Bipolarons in a Bose-Einstein Condensate

Mobile impurities in a Bose-Einstein condensate form quasiparticles called polarons. Here, we show that two such polarons can bind to form a bound bipolaron state. Its emergence is caused by an induced nonlocal interaction mediated by density oscillations in the condensate, and we derive using field theory an effective Schrödinger equation describing this for an arbitrarily strong impurity-boson interaction. We furthermore compare with quantum Monte Carlo simulations finding remarkable agreement, which underlines the predictive power of the developed theory. It is found that bipolaron formation typically requires strong impurity interactions beyond the validity of more commonly used weak-coupling approaches that lead to local Yukawa-type interactions. We predict that the bipolarons are observable in present experiments, and we describe a procedure to probe their properties.

Figure 1

(a) The cartoon shows Bose polarons forming a bipolaron as a consequence of a mediated interaction. (b) Binding energy $E_{\mathrm{BP}}$ of the bipolaron as a function of the impurity-boson interaction strength for two bosonic impurities with $m=m_B$. The solid red and dashed black lines are solutions to Eq. (3) with the induced interaction given by Eq. (4) for the gas parameters $n_B{a}_B^3={10}^{-6}$ and $n_B{a}_B^3={10}^{-5}$. The red squares and black circles are the results of the DMC calculations for the same two gas parameters. The long dashed blue line is the ground state energy of the Yukawa interaction Eq. (5) for $n_B{a}_B^3={10}^{-6}$. (c) The corresponding inverse size $1/\sigma ={\xi }_B/\sqrt{⟨r^2⟩}$ of the bipolaron wave function, where ${\xi }_B=1/\sqrt{8\pi n_B{a}_B}$, is the BEC coherence length. Vertical arrows denote the critical strength to form a bound state.

Figure 2

(a) Diagrammatic representation of the Bethe-Salpeter equation for impurity-impurity scattering. The red lines are the impurity Green’s function, and the double wavy line is the induced interaction. (b) The induced interaction. The black lines are normal and anomalous BEC Green’s functions, the dashed lines are condensate bosons, and $\mathcal{T}$ is the impurity-boson scattering matrix in the ladder approximation.

Figure 3

The local part $U(R)$ of $V_{\text{eff}}({\mathbf{r}}_1,{\mathbf{r}}_2)$ (solid black) and the Yukawa interaction (dashed green) for $n_B{a}_B^3={10}^{-6}$ and $1/k_{n}a=-0.4$. The corresponding $s$-wave binding energy $E_{\mathrm{BP}}$ and wave function are shown by solid red and dashed orange lines. Inset: the nonlocal part $u(r)$ for $1/k_{n}a=-10$ (dashed blue), $-1.5$ (solid gray), and $-0.4$ (short dashed purple).

Figure 4

Binding energy $E_{\mathrm{BP}}$ of two bosonic (solid black line) and fermionic (dashed red line) impurities with the mass ratio $m/m_B=40/23$ for $n_B{a}_B^3={10}^{-6}$. The dashed blue line is to the Yukawa binding energy for the $p$-wave bipolaron. Inset: the radial parts of the $s$- and $p$-wave functions (solid black and dashed red, respectively) for $1/k_{n}a=-0.4$.

Figure 5

The critical interaction strength $k_n{a}_c$ for the formation of bipolarons as a function of $k_n{a}_B$ (or $n_B{a}_B^3$) for bosonic (solid black line) and fermionic impurities (dashed red line). The black triangles and red squares are the Yukawa result for bosonic and fermionic impurities, respectively.