Coherent manipulation of spontaneous emission spectra in coupled semiconductor quantum well structures

Aixi Chen

doi:10.1364/OE.22.026991

1. Introduction

In recent decades, quantum coherent control in atomic system has attracted the eyes of researchers around the world. Via quantum coherent control, some interesting phenomena such as electromagnetically induced transparency (EIT) [1–4], coherent population trapping (CPT) [5, 6], gain without population inversion [7,8] are achieved in atomic system. In addition, based on skillful application of this kind of coherent technique, propagation of ultra-slow light [9, 10], switch between optical bistability and multistability [11, 12], and preparation of entangled state [13, 14] are reported. Recently, quantum coherent control has been extended to semiconductor quantum well structures (SQWs). These phenomena of atomic system mentioned above can also observed in quantum well [15–25]. We know modern crystal growth techniques make it possible to grow layers of semiconductor material which are narrow enough to confine the electron motion in low-dimension. Taking the SQWs as an example, due to confinement of electron motion, its states are quantized like the standing waves of a particle in a square well potential, so SQWs is called an “artificial” atom. Coherent phenomena originates from quantum coherence and interference produced due to intersubband transitions, which is connected with optical properties of SQWs. Comparing with the atomic medium, SQWs has a remarkable advantage that is its distribution of energy level can be controlled by changing the width of well, which shows clearly that the optical properties can be changed through the design of structure. Quantum coherence and interference is expected to have potential application in design of optical device.

In the field of quantum optics, spontaneous emission spectrum is an important research content. Atoms locating in high-level can spontaneously radiate electromagnetic fields. In the last few decades, intensive attention has been paid to the spontaneous emission spectra coming from the atom interacting with the environmental modes [26–29]. The effective control and modulation of spontaneous emission is helpful in development of high-precision spectroscopy and magmetometry, and transparent high-index materials. In the SQWs, there exist a transverse relaxation and longitudinal relation, so spontaneous emission spectra can also easily observed in SQWs. Studies show enhancing Kerr nonlinearity [30], propagation of slow light [31] and change of threshold of optical bistabiltiy [32] can be obtained via spontaneously generated coherence [33, 34]. In this paper, we consider a triple SQWs [20] where intersubband transitions are realized by three light fields with different frequency. Under the interaction of the coherent driving field and coupling field, spontaneous emission spectra show some interesting phenomena such as spectral-line enhancement/suppression, spectral-line narrowing and fluorescence quenching will occur. The manipulation of spontaneous emission spectra originate from quantum coherence and interference which can also be achieved in Alkali-metal atomic gases. The semiconductor quantum well structures has the advantages of flexibility and easy integration, so our scheme, where we expand the gas of atom to SQWs, has more practical value.

2. Theoretical model and evolution of system

In this paper, we consider a triple SQWs model that is shown in Fig. 1. The model [20] consists of a deep GaAs well whose thickness is 7.1-nm and that is adjacent to two shallow 6.8-nm-thick Al_0.2Ga_0.8As wells separated by two Al_0.4Ga_0.6As barriers of thicknesses 2.0 nm and 2.5 nm with the thick barrier neighboring the deep well. Between the deep well and the continuum (Al_0.165Ga_0.835As) there is an Al_0.4Ga_0.6As barrier with 0.7 nm thickness. The left side of the shallow wells is in contact with an Al_0.4Ga_0.6As barrier of 10 nm. Where we assume the thickness $L_{c}$ of quantum wells is in the $z$ direction, and the $x - y$ plane there is no quantum confinement and the electrons can move freeely. For the idealized quantum well, there exists a standing-wave envelope $φ (z)$ which satisfies $[- \frac{ℏ^{2}}{2 m_{e}} \frac{\partial^{2}}{\partial z^{2}} + V_{c o n} (z)] φ (z) = E_{z} φ (z)$ . According to the theory of quantum mechanics, we know the $E_{z} = π^{2} ℏ^{2} n^{2} / (2 m_{e} L_{c}^{2})$ (for $n = 1, 2, 3, \dots$ ), that is, we can obtain the discrete levels in every quantum well, where these discrete levels are called subbands. In our model, structures of the four-level model are expressed as follows: the first excited state of the deep well and the respective ground state in two shallow wells recombine and form three new excited states through tunneling coupling. Here, the three subbands are expressed as $| 2 〉$ , $| 3 〉$ , and $| 4 〉$ , respectively, and state $| 1 〉$ is the ground state of the deep well. Where the occurrence of tunneling coupling is due to the depth of the well is not infinitely deep, so the wave functions of neighboring wells can superpose and form new subbands via quantum tunneling. In our paper, how to prove the existences of subbands $| 1 〉$ , $| 2 〉$ , $| 3 〉$ and $| 4 〉$ are constant concern. In Ref [35], linear absorption spectrum can be used to detect the existences of subbands. Our model is from the Ref [20], where absorption spectra show energies of subbands $| 1 〉$ , $| 2 〉$ , $| 3 〉$ and $| 4 〉$ are 52.8 meV, 197.1 meV, 206.2 meV, and 219.4 meV, respectively. In this model, transition between states $| 2 〉$ and $| 4 〉$ is performed by a driving field with frequency $ω_{d}$ and amplitude $E_{d}$ , the coupling between $| 2 〉$ and $| 3 〉$ is realized by a controlling field with frequency $ω_{c}$ and amplitude $E_{c}$ , and the weak probe field with frequency $ω_{p}$ and amplitude $E_{p}$ couples with states $| 1 〉$ and $| 3 〉$ . Our model is similar to a N-type structure. Furthermore, subband state $| 3 〉$ can spontaneously decay into the continuum whose state is denoted by $| e 〉$ .

Fig. 1 Subband diagram of a triple semiconductor quantum well.

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In interaction picture, the evolution of state vector of the system obeys the Schrödinger equation $i | \dot{ψ} 〉 = V_{I} | ψ 〉$ (we let $ℏ = 1$ ), where

| ψ (t) 〉 = [C_{1} (t) | 1 〉 + C_{2} (t) | 2 〉 + C_{3} (t) | 3 〉 + C_{4} (t) | 4 〉] | {0} 〉 + \sum_{k} C_{k} (t) | e 〉 | 1_{k} 〉,

and

\begin{array}{l} V_{I} = - (Δ_{p} - Δ_{c}) | 2 〉 〈 2 | - Δ_{p} | 3 〉 〈 3 | - (Δ_{p} - Δ_{c} + Δ_{d}) | 4 〉 〈 4 | \\ + Ω_{p} | 3 〉 〈 1 | + Ω_{c} | 3 〉 〈 2 | + Ω_{d} | 4 〉 〈 2 | + \sum_{k} g_{k} a_{k} e^{i (Δ_{p} + δ_{k}) t} | 3 〉 〈 e | + H . c ., \end{array}

where,

C_{j} (t)

is corresponding to the probability amplitude of state

| j 〉

,

| {0} 〉

denotes the absence of photons in all vacuum modes,

| 1_{k} 〉

represents that there is one photon in the kth vacuum mode;

Δ_{p}

=

ω_{31} - ω_{p}

,

Δ_{c} = ω_{32} - ω_{c}

, and

Δ_{d} = ω_{42} - ω_{d}

are the detunings of the probe field, the coherent coupling field and the coherent driving field, respectively, and

δ_{k} = ω_{3 e} - ω_{k}

is the detuning of the kth vacuum mode;

a_{k}^{†}

(

a_{k}

) is the creation (annihilation) operator for the kth vacuum mode,

g_{k}

denotes the coupling constant between the kth vacuum mode and the transition

| e 〉 \leftrightarrow | 3 〉

;

Ω_{p}

,

Ω_{d}

, and

Ω_{c}

are the Rabi frequencies of the probe, the driving and the controlling field, respectively.

Applying the amplitude method and Wisskopf-Winger theory of spontaneous emission, the evolution equation of the system is written as

\begin{array}{l} {\dot{C}}_{1} (t) = - i Ω_{p}^{*} C_{3} (t), \\ {\dot{C}}_{2} (t) = i (Δ_{p} - Δ_{c}) C_{2} (t) - {iΩ}_{c}^{*} C_{3} (t) - {iΩ}_{d}^{*} C_{4} (t), \\ {\dot{C}}_{3} (t) = - i Ω_{p} C_{1} (t) - i Ω_{c} C_{2} (t) + i (Δ_{p} + \frac{i Γ}{2}) C_{3} (t), \\ {\dot{C}}_{4} (t) = - i Ω_{d} C_{2} (t) + i (Δ_{p} - Δ_{c} + Δ_{d}) C_{4} (t), \\ \dot{{\dot{C}}_{k}} (t) = - i g_{k}^{*} e^{- i (Δ_{p} + δ_{k}) t} C_{3} (t), \end{array}

where,

Γ = 2 π {| g_{k} |}^{2} D (ω_{k})

denotes the decay rate of spontaneous emission from the state

| 3 〉

to continuum

| e 〉

, and

D (ω_{k})

is the density of vacuum mode at the frequency

ω_{k}

in the free space.

For Eq. (3), the Laplace transform ${\tilde{C}}_{j} (s) = \int_{0}^{\infty} e^{- s t} C_{j} (t) d t$ is carried out, and this linear operator transforms function $C_{j} (t)$ with a real argument t to function ${\tilde{C}}_{j} (s)$ with complex argument. We can get

s {\tilde{C}}_{1} (s) + i Ω_{p}^{*} {\tilde{C}}_{3} (s) = C_{1} (0),

(s - i ξ_{1}) {\tilde{C}}_{2} (s) + i Ω_{c}^{*} {\tilde{C}}_{3} (s) + i Ω_{d}^{*} {\tilde{C}}_{4} (s) = C_{2} (0),

i Ω_{p} {\tilde{C}}_{1} (s) + i Ω_{c} {\tilde{C}}_{2} (s) + (s - i ξ_{2}) {\tilde{C}}_{3} (s) = C_{3} (0),

i Ω_{d} {\tilde{C}}_{2} (s) + (s - i ξ_{3}) {\tilde{C}}_{4} (s) = C_{4} (0),

C_{k} (t) = - i g_{k}^{*} \int_{0}^{t} e^{i ξ_{4} t^{'}} C_{3} (t^{'}) d t^{'},

where parameters

ξ_{1} = Δ_{p} - Δ_{c}

,

ξ_{2}

=

Δ_{p} + i Γ / 2

,

ξ_{3} = Δ_{p} - Δ_{c} + Δ_{d}

,

ξ_{4} = - Δ_{p} - δ_{k}

, and

C_{j} (0) (j = 1 - 4)

is the probability amplitude at the initial time t = 0.

Here, we are interested in the spontaneous emission spectra of this system. It is generally known that the spontaneous emission spectra is the Fourier transform of the first-order correlation function $〈 E^{-} (t) E^{+} {(t + τ)}_{t \leftrightarrow \infty}$ . After the Fourier transform, the spontaneous emission spectra can be expressed as

S (δ_{k}) = \frac{Γ}{2 π {| g_{k} |}^{2}} {| C_{k} (t \to \infty) |}^{2} .

According to Eq. (4e),

S (δ_{k})

can be rewritten as

S (δ_{k}) = \frac{Γ}{2 π {| g_{k} |}^{2}} {| - i g_{k}^{*} \int_{0}^{t} e^{i ξ_{4} t^{'}} C_{3} (t^{'}) d t^{'} |}^{2} = \frac{Γ}{2 π} {| {\tilde{C}}_{3} (s = - i ξ_{4}) |}^{2} .

From Eqs. (4a)–(4d), we can obtain the concrete form of

{\tilde{C}}_{3} (s)

, and substitute it into the expression (6), the final form of the spontaneous emission spetra is given by

S (δ_{k}) = \frac{Γ}{2 π} {| \frac{f_{1} (δ_{k}) C_{1} (0) + f_{2} (δ_{k}) C_{2} (0) + f_{3} (δ_{k}) C_{3} (0) + f_{4} (δ_{k}) C_{4} (0)}{f (δ_{k})} |}^{2},

where

\begin{array}{l} f_{1} (δ_{k}) = i (ξ_{4} + ξ_{1}) (ξ_{4} + ξ_{3}) Ω_{p} - i Ω_{d}^{2} Ω_{p}, \\ f_{2} (δ_{k}) = i ξ_{4} (ξ_{4} + ξ_{3}) Ω_{c}, \\ f_{3} (δ_{k}) = - i ξ_{4} (ξ_{4} + ξ_{1}) (ξ_{4} + ξ_{3}) - i ξ_{4} Ω_{d}^{2}, \\ f_{4} (δ_{k}) = i ξ_{4} Ω_{c} Ω_{d}^{*}, \end{array}

\begin{array}{l} f (δ_{k}) = ξ_{4} (ξ_{4} + ξ_{1}) (ξ_{4} + ξ_{2}) (ξ_{4} + ξ_{3}) - ξ_{4} (ξ_{4} + ξ_{3}) Ω_{c}^{2} - \\ ξ_{4} (ξ_{4} + ξ_{2}) Ω_{d}^{2} - (ξ_{4} + ξ_{1}) (ξ_{4} + ξ_{3}) Ω_{p}^{2} + Ω_{p}^{2} Ω_{d}^{2} . \end{array}

3. Numerical results and discussions

In section 2, we have got the expression of the spontaneous emission spectra $S (δ_{k})$ which is a function of physical parameters of the system. For different values of physical parameters, we will obtain the change characteristics of spectral line by the method of numerical calculation. These results will show that the change of spectral line results from quantum coherence and interference in SQWs driven by the coherent fields. In the process of the following numerical calculations, all parameters are scaled by decay rate $Γ$ which should be in the order of meV in our SQWs.

In the first place, we consider the influences of the intensity of the driving field on the spontaneous emission spectra. Here, we assume that the state of our SQWs is in the ground state $| 1 〉$ at the initial time t = 0, all subbands resonantly interact with three radiation fields, and the Rabi frequencies $Ω_{p}$ and $Ω_{c}$ are equal to 0.6 $Γ$ and 0.4 $Γ$ , respectively. For different the Rabi frequency $Ω_{d}$ of the coherent driving field, the curves of the spontaneous emission spectra $S (δ_{k})$ are plotted in Fig. 2. In Fig. 2(a), choosing the $Ω_{d} = 0$ , spectral line shows symmetrical double-peak structure which occurs in the place of $δ_{k} = \pm 1 / \sqrt{2}$ . In this case with $Ω_{d} = 0$ , the SQWs is a $Λ$ -tpye system, the double-peak structure originates in the dynamical Stark effect [35] produced by the coupling of the coherent controlling field. If the intensity of the driving field is not equal to zero, that is, the channel of transition between $| 2 〉 \leftrightarrow | 4 〉$ is turned on, the spectral line with four-peak structure occurs (see Figs. 2(a)–2(d)) and its symmetry remains unchanged. Four-peak structure includes two ultra-narrow central spectral lines and two normal width sideband lines. Between the ultra-narrow peak and wide peak, there exist two fluorescence quenching points which locate in the place of $δ_{k} = \pm | Ω_{d} |$ . Compared with Fig. 2(a), the increase of number of peak is due to recombination of levels of the SQWs under the coupling of coherent driving and controlling fields. The theory of dressed state can be used to explain this kind of phenomenon [28]. The fluorescence quenching denotes the power spectra of spontaneous emission can reach zero, which means there exists spontaneously generated coherence or vacuum-induced coherence [33, 34]. Spontaneously generated coherence produces destructive interference, so emission quenching occurs in our system. In addition, from Figs. 2(b)–2(d), with the increase of intensity of the coherent driving field, separation between two quenching points becomes wide and the intensity of emission spectra becomes weak, which means constructive interference is weaker than destructive interference in this system. Change of the intensity of the driving field can obviously affect the intensity and location of quenching of spontaneous emission.

Fig. 2 Curves of spontaneous emission spectra $S (δ_{k})$ for different $Ω_{d}$ under the condition with other parameters with concrete values (see text).

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Next, we consider the influences of the intensity of the controlling field on the spontaneous emission spectra. We let $Ω_{p =} 0.6 Γ$ and $Ω_{d} = 0.4 Γ$ and assume the system is in resonant condition. For different $Ω_{c}$ , the curves of emission spectra are shown in Fig. 3. The four-peak structure are also observed in this case. Under the condition that the controlling field is relatively weak (see Fig. 3(a)), two ultra-narrow central lines become very sharp. Ultra-narrow spectral line has potential application in energy resolution. With the increase of the intensity of the controlling field, both ultra-narrow central lines and normal width sideband lines gradually become wide, at the same time, the intensity of ultra-narrow central lines get enhancement while the intensity of normal width sideband lines become weak little by little. The changes of width and intensity of spectral lines can be explained as the change of quantum coherence and interference produced by the coupling field which is in the process of gradual increase. The SQWs is driven by the controlling field, and its subbands can mix to form new dressed states. New transitions can be reconstructed between the new dressed states. There exist constructive and destructive interference between channels of different transitions. This kind of quantum coherence and interference induced by the controlling field can obviously affect the spontaneous emission spectra. From Fig. 3, we also find the controlling field can affect the intensity and width of spectral line and does not affect the locations of the fluorescence quenching which are in the place with $δ_{k} = \pm 0.4$ .

Fig. 3 Curves of spontaneous emission spectra $S (δ_{k})$ for different $Ω_{c}$ under the condition with other parameters with concrete values (see text).

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Finally, the influences of the detuning of the driving field on the spontaneous emission spectra are considered. For the given Rabi frequencies $Ω_{p}$ with $0.6 Γ$ , $Ω_{c}$ with $0.4 Γ$ , $Ω_{d}$ with $0.4 Γ$ , and $Δ_{p} = Δ_{c} = 0$ , we plot the curves of spontaneous emission spectra under the condition of changing the detuning of the driving field (see Fig. 4). Compared with Fig. 2 and Fig. 3, the distribution of spectral lines is not symmetric when the SQWs does not resonantly interact with the coherent driving field. With the increase of the detuning of the driving field, the left part of ultra-narrow central lines is enhanced and their right part is suppressed, and the intensities of sideband lines are not obviously affected. Enhancement and suppression of spontaneous emission spectra can be explained as follow: in the condition of detuning, the coupling resulting from the driving field will change, which can affect the quantum coherence and interference, finally intensities of emission spectra are affected. In addition, increasing the detuning, the locations of the fluorescence quenching move a tiny distance to the right. According to calculation, we can obtain the quenching points occur in place of $δ_{k} = \frac{Δ_{d} \pm \sqrt{Δ_{d}^{2} + 4 {| Ω_{d} |}^{2}}}{2}$ .

Fig. 4 Curves of spontaneous emission spectra $S (δ_{k})$ for different $Δ_{d}$ under the condition with other parameters with concrete values (see text).

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4. Conclusions

In summary, in this paper, the spontaneous emission spectra are investigated in the triple SQWs. By Wisskopf-Winger theory and traditional calculation method, we obtain the expression of the spontaneous emission spectra $S (δ_{k})$ which depends on the intensities of two coherent light fields and their detunings. Under the resonant condition, we study the change of spectral line when we increase gradually the intensity of the driving field or the controlling field. In this case, the symmetric emission spectra can be obtained. We also find, choosing appropriate intensity of the controlling field, ultra-narrow emission spectra become very sharp, which might be a potential application in high-precision spectroscopy. When the driving light field nonresonantly couple with the SQWs, the symmetry of the spontaneous emission spectra is not maintained, and enhanced and suppressive ultra-narrow central spectral line can be achieved. In our system, due to the existence of spontaneously generated coherence, fluorescence quenching points can appear in the spectral line. Spontaneous generated coherence plays an important role in lasing without inversion, refractive index enhancement without absorption, and high-precision spectroscopy. But realization of spontaneously generated coherence is very difficult in atomic system [29]. In this paper, occurrence of fluorescence quenching points might provide a method for observing the spontaneously generated coherence. In addition, compared with atomic systems, the semiconductor quantum well structure has the advantages of flexibility and easy integration, so the coherent manipulation of spontaneous emission spectra in this system has more practical value.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11165008 and 11365009), the Natural Science Foundation of Jiangxi Province, China (Grant No. 20122BAB202007), and China Scholarship Council.

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Coherent manipulation of spontaneous emission spectra in coupled semiconductor quantum well structures

Abstract

1. Introduction

2. Theoretical model and evolution of system

3. Numerical results and discussions

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (13)

Optics Express