Two-Body and Three-Body Contacts for Three Bosons in the Unitary Regime: Analytic Expressions and Limiting Forms

Round 1

Reviewer 1 Report

This is a very appealing, sort of self-contained set of results that give analytical results for two- and three-body contacts for a system consisting of three harmonically trapped bosons.  I find it well-written and likely quite useful in guiding calculations and estimates to be made for bosons in other contexts.  Particularly appealing is that the article starts essentially from the beginning, reviewing in Sec. 2 a lot of the usual Efimov results, so they are present and useful in the rest of the article, yet without completely re-deriving everything.  I recommend to publish this.

I have only a very few comments that the author may wish to address:

After eqn. (10), the three-body parameter R_t is said to be chosen to conform with the wave number of the bound state “in the sense of Ref. [23].”  Can the author elaborate on what sense this is?

In equations (27) and (47), the letter C appears, and this seems to be the definition of contact in the two cases.  Can this be made explicit?

Section  3.1 has two cases that appear to be labeled as the same thing.  My guess is that in Case II the limit taken is to +\infty rather than -\infty. Please check.

The matrix elements in (47) are referred to as a “tensor.” Does this really imply transformation properties of the elements under coordinate rotations?

Author Response

Reviewer 1:

We thank the reviewer for their thoughtful feedback and praise of our manuscript. We have answered each of their comments and questions below as indicated.

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Q1:  After eqn. (10), the three-body parameter R_t is said to be chosen to conform with the wave number of the bound state “in the sense of Ref. [23].”  Can the author elaborate on what sense this is?             

A1:  In Ref. [23], the universality of the three-body parameter for neutral atoms is analyzed.  Here, “universality” is used to indicate the scaling of the three-body parameter solely with the van der Waals length, r_vdW.  In Ref. [23], the authors found a value for the wavenumber of the ground-state trimer at uniarity \kappa_*=0.226/r_{vdW}.  In our manuscript, we relate R_t with \kappa_* below Eq. (12), and this is how R_t is chosen.  To make this clearer in our manuscript we have moved all discussion of the choice of R_t to below Eq. (12) and added:

“Setting $j=0$ in the above equation gives $R_t=\sqrt{2}\exp(\text{Im}\ln[\Gamma(1+is_0)]/s_0)     /\kappa_*$ in terms of the wave number of the $j=0$ trimer $E_\mathrm{3b}^{(0)}=-\hbar^2\kappa_*^2/m$. We now choose the three-body parameter $R_t$ such that $E_\mathrm{3b}^{(0)}$  matches the ground-state trimer energy at unitarity for neutral atoms studied in Ref.~\cite{PhysRevLett.108.263001}. In that work, the universality of $\kappa_*$ with the van der Waals length was established, finding $\kappa_*=0.226/r_\mathrm{vdW}$.”

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Q2:  In equations (27) and (47), the letter C appears, and this seems to be the definition of contact in the two cases.  Can this be made explicit?

A2:  We assume that the reviewer is referencing Eq. (46).  In both cases, the additional subscripts indicate to which eigenstate and channel the calculated C2 or C3 corresponds to.  This was indicated explicitly above Eq. (30) for C3 but not for C2.  To clarify matters, we have added text to explain what the superscripts for C3 and C2 mean immediately after their first use in Eq. (27) and (46):

Below Eq. (27):  “In the above equation, we have indexed the extensive three-body contact specific to the eigenstate $(s,j)$ as $C^{(s,j)}$.”

Below Eq. (46):  “where we have indexed the extensive two-body contact specific to the eigenstate $(s,j)$ as $C^{(s,j)}$.”

Additionally, we have updated all instances of C_3^{(j)} to C_3^{(s_0,j)} and E_3b^{(j)} to E_3b^{(s,j)} to emphasize that the superscripts are an index.

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Q3:  Section  3.1 has two cases that appear to be labeled as the same thing.  My guess is that in Case II the limit taken is to +\infty rather than -\infty. Please check.

A3:  The referee is correct, and we thank them for catching this typo.  Case II has been updated accordingly.

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Q4:  The matrix elements in (47) are referred to as a “tensor.”  Does this really imply transformation properties of the elements under coordinate rotations?

A4:  We do not imply any special transformation properties of the matrix elements in (47). The A^{s,s’}_{j,j’} are multi-dimensional arrays where the overlaps of various hyperradial eigenfunctions are tabulated.  We have updated the text to refer to the A’s as “arrays” and not “tensors.”

Reviewer 2 Report

In this manuscript, Colussi first reviews the derivation of three-boson wavefunctions in a harmonic trap at unitarity, given in Werner et al. Subsequently, analytic expressions for the two- and three-body contacts as a function of the ratio of eigenenergy to trap frequency are obtained from the spatial form of these wavefunctions, before being compared with numerical results from Blume et al. which incorporate finite-range corrections. Good agreement is found. 

The paper makes a useful contribution to the literature. I have several comments on how the manuscript might be improved. 

- In general, the description of the Figures is too brief. For example, in Fig 1, what does the gray shaded rectangle in the left panel indicate? Furthermore, it is stated in the caption that j=0 labels the ground Efimov trimer, but in the figure this label is not placed next to the lowest energy data point. This is confusing. 

- The agreement between the trapped-system and free-space results should be good when the spatial extent of the trimers is much smaller than the trap lengthscale - this is indeed found. However, too little attention is given in Figure 1 to the more relevant (to this paper) regime of -1<E_b/\omega<0. Only one data point visibly lies away from the free-space limit. Could the plotted range not be adjusted to give more information on the limit of trimer size > trap lengthscale? Additionally, some more discussion of the influence of the trap on the strength of three-body correlations in this regime would be interesting (e.g. why the trapped system shows stronger correlations than the free space system). 

- It would be helpful for readers less familiar with the literature to more clearly describe the role of different lengthscales. E.g. the potential range is introduced to the zero-range model in order to fix the trimer energy. But the model remains zero-range in the sense that one considers only contact potentials. In contrast, the results from Ref [18] also modify the wavefunctions themselves, by using a more realistic finite-ranged potential. 

- Could the results for C2 obtained for three bosons be compared with the results for two bosons in a trap? The discrepancy between the two would directly reveal the first order ‘beyond two body’ contribution to the two-body correlations. 

- A clearer description of what is meant by ‘universal’ and ‘Efimov’ channels would be helpful. These expressions first appear in line 106 with little explanation. 

- The universal relations governed by the contact rely on a zero-range model. Therefore the discrepancy between the analytic zero-range results of the manuscript and the finite-range numerical results would depend on exactly how the contacts are defined and obtained in Blume et al. Some comment on this might be worthwhile. 

Author Response

Reviewer 2:

We thank the reviewer for their thoughtful feedback and insightful questions that we feel have helped improve the manuscript considerably.  We have answered their comments and questions below as indicated.

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Q1:  In general, the description of the Figures is too brief. For example, in Fig 1, what does the gray shaded rectangle in the left panel indicate? Furthermore, it is stated in the caption that j=0 labels the ground Efimov trimer, but in the figure this label is not placed next to the lowest energy data point. This is confusing.

A1:  In both Figs. 1 and 2, the grey shaded region indicates the j=0 eigenstates for the four different traps studied, and this information is absent in caption by mistake. We thank the referee for their careful reading.  We have therefore added to the captions of Figs 1 and 2 the following:

In Fig. 1:  “The $j=0$ indicates the ground-state trimer in the free-space limit in the sense of Eq.~\eqref{eq:jnegative} and are contained in the grey shaded region.”

In Fig. 2:  “The grey shaded region indicates the $j=0$ eigenstates.”

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Q2:  The agreement between the trapped-system and free-space results should be good when the spatial extent of the trimers is much smaller than the trap lengthscale - this is indeed found. However, too little attention is given in Figure 1 to the more relevant (to this paper) regime of -1<E_b/\omega<0. Only one data point visibly lies away from the free-space limit. Could the plotted range not be adjusted to give more information on the limit of trimer size > trap lengthscale? Additionally, some more discussion of the influence of the trap on the strength of three-body correlations in this regime would be interesting (e.g. why the trapped system shows stronger correlations than the free space system).

A2:  We thank the referee for this insightful question.  To highlight this regime -1<E_b/\omega<0, we have added an additional panel to Fig. 1 showing the behavior of the three-body contact over this range.  Additionally, we have updated the text below Eq. (37):

“For $|E_\mathrm{3b}|/\hbar\omega<1$, we find that $a_\mathrm{ho}^2C_3$ as a function of $E_\mathrm{3b}/\hbar\omega$ interpolates smoothly between the limiting behaviors as shown in Fig.~\ref{fig:c3}(b).”

We have also added an additional panel to Fig. 2 to show the behavior of the two-body contact in the Efimov channel over this range.  The following text has been added to the 2^nd to last paragraph of Sec. 4.2:

“Additionally, in \ref{fig:c2}(d) we show the behavior of $a_\mathrm{ho}C_2^{s_0,j}$ as a function of $E_\mathrm{3b}/\hbar\omega$ in the region $| E_\mathrm{3b}|/\hbar\omega<1$. Here we find, as in Fig.~\ref{fig:c3}(b), a smooth interpolation between limiting behaviors.”

The referee is correct that the trapped system leads to a larger three-body contact than in free space. Unsurprisingly, we also find this to be the case for the two-body contact.  The following discussion has been added to the manuscript:

Two paragraphs before Sec. 4:  “In general we find that $C_3^{(s_0,j)}$ for a particular Efimov trimer is increased compared to the free space limit [see Eq.~\eqref{eq:c3deepbound}]. We echo the conclusion of Ref.~\cite{PhysRevA.97.033621} that the restriction of a trimer to a reduced region of space should correspond to an increase in the probability to find three bosons at short distances as encoded in $C_3^{(s_0,j)}$.”

Final paragraph of Sec. 4.2:  “When comparing to the free-space result in Eq.~\eqref{eq:c2efimov} for an Efimov trimer, we find that the trap leads to a relative increase of $C_2$. Tighter traps increasingly confine the atoms, and therefore it makes sense that the probability to find correlated pairs increases accordingly.”

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Q3:  It would be helpful for readers less familiar with the literature to more clearly describe the role of different lengthscales. E.g. the potential range is introduced to the zero-range model in order to fix the trimer energy. But the model remains zero-range in the sense that one considers only contact potentials. In contrast, the results from Ref [18] also modify the wave functions themselves, by using a more realistic finite-ranged potential.

A3:  To make this clearer, we have added the following passage under Eq. (13) in order to distinguish between the two approaches:

“Furthermore, although $R_t$ depends on $r_\mathrm{vdW}$, the model remains zero- range involving only contact potentials. We contrast this with the approach of Ref.~\cite{PhysRevA.97.033621}, which used more realistic finite-ranged potentials.”

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Q4:  Could the results for C2 obtained for three bosons be compared with the results for two bosons in a trap? The discrepancy between the two would directly reveal the first order ‘beyond two body’ contribution to the two-body correlations.

A4:  This is a very interesting question.  However, here we study the two-body contact in a different basis than, for instance, the Busch wave functions (Found. Phys. 28, 549 (1998)), and so we cannot compare directly with the two-body contacts in that base.  It would be interesting to investigate this in the future perhaps for the ground-state energy of a many-body system or three-body corrections to the growth of the dynamical two-body contact for a quenched Bose gas.  The author has just submitted a preprint on this latter topic, which is now referenced in the conclusion of the present manuscript.

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Q5:  A clearer description of what is meant by ‘universal’ and ‘Efimov’ channels would be helpful. These expressions first appear in line 106 with little explanation.

A5:  We agree that this language should be made clearer and defined earlier in the text.  Therefore, in Sec. 2, we have added the following:

Below Eq. (8):  “We label the channels with $s^2>0$ as universal because they depend solely on the trapping parameters and not on any microscopic details.”

Below Eq. (9):  “The Efimov channel depends on the trapping parameters and also on $R_t$, which is sensitive to the microscopic length scale $r_\mathrm{vdW}$. The Efimov channel is therefore not universal.”

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Q6:  The universal relations governed by the contact rely on a zero-range model. Therefore the discrepancy between the analytic zero-range results of the manuscript and the finite-range numerical results would depend on exactly how the contacts are defined and obtained in Blume et al. Some comment on this might be worthwhile.

A6:  We have updated our discussion of the results of Blume et al. to indicate how those results were obtained.  The first paragraph of the conclusion now ends with the following discussion:

“In Ref.[18], however, the contacts were obtained from their operational definitions [2,3]

\begin{align}

\left(a\partial_{a}E_\mathrm{3b}^{(s,j)}\right)|_{\kappa_*}&=\frac{\hbar^2}{8\pi m a}C_2^{(s,j)}, \\

\left(\kappa_*\partial_{\kappa_*}E_\mathrm{3b}^{(s,j)}\right)|_a&=-\frac{2\hbar^2} {m}C_3^{(s,j)},

\end{align}

by taking the derivatives of three-body eigenenergies with variations of $a$ and $\kappa_*$. This amounted to changing the underlying parameters of the finite-range Gaussian potentials and approximating the derivatives via finite differencing. In Table 1, we compare against results in that work, finding good agreement for the eigenenergies and contacts to within a few percent or less. In Ref. [18], the convergence of eigenenergies was estimated as $10^{-3}E_\mathrm{ho}$ or better. The eigenenergies calculated in the present work can be calculated from Eq. (10) to arbitrary precision. We attribute the disagreement therefore to the absence of finite-range effects in the zero-range model in the present work.”