Systematic analysis of relative phase extraction in one-dimensional Bose gases interferometry

- The first two paragraphs of the introduction have been slightly rewritten for clarity
- Eq. (9) has been slightly modified, with some of the constant prefactors absorbed into the definition of $\ell_t$. More importantly, the sign of the second term is changed into negative. Hence, the theoretical derivation in Appendix C has been updated to include the fourth order Taylor expansion, which is important for getting the right constant prefactor in the second term of Eq. (9). Other results are unaffected by this change.
- Changed title of Sec. 4.1 from “Two-point phase correlation function” to “Vertex correlation functions” to distinguish it from the two-point correlation function discussed in Sec. 4.5. The relevant inline text and figure caption have been changed accordingly.
- Added in Sec. 4.3 last paragraph,
“To resolve such an effect, one must resolve fluctuation with a length scale comparable to the length scale of TOF dynamics $\ell_t = \sqrt{\hbar t/(2m)}$. Thus, this effect might not be captured by present experiments and by our perturbative treatment in Eq. (9). However, our numerical results point to the necessity of calibrating the results of dynamical propagation of velocity-velocity correlation such as in Ref. [19] to the measurement background in future experiments with enhanced resolution.”
- Added further elaboration in to discussions in the first paragraph of Sec. 6
- Cited Ref [29] in response to referee’s comment
- Changed the sentence on page 2, “If the trap is switched off rapidly, ... free evolution” into “If the trap is switched off rapidly, the initial tight confinement in transverse directions leads to rapidly expanding density, which allows one to neglect the effect of interactions during the expansion. Consequently, the dynamics are well approximated by a quench into free evolution.”
- Removed the following sentence below Eq. (7) “The above implies that, at least up to the second order, longitudinal expansion does not change the functional relationship between $\rho_{\rm TOF}(x,z,t)$ and $\phi_-(z)$. This demonstrates the robustness of the transversal fit formula; nevertheless, longitudinal expansion still influences the extracted fit parameters. “
and replace it with
“The form of Eq. (7) is expected from Eq. (3). Using our integrand expansion technique, we are able to express the fit parameters in terms of in-situ field variables.”
- Removed sentence in Fig. 3 of the updated submission. “Numerical errors have been accounted for by subtracting the phase error using the transversal model in both encoding and decoding.”
- Added “thermal state of the sine-Gordon Hamiltonian….” below the title of section 4
- Text in Sec 4.2 updated.
- Changed “mean occupation number” to “mean Fourier spectrum” or “mean Fourier coefficients” in the relevant places.
- Changed the notation for momentum modes of the relative phase to $q$ instead of $k$, and also changed the symbol for the sine-Gordon scaling factor $\lambda_T/l_j$ from $q$ to $\chi$ in all the relevant places throughout the manuscript.
- Changed the scaling factor in the definition to $1/\sqrt{L}$ and we have adjusted the scale in Fig. 8 and Fig. 21.
- Provided more context for the similarities and differences between quantities in Sec. 4.1 and Sec. 4.4.
- Added the following sentence in Sec. 4.4: “We note that this effect is due to dynamics in the high momentum modes which goes beyond the correction in Eq. (9). Indeed, for $t = 15\; \rm ms$ the expansion length scale is $\ell_t = \sqrt{\hbar t/(2m)} = 2.3 \mu m$ giving a momentum cutoff $q \sim \ell_t^{-1} \approx 0.43 \mu m$ which is smaller than the typical momenta where this oscillation is observed.”
- Included the value of $\chi = 3$ in the captions of Figs. 10 - 11.
- Changed the section 5 title from “The effects of image processing” to “The effects of imaging”.