# Projective phase measurements in one-dimensional Bose gases

### Submission summary

 As Contributors: Fabian Essler · Jörg Schmiedmayer · Yuri Daniel van Nieuwkerk Arxiv Link: https://arxiv.org/abs/1806.02626v1 (pdf) Date submitted: 2018-06-13 02:00 Submitted by: van Nieuwkerk, Yuri Daniel Submitted to: SciPost Physics Academic field: Physics Specialties: Quantum Physics Approaches: Experimental, Theoretical

### 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 a compact phase field $\hat{\phi}(x)$. By elaborating on existing results in the literature we discuss how a projective measurement of the particle density after trap release is related to the eigenvalues of the vertex operator $e^{i\hat{\phi}(x)}$. We emphasize the theoretical assumptions underlying the analysis of interference patterns and show how, in certain limits, such measurements give direct access to multi-point correlation functions of $e^{i\hat{\phi}(x)}$.

### Ontology / Topics

See full Ontology or Topics database.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 1806.02626v2 on 5 September 2018

Submission 1806.02626v1 on 13 June 2018

## Reports on this Submission

### Report 2 by Karen Kheruntsyan on 2018-8-13 (Invited Report)

• Cite as: Karen Kheruntsyan, Report on arXiv:1806.02626v1, delivered 2018-08-13, doi: 10.21468/SciPost.Report.553

### Strengths

1. Pedagogical
2. Clearly written
3. Clarifies aspects of previous works in a stand-alone article, which will be useful to nonspecialists

### Weaknesses

1. No major weaknesses, except for some minor issues with definitions.

### Report

The authors have revisited the theoretical description of observables involved in time-of-flight measurements of particle density of coherently split one-dimensional Bose gases. They provide a nice piece of pedagogical account of how such measurements relate to the eigenvalues of the vertex operators of the phase filed and how they can be used to extract multi-point correlation functions of vertex operators. The authors also clearly elaborate on the theoretical assumptions underlying the analysis of interference patterns emerging in these experiments. The paper is clearly written and in my opinion warrants publication in SciPost. I feel, however, that it can be further strengthened if the authors address the following comments and suggestions:

### Requested changes

1. The authors use units in which $\hbar=1$ right from the start of the article. It would be nice if this was clearly stated.

2. After Eq. (15), the authors express the healing length $\xi$ in terms of $v$ which is not defined. The speed (presumably of sound) v is defined later in the appendix, but it would be good if it was also defined (at least spelled out) in the main text as well.

3. Similar suggestion applies to the Luttinger liquid parameter K in Eq. (16), and the tunnel-coupling strengths $\lambda$ and $\lambda'$ in Eqs. (18) and (19), which are not defined.

4. The authors carefully examine the cases when the longitudinal expansion can be neglected or when it is included in the analysis. The term "longitudinal expansion" is used throughout the papers, however, the authors never spell out what such an expansion physically actually amounts to in their model, given that the original Hamiltonian, Eq. (13), is formulated as a uniform 1D system of FIXED length L with periodic boundary condition. Given that in real experiments, "longitudinal expansion" refers to an actual physical expansion of a finite-length inhomogeneous gas (initially trapped longitudinally as well, by a harmonic trap), it would be good to clarify for nonspecialist readers what "longitudinal expansion" (phase dynamics, current flow along $x$?) refers to in here, and how this can be incorporated within the framework of the local density approximation which the authors mention towards the end of the paper.

• validity: high
• significance: high
• originality: ok
• clarity: high
• formatting: excellent
• grammar: excellent

### Author:  Yuri Daniel van Nieuwkerk  on 2018-09-03  [id 315]

(in reply to Report 2 by Karen Kheruntsyan on 2018-08-13)

We thank the referees for their helpful comments. We have addressed the various points raised by the referees as follows:

(1) We have added a statement that we use units in which $\hbar=1$.

(2) and (3) We have clarified our definitions of the speed of sound $v$, the Luttinger parameter $K$ and the tunnelling strength $\lambda$.

(4) We have clarified what we mean by “longitudinal expansion” and added a comment explaining why this remains a meaningful concept even though we use periodic boundary conditions for simplicity.

### Anonymous Report 1 on 2018-7-18 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1806.02626v1, delivered 2018-07-18, doi: 10.21468/SciPost.Report.538

### Strengths

1 - very well written

2 - pedagogical

3 - practical

### Weaknesses

1 - original results not clearly stated

### Report

The paper describes interference of two quasicondensates after their release from parallel one-dimensional traps assuming no interaction during the time of flight and no trapping in the longitudinal direction. The authors relate the density profile after the expansion with the initial state of the phase and density fluctuations. They argue that such an interference experiment is a tool for extracting multi-point correlation functions. Effects of the longitudinal dynamics, which complicate this extraction, are analyzed in detail.

The paper is easy to read. The authors present their derivation step-by-step, clearly explaining their motivation, methods, and underlying assumptions. I recommend publication after minor revision. My comments and suggestions are listed below.

### Requested changes

1) My main concern is that although the paper reads smoothly, it is difficult to tell new results from the state of the art. Can the authors be more specific about their original contribution? The abstract is particularly not informative in this respect.

2) How good is the assumption that the atoms are noninteracting after the release? Is there a quantitative condition for this? I think that for excitations at a sufficiently high momentum the drop of the mean-field interaction may seem almost adiabatic.

3) Choose either $\mathfrak{R}$ or $\rm Re$ to denote the real part. Right now both are used (see, for example, Eqs.(45) and (49)).

• validity: top
• significance: high
• originality: ok
• clarity: top
• formatting: excellent
• grammar: perfect

### Author:  Yuri Daniel van Nieuwkerk  on 2018-09-03  [id 314]

(in reply to Report 1 on 2018-07-18)

We thank the referees for their helpful comments. We have addressed the various points raised by the referees as follows:

1) We have changed the abstract and conclusion section in order to make clear what our original contributions are. Our main new results are (i) 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; and (ii) obtaining and presenting results for single-shot projective measurements.

2) The question why interactions after release can be neglected is addressed in Imambekov et al. (2009), to which we have added a reference in the appropriate section.

3) We have uniformized our notations for real parts.