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Large Kerr nonlinearity in a crystal of molecular magnets system
Optics Communications 315 (2014) 116–121
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Large Kerr nonlinearity in a crystal of molecular magnets system Sahar Feili a, Hamid Reza Hamedi b,n a b
Department of Physics, Bandar Abbas Branch, Islamic Azad University, Bandar Abbas, Iran Institute of Theoretical Physics and Astronomy, Vilnius University, A. Gostauto 12, LT-01108 Vilnius, Lithuania
art ic l e i nf o
a b s t r a c t
Article history: Received 9 August 2013 Received in revised form 24 October 2013 Accepted 25 October 2013 Available online 7 November 2013
A novel scheme is proposed to investigate the possible giant Kerr nonlinearity in a crystal of molecular magnets. The crystal is subjected to one dc magnetic field and two probe and coupling ac magnetic fields. By studying the steady-state behavior of the medium, we show that an enhanced Kerr nonlinearity with negligible absorption can be achieved under condition of slow light levels, just by properly adjusting the coupling field. Also, the transient evolution of nonlinear dispersion is proposed. It is found that the frequency detunings of probe and coupling fields, as well as the intensity of coupling field, lead to the large Kerr nonlinearity. Our results can be used as a guideline for optimizing and controlling the switching process in the crystal of molecular magnets, which is much more practical than that in the atomic system because of its flexible design and the long relaxation times. & 2013 Elsevier B.V. All rights reserved.
Keywords: Kerr nonlinearity Crystal of molecular magnets Electromagnetically induced transparency
1. Introduction For many years, the weak nonlinear response of even the best materials has been a predominant limitation in research on quantum nonlinear optics. Fortunately, the field of electromagnetically induced transparency (EIT) [1,2] has opened up a completely new route to achieving large optical nonlinearity [3–7]. Two features of EIT i.e. vanishing resonant absorption and, simultaneously, a refractive index curve with a very steep gradient [1] can significantly enhance the nonlinear interaction strength in multilevel atomic systems. Therefore, it is desirable to use an EIT medium to achieve nonlinear optical processes at very low light intensities [6,7]. Kerr nonlinearity, corresponding to the refractive part of the third-order susceptibility, which results in an intensitydependent refractive index, is one of the most interesting nonlinear optical phenomena. Kerr nonlinearity has many useful applications, such as controlling light [8] and quantum information processing [9]. Several atomic configurations have been proposed theoretically and experimentally [10–13] for achieving giant Kerr nonlinearity with negligible absorption. Xiao et al. [14] present an experimental setup to study the enhanced Kerr nonlinear coefficient in a three level Λ-type atomic medium for different coupling beam powers. They show that the Kerr nonlinear coefficient behaves very differently in the strong and the weak coupling power regions and changes sign when the coupling or probe frequency detuning changes sign. The effect of SGC on
n
Corresponding author. Tel.: þ 98 9369341105; fax: þ98 4113347050. E-mail addresses: [email protected], [email protected] (H.R. Hamedi). 0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.10.077
Kerr nonlinearity behavior of Λ-type, V-type, and Ξ-type threelevel atomic system has been studied [15]. Four-level atomic schemes have been the subject of many works to achieve large Kerr nonlinearities [16–19]. For instance, in Ref. [20] a scheme for giant enhancement of the Kerr nonlinearity with negligible absorption in a four-level system with double dark resonances is also proposed. In one of our recent works [21], nonlinear response of a four-level N-type atomic system for a weak probe field was studied. It is demonstrated that the giant Kerr nonlinearity with reduced absorption can be achieved by spontaneously generated coherence. In addition, the effect of a relative phase between coupling fields on linear and nonlinear absorptions as well as Kerr nonlinearity is then discussed. Besides atomic systems, giant Kerr nonlinearity can occur in other kinds of systems, such as semiconductor structures that include quantum wells and quantum dots [22–28]. These studies have shown that electromagnetically induced transparency (EIT) [27] and tunneling induced transparency (TIT) [28] effects could significantly enhance the Kerr nonlinearity of the medium. On the other hand, a crystal of molecular magnets is characterized by strong crystal field anisotropy. Magnetic molecular clusters [29,30] nowadays attract intense experimental and theoretical investigations. The quantum coherence and interference effect can be realized in a magnetic molecule. In particular, quantum tunneling of magnetization, quantum magnetic phenomena at macroscopic scale, and quantum steps have been theoretically and experimentally studied [31–34]. There have been studies on the oscillations and wave propagations in systems of molecular magnets such as the phonon superradiance and phonon laser effect [35], the nonstationary behavior of a high-spin molecule in a bifrequency ac magnetic field [36], or in an acoustic wave and an
S. Feili, H.R. Hamedi / Optics Communications 315 (2014) 116–121
ac magnetic field [37], the parametric interaction of two acoustic waves in the presence of a strong ac magnetic field [38], the EIT [39], the nonlinear propagation of acoustic wave [40], and fourwave mixing via EIT [41]. The formation of optical bistability (OB) in a crystal of molecular magnets contained in a unidirectional ring cavity is investigated in Ref. [42]. Investigation on nonlinear optical properties of crystal of molecular magnets has been another interesting subject for researchers in this field of research. As the pioneering papers, two interesting articles [7,43] dealing with giant Kerr nonlinearities and solitons in systems of molecular magnets can be mentioned. For instance, Wu et al. investigated the nonlinear dynamics of four-wave mixing in molecular magnets, and showed that matched and coupled electromagnetic soliton pairs could be formed in molecular magnets via a four-wave mixing. It is shown that both bright and dark soliton pairs can propagate through a crystal of molecular magnets and their carrier frequencies are adjustable within the terahertz and sub-terahertz frequency regimes. Now, in this paper, we intend to study the Kerr nonlinearity behavior of such crystal of a molecular magnets structure, which interacts with a coupling field and a probe magnetic field. It is shown that by increasing the intensity of the coupling field, an enhanced Kerr nonlinearity with reduced absorption can be obtained under the condition of slow light levels. In addition, we theoretically calculate the nonlinear transient properties of the weak magnetic field of the medium. This letter is arranged as follows. In Section 2 we present the model and equations of the model. In Section 3 we provide an analytical solution. In Section 4, we present the calculation result of linear and nonlinear optical susceptibilities of the system. Finally, in the last section we give a discussion and summary of our main results.
2. Model and solution Consider a magnetic molecule (for instance, Mn12 acetate or Fe8 ), which is subjected to three fields: one dc magnetic field and two ac magnetic fields with different frequencies. The dc magnetic field H 0 and the first ac magnetic field (probe field) H p e iωp t are directed along the x-axis (medium anisotropy) of the magnetic field, while the second ac magnetic field (coupling field) H c e iωc t is parallel to the y-axis (hard anisotropy) of the magnetic molecule. H c ðH p Þ and ωc ðωp Þ are the amplitude and angular frequency, respectively, of the coupling (probe) ac magnetic field. Hamiltonian for the single molecular magnets can be selected as (ℏ ¼ 1) [44,45] ^ ¼H ^ tr DS^ 2 g μ H 0 S^ x þ H ^ I; H B z
ð1Þ
where z is the easy anisotropy axis, and S^ x ; S^ y and S^ z are the x; y ^ tr and z projections of the spin operator, respectively. Here, H denotes the operator of the transverse anisotropy energy, while ^ I corresponds the operator of interaction of the molecule with H the ac fields, and H 0 is the dc magnetic field. In addition, D is the longitudinal anisotropy constant, g is the Landé factor and μB is the Bohr magneton. Although the transverse anisotropy is weaker than the longitudinal one in magnetic molecules, in the general case, the molecule levels form doublets splits because of the dc magnetic field and the transverse anisotropy [46,47]. The molecule levels are defined as ε1 ; ε2 ; ε3 ; ::: and the corresponding eigenfunctions are Φ1 ; Φ2 ; Φ3 ; :::. We assume that the sample temperature is considerably low, T ⪡ ðε2 ε1 Þ=K B (where K B is the Boltzman constant); thus the three lowest levels of the magnetic molecule only are of interest. We suppose jωc ω21 j ⪡ ω21 and ωp ω20 ⪡ ω20 , respectively, which means that the coupling magnetic field frequency ωc and the probe magnetic field frequency ωp are close to the transition frequencies ω21 ¼ ðε2 ε1 Þ=ℏ and ω20 ¼ ðε2 ε0 Þ=ℏ.
117
Ω
Ω
Fig. 1. Energy level diagram and excitation scheme in a crystal of three-level molecular magnets interacting with coupling probe fields.
Therefore the magnetic molecule can be recognized as a threelevel system (Fig. 1) and described by the density matrix equation [48] i i ρ_ mn þ ½γ mn ðεn εm Þρmn þ ½H^ I ; ρ^ mn ¼ 0 ℏ
ℏ
2
i ρ_ nn þ ½H^ I ; ρ^ mn ¼ ∑ ½W nk ρkk W kn ρnn ℏ
k¼0
ð2Þ ð3Þ
where m a n; and m; n ¼ 0; 1; 2: gμ Here, H I ¼ 2 B ðH c S^ y e iωc t H p S^ x e iωp t Þ þ H c shows the operator of interaction of the magnetic molecule with the coupling and probe magnetic fields [49]. The density matrix elements and the relaxation constant of the matrix elements are characterized by ρmn and γ mn , respectively. By adopting the standard procedures [50], the density matrix equations of motion in rotating-wave approximations for the system can be written as follows:
ρ_ 20 ¼ ðγ 20 iΔp Þρ20 iΩp ðρ00 ρ22 Þ iΩc ρ10 ; ρ_ 10 ¼ ðγ 10 þ iðΔc Δp ÞÞρ10 iΩc ρ20 þ iΩp ρ12 ; ρ_ 21 ¼ ðγ 21 iΔc Þρ21 iΩp ρ01 iΩc ðρ11 ρ22 Þ; ρ_ 11 ¼ W 10 ρ00 þ W 12 ρ22 ðW 01 þ W 21 Þρ11 iΩc ðρ21 ρ12 Þ; ρ_ 22 ¼ W 20 ρ00 þ W 21 ρ11 ðW 02 þ W 12 Þρ22 þ iΩp ðρ20 ρ02 Þ þ iΩc ðρ21 ρ12 Þ; ρ00 þ ρ11 þ ρ22 ¼ 1:
ð4Þ
In this set of equations, Δp ¼ ωp ω20 is the detuning between the frequency of probe magnetic field and the transition frequency ω20 (taking ε0 ¼ ℏω0 ¼ 0 as energy origin) and Δc ¼ ωc ω21 is the detuning between the frequency of coupling magnetic field and the transition frequency ω21 . Here, the Rabi-frequencies of the probe and coupling fields are defined as Ωc ¼ g μB H c S^ 12 =2ℏ and Ωp ¼ gμB Hp S^ 02 =2ℏ, where S^ 12 ¼ 〈Φ1 jS^ y jΦ2 〉 and S^ 02 ¼ 〈Φ0 jS^ y jΦ2 〉.
3. Analytical solution Now, we derive analytical expressions for the first and third order susceptibilities. Using Maxwell's equations, one obtains the wave equation of the magnetic field in magnetic molecules: ! ∂2 H 1 ∂2 H 4π ∂2 M ¼ ð5Þ ∂z2 ν2 ∂t 2 ν2 ∂t 2 ! where M is the magnetization of the crystal, and ν is the wave velocity of the electromagnetic wave without taking into account ! the back action of the magnetization M . And the magnetization can be calculated as follows: ! ! M ¼ g μB NTrðρ^ S Þ
ð6Þ
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S. Feili, H.R. Hamedi / Optics Communications 315 (2014) 116–121
where N is the number of molecule per unit volume. Extracting the component of the magnetization at the frequency of the weak magnetic field ωp , we have ! M ¼ g μB N S^ 02 ρ20
ρð1Þ 20 ¼
iΩp ½γ þ iðΔc Δp Þ; Z 10
ð12Þ
ρð3Þ 20 ¼
iΩp ð2Þ ð2Þ ½Ωc ρð2Þ 12 þ ðγ 10 þ iðΔc Δp ÞÞðρ00 ρ22 Þ; Z
ð13Þ
ð7Þ
The magnetic susceptibility χ M is defined as follows: ! ! M ¼ χM H p
term ρ20 can be found as follows:
ð8Þ
An approximate expression for the weak magnetic field susceptibility can be written as follows: ð1Þ ð3Þ χM C χM þ χM jH p j2
ð9Þ
Using Eqs. (7)–(9), the first and third order magnetic susceptibilities can be defined as follows: ð1Þ χM ¼
ð3Þ χM ¼
Ng 2 μ2B S202 ð1Þ ρ20 2ℏΩp Ng 4 μ4B S402 8ℏ3 Ωp
3
ρð3Þ 20
ð10Þ
ð11Þ
We suppose that the probe magnetic field is much weaker than the coupling magnetic field; therefore, under weak probe field approximation, a simple analytical expression for the coherence Fig. 3. Transient evolution of Kerr nonlinearity for different values of Ωc . The other parameters values are the same as in Fig. 2.
Fig. 2. Transient evolution of probe absorption (a) and dispersion (b) for different values of Ωc . The parameters values are γ 21 ¼ γ 20 ¼ γ 10 ¼ 0:5γ, W 02 ¼ 3γ; W 12 ¼ γ, W 10 ¼ W 01 ¼ W 20 ¼ W 21 ¼ 0:05γ, and Ωp ¼ 0:01γ.
Fig. 4. Transient evolution of Kerr nonlinearity for different values of Δc (a) and Δp (b). The other parameters values are the same as in Fig. 2.
S. Feili, H.R. Hamedi / Optics Communications 315 (2014) 116–121
Z ¼ Δp þiΔp ðγ 10 þ γ 20 þ iΔc Þ γ 20 ðγ 10 þ iΔc Þ þ Ωc : 2
2
ð14Þ
Linear and nonlinear magnetic susceptibilities can be obtained by substituting Eqs. (12)–(14) into Eqs. (10) and (11). The real and ð1Þ imaginary parts of χ M correspond to the linear dispersion and absorption, respectively. In addition, the Kerr nonlinearity corresponds to the refraction part of the third-order susceptibility χ ð3Þ M , while the imaginary part of χ ð3Þ determines the nonlinear M absorption.
4. Results and discussion In the following, by using the numerical result from the density matrix equation of motions, we discuss the linear and nonlinear responses of the magnetic molecule medium by adjusting the coupling field strength. In the following numerical calculations, based on Ref. [42] we assume that γ 21 ¼ γ 20 ¼ γ 10 ¼ 0:5γ ; and W 02 ¼ 3γ ; W 12 ¼ γ ; W 10 ¼ W 01 ¼ W 20 ¼ W 21 ¼ 0:05γ , and all the parameters are scaled with γ . The numerical results are displayed ð3Þ in Figs. 2–7. First, we study the transient properties of χ ð1Þ M andχ M for various values of the intensity of coupling magnetic field. Fig. 2 shows the transient evolution of the linear probe absorption and dispersion for various values of Ωc . We find that linear absorption–dispersion curves have an oscillatory behavior for a short time and finally reach the steady state as time increases. In
Fig. 5. Linear (a) and nonlinear susceptibilities (b) versus Δp . Here, Ωc ¼ 0:3γ and the other parameters values are the same as in Fig. 2.
119
addition, when Ωc ¼ 0:5γ (dash line), the steady state value of absorption coefficient is positive which suggests a probe absorption, while if the steady state value of linear dispersion is negative then it corresponds to superluminal light propagation. However, increasing the intensity of magnetic coupling field to Ωc ¼ 5γ , the probe absorption vanishes at its steady state values and EIT appears. The linear dispersion curve in this case is plotted in Fig. 2(b). It is obvious that the steady-state value for linear dispersion becomes positive which corresponds to subluminal light propagation. Thus, we are able to convert superluminal light propagation to subluminal light propagation and reach EIT, just via the intensity of coupling field. The transient behavior of Kerr nonlinearity in this case is displayed in Fig. 3. Investigation in Fig. 3 shows that the oscillatory frequency of the Kerr nonlinearity curves increases, but the oscillatory amplitude decreases, and then it oscillates rapidly to a steady state value. Moreover, it can be seen that increasing the intensity of Ωc causes an exclamatory enhancement in the steady-state value of Kerr nonlinearity (see solid-line for Ωc ¼ 5γ ). Thus, an enhanced non-absorptive Kerr nonlinearity can be achieved in the condition of slow light levels, just via the intensity of magnetic coupling field. The effects of probe and coupling fields detunings Δp and Δc on transient evolution of Kerr nonlinearity are plotted in Fig. 4. It can be realized that by increasing the time, the oscillatory frequency and oscillatory amplitude of the Kerr nonlinearity curves increase. In addition, we observe that Kerr nonlinear coefficient attains larger values when increasing Δp and Δc to the larger values. Therefore, the
Fig. 6. Linear (a) and nonlinear susceptibilities (b) versus Δp . Here, Ωc ¼ γ and the other parameters values are the same as in Fig. 2.
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S. Feili, H.R. Hamedi / Optics Communications 315 (2014) 116–121
Fig. 7. Three-dimensional plot of the steady-state Kerr nonlinearity (a), linear (b) and nonlinear absorptions (c) spectra versus Ωc and Δp . The other parameters values are the same as in Fig. 2.
detuning parameters Δp and Δc have significant influence on controlling the Kerr nonlinearity. Now, we investigate the steady state properties of the medium through the controlling parameter of the system i.e. Ωc . Typical linear and nonlinear magnetic susceptibilities are displayed in Fig. 5. We focus on the region around zero probe field detuning Δp ¼ 0. It is found that the linear and nonlinear absorptions are very strong at Δp ¼ 0, and the slope of linear dispersion is negative, which corresponds to superluminal light propagation. Also, the Kerr nonlinearity value at Δp ¼ 0 is very small. This condition is not suitable for application of low intensity nonlinear optics. We again increase the intensity of Ωc . The new figures are plotted in Fig. 6. It can be realized that the Kerr nonlinearity enhances remarkably with respect to that in Fig. 5(b). In this case, the linear and nonlinear absorptions of probe field decrease at Δp ¼ 0 and the slope of linear dispersion switches to positive. This means that an enhanced subluminal Kerr nonlinearity with reduced absorption can be obtained. In order to further illustrate explicitly the dependence of the first and third order susceptibilities on the intensity of coupling field Ωc , we carry out a three-dimensional plot of the steady-state nonlinearity as well as linear and nonlinear absorptions spectra as a function of Ωc and Δp in Fig. 6. Figures show a good agreement with the above analysis. In other words, increasing Ωc leads to an efficient enhancement of Kerr nonlinearity. At the same time, the linear and nonlinear absorptions decrease significantly.
5. Conclusion With the help of density matrix equation, the transient and the steady-state behaviors of nonlinear susceptibility of a crystal of molecular magnets driven by the probe field and the coupling field were investigated. We found that large Kerr nonlinearity with negligible absorption can be obtained for weak magnetic field. Also, it is found that the frequency detunings of probe and coupling fields are very important parameters to manipulate and even enhance the Kerr nonlinearity of the crystal of molecular magnets system.
Acknowledgment This work has been supported by the Lithuanian Research Council. References [1] [2] [3] [4] [5] [6]
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Large Kerr nonlinearity in a crystal of molecular magnets system Sahar Feili a, Hamid Reza Hamedi b,n a b
Department of Physics, Bandar Abbas Branch, Islamic Azad University, Bandar Abbas, Iran Institute of Theoretical Physics and Astronomy, Vilnius University, A. Gostauto 12, LT-01108 Vilnius, Lithuania
art ic l e i nf o
a b s t r a c t
Article history: Received 9 August 2013 Received in revised form 24 October 2013 Accepted 25 October 2013 Available online 7 November 2013
A novel scheme is proposed to investigate the possible giant Kerr nonlinearity in a crystal of molecular magnets. The crystal is subjected to one dc magnetic field and two probe and coupling ac magnetic fields. By studying the steady-state behavior of the medium, we show that an enhanced Kerr nonlinearity with negligible absorption can be achieved under condition of slow light levels, just by properly adjusting the coupling field. Also, the transient evolution of nonlinear dispersion is proposed. It is found that the frequency detunings of probe and coupling fields, as well as the intensity of coupling field, lead to the large Kerr nonlinearity. Our results can be used as a guideline for optimizing and controlling the switching process in the crystal of molecular magnets, which is much more practical than that in the atomic system because of its flexible design and the long relaxation times. & 2013 Elsevier B.V. All rights reserved.
Keywords: Kerr nonlinearity Crystal of molecular magnets Electromagnetically induced transparency
1. Introduction For many years, the weak nonlinear response of even the best materials has been a predominant limitation in research on quantum nonlinear optics. Fortunately, the field of electromagnetically induced transparency (EIT) [1,2] has opened up a completely new route to achieving large optical nonlinearity [3–7]. Two features of EIT i.e. vanishing resonant absorption and, simultaneously, a refractive index curve with a very steep gradient [1] can significantly enhance the nonlinear interaction strength in multilevel atomic systems. Therefore, it is desirable to use an EIT medium to achieve nonlinear optical processes at very low light intensities [6,7]. Kerr nonlinearity, corresponding to the refractive part of the third-order susceptibility, which results in an intensitydependent refractive index, is one of the most interesting nonlinear optical phenomena. Kerr nonlinearity has many useful applications, such as controlling light [8] and quantum information processing [9]. Several atomic configurations have been proposed theoretically and experimentally [10–13] for achieving giant Kerr nonlinearity with negligible absorption. Xiao et al. [14] present an experimental setup to study the enhanced Kerr nonlinear coefficient in a three level Λ-type atomic medium for different coupling beam powers. They show that the Kerr nonlinear coefficient behaves very differently in the strong and the weak coupling power regions and changes sign when the coupling or probe frequency detuning changes sign. The effect of SGC on
n
Corresponding author. Tel.: þ 98 9369341105; fax: þ98 4113347050. E-mail addresses: [email protected], [email protected] (H.R. Hamedi). 0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.10.077
Kerr nonlinearity behavior of Λ-type, V-type, and Ξ-type threelevel atomic system has been studied [15]. Four-level atomic schemes have been the subject of many works to achieve large Kerr nonlinearities [16–19]. For instance, in Ref. [20] a scheme for giant enhancement of the Kerr nonlinearity with negligible absorption in a four-level system with double dark resonances is also proposed. In one of our recent works [21], nonlinear response of a four-level N-type atomic system for a weak probe field was studied. It is demonstrated that the giant Kerr nonlinearity with reduced absorption can be achieved by spontaneously generated coherence. In addition, the effect of a relative phase between coupling fields on linear and nonlinear absorptions as well as Kerr nonlinearity is then discussed. Besides atomic systems, giant Kerr nonlinearity can occur in other kinds of systems, such as semiconductor structures that include quantum wells and quantum dots [22–28]. These studies have shown that electromagnetically induced transparency (EIT) [27] and tunneling induced transparency (TIT) [28] effects could significantly enhance the Kerr nonlinearity of the medium. On the other hand, a crystal of molecular magnets is characterized by strong crystal field anisotropy. Magnetic molecular clusters [29,30] nowadays attract intense experimental and theoretical investigations. The quantum coherence and interference effect can be realized in a magnetic molecule. In particular, quantum tunneling of magnetization, quantum magnetic phenomena at macroscopic scale, and quantum steps have been theoretically and experimentally studied [31–34]. There have been studies on the oscillations and wave propagations in systems of molecular magnets such as the phonon superradiance and phonon laser effect [35], the nonstationary behavior of a high-spin molecule in a bifrequency ac magnetic field [36], or in an acoustic wave and an
S. Feili, H.R. Hamedi / Optics Communications 315 (2014) 116–121
ac magnetic field [37], the parametric interaction of two acoustic waves in the presence of a strong ac magnetic field [38], the EIT [39], the nonlinear propagation of acoustic wave [40], and fourwave mixing via EIT [41]. The formation of optical bistability (OB) in a crystal of molecular magnets contained in a unidirectional ring cavity is investigated in Ref. [42]. Investigation on nonlinear optical properties of crystal of molecular magnets has been another interesting subject for researchers in this field of research. As the pioneering papers, two interesting articles [7,43] dealing with giant Kerr nonlinearities and solitons in systems of molecular magnets can be mentioned. For instance, Wu et al. investigated the nonlinear dynamics of four-wave mixing in molecular magnets, and showed that matched and coupled electromagnetic soliton pairs could be formed in molecular magnets via a four-wave mixing. It is shown that both bright and dark soliton pairs can propagate through a crystal of molecular magnets and their carrier frequencies are adjustable within the terahertz and sub-terahertz frequency regimes. Now, in this paper, we intend to study the Kerr nonlinearity behavior of such crystal of a molecular magnets structure, which interacts with a coupling field and a probe magnetic field. It is shown that by increasing the intensity of the coupling field, an enhanced Kerr nonlinearity with reduced absorption can be obtained under the condition of slow light levels. In addition, we theoretically calculate the nonlinear transient properties of the weak magnetic field of the medium. This letter is arranged as follows. In Section 2 we present the model and equations of the model. In Section 3 we provide an analytical solution. In Section 4, we present the calculation result of linear and nonlinear optical susceptibilities of the system. Finally, in the last section we give a discussion and summary of our main results.
2. Model and solution Consider a magnetic molecule (for instance, Mn12 acetate or Fe8 ), which is subjected to three fields: one dc magnetic field and two ac magnetic fields with different frequencies. The dc magnetic field H 0 and the first ac magnetic field (probe field) H p e iωp t are directed along the x-axis (medium anisotropy) of the magnetic field, while the second ac magnetic field (coupling field) H c e iωc t is parallel to the y-axis (hard anisotropy) of the magnetic molecule. H c ðH p Þ and ωc ðωp Þ are the amplitude and angular frequency, respectively, of the coupling (probe) ac magnetic field. Hamiltonian for the single molecular magnets can be selected as (ℏ ¼ 1) [44,45] ^ ¼H ^ tr DS^ 2 g μ H 0 S^ x þ H ^ I; H B z
ð1Þ
where z is the easy anisotropy axis, and S^ x ; S^ y and S^ z are the x; y ^ tr and z projections of the spin operator, respectively. Here, H denotes the operator of the transverse anisotropy energy, while ^ I corresponds the operator of interaction of the molecule with H the ac fields, and H 0 is the dc magnetic field. In addition, D is the longitudinal anisotropy constant, g is the Landé factor and μB is the Bohr magneton. Although the transverse anisotropy is weaker than the longitudinal one in magnetic molecules, in the general case, the molecule levels form doublets splits because of the dc magnetic field and the transverse anisotropy [46,47]. The molecule levels are defined as ε1 ; ε2 ; ε3 ; ::: and the corresponding eigenfunctions are Φ1 ; Φ2 ; Φ3 ; :::. We assume that the sample temperature is considerably low, T ⪡ ðε2 ε1 Þ=K B (where K B is the Boltzman constant); thus the three lowest levels of the magnetic molecule only are of interest. We suppose jωc ω21 j ⪡ ω21 and ωp ω20 ⪡ ω20 , respectively, which means that the coupling magnetic field frequency ωc and the probe magnetic field frequency ωp are close to the transition frequencies ω21 ¼ ðε2 ε1 Þ=ℏ and ω20 ¼ ðε2 ε0 Þ=ℏ.
117
Ω
Ω
Fig. 1. Energy level diagram and excitation scheme in a crystal of three-level molecular magnets interacting with coupling probe fields.
Therefore the magnetic molecule can be recognized as a threelevel system (Fig. 1) and described by the density matrix equation [48] i i ρ_ mn þ ½γ mn ðεn εm Þρmn þ ½H^ I ; ρ^ mn ¼ 0 ℏ
ℏ
2
i ρ_ nn þ ½H^ I ; ρ^ mn ¼ ∑ ½W nk ρkk W kn ρnn ℏ
k¼0
ð2Þ ð3Þ
where m a n; and m; n ¼ 0; 1; 2: gμ Here, H I ¼ 2 B ðH c S^ y e iωc t H p S^ x e iωp t Þ þ H c shows the operator of interaction of the magnetic molecule with the coupling and probe magnetic fields [49]. The density matrix elements and the relaxation constant of the matrix elements are characterized by ρmn and γ mn , respectively. By adopting the standard procedures [50], the density matrix equations of motion in rotating-wave approximations for the system can be written as follows:
ρ_ 20 ¼ ðγ 20 iΔp Þρ20 iΩp ðρ00 ρ22 Þ iΩc ρ10 ; ρ_ 10 ¼ ðγ 10 þ iðΔc Δp ÞÞρ10 iΩc ρ20 þ iΩp ρ12 ; ρ_ 21 ¼ ðγ 21 iΔc Þρ21 iΩp ρ01 iΩc ðρ11 ρ22 Þ; ρ_ 11 ¼ W 10 ρ00 þ W 12 ρ22 ðW 01 þ W 21 Þρ11 iΩc ðρ21 ρ12 Þ; ρ_ 22 ¼ W 20 ρ00 þ W 21 ρ11 ðW 02 þ W 12 Þρ22 þ iΩp ðρ20 ρ02 Þ þ iΩc ðρ21 ρ12 Þ; ρ00 þ ρ11 þ ρ22 ¼ 1:
ð4Þ
In this set of equations, Δp ¼ ωp ω20 is the detuning between the frequency of probe magnetic field and the transition frequency ω20 (taking ε0 ¼ ℏω0 ¼ 0 as energy origin) and Δc ¼ ωc ω21 is the detuning between the frequency of coupling magnetic field and the transition frequency ω21 . Here, the Rabi-frequencies of the probe and coupling fields are defined as Ωc ¼ g μB H c S^ 12 =2ℏ and Ωp ¼ gμB Hp S^ 02 =2ℏ, where S^ 12 ¼ 〈Φ1 jS^ y jΦ2 〉 and S^ 02 ¼ 〈Φ0 jS^ y jΦ2 〉.
3. Analytical solution Now, we derive analytical expressions for the first and third order susceptibilities. Using Maxwell's equations, one obtains the wave equation of the magnetic field in magnetic molecules: ! ∂2 H 1 ∂2 H 4π ∂2 M ¼ ð5Þ ∂z2 ν2 ∂t 2 ν2 ∂t 2 ! where M is the magnetization of the crystal, and ν is the wave velocity of the electromagnetic wave without taking into account ! the back action of the magnetization M . And the magnetization can be calculated as follows: ! ! M ¼ g μB NTrðρ^ S Þ
ð6Þ
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where N is the number of molecule per unit volume. Extracting the component of the magnetization at the frequency of the weak magnetic field ωp , we have ! M ¼ g μB N S^ 02 ρ20
ρð1Þ 20 ¼
iΩp ½γ þ iðΔc Δp Þ; Z 10
ð12Þ
ρð3Þ 20 ¼
iΩp ð2Þ ð2Þ ½Ωc ρð2Þ 12 þ ðγ 10 þ iðΔc Δp ÞÞðρ00 ρ22 Þ; Z
ð13Þ
ð7Þ
The magnetic susceptibility χ M is defined as follows: ! ! M ¼ χM H p
term ρ20 can be found as follows:
ð8Þ
An approximate expression for the weak magnetic field susceptibility can be written as follows: ð1Þ ð3Þ χM C χM þ χM jH p j2
ð9Þ
Using Eqs. (7)–(9), the first and third order magnetic susceptibilities can be defined as follows: ð1Þ χM ¼
ð3Þ χM ¼
Ng 2 μ2B S202 ð1Þ ρ20 2ℏΩp Ng 4 μ4B S402 8ℏ3 Ωp
3
ρð3Þ 20
ð10Þ
ð11Þ
We suppose that the probe magnetic field is much weaker than the coupling magnetic field; therefore, under weak probe field approximation, a simple analytical expression for the coherence Fig. 3. Transient evolution of Kerr nonlinearity for different values of Ωc . The other parameters values are the same as in Fig. 2.
Fig. 2. Transient evolution of probe absorption (a) and dispersion (b) for different values of Ωc . The parameters values are γ 21 ¼ γ 20 ¼ γ 10 ¼ 0:5γ, W 02 ¼ 3γ; W 12 ¼ γ, W 10 ¼ W 01 ¼ W 20 ¼ W 21 ¼ 0:05γ, and Ωp ¼ 0:01γ.
Fig. 4. Transient evolution of Kerr nonlinearity for different values of Δc (a) and Δp (b). The other parameters values are the same as in Fig. 2.
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Z ¼ Δp þiΔp ðγ 10 þ γ 20 þ iΔc Þ γ 20 ðγ 10 þ iΔc Þ þ Ωc : 2
2
ð14Þ
Linear and nonlinear magnetic susceptibilities can be obtained by substituting Eqs. (12)–(14) into Eqs. (10) and (11). The real and ð1Þ imaginary parts of χ M correspond to the linear dispersion and absorption, respectively. In addition, the Kerr nonlinearity corresponds to the refraction part of the third-order susceptibility χ ð3Þ M , while the imaginary part of χ ð3Þ determines the nonlinear M absorption.
4. Results and discussion In the following, by using the numerical result from the density matrix equation of motions, we discuss the linear and nonlinear responses of the magnetic molecule medium by adjusting the coupling field strength. In the following numerical calculations, based on Ref. [42] we assume that γ 21 ¼ γ 20 ¼ γ 10 ¼ 0:5γ ; and W 02 ¼ 3γ ; W 12 ¼ γ ; W 10 ¼ W 01 ¼ W 20 ¼ W 21 ¼ 0:05γ , and all the parameters are scaled with γ . The numerical results are displayed ð3Þ in Figs. 2–7. First, we study the transient properties of χ ð1Þ M andχ M for various values of the intensity of coupling magnetic field. Fig. 2 shows the transient evolution of the linear probe absorption and dispersion for various values of Ωc . We find that linear absorption–dispersion curves have an oscillatory behavior for a short time and finally reach the steady state as time increases. In
Fig. 5. Linear (a) and nonlinear susceptibilities (b) versus Δp . Here, Ωc ¼ 0:3γ and the other parameters values are the same as in Fig. 2.
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addition, when Ωc ¼ 0:5γ (dash line), the steady state value of absorption coefficient is positive which suggests a probe absorption, while if the steady state value of linear dispersion is negative then it corresponds to superluminal light propagation. However, increasing the intensity of magnetic coupling field to Ωc ¼ 5γ , the probe absorption vanishes at its steady state values and EIT appears. The linear dispersion curve in this case is plotted in Fig. 2(b). It is obvious that the steady-state value for linear dispersion becomes positive which corresponds to subluminal light propagation. Thus, we are able to convert superluminal light propagation to subluminal light propagation and reach EIT, just via the intensity of coupling field. The transient behavior of Kerr nonlinearity in this case is displayed in Fig. 3. Investigation in Fig. 3 shows that the oscillatory frequency of the Kerr nonlinearity curves increases, but the oscillatory amplitude decreases, and then it oscillates rapidly to a steady state value. Moreover, it can be seen that increasing the intensity of Ωc causes an exclamatory enhancement in the steady-state value of Kerr nonlinearity (see solid-line for Ωc ¼ 5γ ). Thus, an enhanced non-absorptive Kerr nonlinearity can be achieved in the condition of slow light levels, just via the intensity of magnetic coupling field. The effects of probe and coupling fields detunings Δp and Δc on transient evolution of Kerr nonlinearity are plotted in Fig. 4. It can be realized that by increasing the time, the oscillatory frequency and oscillatory amplitude of the Kerr nonlinearity curves increase. In addition, we observe that Kerr nonlinear coefficient attains larger values when increasing Δp and Δc to the larger values. Therefore, the
Fig. 6. Linear (a) and nonlinear susceptibilities (b) versus Δp . Here, Ωc ¼ γ and the other parameters values are the same as in Fig. 2.
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Fig. 7. Three-dimensional plot of the steady-state Kerr nonlinearity (a), linear (b) and nonlinear absorptions (c) spectra versus Ωc and Δp . The other parameters values are the same as in Fig. 2.
detuning parameters Δp and Δc have significant influence on controlling the Kerr nonlinearity. Now, we investigate the steady state properties of the medium through the controlling parameter of the system i.e. Ωc . Typical linear and nonlinear magnetic susceptibilities are displayed in Fig. 5. We focus on the region around zero probe field detuning Δp ¼ 0. It is found that the linear and nonlinear absorptions are very strong at Δp ¼ 0, and the slope of linear dispersion is negative, which corresponds to superluminal light propagation. Also, the Kerr nonlinearity value at Δp ¼ 0 is very small. This condition is not suitable for application of low intensity nonlinear optics. We again increase the intensity of Ωc . The new figures are plotted in Fig. 6. It can be realized that the Kerr nonlinearity enhances remarkably with respect to that in Fig. 5(b). In this case, the linear and nonlinear absorptions of probe field decrease at Δp ¼ 0 and the slope of linear dispersion switches to positive. This means that an enhanced subluminal Kerr nonlinearity with reduced absorption can be obtained. In order to further illustrate explicitly the dependence of the first and third order susceptibilities on the intensity of coupling field Ωc , we carry out a three-dimensional plot of the steady-state nonlinearity as well as linear and nonlinear absorptions spectra as a function of Ωc and Δp in Fig. 6. Figures show a good agreement with the above analysis. In other words, increasing Ωc leads to an efficient enhancement of Kerr nonlinearity. At the same time, the linear and nonlinear absorptions decrease significantly.
5. Conclusion With the help of density matrix equation, the transient and the steady-state behaviors of nonlinear susceptibility of a crystal of molecular magnets driven by the probe field and the coupling field were investigated. We found that large Kerr nonlinearity with negligible absorption can be obtained for weak magnetic field. Also, it is found that the frequency detunings of probe and coupling fields are very important parameters to manipulate and even enhance the Kerr nonlinearity of the crystal of molecular magnets system.
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