Search for variations of the fine-structure constant via the hyperfine electronic bridge in highly charged <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Th</mml:mi><mml:mprescripts/><mml:none/><mml:mn>229</mml:mn

The search for variations of the fine-structure constant <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>α</a:mi></a:math> by using atomic clocks has driven the development of novel atomic clock technologies. Among these, the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mmultiscripts><b:mi>Th</b:mi><b:mprescripts/><b:none/><b:mn>229</b:mn></b:mmultiscripts></b:math> nuclear clock based on nuclear transition stands out due to its high sensitivity. Previous studies on the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>α</c:mi></c:math> variation in atomic clocks have typically focused purely either on a nuclear or electronic transition but not on the hyperfine electronic bridge (HEB) transitions, which involve simultaneous changes in both nuclear and electronic structures. In this work, we propose to search for the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>α</d:mi></d:math> variation by measuring the 3.14-eV HEB transition between the hyperfine levels <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mo>[</e:mo><e:msub><e:mi>I</e:mi><e:mi>e</e:mi></e:msub><e:mo>,</e:mo><e:mrow><e:mo>(</e:mo><e:mn>4</e:mn><e:msup><e:mi>f</e:mi><e:mn>12</e:mn></e:msup><e:mo>)</e:mo></e:mrow><e:msub><e:mi>J</e:mi><e:mi>e</e:mi></e:msub><e:mo>=</e:mo><e:mn>4</e:mn><e:mo>,</e:mo><e:msub><e:mi>F</e:mi><e:mi>e</e:mi></e:msub><e:mo>]</e:mo></e:mrow></e:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mo>[</f:mo><f:msub><f:mi>I</f:mi><f:mi>g</f:mi></f:msub><f:mo>,</f:mo><f:mrow><f:mo>(</f:mo><f:mn>4</f:mn><f:msup><f:mi>f</f:mi><f:mn>12</f:mn></f:msup><f:mo>)</f:mo></f:mrow><f:msub><f:mi>J</f:mi><f:mi>e</f:mi></f:msub><f:mo>=</f:mo><f:mn>2</f:mn><f:mo>,</f:mo><f:msub><f:mi>F</f:mi><f:mi>g</f:mi></f:msub><f:mo>]</f:mo></f:mrow></f:math> of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mmultiscripts><g:mi>Th</g:mi><g:none/><g:mrow><g:mn>32</g:mn><g:mo>+</g:mo></g:mrow><g:mprescripts/><g:none/><g:mn>229</g:mn></g:mmultiscripts></g:math> ions, and the 2.39-eV HEB transition between <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mo>[</h:mo><h:msub><h:mi>I</h:mi><h:mi>e</h:mi></h:msub><h:mo>,</h:mo><h:mrow><h:mo>(</h:mo><h:mn>4</h:mn><h:msup><h:mi>f</h:mi><h:mn>4</h:mn></h:msup><h:mo>)</h:mo></h:mrow><h:msub><h:mi>J</h:mi><h:mi>g</h:mi></h:msub><h:mo>=</h:mo><h:mn>4</h:mn><h:mo>,</h:mo><h:msub><h:mi>F</h:mi><h:mi>e</h:mi></h:msub><h:mo>]</h:mo></h:mrow></h:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mo>[</i:mo><i:msub><i:mi>I</i:mi><i:mi>g</i:mi></i:msub><i:mo>,</i:mo><i:mrow><i:mo>(</i:mo><i:mn>4</i:mn><i:msup><i:mi>f</i:mi><i:mn>4</i:mn></i:msup><i:mo>)</i:mo></i:mrow><i:msub><i:mi>J</i:mi><i:mi>e</i:mi></i:msub><i:mo>=</i:mo><i:mn>2</i:mn><i:mo>,</i:mo><i:msub><i:mi>F</i:mi><i:mi>g</i:mi></i:msub><i:mo>]</i:mo></i:mrow></i:math> of <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mmultiscripts><j:mi>Th</j:mi><j:none/><j:mrow><j:mn>40</j:mn><j:mo>+</j:mo></j:mrow><j:mprescripts/><j:none/><j:mn>229</j:mn></j:mmultiscripts></j:math> ions. These two HEB transitions exhibit remarkably large sensitivity factors <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:msub><k:mi>K</k:mi><k:mi>α</k:mi></k:msub></k:math> of about <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mrow><l:mo>−</l:mo><l:mn>2.2</l:mn><l:mo>×</l:mo><l:msup><l:mn>10</l:mn><l:mn>4</l:mn></l:msup></l:mrow></l:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>−</m:mo><m:mn>2.9</m:mn><m:mo>×</m:mo><m:msup><m:mn>10</m:mn><m:mn>4</m:mn></m:msup></m:mrow></m:math>, respectively. Compared to the nuclear-clock transition <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mo>(</n:mo><n:msub><n:mi>I</n:mi><n:mi>e</n:mi></n:msub><n:mo> </n:mo><n:mi>to</n:mi><n:mo> </n:mo><n:msub><n:mi>I</n:mi><n:mi>g</n:mi></n:msub><n:mo>)</n:mo></n:math> in bare ions, the sensitivity factors are significantly enhanced by factors of 2.7 and 3.5. Moreover, both the 3.14- and 2.39-eV HEB transitions are well supported by current high-precision laser technology, making our approach promising for improving the precision of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mi>α</o:mi></o:math> variation detection.