  Quantum theory is based on a mathematical structure totally different from
conventional arithmetic. Due to the symmetric nature of bosonic particles,
annihilation or creation of single particles translates a quantum state
depending on how many bosons are already in the given quantum system. This
proportionality results in a variety of non-classical features of quantum
mechanics including the bosonic commutation relation. The annihilation and
creation operations have recently been implemented in photonic systems.
However, this feature of quantum mechanics does not preclude the possibility of
realizing conventional arithmetic in quantum systems. We implement conventional
addition and subtraction of single phonons for a trapped \Yb ion in a harmonic
potential. In order to realize such operations, we apply the transitionless
adiabatic passage scheme on the anti-Jaynes-Cummings coupling between the
internal energy states and external motion states of the ion. By performing the
operations on superpositions of Fock states, we realize the hybrid computation
of classical arithmetic in quantum parallelism, and show that our operations
are useful to engineer quantum states. Our single-phonon operations are nearly
deterministic and robust against parameter changes, enabling handy repetition
of the operations independently from the initial state of the atomic motion. We
demonstrate the transform of a classical state to a nonclassical one of highly
sub-Poissonian phonon statistics and a Gaussian state to a non-Gaussian state,
by applying a sequence of the operations. The operations implemented here are
the Susskind-Glogower phase operators, whose non-commutativity is also
demonstrated.
