научная статья по теме QUANTUM OPTICS PHENOMENA IN ATOMICALLY DOPED CARBON NANOTUBES Физика

Текст научной статьи на тему «QUANTUM OPTICS PHENOMENA IN ATOMICALLY DOPED CARBON NANOTUBES»

ОПТИКА И СПЕКТРОСКОПИЯ, 2007, том 103, № 3, с. 381-388

НАНОФОТОНИКА. МОДИФИКАЦИЯ СПОНТАННОГО ИСПУСКАНИЯ

YflK 535.14

QUANTUM OPTICS PHENOMENA IN ATOMICALLY DOPED CARBON NANOTUBES

© 2007 r. I. V. Bondarev

Physics Department, North Carolina Central University, Durham, NC 27707, USA Institute for Nuclear Problems, Belarusian State University, 220050 Minsk, Belarus

e-mail: ibondarev@nccu.edu Received October 12, 2006

Abstract—Quantum optics phenomena, including light absorbtion and atomic states entanglement, are discussed for carbon nanotubes doped with atoms (ions). It has been shown that, similar to semiconductor micro-cavities and photonic band-gap materials, carbon nanotubes may qualitatively change the character of the atom-electromagnetic-field interactions, yielding strong atom-field coupling regime with the formation of quasi-one-dimensional atomic polaritons. These may be observed experimentally via the effect of the absorbtion line splitting (Rabi splitting) in the frequency range close to the atomic transition frequency. A scheme for entangling atomic polaritons is investigated using the photon Green function formalism for quantizing electromagnetic fields in the presence of quasi-one-dimensional absorbing and dispersing media. Small-diameter metallic nanotubes are shown to result in sizable amounts of the two-qubit atomic entanglement with no damping for sufficiently long times, thus challenging novel applications of atomically doped carbon nanotubes in quantum information science.

PACS: 78.40.Ri, 73.22.-f, 73.63.Fg, 78.67.Ch

1. INTRODUCTION

Carbon nanotubes (CNs) are graphene sheets rolled-up into cylinders of approximately 1 nm in diameter. Extensive work carried out worldwide in recent years has revealed intriguing physical properties of these novel molecular scale wires [1, 2]. Nanotubes have been shown to be useful for miniaturized electronic, mechanical, electromechanical, chemical and scanning probe devices and as materials for macroscopic composites [3, 4]. Important is that their intrinsic properties may be substantially modified in a controllable way by doping with extrinsic impurity atoms, molecules and compounds [5-9]. Recent successful experiments on encapsulation of single atoms into single-wall CNs [9] and their intercalation into single-wall CN bundles [7, 8], along with numerous studies of monoatomic gas absorbtion by the CN bundles (see [10] for a review) and the progress in growth techniques of centimeter-long small-diameter single-walled CNs [11], stimulate an in-depth theoretical analysis of dynamic quantum coherent processes in atomically doped CNs.

Typically, there may be two qualitatively different regimes of the interaction of an atomic state with a vacuum electromagnetic field in the vicinity of a CN [1215]. They are the weak coupling regime and the strong coupling regime. The former is characterized by the monotonic exponential decay dynamics of the upper atomic state with the decay rate altered compared with the free-space value. The latter is, in contrast, strictly non-exponential and is characterized by reversible Rabi oscillations where the energy of the initially excited at-

om is periodically exchanged between the atom and the field. 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 [12]. In small-diameter CNs the strong atom-field coupling may occur [13-15]. 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 dA and frequency roA) to a vacuum field is given by

2 ~ 1/2 hg = (2 ndA hrnJV)

with V being the effective volume of the field mode the atom interacts with (see, e.g., Ref. [16]). For the atom

(ion) encapsulated into the CN of radius Rcn, V ~

~ nRcn(XA/2) that is ~102 nm3 for CNs with diameters ~1 nm in the optical range of XA ~ 600 nm. Approximating dA ~ er ~ e(e2/hroA) [17], one obtains hg ~ 0.3 eV. On the other hand, the "cavity" linewidth is given for roA in resonance with the cavity mode by

hjc = 6 nh c3/®A )V,

where is the atomic spontaneous emission enhance-ment/dehancement (Purcell) factor [16]. Taking into account large Purcell factors ~107 close to CNs [12], one arrives at hyc ~ 0.03 eV for 1 nm-diameter CNs in

the optical spectral range. Thus, for the atoms (ions) encapsulated into small-diameter CNs the strong atom-field coupling condition g/jc > 1 is supposed to be satisfied, giving rise to the rearrangement ("dressing") of atomic levels and formation of atomic quasi-one-dimensional (1D) cavity polaritons [13-15]. The latter ones are similar to quasi-OD excitonic polaritons in quantum dots in semiconductor microcavities [18], that are currently being considered a possible way to produce the excitonic states entanglement [19].

A critical research problem for the nearest future is the development of materials that may host quantum coherent states with coherence lifetimes long enough to enable the nearest neighbour entangling operations over short distances, followed by quantum information transfer over longer distances [20, 21]. In spite of impressive experimental demonstrations of basic quantum information effects in a number of different meso-scopic systems, such as quantum dots in semiconductor microcavities, cold ions in traps, nuclear spin systems, atoms in optical resonators, Josephson junctions, etc., their concrete implementation is still at the proof-of-principle stage (see, e.g., Ref. [21] for a review). There is still a need for the fabrication of quantum bits (qu-bits) with coherence lifetimes at least three-four orders of magnitude longer than it takes to perform a bit flip. It is thus of vital importance to pursue a variety of different strategies and approaches towards physically implementing novel non-trivial applications in modern nanotechnology.

Here we suggest an 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 carbon nanotubes. In Sec. 2, the optical absorbtion is analyzed by atomically doped CNs with a special focus on the frequency range close to the atomic transition frequency. The effect of absorbtion line splitting (Rabi splitting) is demonstrated for small-diameter (~1 nm) nanotubes. Rabi splitting is a clear signature of the strong atom-field coupling regime - the property that is known to facilitate the entanglement of spatially separated quantum bits [22, 23]. As such, Sec. 3 analyzes the entanglement of the two strongly coupled quasi-1D atomic polariton states in CNs. Small-diameter metallic nanotubes are shown to result in sizable amounts of the two-qubit atomic entanglement for sufficiently long times. Envisaged applications of this scheme range from quantum information transfer over long distances (centimeter-long distances, as a matter of fact, since centimeter-long small-diameter singlewalled CNs are now technologically available [11]) to novel sources of coherent light emitted by dopant atoms in CNs. Section 4 concludes the article.

2. OPTICAL ABSORBTION BY ATOMICALLY DOPED CNs

We use our previously developed photon Green function formalism for quantizing an electromagnetic

field close to quasi-1D absorbing and dispersing media [14, 15]. Representing such a medium, the (achiral) CN is considered to be an infinitely long, infinitely thin, anisotropically conducting cylinder. Its (axial) surface conductivity is taken to be that given by the n-electron band structure in the tight-binding approximation with the azimuthal electron momentum quantization and axial electron momentum relaxation taken into account. A two-level atom is supposed to be positioned at the point rA near an infinitely long achiral single-wall CN. The orthonormal cylindric basis {er, ep, ez} is chosen in such a way that ez is directed along the nanotube axis and, without loss of generality, rA = rAer = {rA, 0, 0}. The atom interacts with the quantum electromagnetic field via its transition dipole moment that is assumed to be directed along the CN axis, dA = dzez. The contribution of the transverse dipole moment orientations is suppressed because of the strong depolarization of the transverse field in an isolated CN (the so-called dipole antenna effect [24]). The total secondly quantized Hamiltonian of the system is given in Gaussian units by [14, 15]

H = JdrohroJdR/(R, ro)f(R, ro) + ^&

+

+

JdroJdR[g(+)(rA, R, ro)cc1

(1)

-g(-)(rA, R,ro)c]f (R, ro) + h.c.,

with the three items representing the medium-assisted electromagnetic field (modified by the presence of the CN), the two-level atom and their interaction, respectively. The operators f 1(R, ro) and f (R, ro) are the scalar bosonic field operators defined on the CN surface assigned by the radius-vector R = {Rcn, Z} with Rcn being the radius of the CN. These operators create and annihilate the single-quantum bosonic-type electromagnetic medium excitation of the frequency ro at the point R of the CN surface. The Pauli operators, cz = \u){u | - |/)<l1,

c = |l }<u | and c1 = |u)</1, describe the atomic subsystem and electric dipole transitions between the two atomic states, upper |u) and lower |l), separated by the transition frequency roA. This (bare) frequency is modified by the diamagnetic (~A2) atom-field interaction yielding the new renormalized frequency

(0 A = ro A

1 - 2/(h ro A)2 j droJ dR| g1( rA, R, ro)|

0

0

2

0

in the second term of the Hamiltonian. The dipole atom-field interaction matrix elements g(±)(rA, R, ro) are given by g(±) = g1 ± (ro/roA)g|1, where

g1(||)(rA, R,ro) = = —i(4roa/c2 )-AhroRe azz(ro)dtmGzz(rA, R, ro)

with 1(||)Gzz being the zz-component of the transverse (longitudinal) Green tensor (with respect to the

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