Abstract
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 \hat{\phi}_{a,s}(x)ϕ̂a,s(x). 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 e^{i\hat{\phi}_a(x)}eiϕ̂a(x). 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 e^{i\hat{\phi}_a(x)}eiϕ̂a(x) 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.
Highlights
In this work we have revisited the theoretical description of the measurement process involved in time-of-flight recombination of split one-dimensional Bose gases
We have derived the relation between the measured density operator after expansion and local operators in the Luttinger liquid theory describing the low energy degrees of freedom in such systems
We have described how multi-point correlation functions of vertex operators can be extracted from projective measurements of the boson density in time of flight experiments
Summary
The purpose of this manuscript is to revisit the theoretical basis for the analysis of matter-wave interferometry experiments on split one-dimensional Bose gases [1,2,3,4,5,6,7,8,9,10,11]. Like previous work our approach is based on the Luttinger liquid description of the phase degrees of freedom We discuss why this analysis is restricted to the weakly interacting regime, and what modifications emerge for stronger interactions. Our derivation makes it clear why such measurements provide access to equal time multi-point correlation functions of vertex operators of the phase field. We consider the case of coherently split bose gases without tunnel coupling, cf. Refs. [35, 36]
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