Coherent Control Over the High-Dimensional Space of the Nuclear Spin of Alkaline-Earth Atoms

We demonstrate coherent manipulation of the nuclear degrees of freedom of ultracold ground-state strontium-87 atoms, thus providing a toolkit for fully exploiting the corresponding large Hilbert space as a quantum resource and for quantum simulation experiments with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mrow><a:mrow><a:mi mathvariant="fraktur">s</a:mi></a:mrow><a:mi mathvariant="normal">u</a:mi></a:mrow></a:mrow><a:mo stretchy="false">(</a:mo><a:mi>N</a:mi><a:mo stretchy="false">)</a:mo></a:math>-symmetric matter. By controlling the resonance conditions of Raman transitions with a tensor light shift, we can perform rotations within a restricted Hilbert space of two isolated spin states among the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mn>2</g:mn><g:mi>F</g:mi><g:mo>+</g:mo><g:mn>1</g:mn><g:mo>=</g:mo><g:mn>10</g:mn></g:math> possible states. These manipulations correspond to engineering unitary operations derived from generators of the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:mrow><i:mrow><i:mi mathvariant="fraktur">s</i:mi></i:mrow><i:mi mathvariant="normal">u</i:mi></i:mrow></i:mrow><i:mo stretchy="false">(</i:mo><i:mi>N</i:mi><i:mo stretchy="false">)</i:mo></i:math> algebra beyond what can be done by simple spin precession. We present Ramsey interferometers involving an isolated pair of Zeeman states with no measurable decoherence after 3 s. We also demonstrate that one can harness the large spin degrees of freedom as a qudit resource by implementing two interferometer schemes over four states. The first scheme senses in parallel multiple external fields acting on the atoms, and the second scheme simultaneously measures multiple observables of a collective atomic state—including noncommuting ones. Engineering unitary transformations of the large spin driven by other generators than the usual spin-<o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>F</o:mi></o:math> representation of the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mrow><q:mrow><q:mrow><q:mi mathvariant="fraktur">s</q:mi></q:mrow><q:mi mathvariant="normal">u</q:mi></q:mrow></q:mrow><q:mo stretchy="false">(</q:mo><q:mn>2</q:mn><q:mo stretchy="false">)</q:mo></q:math> group offers new possibilities from the point of view of quantum metrology and quantum many-body physics, notably for the quantum simulation of large-spin <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mrow><w:mrow><w:mrow><w:mi mathvariant="fraktur">s</w:mi></w:mrow><w:mi mathvariant="normal">u</w:mi></w:mrow></w:mrow><w:mo stretchy="false">(</w:mo><w:mi>N</w:mi><w:mo stretchy="false">)</w:mo></w:math>-symmetric quantum magnetism with fermionic alkaline-earth atoms.