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

*This report was written jointly by Marc Cheneau and Isabelle Bouchoule (Laboratoire Charles Fabry).*

In this paper, the authors investigate the extraction of the relative phase between two 1D Bose gases from the analysis of the interference pattern which appears after a time of free expansion, when the two matter waves overlap.
This technique has been used in many previous works of the Vienna group led by J. Schmiedmayer, coauthor of of the present article. In these works, the longitudinal motion during the time of flight was neglected, and the transverse position of the fringes at a given longitudinal position $z$ was identified with the initial relative phase between the two gases at the same position $z$.

Here, the authors instead focus on this longitudinal motion. The authors first explain how they can analytically account for the longitudinal motion, and they clearly explain the underlying assumptions. Then, making an expansion in the length associated with the time-of-flight, they derive a formula showing that the first dominant effect of the longitudinal motion is a coupling between the relative and common phases, while the second sub-dominant effect involves the first and second order derivatives of the relative phase along $z$.
The rest of the paper is a numerical investigation based on the aforementioned formula. More precisely, the authors quantify the error made if the interference pattern is analysed ignoring the longitudinal motion during the time-of-flight---as was done previously. For this analysis, the authors consider two clouds initially at thermal equilibrium, both with and without Josephson coupling. Different situations are considered, which correspond to different experimental studies published by the Vienna group in the past. In most cases, the effect of longitudinal motion during the time-of-flight is found to be negligible, unless very small length scale are considered---which are usually out of reach anyway because of the finite resolution of the imaging system.

Our assessment is that this article is sound, pedagogical (for the sections 2 and 3), and addresses an effect which had not been previously studied.
Given that the analysis of matterwave interference patterns is a widely spread tool to access subtle physical phenomena, investigating in detail the limitations and the domain of validity of the method employed previously is of high importance. We therefore think that this article deserves publication in SciPost.

Still, we have several comments which we would like to share with the authors.
We would also like to draw the authors attention on the presence of some grammatical mistakes, among which the frequent absence of a definite article (“the”). Also, the discussion in section 4 is sometimes hard to follow.

1. On p. 2, the statement “If the trap is switched off rapidly, the dynamics are well approximated by a quench into free evolution” should probably include the requirement of a rapidly decreasing density, which allows one neglecting the effect of interactions during the expansion.

2. Before Eq. (6), the authors say that Eq. (5) can hardly be used as a fit function. However, intuitively, on can a priori extract the phases $\phi_+$ and $\psi_-$ from the interference pattern. The reason is that the interference pattern contains 3 z-depend functions---a linear total density, a contrast $C'$ and a relative phase $\phi_{\rm mes}(z)$---while, assuming that the initial density fluctuations are negligible, there are only two z-dependent functions to retrieve. This issue is discussed a bit in the conclusion but the authors might want to elaborate on this a bit more.

3. From Eq.(5), it is clear that for a given $z$, one expects a density pattern of the form
given by Eq.(7), although $\Delta \phi_-$ might not be given by Eq. (9). This form is not restricted to the second order expansion made to derive Eq.(9). We therefore don't think that it is appropriate to write, after Eq. (7): “This demonstrates the robustness of the transverse fit formula.” This transverse fit formula is exact, within the approximations made to derive Eq. (5), and it does not depend on the expansion of $I$.

4. The formula in Eq. (9) is very nice and it is a that pity the authors do not propose to use it to give analytical predictions for clouds at thermal equilibrium. Then one could fit the interference pattern with 2 temperatures: one for the common modes and one for the relative modes. It seems that predictions for the situations considered in this paper could be done using this formula. Such a prediction could be compared to numerical calculation to test the second order expansion approximation.

4bis. The caption of Fig. 3 says that the solid lines are fitting curves based on Eq. (9). Why fitting the data instead of computing the prediction from Eq. (9) using the known input parameters?

4ter. As explained in the text, Eq.(9) is only valid if one considers wavelengths much larger than $\hbar/\sqrt{t}$. It does not capture phenomena involving wavelengths of the order of, or smaller than
$\hbar/\sqrt{t}$. In particular it does not capture the oscillating feature investigated in section 4.4.

4four. Eq. (9) is tested numerically in Fig. 3. However, it is not used in the rest of the paper, when the reconstruction of physical quantities are considered. We think it would be instructive to compar...

5. In the caption of Fig. 3, we are not sure to understand the meaning of the sentence “Numerical errors have been accounted for by subtracting the phase error using the transversal model in both encoding and decoding.” Could the authors maybe rephrase their remark?

6. Below the title of section 4, the initial state of the system is said to be given by a certain Hamiltonian. The authors probably means that it is given by the ground state of this Hamiltonian.

7. In Eq. (11), $\delta \rho$ is not defined.

8. On p. 8, we found the description of the method used to independently sample the relative and common phase profiles difficult to understand without looking into the given references.

9. In section 4.2, the authors discuss the retrieval of the the full distribution function $P(\xi)$, as done in ref. [13]. Following this reference, they claim that $P(\xi)$ “provides unambiguous signatures of quantum fluctuations”. However we know from [Mazets, Phys. Rev. Lett. 105, 015301 (2010)] that the measurements reported in [13] are actually compatible with thermal expectations.

10. In section 4.4, the authors use the terminology “mean occupation number” for the Fourier spectrum of the phase. We think that this terminology is misleading since it conveys the idea of an occupation number for a quantum bosonic mode, which is not the case here: the model is fully classical without any zero-point of motion and quantization of its energy. This approach is relevant only for experiments where the modes which are probed are highly occupied.

11. In section 4.4, the notation $k$ for the longitudinal Fourier space coordinate of the phase might be confusing because this symbol is already used to denote the wavevector of the interference pattern in the vertical direction. The authors should probably use another symbol, like $q$.

12. The right-hand side of Eq. (15) has the dimension of the a length, whereas, according to the definition given above Eq. (15), $\langle|\phi_k|^2\rangle$ is dimensionless. In this definition, the factor $1/L$ should probably be replaced by $1/\sqrt{L}$.

13. If the phase fluctuations present in the initial system are Gaussian, then the same information is contained in $\langle\cos(\psi_-(z)-\psi_-(z'))\rangle$ (section 4.1) and in $\langle|\phi_k|^2\rangle$ (section 4.4). The main difference between the two observables is that the first one mixes different modes, while the second one treats each mode separately. The authors might want to mention this subtlety in their article, and comment on the relative merit of both observables..

14. As far as we understand, the oscillations of a given Fourier component discussed in section 4.4 are not captured by the correction to the phase given in Eq.(9). Indeed, let us assume that initially only a single mode of wavevector $q$ is present, with a low amplitude, such that the semiclassical field is approximately given by the sum of a constant and a term proportional to the phase:
$$\psi_{1/2}(z )\simeq \sqrt{n} \pm i\sqrt{n}\Theta\cos(qz),$$
with $\Theta \ll 1$.
The relative phase pattern before the time-of-flight is thus
$$\theta_-(z,t=0)=\Theta \cos(qz).$$
Then, one finds that the relative phase between the two clouds after the time-of-fight is
$$\theta_-(z,t)=\Theta \cos(qz)\cos(\hbar^2q^2/(2t)).$$
Thus one finds that $|\phi_q|^2$ oscillates as a function of the time-of-flight, similar to the Talbot effect in optics. Such an oscillation is not captured by Eq.(9), which is of higher order in the phase, and which results from the non linearity of the atomic field with respect to the phase. Eq.(9) is only valid if one consider wavelengths much larger than $\hbar/\sqrt{t}$. We think that this could be explained in the text.

15. In section 4.5, the authors investigate the extraction of higher order correlation functions of the relative phase for a system described by the Sine-Gordon Hamiltonian. The authors speak about “many-body problem”. We don't think that this terminology is appropriate here since, in the context of this work, the physics is that of a classical field whose Hamiltonian---the Sine-Gordon Hamiltonian---is non linear in the field $\phi_-$. We would therefore rather say “non linear field theory”.

16. In section 4.5, when analyzing the 4th-order correlation function, the authors explain that “the disconnected part appears to be considerably modified by TOF”. We don't understand how this disconnected part, which involves only a second-order correlator, can be “considerably modified” when the changes in the second-order correlation itself are modest.

17. The value of the parameter $q$ should be indicated in the legend of Figs. 10 and 11.

18. We find the term “image processing” (title of section 5 plus a few other places) misleading. This term usually refers to the analysis or treatment of a digital image, while the authors rather think of how the the imaging process affects the measured density distribution.

Ask for minor revision

1.This paper revisits the study of extraction of the relative phase between two atomic Bose-Einstein condensates under expansion, characterizing the role of longitudinal dynamics during the process.
2.The manuscript is clearly presented, sound and rather extensive.
3.The manuscript is presented in a manner which could be useful to experimental group analysis in the future.

None

This is an interesting and detailed manuscript investigating the role of longitudinal dynamics during the extraction of the relative phase of two one-dimensional Bose gases. The papers is clearly set in the existing literature, and its aims are very clear. The paper performs a detailed analysis of the impact on different measurable quantities, and also comments on issues related to image processing, making it a very thorough study.

I recommend acceptance of this manuscript.

The only comments I have (after having also seen the other referee report and author response) are the following:

1.How important would the authors expect the role of density fluctuations (ignored in this work) be, and how far do they believe the current findings can be pushed under experimental conditions in the presence of density fluctuations? Perhaps some further discussion on this would be useful.

2.Given that expansion imaging is the standard method of extraction of information, could the authors perhaps comment (more) on the relevance of their findings to:
(i) Ref. [32] (which is duplicated as Ref. [42]): in particular, and noting the existing analysis, would the authors still expect the originally published results to hold reasonably (for all probed observables, most notably the higher ones)? I think the discussion / re-interpretation of that work could have been more extensive/transparent (i.e. whether any key findings are now being cast into question).
(ii) https://doi.org/10.1103/PhysRevLett.130.123401 studied the 1D-3D crossover through expansion imaging and extremely thorough numerics. Can the present analysis also say something about such work? Is there anything that might, for example require re-interpretation?

3. Finally, I was wondering whether the authors were aware of the below manuscript, and -- if not -- whether they thought that some of the presented analysis might be useful in their work:
Piotr Deuar, Comput. Phys. Commun. 208, 92 (2016)
10.1016/j.cpc.2016.08.004 (arXiv:1602.03395 )
A tractable prescription for large-scale free flight expansion of wavefunctions

Publish (easily meets expectations and criteria for this Journal; among top 50%)

1. Introduces improvements of currently used method of relative phase measurement.
2. Provides a detailed derivation and strong arguments supporting the new formula for phase estimation.
3. Reveals differences in using the full expansion formula with respect to transversal fit formula in a number of physical quantities.
4. Identifies approximations assumed in the new formula, whose impact will hopefully be addressed in a future work.

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This work addresses one of the fundamental measurement in the field of ultracold gases, namely the relative phase extraction from interference patterns between freely expanded matter waves. The authors revised currently used methods used to estimate the phase and proposed an extended, more accurate formula, which includes previously neglected influence of longitudinal expansion on the result of measurement. The 'full expansion formula' (eq. 5) derived in this work is then compared to the 'transversal fit formula' in a reconstruction of different physical quantities.
In general I find the manuscript to be well written and organized. Results of this work are of a high importance, especially for improvement of experimental measurements. Therefore, I recon this work should be published in a high-impact journal.

Publish (easily meets expectations and criteria for this Journal; among top 50%)