The most important properties of a Bose-Einstein condensate subject to balanced gain and loss can be modelled by a Gross-Pitaevskii equation with an external PT -symmetric double-delta potential. We study its linear variant with a supersymmetric extension. It is shown that both in the PT -symmetric as well as in the PT -broken phase arbitrary stationary states can be removed in a supersymmetric partner potential without changing the energy eigenvalues of the other state. The characteristic structure of the singular delta potential in the supersymmetry formalism is discussed, and the applicability of the formalism to the nonlinear Gross-Pitaevskii equation is analysed. In the latter case the formalism could be used to remove PT -broken states introducing an instability to the stationary PT -symmetric states.

The quantum Fourier transformation (QFT) is a key building block for a whole wealth of quantum algorithms. Despite its proven efficiency, only a few proof-of-principle demonstrations have been reported. Here we utilize QFT to enhance the performance of a quantum sensor. We implement the QFT algorithm in a hybrid quantum register consisting of a nitrogen-vacancy (NV) center electron spin and three nuclear spins. The QFT runs on the nuclear spins and serves to process the sensor—i.e., the NV electron spin signal. Specifically, we show the application of QFT for correlation spectroscopy, where the long correlation time benefits the use of the QFT in gaining maximum precision and dynamic range at the same time. We further point out the ability for demultiplexing the nuclear magnetic resonance (NMR) signals using QFT and demonstrate precision scaling with the number of used qubits. Our results mark the application of a complex quantum algorithm in sensing which is of particular interest for high dynamic range quantum sensing and nanoscale NMR spectroscopy experiments.

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