Temperature-induced miscibility of impurities in trapped Bose gases

We study the thermal properties of impurities embedded in a repulsive Bose gas under a harmonic trapping potential. In order to obtain exact structural properties in this inhomogeneous many-body system, we resort to the path-integral Monte Carlo method. We find that, at low temperatures, a single impurity is expelled to the edges of the bath cloud if the impurity-boson coupling constant is larger than the boson-boson one. However, when the temperature is increased, but still in the Bose-condensed phase, the impurity occupies the core of the trap and, thus, the system becomes miscible. This thermal-induced miscibility transition is also observed for a finite concentration of impurities in this inhomogeneous system. We find that the transition temperature for miscibility depends on the impurity-boson interaction and we indicate a novel nondestructive method to measure the temperature of a system based on the studied phenomenon.

Introduction.The Bose polaron has been extensively studied both at zero temperature  as well as at finite temperatures [23][24][25][26][27][28].Such a system presents a significant interest not only due to fundamental questions about quasi-particle formation, but also the mobility of the impurity inside the Bose-Einstein condensate (BEC), as was recently shown, can serve as a nondestructive indicator of the temperature of the whole system [29].However, the question of how a single impurity behaves in an inhomogeneous BEC at finite temperature was not addressed.
In homogeneous repulsive Bose-Bose mixtures, the mean-field approximation provides a robust criterion to distinguish miscible and immiscible phases at zero temperature.In particular, if g BB , g II , and g BI denote the intra-and inter-species coupling constants respectively, the two components of the mixture are phase separated when g BI > √ g BB g II and they are mixed when g BI < √ g BB g II [30].Quantum Monte Carlo (QMC) techniques have verified this criterion and, moreover, they have extended the analysis to finite temperatures revealing that Popov theory fails in describing repulsive Bose mixtures at finite polarization [31][32][33][34].Nevertheless, an inhomogeneous thermal mixture has only been studied, using QMC techniques, in [35] with the same number of particles per species.The behavior of an impurity in an inhomogeneous gas at finite temperature and the effect of the number of impurities in the miscibility of the system are topics that have not been addressed in the field yet.QMC techniques can shed some light on it and provide novel methods for thermometry.
In the present work, our main goal is to study thermal properties of impurities immersed into an inhomogeneous Bose gas.We use the Path Integral Monte Carlo method (PIMC) which provides an exact technique to find structural properties such as the density profile in an exact way, within controllable statistical errors.We analyze the system for different interaction strengths between the impurity and the rest of the particles, and we observe that above a certain interaction g BI the impurity remains outside the Bose gas (see Fig. 1).Finally, we observe the existence of a mixing temperature above which the impurity penetrates the center of the gas.
Model.The Hamiltonian of the (N + N I )-particle system arXiv:2401.03926v2[cond-mat.quant-gas]16 Apr 2024 in a harmonic trap in three dimensions is given by where lower (upper) case indices denote particles (impurities) with positions r α and distances r αβ = |r α − r β | between them, with {α, β} = {i, I}.We consider an experimentally relevant case where all the particles have the same mass m.We model the repulsive boson-boson (impurity-impurity) interaction potential by V BB (r) = V II (r) = V 0 /r 12 and boson-impurity by V BI = V BI 0 /r 12 with amplitudes V 0 and V BI 0 chosen to reproduce [36,37] the desired values of the s-wave scattering length a and a BI , correspondingly.The coupling constants, which appear in the mean-field theory, g = g BB = g II and g BI are related to a and a BI within the first Born approximation as g = 4πℏ 2 a/m, and similarly for g BI .The external confinement is taken in the form of a harmonic trap with frequency ω, i.e., V ext (r) = mω 2 r 2 /2, which defines a characteristic oscillator length scale set by a ho = ℏ/(mω).
For sufficiently low values of the gas parameter, na 3 , the specific shape of the interaction potential is no longer important and the description in terms of the s-wave scattering length becomes universal [38].To ensure that the simulations are performed in this universal regime, we ascertain that the density n 0 at the center of the trap in our simulations is sufficiently low, n 0 a 3 < 10 −5 .This is achieved by setting a ho /a = 15 while using around one hundred particles.The temperature scale is set by the degeneracy temperature of an ideal BEC in a harmonic trap, which depends solely on the number of particles N and the trap frequency ω, i.e., where ζ(x) is the Riemann zeta function.
Method.We perform PIMC simulations of N bosons and N I impurities in a 3D harmonic trap, see Eq. ( 1).In this method, the kinetic and the potential parts of the temperature density matrix are decoupled and a trotterization algorithm is applied [39].In this scheme, the exponential of the Hamiltonian is divided into small divisions (called beads) in such a way that the thermal density matrix can be approximated into decoupled terms by using fourth-order expansions such as the Chin action [40,41].This technique has been used in other works exploring dilute mixtures at finite temperature in which structural properties of the system have been computed accurately [28,34].The indistinguishability of the bosonic particles is imposed by sampling permutations using the worm algorithm [42].In this algorithm, particles, represented as polymers, where each subparticle corresponds to a different bead (i.e., the particle at a particular imaginary time), can be cut, bound and swapped to other polymers making the overall chain a cluster of indistinguishable particles.We note that we have verified that doubling the number of particles does not lead to any significant change in the observed phenomena.
Results.In order to study the effects of the temperature, we focus on experimentally observable structural properties, such as the density profile of the impurity.
The density profiles shown in the following figures have been all computed using the PIMC method explained above.In Fig. 2, we report for a system with a single impurity (N I = 1) various density profiles obtained by changing the temperature while keeping the ratio of the coupling constants fixed to g BI /g = 1.5.Remarkably, at low temperatures, the impurity remains mostly outside of the central region of the trap in which bath density is maximal.This effect emerges from a delicate balance between the repulsive interactions and potential energy of the external harmonic trapping, since by expelling the impurity, its energy associated with the external potential is increased, while the energy coming from the interaction with bath particles is decreased.The contribution of the impurity to the potential energy can be roughly estimated by assuming that the boson-impurity interaction is weak and the impurity profile ρ I (r) does not modify where ρ B (r) can be taken as the ideal Bose gas density at T = 0, i.e., ρ IG B (r) = N/(π 3/2 a 3 ho ) exp(−r 2 /a 2 ho ).Equation (3) can be further simplified by assuming that the boson-impurity interaction is a contact pseudopotential, V BI (r) → g BI δ(r) and the impurity is located at a certain distance from the center, ρ I (r) ∝ δ(x − r I )δ(y)δ(z), where we arbitrarily choose the direction of the impurity as the x−axis.Within these approximations, one gets The most probable position of the impurity, r I , is then defined by minimization of its potential energy (4), This prediction for the most probable position of the impurity is shown in Fig. 2 with a vertical dashed line.A reasonably good agreement is found in the cases when the impurity is expelled from the center, which happens for low T .For strong boson-impurity interactions (i.e.g BI /g ≫ 1), Eq. ( 5) loses accuracy because it is derived by assuming an absence of correlations and, moreover, the impurity starts to significantly modify the density profile of the bath.
Another limiting factor to our approximation is that in Eq. ( 5) we assumed a non-interacting bath.However, for a significantly larger number of particles, such that (N a/a ho ) 1/5 ≫ 1, the local density approximation (LDA) is expected to be applicable and the zero-temperature density profile of the bath would instead take a form of an inverted parabola [43].
As the temperature of the system is increased, we find that at a mixing temperature (T = T m ) the impurity fully penetrates the center of the trap.We define T m as the temperature at which the peak in the density profile of the impurity moves from a finite value to the center of the trap.The mixing temperature T m depends on the coupling constant g BI ; for example, we find T m /T BEC = 0.6 ± 0.1 for g BI /g = 1.5, and T m /T BEC = 0.8±0.1 for g BI /g = 4.5.We emphasize that in all our calculations T m is lower than the degeneracy temperature T BEC , which means that this phenomenon appears in the condensed phase.
Since the mean-field criterion for a miscible to immiscible transition, expressed as g BI = √ g BB g II , is not applicable to a single impurity due to the absence of a defined value of g II , we investigate miscibility as a function of the ratio g BI and g = g BB .To this end, in Fig. 3 we plot the density profiles of the impurity for various T and g BI /g.We find that when g BI /g < 1 the impurity remains miscible with the rest of the particles at the center of the trap while, for g BI /g > 1, we observe the same immiscible behavior as in Fig. 2, i.e. at low T the impurity is repelled from the central region of the trap and it penetrates the center when the temperature increases.We note that the estimated position of the impurity with Eq. ( 5) works well in the immiscible phase while it fails in the miscible one.
To explore the influence of multiple impurities on temperature-induced miscibility, we study N I > 1 impurities interacting with each other with the same inter-atomic potential as the particles in the bath, i.e., V II (r) = V BB (r).In Appendix B, we study the effect of N on the system.In Fig. 4, we show characteristic density profiles of the impurities for various number of impurities N I , focusing on the case of strong atomimpurity interactions (g BI /g = 3.5).At low temperatures the immiscible phase is formed.For small impurity concentration, the impurities are expelled outside creating a spherically symmetric shell and completely voiding the central region at r = 0 (see Fig. 4a), similarly to the single-impurity case.As the impurity concentration is increased, the outer shell broadens due to impurity-impurity interactions, as can be seen by comparing Fig. 4a and Fig. 4b, while the spherical symmetry is still preserved.Instead, in the balanced case, N I = N , a phase with a different symmetry is realized [35,44], composed of two blobs of pure phase, each occupying one half of the trap.The interface surface between impurities and particles is minimized as it corresponds to a flat circle rather than to a sphere.A typical snapshot of the two-blob phase is shown in Fig. 1a and it has r = 0 region fully accessible for the impurities as manifested by large values of ρ I (r = 0) (Fig. 4c).As the temperature is increased, the system changes its symmetry and undergoes a crossover to a miscible phase.
Finally, in Fig. 5 we study strongly repulsive interac- tions between the impurity and the rest of the particles.For large g BI /g ratios, the strong boson-impurity repulsion causes the impurity to be expelled to a larger radius, as can be expected from Eq. (5).At the same time, more energy is needed to overcome the interaction with the bath (i.e. to occupy the center of the trap) and, as a consequence, the mixing temperature T m grows.This result indicates that T m has a significant dependence on g BI .Conclusions and Discussion.In this work, we address the polaron problem by performing PIMC simulations of impurities immersed in a bosonic gas within a harmonic trap.We compute the density profile of the impurity at finite temperatures.We find that even though the mean-field criterion for homogeneous mixtures [30] is not defined in the single-impurity case, we observe miscible and immiscible regimes.At low temperatures, the impurity fully penetrates the central region of the bath cloud for g BI /g < 1, while the bosonic bath expels the impurity to the outer shell for g BI /g > 1. Notably, this result goes beyond the quasi-particle picture where the impurity is surrounded and dressed by excitations of the bath.The observed phenomenon recalls what happens in droplets of 4 He when impurities of 3 He are added to the system [45,46].These impurities are also expelled to the surface of the droplet creating what is known as the Andreev state [47,48].Furthermore, we have found a closed expression that approximates the most probable distance of the impurity from the center.This prediction holds well for the immiscible phase (g BI /g > 1) as long as the repulsion is not very strong.Instead, for strong impurity-atom interactions and large impurity concentration the system loses spherical symmetry and splits into two blobs.
As the temperature of the system is increased, we observe a critical temperature T m above which the mixing occurs and the impurity fully penetrates the center of the trap.We discover a noticeable dependence of T m on the coupling constant g BI .The more repulsive is the impurity, the higher is the mixing temperature T m .This phenomenon can be used in experiments to estimate the temperature of the system without disturbing it.Notice that, as it has been shown in our work, the insertion of the impurity to the system barely modifies the properties of the bath and, thus, this method is nondestructive.The relation between impurity properties T m can be obtained using QMC techniques and, if it is necessary, one can extrapolate the results to systems with a larger number of particles.
The phenomenon identified in this study as well as its relation to thermometry can be experimentally observed since, with the current technologies, the properties of the impurities, e.g., its mean position, can be reliably measured.A possible challenge is to introduce only a single impurity to the system but, as we have shown, even with the presence of a few more impurities, the temperatureinduced miscibility still persists making this approach a promising technique for measurements of the temperature in trapped ultracold gases.

FIG. 1 .
FIG. 1. Snapshots of PIMC positions of the beads of particles (blue) and impurities (red) for different characteristic parameters.Panels (a) and (b) correspond to a binary mixture with the number of particles equal to the number of impurities, NI = N , gBI /g = 3.5 and gBB = gII = g.Panels (c) and (d) show a single impurity with gBI /g = 4.5 in a bath.The systems are immiscible at T /TBEC = 0.1 and are miscible at T /TBEC = 0.9, being TBEC defined in Eq. (2).

FIG. 2 .
FIG.2.Single impurity density profile in a bath containing N = 64 bosons with gBI /g = 1.5 and a ho /a = 15 is depicted using various colors, each corresponding to a different temperature.The density profile is normalized to the number of impurities (NI = 1 in this case).The dashed vertical line shows the most probable impurity position as approximated by Eq. (5).The gray area shows where half of the particles of the bath are located at T = 0.1TBEC (in Appendix A we show the density profiles).Notice that the impurity starts penetrating the center of the trap as the temperature is increased.

FIG. 3 .
FIG.3.Single-impurity density profile, similar to Fig.2but for various values of gBI /g, including values both larger and smaller than 1.

FIG. 5 .
FIG. 5. Density profiles of the impurity interacting repulsively with a bath of 64 bosons at different gBI /g and with a ho /a = 15.The density profile is normalized to the number of impurities (NI = 1 in this case).The gray area shows where half of the particles of the bath are located at T = 0.1TBEC.