An ultradilute quantum droplet is a self-bound liquidlike state recently observed in ultracold Bose-Einstein condensates. In most previous theoretical studies, it is described by a phenomenological low-energy effective theory, termed as the extended Gross-Pitaevskii equation. Here, we microscopically derive the Gross-Pitaevskii equation for the condensate and also for a pairing field in an inhomogeneous quantum droplet realized by Bose-Bose mixtures with attractive interspecies interaction. We show that the inclusion of the pairing field is essential, in order to have a consistent description of the droplet state. We clarify that the extended Gross-Pitaevskii equation used earlier should be understood as the equation of motion for the pairing field, rather than the condensate. The fluctuations of the pairing field give rise to low-energy collective excitations of the droplet. We also present the Bogoliubov equations for gapless phonon modes and gapped modes due to pairing in real space, which characterizes single-particle-like excitations of the droplet. The equations of motion derived in this work for the condensate and the pairing field serve as an ideal starting point to understand the structure and collective excitations of nonuniform ultradilute quantum droplets in ongoing cold-atom experiments.
An ultradilute quantum droplet is a self-bound liquidlike state recently observed in ultracold Bose-Einstein condensates. In most previous theoretical studies, it is described by a phenomenological low-energy effective theory, termed as the extended Gross-Pitaevskii equation. Here, we microscopically derive the Gross-Pitaevskii equation for the condensate and also for a pairing field in an inhomogeneous quantum droplet realized by Bose-Bose mixtures with attractive interspecies interaction. We show that the inclusion of the pairing field is essential, in order to have a consistent description of the droplet state. We clarify that the extended Gross-Pitaevskii equation used earlier should be understood as the equation of motion for the pairing field, rather than the condensate. The fluctuations of the pairing field give rise to low-energy collective excitations of the droplet. We also present the Bogoliubov equations for gapless phonon modes and gapped modes due to pairing in real space, which characterizes single-particle-like excitations of the droplet. The equations of motion derived in this work for the condensate and the pairing field serve as an ideal starting point to understand the structure and collective excitations of nonuniform ultradilute quantum droplets in ongoing cold-atom experiments.
Ultradilute quantum droplets are an intriguing new state of matter, in which the attractive mean-field force can be balanced by the repulsive force from quantum fluctuations to avoid collapse. Here, we present a microscopic theory of ultradilute quantum droplets in three-, one-, and two-dimensional two-component Bose-Bose mixtures by generalizing the conventional Bogoliubov theory to include the bosonic pairing arising from the interspecies attraction. Our pairing theory is fully equivalent to a variational approach and hence gives an upper bound for the energy of quantum droplets. In three dimensions, we predict the existence of a strongly interacting Bose droplet at the crossover from Bose-Einstein condensates (BECs) to BCS superfluids and map out the bosonic BEC-BCS crossover phase diagram. In one dimension, we find that the energy of the one-dimensional Bose droplet calculated by the pairing theory is in excellent agreement with the latest diffusion Monte Carlo simulation [L. Parisi et al., Phys. Rev. Lett. 122, 105302 (2019)] for nearly all the interaction strengths at which quantum droplets exist. In two dimensions, we show that Bose droplets disappear and may turn into a solitonlike many-body bound state, when the interspecies attraction exceeds a critical value. Below the threshold, the pairing theory predicts more or less the same results as the Bogoliubov theory derived by D. S. Petrov and G. E. Astrakharchik [Phys. Rev. Lett. 117, 100401 (2016)]. The predicted energies from both theories are higher than the diffusion Monte Carlo results due to the weak interspecies attraction and the increasingly important role played by the beyond-Bogoliubov-approximation effect in two dimensions. Our pairing theory provides an ideal starting point to understand interesting ground-state properties of quantum droplets in various dimensions, including their shape and collective oscillations.
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 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. 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 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, 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.
In a binary quantum droplet, the interspecies attraction dominates over the intraspecies repulsions and the mean-field energy is unstable. The mechanical stability is restored by the repulsive Lee-Huang-Yang (LHY) energy [D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015)]. In the Bogoliubov theory of the binary quantum droplet, there are two branches of gapless excitations. The lower branch describes the phonon excitation and its energy is imaginary in the long-wavelength limit, implying dynamical instability. Recently, it was found that the phonon energy is renormalized by higher-order quantum fluctuations and the dynamical instability is removed [Q. Gu and L. Yin, Phys. Rev. B 102, 220503(R) (2020)]. In this paper, we study a binary quantum droplet in the path-integral formalism to construct an effective model with the correct phonon energy. By integrating out the upper excitation branch, we obtain an effective single-mode model describing density fluctuations, and derive the extended Gross-Pitaevskii equation (EGPE). In this approach, the LHY energy in the EGPE is purely positive without any assumption of neglecting the imaginary part. This effective single-mode model can be also used outside and close to the quantum droplet region such as in the LHY gas. Based on this effective model, we obtain the phase diagram of a uniform binary Bose system near the mean-field unstable point, which can be tested in experiments.
Ultradilute quantum droplets are intriguing new state of matter, in which the attractive mean-field force can be balanced by the repulsive force from quantum fluctuations to avoid collapse. Here, we present a microscopic theory of ultradilute quantum droplets in low-dimensional two-component Bose-Bose mixtures, by generalizing the conventional Bogoliubov theory to include the bosonic pairing arising from the inter-species attraction. Our pairing theory is fully equivalent to a variational approach and hence gives an upper bound for the energy of quantum droplets. In one dimension, we find that the energy calculated by the pairing theory is in an excellent agreement with the latest diffusion Monte Carlo simulation {[}Phys. Rev. Lett. \textbf{122}, 105302 (2019){]}, for nearly all the interaction strengths at which quantum droplets exist. In two dimensions, we show that quantum droplets disappear and may turn into a soliton-like many-body bound state, when the inter-species attraction exceeds a critical value. Below the threshold, the pairing theory predicts more or less the same results as the Bogoliubov theory derived by Petrov and Astrakharchik {[}Phys. Rev. Lett. \textbf{117}, 100401 (2016){]}. The predicted energies from both theories are higher than the diffusion Monte Carlo results, due to the weak inter-species attraction and the increasingly important role played by the beyond-Bogoliubov-approximation effect in two dimensions. Our pairing theory provides an ideal starting point to understand interesting ground-state properties of quantum droplets, including their shape and collective oscillations.
Stabilized by quantum fluctuations, dipolar Bose–Einstein condensates can form self-bound liquid-like droplets. However in the Bogoliubov theory, there are imaginary phonon energies in the long-wavelength limit, implying dynamical instability of this system. A similar instability appears in the Bogoliubov theory of a binary quantum droplet, and is removed due to higher-order quantum fluctuations as shown recently [Gu Q and Yin L 2020 Phys. Rev. B 102 220503(R)]. We study the excitation energy of a dipolar quantum droplet in the Beliaev formalism, and find that quantum fluctuations significantly enhance the phonon stability. We adopt a self-consistent approach without the problem of complex excitation energy in the Bogoliubov theory, and obtain a stable anisotropic sound velocity which is consistent with the superfluid hydrodynamic theory, but slightly different from the result of the extended Gross–Pitaevskii equation due to quantum depletion. A modified Gross–Pitaevskii equation in agreement with the Beliaev theory is proposed, which takes the effect of quantum fluctuations into account more completely.
We theoretically investigate the finite-temperature structure and collective excitations of a self-bound ultradilute Bose droplet in a flat space realized in a binary Bose mixture with attractive inter-species interactions on the verge of mean-field collapse. As the droplet formation relies critically on the repulsive force provided by Lee–Huang–Yang quantum fluctuations, which can be easily compensated by thermal fluctuations, we find a significant temperature effect in the density distribution and collective excitation spectrum of the Bose droplet. A finite-temperature phase diagram as a function of the number of particles is determined. We show that the critical number of particles at the droplet-to-gas transition increases dramatically with increasing temperature. Towards the bulk threshold temperature for thermally destabilizing an infinitely large droplet, we find that the excitation-forbidden, self-evaporation region in the excitation spectrum, predicted earlier by Petrov using a zero-temperature theory, shrinks and eventually disappears. All the collective excitations, including both surface modes and compressional bulk modes, become softened at the droplet-to-gas transition. The predicted temperature effects of a self-bound Bose droplet in this work could be difficult to measure experimentally due to the lack of efficient thermometry at low temperatures. However, these effects may already present in the current cold-atom experiments.