Issue 
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 1, February 2024



Page(s)  67  73  
DOI  https://doi.org/10.1051/wujns/2024291067  
Published online  15 March 2024 
Physics
CLC number: O431.2
Influence of Kerr Effect on SecondOrder Nonlinearity Induced Transparency
^{1}
College of Architectural Engineering, Zhengzhou Business University, Zhengzhou 451200, Henan, China
^{2}
Quantum Information Research Center, Shangrao Normal University, Shangrao 334001, Jiangxi, China
Received:
9
May
2023
We theoretically study the effect of Kerr effect on the secondorder nonlinearity induced transparency in a doubleresonant optical cavity system. We show that in the presence of the Kerr effect, as the strength of the Kerr effect increases, the absorption curve exhibits an asymmetricsymmetricasymmetric transition, and the zero absorption point shifts with the increase of the Kerr effect. Furthermore, by changing the strength of the Kerr effect, we can control the width of the transparent window, and the position of the zeroabsorption point and meanwhile change the left and right width of the absorption peak. The asymmetry absorption curve can be employed to improve the quality factor of the cavity when the frequency detuning is tuned to be around the right peak. The simple dependence of the zeroabsorption point on the strength of Kerr effect suggests that the strength of Kerr effect can be measured by measuring the position of the zeroabsorption point in a possible application.
Key words: Kerr effect / secondorder nonlinearity induced transparency / nonlinear optics / quantum optics
Cite this article: ZHANG Zhiqiang, ZHOU Yanhui. Influence of Kerr Effect on SecondOrder Nonlinearity Induced Transparency[J]. Wuhan Univ J of Nat Sci, 2024, 29(1): 6773.
Biography: ZHANG Zhiqiang, male, Associate professor, research direction: computational physics and nonlinear physics. Email: zhangzhiqiang08@gmail.com
Fundation item: Supported by the Key Scientific Research Plan of Colleges and Universities in Henan Province (23B140006), and the National Natural Science Foundation of China (11965017)
© Wuhan University 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
Electromagnetically induced transparency (EIT)^{[1]} is a technique for turning an opaque medium into a transparent one as a result of quantum interference between two excited processes from independent states. Since it is proposed in the context of atomic systems, the first experimental realization of EIT is also in atomic systems^{[2, 3]}. Recently, Weis et al^{[4]} realized induced transparency in an optomechanical system for controlling the transport of light through the cavity by coupling a cavity with a light pulse. This technology is called optomechanically induced transparency (OMIT)^{[58]}. Therefore, searching for possible induced transparent systems has become an active research topic in quantum optics^{[9, 10]}.
With recent technological advances, secondorder nonlinearities have become a hot research area in modern quantum optics and quantum information processing. Secondorder nonlinearity can be achieved with high nonlinear materials such as IIIV semiconductors (e.g., GaAs, GaN, GaP, etc.)^{[11, 12]}. In 2017, Zhou et al^{[13]} theoretically predicted another form of induced transparency caused by secondorder nonlinearity, which is called secondorder nonlinearity induced transparency (SONIT). They discussed the impact of secondorder nonlinearity on SONIT and found that, unlike EIT and OMIT, the absorption spectrum of SOINT exhibits asymmetric features due to the presence of photon asymmetric switching induced by secondorder nonlinearity, and a potential application is the measurement of secondorder nonlinear coefficients.
In particular, it should be pointed out that introduction of Kerr medium in the optomechanical system can not only enhance the optomechanical coupling, but also prevent the suppression of nonlinearity^{[14]}, which makes the system more excellent in multiple dimensions^{[1517]}. Jiao et al^{[18]} found that the frequency shift induced by the Kerr effect leads to an asymmetric OMIT signal absorption window, which may enhance the performance of OMIT devices with various nonlinear optical cavities. Xiong et al^{[15]} introduced the crossKerr effect in the optomechanical system, and found that the crossKerr effect leads to the asymmetry of the absorption curve, which can be used to design a cavity with a higher quality factor.
Inspired by these studies, we introduce the Kerr medium in the doubleresonant optical cavity system to study the influence of the Kerr effect on SONIT. Firstly, the Hamiltonian of the SOINT system including the Kerr effect is given. Secondly, the dispersion and absorption coefficients of the SOINT system are solved by analytical methods. Finally, the influence of the Kerr effect on SOINT is discussed. We find that the absorption curve exhibits an asymmetricsymmetricasymmetric transition, and the zero absorption point shifts as the Kerr effect increases. The width of the transparent window and the position of the zeroabsorption point can be controlled by changing the strength of the Kerr effect. Possible applications of these results are to improve the quality factor of the cavity and to measure the strength of the Kerr effect.
The structure of this paper is as follows. In Section 1, the theoretical model is introduced, the absorption coefficient and dispersion coefficient of the system is obtained by theoretical methods. In Section 2, the influence of the Kerr effect on the SOINT, the position of the zeroabsorption point and the width of the transparent window are discussed. In Section 3, a short summary is given.
1 Model and Theoretically Analysis
We consider two singlemode cavities A and B, coupled through a secondorder nonlinear medium , where the secondorder nonlinear medium mediates the conversion of two photons in cavity mode A into one photon in cavity mode B. The frequencies of the two cavity modes are and , respectively, and we have . The scheme of a doubly resonant cavity is shown in Fig. 1.
Fig. 1 Scheme of a doubly resonant cavity mode of nonlinearity material is the driving strength of strong control light, the frequency is , and is the driving strength of weak probe light, and the frequency is 
Considering the Kerr effect coefficient of the medium is , the Hamiltonian of the system can be written as
where denotes the creation (annihilation) operator of cavity mode A, denotes the creation (annihilation) operator of cavity mode B, denotes the coefficient of secondorder nonlinear interactions, which can be calculated directly from ^{[13]}, denotes the Kerr effect coefficient of the medium. The last term of Eq. (1) represents the two drives of the external light to the cavity, one is the strong control light, the intensity is , the frequency is , and the other is the weak probe light, the intensity is , and the frequency is .
To study the dynamics of the system, we want to turn to a rotating framework dominated by operators , then we get an effective Hamiltonian . The effective Hamiltonian can be written as follows:
where , .
Using the Heisenberg equations for the cavity variables, here . For convenience, we set , then we get
where and are decay rates of cavity mode A and cavity mode B, respectively, and are the corresponding input fields. It is important to point out that the input field has a mean value of zero, that is , and .
We take the mean values of Eq. (3), and take the notation , , and , here the asterisk "^{*}" represents complex conjugation. Then Eq. (3) can be rewritten as
Then we apply the linearization approach by assuming that each operator mean value in the system can be written as a sum of its steadystate mean value and deviations from the stationary state, i.e., and . We should note that both and are small quantity. Considering the control field is much stronger than the probe field, which is , is regarded as a small quantity, too.
The steadystate mean values of the system can be obtained from Eq. (4) as
when the highorder small quantity is ignored, the deviations from the stationary state can be obtained as
For Eq. (5), it is assumed that the solution can be expressed as and , where , , , and are real numbers, and substitute them into Eq. (5), then separate the real and imaginary part, we can get
To our knowledge, the analytical solution of Eq. (7) is difficult to obtain, we can solve it numerically.
For Eq. (6), we suppose the solutions take the form of
By substituting Eq. (8) into Eq. (6), we obtain
The transmission rate of the probe field depends only on . By solving Eq. (9), we get
where , , , , , , and .
To simplify the discussion, we can consider that the output of probe light in the cavity is equal to the input probe light minus the light dissipated in the cavity. Applying the inputoutput relationship of the cavity^{[4, 19]}, we can get the output field
Here, we refer to the probe transmission in the general sense of the ratio of the probe field returned from the system divided by the sent probe field. The transmission rate of the probe field is given by^{[4, 19]}
which could be measured by the homodyne technique^{[20]}. The corresponding quadrature field can be defined as^{[21]}
Its real part is the absorption coefficient, and the imaginary part is the dispersion coefficient.
2 Results and Discussions
We have learned that, as the Kerr coefficient for thirdorder nonlinear effects in nonlinear materials^{[2228]}, it is usually a small quantity. In order to discuss its effect on secondorder nonlinearity induced transparency, we use available experimental data to estimate the magnitude of the Kerr nonlinearity coefficient. The Kerr coefficient can be expressed as ^{[29]}, where is the effective cavity mode volume, is the frequency of the optical mode, and is the relative dielectric permittivity.
The highly nonlinear materials currently used in experiments mainly include silicon (Si), germanium (Ge), silicon dioxide (SiO_{2}) or gallium arsenide (GaAs)^{[30]}. Typical orders of magnitude for of these materials are in the range of m^{2}/V^{2}. In addition, larger values can be found in certain nanoparticlesdoped glasses, chalcogenide glasses, or other polymeric materials^{[31, 32]}. For a typical parameter setting, such as the data used in Ref. [29], , , , m^{2}/V^{2}, then the Kerr coefficient is about . As a typical value for the decay rate of cavity mode A, taking , we get . Considering that the smaller the value of is, the better it can be realized experimentally, we set the value of in the range of 0 in the followup discussion of this article.
Now, we can discuss the influence of Kerr effect on the secondorder nonlinearity induced transparency. We plot absorption coefficient and dispersion coefficient as a function of frequency detuning with different values of Kerr effect coefficient , as shown in Fig. 2.
Fig. 2 The absorption coefficient and dispersion coefficient as a function of with different Kerr effect coefficient (a) , (b) , (c) , (d) , (e) , and (f) . Values of other parameters are , , , and . For convenience, we rescale all the parameters with respect to the decay rate 
First of all, we set , that is to say, there is no Kerr effect, the corresponding graph of absorption and dispersion coefficients as a function of is shown in Fig. 2(a). From Fig. 2(a), it can be concluded that the secondorder nonlinearity induced transparency can be observed at point in the system without Kerr effect. At that point, both absorption coefficient and dispersion coefficient are equal to zero, which means that the absorption of the medium is almost zero, and the refractive index of the medium is close to 1, so the medium becomes transparent under the action of a strong coherent field. We have also found that, unlike electromagnetically induced transparency and optomechanically induced transparency, the absorption curve of secondorder nonlinearity induced transparency is asymmetric. This is due to the presence of asymmetric switching of photons caused by the secondorder nonlinearity.
When Kerr effect is introduced, we set , the position of the zeroabsorption point is shifted to and the absorption curve becomes almost symmetric about the zeroabsorption point, as shown in Fig. 2(b). This suggests that the Kerr effect can balance the secondorder nonlinearity, resulting in a leftright symmetric absorption curve. When the strength of Kerr effect increases, we set , the position of the zeroabsorption point moves to , as shown in Fig. 2(c), and the absorption peak on the left becomes wider and the absorption peak on the right becomes narrower, the absorption curve becomes asymmetric again. When we further increase the strength of Kerr effect to , the corresponding absorption and dispersion curves are shown in Fig. 2(d). At this time, the position of the zeroabsorption point moves to , and the absorption peak on the left side continues to widen and the one on the right side further narrows.
We continue to increase the strength of Kerr effect to , in which case the position of the zeroabsorption point is moved to . When we set Kerr effect , the zeroabsorption point is at . The corresponding images are shown in Fig. 2(e) and 1(f), respectively. At the same time, we can also see from the Fig. 2(e) and 1(f) that the left absorption peak is getting wider and wider, while the right absorption peak is getting narrower and narrower. Many interesting applications of EIT arise from the spectral properties of dark resonances^{[33]}, i.e., narrow transmission windows in opaque media. This result suggests that the Kerr effect may be employed to improve the quality factor of the cavity when the frequency detuning is tuned to be around the right peak.
Based on the above discussion, we find that the zeroabsorption point moves to the right side with the increase of the strength of Kerr effect . Therefore, we plot the zeroabsorption point as a function of in Fig. 3. Here, we define the point where the absorption coefficient and the dispersion coefficient are both equal to zero as zeroabsorption point . It can be seen from Fig. 3 that the zero absorption point first increases rapidly with the enhancement of the Kerr effect strength , and then slowly increases with the increase of . These results show that by adjusting the strength of Kerr effect, the position of the zeroabsorption point can be controlled. The simple dependence shown in Fig. 3 indicates that the strength of Kerr effect can be measured by measuring the position of the zeroabsorption point in a possible application.
Fig. 3 The zeroabsorption point as a function of The other parameters used are the same as those in Fig. 2 
Observing Fig. 2, we can see that the left and right absorption peaks have a maximum value respectively, and the positions of these two maximum values which can be evaluated by
where and are the two points corresponding to the maximums of the left and right absorption peaks. Here, we define the width of transparent window . In order to discuss the variation of the width of transparent window with the Kerr effect strength, we plot the width of transparent window as a function of in Fig. 4. We can see from Fig. 4 that, at the beginning, the width of transparent window decreases rapidly with the increase of . The minimum value of the width of transparent window is , and the corresponding Kerr effect strength is . After that, the width of transparent window increases almost linearly with Kerr effect strength . This result also reminds us of the possibility of detecting the Kerr effect strength by measuring the width of transparent window in the absorption spectrum of the output field, which may have important applications in the future.
Fig. 4 The width of transparent window as a function of The other parameters used are the same as those in Fig. 2 
3 Conclusion
In conclusion, we theoretically investigate the impacts of the Kerr effect on the secondorder nonlinearity induced transparency in a doubleresonant optical cavity coupled by a secondorder nonlinearity material. Concerning the secondorder nonlinearity induced transparency phenomenon, we have predicted an asymmetricsymmetricasymmetric absorption curve transition and a shift of the zeroabsorption point with the increasing of Kerr effect strength. This asymmetry absorption curve can be employed to improve the quality factor of the cavity when the frequency detuning is tuned to be around the right peak. The strength of Kerr effect can be measured by measuring the position of the zeroabsorption point. Therefore, by changing the strength of the Kerr effect, we can control the width of the transparent window, the position of the zeroabsorption point and change the left and right width of the absorption peak, which may have potential practical applications in the future.
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All Figures
Fig. 1 Scheme of a doubly resonant cavity mode of nonlinearity material is the driving strength of strong control light, the frequency is , and is the driving strength of weak probe light, and the frequency is 

In the text 
Fig. 2 The absorption coefficient and dispersion coefficient as a function of with different Kerr effect coefficient (a) , (b) , (c) , (d) , (e) , and (f) . Values of other parameters are , , , and . For convenience, we rescale all the parameters with respect to the decay rate 

In the text 
Fig. 3 The zeroabsorption point as a function of The other parameters used are the same as those in Fig. 2 

In the text 
Fig. 4 The width of transparent window as a function of The other parameters used are the same as those in Fig. 2 

In the text 
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