Constructing and characterising anharmonic potentials in a harmonic oscillator

Continuous-variable quantum systems offer a natural platform for quantum information processing and analogue quantum simulation of real-world processes. Realising these applications, however, requires anharmonic potentials with programmable functional forms, which are challenging to engineer on demand. Here, in a superconducting harmonic oscillator coupled to a transmon qubit, we construct programmable non-Gaussian phase gates e-iV, corresponding to the impulsive action of an anharmonic potential V, and characterise them by reconstructing their corresponding force pointwise. Using bosonic quantum signal processing, we compile target potentials into modular circuits of qubit rotations interleaved with a fixed conditional-displacement primitive. We first demonstrate a cubic phase gate whose cubic nature is jointly validated by the resulting non-Gaussian states and the force reconstructed pointwise from the gate's action on coherent probe states. We then construct a family of double-well potentials and use the reconstructed force to witness the double-well topology and tunable asymmetry. Finally, we engineer an approximate Morse potential and obtain its force profile using position-compressed probes, which extend reliable reconstruction towards the steep repulsive wall. These results establish a programmable route to constructing the potential landscape of quantum harmonic oscillators and a practical method for reconstructing its spatial structure, providing a building block for quantum simulation of anharmonic dynamics.