Teleportation and spin squeezing utilizing multimode entanglement of light with atoms

We present a protocol for the teleportation of the quantum state of a pulse of light onto the collective spin state of an atomic ensemble. The entangled state of light and atoms employed as a resource in this protocol is created by probing the collective atomic spin, Larmor precessing in an external magnetic field, off resonantly with a coherent pulse of light. We take here full account of the effects of Larmor precession and show that it gives rise to a qualitatively different type of multimode entangled state of light and atoms. The protocol is shown to be robust against the dominating sources of noise and can be implemented with an atomic ensemble at room temperature interacting with free-space light. We also provide a scheme to perform the readout of the Larmor precessing spin state enabling the verification of successful teleportation as well as the creation of spin squeezing.

Figure 1

von Neumann entropy of the reduced state of atoms versus coupling strength $\kappa$ for the state of Eq. (5) (full line) and for the state generated without magnetic field in a pure QND interaction (dashed line) with the same coupling strength. Application of a magnetic field significantly enhances the amount of light-atom entanglement.

Figure 2

Scheme for teleportation of light onto atoms. As described in Sec. 2, a classical pulse (linearly polarized along $x$) propagating along the positive $z$ direction is scattered off an atomic ensemble contained in a glass cell and placed in a constant magnetic field $B$ along $x$. Classical pulse and scattered light (linearly polarized along $y$) are overlapped with a coherent pulse (linearly polarized along $z$) at beam splitter BS. By means of standard polarization measurements Stokes vector components $S_y$ and $S_z$ are measured at one and the other port, respectively, realizing the Bell measurement. The Fourier components at Larmor frequency $\Omega$ of the corresponding photocurrents determine the amount of conditional displacement of the atomic spin which can be achieved by applying a properly timed transverse magnetic field $b\left(t\right)$. See Sec. 3A for details.

Figure 3

(a) Theoretical limit on the achievable fidelity $F$ versus entanglement between atoms and light measured by the von Neumann entropy $E_{\mathrm{vN}}$ of the reduced state of atoms. The gray area is unphysical. For moderate amounts of entanglement our protocol is close to optimal. (b) Coupling strength $\kappa$ versus entanglement. The dashed lines indicate the maximal fidelity of $F=0.77$ which is achieved for $\kappa =1.64$.

Figure 4

(a) Average fidelity achievable in the presence of atomic decay $\beta$, reflection, and light absorption losses $ϵ=8\%$, 0.12%, 0.16%, coupling $\kappa =0.96$, and Gaussian-distributed input states with mean photon number $\overline{n}=4$. The fidelity benchmark is in this case $5∕9$ (dashed line). (b) Corresponding optimal values for gains $g_x$ (solid lines) and $g_q$ (dashed lines).

Figure 5

Scheme for spin measurement. After the scattering a $\pi ∕2$ rotation is performed on the scattered light modulated at the Larmor frequency such as to affect only the sine (cosine) component. Standard polarization measurement of $S_y$ and appropriate postprocessing allow one to read out the mean of $X\left(P\right)$, leaving the atoms eventually in a spin-squeezed state.