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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 interms of two compact phase fields $\hat{\phi}_{a,s}(x)$. Building on existingresults in the literature we discuss how a single projective measurement of theparticle density after trap release is in a certain limit related to theeigenvalues of the vertex operator $e^{i\hat{\phi}_a(x)}$. We emphasize thetheoretical assumptions underlying the analysis of "single-shot" interferencepatterns and show that such measurements give direct access to multi-pointcorrelation functions of $e^{i\hat{\phi}_a(x)}$ in a substantial parameterregime. For experimentally relevant situations, we derive an expression for themeasured particle density after trap release in terms of convolutions of theeigenvalues of vertex operators involving both sectors of the two-componentLuttinger liquid that describes the low-energy regime of the split condensate.This opens the door to accessing properties of the symmetric sector via anappropriate analysis of existing experimental data.

Projective phase measurements in one-dimensional Bose gases