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Optical transmission gratings by one driven three-level atom and a microtoroidal resonator
Optics Communications 284 (2011) 5263–5268
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Optical transmission gratings by one driven three-level atom and a microtoroidal resonator Rong Yu a,⁎, Jiahua Li b, Min Liu c, Chunling Ding b, Xiaoxue Yang b a b c
School of Science, Hubei Province Key Laboratory of Intelligent Robot, Wuhan Institute of Technology, Wuhan 430073, People's Republic of China Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People's Republic of China School of Computer Science and Technology, Xiantao Vocational College, Xiantao 433000, People's Republic of China
a r t i c l e
i n f o
Article history: Received 26 May 2011 Received in revised form 30 June 2011 Accepted 28 July 2011 Available online 16 August 2011 Keywords: Transmission gratings Microtoroidal resonator Tapered fiber Strong interaction
a b s t r a c t Based on the strong coherent interaction between a three-level ladder-type atom and a fiber-taper-coupled microtoroidal resonator, we present a scheme for optical transmission gratings. Using experimentally accessible parameters, it is shown that alternating regions of high transmission and absorption can be created in the fibertaper channel by spatially modulating an external coupling field. The model shows an obvious effect which has a direct analogy with the phenomenon of electromagnetically induced grating (EIG) in quantum systems. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Electromagnetically induced grating (EIG) [1], which is attributed to destructive quantum interference, is an interesting phenomenon where the absorption of a weak probe beam coupled to an atomic transition can be spatially modulated (reduced absorption at the peaks of the standing-wave field and high absorption at the nodes) by applying a strong standing wave that is coupled to another atomic transition in a three-level atomic system. EIG was originally observed in cold three-level atomic vapors [2]. Afterward, it was demonstrated that such an EIG effect can be used to store probe pulses in a vapor of rubidium atoms [3], to achieve tunable photonic band gap [4], to devise a dynamic controlled cavity [5], to implement optical routing [6,7], and so on. However, as shown in [1], low efficiency of the EIG due to weak interactions between single atoms and photons in a three-level atomic medium restricts its practical uses. It is worth pointing out that, on account of both highly confined microscale mode volume and ultrahigh quality factors [8–10], optical microresonators enable strong light-matter interactions as well as drastic reductions of the power necessary to observe strong nonlinear effects [11–14]. As a result, in the past few years optical microresonators are increasingly gaining interest in many diverse areas of research, ranging from nanophotonics [15,16] and biochemical sensing [17–19] to cavity quantum electrodynamics (CQED) [20,21]. Optical microresonators support whispering-gallery modes (WGMs). Different from the standing modes in a conventional Fabry–Perot (F–P) cavity, WGMs are a ⁎ Corresponding author. Tel./fax: + 86 2787557477. E-mail addresses: [email protected] (R. Yu), [email protected] (J. Li). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.07.066
type of traveling modes. In other words, WGM microresonators typically support two counterpropagating modes, i.e., clockwise (CW) and counterclockwise (CCW) propagating modes, with the same polarization and a degenerate frequency. This degeneracy can be lifted, and it can form a doublet through backscattering coupling induced by internal defect centers or surface roughness [22,23]. This phenomenon is known as modal coupling. Atoms in the vicinity of the resonator are able to interact with the two WGMs via the evanescent field. With the help of the fiber taper, the efficiency for coupling the quantum fields into and out of the microtoroidal resonator can approach 0.99–0.999 [24,25]. Also, strong coupling between a single cesium atom and the electromagnetic mode in a microtoroid has been theoretically investigated and experimentally observed [26]. Under a certain condition, the atom can transfer its excitation to the CW or CCW mode which is intrinsic in the microtoroidal resonators. In view of this, making use of the above-mentioned coherent interactions between the microtoroidal resonator and atoms, some schemes about photon turnstiles [27], photon routers [28], singlephoton transistors [29] and quantum controlled-phase-flip gates [30,31] have been put forward. It is worth pointing out that, in the investigation of Ref. [27], Dayan et al. have addressed that, with quantum critical coupling of input lights into and out of a microtoroidal resonator, a single cesium atom near the surface of the resonator can dynamically control the cavity output depending on the photon number at the input. Based on these achieved advances, we demonstrate that such a system can be employed to act as a kind of optical transmission gratings, analogous to an EIG in a three-level atomic system [1]. Using experimentally accessible parameters, alternating regions of high transmission and absorption can be caused by an external standing-wave coupling field.
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R. Yu et al. / Optics Communications 284 (2011) 5263–5268
exclusively) to only one of the two normal modes [26,27]. Hereafter, we assume the position of the atom such that kz =nπ (n is integer), i.e., gB =0, then the normal mode Bˆ is decoupled from the interaction with the atom. Alternatively, the upmost state |e〉 of the three-level ladder-type atom (with the energy ωe) is coupled to the state |r〉 by a classical standing-wave coupling field with central frequency ωsw and position-dependent Rabi frequency Ωsw(x). The interactions between a high-Q microresonators (or microcavities) and a three-level system have been intensively studied previously [29,32–36] but in different contexts. Following the method developed in Refs. [26,27], the total Hamiltonian describing the interaction of a microtoroidal resonator with a three-level ladder-type atom and with a tapered fiber (see Fig. 1) can be written in the form
Due to its robustness against major experimental imperfections together with the high efficiency for coupling of the quantum fields into and out of the resonator with a fiber taper [24,25], this work might broaden variety applications in imaging techniques and precise measurements compared with EIG in a three-level atomic system [1]. On the other hand, in contrast to traditional F–P cavities, microtoroidal resonator structures possess advantages in easy integration and fabrication. They will likely be valuable for potential device applications. The remainder of this paper is arranged into three parts as follows. In Section 2, we establish the physical model and its theoretical description. By solving the coupled amplitude equations of motion for the microtoroidal resonator, the tapered fiber and the three-level ladder-type atom in the frequency domain, we derive explicit analytical expressions of the forward- and backward-propagating transmission functions for the output fields. In Section 3, we devote to analyzing and demonstrating in details optical transmission gratings in this device. At the same time, we also present the principal mechanism behind the transmission grating. Finally, our main conclusions are summarized in Section 4.
ˆ = ˆ ˆ †A H h = ωr jr〉〈r j + ωe je〉〈ej + ðωc + hÞ A +∞ † † ˆ −i jg〉〈r j A ˆ† +∫ ω aˆ1ω aˆ1ω + aˆ2ω aˆ2ω dω + gA i jr〉〈g j A h −∞ i + iΩsw ðxÞe−iωsw t je〉〈r j−iΩsw ðxÞeiωsw t jr〉〈ej ð1Þ + ∞ rffiffiffiffiffiffiffi κex † † ˆ ˆ i aˆ1ω A−i A aˆ1ω dω +∫ 2πffi −∞ rffiffiffiffiffiffi +∞ κex † ˆ −i A ˆ †aˆ2ω dω; +∫ i aˆ2ω A 2π −∞
2. Model and equations Fig. 1 is a schematic description of the composite system, which consists of a microtoroidal resonator, a tapered fiber, and a three-level ladder-type atom. A microtoroidal resonator has two internal counterpropagating modes which are described in terms of the annihilation ˆ with a common frequency ωc in the absence of operators aˆ and b scattering [27]. These two modes are coupled to each other in the presence of scattering with a strength that is parameterized by h. The intracavity field decays at a rate κ=κi +κex, where κi and κex describe intrinsic losses and extrinsic loss due to adjustable interaction with the modes of a tapered fiber. The intracavity fields are coupled to a tapered fiber with ˆ high efficiency ηN 0.99 [24,25]. The evanescent fields of modes aˆ and b
_ _ where the energy of state |g〉 has been set as zero. hωr and hωe are the energies of the atomic states |r〉 and |e〉, respectively. The symbols |m〉〈n| (m, n = g, e, r) for m ≠ n, are the atomic transition or projection operators between the states |m〉 and |n〉 involving the levels of the atom while |m〉〈m| represent the atomic population operators [see also the † bubble of Fig. 1]. Aˆ and Aˆ are the bosonic annihilation and creation operators of the normal mode with the frequency ωc + h, where ωc is the † “bare” cavity mode frequency. aˆ jω and aˆ jω ( j = 1, 2) are the annihilation and creation operators for the two modes of frequency ωiin the tapered h † ˆ jω ; aˆ ′ = δðω−ω′ Þ. fiber channel, with the commutation relations a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jω κex = 2π describes the coupling strength between the toroidal resonator mode and the tapered fiber mode. According to the spirit of Refs. [37,38], we choose the proper free Hamiltonian to transform to the interaction picture, that is,
have the coherent interactions with a ground state |g〉 and an upper excited state |r〉 with the energy ωr of a three-level ladder-type atom near the external surface of the microtoroidal resonator. It is convenient to describe the interaction by the normal modes of the microtoroidal ˆ and Bˆ = p1ffiffi a− ˆ with the coupling rates gA ˆ b resonator Aˆ = p1ffiffi aˆ + b 2
2
and gB, respectively, where gA, B ∼g0e− δρ{cos kz, sin kz} (δ∼ λ1) with ρ being the radial distance from the surface of the toroid to the atom, z being the position around the circumference of the resonator, and k being the vacuum wave vector. As a result, depending on the position of the threelevel ladder-type atom, coupling may occur predominantly (or even
α1in
ˆ
ˆ
ð2Þ
ˆA ˆ + ðωc + hÞ jr〉〈r j + ðωc + h + ωsw Þje〉〈ej ˆ0 = H h = ðωc + hÞ A †
α1out
Tapered fiber
α 2out
ˆ res e−i H0 t = h ; ˆ int = ei H0 t = h H H
κ ex
α 2in
x ⊗
y
z Three-level atom
a
h
ρ
e
Δ sw Atom
Ω sw ( x ) r
ΔA
κi b
gA g
Microtoroidal resonator Fig. 1. A schematic of the microtoroidal resonator and fiber taper system. The tapered fiber and the resonator are in the critical coupling and the atom is located with well-defined ^ of the microtoroidal resonator with strength gA and azimuthal phase kz = nπ (n is integer) [18,27]. In this case, the atomic transition |g〉 ↔ |r〉 is coupled to the normal mode A frequency ωc + h (ωc is the “bare” cavity mode frequency). At the same time, the |r〉 ↔ |e〉 transition is driven by a classical standing-wave coupling field with position-dependent Rabi frequency Ωsw(x) and central frequency ωsw. The standing-wave coupling field is aligned along the x axis perpendicular to the y–z plane.
R. Yu et al. / Optics Communications 284 (2011) 5263–5268 +∞
+
∫−∞
ˆ†1ω aˆ1ω + a ˆ†2ω aˆ2ω dω; ω a
ˆ + i j g〉〈r jA ˆ† ˆ res = h = ΔA j r〉〈r j + ðΔA + Δsw Þ j e〉〈ej + gA i jr〉〈g jA H ð3Þ h i −iω t iω t + iΩsw ðxÞe sw j e〉〈r j−iΩsw ðxÞe sw j r〉〈e j +∞rffiffiffiffiffiffiffi κex † ˆ ˆ †aˆ1ω dω ˆ 1ω A−i A ia +∫ 2π −∞
+
+ ∞ rffiffiffiffiffiffiffi κex
∫−∞
2π
†
ˆ A ˆ aˆ ˆ†2ω A−i ia 2ω dω;
ˆ ˆ † ð5Þ ˆ int = h = ΔA jr〉〈r j + ðΔA + Δsw Þje〉〈ej + gA ijr〉〈g jA−ijg〉〈r jA H + iΩsw ðxÞje〉〈r j−iΩsw ðxÞjr〉〈ej r ffiffiffiffiffiffi ffi +∞ i κex h † ˆ iðω−ωc −hÞt ˆ †aˆ e−iðω−ωc −hÞt dω iaˆ1ω Ae +∫ −i A 1ω 2π −∞ + ∞ rffiffiffiffiffiffiffih i κex † ˆ iðω−ωc −hÞt ˆ †a ˆ Ae ˆ2ω e−iðω−ωc −hÞt dω; ia +∫ −i A 2π 2ω −∞ It should be pointed out that the coupled system described by this Hamiltonian, under optical excitation and with the resonator-atom and resonator-fiber interaction, has two invariant Hilbert subspaces, with the bases |g, 0〉|vac〉 and |r, 0〉|vac〉, |e, 0〉|vac〉, |g, 1〉|vac〉, |g, 0〉|ω〉, respectively, where in |s〉|n〉, s = g, e, r denotes the state of three-level ladder-type atom and n denotes the number of photons in the normal ˆ |ω〉 denotes the one-photon Fock state of the tapered fiber mode A, channel mode of frequency ω, |vac〉 describes the vacuum state of the fiber mode. So the evolution of the whole system can be generally described by the wave function |Ψ(t)〉 = Cg|g, 0〉|vac〉 + Ce|Ψ e(t)〉 in the interaction picture, where j Ψe ðt Þ〉 = βr ðt Þ jr; 0〉jvac〉 + βe ðt Þje; 0〉jvac〉 + βc ðt Þjg; 1〉jvac〉 + ∞h i +∫ α1ω ðt Þaˆ†1ω + α2ω ðt Þaˆ†2ω jg; 0〉jvac〉dω:
ð6Þ
−∞
Firstly, by making use of the interaction Hamiltonian operator (5) e ∂jΨ ðt Þ〉 = H ˆ int jΨe ðt Þ〉 and the well-known Schrödinger equation ih ∂t in the interaction picture, the time evolution of the amplitudes for the microtoroidal resonator and tapered fiber modes are
α˙ jω ðt Þ =
rffiffiffiffiffiffiffi κex 2π
+∞
∫−∞
rffiffiffiffiffiffiffi κex iðω−ωc −hÞt e βc ðt Þ 2π
−iðω−ωc −hÞt
e
½α1ω ðt Þ + α2ω ðt Þdω;
ð11Þ
κ pffiffiffiffiffiffiffi i −κex βc ðt Þ− κex ½α1out ðt Þ + α2out ðt Þ; 2
ð12Þ
ð8Þ
ð j = 1; 2Þ;
rffiffiffiffiffiffiffi κex t iðω−ωc −hÞt ′ ∫ e α jω ðt Þ = α jω ðt0 Þ + βc ðt ′ Þdt ′ 2π t0 rffiffiffiffiffiffiffi κex t1 iðω−ωc −hÞt ′ ∫ e βc ðt ′ Þdt ′ 2π t
ðt0 < t Þ;
ðt1 > t Þ;
+∞
1
where α jin ðt Þ=pffiffiffiffiffiffi ∫−∞ e−iΔωt αjω ðt0 ÞdΔω with t0 → − ∞ and αjout ðt Þ= 2π
+ ∞ −iΔωt 1 pffiffiffiffiffiffi ∫ e αjω ðt1 ÞdΔω with t1 →+∞ (j=1, 2, Δω=ω−ωc −h) are 2π −∞
the input and output field operators in the tapered fiber channel, respectively. Combining Eqs. (11), (12) and the symmetry of the tapered fiber modes, we can achieve the input–output formalism α1out −α1in = pffiffiffiffiffiffi pffiffiffiffiffiffi κex βc and α2out −α2in = κex βc [40,41]. Secondly, by considering the spontaneous emission rates of the three-level ladder-type atom within the Weisskopf–Wigner approximation [39–41], similarly we can obtain the time evolution of the amplitudes for the three-level atomic states as follows h γi β˙ e ðt Þ = − iðΔA + Δsw Þ + e βe ðt Þ + Ωsw ðxÞβr ðt Þ; 2
ð13Þ
γ β˙ r ðt Þ = − iΔA + r βr ðt Þ + gA βc ðt Þ−Ωsw ðxÞβe ðt Þ; 2
ð14Þ
where γr and γe denote the decay rates of the atomic states |r〉 and |e〉, respectively. In what follows, we are interested in the transmission properties in the frequency domain of the coupled microresonator system, therefore we 1 + ∞ perform the Fourier transformations F ðΔωÞ = pffiffiffiffiffiffi ∫−∞ F ðt ÞeiΔωt dt on 2π the amplitudes βc, βe and βr in Eqs. (11), (13) and (14), respectively. After carrying out some algebraic calculations, the analytical solution of the amplitude βc can be found as pffiffiffiffiffiffi κex ðα1in + α2in Þ
βc ðxÞ = iΔω−κex −
κi + 2
ð7Þ
Integrating Eq. (8) formally yields
α jω ðt Þ = α jω ðt1 Þ−
κ pffiffiffiffiffiffi β˙ c ðt Þ = −g A βr ðt Þ− i + κex βc ðt Þ− κex ½α1in ðt Þ + α2in ðt Þ; 2
ð4Þ
where ΔA = ωr − ωc − h and Δsw = ωe − ωr − ωsw are the detunings of the normal mode Aˆ and the standing-wave coupling field from the corresponding atomic transitions |g〉 ↔ |r〉 and |r〉 ↔ |e〉, respectively. The resulting interaction Hamiltonian in the interaction picture can be then reexpressed as follows
β˙ c ðt Þ = −gA βr ðt Þ−
procedures of the Weisskopf–Wigner approximation [39–41]), we can obtain
= −gA βr ðt Þ−
ð9Þ
ð10Þ
Now, by substituting the expressions of αjω(t) from Eqs. (9) and (10) into Eq. (7) as well as by taking the intrinsic decay rates κi of the normal mode Aˆ into account (which can be obtained by following the established
5265
igA2
2 γ jΩsw ðxÞj − Δω−ΔA + i r + γ 2 Δω−ΔA −Δsw + i e 2 ð15Þ
pffiffiffiffiffiffiffi According to the input–output formalism α1out −α1in = κex βc pffiffiffiffiffiffiffi and α2out −α2in = κex βc , finally we can readily derive the relationship between the input optical field and the output optical field κi igA2 + γ 2 − Δω−ΔA + i r + Ssw ðxÞ 2 α1out ðxÞ = α1in κi igA2 + iΔω−κex − γ 2 − Δω−ΔA + i r + Ssw ðxÞ 2 κex α2in ; + κ igA2 iΔω−κex − i + γ 2 − Δω−ΔA + i r + Ssw ðxÞ 2 ð16Þ iΔω−
:
κex α1in 2 κi igA + iΔω−κex − γ 2 − Δω−ΔA + i r + Ssw ðxÞ 2 2 κi igA + iΔω− γ 2 − Δω−ΔA + i r + Ssw ðxÞ 2 α2in ; + κi igA2 + iΔω−κex − γ 2 − Δω−ΔA + i r + Ssw ðxÞ 2 ð17Þ
where we have introduced the new definition Ssw ðxÞ=
jΩsw ðxÞj
2
Δω−ΔA −Δsw + i
γe : 2
The real part of Ssw stands for the optical Stark shift created by the coherent interaction between the standing-wave coupling field and the atomic transition |e〉 ↔ |r〉. However, the imaginary part of Ssw is the noise term which arises from the decay of the atom. The imaginary part would disappear when γe = 0 and in this case the component of Ssw reduces the Stark shift. Consequently, when the coupling field is applied, the atom would be detuned from the microtoroidal resonator by an optical Stark shift Ssw, and it leads to a significant change of the forward- and backward-propagating output fields α1out and α2out, as will be shown in Section 3. For the initial given input fields α1in ≠0 and α2in =0 in the tapered fiber channel, from Eqs. (16) and (17) the transmission functions of forward- and backward-propagating output fields can be respectively defined by 2 2 α α T1 ðxÞ = 1out ; T2 ðxÞ = 2out : α1in α1in
ð18Þ
Since the transmission functions of forward- and backwardpropagating output fields in the tapered fiber depend strongly on the strength of the coupling field, they are expected to change periodically as the standing wave changes from nodes to antinodes across x dimension. 3. Realization of optical transmission gratings In the following calculations, according to Refs. [27,29], we consider an impedance-matched input (that is, critical coupling) for which κex is set to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cr κex = κi2 + h2 and the case of small intrinsic losses, i.e., κi ≪κ≃κex ≃h (κ=κi +κex). In this case, the system parameters for the microtoroidal resonator, fiber taper and cesium atom under study are taken as: cr gA =70 MHz, κi =5 MHz, h=250 MHz, κex =κ ex ≃250 MHz, γr =1 MHz, and γe =0.5 MHz, respectively. Before proceeding further, it is instructive to briefly analyze the principal mechanism behind the transmission gratings in our studied system. It follows from Eqs. (16) and (17) that the laser-induced Stark shift Ssw plays an important role in forming the alternating regions of high transmission and absorption in the fiber-taper channel that act as a type of transmission gratings. To illustrate this point, in Fig. 2(a) and (b) we compare the forward and backward transmission line shapes in the absence of the coupling field (solid lines) with the ones in the presence of a strong resonant coupling field (dashed lines). We further assume that the intensity of the coupling field which is used for producing the dashed lines in Fig. 2(a) and (b), corresponds to the peak intensity of a standing-wave field. The forward transmission T1 increases rapidly and the backward transmission T2 decreases quickly when the standingwave coupling field Ωsw(x) is located at the node (Ωsw(x) = 0, see solid lines) for frequencies lying near the resonance point Δω = 0, with T1(Δω= 0)≈ 0.95 whereas T2(Δω= 0)≈ 0. Instead, when the standingwave coupling field Ωsw(x) is located at the antinode (Ωsw(x) = 10 MHz, see dashed lines), the transmission T1 decreases rapidly and T2 increases quickly near the resonance point, where T1(Δω = 0) ≈ 0 whereas T2(Δω= 0)≈ 0.90. Overall, for the situations considered in Fig. 2(a)
(a) 1.0
Ω (x)=0 sw
Normalized transmission T 1
α2out ðxÞ =
R. Yu et al. / Optics Communications 284 (2011) 5263–5268
Ωsw(x)=10 MHz
0.8
0.6
0.4
0.2
0.0
0
20
40
60
20
40
60
Δω (MHz)
(b) 1.0
Normalized transmission T 2
5266
0.8
0.6
0.4
0.2
0.0
0
Δω (MHz) Fig. 2. Normalized forward- and backward-propagating transmissions T1 and T2 for the output fields versus Δω≡ω−ωc −h. The solid lines are produced without the coupling field (Ωsw(x)=0). The dashed ones are produced with the coupling field (Ωsw(x)=10 MHz). Here, the intensity of the coupling field which is used for producing the dashed lines in (a) and (b), corresponds to the peak intensity of a standing-wave field. The other system parameters are chosen as gA = 70 MHz, κi = 5 MHz, κex = 250 MHz, γr = 1 MHz, γe = 0.5 MHz, and ΔA =Δsw =0, respectively.
and (b), conditioned on the position of the standing-wave coupling field Ωsw(x), the transmission characteristics of the coupled resonator system can transmit from those (solid lines) for the case Ωsw(x) = 0 at the position of node to those (dashed lines) for Ωsw(x) = 10 MHz at the position of antinode. That is to say, a standing-wave coupling field triggers the change {T1(Δω= 0)≈ 0.95, T2(Δω= 0)≈ 0} for Ωsw(x) = 0 ⇔ {T1(Δω= 0)≈ 0, T2(Δω = 0) ≈ 0.90} for Ωsw(x) ≠ 0. Consequently, the position-dependent coupling field is sufficient to well control the propagation of the input signal field in the tapered fiber channel, and the system can be used as a substantial intensity modulation. In addition, the mechanism behind the transmission gratings in our studied system shows an obvious effect which has a direct analogy with the phenomenon of EIG in a three-level atomic medium. The forward and backward transmission functions T1(x) and T2(x) as a function of position x are plotted in Fig. 3 for the case of resonant coupling of the standing wave (Δsw =0). At the transverse locations around the nodes of the standing wave, the coupling field intensity is very weak. Under this circumstance, the intensity transmission T1 for the forward flux reaches to 0.95 while the intensity transmission T2 for the backward flux reaches to 0. In contrast, at the transverse locations around the antinodes of the standing wave, the coupling field intensity is quite strong, and the intensity transmission T1 for the forward flux quickly decreases to a zero value (T1 =0) due to the optical Stark shift Ssw. On the other hand, the
R. Yu et al. / Optics Communications 284 (2011) 5263–5268
(a)
(a)
1.0
Normalized transmission T 1
Normalized transmission T 1
1.0
0.8
0.6
0.4
0.2
0.0
5267
0
10
20
30
40
0.8
0.6
0.4
0.2
0.0
50
0
(b)
30
40
50
20
30
40
50
1.0
Normalized transmission T 2
Normalized transmission T 2
20
(b)
1.0
0.8
0.6
0.4
0.2
0.0
10
Position x (μm)
Position x (μm)
0
10
20
30
40
0.8
0.6
0.4
0.2
0.0
50
0
Position x (μm)
4. Conclusion In summary, we have theoretically studied optical transmission characteristics of the composite system, which are composed of a microtoroidal resonator, a tapered fiber, and a three-level ladder-type atom driven by an external standing-wave coupling field. Due to interaction with the coupling field in this system, the phenomenon of the optical Stark shift can arise. We have shown that, by spatially modulating the on-resonant coupling field, alternating regions of high transmission and absorption can be produced in the fiber-taper
(a)
(b)
1.0
1.0
0.8
0.8
Normalized transmission T 2
intensity transmission T2 for the backward flux rapidly increases to a peak value of T2 =0.90. This leads to a periodic intensity modulation across the beam profiles of forward- and backward-propagating output fields, a phenomenon reminiscent of optical intensity transmission gratings. Fig. 4 compares the modulation depth of the transmission functions in the situation where the detuning of the standing-wave coupling field Δsw is 5 MHz with the one in Fig. 3 where Δsw is zero (other parameters are the same). It can be seen from Fig. 4 that the modulation depth of the transmission functions is considerably decreased when the standingwave coupling field is detuned off the resonance. To illustrate the influences of the detuning Δsw on the transmission functions T1(x) and T2(x) in the forward and backward fluxes, the result is plotted in Fig. 5. By analyzing the result of Fig. 5(a) and (b), a small intensity modulation of the transmission functions appearing in Fig. 4 can be well understood when the detuning of the standing-wave coupling field is introduced.
Fig. 4. Normalized forward- and backward-propagating transmission functions T1(x) and T2(x) for the output fields as a function of position x. The system parameters used are the same as in Fig. 3 except for Δsw = 5 MHz.
Normalized transmission T 1
Fig. 3. Normalized forward- and backward-propagating transmission functions T1(x) and T2(x) for the output fields as a function of position x. The standing-wave coupling πx , where Λ is the spatial frequency of the standing wave. The field: Ωsw ðxÞ = Ωsw 0 sin Λ system parameters are chosen as Ω0sw = 10 MHz, Λ = 10 μm, gA = 70 MHz, κi = 5 MHz, κex = 250 MHz, γr = 1 MHz, γe = 0.5 MHz, Δω = 0, and ΔA = Δsw = 0, respectively.
10
Position x (μm)
0.6
0.4
0.2
0.0
Ω (x)=0 sw
Ωsw(x)=10 MHz
0.6
0.4
0.2
0.0 0
20 40 60
Δsw (MHz)
0
20 40 60
Δsw (MHz)
Fig. 5. Normalized forward- and backward-propagating transmissions T1 and T2 for the output fields versus the frequency detuning Δsw of the standing-wave coupling field. The solid lines are produced without the coupling field (Ωsw(x) = 0). The dashed ones are produced with the coupling field (Ωsw(x) = 10 MHz). Note that, the intensity of the coupling field which is used for producing the dashed lines in (a) and (b), corresponds to the peak intensity of a standing-wave field. The other system parameters are chosen as gA = 70 MHz, κi = 5 MHz, κex = 250 MHz, γr = 1 MHz, γe = 0.5 MHz, and ΔA = 0, respectively.
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channel which can be used for implementing a kind of optical intensity transmission gratings. Combined with the strong coherent interaction between a three-level ladder-type atom and a fiber-tapercoupled microtoroidal resonator as well as its robustness against major experimental imperfections [27], the microtoroidal resonatorbased scheme proposed here may provide a way to achieve the design of microresonator-based devices for applications requiring the grating effect. Acknowledgments We would like to thank Professor Ying Wu for his encouragement and helpful discussion. This research was supported in part by the National Natural Science Foundation of China under grant nos. 11004069, 10975054 and 91021011, by the Doctoral Foundation of the Ministry of Education of China under grant no. 20100142120081 and by the Fundamental Research Funds from Huazhong University of Science and Technology (HUST) under grant no. 2010MS074. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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[15] D.K. Armani, T.J. Kippenberg, S.M. Spillane, K.J. Vahala, Nature 421 (2003) 925. [16] S. Noda, M. Fujita, T. Asano, Nat. Photonics 1 (2007) 449. [17] F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L.C. Andreani, E. Di Fabrizio, Nano Lett. 8 (2008) 2321. [18] F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L.C. Andreani, E. Di Fabrizio, Nat. Nanotechnol. 5 (2010) 67. [19] A.M. Armani, R.P. Kulkarni, S.E. Fraser, R.C. Flagan, K.J. Vahala, Science 317 (2007) 783. [20] H.J. Kimble, Phys. Scr. T76 (1998) 127. [21] K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E.L. Hu, A. Imamoğlu, Nature (London), 445, 2007, p. 896. [22] S.M. Spillane, Fiber-coupled ultra-high-Q microresonators for nonlinear and quantum optics. Ph.D. Thesis, California Institute of Technology, 2004. [23] X. Yi, Y.F. Xiao, Y.C. Liu, B.B. Li, Y.L. Chen, Y. Li, Q. Gong, Phys. Rev. A 83 (2011) 023803. [24] S.M. Spillane, T.J. Kippenberg, K.J. Vahala, K.W. Goh, E. Wilcut, H.J. Kimble, Phys. Rev. A 71 (2005) 013817. [25] S.M. Spillane, T.J. Kippenberg, O.J. Painter, K.J. Vahala, Phys. Rev. Lett. 91 (2003) 043902. [26] T. Aoki, B. Dayan, E. Wilcut, W.P. Bowen, A.S. Parkins, T.J. Kippenberg, K.J. Vahala, H.J. Kimble, Nature (London) 443 (2006) 671.Also see supplementary information for observation of strong coupling between one atom and a monolithic microresonator [27] B. Dayan, A.S. Parkins, T. Aoki, E.P. Ostby, K.J. Vahala, H.J. Kimble, Science 319 (2008) 1062.Also see supporting online material for a photon turnstile dynamically regulated by one atom. [28] T. Aoki, A.S. Parkins, D.J. Alton, C.A. Regal, B. Dayan, E. Ostby, K.J. Vahala, H.J. Kimble, Phys. Rev. Lett. 102 (2009) 083601. [29] F.Y. Hong, S.J. Xiong, Phys. Rev. A 78 (2008) 013812. [30] Y.F. Xiao, Z.F. Han, G.C. Guo, Phys. Rev. A 73 (2006) 052324. [31] J.S. Jin, C.S. Yu, P. Pei, H.S. Song, Phys. Rev. A 81 (2010) 042309. [32] W. Yao, R.B. Liu, L.J. Sham, Phys. Rev. Lett. 95 (2005) 030504. [33] W. Yao, R.B. Liu, L.J. Sham, J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S318. [34] J.I. Cirac, P. Zoller, H.J. Kimble, H. Mabuchi, Phys. Rev. Lett. 78 (1997) 3221. [35] L.M. Duan, A. Kuzmich, H.J. Kimble, Phys. Rev. A 67 (2003) 032305. [36] E. Waks, J. Vuckovic, Phys. Rev. A 73 (2006) 041803(R). [37] Y. Wu, J. Saldana, Y. Zhu, Phys. Rev. A 67 (2003) 013811. [38] Y. Wu, Phys. Rev. A 71 (2005) 053820. [39] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, 1997. [40] D. Walls, G. Milburm, Quantum Optics, Springer, Berlin, 1994. [41] C.W. Gardiner, P. Zoller, Quantum Noise, 3rd ed. Springer, Berlin, 2004.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Optical transmission gratings by one driven three-level atom and a microtoroidal resonator Rong Yu a,⁎, Jiahua Li b, Min Liu c, Chunling Ding b, Xiaoxue Yang b a b c
School of Science, Hubei Province Key Laboratory of Intelligent Robot, Wuhan Institute of Technology, Wuhan 430073, People's Republic of China Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People's Republic of China School of Computer Science and Technology, Xiantao Vocational College, Xiantao 433000, People's Republic of China
a r t i c l e
i n f o
Article history: Received 26 May 2011 Received in revised form 30 June 2011 Accepted 28 July 2011 Available online 16 August 2011 Keywords: Transmission gratings Microtoroidal resonator Tapered fiber Strong interaction
a b s t r a c t Based on the strong coherent interaction between a three-level ladder-type atom and a fiber-taper-coupled microtoroidal resonator, we present a scheme for optical transmission gratings. Using experimentally accessible parameters, it is shown that alternating regions of high transmission and absorption can be created in the fibertaper channel by spatially modulating an external coupling field. The model shows an obvious effect which has a direct analogy with the phenomenon of electromagnetically induced grating (EIG) in quantum systems. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Electromagnetically induced grating (EIG) [1], which is attributed to destructive quantum interference, is an interesting phenomenon where the absorption of a weak probe beam coupled to an atomic transition can be spatially modulated (reduced absorption at the peaks of the standing-wave field and high absorption at the nodes) by applying a strong standing wave that is coupled to another atomic transition in a three-level atomic system. EIG was originally observed in cold three-level atomic vapors [2]. Afterward, it was demonstrated that such an EIG effect can be used to store probe pulses in a vapor of rubidium atoms [3], to achieve tunable photonic band gap [4], to devise a dynamic controlled cavity [5], to implement optical routing [6,7], and so on. However, as shown in [1], low efficiency of the EIG due to weak interactions between single atoms and photons in a three-level atomic medium restricts its practical uses. It is worth pointing out that, on account of both highly confined microscale mode volume and ultrahigh quality factors [8–10], optical microresonators enable strong light-matter interactions as well as drastic reductions of the power necessary to observe strong nonlinear effects [11–14]. As a result, in the past few years optical microresonators are increasingly gaining interest in many diverse areas of research, ranging from nanophotonics [15,16] and biochemical sensing [17–19] to cavity quantum electrodynamics (CQED) [20,21]. Optical microresonators support whispering-gallery modes (WGMs). Different from the standing modes in a conventional Fabry–Perot (F–P) cavity, WGMs are a ⁎ Corresponding author. Tel./fax: + 86 2787557477. E-mail addresses: [email protected] (R. Yu), [email protected] (J. Li). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.07.066
type of traveling modes. In other words, WGM microresonators typically support two counterpropagating modes, i.e., clockwise (CW) and counterclockwise (CCW) propagating modes, with the same polarization and a degenerate frequency. This degeneracy can be lifted, and it can form a doublet through backscattering coupling induced by internal defect centers or surface roughness [22,23]. This phenomenon is known as modal coupling. Atoms in the vicinity of the resonator are able to interact with the two WGMs via the evanescent field. With the help of the fiber taper, the efficiency for coupling the quantum fields into and out of the microtoroidal resonator can approach 0.99–0.999 [24,25]. Also, strong coupling between a single cesium atom and the electromagnetic mode in a microtoroid has been theoretically investigated and experimentally observed [26]. Under a certain condition, the atom can transfer its excitation to the CW or CCW mode which is intrinsic in the microtoroidal resonators. In view of this, making use of the above-mentioned coherent interactions between the microtoroidal resonator and atoms, some schemes about photon turnstiles [27], photon routers [28], singlephoton transistors [29] and quantum controlled-phase-flip gates [30,31] have been put forward. It is worth pointing out that, in the investigation of Ref. [27], Dayan et al. have addressed that, with quantum critical coupling of input lights into and out of a microtoroidal resonator, a single cesium atom near the surface of the resonator can dynamically control the cavity output depending on the photon number at the input. Based on these achieved advances, we demonstrate that such a system can be employed to act as a kind of optical transmission gratings, analogous to an EIG in a three-level atomic system [1]. Using experimentally accessible parameters, alternating regions of high transmission and absorption can be caused by an external standing-wave coupling field.
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exclusively) to only one of the two normal modes [26,27]. Hereafter, we assume the position of the atom such that kz =nπ (n is integer), i.e., gB =0, then the normal mode Bˆ is decoupled from the interaction with the atom. Alternatively, the upmost state |e〉 of the three-level ladder-type atom (with the energy ωe) is coupled to the state |r〉 by a classical standing-wave coupling field with central frequency ωsw and position-dependent Rabi frequency Ωsw(x). The interactions between a high-Q microresonators (or microcavities) and a three-level system have been intensively studied previously [29,32–36] but in different contexts. Following the method developed in Refs. [26,27], the total Hamiltonian describing the interaction of a microtoroidal resonator with a three-level ladder-type atom and with a tapered fiber (see Fig. 1) can be written in the form
Due to its robustness against major experimental imperfections together with the high efficiency for coupling of the quantum fields into and out of the resonator with a fiber taper [24,25], this work might broaden variety applications in imaging techniques and precise measurements compared with EIG in a three-level atomic system [1]. On the other hand, in contrast to traditional F–P cavities, microtoroidal resonator structures possess advantages in easy integration and fabrication. They will likely be valuable for potential device applications. The remainder of this paper is arranged into three parts as follows. In Section 2, we establish the physical model and its theoretical description. By solving the coupled amplitude equations of motion for the microtoroidal resonator, the tapered fiber and the three-level ladder-type atom in the frequency domain, we derive explicit analytical expressions of the forward- and backward-propagating transmission functions for the output fields. In Section 3, we devote to analyzing and demonstrating in details optical transmission gratings in this device. At the same time, we also present the principal mechanism behind the transmission grating. Finally, our main conclusions are summarized in Section 4.
ˆ = ˆ ˆ †A H h = ωr jr〉〈r j + ωe je〉〈ej + ðωc + hÞ A +∞ † † ˆ −i jg〉〈r j A ˆ† +∫ ω aˆ1ω aˆ1ω + aˆ2ω aˆ2ω dω + gA i jr〉〈g j A h −∞ i + iΩsw ðxÞe−iωsw t je〉〈r j−iΩsw ðxÞeiωsw t jr〉〈ej ð1Þ + ∞ rffiffiffiffiffiffiffi κex † † ˆ ˆ i aˆ1ω A−i A aˆ1ω dω +∫ 2πffi −∞ rffiffiffiffiffiffi +∞ κex † ˆ −i A ˆ †aˆ2ω dω; +∫ i aˆ2ω A 2π −∞
2. Model and equations Fig. 1 is a schematic description of the composite system, which consists of a microtoroidal resonator, a tapered fiber, and a three-level ladder-type atom. A microtoroidal resonator has two internal counterpropagating modes which are described in terms of the annihilation ˆ with a common frequency ωc in the absence of operators aˆ and b scattering [27]. These two modes are coupled to each other in the presence of scattering with a strength that is parameterized by h. The intracavity field decays at a rate κ=κi +κex, where κi and κex describe intrinsic losses and extrinsic loss due to adjustable interaction with the modes of a tapered fiber. The intracavity fields are coupled to a tapered fiber with ˆ high efficiency ηN 0.99 [24,25]. The evanescent fields of modes aˆ and b
_ _ where the energy of state |g〉 has been set as zero. hωr and hωe are the energies of the atomic states |r〉 and |e〉, respectively. The symbols |m〉〈n| (m, n = g, e, r) for m ≠ n, are the atomic transition or projection operators between the states |m〉 and |n〉 involving the levels of the atom while |m〉〈m| represent the atomic population operators [see also the † bubble of Fig. 1]. Aˆ and Aˆ are the bosonic annihilation and creation operators of the normal mode with the frequency ωc + h, where ωc is the † “bare” cavity mode frequency. aˆ jω and aˆ jω ( j = 1, 2) are the annihilation and creation operators for the two modes of frequency ωiin the tapered h † ˆ jω ; aˆ ′ = δðω−ω′ Þ. fiber channel, with the commutation relations a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jω κex = 2π describes the coupling strength between the toroidal resonator mode and the tapered fiber mode. According to the spirit of Refs. [37,38], we choose the proper free Hamiltonian to transform to the interaction picture, that is,
have the coherent interactions with a ground state |g〉 and an upper excited state |r〉 with the energy ωr of a three-level ladder-type atom near the external surface of the microtoroidal resonator. It is convenient to describe the interaction by the normal modes of the microtoroidal ˆ and Bˆ = p1ffiffi a− ˆ with the coupling rates gA ˆ b resonator Aˆ = p1ffiffi aˆ + b 2
2
and gB, respectively, where gA, B ∼g0e− δρ{cos kz, sin kz} (δ∼ λ1) with ρ being the radial distance from the surface of the toroid to the atom, z being the position around the circumference of the resonator, and k being the vacuum wave vector. As a result, depending on the position of the threelevel ladder-type atom, coupling may occur predominantly (or even
α1in
ˆ
ˆ
ð2Þ
ˆA ˆ + ðωc + hÞ jr〉〈r j + ðωc + h + ωsw Þje〉〈ej ˆ0 = H h = ðωc + hÞ A †
α1out
Tapered fiber
α 2out
ˆ res e−i H0 t = h ; ˆ int = ei H0 t = h H H
κ ex
α 2in
x ⊗
y
z Three-level atom
a
h
ρ
e
Δ sw Atom
Ω sw ( x ) r
ΔA
κi b
gA g
Microtoroidal resonator Fig. 1. A schematic of the microtoroidal resonator and fiber taper system. The tapered fiber and the resonator are in the critical coupling and the atom is located with well-defined ^ of the microtoroidal resonator with strength gA and azimuthal phase kz = nπ (n is integer) [18,27]. In this case, the atomic transition |g〉 ↔ |r〉 is coupled to the normal mode A frequency ωc + h (ωc is the “bare” cavity mode frequency). At the same time, the |r〉 ↔ |e〉 transition is driven by a classical standing-wave coupling field with position-dependent Rabi frequency Ωsw(x) and central frequency ωsw. The standing-wave coupling field is aligned along the x axis perpendicular to the y–z plane.
R. Yu et al. / Optics Communications 284 (2011) 5263–5268 +∞
+
∫−∞
ˆ†1ω aˆ1ω + a ˆ†2ω aˆ2ω dω; ω a
ˆ + i j g〉〈r jA ˆ† ˆ res = h = ΔA j r〉〈r j + ðΔA + Δsw Þ j e〉〈ej + gA i jr〉〈g jA H ð3Þ h i −iω t iω t + iΩsw ðxÞe sw j e〉〈r j−iΩsw ðxÞe sw j r〉〈e j +∞rffiffiffiffiffiffiffi κex † ˆ ˆ †aˆ1ω dω ˆ 1ω A−i A ia +∫ 2π −∞
+
+ ∞ rffiffiffiffiffiffiffi κex
∫−∞
2π
†
ˆ A ˆ aˆ ˆ†2ω A−i ia 2ω dω;
ˆ ˆ † ð5Þ ˆ int = h = ΔA jr〉〈r j + ðΔA + Δsw Þje〉〈ej + gA ijr〉〈g jA−ijg〉〈r jA H + iΩsw ðxÞje〉〈r j−iΩsw ðxÞjr〉〈ej r ffiffiffiffiffiffi ffi +∞ i κex h † ˆ iðω−ωc −hÞt ˆ †aˆ e−iðω−ωc −hÞt dω iaˆ1ω Ae +∫ −i A 1ω 2π −∞ + ∞ rffiffiffiffiffiffiffih i κex † ˆ iðω−ωc −hÞt ˆ †a ˆ Ae ˆ2ω e−iðω−ωc −hÞt dω; ia +∫ −i A 2π 2ω −∞ It should be pointed out that the coupled system described by this Hamiltonian, under optical excitation and with the resonator-atom and resonator-fiber interaction, has two invariant Hilbert subspaces, with the bases |g, 0〉|vac〉 and |r, 0〉|vac〉, |e, 0〉|vac〉, |g, 1〉|vac〉, |g, 0〉|ω〉, respectively, where in |s〉|n〉, s = g, e, r denotes the state of three-level ladder-type atom and n denotes the number of photons in the normal ˆ |ω〉 denotes the one-photon Fock state of the tapered fiber mode A, channel mode of frequency ω, |vac〉 describes the vacuum state of the fiber mode. So the evolution of the whole system can be generally described by the wave function |Ψ(t)〉 = Cg|g, 0〉|vac〉 + Ce|Ψ e(t)〉 in the interaction picture, where j Ψe ðt Þ〉 = βr ðt Þ jr; 0〉jvac〉 + βe ðt Þje; 0〉jvac〉 + βc ðt Þjg; 1〉jvac〉 + ∞h i +∫ α1ω ðt Þaˆ†1ω + α2ω ðt Þaˆ†2ω jg; 0〉jvac〉dω:
ð6Þ
−∞
Firstly, by making use of the interaction Hamiltonian operator (5) e ∂jΨ ðt Þ〉 = H ˆ int jΨe ðt Þ〉 and the well-known Schrödinger equation ih ∂t in the interaction picture, the time evolution of the amplitudes for the microtoroidal resonator and tapered fiber modes are
α˙ jω ðt Þ =
rffiffiffiffiffiffiffi κex 2π
+∞
∫−∞
rffiffiffiffiffiffiffi κex iðω−ωc −hÞt e βc ðt Þ 2π
−iðω−ωc −hÞt
e
½α1ω ðt Þ + α2ω ðt Þdω;
ð11Þ
κ pffiffiffiffiffiffiffi i −κex βc ðt Þ− κex ½α1out ðt Þ + α2out ðt Þ; 2
ð12Þ
ð8Þ
ð j = 1; 2Þ;
rffiffiffiffiffiffiffi κex t iðω−ωc −hÞt ′ ∫ e α jω ðt Þ = α jω ðt0 Þ + βc ðt ′ Þdt ′ 2π t0 rffiffiffiffiffiffiffi κex t1 iðω−ωc −hÞt ′ ∫ e βc ðt ′ Þdt ′ 2π t
ðt0 < t Þ;
ðt1 > t Þ;
+∞
1
where α jin ðt Þ=pffiffiffiffiffiffi ∫−∞ e−iΔωt αjω ðt0 ÞdΔω with t0 → − ∞ and αjout ðt Þ= 2π
+ ∞ −iΔωt 1 pffiffiffiffiffiffi ∫ e αjω ðt1 ÞdΔω with t1 →+∞ (j=1, 2, Δω=ω−ωc −h) are 2π −∞
the input and output field operators in the tapered fiber channel, respectively. Combining Eqs. (11), (12) and the symmetry of the tapered fiber modes, we can achieve the input–output formalism α1out −α1in = pffiffiffiffiffiffi pffiffiffiffiffiffi κex βc and α2out −α2in = κex βc [40,41]. Secondly, by considering the spontaneous emission rates of the three-level ladder-type atom within the Weisskopf–Wigner approximation [39–41], similarly we can obtain the time evolution of the amplitudes for the three-level atomic states as follows h γi β˙ e ðt Þ = − iðΔA + Δsw Þ + e βe ðt Þ + Ωsw ðxÞβr ðt Þ; 2
ð13Þ
γ β˙ r ðt Þ = − iΔA + r βr ðt Þ + gA βc ðt Þ−Ωsw ðxÞβe ðt Þ; 2
ð14Þ
where γr and γe denote the decay rates of the atomic states |r〉 and |e〉, respectively. In what follows, we are interested in the transmission properties in the frequency domain of the coupled microresonator system, therefore we 1 + ∞ perform the Fourier transformations F ðΔωÞ = pffiffiffiffiffiffi ∫−∞ F ðt ÞeiΔωt dt on 2π the amplitudes βc, βe and βr in Eqs. (11), (13) and (14), respectively. After carrying out some algebraic calculations, the analytical solution of the amplitude βc can be found as pffiffiffiffiffiffi κex ðα1in + α2in Þ
βc ðxÞ = iΔω−κex −
κi + 2
ð7Þ
Integrating Eq. (8) formally yields
α jω ðt Þ = α jω ðt1 Þ−
κ pffiffiffiffiffiffi β˙ c ðt Þ = −g A βr ðt Þ− i + κex βc ðt Þ− κex ½α1in ðt Þ + α2in ðt Þ; 2
ð4Þ
where ΔA = ωr − ωc − h and Δsw = ωe − ωr − ωsw are the detunings of the normal mode Aˆ and the standing-wave coupling field from the corresponding atomic transitions |g〉 ↔ |r〉 and |r〉 ↔ |e〉, respectively. The resulting interaction Hamiltonian in the interaction picture can be then reexpressed as follows
β˙ c ðt Þ = −gA βr ðt Þ−
procedures of the Weisskopf–Wigner approximation [39–41]), we can obtain
= −gA βr ðt Þ−
ð9Þ
ð10Þ
Now, by substituting the expressions of αjω(t) from Eqs. (9) and (10) into Eq. (7) as well as by taking the intrinsic decay rates κi of the normal mode Aˆ into account (which can be obtained by following the established
5265
igA2
2 γ jΩsw ðxÞj − Δω−ΔA + i r + γ 2 Δω−ΔA −Δsw + i e 2 ð15Þ
pffiffiffiffiffiffiffi According to the input–output formalism α1out −α1in = κex βc pffiffiffiffiffiffiffi and α2out −α2in = κex βc , finally we can readily derive the relationship between the input optical field and the output optical field κi igA2 + γ 2 − Δω−ΔA + i r + Ssw ðxÞ 2 α1out ðxÞ = α1in κi igA2 + iΔω−κex − γ 2 − Δω−ΔA + i r + Ssw ðxÞ 2 κex α2in ; + κ igA2 iΔω−κex − i + γ 2 − Δω−ΔA + i r + Ssw ðxÞ 2 ð16Þ iΔω−
:
κex α1in 2 κi igA + iΔω−κex − γ 2 − Δω−ΔA + i r + Ssw ðxÞ 2 2 κi igA + iΔω− γ 2 − Δω−ΔA + i r + Ssw ðxÞ 2 α2in ; + κi igA2 + iΔω−κex − γ 2 − Δω−ΔA + i r + Ssw ðxÞ 2 ð17Þ
where we have introduced the new definition Ssw ðxÞ=
jΩsw ðxÞj
2
Δω−ΔA −Δsw + i
γe : 2
The real part of Ssw stands for the optical Stark shift created by the coherent interaction between the standing-wave coupling field and the atomic transition |e〉 ↔ |r〉. However, the imaginary part of Ssw is the noise term which arises from the decay of the atom. The imaginary part would disappear when γe = 0 and in this case the component of Ssw reduces the Stark shift. Consequently, when the coupling field is applied, the atom would be detuned from the microtoroidal resonator by an optical Stark shift Ssw, and it leads to a significant change of the forward- and backward-propagating output fields α1out and α2out, as will be shown in Section 3. For the initial given input fields α1in ≠0 and α2in =0 in the tapered fiber channel, from Eqs. (16) and (17) the transmission functions of forward- and backward-propagating output fields can be respectively defined by 2 2 α α T1 ðxÞ = 1out ; T2 ðxÞ = 2out : α1in α1in
ð18Þ
Since the transmission functions of forward- and backwardpropagating output fields in the tapered fiber depend strongly on the strength of the coupling field, they are expected to change periodically as the standing wave changes from nodes to antinodes across x dimension. 3. Realization of optical transmission gratings In the following calculations, according to Refs. [27,29], we consider an impedance-matched input (that is, critical coupling) for which κex is set to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cr κex = κi2 + h2 and the case of small intrinsic losses, i.e., κi ≪κ≃κex ≃h (κ=κi +κex). In this case, the system parameters for the microtoroidal resonator, fiber taper and cesium atom under study are taken as: cr gA =70 MHz, κi =5 MHz, h=250 MHz, κex =κ ex ≃250 MHz, γr =1 MHz, and γe =0.5 MHz, respectively. Before proceeding further, it is instructive to briefly analyze the principal mechanism behind the transmission gratings in our studied system. It follows from Eqs. (16) and (17) that the laser-induced Stark shift Ssw plays an important role in forming the alternating regions of high transmission and absorption in the fiber-taper channel that act as a type of transmission gratings. To illustrate this point, in Fig. 2(a) and (b) we compare the forward and backward transmission line shapes in the absence of the coupling field (solid lines) with the ones in the presence of a strong resonant coupling field (dashed lines). We further assume that the intensity of the coupling field which is used for producing the dashed lines in Fig. 2(a) and (b), corresponds to the peak intensity of a standing-wave field. The forward transmission T1 increases rapidly and the backward transmission T2 decreases quickly when the standingwave coupling field Ωsw(x) is located at the node (Ωsw(x) = 0, see solid lines) for frequencies lying near the resonance point Δω = 0, with T1(Δω= 0)≈ 0.95 whereas T2(Δω= 0)≈ 0. Instead, when the standingwave coupling field Ωsw(x) is located at the antinode (Ωsw(x) = 10 MHz, see dashed lines), the transmission T1 decreases rapidly and T2 increases quickly near the resonance point, where T1(Δω = 0) ≈ 0 whereas T2(Δω= 0)≈ 0.90. Overall, for the situations considered in Fig. 2(a)
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and (b), conditioned on the position of the standing-wave coupling field Ωsw(x), the transmission characteristics of the coupled resonator system can transmit from those (solid lines) for the case Ωsw(x) = 0 at the position of node to those (dashed lines) for Ωsw(x) = 10 MHz at the position of antinode. That is to say, a standing-wave coupling field triggers the change {T1(Δω= 0)≈ 0.95, T2(Δω= 0)≈ 0} for Ωsw(x) = 0 ⇔ {T1(Δω= 0)≈ 0, T2(Δω = 0) ≈ 0.90} for Ωsw(x) ≠ 0. Consequently, the position-dependent coupling field is sufficient to well control the propagation of the input signal field in the tapered fiber channel, and the system can be used as a substantial intensity modulation. In addition, the mechanism behind the transmission gratings in our studied system shows an obvious effect which has a direct analogy with the phenomenon of EIG in a three-level atomic medium. The forward and backward transmission functions T1(x) and T2(x) as a function of position x are plotted in Fig. 3 for the case of resonant coupling of the standing wave (Δsw =0). At the transverse locations around the nodes of the standing wave, the coupling field intensity is very weak. Under this circumstance, the intensity transmission T1 for the forward flux reaches to 0.95 while the intensity transmission T2 for the backward flux reaches to 0. In contrast, at the transverse locations around the antinodes of the standing wave, the coupling field intensity is quite strong, and the intensity transmission T1 for the forward flux quickly decreases to a zero value (T1 =0) due to the optical Stark shift Ssw. On the other hand, the
R. Yu et al. / Optics Communications 284 (2011) 5263–5268
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4. Conclusion In summary, we have theoretically studied optical transmission characteristics of the composite system, which are composed of a microtoroidal resonator, a tapered fiber, and a three-level ladder-type atom driven by an external standing-wave coupling field. Due to interaction with the coupling field in this system, the phenomenon of the optical Stark shift can arise. We have shown that, by spatially modulating the on-resonant coupling field, alternating regions of high transmission and absorption can be produced in the fiber-taper
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intensity transmission T2 for the backward flux rapidly increases to a peak value of T2 =0.90. This leads to a periodic intensity modulation across the beam profiles of forward- and backward-propagating output fields, a phenomenon reminiscent of optical intensity transmission gratings. Fig. 4 compares the modulation depth of the transmission functions in the situation where the detuning of the standing-wave coupling field Δsw is 5 MHz with the one in Fig. 3 where Δsw is zero (other parameters are the same). It can be seen from Fig. 4 that the modulation depth of the transmission functions is considerably decreased when the standingwave coupling field is detuned off the resonance. To illustrate the influences of the detuning Δsw on the transmission functions T1(x) and T2(x) in the forward and backward fluxes, the result is plotted in Fig. 5. By analyzing the result of Fig. 5(a) and (b), a small intensity modulation of the transmission functions appearing in Fig. 4 can be well understood when the detuning of the standing-wave coupling field is introduced.
Fig. 4. Normalized forward- and backward-propagating transmission functions T1(x) and T2(x) for the output fields as a function of position x. The system parameters used are the same as in Fig. 3 except for Δsw = 5 MHz.
Normalized transmission T 1
Fig. 3. Normalized forward- and backward-propagating transmission functions T1(x) and T2(x) for the output fields as a function of position x. The standing-wave coupling πx , where Λ is the spatial frequency of the standing wave. The field: Ωsw ðxÞ = Ωsw 0 sin Λ system parameters are chosen as Ω0sw = 10 MHz, Λ = 10 μm, gA = 70 MHz, κi = 5 MHz, κex = 250 MHz, γr = 1 MHz, γe = 0.5 MHz, Δω = 0, and ΔA = Δsw = 0, respectively.
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Fig. 5. Normalized forward- and backward-propagating transmissions T1 and T2 for the output fields versus the frequency detuning Δsw of the standing-wave coupling field. The solid lines are produced without the coupling field (Ωsw(x) = 0). The dashed ones are produced with the coupling field (Ωsw(x) = 10 MHz). Note that, the intensity of the coupling field which is used for producing the dashed lines in (a) and (b), corresponds to the peak intensity of a standing-wave field. The other system parameters are chosen as gA = 70 MHz, κi = 5 MHz, κex = 250 MHz, γr = 1 MHz, γe = 0.5 MHz, and ΔA = 0, respectively.
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channel which can be used for implementing a kind of optical intensity transmission gratings. Combined with the strong coherent interaction between a three-level ladder-type atom and a fiber-tapercoupled microtoroidal resonator as well as its robustness against major experimental imperfections [27], the microtoroidal resonatorbased scheme proposed here may provide a way to achieve the design of microresonator-based devices for applications requiring the grating effect. Acknowledgments We would like to thank Professor Ying Wu for his encouragement and helpful discussion. This research was supported in part by the National Natural Science Foundation of China under grant nos. 11004069, 10975054 and 91021011, by the Doctoral Foundation of the Ministry of Education of China under grant no. 20100142120081 and by the Fundamental Research Funds from Huazhong University of Science and Technology (HUST) under grant no. 2010MS074. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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