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All condensate atoms in a BEC (say like Rb, etc) effectively occupy the lowest energy-state. If it is that the case, then how are such bosons in a BEC encoded as a qubit? In particular, when Grover Search is done using a BEC, it is said in arxiv.org/abs/1303.0371: that each condensate atom does an effective log(N) bit (or n-bit) query on the oracle in parallel. That is, each condensate atom is modelled as a n-qubit system (n=log(N)).

But, since a BEC has all the condensate atoms in the ground state, how is this log(N) bit query implemented by each of the atoms? Being an undergrad, I didn't come across bosonic qubit states until now. How are these realized in comparison with the "usual" qubits with many energy "levels"?

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BEC qubits are set up by splitting one harmonic condensate trap into a double-well trap by means of a suitable laser or microwave field. See for instance pg.6 on these LANL slides: Quantum dynamics, measurement and decoherence in Bose-Einstein condensates. Under strong enough separation the overall ground state becomes two-fold degenerate and the two distinct levels can be used to encode one qubit. Several juxtaposed double-traps will provide a multi-qubit system. The paper simply assumes a system of $log_2N$ such qubits, evolving overall according to the Gross-Pitaevskii nonlinear dynamics Schroedinger equation.

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