Fate of the Bose polaron at finite temperature

We consider an impurity immersed in a Bose-Einstein condensate with tunable boson-impurity interactions. Such a Bose polaron has recently been predicted to exhibit an intriguing energy spectrum at finite temperature, where the ground-state quasiparticle evenly splits into two branches as the temperature is increased from zero [Guenther et al., Phys. Rev. Lett. 120, 050405 (2018)]. To investigate this theoretical prediction, we employ a recently developed variational approach that systematically includes multibody correlations between the impurity and the finite-temperature medium, thus allowing us to go beyond previous finite-temperature methods. Crucially, we find that the number of quasiparticle branches is simply set by the number of hole excitations of the thermal cloud, such that including up to one hole yields one splitting, two holes yields two splittings, and so on. Moreover, this effect is independent of the impurity mass. We thus expect that the exact ground-state quasiparticle will evolve into a single broad peak for temperatures $T>0$, with a broadening that scales as $T^{3/4}$ at low temperatures and sufficiently weak boson-boson interactions. In the zero-temperature limit, we show that our calculated ground-state polaron energy is in excellent agreement with recent quantum Monte Carlo results and with experiments.

Figure 1

Schematic depiction of the Efimov trimer spectrum. The Efimov trimers are shown as dashed lines and the dimer state as a solid line. The spectrum is invariant under a scaling transformation that takes $1/a\to 1/\left({\lambda }_{0}a\right)$ and $E\to E/{\lambda }_0^2$.

Figure 2

Ground-state polaron energy for the equal-mass case calculated within our variational approach (lines) and in a recent QMC study [42] (symbols). We show the results of dressing the impurity by up to one, two, or three excitations of the Bose gas (dotted, dashed, and solid lines, respectively), and we take $n^{1/3}{R}^{*}=0.015$ such that we have three-body parameter $n^{1/3}\left|a_{-}\right|\simeq 70$, which is the same as in the QMC (see text).

Figure 3

Ground-state polaron energy for the equal-mass case calculated within our variational approach (lines) using three Bogoliubov excitations, compared with the experimentally measured ground-state energy (symbols) [19] (see also Ref. [42]). We take $n=4\times {10}^{14}{\mathrm{cm}}^{-3},R^{*}=60a_0$, and $a_B=9a_0$ as in the experiment, which corresponds to $n^{1/3}{R}^{*}\simeq 0.02$ and $n^{1/3}{a}_B\simeq 3.5\times {10}^{-3}$ (solid line). We also include the results where we take the ideal gas limit $a_B\to 0$ (dotted line).

Figure 4

Impurity spectral functions obtained from the two-body ansatz Eq. (32) for different temperatures (a) $T=0$, (b) $T=0.5T_c$, and (c) $T=T_c$. We use a broadening of $\sigma =0.4n^{2/3}/m_B$, which is comparable to the Fourier broadening in the Aarhus experiment [19], and we take $n^{1/3}{R}^{*}=0.02$ and $m=m_B$. The solid red lines are the mean-field energy Eq. (36), while the dashed orange lines are the dimer energy $-E_{\mathrm{B}}$.

Figure 5

Impurity spectra obtained from the two-body ansatz Eq. (32) at unitarity $1/a=0$ and plotted with respect to temperature. (a) The equal-mass case, $m=m_B$, with broadening $\sigma =0.4n^{2/3}/m_B$. (b) The fixed impurity, $m\to \infty$, with smaller broadening $\sigma =0.2n^{2/3}/m_B$ due to the smaller polaron energy scale. (c) The light impurity with broadening $\sigma =0.4n^{2/3}/m_B$ and $m/m_B=40/87$, corresponding to the mass ratio in the JILA experiment [20]. In all plots, we take $n^{1/3}{R}^{*}=0.02$. The orange dashed lines are the predicted energies Eq. (42) from a low-temperature analysis.

Figure 6

(a) Impurity spectral function at unitarity produced by the three-body ansatz Eq. (37) with $m=m_B,n^{1/3}{R}^{*}=0.02$, and $\sigma =0.4n^{2/3}/m_B$. Only the energy range around the attractive polaron is plotted. The orange dashed lines are the predicted energies Eq. (46) from a low-temperature analysis. (b) Slices through the spectrum (a) at several low temperatures, with a narrower broadening of $\sigma =0.05n^{2/3}/m_B$.

Figure 7

Impurity spectral function at unitarity produced by the three-body ansatz Eq. (37) with $m/m_B\to \infty ,n^{1/3}{R}^{*}=0.02$, and $\sigma =0.05n^{2/3}/m_B$. The orange dashed lines are the predicted energies Eq. (46) from a low-temperature analysis.

Figure 8

Connections between the various coefficients in the three-body ansatz Eq. (37). The various terms are illustrated by the presence of the impurity (blue circle), particles (dark gray circles), holes (light gray circles), and the closed-channel dimer (dark gray and blue circles). The blue shaded area contains those terms responsible for the splitting into three branches, while the orange area are those terms responsible for Efimov physics at zero temperature.

Figure 9

Diagrammatic approach to quasiparticle branch splitting. (a) Dyson equation for the impurity Green's function (double line) in terms of the bare impurity Green's function (single line) and the self energy (shaded box). (b) Dyson equation at $T=0$. (c) Dyson equation at $0<T\ll T_c$, where we only include the most important diagrams for the splitting with one hole excitation. (d) Lowest order diagrams where we dress the bare impurity propagator inside the $T=0$ impurity propagator (double dashed line) of panel (c). Diagrams where hole propagators cross each other, as depicted in (e), also appear at the same order.

Figure 10

Impurity spectral function produced by the ansatz Eq. (48) plotted with respect to temperature at $1/n^{1/3}a=0$. Here we take $n^{1/3}{a}_B=0.003,n^{1/3}{R}^{*}=0.02,m=m_B$ and $\sigma =0.4n^{2/3}/m_B$, which correspond to the parameters of the Aarhus experiment [19].

Figure 11

Impurity spectra produced by the ansatz Eq. (48) at $T=0.5T_c$ with boson-boson scattering length of (a) $n^{1/3}{a}_B=0$, (b) $n^{1/3}{a}_B=0.01$, and (c) $n^{1/3}{a}_B=0.05$. The solid red lines are the mean-field energy Eq. (36), while the dashed orange lines are the dimer energy $-E_{\mathrm{B}}$. Here we take $n^{1/3}{R}^{*}=0.02,m=m_B$, and $\sigma =0.4n^{2/3}/m_B$.

Figure 12

Energy of the attractive polaron in the weak-coupling regime $n^{1/3}\left|a\right|\ll 1$, measured with respect to the energy at zero temperature. Points indicate values calculated using the variational ansatz Eq. (48), while lines indicate the prediction of low-temperature perturbation theory Eq. (49). Here we take $n^{1/3}{R}^{*}=0.001$ and $m=m_B$. Note that both axes are logarithmically scaled, with a small region around zero excluded.

Figure 13

Spectral functions produced using the interacting gas ansatz Eq. (48), comparing the $s$-wave approximation (red, solid) with the result of the full angular calculation (blue, dashed) for a variety of different values of $1/a$ and $T$. We take $a_B=0,\sigma =0.4n^{2/3}/m_B,R^{*}=0.02n^{-1/3}$, and $m=m_B$.

Figure 14

(a) Omitting from the two-body ansatz a term involving scattering of the impurity off thermal bosons makes the splitting disappear, indicating that the splitting arises from scattering off thermal bosons. (b) Omitting all terms involving scattering off thermal bosons does not make the zero-energy peak disappear, indicating that the zero-energy peak is created by some other temperature effect. We take $1/a=0,R^{*}=0.02n^{-1/3},T=0.5T_c,m=m_B$, and $\sigma =0.2n^{2/3}/m_B$.

Figure 15

The attractive polaron in the three-body ansatz, where terms involving either two-particle or two-hole excitations have been omitted. We take $1/a=0,R^{*}=0.02n^{-1/3},T=0.01T_c,m=m_B$, and $\sigma =0.05n^{2/3}/m_B$.

Figure 16

Spectra with respect to temperature for (a) the full three-body ansatz, (b) the three-body ansatz without two-particle terms, and (c) the three-body ansatz without two-hole terms. The solid red lines are the predicted splitting with $Z_{\mathrm{att}}$ [Eq. (D2) for (a) and (b); Eq. (D1) for (c)]. The dashed orange lines are the predicted splitting with $\sqrt{Z_{\mathrm{att}}}$ [Eq. (D4) for (a) and (b); Eq. (D3) for (c)]. These spectra were obtained using $1/a=0,R^{*}=0.02n^{-1/3},m=m_B$, and $\sigma =0.025n^{2/3}/m_B$.