Field-theoretical study of the Bose polaron

We study the properties of the Bose polaron, an impurity strongly interacting with a Bose-Einstein condensate, using a field-theoretic approach and make predictions for the spectral function and various quasiparticle properties that can be tested in experiment. We find that most of the spectral weight is contained in a coherent attractive and a metastable repulsive polaron branch. We show that the qualitative behavior of the Bose polaron is well described by a non-self-consistent $T$-matrix approximation by comparing analytical results to numerical data obtained from a fully self-consistent $T$-matrix approach. The latter takes into account an infinite number of bosons excited from the condensate.

Figure 1

Diagrams corresponding to the non-self-consistent $T$-matrix approximation. (a) Dyson's equation for the impurity Green's function. (b) Impurity self-energy. (c) In-medium $T$-matrix equation. Solid thin lines stand for impurity propagators while dashed lines represent normal Bose propagators. The zigzag lines denote in- and outgoing condensate particles. In the case of the self-consistent $T$-matrix approximation discussed in Sec. 4, the solid thin line (bare fermion propagator) in the particle-particle loop in (c) is replaced by the thick line from (a) representing the dressed fermion propagator.

Figure 2

Impurity spectral function $A_{\mathrm{pol}}(\Omega ,\mathbf{p}=\mathbf{0})$ in the non-self-consistent $T$-matrix approximation with $n^{1/3}{a}_{\varphi \varphi }=0.1$. Solid lines: weak-coupling limit (valid for both branches) $E\sim g_{\varphi \psi }n$. Dashed line: energy of the universal dimer in vacuum, $E_{\dim }\sim -1/m_{\varphi }{a}_{\varphi \psi }^2$. Here and in all following plots, zero-width peaks are given a small artificial width to be visible on the graph.

Figure 3

Quasiparticle energies and width obtained in the non-self-consistent $T$-matrix (NSCT) approach with both a finite and a vanishing boson-boson interaction, and the self-consistent (SCT) approach. In the latter case, the dotted line indicates the result after the first iteration of the self-consistency loop while the individual dots denote the final value for 15 iterations when full self-consistency of the $T$-matrix equations is reached. (a) Attractive polaron energy with the universal dimer binding energy $E_{\dim }=-\theta \left(a_{\varphi \psi }\right)/m_{\varphi }{a}_{\varphi \psi }^2$ subtracted. This representation accentuates the absolute difference between the values obtained in the various approximation schemes. The relative deviation is far smaller on the right of the minimum due to the large modulus of $E_{\dim }$. (b) Energy of the repulsive polaron. The plot excludes low values of $a_{\varphi \psi }^{-1}$ where the repulsive polaron is not properly defined. (c) Quasiparticle width of the repulsive polaron for the same range of values of $a_{\varphi \psi }^{-1}$.

Figure 4

(a) Quasiparticle weights for both branches (the lines on the upper right of the graph correspond to the repulsive polaron). (b) Effective mass of the attractive polaron. Parameters and line styles are as in Fig. 3.

Figure 5

Impurity spectral function $A_{\mathrm{pol}}(\Omega ,\mathbf{p})$ as a function of frequency and momentum for (a) ${\left(n^{1/3}{a}_{\varphi \psi }\right)}^{-1}=1$ and (b) ${\left(n^{1/3}{a}_{\varphi \psi }\right)}^{-1}=-5$. In both graphs, $n^{1/3}{a}_{\varphi \varphi }=0.1$. Solid line: free-impurity dispersion. Dashed line: free-molecule-like dispersion. Dash-dotted line: dispersion according to the effective mass of the attractive polaron at $p=0$. For positive $a_{\varphi \psi }$, the attractive polaron peak gradually bends away from its dispersion at vanishing momentum, reflecting an increase in the momentum-dependent effective mass. At negative $a_{\varphi \psi }$, the effective mass stays approximately constant as a function of momentum.

Figure 6

Spectral function $A_{\mathrm{pol}}(\Omega ,\mathbf{p})$ for $a_{\varphi \varphi }=0$; all other parameters are as in Fig. 5b. Note that here the continuum onset coincides with the dashed line indicating the free-molecule-like dispersion. The point where the polaron peak acquires a finite width is shifted accordingly.

Figure 7

Polaron spectral function calculated using the self-consistent scheme discussed in Sec. 4. The black lines are identical to the ones in Fig. 2. Note how the continuum of excitations now follows the attractive polaron peak.

Figure 8

Polaron spectral function calculated using the self-consistent scheme discussed in Sec. 4 as a function of momentum and frequency. The black lines have the same meaning as in Fig. 5. (a) ${\left(n^{1/3}{a}_{\varphi \psi }\right)}^{-1}=1$. Note how the attractive polaron peak is “annihilated” when it touches the continuum. (b) ${\left(n^{1/3}{a}_{\varphi \psi }\right)}^{-1}=-5$. The continuum almost merges with the peak for higher momenta.

Figure 9

(a) Dyson's equation for the molecule propagator in the two-channel model (A2) in a $T$-matrix-like approximation. Double lines denote the molecule propagator and open circles represent the atom-molecule conversion coupling $h$. (b) Diagrammatic representation of the molecule propagator within a single-channel description of the Bose polaron. Note that in order to establish the correspondence ${\mathcal{G}}_{\xi }\leftrightarrow {\Gamma }_{\varphi \psi }$, one has to assume implicit conversion coupling vertices at both ends of the propagator line on the left-hand side.

Figure 10

Real and imaginary parts of the pair propagator $L$ for $\alpha =1$ and different values of the external momentum $p$. $\mathrm{Im}L$ is strictly positive while $\mathrm{Re}L$ is mostly negative. Note that this figure uses units in which $g_{\varphi \varphi }{\rho }_0=1$.