Coherent Interaction of a Single Fermion with a Small Bosonic Field

We have experimentally studied few-body impurity systems consisting of a single fermionic atom and a small bosonic field on the sites of an optical lattice. Quantum phase revival spectroscopy has allowed us to accurately measure the absolute strength of Bose-Fermi interactions as a function of the interspecies scattering length. Furthermore, we observe the modification of Bose-Bose interactions that is induced by the interacting fermion. Because of an interference between Bose-Bose and Bose-Fermi phase dynamics, we can infer the mean fermionic filling of the mixture and quantify its increase (decrease) when the lattice is loaded with attractive (repulsive) interspecies interactions.

Figure 1

Quantum phase revivals in a few-body Bose-Fermi system. (a) In a shallow optical lattice both the bosonic and the fermionic species are delocalized and show atom number fluctuations. The cartoon picture displays a possible result of a projection measurement assuming a mean fermionic filling of $\overline{m}=0.5$. Blue (red) balls indicate bosons (fermions). (b) The local quantum state at a lattice site corresponds to coherent superpositions of bosonic atom number states with and without the presence of a fermion with eigenenergies $E_{n,m}$ (see text). (c) Suppression of odd quantum phase revivals in a Bose-Fermi mixture at $|U^{\mathrm{BF}}/U^{\mathrm{BB}}|=0.5$ for variable fermionic fillings: $\overline{m}=0$, 0.2, 0.35 and 0.5 (blue solid lines, darker color for larger $\overline{m}$). The right panel displays the suppression factor $s$ as a function of the filling $\overline{m}$.

Figure 2

Interference of Bose-Bose and Bose-Fermi phase dynamics. The initial quantum phase revivals are modulated as a function of the interspecies scattering length $a_{\mathrm{BF}}$; the intraspecies scattering length $a_{\mathrm{BB}}$ is fixed at $+102(2)a_0$ [9]. Suppressions of the first revival happen at $a_{\mathrm{BF}}=-126(2)a_0$, $-40(3)a_0$ and $+41(3)a_0$, as marked by the arrows. Black lines indicate the traces recorded in the experiment.

Figure 3

Quantum phase revival traces of the Bose-Fermi system for vanishing, (a), and strong interspecies attraction, (b). In the later case, every second revival is suppressed by an envelope (gray dashed line), which corresponds to the spectral components of order $|U^{\mathrm{BB}}+U^{\mathrm{BF}}|$. Each data point represents a single run of the experiment. The solid lines interpolate the data and serve as a guide to the eye.

Figure 4

Fourier spectra of the time traces for vanishing, (a), and linearly increasing Bose-Fermi attraction, (b) to (e). The shaded ovals highlight the contributions involving direct Bose-Fermi interactions. The inset of panel (e) shows the $|U^{\mathrm{BB}}+U^{\mathrm{BF}}|$ components as a function of $a_{\mathrm{BF}}$. Dashed vertical lines indicate the spectral contributions of orders $U^{\mathrm{BB}}$ (1 to 3) and $2U^{\mathrm{BB}}$ (4 to 6) for a purely bosonic system.

Figure 5

Modification of the Bose-Bose interaction strength induced by an interacting fermion. (a) Repulsive [attractive] interspecies interactions broaden [shrink] the on-site orbitals ${\varphi }_{\mathrm{B}}(\mathbf{r})$ and ${\varphi }_{\mathrm{F}}(\mathbf{r})$ and thereby affect the Bose-Bose interaction strengths $U_{n,1}^{\mathrm{BB}}$. (b) The highest spectral components of order $U^{\mathrm{BB}}$ [$2U^{\mathrm{BB}}$] are shown as a function of $a_{\mathrm{BF}}$ in the lower [upper] panel. The underlying energies $E_{2,0}$ and $E_{2,1}-2E_{1,1}$ [$E_{3,0}-E_{2,0}$ and $E_{3,1}-E_{2,1}-E_{1,1}$] cannot be resolved individually and appear as a single superposition peak in the spectra. The dashed line is a linear fit with a slope of $-0.46(4)\mathrm{Hz}/a_0$ [$-0.87(5)\mathrm{Hz}/a_0$]. Shaded areas show the shift calculated within a variational model [12], which takes into account the measured values for the fermionic filling fitted by an exponential function, $\overline{m}(a_{\mathrm{BF}})$ (inset). The data at $a_{\mathrm{BF}}=0a_0$ have been obtained in a sample without fermions.