Single-Atom Quantum Probes for Ultracold Gases Boosted by Nonequilibrium Spin Dynamics

Quantum probes are atomic sized devices mapping information of their environment to quantum-mechanical states. By improving measurements and at the same time minimizing perturbation of the environment, they form a central asset for quantum technologies. We realize spin-based quantum probes by immersing individual Cs atoms into an ultracold Rb bath. Controlling inelastic spin-exchange processes between the probe and bath allows us to map motional and thermal information onto quantum-spin states. We show that the steady-state spin population is well suited for absolute thermometry, reducing temperature measurements to detection of quantum-spin distributions. Moreover, we find that the information gain per inelastic collision can be maximized by accessing the nonequilibrium spin dynamic. Keeping the motional degree of freedom thermalized, individual spin-exchange collisions yield information about the gas quantum by quantum. We find that the sensitivity of this nonequilibrium quantum probing effectively beats the steady-state Cramér-Rao limit by almost an order of magnitude, while reducing the perturbation of the bath to only three quanta of angular momentum. Our work paves the way for local probing of quantum systems at the Heisenberg limit, and moreover, for optimizing measurement strategies via control of nonequilibrium dynamics.

Popular Summary

Quantum properties of materials promise many improved technological applications, among them probing of microscopic processes. The most elementary quantum probe is a single atom, which employs quantum properties to store information about its environment. While individual ions or ultracold atomic clouds have been used to measure surrounding fields, an atomic probe storing information in its quantum states about a surrounding atomic many-body system remains elusive. Here, we report on not only the realization of such a probe, but one that outperforms a fundamental quantum limit.

We realize individual atomic quantum probes by controlling microscopic inelastic collisions that couple the motion of cesium atoms to their spin. We find that the width of the spin distribution, upon reaching a steady state, provides an unambiguous measure of the temperature of a surrounding cold bath of rubidium atoms. This enables local absolute quantum thermometry, reducing temperature measurements to simply measurements of the spin distribution of the probe.

We also find that, surprisingly, we can outperform the so-called Cramer-Rao bound, a fundamental quantum limit on the precision of atomic probes. Since our coupling to spin is quantized, we obtain information about the system quantum by quantum, which optimizes the information flow. For only three spin-exchanging collisions, that is, perturbing the many-body system by three quanta of angular momentum, we obtain the maximum amount of information in the probe.

Our work opens the door to experimental quantum probing and paves the way for new optimization strategies using nonequilibrium dynamics.

Figure 1

Single-atom quantum probing of an ultracold gas. A single atom is coupled to the bath, interacting via two processes. First, elastic collisions at rate ${\mathrm{\Gamma }}_{\mathrm{el}}$ thermalize the probe to the temperature of the bath, where the kinetic energy distribution allows for classical thermometry (top). Second, spin-exchange collisions at rate ${\mathrm{\Gamma }}_{\mathrm{SE}}$ ensure motion-spin mapping and allow us to store information on the bath energy in the quantum states of the probe, here the quasispin states visualized by a macroscopic Bloch sphere (bottom).

Figure 2

Mechanism of motion-spin mapping onto quantum states. (a) Sketch of the endoergic process. Cs atoms go to a lower magnetic substate and release $\mathrm{\Delta }E/2$ of energy, whereas Rb atoms go to a higher magnetic substate and require $\mathrm{\Delta }E$ of energy. Therefore, this process can occur only if the missing Zeeman energy can be provided by kinetic energy $E_c$ during the collision, which is only possible for a fraction $p(B,T)$ of the probe atoms. (b) Experimental spin population of Cs atoms for a magnetic field of $B=10\mathrm{mG}$ before (green) and after 350-ms interaction time in a Rb bath at ${366}_{-40}^{+60}\mathrm{nK}$ (blue). We measure a nonzero population in $m_{F,\mathrm{Cs}}=+3$ (endoergic SE) and $m_{F,\mathrm{Cs}}=-3$ (exoergic SE). (c) Fraction of Cs atoms allowing to undergo an endoergic process as a function of the magnetic field $B$, assuming a Maxwell-Boltzmann distribution at temperature $T$. The indicated values correspond to measurements in (b) $p(B,T)=0.8$ and (d) $p(B,T)=0.1$. (d) Same as (b) but for $B=60\mathrm{mG}$. Here, exoergic processes dominate, yielding a measured population (in blue) of Cs atoms in $m_{F,\mathrm{Cs}}=-3$. (e) Sketch of the exoergic process. Rb atoms are promoted to a lower magnetic state and release $\mathrm{\Delta }E$, while Cs atoms are left in a higher magnetic state and need only $\mathrm{\Delta }E/2$. As a consequence, this process has no energetic threshold and releases $\mathrm{\Delta }E/2$ of energy into the system.

Figure 3

Information gain from the spin distribution of quantum probes. (a) Quasispin-state dynamics of Cs atoms immersed in a Rb bath at $\mathrm{T}={366}_{-40}^{+60}\mathrm{nK}$ at a fixed magnetic field $B=25\mathrm{mG}$ [expected endoergic fraction $p(B,T)=0.54$]. Experimental time trace (dots) and theoretical predictions of our rate model (solid line, for details see Appendix pp4) are shown. Each experimental point is an average over approximately 200 measurements, and the error bars indicate the statistical uncertainties in the atom number determination. The histogram projected in the back plane shows the steady-state spin distribution. (b) Theoretically calculated probe-energy fluctuations ${\sigma }_E^2$ of the steady state for $B=25\mathrm{mG}$ calculated from modeled quantum-state distributions as $⟨E^2⟩={\sum }_{m_F}{E}_{m_F}^2{P}_{m_F}$, with $E_{m_F}$ and $P_{m_F}$ being the energy and population probability of quantum state $m_F$. The $P_{m_F}$ are the steady-state solutions of our rate model. Dashed lines direct to $T=366\mathrm{nK}$. Inset: Slope of the linear trend between ${\sigma }_E$ and $T$ for different values of the magnetic field. (c) Time evolution of the probe entropy calculated from the spin populations of (a). The time of maximal entropy corresponds to an average of three SE collisions (0.5 endoergic and 2.5 exoergic).

Figure 4

Nonequilibrium quantum probing. (a) Comparison of the temperatures extracted from the spin population of Cs atoms $T_{\mathrm{spin}}$ after 350-ms interaction time to time-of-flight temperatures of the Rb cloud $T_{\mathrm{TOF}}$ for a fixed magnetic field $B=10\mathrm{mG}$. The error bars of $T_{\mathrm{spin}}$ originate from the statistical errors on the ${\chi }^2$ analysis, and the error bars of $T_{\mathrm{TOF}}$ reflect the shot-to-shot fluctuations in the experiment (around 10%–15%). The red line serves as a guide to the eye $T_{\mathrm{spin}}=T_{\mathrm{TOF}}$. Inset: Example of spin population. The dots represent the data and the histogram the theory with the best-fitting temperature, here $T_{\mathrm{spin}}={1008}_{-160}^{+140}\mathrm{nK}$. (b) Same as (a) but with the magnetic field $B$ for a fixed temperature $T=1\mu \mathrm{K}$. Inset: Spin population with $B_{\mathrm{spin}}={20.7}_{-6.2}^{8.9}\mathrm{mG}$.

Figure 5

Nonequilibrium boost of quantum-probe sensitivity. Time dependence of the magnetic sensitivity $\sqrt{F_B}$ (for $T=590\mathrm{nK}$, centered at $B=40\mathrm{mG}$) (a) and thermal sensitivity $\sqrt{F_T}$ (for $B=40\mathrm{mG}$, centered at $T=640\mathrm{nK}$) (b) of the quantum probe. Red lines correspond to theoretical calculations, blue dots or triangles are experimental data; dots are sensitivities extracted comparing only experimental populations, whereas triangles indicate sensitivities extracted comparing measured populations to theoretical ones. Dashed lines represent the respective sensitivity value expected for the steady state. (c) Thermal sensitivity (centered at $T=640\mathrm{nK}$) as a function of the $B$ field for an interaction time that fixes the number of SE collisions to approximately 4.

Figure 6

Scattering cross sections for endoergic SE (top) and exoergic SE (bottom) for Cs in the state $m_{F,\mathrm{Cs}}=2$ and Rb in the state $m_{F,\mathrm{Rb}}=0$. The cross sections are plotted for four different fixed collision energies $E_c$ (250, 450, 750, and 1500 nK). In the endoergic SE, the energy condition is underlined: If $E_c\le {\mu }_{B}B/4$, the collision is forbidden, and therefore, the cross section drops to 0.

Figure 7

Scattering cross sections for endoergic SE (top) and exoergic SE (bottom) for Cs in the state $m_{F,\mathrm{Cs}}=2$ and Rb in $m_{F,\mathrm{Rb}}=0$. The cross sections shown here include the effect of finite temperature and are plotted for four different temperatures $T$ (250, 450, 750, and 1500 nK).

Figure 8

Sketch of the rate model used for calculating the spin dynamic. For each spin state $m_{F,\mathrm{Cs}}$, endoergic processes lead to a decay to the $m_{F,\mathrm{Cs}}+1$ state and a gain from the $m_{F,\mathrm{Cs}}-1$ state. Likewise, exoergic SE processes lead to a decay to the $m_{F,\mathrm{Cs}}-1$ state and a gain from the $m_{F,\mathrm{Cs}}+1$ state.

Figure 9

Simulation of the spin-states’ dynamics of Cs immersed in a Rb bath with $N_{\mathrm{Rb}}=7\times {10}^3$ atoms at $T=400\mathrm{nK}$ and $B=10\mathrm{mG}$ [expected endoergic fraction $p(B,T)=0.84$]. The seven spin states ($m_{F,\mathrm{Cs}}\in [-3,-2,-1,0,1,2,3]$) are plotted. The steady state is reached beyond 3-s interaction time.

Figure 10

Evolution of the entropy $S$ of the probe. The Cs atoms are initially prepared in $m_{F,\mathrm{Cs}}=2$, immersed in a Rb bath at $T=366\mathrm{nK}$ with $N_{\mathrm{Rb}}=6.7\times {10}^3$ atoms. The magnetic field is $B=25\mathrm{mG}$. Experimental points (circle) and theoretical predictions (solid lines) are shown. (a) Time evolution of the entropy of the probe $S$. (b) Time evolution of the mean number of spin collisions: in red the mean number of endoergic spin collisions and in blue the mean number of exoergic spin collisions. (c) The entropy of the probe as a function of the total number of spin collisions is shown.

Figure 11

Fluctuation of energy ${\sigma }_E$ of the steady state of the spin population of the probe as a function of the temperature $T$ for a fixed magnetic field $B$: (a) $B=25\mathrm{mG}$, (b) $B=40\mathrm{mG}$, (c) $B=60\mathrm{mG}$, (d) $B=80\mathrm{mG}$. The dots are the theoretical points, and the dashed lines represent the linear fit.

Figure 12

Extraction of the spin temperature $T_{\mathrm{spin}}$ and the spin magnetic field $B_{\mathrm{spin}}$ with a ${\chi }^2$ analysis. (a) shows a ${\chi }_{\nu }^2(T)$ curve (the magnetic field is fixed at $B=10\mathrm{mG}$) and (c) represents a ${\chi }_{\nu }^2(B)$ curve (the temperature is fixed at $T=1$ $\mu K$). In each curve, the vertical dashed line represents the minimum value of the ${\chi }_{\nu }^2$, which marks the extracted $T_{\mathrm{spin}}$ or $B_{\mathrm{spin}}$. The slightly gray band represents the standard deviation corresponding to an increase of ${\chi }_{\nu }^2$ of 1. We find $T_{\mathrm{spin}}=702_{-76}^{+60}$ for (a) and $B_{\mathrm{spin}}={21.6}_{-6.1}^{+5.7}$ for (c). (b) and (d) represent the measured populations (dots) and the best parameter of interest with the model (histogram).

Figure 13

Experimental extraction of the quantum Fisher information. The temperature is fixed at $T=280\mathrm{nK}$. In (a)–(c), we represent the spin populations of the Cs atoms for different magnetic fields $B$: The dots are the experimental points, and the histograms are the theoretical expectation values. In (d), we represent the Bures distance between the different states centered at $B_0=25\mathrm{mG}$. In red, the Bures distance is inferred by the theoretical spin populations and in blue by the experimental populations.