A novel phase sensitive quantum well nanostructure scheme for controlling optical bistability

Brazilian Journal of Physics https://doi.org/10.1007/s13538-018-0556-x ATOMIC PHYSICS A Novel Phase Sensitive Quantum Well Nanostructure Scheme for Controlling Optical Bistability Ali Raheli Received: 24 October 2017 / # Sociedade Brasileira de Física 2018 Abstract A novel four-level lambda-type quantum well (QW) nanostructure is proposed based on phase sensitive optical bistability (OB) and multistability (OM) with a closed-loop configuration The influence of controlling parameters of the system on OB and OM is investigated In particular, it is found that the OB behavior is strongly sensitive to the relative phase of applied fields It is also shown that under certain parametric conditions, the OB can be switched to OM or vice versa The controllability of OB/OM in such a QW nanostructure may bring some new possibilities for technological applications in solid-state quantum information science and optoelectronics Keywords Optical bistability Optical multistability Probe absorption Introduction During the recent years, several kinds of quantum optical phenomena have been extensively explored based on the quantum interference and coherence, such as lasing without inversion [1, 2], electromagnetically induced transparency (EIT) [3, 4], atom localization [5, 6], large Kerr index [7–11], multiwave mixing [12–15], optical soliton [16–18], optical bi(multi)stability [19–27], and so on [28–31] The phenomenon of optical bistability, which refers to a system that two stable output intensities are often accessible for a single input intensity, has been widely studied from both experimental and theoretical points of view in the past years The basic mechanism for OB is atomic coherence and interference Harshawardhan and Agarwal [19] have shown schemes which one can decrease substantially the threshold intensity required for a bistable device using optical field induced transparency and through atomic coherence and interference effects Moreover, there has been an intense interest on quantum optical phenomena based on the quantum interference and coherence effects for the semiconductor quantum wells and * Ali Raheli raheliali.b@gmail.com Department of Physics, Bonab Branch, Islamic Azad University, Bonab, Iran quantum dots [32–41] Practically, it is more advantageous to employ a solid-state medium to realize different optical properties Semiconductor quantum wells have drawn significant attention because they can be considered as a two-dimensional (2D) electron gas, possessing properties similar to those of atomic vapors such as the discrete levels, but with the advantages of high nonlinear optical coefficients and large electric dipole moments, due to the small effective electron mass Several typical interesting phenomena have been investigated in solid media For instance, the hybrid absorptive–dispersive OB/OM in a four-level inverted-Y quantum well system inside a unidirectional ring cavity was investigated by Wang et al [24] They found that the coupling and the pumping fields, as well as the cycling field, can affect the OB and OM dramatically, leading to efficient manipulation of the threshold intensity and the hysteresis loop The switching feature of EIT-based slow light giant phasesensitive Kerr nonlinearity in a semiconductor quantum well is also investigated [42] In this paper, we employ a novel four-level lambda-type quantum well (QW) nanostructure to explore the possibility to realize the OB and OM This system includes a closed-loop level structure; as a result, the OB and OM are strongly phase sensitive, which will be discussed in more detail in the paper In addition, we show that that this system can change the OB to OM or vice versa under certain parametric conditions It should be pointed out that recently the OB and OM behaviors are investigated in a similar level structure with tripod four- Braz J Phys level atomic configuration [27] However, the level structure we proposed is a quantum well nanostructure which may provide some new possibilities for technological applications in optoelectronics and solid-state quantum information science The organization of the paper is as follows: the theoretical model and equations are presented in Sect 2, the numerical results are discussed in Sect 3, and the conclusions are given in Sect Model and Equations of the Motion Figure 1a shows the proposed quantum well system with four energy levels in Lambda—the configuration illustrated in Fig 1b The considered quantum well sample is very similar to the one employed in [35, 43, 44] It is possible to select the suitable parametric conditions for this QW sample For example, it can be designed to possess transition energies in the range of 120–170 meV and desired dipole moments, and these quantum well samples can be grown by the molecular-beam epitaxy (MBE) method with 40–80 symmetric 10-nm n-doped (ne = × 1011cm−2) InxGa1 − xAs (y = 0.47) wells and 10-nm In y Ga − y As (y = 0.48) barriers supported on a latticematched undoped InP substrate containing a 1–2-mmdiameter etched hole for optical access [45] A weak tunable probe field Ep with frequency ωp couples the transition |4〉 ↔ |1〉 with a Rabi frequency Ωp = Ep ℘41/2ℏ Transitions |4〉 ↔ |3〉, |2〉 ↔ |3〉 and |4〉 ↔ |2〉 are mediated by three coupling fields E3, E2, E4, with frequencies ω3, ω2, ω4(ω3 + ω2 = ω4) The corresponding Rabi frequencies are a b 4 3 2 p c E PI E PT medium M1 R M4 L E2 E3 E4 M2 R M3 Fig a Schematic diagram of the four level QW system b The energy level arrangement for system under study c Schematic setup of unidirectional ring cavity containing a QW sample Ω3 = E3 ℘43/2ℏ, Ω2 = E2 ℘32/2ℏ, Ω4 = E4 ℘42/2ℏ, where ℘ij denotes the induced electric dipole moment of the corresponding transition The density matrix equation of the motion for the above QW system can be obtained under the rotating-wave approximation as   iϕ ρ˙ 21 ¼ i Δp ỵ i dph 21 =2 21 ỵ i2 e 31 ỵ i4 41 ip 24 ;   i 31 ẳ i p ỵ i dph 31 =2 31 ỵ i2 e 21 ỵ i3 41 ip ρ34 ;     ρ˙ 41 ¼ i p ỵ i ỵ dph 41 =2 41 ỵ ip 11 44 ị ỵ i4 21 ỵ iΩ3 ρ31 ;   −iϕ ρ˙ 32 ¼ i ỵ i dph 32 =2 32 ỵ i2 e 22 33 ị ỵ i3 42 i4 34 ;     −iϕ ρ˙ 42 ¼ i Δ4 þ i γ þ γ dph 42 =2 ρ42 þ iΩ4 ðρ22 −ρ44 Þ þ iΩp ρ12 þ iΩ3 ρ32 −iΩ2 e ρ43 ;     iϕ 43 ẳ i ỵ i ỵ dph 43 =2 43 ỵ i3 33 44 ị þ iΩp ρ13 þ iΩ4 ρ23 −iΩ2 e ρ42 ; 1ị 11 ẳ 41 44 ỵ ip 41 14 ị; 22 ẳ 42 44 ỵ i2 32 ei 23 ei ỵ i4 42 24 ị; À Á ρ˙ 33 ¼ γ 43 ρ44 −iΩ2 ρ32 ei 23 ei ỵ i3 43 34 ị; 11 ỵ 22 ỵ 33 ỵ 44 ẳ 1; where p = ω p − ω 41 , Δ = ω − ω 23 , Δ = ω − ω 43, and Δ4 = ω4 − ω42 are the detuning parameters (Δ3 + Δ2 = Δ4) Also, the parameter ϕ = φ2 + φ3 − φ4 is the relative phase of applied fields, with φi being the individual phase of each applied field The decay rates for the transitions |4〉 ↔ |1〉, |4〉 ↔ |2〉, |4〉 ↔ |3〉 are γ41, γ42, γ43, respectively We neglect the decay rates for the transitions |3〉 ↔ |1〉, |3〉 ↔ |2〉, |2〉 ↔ |1〉 The dephasing rates are γ dph ij The total decay rates including population decay contribution as well as dephasing contribution are added phenomenologically in the above Braz J Phys Fig The effect of a Ω2, b Ω3, c Ω4 on the OB behavior The selected parameters are a Ω3 = Ω4 = γ, b Ω2 = Ω4 = γ, and c Ω3 = Ω2 = γ, and the other dph parameters are C ¼ 100; γ 42 ¼ γ 41 ¼ γ 43 ¼ γ; γ ij ¼ 0:01γ Ωp = 0.01γ Δp = Δ3 = Δ4 = 0, ϕ = density matrix equations The first contribution is mainly due to longitudinal–optical (LO) phonon emission events at low temperature, and the dephasing contribution may originate from electron–electron scattering and electron–phonon scattering as well as inhomogeneous broadening due to scattering on interface roughness Now, we put a medium of length L composed of the proposed QW sample in a unidirectional ring cavity as shown in Fig 1c The intensity of reflection and transmission coefficients of mirrors and are R and T (with R + T = 1), respectively We assume that both mirrors and are perfect reflectors The probe field Ep is circulating in the ring cavity Under slowly varying envelope approximation, the dynamic response of the probe field is governed by Maxwell’s equations ∂E p ∂E p iωp À Á ỵc ẳ P p t z 20 2ị where the induced polarization P(ω p) in the transition |4〉 ↔ |1〉 is given by À Á P ωp ¼ N ℘41 ðρ41 Þ: ð3Þ ∂E In the steady state, the term ∂tp in Eq (2) is equal to zero Substituting Eq (3) into Eq (2), we obtain the field amplitude relation E p N p 41 ẳi 41 ị: z 2c0 ð4Þ when N is the number density of the electrons in the sample For a perfectly tuned ring cavity, in the steady-state limit, the boundary conditions impose the following conditions between the incident field EIP and the transmitted field ETP [46] ETp E p Lị ẳ p T 5:aị Braz J Phys Fig The effect of a Ω2, b Ω3, and c Ω4 on the probe absorption versus probe detuning The selected parameters are the same as Fig p 0ị ẳ p I T p ỵ Rp Lị; ð5:bÞ where L is the length of the QW sample Note that R is the feedback mechanism and it is responsible for the bistable behavior, so we not expect any bistability when R = in Eq 5(b) According to the mean field limit [47] and by using the boundary condition, the steady-state behavior of transmitting field is given by y ẳ 2xiC 41 ị; where y ẳ 41 E Ip p T 6ị and x ẳ 41 E Tp pffiffiffi 2ℏ T are the normalized input and MLω ℘2 output field, respectively The parameter C ¼ 2ℏcεp0 T41 is the cooperatively parameter in a ring cavity Transmitted field depends on the incident probe field and the coherence term ρ41 via Eq (6) So, the bistable behavior of the medium can be determined by the QW sample variables through ρ41 We set the time derivatives of ρij equals to zero, i.e., ∂ρij ∂t ¼ in Eq (1) and solve the corresponding density matrix equations together with the coupled field in Eq (6) Assuming that the atom is initially in its ground level, the steady-state analytical solution for the coherence term 41 reads ip 22 ỵ y2 y3 ; 7ị 41 ẳ w where w ẳ 22 y3 ỵ 23 y2 ỵ 22 y1 ỵ y1 y2 y3 ỵ 22 cos, and       dph y1 ẳ i p ỵ i ỵ dph 41 =2 , y2 ẳ i p ỵ i 21 =2 , and   =2 y3 ẳ i p ỵ iγ dph 31 Results and Discussion Now, we start to analyze the steady-state behaviors of the output field intensity versus the input field intensity for Braz J Phys various parameters illustrated in Figs 2, 3, 4, 5, and First, we study the influence of coupling fields Ω2, Ω3, Ω4 on the shift of the OB hysteresis curve The plots of the output field intensity against the input field intensity for different values of the coupling fields are displayed in Fig and for the parametric condition γ42 = γ41 = γ43 = γ, γ dph ij ¼ 0:01 It can be seen from Fig 2a that with the gradual increase of the Rabi frequency Ω2 from γ to 3γ, the threshold of OB increases progressively and the area of the hysteresis loop becomes larger In contrast, as shown in Fig 2b, c, the slight increase on Rabi frequencies Ω3 and Ω4 results in an opposite behavior with respect to Fig 2a In these cases, the OB threshold intensity reduces remarkably when we increase each Rabi frequency Ω3 or Ω4 from γ to 3γ The main reason for such behaviors of OB under the effect of different Rabi frequencies can be qualitatively interpreted as follows The gradual increase of the Rabi Fig Switching mode from OB to OM The selected parameters are Ω2 = 4γ, Ω3 = 0.6γ, Ω4 = 0.7γ, and the other parameters are the same as Fig frequency Ω2 (Ω3 or Ω4) from γ to 3γgives rise to the enhancement (reduction) of the absorption for the probe laser field around zero probe detuning as plotted in Fig 3a (Fig 3b, c), which makes the cavity field harder (easier) to reach saturation The coupling fields of this QW system make a closed loop structure; therefore, their relative phase can affect the optical properties of the medium In the following, the influence of the relative phase of applied fields on OB behaviors is discussed As illustrated in Fig 4a, when we change ϕ from to π/2, the OB threshed increases correspondingly In addition, by modulating ϕ = π, the threshold intensity reduces again and arrives at the state plotted for ϕ = In order to explain the origin of this phase sensitivity, we display here the probe susceptibility Fig The effect of the relative phase of applied fields on the a OB behavior and b probe absorption Here, Ω2 = Ω3 = Ω4 = γ, and the other parameters are the same as Fig Fig The effect of the electronic cooperation parameter on the OB behavior Here, Ω2 = Ω3 = Ω4 = γ, and the other parameters are the same as Fig Braz J Phys curves for different values of the relative phase (ϕ = 0, π/ 2, π) in Fig 4b We observe that two absorption peaks appear for ϕ = (π), one is narrower (wider) located on the left side of the line center, and another one wider (narrower) is being located on the right side of the line center It is obvious that the value of probe absorption is the same at zero probe detuning for both ϕ = and π(see point A at Fig 4b), which is the main reason for the same OB threshold intensity observed for ϕ = and π in Fig 4a The probe absorption experiences three peaks (Fig 4b) by adjusting the relative phase to π/2, leading to enhancement of OB threshold intensity for this case (see Fig 4a) The origin of such phase-sensitive OB behavior is due to the closed loop structure of the system It is clear from the level structure of the system (Fig 1b) that there exists two different possible pathways from the ground level |1〉 to the lower level |2〉; the direct one j1i →Ωp j4i →Ω4 j2i and the indirect one j1i →Ωp j4i →Ω3 j3i →Ω2 j2i The effect of the relative phase ϕ on the OB behaviors in such a closed-loop four-level system can be understood through the quantum interference induced by these two channels a Due to its practical application in coding elements or alloptical switching, it is worth exploring the possibility to convert OB to OM or vice versa Illustrated in Fig 5, the OB can be switched to OM by modulating the relative phase ϕ from to π/ The main reason for switching between OB and OM comes from Eq (6) By manipulating the relative phase of applied fields, the order of the variable x in Eq (6) changes for the QW system under consideration, which means that the observed OM may depend on the complicated form of variable x Physically, the coherence of the medium may or may not be high enough, depending on the values of ϕ Modulating ϕ can increase or decrease the coherence of the system which may lead to enhancement or reduction of Kerr nonlinearity for the probe field; thus, the transition between OB to OM or vice versa can be realized In order to have a better understanding about how the bistable threshold value varies with the electronic cooperation parameter C, we display the input-output field curves for different values of C in Fig It is realized that the threshold and the hysteresis cycle shape are changed by increasing C The threshold goes up as the cooperation parameter C becomes larger In fact, the cooperation parameter is directly proportional to the electronic number density, and the increase in the number b 0.015 0.01 1000 0.005 Probe intensity Probe intensity 0.015 0.01 1000 0.005 600 800 Retarded time 600 400 200 Distance Retarded time 400 Distance c 200 0 d 0.015 0.015 0.01 1000 0.005 800 500 600 Probe intensity Probe intensity 500 800 500 0.01 1000 0.005 500 800 600 400 Distance Retarded time 200 0 Retarded time 400 Distance 200 0 Fig Plots of probe field intensity in the medium against retarded time and distance for σ = 30/γ, τ0 = 30/γ, and a Ω2 = Ω3 = Ω4 = γ, b Ω2 = 4γ, Ω3 = Ω4 = γ, c Ω3 = 4γ, Ω2 = Ω4 = γ, and d Ω4 = 4γ, Ω2 = Ω3 = γ Here, Δp = and the other parameters are the same as in Fig Braz J Phys density of electrons will enhance the absorption of the sample, which accounts for the raise of the bistable threshold Finally, we investigate the propagation of probe pulse in a realistic situation where the incident wave has a Gaussian profile, and its propagation is controlled by three control fields of larger intensity The propagation dynamics of the probe field inside the QW medium and along the z direction can be described by the Maxwell equation, which under the slowly varying envelope approximation can be expressed as ∂Ωp ðz; t Þ p z; t ị ỵ ẳ ik41 z; t ị; ∂z ∂t c ð8Þ N ω jμ j2 p 41 characterizes the strength light coupling with where k ¼ 4cℏε the atomic medium Going to the retarded coordinates ξ = z and τ = z − t/c, we shall consider the propagation of a Gaussianshaped probe pulse of the form Ωp ð0; τ ị ẳ 0p eẵ ị = ; where Ω0p describes the peak value of the Rabi frequency before the probe pulse enters the medium, τ0 gives the peaks location, and σ denotes the temporal width of the input pulse The propagation of a Gaussian pulse through the four-level QW nanostructure is shown in Fig As illustrated in Fig 7a, when Ω2 = Ω3 = Ω4, the probe pulse suffers the absorption losses during its propagation inside the medium Increasing the control field Ω2, the weak probe pulse propagates with the maximum losses inside the medium, as can be seen in Fig 7b Increasing Ω3 or Ω4 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