Thermal destabilization of self-bound ultradilute quantum droplets

We theoretically investigate the temperature effect in a Bose-Bose mixture with attractive inter-species interactions, in the regime where a self-bound ultradilute quantum droplet forms due to the subtle balance between the attractive mean-field force and the repulsive force provided by Lee-Huang-Yang quantum fluctuations. We find that in contrast to quantum fluctuations, thermal fluctuations destabilize the droplet state and completely destroy it above a threshold temperature. We show that the threshold temperature is determined by the intra-species interaction energy. For a three-dimensional Bose-Bose mixture, the threshold temperature is less than one-tenth of the Bose-Einstein condensation temperature under the typical experimental conditions. With increasing temperature, the droplet's equilibrium density gradually decreases and can be reduced by several tens of percent upon reaching the threshold temperature. We also consider a one-dimensional quantum droplet and find a similar destabilization effect due to thermal fluctuations. The threshold temperature in one dimension is roughly set by the binding energy of the inter-species dimer. The pronounced thermal instability of a self-bound quantum droplet predicted in our work could be examined in future experiments, by measuring the temperature dependence of its central density and observing its sudden disappearance at the threshold temperature.


I. INTRODUCTION
In his classical textbook "The Universe in a Helium Droplet " [1], Volovik described an interesting autonomously isolated quantum system of helium nanodroplet, without any interaction with the surrounding environment. It can be in equilibrium at zero external pressure (i.e., P = 0) in empty space, with an equilibrium particle density n eq ∼ 2 × 10 22 cm −3 [2][3][4] determined by the balance between the attractive interatomic interaction (∝ n) and the repulsive zero-point quantum motions of helium atoms (∝ n 3/2 ). As a negative chemical potential µ < 0 is needed to prevent self-evaporation, the droplet formation is generally impossible in the wellstudied single-component weakly interacting Bose gas of utracold alkali-metal atoms [5], where the attractive short-range inter-particle interactions lead to a meanfield collapse [6].
In this work, we would like to understand how quantum droplets' bulk properties are affected by a small but nonzero temperature, which always exists experimentally. This issue is rarely addressed in the past literature, presumably due to the lack of a useful microscopic theory of quantum droplets from the first principle. Here, we overcome such a difficulty by extending a most recently developed pairing description of the droplet state [35][36][37] to finite temperatures. For simplicity, we follow Petrov's binary Bose mixture model of quantum droplets [7], to avoid subtly treating the long-range anisotropic dipolar interactions in dipolar droplets.
We find that thermal fluctuations generally destabilize the droplet. This tendency is natural to understand. As temperature increases, the atomic motion becomes increasingly significant, and the droplet may fail to maintain its zero-pressure state. We observe that the threshold temperature for the complete destruction of the quantum droplet is typically set by the intra-species interaction energy. For a weakly interacting Bose-Bose mixture in three dimensions, the interaction energy is small, so the threshold temperature could be less than one-tenth of the condensation temperature under the current experimental conditions [9,11]. Remarkably, despite this low threshold temperature, the equations of state of quantum droplets still show a strong temperature dependence. In particular, with increasing temperature, the droplet's equilibrium density can be reduced by several tens of percent upon reaching the threshold temperature. We also consider the droplet state in a Bose-Bose mixture in one dimension and find similar thermal destabilization due to thermal fluctuations. The rest of the paper is organized as follows. In the next section (Sec. II), we introduce the model Hamiltonian and present a microscopic pairing theory of quantum droplets at low but finite temperature based on the conventional Bogoliubov theory [38]. In Sec. III, we discuss the thermal destabilization of quantum droplets in three dimensions. The quantum depletion and thermal depletion are calculated to validate the applicability of the Bogoliubov theory. In Sec. IV, we consider one-dimensional quantum droplets and show that the thermal destabilization effect is universal for the droplet state in different dimensions. Finally, Sec. V is devoted to the conclusions and outlooks.

II. PAIRING THEORY AT FINITE TEMPERATURE
To highlight the essential temperature effect, we consider the simplest possible homonuclear Bose-Bose mixture, with equal repulsive intra-species interactions (i.e., g 11 = g 22 = g) and attractive inter-species interactions (g 12 ). In this case, we have equal population in each species and equal chemical potential µ. In real space, the system is described by a grand canonical model Hamil- where in the last Hamiltonian density for inter-species interactions, we have used the Hubbard-Stratonovich transformation to decouple the four-fermion interaction term through the introduction of a pairing field∆(x) [36,37].φ i (x) andφ † i (x) (i = 1, 2) are annihilation and creation field operators for the i-species bosons with mass m 1 = m 2 = m.
In this work, we consider both three-dimensional and one-dimensional binary mixtures. In three dimensions, the short-range contact inter-particle interactions used in the model Hamiltonian are unphysical in the largemomentum and high-energy limit, as reflected by the well-known ultraviolet divergence. The divergence can be removed by the standard regularization procedure: we simply re-express the bare interaction strengths g ij in terms of the s-wave scattering lengths a ij , i.e., where V is the volume of the system (or the length of the system in the one-dimensional case). The interaction regularization is not required in one dimension. There, the interaction strengths g ij are related to the s-wave scattering length via A. Bogoliubov theory with pairing The pairing theory of quantum droplets in a binary Bose mixture has been discussed in detail in the previous works [35][36][37]. Here, for self-containedness we briefly review the theory and extend it to the finite temperature case. In the weakly interacting regime, we take the Bogoliubov approximation to rewrite the bosonic field operators [38][39][40], where δφ i is considered as small fluctuation around the condensate wave-function φ c (x). At the same level of approximation, we also take a a static c-number function for the pairing field [35][36][37], i.e., and determine it variationally. As we focus on the ground state, both the condensate wave-function φ c (x) and the pairing function ∆(x) can be chosen as real and nonnegative functions. By expanding the model Hamiltonian in terms of δφ † i and δφ i and truncate it to the second order, we find that a quadratic form, provided that φ c (x) and ∆(x) satisfy a Gross-Pitaevskii (GP) equation, where we have defined the short-hand notations, The function C(x) can be simply viewed as the intraspecies interaction energy density. The quadratic form of the model Hamiltonian is straightforward to diagonalize, by a linear real-space Bogoliubov transformation [39,40], whereα † n andα n are creation and annihilation field operators of Bogoliubov quasiparticles, and u ni (x) and v ni (x) are the corresponding quasiparticle wave-functions. The truncated Bogoliubov Hamiltonian then becomes [37], if u ni (x) and v ni (x) obey the Bogoliubov equations, where E n ≥ 0 is the energy of Bogoliubov quasi-particles. It is easy to check that the zero-mode with E = 0 has the form u 1 = u 2 = +φ c (x) and v 1 = v 2 = −φ c (x), which is precisely the condensate mode of the GP equation and hence should be excluded. From the diagonalized Hamiltonian (14), it is straightforward to write down the thermodynamic potential at finite temperature (β = 1/k B T ), from which, we determine the variational pairing function ∆(x) through the functional minimization, i.e., Once ∆(x) is found, we calculate the total number of atoms, N = −∂Ω/∂µ, and consequently the free energy F = Ω+µN . In the thermodynamic limit, the free energy per particle takes a global minimum as a function of the density n = N/V when the system is in the self-bound droplet state, i.e., ∂(F/N )/∂n = 0, which is equivalent to the zero-pressure condition, It is worth noting that the above Bogoliubov theory is applicable at low temperatures, where the depletion from the condensate, due to either quantum fluctuations or thermal fluctuations, should be sufficiently small. For this purpose, one may need to explicitly examine the quantum depletion N qd and thermal depletion N th . These two quantities can be evaluated by taking the average of the number operators (i = 1, 2), and we obtain, where f B (E) = 1/(e βE − 1) is the Bose-Einstein distribution function.

B. Bulk properties of quantum droplets in the thermodynamic limit
For simplicity, in this work we consider a sufficiently large droplet, where the edge effect can be safely neglected. We therefore have constant condensate wavefunction φ c , intra-species interaction energy C, and pairing parameter ∆. The GP equation (9) then leads to the relation C = µ + ∆. In this case, the quasi-particle wave-functions, are plane waves with momentum k and energy E k . The Bogoliubov equations (16) and (17) in momentum space take the form, where is a 2 by 2 matrix. It is straightforward to show that the quasi-particle wave-functions satisfy [37], where the dispersion relation E k can take two branches, E k− is the gapless phonon spectrum, while E k+ becomes gapped due to the bosonic pairing. As a result and Eq. (18) the thermodynamic potential at finite temperature, The quantum and thermal depletions are given by, respectively.

III. THREE-DIMENSIONAL DROPLETS
In three dimensions, the zero-temperature thermodynamic potential Eq. (33) has been discussed in detail in the previous works [35,36]. By replacing the bare interaction strengths g and g 12 with the s-wave scattering lengths a and a 12 , one obtains [35,36], where To carry out the momentum integration in the finite-temperature contribution to the thermodynamic potential Eq. (34), we introduce t = [ 2 k 2 /(2m)]/(2C), α = ∆/C and γ = k B T /C to write the two dispersion relations into the dimensionless forms, Hence, we find that, where At very low temperature, i.e., k B T ≪ C or γ ≪ 1, where the gapless phonon spectrum can be well-approximated asÊ − (t) ≃ (2 √ 1 + α/γ) √ t and the gapped modeẼ + (t) does not contribute to the integral, we obtain the lowtemperature result, where in the last step, we have introduced the variable x = (2 √ 1 + α/γ) √ t and have used the identitý ∞ 0 dxx 2 ln(1 − e −x ) = −π 4 /45. Therefore, it is useful to rewrite s 3 (α, γ) into the form, where the functions 3 (α, γ) accounts for the high-order correction at nonzero temperature. In Fig. 1, we show the functions s 3 (α, γ) ands 3 (α, γ) as a function of the reduced temperature γ at three typical ratios α = ∆/C = 0.5 (red dashed line), 1.0 (black solid line), and 1.5 (blue dot-dashed line). We find that s 3 (α, γ) follows closely its low-temperature approximate result Eq. (42) up to k B T ∼ C, i.e., when the thermal energy k B T becomes comparable to the intraspecies interaction energy C. At this temperature scale, s 3 (α, γ) ∼ −O(1) is at the same order of the function G 3 (α) but has an opposite sign, indicating that the repulsive force provided by LHY quantum fluctuations might be compensated by thermal fluctuations. More quantitatively, at α = 1 where G 3 (α = 1) = 4 √ 2, the combined contribution to the thermodynamic potential from quantum and thermal fluctuations, vanishes at about γ ≃ 2.

A. Equation of state
For a given chemical potential µ, we numerically calculate the thermodynamic potential Ω = Ω 0 + Ω T as a function of the pairing gap ∆, by using Eq.  Quite generally, in three dimensions the saddle-point pairing gap is much large than the chemical potential ∆ 0 ≫ |µ| and hence C = µ + ∆ 0 ≫ |µ| [35,36]. Therefore, to a very good approximation, around the saddle point (∆ ∼ ∆ 0 ) we obtain, where we have set C = ∆ in the LHY term (i.e., the second line) and thereby γ = k B T /∆. By taking the derivatives with respect to the chemical potential µ and the pairing gap ∆, i.e., n = N/V = −∂(Ω/V)/∂µ and ∂Ω/∂∆ = 0 at ∆ = ∆ 0 , we find that, and respectively. It is then straightforward to obtain the analytic expressions for the pressure P and the free energy F , P = − π 2 m a + a 2 a 12 n 2 + 128 √ π 5 2 a 5/2 m n 5/2 1 + 5π 4 2304 γ 4s 3 (1, γ) + where the parameter γ should now be understood as, In the second step of the above equation, we have reexpressed the density n in terms of the small gas parameter na 3 ≪ 1 and the Bose-Einstein condensation temperature of an ideal gas k B T c = 2π 2 [n/(2ζ(3/2))] 3/2 /m, where ζ(3/2) ≃ 2.6124 is the Riemann zeta function. In the zero-temperature limit, the free energy Eq. (49) re-covers the analytical expression for the ground-state energy found in earlier works [35,36]. In Fig. 2, we report the numerical and analytical results of the pressure per particle P/n = −Ω/N (a) and the free energy per particle F/N (b) as a function of the density n, at the inter-species interaction strength a 12 = −1.05a and at three typical temperatures k B T = 0.5, 1.0 and 1.5, measured in units of 10 −4 2 /(2ma 2 ). There is an excellent agreement between numerical and analytical predictions, as we anticipate. At finite temperature, we find that the density depen-dence of the pressure and free energy develop non-trivial features in the low-density limit (n → 0). Instead of becoming vanishingly small as in the zero-temperature case, both equations of state become divergent when the density decreases to zero. This divergence is purely a temperature effect and can be easily understood from the last term in Eq. (48) and Eq. (49), i.e., δP, δF ∝ n 5/2 γ 4 = n 5/2 mk B T 2π 2 an The temperature correction in the pressure and free energy therefore diverges like ±n −3/2 as the density approaches zero and vanishes like T 4 with decreasing temperature. As a consequence of such a divergent low-density dependence, at low temperature (i.e., at k B T = 0.5 × 10 −4 2 /(2ma 2 ) as indicated by the red dashed line) we find there are two solutions for the self-bound condition P = 0, which correspond to a local maximum and a local minimum in the free energy, respectively. However, the lower density self-bound solution (i.e., the local maximum in the free energy) is not mechanically stable and should be discarded, since one can readily identify that its inverse compressibility, becomes negative so the system collapses. As we increase temperature, we observe that the two self-bound solutions start to merge and eventually disappear at a threshold temperature k B T th ∼ 10 −4 2 /(2ma 2 ). Above this threshold temperature, the pressure is always positive and there is no longer a local minimum in the free energy (see, i.e., the blue dot-dashed lines at k B T = 1.5×10 −4 2 /(2ma 2 )). Thus, the self-bound droplet state is completely destabilized by the temperature effect.

B. Threshold temperature for destabilization
We may determine the threshold temperature for destabilization from the analytic expression for the pressure. By setting P = 0 in Eq. (48) and using Eq. (50), we obtain that where we have defined the function, (54) As shown in the inset of Fig. 3(a), f 3 (γ) is a nonmonotonous function and reaches its maximum at γ th ≃ 0.9835. Therefore, we find which is shown in Fig. 3(a) as a function of the interaction strength ratio a 12 /a. By recalling that the zerotemperature equilibrium density n eq is given by [35], we can measure the threshold temperature in units of the condensation temperature T c,eq at the equilibrium density n eq , T th T c,eq ≃ 0.3742 1 + a a 12 As can be seen from Fig. 3(b), under the typical experimental conditions (i.e., a 12 ∼ [−1.10, −1.05]a as in Refs. [9,11]), the threshold temperature is less than one-tenth of the condensation temperature. This small threshold temperature can alternatively be understood from Eq. (50). At the threshold reduced temperature, γ th ∼ 1, we find that T th /T c,eq ∼ (na 3 ) 1/3 ∼ 0.1 for the small interaction parameter na 3 ∼ 10 −5 − 10 −4 in the experiments [9,11]. In Fig. 4, we present the temperature dependence of the equilibrium pairing parameter ∆ eq (T ) at the interspecies interaction strength a 12 = −1.05a. As the the pairing parameter is proportional to the density, this figure also show the equilibrium density n eq as a function of temperature. In general, with increasing temperature there are two branches in ∆ eq (T ) and n eq (T ). The lower branch, which is shown by red crosses, corresponds to the unstable low-density self-bound solution we discussed earlier and therefore should be neglected. We find that for T < 0.7T th the temperature dependence in the upper branch n eq is relatively weak. However, towards and upon reaching the threshold temperature, the equilibrium density n eq can decrease significantly, by several tens of percent in relative.

C. Quantum and thermal depletions
The results we presented so far are all obtained within the Bogoliubov theory, which is valid at sufficiently low temperature in the weakly interacting regime. To have a self-consistent check, in Fig. 5, we show the quantum depletion (a) and thermal depletion (b) as a function of the density at the inter-species interaction strength a 12 = −1.05a, obtained by using Eq. (35) and Eq. (36). At the typical temperatures considered in this figure, the quantum depletion is less than one percent and is essentially temperature dependent. The thermal depletion is even smaller and is about 0.1% close to the threshold temperature for the thermal destabilization of the droplet state. Therefore, we conclude that the conditions for the application of the Bogoliubov theory are well satisfied.

IV. ONE-DIMENSIONAL DROPLETS
We now turn to consider one-dimensional quantum droplets. In this case, counterintuitively, the droplet formation is driven by LHY quantum fluctuations [18,28], which provide an attractive force to the system. It is then balanced by the repulsive mean-field force under the condition g > −g 12 . As given in our earlier work [36], the zero-temperature thermodynamic potential takes the form (V is now the length of the system), where G 1 (α) ≡ (1 + α) 3/2 + h 1 (α) with h 1 ≡ (3/2)´∞ 0 dtt −1/2 [t + (1 + α)/2 − (t + 1)(t + α)]. As in the three-dimensional case, we introduce the variables t, α and γ to rewrite the finite-temperature contribution to the thermodynamic potential, where At very low temperature γ ≪ 1, where only the gapless phonon mode contributes to the integral for s 1 (α, γ), we obtain, Therefore, it is convenient to rewrite s 1 (α, γ) in the form, The temperature-or γ-dependence of s 1 (α, γ) and s 1 (α, γ) at three selected values of α is shown in Fig.  6. In contrast to the three-dimensional case, we find that the higher-order correction factors 1 is generally larger than 1.0 and does not change too significantly as we increase the temperature. By collecting the contributions from quantum and thermal fluctuations to the thermodynamic potential, we obtain (α = C/∆ and γ = k B T /C), become comparable at the reduced temperature γ ∼ O(1). NaÃ¯vely, one may think that thermal fluctuations enhance the stability of the the one-dimensional droplet state, contrary to its three-dimensional counterpart. However, it turns out to be an incorrect picture. We have performed numerical calculations for the equations of state, by minimizing Ω = Ω 0 + Ω T in Eq. (59) and Eq. (60) at a given chemical potential µ and consequently calculating the pressure and the free energy. Their numerical results at the inter-species interaction strength g 12 = −0.75g is reported in Fig. 7. As shown in (a) for pressure, at sufficiently low temperature, i.e., k B T = 0.5ε B , where ε B = 2 /(ma 2 12 ) is the binding energy for an inter-species dimer, we find a droplet state satisfying the self-bound condition P = 0 (or equivalently a local minimum F/N ) at the density n eq ∼ 3 |a| −1 . With increasing temperature (k B T = ε B ), the equilibrium density n eq becomes smaller. At the largest temperature considered in the figure, i.e., k B T = 1.5ε B , it seems that the pressure P is always positive and at the same time the free energy per particle decreases monotonically as the density decreases to zero. Therefore, the self-bound droplet state disappears at a threshold temperature ε B < k B T th < 1.5ε B . To understand the thermal destabilization of the droplet state, it is useful to derive an analytic expression for the pressure, following the same steps in the threedimensional case. However, we note that, the condition |µ| ≪ C, ∆ 0 is not so well-satisfied in one dimension [36]. As a result, the analytic expressions derived are of qualitative use only (so we do not show them in Fig. 7). For the pressure P , it is straightforward to obtain that, where the last two terms come from quantum fluctuations and thermal fluctuations, respectively. It is easy to see that, while the contribution from quantum fluctuations to the pressure is negative, the contribution from thermal fluctuations is always positive and increases with increasing temperature, consistent with our numerical results shown in Fig. 7(a). Therefore, thermal fluctuations eventually destroy the droplet state. By setting P = 0 in Eq. (65), we find the relation, where f 1 (γ) ≡ γ 1 − The function f 1 (γ) is shown in the inset of Fig. 8 at γ th ≃ 0.330. Therefore, the threshold temperature in one dimension is given by, This analytical prediction is shown in Fig. 8 by using a black solid line. We have also numerically determined the threshold temperature by repeating the calculations in Fig. 7 at different inter-species interaction strengths and show the results in the figure by empty circles. It seems that there is a good agreement between numerical and analytical results. In the interval of interest, i.e., g 12 ∼ [−0.9, −0.5]g, the threshold temperature is roughly at the order of the binding energy ε B of the inter-species dimer.

V. CONCLUSIONS
In summary, we have presented a systematic investigation of the temperature effect in self-bound ultradilute quantum droplets, both in three dimensions and in one dimension, by extending a recently developed microscopic pairing theory [35,36] to nonzero temperature. We have shown that thermal fluctuations generally destabilize the droplet state and destroy it above a threshold temperature. The energy scale of the threshold temperature is at the order of the intra-species interaction energy and is therefore very small in the weakly interacting regime. For a three-dimensional quantum droplet, the threshold temperature is less than one-tenth of the Bose-Einstein condensation temperature under current experimental conditions [9,11]. We have also predicted the temperature dependence of the equilibrium density and have found that it can decrease by several tens of percent upon reaching the threshold temperature.
Our predictions could be readily examined in future experiments on quantum droplets realized by a binary Bose mixture with attractive inter-species interactions. The sensitive temperature dependence of the droplet state may alternatively provide us good thermometry in ultracold atomic experiments, where the low temperature at the scale of one-tenth of the condensation temperature is often challenging to measure.