Self-bound droplets in quasi-two-dimensional dipolar condensates

The general analysis based on Gaussian state theory is interesting. The ability to discuss the nature of the condensate is also important. Many of the predictions made make a lot of sense.

The prediction of the critical number for droplet formation is not consistent with the assumed hypothesis, and overall is not really good.

Self-bound droplets in quasi-two-dimensional dipolar condensates
by Yuqi Wang, Tao Shi, Su Yi

Referee Report
==============

Here we have another round of referee reports regarding this work,
which is I said before is interesting although not absent of somewhat
not completely clear points. In any case I summarize my new
comments and suggestions to the authors:

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Original question:

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.

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?

Authors reply:

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:

Nρ1(r,r′)=⟨^ψ†(r′)^ψ (r)⟩=∑iNi¯ϕ∗i(r′)¯ϕi(r),(1)

where ¯ϕi is a set of orthnormal single-particle wave functions well defined
for both ideal and interacting gases and Ni is the single-particle
occupation numbers relative to ¯ϕi satisfying ∑iNi=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
Ni=0=N0 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
|Φ⟩. We then decompose the field operator into

ψ(r)=ϕc(r)+δ^ψ(r),(2)

where ϕc(r)=⟨Φ|^ψ(r)|Φ⟩ and δ^ψ(r) represents the fluctuation
satisfying ⟨δ^ψ(r)⟩=0. The one-body reduced density matrix then becomes

Nρ1(r,r′)=⟨^ψ†(r′)⟩⟨^ψ(r)⟩+⟨δ^ψ†(r′)δ^ψ(r)⟩=ϕ∗c(r′)ϕc(r)+G(r,r′),(3)

where G(r,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.(3) dominates, i.e.,

Nc=∫dr|ϕc(r)|2≈N,

we assume that G(r,r′)=0, which immediately leads to δ^ψ|Φ⟩=0⇒^ψ(r)|Φ⟩=ϕc
(r)|Φ⟩,(4)

indicating that |Φ⟩ is the eigenstate of the field operator ^ψ(r), i.e., a
coherent state.

In the opposite limit where Nc≈0, the one-body reduced density matrix
simplifies to

Nρ1(r,r′)≈G(r,r′)=∑jNs,j¯ϕ∗s,j(r′)¯ϕs,j(r).(5)

If Ns,1≈N, the gas is clearly Bose condensed as the ¯ϕs,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.

If both Nc and Ns,1 are of order N, two spatial modes, ¯ϕc and ¯ϕs,1, are
macroscopically occupied. The corresponding quantum state is a squeezed
coherent state. Interestingly, since the spatial modes for the coherent
and squeezed states are generally different, i.e., ¯ϕc≠¯ϕs,1, the
condensate is then fragmented.

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.

Referee's reply:

At this point the authors have managed to make their point much more clear, so
everything following in the manuscript makes way better sense for the
non-specialist in Gaussian state theory. For that reason I urge the authors
to include exactly this same explanation in the manuscript, maybe at the
beginning of the theory section. The authors should not assume this is fully
understood by the main reader, so they must include that part in the
text -mostly considering there are no (strong) space restrictions in the
Scipost Physics journal.

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Original question:

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 \sigma_{EGPE} and the \sigma^{(c)} curve giving the
same critical number. This is something that should be written in a more
clear way...

Author's reply:

In Fig. 4(b), there are four curves. Among them, σ(N) is obtained by full
numerical calculations of the Gaussian-state theory. Then, to understand
the origin of the W shape on σ(N), we plot σ(s)(N) and σ(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 σEGPE(N).

Reply:

Ok that's good.

Author's reply:

In the manuscript, we showed that σ(N) and σ(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,
σEGPE(N) and σ(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, Ncri, in the manuscript. To start, let us first write down total
energy [Eq. (21) in the manuscript]

ϵ(σ)∝1σ2+[~gs+~gdf(σ)]Nσ2+~g3N2σ4, ...

which consists of the kinetic, two-body interaction, and three-body
interaction energies. The equilibrium radial size can be obtained by
minimizing ϵ(σ). To carry out the analysis, we take into account two
simplifications: i) By noting that Ncri represents the minimal atom number
that can sustain a self-bound state, we may first ignore the last term in
Eq.(6) since it is repulsive and always leads to a larger Ncri; ii) Making
use of the fact that σ is much larger than az (the size of the harmonic
trap along the z direction) for quasi-2D geometry, we may assume that
σ/az→∞ which leads to f(σ)→−2. Eventually, we will find that these
assumptions are consistent with the geometry of the self-bound droplet with
N=Ncri.

Now, under simplifications i) and ii), the kinetic energy exactly cancel out
the interaction energy when the number of atoms satisfies N=12~gd−~gs,(7)

for which ϵ(σ) is invariant with respect to σ, i.e., the self-bound droplet
is in equilibrium for any radial size. We are particularly interested in
the case with σ=∞, for which both conditions i) and ii) are satisfied.
Consequently, Eq.(7) indeed represents the minimal number of atoms in the
self-bound droplet, i.e.,

Ncri=12~gd−~gs, ...

which is independent of the three-body interaction. Nevertheless, Ncri still
depends on the quantum state of the droplet through the reduced interaction
parameters ~gs and ~gd. Particularly, because the quantum state of σ(N) at
the lower N limit is a pure squeezed state, we see that σ(N) and σ(s)
(N) give rise to the same critical atom number.

More generally, because the energy associated all stabilization forces must
decay faster than 1/σ2 as σ 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,
σEGPE(N) and σ(c)(N) lead to the same critical atom number.

In the revised manuscript, we have made it clearer that σEGPE(N) and σ(c)
(N) lead to the same critical atom number.

Referee's reply:

At this point I can only say that this analysis would be very interesting -if
it made sense to me. But it does not make much, unfortunately.
The statement
'To carry out the analysis, we take into account two
simplifications: i) By noting that Ncri represents the minimal atom number
that can sustain a self-bound state, we may first ignore the last term in
Eq.(6) since it is repulsive and always leads to a larger Ncri;'
is unclear to say the least. In principle you can't discard terms like that.
Yes it is repulsive, and yes it tends to increase Ncri, but that only means
the prediction for this value will be larger. Otherwise the authors must show
that their prediction for Ncri including and excluding this term remains
unaffected, which I don't believe is the case. Actually I don't think this is
the right way to proceed...

And of course if you discard this term, the 1/sigma^2 in all other terms can
be canceled in the optimization. That, together with assuming a constant f
(\sigma), leads immediately to the reported prediction for Ncri. But the
authors indicate that f(\sigma) -> -2 when the ratio \sigma/a_z tends to
infinity which, for fixed a_z (that is, fixing the trap), means that \sigma
is very large, which means the particle number must be very large. But this
is not consistent with the main assumption that the starting profile is
Gaussian, which the authors state at the beginning, while they admitted this
approximation does not hold for large, saturated droplets.

All in all I understand the need to provide a theoretical ground for the
critical atom number, but this one is definitely not convincing. I think at
this point that authors should remove that part from their manuscript.

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Original question:

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...

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.

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.

Author's reply:

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.)

Referee's reply:

Ok, that sounds good to me.

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Original question:

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.

Author's reply:

We have added this discussion into the revised manuscript.

Referee's reply:

Ok that's also good.

Remove the discussion of the critical atom number from the manuscript.