Creating NOON states with ultracold atoms via counterdiabatic driving

We theoretically investigate quantum control protocols for the creation of NOON states using ultracold bosonic atoms on two modes, corresponding to the coherent superposition $\vert N,0\rangle + \vert 0,N \rangle $, for a small number $N$ of bosons. One possible method to create this state is to consider a third mode where all bosons are initially placed, which is symmetrically coupled to the two other modes. Tuning the energy of this third mode across the energy level of the other two modes allows the adiabatic creation of the NOON state. The main issue with this method is that it requires a large amount of time to reach the NOON state. However, this problem can be addressed by the application of a counterdiabatic Hamiltonian, which allows one to significantly reduce the time required to achieve these entangled states. We demonstrate that such a counterdiabatic protocol is feasible and effective for a single particle, and then discuss how to extend its application to a larger number of bosons.