Simultaneous Readout of Noncommuting Collective Spin Observables beyond the Standard Quantum Limit

We augment the information extractable from a single absorption image of a spinor Bose-Einstein condensate by coupling to initially empty auxiliary hyperfine states. Performing unitary transformations in both the original and auxiliary hyperfine manifold enables the simultaneous measurement of multiple spin-1 observables. We apply this scheme to an elongated atomic cloud of ${}^{87}\mathrm{Rb}$ to simultaneously read out three orthogonal spin directions and with that directly access the spatial spin structure. The readout even allows the extraction of quantum correlations which we demonstrate by detecting spin-nematic squeezing without state tomography.

Figure 1

Schematics of the readout technique. (a) Coupling (red squares) the system (blue) to auxiliary states (green) enlarges the Hilbert space. Applying independent unitaries in both subspaces (red circles) allows choosing the measurement bases individually. From the resulting probabilities ($p_0,\dots ,p_7$) one infers the corresponding observables of the original system. (b) Level scheme of the electronic ground state of ${}^{87}\mathrm{Rb}$ in a magnetic field. Coupling the two manifolds with microwave pulses we experimentally realize the extension of the Hilbert space. Rotating magnetic fields selectively couple the magnetic substates in each manifold. (c) Using two rf coils we generate a rotating magnetic field. For a relative phase of $\mathrm{\Delta }\theta =-0.7\pi$ we induce spin rotations only in $F=1$ [10]. For this we measure the $z$ projections $S_z$ and $S_z^{F=2}$ of the spin normalized to the atom numbers $N_1$ and $N_2$ detected in the $F=1$ and $F=2$ manifold, respectively.

Figure 2

Spatially resolved readout of three spin components. Using magnetic field gradients and spin rotations we generate a spin wave along the longitudinal direction of an elongated BEC. (a) Populations of the magnetic substates measured via absorption imaging after Stern-Gerlach separation. (b) All three spin directions inferred from the measured populations at each position. The spin observables are normalized to the local atom number $n_F(y)$ in the corresponding hyperfine manifold. (c) Reconstructed spin vector in space and its distribution on a spin sphere.

Figure 3

Efficient detection of spin-mixing dynamics. (a) Using off-resonant MW dressing we tune spin mixing into resonance between the ground mode of (1,0) (gray) and the first excited spatial mode of $(1,\pm 1)$ (red). The lower panel shows an absorption image of the density distributions after 2 s of spin-mixing evolution time where the populations in $(1,\pm 1)$ feature the characteristic double-peak structure. (b) After the readout sequence the observables $S_x$ and $Q_{yz}$ are extracted from the population differences of the states $(2,\pm 2)$ and $(1,\pm 1)$, respectively. Here, the absorption images have been taken after 800 ms of evolution time for better visibility of the mode structure. (c) By plotting the two values for each experimental realization we directly visualize the spin-mixing dynamics in the spin-nematic phase space. The lines correspond to the mean field energy contours of the phase space. For an evolution time of 500 ms a non-Gaussian shape has emerged.

Figure 4

Detection of spin-nematic squeezing. (a) In each experimental run we obtain a pair of values $S_x/N_2$ and $Q_{yz}/N_1$. After 100 ms of evolution time the resulting scatter plot clearly indicates correlated fluctuations between the two observables compared to the initial state (inset). The blue and black line depict the 2 standard deviation (s.d.) interval. (b) For each projection axis parametrized by the angle $\varphi$ we compute the variance. We infer $\mathrm{\Delta }{}^{2}F(\varphi )$ by subtracting imaging noise and normalize to the expected coherent state fluctuations $\mathrm{\Delta }{}^{2}F(\varphi {)}_{\mathrm{SN}}$ resulting in the blue line and 1 s.d. error band. The prepared state shows reduced fluctuations of $0.62\pm 0.07$ (blue point) along the maximally squeezed direction. The red shading indicates the fundamental limit of 0.5 of the readout scheme. The two gray points correspond to the imaging calibration of the $F=1$ and $F=2$ manifolds using a coherent spin state. The error bars correspond to the 1 s.d. interval.