Coherent and Dephasing Spectroscopy for Single-Impurity Probing of an Ultracold Bath

We report Ramsey spectroscopy on the clock states of individual Cs impurities immersed in an ultracold Rb bath. We record both the interaction-driven phase evolution and the decay of fringe contrast of the Ramsey interference signal to obtain information about bath density or temperature nondestructively. The Ramsey fringe is modified by a differential shift of the collisional energy when the two Cs states superposed interact with the Rb bath. This differential shift is directly affected by the mean gas density and the details of the Rb-Cs interspecies scattering length, affecting the phase evolution and the contrast of the Ramsey signal. Additionally, we enhance the temperature dependence of the phase shift preparing the system close to a low-magnetic-field Feshbach resonance where the $s$-wave scattering length is significantly affected by the collisional (kinetic) energy. Analyzing coherent phase evolution and decay of the Ramsey fringe contrast, we probe the Rb cloud’s density and temperature. Our results point at using individual impurity atoms as nondestructive quantum probes in complex quantum systems.

Figure 1

(a) Single impurity (green) immersed in Rb bath (gray) interacting. The coherent superposition of Cs states is visualized on a Bloch sphere showing the interaction-induced phase evolution $\mathrm{\Phi }$ and phase dispersion ${\sigma }_{\mathrm{\Phi }}$. (b) Scattering length $a$ (black solid) for the Cs ground state $|g⟩$ interacting with the Rb bath for different collision energies (inset) in the vicinity of a Feshbach resonance at 198.5 mG. On the same graph, we also plot the Maxwell-Boltzmann distribution of collision energies in green for 200, 600, and 1000 nK from light to dark green in solid, dotted, and dashed lines, respectively. (c) Probability distribution of the scattering length $a$ for different bath temperatures; colors and line styles as in (b). Inset: a temperature-averaged scattering length $\overline{a}$ versus $B$ field. The black dashed line marks the experimentally fixed magnetic field $B$ at $198.50\pm 0.05\mathrm{mG}$, at the peak of the Feshbach resonance. (d) Mean scattering length $\overline{a}$ (blue) as well as the variance of the probability distribution of scattering length ${\sigma }_a^2$ (red) as a function of temperature.

Figure 2

(a) Ramsey sequence with waiting time $t$ and adjustable phase $\phi$ of the last pulse. Ramsey fringes for (b) $t=0.1$, (c) $t=6$, and (d) $t=12\mathrm{ms}$ for the case without (black, open circles) and with (green, filled circles) Rb bath. (e) Time evolution of the color-coded Ramsey signal for a bath peak density of $n_{\mathrm{Rb}}=0.195\times {10}^{13}{\mathrm{cm}}^{-3}$ and temperature of $T=980\mathrm{nK}$, showing the phase evolution. The phase shift is directly proportional to evolution time, where the slope is given by the energy difference between the two Cs clock states. The dashed line is a linear fit to the fringe maxima; the negative phase is a copy of positive phase data for better illustration. (f) Decay of Ramsey-fringe contrast with time. The decoherence time $T_2$ is fitted by a Gaussian function.

Figure 3

(a) Interaction-induced phase shift $\mathrm{\Phi }(t)$ as a function of free-evolution time $t$. The data are fitted with a line (dashed blue line) to extract the slope $\delta$ up to the dephasing time $T_2$, indicated by the vertical dashed line. Inset: a Ramsey fringe for $t=6\mathrm{ms}$ with (green, filled circles) and without (black, open circles) Rb cloud. (b) Measured interaction-induced detuning $\delta$ (blue data points) and $T_2$ times (red data points) as a function of gas density. The red solid and blue dashed lines are predictions of Eq. (5) without free parameter. The fixed temperature $T=850\mathrm{nK}$ with changing density leads, in this dataset, to a range of $T/T_c=2.2,\dots ,5$ for the Rb gas.

Figure 4

(a) Decay of the Ramsey-fringe contrast as a function of evolution time in the Rb bath. The dashed line is a Gaussian fit. Inset: Ramsey fringes for $t=0.6\mathrm{ms}$ for Cs atom without (black, open) and with (green, filled) Rb bath present. (b) Dephasing time of impurities in the Rb bath as a function of gas temperature. The data (red points) show an increase with temperature; the red line is a prediction of our model. Data are taken for a fixed Rb peak density of $n_{\mathrm{Rb}}\approx 1.5\times {10}^{13}{\mathrm{cm}}^{-3}$. Error bars and shaded area indicate $1\sigma$ uncertainties of the experimental data and numerical model, respectively. Scaled with the critical temperature, the temperature range is $T/T_c=1.2$–2.3.