Control of light in a quantized four level graphene atomic system via self and cross-Kerr nonlinearity

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Control of light in a quantized four level graphene atomic system via self and cross-Kerr nonlinearity Usman Khan a , Bakht Amin a , Kefayat Ullah b , Arif Ullah a,∗ a b

Quantum Optics and Quantum Information Research Group, Department of Physics, University of Malakand, Khyber Pakhtunkhwa, Pakistan Department of Applied Physical and Material Sciences, University of Swat, Khyber Pakhtunkhwa, Pakistan

a r t i c l e

i n f o

Article history: Received 8 June 2019 Received in revised form 17 September 2019 Accepted 17 September 2019 Available online xxxx Communicated by R. Wu Keywords: Nonlinear optics Optical Kerr effect Graphene

a b s t r a c t We investigate self and cross-Kerr nonlinearity in a four level quantized graphene atomic medium. The absorption, dispersion, transmission and subluminal/superluminal behaviors of a probe light field are studied. An amplification of the probe light field is observed in the absorption spectrum. The normal and anomalous slope of dispersion is also investigated at the positive/negative absorption region. It is shown that Kerr nonlinearity invert and enhance the subluminal/superluminal behaviors of the pulse and self-Kerr effect is found to be more subluminal/superluminal as compared to cross-Kerr effect. The results show significant applications in information storage, self and cross phase modulation and lasing without inversion. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Light-matter interaction has been studied extensively over the years and is the key to fundamental and applied quantum physics research [1–3]. The absorption and scattering processes, in which two or more than two light quanta are involved have been of great interest in the last few decades [4–7]. Moreover, the intense and coherent light sources available from lasers have made possible the experimental observation of emerging ideas in quantum optics such as optical sensing, optical metrology, quantum information processing and many more [8–10]. Sources of light having higher intensities can dramatically change the properties of a medium and leads to nonlinear effects. In nonlinear optics, the response of a material to incident electromagnetic wave is usually described in terms of polarization. Polarization is related to the incoming fields by nonlinear susceptibility. Consequently, the susceptibility is a measure of how much polarization is built up in the medium by the incident light fields. Recently, nonlinear optical techniques have become well established tools to study the properties of various media [11]. Using the absorption and dispersion effects, Asadpour et al. [12] have proposed a model for investigating nonlinear properties of light in graphene nanostructure. Graphene has got some outstanding optical properties that include transmission, absorption and reflection

*

Corresponding author. E-mail address: [email protected] (A. Ullah).

https://doi.org/10.1016/j.physleta.2019.125998 0375-9601/© 2019 Elsevier B.V. All rights reserved.

characteristics [13–29]. In graphene, these effects lead to some of the important optical phenomena e.g. electromagnetically induced transparency (EIT) of atomic system, lasing without inversion and four wave mixing [30–35]. The feature of EIT medium which is transparent to the applied fields makes it conditionally applicable in nonlinear optical science. In the field of nonlinear optics, Kerr non-linearity plays an important role in the investigation of other advanced quantum phenomena such as quantum non-demolition measurements [36], quantum bit re-generation [37], quantum state teleportation [38], and the generation of the optical solitons [39]. Kerr non-linearity of the medium corresponds to the real part of third-order susceptibility of the probe light which propagates through it. In this work, which is based on a quantized four level graphene structure, we study the effect of the medium on an applied field in the presence of optical Kerr effect. 2. Model and the basic formalism In order to investigate self and cross Kerr effect, we consider a four-level atomic configuration of graphene medium as discussed in [40–42] and is shown in Fig. 1. In this configuration the probe field E p , having Rabi frequency  p and detuning  p is coupled with the states |1 and |4. The states |2 and |3 are coupled by control field E 1 having Rabi frequency 1 and detuning 1 . Furthermore, the |3 and |4 states are coupled by control field E 2 having Rabi frequency 2 and detuning 2 . The control fields are elliptically polarized fields formed by left and right circularly polarized components as discussed in [42]. The detuning of these fields

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2

Where, χ is the linear electric susceptibility, χck represent crossKerr nonlinear susceptibility while χsk is the self-Kerr nonlinear susceptibility, N represent the number of atoms and μ14 is the dipole moment. Self Kerr nonlinearity is due to the intensity of the probe field itself, while the cross Kerr nonlinearity is due to the intensity of other control field. The self Kerr nonlinearity can be obtained from the third order susceptibility as χsk = χ (1) + I s χ (3) , with I s = | E p |2 [43]. Cross Kerr nonlinearity on the other hand is due to the control field and can be obtained from the rela∂χ tion χck = χ 0 + I c ∂ I , where I c = |c |2 [44,45]. c is the Rabi c frequency of the elliptically polarized coupling fields that drives the transitions |4 to |3 and |3 to |2 and can be defined as 1 = c (cos θ − sin θ)e −i θ and 2 = c (cos θ + sin θ)e i θ , where c = μ√E 0 , θ is ellipticity parameter and E 0 = E + σ + + E − σ − . E +

Fig. 1. Model system for a four level atomic system of graphene.

2 2h¯

are related to their corresponding angular frequencies and atomic states resonance frequencies as:  p = ω14 − ω p , 2 = ω34 − ω2 and 1 = ω23 − ω1 . In order to study dynamics of the system, we consider the standard time evolution equation for Dirac electrons in graphene as:

∂ ρˆ i = − [ H I , ρˆ ] + Rˆ (ρˆ ), ∂t h¯

(1)

where, H I is interaction Hamiltonian and Rˆ (ρˆ ) represent incoherent relaxations as described in [41,42]. The resulting coupled rate equations in the quantum mechanical density matrix approach are given as: · ∼

1

i

i

2

2

2

31 , ρ 41 = −[i  p + γ4 ]ρ41 +  p (ρ11 − ρ44 ) + 2 ρ · ∼

1

i

i

2

2

2

2

+

1 4

1 4

A 3 exp(−2i θ)c2 (cos θ − sin θ)2 ,

and A 1 , A 2 , A 3 are given as:

A 1 = −(i  p +

1 2

γ4 ),

2

31 −  p ρ 24 , + 2 ρ

linear and nonlinear susceptibility for the system is calculated as described in [40]:

2N μ

0 h¯

[

i ( A 2 A 3 + 14 c2 (cos θ +sin θ )2 )

2( A 1 A 2 A 3 + 14 A 1 c2 (cos θ +sin θ )2 ))+ 14 A 3 c2 (cos θ +sin θ )2 ))

0 h¯ +

[

−i ( A 2 A 3 + 14 exp(−2i θ)c2 (cos θ − sin θ)2 ) 2( A 1 A 2 A 3 +

1 4

χsk =

2N μ

3 0 h¯

+

[

−i ( A 2 exp(−i θ)c (cos θ − sin θ)) 2( A 1 A 2 +

1 4

32( A 1 A 2 +

1 4

exp(−2i θ)c2 (cos θ − sin θ)2 ) W

].

(11)

2π

Im(χ , χck , χsk )).

(7)

(12)

∂ Re (χ , χck , χsk ). ∂ p

(13)

c 1 + 2π Re (χ , χ

k c,

χ

k s)+2

πω ∂ ∂ p Re(χ , χck , χsk )

.

(14) The delay or advance times can be written as:

td , tdc , tds =

exp(−2i θ)c2 (cos θ − sin θ)2 )

i exp(−3i θ)c3 2p (cos θ − sin θ)3

γ2 .

],

(6) 4 14

2

The group velocity v g = c /n g and can be expressed as:

A 1 exp(−2i θ)c2 (cos θ − sin θ)2 )

8( A 1 A 2 A 3 + 14 A 1 exp(−2i θ )c2 (cos θ −sin θ )2 )2

1

n g , n gc , n gs = 1 + 2π Re (χ , χck , χsk )

v g , v gc , v gs =

i A 3 exp(−2i θ )c2 ( A 2 A 3 + 14 exp(−2i θ )c2 (cos θ −sin θ )2 (cos θ −sin θ )2 )

3N γ3 L λ p

+ 2πω

(5) 2N μ214

(10)

Where, γ3 is the decay from state 3, L is the length of the medium and λ p is the wavelength of the probe field. √ Further the refractive index is equal to εr , where ε = 1 + 4πχ . The group index is calculated from refractive index by the relation n g = nr + ωdnr /dω and is written as:

],

and

χck =

γ3 ,

T , T ck , T sk = exp(−

(0) 24 ρ ,34,44 = 0. After straight forward algebraical manipulation the

χ=

2

(4)

where,  P is taken in the first order, while 1,2 are taken in all order of perturbations. The atoms are initially assumed to be in the ground states |1, while at other states |2, |3 and |4 (0) 11 the probability is zero. Under these conditions, we have ρ = 1,

2 14

1

A 2 = −i ( p − 3 ) −

2

2

(9)

The real part of susceptibility is related to dispersion spectrum and imaginary part of susceptibility is related to the absorption or gain spectrum of the medium. The transmission of the probe field is related to the imaginary part by the relation as:

1

i

(8)

(3)

ρ 21 = −[i  p − i 1 − i 2 + γ2 ]ρ21 i

A 1 exp(−2i θ)c2 (cos θ − sin θ)2

A 3 = −i ( p − 2 − 3 ) −

34 + 2 ρ 41 , − p ρ · ∼

W = A1 A2 A3 +

(2)

ρ 31 = −[i  p − i 2 + γ3 ]ρ31 + 2 ρ21 i

and E − are the field components with corresponding unit vectors σ + and σ − as discussed in [42]. Also, W is defined as

L c

(n g , n gc , n gs − 1)

(15)

when td , tdc , tds is positive, it is called delay time and when td , tdc , tds is negative, then it is known as advance time.

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Fig. 2. Imaginary part of electric susceptibility of the absorption spectrum. (a) Absorption of light in absence of Kerr nonlinearity at c = 5γ while (b) shows in the presence of cross Kerr nonlinearity at c = 3γ . The (c) are for the self Kerr nonlinearity without cross effect at c = 1γ . Other parameters are  p = 0γ , θ = 0,  P = 2γ where γ = 10 MHz and 1 = 2 = 0γ .

3. Results and discussions The results are presented for absorption, dispersion and transmission of light for the graphene medium. The atomic decay rate γ is assumed to be 10 MHz and other parameters are scaled to this γ . The parameters h¯ , μ0 , 0 = 1 are taken in atomic units. In Fig. 2, we show the absorption behavior of light in graphene medium as a function of probe detuning and control fields  p /γ , c /γ . The absorption of light is shown without Kerr nonlinearity (Fig. 2a), in the presence of cross-Kerr nonlinearity (Fig. 2b), and in the presence of self-Kerr nonlinearity (Fig. 2c). It is observed that absorption without the Kerr effect is negative at the resonance point  p = 0γ , which show that the pulse is amplified at the resonance point. The splitting of absorption increases with the intensity of the control field Rabi frequency c . At c = 5γ four absorption peaks are observed, in which there are three negative absorption regions as shown in Fig. 2a. There is maximum positive absorption of light in the presence of crossKerr nonlinearity at resonance  p = 0γ in the absence of control field. The absorption then splits into two peaks at c = 3γ and its width increases with an increase in strength of the control field intensity as shown in Fig. 2b. A similar behavior is seen in Fig. 2c,

3

Fig. 3. Dispersion of the electric susceptibility. The parameters are γ1 = γ2 = γ3 = 0.5γ where γ = 10 MHz, θ = 0, 1 = 2 = 0γ ,  p = −10γ , 10γ , c = 0 to 10γ and  P = 2γ .

where the absorption is narrow and goes to negative value in the presence of self-Kerr nonlinearity at resonance  p = 0. In Fig. 3, the dispersive behavior of light in graphene medium is shown as a function of probe detuning  p /γ and control field c /γ . The normal and anomalous slopes of dispersion are observed at the region of positive/negative absorption regions. It is noticed that the dispersion slope is normal at the region of positive absorption and anomalous in the negative absorption region. There are four normal slope of dispersion and three anomalous slope of dispersion regions in the absence of Kerr nonlinearity as shown in Fig. 3a. The dispersion slopes are reduced to two normal and one anomalous dispersion peaks in the presence of cross-Kerr effect as shown in Fig. 3b. In Fig. 3c, the dispersion peak split into two anomalous and two normal dispersion slopes in the presence of self-Kerr nonlinearity. Fig. 4, elucidates the transmission spectrum without Kerr nonlinearity and in the presence of cross as well as self-Kerr nonlinearity. It is observed that at zero absorption region there is 100% transmission of the pulse and at negative absorption region the transmission probability increases from 100% due to negative absorption as shown in Fig. 2. This shows that negative absorption enhances the transmission intensity and the pulse is amplified within the medium. In Fig. 4a, transmission intensity peaks at positive and negative detuning regions with an increase of the control field intensity are shown. The transmission is zero at resonance

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Fig. 4. Transmission spectrum of the probe field. The parameters are γ1 = γ2 = γ3 = 0.5γ where γ = 1, θ = 0, c = 0 to 10γ ,  P = 2γ , 1 = 2 = 0γ and  p = −10γ , 10γ .

point at low control field and increases as the control field increases in the presence of cross-Kerr effect as shown in the Fig. 4b. In the presence of self-Kerr effect, the transmission is maximum at resonance point and decreases with increase in strength of the control field as shown in Fig. 4c. In Fig. 5 the plots reveals the spectrum of group index without Kerr effect (Fig. 5a) and in the presence of cross-Kerr effect (Fig. 5b) as will as self Kerr effect (Fig. 5c). The group index has negative value of −3000 at the resonance point in the absence of Kerr effect in the region of anomalous dispersion. It is noticed that the group index is maximum (both positive and negative) close to resonance and decreases with an increase in detuning. Also the effect of control field intensity on the group index is shown. These results confirm the validity of our previous plots for absorption in the presence of self and cross-Kerr effect and without the Kerr effect. It is further noticed that the group index enhances in the presence of Kerr nonlinearity, and subluminal/superluminal behavior of the probe light is revealed. It is shown that Kerr nonlinearity invert and enhances the subluminal/superluminal behavior of the pulse. Also the effect of self-Kerr nonlinearity on pulse propagation is found to be more as compared to cross-Kerr nonlinearity. Similarly, in Fig. 6 we investigate the delay and advance time of the probe light without Kerr effect (Fig. 6a) and in the presence of Kerr nonlinearity (Figs. 6b and 6c). We observed maximum pos-

Fig. 5. Group index in the presence of Kerr effect, cross Kerr effect and without Kerr effect. The chosen parameters are γ1 = γ2 = γ3 = 0.5γ where γ = 1, 1 = 2 = 0γ ,  p = −10γ , 10γ , θ = 0, c = 3γ and  P = 2γ .

itive group delay time 1.5 μs and maximum negative group index or advance time of 0.5 μs in the absence of Kerr effect. In the presence of cross-Kerr nonlinearity 1 μs delay and advance time of 2 μs are measured near the resonance point. Furthermore large negative group delay is observed in the presence of self-Kerr nonlinearity. 4. Conclusions In conclusion, we investigate the behavior of a probe light field in a quantised four level atomic system of graphene in the presence of self and cross-Kerr nonlinearity as well as in the absence of Kerr effect. The light field signal is controlled and modified as a function of probe detuning and strength of the control field. The absorption, dispersion, transmission, group index and group delay/advance times of light are investigated. In the absorption spectrum, negative absorption (prob amplification) is measured, which is significant for lasing without inversion. The amplification could be due to the increase in population of the lower states and have substantial applications in communication channels [46]. At the positive absorption region the transmission probability is less than 1 and in the negative absorption region the transmission probability is found to be larger than 1. Hence, we observed 100%

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Fig. 6. Time delay of the probe filed in the medium. The parameters are γ1 = γ2 = γ3 = 0.5γ where γ = 1, θ = 0,  P = 2γ , c = 3γ ,  p = −10γ , 10γ and 1 = 2 = 0γ .

transmission in the lossless absorption regions and even more than that in the amplification regions. Further the normal and anomalous slopes of dispersion are examined at the positive/negative absorption regions. The positive group index of 4000 and negative group index of −2000 is measured without Kerr effect. In the presence of cross-Kerr nonlinearity the values are inverted to −4000 and 2000, while in the presence of self-Kerr nonlinearity the values of positive/negative index enhances to ±10, 000. It is observed that Kerr nonlinearity invert and enhance the subluminal/superluminal behaviors of the pulse and self-Kerr nonlinearity is found to be more subluminal/superluminal than cross Kerr nonlinearity in a four level quantized graphene atomic medium. The results presented could have potential applications in information storage devices, self and cross phase modulations and lasing without inversion. References [1] P.A.M. Dirac, The quantum theory of dispersion, in: Special Relativity and Quantum Theory, Springer, Netherland, 1988, pp. 180–198. [2] B.R. Masters, P. So, Handbook of Biomedical Nonlinear Optical Microscopy, Oxford University Press, 2008. [3] J.I. Frenkel, Wave Mechanics: Advanced General Theory, Oxford University Press, 1934, p. 266.

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