Mesoscopic Rydberg Gate Based on Electromagnetically Induced Transparency

We demonstrate theoretically a parallelized C-NOT gate which allows us to entangle a mesoscopic ensemble of atoms with a single control atom in a single step, with high fidelity and on a microsecond time scale. Our scheme relies on the strong and long-ranged interaction between Rydberg atoms triggering electromagnetically induced transparency. By this we can robustly implement a conditional transfer of all ensemble atoms between two logical states, depending on the state of the control atom. We outline a many-body interferometer which allows a comparison of two many-body quantum states by performing a measurement of the control atom.

Figure 1

In the envisioned setup the quantum state of an atomic ensemble is manipulated depending on the state of a single control atom. The atomic ensemble can consist of atoms in a single trap or of atoms being confined in a lattice.

Figure 2

(a) Sequence of laser pulses (not to scale). (b) Electronic level structure of the control and ensemble atoms. The ground state $|1⟩$ is resonantly coupled to the Rydberg state $|r⟩$. The states $|A⟩$ and $|B⟩$ are off-resonantly coupled (detuning $\Delta$, Rabi frequency ${\Omega }_p$) to $|P⟩$. A strong laser with Rabi frequency ${\Omega }_c\gg {\Omega }_p$ couples the Rydberg level $|R⟩$ to $|P⟩$ such that $|R⟩$ is in two-photon resonance with $|A⟩$ and $|B⟩$. In this situation (known as EIT) Raman transfer from $|A⟩$ to $|B⟩$ is inhibited. (c) With the control atom excited to $|r⟩$ the two-photon resonance condition is lifted as the level $|R⟩$ is shifted due to the interaction energy $V$ between the Rydberg states, thereby enabling off-resonant Raman transfer from $|A⟩$ to $|B⟩$.

Figure 3

(a) Linear susceptibility (not to scale) with respect to the Raman laser as a function of its detuning $\delta$ from the $|P⟩$-level for blocked transfer (solid curve) and in the unblocked case (dashed curve). (b) Efficiency of the blocking (I) as a function of ${\Omega }_c/{\Omega }_p$. For ${\Omega }_c/{\Omega }_p>2$ the transfer of the ensemble atoms from $|A⟩$ to $|B⟩$ is blocked with more than 99% fidelity. (c) Transfer efficiency in the unblocked case (II) as a function of the interaction between the control and one ensemble atom (${\Omega }_c=6{\Omega }_p$). (d) Fidelity of the process $1/\sqrt{2}(|0⟩+|1⟩)|AAA⟩\to 1/\sqrt{2}(|0⟩|AAA⟩+|1⟩|BBB⟩)$ for three ensemble atoms as a function of their interaction strength $V_{jk}$ and the ratio $x_{\max }=\sqrt{2}\max ({\Omega }_p)/{\Omega }_c$. We chose the worst case scenario; i.e., all $V_{jk}$ are equal. The interaction between control atom and ensemble atoms was $V_k=10\hslash {\Omega }_c^2/\Delta$ for all four curves, giving rise to a maximal allowed distance between the control atom and the ensemble atoms of $2.2\mu \mathrm{m}$ ($1.4\mu \mathrm{m}$) for the ratio $x=0.4$ ($x=0.1$). We have chosen $\max ({\Omega }_p)=2\pi \times 70\mathrm{MHz}$, $\Delta =2\pi \times 1.2\mathrm{GHz}$ in (b)–(d), and atomic parameters of ${}^{87}\mathrm{Rb}$ (see text).

Figure 4

The gate (1) is the fundamental building block of a many-particle interferometer where the overlap of two many-particle wave functions can be measured. (i) Initial state preparation. (ii) The control atom is prepared in $(1/\sqrt{2})(|0⟩+|1⟩)$. (iii) Gate [Eq. (1)]. (iv) Internal-state-dependent evolution, governed by $U_A$ and $U_B$. (v) Recombination of the interferometer arms by (1). (vi) Measurement of the control atom in the basis $|c_{\pm }⟩=(1/\sqrt{2})(|0⟩\pm |1⟩)$ yields access to the wave function overlap $⟨{\Phi }_A\mid {\Phi }_B⟩=⟨\Phi |U_A^{†}{U}_B|\Phi ⟩$.