Anomalous to normal dispersion nonlinear optical dephasing switch in electromagnetically induced transparency using a Kerr effect

The atomic decoherence effect (DE) on a Kerr nonlinear (KNL) electromagnetically induced transparency (EIT)is studied in a Δ system. The DE between the ground state hyperfine levels is caused by the dephasing rate γ d which dramatically modifies the medium response. It controls the normal dispersive region which shows steep positive slopes for linear response at the line center while the nonlinear response experiences steep negative slopes for low γ d . The microwave field strength and γ d modify the nonlinear response from the anomalous dispersion to normal dispersion. The calculations show that room-temperature atoms are used to quantify the quantum interference (QI) on linear and nonlinear absorption with γ d . The EIT spectrum explores the understanding of the subluminal and superluminal wave propagation of probe signal and this study opens a new pathway for the understanding of the QI devices and their nonlinearities based on EIT.


Introduction
The EIT is a quantum coherence effect [1,2] that has a considerable interest in the field of quantum optics and quantum information sciences, because of its wider applications.Quantum technological instruments use such coherence effect to enhance the performance in various applications such as transmission, sensing, generation of nonlinear photons, and information processing [3].The performance and high sensitivity of such an instrument are affected by the environmental interaction which causes atomic decoherence [4][5][6].The decoherence-free interaction can avoid these issues by preserving the quantum state coherence (QSC) against environmental noise [7,8].In recent years, the coherent conversion of microwave-to-optical (MWTO) frequencies has been crucial for transferring quantum information [9][10][11][12].The pure dephasing contribution in QSC causes the loss of quantum information in the MWTO frequency conversion.This quantum transducer's conversion efficiency is highly limited by environmental noise [4,13].In the presence of buffer gas [14] and an anti-reflection coating [15], the quantum information against the dephasing rate is preserved.In this condition, the ultra-narrow EIT width is achieved.For the MWTO conversion, the Δ system is used because of its high symmetry, and phasesensitive nature, and it doesn't require the ultra-cold temperature.There are many theoretical and experimental studies reported in the last two decades [16][17][18][19][20][21][22].In general, the EIT system is used to harvest the nonlinear photons with ultra-low light levels [23,24].The nonlinear studies have concentrated on the third-order self-Kerr [25][26][27][28] and cross-Kerr nonlinear response [29,30].The self-Kerr effect has been studied in EIT at low light levels for its vanishing linear absorption and nonlinear absorption is non-vanishing at the line center [20].The resultant phase of the applied fields [31], spontaneously generated coherence [32], doppler broadening [20], and frequency and intensity of the strong coupling field [33,34] are explored in the KNL medium.The three-level cascaded system experiences normal and anomalous dispersion which depends on the polarization of various orientations of the Laguerre-Gaussian (LG) light as a coupling field [35].These systems are open to the environment and the environment hampers the KNL coefficient of the EIT.Surprisingly no theoretical and experimental studies have concentrated on the dephasing effect in the Kerr nonlinear Δ system.The dephasing is the crucial parameter in determining the amplification of the output laser field.The amplification mainly depends on the linear and nonlinear response of the medium.So, it is important to know how the dephasing affects the individual medium responses in an EIT-based system.In this paper, we analyzed the dephasing effect in the KNL response.The dephasing rate and microwave field strength switch the nonlinear response from anomalous dispersion to normal dispersion and vice versa.The right value of the dephasing rate leads to a giant self-Kerr nonlinearity with vanishing linear absorptions [20].This property of the EIT system can be used as an optical dephasing switch from normal to anomalous dispersion and vice versa in linear and nonlinear responses.
The paper is organized as follows.Section 2 explains the theoretical developments on the KNL responses.In section 3, we explain the effect of the dephasing rate and the medium response on KNL.The last section explains the conclusion part.

The equations of motion of KNL EIT in the Δ system
The 85 Rb atomic energy levels are taken for the theoretical calculations which are shown in figure 1 , the ground state |b〉 represents the 85 Rb hyperfine level 5S 1/2 , F = 2, and the metastable state |c〉 represents hyperfine level 5S 1/2 , F = 3.The energy of each level is represented by ÿω i (i = a, b, c).The transitions |b〉 → |a〉and|c〉 → |a〉 are mediated by the probe and coupling laser fields respectively.They are electric dipoleallowed transitions.The transition |b〉 → |c〉 is an electric dipole forbidden which is a magnetically allowed transition.This is coupled by the microwave magnetic field using a microwave cavity [11,21,22,36].
In the Δ system, it is assumed that the probe laser field  p and coupling laser field  c are propagating fields and microwave field m  forms a standing wave inside a microwave cavity [21,22].The field equations are, The E p , E c , and E μ are the amplitudes of the probe, coupling, and microwave fields respectively.The ν p , ν c are the frequencies of the probe and coupling fields, k p , k c are wave numbers of the probe, and coupling laser fields, and ν μ is the microwave field frequency.The f p , f c are phases of the probe and coupling fields.The f μ is the phase of the microwave field inside a microwave cavity [21,22].
The semiclassical Hamiltonian of the Δ system under the interaction and rotating wave approximation can be written as [20], where δ p , δ c are the probe and coupling laser field detunings respectively.The ) )

E e and E e c ac c i cb
are the probe, coupling, and microwave field Rabifrequencies.These Rabi-frequencies define the transition strengths in the Δ system.The μ ij = 〈i|μ|j〉 represents the transition dipole moment for a transition from i to j level(i, j = a, b, c).
The evaluation of the Δ system is given by the density matrix equation [37][38][39], Here, the ρ is the density operator.The first term on the right-hand side of the equation (5) represents the commutator of the Hamiltonian with the density operator ρ.The second term describes the natural relaxation rates γ aj ( j = b, c) of the ensembles of atoms from the excited state |a〉 to the metastable state |c〉 and a ground state |b〉.The third term represents the phase damping between the metastable and the ground levels with the dephasing rate g ( ) d .The expansion for the Lindblad operators are r r rr r r r rr r r r rr r r r rr r   , 1 3 at the probe laser field are derived based on the perturbative method [27,28].For this calculation, the iteration method [20,41] is used to solve the system of equations (6).The perturbation expansion of the density matrix element is, The subscripts l, and m stand for the states a, b, c.The population is conserved i.e., r r r .But, in an actual experiment [22], the Rabifrequency field strength satisfies the condition Ω p 〈Ω μ 〈Ω c such that the populations are assumed to be equally distributed in the ground and metastable state at room temperature r r = = ( ) ( ) . Since the probe Rabifrequency is small, the Ω p terms are approximated as r r . Here, n is the order of perturbation, and ij = ab, ba, bc, cb, ac, ca.The higher-order populations satisfy the condition, where n 0, 1, 2, 3 , 8   Here, n = 0, 1, 2, 3,K.represents the zeroth order, first order, second order, third order, etc., All the zeroth order density matrix elements are zero except r  The first-order population is obtained by substituting n = 1 in the equation (10).It gives r r r , , 1 which is substituted in the equation (11) to obtain the first-order coherence r ( ) , ab 1 the procedure is continued up to the third order to obtain the third-order coherence r ( ) ab 3 .The χ (1) ,χ (3) is related to the first order and third order coherent terms r r , ab ab Here, N represents the number of atomic ensembles, and μ ab is the atomic transition dipole for the transition |a〉 → |b〉.The Kerr Non-linearity is the real part of χ (3) .The imaginary part of χ (3) defines the nonlinear absorption [42].The real χ (1) is the linear dispersion and imaginary parts of χ (1) define the linear absorption.In general, at room temperature, the ensembles of atoms are not at rest.The temperature effect can be included in the Kerr-nonlinearity by the following procedures.The kinetic energy of the ensembles of atoms is not zero at room temperature.The velocity-averaged coherent term defines the Maxwellian velocity distribution experienced by the atoms, The average value is calculated by replacing the detunings parameters as in equations (10), (11).The velocity of the 85 Rb atoms at T = 300 K varies from −170 m/s to + 170 m/s.The most probable speed of the atoms is max are the maximum velocity of Rb atoms in parallel and anti-parallel directions to the laser fields respectively.The k p and k c values are 8.05291 × 10 4 cm −1 , and 8.05285 × 10 4 cm −1 which are taken from D line data [43] and correspond to the 85 Rb D2 transition.

Results and discussion
In a Δ-type EIT system, the impact of ground state decoherence on the probe signal is studied by many research groups.These studies demonstrated that with the increase in ground state decoherence (GSDC), the EIT width increases with a decrease in peak height.The presence or absence of GSDC significantly influences the amplification of EIT.In the last two decades, it has been found that the amplification in the Δ system is possible when the Λ-type EIT system is coupled with an additional microwave field, i.e., when the system becomes an Δ-type system.The variation of GSDC between the ground state and the metastable state has an influence on the EIT or amplification process in the Δ-type system.The impact of variation of the GSDC parameter g ( ) d on the optical susceptibility of the medium and the probe transmission has a significant contribution to quantum optics.The effect of GSDC in the linear and nonlinear EIT is established in the Δ system.
The role of decoherence in the nonlinear absorption of the Δ system is theoretically solved using the equations ( 10), (11), (13).The results are discussed in detail in this section.Initially, the investigation is focused on nonlinear absorption under steady-state conditions with a zero dephasing rate g = ( ) 0 d . Our previous study [20] shows that the linear absorption experiences destructive QI at the line center and causes the probe field propagation without any absorption.But, the nonlinear response experiences constructive QI at the line center d = ( ) 0 p [20].In this study, we examine the influence of γ d on the nonlinear EIT profile.Im 3 increases as microwave field strength increases.This is due to the power-broadening mechanisms [2].The maximum value of the c ( ) Im 3 varies and the symmetry of the profile is distorted at the line center for the small value of Ω μ .In most of the EIT systems, the linear profile is zero at δ p = 0 causing a positive slope on c ( ) Re 1 which results in slow light [2,18].Figure 2(b) shows the real part of χ (3) vs probe detuning δ p .This is also plotted for the three different values of Ω μ .As the Ω μ increases the slope switches from negative (anomalous dispersion) to a positive value (normal dispersion) and vice versa.The nonlinear dispersion c ( ) Re 3 shows the negative slope defines the superluminal wave propagation for the nonlinear photons generated by the nonlinear process.This superluminal wave propagation can be controlled and shaped by the microwave field strength Ω μ .
In this section, the non-zero dephasing rate is included in the equations (5), and the system of equations is solved.The obtained linear response χ (1) is plotted against the probe detuning for different γ d .The used parameters for this figures are Ω p = 0.1 MHz, Ω μ = 3.0 MHz, γ ab = 6.0 MHz, γ ac = 6.0 MHz, γ cb = 0.001 MHz, f r = 0. Figure 3(a) shows the real part of χ (1) vs δ p for different values of γ d .The slope of the curve at the line center is positive for the γ d values 0.001 MHz, 0.01 MHz, and it is decreasing as γ d increases.This is because the probe photon experiences the drag by the linear response of the medium which results in the slow light EIT.As γ d values reach the spontaneous decay rates of the excited states as well as the coupling field strength the response of the medium switches from positive to negative for the γ d = 5 MHz.This shows the anomalous dispersion.  .The symmetry profile is obtained at δ p = 0 for γ d = 0.001 MHz.It conveys that there is a perfect constructive QI in the third-order response.For the dephasing rate γ d = 1 MHz, the symmetry of the profile is completely broken, and the system is no longer in a perfect QI state.Figure 4(c) also depicts the same broken symmetry in the nonlinear response with a larger negative value for γ d = 1 MHz and Ω μ = 7 MHz.As a result, the system deviates from the perfect QI state to no interference state for the large value of γ d , and Ω μ .For the Ω μ Ω c , the power broadening happens in the nonlinear EIT.In addition to that, the dephasing rate destroys the nonlinear constructive QI state.Hence, the ground state decoherence between the two lower hyperfine levels is responsible for the control of non-linear photons in Δ system.
Next, we study the behavior of the real part of the nonlinear response c ( ) Re 3 with three different dephasing rates.Figure 5(a) shows c ( ) Re 3 versus the δ p for γ d = 0.001 MHz, 0.01 MHz, 1 MHz and Ω μ = 1.0 MHz.The negative slope is observed around the δ p = 0 which is known as nonlinear anomalous dispersion and it is the signature of the superluminal wave propagation (SLW) for the nonlinear photons.As γ d increases to 1 MHz, the slope decreases.Figures 5(b), (c) are plotted for Ω μ = 3.5 MHz, 7 MHz.This shows that the slope at δ p = 0 changes from negative to positive as γ d increases from 0.001 MHz to 1 MHz.So the γ d changes the dispersion from negative to positive value.Hence, the nonlinear wave propagation varies from superluminal to subluminal for nonlinear photons in the EIT system.In the Δ system, lower hyperfine levels are connected by the microwave field which decides whether the dispersion is anomalous or normal.This results in superluminal propagation of a weak probe light [17].  (1is plotted against the probe detuning (δ p ).The left-hand side green curves show the real part of χ (1) for the dephasing rate values γ d = 0.001 MHz, 0.01 MHz, 0.1 MHz, 1 MHz, 5 MHz.The right-hand side red curves show the imaginary part of χ (1) .The other parameters are the same as mentioned in figure 2. However, our study shows that the dephasing rate is an additional effect contributing to anomalous dispersion in the same system.The control of the dephasing rate decides whether the nonlinear wave propagation is superluminal or subluminal in the EIT systems.
An interesting experiment showed that the source of decoherence in the EIT window of atomic rubidium vapor cells is dephasing rather than population relaxation [44].So, the dephasing rate γ d plays a major role in EIT-based linear and nonlinear study.No theoretical study developed the effect of γ d on the nonlinear response of EIT in the Δ system.Our study paves a new way to understand the γ d effect in EIT-based experiments at low light levels.

Conclusion
In summary, this article has developed an EIT-based nonlinear effect for realistic experiments including the dephasing rate in a Δ system.In this article, it is assumed that the decoherence is because of pure dephasing mechanisms.The theoretical results quantify the decoherence sensitivity on the nonlinear absorption and dispersion.As the dephasing rate increases the nonlinear anomalous dispersion is turned into normal dispersion.The obtained analytical results found that the nonlinear EIT region is largely affected by both the dephasing rate and the microwave field strength.Therefore, the dephasing rate is as significant as the microwave field in shaping the EIT nonlinear spectrum.
40].The elements of the population and the coherent terms dynamical equations is derived from equation (5),

.
The ρ aa , ρ bb , ρ cc represent the diagonal density matrix elements known as populations, and the ρ ab , ρ ac , and ρ cb are the offdiagonal density matrix element operators known as coherent terms, the f = (k p − k c )z − f μ is the resultant phase[21,22] of three fields.k p z, k c z represents the propagation phase of the laser fields and f μ is a standing wave phase of the microwave field.The detunings of the probe and coupling laser fields are d n

.
Under the weak probe and microwave field Rabi-frequency Ω p , Ω μ = Ω c , the population is settled down in the ground state for the zeroth order r r Under the above assumptions, the time evolution of the populations and coherent terms density matrix equations are,

2 bb cc 0 0 .
The solutions of the coherent density matrix element correspond to the transition |b〉 → |a〉,

Figure 2 Im 3
versus the probe detuning δ p for the three different values of microwave field strength W

Figure 3 (
b) shows the imaginary part of χ(1) vs δ p for γ d = 0.001 MHz, 0.01 MHz, 0.1 MHz, 1 MHz, 5 MHz.The lower dephasing rate does not disturb the perfect QI effect[21] and causes EIT for the probe signal.As γ d increases the constructive QI effect increases and kills the EIT spectrum where the probe signal experiences the complete absorption for γ d = 5 MHz.Figures 4(a)-(c) show the imaginary part of the nonlinear response c ( ) ( ) Im 3 versus the probe field detuning δ p . Figure 4(a) is plotted for the three different values of γ d .The parameters for this plot are Ω p = 0.1 MHz, Ω c = 4 MHz, Ω μ = 1.0 MHz, γ ab = 6 MHz, γ ac = 6 MHz, γ cb = 0.001 MHz.The resultant phase f r is assumed to be zero.It is observed that the nonlinear response is enlarged other than the line center d =

Figure 3 .
Figure3.The linear χ(1) is plotted against the probe detuning (δ p ).The left-hand side green curves show the real part of χ(1) for the dephasing rate values γ d = 0.001 MHz, 0.01 MHz, 0.1 MHz, 1 MHz, 5 MHz.The right-hand side red curves show the imaginary part of χ(1) .The other parameters are the same as mentioned in figure 2.

Figure 4 . 7 JFigure 5 .
Figure 4. Figures 4(a)-(c) show a plot of the third-order nonlinear absorption c ( ) ( ) Im 3 vs probe detuning δ p for the Ω μ values 1 MHz, 3.5 MHz, and 7 MHz respectively.The other parameters are the same as mentioned in figures 2(a), (b).Each figure is plotted for three different values of γ d .
. It contains one excited state |a〉, metastable state |c〉, and a ground state |b〉.The excited state |a〉 represents the 85 Rb hyperfine Figure1.The energy level diagram of the hyperfine levels of the Δ system is shown.The Ω p , Ω c , and Ω μ represent the Rabi frequencies of the probe, coupling, and microwave field respectively.The ν p , ν c are the frequencies of the probe and coupling laser field and ν μ is the frequency of the microwave field.The δ p , δ c , and δ μ are detunings of the probe, coupling, and microwave field.The natural decay rates from the level |a〉 to |b〉, |a〉 to |c〉 are represented by γ ab , γ ac , and γ cb is the decay rate from metastable state |c〉 to ground state |b〉.
First, the populations are solved in a steady state using the conditions 8. The obtained solutions are,