Quantum phase transitions in spin systems are supposed to be accompanied by a soft collective mode, which has not been seen in experiments. Here, we directly measure the low energy excitation modes of a well-known realization of the Ising model in transverse field, LiHoF 4 , using microwave spectroscopy techniques to probe energies well below what is accessible via neutron scattering experiments. Instead of the single excitation expected for a simple quantum Ising system, we find and characterize a remarkable array of 'electronuclear' modes, arising from coupling of the spin-1/2Ising electronic spins to a bath of spin-7/2 Ho nuclear spins. The lowest-lying electronuclear mode softens at the approach to the quantum critical point from below and above, a softening that can be quenched with the application of a longitudinal magnetic field. The electronuclear mode structure has direct implications for the Ising systems that serve as the building blocks of adiabatic quantum computers and quantum annealers.

The loop-gap resonator (LGR) was originally developed to provide a uniform microwave magnetic field on a sample for electron spin resonance (ESR) experiments. The LGR is composed of one or more loops and gaps acting as inductances and capacitances respectively. Typical LGR designs produce a uniform field on a sample at a single resonant frequency, but for certain experiments it is necessary to study the response of a material to uniform fields at multiple frequencies applied simultaneously. In this work we develop an empirical design procedure using finite element method calculations to design an asymmetric loop-gap resonator with uniform fields at two frequencies in the same sample volume and analyze the field uniformity, frequency tunability and filling factors, providing comparison to a manufactured device.

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