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Hybrid transparency effect in the drop-filter cavity–waveguide system
Optics Communications 427 (2018) 363–368
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Hybrid transparency effect in the drop-filter cavity–waveguide system Zhao-Hui Peng a, *, Chun-Xia Jia a,b , Yu-Qing Zhang a , Zhong-Hua Zhu a , Xiao-Juan Liu a a
Institute of Modern Physics and Department of Physics, Hunan University of Science and Technology, Xiangtan 411201, PR China Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, and Department of Physics, Hunan Normal University, Changsha 410081, PR China b
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Keywords: Electromagnetically induced transparency Dipole induced transparency Hybrid transparency
ABSTRACT We theoretically investigate hybrid transparency effect, which incorporates both dipole-induced transparency (DIT) and all-optical electromagnetically induced transparency (EIT)-like effects, in the drop-filter cavity– waveguide system. It is shown that the hybrid transparency effect originates from not only destructive interference of the cavity fields but also interference between DIT and all-optical EIT-like effects, thus the disappearance and revival phenomena of transparency windows occur. Especially, the transmission amplitude of hybrid transparency effect is even larger than that of all-optical EIT-like effect in the case of large intrinsic loss of microcavity. Benefiting from the hybrid transparency effect, photonic Stern–Gerlach-like and Faraday rotation effects can be realizable in multiple regimes, which may be useful for hybrid quantum information processing with photon and solid state qubit.
1. Introduction Electromagnetically induced transparency (EIT) effect [1], which was firstly found in the atomic system, is crucial for coherent manipulation of light. Up to now, the analog transparency effects have already been theoretically proposed and experimentally demonstrated in many other systems, e.g. quantum dots [2], superconducting circuits [3], metamaterials [4], nanoplasmonics [5], optomechanics [6] and alloptical coupled resonators [7–9]. Drop-filter cavity–waveguide system, which have been experimentally demonstrated in two-dimensional photonic crystal [10] and whispering-gallery microcavity [11–13], is also one of the most promising platforms for studying photon transport, quantum sensing and quantum information processing (QIP) etc. In Ref. [14], Waks and Vuckovic have proposed dipole induced transparency (DIT) effect in the drop-filter cavity–waveguide system. The distinct advantage of DIT effect is that it only requires large Purcell factor of cavity quantum electrodynamics (QED) system, thus allows the system work in the bad cavity regime, which greatly relaxes the experimental requirement of cavity QED system. DIT effect is feasible with present technology and has already been experimentally demonstrated in the photonic crystal system [15,16]. If we consider the polarization degree of freedom (DoF) of incident photon, the drop-filter cavity–waveguide system with dipole emitter can readily modulate both the amplitude and phase of incident photon. The amplitude modulation feature can be utilized to split polarized light beam, thus functions as polarized *
beam splitter or photonic Stern–Gerlach apparatus [17], which may be useful for generating spatial entanglement of photon and related QIP [18,19]. In the case of perfect or balanced transmission and reflection of incident photon, photonic conditional phase shift or photonic Faraday rotation effect [20,21] may be obtained, which can also be useful for QIP [22] and quantum computation [23,24]. On the other hand, all-optical EIT-like effect has also been theoretically investigated with cascaded resonators in the drop-filter cavity–waveguide system [25,26], then experimentally demonstrated [7–9]. The all-optical EIT-like effect can overcome much of the limitations on decoherence and bandwidth from atomic states for EIT, which may be useful for stopping and trapping light at room temperature. The waveguide-based devices are convenient for on-chip integration and can also exhibit strong light–matter interaction [27], thus onchip quantum circuits with solid state resonator and qubit are highly desirable for coherent manipulation of photon and QIP. Based on DIT [14] and all-optical EIT-like effects [25,26], we theoretically investigate hybrid transparency phenomenon in the drop-filter cavity–waveguide system. It is shown that the locations of hybrid transparency windows have obvious shifts and the full-widths at half maximum of the dip (FWHMs) of hybrid transparency windows are narrower than those of DIT and all-optical EIT-like effects. Moreover, the amplitude of hybrid transparency peak behaves as the property of collapse and revival with increasing Purcell factor and decreasing detuning of cavity resonance frequency, which cannot be explained within the framework of DIT
Corresponding author. E-mail address: [email protected] (Z. Peng).
https://doi.org/10.1016/j.optcom.2018.06.081 Received 27 February 2018; Received in revised form 18 June 2018; Accepted 28 June 2018 Available online 3 July 2018 0030-4018/© 2018 Elsevier B.V. All rights reserved.
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Optics Communications 427 (2018) 363–368
state, which allows us to replace dipole population operator with its expectation value ⟨𝜎𝑧 ⟩ ≈ −1. Furthermore, we assume that the external noise input fields of dipole emitter and microcavities are in the vacuum state and the contributions of noise input operators are negligible (i.e., ⟨𝑏𝑖𝑛 (𝑡)⟩ = 0, ⟨𝑒1,𝑖𝑛 (𝑡)⟩ = ⟨𝑒2,𝑖𝑛 (𝑡)⟩ = 0). The output fields into the waveguides are related to the input fields by the input–output relations [31] √ 𝜅1 𝑎1 (𝑡),
(4)
√ 𝑎3,𝑖𝑛 (𝑡) = 𝑎2,𝑖𝑛 (𝑡) + 𝜅1 𝑎2 (𝑡),
(5)
√ 𝑎1,𝑜𝑢𝑡 (𝑡) = 𝑎2,𝑜𝑢𝑡 (𝑡) + 𝜅2 𝑎1 (𝑡),
(6)
√ 𝑎2,𝑜𝑢𝑡 (𝑡) = 𝑎3,𝑜𝑢𝑡 (𝑡) + 𝜅2 𝑎2 (𝑡).
(7)
𝑎2,𝑖𝑛 (𝑡) = 𝑎1,𝑖𝑛 (𝑡) + Fig. 1. (a) The schematic of hybrid transparency effect in the drop-filter cavity– waveguide system. (b) The energy-level configuration of dipole emitter. The transition |𝑔𝐿(𝑅) ⟩ ↔ |𝑒⟩ is driven by left (right) circularly polarized light.
effect or all-optical EIT-like effect. In the case of large intrinsic loss of microcavity, the transmission amplitude of hybrid transparency effect is even larger than that of all-optical EIT-like effect. Therefore, the hybrid transparency effect originates from not only destructive interference of the cavity fields but also interference between DIT and all-optical EITlike effects. Benefiting from the hybrid transparency effect, photonic Stern–Gerlach-like effect and photonic Faraday rotation effect can be realized in multiple regimes, which may be useful for QIP and quantum networks in the future. This paper is organized as follows. In Section 2, we present the theoretical model and its solution, then discuss the hybrid transparency phenomenon in Section 3. In Section 4, we investigate the potential application of hybrid transparency effect, e.g. photonic Stern–Gerlach-like effect and photonic Faraday rotation effect. Finally, we briefly discuss the feasibility of hybrid transparency effect.
We assume that a weak monochromatic field with frequency 𝜔𝑝 input from the left port of upper waveguide as shown in Fig. 1(a), and the input fields from other ports of waveguides are discarded. After neglecting the contribution of 𝑎3,𝑜𝑢𝑡 (𝑡), we can obtain the output fields into the waveguides as follows 𝑎3,𝑖𝑛 (𝑡) = {1 +
𝑎1,𝑜𝑢𝑡 (𝑡) =
The schematic of hybrid transparency effect is illustrated in Fig. 1(a), where two microcavities 1 and 2 with single modes 𝑎1 and 𝑎2 are evanescently coupled to two drop-filter waveguides. The dipole emitter with transition frequency 𝜔𝑎 is evanescently coupled to the first microcavity, and their interaction is descried by the Jaynes–Cummings model Hamiltonian. The Heisenberg–Langevin equations of cavity modes are (ℏ = 1)
𝜅 + 𝜅2 + 𝜅0 𝑑𝑎2 (𝑡) = −𝑖(𝜔2 − 𝜔𝑝 )𝑎2 (𝑡) − 1 𝑎2 (𝑡) 𝑑𝑡 2 √ √ √ − 𝜅1 𝑎2,𝑖𝑛 (𝑡) − 𝜅2 𝑎3,𝑜𝑢𝑡 (𝑡) − 𝜅0 𝑒2,𝑖𝑛 (𝑡),
𝑡= (1) 𝑟=
−
√ 𝜅1 𝜅2 [(𝑖𝛥′1 − (𝑖𝛥′1
−
+ 𝜅1 ) + (𝑖𝛥2 − 𝜅 )(𝑖𝛥2 2
−
𝜅 2
+ 𝜅2 )]
𝜅 ) − 𝜅1 𝜅2 2
𝜅 + 𝜅1 ) + (𝑖𝛥2 − 𝜅2 2 𝜅 )(𝑖𝛥2 − 𝜅2 ) − 𝜅1 𝜅2 2
+ 𝜅2 )]
}𝑎1,𝑖𝑛 (𝑡),
(8)
𝑎1,𝑖𝑛 (𝑡),
(9)
1 , 1 − 𝜅1 [(𝑖𝛥′1 − 𝜅0 ∕2)−1 + (𝑖𝛥2 − 𝜅0 ∕2)−1 ] 𝜅1 [(𝑖𝛥′1 − 𝜅0 ∕2)−1 + (𝑖𝛥2 − 𝜅0 ∕2)−1 ] 1 − 𝜅1 [(𝑖𝛥′1 − 𝜅0 ∕2)−1 + (𝑖𝛥2 − 𝜅0 ∕2)−1 ]
.
(10)
(11)
(2) 3. Hybrid transparency effect
where 𝜔𝑝 , 𝜔𝑖 are the frequency of input field and the central frequencies of two microcavities respectively. 𝜅𝑖 is the cavity–waveguide coupling strength, and 𝜅0 is the intrinsic loss of microcavity. 𝑎𝑖,𝑖𝑛 and 𝑎𝑖,𝑜𝑢𝑡 describe the input and output fields of 𝑖th microcavity. 𝜙1 denotes phase delay along the waveguides between microcavities 1 and 2, and it determines the indirectly coupling strength of adjacent microcavities [28,29]. The asymmetric Fano lineshape can be obtained for 𝜙1 ≠ 𝑚𝜋∕2 with 𝑚 being an integral number [28,29], and here we only consider exp(𝑖𝜙1 ) = 1 by choosing appropriate distance of microcavities as that in the alloptical EIT-like effect [26]. 𝑔 and 𝜎− are the vacuum Rabi frequency and lower operator of dipole emitter. 𝑒𝑖,𝑖𝑛 (𝑡) is the noise input operator of 𝑖th microcavity which preserves the canonical commutation relation. The motion equation of dipole lower operator is 𝛾 𝑑𝜎− (𝑡) = −𝑖(𝜔𝑎 − 𝜔𝑝 )𝜎− (𝑡) − 𝑎 𝜎− (𝑡) 𝑑𝑡 2 √ − 𝑔𝑎1 (𝑡)𝜎𝑧 (𝑡) + 𝛾𝑎 𝜎𝑧 (𝑡)𝑏𝑖𝑛 (𝑡),
(𝑖𝛥′1
𝜅 2
where 𝜅 = 𝜅0 + 𝜅1 + 𝜅2 , 𝑖𝛥′1 = 𝑖𝛥1 + 𝑔 2 ∕(𝑖𝛥𝑎 − 𝛾𝑎 ∕2), 𝛥𝑖 = 𝜔𝑝 − 𝜔𝑖 and 𝛥𝑎 = 𝜔𝑝 − 𝜔𝑎 . Analog to the double-sided microcavity, we denote the output fields into upper and lower waveguides as transmission and reflection fields respectively, thus can obtain the transmission coefficient 𝑡 = 𝑎3,𝑖𝑛 (𝑡)∕𝑎1,𝑖𝑛 (𝑡) and reflection coefficient 𝑟 = 𝑎1,𝑜𝑢𝑡 (𝑡)∕𝑎1,𝑖𝑛 (𝑡). We must point out that the same transmission and reflection coefficients will be obtained if the incident photon inputs from the right port of upper waveguide. In other words, the system is reciprocal when the phase delay along the waveguides satisfies the condition exp(𝑖𝜙1 ) = 1. We assume that two microcavities have equal coupling rates with the waveguides (i.e., 𝜅1 = 𝜅2 ), which is known as the critical coupling condition. In this case, the transmission and reflection coefficients are
2. The model and solution
𝜅 + 𝜅2 + 𝜅0 𝑑𝑎1 (𝑡) = −𝑖(𝜔1 − 𝜔𝑝 )𝑎1 (𝑡) − 1 𝑎1 (𝑡) − 𝑔𝜎− (𝑡) 𝑑𝑡 2 √ √ 𝑖𝜙 √ − 𝜅1 𝑎1,𝑖𝑛 (𝑡) − 𝜅2 𝑒 1 𝑎2,𝑜𝑢𝑡 (𝑡) − 𝜅0 𝑒1,𝑖𝑛 (𝑡),
𝜅1 [(𝑖𝛥′1 −
In the following, we investigate the transmission properties of incident field from the left port of upper waveguide as shown in Fig. 1(a), and the numerical calculation is based on Eqs. (10) and (11). We consider the case where dipole emitter is resonant with the first microcavity (i.e., 𝜔1 = 𝜔𝑎 ) and the central frequencies of two microcavities are assumed to satisfy 𝜔2 − 𝜔1 = 𝛿 = 𝜅1 ∕2. The losses of microcavities to waveguides are far larger than their intrinsic loss, thus it is reasonable of considering the overcoupling regime 𝜅1 ≫ 𝜅0 , where the factor of 103 decrease in the quality factor relative to the original microcavity can be realizable (i.e., 𝜅0 ≈ 10−3 𝜅1 ) [11]. The dipole–microcavity coupling strength and dipole decay rate (𝑔, 𝛾𝑎 ) = (0.33, 10−3 )𝜅1 are taken into account [14]. In Fig. 2(a), it is shown that there are two sharp transparency windows (red solid) which incorporate both DIT (black dashdotted) and all-optical EIT-like effects (blue dotted). Compared with transparency windows in the DIT and all-optical EIT-like effects, the locations of transmission peaks, which are originally at 𝜔𝑝 − 𝜔1 = 0 √ and 𝜔𝑝 − 𝜔1 = 𝛿∕2, have obvious shifts to 𝜔𝑝 − 𝜔1 = (𝛿 − 𝛿 2 + 8𝑔 2 )∕4 √ and 𝜔𝑝 − 𝜔1 = (𝛿 + 𝛿 2 + 8𝑔 2 )∕4 respectively. In the bad cavity limit (𝜅1 , 𝛿 ≫ 𝑔), the locations of transmission peaks are 𝜔𝑝 − 𝜔1 = 𝑔 2 ∕𝛿 and
(3)
where 𝛾𝑎 and 𝑏𝑖𝑛 (𝑡) are spontaneous emission rate of dipole emitter and its noise input operator. In what follows, we consider the weak excitation limit [30], thus the dipole emitter is always in its ground 364
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expanding the FWHMs within the first order of 𝑔∕𝜅 in the bad cavity limit, we can find that they are also closely related with Purcell factor 𝐹𝑝 and detuning of cavity resonance frequency 𝛿. In Fig. 2(a), it is shown that the FWHMs of hybrid transparency windows are obviously narrower than those of DIT effect and all-optical EIT-like effect. As shown in Refs. [8,26], the FWHMs of transparency windows will become narrower when the interference increases, thus it is believed that the interference between DIT effect and all-optical EIT-like effect results in the narrowness of FWHMs of hybrid transparency effect. To investigate the effect of Purcell factor on hybrid transparency effect, we plot the power transmission |𝑡|2 for different Purcell factors in Fig. 3, and it is shown that the FWHMs of DIT window (black dashdotted) and the left window of hybrid transparency (red solid) improve with increasing Purcell factor. However, the amplitude of the right window of hybrid transparency (red solid) decreases with increasing Purcell factor as shown in Fig. 3(a)–(b). Especially, the right transparency window whose location of peak is supposed to be at 𝜔𝑝 − 𝜔1 = 𝛿, corresponding to the Purcell factor 𝐹𝑝 = 1000, disappears as shown in Fig. 3(b), because the term 𝜅1 (𝑖𝛥2 − 𝜅0 ∕2)−1 in Eq. (10) is far larger than 1 in this case. It is clear that the incident photon which is resonant with the second microcavity will be completely reflected by the dropfilter cavity–waveguide system. For Purcell factors 𝐹𝑝 = 1200, 1800, the right window of hybrid transparency is gradually reviving as shown in Fig. 3(c)–(d). The reviving phenomena of hybrid transparency window cannot be explained within the framework of DIT effect or all-optical EIT-like effect. Because the dipole–microcavity system is linearized in the weak excitation limit, it is believed that the strong destructive interference between DIT effect and all-optical EIT-like effect plays the key role in the revival phenomena when the transparency windows of DIT effect and all-optical EIT-like effect overlap as shown in Fig. 3(c)– (d). In Fig. 4, we plot the power transmission |𝑡|2 for different values of 𝛿, and investigate the effect of detuning of cavity resonance frequency on hybrid transparency effect. It is shown that all-optical EIT-like effect will be less obvious with the decrease of 𝛿 because the ratio of 𝜅0 and 𝜅1 becomes larger, which is consistent with the result in Ref. [26]. The behavior of right window of hybrid transparency effect is just the same as that of all-optical EIT-like effect as shown in Fig. 4(a)–(b). Especially, in Fig. 4(b) the right window of hybrid transparency effect disappears when the incident photon is resonant with the second microcavity. From Fig. 4(b)–(d), we can find that the right window of hybrid transparency
Fig. 2. (a) The power transmission |𝑡|2 and (b) phase 𝛷 for hybrid transparency effect (red solid), all-optical EIT-like effect (blue dotted) and DIT effect (black dashdotted) respectively. The dipole–microcavity parameters (𝑔, 𝛿, 𝜅𝑠 , 𝛾𝑎 ) = (0.33, 0.5, 10−3 , 10−3 )𝜅1 are taken into account.
𝜔𝑝 − 𝜔1 = 𝛿∕2 + 𝑔 2 ∕𝛿 in the first order of 𝑔∕𝛿. The shift of locations of peaks 𝑔 2 ∕𝛿 is closely related to both Purcell factor 𝐹𝑝 = 2𝑔 2 ∕(𝛾𝑎 𝜅) and detuning of cavity resonance frequency 2𝛿∕(𝛾𝑎 𝜅). In Refs. [14,26], it is shown that the transparency conditions for DIT and all-optical EITlike effects are simply because of large Purcell factor 𝐹𝑝 and detuning of cavity resonance frequency 𝛿. On the other hand, the transmission amplitude as shown in Eq. (10) depends on interference between the terms 𝜅1 (𝑖𝛥′1 − 𝜅0 ∕2)−1 and 𝜅1 (𝑖𝛥2 − 𝜅0 ∕2)−1 , which account for DIT and all-optical EIT-like effects respectively. The interference property relies on their relative amplitude and phase, and their amplitudes are just proportional to Purcell factor 𝐹𝑝 and detuning of cavity resonance frequency 𝛿 respectively. Therefore, the hybrid transparency effect not only originates from the destructive interference of the cavity fields as those in DIT and all-optical EIT-like effects, but also the interference between them. The √ FWHMs for DIT and√all-optical EIT-like effects are √ 𝜅 2 + 4𝑔 2 − 𝜅 and [ (𝛿 − 2𝜅)2 + 4𝜅𝛿 + (𝛿 + 2𝜅)2 − 4𝜅𝛿 − 4𝜅]∕2. After
Fig. 3. The power transmission |𝑡|2 for hybrid transparency effect (red solid), all-optical EIT-like effect (blue dotted) and DIT effect (black dashdotted) for Purcell factors 𝐹𝑝 = 200(𝑎), 1000(𝑏), 1200(𝑐), 1800(𝑑). The dipole–microcavity parameters (𝛿, 𝜅𝑠 , 𝛾𝑎 ) = (0.5, 10−3 , 10−3 )𝜅1 are taken into account. 365
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Fig. 4. The power transmission |𝑡|2 for hybrid transparency effect (red solid), all-optical EIT-like effect (blue dotted) and DIT effect (black dashdotted) for detuning of cavity resonance frequency 𝛿 = 𝜅1 (a), 0.5𝜅1 (b), 0.1𝜅1 (c), 0.05𝜅1 (d). The dipole–microcavity parameters (𝑔, 𝜅𝑠 , 𝛾𝑎 ) = (0.33, 10−3 , 10−3 )𝜅1 are taken into account.
effect will also revive with the decrease of cavity detuning 𝛿 which is similar to that of Fig. 3. Interestingly, the transmission amplitude of hybrid transparency effect is even larger than that of all-optical EIT-like effect as shown in Fig. 4(d), which is ascribed to destructive interference between DIT effect and all-optical EIT-like effect. Therefore, the revival phenomena of transparency window for large intrinsic loss of microcavity has potential application in the case of solid state resonator with low quality factor, which may greatly relax the experimental requirement. Furthermore, the FWHMs of second transparency peaks in Figs. 3 and 4 will also become narrower with increasing Purcell factor and detuning of cavity resonance frequency, which also reflects the interference between DIT effect and all-optical EIT-like effect.
the ground state |𝑔𝐿 ⟩. If the photons in the upper and lower waveguides are regarded as the spatial qubits |1⟩ and |2⟩, the evolution of system is
4. Photonic Stern–Gerlach effect and photonic Faraday rotation effect
Thus, the distinct advantage of photonic Stern–Gerlach effect in this system is that it can be realized in the multiple regimes, which results from the interference between DIT effect and all-optical EIT-like effect. However, photonic Stern–Gerlach effect may be realized only for single frequency of photon in the DIT effect. As a result, hyperentanglement [32] of photon as shown in Eqs. (12) and (13) can be generated for multiple frequencies of incident photon. Because hyperentanglement exists in more than one DoF of photon (i.e., polarization and spatial DoFs here), it may be useful for hybrid QIP [18,19]. The complex reflection and transmission coefficients mean that both reflection and transmission photons experience a phase shift, which is not only related with the frequency of incident photon but also its polarization, and the well-known photonic Faraday rotation effect [20,21] can be obtained. To explain the mechanism of photonic Faraday rotation effect in the drop-filter cavity–waveguide system, we consider the dipole emitter with two degenerate ground states (|𝑔𝐿 ⟩ and |𝑔𝑅 ⟩) as shown in Fig. 1(b). The transition of dipole emitter |𝑒⟩ ↔ |𝑔𝐿(𝑅) ⟩ is driven by left (right) circularly polarized photon, and the evolution of system can be described with transmission and reflection operators [21]. If the incident photon and dipole emitter are initially in the state √ (|𝐿⟩+|𝑅⟩)|𝑔𝐿 ⟩∕ 2, the states of transmission and reflection photons will be 1 𝑖𝜙 𝑖𝜙 |𝛹𝑡 ⟩ = √ (𝑒 𝑡1 |𝑡1 (𝜔)||𝐿⟩ + 𝑒 𝑡2 |𝑡2 (𝜔)||𝑅⟩), (14) 2 1 𝑖𝜙 𝑖𝜙 |𝛹𝑟 ⟩ = √ (𝑒 𝑟1 |𝑟1 (𝜔)||𝐿⟩ + 𝑒 𝑟2 |𝑟2 (𝜔)||𝑅⟩), (15) 2
1 1 √ (|𝐿⟩ + |𝑅⟩)|𝑔𝐿 ⟩|1⟩ ⟶ √ (|𝐿⟩|1⟩ + |𝑅⟩|2⟩)|𝑔𝐿 ⟩. 2 2
(12)
Evidently, the incident photon has been split into two waveguides according to its polarization, and photonic Stern–Gerlach effect is realized. If the frequency of incident photon satisfies 𝜔𝑝 = 𝜔1 + 𝛿∕2, we may also realize photonic Stern–Gerlach effect with the similar way which can be seen from Figs. 3(a) and 4(a), and the evolution can be described as follows 1 1 √ (|𝐿⟩ + |𝑅⟩)|𝑔𝐿 ⟩|1⟩ ⟶ √ (|𝐿⟩|2⟩ + |𝑅⟩|1⟩)|𝑔𝐿 ⟩. (13) 2 2
In this section, we investigate the potential application of hybrid transparency effect by considering the polarization DoFs of photon. From Eqs. (10) and (11), we can see that the transmission and reflection coefficients are complex, and they are closely related with both the frequency and polarization of incident photon. Thus, the drop-filter cavity–waveguide system can readily modulate the phase and amplitude of incident photon by its frequency and polarization. The transmission spectrum of hybrid transparency effect is related to the polarization of incident photon, and the well-known photonic Stern–Gerlach effect [17] and photonic Faraday rotation [20,21] can be obtained. The original photonic Stern–Gerlach effect is just the birefringence of quantized polarized light in a magneto-optically manipulated atomic ensemble as a generalized Stern–Gerlach effect of light [17]. In the dropfilter cavity–waveguide system, we can also obtain the polarizationdependent splitting of incident photon which is analogy to photonic Stern–Gerlach effect. If the incident photon with the frequency 𝜔𝑝 = √ 𝜔1 + (𝛿 − 𝛿 2 + 8𝑔 2 )∕4 is coupling with dipole emitter, the transmission and reflection amplitudes of hybrid transparency effect are |𝑡| ≈ 1 and |𝑟| ≈ 0 as shown in Figs. 2(a) and 5(a). However, the transmission and reflection amplitudes of all-optical EIT-like effect are |𝑡| ≈ 0 and |𝑟| ≈ 1 for the incident photon with the same frequency but orthogonal polarization. To explain the mechanism of photonic Stern–Gerlach effect in the drop-filter cavity–waveguide system, we assume√ that the incident photon is in the linearly polarized state (|𝐿⟩ + |𝑅⟩)∕ 2 (|𝐿⟩ and |𝑅⟩ denote left and right circularly polarized state) and dipole emitter is in
where subscripts 1 and 2 denote the cases of hybrid transparency effect and all-optical EIT-like effect respectively. In the balanced reflection or 366
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Fig. 7. The fidelities 𝐹1 (black) and 𝐹2 (red) of hyperentanglement against Purcell factor of system respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2
Fig. 5. The power reflection |𝑟| and phase 𝛷 for hybrid transparency effect (red solid) and all-optical EIT-like effect (blue dotted). The dipole–microcavity parameters (𝑔, 𝛿, 𝜅𝑠 , 𝛾𝑎 ) = (0.33, 0.5, 10−3 , 10−3 )𝜅1 are taken into account.
microcavity and fiber taper system [11,35,36]). For photonic crystal system, two linear three-hole defect cavities (L3) [37] but coupled to two photonic crystal waveguides are experimentally realizable [8]. The detuning of cavity resonance frequency 𝛿 and phase difference 𝜙1 can be controlled independently with two pump beams [8]. The negatively charged quantum dot, whose states include two ground states and two excited trion states, can be chosen as the dipole emitter embedded in the first microcavity [22,23]. Therefore, the schematic of hybrid transparency effect in Fig. 1(a) can be realizable with present experimental technology in the photonic crystal system. To evaluate the performance of hybrid transparency effect, we consider the dipole– microcavity parameters (𝜅, 𝜅𝑠 , 𝑔, 𝛾𝑎 ) = (35.9, 3.59, 10.2, 2.9) × 2𝜋 GHz which are achievable with the present experiment technology as shown in Ref. [22]. The Purcell factor of the dipole–microcavity system is 𝐹𝑝 ≈ 2, thus DIT effect and hybrid transparency effect can be realized. However, the amplitude of hybrid transparency window will be far less than 1. Moreover, the large dephasing in the negatively charged quantum dot may further result in the difficulty for experimental realization of hybrid transparency effect, although its coherence time may be improved with spin echo technology [22,23]. On the other hand, large Purcell factor of system is required for exhibiting the disappearance and revival phenomena of transparency window. The high quality whispering-gallery microcavity may be the potential candidate for implementing large Purcell factor because of its ultrahigh quality factor (∼ 108 ) and small modal volume [35,36]. The drop-filter cavity–waveguide systems can be constructed with two ultrahigh quality factor microtoroid cavities evanescently coupled to a pair of tapered optical fibers [27]. The nitrogen-vacancy (NV) center in a diamond, which is chosen as the dipole emitter, can be located on the surface of microtoroid cavity, and the parameters of system (𝑔, 𝜅, 𝜅𝑠 , 𝛾𝑎 ) = (180, 4.7, 2.35, 13)×2𝜋 MHz are reachable [36]. The related Purcell factor 𝐹𝑝 is on the order of 103 , thus the experimental conditions for photonic Stern–Gerlach effect (large Purcell factor) and photonic Faraday rotation effect (𝐹𝑝 ∼ 100 and 𝛿 ∼ 𝜅1 ) can also be fulfilled with present experimental technology. In Section 4, we have assumed that the incident photon will be completely transmitted or reflected in the case of large Purcell factor without considering the noise of system, and perfect hyperentanglement of photon can be generated. However, there are still some sources of noise, e.g. spontaneous emission of dipole emitter and photon loss in the microcavities and waveguides. After considering the experimental noise, the evolution of photon and dipole emitter in the realistic process may be approximately expressed as |𝑖⟩|𝑔𝐿 ⟩|1⟩ → 𝑡|𝑖⟩|𝑔𝐿 ⟩|1⟩ + 𝑟|𝑖⟩|𝑔𝐿 ⟩|2⟩ (𝑖 =
Fig. 6. The Faraday rotation angle 𝜃𝐹 against Purcell factor 𝐹𝑝 (a) (𝛿 = 𝜅1 ) and detuning of cavity resonance frequency 𝛿∕𝜅1 (b) (𝐹𝑝 = 100) for transmission photon (red solid) and reflection photon (blue dotted). The dipole–microcavity parameters (𝛿, 𝜅𝑠 , 𝛾𝑎 ) = (0.5, 10−3 , 10−3 )𝜅1 are taken into account.
transmission (i.e., |𝑡1 (𝜔)| = |𝑡2 (𝜔)| or |𝑟1 (𝜔)| = |𝑟2 (𝜔)|), the polarized directions of reflection and transmission photons occur a Faraday rotation angle 𝜃𝐹 = 𝜙𝑖2 −𝜙𝑖1 (𝑖 = 𝑡, 𝑟), which is a function of parameters of system. The key parameters of system are Purcell factor 𝐹𝑝 and detuning of cavity resonance frequency 𝛿, thus we plot Faraday rotation angle 𝜃𝐹 against 𝐹𝑝 and 𝛿 in Fig. 6. From Fig. 6(a) and (b), it is shown that Faraday rotation angles 𝜃𝐹 ≈ −𝜋∕2, −3𝜋∕2 can be obtained in the regimes of 𝐹𝑝 ∼ 100 and 𝛿 ∼ 𝜅1 for transmission and reflection photons respectively. Faraday rotation angles attained in the hybrid transparency effect are large enough to generate entanglement of photon and dipole emitter [21,22], and may be useful for hybrid entanglement concentration [33] and quantum logic gate [34]. 5. Discussions and conclusion We briefly discuss the experimental feasibility of hybrid transparency effect based on the recent experimental accessible technology. The drop-filter cavity–waveguide system may be realizable in the photonic crystal [8,15] or other solid state system (e.g. whispering-gallery 367
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Optics Communications 427 (2018) 363–368
𝐿, 𝑅). To calculate the fidelity of hyperentanglement, we define 𝐹 = |⟨𝜓𝑟𝑒𝑎𝑙 |𝜓𝑖𝑑𝑒𝑎𝑙 ⟩|2 where |𝜓𝑖𝑑𝑒𝑎𝑙 ⟩ and |𝜓𝑟𝑒𝑎𝑙 ⟩ are the output states in the ideal and realistic cases respectively. The fidelities of hyperentanglement as shown in Eqs. (12) and (13) are 𝐹1 = (|𝑡1 |2 + |𝑟2 |2 )∕2 and 𝐹2 = (|𝑟1 |2 + |𝑡2 |2 )∕2 where subscripts 1 and 2 denote the cases of hybrid transparency effect and all-optical EIT-like effect respectively. As the transmission and reflection amplitudes depend on Purcell factor of system, we plot the fidelities of hyperentanglement against Purcell factor in Fig. 7. It is shown that the fidelities of both hyperentanglement improve with the increase of Purcell factor and will be close to one, thus photonic Stern– Gerlach effect may be feasible for system with large Purcell factor. In conclusion, we have theoretically investigated the hybrid transparency effect in the drop-filter cavity–waveguide system. It is shown the interference between DIT effect and all-optical EIT-like effect plays the key role in the hybrid transparency effect. The amplitude of hybrid transparency window behaves as the property of disappearance and revival with increasing Purcell factor and decreasing detuning of cavity resonance frequency, which cannot be explained within the framework of DIT effect or all-optical EIT effect. In the case of large intrinsic loss of microcavity, the amplitude of hybrid transparency peak is even larger than that of all-optical EIT-like effect. Benefiting from the hybrid transparency effect, photonic Stern–Gerlach effect and photonic Faraday rotation effect can be realized in multiple regimes. In the future, the hybrid system we propose may be chosen as the building block for coherent manipulation of photon and hybrid QIP, and it may also be interesting to investigate the hybrid transparency effect in other coupled cavity QED systems.
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Acknowledgments Z. H. Peng , C. X. Jia and X. J. Liu were supported by the National Science Foundation of China (NSFC) under Grants No. 11405052 and Key Laboratory of Low Dimensional Quantum Structures and Quantum Control under Grants No. QSQC1409. Y. Q. Zhang and Z. H. Zhu were supported by NSFC under Grants Nos. 11504104 and 11704115. References [1] M. Fleischhauer, A. Imamoglu, J.P. Marangos, Rev. Modern Phys. 77 (2005) 633; S.E. Harris, J.E. Field, A. Imamoglu, Phys. Rev. Lett. 64 (1990) 1107. [2] X. Xu, B. Sun, P.R. Berman, D.G. Steel, A.S. Bracker, D. Gammon, L.J. Sham, Nat. Phys. 4 (2008) 692. [3] W.R. Kelly, Z. Dutton, J. Schlafer, B. Mookerji, T.A. Ohki, J.S. Kline, D.P. Pappas, Phys. Rev. Lett. 104 (2010) 163601.
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Hybrid transparency effect in the drop-filter cavity–waveguide system Zhao-Hui Peng a, *, Chun-Xia Jia a,b , Yu-Qing Zhang a , Zhong-Hua Zhu a , Xiao-Juan Liu a a
Institute of Modern Physics and Department of Physics, Hunan University of Science and Technology, Xiangtan 411201, PR China Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, and Department of Physics, Hunan Normal University, Changsha 410081, PR China b
ARTICLE
INFO
Keywords: Electromagnetically induced transparency Dipole induced transparency Hybrid transparency
ABSTRACT We theoretically investigate hybrid transparency effect, which incorporates both dipole-induced transparency (DIT) and all-optical electromagnetically induced transparency (EIT)-like effects, in the drop-filter cavity– waveguide system. It is shown that the hybrid transparency effect originates from not only destructive interference of the cavity fields but also interference between DIT and all-optical EIT-like effects, thus the disappearance and revival phenomena of transparency windows occur. Especially, the transmission amplitude of hybrid transparency effect is even larger than that of all-optical EIT-like effect in the case of large intrinsic loss of microcavity. Benefiting from the hybrid transparency effect, photonic Stern–Gerlach-like and Faraday rotation effects can be realizable in multiple regimes, which may be useful for hybrid quantum information processing with photon and solid state qubit.
1. Introduction Electromagnetically induced transparency (EIT) effect [1], which was firstly found in the atomic system, is crucial for coherent manipulation of light. Up to now, the analog transparency effects have already been theoretically proposed and experimentally demonstrated in many other systems, e.g. quantum dots [2], superconducting circuits [3], metamaterials [4], nanoplasmonics [5], optomechanics [6] and alloptical coupled resonators [7–9]. Drop-filter cavity–waveguide system, which have been experimentally demonstrated in two-dimensional photonic crystal [10] and whispering-gallery microcavity [11–13], is also one of the most promising platforms for studying photon transport, quantum sensing and quantum information processing (QIP) etc. In Ref. [14], Waks and Vuckovic have proposed dipole induced transparency (DIT) effect in the drop-filter cavity–waveguide system. The distinct advantage of DIT effect is that it only requires large Purcell factor of cavity quantum electrodynamics (QED) system, thus allows the system work in the bad cavity regime, which greatly relaxes the experimental requirement of cavity QED system. DIT effect is feasible with present technology and has already been experimentally demonstrated in the photonic crystal system [15,16]. If we consider the polarization degree of freedom (DoF) of incident photon, the drop-filter cavity–waveguide system with dipole emitter can readily modulate both the amplitude and phase of incident photon. The amplitude modulation feature can be utilized to split polarized light beam, thus functions as polarized *
beam splitter or photonic Stern–Gerlach apparatus [17], which may be useful for generating spatial entanglement of photon and related QIP [18,19]. In the case of perfect or balanced transmission and reflection of incident photon, photonic conditional phase shift or photonic Faraday rotation effect [20,21] may be obtained, which can also be useful for QIP [22] and quantum computation [23,24]. On the other hand, all-optical EIT-like effect has also been theoretically investigated with cascaded resonators in the drop-filter cavity–waveguide system [25,26], then experimentally demonstrated [7–9]. The all-optical EIT-like effect can overcome much of the limitations on decoherence and bandwidth from atomic states for EIT, which may be useful for stopping and trapping light at room temperature. The waveguide-based devices are convenient for on-chip integration and can also exhibit strong light–matter interaction [27], thus onchip quantum circuits with solid state resonator and qubit are highly desirable for coherent manipulation of photon and QIP. Based on DIT [14] and all-optical EIT-like effects [25,26], we theoretically investigate hybrid transparency phenomenon in the drop-filter cavity–waveguide system. It is shown that the locations of hybrid transparency windows have obvious shifts and the full-widths at half maximum of the dip (FWHMs) of hybrid transparency windows are narrower than those of DIT and all-optical EIT-like effects. Moreover, the amplitude of hybrid transparency peak behaves as the property of collapse and revival with increasing Purcell factor and decreasing detuning of cavity resonance frequency, which cannot be explained within the framework of DIT
Corresponding author. E-mail address: [email protected] (Z. Peng).
https://doi.org/10.1016/j.optcom.2018.06.081 Received 27 February 2018; Received in revised form 18 June 2018; Accepted 28 June 2018 Available online 3 July 2018 0030-4018/© 2018 Elsevier B.V. All rights reserved.
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Optics Communications 427 (2018) 363–368
state, which allows us to replace dipole population operator with its expectation value ⟨𝜎𝑧 ⟩ ≈ −1. Furthermore, we assume that the external noise input fields of dipole emitter and microcavities are in the vacuum state and the contributions of noise input operators are negligible (i.e., ⟨𝑏𝑖𝑛 (𝑡)⟩ = 0, ⟨𝑒1,𝑖𝑛 (𝑡)⟩ = ⟨𝑒2,𝑖𝑛 (𝑡)⟩ = 0). The output fields into the waveguides are related to the input fields by the input–output relations [31] √ 𝜅1 𝑎1 (𝑡),
(4)
√ 𝑎3,𝑖𝑛 (𝑡) = 𝑎2,𝑖𝑛 (𝑡) + 𝜅1 𝑎2 (𝑡),
(5)
√ 𝑎1,𝑜𝑢𝑡 (𝑡) = 𝑎2,𝑜𝑢𝑡 (𝑡) + 𝜅2 𝑎1 (𝑡),
(6)
√ 𝑎2,𝑜𝑢𝑡 (𝑡) = 𝑎3,𝑜𝑢𝑡 (𝑡) + 𝜅2 𝑎2 (𝑡).
(7)
𝑎2,𝑖𝑛 (𝑡) = 𝑎1,𝑖𝑛 (𝑡) + Fig. 1. (a) The schematic of hybrid transparency effect in the drop-filter cavity– waveguide system. (b) The energy-level configuration of dipole emitter. The transition |𝑔𝐿(𝑅) ⟩ ↔ |𝑒⟩ is driven by left (right) circularly polarized light.
effect or all-optical EIT-like effect. In the case of large intrinsic loss of microcavity, the transmission amplitude of hybrid transparency effect is even larger than that of all-optical EIT-like effect. Therefore, the hybrid transparency effect originates from not only destructive interference of the cavity fields but also interference between DIT and all-optical EITlike effects. Benefiting from the hybrid transparency effect, photonic Stern–Gerlach-like effect and photonic Faraday rotation effect can be realized in multiple regimes, which may be useful for QIP and quantum networks in the future. This paper is organized as follows. In Section 2, we present the theoretical model and its solution, then discuss the hybrid transparency phenomenon in Section 3. In Section 4, we investigate the potential application of hybrid transparency effect, e.g. photonic Stern–Gerlach-like effect and photonic Faraday rotation effect. Finally, we briefly discuss the feasibility of hybrid transparency effect.
We assume that a weak monochromatic field with frequency 𝜔𝑝 input from the left port of upper waveguide as shown in Fig. 1(a), and the input fields from other ports of waveguides are discarded. After neglecting the contribution of 𝑎3,𝑜𝑢𝑡 (𝑡), we can obtain the output fields into the waveguides as follows 𝑎3,𝑖𝑛 (𝑡) = {1 +
𝑎1,𝑜𝑢𝑡 (𝑡) =
The schematic of hybrid transparency effect is illustrated in Fig. 1(a), where two microcavities 1 and 2 with single modes 𝑎1 and 𝑎2 are evanescently coupled to two drop-filter waveguides. The dipole emitter with transition frequency 𝜔𝑎 is evanescently coupled to the first microcavity, and their interaction is descried by the Jaynes–Cummings model Hamiltonian. The Heisenberg–Langevin equations of cavity modes are (ℏ = 1)
𝜅 + 𝜅2 + 𝜅0 𝑑𝑎2 (𝑡) = −𝑖(𝜔2 − 𝜔𝑝 )𝑎2 (𝑡) − 1 𝑎2 (𝑡) 𝑑𝑡 2 √ √ √ − 𝜅1 𝑎2,𝑖𝑛 (𝑡) − 𝜅2 𝑎3,𝑜𝑢𝑡 (𝑡) − 𝜅0 𝑒2,𝑖𝑛 (𝑡),
𝑡= (1) 𝑟=
−
√ 𝜅1 𝜅2 [(𝑖𝛥′1 − (𝑖𝛥′1
−
+ 𝜅1 ) + (𝑖𝛥2 − 𝜅 )(𝑖𝛥2 2
−
𝜅 2
+ 𝜅2 )]
𝜅 ) − 𝜅1 𝜅2 2
𝜅 + 𝜅1 ) + (𝑖𝛥2 − 𝜅2 2 𝜅 )(𝑖𝛥2 − 𝜅2 ) − 𝜅1 𝜅2 2
+ 𝜅2 )]
}𝑎1,𝑖𝑛 (𝑡),
(8)
𝑎1,𝑖𝑛 (𝑡),
(9)
1 , 1 − 𝜅1 [(𝑖𝛥′1 − 𝜅0 ∕2)−1 + (𝑖𝛥2 − 𝜅0 ∕2)−1 ] 𝜅1 [(𝑖𝛥′1 − 𝜅0 ∕2)−1 + (𝑖𝛥2 − 𝜅0 ∕2)−1 ] 1 − 𝜅1 [(𝑖𝛥′1 − 𝜅0 ∕2)−1 + (𝑖𝛥2 − 𝜅0 ∕2)−1 ]
.
(10)
(11)
(2) 3. Hybrid transparency effect
where 𝜔𝑝 , 𝜔𝑖 are the frequency of input field and the central frequencies of two microcavities respectively. 𝜅𝑖 is the cavity–waveguide coupling strength, and 𝜅0 is the intrinsic loss of microcavity. 𝑎𝑖,𝑖𝑛 and 𝑎𝑖,𝑜𝑢𝑡 describe the input and output fields of 𝑖th microcavity. 𝜙1 denotes phase delay along the waveguides between microcavities 1 and 2, and it determines the indirectly coupling strength of adjacent microcavities [28,29]. The asymmetric Fano lineshape can be obtained for 𝜙1 ≠ 𝑚𝜋∕2 with 𝑚 being an integral number [28,29], and here we only consider exp(𝑖𝜙1 ) = 1 by choosing appropriate distance of microcavities as that in the alloptical EIT-like effect [26]. 𝑔 and 𝜎− are the vacuum Rabi frequency and lower operator of dipole emitter. 𝑒𝑖,𝑖𝑛 (𝑡) is the noise input operator of 𝑖th microcavity which preserves the canonical commutation relation. The motion equation of dipole lower operator is 𝛾 𝑑𝜎− (𝑡) = −𝑖(𝜔𝑎 − 𝜔𝑝 )𝜎− (𝑡) − 𝑎 𝜎− (𝑡) 𝑑𝑡 2 √ − 𝑔𝑎1 (𝑡)𝜎𝑧 (𝑡) + 𝛾𝑎 𝜎𝑧 (𝑡)𝑏𝑖𝑛 (𝑡),
(𝑖𝛥′1
𝜅 2
where 𝜅 = 𝜅0 + 𝜅1 + 𝜅2 , 𝑖𝛥′1 = 𝑖𝛥1 + 𝑔 2 ∕(𝑖𝛥𝑎 − 𝛾𝑎 ∕2), 𝛥𝑖 = 𝜔𝑝 − 𝜔𝑖 and 𝛥𝑎 = 𝜔𝑝 − 𝜔𝑎 . Analog to the double-sided microcavity, we denote the output fields into upper and lower waveguides as transmission and reflection fields respectively, thus can obtain the transmission coefficient 𝑡 = 𝑎3,𝑖𝑛 (𝑡)∕𝑎1,𝑖𝑛 (𝑡) and reflection coefficient 𝑟 = 𝑎1,𝑜𝑢𝑡 (𝑡)∕𝑎1,𝑖𝑛 (𝑡). We must point out that the same transmission and reflection coefficients will be obtained if the incident photon inputs from the right port of upper waveguide. In other words, the system is reciprocal when the phase delay along the waveguides satisfies the condition exp(𝑖𝜙1 ) = 1. We assume that two microcavities have equal coupling rates with the waveguides (i.e., 𝜅1 = 𝜅2 ), which is known as the critical coupling condition. In this case, the transmission and reflection coefficients are
2. The model and solution
𝜅 + 𝜅2 + 𝜅0 𝑑𝑎1 (𝑡) = −𝑖(𝜔1 − 𝜔𝑝 )𝑎1 (𝑡) − 1 𝑎1 (𝑡) − 𝑔𝜎− (𝑡) 𝑑𝑡 2 √ √ 𝑖𝜙 √ − 𝜅1 𝑎1,𝑖𝑛 (𝑡) − 𝜅2 𝑒 1 𝑎2,𝑜𝑢𝑡 (𝑡) − 𝜅0 𝑒1,𝑖𝑛 (𝑡),
𝜅1 [(𝑖𝛥′1 −
In the following, we investigate the transmission properties of incident field from the left port of upper waveguide as shown in Fig. 1(a), and the numerical calculation is based on Eqs. (10) and (11). We consider the case where dipole emitter is resonant with the first microcavity (i.e., 𝜔1 = 𝜔𝑎 ) and the central frequencies of two microcavities are assumed to satisfy 𝜔2 − 𝜔1 = 𝛿 = 𝜅1 ∕2. The losses of microcavities to waveguides are far larger than their intrinsic loss, thus it is reasonable of considering the overcoupling regime 𝜅1 ≫ 𝜅0 , where the factor of 103 decrease in the quality factor relative to the original microcavity can be realizable (i.e., 𝜅0 ≈ 10−3 𝜅1 ) [11]. The dipole–microcavity coupling strength and dipole decay rate (𝑔, 𝛾𝑎 ) = (0.33, 10−3 )𝜅1 are taken into account [14]. In Fig. 2(a), it is shown that there are two sharp transparency windows (red solid) which incorporate both DIT (black dashdotted) and all-optical EIT-like effects (blue dotted). Compared with transparency windows in the DIT and all-optical EIT-like effects, the locations of transmission peaks, which are originally at 𝜔𝑝 − 𝜔1 = 0 √ and 𝜔𝑝 − 𝜔1 = 𝛿∕2, have obvious shifts to 𝜔𝑝 − 𝜔1 = (𝛿 − 𝛿 2 + 8𝑔 2 )∕4 √ and 𝜔𝑝 − 𝜔1 = (𝛿 + 𝛿 2 + 8𝑔 2 )∕4 respectively. In the bad cavity limit (𝜅1 , 𝛿 ≫ 𝑔), the locations of transmission peaks are 𝜔𝑝 − 𝜔1 = 𝑔 2 ∕𝛿 and
(3)
where 𝛾𝑎 and 𝑏𝑖𝑛 (𝑡) are spontaneous emission rate of dipole emitter and its noise input operator. In what follows, we consider the weak excitation limit [30], thus the dipole emitter is always in its ground 364
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expanding the FWHMs within the first order of 𝑔∕𝜅 in the bad cavity limit, we can find that they are also closely related with Purcell factor 𝐹𝑝 and detuning of cavity resonance frequency 𝛿. In Fig. 2(a), it is shown that the FWHMs of hybrid transparency windows are obviously narrower than those of DIT effect and all-optical EIT-like effect. As shown in Refs. [8,26], the FWHMs of transparency windows will become narrower when the interference increases, thus it is believed that the interference between DIT effect and all-optical EIT-like effect results in the narrowness of FWHMs of hybrid transparency effect. To investigate the effect of Purcell factor on hybrid transparency effect, we plot the power transmission |𝑡|2 for different Purcell factors in Fig. 3, and it is shown that the FWHMs of DIT window (black dashdotted) and the left window of hybrid transparency (red solid) improve with increasing Purcell factor. However, the amplitude of the right window of hybrid transparency (red solid) decreases with increasing Purcell factor as shown in Fig. 3(a)–(b). Especially, the right transparency window whose location of peak is supposed to be at 𝜔𝑝 − 𝜔1 = 𝛿, corresponding to the Purcell factor 𝐹𝑝 = 1000, disappears as shown in Fig. 3(b), because the term 𝜅1 (𝑖𝛥2 − 𝜅0 ∕2)−1 in Eq. (10) is far larger than 1 in this case. It is clear that the incident photon which is resonant with the second microcavity will be completely reflected by the dropfilter cavity–waveguide system. For Purcell factors 𝐹𝑝 = 1200, 1800, the right window of hybrid transparency is gradually reviving as shown in Fig. 3(c)–(d). The reviving phenomena of hybrid transparency window cannot be explained within the framework of DIT effect or all-optical EIT-like effect. Because the dipole–microcavity system is linearized in the weak excitation limit, it is believed that the strong destructive interference between DIT effect and all-optical EIT-like effect plays the key role in the revival phenomena when the transparency windows of DIT effect and all-optical EIT-like effect overlap as shown in Fig. 3(c)– (d). In Fig. 4, we plot the power transmission |𝑡|2 for different values of 𝛿, and investigate the effect of detuning of cavity resonance frequency on hybrid transparency effect. It is shown that all-optical EIT-like effect will be less obvious with the decrease of 𝛿 because the ratio of 𝜅0 and 𝜅1 becomes larger, which is consistent with the result in Ref. [26]. The behavior of right window of hybrid transparency effect is just the same as that of all-optical EIT-like effect as shown in Fig. 4(a)–(b). Especially, in Fig. 4(b) the right window of hybrid transparency effect disappears when the incident photon is resonant with the second microcavity. From Fig. 4(b)–(d), we can find that the right window of hybrid transparency
Fig. 2. (a) The power transmission |𝑡|2 and (b) phase 𝛷 for hybrid transparency effect (red solid), all-optical EIT-like effect (blue dotted) and DIT effect (black dashdotted) respectively. The dipole–microcavity parameters (𝑔, 𝛿, 𝜅𝑠 , 𝛾𝑎 ) = (0.33, 0.5, 10−3 , 10−3 )𝜅1 are taken into account.
𝜔𝑝 − 𝜔1 = 𝛿∕2 + 𝑔 2 ∕𝛿 in the first order of 𝑔∕𝛿. The shift of locations of peaks 𝑔 2 ∕𝛿 is closely related to both Purcell factor 𝐹𝑝 = 2𝑔 2 ∕(𝛾𝑎 𝜅) and detuning of cavity resonance frequency 2𝛿∕(𝛾𝑎 𝜅). In Refs. [14,26], it is shown that the transparency conditions for DIT and all-optical EITlike effects are simply because of large Purcell factor 𝐹𝑝 and detuning of cavity resonance frequency 𝛿. On the other hand, the transmission amplitude as shown in Eq. (10) depends on interference between the terms 𝜅1 (𝑖𝛥′1 − 𝜅0 ∕2)−1 and 𝜅1 (𝑖𝛥2 − 𝜅0 ∕2)−1 , which account for DIT and all-optical EIT-like effects respectively. The interference property relies on their relative amplitude and phase, and their amplitudes are just proportional to Purcell factor 𝐹𝑝 and detuning of cavity resonance frequency 𝛿 respectively. Therefore, the hybrid transparency effect not only originates from the destructive interference of the cavity fields as those in DIT and all-optical EIT-like effects, but also the interference between them. The √ FWHMs for DIT and√all-optical EIT-like effects are √ 𝜅 2 + 4𝑔 2 − 𝜅 and [ (𝛿 − 2𝜅)2 + 4𝜅𝛿 + (𝛿 + 2𝜅)2 − 4𝜅𝛿 − 4𝜅]∕2. After
Fig. 3. The power transmission |𝑡|2 for hybrid transparency effect (red solid), all-optical EIT-like effect (blue dotted) and DIT effect (black dashdotted) for Purcell factors 𝐹𝑝 = 200(𝑎), 1000(𝑏), 1200(𝑐), 1800(𝑑). The dipole–microcavity parameters (𝛿, 𝜅𝑠 , 𝛾𝑎 ) = (0.5, 10−3 , 10−3 )𝜅1 are taken into account. 365
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Fig. 4. The power transmission |𝑡|2 for hybrid transparency effect (red solid), all-optical EIT-like effect (blue dotted) and DIT effect (black dashdotted) for detuning of cavity resonance frequency 𝛿 = 𝜅1 (a), 0.5𝜅1 (b), 0.1𝜅1 (c), 0.05𝜅1 (d). The dipole–microcavity parameters (𝑔, 𝜅𝑠 , 𝛾𝑎 ) = (0.33, 10−3 , 10−3 )𝜅1 are taken into account.
effect will also revive with the decrease of cavity detuning 𝛿 which is similar to that of Fig. 3. Interestingly, the transmission amplitude of hybrid transparency effect is even larger than that of all-optical EIT-like effect as shown in Fig. 4(d), which is ascribed to destructive interference between DIT effect and all-optical EIT-like effect. Therefore, the revival phenomena of transparency window for large intrinsic loss of microcavity has potential application in the case of solid state resonator with low quality factor, which may greatly relax the experimental requirement. Furthermore, the FWHMs of second transparency peaks in Figs. 3 and 4 will also become narrower with increasing Purcell factor and detuning of cavity resonance frequency, which also reflects the interference between DIT effect and all-optical EIT-like effect.
the ground state |𝑔𝐿 ⟩. If the photons in the upper and lower waveguides are regarded as the spatial qubits |1⟩ and |2⟩, the evolution of system is
4. Photonic Stern–Gerlach effect and photonic Faraday rotation effect
Thus, the distinct advantage of photonic Stern–Gerlach effect in this system is that it can be realized in the multiple regimes, which results from the interference between DIT effect and all-optical EIT-like effect. However, photonic Stern–Gerlach effect may be realized only for single frequency of photon in the DIT effect. As a result, hyperentanglement [32] of photon as shown in Eqs. (12) and (13) can be generated for multiple frequencies of incident photon. Because hyperentanglement exists in more than one DoF of photon (i.e., polarization and spatial DoFs here), it may be useful for hybrid QIP [18,19]. The complex reflection and transmission coefficients mean that both reflection and transmission photons experience a phase shift, which is not only related with the frequency of incident photon but also its polarization, and the well-known photonic Faraday rotation effect [20,21] can be obtained. To explain the mechanism of photonic Faraday rotation effect in the drop-filter cavity–waveguide system, we consider the dipole emitter with two degenerate ground states (|𝑔𝐿 ⟩ and |𝑔𝑅 ⟩) as shown in Fig. 1(b). The transition of dipole emitter |𝑒⟩ ↔ |𝑔𝐿(𝑅) ⟩ is driven by left (right) circularly polarized photon, and the evolution of system can be described with transmission and reflection operators [21]. If the incident photon and dipole emitter are initially in the state √ (|𝐿⟩+|𝑅⟩)|𝑔𝐿 ⟩∕ 2, the states of transmission and reflection photons will be 1 𝑖𝜙 𝑖𝜙 |𝛹𝑡 ⟩ = √ (𝑒 𝑡1 |𝑡1 (𝜔)||𝐿⟩ + 𝑒 𝑡2 |𝑡2 (𝜔)||𝑅⟩), (14) 2 1 𝑖𝜙 𝑖𝜙 |𝛹𝑟 ⟩ = √ (𝑒 𝑟1 |𝑟1 (𝜔)||𝐿⟩ + 𝑒 𝑟2 |𝑟2 (𝜔)||𝑅⟩), (15) 2
1 1 √ (|𝐿⟩ + |𝑅⟩)|𝑔𝐿 ⟩|1⟩ ⟶ √ (|𝐿⟩|1⟩ + |𝑅⟩|2⟩)|𝑔𝐿 ⟩. 2 2
(12)
Evidently, the incident photon has been split into two waveguides according to its polarization, and photonic Stern–Gerlach effect is realized. If the frequency of incident photon satisfies 𝜔𝑝 = 𝜔1 + 𝛿∕2, we may also realize photonic Stern–Gerlach effect with the similar way which can be seen from Figs. 3(a) and 4(a), and the evolution can be described as follows 1 1 √ (|𝐿⟩ + |𝑅⟩)|𝑔𝐿 ⟩|1⟩ ⟶ √ (|𝐿⟩|2⟩ + |𝑅⟩|1⟩)|𝑔𝐿 ⟩. (13) 2 2
In this section, we investigate the potential application of hybrid transparency effect by considering the polarization DoFs of photon. From Eqs. (10) and (11), we can see that the transmission and reflection coefficients are complex, and they are closely related with both the frequency and polarization of incident photon. Thus, the drop-filter cavity–waveguide system can readily modulate the phase and amplitude of incident photon by its frequency and polarization. The transmission spectrum of hybrid transparency effect is related to the polarization of incident photon, and the well-known photonic Stern–Gerlach effect [17] and photonic Faraday rotation [20,21] can be obtained. The original photonic Stern–Gerlach effect is just the birefringence of quantized polarized light in a magneto-optically manipulated atomic ensemble as a generalized Stern–Gerlach effect of light [17]. In the dropfilter cavity–waveguide system, we can also obtain the polarizationdependent splitting of incident photon which is analogy to photonic Stern–Gerlach effect. If the incident photon with the frequency 𝜔𝑝 = √ 𝜔1 + (𝛿 − 𝛿 2 + 8𝑔 2 )∕4 is coupling with dipole emitter, the transmission and reflection amplitudes of hybrid transparency effect are |𝑡| ≈ 1 and |𝑟| ≈ 0 as shown in Figs. 2(a) and 5(a). However, the transmission and reflection amplitudes of all-optical EIT-like effect are |𝑡| ≈ 0 and |𝑟| ≈ 1 for the incident photon with the same frequency but orthogonal polarization. To explain the mechanism of photonic Stern–Gerlach effect in the drop-filter cavity–waveguide system, we assume√ that the incident photon is in the linearly polarized state (|𝐿⟩ + |𝑅⟩)∕ 2 (|𝐿⟩ and |𝑅⟩ denote left and right circularly polarized state) and dipole emitter is in
where subscripts 1 and 2 denote the cases of hybrid transparency effect and all-optical EIT-like effect respectively. In the balanced reflection or 366
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Fig. 7. The fidelities 𝐹1 (black) and 𝐹2 (red) of hyperentanglement against Purcell factor of system respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2
Fig. 5. The power reflection |𝑟| and phase 𝛷 for hybrid transparency effect (red solid) and all-optical EIT-like effect (blue dotted). The dipole–microcavity parameters (𝑔, 𝛿, 𝜅𝑠 , 𝛾𝑎 ) = (0.33, 0.5, 10−3 , 10−3 )𝜅1 are taken into account.
microcavity and fiber taper system [11,35,36]). For photonic crystal system, two linear three-hole defect cavities (L3) [37] but coupled to two photonic crystal waveguides are experimentally realizable [8]. The detuning of cavity resonance frequency 𝛿 and phase difference 𝜙1 can be controlled independently with two pump beams [8]. The negatively charged quantum dot, whose states include two ground states and two excited trion states, can be chosen as the dipole emitter embedded in the first microcavity [22,23]. Therefore, the schematic of hybrid transparency effect in Fig. 1(a) can be realizable with present experimental technology in the photonic crystal system. To evaluate the performance of hybrid transparency effect, we consider the dipole– microcavity parameters (𝜅, 𝜅𝑠 , 𝑔, 𝛾𝑎 ) = (35.9, 3.59, 10.2, 2.9) × 2𝜋 GHz which are achievable with the present experiment technology as shown in Ref. [22]. The Purcell factor of the dipole–microcavity system is 𝐹𝑝 ≈ 2, thus DIT effect and hybrid transparency effect can be realized. However, the amplitude of hybrid transparency window will be far less than 1. Moreover, the large dephasing in the negatively charged quantum dot may further result in the difficulty for experimental realization of hybrid transparency effect, although its coherence time may be improved with spin echo technology [22,23]. On the other hand, large Purcell factor of system is required for exhibiting the disappearance and revival phenomena of transparency window. The high quality whispering-gallery microcavity may be the potential candidate for implementing large Purcell factor because of its ultrahigh quality factor (∼ 108 ) and small modal volume [35,36]. The drop-filter cavity–waveguide systems can be constructed with two ultrahigh quality factor microtoroid cavities evanescently coupled to a pair of tapered optical fibers [27]. The nitrogen-vacancy (NV) center in a diamond, which is chosen as the dipole emitter, can be located on the surface of microtoroid cavity, and the parameters of system (𝑔, 𝜅, 𝜅𝑠 , 𝛾𝑎 ) = (180, 4.7, 2.35, 13)×2𝜋 MHz are reachable [36]. The related Purcell factor 𝐹𝑝 is on the order of 103 , thus the experimental conditions for photonic Stern–Gerlach effect (large Purcell factor) and photonic Faraday rotation effect (𝐹𝑝 ∼ 100 and 𝛿 ∼ 𝜅1 ) can also be fulfilled with present experimental technology. In Section 4, we have assumed that the incident photon will be completely transmitted or reflected in the case of large Purcell factor without considering the noise of system, and perfect hyperentanglement of photon can be generated. However, there are still some sources of noise, e.g. spontaneous emission of dipole emitter and photon loss in the microcavities and waveguides. After considering the experimental noise, the evolution of photon and dipole emitter in the realistic process may be approximately expressed as |𝑖⟩|𝑔𝐿 ⟩|1⟩ → 𝑡|𝑖⟩|𝑔𝐿 ⟩|1⟩ + 𝑟|𝑖⟩|𝑔𝐿 ⟩|2⟩ (𝑖 =
Fig. 6. The Faraday rotation angle 𝜃𝐹 against Purcell factor 𝐹𝑝 (a) (𝛿 = 𝜅1 ) and detuning of cavity resonance frequency 𝛿∕𝜅1 (b) (𝐹𝑝 = 100) for transmission photon (red solid) and reflection photon (blue dotted). The dipole–microcavity parameters (𝛿, 𝜅𝑠 , 𝛾𝑎 ) = (0.5, 10−3 , 10−3 )𝜅1 are taken into account.
transmission (i.e., |𝑡1 (𝜔)| = |𝑡2 (𝜔)| or |𝑟1 (𝜔)| = |𝑟2 (𝜔)|), the polarized directions of reflection and transmission photons occur a Faraday rotation angle 𝜃𝐹 = 𝜙𝑖2 −𝜙𝑖1 (𝑖 = 𝑡, 𝑟), which is a function of parameters of system. The key parameters of system are Purcell factor 𝐹𝑝 and detuning of cavity resonance frequency 𝛿, thus we plot Faraday rotation angle 𝜃𝐹 against 𝐹𝑝 and 𝛿 in Fig. 6. From Fig. 6(a) and (b), it is shown that Faraday rotation angles 𝜃𝐹 ≈ −𝜋∕2, −3𝜋∕2 can be obtained in the regimes of 𝐹𝑝 ∼ 100 and 𝛿 ∼ 𝜅1 for transmission and reflection photons respectively. Faraday rotation angles attained in the hybrid transparency effect are large enough to generate entanglement of photon and dipole emitter [21,22], and may be useful for hybrid entanglement concentration [33] and quantum logic gate [34]. 5. Discussions and conclusion We briefly discuss the experimental feasibility of hybrid transparency effect based on the recent experimental accessible technology. The drop-filter cavity–waveguide system may be realizable in the photonic crystal [8,15] or other solid state system (e.g. whispering-gallery 367
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𝐿, 𝑅). To calculate the fidelity of hyperentanglement, we define 𝐹 = |⟨𝜓𝑟𝑒𝑎𝑙 |𝜓𝑖𝑑𝑒𝑎𝑙 ⟩|2 where |𝜓𝑖𝑑𝑒𝑎𝑙 ⟩ and |𝜓𝑟𝑒𝑎𝑙 ⟩ are the output states in the ideal and realistic cases respectively. The fidelities of hyperentanglement as shown in Eqs. (12) and (13) are 𝐹1 = (|𝑡1 |2 + |𝑟2 |2 )∕2 and 𝐹2 = (|𝑟1 |2 + |𝑡2 |2 )∕2 where subscripts 1 and 2 denote the cases of hybrid transparency effect and all-optical EIT-like effect respectively. As the transmission and reflection amplitudes depend on Purcell factor of system, we plot the fidelities of hyperentanglement against Purcell factor in Fig. 7. It is shown that the fidelities of both hyperentanglement improve with the increase of Purcell factor and will be close to one, thus photonic Stern– Gerlach effect may be feasible for system with large Purcell factor. In conclusion, we have theoretically investigated the hybrid transparency effect in the drop-filter cavity–waveguide system. It is shown the interference between DIT effect and all-optical EIT-like effect plays the key role in the hybrid transparency effect. The amplitude of hybrid transparency window behaves as the property of disappearance and revival with increasing Purcell factor and decreasing detuning of cavity resonance frequency, which cannot be explained within the framework of DIT effect or all-optical EIT effect. In the case of large intrinsic loss of microcavity, the amplitude of hybrid transparency peak is even larger than that of all-optical EIT-like effect. Benefiting from the hybrid transparency effect, photonic Stern–Gerlach effect and photonic Faraday rotation effect can be realized in multiple regimes. In the future, the hybrid system we propose may be chosen as the building block for coherent manipulation of photon and hybrid QIP, and it may also be interesting to investigate the hybrid transparency effect in other coupled cavity QED systems.
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Acknowledgments Z. H. Peng , C. X. Jia and X. J. Liu were supported by the National Science Foundation of China (NSFC) under Grants No. 11405052 and Key Laboratory of Low Dimensional Quantum Structures and Quantum Control under Grants No. QSQC1409. Y. Q. Zhang and Z. H. Zhu were supported by NSFC under Grants Nos. 11504104 and 11704115. References [1] M. Fleischhauer, A. Imamoglu, J.P. Marangos, Rev. Modern Phys. 77 (2005) 633; S.E. Harris, J.E. Field, A. Imamoglu, Phys. Rev. Lett. 64 (1990) 1107. [2] X. Xu, B. Sun, P.R. Berman, D.G. Steel, A.S. Bracker, D. Gammon, L.J. Sham, Nat. Phys. 4 (2008) 692. [3] W.R. Kelly, Z. Dutton, J. Schlafer, B. Mookerji, T.A. Ohki, J.S. Kline, D.P. Pappas, Phys. Rev. Lett. 104 (2010) 163601.
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