Effect of Quantum Interference from Incoherent Pumping Field and Spontaneous Emission on Controlling the Optical Bistability and Multi-Stability

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1 Commun. Theor. Phys. 59 (2013) Vol. 59, No. 2, February 15, 2013 Effect of Quantum Interference from Incoherent Pumping Field and Spontaneous Emission on Controlling the Optical Bistability and Multi-Stability H.R. Hamedi, 1, Ali Sari, 2 M. Sahrai, 1 and S.H. Asadpour 3 1 Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz, Iran 2 Department of Electrical Engineering, Islamic Azad University, Shadegan Branch, Shadegan, Iran 3 Young Researchers Club, Islamic Azad University, Bandar Anzali, Iran (Received August 17, 2012; revised manuscript received November 7, 2012) Abstract Optical bistability (OB) and optical multi-stability (OM) of a four-level Λ-type atomic system with two fold lower levels inside a unidirectional ring cavity is investigated. The effect of quantum interference arising from spontaneous emission and incoherent pumping on OB and OM is discussed. It is found that the threshold of OB and OM can be controlled by quantum interference mechanisms. In addition intensity of coupling field and the rate of an incoherent pumping field on behavior of OB and OM are then discussed. PACS numbers: Pc, Md, Hz Key words: optical bistability, optical multistability, quantum interference, incoherent pumping 1 Introduction Nonlinear response of an atomic system has been attracted a lot of attention in few past decades. Quantum coherence and quantum interference are the basic mechanism for controlling the linear and nonlinear response of the medium. The effect of electromagnetically induced transparency (EIT) [1 3] has led to many interesting nonlinear optical phenomena such as Kerr nonlinearity [4] and optical bistability. [5] Due to its potential application in many area of optics, such as quantum computing, quantum communication, and fast optical switching, the optical bistability has recently been developed in multi-level gas systems. Many experimental and theoretical proposals have been performed to investigation the bistable and multi-stable behavior of two- and multilevel atomic systems. [6 8] In recent years, optical bistability (OB) has extensively been studied in a two-level atomic system. [9 10] One important advantage of threelevel systems, instead of two-level atoms, as a nonlinear medium inside an optical cavity is to make of the atomic coherence induced by three-level atomic systems, which can modify the dispersion, absorption and nonlinearity of the medium. [11] The bistable behavior of Λ-type, [12] V-type, [13] ladder type, [14] and microwave-driven threelevel atomic system has been studied. [15] Harshawadhan and Agarwal [16] studied the influence of quantum interference on OB and showed that the threshold required for bistable device can be decreased substantially due to quantum interference effects. Cheng et al. [17] investigated the OB behavior of a nearly equispaced ladder type threelevel atomic system considering the spontaneously generated coherence effect and showed that the OM can be observed by adjusting the relative phase of two applied fields. Controlling the OB and OM behavior of a threelevel atomic system via incoherent pumping field is also presented. [18] Gong et al. [19] investigated the bistable behavior of a three-level atomic system via initial coherence and demonstrated that the OB can be realized by the initial coherence. They have also shown that increasing the initial coherence makes the bistable hysteresis cycle to be larger. In viewing many proposals, we note that the optical bistability and multi-stability in a four-level atomic system in a ring cavity is easy to implement. This is due to the existence of many controlling parameters which cannot be found in simple two-level or three-level atomic systems. Recently, the bistable and mutistable behavior in a four-level atomic system via interacting dark resonance [20] and tripod-type medium [21] was also investigated. In recent years many kinds of nonlinear quantum optical phenomena based on the quantum interference and coherence have also been extensively studied in the semiconductor quantum wells (SQWs), such as electromagnetically induced transparency, [22 25] optical bistability, [26 27] and Kerr nonlinearity. [28] It is worth note that the SQWs have many potential applications in optoelectronics and solid-state quantum information science. In fact, the devices based on intersubband transitions in the SQWs have many inherent advantages that the atomic systems do not have. Some important advantages include the large electric dipole moments due to the small effective electron mass, the great flexibilities in devices designed by choosing the materials and structure dimensions, the high nonlinear optical coefficients, controllable transition energies and the dipoles. Optical bistability (OB) based on intersubband transitions in an asymmetric coupled-quantum well (CQW) driven coherently by a probe laser field and a Hamid.R.Hamedi@gmail.com c 2013 Chinese Physical Society and IOP Publishing Ltd

2 200 Communications in Theoretical Physics Vol. 59 control laser field was investigated by Li. [29] It is demonstrated that OB can be controlled by tuning the energy splitting between two tunnel-coupled electronic levels and the intensity of the control field. This may be used for optimizing and controlling the optical switching process in the CQW solid-state system, which is much more practical than that in atomic system because of its flexible design and the controllable interference strength. In Ref. [30] it is shown that the OB can be controlled by adjusting the intensity and the frequency detuning of the coupling field in an N-type four-level atomic system. Moreover, the OM can also be observed under appropriate detuning condition. Kim [31] demonstrated that the OB can be created in a four-level atomic system by means of a microwave field in an optical cavity. Under the steady state condition, they controlled the occurrence of the OB by modulating the atomic parameters. In transient response, it is found that the stable output field of the optical cavity can be controlled by adjusting the intensity of the input field. Recently, we showed that the OB and OM behavior of a four-level N-type atomic system can be controlled by spontaneously generated coherence (SGC) and the relative phase of applied fields. [32] We found that the OB and OM can be created just by adjusting the intensity and frequency detuning of the probe and coupling laser fields. Among the realization of OB and OM in different four-level atomic configurations, the study of a fourlevel Λ-type atomic system with two fold lower-levels is of interest to implement. In Ref. [33] the absorption and dispersion properties of a Λ-type atomic system with two fold lower-levels has been investigated. It is found that the interaction of double-dark states lead to controllable group velocity of the weak probe fields by the intensity of driving fields. In addition, two closely lying lower levels couple to the upper level by three coherent driving fields; thus the applied fields build a closed loop, and the phase control of group velocity becomes possible. More recently, the dispersive-absorptive properties of such atomic system when a weak probe field couples both the two lower levels to the upper level were investigated. [34] In this case, with the quantum interference induced by incoherent pumping field and decay process, the steady state and transient behavior of the dispersion and absorption were analyzed. It is shown that the probe dispersion and absorption can be controlled by the rate of an incoherent pump field and the quantum interference induced by incoherent pumping field and decay processes. Now, in this paper, the possibility of creation the OB and OM in a four-level Λ-type atomic system with two fold lower-level is presented. The effect of quantum interference due to spontaneous emission and incoherent pumping field on novel properties of OB and OM are then discussed. We find that the threshold intensity of OB and OM can effectively be reduced by the effect of quantum interference. In addition, existence of spontaneously generated coherence (SGC) may provide an extra controlling parameter which can control the threshold of OB and OM. The OB and OM can be described by the means of dark states. 2 Model and Equations Consider a four-level Λ-type atomic system with two fold lower-levels 1, 2, upper level 3 and intermediate level 4 as depicted in Fig. 1(a). Both the lower levels 1, 2 and upper level 3 are coupled by a probe laser field. An incoherent pump field ε pumps the population from two lower levels 1 and 2 to upper level 3 with the pumping rates r 1 and r 2. Correspondingly, upper level 3 decays to lower levels 1, 2 and 4 with the rates of γ 1, γ 2 and γ 3 respectively. In addition, the intermediate level 4 is coupled to the upper level 3 by a coherent coupling laser field. The density matrix equations of motion under the rotating wave approximation are: [34 35] ρ 11 = (γ 1 + r 1 )ρ 11 + r 1 ρ k( r 1 r 2 )(ρ 12 + ρ 21 ) + i(ω p 1 ρ 31 Ω p1 ρ 13 ), ρ 22 = (γ 2 + r 2 )ρ 22 + r 2 ρ k( r 1 r 2 )(ρ 12 + ρ 21 ) + i(ω p 2 ρ 32 Ω p2 ρ 23 ), ρ 33 = r 1 ρ 11 + r 2 ρ 22 (γ 3 + r 1 + r 2 )ρ 33 + k( r 1 r 2 )(ρ 12 + ρ 21 ) + i(ω p1 ρ 13 Ω p 1 ρ 31 ) + i(ω p2 ρ 23 Ω p 2 ρ 32 ) iω c ρ 34 + iω c ρ 43, ρ 12 = 1 2 k( r 1 r 2 )(ρ 11 + ρ 22 ) + k( r 1 r [ 1 ] γ 1 γ 2 )ρ 33 2 (γ 1 + γ 2 + r 1 + r 2 ) i ρ 12 iω p2 ρ 13 + iω p 1 ρ 32, [( 1 ) ρ 13 = iω p 1 (ρ 33 ρ 11 ) iω p 2 ρ 12 2 γ 1 + 2r 1 + r 2 i (δ 1 )] 2 ρ 13 iω c ρ k( r 1 r 2 )ρ 23, [ 1 ρ 14 = iω c ρ 13 2 (γ 1 + γ 3 + r 1 ) i ((δ 1 ) )] 2 δ c ρ k( r 1 r 2 )ρ 24 + iω p 1 ρ 34, ρ 23 = iω p 2 (ρ 33 ρ 22 ) iω p 1 ρ 21 1 [( 1 ) 2 k( r 1 r 2 )ρ 13 2 γ 2 + 2r 2 + r 1 i (δ + 1 )] 2 ρ 23 iω cρ 24, ρ 24 = 1 [ 1 2 k( r 1 r 2 )ρ 14 iω c ρ 23 2 (γ 2 + γ 3 + r 2 ) i ((δ + 1 ) )] 2 δ c ρ 24 + iω p 2 ρ 34, [ 1 ] ρ 34 = iω c (ρ 44 ρ 33 ) + iω p1 ρ 14 + iω p2 ρ 24 2 (γ 3 + r 1 + r 2 ) + iδ c ρ 34, ρ 11 + ρ 22 + ρ 33 + ρ 44 = 1. (1)

3 No. 2 Communications in Theoretical Physics 201 The last equation expresses the conservation of probability for the closed four-level system. Here, γ i = 3i ν 3 3i /3π ε 0c 3 (i = 1, 2, 4) are the spontaneous decay rates in transition 3 j. ν 3i is the frequency deference between level i and level 3, ε 0 is the vacuum permittivity, and c is the speed of light in vacuum. Electric dipole moments of these transitions are characterized by 3i. The terms r j (j = 1, 2) are the incoherent pumping rate. Here, Ω pj = ε p j3 /2 (j = 1, 2) are the corresponding Rabi-frequencies of the probe laser field in transitions 3 j, and Ω c = ε c 43 /2 denotes the Rabi-frequency of the coherent coupling field in transition 3 4. ε c and ε p denote the amplitude of the coherent coupling field and probe laser field, respectively. The frequency detuning are also defined as δ c = ω 34 ν c, δ p1 = ω 13 ν p = δ ω 12 2, δ p2 = ω 23 ν p = δ + ω 12 2, where ( ω23 + ω ) ( 13 δp1 + δ p2 δ = ν p = 2 2 and = δ p1 δ p2. ), Fig. 1 (a) A four-level Λ-type atomic system. A coherent laser field couples upper levels 3 to 4. A weak tunable probe field couples level 3 to both the levels 1 and 2. (b) Unidirectional ring cavity containing an atomic sample of length L. ε I p and ε T p are the incident and transmitted fields, respectively, while ε c represents the coherent laser field which is non-circulating in the cavity. The quantum interference terms k p and k s appear due to cross coupling of pump rates and decay constants, respectively. These terms are central to our discussion that are considered k p = k s = k. These factors are the normalized inner product of corresponding dipole matrix elements as k = = cosθ, (2) where θ is the alignment of two dipole moments 31 and 32. In fact, k represents the strength of the interference in spontaneous emission or incoherent pumping field. According to its definition, the alignment factor takes the value 1 for parallel dipole moments, 1 for anti-parallel, and 0 for orthogonal. Maximal coherence corresponds to parallel or anti-parallel dipole moments, while zero coherence corresponds to orthogonal dipole moments. These two extremes of maximal and minimal coherence deserve special attention. In real experiment the parameter k can be determined by the intensity of applied fields. In fact, the coefficient k depends on the angle between two electric dipole moments, which can be controlled by the intensity of applied fields. Now, we put the ensemble of N homogeneously Λ-type four-level atoms in a unidirectional ring cavity as shown in Fig. 1(b). The intensity reflection and transmission coefficients of mirrors 1 and 2 are R and T (with R + T = 1), respectively. We assume that both the mirrors 3 and 4 are perfect reflectors. The total electromagnetic field seen by N homogenously four-level atoms contained in cell of length L can be written as: ε = ε c e iνct + ε p e iνpt + c.c. (3) The probe field ε p is circulating in the ring cavity, but the coupling field ε c is not circulate in the cavity. Under slowly varying envelop approximation, the dynamical response of the probe field is governed by Maxwell s equations: [36] ε p t + c ε p z = iν p 2ε 0 P(ν p ). (4) P(ν p ) is the induced polarization in the transition j 3 (j = 1, 2) and is given by P(ν p ) = N (ρ 31 + ρ 32 ), (5) where 31 = 32 =. At the steady state, the term ε p / t in Eq. (4) is set to be zero. Substituting Eq. (5) into Eq. (4), we obtain the field amplitude relation as: ε p z = i Nω p (ρ 31 + ρ 32 ). (6) 2cε 0 The coherent field ε I p enters through mirror M 1, interacts with the atomic sample of length L, circulates in the cavity, and partially comes out of the mirror M 2 as E T p. The probe field at the start of atomic sample is ε p (0) that propagates to the end of the atomic sample to be ε L p in the single pass transition. The coupling fields can also

4 202 Communications in Theoretical Physics Vol. 59 enter the cavity through the polarizing beam splitter and co-propagates with the cavity field in the atomic cell. For a perfectly tuned cavity, the boundary conditions between the incident field ε I p and transmitted field ε T p are ε p (L) = εt p T, (7) ε p (0) = Tε I p + Rε p (L), (8) where L is the length of the atomic sample. Note that R is the feedback mechanism due to the reflection from mirror M 2. It is responsible for the bistable behavior, so we do not expect any bistability when R = 0 in Eq. (8). According to the mean-field limit [18] and by using the boundary condition, the steady state behavior of transmitted field is given by y = x icγ(ρ 31 + ρ 32 ), (9) where y = ε I p/ T and x = ε T p/ T are the normalized input and output field, respectively. The parameter C = Nω p L 2 /2 cε 0 T is the cooperatively parameter for atoms in a ring cavity. Transmitted field depends on the incident probe field and the coherence term ρ 31 and ρ 32 via Eq. (9). So, the bistable behavior of the medium can be determined by the atomic variables through ρ 32. We set the time derivatives of ρ ij equals zero, i.e. ρ ij / t = 0 (i, j = 1, 2, 3, 4), in Eq. (1) and solve the corresponding density matrix equations together with the coupled field Eq. (9). 3 Results and Discussion We summarize our results for the steady state behavior of the output field intensity versus the input field intensity for various parameters illustrated in Figs We assume γ 1 = γ 2 = 0.5γ, γ 3 = γ and all the figures are plotted in the unit of γ. It is desirable to decrease the threshold of OB and OM via quantum interference arising from decay process and incoherent pumping. We show the behavior of output-input fields for various quantum interference mechanisms. The effect of quantum interference induced by incoherent pumping and decay processes is displayed in Fig. 2. An investigation on Fig. 2 shows that for k = 0 (without interference), the threshold of optical bistability reaches to its maximal value. This is due to the existence of a large absorption in the atomic medium, which causes the field hard to reach saturation. [34] It is obvious that in presence of quantum interference, i.e. k = 0.5, the threshold of OB decreases and eventually for the maximal interference i.e. k = 1, OM with reduced threshold appear. Physically, in the presence of quantum interference mechanism induced by spontaneous emission and incoherent pumping field, i.e. k = 1, the population will be trapped in lower levels 1 and 2. So, the dark state will be created, and the probe field absorption will be reduced. The effect of incoherent pumping rates in behavior of OB and OM is displayed in Fig. 3. Investigation on Fig. 3(a) shows that for k = 0 and for the given parameters, by increasing the rate of incoherent pumping field the threshold of OB decreases. In addition, by more attention on Fig. 3(b) we realize that by increasing the rate of an incoherent pumping field the threshold of OM decreases, and finally OM converts to OB. The reason can be qualitatively explained as follow. Increasing an incoherent pumping field dramatically reduces the absorption of probe field on transition from levels 1 and 2 to level 3. This may lead to enhancement of the Kerr nonlinearity and make the cavity field easier reaches to saturation. Figures 4(a) and 4(b) show the effect of coupling field on OB and OM. It is observed that without quantum interference (Fig. 4(a)) the system experiences the bistable behavior, whereas in the presence of quantum interference (Fig. 4(b)) OM appears. Increasing the intensity of coupling field from Ω c = 1γ to Ω c = 5γ leads to a significant decrease of the bistable and multi-stable threshold. Physically, applying strong coupling field between levels 3 and 4 dramatically reduces the absorption for probe field transition and thus enhances the Kerr nonlinearity of the atomic medium. This may be useful to control the threshold value and the hysteresis cycle width of the bistable curve simply by adjusting the intensity of the coupling field. The input-output field curves for different values of frequency detunings δ are also shown in Fig. 5. It is found that the threshold and the hysteresis cycle are changed by increasing the frequency detuning. Here, the OB is converted to OM just by increasing the probe field detuning. Moreover, the threshold of OM is smaller than OB. This is to say that a four-level Λ-type atomic medium with two fold lower levels is a convince atomic sample to convert the OB to OM (or vice versa) just simply by adjusting the probe field detuning. Fig. 2 Output field versus input field for different values of quantum interference parameter k. Other parameters are: C = 200γ, γ 1 = γ 2 = 0.5γ, γ 3 = 1γ, Ω c = 0.5γ, r 1 = r 2 = 0.1γ, δ c = 0.8γ, = 1γ, δ = 1.5γ.

5 No. 2 Communications in Theoretical Physics 203 Fig. 3 Output field versus input field for different values of incoherent pumping (r 1 = r 2 = r) for k = 0 (a) and k = 1 (b). Other parameters are the same as Fig. 2. Fig. 4 Output field versus input field for different values of Ω c with k = 0 (a) and k = 1 (b). Other parameters are: C = 200 γ, γ 1 = γ 2 = 0.5γ, γ 3 = 1γ, δ c = 0.8γ, = 1γ, δ = 1γ. Fig. 5 Output field versus input field for different values of frequency detuning δ. Other parameters are: C = 200 γ, γ 1 = γ 2 = 0.5γ, γ 3 = 1γ, δ c = 0.8γ, = 1γ, k = 1, and Ω c = 1γ. 4 Conclusion The OB and OM behavior of a four-level atomic system with two fold lower levels is investigated. The effect of quantum interference of spontaneous emission and incoherent pumping process on OB and OM is then discussed. We find that the threshold of OB and OM can be controlled and even reduced with the effect of quantum interference and intensity of coupling field. In addition, we demonstrate that OB can be converted to OM only by the rate of incoherent pumping field and frequency detuning of probe field. Optical bistability and multi-stability is an implement technique in fast optical switch that is needed in optical devices.

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