Quantum Spin Dynamics of Individual Neutral Impurities Coupled to a Bose-Einstein Condensate

We report on spin dynamics of individual, localized neutral impurities immersed in a Bose-Einstein condensate. Single cesium atoms are transported into a cloud of rubidium atoms and thermalize with the bath, and the ensuing spin exchange between localized impurities with quasispin $F_i=3$ and bath atoms with $F_b=1$ is resolved. Comparing our data to numerical simulations of spin dynamics, we find that, for gas densities in the Bose-Einstein condensate regime, the dynamics is dominated by the condensed fraction of the cloud. We spatially resolve the density overlap of impurities and gas by the spin population of impurities. Finally, we trace the coherence of impurities prepared in a coherent superposition of internal states when coupled to a gas of different densities. For our choice of states, we show that, despite high bath densities and, thus, fast thermalization rates, the impurity coherence is not affected by the bath, realizing a regime of sympathetic cooling while maintaining internal state coherence. Our work paves the way toward the nondestructive probing of quantum many-body systems via localized impurities.

Figure 1

Experiment overview. (a) Sketch of Cs impurities (blue dots and fluorescence images), immersed in a Rb BEC (red dots and time-of-flight image) by a species-selective optical lattice (blue). (b) Sketch of possible interaction paths: Elastic (spin-exchange) collisions occur at rate ${\mathrm{\Gamma }}_{\mathrm{el}}$ (${\mathrm{\Gamma }}_{\mathrm{se}}$). (c)–(e) Spin evolution of the impurity, prepared in $m_{F,i}=3$ in a thermal Rb bath in states $m_{F,b}=-1$, 0, 1 (populations normalized columnwise) with density overlap $⟨n⟩=2.6\times {10}^{12}{\mathrm{cm}}^{-3}$ and a magnetic field of 250 mG. The Zeeman energy of Cs and Rb determines the direction of spin exchange (SE). For $m_{F,b}=0,-1$ (c),(d), SE between the impurity and bath atoms lowers $m_{F,i}$, so impurity atoms are eventually pumped to $m_{F,i}=-3$. For Rb in $m_{F,b}=1$ (e), this process is energetically forbidden. For $m_{F,b}=-1$ (c), we expect SE at a timescale of $1/{\mathrm{\Gamma }}_{\mathrm{se}}=1/(G⟨n⟩)=22\mathrm{ms}$, leading to SE pumping of Cs from $m_{F,i}=3$ to $m_{F,i}=-3$ within $6/{\mathrm{\Gamma }}_{\mathrm{se}}\approx 130\mathrm{ms}$.

Figure 2

(a) Measured SE dynamics of Cs impurities (initially in $m_{F,i}=3$) in an expanding BEC ($m_{F,b}=-1$) for a magnetic background field of 250 mG. The initial population of Cs atoms in $m_{F,i}=3$ is 136. (b) The calculated averaged density overlap $⟨n⟩$ of impurities with a condensate fraction (red line) and thermal background (orange, dashed line) of the BEC shows enhanced interaction with the condensate fraction (${⟨n⟩}_{\max }=2.7\times {10}^{13}{\mathrm{cm}}^{-3}$). $⟨n⟩$ decreases due to the BEC expansion. The inset shows the radial BEC line density ${\rho }_r$ (${\rho }_0=440\mu {\mathrm{m}}^{-1}$) at $t_i=0$ (marked as 1) and $t_i=20\mathrm{ms}$ (2). From this, we calculate the spin evolution of Cs in the bath (no free parameters; $G=1.71\times {10}^{-11}{\mathrm{cm}}^3\mathrm{Hz}$) with (c) BEC properties, from (a), and (d) a purely thermal Rb cloud at the same temperature and atom number (model details in Ref. [35]). Data were taken in 3200 experimental runs.

Figure 3

(a) Position-resolved impurity spin dynamics (fluorescence image) in $m_{F,i}=3$ immersed into a BEC ($m_{F,b}=-1$) from right to left, without BEC expansion. Impurity positions are sorted into bins of $\approx 8\mu \mathrm{m}$ in order to enhance statistics (optical resolution, $2\mu \mathrm{m}$). (b) $m_{F,i}$ populations are read out after $t_i=10\mathrm{ms}$ (columnwise normalized). Effective density overlap $⟨n⟩$ extracted for each bin from our model (blue dots), using a ${\chi }^2$ optimization (fitting uncertainty yields error bars). The BEC position is extracted in a simple Gaussian fit (red, dashed line), where $z=0$ is given by the experimentally determined BEC position via absorption imaging on a different imaging axis, with a possible systematic error of $\approx 10\mu \mathrm{m}$. Impurity transport through the Rb BEC takes $\approx 2\mathrm{ms}$ and can lead to SE prior to the beginning of interaction.

Figure 4

(a) Expected collisional dephasing timescale of the $|g⟩=|F_i=3,m_{F,i}=3⟩-|e⟩=|4,3⟩$ qubit (${\tau }_{3-3}$), as used for measurement (b), and the $|g⟩=|3,0⟩-|e⟩=|4,0⟩$ qubit (${\tau }_{0-0}$) for comparison, given in multiples of the inverse elastic $s$-wave collision rate (${\tau }_{\mathrm{el}}$). (b) Pulse-echo sequence applied to Cs impurities, while immersed in a Rb bath, which has been expanded for 6 ms prior to impurity immersion. Visibility of Ramsey fringes for impurities in a BEC (green circles, $⟨n⟩=4.0\times {10}^{13}{\mathrm{cm}}^{-3}$), in a thermal Rb bath (blue triangles, $⟨n⟩=1.5\times {10}^{13}{\mathrm{cm}}^{-3}$), and without Rb (gray squares). The latter data sets are offset by a constant factor of 5 and 10, respectively. Extracted $T_2$ coherence times are given in the text, and error bars give statistical uncertainties. (c) Ratio of the experimentally obtained coherence times from (b) and the elastic collision time ${\tau }_{\mathrm{el}}$ as a function of the density overlap $⟨n⟩$, showing coherent cooling with $T_2/{\tau }_{\mathrm{el}}\gg 1$. The green solid line gives the averaged $T_2=1.1\mathrm{ms}$ as a constant. Three-body loss (here, the ratio of three-body loss and elastic collision time) is characterized independently (black data points) and does not limit the coherence measurement.