Three-dimensional light-matter interface for collective spin squeezing in atomic ensembles

We study the three-dimensional nature of the quantum interface between an ensemble of cold, trapped atomic spins and a paraxial laser beam, coupled through a dispersive interaction. To achieve strong entanglement between the collective atomic spin and the photons, one must match the spatial mode of the collective radiation of the ensemble with the mode of the laser beam while minimizing the effects of decoherence due to optical pumping. For ensembles coupling to a probe field that varies over the extent of the cloud, the set of atoms that indistinguishably radiates into a desired mode of the field defines an inhomogeneous spin wave. Strong coupling of a spin wave to the probe mode is not characterized by a single parameter, the optical density, but by a collection of different effective atom numbers that characterize the coherence and decoherence of the system. To model the dynamics of the system, we develop a full stochastic master equation, including coherent collective scattering into paraxial modes, decoherence by local inhomogeneous diffuse scattering, and backaction due to continuous measurement of the light entangled with the spin waves. This formalism is used to study the squeezing of a spin wave via continuous quantum nondemolition measurement. We find that the greatest squeezing occurs in parameter regimes where spatial inhomogeneities are significant, far from the limit in which the interface is well approximated by a one-dimensional, homogeneous model.

Figure 1

Schematic of a linearly polarized laser probe with a Gaussian spatial mode (solid blue lines entering from the left of the cloud) interacting dispersively with a cold, trapped atomic ensemble. The light that is indistinguishably scattered by the average atomic density distribution defines the radiated paraxial mode (solid red arrows emanating from the cloud to the right). The interference of the radiated and probe modes leads to a rotation of the field polarization according to the Faraday effect. When measured in a polarimeter, this can be used to generate spin squeezing in the ensemble. The spatial overlap of the collectively scattered field and the probe, a measure of the strength of the atom-light coupling, depends highly on geometry. In addition, diffusely scattered photons due to density fluctuations in the ensemble lead to local decoherence in the collective spin variables.

Figure 2

Diagrams of the scattered modes for various atomic cloud and beam geometries. The spatial profile of the probe mode is indicated with solid blue lines and that of the field scattered by a given dielectric distribution is indicated by solid red lines with arrows. (a) A pointlike atomic ensemble scatters light isotropically. (b) A pancake-shaped cloud at a fixed $z$ plane radiates nearly perfectly into the probe mode. Extended pencil-shaped clouds can radiate into the probe mode well, as in (c), or poorly, as in (d), depending on the geometry of the probe.

Figure 3

Squeezing dynamics for a fixed ${\text{OD}}_{\mathrm{eff}}=50$ and different atomic cloud geometries. The laser probe is a TEM${}_{00}$ mode with beam waist $w_0=20$ $\mu$m. (a) Peak squeezing denoted as the inverse of the squeezing parameter, ${\zeta }^{-1}$ in dB, as a function of aspect ratio of the cloud. The inset shows effective atom numbers as a function of aspect ratio; $N_{\mathrm{eff}}^{\left(2\right)}$ is constant by design. (b) Comparison of squeezing dynamics for clouds with $\text{AR}=0.1$ (solid green line) and $\text{AR}=316$ (dashed red line). The behavior in the absence of decoherence [Eq. (55)] (dotted black line) is plotted at the same ${\text{OD}}_{\mathrm{eff}}$, showing agreement for short times. (c) Dynamics of the spin wave variance for the two clouds, normalized by dividing each by its initial variance, $N_{\mathrm{eff}}^{\left(2\right)}/4$. (d) Dynamics of the mean spin for the two clouds, normalized by dividing each by its initial mean spin, $N_{\mathrm{eff}}^{\left(1\right)}/2$. For fixed ${\text{OD}}_{\mathrm{eff}}$, the superior squeezing of the pencil-shaped cloud over the pancake-shaped cloud is attributed to slower decay of the mean spin.

Figure 4

Squeezing for different cloud geometries with Gaussian atomic density distribution [Eq. (70)] and a fixed total atom number $N=9.84\times {10}^6$. (a) Contours of peak squeezing, ${\zeta }^{-1}$ in dB, as a function of cloud aspect ratio and laser probe beam waist. (b) Contours of the coupling strength, ${\text{OD}}_{\mathrm{eff}}$. The difference between the optimal coupling strength and the resulting squeezing depends on the balance between coherent interactions and decoherence, characterized by different effective atom numbers, $N_{\mathrm{eff}}^{\left(1\right)}$, $N_{\mathrm{eff}}^{\left(2\right)}$, and $N_{\mathrm{eff}}^{\left(3\right)}$, shown in (c).

Figure 5

Squeezing for a fixed peak density ${\eta }_0=5\times {10}^{11}$ cm${}^{-3}$ and variable atom number that fills a dipole trap for cold atoms. In (a)–(c) the transverse cloud width is fixed at ${\sigma }_{\perp }=100$ $\mu$m and cloud length is taken to be variable. (a) Contours of peak squeezing, ${\zeta }^{-1}$ in dB. (b) Contours of ${\text{OD}}_{\mathrm{eff}}$. (c) Optimal beam waist for maximizing ${\text{OD}}_{\mathrm{eff}}$ (upper, red dots) and for maximizing peak squeezing (lower, blue dots). For a given atomic geometry, the beam waist that optimizes the ${\text{OD}}_{\mathrm{eff}}$ is not the same as that which optimizes peak squeezing. (d) Peak squeezing as a function of cloud size for the optimal beam waist at each point.

Figure 6

Comparison between the symmetric one-dimensional model and the three-dimensional spin wave model. (a) Peak squeezing, ${\zeta }^{-1}$ in dB, for a spherical cloud with ${\text{OD}}_{\mathrm{eff}}=50$ as the beam waist is increased. The inset shows the convergence of $N_{\mathrm{eff}}^{\left(1\right)}$ (upper, blue line), $N_{\mathrm{eff}}^{\left(2\right)}$ (middle, green line), and $N_{\mathrm{eff}}^{\left(3\right)}$ (lower, red line) as $w_0$ increases. (b) Comparison of squeezing dynamics for the extremal waists from (a): the smallest, $w_0=10$ $\mu$m, and the largest, $w_0={10}^4$ $\mu$m. For comparison, the symmetric one-dimensional case using Eq. (71) is plotted with decoherence (dotted black line) and without (dashed black line).

Figure 7

Squeezing of a spin-4 ensemble. (a) Contours of peak squeezing for an ensemble of spin-4 atoms for the same fixed atom number as in Fig. 4 as function of beam waist and cloud aspect ratio. The largest peak squeezing, ${\zeta }^{-1}=7.8$ dB, is achieved for the optimal geometry (indicated by a white “x”): $\text{AR}=300$, $w_0=28$ $\mu$m. (b) Dynamics of ${\zeta }^{-1}$ in dB for the optimal geometry, for spin-4 and for spin-$\frac{1}{2}$.