Multiatom entanglement in cold Rydberg mixtures

Daniel Cano and József Fortágh

Phys. Rev. A 89, 043413 – Published 15 April 2014

Abstract

We present an efficient method for generating maximum entanglement in one-dimensional atomic lattices. The proposed method relies on adiabatic rapid transfer into two Rydberg states with strongly asymmetric interactions. The method is suitable for Rydberg $S$ states in the absence of applied electrostatic fields. We show numerical simulations of entanglement generation in rubidium atoms using calculated van der Waals potentials under realistic experimental conditions. We study the effect of the chosen Rydberg states on the final entanglement.

Received 11 February 2014

DOI:https://doi.org/10.1103/PhysRevA.89.043413

Authors & Affiliations

Daniel Cano^* and József Fortágh

CQ Center for Collective Quantum Phenomena and Their Applications, Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Auf der Morgenstelle 14, D-72076 Tübingen, Germany

^*daniel.cano@imdea.org

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Vol. 89, Iss. 4 — April 2014

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Figure 1
(a) Scheme of a four-level atom. Three laser fields are used to coherently couple the atomic states. The Rabi frequencies $Ω_{g e}$ , $Ω_{e r}$ , and $Ω_{e v}$ correspond to the atomic transitions $| g 〉 \leftrightarrow | e 〉$ , $| e 〉 \leftrightarrow | r 〉$ , and $| e 〉 \leftrightarrow | v 〉$ , respectively. The population in $| e 〉$ quickly decays to the ground state at a rate of $Γ_{e g}$ . In some cases, we introduce a frequency detuning $Δ$ from state $| e 〉$ . (b) One-dimensional lattice with $N$ sites evenly spaced at a distance $d$ . Each site is occupied by a single atom. Atoms are labeled by numbers. (c) Laser pulse sequence used for adiabatic transfer into Rydberg states. In the first laser pulse the state $| e 〉$ is simultaneously coupled to $| r 〉$ and $| v 〉$ with equal Rabi frequencies $Ω_{e r} = Ω_{e v}$ . The second laser pulse couples $| g 〉$ with $| e 〉$ .
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Figure 2
(a) van der Waals coefficients of $| 70 S, n S 〉$ . (b) Energies of the state-mixing channels that contribute to the van der Waals coefficients, with the zero-energy level in $| 70 S, n S 〉$ . The relation between the state-mixing channels and the van der Waals coefficients is explained in Refs. [29, 32]. (c–f) Numerical simulations of two-atom entanglement. The parameters of the laser pulse sequence are $Ω_{0, e r} = 2 π \times 40$ MHz, $Δ = 0$ , and three values of $β$ : 0.4, 1, and 2.5 $μ$ s. (c) Probability of finding the system in the entangled state $| ψ_{+} 〉 = (1 / \sqrt{2}) (| 70 S, n S 〉 + | n S, 70 S 〉)$ for $n = 100$ . Solid lines correspond to quantum trajectories that evolve without spontaneous emission. The dotted line is a quantum trajectory with one spontaneous emission event. (d) Time evolution of the populations in $| e 〉$ . (e) Probability of finding the system in the entangled state $| ψ_{+} 〉$ at the end of the adiabatic transfer for $n = 85$ and $n = 100$ . This graph considers only quantum trajectories without spontaneous decay. (f) Accumulated probability of spontaneous decay from $| e 〉$ during the whole process.
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Figure 3
(a) Solid (red) curves are the interatomic potentials of states $| 40 S, 40 S 〉$ and $| 41 S, 41 S 〉$ . The dotted (blue) curve labeled ANT is the potential $V_{-}$ of the antisymmetric state $| ψ_{-} 〉 = (1 / \sqrt{2}) (| 40 S, 41 S 〉 - | 41 S, 40 S 〉)$ . The three dotted (blue) curves labeled SYM are the three branches of the potential $V_{+}$ of the symmetric state $| ψ_{+} 〉 = (1 / \sqrt{2}) (| 40 S, 41 S 〉 + | 41 S, 40 S 〉)$ . The potential $V_{+}$ splits into three curves because the degeneracy between Zeeman sublevels is slightly lifted by the interaction. (b–e) Numerical simulations of the adiabatic transfer into $| ψ_{r r, v v} 〉$ . The parameters of the laser pulse sequence are $Ω_{0, e r} = 2 π \times 40$ MHz, $Δ = 0$ , and three values of $β$ : 0.4, 1, and 2.5 $μ$ s. (b) Population $P_{r r}$ of $| 40 S, 40 S 〉$ for a quantum trajectory without spontaneous emission. The curve for $P_{v v}$ is the same. (c) Phase difference between the coefficients $c_{r r}$ and $c_{v v}$ for the same quantum trajectory as in (b). (d) Populations $P_{r r}$ (solid lines) and $P_{v v}$ (dotted lines) at the end of the laser pulse sequence for quantum trajectories without spontaneous decay. (e) Accumulated probability of spontaneous decay from $| e 〉$ during the whole process as a function of the interatomic distance.
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Figure 4
(a) Functions $F_{4}$ and $F_{6}$ for $| r 〉 = | 70 S 〉$ and $| v 〉 = | n S 〉$ . Populations of $| Ψ_{+}^{(4)} 〉$ in a four-atom system for two quantum trajectories without spontaneous emission. The parameters are shown in the legend. (c) Probability of finding the system in the entangled state at the end of the laser pulse sequence for systems with four and six atoms. Only quantum trajectories without spontaneous decay are considered. (d) Accumulated probability of spontaneous decay from $| e 〉$ during the whole process. The legend shows the parameters for graphs (b–d).
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Figure 5
Adiabatic transfer from entangled Rydberg states into entangled ground states in a four-atom system. (a) Populations of the important collective states. (b) Rabi frequencies as a function of time. The first two laser pulses transfer $| v 〉$ into $| g 〉$ . The second pair of laser pulses transfers $| r 〉$ into $| \hat{g} 〉$ . The peak amplitudes, detunings, and temporal widths are the same for all laser pulses: $Ω_{0, g e} = Ω_{0, e r} = Ω_{0, e v} = 2 π \times 100$ MHz, $Δ = 2 π \times 100$ MHz, and $β = 0.4 μ$ s.
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