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Quantum Bose-Fermi droplets

by Debraj Rakshit, Tomasz Karpiuk, Mirosław Brewczyk, Mariusz Gajda

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Submission summary

As Contributors: Miroslaw Brewczyk · Mariusz Gajda · Tomasz Karpiuk · Debraj Rakshit
Arxiv Link:
Date submitted: 2018-11-30
Submitted by: Brewczyk, Miroslaw
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: Quantum Physics


We study 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 of the system can lead to a formation of Bose-Fermi liquid droplets -- self-bound incompressible systems in a three-dimensional space. The stability analysis of the homogeneous system 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:
Has been resubmitted

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Anonymous Report 2 on 2018-12-21 Invited Report


1. The current manuscript certainly touches on a highly topical and interesting area. A feasible approach to dilute quantum droplets in free space was presented in just 2015 (for Bose-Bose mixtures [1]). Dilute dipolar gas droplets were observed in 2016 [2], and droplets of Bose-Bose mixtures very recently [3-4].

2. This submission appears to be the first work considering self-bound Bose-Fermi mixtures stabilized by Lee-Huang-Yang effects with only contact interactions.

3. The submission is clearly laid out.


1. The use of a single complex function to represent all of the fermions in this system is not physically reasonable. There are significant issues both for the ground state and for dynamics.


1. Ground state

In a dilute self-bound system there do not tend to be many bound states. For example Fig. 3(a) of [5] identifies in the order of 10 modes for fermionic impurities in a dipolar Bose droplet, and [1] identifies only a few modes for a Bose-Bose mixture. At most one fermion can occupy each bound state. Any additional fermions in addition to those in bound modes will occupy the continuum and escape from the mixture. Since the number of bound modes is small, each of them should be treated discretely, as [5] did for fermions trapped in a dipolar Bose droplet.

There will be a balance. The system needs to be deeply self-bound to have enough discrete modes below the energy cutoff at the continuum, but not so deeply bound that the density is too high. The loss from three-body interactions would then make the time-scales unfeasible in experiment.

2. Dynamics

The phase of a single Fermi wavefunction cannot generally be used to represent the dynamics of the fermions, as shown by Ref. [6] (with mapping from bosons).

The submission cites three papers related to the single Fermi wavefunction, two on very different systems (metal clusters and a single helium atom) and Ref. [7]. The only comparison to experiment or other methods in Ref. [7] seems to be Fig. 1. The difference of the spin dipole mode from one (the non-interacting limit) is either small or the difference from one is comparable to the difference from experiment.

[1] D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015)
[2] M. Schmitt et al, Nature 539, 259 (2016)
[3] C. R. Cabrera et al, Science 359, 301 (2018)
[4] G. Semeghini et al, Phys. Rev. Lett. 120, 235301 (2018)
[5] M. Wenzel et al, Phys. Scr. 93 104004 (2018)
[6] K. K. Das, G. J. Lapeyre, and E. M. Wright, Phys. Rev. A 65, 063603 (2002)
[7] P. T. Grochowski et al, Phys. Rev. Lett. 119, 215303 (2017)

Requested changes

1. Use a methodology that allows for the small number of discrete modes available to the fermions.

2. With that methodology show the bounds on the number of bosons and fermions for a droplet to exist and how that varies with interaction strength.

3. For reasonable interaction strengths, then give corresponding peak densities in SI units to allow comment on three-body loss.

  • validity: low
  • significance: good
  • originality: ok
  • clarity: good
  • formatting: good
  • grammar: reasonable

Author Mariusz Gajda on 2018-12-28 (in reply to Report 2 on 2018-12-21)

The referee admits that our manuscript touches on a highly topical and interesting area, but the use of a single complex function to represent all of the fermions in this system is not physically reasonable. There are significant issues both for the ground state and for dynamics.

We agree that our approach might have weaknesses in a case of small systems of tens fermions as studied in [1]. Discreetness of fermionic states might be an issue then. We have done orbital calculations in such a situation recently. The differences are not big, and we are now able to revise the manuscript in this direction. However, orbital calculations are numerically very demanding, thus must be limited to small droplets only.

But as shown in [2] not only few fermions, but also about 400 fermionic atoms can be trapped in a bosonic cloud. Then the mean field approach is applicable.

The mean field approach used by us and rigorously transformed to the pseudo wave-function formalism (with some limitations on the velocity field) corresponds to the limit of a large number of particles. Our goal was to compare results of the first part of our manuscript, where we consider infinite system, to the results which account for the surface effects. We choose big systems then.

In a stationary case, the velocity field is zero, and there is no problem in transforming the hydrodynamic description into the pseudo wave-function formalism. As the dynamic situation is concerned, we study adiabatic opening of trap. As can be checked a posteriori, no vorticity is created then. Hydrodynamic approach and pseudo wave-function formalisms are equivalent.

In our description the fermion contribution to the mean field energy is given by the Thomas-Fermi functional with the Weizscaker term. The functional used is very similar to the one introduced by Thomas and Fermi. It is well known that binding energies of multielectron atoms given by the Thomas-Fermi model differ from the `exact ones’ by about 10%. Differences are the largest for atoms with closed shells, and become smaller with increasing atomic number. Thus, by analogy, we expect that our approach is quite accurate for larger system such as we study in the manuscript. Orbital based calculations are beyond our reach for these number of atoms.

[1] Matthias Wenzel, Tilman Pfau and Igor Ferrier-Barbut, Physica Scripta 93, 104004 (2018)
[1] B. J. DeSalvo, Krutik Patel, Jacob Johansen, and Cheng Chin, PRL 119, 233401 (2017)

Author Miroslaw Brewczyk on 2019-01-17 (in reply to Report 2 on 2018-12-21)
reply to objection

We performed further calculations by using the Hartree-Fock method (in reply to the requested change number 1) and in the attached file we compare the results with the ones included already in the main text, obtained within the hydrodynamic approach.



Matthew Davis on 2019-02-05 (in reply to Miroslaw Brewczyk on 2019-01-17)
suggestion for further work

Following on from your additional calculations, below I relay further specific comments from one of the referees of your initial submission. I will hope this will help expedite the next round of refereeing:

1. As the self-bound droplet is not trapped, the effective potential is finite and the number of bound states is limited before the continuum is reached. Can you please calculate the number of bound states of the bosons (i.e. the states available to the fermions)? A full Bogoliubov-de Gennes calculation would be time consuming, but a possible simple and approximate approach would be to take the effective potential of the bosons in the self-bound Bose+Fermi droplet from your calculations and find the number of non-interacting modes supported by that numerical effective potential.

2. Can you please perform the same calculation as in your reply of 17 Jan, but applied to the solid curve in Fig. 3 of your manuscript, where there is expansion of the gas. A smaller atom number would be OK.

Anonymous Report 1 on 2018-12-18 Invited Report


1- The topic is interesting and timely.


1-The methods used are not valid to describe fermion dynamics.
2-The methods used are not valid to describe fermions in self-bound droplets.


This manuscript theoretically investigates the possibility of self-bound Bose-Fermi droplets. It begins with an infinite uniform system analysis then moves to the finite self-bound droplet. Later, dynamic simulations are considered to test droplet stability, and droplet preparation. I do not recommend this manuscript for publication because the methods employed are inappropriate to describe the system studied.

Self-bound droplets typically support only a small number of bound states, usually a tiny percentage of the total atom number, see Petrov, PRL 115, 155302 (2015), Wenzel et al., Physica Scripta 93, 104004 (2018), and Baillie et al., PRL 119, 255302 (2017). While this is not a problem for the bosons, only one fermion can occupy each bound state. If the number of fermions exceeds the number of bound states then the excess fermions must be unbound, which will affect the stability of the entire droplet. The model in the present manuscript doesn’t account for this crucial physics, nor do they consider how the ratio of fermion number to bound state number behaves in the thermodynamic limit. Therefore, the authors cannot claim that their results are physically relevant for either the finite or uniform systems. Such considerations must be a central focus for any paper considering the self-bound nature of Bose-Fermi droplets, as was demonstrated in the paper on Bose-Fermi droplets with dipolar interactions by the Pfau group: Physica Scripta 93, 104004 (2018).

Furthermore, for the section on finite systems, the small numbers of fermions expected (typically 10s of bound states) should instead be treated as discrete modes, as was done in the paper by the Pfau group: Physica Scripta 93, 104004 (2018). Even if the fermion number was large, any continuous treatment (e.g. local density approximation) of the fermions would require a high energy cutoff (at zero energy) so that only the bound fermion states are included within the droplet.

Another significant problem is that Eq. (10) is inappropriate to describe the dynamics of fermions. Equation (10) attempts to describe all fermions within a single pseudo wavefunction. However, it is well-known that this is unphysical for describing dynamics. What would be the physical interpretation of the ‘phase’ of the fermion wavefunction? The unphysical effects on the dynamics from assigning a ‘phase’ to a single wavefunction to describe many fermions has already been clearly addressed in the literature. For example, the paper by Girardeau et al. PRL 84, 5239 (2000), which considers fermionized bosons, focusses quite a lot on this point. For another example see Kunal et al. PRA 65, 063603 (2002).

Requested changes

(see report)

  • validity: poor
  • significance: poor
  • originality: low
  • clarity: good
  • formatting: excellent
  • grammar: good

Author Miroslaw Brewczyk on 2018-12-26 (in reply to Report 1 on 2018-12-18)

Below we shortly reply to the referee's criticism:

1. "The methods used are not valid to describe fermion dynamics"

Our explanation must be not clear enough. Contrary to what the referee says, we do not assign a single wave function to a many-fermion system.

We use a hydrodynamic approach originated in the density functional method. The hydrodynamic equations for fermionic system can be rigorously derived based on quantum kinetic equations for reduced density matrices. A 3-component velocity field and an atomic density are used to describe the system.

If the velocity field is a gradient of a scalar potential then problem can be simplified because of reduction of the number of dynamical variables. The density (in fact its square root) and the velocity potential can be combined into a single complex function. This way the hydrodynamic equations, under some assumptions, can be mapped to a pseudo-wave function dynamics.

In a case of violent dynamics and turbulent flow one must use all components of the velocity field. This is because no potential can be assigned to the velocity field then. We do not study such a situation.

We show that the Bose-Fermi droplet can be created dynamically by slow opening of a trap. We have checked (and results might be included into the manuscript) that the atomic-orbital approach and a hydrodynamic description give very similar results already for a droplet with tens of fermions and ten times larger number of bosons. Larger systems are not tractable by this method.

2. "The methods used are not valid to describe fermions in self-bound droplets."

In the first part we study infinite system, and we simply minimize the energy functional under physical constraints.

To account for the surface effects we modified the energy functional. Moreover in stationary case the velocity field vanishes, and our hydrodynamic description of fermions becomes the Thomas-Fermi model with the Weizsacker correction with all necessary modifications like replacing the Coulomb potential of nucleus by the effective potential produced by bosons.

The Thomas-Fermi model, used to describe multielectron atoms, has some drawbacks. It does not account for shell effects, but gives quite reasonable estimations of basic characteristics of atoms with atomic number Z. There is no better approach in the limit of infinite system than the density functional method. This is what we do for large system, being aware of limitations. We cannot do any better.

Referee correctly noticed that in a case of small systems the orbital approach is the one which is more adequate.

We have recently applied the atomic-orbital approach to a small system of Bose-Fermi mixture and confirmed the existence of Bose-Fermi droplets. These results can be incorporated into the present paper if we are given a chance.

3. "Another significant problem is that Eq. (10) is inappropriate to describe the dynamics of fermions."

We claim that Eqs. (10) are good enough to describe adiabatic formation of the Bose-Fermi droplet. We do not claim that every dynamics can be described by Eqs. (10).

As we have already mentioned, the Madelung approach is equivalent to the hydrodynamic description if the velocity field is irrotational. However, this is not a general feature. For example, when vortices are present in the system, the phase is not differentiable at the vortices cores and the equivalence between the Madelung and hydrodynamic approaches is broken.

Finally, the paper of Girardeau et al. mentioned by the referee is related to the interference effect studied for fermionized bosons. We are well aware of this paper. Here, two interfering clouds of fermions moving irrotationally can not be, in general, described as a single fluid with a potential flow since the total velocity is calculated as a ratio of the total density current to the total density.

However, the formation of the Bose-Fermi droplets has nothing to do with creation of vortices nor with any subtleties of non-potential flow as the one studied by Girardeau et al. (the fermionic cloud is simply connected).

Author Debraj Rakshit on 2019-03-02 (in reply to Miroslaw Brewczyk on 2018-12-26)

We thank for the comment. Indeed the paper of S. K. Adhikari considers related, but different problem, and if we have an opportunity we shall add some critical comments in the present manuscript. We did not include the corresponding citation in the first version of our paper simply because our manuscript appeared on arXiv on 01 January 2018, well ahead of the S.
K. Adhikari's paper - sumbitted for publication much later.

Anonymous on 2019-02-26 (in reply to Miroslaw Brewczyk on 2018-12-26)

I am interested in the topic of this study and I find the comments by the referees to this manuscript of interest. One of the referees commented that
" This submission appears to be the first work considering self-bound Bose-Fermi mixtures stabilized by Lee-Huang-Yang effects with only contact interactions."
This statement is not to the point. I find a very similar work (S K Adhikari 2018 Laser Phys. Lett. 15 095501) demonstrating the existence and studying the properties of self-bound Bose-Fermi mixtures stabilized by Lee-Huang-Yang effects with only contact interactions using essentially the same Hamiltonian and same analytical and numerical procedures. Moreover, that paper goes beyond and considers the effect of three-body force on these states. This paper was published long before this manuscript was submitted for publication. The authors should clearly acknowledge this fact in the Abstract and Introduction of the manuscript and point out what is new or different in their study for the benefit of the reader.

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