Polaronic mass renormalization of impurities in Bose-Einstein condensates: Correlated Gaussian-wave-function approach

We propose a class of variational Gaussian wave functions to describe Fröhlich polarons at finite momenta. Our wave functions give polaron energies that are in excellent agreement with the existing Monte Carlo results for a broad range of interactions. We calculate the effective mass of polarons and find smooth crossover between weak- and intermediate-coupling strength. Effective masses that we obtain are considerably larger than those predicted by the mean-field method. A prediction based on our variational wave functions is a special pattern of correlations between host atoms that can be measured in time-of-flight experiments.

Figure 1

Polaron effective mass computed using different approaches as a function of the dimensionless interaction parameter $\alpha$ defined by Eq. (2). Our result (CGWs) is compared with Feynman's variational method [18], mean-field [20], and renormalization-group approach (RG) [33]. For the comparison with Refs. [18, 34] we fix the mass imbalance to be $M/m_B=0.23({}^6\mathrm{Li}/{}^{23}\mathrm{Na})$.

Figure 2

Ground-state energy of the Fröhlich-Bogoliubov model $E_p+E_{\mathrm{reg}}^{\mathrm{LO}}$ for zero-momentum impurity ($P=0$) predicted by different theoretical approaches as a function of the dimensionless coupling constant $\alpha$. Our result (CGWs) is compared with MC calculations [34], Feynman's variational method [18], mean field [20], and RG [33]. For the comparison with Refs. [18, 34] we fix the mass imbalance to be $M/m_B=0.23({}^6\mathrm{Li}/{}^{23}\mathrm{Na})$ and the UV cutoff $\mathrm{\Lambda }=2\times {10}^3{\xi }^{-1}$.

Figure 3

Polaron mass for the mass imbalance $M/m_B=0.46({}^{41}\mathrm{K}/{}^{87}\mathrm{Rb})$ and $M/m_B=1.53\left({}^{133}\mathrm{Cs}/{}^{87}\mathrm{Rb}\right)$ in units of bare impurity mass $M$. Increase of the interaction strength $\alpha$ between impurity atom and BEC enhances quantum fluctuations and results in stronger renormalization of $M_{\mathrm{p}}$.

Figure 4

Inset in the panel (a) shows the typical experimental setup for measuring noise correlation functions $g^{\left(2\right)}(k,k^{\prime })$ [see Eq. (13)]. TOF measurement should be performed with two detectors placed at relative angle $\theta$ between two directions of measurement. Panels (a) and (b) show noise correlations for (a) $\theta =0$ and (b) $\theta =\pi$ for the mass imbalance $M/m_B=0.46\left({}^{41}\mathrm{K}/{}^{87}\mathrm{Rb}\right)$ and $M/m_B=1.53\left({}^{133}\mathrm{Cs}/{}^{87}\mathrm{Rb}\right)$ at $\alpha =4$. In the case $\theta =0$ BEC atoms show antibunching. In the case of $\theta =\pi$ we find atom bunching. For the ${}^{41}\mathrm{K}/{}^{87}\mathrm{Rb}$ mixture this bunching has a peak at $k/\xi =3$. Large momentum asymptotics can be computed analytically and are shown with dashed lines.

Figure 5

Schematic of the respect direction of vectors: total momentum $\vec{P}$, momentum of given mode $\vec{k}$, and vector $\vec{F}\left(k\right)$. Vectors $\vec{P}$ and $\vec{k}$ define a plane $\left(\vec{P},\vec{k}\right)$ and vector $\vec{F}\left(k\right)$ is in this plane.

Figure 6

Polaron energy $E_p+E_{\mathrm{g}}^{\mathrm{O}}$ for static ${}^6\mathrm{Li}$ impurity ($P=0$) in ${}^{23}\mathrm{Na}$ BEC predicted by different theoretical approaches as a function of the dimensionless coupling constant $\alpha$ in the strong-coupling regime for different values of cutoff parameter: (a) $\mathrm{\Lambda }=3000{\xi }^{-1}$, (b) $\mathrm{\Lambda }=100{\xi }^{-1}$, and (c) $\mathrm{\Lambda }=10{\xi }^{-1}$. Our result (CGWs) is compared with MC calculations [34], Feynman's variational method [18], mean field [20], and renormalization group [33].

Figure 7

Measure of the correlation strength, $\sqrt{\mathrm{Tr}\left[QQ\right]}$, as a function of the dimensionless coupling constant $\alpha$ for systems with various mass imbalance: $M/m_B=0.23$ (Li/Na), $M/m_B=0.47$ (K/Rb), and $M/m_B=1.53$ (Cs/Rb).