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A large time delay of light via two dimensional atom localization in a coupled cavity waveguide
Optics Communications 448 (2019) 55–59
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
A large time delay of light via two dimensional atom localization in a coupled cavity waveguide Ronggang Liu a ,∗, Tong Liu b , Yujie Li c ,∗ a
Department of Civil Engineering, Harbin Institute of Technology, Weihai 264209, China Aerospace Research Institute of Materials and Processing Technology, Beijing, 10076, China c School of Materials Science and Engineering, Harbin Institute of Technology, Weihai 264209, China b
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Keywords: Atom localization Time delay Coupled cavity waveguide
ABSTRACT We investigate the time delay behavior of light through a 𝛬-type atomic system interacting with two orthogonal standing-wave fields in a modified reservoir. Results show that transmission spectra splitting occurs in a coupled cavity waveguide (CCW) and the time delay of light relies remarkably on the strong coupling effect between a localized atom and a CCW. The atom localization can even be larger than 𝜆∕100 in two orthogonal directions, which arises from field-induced interference effect in a modified reservoir. The proposed scheme not only provides a high-precision atom localization but also achieves a large time delay of the probe light.
1. Introduction Motivated by the need for optically controllable pulse delays for applications such as tunable delay lines [1,2], specific interferometers [3], flight imaging [4] and quantum memory [5], researchers have made a great effort to slow down the speed of light [6–12] through slow-light media [13,14] or modified reservoir [15,16]. On one hand, it is well known that the slow-light effect can be preserved while the effects of absorption are simultaneously canceled using a coherent optical effect occurring in a gas of atoms. Without a limitation to the maximum achievable time delay for the light pulses through material systems, for example, electromagnetically induced transparency (EIT) [5,14] medium, Hau and coworkers have demonstrated that light speed can be reduced to 17 m/s in an ultracold gas of sodium atoms [13]. They have found that the optical properties of the medium can be controlled by quantum interference effect, in which the probe light speed reduction is achieved by adjusting Rabi frequency of the control light according to the EIT mechanism. Furthermore, The room temperature slow light has been studied by two-beam coupling in a photorefractive crystal [17], where the spectral components of an input optical pulse are simultaneously slowed. On the other hand, there are also a number of slow light studies related to photonic crystal [16,18–20]. Particularly, the coupled cavity waveguide (CCW) fabricated in a photonic crystal can form an effective system to achieve slow light [15]. The dispersion relation and group velocity are dependent on the coupled factor, which is influenced by the structure of photonic crystal and the intercavity distance. Though the aforementioned two methods to obtain slow light are quite different, we have combined the two mechanisms to get slower light in our previous ∗
Corresponding authors. E-mail addresses: [email protected] (R. Liu), [email protected] (Y. Li).
https://doi.org/10.1016/j.optcom.2019.05.024 Received 12 April 2019; Accepted 12 May 2019 Available online 15 May 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
work [21]. We have found that the group velocity delays induced by the EIT mechanism and the CCW can be superimposed, when the probe light propagates in the transmission mode. However, we still have a significant problem to solve, that is, for a two dimensional CCW, how to realize high-precision and high-resolution atom localization in two dimensions. For a four-level atom with tripod configuration [22] and a five-level M-type atomic system [23], the schemes have been proposed for two-dimensional atom localization in the subwavelength domain via controlled spontaneous emission. The phenomenon originates from the interference effect between the spontaneous decay channels and induced quantum interference effect generated by the two standing-wave fields. In this paper, we propose a 𝛬-type atomic system interacting with two orthogonal standing-wave fields aimed to localize atom effectively, and in the mean time investigate time delay behavior of the probe light in this 2D system. It is demonstrated that the slow-light medium in the environment of coupled cavity structure may pave the way to achieve a realistic system for slow light, and benefit potential applications in the integrated optical devices [24]. 2. Model and equations The system proposed is composed by slow-light medium and a CCW as shown in Fig. 1(a). The CCW is realized by the coupling of individual defect cavities in a two-dimensional photonic crystal, in which the horizontal coordinate of the center of the 𝑛th cavity is 𝑥 = 𝑛𝑅. The slowlight medium adopts a 𝛬-type atomic system (blue circle) embedded in the first cavity of the CCW. The transition from excited level |2⟩ to the
R. Liu, T. Liu and Y. Li
Optics Communications 448 (2019) 55–59
Fig. 3. The field distribution (normalized with the maximum of electric field density) of an incident light with 𝜔 = 0.3954(2𝜋𝑐∕𝑎) in a five-cavity CCW, where both 𝑥 and 𝑦 coordinates are scaled with the number of the lattice constant 𝑎. Fig. 1. Schematic sketch of a 𝛬-type atomic system interacting with two orthogonal standing-wave fields in a CCW. (a) The coupled cavity waveguide of photonic crystals. (b) The 𝛬-type atomic system in the cavities. (c) Atom moving along the 𝑧 axis and interacting with two orthogonal standing-wave fields aligned along the 𝑥 and 𝑦 directions, respectively. (d) The decomposed energy level diagram . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
and the driving field is space dependent on the 𝑥–𝑦 plane. It is assumed that the center-of-mass position of the atom along the directions of the standing-wave laser fields is nearly constant. Therefore, using the Raman–Nath approximation, we can neglect the kinetic part of the atom in the Hamiltonian. Then the interaction Hamiltonian of the system is given by ∑ ℏ 𝐻 = ℏ𝜔𝑖 |𝑖⟩⟨𝑖| − [𝛺𝑝 𝑒−𝑖𝜔𝑝 𝑡 |2⟩⟨1|+𝛺1 sin (𝑘1 𝑥)𝑒−𝑖𝜔1 𝑡 |2⟩⟨3| 2 𝑖 +
∫
𝑑𝜔𝜆 𝐴(𝜔𝜆 )𝜌(𝜔𝜆 ) sin (𝑘𝜆 𝑦)𝑒−𝑖𝜔𝜆 𝑡 |3⟩⟨1| + H.c.],
(1)
where
√ 𝐴(𝜔𝜆 ) = 𝐿(𝜔𝜆 ) 𝛺22 + (𝜔𝜆 − 𝜔31 )2 ,
(2)
and 𝐿(𝜔𝜆 ) is the localization factor of the CCW in the system. The density of states of the CCW is given by 𝜌(𝜔𝜆 ) =
1 , √ 𝜋𝑅 𝑄2 − (𝜔𝜆 − 𝜔0 )2
(3)
where 𝑅 is the inter-cavity distance, 𝜔0 = (1 − 𝛥𝛼∕2)𝛺 ≈ 𝛺, 𝑄 = 𝜅𝛺, and 𝛺 is the resonance frequency of the single high-Q cavity without coupling. 𝜅 is the coupling factor depending on the geometrical properties of a CCW, and 𝛥𝛼 is a parameter related to the coupled cavity, which decreases rapidly with inter-cavity distance R [15]. Here, we set a CCW with an inter-cavity distance 𝑅 = 4𝑎 made from a two-dimensional photonic crystal formed by dielectric cylinders placed in vacuum according to a square lattice. The dielectric constant and radius of the cylinders are 𝜖 = 8.9 and 𝑟 = 0.2𝑎, respectively. In order to obtain the transmission and localization properties of the CCW, we simulate the target system by using finite-difference time-domain (FDTD) method [27,28]. It is found a resonance splitting effect occurs near 𝜔 = 0.3954(2𝜋𝑐∕𝑎) for the aimed system, and the number of peaks just equals to the number of cavities as shown in Fig. 2. This splitting property presents the form of conduction band from the forbidden band of photonic crystals [29]. We know that for a three-cavity and a fivecavity structure 𝜔 = 0.3954(2𝜋𝑐∕𝑎) is an eigenmode. Hence we use 𝜔 = 0.3954(2𝜋𝑐∕𝑎) as a work frequency to observe localization properties of the system, and the results show that if the number of the cavities is odd (even), the strong localized states occur in the odd (even) indexed cavities, as depicted in Fig. 3. For a four-cavity, 𝜔 = 0.3954(2𝜋𝑐∕𝑎) is not a eigenmode, the strong localization can also be observed, but it is weaker than a three-cavity and a five-cavity system. This also can be
Fig. 2. Transmission spectrum of a five-cavity [dotted (blue) curve], four-cavity [dashed (red) curve], and three-cavity [solid (yellow) curve] CCW, in which a CCW composed by dielectric columns with the radius 𝑟 = 0.2𝑎, the dielectric constant 𝜖 = 8.9, and the size of a unit cell is 𝑅 = 4𝑎 as shown in Fig. 1(a).
ground level |1⟩ is coupled by a probe laser with a carry frequency 𝜔𝑝 and Rabi frequency 𝛺𝑝 . The excited level |2⟩ is coupled to the lower level |3⟩ and the level |3⟩ is coupled to the ground level |1⟩ by two orthogonal standing-wave laser fields (aligned along the 𝑥 direction with a carry frequency 𝜔1 and 𝑦 direction with a carry frequency 𝜔2 ) with position-dependent Rabi frequencies 𝛺1 sin (𝑘1 𝑥) and 𝛺2 sin (𝑘2 𝑦), respectively, where 𝑘1 and 𝑘2 are the corresponding wave vectors of the two fields, as shown in Fig. 1(b) and (c). Here standing-wave laser field 𝛺2 sin (𝑘2 𝑦) is coupled by a CCW, hence the modified reservoir can be viewed as a background environment of the coupling beam. Since the transition |3⟩ to |1⟩ is coupled to the structured CCW mode ∑ continuum 𝜆, we have 𝜆 → ∫ 𝑑𝜔𝜆 𝜌(𝜔𝜆 ), where 𝜌(𝜔𝜆 ) is the density of states [15,25,26] of a CCW. We set the target atom passes through the classical standing-wave laser fields along the 𝑧 direction, then the interaction between the atom 56
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Optics Communications 448 (2019) 55–59
Fig. 5. Im(𝜒) as a function of (𝑘1 𝑥, 𝑘2 𝑦) for Rabi frequencies 𝛺1 = 0 and the detuning 𝛥 = 2𝛾: (a) 𝐵 = 4𝛾, (b) 𝐵 = 6𝛾, (c) 𝐵 = 10𝛾 and (d) 𝐵 = 40𝛾.
Fig. 4. Im(𝜒) as a function of (𝑘1 𝑥, 𝑘2 𝑦) for the parameters 𝐵 = 0 and the detuning 𝛥 = 2𝛾: (a) 𝛺1 = 4𝛾, (b) 𝛺1 = 6𝛾, (c) 𝛺1 = 10𝛾 and (d) 𝛺1 = 40𝛾.
can be related to the real part of the susceptibility. Based on these characteristics, we will discuss the atom localization and time delay properties in detail in the following section.
found from the weaker transmission spectra with a four-cavity as shown in Fig. 2. This means photons with certain frequencies with a stronger coupling-field intensity can be trapped inside the defect cavities, which help more to accomplish a large time delay in the system. In order to achieve the precise information about the atomic position from the susceptibility of the system at the probe field frequency, Qamar has presented a scheme for subwavelength 2D atom localization through Raman gain process [30]. Since the nonlinear Raman susceptibility is related to density matrix elements, by using the density matrix approach [31], the equations of motion for the density matrix elements are given by the following:
3. Results and discussions In this section, we present atom localization information via a few numerical calculations based on the maximum of the imaginary part of the susceptibility from Eq. (9), and then demonstrate strong 2D atom localization by adjusting the Rabi frequency 𝛺1 and parameter 𝐵 of both control lights. In order to see the atom localization in CCW clearly, three specific cases are analyzed below. Firstly, we set the parameter 𝐵 = 0, and only change Rabi frequency 𝛺1 to observe the Raman gain. The plot of the Raman gain in Fig. 4 exhibits localization maxima centered at 𝑘1 𝑥 = ± sin−1 (2𝛥∕𝛺1 ) + 𝑛𝜋, where 𝑛 is an integer. Therefore, the degree of localization depends on the detuning 𝛥 and Rabi frequency 𝛺1 . In Fig. 4(a), the peaks of atom localization along 𝑥 direction have a wavelike pattern. It can be clearly seen that two localization peaks occur at 𝑘1 𝑥 = ±𝜋∕2 for 𝛺1 = 4𝛾 case, and then the two peaks are split into four peaks when 𝛺1 = 6𝛾 as in Fig. 4(b). With Rabi frequency 𝛺1 increasing, the two peaks in the middle approach gradually shown by Fig. 4(c). When 𝛺1 = 40𝛾, the atom is localized near 𝑘1 𝑥 = 0 and the peaks are more pronounced as shown Fig. 4(d). The full width at half maximum (FWHM) of the peaks in Fig. 4(d) can easily be obtained numerically and it comes out to be 0.05, which means the localization is larger than 𝜆∕100. Secondly, we set 𝛺1 = 0, and observe the atom localization by adjusting the parameter 𝐵. We know that the strong localization occurs in the cavity at some specific frequencies, with the parameter B of the system significantly dependent on geometrical characteristics of the CCW. It can be√calculated the atom localization maxima centered at 𝑘2 𝑦 = ± sin−1 (2 (𝛾(𝛥 − 𝛾))∕𝐵)+𝑛𝜋. In Fig. 5 we show the similar results as in Fig. 4 but with 𝛺1 = 0 to reveal the atom localization along 𝑦 direction. It is clear that the atom localization along 𝑦 axis takes a four-peak structure. The localization of the atom is much stronger when choosing a larger value of parameter 𝐵. We note that in Fig. 5 the atom localization is larger than 𝜆∕200. Finally, we analyze the atom localization based on adjusting both Rabi frequency 𝛺1 and parameter 𝐵 along the two orthogonal directions. In order to acquire position information of the atom clearly, we choose the same parameters as in Figs. 4 and 5. In Fig. 6(a), we set 𝛺1 = 𝐵 = 4𝛾 subwavelength atom localization pattern show a six peak structures. With the increase of the coupling-field intensity,
𝑖 𝑖 𝛺 𝑒−𝑖𝜔𝑝 𝑡 (𝜌11 − 𝜌22 ) + 𝛺1 sin (𝑘1 𝑥)𝑒−𝑖𝜔1 𝑡 𝜌31 2 𝑝 2 𝑖 − 𝐵 sin (𝑘2 𝑦)𝑒−𝑖𝜔2 𝑡 𝜌23 − (𝛾1 + 𝑖𝜔21 )𝜌21 , (4) 2 𝑖 𝑖 𝜌̇ 23 = 𝛺𝑝 𝑒−𝑖𝜔𝑝 𝑡 𝜌13 − 𝛺1 sin (𝑘1 𝑥)𝑒−𝑖𝜔1 𝑡 (𝜌22 − 𝜌33 ) 2 2 𝑖 − 𝐵 sin (𝑘2 𝑦)𝑒𝑖𝜔2 𝑡 𝜌21 − (𝛾2 + 𝑖𝜔23 )𝜌23 , (5) 2 𝑖 𝑖 𝜌̇ 31 = − 𝛺𝑝 𝑒−𝑖𝜔𝑝 𝑡 𝜌32 + 𝛺1 sin (𝑘1 𝑥)𝑒𝑖𝜔1 𝑡 𝜌21 2 2 𝑖 − 𝐵 sin (𝑘2 𝑦)𝑒−𝑖𝜔2 𝑡 (𝜌33 − 𝜌11 ) − (𝛾3 + 𝑖𝜔21 )𝜌31 . (6) 2 The off-diagonal decay rates for 𝜌21 , 𝜌23 and 𝜌31 are denoted by 𝛾1 , 𝛾2 and 𝛾3 , respectively. The parameter 𝐵 takes the form √ 𝜉 𝜔0 + 2 𝑄 𝛺22 + (𝜔 − 𝜔31 )2 1 𝐵= 𝑑𝜔𝐿(𝜔) , (7) 𝜉 𝜋𝑅 ∫𝜔0 − 𝑄 𝑄2 − (𝜔 − 𝜔0 )2 𝜌̇ 21 =
2
where 𝜉 is the line width factor of the coupling laser. As the atoms are initially in the ground level |1⟩, 𝜌(0) = 1, 𝜌(0) = 𝜌(0) = 𝜌(0) = 0. 11 22 33 32 For simplicity, we assume that 𝛾1 = 𝛾2 = 𝛾 ≫ 𝛾3 . Using the same calculation method as Ref. [21], the real part and imaginary part of the susceptibility [21,31] read 𝜒′ = −
𝜁[2𝐵𝛥 + 𝛺1 𝐵sin(𝑘1 𝑥)sin(𝑘2 𝑦)][𝛺12 sin2 (𝑘1 𝑥) − 4𝛥2 ]∕𝛺𝑝 [𝛺12 sin2 (𝑘1 𝑥) − 4𝛥2 ]2 + 𝛥2 [𝐵 2 sin2 (𝑘2 𝑦)∕𝛾 + 4𝛾]2
(8)
,
and 𝜒 ′′ =
𝜁 𝛥[2𝐵𝛥 + 𝛺1 𝐵sin(𝑘1 𝑥)sin(𝑘2 𝑦)][𝐵 2 sin2 (𝑘2 𝑦)∕𝛾 + 4𝛾]∕𝛺𝑝 [𝛺12 sin2 (𝑘1 𝑥) − 4𝛥2 ]2 + 𝛥2 [𝐵 2 sin2 (𝑘2 𝑦)∕𝛾 + 4𝛾]2
,
(9)
respectively, where 𝛥 = 𝜔21 − 𝜔𝑝 is the detuning of the probe field. The information about the atomic position is directly proportional to the imaginary part of the susceptibility, and the time delay of light 57
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Optics Communications 448 (2019) 55–59
The subwavelength atom localization has been explored above, we will discuss the time delay in a CCW below. The delay time per unit length is given by 𝑇𝑑𝑒𝑙 =
𝜕𝑅𝑒(𝜒) . 𝜕𝜔𝑝
(10)
In Fig. 7, we present the tendency of 𝑇𝑑𝑒𝑙 with detuning 𝛥 in the system. We know that the probe light speed reduction is achieved by decreasing Rabi frequency of the control light according to the EIT mechanism. This can be observed from Fig. 7(a) and (b), which is a disadvantage case for the high-precision atom localization. The phenomenon is still kept with parameter 𝐵 increasing from 6𝛾 to 10𝛾 as shown in Fig. 7(b) and (c). Fortunately, in the system one can slow down the group velocity just by increasing the parameter 𝐵 continually. The numerical results show that a marked delay time can still be obtained even when enhancing Rabi frequency 𝛺1 . When 𝛺1 = 40𝛾, 𝐵 = 100𝛾, the curve becomes very sharp and a large time delay occurs near 𝛥 = 0 as shown in Fig. 7(d). The above analyses imply that, the scheme in which the Rabi frequency 𝛺1 and parameter 𝐵 are simultaneously enhanced not only provides strong atom localization but also achieves a large time delay. The mechanism of the slow light can be demonstrated by Fig. 1(d), in which we show when a probe light passes through the system, the energy is transferred with the transition between the ground state |1⟩ and the excited state |2⟩. Since the strong coupling field exists between the level |1⟩ (|3⟩) and the level |3⟩ (|2⟩), the energy coming from the probe light can be transferred between the level |1⟩ and the level |3⟩, or between the level |3⟩ and the level |2⟩ rapidly. This means the forming mechanism of slow light can be attributed to the probe light stored in between the level |1⟩ (|3⟩) and the level |3⟩ (|2⟩) in most of the interaction time.
Fig. 6. Im(𝜒) as a function of (𝑘1 𝑥, 𝑘2 𝑦) for different coupling-field intensity of the two orthogonal standing-wave fields. (a) 𝛺1 = 𝐵 = 4𝛾; (b) 𝛺1 = 𝐵 = 6𝛾; (c) 𝛺1 = 𝐵 = 10𝛾; (d) 𝛺1 = 𝐵 = 40𝛾. The system parameters used are the same as in Figs. 4 and 5.
4. Conclusion We have investigated the time delay of light through a 𝛬-type atomic system interacting with two orthogonal standing-wave fields in the environment of coupled cavity structure. The numerical results reveal a hierarchic distribution of the field mode when the system is driven by plane waves at a specific frequency. 2D atom localization can be improved significantly due to the field-induced interference effect in a modified reservoir. It is shown that for our considered system, the localization is larger than 𝜆∕100. 2D atom localization requires a strong coupling-field intensity, which cannot help for a large time delay of light according to the EIT mechanism. However, we discover an effective scheme not only providing a high-precision and high-resolution 2D atom localization but also obtaining a large time delay of light. The scheme is realized via an atomic system driven by two orthogonal standing-wave fields in a modified reservoir, which potentially facilitates a number of future applications in all-optical switch and optical signal processing.
Fig. 7. The delay time per unit length 𝑇𝑑𝑒𝑙 as a function of the detuning 𝛥, where 2
𝜁=
𝑁𝑎 |𝜇⃗21 | 𝜖0 ℏ
. (a) 𝛺1 = 𝐵 = 6𝛾; (b) 𝛺1 = 40𝛾, 𝐵 = 6𝛾; (c) 𝛺1 = 40𝛾, 𝐵 = 10𝛾; (d) 𝛺1 = 40𝛾,
𝐵 = 100𝛾.
localization pattern presents a twelve-peak structure as in Fig. 6(b) and (c). Moreover, when the Rabi frequency is tuned to 𝛺1 = 𝐵 = 40𝛾, the peaks of atom localization become quite sharp as shown in Fig. 6(d). For the given detuning 𝛥, high-precision and high-resolution localization patterns can be obtained by adjusting Rabi frequency 𝛺1 and the parameter 𝐵, and the spatial resolution is reached to 𝜆∕100. The above results show that the atom localization can be attributed to the field-induced quantum interference effect from two channels: the first one is, between the two pathways |+⟩2 → |1⟩ and |−⟩2 → |1⟩, in which the levels |±⟩2 are the split dressed-state sublevels generated by standing-wave field 𝛺1 sin (𝑘1 𝑥) aligned along the 𝑥 direction; the second one is, between the two pathways |+⟩3 → |1⟩ and |−⟩3 → |1⟩, in which the levels |±⟩3 are the split dressed-state sublevels generated by standing-wave field 𝛺2 sin (𝑘2 𝑦) coupled to a CCW aligned along the 𝑦 direction. These analyses imply that the field-induced quantum interference effect from two channels can be superimposed as shown in Fig. 6. Hence the high-precision atom localization can originate from the field-induced interference effect in a modified reservoir, and the increase of the coupling-field intensity leads to stronger localization of the atom.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
A large time delay of light via two dimensional atom localization in a coupled cavity waveguide Ronggang Liu a ,∗, Tong Liu b , Yujie Li c ,∗ a
Department of Civil Engineering, Harbin Institute of Technology, Weihai 264209, China Aerospace Research Institute of Materials and Processing Technology, Beijing, 10076, China c School of Materials Science and Engineering, Harbin Institute of Technology, Weihai 264209, China b
ARTICLE
INFO
Keywords: Atom localization Time delay Coupled cavity waveguide
ABSTRACT We investigate the time delay behavior of light through a 𝛬-type atomic system interacting with two orthogonal standing-wave fields in a modified reservoir. Results show that transmission spectra splitting occurs in a coupled cavity waveguide (CCW) and the time delay of light relies remarkably on the strong coupling effect between a localized atom and a CCW. The atom localization can even be larger than 𝜆∕100 in two orthogonal directions, which arises from field-induced interference effect in a modified reservoir. The proposed scheme not only provides a high-precision atom localization but also achieves a large time delay of the probe light.
1. Introduction Motivated by the need for optically controllable pulse delays for applications such as tunable delay lines [1,2], specific interferometers [3], flight imaging [4] and quantum memory [5], researchers have made a great effort to slow down the speed of light [6–12] through slow-light media [13,14] or modified reservoir [15,16]. On one hand, it is well known that the slow-light effect can be preserved while the effects of absorption are simultaneously canceled using a coherent optical effect occurring in a gas of atoms. Without a limitation to the maximum achievable time delay for the light pulses through material systems, for example, electromagnetically induced transparency (EIT) [5,14] medium, Hau and coworkers have demonstrated that light speed can be reduced to 17 m/s in an ultracold gas of sodium atoms [13]. They have found that the optical properties of the medium can be controlled by quantum interference effect, in which the probe light speed reduction is achieved by adjusting Rabi frequency of the control light according to the EIT mechanism. Furthermore, The room temperature slow light has been studied by two-beam coupling in a photorefractive crystal [17], where the spectral components of an input optical pulse are simultaneously slowed. On the other hand, there are also a number of slow light studies related to photonic crystal [16,18–20]. Particularly, the coupled cavity waveguide (CCW) fabricated in a photonic crystal can form an effective system to achieve slow light [15]. The dispersion relation and group velocity are dependent on the coupled factor, which is influenced by the structure of photonic crystal and the intercavity distance. Though the aforementioned two methods to obtain slow light are quite different, we have combined the two mechanisms to get slower light in our previous ∗
Corresponding authors. E-mail addresses: [email protected] (R. Liu), [email protected] (Y. Li).
https://doi.org/10.1016/j.optcom.2019.05.024 Received 12 April 2019; Accepted 12 May 2019 Available online 15 May 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
work [21]. We have found that the group velocity delays induced by the EIT mechanism and the CCW can be superimposed, when the probe light propagates in the transmission mode. However, we still have a significant problem to solve, that is, for a two dimensional CCW, how to realize high-precision and high-resolution atom localization in two dimensions. For a four-level atom with tripod configuration [22] and a five-level M-type atomic system [23], the schemes have been proposed for two-dimensional atom localization in the subwavelength domain via controlled spontaneous emission. The phenomenon originates from the interference effect between the spontaneous decay channels and induced quantum interference effect generated by the two standing-wave fields. In this paper, we propose a 𝛬-type atomic system interacting with two orthogonal standing-wave fields aimed to localize atom effectively, and in the mean time investigate time delay behavior of the probe light in this 2D system. It is demonstrated that the slow-light medium in the environment of coupled cavity structure may pave the way to achieve a realistic system for slow light, and benefit potential applications in the integrated optical devices [24]. 2. Model and equations The system proposed is composed by slow-light medium and a CCW as shown in Fig. 1(a). The CCW is realized by the coupling of individual defect cavities in a two-dimensional photonic crystal, in which the horizontal coordinate of the center of the 𝑛th cavity is 𝑥 = 𝑛𝑅. The slowlight medium adopts a 𝛬-type atomic system (blue circle) embedded in the first cavity of the CCW. The transition from excited level |2⟩ to the
R. Liu, T. Liu and Y. Li
Optics Communications 448 (2019) 55–59
Fig. 3. The field distribution (normalized with the maximum of electric field density) of an incident light with 𝜔 = 0.3954(2𝜋𝑐∕𝑎) in a five-cavity CCW, where both 𝑥 and 𝑦 coordinates are scaled with the number of the lattice constant 𝑎. Fig. 1. Schematic sketch of a 𝛬-type atomic system interacting with two orthogonal standing-wave fields in a CCW. (a) The coupled cavity waveguide of photonic crystals. (b) The 𝛬-type atomic system in the cavities. (c) Atom moving along the 𝑧 axis and interacting with two orthogonal standing-wave fields aligned along the 𝑥 and 𝑦 directions, respectively. (d) The decomposed energy level diagram . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
and the driving field is space dependent on the 𝑥–𝑦 plane. It is assumed that the center-of-mass position of the atom along the directions of the standing-wave laser fields is nearly constant. Therefore, using the Raman–Nath approximation, we can neglect the kinetic part of the atom in the Hamiltonian. Then the interaction Hamiltonian of the system is given by ∑ ℏ 𝐻 = ℏ𝜔𝑖 |𝑖⟩⟨𝑖| − [𝛺𝑝 𝑒−𝑖𝜔𝑝 𝑡 |2⟩⟨1|+𝛺1 sin (𝑘1 𝑥)𝑒−𝑖𝜔1 𝑡 |2⟩⟨3| 2 𝑖 +
∫
𝑑𝜔𝜆 𝐴(𝜔𝜆 )𝜌(𝜔𝜆 ) sin (𝑘𝜆 𝑦)𝑒−𝑖𝜔𝜆 𝑡 |3⟩⟨1| + H.c.],
(1)
where
√ 𝐴(𝜔𝜆 ) = 𝐿(𝜔𝜆 ) 𝛺22 + (𝜔𝜆 − 𝜔31 )2 ,
(2)
and 𝐿(𝜔𝜆 ) is the localization factor of the CCW in the system. The density of states of the CCW is given by 𝜌(𝜔𝜆 ) =
1 , √ 𝜋𝑅 𝑄2 − (𝜔𝜆 − 𝜔0 )2
(3)
where 𝑅 is the inter-cavity distance, 𝜔0 = (1 − 𝛥𝛼∕2)𝛺 ≈ 𝛺, 𝑄 = 𝜅𝛺, and 𝛺 is the resonance frequency of the single high-Q cavity without coupling. 𝜅 is the coupling factor depending on the geometrical properties of a CCW, and 𝛥𝛼 is a parameter related to the coupled cavity, which decreases rapidly with inter-cavity distance R [15]. Here, we set a CCW with an inter-cavity distance 𝑅 = 4𝑎 made from a two-dimensional photonic crystal formed by dielectric cylinders placed in vacuum according to a square lattice. The dielectric constant and radius of the cylinders are 𝜖 = 8.9 and 𝑟 = 0.2𝑎, respectively. In order to obtain the transmission and localization properties of the CCW, we simulate the target system by using finite-difference time-domain (FDTD) method [27,28]. It is found a resonance splitting effect occurs near 𝜔 = 0.3954(2𝜋𝑐∕𝑎) for the aimed system, and the number of peaks just equals to the number of cavities as shown in Fig. 2. This splitting property presents the form of conduction band from the forbidden band of photonic crystals [29]. We know that for a three-cavity and a fivecavity structure 𝜔 = 0.3954(2𝜋𝑐∕𝑎) is an eigenmode. Hence we use 𝜔 = 0.3954(2𝜋𝑐∕𝑎) as a work frequency to observe localization properties of the system, and the results show that if the number of the cavities is odd (even), the strong localized states occur in the odd (even) indexed cavities, as depicted in Fig. 3. For a four-cavity, 𝜔 = 0.3954(2𝜋𝑐∕𝑎) is not a eigenmode, the strong localization can also be observed, but it is weaker than a three-cavity and a five-cavity system. This also can be
Fig. 2. Transmission spectrum of a five-cavity [dotted (blue) curve], four-cavity [dashed (red) curve], and three-cavity [solid (yellow) curve] CCW, in which a CCW composed by dielectric columns with the radius 𝑟 = 0.2𝑎, the dielectric constant 𝜖 = 8.9, and the size of a unit cell is 𝑅 = 4𝑎 as shown in Fig. 1(a).
ground level |1⟩ is coupled by a probe laser with a carry frequency 𝜔𝑝 and Rabi frequency 𝛺𝑝 . The excited level |2⟩ is coupled to the lower level |3⟩ and the level |3⟩ is coupled to the ground level |1⟩ by two orthogonal standing-wave laser fields (aligned along the 𝑥 direction with a carry frequency 𝜔1 and 𝑦 direction with a carry frequency 𝜔2 ) with position-dependent Rabi frequencies 𝛺1 sin (𝑘1 𝑥) and 𝛺2 sin (𝑘2 𝑦), respectively, where 𝑘1 and 𝑘2 are the corresponding wave vectors of the two fields, as shown in Fig. 1(b) and (c). Here standing-wave laser field 𝛺2 sin (𝑘2 𝑦) is coupled by a CCW, hence the modified reservoir can be viewed as a background environment of the coupling beam. Since the transition |3⟩ to |1⟩ is coupled to the structured CCW mode ∑ continuum 𝜆, we have 𝜆 → ∫ 𝑑𝜔𝜆 𝜌(𝜔𝜆 ), where 𝜌(𝜔𝜆 ) is the density of states [15,25,26] of a CCW. We set the target atom passes through the classical standing-wave laser fields along the 𝑧 direction, then the interaction between the atom 56
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Optics Communications 448 (2019) 55–59
Fig. 5. Im(𝜒) as a function of (𝑘1 𝑥, 𝑘2 𝑦) for Rabi frequencies 𝛺1 = 0 and the detuning 𝛥 = 2𝛾: (a) 𝐵 = 4𝛾, (b) 𝐵 = 6𝛾, (c) 𝐵 = 10𝛾 and (d) 𝐵 = 40𝛾.
Fig. 4. Im(𝜒) as a function of (𝑘1 𝑥, 𝑘2 𝑦) for the parameters 𝐵 = 0 and the detuning 𝛥 = 2𝛾: (a) 𝛺1 = 4𝛾, (b) 𝛺1 = 6𝛾, (c) 𝛺1 = 10𝛾 and (d) 𝛺1 = 40𝛾.
can be related to the real part of the susceptibility. Based on these characteristics, we will discuss the atom localization and time delay properties in detail in the following section.
found from the weaker transmission spectra with a four-cavity as shown in Fig. 2. This means photons with certain frequencies with a stronger coupling-field intensity can be trapped inside the defect cavities, which help more to accomplish a large time delay in the system. In order to achieve the precise information about the atomic position from the susceptibility of the system at the probe field frequency, Qamar has presented a scheme for subwavelength 2D atom localization through Raman gain process [30]. Since the nonlinear Raman susceptibility is related to density matrix elements, by using the density matrix approach [31], the equations of motion for the density matrix elements are given by the following:
3. Results and discussions In this section, we present atom localization information via a few numerical calculations based on the maximum of the imaginary part of the susceptibility from Eq. (9), and then demonstrate strong 2D atom localization by adjusting the Rabi frequency 𝛺1 and parameter 𝐵 of both control lights. In order to see the atom localization in CCW clearly, three specific cases are analyzed below. Firstly, we set the parameter 𝐵 = 0, and only change Rabi frequency 𝛺1 to observe the Raman gain. The plot of the Raman gain in Fig. 4 exhibits localization maxima centered at 𝑘1 𝑥 = ± sin−1 (2𝛥∕𝛺1 ) + 𝑛𝜋, where 𝑛 is an integer. Therefore, the degree of localization depends on the detuning 𝛥 and Rabi frequency 𝛺1 . In Fig. 4(a), the peaks of atom localization along 𝑥 direction have a wavelike pattern. It can be clearly seen that two localization peaks occur at 𝑘1 𝑥 = ±𝜋∕2 for 𝛺1 = 4𝛾 case, and then the two peaks are split into four peaks when 𝛺1 = 6𝛾 as in Fig. 4(b). With Rabi frequency 𝛺1 increasing, the two peaks in the middle approach gradually shown by Fig. 4(c). When 𝛺1 = 40𝛾, the atom is localized near 𝑘1 𝑥 = 0 and the peaks are more pronounced as shown Fig. 4(d). The full width at half maximum (FWHM) of the peaks in Fig. 4(d) can easily be obtained numerically and it comes out to be 0.05, which means the localization is larger than 𝜆∕100. Secondly, we set 𝛺1 = 0, and observe the atom localization by adjusting the parameter 𝐵. We know that the strong localization occurs in the cavity at some specific frequencies, with the parameter B of the system significantly dependent on geometrical characteristics of the CCW. It can be√calculated the atom localization maxima centered at 𝑘2 𝑦 = ± sin−1 (2 (𝛾(𝛥 − 𝛾))∕𝐵)+𝑛𝜋. In Fig. 5 we show the similar results as in Fig. 4 but with 𝛺1 = 0 to reveal the atom localization along 𝑦 direction. It is clear that the atom localization along 𝑦 axis takes a four-peak structure. The localization of the atom is much stronger when choosing a larger value of parameter 𝐵. We note that in Fig. 5 the atom localization is larger than 𝜆∕200. Finally, we analyze the atom localization based on adjusting both Rabi frequency 𝛺1 and parameter 𝐵 along the two orthogonal directions. In order to acquire position information of the atom clearly, we choose the same parameters as in Figs. 4 and 5. In Fig. 6(a), we set 𝛺1 = 𝐵 = 4𝛾 subwavelength atom localization pattern show a six peak structures. With the increase of the coupling-field intensity,
𝑖 𝑖 𝛺 𝑒−𝑖𝜔𝑝 𝑡 (𝜌11 − 𝜌22 ) + 𝛺1 sin (𝑘1 𝑥)𝑒−𝑖𝜔1 𝑡 𝜌31 2 𝑝 2 𝑖 − 𝐵 sin (𝑘2 𝑦)𝑒−𝑖𝜔2 𝑡 𝜌23 − (𝛾1 + 𝑖𝜔21 )𝜌21 , (4) 2 𝑖 𝑖 𝜌̇ 23 = 𝛺𝑝 𝑒−𝑖𝜔𝑝 𝑡 𝜌13 − 𝛺1 sin (𝑘1 𝑥)𝑒−𝑖𝜔1 𝑡 (𝜌22 − 𝜌33 ) 2 2 𝑖 − 𝐵 sin (𝑘2 𝑦)𝑒𝑖𝜔2 𝑡 𝜌21 − (𝛾2 + 𝑖𝜔23 )𝜌23 , (5) 2 𝑖 𝑖 𝜌̇ 31 = − 𝛺𝑝 𝑒−𝑖𝜔𝑝 𝑡 𝜌32 + 𝛺1 sin (𝑘1 𝑥)𝑒𝑖𝜔1 𝑡 𝜌21 2 2 𝑖 − 𝐵 sin (𝑘2 𝑦)𝑒−𝑖𝜔2 𝑡 (𝜌33 − 𝜌11 ) − (𝛾3 + 𝑖𝜔21 )𝜌31 . (6) 2 The off-diagonal decay rates for 𝜌21 , 𝜌23 and 𝜌31 are denoted by 𝛾1 , 𝛾2 and 𝛾3 , respectively. The parameter 𝐵 takes the form √ 𝜉 𝜔0 + 2 𝑄 𝛺22 + (𝜔 − 𝜔31 )2 1 𝐵= 𝑑𝜔𝐿(𝜔) , (7) 𝜉 𝜋𝑅 ∫𝜔0 − 𝑄 𝑄2 − (𝜔 − 𝜔0 )2 𝜌̇ 21 =
2
where 𝜉 is the line width factor of the coupling laser. As the atoms are initially in the ground level |1⟩, 𝜌(0) = 1, 𝜌(0) = 𝜌(0) = 𝜌(0) = 0. 11 22 33 32 For simplicity, we assume that 𝛾1 = 𝛾2 = 𝛾 ≫ 𝛾3 . Using the same calculation method as Ref. [21], the real part and imaginary part of the susceptibility [21,31] read 𝜒′ = −
𝜁[2𝐵𝛥 + 𝛺1 𝐵sin(𝑘1 𝑥)sin(𝑘2 𝑦)][𝛺12 sin2 (𝑘1 𝑥) − 4𝛥2 ]∕𝛺𝑝 [𝛺12 sin2 (𝑘1 𝑥) − 4𝛥2 ]2 + 𝛥2 [𝐵 2 sin2 (𝑘2 𝑦)∕𝛾 + 4𝛾]2
(8)
,
and 𝜒 ′′ =
𝜁 𝛥[2𝐵𝛥 + 𝛺1 𝐵sin(𝑘1 𝑥)sin(𝑘2 𝑦)][𝐵 2 sin2 (𝑘2 𝑦)∕𝛾 + 4𝛾]∕𝛺𝑝 [𝛺12 sin2 (𝑘1 𝑥) − 4𝛥2 ]2 + 𝛥2 [𝐵 2 sin2 (𝑘2 𝑦)∕𝛾 + 4𝛾]2
,
(9)
respectively, where 𝛥 = 𝜔21 − 𝜔𝑝 is the detuning of the probe field. The information about the atomic position is directly proportional to the imaginary part of the susceptibility, and the time delay of light 57
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Optics Communications 448 (2019) 55–59
The subwavelength atom localization has been explored above, we will discuss the time delay in a CCW below. The delay time per unit length is given by 𝑇𝑑𝑒𝑙 =
𝜕𝑅𝑒(𝜒) . 𝜕𝜔𝑝
(10)
In Fig. 7, we present the tendency of 𝑇𝑑𝑒𝑙 with detuning 𝛥 in the system. We know that the probe light speed reduction is achieved by decreasing Rabi frequency of the control light according to the EIT mechanism. This can be observed from Fig. 7(a) and (b), which is a disadvantage case for the high-precision atom localization. The phenomenon is still kept with parameter 𝐵 increasing from 6𝛾 to 10𝛾 as shown in Fig. 7(b) and (c). Fortunately, in the system one can slow down the group velocity just by increasing the parameter 𝐵 continually. The numerical results show that a marked delay time can still be obtained even when enhancing Rabi frequency 𝛺1 . When 𝛺1 = 40𝛾, 𝐵 = 100𝛾, the curve becomes very sharp and a large time delay occurs near 𝛥 = 0 as shown in Fig. 7(d). The above analyses imply that, the scheme in which the Rabi frequency 𝛺1 and parameter 𝐵 are simultaneously enhanced not only provides strong atom localization but also achieves a large time delay. The mechanism of the slow light can be demonstrated by Fig. 1(d), in which we show when a probe light passes through the system, the energy is transferred with the transition between the ground state |1⟩ and the excited state |2⟩. Since the strong coupling field exists between the level |1⟩ (|3⟩) and the level |3⟩ (|2⟩), the energy coming from the probe light can be transferred between the level |1⟩ and the level |3⟩, or between the level |3⟩ and the level |2⟩ rapidly. This means the forming mechanism of slow light can be attributed to the probe light stored in between the level |1⟩ (|3⟩) and the level |3⟩ (|2⟩) in most of the interaction time.
Fig. 6. Im(𝜒) as a function of (𝑘1 𝑥, 𝑘2 𝑦) for different coupling-field intensity of the two orthogonal standing-wave fields. (a) 𝛺1 = 𝐵 = 4𝛾; (b) 𝛺1 = 𝐵 = 6𝛾; (c) 𝛺1 = 𝐵 = 10𝛾; (d) 𝛺1 = 𝐵 = 40𝛾. The system parameters used are the same as in Figs. 4 and 5.
4. Conclusion We have investigated the time delay of light through a 𝛬-type atomic system interacting with two orthogonal standing-wave fields in the environment of coupled cavity structure. The numerical results reveal a hierarchic distribution of the field mode when the system is driven by plane waves at a specific frequency. 2D atom localization can be improved significantly due to the field-induced interference effect in a modified reservoir. It is shown that for our considered system, the localization is larger than 𝜆∕100. 2D atom localization requires a strong coupling-field intensity, which cannot help for a large time delay of light according to the EIT mechanism. However, we discover an effective scheme not only providing a high-precision and high-resolution 2D atom localization but also obtaining a large time delay of light. The scheme is realized via an atomic system driven by two orthogonal standing-wave fields in a modified reservoir, which potentially facilitates a number of future applications in all-optical switch and optical signal processing.
Fig. 7. The delay time per unit length 𝑇𝑑𝑒𝑙 as a function of the detuning 𝛥, where 2
𝜁=
𝑁𝑎 |𝜇⃗21 | 𝜖0 ℏ
. (a) 𝛺1 = 𝐵 = 6𝛾; (b) 𝛺1 = 40𝛾, 𝐵 = 6𝛾; (c) 𝛺1 = 40𝛾, 𝐵 = 10𝛾; (d) 𝛺1 = 40𝛾,
𝐵 = 100𝛾.
localization pattern presents a twelve-peak structure as in Fig. 6(b) and (c). Moreover, when the Rabi frequency is tuned to 𝛺1 = 𝐵 = 40𝛾, the peaks of atom localization become quite sharp as shown in Fig. 6(d). For the given detuning 𝛥, high-precision and high-resolution localization patterns can be obtained by adjusting Rabi frequency 𝛺1 and the parameter 𝐵, and the spatial resolution is reached to 𝜆∕100. The above results show that the atom localization can be attributed to the field-induced quantum interference effect from two channels: the first one is, between the two pathways |+⟩2 → |1⟩ and |−⟩2 → |1⟩, in which the levels |±⟩2 are the split dressed-state sublevels generated by standing-wave field 𝛺1 sin (𝑘1 𝑥) aligned along the 𝑥 direction; the second one is, between the two pathways |+⟩3 → |1⟩ and |−⟩3 → |1⟩, in which the levels |±⟩3 are the split dressed-state sublevels generated by standing-wave field 𝛺2 sin (𝑘2 𝑦) coupled to a CCW aligned along the 𝑦 direction. These analyses imply that the field-induced quantum interference effect from two channels can be superimposed as shown in Fig. 6. Hence the high-precision atom localization can originate from the field-induced interference effect in a modified reservoir, and the increase of the coupling-field intensity leads to stronger localization of the atom.
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