Publications associated with Condensed Matter Theory


Projective phase measurements in one-dimensional Bose gases

SciPost Physics Stichting SciPost 5 (2018) 046

YD van Nieuwkerk, J Schmiedmayer, F Essler

<jats:p>We consider time-of-flight measurements in split one-dimensional Bose gases. It is well known that the low-energy sector of such systems can be described in terms of two compact phase fields <jats:inline-formula><jats:alternatives><jats:tex-math>\hat{\phi}_{a,s}(x)</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mover><mml:mi>ϕ</mml:mi><mml:mo accent="true">̂</mml:mo></mml:mover><mml:mrow><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>. Building on existing results in the literature we discuss how a single projective measurement of the particle density after trap release is in a certain limit related to the eigenvalues of the vertex operator <jats:inline-formula><jats:alternatives><jats:tex-math>e^{i\hat{\phi}_a(x)}</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:msub><mml:mover><mml:mi>ϕ</mml:mi><mml:mo accent="true">̂</mml:mo></mml:mover><mml:mi>a</mml:mi></mml:msub><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:msup></mml:math></jats:alternatives></jats:inline-formula>. We emphasize the theoretical assumptions underlying the analysis of “single-shot” interference patterns and show that such measurements give direct access to multi-point correlation functions of <jats:inline-formula><jats:alternatives><jats:tex-math>e^{i\hat{\phi}_a(x)}</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:msub><mml:mover><mml:mi>ϕ</mml:mi><mml:mo accent="true">̂</mml:mo></mml:mover><mml:mi>a</mml:mi></mml:msub><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:msup></mml:math></jats:alternatives></jats:inline-formula> in a substantial parameter regime. For experimentally relevant situations, we derive an expression for the measured particle density after trap release in terms of convolutions of the eigenvalues of vertex operators involving both sectors of the two-component Luttinger liquid that describes the low-energy regime of the split condensate. This opens the door to accessing properties of the symmetric sector via an appropriate analysis of existing experimental data.</jats:p>


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