Active fuel management is a technology developed by General Motors, which is used to improve engine efficiency in the times when the engine is operating under loads which are considerably less than the rated load, as shown in this link. This technology exploits the property of piston engines where some of the cylinders are turned OFF while only few of them work, and hence improving the overall efficiency of the engine. Can a similar technique be employed on an aircraft's engines, where some of the turbine stages (Compressor or Turbine) are switched off in order to improve the performance efficiency of the engine?
You cannot "switch off" a stage of a turbine engine.
As seen elsewhere jet engines are a single block mechanically locked together: all the stages (compressor AND turbine) connected to the same shaft. If the shaft turns, all connected stages turn with it.
You can have designs with multiple shafts, but this increases the complexity (design and maintenance) heavily.
Even if we assume for a moment that you could have 1 shaft per stage, stopping one of these would be generally a bad idea. For acoustic reasons the number of blades on the stationary part of the stage and those on the rotary part must be prime to each other (example: 37 blades on the stationary side and 39 on the rotary one); this means that when the rotary part stops you cannot align all the blades with the stationary ones, creating drag and losses within the engine, thus reducing the efficiency.
How would efficiency of a Jet Engine change, if we assume for a second that its a possibility, to stop a few stages(turn them into free instead of locked)?
You would still decrease the efficiency: compressor stages are the ones actually pulling the aircraft forward (you would then produce less thrust*) and the turbine stages are the ones extracting energy from the exahust.
*: you could even risk having the flame going upstream if you stop too many compressor stages and not having enough pressure at the combustion chamber inlet.
Can it be done?
Has it been done?
Wait, what?! Where what how?
The Pratt and Whitney J58 was designed to allow partial shutdown of the compressor. It does this by placing a distinctive spike in front of the compressor assembly.
However the purpose of the transition to ramjet that is distinctive of the J58 was to allow it to avoid overheating the turbine during supersonic flight, where the increased inlet velocity translated to additional heating of the air.
However this does not increase the efficiency of the jet engine at all.
Overall you actually want as many stages as possible for subsonic flight. This is because multiple stages allow the gas to compress/expand more adiabatically, leading to better Carnot efficiency.
Short answer: No.
Longer answer: Jet engines are designed with multiple compressor and turbine discs mechanically attached to a single shaft, called a spool. Usually, there are 2 (or sometimes 3) of these spools within the engine, due to the need of the high pressure compressor and the main intake fan to spin at different speeds. Each spool spins freely and is driven by the compressed air in the engine core being pushed through the turbine stages attached to that spool.
Diagram of a Turbofan Jet Engine
There is no way to "shut down" a particular spool. As long as the pressure in the engine core is higher than the pressure behind the nozzle, the turbines will spin and drive the spool. If you detach either one of the turbine stages or one of the compressor stages from a spool such that it 'freewheels,' the turbine discs will likely overspeed due to decreased load. Additionally, an undriven compressor stage would experience compressor stall, which is essentially an aerodynamic stall of the blades of the compressor stage.
Finally, this is entirely unneeded for a couple of reasons:
First, aside from the problems it would cause, there would be nothing to gain from shutting down a compressor stage. As Andy correctly mentioned in a comment, jet engines do not have a constant fuel burn per revolution, unlike piston engines. If a jet engine doesn't need to produce as much power, it can simply inject less fuel in the combustion chamber. As the pressure in the combustion chamber drops, the spools will spin down accordingly on their own (due to less pressure driving the turbine stages.)
Second, the power output profiles of jet aircraft operation are nothing like those of a car. For a car, high engine power is usually used only in brief spurts for acceleration and cruise uses a small fraction of that amount of power. In a jet airplane, on the other hand, the engines will be operated at a rather high percentage of their maximum RPMs for most of the flight. This is because additional power allows them to fly higher, which actually increases efficiency. See the answers to the previous question How far can a 777 fly with just one engine at altitude? Unlike a car, significantly reducing the power during cruise flight actually reduces the fuel efficiency of an airliner.
If you extend the idea of "switch off" a bit, this is already done in several different ways.
Surveillance aircraft may be designed to loiter over their target area with some engines shut down completely. That is not just a matter of shutting down the engines - for example if you are going to loiter at high altitude for several hours at temperatures below -50C, you are likely to need special systems to heat the oil in the shut-down engines, to avoid damage when you restart them.
Many jet engines contain variable stator vanes in the compressors, which are automatically adjusted to improve the efficiency under different working conditions.
In multi-shaft engines, the relative speeds of the different shafts changes depending on the working conditions. That is more efficient that forcing all the compressor and turbine stages to run at the same speed on a single shaft. In a multi-shaft turbofan, the bypass ratio (i.e. the fraction of the air being compressed in the core engine and used for combustion, rather than just being blown by the fan to generate thrust) will also change.
Fuel efficiency is not improved by shutting off stages of a compressor or turbine. Fuel efficiency is a function of overall pressure ratio, reduce this and efficiency is reduced.