Polaron physics in optical lattices

We investigate the effects of a nearly uniform Bose-Einstein condensate (BEC) on the properties of immersed trapped impurity atoms. Using a weak-coupling expansion in the BEC-impurity interaction strength, we derive a model describing polarons, i.e., impurities dressed by a coherent state of Bogoliubov phonons, and apply it to ultracold bosonic atoms in an optical lattice. We show that, with increasing BEC temperature, the transport properties of the impurities change from coherent to diffusive. Furthermore, stable polaron clusters are formed via a phonon-mediated off-site attraction.

Figure 1

(Color online) A quantum degenerate gas confined to an optical lattice is immersed in a much larger BEC. For increasing BEC temperature $T$, a crossover from coherent to diffusive hopping, characterized by $\tilde{J}$ and $E_a$, respectively, can be observed. The phonon-induced interaction potential $V_{i,j}$ leads to the formation of off-site polaron clusters, separated by a gap $E_b$ from the continuum of unbound states.

Figure 2

(a) Memory function ${\hslash }^2{W}_{i,j}\left(t\right)∕\left(2J^2\right)$ versus time for $k_{B}T=0$ (dotted line), $2E_p$ (dashed line), and $10E_p$ (full line). It drops off rapidly for $k_{B}T\gg E_p$, indicating the dominance of incoherent hopping. (b) Coefficients $\hslash A∕J$ (dashed line) and $\hslash \sqrt{B}∕J$ (full line) versus temperature, obtained from the numerical solution of the GME in Eq. (4) for an evolution time $\tau =10\hslash ∕J\approx 0.8\times {10}^{-3}\mathrm{s}$. The condition $A=\sqrt{B}$ is satisfied at $k_{B}T∕E_p\approx 2.3$, with $E_p∕k_B\approx 11\mathrm{nK}$. The lattice (wavelength $\lambda =790\mathrm{nm}$) contains a single ${}^{41}\mathrm{K}$ atom with $J=2.45\times {10}^{-2}{E}_R$, $\kappa ∕E_R\lambda =2.3\times {10}^{-2}$, the recoil energy $E_R=(2\pi \hslash {)}^2∕2m_a{\lambda }^2$, and $m_a$ the mass of the lattice atom. The BEC consists of ${}^{87}\mathrm{Rb}$ atoms with $n_0=5\times {10}^6{\mathrm{m}}^{-1}$ and $g∕E_R\lambda =8.9\times {10}^{-3}$.

Figure 3

(Color online) (a) Energy spectrum of three polarons in a 1D optical lattice. The solid line shows the lowest energy band $E_3\left(k\right)$ characterizing off-site three-polaron clusters. The lattice (wavelength $\lambda =790\mathrm{nm}$, $M=31$ sites, periodic boundary conditions) contains ${}^{133}\mathrm{Cs}$ atoms with $\tilde{J}=7.5\times {10}^{-3}{E}_R$, $U=50E_p$, and $\kappa ∕E_R\lambda =1.05\times {10}^{-1}$. The BEC consists of ${}^{87}\mathrm{Rb}$ atoms with $n_0=5\times {10}^6{\mathrm{m}}^{-1}$ and $g∕E_R\lambda =4.5\times {10}^{-2}$. (b) Probability of finding a three-polaron cluster (solid line) or a two-polaron cluster (dashed line) versus temperature. The parameters are $g∕E_R\lambda =6.5\times {10}^{-2}$, $\kappa ∕E_R\lambda =1.32\times {10}^{-1}$, $U=2.2E_p$, $M=27$, and the rest as for (a).

Figure 4

(Color online) (a) Density-density correlation in a system of three polarons indicating the formation of off-site three-polaron clusters. (b) Corresponding momentum distribution $⟨n_k⟩$. $\kappa ∕E_R\lambda =\{4.0,6.1,8.1,10.1,12.1\}\times {10}^{-2}$ where a higher value corresponds to a more localized correlation in (a) and a more spread distribution in (b). The other parameters are $g∕E_R\lambda =6.5\times {10}^{-2}$, $M=27$, $U=3E_p$, and the rest as in Fig. 3a.