Strong-coupling Bose polarons out of equilibrium: Dynamical renormalization-group approach

When a mobile impurity interacts with a surrounding bath of bosons, it forms a polaron. Numerous methods have been developed to calculate how the energy and the effective mass of the polaron are renormalized by the medium for equilibrium situations. Here, we address the much less studied nonequilibrium regime and investigate how polarons form dynamically in time. To this end, we develop a time-dependent renormalization-group approach which allows calculations of all dynamical properties of the system and takes into account the effects of quantum fluctuations in the polaron cloud. We apply this method to calculate trajectories of polarons following a sudden quench of the impurity-boson interaction strength, revealing how the polaronic cloud around the impurity forms in time. Such trajectories provide additional information about the polaron's properties which are challenging to extract directly from the spectral function measured experimentally using ultracold atoms. At strong couplings, our calculations predict the appearance of trajectories where the impurity wavers back at intermediate times as a result of quantum fluctuations. Our method is applicable to a broader class of nonequilibrium problems. As a check, we also apply it to calculate the spectral function and find good agreement with experimental results. At very strong couplings, we predict that quantum fluctuations lead to the appearance of a dark continuum with strongly suppressed spectral weight at low energies. While our calculations start from an effective Fröhlich Hamiltonian describing impurities in a three-dimensional Bose-Einstein condensate, we also calculate the effects of additional terms in the Hamiltonian beyond the Fröhlich paradigm. We demonstrate that the main effect of these additional terms on the attractive side of a Feshbach resonance is to renormalize the coupling strength of the effective Fröhlich model.

Figure 1

(a) A mobile impurity atom in a noninteracting spin state can be transferred into an interacting spin state by a radio-frequency (RF) pulse. We describe the subsequent polaron dynamics using a time-dependent renormalization-group approach (tRG). (b) When the RF pulse is weak, detecting the spin-flip probability allows measurement of the spectral function of the impurity. (c) When a strong RF $\pi$ pulse is used, interesting polaron formation dynamics can be observed. In this case, the trajectories of impurities at strong couplings and with a finite initial momentum can waver back at intermediate times. In (b), we used the Bogoliubov approximation but included two-phonon terms in the polaron Hamiltonian as in Refs. [25, 26, 28]. We used the same parameters as in the experiment by Hu et al. [18] with a mixture of ${}^{40}\mathrm{K}$ impurities in a ${}^{87}\mathrm{Rb}$ BEC: $n_0=1.8\times {10}^{14}{\mathrm{cm}}^{-3}$, $a_{\mathrm{BB}}=100a_0$, and we employed a UV cutoff ${\mathrm{\Lambda }}_0={10}^3/\xi$ where $\xi$ is the corresponding healing length of the BEC. We performed calculations at vanishing polaron momentum $P=0$. In (c), we solve the problem of an impurity with finite momentum $P>0$ (in the subsonic regime). The polaron trajectory $x_{\mathrm{imp}}\left(t\right)$ is shown after an interaction quench of the polaronic coupling constant from $\alpha =0$ to 2.1. The initial impurity velocity was $P/M=0.5c$, and the mass ratio was $M/m_{\mathrm{B}}=0.5$. We used a sharp UV cutoff at ${\mathrm{\Lambda }}_0=20/\xi$ in the calculations, and $c$ denotes the speed of sound in the BEC.

Figure 2

Polaron trajectories are shown after an interaction quench at time $t=0$, from noninteracting to different values of $\alpha$. The initial impurity velocity was $P/M=0.5c$, and we used a sharp UV cutoff at ${\mathrm{\Lambda }}_0=20/\xi$ in the calculations. In (a) we compare the result for different values of the final coupling $\alpha$, at a mass ratio of $M/m_{\mathrm{B}}=0.5$. In (b) we set $\alpha =1$ for all curves and varied the mass ratio $M/m_{\mathrm{B}}$. We compare tRG simulations and MF results for the Fröhlich model with MF calculations including beyond-Fröhlich effects.

Figure 3

The polaron trajectory is shown after an interaction quench from $\alpha =0$ to 2 at time $t=0$. The initial impurity velocity was $P/M=0.5c$, and the mass ratio is $M/m_{\mathrm{B}}=0.5$. We used a sharp UV cutoff at ${\mathrm{\Lambda }}_0=20/\xi$ in the calculations. In (b), the same data as in (a) are shown, but on a double-logarithmic scale.

Figure 4

Polaron trajectories after an interaction quench from $\alpha =0$ to 1.5 (a) and $\alpha =2$ (b) at time $t=0$ are shown for different initial impurity momenta $P$. The mass ratio is $M/m_{\mathrm{B}}=0.5$ and we used a sharp UV cutoff at ${\mathrm{\Lambda }}_0=20/\xi$ in the calculations.

Figure 5

(a) The polaron quasiparticle weight $Z$ is calculated as a function of the coupling strength $\alpha$ in the Fröhlich Hamiltonian. In (b) and (c) the incoherent part of the spectral function is shown, calculated from tRG (a) and MF (b), respectively. Energies are measured as a difference $\mathrm{\Delta }\omega$ from the polaron ground-state energy $E_0$ where the coherent delta peak $I_{\mathrm{coh}}\left(\omega \right)=Z\delta (\omega -E_0)$ is located. Note the different scales in (b) and (c). In all curves, we have chosen $M/m_{\mathrm{B}}=0.26$.

Figure 6

The same data as in Fig. 5 are shown, but calculated for a larger mass ratio of $M/m_{\mathrm{B}}=1$. (b) Corresponds to tRG results and (c) to MF theory. They resemble each other much more closely in this case of a heavier impurity than in Fig. 5.

Figure 7

The time-dependent overlap $A\left(t\right)$, defined in Eq. (11), is shown for $M/m_{\mathrm{B}}=0.26$ at $\alpha =5$ in (a) and for $M/m_{\mathrm{B}}=1$ at $\alpha =30$ in (b). For long times, the complex phase $\arg A\left(t\right)$ approaches an asymptotic form $E_{0}t$, where $E_0$ is the polaron ground-state energy. This part was subtracted and we only show how the asymptotic behavior is approached.

Figure 8

(a) The spectral function is calculated after including two-phonon terms beyond the Fröhlich Hamiltonian, but still using the Bogoliubov approximation of noninteracting phonons (see Ref. [26]). We used the same parameters as in the experimental observation of Bose polarons [17], $M/m_{\mathrm{B}}=1$ and $n_0=2.3\times {10}^{14}{\mathrm{cm}}^{-3}$. A UV cutoff ${\mathrm{\Lambda }}_0=1/60a_0$ corresponding to the inverse effective range estimated in Ref. [17] was used ($a_0$ is the Bohr radius). We performed calculations at vanishing polaron momentum $P=0$, and included the same Fourier broadening as discussed in Ref. [17]. For comparison, the measured spectrum is shown in (b), taken from Ref. [17].