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Optical absorption by atomically doped carbon nanotubes under strong atom–field coupling
ARTICLE IN PRESS
Physica E 37 (2007) 105–108 www.elsevier.com/locate/physe
Optical absorption by atomically doped carbon nanotubes under strong atom–field coupling I.V. Bondarev, B. Vlahovic Physics Department, North Carolina Central University, 1801 Fayetteville Street, Durham, NC 27707, USA Available online 21 August 2006
Abstract We analyze optical absorption by atomically doped carbon nanotubes with a special focus on the frequency range close to the atomic transition frequency. We derive the optical absorption line-shape function and, having analyzed particular achiral nanotubes of different diameters, predict the effect of absorption line splitting due to strong atom–vacuum–field coupling in small-diameter nanotubes. r 2006 Elsevier B.V. All rights reserved. PACS: 78.40.Ri; 73.22.f; 73.63.Fg; 78.67.Ch Keywords: Carbon nanotubes; Atomic doping; Optical absorption; Strong coupling
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 [8] and their intercalation into single-wall CN bundles [5,9], along with numerous studies of monoatomic gas absorption by the CN bundles (see Ref. [10] for a review), stimulate an in-depth theoretical analysis of near-field electrodynamical properties of atomically doped CNs systems. Typically, there may be two qualitatively different regimes of the interaction of an atomic state with a Corresponding author. Institute for Nuclear Problems, Belarusian State University, Bobruiskaya Street 11, 220050 Minsk, Belarus. Tel.: +1 919 530 6623; fax: +1 919 530 7472. E-mail address: [email protected] (I.V. Bondarev).
1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.06.015
vacuum electromagnetic field in the vicinity of a CN [11–14]. 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 nonexponential and is characterized by reversible Rabi oscillations where the energy of the initially excited atom is periodically exchanged between the atom and the field. We have recently shown that the relative density of photonic states (DOS) near the CN effectively increases due to the presence of additional surface photonic states coupled to CN electronic quasiparticle excitations [11]. This causes the atom–vacuum–field coupling constant, which is determined by the local (distance-dependent) photonic DOS, to be very sensitive to the atom–CNsurface distance. At small enough distances, the system may exhibit strong atom–field coupling [12,13], giving rise to rearrangement (‘‘dressing’’) of atomic levels by the vacuum–field interaction with formation of atomic quasi1D cavity polaritons. The problem of stability of quasi-1D polaritonic states in atomically doped CNs has been partly addressed in Refs. [13,14] by investigating the atom–nanotube van der Waals interactions and demonstrating the inapplicability of standard weak-couplingbased van der Waals coupling models close to the CN
ARTICLE IN PRESS I.V. Bondarev, B. Vlahovic / Physica E 37 (2007) 105–108
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surface. Here, we analyze optical absorption properties of atomically doped CNs with a special focus on the frequency range close to the atomic transition frequency and predict the effect of the absorption line splitting for quasi-1D atomic polaritons in small-diameter CNs. 2. Model Consider a two-level atom at the point rA near an infinitely long achiral single-wall CN. The atom interacts with a quantum electromagnetic field via an electric dipole transition of frequency oA. The atomic dipole moment, d ¼ d z ez , is assumed to be directed along the CN axis (assigned by the unit vector ez) which is chosen to be the z-quantization axis of the system. The contribution of the transverse dipole moment orientation is suppressed because of the strong depolarization of the transverse field in an isolated CN (the so-called dipole antenna effect [15]). The total secondly quantized Hamiltonian of the system is given by [13,14] Z 1 Z ~A y _o ^ H¼ s^ z do _o dR f^ ðR; oÞf^ðR; oÞ þ 2 0 Z 1 Z þ do dR½gðþÞ ðrA ; R; oÞs^ y g
0 ðÞ
^ f^ðR; oÞ þ h:c. ðrA ; R; oÞs
ð1Þ
with the three items representing the electromagnetic field (modified by the presence of the CN), the two-level atom y and their interaction, respectively. The operators f^ ðR; oÞ and f^ðR; oÞ are the scalar bosonic field operators defined on the surface of the CN (R is the radius-vector of an arbitrary point of the CN surface). They play the role of the fundamental dynamical variables of the field subsystem and satisfy the standard bosonic commutation relations. The Pauli operators, s^ z ¼ juihuj jlihlj, s^ ¼ jlihuj and s^ y ¼ juihlj, describe the atomic subsystem and electric dipole transitions between the two atomic states, upper jui and lower jli, separated by the transition frequency oA. This (bare) frequency is modified by the diamagnetic (A2) atom–field interaction yielding the new ~ A ¼ oA ½1 2=ð_oA Þ2 renormalized transition frequency o R R1 2 ? 0 do dRjg ðrA ; R; oÞj in the second term of the Hamiltonian. The dipole atom–field interaction matrix elements gðÞ ðrA ; R; oÞ are given by gðÞ ¼ g? ðo=oA Þgk pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where g?ðkÞ ðrA ; R; oÞ ¼ ið4oA =c2 Þ p_o Re szz ðoÞ d ?ðkÞ G zz z ?ðkÞ ðrA ; R; oÞ with G zz being the zz-component of the transverse (longitudinal) Green tensor (with respect to the first variable) of the electromagnetic subsystem and szz(o) representing the CN surface axial conductivity. Matrix R elements g?ðkÞ have the property of dRjg?ðkÞ ðrA ; R; oÞj2 ¼ ð_2 =2pÞðoA =oÞ2 G0 ðoÞx?ðkÞ ðrA ; oÞ with x?ðkÞ ðrA ; oÞ ¼ Im?ðkÞ 0 G ?ðkÞ zz ðrA ; rA ; oÞ=Im G zz ðoÞ being the transverse (longitudinal) distance-dependent (local) photonic DOS functions and G0 ðoÞ ¼ 8po2 d 2z Im G 0zz ðoÞ=3_c2 representing
the atomic spontaneous decay rate in vacuum where Im G 0zz ðoÞ ¼ o=6pc is the vacuum imaginary Green tensor zz-component. Hamiltonian (1) involves only two standard approximations: the electric dipole approximation and two-level approximation. The rotating wave approximation commonly used is not applied, and the diamagnetic term of the atom–field interaction is not neglected (as opposed to, e.g., Refs. [11,12,16]). Quantum electrodynamics (QED) of the two-level atom close to the CN is thus described in terms of only two intrinsic characteristics of the electromagnetic subsystem—the transverse and longitudinal local photonic DOS functions. In deriving the absorption/emission spectral line shape, we follow the general quantum approach of Ref. [17]. (Obviously, the absorption line shape coincides with the emission line shape if the monochromatic incident light beam is being used in the absorption experiment.) When the atom is initially in the upper state and the field subsystem is in vacuum, the time-dependent wave function of the whole system can be written as Z jcðtÞi ¼ C u ðtÞeiðo~ A =2Þt ju4jf0g4 þ dR Z 1 do C l ðR; o; tÞ eiðoo~ A =2Þt jlij1ðR; oÞi, ð2Þ 0
where jf0gi is the vacuum state of the field subsystem, jf1ðR; oÞgi is its excited state with the field being in the single-quantum Fock state, and Cu and Cl are the population probability amplitudes of the upper state and the lower state of the whole system, respectively. Applying Hamiltonian (1) to the wave function (2), one has Z Z i 1 Cu ðtÞ ¼ do dR gðþÞ ðrA ; R; oÞC l ðR; o; tÞ eiðoo~ A Þt , _ 0 (3) i (4) C l ðR; o; tÞ ¼ ½gðþÞ ðrA ; R; oÞ C u ðtÞ eiðoo~ A Þt . _ In terms of the probability amplitudes above, the emission/absorption spectral line shape I(o) is given by R dRjC l ðR; o; t ! 1Þj2 , yielding, after the substitution of Cl obtained by integration [under initial conditions C l ðR; o; 0Þ ¼ 0, C u ð0Þ ¼ 1] of Eq. (4), the result Z 1 2 iðxx~ A Þt ~IðxÞ ¼ I~0 ðxÞ dt C u ðtÞe (5) , 0
~ 0 ðxÞ=2p½ðxA =xÞ2 x? ðxÞ þ xk ðxÞ and we where I~0 ðxÞ ¼ ½G have, for convenience, introduced the dimensionless ~ 0 ¼ _G0 =2g0 , x ¼ _o=2g0 , and variables I~ ¼ 2g0 I=_, G t ¼ 2g0 t=_, with g0 ¼ 2:7 eV being the carbon nearest neighbor hopping integral appearing in the CN surface axial conductivity szz. The upper state population probability amplitude Cu(t) in Eq. (5) is given by the Volterra integral equation [obtained by substituting the result of the formal integration of Eq. (4) into Eq. (3)] with the kernel determined by the local photonic DOS functions x?ðkÞ ðrA ; xÞ [12,13]. Thus,
ARTICLE IN PRESS I.V. Bondarev, B. Vlahovic / Physica E 37 (2007) 105–108
the numerical solution is only possible for the line shape ~ IðxÞ, strictly speaking. This, however, offers very little physical insight into the problem of optical absorption by atomically doped CNs under different atom–field coupling regimes. Therefore, we choose a simple analytical approach valid for those atomic transition frequencies x~ A which are located in the vicinity of the resonance frequencies xr of the DOS functions x?ðkÞ . Within this approach, x?ðkÞ ðrA ; xxr Þ are approximated by the Lorentzians of the same halfwidth-at-half-maxima dxr, thus making it possible to solve the integral equation for Cu analytically to obtain [12,13] 0 1 pffiffiffiffiffiffiffiffiffiffiffiffi2 1B dxr 2 C C u ðtÞ @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAeð1=2Þ dxr dxr X t 2 2 dx2r X 0 1 pffiffiffiffiffiffiffiffiffiffiffiffi 1B dxr C ð1=2Þ dxr þ dx2r X 2 t þ @1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAe ð6Þ 2 dx2r X 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with X ¼ 2dxr G~ 0 ðx~ A Þx? ðrA ; x~ A Þ. This solution is valid for x~ A xr whatever the atom–field coupling strength is, yielding the exponential decay of the upper state popula~ x~ A Þt with the rate G~ G ~ 0 x? , in tion, jC u ðtÞj2 exp½Gð 2 the weak coupling regime where ðX =dxr Þ 51, and the decay via damped Rabi oscillations, jC u ðtÞj2 expðdxr tÞ cos2 ðX t=2Þ, in the strong coupling regime where ðX = dxr Þ2 b1. Substituting Eq. (6) into Eq. (5) and integrating over t, one arrives at the expression ~ ¼ I~0 ðx~ A Þ IðxÞ
ðx x~ A Þ2 þ dx2r 2 ðx x~ A Þ2 X 2 =4 þ dx2r ðx x~ A Þ2
(7)
representing the emission/absorption spectral line shape for the frequencies xxr x~ A regardless of the atom–field coupling strength. The line shape is clearly seen to be of a symmetric two-peak structure in the strong coupling regime where ðX =dxr Þ2 b1. The exact peak positions rq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 8ðdxr =X Þ2 4ðdxr =X Þ2 , seare x1;2 ¼ x~ A ðX =2Þ parated from each other by x1 x2 X with X being the Rabi frequency defined in Eq. (6). In the weak coupling
107
regime ðX =dxr Þ2 51, and x1,2 become complex, indicating that there are no longer peaks at these frequencies. As this takes place, Eq. (7) is approximated with the weak coupling condition, the explicit form for X, and the fact ~ ¼ that xx~ A , to give the well-known Lorentzian IðxÞ ~I 0 ðx~ A Þ=½ðx x~ A Þ2 þ G ~ 2 ðx~ A Þ=4 of the half-width-at-half~ x~ A Þ=2, peaked at x ¼ x~ A . maximum Gð 3. Numerical results and discussion To compute the absorption line shapes of particular CNs, one needs to know their transverse local photonic DOS functions x? ðrA ; x~ A Þ appearing in the Rabi frequency X in Eq. (7). These are basically determined by the CN surface axial conductivity szz. The latter one was calculated in the relaxation-time approximation (relaxation time 3 1012 s [18]) at temperature 300 K. The functions x? were calculated thereafter as described in Refs. [11–14]. The vacuum spontaneous decay rate was estimated from ~ 0 ¼ a3 x (a ¼ 1=137 is the fine-structure the expression G constant) valid for atomic systems with Coulomb interaction [19]. Fig. 1(a) shows x?(x) for the atom in the center of the (5,5), (10,10) and (23,0) CNs. It is seen to decrease with increasing CN radius, representing the decrease of the atom–field coupling strength as the atom moves away from the CN wall. The corresponding normalized absorp~ tion curves, AðxÞ ¼ IðxÞ= I~peak , are shown in Fig. 1(b). The curves were calculated from Eq. (7) for x~ A ¼ 0:45 [indicated by the vertical dashed line in Fig. 1(a)] as this is the approximate peak position of x? for all the three CNs. For the small radius (5,5) CN, the absorption line is split into two components, indicating the strong atom–field coupling with the Rabi splitting frequency X 0:2 corresponding to the energy of 0:2 2g0 ¼ 1:08 eV. The splitting is disappearing with increasing CN radius, and the absorption line shape becomes Lorentzian with the narrower widths for the larger radii CNs. Recently, Rabi splitting of 140 400 meV was detected in the photoluminescence experiments for quasi-0D excitonic polaritons in quantum dots in semiconductor microcavities [20–22]. The effect we report here is at least
107
1.0
106
0.8
104
A (x)
ξ⊥ (x)
105 103 102
0.4 0.2
101 100
0.0 0
(a)
0.6
0.45
1
2 x
3
0.2 (b)
0.3
0.4
0.5
0.6
0.7
x
Fig. 1. Transverse local photonic DOS (a) and normalized absorption line shapes (b) for the atom in the center of the CNs of increasing radii.
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3 orders of magnitude larger. This comes from the fact that the typical atomic binding energies, are at least 3 orders of magnitude larger than the typical excitonic binding energies in solids. In terms of the cavity-QED (see, e.g., Ref. [23]), the coupling constant of the atom–cavity interaction is given in our notations by _g ¼ d z ð2p_oA =V~ Þ1=2 with V~ being the effective volume of the field mode the atom interacts with. In our case V~ pR2cn ðlA =2Þ, which is 102 nm3 ¼ 107 mm3 for the CNs with diameters 1 nm in the optical spectral region (as apposed to 101 to 102 mm3 for the semiconductor microcavities in Refs. [20–22]). Small mode volumes yield large atom–cavity coupling constants in the small-diameter CNs. For example, in the situation shown in Fig. 1(a) one has _g0:3 eV for the atom in the center of the (5,5) CN [Rcn ¼ 0:34 nm, d z ereðe2 =_oA Þ], whereas the ‘‘cavity’’ line width, which is related to the Purcell factor x? ðoA Þ via the mode quality factor when oA or [22], is _gc ¼ 6p_c3 =o2A x? ðoA ÞV~ 0:03, so that the strong atom– field coupling condition g=gc b1 has been well satisfied. Acknowledgments This work was supported by the US National Science Foundation (Grant no. ECS-0631347), Department of Defense (Grant no. W911NF-05-1-0502) and NASA (Grant no. NAG3-804).
References [1] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Science of Fullerens and Carbon Nanotubes, Imperial College Press, London, 1998. [2] H. Dai, Surf. Sci. 500 (2002) 218. [3] R.H. Baughman, A.A. Zakhidov, W.A. de Heer, Science 297 (2002) 787. [4] S.G. Chou, et al., Phys. Rev. Lett. 94 (2005) 127402. [5] L. Duclaux, Carbon 40 (2002) 1751. [6] S. Latil, F. Triozon, S. Roche, Phys. Rev. Lett. 95 (2005) 126802. [7] Y.-W. Son, et al., Phys. Rev. Lett. 95 (2005) 216602. [8] G.-H. Jeong, et al., Phys. Rev. B 68 (2003) 075410. [9] H. Shimoda, et al., Phys. Rev. Lett. 88 (2002) 015502. [10] M.M. Calbi, et al., Rev. Mod. Phys. 73 (2001) 857. [11] I.V. Bondarev, et al., Phys. Rev. Lett. 89 (2002) 115504. [12] I.V. Bondarev, Ph. Lambin, Phys. Rev. B 70 (2004) 035407; I.V. Bondarev, Ph. Lambin, Phys. Lett. A 328 (2004) 235. [13] I.V. Bondarev, Ph. Lambin, in: Trends in Nanotubes Research, Nova Science, New York, 2006. [14] I.V. Bondarev, Ph. Lambin, Phys. Rev. B 72 (2005) 035451; I.V. Bondarev, Ph. Lambin, Solid State Commun. 132 (2004) 203. [15] A. Jorio, et al., Phys. Rev. B 65 (2002) 121402R. [16] H.T. Dung, L. Kno¨ll, D.-G. Welsch, Phys. Rev. A 62 (2000) 053804. [17] W. Heitler, The Quantum Theory of Radiation, Clarendon, Oxford, 1954. [18] S.J. Tans, et al., Nature 386 (1997) 474. [19] A.S. Davydov, Quantum Mechanics, NEO, Ann Arbor, MI, 1967. [20] J.P. Reithmaier, et al., Nature 432 (2004) 197. [21] T. Yoshie, et al., Nature 432 (2004) 200. [22] E. Peter, et al., Phys. Rev. Lett. 95 (2005) 067401. [23] L.C. Andreani, G. Panzarini, J.-M. Ge´rard, Phys. Rev. B 60 (1999) 13276.
Physica E 37 (2007) 105–108 www.elsevier.com/locate/physe
Optical absorption by atomically doped carbon nanotubes under strong atom–field coupling I.V. Bondarev, B. Vlahovic Physics Department, North Carolina Central University, 1801 Fayetteville Street, Durham, NC 27707, USA Available online 21 August 2006
Abstract We analyze optical absorption by atomically doped carbon nanotubes with a special focus on the frequency range close to the atomic transition frequency. We derive the optical absorption line-shape function and, having analyzed particular achiral nanotubes of different diameters, predict the effect of absorption line splitting due to strong atom–vacuum–field coupling in small-diameter nanotubes. r 2006 Elsevier B.V. All rights reserved. PACS: 78.40.Ri; 73.22.f; 73.63.Fg; 78.67.Ch Keywords: Carbon nanotubes; Atomic doping; Optical absorption; Strong coupling
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 [8] and their intercalation into single-wall CN bundles [5,9], along with numerous studies of monoatomic gas absorption by the CN bundles (see Ref. [10] for a review), stimulate an in-depth theoretical analysis of near-field electrodynamical properties of atomically doped CNs systems. Typically, there may be two qualitatively different regimes of the interaction of an atomic state with a Corresponding author. Institute for Nuclear Problems, Belarusian State University, Bobruiskaya Street 11, 220050 Minsk, Belarus. Tel.: +1 919 530 6623; fax: +1 919 530 7472. E-mail address: [email protected] (I.V. Bondarev).
1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.06.015
vacuum electromagnetic field in the vicinity of a CN [11–14]. 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 nonexponential and is characterized by reversible Rabi oscillations where the energy of the initially excited atom is periodically exchanged between the atom and the field. We have recently shown that the relative density of photonic states (DOS) near the CN effectively increases due to the presence of additional surface photonic states coupled to CN electronic quasiparticle excitations [11]. This causes the atom–vacuum–field coupling constant, which is determined by the local (distance-dependent) photonic DOS, to be very sensitive to the atom–CNsurface distance. At small enough distances, the system may exhibit strong atom–field coupling [12,13], giving rise to rearrangement (‘‘dressing’’) of atomic levels by the vacuum–field interaction with formation of atomic quasi1D cavity polaritons. The problem of stability of quasi-1D polaritonic states in atomically doped CNs has been partly addressed in Refs. [13,14] by investigating the atom–nanotube van der Waals interactions and demonstrating the inapplicability of standard weak-couplingbased van der Waals coupling models close to the CN
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surface. Here, we analyze optical absorption properties of atomically doped CNs with a special focus on the frequency range close to the atomic transition frequency and predict the effect of the absorption line splitting for quasi-1D atomic polaritons in small-diameter CNs. 2. Model Consider a two-level atom at the point rA near an infinitely long achiral single-wall CN. The atom interacts with a quantum electromagnetic field via an electric dipole transition of frequency oA. The atomic dipole moment, d ¼ d z ez , is assumed to be directed along the CN axis (assigned by the unit vector ez) which is chosen to be the z-quantization axis of the system. The contribution of the transverse dipole moment orientation is suppressed because of the strong depolarization of the transverse field in an isolated CN (the so-called dipole antenna effect [15]). The total secondly quantized Hamiltonian of the system is given by [13,14] Z 1 Z ~A y _o ^ H¼ s^ z do _o dR f^ ðR; oÞf^ðR; oÞ þ 2 0 Z 1 Z þ do dR½gðþÞ ðrA ; R; oÞs^ y g
0 ðÞ
^ f^ðR; oÞ þ h:c. ðrA ; R; oÞs
ð1Þ
with the three items representing the electromagnetic field (modified by the presence of the CN), the two-level atom y and their interaction, respectively. The operators f^ ðR; oÞ and f^ðR; oÞ are the scalar bosonic field operators defined on the surface of the CN (R is the radius-vector of an arbitrary point of the CN surface). They play the role of the fundamental dynamical variables of the field subsystem and satisfy the standard bosonic commutation relations. The Pauli operators, s^ z ¼ juihuj jlihlj, s^ ¼ jlihuj and s^ y ¼ juihlj, describe the atomic subsystem and electric dipole transitions between the two atomic states, upper jui and lower jli, separated by the transition frequency oA. This (bare) frequency is modified by the diamagnetic (A2) atom–field interaction yielding the new ~ A ¼ oA ½1 2=ð_oA Þ2 renormalized transition frequency o R R1 2 ? 0 do dRjg ðrA ; R; oÞj in the second term of the Hamiltonian. The dipole atom–field interaction matrix elements gðÞ ðrA ; R; oÞ are given by gðÞ ¼ g? ðo=oA Þgk pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where g?ðkÞ ðrA ; R; oÞ ¼ ið4oA =c2 Þ p_o Re szz ðoÞ d ?ðkÞ G zz z ?ðkÞ ðrA ; R; oÞ with G zz being the zz-component of the transverse (longitudinal) Green tensor (with respect to the first variable) of the electromagnetic subsystem and szz(o) representing the CN surface axial conductivity. Matrix R elements g?ðkÞ have the property of dRjg?ðkÞ ðrA ; R; oÞj2 ¼ ð_2 =2pÞðoA =oÞ2 G0 ðoÞx?ðkÞ ðrA ; oÞ with x?ðkÞ ðrA ; oÞ ¼ Im?ðkÞ 0 G ?ðkÞ zz ðrA ; rA ; oÞ=Im G zz ðoÞ being the transverse (longitudinal) distance-dependent (local) photonic DOS functions and G0 ðoÞ ¼ 8po2 d 2z Im G 0zz ðoÞ=3_c2 representing
the atomic spontaneous decay rate in vacuum where Im G 0zz ðoÞ ¼ o=6pc is the vacuum imaginary Green tensor zz-component. Hamiltonian (1) involves only two standard approximations: the electric dipole approximation and two-level approximation. The rotating wave approximation commonly used is not applied, and the diamagnetic term of the atom–field interaction is not neglected (as opposed to, e.g., Refs. [11,12,16]). Quantum electrodynamics (QED) of the two-level atom close to the CN is thus described in terms of only two intrinsic characteristics of the electromagnetic subsystem—the transverse and longitudinal local photonic DOS functions. In deriving the absorption/emission spectral line shape, we follow the general quantum approach of Ref. [17]. (Obviously, the absorption line shape coincides with the emission line shape if the monochromatic incident light beam is being used in the absorption experiment.) When the atom is initially in the upper state and the field subsystem is in vacuum, the time-dependent wave function of the whole system can be written as Z jcðtÞi ¼ C u ðtÞeiðo~ A =2Þt ju4jf0g4 þ dR Z 1 do C l ðR; o; tÞ eiðoo~ A =2Þt jlij1ðR; oÞi, ð2Þ 0
where jf0gi is the vacuum state of the field subsystem, jf1ðR; oÞgi is its excited state with the field being in the single-quantum Fock state, and Cu and Cl are the population probability amplitudes of the upper state and the lower state of the whole system, respectively. Applying Hamiltonian (1) to the wave function (2), one has Z Z i 1 Cu ðtÞ ¼ do dR gðþÞ ðrA ; R; oÞC l ðR; o; tÞ eiðoo~ A Þt , _ 0 (3) i (4) C l ðR; o; tÞ ¼ ½gðþÞ ðrA ; R; oÞ C u ðtÞ eiðoo~ A Þt . _ In terms of the probability amplitudes above, the emission/absorption spectral line shape I(o) is given by R dRjC l ðR; o; t ! 1Þj2 , yielding, after the substitution of Cl obtained by integration [under initial conditions C l ðR; o; 0Þ ¼ 0, C u ð0Þ ¼ 1] of Eq. (4), the result Z 1 2 iðxx~ A Þt ~IðxÞ ¼ I~0 ðxÞ dt C u ðtÞe (5) , 0
~ 0 ðxÞ=2p½ðxA =xÞ2 x? ðxÞ þ xk ðxÞ and we where I~0 ðxÞ ¼ ½G have, for convenience, introduced the dimensionless ~ 0 ¼ _G0 =2g0 , x ¼ _o=2g0 , and variables I~ ¼ 2g0 I=_, G t ¼ 2g0 t=_, with g0 ¼ 2:7 eV being the carbon nearest neighbor hopping integral appearing in the CN surface axial conductivity szz. The upper state population probability amplitude Cu(t) in Eq. (5) is given by the Volterra integral equation [obtained by substituting the result of the formal integration of Eq. (4) into Eq. (3)] with the kernel determined by the local photonic DOS functions x?ðkÞ ðrA ; xÞ [12,13]. Thus,
ARTICLE IN PRESS I.V. Bondarev, B. Vlahovic / Physica E 37 (2007) 105–108
the numerical solution is only possible for the line shape ~ IðxÞ, strictly speaking. This, however, offers very little physical insight into the problem of optical absorption by atomically doped CNs under different atom–field coupling regimes. Therefore, we choose a simple analytical approach valid for those atomic transition frequencies x~ A which are located in the vicinity of the resonance frequencies xr of the DOS functions x?ðkÞ . Within this approach, x?ðkÞ ðrA ; xxr Þ are approximated by the Lorentzians of the same halfwidth-at-half-maxima dxr, thus making it possible to solve the integral equation for Cu analytically to obtain [12,13] 0 1 pffiffiffiffiffiffiffiffiffiffiffiffi2 1B dxr 2 C C u ðtÞ @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAeð1=2Þ dxr dxr X t 2 2 dx2r X 0 1 pffiffiffiffiffiffiffiffiffiffiffiffi 1B dxr C ð1=2Þ dxr þ dx2r X 2 t þ @1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAe ð6Þ 2 dx2r X 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with X ¼ 2dxr G~ 0 ðx~ A Þx? ðrA ; x~ A Þ. This solution is valid for x~ A xr whatever the atom–field coupling strength is, yielding the exponential decay of the upper state popula~ x~ A Þt with the rate G~ G ~ 0 x? , in tion, jC u ðtÞj2 exp½Gð 2 the weak coupling regime where ðX =dxr Þ 51, and the decay via damped Rabi oscillations, jC u ðtÞj2 expðdxr tÞ cos2 ðX t=2Þ, in the strong coupling regime where ðX = dxr Þ2 b1. Substituting Eq. (6) into Eq. (5) and integrating over t, one arrives at the expression ~ ¼ I~0 ðx~ A Þ IðxÞ
ðx x~ A Þ2 þ dx2r 2 ðx x~ A Þ2 X 2 =4 þ dx2r ðx x~ A Þ2
(7)
representing the emission/absorption spectral line shape for the frequencies xxr x~ A regardless of the atom–field coupling strength. The line shape is clearly seen to be of a symmetric two-peak structure in the strong coupling regime where ðX =dxr Þ2 b1. The exact peak positions rq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 8ðdxr =X Þ2 4ðdxr =X Þ2 , seare x1;2 ¼ x~ A ðX =2Þ parated from each other by x1 x2 X with X being the Rabi frequency defined in Eq. (6). In the weak coupling
107
regime ðX =dxr Þ2 51, and x1,2 become complex, indicating that there are no longer peaks at these frequencies. As this takes place, Eq. (7) is approximated with the weak coupling condition, the explicit form for X, and the fact ~ ¼ that xx~ A , to give the well-known Lorentzian IðxÞ ~I 0 ðx~ A Þ=½ðx x~ A Þ2 þ G ~ 2 ðx~ A Þ=4 of the half-width-at-half~ x~ A Þ=2, peaked at x ¼ x~ A . maximum Gð 3. Numerical results and discussion To compute the absorption line shapes of particular CNs, one needs to know their transverse local photonic DOS functions x? ðrA ; x~ A Þ appearing in the Rabi frequency X in Eq. (7). These are basically determined by the CN surface axial conductivity szz. The latter one was calculated in the relaxation-time approximation (relaxation time 3 1012 s [18]) at temperature 300 K. The functions x? were calculated thereafter as described in Refs. [11–14]. The vacuum spontaneous decay rate was estimated from ~ 0 ¼ a3 x (a ¼ 1=137 is the fine-structure the expression G constant) valid for atomic systems with Coulomb interaction [19]. Fig. 1(a) shows x?(x) for the atom in the center of the (5,5), (10,10) and (23,0) CNs. It is seen to decrease with increasing CN radius, representing the decrease of the atom–field coupling strength as the atom moves away from the CN wall. The corresponding normalized absorp~ tion curves, AðxÞ ¼ IðxÞ= I~peak , are shown in Fig. 1(b). The curves were calculated from Eq. (7) for x~ A ¼ 0:45 [indicated by the vertical dashed line in Fig. 1(a)] as this is the approximate peak position of x? for all the three CNs. For the small radius (5,5) CN, the absorption line is split into two components, indicating the strong atom–field coupling with the Rabi splitting frequency X 0:2 corresponding to the energy of 0:2 2g0 ¼ 1:08 eV. The splitting is disappearing with increasing CN radius, and the absorption line shape becomes Lorentzian with the narrower widths for the larger radii CNs. Recently, Rabi splitting of 140 400 meV was detected in the photoluminescence experiments for quasi-0D excitonic polaritons in quantum dots in semiconductor microcavities [20–22]. The effect we report here is at least
107
1.0
106
0.8
104
A (x)
ξ⊥ (x)
105 103 102
0.4 0.2
101 100
0.0 0
(a)
0.6
0.45
1
2 x
3
0.2 (b)
0.3
0.4
0.5
0.6
0.7
x
Fig. 1. Transverse local photonic DOS (a) and normalized absorption line shapes (b) for the atom in the center of the CNs of increasing radii.
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3 orders of magnitude larger. This comes from the fact that the typical atomic binding energies, are at least 3 orders of magnitude larger than the typical excitonic binding energies in solids. In terms of the cavity-QED (see, e.g., Ref. [23]), the coupling constant of the atom–cavity interaction is given in our notations by _g ¼ d z ð2p_oA =V~ Þ1=2 with V~ being the effective volume of the field mode the atom interacts with. In our case V~ pR2cn ðlA =2Þ, which is 102 nm3 ¼ 107 mm3 for the CNs with diameters 1 nm in the optical spectral region (as apposed to 101 to 102 mm3 for the semiconductor microcavities in Refs. [20–22]). Small mode volumes yield large atom–cavity coupling constants in the small-diameter CNs. For example, in the situation shown in Fig. 1(a) one has _g0:3 eV for the atom in the center of the (5,5) CN [Rcn ¼ 0:34 nm, d z ereðe2 =_oA Þ], whereas the ‘‘cavity’’ line width, which is related to the Purcell factor x? ðoA Þ via the mode quality factor when oA or [22], is _gc ¼ 6p_c3 =o2A x? ðoA ÞV~ 0:03, so that the strong atom– field coupling condition g=gc b1 has been well satisfied. Acknowledgments This work was supported by the US National Science Foundation (Grant no. ECS-0631347), Department of Defense (Grant no. W911NF-05-1-0502) and NASA (Grant no. NAG3-804).
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