Universality of an Impurity in a Bose-Einstein Condensate

We consider the ground-state properties of an impurity particle (“polaron”) resonantly interacting with a Bose-Einstein condensate (BEC). Focusing on the equal-mass system, we use a variational wave function for the polaron that goes beyond previous work and includes up to three Bogoliubov excitations of the BEC, thus allowing us to capture both Efimov trimers and associated tetramers. We find that the length scale associated with Efimov trimers (i.e., the three-body parameter) can strongly affect the polaron’s behavior, even at densities where there are no well-defined Efimov states. However, by comparing our results with recent quantum Monte Carlo calculations, we argue that the polaron energy is a universal function of the Efimov three-body parameter for sufficiently low boson densities. We further support this conclusion by showing that the energies of the deepest bound Efimov trimers and tetramers at unitarity are universally related to one another, regardless of the microscopic model. On the other hand, we find that the quasiparticle residue and effective mass sensitively depend on the coherence length $\xi$ of the BEC, with the residue tending to zero as $\xi$ diverges, in a manner akin to the orthogonality catastrophe.

Popular Summary

Universality is a powerful concept, allowing one to construct physical descriptions of systems that are independent of their scale. A prime example is the fact that trapped atomic gases at nanokelvin temperatures can simulate the behavior within a neutron star, which is at one million kelvin and contains some of the densest matter in the Universe. This remarkable universality relies on the particles having short-range interactions that lie within the so-called unitarity regime, where there is no length scale associated with the interactions. However, an outstanding question is whether a similar universality exists for bosonic particles such as ${}^4\mathrm{He}$ atoms, which are known to cluster strongly together and condense. To address this question, we consider a single impurity atom interacting with a Bose condensed gas via unitarity-limited interactions. By comparing the results of different models, we demonstrate the existence of universal features that are model independent.

Because of the propensity of bosons to cluster, there exist the well-known Efimov trimers (trios of bound particles), which yield an additional interaction length scale $a_{-}$ that exists even at unitarity. However, we find that the ground-state energy of the impurity is a universal function of $a_{-}$ when the boson density is sufficiently low. By contrast, the characteristics of the impurity (such as its mass) themselves sensitively depend on the details of the Bose condensed medium, such as the coherence length.

Our results on the universality of impurities have broad implications since the scenario of an impurity interacting with bosons appears in a diverse range of solid-state systems and quantum fluids.

Figure 1

Few-body energy spectrum calculated from Eqs. (7)–(9). The ground (first excited) trimer is shown as a red solid curve, which first appears at a negative scattering length $a_{-}$ $(a_{-}^{(1)})$ and then dissociates into the atom-dimer continuum (delimited by the black solid line) at a positive scattering length $a_{*}$ $(a_{*}^{(1)})$, where $a_{*}$ is outside the plotted region. We only show the ground and first excited trimers, but there exists an infinite series of higher excited trimers with an accumulation point at unitarity, $1/a=0$. We also find two tetramer states (blue dashed lines) tied to the ground Efimov trimer. The excited tetramer is very weakly bound, but it persists at unitarity and disappears into the atom-trimer continuum at the positive scattering length $a\simeq 9\times {10}^3|r_0|$.

Figure 2

Energy equation of state for the Bose polaron at unitarity $1/a=0$, obtained using the variational wave function with one (gray lines), two (red lines), and three (blue lines) Bogoliubov excitations for vanishing boson-boson interactions $a_B\to 0$. We show the results of the $r_0$ model (dashed lines) and the $\mathrm{\Lambda }$ model (solid lines). In the low-density limit, the dotted straight lines correspond to the Efimov trimer and associated tetramer energies, with $E_{-}=-9.91/m|a_{-}{|}^2$ and $E_{4B,-}^{(1)}=-92.7/m|a_{-}{|}^2$, respectively. The black dotted line in the high-density regime shows the result of a perturbative expansion, Eq. (14). The gray shaded region depicts the low-density few-body dominated regime, while the blue shaded region corresponds to the high-density regime $|r_0|$, ${\mathrm{\Lambda }}^{-1}>n^{-1/3}$, where Efimov physics is suppressed and the system is sensitive to the microscopic model.

Figure 3

Lowest-order diagrams for the unitary ground-state polaron in the high-density limit $n^{1/3}|r_0|\gg 1$, where (a) and (b) give, respectively, the first- and second-order terms for the polaron energy in a weakly interacting BEC. Here, the dashed lines correspond to particles emitted or absorbed from the condensate, the blue solid line is the boson propagator, the wavy lines represent closed-channel molecule propagators, and the filled circles correspond to the interchannel coupling $g$. The black double line, defined in (c), represents the dressed impurity to lowest order, where the black single line is the bare impurity propagator.

Figure 4

Comparison of different models for the unitary Bose polaron in the regime $r\ll n^{-1/3}\ll |a_{-}|$, where $r$ is the range of the interactions. We show the results for the polaron energy from the $\mathrm{\Lambda }$ model (solid line) and the $r_0$ model (dashed line), which include up to three excitations of the condensate and which take $a_B\to 0$. The QMC results of Ref. [36] are shown as the gray dots. The recent Aarhus experiment [16] is estimated to have $n^{1/3}|a_{-}|\simeq 100$.

Figure 5

Polaron residue $Z$ at unitarity $1/a=0$ as a function of the dimensionless three-body parameter. We show the results for the $\mathrm{\Lambda }$ model (solid line) and the $r_0$ model (dashed line) for $a_B\to 0$. In order of decreasing residue, the results are obtained for wave functions including up to one, two, or three excitations of the condensate. The grey and blue shaded regions are the same as in Fig. 2, i.e., the low- and high-density regimes, respectively.

Figure 6

Inverse effective mass of the Bose polaron at unitarity, $1/a=0$, for $a_B=0$. We calculate it with the $\mathrm{\Lambda }$ model (solid lines) and the $r_0$ model (dashed lines), taking into account one (gray lines) and two (red lines) Bogoliubov excitations. The grey and blue shaded regions are the same as in Fig. 2.

Figure 7

Contact of the Bose polaron at unitarity, $1/a=0$, for $a_B=0$. The solid and dashed curves correspond to the $\mathrm{\Lambda }$ and $r_0$ models, respectively. We show the results obtained using wave functions with up to one (gray line), two (red line), and three (blue line) excitations. The grey and blue shading indicates the same regions as in Fig. 2.

Figure 8

Polaron energy calculated within the $\mathrm{\Lambda }$ model as a function of $n^{1/3}|a_{-}|$ for $a_B=0$ (solid line) and for $a_B=2.1\times {10}^4|a_{-}|$ (dashed line). The latter choice of $a_B$ ensures that we take the same ratio of $a_{-}$ and $a_B$ as in the QMC calculation (gray dots) [see Eq. (12)], and the corresponding scale for $a_B$ is shown on the top $x$ axis. Inset: Polaron energy at unitarity as a function of inverse number $N$ of Bogoliubov excitations for $a_B=0$. We present the results of the $\mathrm{\Lambda }$ model (solid) and $r_0$ model (open) for $n^{1/3}|a_{-}|=100$ (blue circles) and $n^{1/3}|a_{-}|=500$ (red triangles).