# Quantum Bose-Fermi droplets

### Submission summary

 As Contributors: Miroslaw Brewczyk · Mariusz Gajda · Tomasz Karpiuk · Debraj Rakshit Arxiv Link: https://arxiv.org/abs/1801.00346v5 Date submitted: 2019-04-25 Submitted by: Brewczyk, Miroslaw Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Quantum Physics

### Abstract

We study the stability of a zero temperature mixture of attractively interacting degenerate bosons and spin-polarized fermions in the absence of confinement. We demonstrate that higher order corrections to the standard mean field energy can lead to a formation of Bose-Fermi liquid droplets -- self-bound systems in three-dimensional space. The stability analysis of the homogeneous case is supported by numerical simulations of finite systems by explicit inclusion of surface effects. Our results indicate that Bose-Fermi droplets can be realized experimentally.

###### Current status:
Editor-in-charge assigned

Dear Editor,

We want to thank you for handling our manuscript.

We are very grateful to the referees who found our work interesting and physically sound.

The present version of our manuscript is modified according to the requests of the Referees.

In particular we performed dynamical simulations showing behavior of the mixture released from the trap for different parameters. The calculations, seemingly simple were quite demanding so we were not able to respond shortly.

Below we give detailed answers to all requests of the three referees.
We hope that the present version of our manuscript will be accepted for publication.

Sincerely Yours,

Debraj Rakshit
Tomasz Karpiuk
Miroslaw Brewczyk
Mariusz Gajda

Answer to the editor’s and referees’ requests and a list of changes

Editor’s requests:
1. Please carefully consider the changes recommended in each of the three referee reports.

Our response:
Ad. 1. Bellow we carefully answer to the all requests and recommendations of every of three referees.
Ad. 2 As suggested by the editor we added the following paragraph commenting on the recent work on Bose-Fermi droplets by S.K. Adhikari:
In a recently published work, Ref. \cite{Adhikari18}, a repulsive short-range three-bosons interactions are added to stabilize the Bose-Fermi mixture. This mechanism was previously suggested in \cite{Blakie16} to stabilize a dipolar condensate. Unfortunately the mechanism is rather difficult to implement because large three-body elastic collisions are typically accompanied by large three-body losses. In addition, in \cite{Adhikari18} it is assumed that fermions are in a fully-paired superfluid state, what in fact makes the system similar to a Bose-Bose rather than to a Bose-Fermi mixture. And finally, the Ref. \cite{Adhikari18} shows that the droplets may consist of bosonic and fermionic atoms in almost equal ratio in contrary to our results indicating a significant domination of the Bose component.

Referee 1 requests the following changes:

The authors should either admit that the term in Eqs. (10) which involves the ‘phase’ of the single fermion pseudo-wave function (which they use to describe the non-adiabatic dynamics of many fermions) is unphysical, or they should explain what physical process it actually represents, and give evidence for this.

In order to meet the referee's demand we added the paragraph:

We would like to emphasize that the fermionic pseudo-wave function has no direct physical meaning. Only the quantities which are the square of modulus of $\psi_F({\bf r},t)$ and the gradient of its phase can be interpreted as physical quantities. The Madelung transformation itself is supported by the Stokes' theorem. Provided that in a given region the condition $\nabla \times \vec v_F=0$ is fulfilled, then the phase of the pseudo-wave function is defined as a curvilinear integral of the velocity $\vec v_F$.

The Referee 2 has the following concerns:

1. Dynamics with one Fermi wave function

I am still concerned that the solid curve in Fig. 3(left) may not be approximately correct, as it uses a single wave function with a single phase for an expanding gas. The behaviour of the solid line Fig. 3(left) cannot be easily compared to the black curve in Fig. 5(right) as the quench, timescales and number of atoms are quite different. In Fig. 5(right), please include dynamical results from the hydrodynamical approach with the same parameters as for the atomic-orbital approach. This will allow easy comparison, as we have for equilibrium results in Fig. 5(left).

We performed additional calculations. We want to add that even the simplistic hydrodynamic approach is quite demanding and time of computation takes several weeks. In order to make a required comparison we decided to show results for shorter than originally presented time of opening and subsequent dynamics. In fig. 4, being a new version of the old figure 5, we show results for a total time of evolution being about 170 times shorter than in the previous one. We substituted the paragraph commenting the numerical results by the following one:

On the other hand, in the right frame of Fig. 4 we compare the dynamical properties of the Bose-Fermi mixture for different values of the mutual scattering length aBF, obtained within the atomic-orbital and hydrodynamic approaches. Here, the trapping potential is removed in 1ms (marked by a vertical line) as in the case of Fig. 3. Similarly to the previous analysis, only for large enough |aBF|/aB (>2.8) a droplet is formed, otherwise we observe an expansion of both atomic clouds. Note however that for |aBF|/aB close to the critical value, there appears a small discrepancy between both descriptions. But it only means that the critical values of |aBF|/aB found within the atomic-orbital and hydrodynamic analyses are slightly different. This is because of relatively large contribution of the surface terms to the total energy for such small systems. These terms are treated on a different footing in both compared methods. Away of the critical value of |aBF|/aB both approaches match perfectly.

2. Three body loss

The calculation for Fig. 4 is too approximate to be useful. The quench of scattering lengths and trap in an experiment will lead to oscillations which will affect loss, there being a trade-off between a fast quench (desirable due to short lifetimes) but stronger oscillations. The experiment will also have noise which has not been added in the calculations. I suggest removing Fig. 4 and coming up with a broad estimate based on the rate coefficient.
Also the paragraph on three body loss is confusing. Please state clearly what you get from where. You are considering a case of aB=250a0 and aBF=−3.6aB=−900a0? The bosonic density of nB=0.0009/a3B=3×1014cm−3 is taken from your calculations (the manuscript says 'From the rate equation')? I see you have used K3=Γ/n2B, but where did your Γ=10/s come from? Please do not state K3 to three significant figures. Please state clearly how you get Γ=50/s from Fig. 4(b) of [29] including how you allow for your values of aB and aBF and for your increased density.

We have added the following discussion:

Finally, we address the issue of a life-time of a Bose-Fermi droplet due to only three-body recombination processes, neglecting all the other sources of atomic loss. A crude estimation of losses could be done based on the loss dynamics of Cs condensate atoms immersed in Li degenerate fermions as observed in Ref. [29], see Fig. 4. For example, for aB= 250 a0 and the range of |aBF|/aB (2.8,3.6), the loss rate Γ=(1/NB) dNB/dt can be extracted from Fig. 4b of [29] assuming the expected Γ ~ aBF^4 dependence. The resulting life-time is in the range 15 ms-30 ms. The finer analysis could rely on solving Eqs. (10) with additional terms, representing losses resulting from Li-Cs-Cs collisions. The right-hand side of Eqs. (10) is then modified by adding (-iK3/2 nBnF) and (-iK3/2 nB^2) terms for bosonic and fermionic components, respectively. Here, K3 is the rate coefficient which is estimated from the rate equation (1/NB) dNB/dt = -K3. For example, for aBF/aB = -3.6 and aB= 250 a0 one has Γ = (1/NB) dNB/dt = 50/s (from Fig. 4b in [29]) and for nB = 4 \times 10^{14}cm^{-3} we find K3 = 3 \times 10^{-28} cm^{6}/s. Now, solving Eqs. (10) gives the life-time of the droplet to be about 45 ms.

Please also discuss how 'The loss rate exceeds the thermalization rate at aBF=−520a0, above which the system no longer reaches thermal equilibrium' [29] relates to your system.

The case of nonequilibrium dynamics requires different approach. We added the following comment:

Our calculations assume a zero temperature case and no heating due to the atom loss. For nonzero temperature, however, in particular when the loss rate exceeds the thermalization rate a dynamical nonequilibrium approach is required which is highly nontrivial and is beyond the scope of the present study.

Referee 3 requests the following changes:

1) The author state in the abstract that "Bose-Fermi liquid droplets -- self-bound incompressible systems" are formed, but then the (in)compressibility of the system is never discussed. The authors should clarify this point, are these systems really incompressible?

We agree with the referee. Compressibility of the BF droplets requires detailed studies. We modified abstract and removed expressions suggesting incompressible character of droplets.

2) The choice of the time unit -- ħ/(mBaB2) -- is very uncommon. Why not using milliseconds or something more immediate to read? I would recommend to change it or explain clearly in the text the reason of this choice.

We decided to express time in units related to the energy related to the scattering length a_B^2 rather than milliseconds. To use milliseconds we should chose some particular value of the Bose-Bose s-wave scattering length. For the exemplary case of Cs-Li mixture used here there are two different regimes of parameters used by the Cheng Chin's group, namely aB=250 (Ref. [29]) or 4 (Ref. [26]) Bohr radii. These two values lead to the very different time scales. This is why we decided not to translate the dimensionless values of time to milliseconds. However, in Figs. 3 and 4b (new version) we included a vertical solid lines which show the moment of time equal to 1ms, so one could easily read the typical time scales
(for aB=250 a0).'

3) The content of Fig. 4 does not justify the need of a figure. Its meaning could be easily explained by adding a text line.

4) Which is the physical origin of the oscillations in Fig. 4right? Have the authors checked that this is not just a numerical effect? Please explain in the text.

The effect is physical. These oscillations are caused by the mismatch of the rate with which the trap is open and the rate of mixture's expansion. The trap is opened very slowly while expansion is very fast, so expanding atoms are back-reflected by the slowly softened trap. This scenario is repeated then.

The Referee 2 asked us to perform the additional calculations allowing for a direct comparison of the expansion obtained on the ground of the pseudo-wave function formalism to the one given by the orbital method. The calculations based on the pseudo-wave function, are quite demanding and require weeks of computations (the advantages of the pseudo-wave function approach becomes evident for large systems). To save time we decided to show results for a fast opening of the trap. Therefore in Fig. 4 (which substitutes for the old figure 5) the oscillations of the radius of the cloud are not present. Opening of the trap is so fast that expanding cloud is not back-reflected from the “trap walls”.

5) The English may need some revision. Please check carefully the (missing) articles.
We made our best to correct English.

### List of changes

List of changes:

1. The paper by S K Adhikari 2018 Laser Phys. Lett. 15 095501 is discussed.

2. We discuss the mathematical foundations of Eqs. (10).

3. We compare the dynamics of the Bose-Fermi mixture obtained by the hydrodynamic and atomic-orbital approaches, Fig. 4b (which replaced Fig. 5b from the previous version).

4. We removed Fig. 4 (old version) and discuss losses based on experimentally measured loss dynamics and on the rate coefficient, K3, deduced from the experiment.

5. We comment the referee's question about experimentally measured loss rate at aBF=−520a0 and its relation to the thermalization rate (Ref. [29]).

6. We removed the word 'incompressible' from the abstract.

7. In Figs. 3 and 4b there are vertical solid lines now, which show the moment of time equal to 1ms to allow the reader to immediately read the time scales.

8. We improved English.

### Submission & Refereeing History

Resubmission 1801.00346v5 on 25 April 2019
Resubmission 1801.00346v4 on 20 February 2019
Submission 1801.00346v3 on 30 November 2018

## Reports on this Submission

### Strengths

1. The work is very timely. Currently, self-bound quantum droplets represent
one the most exciting topics in the field of ultracold atoms.

2. The results are interesting and they could inspire new experiments, and also further theoretical investigation.

None

### Report

I'm fully satisfied with the authors' reply and with the new version of the manuscript. I am pleased to recommend it for publication.

### Requested changes

None

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

### Report

The authors have not adequately addressed my remaining concern. In the last round of refereeing this was summarized by the request:

“The authors should either admit that the term in Eqs. (10) which involves the ‘phase’ of the single fermion pseudo-wave function (which they use to describe the non-adiabatic dynamics of many fermions) is unphysical, or they should explain what physical process it actually represents, and give evidence for this.”

“We would like to emphasize that the fermionic pseudo-wave function has no direct physical meaning. Only the quantities which are the square of modulus of ψ_F(r,t) and the gradient of its phase can be interpreted as physical quantities. The Madelung transformation itself is supported by the Stokes' theorem. Provided that in a given region the condition ∇×v_F=0 is fulfilled, then the phase of the pseudo-wave function is defined as a curvilinear integral the velocity v_F.”

However, this text does not admit that the term in Eq. (10) is unphysical, nor does it adequately explain or give evidence for how this could possibly describe a physical dynamic process. I understand that you’re saying v_F is related to the phase of the pseudo-wavefunction since, essentially, you are treating the fermions a bit like bosons. But how is it justified to treat the dynamics of a many-fermion cloud with a single pseudo-wavefunction phase in this way?

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

Author Mariusz Gajda on 2019-05-17 (in reply to Report 1 on 2019-05-02)

Evidently, we have a problem to understand the referees worry:
“But how is it justified to treat the dynamics of a many-fermion cloud with a single pseudo-wavefunction phase in this way?”

We believe that we explained in our numerous responses to the referees that Eqs.(10) are as physical as the hydrodynamic equations they originate in. Both formalisms are connected by a rigorous mathematical transformation only.

One can argue with the physical assumptions leading the hydrodynamic equations, but cannot argue with mathematical theorems.

To help the referee to understand our reasoning we want to stress again that we use a mean-field description based on the effective ONE-PARTICLE FORMALISM in the hydrodynamic form (see appendix B in arXiv:1808.04793 to get to know details on derivation of hydrodynamic equations). At the mean field level, i.e. at an effective one-particle description, a main difference between bosons and fermions is the presence of the fermionic quantum pressure, the gradient correction and A VERY STRONG CONSTRAINT ON THE VELOCITY FIELD OF FERMIONS. The pseudo-wavefucntion formalism can be transformed into hydrodynamics equations for fermions only if the velocity field is irrotational. This constraint excludes quantized vortices, for instance.

Similarity between description of condensed bosons and ultracold fermions (Eqs. (10)) is very misleading. Indeed, both equations can be classified as nonlinear Schroedinger equations. However, the similarities are stopped at this point. Let us give an example. After imprinting a phase step on a 1D BEC, a dark soliton is formed. For the uniform system, the analytical solution (Zakharov solution) for dark solitons is known. If the same phase-step is imprinted on the fermionic pseudo-wavefunction the response of the system is qualitatively different (see Phys. Rev. A 66, 023612 (2002)). After the phase imprinting, two quasisolitons, the bright and the dark ones, propagating in opposite directions are generated. Such a result is supported by the atomic-orbital calculations as well (see above mentioned paper).