Quasiparticle properties of long-range impurities in a Bose condensate

An impurity immersed in a Bose condensate can form a quasiparticle known as a Bose polaron. When the impurity-boson interaction is short ranged and three-body Efimov correlations are neglected, the quasiparticle properties can be characterized in terms of the impurity-boson scattering length <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msub> <a:mi>a</a:mi> <a:mi>IB</a:mi> </a:msub> </a:math> and the condensate coherence length <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>ξ</b:mi> </b:math> . This description remains valid irrespective of the bath density <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msub> <c:mi>n</c:mi> <c:mn>0</c:mn> </c:msub> </c:math> . Long-ranged interactions—such as provided by Rydberg or ionic impurities—introduce an effective interaction range <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:msub> <d:mi>r</d:mi> <d:mi>eff</d:mi> </d:msub> </d:math> as the third length scale. The interplay among these competing length scales raises the question of whether a robust description across different bath densities can still be maintained. In this study, we discuss the quasiparticle nature of long-range impurities and its dependence on the length scales <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msubsup> <e:mi>n</e:mi> <e:mn>0</e:mn> <e:mrow> <e:mo>−</e:mo> <e:mn>1</e:mn> <e:mo>/</e:mo> <e:mn>3</e:mn> </e:mrow> </e:msubsup> <e:mo>,</e:mo> <e:mo> </e:mo> <e:msub> <e:mi>r</e:mi> <e:mi>eff</e:mi> </e:msub> </e:math> , and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mi>ξ</f:mi> </f:math> . We employ two complementary theories—the coherent-state ansatz and the perturbative Gross-Pitaevskii theory—which incorporate beyond-Fröhlich interactions. We derive an analytical expression for the beyond-Fröhlich effective mass for a contact interaction and numerically compute the effective mass for long-range impurities. We argue that the coupling parameter <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mrow> <g:mo>|</g:mo> </g:mrow> <g:msub> <g:mi>a</g:mi> <g:mi>IB</g:mi> </g:msub> <g:mrow> <g:mo>|</g:mo> </g:mrow> <g:msubsup> <g:mi>n</g:mi> <g:mn>0</g:mn> <g:mrow> <g:mn>1</g:mn> <g:mo>/</g:mo> <g:mn>3</g:mn> </g:mrow> </g:msubsup> </g:mrow> </g:math> remains the principal parameter governing the properties of the polaron. For weak ( <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mrow> <h:mo>|</h:mo> </h:mrow> <h:msub> <h:mi>a</h:mi> <h:mi>IB</h:mi> </h:msub> <h:mrow> <h:mo>|</h:mo> </h:mrow> <h:msubsup> <h:mi>n</h:mi> <h:mn>0</h:mn> <h:mrow> <h:mn>1</h:mn> <h:mo>/</h:mo> <h:mn>3</h:mn> </h:mrow> </h:msubsup> <h:mo>≪</h:mo> <h:mn>1</h:mn> </h:mrow> </h:math> ) and intermediate ( <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mrow> <i:mo>|</i:mo> </i:mrow> <i:msub> <i:mi>a</i:mi> <i:mi>IB</i:mi> </i:msub> <i:mrow> <i:mo>|</i:mo> </i:mrow> <i:msubsup> <i:mi>n</i:mi> <i:mn>0</i:mn> <i:mrow> <i:mn>1</i:mn> <i:mo>/</i:mo> <i:mn>3</i:mn> </i:mrow> </i:msubsup> <i:mo>≃</i:mo> <i:mn>1</i:mn> </i:mrow> </i:math> ) values of the coupling parameter, long-range impurities in a Bose condensate are well described as quasiparticles with a finite quasiparticle weight and a well-defined effective mass. However, the quasiparticle weight becomes significantly suppressed as the effective impurity volume is occupied by an increasing number of bath particles ( <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:msub> <j:mi>r</j:mi> <j:mi>eff</j:mi> </j:msub> <j:msubsup> <j:mi>n</j:mi> <j:mn>0</j:mn> <j:mrow> <j:mn>1</j:mn> <j:mo>/</j:mo> <j:mn>3</j:mn> </j:mrow> </j:msubsup> <j:mo>≫</j:mo> <j:mn>1</j:mn> </j:mrow> </j:math> ).