Decoherence induced by a sparse bath of two-level fluctuators: Peculiar features of 1/f noise in high-quality qubits

Progress in fabrication of semiconductor and superconductor qubits has greatly diminished the number of decohering defects, thus decreasing the devastating low-frequency <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mn>1</a:mn><a:mo>/</a:mo><a:mi>f</a:mi></a:mrow></a:math> noise and extending the qubits' coherence times (dephasing time <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msubsup><b:mi>T</b:mi><b:mn>2</b:mn><b:mo>*</b:mo></b:msubsup></b:math> and the echo decay time <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>T</c:mi><c:mn>2</c:mn></c:msub></c:math>). However, large qubit-to-qubit variation of the coherence properties remains a problem, making it difficult to produce a large-scale register where all qubits have a uniformly high quality. In this work, we show that large variability is a characteristic feature of a qubit dephased by a sparse bath made of many (<d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>n</d:mi><d:mo>≫</d:mo><d:mn>1</d:mn></d:mrow></d:math>) decohering defects, coupled to the qubit with similar strength. We model the defects as two-level fluctuators (TLFs) whose transition rates <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>γ</e:mi></e:math> are sampled from a log-uniform distribution over an interval <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mo>[</f:mo><f:msub><f:mi>γ</f:mi><f:mi>m</f:mi></f:msub><f:mo>,</f:mo><f:msub><f:mi>γ</f:mi><f:mi>M</f:mi></f:msub><f:mo>]</f:mo></f:mrow></f:math>, which is a standard model for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mn>1</g:mn><g:mo>/</g:mo><g:mi>f</g:mi></g:mrow></g:math> noise. We investigate decoherence by such a bath in the limit of high-quality qubit, i.e., when the TLF density <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>d</h:mi></h:math> is small (the limit of sparse bath, with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mi>d</i:mi><i:mo>=</i:mo><i:mi>n</i:mi><i:mo>/</i:mo><i:mi>w</i:mi><i:mo>≪</i:mo><i:mn>1</i:mn></i:mrow></i:math>, where <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi>n</j:mi></j:math> is the number of TLFs and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mi>w</k:mi><k:mo>=</k:mo><k:mo form="prefix">ln</k:mo><k:mrow><k:mo>[</k:mo><k:msub><k:mi>γ</k:mi><k:mi>M</k:mi></k:msub><k:mo>/</k:mo><k:msub><k:mi>γ</k:mi><k:mi>m</k:mi></k:msub><k:mo>]</k:mo></k:mrow></k:mrow></k:math> is the log-width of the distribution). We show that different realizations of the bath produce very similar noise power spectra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>S</m:mi><m:mo>(</m:mo><m:mi>f</m:mi><m:mo>)</m:mo><m:mo>∼</m:mo><m:mn>1</m:mn><m:mo>/</m:mo><m:mi>f</m:mi></m:mrow></m:math>, but lead to drastically different coherence times <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:msubsup><n:mi>T</n:mi><n:mn>2</n:mn><n:mo>*</n:mo></n:msubsup></n:math> and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:msub><o:mi>T</o:mi><o:mn>2</o:mn></o:msub></o:math>. Thus the spectral density <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"><p:mrow><p:mi>S</p:mi><p:mo>(</p:mo><p:mi>f</p:mi><p:mo>)</p:mo></p:mrow></p:math> does not determine coherence of a qubit coupled to a sparse TLF bath, as opposed to a dense bath; instead, decoherence is controlled by only a few exceptional fluctuators, determined by their value of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mi>γ</q:mi></q:math>. We show that removing only two of these TLFs greatly increases <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"><r:msub><r:mi>T</r:mi><r:mn>2</r:mn></r:msub></r:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"><s:msubsup><s:mi>T</s:mi><s:mn>2</s:mn><s:mo>*</s:mo></s:msubsup></s:math> times. Our findings help theoretical understanding and further improvements in the coherence properties of semiconductor and superconductor qubits, battling the <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"><t:mrow><t:mn>1</t:mn><t:mo>/</t:mo><t:mi>f</t:mi></t:mrow></t:math> noise in these platforms. Published by the American Physical Society 2024