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Self-bound droplets in quasi-two-dimensional dipolar condensates
by Yuqi Wang, Tao Shi, Su Yi
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Yuqi Wang |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2112.09314v4 (pdf) |
| Date submitted: | 2023-09-07 04:06 |
| Submitted by: | Wang, Yuqi |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We study the ground-state properties of self-bound dipolar droplets in quasi-two-dimensional geometry by using the Gaussian state theory. We show that there exist two quantum phases corresponding to the macroscopic squeezed vacuum and squeezed coherent states. We further show that the radial size versus atom number curve exhibits a double-dip structure, as a result of the multiple quantum phases. In particular, we find that the critical atom number for the self-bound droplets is determined by the quantum phases, which allows us to distinguish the quantum state and validates the Gaussian state theory.
List of changes
1. We have removed several acronyms such as CS and HFBT.
2. We have used Sec. II.C, especially the last two paragraphs, to discuss the properties of the quantum phases.
3. In Fig.4(b), we have added the $\sigma_{\rm EGPE}$-$N$ curve obtained by numerically solving EGPE.
4. We have added one paragraph at the end of Sec. III.C to discuss the EGPE results for the radial size.
5. Reference [49] is added for the validity of the Gaussian density profile.
Current status:
Reports on this Submission
Report
> Self-Bound Droplets in Quasi-Two Dimensional Dipolar Condensates
> by: Yuqi Wang et al.
Referee Report
==============
Here we have a second, revised version of the original work submitted to
Scipost Physics. After some recommendations made by the referees, the
authors have performed a series of changes in order to improve their
original work.
While as before I find their work interesting, there are still points that
have to be commented in a more convincing way. I go through the original
raised comments/questions and the response received from the authors:
>> 1- A frst initial comment is that I think the authors use too
>> many acronyms in the text: SVS, SGS, CAN, GST, CS, SCS,
>> HFBT... I understand acronyms are often used in the scientific
>> literature, but I have found this to be a little bit excessive in
>> this work, making the manuscript di�cult to read at some
>> points. I think the authors can improve that.
>>
> We thank the referee for this advise. In the revised manuscript,
> we have remove several acronyms and only retained SVS, SCS,
> CAN, GST, and EGPE.
Ok that's good.
>> 2- Regarding the Gaussian States theory and its review in
>> section II.B, the authors develop the mathematics of the
>> model but do not discuss the physics behind it, as said above.
>> For that reason it is dificult for a non-specialist in Gaussian
>> State Theory to appreciate most of the results derived on the
>> draft. Since there is no a priori restriction on the length of the
>> works being published in this journal, I urge the authors to at
>> least explain what the physics behind each of the relevant
>> states being discussed is.
>>
> We thank the referee for the suggestion. In the revised
> manuscript, we have used Sec. IIC to explain the physics behind
> the quantum states. Speci�cally, we present the criterions for the
> quantum phases and summarize their properties (e.g., the
> second-order correlation function and atom number
> distribution). These discussions can help the readers to capture
> the physics of the quantum phases.
This is a very interesting discussion that in any case raises also some
relevant questions that were already unanswered in the first version. The
authors discuss the nature of the Squeezed States and state that these
contribute to the second order correlation functions of the fluctuation field
operators, and have Nc/N approx 0. On the other hand, coherent states have
Nc/N aapprox 1. In physical terms, what are really squeezed states?
Or put in a better way: what do squeezed states have to do with a
Bose condensate? If Nc/N approx 0, then there is no condensate at all and
I don't see the point in the discussion. This discussion is what I meant
when I asked for the 'physics behind it'. Stating that the difference
appears in the contribution to the correlation functions is interesting
but not enough to my understanding.
>> 3- Related to the previous point, for instance in Fig.1 the authors present
>> the dependence of the coherent fraction fc as a function of the number of
>> particles N, where fc = Nc/N with Nc the occupation of the coherent mode.
>> What is the relevance of that plot exactly, what are the physical
>> implications behind that? Actually the most relevant question would be: what
>> does a larger or lower Nc tells us about the physical structure of the Bose
>> condensate? Is it related in any way to the number of particles occupying it?
>>
> We thank the referee for the careful reading of our manuscript. Before we
> explain the physical implication of fc, we would like to point out that the
> purpose of Fig. 1 is to validate the quasi-2D model, which can be used to
> dramatically reduce computational complicity. The results regarding the
> physical implication of fc are presented in Fig. 2 where fc acts as the
> order parameter that determines the quantum phases of the system. For the
> question about Nc, we frst note that it is indeed the occupation number in
> the coherent mode ϕc.
>
> Now, let us explain how fc reveals the structure of a Bose condensate.
> • For a conventional condensate, we have Nc ≈ N implying
> that majority of the atoms are condensed in the coherent
> mode. The Ns (≪N) uncondensed atoms are depletion
> described by a multimode squeezed state with mode ϕs,j
> being occupied by Ns,j atoms. Here the total number of
> squeezed atoms is Ns = ∑_j Ns,j .
> • In the opposite limit with a single squeezed mode being
> macroscopically occupied, i.e., Nc/N ≈ 0 and
> Ns,1/N ≈ 1 , the condensate is in a macroscopic squeezed
> vacuum state which has a completely diferent statistical
> property compared to the coherent state, e.g., the second-
> order correlation function and the atom number
> distribution [1-3].
> • For the intermediate case with the coherent and a single
> squeezed modes being macroscopically occupied (i.e.,
> Nc/N ≠ 0 , Ns,1/N ≠ 0, and Ns,j>1 ≈ 0 ), the
> condensate is in the squeezed coherent state.
> In the revised manuscript, we have summarized these results in
> Sec. IIC.
Ok at this point it looks like we are reaching something,
as this answer is also related to the point 2 I raised before.
To summarize my questions 2) and 3), the authors must
discuss a very relevant point:
what is the exact relation between a standard Bose condensate,
and coherent vs squeezed states? A typical Bose condensate
contains a macroscopic number of particles in the same
quantum states, so does one have to identify a standard
Bose condensed system with a coherent state?
>> 4- In the same fgure the authors show the radial size \sigma
>> of the system as a function of the particle number. How does
>> that result compare to what one would get with a standard
>> (extended)-Gross Pitaevskii calculation? The later is the
>> reference for most of the calculations on these kind of
>> systems today, so an actual comparison is in order. Not just for
>> this quantity, but also for many others along the paper that
>> can be computed in both ways, for instance the Critical Atom
>> Number, which is known to be a rather delicate quantity to
>> reproduce accurately, see point 6
>>
> We thank the referee for the suggestion. Since the results
> presented in Fig. 1 are merely the demonstrations for the
> agreement between the full 3D and reduced 2D calculations, we
> have added the results for σEGPE (radial size numerically
> computed using EGPE) versus N in Fig. 4(b) where the physics of
> the results are also discussed.
> From the σEGPE - N curve, one may determine the corresponding
> critical atom number which, as shown in Fig. 4(b), is the same as
> that obtained from the σ(c) - N curve. The underlying reason for
> this is because the many-body wave function associated with
> EGPE is also a coherent state.
> In the revised manuscript, we have added one paragraph at the
> end of Sec. IIIC to discuss the EGPE results for the radial size.
I do not follow very well the explanations provided here. By looking at fig4
(b) I see several curves that give a different radial size, but ultimately
they all state that the system has a well defined radial size. Now this size
can be imposed by the external trap confinement, or can be the product of
self-organization. But this is not visible in the plot, so do not see how
that statement about the σGPE and the σ(c) curve giving the same critical
number. This is something that should be written in a more clear way...
>> 5- In the same line of point 4, in Eq.(21) the authors provide an
>> expression for the energy per particle of the system, assuming
>> the density profile is a Gaussian. Once again, they do not
>> provide any plot or evidence that this is indeed the case in the
>> range of parameter values where they use it. A plot of the
>> density profiles, compared to the equivalent GPE result, is
>> required here.
>>
> We thank the referee for the careful reading of the manuscript.
> First of all, we would like to clarify that the validity of the Gaussian
> density profile for dipolar condensates was checked as early as in 2001
> (see, e.g., Ref.[4]). And we also used this assumption for binary droplets of
> K atoms [3]. The general conclusion is that the Gaussian-density-profile
> assumption holds when interactions are weak, or equivalently, when the number
> of atoms is small. As a proof for this, we compare the numerically obtained
> normalized densities n̄(ρ) (i.e., 2π √ πa z ∫ ρdρn̄(ρ) = 1 ) with their
> Gaussian fits in the figure attached to this reply for two direerent atom
> numbers. Indeed, nearly perfect agreement is achieved when the number of
> atoms in the condensate is small. While, for large N , obvious deviation is
> found.
>
The authors were then assuming too many things that were not written in the
original text. It is true that these profiles are Gaussian -but only when the
number of particles is small, as stated in the reply. This should be written
explicitly in the text as it limits the discussion of what comes next.
The provided figure must be included in the text in order for the reader to
understand what is the regime of validity of their approximation. BTW, you
see clear discrepancies between the calculation and the gaussian fit
for N=2x10^4 particles...
> Secondly, in this work, the Gaussian density profile is only introduced
> to qualitatively understand the W-shape σ-N curve, which does not require
> high agreement between the actual and variational density profiles. In fact,
> for our purpose, we only need to show that the σ-N curve is of V shape
> within the range of parameter values for a single quantum phase. Since the
> validity the Gaussian-density-profile assumption is rather clear, it is
> cumbersome to add the comparisons in the present paper. Instead, we cite
> Ref. [49] in the revised manuscript to avoid any confusion. In addition, we
> also emphasize before Eq. (21) that we only perform a qualitative analysis.
>
Once again all that should be explicitely written so as to prevent the
reader from getting the wrong conclusions, probably implied by the absence
of precise information.
> Since the validity the Gaussian-density-profile assumption is
> rather clear, it is cumbersome to add the comparisons in the
> present paper. Instead, we cite Ref. [49] in the revised
> manuscript to avoid any confusion. In addition, we also
> emphasize before Eq. (21) that we only perform a qualitative
> analysis.
>
I can't agree with the sentence 'Since the validity the
Gaussian-density-profile assumption is rather clear' as a general statement,
as I have mentioned above. That only applies to low particle numbers. As I
said before, all that must be explicitly mentioned.
>> 6- The authors finally use this gaussian ansatz to derive a rather simple
>> expression for the Critical Atom Number (CAN) for droplet formation. This
>> is quite an interesting result because there are actual measurements of
>> the CAN for 162Dy and 164Dy atom droplets, performed by the group of
>> Prof.T.Pfau in Stuttgart (see for instance Ref[23] of the manuscript).
>> This is an interesting benchmark because current theoretical predictions
>> from the GPE show orders of magnitude deviations from the measured data.
>> In this sense, a comparison between the actual data, the author's results
>> and the GPE prediction would be a very interesting result.
>>
> We thank the referee for the comment. In the earlier works of ours [2,3], we
> performed systematical comparisons between the calculated critical atom
> number with the experimental data for both three-dimensional dipolar
> droplets of Dy atoms [5] and quasi-two-dimensional binary droplets of K
> atoms [6]. In both works, reasonable agreements were found.
Ok, that can be mentioned in the manuscript.
> In the present work, we propose a new con�guration to generate
> quasi-two-dimensional dipolar droplets which has not been experimentally
> realized yet. Therefore, a direct comparison between our results and the
> experimental data is not applicable. Nevertheless, it would be very
> interesting to check our calculations when the experimental data become
> available.
That's also ok.
I hope all these comments help the authors to produce a final version
that can be published in the Scipost Physics journal. As I understand it,
the work is worth publishing, but only when the comments raised are
properly addressed.
Author: Yuqi Wang on 2023-11-15 [id 4114]
(in reply to Report 1 on 2023-10-16)List of Changes
Response to the comments and questions
To answer the above questions, let us briefly recall the criterion for the Bose-Einstein condensation. For an ideal Bose gas, the Bose-Einstein condensation occurs when a macroscopic number of atoms occupies the lowest-energy single-particle state. While in the presence of atom-atom interactions, the lowest-energy single-particle state is not well defined. To characterize the Bose-Einstein condensation of an $N$-particle system, one has to resort to the one-body reduced density matrix:
$$ N\rho_1(\mathbf{r},\mathbf{r}')=\langle\hat{\psi}^{\dagger}(\mathbf{r}')\hat{\psi}(\mathbf{r})\rangle=\sum_iN_i\bar\phi_i^*({\mathbf r}')\bar\phi_i({\mathbf r}), \tag{1} $$where ${\bar\phi_i}$ is a set of orthnormal single-particle wave functions well defined for both ideal and interacting gases and $N_i$ is the single-particle occupation numbers relative to $\bar\phi_i$ satisfying $\sum_iN_i=N$. Bose–Einstein condensation occurs when one of the single-particle states (hereafter called the condensate, $i = 0$) is occupied in a macroscopic way [1], i.e. when $N_{i=0}=N_0$ is a number of order $N$, while the other single-particle states have a microscopic occupation of order $1$.
Now, to see how a Bose-Einstein condensate is connected to the coherent or squeezed state, we assume the system is in a pure state with wave function $|\Phi\rangle$. We then decompose the field operator into
$$ \hat{\psi}({\mathbf r})=\phi_c({\mathbf r})+\delta \hat{\psi}({\mathbf r}), \tag{2} $$where $\phi_c({\mathbf r})=\langle\Phi|\hat{\psi}({\mathbf r})|\Phi\rangle$ and $\delta \hat{\psi}({\mathbf r})$ represents the fluctuation satisfying $\langle\delta\hat{\psi}({\mathbf r})\rangle=0$. The one-body reduced density matrix then becomes
$$ \begin{align} N\rho_1(\mathbf{r},\mathbf{r}')&=\langle\hat{\psi}^\dagger({\mathbf r}')\rangle\langle\hat{\psi}({\mathbf r})\rangle+\langle\delta\hat{\psi}^\dagger({\mathbf r}')\delta\hat{\psi}({\mathbf r})\rangle\nonumber\\ &=\phi_c^*({\mathbf r}')\phi_c({\mathbf r})+G({\mathbf r},{\mathbf r}'), \tag{3}\label{rho12} \end{align} $$where $G({\mathbf r},{\mathbf r}')$ is the first-order correlation function as defined in the manuscript. Below, we consider three typical cases for the quantum states of a condensate.
- In case the first term in Eq.\eqref{rho12} dominates, i.e.,
$$N_c=\int d{\mathbf r}|\phi_c({\mathbf r})|^2\approx N,$$we assume that $G({\mathbf r},{\mathbf r}')=0$, which immediately leads to
$$ \delta\hat \psi|\Phi\rangle=0\quad\Rightarrow\quad \hat\psi({\mathbf r})|\Phi\rangle=\phi_c({\mathbf r})|\Phi\rangle,\tag{4}\label{cohst} $$indicating that $|\Phi\rangle$ is the eigenstate of the field operator $\hat\psi({\mathbf r})$, i.e., a coherent state.
- In the opposite limit where $N_c\approx0$, the one-body reduced density matrix simplifies to
$$ N\rho_1(\mathbf{r},\mathbf{r}')\approx G({\mathbf r},{\mathbf r}') = \sum_{j}N_{s,j}\bar{\phi}_{s,j}^*({\mathbf r}')\bar{\phi}_{s,j}({\mathbf r}). \tag{5} $$If $N_{s,1}\approx N$, the gas is clearly Bose condensed as the $\bar\phi_{s,1}$ mode is occupied by a macroscopic number of atoms. Moreover, since, according to the Eq. (16) in the manuscript, the corresponding many-body wave function of the gas is a squeezed vacuum state, the condensate is a squeezed one whose statistical properties significantly differ from the coherent condensates.
From above discussion, it is now clear that, in addition to the single-particle spatial mode to which a macroscopic number of atom condense, one should also specify the many-body quantum state of these atoms to fully characterize a condensate.
First of all, we would like to point out that since there is no trapping potential on the $xy$ plane in our configuration, those well-defined radial sizes are all due to self bounding.
In Fig. 4(b), there are four curves. Among them, $\sigma(N)$ is obtained by full numerical calculations of the Gaussian-state theory. Then, to understand the origin of the W shape on $\sigma(N)$, we plot $\sigma^{(s)}(N)$ and $\sigma^{(c)}(N)$ through minimizing Eq. (21) for two different sets of parameters corresponding, respectively, to the pure squeezed and coherent states. Moreover, in order to compare our results with that from EGPE, we plot the numerically computed $\sigma_{\rm EGPE}(N)$.
In the manuscript, we showed that $\sigma(N)$ and $\sigma^{(s)}(N)$ give rise to the same critical atom number. This result, as shown in the manuscript, only depends on the quantum phases (i.e., squeezed or coherent) of the gas and is independent of the stabilization force (the last term in Eq. (21)). The underlying reason is that the radial size of a two-dimensional self-bound droplet can be infinite, for which the energy associated with the stabilization force is negligible as it decays with radial size faster than the kinetic and two-body interaction energies. Following the same argument, $\sigma_{\rm EGPE}(N)$ and $\sigma^{(c)}(N)$ also give rise to the same critical number since they both assume that the condensate is in the coherent state.
For convenience, here we recapitulate the analysis for the critical atom number, $N_{\rm cri}$, in the manuscript. To start, let us first write down total energy [Eq. (21) in the manuscript]
$$ \epsilon(\sigma)\propto \frac{1}{\sigma^2}+\frac{[\tilde g_s+\tilde g_df(\sigma)]N}{\sigma^2}+\frac{\tilde{g}_3 N^{2}}{\sigma^{4}},\tag{6}\label{tote} $$which consists of the kinetic, two-body interaction, and three-body interaction energies. The equilibrium radial size can be obtained by minimizing $\epsilon(\sigma)$. To carry out the analysis, we take into account two simplifications: i) By noting that $N_{\rm cri}$ represents the minimal atom number that can sustain a self-bound state, we may first ignore the last term in Eq.\eqref{tote} since it is repulsive and always leads to a larger $N_{\rm cri}$; ii) Making use of the fact that $\sigma$ is much larger than $a_z$ (the size of the harmonic trap along the $z$ direction) for quasi-2D geometry, we may assume that $\sigma/a_z\rightarrow\infty$ which leads to $f(\sigma)\rightarrow-2$. Eventually, we will find that these assumptions are consistent with the geometry of the self-bound droplet with $N=N_{\rm cri}$.
Now, under simplifications i) and ii), the kinetic energy exactly cancel out the interaction energy when the number of atoms satisfies
$$ N=\frac{1}{2\tilde g_d-\tilde g_s},\tag{7}\label{minum} $$for which $\epsilon(\sigma)$ is invariant with respect to $\sigma$, i.e., the self-bound droplet is in equilibrium for any radial size. We are particularly interested in the case with $\sigma=\infty$, for which both conditions i) and ii) are satisfied. Consequently, Eq.\eqref{minum} indeed represents the minimal number of atoms in the self-bound droplet, i.e.,
$$ N_{\rm cri}=\frac{1}{2\tilde g_d-\tilde g_s},\tag{8} $$which is independent of the three-body interaction. Nevertheless, $N_{\rm cri}$ still depends on the quantum state of the droplet through the reduced interaction parameters $\tilde g_s$ and $\tilde g_d$. Particularly, because the quantum state of $\sigma(N)$ at the lower $N$ limit is a pure squeezed state, we see that $\sigma(N)$ and $\sigma^{(s)}(N)$ give rise to the same critical atom number.
More generally, because the energy associated all stabilization forces must decay faster than $1/\sigma^2$ as $\sigma$ increases, it is then proved that the critical atom number is independent of any stabilization mechanism, including the LHY correction. Finally, because the EGPE is a coherent-state-based theory, $\sigma_{\rm EGPE}(N)$ and $\sigma^{(c)}(N)$ lead to the same critical atom number.
In the revised manuscript, we have made it clearer that $\sigma_{\rm EGPE}(N)$ and $\sigma^{(c)}(N)$ lead to the same critical atom number.
We thank the referee for the suggestion. In the revised manuscript, we have explicitly stated that the assumption of the Gaussian density profile work well only in the weak interaction regime or when atom number is small. In addition, we have also compared the density profiles obtained from the numerical calculations with the corresponding Gaussian fits in Fig. 5. (This figure is also attached to the reply.)
We have added this discussion into the revised manuscript.
Reference
[1] L. Pitaevskii and S. Stringari, Bose-Einstein condensation and Superfluidity (Oxford University Press, Oxford, 2016), Vol. 164. [2] Y. Wang, L. Guo, S. Yi, and T. Shi, Phys. Rev. Research 2, 043074 (2020). [3] J. Pan, S. Yi and T. Shi, Phys. Rev. Research 4, 043018 (2022). [4] S. Yi and L. You, Phys. Rev. A 63, 053607 (2001). [5] M. Schmitt et al., Nature 539, 259 (2016). [6] C. R. Cabrera et al., Science 359, 301 (2018).
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