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Physics Today

Getting the drop on quantum droplets

Aurel Bulgac

Published Jun 1, 2020• 2 citations
Publisher

Abstract

The article "Ultradilute quantum droplets" by Igor Ferrier-Barbut (Physics Today, April 2019, page 46) was really nice to see. It reported on the creation, at last, of real ultradilute liquid droplets and on the tremendous progress that has been made in that area. However, I was disappointed to see no mention or discussion that ultradilute liquid quantum droplets were predicted a long time ago.11. A. Bulgac, Phys. Rev. Lett. 89, 050402 (2002). https://doi.org/10.1103/PhysRevLett.89.050402,22. P. F. Bedaque, A. Bulgac, G. Rupak, Phys. Rev. A. 68, 033606 (2003). https://doi.org/10.1103/PhysRevA.68.033606 That early work was a major source of inspiration for Dmitry Petrov's 2015 paper on the subject,33. D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015). https://doi.org/10.1103/PhysRevLett.115.155302 at least according to what he told me years ago.In his discussion of mean-field quantum gases, Ferrier-Barbut doesn't note that the Efimov effect, which involves the creation of an infinite number of three-body bound states, can also allow Bose liquids to exist when the scattering length is negative (that is, when the two-particle interaction is attractive). Such a system is not always unstable. Tsung-Dao Lee, Kerson Huang, and Chen Ning Yang (LHY) proposed in 1957 a leading-order correction to the mean-field approximation due to two-body collisions, which was used by Petrov;33. D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015). https://doi.org/10.1103/PhysRevLett.115.155302 that "game-changing correction," as Ferrier-Barbut calls it, can alter the nature of the Bose–Einstein condensate.However, as was discussed quite some time ago,11. A. Bulgac, Phys. Rev. Lett. 89, 050402 (2002). https://doi.org/10.1103/PhysRevLett.89.050402,22. P. F. Bedaque, A. Bulgac, G. Rupak, Phys. Rev. A. 68, 033606 (2003). https://doi.org/10.1103/PhysRevA.68.033606 the strength of three-body interactions can dominate over LHY corrections and can be infinite even if the two-body scattering length is finite.11. A. Bulgac, Phys. Rev. Lett. 89, 050402 (2002). https://doi.org/10.1103/PhysRevLett.89.050402 A liquid model based on the Efimov effect11. A. Bulgac, Phys. Rev. Lett. 89, 050402 (2002). https://doi.org/10.1103/PhysRevLett.89.050402,22. P. F. Bedaque, A. Bulgac, G. Rupak, Phys. Rev. A. 68, 033606 (2003). https://doi.org/10.1103/PhysRevA.68.033606 is more robust than the one Petrov envisioned and much more flexible than the van der Waals model. Unlike the quantum liquid droplets created in mixtures of Bose–Einstein condensates,44. C. R. Cabrera et al., Science 359, 301 (2018). https://doi.org/10.1126/science.aao5686 which have practically the same size for particle numbers up to tens of thousands, the quantum liquid droplets I suggested are truly saturating systems, with basically constant interior density. A droplet can have any size, and it can be formed even from a single element. It is a real liquid, with constant density inside and a well-defined surface, and its density and surface tension can be controlled. Also, it is stable against quantum corrections to the mean field.22. P. F. Bedaque, A. Bulgac, G. Rupak, Phys. Rev. A. 68, 033606 (2003). https://doi.org/10.1103/PhysRevA.68.033606Moreover, in a rather special system—an ensemble of spin-polarized tritium atoms—three-body recombination processes are most likely absent.22. P. F. Bedaque, A. Bulgac, G. Rupak, Phys. Rev. A. 68, 033606 (2003). https://doi.org/10.1103/PhysRevA.68.033606,55. D. Blume et al., Phys. Rev. Lett. 89, 163402 (2002). https://doi.org/10.1103/PhysRevLett.89.163402 Although I did not make the estimates, which should be straightforward, I am sure that by controlling the density and thus the rate of four-body recombination, one could create droplets with basically arbitrarily long lifetimes. A droplet of spin-polarized tritium atoms would be a totally unique object, perhaps as unique as macroscopic superfluid helium, but amenable to precise quantum many-body calculations, both static and time-dependent. Quantum turbulence could be studied in a large class of systems, for which a microscopic theory exists, and unlike in the case of superfluid helium, theory could be directly confronted with experiment.Quantum liquid droplets could be either boselets or fermilets and would undergo at least two types of phase transitions, from superfluid to normal and from liquid to gas. Their physics should be fascinating. Mixing bosons and fermions can lead to even more interesting and complex objects.ReferencesSection:ChooseTop of pageReferences <<CITING ARTICLES1. A. Bulgac, Phys. Rev. Lett. 89, 050402 (2002). https://doi.org/10.1103/PhysRevLett.89.050402, Google ScholarCrossref2. P. F. Bedaque, A. Bulgac, G. Rupak, Phys. Rev. A. 68, 033606 (2003). https://doi.org/10.1103/PhysRevA.68.033606, Google ScholarCrossref3. D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015). https://doi.org/10.1103/PhysRevLett.115.155302, Google ScholarCrossref4. C. R. Cabrera et al., Science 359, 301 (2018). https://doi.org/10.1126/science.aao5686, Google ScholarCrossref5. D. Blume et al., Phys. Rev. Lett. 89, 163402 (2002). https://doi.org/10.1103/PhysRevLett.89.163402, Google ScholarCrossref© 2020 American Institute of Physics.

DOI
10.1063/pt.3.4486

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Ferrier-Barbut replies: I am grateful for Aurel Bulgac’s insight about three-body stabilized quantum droplets. I was aware of his work, but space constraints made it impossible for me to cite the broad swath of related literature. A tritium droplet would certainly be a peculiar object, though as an experimentalist I think making a Bose–Einstein condensate of tritium would be quite challenging. Section:ChooseTop of page <<© 2020 American Institute of Physics.

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The article "Ultradilute quantum droplets" by Igor Ferrier-Barbut (Physics Today, April 2019, page 46) was really nice to see. It reported on the creation, at last, of real ultradilute liquid droplets and on the tremendous progress that has been made in that area. However, I was disappointed to see no mention or discussion that ultradilute liquid quantum droplets were predicted a long time ago.11. A. Bulgac, Phys. Rev. Lett. 89, 050402 (2002). https://doi.org/10.1103/PhysRevLett.89.050402,22. P. F. Bedaque, A. Bulgac, G. Rupak, Phys. Rev. A. 68, 033606 (2003). https://doi.org/10.1103/PhysRevA.68.033606 That early work was a major source of inspiration for Dmitry Petrov's 2015 paper on the subject,33. D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015). https://doi.org/10.1103/PhysRevLett.115.155302 at least according to what he told me years ago.In his discussion of mean-field quantum gases, Ferrier-Barbut doesn't note that the Efimov effect, which involves the creation of an infinite number of three-body bound states, can also allow Bose liquids to exist when the scattering length is negative (that is, when the two-particle interaction is attractive). Such a system is not always unstable. Tsung-Dao Lee, Kerson Huang, and Chen Ning Yang (LHY) proposed in 1957 a leading-order correction to the mean-field approximation due to two-body collisions, which was used by Petrov;33. D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015). https://doi.org/10.1103/PhysRevLett.115.155302 that "game-changing correction," as Ferrier-Barbut calls it, can alter the nature of the Bose–Einstein condensate.However, as was discussed quite some time ago,11. A. Bulgac, Phys. Rev. Lett. 89, 050402 (2002). https://doi.org/10.1103/PhysRevLett.89.050402,22. P. F. Bedaque, A. Bulgac, G. Rupak, Phys. Rev. A. 68, 033606 (2003). https://doi.org/10.1103/PhysRevA.68.033606 the strength of three-body interactions can dominate over LHY corrections and can be infinite even if the two-body scattering length is finite.11. A. Bulgac, Phys. Rev. Lett. 89, 050402 (2002). https://doi.org/10.1103/PhysRevLett.89.050402 A liquid model based on the Efimov effect11. A. Bulgac, Phys. Rev. Lett. 89, 050402 (2002). https://doi.org/10.1103/PhysRevLett.89.050402,22. P. F. Bedaque, A. Bulgac, G. Rupak, Phys. Rev. A. 68, 033606 (2003). https://doi.org/10.1103/PhysRevA.68.033606 is more robust than the one Petrov envisioned and much more flexible than the van der Waals model. Unlike the quantum liquid droplets created in mixtures of Bose–Einstein condensates,44. C. R. Cabrera et al., Science 359, 301 (2018). https://doi.org/10.1126/science.aao5686 which have practically the same size for particle numbers up to tens of thousands, the quantum liquid droplets I suggested are truly saturating systems, with basically constant interior density. A droplet can have any size, and it can be formed even from a single element. It is a real liquid, with constant density inside and a well-defined surface, and its density and surface tension can be controlled. Also, it is stable against quantum corrections to the mean field.22. P. F. Bedaque, A. Bulgac, G. Rupak, Phys. Rev. A. 68, 033606 (2003). https://doi.org/10.1103/PhysRevA.68.033606Moreover, in a rather special system—an ensemble of spin-polarized tritium atoms—three-body recombination processes are most likely absent.22. P. F. Bedaque, A. Bulgac, G. Rupak, Phys. Rev. A. 68, 033606 (2003). https://doi.org/10.1103/PhysRevA.68.033606,55. D. Blume et al., Phys. Rev. Lett. 89, 163402 (2002). https://doi.org/10.1103/PhysRevLett.89.163402 Although I did not make the estimates, which should be straightforward, I am sure that by controlling the density and thus the rate of four-body recombination, one could create droplets with basically arbitrarily long lifetimes. A droplet of spin-polarized tritium atoms would be a totally unique object, perhaps as unique as macroscopic superfluid helium, but amenable to precise quantum many-body calculations, both static and time-dependent. Quantum turbulence could be studied in a large class of systems, for which a microscopic theory exists, and unlike in the case of superfluid helium, theory could be directly confronted with experiment.Quantum liquid droplets could be either boselets or fermilets and would undergo at least two types of phase transitions, from superfluid to normal and from liquid to gas. Their physics should be fascinating. Mixing bosons and fermions can lead to even more interesting and complex objects.ReferencesSection:ChooseTop of pageReferences <<CITING ARTICLES1. A. Bulgac, Phys. Rev. Lett. 89, 050402 (2002). https://doi.org/10.1103/PhysRevLett.89.050402, Google ScholarCrossref2. P. F. Bedaque, A. Bulgac, G. Rupak, Phys. Rev. A. 68, 033606 (2003). https://doi.org/10.1103/PhysRevA.68.033606, Google ScholarCrossref3. D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015). https://doi.org/10.1103/PhysRevLett.115.155302, Google ScholarCrossref4. C. R. Cabrera et al., Science 359, 301 (2018). https://doi.org/10.1126/science.aao5686, Google ScholarCrossref5. D. Blume et al., Phys. Rev. Lett. 89, 163402 (2002). https://doi.org/10.1103/PhysRevLett.89.163402, Google ScholarCrossref© 2020 American Institute of Physics.
Quantum Fluids In recent years, quantum fluids have been studied largely in gaseous form, such as the Bose-Einstein condensates (BECs) of alkali atoms and related species. Quantum liquids, other than liquid helium, have been comparatively more difficult to come by. Cabrera et al. combined two BECs and manipulated the atomic interactions to create droplets of a quantum liquid (see the Perspective by Ferrier-Barbut and Pfau). Because the interactions were not directional, the droplets had a roughly round shape. The simplicity of this dilute system makes it amenable to theoretical modeling, enabling a better understanding of quantum fluids. Science , this issue p. [301][1]; see also p. [274][2] [1]: /lookup/doi/10.1126/science.aao5686 [2]: /lookup/doi/10.1126/science.aar3785
Among settings which have recently been elaborated for the creation of stable multidimensional solitons, most advanced, as concerns the theory and experiment alike, is the work with quantum droplets (QDs) in atomic BEC. Important theoretical predictions for fundamental and topologically structured soliton-like (stably self-trapped) QDs have been published in many papers, starting from the pioneering work of Petrov [Phys. Rev. Lett. 115, 155302 (2015)]. Simultaneously, the experimental advancement in the making of QDs in binary BECs, as well as in single-component BECs of atoms carrying magnetic dipole moments, have been really remarkable and unique. Successful experiments followed the first theoretical prediction of the QDs very quickly, and the progress in the experimental work was much faster than in other areas of physics where stable multidimensional solitons were predicted. The basis of these works is a correction to the MF (mean-field) BEC dynamics produced by quantum fluctuations around MF states. The correction was first predicted in 1957 by Lee, Huang, and Yang (LHY). The breakthrough proposal made by Petrov [Phys. Rev. Lett. 115, 155302 (2015)] was to use the effective higher-order (quartic) self-repulsion, produced by the LHY effect in the atomic BEC, as a mechanism arresting the collapse driven by mutual attraction of two self-repulsive components in a binary quantum bosonic gas. The balance between the inter-species attraction and LHY-induced self-repulsion gives rise to stable 3D and quasi-2D (pancake-shaped) localized bound states in the form of the quantum droplets (QDs). In single-component condensates of magnetic atoms, the attraction is provided by the long-range dipole–dipole interaction (Chomaz et al., 2016; Schmitt et al., 2016). The competition between the MF attraction and LHY repulsion maintains a superfluid state whose density (taking very low values) cannot exceed a certain maximum, thus making it effectively incompressible. This is a reason why this quantum macroscopic state of matter is identified as a superfluid. Accordingly, localized states filled by this quantum fluid are called “droplets.” In addition to the 3D QDs, droplets in the effectively 2D setting, which may exist under the action of tight confinement in one direction, imposed by an external potential, were theoretically predicted [Petrov and Astrakharchik, Phys. Rev. Lett. 117, 100401 (2016)] and experimentally created [Cabrera et al., Science 359, 301-304 (2018)] too. This chapter provides a condensed review of basic theoretical results for stable QDs in binary BECs, maintained through the stable balance between the inter-species attraction and intra-species LHY repulsion. Both fundamental modes and ones with embedded vorticity are considered. The presentation is partly based on a review article by Luo et al. [Front. Phys. 16, 32501 (2021)]. In addition to 2D and 3D self-trapped QDs in free space, the present chapter addresses 2D fundamental and vortex modes trapped in a singular attractive potential, ∼−1/r2, which are stabilized by the LHY repulsion. Two subsequent chapters present a summary of experimental findings for QDs in binary BECs and for QDs in single-component BECs of magnetic atoms.
We revisit the Bogoliubov theory of quantum droplets proposed by Petrov [Phys. Rev. Lett. 115, 155302 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.155302] for an ultracold Bose-Bose mixture, where the mean-field collapse is stabilized by the Lee-Huang-Yang quantum fluctuations. We show that a loophole in Petrov's theory, i.e., the ignorance of the softening complex Bogoliubov spectrum, can be naturally removed by the introduction of bosonic pairing. The pairing leads to weaker mean-field attractions, and also a stronger Lee-Huang-Yang term in the case of unequal intraspecies interactions. As a result, the equilibrium density for the formation of self-bound droplets significantly decreases in the deep droplet regime, in agreement with a recent observation from diffusion Monte Carlo simulations. Our construction of a consistent Bogoliubov theory paves the way to understand the puzzling low critical number of small quantum droplets observed in the experiment [C. Cabrera et al., Science 359, 301 (2018)SCIEAS0036-807510.1126/science.aao5686].
In his seminal proposal of quantum droplets in binary Bose mixtures [D. Petrov, Phys. Rev. Lett. 115, 155302 (2015)], Petrov suggested that the density ratio n2/n1{n}_{2}/{n}_{1}n2​/n1​ of the two bosonic components is locked to an optimal value, which is given by the square root of the ratio of the two intraspecies scattering lengths, i.e., a11/a22\sqrt{{a}_{11}/{a}_{22}}a11​/a22​​. Due to such density locking, quantum droplets can be efficiently described by using an extended Gross-Pitaevskii equation within the single-mode approximation. Here, we find that this single-mode description necessarily breaks down in the deep quantum droplet regime, when the attractive interspecies scattering length a12{a}_{12}a12​ significantly deviates away from the threshold of mean-field collapse (i.e., $\ensuremath{-}\sqrt{{a}_{11}{a}_{22}}$). By applying a bosonic pairing theory, we show that the density ratio is allowed to fluctuate in a sizable interval. Most importantly, the optimal density ratio would be very different from a11/a22\sqrt{{a}_{11}/{a}_{22}}a11​/a22​​, in the case of unequal intraspecies scattering lengths (${a}_{11}\ensuremath{\ne}{a}_{22}$). Our finding might provide a plausible microscopic explanation of the puzzling low critical particle number of quantum droplets, as experimentally observed. Our predicted interval of the density ratio, as a function of the interspecies scattering length, could also be experimentally examined in cold-atom laboratories in the near future.
The unique achievement of the work on multidimensional (quasi-) solitons in the form of quantum droplets (QDs) is the creation of such stable objects in real experiments and, in addition, the observation of collisions between the stable QDs. This chapter, first, offers a brief summary of the results for QDs produced by the experimental work with binary BECs in the ultracold gas of potassium atoms, which was performed by Cabrera et al. (2016), Cheiney et al. [Phys. Rev. Lett. 120, 135301 (2018)], Semeghi et al. [Phys. Rev. Lett. 120, 235301 (2018)], and Ferioli et al. [Phys. Rev. Lett. 122, 090401 (2019)]. Then, the results of D’Errico et al. [Phys. Rev. Res. 1, 033155 (2019)] that demonstrate the creation of QDs with an essentially longer lifetime and a possibility to channel stable motion of these droplets in a waveguide are summarized too. Finally, the chapter includes a brief outline of earlier experiments that had produced robust QDs in single-component dipolar BECs in atomic gases of dysprosium [Kadau et al., Nature 530, 194–197 (2016); Schmitt et al., Nature 539, 259–262 (2016); Ferrier-Barbut et al. Phys. Rev. Lett. 116 215301 (2016); Ferrier-Barbut et al., Phys. Rev. A 97, 011604 (2018); Böttcher et al., 2019] and erbium [Chomaz et al., Phys. Rev. X 6, 041039 (2016)]. Generating droplets in this setting is possible using attraction provided by the long-range dipole–dipole interaction and stabilizing repulsion induced by the contact interaction, including the Lee–Huang–Yang term. The presentation in this chapter partly follows the review of the topic recently presented by Luo et al. [Front. Phys. 16, 32501 (2021)].