Cooperative squeezing of internal and collective spins in an atomic ensemble

Abstract

Creating highly spin-squeezed states for quantum metrology surpassing the standard quantum limit is a topic of great interest. Spin squeezing has been achieved by either entangling different atoms in an ensemble, or by controlling the multilevel internal spin state of an atom. Here, we experimentally demonstrate combined internal and collective spin squeezing in a hot atomic ensemble with $\sim 10^{11}$ rubidium atoms. By synergistically combining these two types of squeezing and carefully aligning their squeezing quadratures, we have achieved a metrologically relevant spin squeezing of $-6.21\pm0.84$ dB, significantly outperforming the results obtained by utilizing either type of squeezing alone. Our approach provides a new perspective on fully harnessing the degrees of freedom inherent in quantum states of an atomic ensemble.

Figures (3)

• Figure 1: (a) Experimental setup schematics. The $^{87} \mathrm{Rb}$ atoms are contained in a $7 \times 7 \times 20 \ \rm{mm}$ vapor cell placed inside a four-layer magnetic shield and coils producing a bias magnetic field along the $x$ direction. The $\sigma^{-}$-polarized pump and repump lasers both propagate along the $x$ direction, with the former tuned to the $D1$ transition $5S_{1/2}, F=2 \rightarrow 5P_{1/2}, F'=2$, and the latter to the $D2$ transition $5S_{1/2}, F=1 \rightarrow 5P_{3/2}, F'=2$. The two probe lasers propagate along the $z$ direction, with linear $y$ polarization. The probe W$_{1}$ is blue detuned by $1.6$ GHz from the $D2$ transition $5S_{1/2}, F=2 \rightarrow 5P_{3/2}, F'=3$, and the probe W$_{2}$ is blue detuned by $2.5$ GHz from $5S_{1/2}, F=2 \rightarrow 5P_{3/2}, F'=3$. The Stokes component $\hat{S}_{y}$ of the probe W$_{2}$ is detected via balanced homodyne detection. HWP, half-wave plate; PBS, polarization beam splitter. (b) Pulse sequence. Atoms are first prepared in the CSS by optical pumping, then interact with two stroboscopic pulses. The first probe W$_{1}$ creates the internal squeezing and the second probe W$_{2}$ produces collective squeezing. The probe W$_{2}$ consists of three parts: the first part creates the spin squeezing, the second part verifies the spin squeezing, and the third part further retrodicts the spin state. The time interval $\Delta \tau = 0.31$ ms between the three probe periods is set to prevent signal correlation due to the lock-in amplifier. (c) Atomic level scheme of the atom-light interactions. The stroboscopic pulse in the frequency domain can be viewed as a frequency comb (see Supplemental Material sm) (right). The central carrier and the first sideband drive the resonant two-photon Raman transitions between magnetic sublevels. Dashed arrows represent the scattering of Stokes and anti-Stokes photons, which have the same frequency. (d) Pictorial representation of the cooperative spin squeezing. Left: Bloch sphere representation of the internal state due to OAT evolution. Middle: the five time-dependent internal states of a single atom after an $x$-axis rotation by angle $\phi$ and the effective spin transition $\ket{\uparrow} \rightarrow\ket{\downarrow}$ (see text). Right: the QND detection process creates pairwise entanglement between atoms, which enhances the overall spin squeezing.
• Figure 2: (a) Rotation angle of the transverse mean spin versus the squeezing pulse duration of W$_{1}$ for $d=0.1$ and mean power $0.8$ mW. Inset: rotation angle versus the amplitude of the rf field (proportional to the created mean spin value) with the W$_{1}$ pulse duration of $1.0$ ms. With a standard deviation of about $\pm0.3^\circ$ (not shown), each data point is the average of five identical experiments each consisting of 1000 repeated measurements. The accuracy of the rotation angle is technically constrained by the minimum pulse delay time $0.02$$\upmu$s, which is equal to $1\%$ of a Larmor period, causing an error of $\pm3.6^\circ$ (shown). (b) Internal spin squeezing versus duty cycle $d$ with pulse duration $\tau_{1}=1.0$ ms and mean laser power $0.8$ mW. Inset: internal spin squeezing versus pulse duration of W$_{1}$ with $d=0.1$, showing that there exists an optimal squeezing. The data in the main figure are pulse-duration optimized. The error bar for each data point represents the standard deviation of five identical experiments each consisting of 10000 pulse (optical pumping $+$ probe) cycles. The black solid line denotes the SQL.
• Figure 3: Spin squeezing versus rotation angle $\theta$ for $d=0.1$. The black dotted and solid lines in both (a) and (b) represent the noise level of the prepared CSS ($0.31\pm0.20$ dB for 97.4$\%$ spin polarization), and the SQL (0 dB), respectively. The green dashed line ($-2.83\pm0.47$ dB) in (a) and the yellow dashed line ($-5.10\pm0.93$ dB) in (b) denote the maximal squeezing achieved solely by the two- and three-pulse QND schemes (without internal squeezing), respectively. The purple, green and yellow solid curves are the theoretical predictions. The mean power is $0.8$ and $1.0$ mW for the probe W$_{1}$ and the probe W$_{2}$, respectively. The error bar for each data point represents the standard deviation of five identical experiments, each consisting of 10000 cycles. The rotation angle is calibrated through the mean spin measurement, and its accuracy is also limited by the minimum pulse delay time. NR denotes the noise reduction caused by the QND measurement.