Background
Hybrid quantum systems are highly interesting both from an applied technological viewpoint and for fundamental physics reasons. Technologically speaking, interfacing a future solid state quantum computer capable of local quantum computations with longdistance quantum links formed by optical photons requires a hybrid quantum system, a ‘quantum modem’ interfacing the ‘quantum computer’ with the ‘quantum internet’.
From a fundamental physics viewpoint, an iconic topic of relevance to this project is that of a heavy mechanical oscillator in its quantum ground state. At what mass does the gravitational force become significant and can we observe this? When entangled, a nuclear spin has been proposed as a ‘witness’ to the quantum behaviour of such a system^{1}.
Physical platform
Nuclear spins in silicon are wellstudied, extremely coherent quantum objects. The same applies to highQ macroscopic mechanical resonators. Thus far these two systems have not been coupled. But a direct coupling mechanism does exist for certain nuclear isotopes: if the nuclear spin is larger than \(1/2\), the nuclear spin couples directly to electric field gradients due to the nuclear quadrupole interaction^{2}. Sizeable electric field gradients are caused by microscopic distortions around the atom, such as strain. Through the nuclear quadrupole interaction, dynamic strain will allow driving transitions of the nuclear spin (known as Nuclear Acoustic Resonance)^{3}. In the quantum limit of a resonator this may allow for coherent coupling of a single quantum mechanical vibrational mode, a phonon, and the nuclear spin. We propose to use a nuclear spin of e.g. 73Ge (spin \(9/2\)) or a groupV donor such as 123Sb (spin \(7/2\)), implanted in a silicon resonator, for this purpose.
High spin nucleus covalently bonded to a silicon lattice (4 silicon atoms shown in black). The larger high spin nucleus locally distorts the lattice. Upon further breaking the lattice symmetry through strain, a significant electric field gradient is present at the nucleus, causing a nonzero nuclear quadrupole interaction term. Dynamical strain in a mechanical resonator may allow for coupling mechanical modes to the nuclear spin. Figure adapted from^{2}
Topics for Master thesis project
 Familiarize with basics of nuclear quadrupole interaction, mechanical resonators in the quantum limit, spinboson coupling Hamiltonian. At a conceptual, small Hamiltonian level.
 Estimate required coupling strengths based on basic Hamiltonian description of the problem.
 Model (using COMSOL) realistic resonator geometry for reaching maximum dynamical strain. Compare different types of resonators (clamped beam, dumbbell, acoustic membrane, …). Estimate dynamical strain for zeropoint motion and use this for comparison to required coupling strength. Assess experimental feasibility
 Optional: consider realistic readout and detection schemes, in particular integration into onchip superconducting circuits used in qubit readout, e.g. through dynamical capacitance of the resonator. Estimate coupling strengths, quality factors etc.
 Optional: Estimate gravitational force between two resonators at a given distance. Estimate maximum noise level of this gravitational force still compatible with quality factors and realistic experimental conditions.
Required Skills
 Selfmotivated, independent working style
 Strong background in quantum mechanics and solidstate physics
 Affinity / experience with modelling software and programming
 Affinity with formula based theoretical calculations
 Ability to communicate efficiently with people of a varied background
What we offer
 Workplace, laptop for duration of the project.
 Student assistant contract via RWTH, or equivalent via FZ Jülich contract
 Young, international, dynamical workplace, located on Campus Melaten (Campus Boulevard 79)
 Exposure to leading research activities in quantum technology
Supervision
 Dr. Vincent Mourik, FZ Jülich, PGI11 v.mourik@fzjuelich.de
 Dr. Pavel Bushev, FZ Jülich, PGI13 (FUNQS) p.bushev@fzjuelich.de
References

S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroš, M. Paternostro, A. A. Geraci, P. F. Barker, M. S. Kim, and G. Milburn, Spin Entanglement Witness for Quantum Gravity, Phys. Rev. Lett. 119, 240401, 2017 ↩

S. Asaad, V. Mourik, B. Joecker, M. A. I. Johnson, A. D. Baczewski, H. R. Firgau, M. T. Mądzik, V. Schmitt, J. J. Pla, F. E. Hudson, K. M. Itoh, J. C. McCallum, A. S. Dzurak, A. Laucht and A. Morello, Coherent electrical control of a single highspin nucleus in silicon, Nature 579, 7798, 205209, 2020 ↩ ↩^{2}

L. A. O’Neill, B. Joecker, A. D. Baczewski, and A. Morello, Engineering local strain for singleatom nuclear acoustic resonance in silicon, Appl. Phys. Lett. 119, 174001, 2021 ↩