Two-component bosonic droplets are commonly described within Bogoliubov theory, where beyond-mean-field quantum fluctuations stabilize the system against mean-field collapse. In the interaction regime where droplets form, however, the Bogoliubov excitation spectrum contains an imaginary branch associated with the underlying instability, which is typically omitted when evaluating the beyond-mean-field energy. Bosonic pairing theory provides an alternative description with a fully real excitation spectrum. In this work, we reformulate bosonic pairing theory within an operator formalism, making its underlying approximations transparent, and compare its predictions with those of Bogoliubov theory for a homonuclear binary Bose mixture across the crossover from a weakly interacting gas to the droplet regime. Working at fixed particle density, we determine the variational parameters of bosonic pairing theory self-consistently and focus on the regime in which both theories possess real excitation spectra, allowing a direct comparison. We find that bosonic pairing theory yields a lower grand potential than Bogoliubov theory and predicts qualitatively distinct correlation signatures, including enhanced long-wavelength interspecies momentum-space correlations and a finite density static structure factor at experimentally accessible low momenta. These differences persist over a broad range of interaction strengths and suggest that correlation and structure-factor measurements can provide direct experimental tests of bosonic pairing across the gas-to-droplet crossover.
Two-component bosonic droplets are commonly described within Bogoliubov theory, where beyond-mean-field quantum fluctuations stabilize the system against mean-field collapse. In the interaction regime where droplets form, however, the Bogoliubov excitation spectrum contains an imaginary branch associated with the underlying instability, which is typically omitted when evaluating the beyond-mean-field energy. Bosonic pairing theory provides an alternative description with a fully real excitation spectrum. In this work, we reformulate bosonic pairing theory within an operator formalism, making its underlying approximations transparent, and compare its predictions with those of Bogoliubov theory for a homonuclear binary Bose mixture across the crossover from a weakly interacting gas to the droplet regime. Working at fixed particle density, we determine the variational parameters of bosonic pairing theory self-consistently and focus on the regime in which both theories possess real excitation spectra, allowing a direct comparison. We find that bosonic pairing theory yields a lower grand potential than Bogoliubov theory and predicts qualitatively distinct correlation signatures, including enhanced long-wavelength interspecies momentum-space correlations and a finite density static structure factor at experimentally accessible low momenta. These differences persist over a broad range of interaction strengths and suggest that correlation and structure-factor measurements can provide direct experimental tests of bosonic pairing across the gas-to-droplet crossover.
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 study spin and density excitations of one-dimensional self-bound Bose-Bose droplets within Bogoliubov theory, and show that spin excitations come alive, especially as the interspecies coupling is made less attractive. We argue that spin excitations are particularly relevant in the one-dimensional droplet regime, where droplets are realized within the mean-field stability regime, as has been confirmed by the quantum Monte Carlo simulations. As the interspecies coupling strength increases within the mean-field stability regime, spin modes ultimately fall below the particle-emission threshold, thus becoming observable in the droplet spectrum. We analyze the Bogoliubov model for both pseudospinor and population-imbalanced scalar mixtures, encompassing both the density and spin sectors. We corroborate our findings through variational analysis of density- and spin-breathing modes, which offers physical insight into the mode structure and independently validates the spectrum, as well as through real-time dynamics. Additionally, we compare our results with both Petrov's original theory, which considers the Lee-Huang-Yang (LHY) correction at the attractive edge of the mean-field stability regime, and a beyond-LHY description of Bose-Bose mixtures.
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 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.
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.
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.