Nhng tin b trong Quang hc, Quang ph vÃ ng dng VI ISSN 1859 - 4271

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Advances in Optics, Photonics, Spectroscopy & Applications VI ISSN 1859 - 4271Here p and M are the momentum and the mass of the atom, respectively,the detuning of the probe field frequencyand∆ = ω −ωisωpfrom the shifted atomic transition frequency2ωcd( ωc) ( ωc)G = (| er| + | eϕ|)cos( βcz)(2)2ε0hL†is the atom-cavity coupling coefficient [19]. The notations a and a stand for the cavity<strong>ph</strong>oton operators. The operator σz=| e〉〈e | − | g〉〈g | characterizes the atomic population†inversion. The notations σ =| g〉〈 e | and σ =| e〉〈g | stand for the atomic downward- andupward-transition operators, respectively. In expression (2), the parameter d is the matrix(c)element of the electric-dipole moment of the atom, and e ω( )rand e ω cϕ are the radial andazimuthal components of the normalized guided-mode profile function [15, 21]. We haveassumed for simplicity that the order of the cavity resonance is even, that is, the center of thecavity corresponds to an an<s<strong>trong</strong>>tin</s<strong>trong</strong>>ode of the cavity standing-wave field.The equation for the mean value 〈O〉of an arbitrary observable O is† ††† ††〈 O& iγκ〉 = 〈[Heff, O]〉 − 〈 Oσσ − 2σOσ+ σ σO〉− 〈 Oaa − 2aOaρ+ a aO〉, (3)h22where γ is the decay rate of the atom in the presence of the nanofiber. Hence, we finddκ〈 a〉= i∆c〈 a〉+ G〈σ 〉 − 〈 a〉+ η,dt2dγ〈 σ 〉 = i∆a〈 σ 〉 + G〈aσz〉− 〈 σ 〉,(4)dt2andd †† † ††〈 a a〉= G〈a σ + aσ〉 −κ〈 a a〉+ η(〈 a 〉 + 〈 a〉),dtd †† ††〈 σ σ 〉 = −G〈a σ + aσ〉 − γ 〈 σ σ 〉,dtd † †† ††† κ + γ † ††〈 a σ + aσ〉 = i∆〈a σ − aσ〉 + 2G〈a aσz+ σ σ 〉 − 〈 a σ + aσ〉 + η(〈 σ 〉 + 〈 σ 〉),dt2d † †† † κ + γ † ††〈 a σ − aσ〉 = i∆〈a σ + aσ〉 − 〈 a σ − aσ〉 + η(〈 σ 〉 − 〈 σ 〉).(5)dt2Here ∆ = ωc−ωais the cavity-atom detuning.To obtain some simple analytical results, we consider the case where the probe field is soweak that the excited state is hardly occupied and that there is at most one <strong>ph</strong>oton in the cavity.In this case, we can linearize Eqs. (??) and (??) by using the approximations 〈 aσ z〉 = −〈 a〉and††〈 a aσz〉= −〈 a a〉. When we solve the resul<s<strong>trong</strong>>tin</s<strong>trong</strong>>g equations in the steady-state regime, we obtain[22]22η〈 a η〉 = − ( i∆a− γ/2),〈 σ 〉 = − GDD, † † η† † η〈 a σ + aσ〉 = − Gγ, 〈 a σ − aσ〉 = −2iG∆,22 a(6)| D || D |and22† η 2 2† η 2N ≡ 〈 a a〉= ( ∆ + γ /4),2 aP e≡ 〈σ σ 〉 = G ,(7)2| D || D |apaωa,64
Những tiến bộ <strong>trong</strong> <strong>Quang</strong> học, <strong>Quang</strong> <strong>ph</strong>ổ và Ứng dụng VI ISSN 1859 - 4271PoutwhereD = G2 + κγ/4− ∆c∆a− i(∆cγ+ ∆aκ)/2.(8)Since the parameters G , γ , and ∆ vary in space, the intensity of the transmitted light= h ω pκN/2depends on the position of the atom.III. FORCE ON AN ATOMaThe total force F on an atom is defined asiF = 〈[Heff, p]〉.(9)hWe find from Eqs. (1) and (9) the expression† ††F = −ih(∇G)〈 a σ − aσ〉 − ( ∇V eg)〈σ σ 〉 − ∇V g.(10)Note that, due to the cylindrical symmetry of the atom-field interaction, the azimuthalcomponent F ϕ of the force is zero. Consequently, the total force F in the cylindrical coordinateshas only two components, Frand Fz.A, Force on a motionless atomWe consider the case where the atom is motionless or its center-of-mass motion is very slow.In this case, the internal state of the atom-field system can be approximated by the local steady(0)state. Using this state, we can calculate the force F on a motionless atom. We find22 2(0)= hη ∆a2 η GF − ∇G− ∇V.2| | | |2 eg− ∇Vg(11)D D(0)(0)Fig. 2: Radial component Frand axial component Fzof the force as functions of the atom-tosurfacedistance r − a (left column) and the axial position z (right column). The fiber radius is a = 2002nm. The FBG cavity length is L = 10 cm. The FBG mirror reflectivity is | R | = 0.9 . The parameters ofthe atom correspond to the D2-line transition 6S 1/2F= 4, M = 4 ↔ 6P3/2F′= 5, M ′ = 5 of cesium,with the wavelength λ0= 852 nm and the natural linewidth γ0/2π= 5.25 MHz. The van der Waalscoefficients are C3g/2πh= 1.56 kHz µ m 3 and C3e/2πh= 3.09 kHz µ m 3 . The probe field, the cavity,and the atom are at exact resonance, that is, ωp= ωc= ω0. The input probe power is Pin= 1 pW. In theleft column, the axial position of the atom is β cz = π/4. In the right column, the distance from the atomto the fiber surface is r − a = 100 nm.65
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