Efficient quantum memory and entanglement between light and an atomic ensemble using magnetic fields

We present two protocols, one for the storage of light in an atomic ensemble and the subsequent retrieval, and another one for the generation of entanglement between light and atoms. They rely on two passes of a single pulse through the ensemble, Larmor precessing in an external field. Both protocols work deterministically and the relevant figures of merit—such as the fidelity or the EPR variance—scale exponentially in the coupling strength. We solve the corresponding Maxwell-Bloch equations describing the scattering process and determine the resulting input-output relations which only involve one relevant light mode that, in turn, can be easily accessed experimentally.

Figure 1

Relevant internal levels with quantization along $\widehat{x}$. Thick arrows represent the strong coherent field in $\widehat{x}$ polarization, thin arrows indicate the quantum field in $\widehat{y}$ polarization.

Figure 2

Setups for having (a) a beam splitter or (b) a two-mode-squeezinglike dynamics; (c) and (d) show the effective transitions.

Figure 3

Schemes for realization of a quantum memory (a) and a source of EPR entanglement (b). (a) In the first pass a $p_L{p}_A$ interaction occurs. Subsequently the pulse is sent through a $\frac{\lambda }{4}$ plate, which interchanges $x_L$ and $p_L$. The pulse is reflected back onto the sample. This happens at a time scale much shorter than the Larmor precession of the atoms. Therefore the transverse components of the collective spin can be assumed to remain in their place to a very good approximation. Finally the pulse passes the atoms along $\widehat{y}$. Due to the changed geometry the atomic quadratures are also interchanged. $p_A\to x_A$ and $x_A\to -p_A$, which means that the light field couples to $x_A$ in its second passage, hence leading to a $x_L{x}_A$ interaction. In (b) the changed geometry introduces a different sign in the exchange of atomic quadratures, which leads to a $-x_L{x}_A$ interaction in the second pass.

Figure 4

Average fidelity for write-in and subsequent read-out of a light state vs coupling ${\kappa }^2$. Crosses indicate the classical limit in each case. (a) Average fidelity for coherent light states according to distributions with different mean photon numbers (solid line: $n=4$; dashed line: $n=8$; dotted line: $n=20$). (b) Average fidelity for light qubits.

Figure 5

Entanglement produced in the two-mode-squeezing scheme vs coupling ${\kappa }^2$. The entanglement is hereby measured by the EPR variance ${\Delta }_{\mathit{EPR}}$.

Figure 6

Atomic squeezing ${\left(\Delta p_a\right)}^2$ in db vs coupling ${\kappa }^2$ for optimal gain factor $g_{\mathit{opt}}$. The inset shows how the optimal gain factor depends on the coupling.

Figure 7

Average fidelity with losses vs coupling. The atomic decay rate $\eta$ and the reflection coefficient $r$ both have a value of 7.5%. Crosses mark the corresponding classical limits. (a) Fidelity for coherent input states according to distributions with different mean photon numbers. Solid line: $n=4$; dashed line: $n=8$; dotted line: $n=20$. (b) Fidelity for light qubits.

Figure 8

Maximal attainable average fidelity for coherent input states according to a distribution with mean photon number $n=8$ (a) and light qubits (b) vs reflection coefficient $r$ for different atomic decay parameters $\eta$. Solid lines: $\eta =5\%$; dashed lines: $\eta =10\%$; dotted lines: $\eta =25\%$. The dash-dotted line and the cross indicate the classical limits.

Figure 9

(a) EPR variance ${\Delta }_{EPR}$ vs coupling ${\kappa }^2$ in the presence of losses. The reflection coefficient $r$ and the atomic decay rate are both chosen to have a value of 10%. (b) Optimized EPR variance vs coupling for different atomic decay parameters. Solid line: $\eta =5\%$; dashed line: $\eta =10\%$; dotted line: $\eta =25\%$. The inset shows how the optimal coupling $k_{\mathit{opt}}$ varies with $r$.

Figure 10

(a) Spin squeezing in $db$ vs coupling ${\kappa }^2$ in the presence of losses. The reflection coefficient $r$ and the atomic decay rate both have a value of 10%. The inset shows how the optimal gain factor $g_{\mathit{opt}}$ depends on the coupling. (b) Maximal spin squeezing vs reflection coefficient $r$ for different atomic decay parameters. Solid line: $\eta =5\%$; dashed line: $\eta =10\%$; dotted line: $\eta =25\%$.