Decoherence induced by a sparse bath of two-level fluctuators: Peculiar features of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> noise in high-quality qubits

Abstract

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

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