Observing Spin-Squeezed States under Spin-Exchange Collisions for a Second

Using the platform of a trapped-atom clock on a chip, we observe the time evolution of spin-squeezed hyperfine clock states in ultracold rubidium atoms on previously inaccessible time scales up to 1 s. The spin degree of freedom remains squeezed after 0.6 s, which is consistent with the limit imposed by particle loss and is compatible with typical Ramsey times in state-of-the-art microwave clocks. The results also reveal a surprising spin-exchange interaction effect that amplifies the cavity-based spin measurement via a correlation between spin and external degrees of freedom. These results open up perspectives for squeezing-enhanced atomic clocks in a metrologically relevant regime and highlight the importance of spin interactions in real-life applications of spin squeezing.

Popular Summary

Atomic clocks are among the most precise measurement instruments in the world but they still have room for improvement. Indeed, all clocks so far use uncorrelated atoms, creating statistical noise called quantum projection noise, which limits the performance of some of the best clocks. It is known that this limit can be removed if the atoms are put into a quantum correlated state called a spin-squeezed state but the lifetime of spin-squeezed states produced in the laboratory used to be orders of magnitude shorter than what is needed in most clocks. Now, an experiment has produced spin squeezed states that live for a whole second—enough for a metrology-grade clock.

Being able to conserve the fragile many-body quantum state for such a long time also leads to new insights into its “real-life” properties and yields a pleasant surprise: the clock signal produced by the state gets stronger over time, growing to more than 4 times its expected value. The effect is explained by extremely small interactions between the atoms—so small that they have remained invisible in the earlier short-lifetime experiments. They conspire with the quantum correlation to amplify the coupling to the optical microcavity that is used for detection.

The microwave clock used in this experiment works in the regime of next-generation compact clocks for global navigation satellites, raising hopes for quantum enhanced compact clocks in the near future.

Figure 1

The concept of the experiment. (a) The experimental setup. ${}^{87}$$\mathrm{Rb}$ atoms are magnetically trapped by the atom chip (the trap potential is illustrated by the dashed orange curve). Transmitted photons are collected by a single-photon counter (SPC). Absorption images are taken along $\hat{y}$. Cavity-locking light at 1560 nm is not shown. (b) The energy-level structure and the cavity-probing scheme. (c) Typical data of the correlation of a cavity measurement with absorption imaging (left) and two consecutive cavity measurements, $M_1$ and $M_2$ (right). The correlation with absorption imaging is limited by imaging noise, $\mathrm{\Delta }(N_{\uparrow }-N_{\downarrow })\sim 100$, comparable to the SQL. (d) An experimental sequence with two composite cavity measurements, $M_1$ and $M_2$, for squeezing verification. The green boxes represent pulses on the clock transition and the red boxes cavity-probe transmission, from which $\delta {\omega }_{\pm }$ are deduced. The delay $t_d$ between measurements varies from a few milliseconds to 1 s. $\tilde{\pi }$ denotes a composite $\pi$ pulse.

Figure 2

The spin-squeezing characterization. (a) Conditional squeezing results for $N=1.8(1)\times {10}^4$ atoms and minimum delay ($t_d=6\mathrm{ms}$) between measurements $M_1$ and $M_2$. The variance values are normalized to the SQL ($N/4$) and expressed in decibels. The cavity measurement without atoms (open circles) approaches the photon-shot-noise (PSN) limit (the boundary of the shaded zone). The number squeezing ${\xi }_N^2$ (purple circles) [Eq. (1)] decreases to below $-12$ dB. After normalizing by the independently measured coherence [panel (b)], we obtain the metrological squeezing (red squares), reaching an optimum of ${8.6}_{-1.3}^{+1.8}$ dB for $13.0(2)\times {10}^3$ photons. The error bars indicate $1\sigma$ confidence and are obtained with a bootstrapping method. (b) The Ramsey contrast $\mathcal{C}$ as a function of the measurement strength. The curve is a fit to $\mathcal{C}=\exp [-⟨n_1⟩/{\gamma }_1-{(⟨n_1⟩/{\gamma }_2)}^2]$ (Appendix 11). Its thickness indicates the fit uncertainty. (c) The spin-noise tomography at $⟨n_1⟩=8.9(2)\times {10}^3$, measured by inserting, between $M_1$ and $M_2$, a rotation $\theta$ around $⟨\hat{\mathbf{S}}⟩$. The gray curve represents the theoretical minimum-uncertainty state, while the pink curve takes into account the phase noise induced by the PSN of $M_1$.

Figure 3

Amplification of the cavity measurement. (a) Raw data (measured cavity detunings) of $M_2$ versus $M_1$ at $t_d=0$, $0.2$, $0.5$, and 1 s, respectively. The initial atom number and the average photon number are $N=2.1(1)\times {10}^4$ and $⟨n_1⟩=9.4(2)\times {10}^3$, respectively. The dashed lines indicate $M_2=M_1$. The slope of a linear fit (solid lines) gives the amplification factor in (b). (b) The amplification factor $\alpha$ (open circles) as a function of time $t_d$ between $M_1$ and $M_2$. The error bars are smaller than the symbol size. The gray curve represents a semiclassical numerical simulation with no fitting (see the text and Appendix 16). (c) The correlation evaluated by $\mathrm{Var}(M_1-M_2/\alpha )$ normalized to the SQL at $t=0$. For all times, $M_2$ remains correlated with $M_1$ to below the SQL of the state prepared by $M_1$. The dashed curve is a lower bound given by our model (see the text and Appendix 14).

Figure 4

The spin-orbit correlation from temperature measurements. (a) The temperatures along $\hat{z}$ of states $|\downarrow ⟩$ (light green dots) and $|\uparrow ⟩$ (dark green dots) versus $M_1$ from the same data set (the offset between the initial temperatures is due to the state-dependent imaging procedure). There is a clear correlation between $T_{z,\uparrow }$ and $M_1$ (anticorrelation for $|\downarrow ⟩$) for $t_d=0.2$ s and 0.5 s, where $\alpha >1$. (b) The time evolution of the temperature fluctuations (standard deviation) of $T_{z,\uparrow }$. The evolution closely resembles that of the amplification factor $\alpha$ (open circles, right axis), further corroborating the amplification mechanism described in the text.

Figure 5

Inferred squeezing in the spin degree of freedom. The purple circles show the inferred upper limit of number squeezing in the spin degree of freedom [Eq. (2)]. The evolution is consistent with the theoretical lower bound given by atom loss (dashed curve). The red squares show the resulting metrological squeezing, obtained as usual by dividing the number squeezing by the measured Ramsey contrast shown in the inset. For comparison, the solid curve is the theoretical limit normalized to the fit of the experimental Ramsey contrast. The inset shows the Ramsey-contrast data (circles) as a function of $t_d$ and their exponential fit (shaded curve), which yields $\tau =7.7(6)$ s.

Figure 6

(a) The layout of the atom chip, which has a two-layer structure and incorporates two fiber Fabry-Perot (FFP) cavities of different finesse. Only the low-finesse cavity (located left in the image) is used in the experiments described here. The red (yellow) wires are on the top (bottom) chip. The central “Omega” wire is used to transfer atoms from the MOT site to the cavity. The green shading indicates the cross section of the vacuum cell. (b) The Zeeman levels of the ${}^{87}$$\mathrm{Rb}$ ground state. The clock states are marked in green. The clock transition is excited with two-photon pulses as indicated.