## Abstract

In a triple semiconductor quantum well structure coupled by two external fields, we investigate the influence of atomic coherence induced by external fields and decay interference on the absorption and dispersion of a weak pulsed light, and slow light can be achieved in this system. Quantum well structure behaves as “artificial atom” and its advantage of easy integration makes it has some practical applications.

© 2011 OSA

In a semiconductor quantum well structure, quantum coherence and interference produced due to intersubband transitions between the states of an electron confined in this structure have induced a lot of interesting and unexpected physical consequences. Some of these consequences are electromagnetically induced transparency (EIT) [1], pulsed-induced quantum interference [2], coherent population trapping [3], carrier-envelope-phase dependent coherence [4], gain without inversion [5]. In addition, the technology of quantum coherence and interference is expected to have potentially important application in various fields such as ultrafast optical switches [6] and quantum switches [7], quantum information storage and retrieval [8], and preparation of entangled state [9].

As a class of quantum coherence and interference, spontaneously generated coherence (SGC) has been studied in atomic system [10] and semiconductor dot system [11]. We know semiconductor quantum well structures possess of intersubband transitions and behave as “artificial atoms” [12]. Due to longitudinal optical phonon emission events at low temperature [3], the population decay occurs. Some research groups have investigated spontaneously generated coherence in quantum well structure [13]. SGC is the interference of spontaneous emission channels, and the existence of such decay interference requires the nonorthogonality of the two dipole moments. The recent studies show that decay interference can affect the EIT, shot-pulse propagation, dark state, etc [14,15]. To the best of our knowledge, no further theoretical and experimental work has been carried out to investigate the influence of SGC on transmission of a pulsed light in a triple semiconductor quantum well driven by external light fields, which motivates us to investigate effect of decay interference.

In this paper, we consider the effect of quantum interference in a triple semiconductor quantum well with one ground subband and three excited subbands when driven by a pump field and a control field. The influence of atomic coherence produced by coherent driving of external fields on the absorption and dispersion of the pulsed field is investigated. In addition, due to longitudinal optical phonon emission, coupling to the lattice phonon spectrum, and interface roughness scattering, these excited subbands take place decay. According to the theory of decay interference, we theoretically study the influence of decay interference on the absorption and dispersion of a pulsed light in a quantum well. The effect of slow light is investigated in this system. Being an attractive quantum coherent medium, semiconductor quantum well structure has its own advantages: strong electron-electron interactions can produce a collective oscillation, which can behave as a single quantum object, and its large transition dipoles give rise to sizeable Rabi frequencies which are large enough to overcome dephasing. In addition, compared with atomic system, semiconductor quantum well structure has an advantage of easy integration, so the manipulation of absorption of light in this system has a more practical value. Our work is based on existing physical model [5,9] and maybe our theoretical studies have some reference value for the future experiments.

We consider a semiconductor quantum well structure that consists of a deep well and two shallow wells. This triple quantum well is shown in Fig. 1(a) , where the first excited state in the deep well and the respective ground states in the two shallow wells mix to form three new excited states of this system, and the ground state in the deep well and three excited states created newly are expressed as $|0\u3009$, $|1\u3009$, $|2\u3009$ and $|3\u3009$, respectively, which refer to the electronic wave functions (subbands). For the excited states, there are two different types of dephasing rates; one is population decay rate which is represented by ${\gamma}_{j}^{p}$ ($j=1,2,3$), the other is pure dipole dephasing rate which is stood for ${\gamma}_{j}^{d}$. For the population decay rate, it is due primarily to longitudinal optical (LO) phonon emission events at low temperature, which is different from atomic system where the excited state spontaneously decays to a lower lying level, that is, population decay rates for atomic system are pure radiative decay. For the pure dipole dephasing rate, it results from a combination of quasielastic interface roughness scattering or acoustic phonon scattering, and this kind of dephasing rate is akin to atomic collisional broadening in gases. The total dephasing rates are expressed as ${\gamma}_{j}$ and are given by ${\gamma}_{j}={\gamma}_{j}^{p}+{\gamma}_{j}^{d}$.

For the dephasing rate in quantum well structure, the dominant phonon contributions come from piezoelectric and Frohlich coupling to acoustic and optical phonon branches, respectively. In the two branches of coupling, the intersubband transition couples via the phonon’s oscillating electric field [16], so the rates scale with ${D}_{ij}^{2}$ (${D}_{ij}$ is transition dipole), just as natural atoms. The dipole dephasing rates of this structure are strongly dependent upon the electronelectron scattering process. For simplicity, we assume our calculation is performed at zero temperature, so the pure dipole dephasing rate due to scattering is negligible compared with the population decay rates due to longitudinal optical phonon emission events at low temperature.

In this paper, we consider the triple quantum well interacts with a weak pulsed laser field (with central frequency ${\omega}_{p}$ and amplitude ${E}_{p}$) that couples transition between states $|0\u3009$ and $|1\u3009$, simultaneously, $|1\u3009\leftrightarrow |2\u3009$ and $|1\u3009\leftrightarrow |3\u3009$ transitions are coupled by a strong control field (with frequency ${\omega}_{c}$ and amplitude ${E}_{c}$) and a pump field (with frequency ${\omega}_{b}$ and amplitude ${E}_{b}$), respectively. The schematic of the system interacting with three light fields is showed in Fig. 1(b).

A deep 7.1-nm-thick GaAs well is coupled, on one side, to two shallow 6.8-nm-thick
Al_{0.2}Ga_{0.8}As wells by a 2.5-nm-thick Al_{0.4}Ga_{0.6}As
barrier. The two shallow wells are separated by a 2.0-nm-thick Al_{0.4}Ga_{0.8}As
barrier. Both sides of quantum well contact with 36 nm Al_{0.4}Ga_{0.6}As. The
electronic wave functions of the ground state of the deep well and the three excited states are
shown with respective energies of 52.8 ($|0\u3009$), 197.1 ($|1\u3009$), 206.2 ($|2\u3009$), and 219.4 ($|3\u3009$) meV. (b) the schematic of the system interacting with three light
fields.

In the interaction picture, the wave function of the triple semiconductor quantum well can be
written in the form (at time *t*)$|\Psi (t)\u3009={C}_{0}(t)|0\u3009+{C}_{1}(t)|1\u3009+{C}_{2}(t)|2\u3009+{C}_{3}(t)|3\u3009$, and the corresponding time-dependent Schrödinger equation
is (we let $\hslash =1$)

*T*denotes transpose, and the Hamiltonian ${V}_{I}$ describes the four-level system interacting with three light fields. By using the completeness relation $\sum _{k=0}^{3}|k\u3009}\u3008k|=I$ and considering the effect of decay interference, the matrix form of the Hamiltonian can be written as

*θ*being the angle between the two dipole elements. In experiment, the decay interference requires the levels $|2\u3009$ and $|3\u3009$ are close in our system, that is, they are two close-lying levels, and the corresponding dipole matrix elements are not orthogonal. Theses rigorous conditions are rarely met in real atoms. In the quantum well, energy spacing between the two higher excited states are easily adjusted by changing the structure of well. Furthermore, in order to obtain significant decay interference, the system satisfies the restriction that each field acts only on one transition. In our paper, we assume choice of parameter

*p*is theoretically unrestricted ($p\in [0,1]$), and our aim is investigating the influence of decay interference on the propagation of a pulsed light. In the slowly varying envelope approximation, Maxwell’s equation that denotes the dynamic response of the pulsed light is described by $\partial {E}_{p}/\partial t+c\partial E/\partial z=iN{\omega}_{p}{D}_{01}{C}_{1}{C}_{0}^{*}$. According to ${\Omega}_{p}={D}_{10}{E}_{p}/(2\hslash )$, the Maxwell’s equation is rewritten as

*N*is the electron density in the coupled quantum well sample.

We now assume that the pulsed light is weak enough so that we can perform a perturbation
expansion of the amplitude, so the first order of the pulsed light ${\Omega}_{p}$ and the amplitudes ${C}_{1,2,3}$ are retained [17]. In
addition, we assume that the system of quantum well is initially in the state
$|0\u3009$ (${C}_{0}\approx 1$). Under the condition of the weak field approximation, we carry out
Fourier transformations ${C}_{1,2,3}(t)=\frac{1}{\sqrt{2\pi}}{\displaystyle {\int}_{-\infty}^{\infty}{\alpha}_{1,2,3}(\omega )}{e}^{-i\omega t}d\omega $, with the Fourier transformation variable *ω* for Eq. (1) and
${\Omega}_{p}(t)=\frac{1}{\sqrt{2\pi}}{\displaystyle {\int}_{-\infty}^{\infty}{\Lambda}_{p}(\omega )}{e}^{-i\omega t}d\omega $ for Eq. (3), we
obtain

*ω*, so we obtainwhere the real part of $K(0)$ denotes the phase shift per unit length, the imaginary part of $K(0)$ stands for energy absorption, Re(1/${K}^{\prime}(0)$) is the group velocity ${v}_{g}$, ${K}^{\u2033}(0)$ is called as the group velocity dispersion which contributes to the input pulsed shape change and additional loss of the pulsed intensity, and $O({\omega}^{3})$ are the terms of high order.

Now, based on the various physical parameters of the system, we investigate the probe absorption and dispersion and propagation of the pulsed light. We first consider influences of atomic coherence induced by external fields. We choose physical parameters ${\Delta}_{b}={\Delta}_{c}=0$, that is, the system is in the two-photon resonance. The curves of the probe absorption and dispersion are shown in Fig. 2(a) . From this figure, we find the system shows one transparent window for the probe field, that is, EIT occurs, which is similar to phenomena produced in cascade-type three-level atomic system [18]. In addition, if the intensity of the control field is kept unchangeable and intensity of the pump field is changed, the width of transparent window changes: decreasing the intensity of the pump field, the window becomes narrow (see black dashed line and red dashed line in Fig. 2(a)). Under the condition of two-photon resonance, if we consider the decay interference simultaneously, where we choose strong effect of decay interference and let $p=0.96$, the plot of absorption is shown in Fig. 2(b). This figure shows transparent window disappears. In the place of zero detuning, a weak absorption peak appears, which can be explained as follows: decay interference destroys the atomic coherence induced by external fields and destructive interference of the system becomes weak, so the medium does not become transparent.

Next, we consider the case where ${\Delta}_{b}=-{\Delta}_{c}\ne 0$ and the control field and pump field have same Rabi frequency.
Firstly, the effect of atomic interference based on induction of external fields is considered.
The curves of absorption and dispersion versus dimensionless detuning ${\Delta}_{p}/{\gamma}_{1}$ may be plotted as shown in Figs.
3(a)
and 3(b). The two figures show that two EIT
windows occur. With the increase of detuning ${\Delta}_{b}$ and $-{\Delta}_{c}$, the central peak of absorption become higher, and both sides of
absorption peaks become lower. In this case, the occurrences of two EIT windows result from the
quantum interference effect induced by the control field and pump field. Similar phenomena can
also occur in the four-level atomic system [19,20]. If we consider the effect of decay interference, the curve
of probe absorption obviously changes (see Figs. 4(a)
and 4(b)),
where the parameter *p* which denotes decay interference has different value without
changing other parameters noted in Fig. 3(b). Comparing
Fig. 4 with Fig.
3(b), we can find locations of transparent windows move significantly and the width of
windows narrows when the decay interference intensifies from $p=0$ to $p=0.96$, and it is also to be seen the central absorption peak increases
obviously and both sides of absorption are inhibited significantly. In fact, in the case of
$p=0.96$ which means strong decay interference, the minimum value of curve
of absorption approaches zero and do not reach zero (see Fig.
4(b)), that is, strong decay interference affects destructive interference produced by
external fields, which makes weak absorption appear.

In this quantum coherence medium, if we choose ${\gamma}_{1}=1.0\times {10}^{5}{s}^{-1}$, $\kappa =0.5{\gamma}_{1}$, ${\gamma}_{2}={\gamma}_{1}$, ${\gamma}_{3}=1.5{\gamma}_{1}$, ${\Omega}_{c}={\Omega}_{b}=6{\gamma}_{1}$, ${\Delta}_{b}=4{\gamma}_{1}=-{\Delta}_{c}$, for different values of decay interference *p*, the
ratio ${v}_{g}/c$ of group velocity of pulsed light to the velocity of light is
plotted in Fig. 5
, where the dashed curve
means that decay interference is not considered ($p=0$), and the influence of decay interference on the group velocity is
shown by solid curve ($p=0.96$). We find, in the places of absorption peaks
(${\Delta}_{p}/{\gamma}_{1}=0,\pm 5$ for solid line, ${\Delta}_{p}/{\gamma}_{1}=0,\pm 10$ for dashed line), the pulsed light propagates in the coherent
medium with minimum velocity that is far less than the velocity of light
(${v}_{g}/c~{10}^{-4}$). In the locations of transparent windows
(${\Delta}_{p}/{\gamma}_{1}~\pm 2.5$ for solid line, ${\Delta}_{p}/{\gamma}_{1}~\pm 5$ for dashed line), we find values of ${v}_{g}/c$ of solid line are less than ones of dashed line, which means we can
get slower light in transparent medium if we consider effect of decay interference.

In summary, we have investigated and analyzed the influence of atomic coherence induced by external fields and decay interference on the probe absorption and dispersion in a triple semiconductor quantum well structure. One EIT window or two EIT windows can be obtained by changing parameters of external fields. Strong decay interference can affect atomic coherence and weakens destructive interference due to atomic coherence, and ultimately transparent windows disappear or weaken. In addition, a new scheme to achieve slow propagation of the pulsed field has been proposed. Compared with the case in atomic system, semiconductor quantum well may easily be designed and integrated, so it has more practical value. Slow light proposed in present work may lead to important applications such high fidelity optical delay lines and optical buffers.

## Acknowledgments

We thank theoretical physics group in Centre for Atom Optics and Ultrafast Spectroscopy of Swinburne University for useful discussions. Financial support by the National Natural Science Foundation of China under Grant No. 11065007 and the Foundation of Talent of Jinggang of Jiangxi Province, China, Grant No. 2008DQ00400, is acknowledged.

## References and links

**1. **M. Phillips and H. Wang,
“Electromagnetically induced transparency due to intervalence band
coherence in a GaAs quantum well,” Opt. Lett. **28**(10), 831–833
(2003). [CrossRef]

**2. **T. Müller, W. Parz, G. Strasser, and K. Unterrainer,
“Influence of carrier-carrier interaction on time-dependent intersubband
absorption in a semiconductor quantum well,” Phys. Rev.
B **70**(15), 155324 (2004). [CrossRef]

**3. **J.
F. Dynes, M.
D. Frogley, J. Rodger, and C.
C. Phillips, “Optically mediated coherent
population trapping in asymmetric semiconductor quantum wells,”
Phys. Rev. B **72**(8), 085323 (2005). [CrossRef]

**4. **W.
X. Yang, X.
X. Yang, and R.
K. Lee, “Carrier-envelope-phase
dependent coherence in double quantum wells,” Opt.
Express **17**(18), 15402–15408
(2009). [CrossRef]

**5. **C.
R. Lee, Y. Li, F.
K. Men, C. Pao, Y. Tsai, and J. Wang,
“Model for an inversionless two-color laser,”
Appl. Phys. Lett. **86**(20), 201112 (2005). [CrossRef]

**6. **H. Schmidt and R.
J. Ram, “All-optical wavelength
converter and switch based on electromagnetically induced
transparency,” Appl. Phys. Lett. **76**(22), 3173–3175
(2000). [CrossRef]

**7. **B. S. Ham,
“Potential applications of dark resonance to subpicosecond optical
switches in hyper-terahertz repetition rates,” Appl. Phys.
Lett. **78**(22), 3382–3384
(2001). [CrossRef]

**8. **J.
M. Tang, J. Levy, and M.
E. Flatté, “All-electrical control of
single ion spins in a semiconductor,” Phys. Rev. Lett. **97**(10), 106803 (2006). [CrossRef]

**9. **X. Lü and J. Wu,
“Three-mode entanglement via tunneling-induced interference in a coupled
triple-semiconductor quantum-well structure,” Phys. Rev.
A **82**(1), 012323 (2010). [CrossRef]

**10. **A. Fountoulakis, A. Terzis, and E. Paspalakis,
“Coherence phenomena due to double-dark states in a system with decay
interference,” Phys. Rev. A **73**(3), 033811 (2006). [CrossRef]

**11. **S.
E. Economou, R.
B. Liu, L.
J. Sham, and D.
G. Steel, “Unified theory of consequences
of spontaneous emission in a *Λ* system,”
Phys. Rev. B **71**(19), 195327 (2005). [CrossRef]

**12. **J.
F. Dynes, M.
D. Frogley, M. Beck, J. Faist, and C.
C. Phillips, “AC stark splitting and
quantum interference with intersubband transitions in quantum wells,”
Phys. Rev. Lett. **94**(15), 157403 (2005). [CrossRef]

**13. **H. Schmidt, K.
L. Campman, A.
C. Gossard, and A. Imamoglu,
“Tunneling induced transparency: Fano interference in intersubband
transitions,” Appl. Phys. Lett. **70**(25), 3455–3457
(1997). [CrossRef]

**14. **E. Paspalakis, N.
J. Kylstra, and P.
L. Knight, “Transparency induced via decay
interference,” Phys. Rev. Lett. **82**(10), 2079–2082
(1999). [CrossRef]

**15. **P. R. Berman,
“Spontaneously generated coherence and dark
states,” Phys. Rev. A **72**(3), 035801 (2005). [CrossRef]

**16. **B. K. Ridley, *Quantum Processes in
Semiconductors* (Clarendon Press, 1988).

**17. **Y. Wu,
“Two-color ultraslow optical solitons via four-wave mixing in cold-atom
media,” Phys. Rev. A **71**(5), 053820 (2005). [CrossRef]

**18. **Y. Wu and X.
X. Yang, “Electromagnetically induced
transparency in V-, Λ-, and cascade-type schemes beyond steady-state
analysis,” Phys. Rev. A **71**(5), 053806 (2005). [CrossRef]

**19. **S. Yelin, V.
A. Sautenkov, M. Kash, G. Welch, and M. Lukin,
“Nonlinear optics via double dark resonances,”
Phys. Rev. A **68**(6), 063801 (2003). [CrossRef]

**20. **M.
D. Lukin, S.
F. Yelin, M. Fleischhauer, and M.
O. Scully, “Quantum interference effects
induced by interacting dark resonances,” Phys. Rev. A **60**(4), 3225–3228
(1999). [CrossRef]