We propose and analyze an efficient way to enhance four-wave mixing (FWM) signals in a four-subband semiconductor quantum well via Fano-type interference. By using Schrödinger-Maxwell formalism, we derive explicitly analytical expressions for the input probe pulse and the generated FWM field in linear regime under the steady-state condition. With the aid of interference between two excited subbands tunneling to the common continuum, the efficiency to generate FWM field is found to be significantly enhanced, up to 35%. More interestingly, a linear growth rate in the FWM efficiency is demonstrated as the strength of Fano-type interference increases in presence of the continuum states, which can be maintained for a certain propagation distance (i.e., 50μm).
© 2014 Optical Society of America
Over the past few decades, quantum interference in the form of electromagnetically induced transparency (EIT) has attracted tremendous interests in optical communications and quantum information processing [1,2]. This concept of quantum interference is also extended to a variety of applications, including lasing without inversion , controlling the group velocity of light pulses [4, 5], and highly efficient schemes for nonlinear optics [6–8]. For examples, nonlinear optical phenomena based on the quantum interference have been demonstrated for optical solitons [9–11], enhanced nonlinearity [12–14], and multi-wave mixing processes in the ultraslow propagation regime [15, 16]. In particular, multi-wave mixing process is expected to have a wide range of applications in diverse fields, such as generation of short-wavelength coherent radiation, nonlinear spectroscopy with low-light intensity, quantum single-photon nonlinear optics, and quantum information science. However, the low conversion efficiency for multi-wave mixing in ultraslow propagation regime limits its applications.
In addition to the atomic media, quantum interference phenomena associated with intersubband transitions (ISBT) in semiconductor quantum wells (SQW) have also been utilized based on the analogue between atomic and electronic energy levels . One advantage in the SQW system is the more accessible Fano-type interference, which arises from the absorption paths of two states coupled to a common continuum and leads to nonreciprocal absorptive and dispersive profiles [18, 19]. In this work, with the aid of interference between two excited subbands tunneling to the common continuum, we propose and analyze an efficient way to enhance four-wave mixing (FWM) signals in SQW with a four -subband configuration. Different from using quantum destructive interference induced by the continuous-wave (cw) control field [20, 21] or the resonant tunneling , our scheme based on one weak probe pulse interacting with two cw pump fields is analyzed for the enhanced FWM generation. By solving the coupled Schrödinger-Maxwell equations, a time-dependent analysis in the ultraslow propagation regime is performed for the generated FWM field. We show the corresponding analytical expressions explicitly, for the probe and generated FWM pulses, as well as their phase shifts, absorption coefficients, group velocities, and the related conversion efficiency, respectively. Owing to the existence of Fano-type interference, our results illustrate the possibility to have a maximum FWM conversion efficiency up to 35%. Last but not least, a linear growth rate in the FWM efficiency is demonstrated as the strength of Fano-type interference increases in presence of the continuum states, which can be maintained for a certain propagation distance (i.e., 50μm).
Our proposed SQW structure includes a deep well and a shallow well, as illustrated in Fig. 1. The left shallow well and right deep well may be made by an Al0.07Ga0.93As layer on the GaAs layer, separated by a Al0.32Ga0.68As potential barrier. In general, one can obtain the eigenenergies for the four conduction subbands by solving the effective mass Schrödinger equations. The energies of two ground subbands |1〉 and |2〉 are assumed to be E1 = 51.53 meV and E2 = 97.78 meV, respectively. Two closely spaced delocalized subbands |3〉 and |4〉, with the eigenenergies E3 = 191.3 meV and E4 = 233.23 meV, are created by mixing the first excited subbands in the shallow well |se〉 and in the deep well |de〉 through the resonant tunneling. The subbands |3〉 and |4〉 can be represented by a coherent superposition of the first excited subbands in shallow and deep wells [17–19]. They are and . In Fig. 1, the dashed-lines represent the corresponding energies; while the solid-curves show the wave functions for four subbands. This SQW system is driven by two continuous-wave (cw) pump laser, with the central frequency ωc1,c2 and wavevector kc1,c2. Simultaneously, a weak probe pulse with the central frequency ωp and wavevector kp, and a FWM pulse field with the central frequency ωm and wavevector km construct the wave-mixing process |1〉 → |3〉 → |2〉 → |4〉 → |1〉.
To simplify the physical picture, in the following analysis, we assume that all the subbands have the same effective mass. The SQW structure is also designed with a low electron sheet density, in order to neglect the electron-electron interactions. In the interaction picture, with the rotating-wave approximation and the electric-dipole approximation, the interaction Hamiltonian of this system can be written as (h̄ = 1):
The state wave function is assumed to be in the form of |Ψ〉 = A1|1〉 + A2ei(kp−kc1)·r |2〉 + A3eikp·r |3〉+ A4ei(kp−kc1+kc2)·r |4〉, where Aj (j = 1, 2, 3, 4) means the time-dependent probability amplitude for finding particles in the corresponding subband. Then, we substitute |Ψ〉 into the Schrödinger equation , and have the equation for the probability amplitudes,19]: upon solving the effective mass Schrodinger equation with outgoing waves at infinity, we obtain a set of complex eigenvalues whose real and imaginary parts yield respectively, the quasibound state energy levels and resonance widths. We should note that the population decay rates γ3l and γ4l are assumed to be the decay rates from the subbands |3〉 and |4〉 to the continuum by tunneling in the presence of the electronic continuum. When the electronic continuum is not included in the asymmetric double SQW structure, the population decay rates γ3l and γ4l are assumed to be the decay rates from the subbands |3〉 and |4〉 to the lower states. For the present asymmetric double quantum well structure, the population decay rates turn out to be γ3l = 1.58meV, γ4l = 1.5meV in presence of the electronic continuum, while the population decay rates are γ3l ≈ γ4l = 1meV in absence of the electronic continuum. For the temperatures up to 10 K, the electric density may be kept as low as 1024m−3 . In this scenario, the dephasing rates can be estimated to be γ3d = 0.32meV, γ4d = 0.3meV. A comprehensive treatment of the decay rates would involve incorporation of the decay mechanisms in the Hamiltonian of the system. However, we have adopted the phenomenological approach of treating the decay mechanisms just as those done in Refs. [23–26]. It is worth to be noted that a cross coupling term between the excited states |3〉 and |4〉 is introduced by [17–19] in the presence of the electronic continuum. We remark that represents the strength of Fano-type interference, for which the values p = 0 and p = 1 correspond to no interference and perfect interference, respectively.
In the limit of slowly varying amplitude approximation, the input probe pulse and the generated FWM field propagate along the ”z” direction (i.e., δk · r = δk · z) according to the 1D wave equations:
In principle, above equations of probability amplitude shown in Eqs. (2)–(5) must be solved simultaneously with Maxwell’s equation shown in Eqs. (6)–(7) in a self-consistent manner. Before solving those nonlinear equations, let us first examine the linear excitation of the system, which provides useful hints for the weak nonlinear theory. As a weak perturbation, we assume that the Rabi frequencies of the probe pulse and FWM field, 2Ωp,m, are much smaller than that of the cw pump fields 2Ωc1,c2. When the electrons are initially populated in the ground state, |1〉 (i.e., A1 ≃ 1), on can perform the Fourier transformations for Eqs. (2)–(7) by definingEqs. (2)–(7) become Eqs. (10)–(14) can be derived as:
With the given initial conditions for the probe pulse and FWM field, i.e., Λp (0, ω) and Λm (0, ω) = 0, we can have the analytic solutions for these two weak fields:Eqs. (24)–(25) give the propagation dynamics for the probe pulse and the generated FWM field in our proposed SQW system, based on which we use practical parameters to demonstrate examples to enhance FWM signals via Fano-type interference.
For the temperatures up to 10 K, the electric density may be kept as low as 1024m−3 . In this scenario, the dephasing rates can be estimated to be γ3d = 0.32meV, γ4d = 0.3meV. The population decay rates are calculated by solving effective mass Schrödinger equation. For our QWs with continuum considered above, the population decay rates turn out to be γ2l = 2.36 × 10−6 μeV, γ3l = 1.58meV, γ4l = 1.5meV, while the other set of decay parameters without continuum are γ3l = γ4l = 1meV. As a result, we obtain p = 0 and p = 0.83 for the strength of Fano-type interference represent without and with continuum states, respectively. In addition, we choose Δc1 = 1meV, Δp = Δc2 = 0, and κm = κp = 9.6 × 103μm−1meV.
Within this practical parameter set, we plot in Fig. 2 the amplitudes of probe and FWM pulses as a function of the propagation distance z for the cases (a) without and (b) with continuum states. From Fig. 2, one can find that the amplitude of probe pulse decreases monotonically as the propagation distance z increases; while the generated FWM field increases monotonically. Subsequently, after a critical propagation distance, the amplitudes of probe pulse and FWM field reach the same value. This can be explained by using the multiphoton quantum destructive interference, through the coupling excitation pathway |1〉 → |3〉 and the feedback excitation pathway |1〉 → |3〉 mediated by |4〉 . Moreover, compared to the absence of continuum sates, as shown in Fig. 2(a), the amplitude remains unchanged with the introduction of the continuum sates, as shown in Fig. 2(b). Above interesting results come from the Fano-type interference caused by tunneling from the excited subbands to the continuum. This interference modifies the neighboring transitions and thus affects the multiphoton quantum destructive interference between the two different coupling pathways. When the amplitudes of probe pulse and FWM field reach the same value, the feedback excitation pathway to the subband |4〉 becomes important, which is π out of phase with respect to the coupling excitation pathway. These processes simultaneously lead to the suppression for the subband |3〉. As the propagation distance z increases, the amplitudes of |3〉 and |4〉 sacrifice from the absorption, resulting in the decay shown in Fig. 2(a). However, the decay in the amplitudes Ωm,p can be suppressed, with the existence of a Fano-type interference, as shown in Fig. 2(b). As a matter of fact, this is a consequence from the competition between the destructive quantum coherence in the multiphoton interference and the constructive quantum coherence in the Fano-type interference.
According to Eqs. (24)– (25), both K+ and K− modes will simultaneously be excited in the SQW medium even only with one input field. However, there exists some parameter regimes where one of the two modes always decays much faster than the other. In Fig. 3, we show one example that the absorption coefficients α± = Im[K±(0)] differ significantly. Here, the ratio between two absorption coefficients α−/α+ = Im[K−(0)]/Im[K+(0)] is plotted versus the amplitude in the cw pump field |Ωc2| for different control field amplitudes |Ωc1| in the absence of the electronic continuum, see Fig. 3(a). In order to examine the effect of Fano-type interference on the absorption coefficients, we plot in Fig. 3(b) the ratio α−/α+ versus the amplitude in the cw pump field |Ωc2| with and without including the continuum states. It can be obviously found that the absorption coefficient ratio decreases as the amplitudes of two pump fields increase. Moreover, even for p = 0.83 in presence of the continuum state, we have α− ≪ α+. In other words, Fig. 3 illustrates that the K+ mode decays much more quickly than the K− mode. Thus, one can safely neglect this faster variable K+ after a characteristic propagation distance. As a consequence, the two weak pulses propagate with the matched group velocity Vg = Vg−. By neglecting the K+ mode, the probe and FWM pulses Ωp,m can be rewritten asEqs. (26)–(27), one can see that the probe pulse and the generated FWM field travel at a ultraslow group velocity.
In Fig. 4, we plot the group velocity Vg/c as a function of pump field amplitude |Ωc2| for different sets of cw amplitude |Ωc1|. The group velocity exhibits the similar trends for different case, i.e., with or without continuum states. However, in the presence of the continuum states (p = 0.83), the slower group velocity can be achieved, i.e., on the order of 10−4c as shown in Fig. 4(b). It should be noted that a further reduced group velocity can be obtained by choosing the related parameter values appropriately.
As illustrated in Fig. 3 and Fig. 4, the probe pulse and the generated FWM field can travel through the SQW medium with the same temporal profile and at a ultraslow group velocity within a characteristic distance. Now, we turn to the conversion efficiency for the FWM field. Based on the definition in Ref. , the efficiency for the generated FWM field can be measured as , where is the electric field Em (|Em|2 = 4h̄2|Ωm|2/|μ31|2) for the generated FWM field at the exit z = L and (|Ep|2 = 4h̄2|Ωp|2/|μ31|2) is the electric field for the probe pulse at the entrance z = 0. According to Eqs. (26) – (27), the efficiency for FWM signal has the form,
To study the dependence on different system parameters for the generated FWM efficiency, first of all, we analyze how the amplitude of cw pump fields modifies the conversion efficiency in Fig. 5. Here, the FWM efficiency ρ is plotted as a function of the amplitude Ωc2 of the cw pump field c2, for the different cw pump fields c1 without and with including the continuum states, shown in Fig. 5(a) and 5(b), respectively. One can see that the amplitude of two cw pump fields and the Fano-type interference caused by the tunneling from the excited subbands to the continuum both have an obvious influence on the FWM conversion efficiency. Moreover, a maximum value in the FWM conversion efficiency can be achieved when |Ωc1| = |Ωc2|. In our simulations, even for a deep penetration distance into the SQW structure, such as L = 6μm, one can still have the conversion efficiency up to ρ > 35%. Direct comparison in Figs. 5(a) and (b) implies that the maximum value in the conversion efficiency can be enhanced when the continuum states are included. In other words, the Fano-type interference results from the resonant tunneling from the excited subbands to the continuum opens the channel for the FWM process. Different from previous FWM schemes based on EIT configuration, parameters for the electronic subbands in SQW systems can be engineered to give a desired strength of interference by utilizing the so-called structure coherent control in design .
To give a better insight on the FWM conversion efficiency from the amplitudes of two cw pump fields, in Fig. 6, we show the conversion efficiency ρ as a function of the amplitudes in two cw pump fields |Ωc1|/|Ωc2| without including the continuum states. The results shown in Fig. 6 clearly verify that the condition to achieve a maximum efficiency happens around |Ωc1| = |Ωc2|. Although, for the same amplitudes of two cw pump fields the highly efficiency can be obtained, the extent of the robustness is small. In the presence of the Fano-type interference caused by tunneling from the excited subbands to the continuum, in Fig. 7, we show the conversion efficiency ρ as a function of the strength of the Fano-type interference p for the different amplitudes of two pump fields. For simplicity, here we have assumed the population decay rates from subbands |3〉 and |4〉 to the continuum states remain unchanged (i.e., γ3l = 1.58 meV and γ4l = 1.5 meV), and the different values of interference strength p can be obtained by varying total decay rates γ3 and γ4. Interestingly enough, as shown in Fig. 7, the conversion efficiency for the generated FWM field increases linearly as the strength of Fano-type interference increases for a certain propagation distance. It is the Fano-type interference that suppresses the population in the subbands |3〉 and |4〉, resulting in the reduction of the corresponding absorption in the generated FWM field. Again, the Fano-type interference leads to the enhancement of conversion efficiency. However, the linear increasing of the maximal conversion efficiency has robustness only in a small region of two cw pump fields intensities. As illustrated in Fig. 7, the increasing effect will be not obviously if the amplitudes of two pump fields increases continuously.
Since the probe field and the generated FWM signal are in the form of optics pulses, here, we also consider the influence of propagation distance on the relevant conversion efficiency. To do this, we assume that the probe field at the entrance of the SQW structure has a Gaussian pulse profile, i.e., Ωp (0, t) = Ωp (0, 0) e−(t2/τ2), with the pulse width τ. Now, the conversion efficiency can be rewritten as ρ = |Ωm(z, t)/Ωp(0, 0)|2. The results from numerical simulations for the corresponding FWM conversion efficiency are shown in Fig. 8, with the pulse width fixed as τ = 10 meV−1. It can be seen clearly that the conversion efficiency ρ decreases significantly as the propagation distance increases when the Fano-type interference caused by tunneling from the excited subbands to the continuum is not included, (a) p = 0; while it decreases slowly for a longer propagation distance due to the existence of the Fano-type interference when the continuum states are included, (b) p = 0.83. From the comparison between Fig. 8(a) and Fig. 8(b), the presence of a Fano-type interference indeed modifies not only the linear optical property, but also the FWM process. As shown in Fig. 4, the Fano-type interference contributes to slow down the matched group velocity. Consequently, such a reduced group velocity ensures a longer interaction time, and leads to maintain a higher FWM conversion efficiency for a longer propagation distance.
In conclusion, we have performed a time-dependent analysis for the four-wave mixing (FWM) process in the ultraslow propagation regime, by considering the intersubband transitions in an asymmetric semiconductor double quantum well system. By including the Fano-type interference caused by the tunneling from the excited subbands to the electronic continuum, we demonstrate an efficient way to generate FWM field by means of two continuous-wave (cw) pump fields and a weak probe pulse. Through the Schrödinger-Maxwell formalism, we also give the corresponding analytical expressions for the input probe pulse and the generated FWM field in the linear regime under the steady-state condition. In presence of the Fano-type interference, the conversion efficiency of FWM field is revealed to be enhanced significantly, up to 35%. Take the advantages of flexibilities in semiconductor structures, our results provide a practical opportunity to implement optical modulated solid-state devices.
The research is supported in part by National Natural Science Foundation of China under Grant Nos. 11374050 and 61372102, by Qing Lan project of Jiangsu, and by the Fundamental Research Funds for the Central Universities under Grant No. 2242012R30011.
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