1. Department of Physics, University of California, Santa Barbara, California 93106, USA

Correspondence to: A. N. Cleland¹ Correspondence and requests for materials should be addressed to A.N.C. (Email: Cleland@physics.ucsb.edu).

The superposition principle is a fundamental tenet of quantum mechanics. It allows a quantum system to be 'in two places at the same time', because the quantum state of a physical system can simultaneously include measurably different physical states. The preparation and use of such superposed states forms the basis of quantum computation and simulation¹. The creation of complex superpositions in harmonic systems (such as the motional state of trapped ions², microwave resonators^{3, 4, 5} or optical cavities⁶) has presented a significant challenge because it cannot be achieved with classical control signals. Here we demonstrate the preparation and measurement of arbitrary quantum states in an electromagnetic resonator, superposing states with different numbers of photons in a completely controlled and deterministic manner. We synthesize the states using a superconducting phase qubit to phase-coherently pump photons into the resonator, making use of an algorithm⁷ that generalizes a previously demonstrated method of generating photon number (Fock) states in a resonator⁸. We completely characterize the resonator quantum state using Wigner tomography, which is equivalent to measuring the resonator's full density matrix.

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