How many physical qubits are needed to encode a logical qubit on an
-- ion trap,
-- neutral atom,
quantum computer based on today's technologies?
The answer depends on several factors:
Just to expand on point (1), getting to increasingly small error rates of the logical qubit requires increasingly more physical qubits. One can create a logical qubit with less than 10 physical qubits, but to get to very low error rates allowing for very deep circuits then you will likely want something like thousands of physical qubits per logical qubit.