<p>We consider two Hamiltonians that are close to each other, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1904_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$H_1 \approx H_2 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mo>≈</mo> <msub> <mi>H</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, and analyze the time decay of the corresponding <i>Loschmidt echo</i> <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1904_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="205" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {M}(t):= |\langle \psi _0, \textrm{e}^{\textrm{i} t H_2} \textrm{e}^{-\textrm{i} t H_1} \psi _0 \rangle |^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="fraktur">M</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mo stretchy="false">|</mo> </mrow> <mrow> <mo stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mn>0</mn> </msub> <mo>,</mo> <msup> <mtext>e</mtext> <mrow> <mtext>i</mtext> <mi>t</mi> <msub> <mi>H</mi> <mn>2</mn> </msub> </mrow> </msup> <msup> <mtext>e</mtext> <mrow> <mo>-</mo> <mtext>i</mtext> <mi>t</mi> <msub> <mi>H</mi> <mn>1</mn> </msub> </mrow> </msup> <msub> <mi>ψ</mi> <mn>0</mn> </msub> <mo stretchy="false">⟩</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> that expresses the effect of an imperfect time reversal on the initial state <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1904_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi _0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ψ</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>. Our model Hamiltonians are deformed Wigner matrices that do not share a common eigenbasis. The main tools are new two-resolvent laws for such <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1904_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$H_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1904_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$H_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>.</p>

Loschmidt echo for deformed Wigner matrices

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2025年1月30日

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László Erdős,
Joscha Henheik,
Oleksii Kolupaiev

摘要

We consider two Hamiltonians that are close to each other, $H_1 \approx H_2 $ $H_{1} \approx H_{2}$ , and analyze the time decay of the corresponding Loschmidt echo $\mathfrak {M}(t):= |\langle \psi _0, \textrm{e}^{\textrm{i} t H_2} \textrm{e}^{-\textrm{i} t H_1} \psi _0 \rangle |^2$ $M (t) : = | ⟨ ψ_{0}, e^{i t H_{2}} e^{- i t H_{1}} ψ_{0} ⟩ |^{2}$ that expresses the effect of an imperfect time reversal on the initial state $\psi _0$ $ψ_{0}$ . Our model Hamiltonians are deformed Wigner matrices that do not share a common eigenbasis. The main tools are new two-resolvent laws for such $H_1$ $H_{1}$ and $H_2$ $H_{2}$ .