## Abstract

We investigate the carrier-wave Rabi flopping effects in an asymmetric semiparabolic semiconductor quantum well (QW) with few-cycle pulse. It is found that higher spectral components of few-cycle ultrashort pulses in the semiparabolic QW depend crucially on the carrier-envelope phase (CEP) of the few-cycle ultrashort pulses: continuum and distinct peaks can be achieved by controlling the CEP. Our results demonstrate that by adjusting the CEP of few-cycle ultrashort pulses, the intersubband dynamics in the asymmetric semiparabolic QW can be controlled in an ultrashort timescale with moderate laser intensity.

©2008 Optical Society of America

## 1. Introduction

The evolution of ultrashort pulse optics has made it routine to generate light pulses with durations comparable to the carrier oscillation cycle [1]. This progress opens new prospects in nonlinear optics [2–7]. For few-cycle ultrashort pulse propagating in a resonant two-level system, Hughes firstly predicted that, when the area under the individual carriers may themselves cause Rabi flopping, carrier-wave Rabi flopping (CWRF) will occur, which manifests in local carrier reshaping and subsequently the generation of higher spectral components [2]. This phenomenon has then been demonstrated experimentally in the semiconductor GaAs [3].

When few-cycle ultrashort pulse is considered, the CEP becomes a physically important parameter for studying nonlinear optics [8–20]. The study the CEP dependent phenomenon is very meaningful. Firstly, by controlling the CEP, the motion of electronic can be controlled or detected in an ultrashort timescale [10–14]. Secondly, the CEP dependent phenomena also provide routes to determine the CEP [15, 16]. Most of the CEP dependent phenomena heretofore are based on the ionization mechanism, and amplified few-cycle ultrashort pulses are usually necessary [10–16]. With nonamplified laser pulse, several CEP dependent schemes using semiconductor have been discussed recently. For example, if the Rabi frequency becomes comparable to the light frequency, the different Rabi sidebands interfere around twice the laser center frequency, giving rise to a signal which depends on the CEP [17]. This CEP dependent phenomenon was observed in experiments on thin GaAs films [18]. Recently, the CEP control of ultrafast optical rectification has also been investigated in resonantly excited semiconductors. A characteristic phase map is predicted using parameters for thin-film GaAs [19]. However, to the best of our knowledge, no corresponding studies about the CEP effects with moderate electric field intensity in asymmetric semiconductor QW have been discussed.

In the past few years, thanks to computer-controlled molecular beam epitaxy technology, many specially-shaped QWs have been synthesized [22, 23]. Especially, the semiparaboic QWs have been grown and studied in the experiments [22, 23]. Because the inversion symmetry is broken, the second-order optical effects have been found in this system [4]. Moreover, the permanent dipole moment (PDM) in these media is nonzero [24, 25]. This parameter plays an essential role in many nonlinear optical effects [20, 26, 29, 30]. For example, due to the existence of PDM, two-photon transition enhancement in polar molecule has been observed in experiment [26]. Very recently, in our previous work, we found the pulse evolution in polar molecule depends sensitively on the CEP of the incident ultrashort pulse due to the effect of PDM, which may be utilized to measure the CEP of ultrashort pulse under the moderate intensity [20].

In this work, we investigate the interaction of few-cycle ultrashort pulse with asymmetric semiparabolic semiconductor QWs. Comparing with the symmetrical parabolic QW models, even-order spectral components can occur due to the asymmetry structure of the medium itself, and the CEPs related phenomena are more evident in asymmetric semiconductor QW. Moreover, continuum and distinct peaks of higher spectral components can be achieved by the control of the CEP from 0 to π. Our results show that by adjusting the CEP of few-cycle pulse in an asymmetric semiconductor QW, the CWRF can be controlled in an ultrashort timescale.

This paper is organized as follows. The interaction of few-cycle ultrashort pulses with the semiconductor QW is described in Sec. 2. The characteristics of the CWRF and the corresponding higher spectral components with the CEP changing in the asymmetric QW are presented in Sec. 3. We summary the results in Sec. 4.

## 2. Theory

Under the effective mass approximation, the Hamiltonian of the semiconductor QW structure can be written as

where

is the part of light-matter interaction, and

is the light-unperturbed Hamiltonian, here *V*(*x*) is the semiconductor confining potential which is illustrated in Fig. 1. For symmetric parabolic QW [21]

while for asymmetric semiparabolic QW [24, 25, 27]

here *x* represents the QW’s grown direction, and *ω _{0}* is the frequency of the confining potential in the QW.

The unperturbed energy levels and wave function for the symmetric QW are given by

with

which for the asymmetric QW, energy levels and wave function are given by

where *k⃑*
_{‖} and *r⃑*
_{‖} are the wave and position vectors in the *y*-*z* plane, respectively, and *U _{c}* (

*r⃑*) is the periodic part of the Bloch function in the conduction band at $k=0.{H}_{{n}_{1}}\left(\beta x\right)$ and ${H}_{2{n}_{2}+1}\left(\beta x\right)$ denote the Hermite polynomials,

*N*and

_{s}*N*are the normalization constants. We should note that

_{a}*d*=

*μ*-

_{22}*μ*=<

_{11}*φ*|

_{2}*ex*|

*φ*>-<

_{2}*φ*|

_{1}*ex*|

*φ*> is the difference in the PDMs between the ground and the excited levels, and

_{1}*d*is nonzero in asymmetric semiparabolic QW.

Consider a hyperbolic secant functional form for the initial electric field polarized along *x* direction, which can be written as [28]

where Ω_{m} is the maximum Rabi frequency and Ω=*μ _{21}*

*E*/ħ,

_{x}*μ*=<

_{21}*φ*|

_{2}*ex*|

*φ*> is the dipole moment,

_{1}*ϕ*is the initial CEP, and

*τ*is the full width at half maximum (FWHM) of the pulse intensity envelope. The pulse area

_{p}*A*=Ω

_{m}

*τ*π/1.76, and Ω

_{p}_{m}=1.0 fs

^{-1}corresponds to the electric field of

*E*=6.9×10

_{x}^{6}V/cm or an intensity of

*I*=6.3×10

^{10}W/cm

^{2}.

We employ an iterative predictor-corrector finite-difference time-domain technique to solve the full-wave Maxwell equations in the system without invoking the slowly-varying-envelope approximation and the rotating-wave approximation [20, 28, 30–37]. The time and space increments *Δt* and *Δz* are chosen to ensure *c*Δ*t*≤*Δz* [38].

## 3. Results and discussion

The above theory is now applied to study the CEP dependent effect of few-cycle ultrashort pulses in GaAs-AlGaAs symmetric QW and asymmetric semiparabolic QW. The material parameters chosen are listed [39–41]: *m** _{GaAs}=0.067*m*
_{0}, where *m _{0}* is the mass of a free-electron.

*N*=1.0×10

^{23}m

^{-3},

*τ*=20 fs,

_{p}*z*=15 µm,

_{0}*ω*=

_{p}*ω*=0.4 fs

_{0}^{-1}. We consider the excited-state lifetime

*τ*and the dephasing time

_{1}*τ*in ps timescale which are much longer than the time duration of the pulse we used in our work (20 fs) [27, 42, 43]. The intensity we used is

_{2}*I*=3.1×10

^{10}W/cm

^{2}, corresponding to

*A*=8π.

We first model the CEP effect on the higher spectral components in the symmetric semiconductor QW. From Fig. 2, it can be seen that, only the odd higher spectral components can be found because of the inversion symmetry of the QW. Moreover, odd higher spectral components do not change with the CEP increasing from 0 to π.

Then we have a further discussion on the influence of the asymmetry on the higher spectral components. When asymmetric semiparabolic QW is considered, both odd and even spectral components occur because inversion symmetry is broken (see Fig. 3(a)). Moreover, there have an obvious modification which depends on the CEP of few-cycle ultrashort pulses in this case. Figure 3(a) presents the spectra produced by the few-cycle pulses at the propagation distance of *z*=120 µm for the three different initial CEPs *ϕ*=0, π/2 and π, respectively. We can find some radical discrepancies among them: for *ϕ*=0, higher spectral components exhibit a continuous feature; when *ϕ* increases, the continuous feature becomes much weaker whereas the oscillatory feature becomes more and more intense, and especially for *ϕ*=π, well-resolved splitting peaks of higher spectral components can be obtained.

To describe the physical mechanism for these phenomena, Fig. 3(b) shows the carrier of the few-cycle ultrashort pulses and the population difference with different CEPs of *ϕ*=0, π/2 and π at the input surface of the nonlinear material. The PDM plays an essential role on the interaction. When the carrier is parallel to the PDM (labeled as elliptical symbol), the incomplete CWRF occurs instead of the integer number. The CWRF which will further induce the production of higher spectral components is clearly discerned. While for the carrier which is antiparallel to the PDM (labeled as quadrate symbol), enhanced intersubband transitions are clearly found. Approximately symmetric transitions between the ground and excited states occur, and the CWRF is much weaker for these carriers. Because the generation of higher spectral components is mainly due to the CWRF, the pattern of the generated higher spectral components is determined predominantly by the carriers paralleling to the PDM. For *ϕ*=0, the CWRF is mainly induced by the single central carrier which has the largest amplitude among the carriers paralleling to the PDM, so the higher spectral components distribution is continuous. When *ϕ*=π/2, the second largest carrier paralleling to the PDM becomes stronger, hence the interference structure of the higher spectral components is enhanced. While for *ϕ*=π, the CWRF is caused by two carriers with equal amplitudes, and the interference between them is most strong, as a result, the spectrum shows distinct peaks of higher spectral components. That means the CWRF can be controlled in an ultrashort timescale in asymmetric semiparabolic QW by adjusting the CEP.

In fact, such CEP dependent phenomena are not limited to the exact two-photon resonance case. When confinement potential *ω _{0}* of asymmetric QW increases, two-photon transition resonance is mismatch. As can be shown in Fig. 4(a) and Fig. 4(b) that the CEP dependent CWRF effects and the corresponding higher spectral components also occur.

## 4. Conclusions

In conclusion, we have studied theoretically the spectra of few-cycle ultrashort pulses in the semiconductor QW by solving the full-wave Maxwell equations. It has been shown that the QW’s formation can significantly modify the behavior of the higher spectral components. In asymmetric semiparabolic QW, the features of higher spectral components depend crucially on the CEP and ultrafast controls of electron dynamics can be achieved by adjusting the CEP of few-cycle ultrashort pulse with moderate intensity. Since the main physics of our results originates from the properties of the permanent dipole moment existing in semiparabolic quantum well, we expect these effects to occur in other asymmetric quantum wells with similar properties.

## Acknowledgments

The work was supported by the National Basic Research Program of China (Grant No. 2006CB806000), the National Natural Science Foundation of China (Grant Nos. 10523003 and 60608001) and the Knowledge Innovation Program of the Chinese Academy of Science.

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