistance. In the interim Rolls is doing a lot of materials research to push engines above 40% efficiency.
For 3, as Nick comments the heli industry (should that be helo? i've never been sure) will continue to improve their product with resulting benefits. I will be interested to see a Sikorsky of 2040, it may be suprisingly different...