Phase control of two-dimensional probe gain-absorption spectra in semiconductor quantum wells

Author’s Accepted Manuscript Phase control of two-dimensional probe gainabsorption spectra in semiconductor quantum wells Haifeng Xu

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S1386-9477(15)00040-5 http://dx.doi.org/10.1016/j.physe.2014.12.034 PHYSE11863

To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 14 November 2014 Accepted date: 9 December 2014 Cite this article as: Haifeng Xu, Phase control of two-dimensional probe gainabsorption spectra in semiconductor quantum wells, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2014.12.034 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Phase control of two-dimensional probe gain-absorption spectra in semiconductor quantum wells Haifeng Xu∗ School of Mechanical and Electronic Engineering, Suzhou University, Suzhou 234000, China December 10, 2014

Abstract We investigate the two-dimensional gain and absorption of a weak probe field via two orthogonal standing-wave lasers in a four-level inverted-Y asymmetric quantum well system. We find that, due to the spatial-dependent quantum interference effect, the spatial distribution of the 2D gain and absorption spectra can be easily controlled by adjusting the system parameters. More importantly, the probe gain-absorption spectrum can be controlled at a particular position and the 2D localization effect is indeed achieved efficiently. Thus, our scheme shows the underlying probability for the formation of the 2D localization effect by using a QW structure. Keywords. standing-wave lasers; quantum well system; gain and absorption spectra

1

Introduction

During the past few decades, phenomena based on quantum interference have attracted lots of attention of many researchers in coherent media [1-6]. One of an interesting phenomena, the precision position measurement of an atom has been a subject of many studies because of its potential wide applications in trapping of neutral atoms, laser cooling [7], atom nano-lithography [8], etc. Based on atomic coherence and interference, many schemes [9-20] have been proposed for realizing one- and two-dimensional (2D) atom localization behaviors via the measurement of the atomic population, double-dark resonances, spontaneous emission, the probe absorption spectrum, and so on. It should be noted that similar quantum-interference-induced phenomena in semiconductor quantum wells have also attracted great attention due to the potentially important applications in solid-state optoelectronics and quantum information science. In fact, the analogies between coherent nonlinear phenomena in atomic systems and semiconductor quantum wells have been successfully exploited over the past few years, various effects including gain without inversion [21-23], electromagnetically induced transparency (EIT) [24], Kerr nonlinearity [25], and other novel phenomena [26-38] have been extensively investigated by many groups. Here, one may naturally ask if such techniques can also be used to facilitate the formation of the localization effect in artificial nanostructures. In this paper for the first time, to ∗ E-mail

address: [email protected]

1

our knowledge, we theoretically investigate the 2D gain and absorption of a weak probe field via two orthogonal standing-wave lasers in a four-level inverted-Y asymmetric quantum well nanostructure. Due to the spatial-dependent quantum interference effect, the spatial distribution of the 2D gain and absorption spectra can be easily controlled by adjusting the system parameters. As a result, our scheme shows the underlying probability for the formation of the 2D localization effect in a solid nanostructure.

2

Model and dynamic equations

We consider an asymmetric quantum well system with four levels as shown in Fig. 1, which has been realized experimentally. Levels |1, |2, and |3 are in a usual three-level ladder-type configuration and level |4 together with levels |1 and |2 forms a three-level Λ-type configuration. So, this composite system consists of two subsystems, i.e., one ladder type and the other Λ-type. The quantum well samples considered here could be very much similar to the one used in Refs. 40, 42 and 50, where we can choose the proper parametric conditions. For instance, these quantum well samples are grown by the molecularbeam epitaxy (MBE) method with 40–80 symmetric 10 nm n-doped (ne = 6 × 1011 cm−2 ) Inx Ga1−x As (x = 0.47) wells and 10 nm Aly In1−y As (y = 0.48) barriers supported on a lattice-matched undoped InP substrate containing a 1–2 mm diameter etched hole for optical access. Following the method described in the Ref. [24], the sample can be designed to have desired transition energies in the range of 120–180 meV and desired dipole moments, and the quantum well substrate may be consisting of 40–60 modulation-doped coupled quantum wells. These GaAs quantum wells (with approximate thicknesses of 6–8 nm) separated by an Al0.33 Ga0.67 As barrier (2 nm thickness) can be grown on the GaAs substrate. The coupled well periods can have separation of 95 nm by inserting another spacer of Al0.33 Ga0.67 As. This structure could be designed to meet the transition energy requirements and desired dipole moments, which can easily be accessible with an available semiconductor diode laser system in the midinfrared range. The transition |2 ↔ |4 is mediated by a composition of two orthogonal standing-wave fields Ωs (x, y) (frequency ω s , phase φs ) with the respective Rabi frequencies 2Ωs (x, y) = 2Ωs [sin(kx)+sin(ky)], while a pumping field E2 (frequency ω 2 , phase φ2 ) and a weak probe field Ep (frequency ω p , phase φp ) are applied to the transitions |3 ↔ |2 and |2 ↔ |1, respectively. The cycling field Ed (frequency ω d , phase φd ) is coupled to the transition |4 ↔ |1. By adopting the standard approach, the density-matrix equations of motion in dipole and rotating-

2

wave approximations for this system can be written as follows iρ˙ 11

=

Ωp ρ21 − Ωp ρ12 + Ωd ρ41 − Ωd ρ14 + 2iγ 4 ρ44 + 2iγ 2 ρ22 ,

iρ˙ 22

=

Ω2 ρ32 − Ω2 ρ23 + Ωp ρ12 − Ωp ρ21 + Ωs (x, y)ρ42 − Ωs (x, y)ρ24 + 2iγ 3 ρ33 − 2i(γ 1 + γ 2 )ρ22 ,

iρ˙ 33

=

Ω2 ρ23 − Ω2 ρ32 − 2iγ 3 ρ33 ,

iρ˙ 12

=

Ωp (ρ22 − ρ11 ) − Ω2 ρ13 − Ωs (x, y)ρ14 eiΦ + Ωd ρ42 eiΦ − (iγ 1 + iγ 2 + Δp )ρ12 ,

iρ˙ 13

=

Ωd ρ43 − Ω2 ρ12 + Ωp ρ23 − (iγ 1 + Δ2 + Δp )ρ13 ,

iρ˙ 14

=

Ωd (ρ44 − ρ11 ) + Ωp ρ24 e−iΦ − Ωs (x, y)ρ12 e−iΦ − (iγ 4 + Δp − Δs − Δd )ρ14 ,

iρ˙ 23

=

Ω2 (ρ33 − ρ22 ) + Ωp ρ13 + Ωs (x, y)ρ43 eiΦ − (iγ 4 + iγ 1 + iγ 2 + Δ2 )ρ23 ,

iρ˙ 24

=

Ωs (x, y)(ρ44 − ρ22 ) + Ωp ρ14 eiΦ + Ω2 ρ34 eiΦ − Ωd ρ21 eiΦ − (iγ 1 + iγ 2 − Δ2 − Δd )ρ24 ,

iρ˙ 34

=

Ω2 ρ24 e−iΦ − Ωs (x, y)ρ32 e−iΦ − Ωd ρ31 − (iγ 3 + iγ 4 − Δs − Δ2 − Δd )ρ34 ,

(1)

in which Φ = φ2 + φd − φs , ρnm = ρ∗mn (m, n = 1, 2, 3, 4), and constrained by ρ11 + ρ22 + ρ33 + ρ44 = 1. Δp = ω p − ω 21 , Δs = ω s − ω 24 , Δ2 = ω 2 − ω 32 , and Δd = ω d − ω 41 are the detunings of the weak probe, the standing-wave, the pumping and the cycling fields. ω 21 , ω 32 , ω 24 , and ω 41 are resonant frequencies which associated with the corresponding optical transitions |2 ↔ |1, |3 ↔ |2, |2 ↔ |4, and |4 ↔ |1, respectively. Ωp = Ep μ21 /2, Ω2 = E2 μ32 /2, Ωs = Es μ24 /2, and Ωd = Ed μ41 /2 are the one-half Rabi frequencies for the respective transitions. 2γ 1 , 2γ 3 , 2γ 2 , and 2γ 4 are the total decay rates associated with the corresponding optical transitions |2 ↔ |1, |3 ↔ |2, |2 ↔ |4, and |4 ↔ |1, respectively. The total decay rates are added phenomenologically in the above density-matrix equations, which are comprised of a population decay contribution as well as a dephasing contribution. The first contribution is mainly due to longitudinal optical (LO) phonon emission at low temperature and the other contribution comes from electron-phonon scattering and scattering on interface roughness. In this paper, the choices of the decay rates are based on the realistic experimental results from Ref. [30,31]. In the limit of a weak probe, under the steady-state condition, the solutions of Eq. (1) for ρ21 to the first order of the probe field and for other ρij to the zero order of the probe field are  Ωs (x, y)Ωd e−iΦ (ρ − ρ11 ) + (ρ44 − ρ22 )× [iΩp (ρ22 − ρ11 ) + ρ21 = γ 1 + i(Δp − Δ2 ) 44   iΩp Ω2s (x, y) Ωs (x, y)Ωd e−iΦ + × γ 4 + i(Δp − Δs − Δd )(γ 1 + γ 2 − iΔs − iΔd ) γ 1 + γ 2 + i(Δs + Δd ) −1  Ω22 Ω2d Ω2s (x, y) + + (2) . γ 1 + γ 2 + iΔp + γ 4 + i(Δp − Δs − Δd ) γ 3 + i(Δp + Δ2 ) γ 1 + γ 2 + i(Δs + Δd )

3

Numerical results and discussion

Now we elaborate the results for this composite EIT system by numerical integration of Eq. (1) in the steady-state condition. For this purpose we will examine the coherence term ρ12 for the probe transition in terms of its imaginary part Im(ρ12 ) as a function of positions (kx, ky). As we know, the imaginary part of ρ12 represent the probe field gain and absorption spectra [24]. Eq. (2) exhibits that Im(ρ12 ) is dependent on the positions (kx, ky), the single phase parameter Φ, the detunings of optical fields Δp , Δs , Δ2 and Δd , as well as the Rabi frequencies Ωs (x, y), Ωp , Ω2 and Ωd . It is, in principle, possible to 3

obtain information about the conditional position probability distribution of the probe gain-absorption spectrum. Thus, the measurement of such a position-dependent quantity can provide information on the 2D localization effect in this solid system. First of all, we investigate the influence of the detuning of the probe field on the probe gain-absorption spectrum. The probe gain-absorption spectrum versus the positions (kx, ky) for different detuning of the probe field is plotted in Fig. 2. As can be seen, the spatial distribution of probe gain-absorption spectrum is very sensitive to the probe detuning Δp . For Δp = 10 meV and Δp = 20 meV, the peak maxima of the probe gain-absorption spectrum are distributed in all four different quadrants (Figs. 2(a) and 2(b)). When the probe detuning is tuned to Δp = 30 meV, the peak maxima of gain-absorption spectrum are situated in the quadrants I and III with two crater-like patterns (Fig. 2(c)). As we further increase the probe detuning (i.e. Δp = 40 meV in Fig. 2(d)), the spatial distribution of the gainabsorption spectrum exhibits two spike-like patterns. Based on the above analysis, it can be easily seen that there is a strong correlation between the spatial distribution of gain-absorption spectrum and probe detuning. The reason for the above results can be qualitatively explained as follows. When Ωp,d  Ωs,2 , this system is a kind of configuration which owns the property of the EIT [24]. As is well known, EIT in a resonant system can affect the linear and nonlinear susceptibilities, which means that the 2D probe gain-absorption spectrum can be controlled in our scheme by adjusting the system parameters (see Eq. (2)). Therefore, due to the quantum interference effect, we can esaily control the spatial distribution of gain-absorption spectrum within a subwavelength region of the standing-wave fields via the detuning of the probe field. The effect of the intensity of cycling field Ωd on probe gain-absorption spectrum is given in Fig. 3. From this figure, one can find gain and absorption spectra depend strongly on the intensity of cycling field Ωd . For a small intensity of the cycling field, i.e. Ωd = 0.2 meV, the probe gain-absorption spectrum shows two crater-like patterns with different height in the quadrants I and III (Fig. 3(a)). Interestingly, when the probe detuning is increased to Ωd = 0.4 meV, the peak maxima of the gainabsorption spectrum are only situated in the third quadrant with a spike-like pattern, as shown in Fig. 3(b), the localization peak with a small crater-like pattern in the first quadrant in Fig. 3(a) is completely disappeared. In such a condition, the probe gain-absorption spectrum can be controlled at a particular position and the 2D localization effect is indeed achieved efficiently. With a further increase of Ωd (Ωd = 0.6 meV in Fig. 3(c) or Ωd = 1 meV in Fig. 3(d)), the probe gain-absorption spectrum has a spike-like pattern in quadrant III and inverted-spike-like pattern in quadrant I. According to the numerical results, we can see that the increasing intensity of cycling field will lead to absorption in quadrant III, but gain in quadrant I (Im(ρ12 ) < 0, the probe field is amplified). In fact, owing to the presence of the quantum perturbation induced by the cycling field Ωd , the spatial-dependent probe gain-absorption spectrum changes remarkably in the x − y plane. Hence, this fact can be exploited to the 2D localization effect in a QW structure. Up to now, we have demonstrated the realization of the underlying probability for the formation of the 2D localization effect in the present quantum well system and discussed the influences of some system parameters. However, one of the most interesting characters of this asymmetric quantum well

4

system is the phase-related property in the presence of the closed-loop configuration. So, in Fig. 4, we plot gain-absorption spectrum versus positions kx (−π  kx  π) and ky (−π  ky  π) for different values of Φ. It is easy to see from Figs. 4(a)-4(c) that there appears a crater-like pattern in the quadrant I, and the value of Im(ρ12 ) becomes negative in quadrant III. However, in the case of Φ = π (Fig. 4(d)), the probe gain-absorption spectrum is the mirror image of the pattern of Φ = 0 in Fig. 3(b) with respect to the beeline y = −x in the x − y plane, as can be easily verified from Eq. (2), i.e., Im[ρ12 (x, y; Φ = 0)] = Im[ρ12 (−x, −y; Φ = π)]. Therefore, on the condition of Φ = π, we expect the reverse of the previous case (Φ = 0), and we can also achieve 2D localization effect.

4

Conclusions

To sum up, we have numerically investigated the 2D gain and absorption of a weak probe field via two orthogonal standing-wave lasers in a four-level inverted-Y asymmetric quantum well system. Due to the spatial-dependent quantum interference effect, the spatial distribution of the 2D gain and absorption spectra can be easily controlled by adjusting the system parameters. As a result, our scheme shows the underlying probability for the formation of the 2D localization effect by using a QW structure.

5

Acknowledgements

This work was supported in part by the Natural Science Foundation of Anhui Province (Grants No.1408085QA20, No.1408085MB40 and No.1308085MA11).

6

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Figure captions Fig. 1. (a) Schematic diagram of the four-level inverted-Y quantum well system. (b) Schematic of

the energy level arrangement for the quantum well system under study. Fig. 2. Gain-absorption spectrum Im(ρ12 ) versus positions kx (−π  kx  π) and ky (−π  ky  π) for different probe detuning Δp : (a) Δp = 10 meV, (b) Δp = 20 meV, (c) Δp = 30 meV, and (d) Δp = 40 meV. The other parameters are Ωd = 0.1 meV, Ωs = 20 meV, Ω2 = 5 meV, Ωp = 0.4 meV, Δs = Δd = Δ2 = 0, 2γ 1 = 2γ 2 = 2γ 3 = 5 meV, Φ = 0, and 2γ 4 = 0.2 meV. Fig. 3. Gain-absorption spectrum Im(ρ12 ) versus positions kx (−π  kx  π) and ky (−π  ky  π) for different values of Ωd : (a) Ωd = 0.2 meV, (b) Ωd = 0.4 meV, (c) Ωd = 0.6 meV, and (d) Ωd = 1 meV. The other parameters are Δp = 40 meV, Ωs = 20 meV, Ω2 = 5 meV, Ωp = 0.4 meV, Δs = Δd = Δ2 = 0, 2γ 1 = 2γ 2 = 2γ 3 = 5 meV, Φ = 0, and 2γ 4 = 0.2 meV. Fig. 4. Gain-absorption spectrum Im(ρ12 ) versus positions kx (−π  kx  π) and ky (−π  ky  π) for different values of Φ: (a) Φ =

π 4,

(b) Φ =

π 3,

(c) Φ =

π 2,

and (d) Φ = π. The other parameters are

Δp = 40 meV, Ωs = 20 meV, Ω2 = 5 meV, Ωp = 0.4 meV, Δs = Δd = Δ2 = 0, 2γ 1 = 2γ 2 = 2γ 3 = 5 meV, Ωd = 0.4 meV, and 2γ 4 = 0.2 meV.

7

Figure 1

3

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4

2

: s ( x, y )

1

(a)

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:p

4 :d

1

(b)

Figure 2





 

 

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(d)

Figure 3





 

 

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