Motional Coherence of Fermions Immersed in a Bose Gas

We prepare a superposition of two motional states by addressing lithium atoms immersed in a Bose-Einstein condensate of sodium with a species-selective potential. The evolution of the superposition state is characterized by the populations of the constituent states as well as their coherence. The latter we extract employing a novel scheme analogous to the spin-echo technique. Comparing the results directly to measurements on freely evolving fermions allows us to isolate the decoherence effects induced by the bath. In our system, the decoherence time is close to the maximal possible value since the decoherence is dominated by population relaxation processes. The measured data are in good agreement with a theoretical model based on Fermi’s golden rule.

Figure 1

Detection of motional coherence with and without the bath. (a) Shaking the position of the optical lattice as indicated in the insets implements motional Ramsey (gray) and spin-echo spectroscopy (blue). Motional coherence is inferred from the interference amplitude which is determined by the population in the excited state after evolution times corresponding to the $\mathrm{\text{zero}}(\pi )$ phase of the last Ramsey pulse. The results are depicted as solid (open) symbols. (b) Motional interference fringes for the spin-echo sequence obtained by a short delay of the second $\pi /2$ pulse. The signal without (with) the bath (present for 2.8 ms) is shown in the left (right) panel. The blue shading in the right panel indicates the signal height without the bath and clearly reveals decoherence.

Figure 2

Systematic measurement of the decoherence time $T_2$. (a) The interference amplitudes for zero ($\pi$) phase shown as solid (open) symbols reveal the faster signal decay in presence of the bath (circles) compared to no bath (squares). Because of the different sampling, the spin-echo signal does not show the oscillatory behavior apparent in Fig. 1a, but only one revival at long evolution times. (b) The decoherence time $T_2=(6.8\pm 0.4)$ ms is deduced from the exponential fit to the ratio of the interference amplitudes with and without the bath.

Figure 3

Systematic investigation of the relaxation time $T_1$. (a) The ${}^6\mathrm{Li}$ atom numbers in the ground (diamonds), first excited (circles), and second excited (squares) state reveal the relaxation to the ground state. The third level is occupied by a negligible number of atoms and does not contribute to the dynamics. (b) For long evolution times, the normalized population in the first excited state (open circles) deviates from an exponential fit (dashed line). Including the inhomogeneous density distribution of the ${}^6\mathrm{Li}$ and ${}^{23}\mathrm{Na}$ atoms (solid line) explains the deviation. The population in the excited state without the bath (squares) is consistent with the expected loss due to spontaneous emission of the species-selective potential. The inset reveals that the population of the excited state decays for short times exponentially. For clarity, we subtract the observed long-term background due to inhomogeneous density distributions. The dashed line shows an exponential fit with a relaxation time $T_1=(4.5\pm 0.4)\mathrm{ms}$.

Figure 4

(a) Connection between relaxation and decoherence time. For a two-level system coupled to a Markovian bath $T_2=2T_1$ as indicated with the dashed line. Additional coupling of the states to other unobserved degrees of freedom represented by the wiggly arrows leads to a reduction of the ratio to $T_2=(1.5\pm 0.1)T_1$. (b) Schematic representation of the microscopic processes leading to $T_{\perp }^{\mathrm{g},\mathrm{e}}$ involve phononlike, low energetic elementary excitations of the bath. The shadings display the chemical potential of ${}^6\mathrm{Li}$ which indicates the occupied transverse modes.