Abstract In 2015 Dmitry Petrov theoretically suggested that, in a binary mixture of bosons a quantum liquid droplet may arise due to the competition between attractive intercomponent and repulsive intracomponent forces. Although this prediction has been confirmed experimentally, the model by itself suffers from a serious conceptual problem: the low—lying excitation spectrum manifests a purely imaginary phonon velocity, c d 2 < 0 . In the present work, we develop a self consistent theory of two-component Bose systems with attractive interspecies interactions, which accurately takes into account pair correlations in terms of anomalous and mixed densities. We have shown that this procedure is able to resolve the problem of c d 2 < 0 . Limiting ourselves with a symmetric Bose mixture at zero temperature, we have found a region of stability in which a droplet can survive.
Abstract In 2015 Dmitry Petrov theoretically suggested that, in a binary mixture of bosons a quantum liquid droplet may arise due to the competition between attractive intercomponent and repulsive intracomponent forces. Although this prediction has been confirmed experimentally, the model by itself suffers from a serious conceptual problem: the low—lying excitation spectrum manifests a purely imaginary phonon velocity, c d 2 < 0 . In the present work, we develop a self consistent theory of two-component Bose systems with attractive interspecies interactions, which accurately takes into account pair correlations in terms of anomalous and mixed densities. We have shown that this procedure is able to resolve the problem of c d 2 < 0 . Limiting ourselves with a symmetric Bose mixture at zero temperature, we have found a region of stability in which a droplet can survive.
Quantum droplets have been realized in experiments on binary boson mixtures and dipolar Bose gases. In these systems, the mean-field energy of the Bose-Einstein condensation is attractive, and the repulsive Lee-Huang-Yang energy is crucial for stability. However, since the Lee-Huang-Yang term is not taken into account when constructing the Bogoliubov Hamiltonian, in the droplet regime one faces the problem of dynamically unstable phonon modes. In this work, we go beyond the Bogoliubov theory to study how the phonon mode is stabilized in the quantum droplet of a binary boson mixture. Similar to Beliaev's approach to a single-component Bose gas, we compute higher-order contributions to the self-energy of the boson propagator. We find that the interaction between spin and phonon excitations is the key for the phonon stability. We obtain the sound velocity which can be tested by measuring the superfluid critical velocity of the droplet in experiments. Beliaev damping of this quantum droplet is also discussed.
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 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.
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].
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.
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.