Deterministic Squeezed States with Collective Measurements and Feedback

We demonstrate the creation of entangled, spin-squeezed states using a collective, or joint, measurement and real-time feedback. The pseudospin state of an ensemble of $N=5\times {10}^4$ laser-cooled ${}^{87}\mathrm{Rb}$ atoms is deterministically driven to a specified population state with angular resolution that is a factor of 5.5(8) [7.4(6) dB] in variance below the standard quantum limit for unentangled atoms—comparable to the best enhancements using only unitary evolution. Without feedback, conditioning on the outcome of the joint premeasurement, we directly observe up to 59(8) times [17.7(6) dB] improvement in quantum phase variance relative to the standard quantum limit for $N=4\times {10}^5\mathrm{atoms}$. This is one of the largest reported entanglement enhancements to date in any system.

Figure 1

(a) A coherent spin state’s spin-projection noise (pink distribution) is projected onto a squeezed state by a measurement of $J_z$. The quantum state randomly collapses within the original distribution, creating a conditionally squeezed state. The premeasurement’s outcome is then used to rotate the spin state’s polar angle to a desired target spin projection (black solid line) $J_z=J_{z\text{tar}}$, creating a deterministically squeezed state. (b) The relevant ${}^{87}\mathrm{Rb}$ energy levels (black) and cavity resonance frequency ${\omega }_c$ (blue). (c) Simplified experimental diagram. The cavity is probed in reflection. Homodyne detection of the probe is sampled by a microcontroller that then applies microwaves at 6.8 GHz to achieve the desired feedback rotation ${\theta }_{\mathrm{fb}}$ to create the deterministically squeezed state in (a). See the Supplemental Material [5] for experimental details.

Figure 2

(a) Measured cavity resonance frequency for a single trial versus time, subtracting a constant 12 MHz frequency offset. (b) The time windows in which the probe is turned on (green) and the populations determined from each window. The fixed microwave rotations are shown in black with the feedback rotation shown in orange. (c) The premeasurements $J_{zp}$ (left) and final measurements $J_{zf}$ (right) of $J_z$ are plotted versus trial number and accumulated into histograms. Five different $J_z$ states are targeted (five distinct colors on the right) and reached with noise below the QPN. The maximum deterministic squeezing is $S=-7.4(6)\mathrm{dB}$ relative to the SQL. (d) Feedback reduces the noise distribution of the final measurement relative to the initial quantum noise in the premeasurement. (e) If no feedback is applied the final measurement and premeasurement are strongly correlated (black), allowing for conditional squeezing [$S=-10.3(6)\mathrm{dB}$] by using the differential quantity $J_{zf}-J_{zp}$ (gold). The increase in noise from feedback is discussed in the Supplemental Material [5].

Figure 3

(a) Experimental sequence for conditional spin squeezing, with labeling mirroring that of Fig. 2. (b) The squared contrast $C^2$ (blue), spin noise $R$ (red), and spin squeezing $S$ (black) are plotted versus the average number of incident photons $M_i$ in a single measurement window. The solid lines are fits, the blue band is the predicted loss of contrast from free-space scattering, and the gray band indicates the total squeezing error bar. (c) The experimental sequence used to observe the backaction spin projection. (d) The measured spin noise $R$ is plotted versus $\psi$ with a fit (purple). (e) The reconstructed conditional probability distribution of the quantum state (red) on a Bloch sphere with the Bloch (black) vector. The distribution is magnified with a 1:1 aspect ratio and plotted with the equivalent coherent spin state (blue) in the lower panel. (f) Thermal radial motion of the atoms causes the spin noise $R$ to oscillate at twice the radial trap frequency as the time separation $T$ between the premeasurement and final measurement is increased.