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Physics Department, North Carolina Central University,. 1801 Fayetteville Str., Durham, NC 27707, USA. Atoms (ions) encapsulated into small-diameter (~1nm) ...
PHYSICS, CHEMISTRY AND APPLICATION OF NANOSTRUCTURES, 2007

QUBIT ENTANGLEMENT FROM A BIPARTITE ATOMIC SYSTEM UNDER STRONG ATOM-VACUUM-FIELD COUPLING IN A CARBON NANOTUBE* I.V.BONDAREV†, H.QASMI AND B.VLAHOVIC Physics Department, North Carolina Central University, 1801 Fayetteville Str., Durham, NC 27707, USA Atoms (ions) encapsulated into small-diameter (~1nm) metallic carbon nanotubes may form quasi-one-dimensional atomic polariton states via strong coupling to the virtual photonic modes of the nanotube. This results in sizable amounts of the two-qubit atomic entanglement that persists with no damping for very long times. We expect this effect to stimulate relevant experimental efforts and thus to open a path to new device applications of atomically doped carbon nanotubes in quantum information technologies.

1. Introduction In spite of impressive experimental demonstrations of basic quantum information effects in a number of different mesoscopic solid state systems, such as quantum dots in semiconductor microcavities, cold ions in traps, nuclear spin systems, Josephson junctions, etc., their concrete implementation is still at the proof-of-principle stage (see Ref.[1] for recent progress in solid state quantum information science). The development of materials that may host quantum coherent states with long coherence lifetimes is a critical research problem for the nearest future. There is a need for the fabrication of quantum bits (qubits) with coherence lifetimes at least three-four orders of magnitude longer than it takes to perform a bit flip. This would involve entangling operations, followed by the nearest neighbor interaction over short distances and quantum information transfer over longer distances. Extensive work carried out worldwide in recent years has revealed intriguing physical properties of carbon nanotubes (CNs). Nanotubes have been shown to be useful for miniaturized electronic, mechanical, electromechanical, chemical and scanning probe devices and as materials for macroscopic composites[2]. Recent experiments on the encapsulation of single atoms (ions) into single-walled CNs[3], along with the progress in growth techniques of centimeter-long small-diameter single-walled CNs[4], stimulate the studies of *

This work was supported by the US National Science Foundation (grant No ECS-0631347), the US Department of Defence (grant No W911NF-05-1-0502) and NASA (grant No NAG3-804). † Also at: Institute for Nuclear Problems, Belarusian State University, Bobruiskaya Str. 11, 220050 Minsk, Belarus. 1

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possible novell applications of atomically doped CNs in quantum information processing technologies[5],[6]. It was shown recently that the relative density of photonic states (DOS) and the atom-vacuum-field coupling, respectively, near CNs effectively increase due to the presence of additional surface photonic modes coupled to CN electronic quasiparticle excitations[7]. In small-diameter CNs the strong atom-field coupling may occur[8]-[10] – the property that is known to facilitate the entanglement of spatially separated qubits[11]. Qualitatively, in terms of the cavity quantum electrodynamics (QED), the coupling constant of an atom (modelled by a two-level system with the transition dipole moment d A and 1/ 2 frequency ω A ) to a vacuum field is given by g = d A (2π ω A /V ) with V being the effective volume of the field mode the atom interacts with[12]. For the 2 atom (ion) encapsulated into the CN of radius Rcn , V ∼ π Rcn (λ A / 2) that is 3 2 ∼ 10 nm for CNs with diameters ~1 nm in the optical spectral range of 2 λ A ~ 600 nm. Approximating d A ∼ er ∼ e(e / ω A ) , one obtains g ∼ 0.3 eV. On the other hand, the "cavity" linewidth is given for ω A in resonance with the 3 2 ⊥ ⊥ cavity mode by γ c = 6π c /ω A ξ (ω A ) V ∼ 0.03 eV, where ξ (ω A ) is the transverse local (distance-dependent) photonic DOS (also called the Purcell 7 factor)[12]. In view of the large Purcell factors ∼ 10 close to CNs [see Ref.[7]; also shown in Fig.1(a)], one arrives at γ c ∼ 0.03 eV for 1 nm-diameter CNs in the optical spectral range. Thus, for the atoms (ions) encapsulated into smalldiameter CNs the strong atom-field coupling condition g /γ c 1 is supposed to be satisfied. Our latest theoretical analysis of the optical absorption by atomically doped CNs[10] confirms this statement by demonstrating the optical absorbtion line splitting for small-diameter CNs in the frequency range close to the atomic transition frequency [an example shown in Fig.1(c)]. The absorption line splitting comes from the strong atom-vacuum-field coupling that gives rise to the rearrangement ("dressing") of atomic levels and formation of new elementary excitations, eigen states of the full photon-matter Hamiltonian, – the quasi-one-dimensional (1D) atomic polaritons[8]-[10]. These are similar to quasi-0D excitonic polaritons in quantum dots in semiconductor microcavities[13], that are currently being considered a possible way to produce the excitonic qubit entanglement[14]. Thus, atomically doped carbon nanotubes offer another, alternative way to generate the qubit entanglement by using quasi-1D atomic polariton states formed by the atoms (ions) located close to or encapsulated inside CNs. Here we show that small-diameter metallic nanotubes indeed result in sizable amounts of the two-qubit atomic entanglement for sufficiently long times.

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Figure 1. (a) Transverse local photonic DOS ξ ( x ) for the two-level atom in the centers of the four + 'zigzag' CNs (x is the dimensionless frequency); (b) two-particle local photonic DOS functions ξ − ⊥ (solid lines) and ξ (dashed lines) taken at the peak frequencies of ξ ( x ) [see (a)], as functions of the distances between the two atoms on the axes of the (10,0) (lines 1; x=0.29), (11,0) (lines 2; x=0.25) and (12,0) (lines 3; x=0.24) CNs (see Ref.[5] for more details). (c) Optical absorbtion lineshapes for the atom at different distances outside the metallic (9,0) CN, demon-strating the formation of the atomic quasi-1D polariton state as the atom approaches the CN surface (see Ref.[10] for more details). (d) Upper-level population decay probability of initially excited atom A (lines 1) and initially unexcited atom B (lines 2), and the two-qubit atomic entanglement (lines 3), as functions of dimensionless time τ for the two atoms in the center of the metallic (9,0) CN separated from each other by the distance of 6.3 Rcn ≈ 22.2 Å (see Ref.[6] for more details).

2. Results and Conclusions

To calculate the qubit entanglement for a bipartite atomic system near a CN, we used our previously developed photon Green function formalism for quantizing an electromagnetic field in the presence of quasi-1D absorbing and dispersing media[8],[9]. Representing such a medium is the (achiral) infinitely long CN. Two identical two-level atoms, A and B, are positioned at their respective equivalent places rA , B close to the CN. The atoms are assumed to interact with the quantum vacuum field via their transition dipole moments directed along the CN axis. Also assumed was that the atoms were located far enough from each

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other, to simplify the problem by ignoring the interatomic Coulomb interaction. Our formalism has an advantage of representing the total system "atom + CN" in terms of intrinsic characteristics of the electromagnetic subsystem, such as the one-particle and two-particle local photonic DOS functions to be calculated beforehand from the electromagnetic Green function of the problem. Our main results are shown in Fig.1. The details of calculations can be found in Ref.[5]. Our analysis shows sizable amounts of the two-qubit entanglement of two two-level atoms (ions) located inside small-diameter metallic CNs [Fig.1(d)]. 3+ We consider Eu complexes to be good candidates for testing our predictions, 5 7 owing to the dominant narrow, easily detectible D0 → F2 transition between two deep-lying electronic levels that essentially create a two-level system[15]. Our entanglement scheme does not require involving spin degrees of freedom. Rather, it operates with an optically allowed atomic dipole transition. Envisaged applications of this scheme range from quantum information transfer over long distances (centimeter-long distances, as a matter of fact, since cm-long smalldiameter single-walled CNs are currently technologically available[4]) to novel sources of coherent light emitted by dopant atoms encapsulated in nanotubes. References

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