Slow light propagation via quantum coherence in a semiconductor quantum well

Optik 124 (2013) 1641–1643

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Slow light propagation via quantum coherence in a semiconductor quantum well Dingan Han a,∗ , Yaguang Zeng a , Yanfeng Bai b a b

Department of Physics, Foshan University, Foshan 528000, China Department of Physics, Southeast University, Nanjing 210096, China

a r t i c l e

i n f o

Article history: Received 29 December 2011 Accepted 3 May 2012

PACS: 42.50.Ct 42.50.Gy 78.67.De

a b s t r a c t ¨ With the Schrodinger equations, we investigate the low-intensity light pulse propagation through a semiconductor quantum wells. Through studying the dispersion and absorption properties of the weak probe field, it is shown that slow light propagation is observed in this system. From the view point of practical purpose, it is more advantageous than its corresponding atomic system. Such investigation of slow light propagation may lead to important practical applications in semiconductor quantum information. © 2012 Elsevier GmbH. All rights reserved.

Keywords: Electromagnetically induced transparency (EIT) Slow light Quantum wells

1. Introduction Semiconductor quantum wells (QWs) have many properties similar to atomic vapors and behave as “artificial atoms”. Due to the small effective electron mass, they have the advantages of high nonlinear optical coefficients and large electric dipole moments of intersubband transitions. Choosing the materials and structure dimensions, we can engineer their transition energies, dipole moments and symmetries [1,2]. As the wide-spread use of semiconductor in optoelectronics and quantum information science, many phenomena such EIT [3], pulsed-induced quantum interference [4], lasing without inversion [5,6], and slow light [7], are investigated carefully. Recently, Fano-type interference has been observed in asymmetric double QW. For example, Wu et al. reported ultrafast all optical switching via tunable Fano interference in asymmetric double quantum wells [8]. With the changed refractive index, the study on optical group velocity and quantum information storage is ignited. In a cooled, sodium Bose–Einstein condensate gas, the group velocity can be as low as 17 m/s [9]. Our group also observed these phenomena [10,11]. The existence of the EIT in systems of semiconductor quantum well structures is of particular significance because EIT-related phenomena have been proven to have a vast number of important applications, for example, ultrafast optical switches [12], entanglement via tunneling-induced interference [13],

∗ Correspondence author. Tel.: +86 757 8388 9687. E-mail address: [email protected] (D. Han). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.06.019

and quantum information storage and retrieval [14]. From the view point of practical purpose, it is more advantageous to find solid-state media to realize the slow light propagation. In this paper, we study theoretically group velocity in a QW structure based on intersubband transitions (ISBTs). The devices have many inherent advantages in quantum information processing. Its easy integration makes it have practical applications. These advantages can hardly be found in the models for cold atom media. It is shown that in the almost zero absorption regions, there is ultraslow laser pulse propagation for the probe field. This kind of properties may lead to potential applications in quantum switches, high-fidelity optical delay lines, optical buffers, etc. 2. Model and analysis We consider a three energy levels quantum well, as shown in Fig. 1. It forms the well-known cascade configuration. The sample consists of 30 periods, each with 4.8 nm In0.47 Ga0.53 As, 0.2 nm Al0.48 In0.52 As and 4.8 nm In0.47 Ga0.53 As coupled quantum wells, separated by modulation-doped 36 nm Al0.48 In0.52 As barriers. The sample can be designed to have desired transition energies. This model has been considered in Ref. [15], where slow optical solitons had been investigated. Here, we especially focus on the group velocity of the probe field interacting with the quantum well, and show the effects of the control field and the probe detuning on the group velocity. ¨ Working in the Schrodinger picture, with the rotating-wave approximation and the electric-dipole approximation, and based ¨ equations, the equations of on coupled Schrodinger–Maxwell

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D. Han et al. / Optik 124 (2013) 1641–1643

c

0.5

|3

(a) 0.4

|3

c

0.3 0.2

p

0.1

|2

|2

0 −0.1 −0.2

p

|1

|1 (a)

−0.3 −20

−15

−10

−5

0

5

10

15

20

0

5

10

15

20

0

5

10

15

20

ω/γ

1

(b)

0.3

(b) 0.25

Fig. 1. (a) Schematic of the energy level arrangement for the quantum wells. The subband transition |1 ↔ |2 and |2 ↔ |3 are coupled by two control fields ωp and ωc , respectively. (b) The corresponding schematic of the three-level cascade electronic system synthesized in a semiconductor quantum well of (a).

motion for the probability amplitude of the electronic wave functions can be obtained

∂c1 = i˝p∗ c2 , ∂t ∂c2 = i(p + i2 )c2 + i˝c∗ c3 + i˝p c1 , ∂t ∂c3 = i(c + i3 ]c3 + i˝c A2 , ∂t ∂˝p 1 ∂˝p + = ic2 c1∗ , c ∂t ∂z

ω (ω + c + i3 ) + c |˝c |2 − (ω + p + i2 )(ω + c + i3 )

0.1 0.05 0 −0.05 −0.1 −0.15 −20

(1)

−15

−10

−5

ω/γ1

0.25

(c)

0.2 0.15 0.1

where ˝p(c) is one-half Rabi frequencies for the relevant laserdriven intersubband transitions,  = Nωp |21 |2 /(2  c), and N is the electron density in the coupled quantum well sample.  i (i = 2, 3, 4) is the corresponding total decay rate of |i. Here it comprises a lifetime broadening contribution  il (at low temperature, it is due primarily to longitudinal optical photon emission events) as well as a dephasing contribution  id (a combination of the quasielastic acoustic phonon scattering and the elastic interface roughness scattering). p = ω21 − ωp and c = ω32 − ωc are the corresponding detunings. Assuming the probe pulsed light is weak enough, the system of the quantum well is initially in the state |1, i.e., c1 ≈ 1, so that we can perform a perturbation expansion to the amplitude of the state. Under the condition of the weak field approximation, with the Fourier transformation, and taking ˝p and cj (j = 1, 2, 3) as being proportional to exp[i(K(ω)z − ωt)], one obtains readily the linear dispersion relation K(ω) =

0.2 0.15

(2)

 K0 + K1 ω + K2 ω2 + o(ω3 ). Here, higher-order derivative terms have been neglected. Here K0 = ˚ + i˛ describes the phase shift ˚ per unit length and absorption coefficient ˛ of the pulsed probe field. K1 is related with the group velocity vg = Re[1/K1 ], K2 represents the group velocity dispersion, which contributes to the probe pulse’s shape change and additional loss of the pulsed probe field intensity. With the method used in Ref. [16], Fig. 2 plots the dispersion (solid line) and absorption (dotted line) curves of the probe field ˝p , which are characterized by Re(K) and Im(K), respectively. It is shown that when the coupling field is small [˝c =  1 , in Fig. 2(a)], there is only one absorption peak with anomalous dispersion occurring. However, when the coupling field is stronger (˝c = 6 1 ), as shown in Fig. 2(b), there are two absorptive peaks. Especially, there

0.05 0 −0.05 −0.1 −0.15 −20

−15

−10

−5

ω/γ1

Fig. 2. The dispersion Re(K) (solid line) and absorption Im(K) curves (dotted line) of the probe field as functions of ω with different ratios of the coupling fields ˝c . (a) ˝c =  1 ; (b) ˝c = 6 1 ; (c) ˝c = 12 1 . Here,  1 = 1 ×105 s−1 , c = 4 1 ,  3 = 1.5 1 , and 2 = 1.

is one EIT window with normal dispersion exhibited. That means, due to the quantum interference effect induced by the coupling field ˝c and ˝p , one EIT transparency windows in the absorption curve Im(K) exhibits. In addition, if the intensity of the control field is changed, the width of transparent window changes. Increasing ˝c to 12 1 [Fig. 2(c)], the windows is large than before. This kind of property is rather interesting, which can be explained as follows: the atomic coherence induced by external fields of the system becomes strong, so the medium becomes transparent. Similar phenomena can also occur in the cascade-type three-level atomic system [17]. According to Fig. 2, the dispersion curves can be divided into normal and anomalous dispersion regimes. Meanwhile, in the normal (anomalous) dispersion regimes the probe field has a negligible (strong) absorption. Thus, the probe field is nearly transparent (opaque) for the slow-light (superluminal) propagation when working in the normal (anomalous) dispersion regimes. 3. Group velocity for the probe field The group velocity vg of the pulse applied is our focus for further consideration. We will investigate the effects of the probe detuning

D. Han et al. / Optik 124 (2013) 1641–1643

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the group velocity is increased. However, it is always positive, i.e., subluminal group velocity can be exhibited in transparent medium.

100 50 0

4. Conclusions

ng

−50 −100 −150 −200 −250 −300 −20

−15

−10

−5

0

5

Δp/γ1

10

15

20

Fig. 3. The group index ng versus the probe detuning p . The parameters are the same as those in Fig. 2(b). 60

In conclusion, we have investigated, in a quantum well structure, the influence of the atomic coherence on the dispersion and the absorption of a weak probe field. It is shown that the probe detuning and the Rabi-c field can affect the optical properties of the QW structure. With proper Rabi-c and probe detuning, the EIT-assisted subluminal phenomenon is established in this QW structure. This kind of semiconductor system is much more practical than its atomic counterpart because of its flexible design and its wide adjustable parameters. These phenomena may be helpful for observation the group velocity and may lead to important applications in semiconductor quantum information and quantum computer.

50

Acknowledgements ng

40

This work is supported by the Project of National Natural Science Foundation of China (Grant Nos. 61008063, 10904015), the Key Project of National Natural Science Foundation of China (Grant No. 60837004), the Project-sponsored by SRF for ROCS, SEM, and Natural Science Foundation of Foshan University.

30 20 10 0

6

8

10

12

14

16

18

References

20

Ωc/γ1 Fig. 4. The group index ng versus the coupling field ˝c . p =  1 , the parameters are the same as those in Fig. 2(b).

and the control c-Rabi on the group velocity of the probe field in this part. In a dispersive medium, the group velocity can be written as Re(K1 ) =

1

vg

=

1 (AC + BD) , + c A2 + B2

(3)

where A =

−22 2c + 2p (2c − 32 ) + (2 3 + ˝c2 )2 −2c p (22 3 + ˝c2 ),

=

−2(3 p + 2 c )(2 3 − p c + ˝c2 );

C

=

2c

D

=

23 c ,

B

+ ˝c2

(4)

− 32 ;

and c is the vacuum light speed. In the normal dispersion regimes, i.e. ∂Re(K)/∂ω > 0, one has vg < c and thus the probe field is a slow light; in the anomalous dispersion regimes, i.e., ∂Re(K)/∂ω < 0, one has vg > c or even negative, and hence the probe field is superluminal propagation. In this quantum coherent medium, we choose the parameters as following [18]:  1 = 1.0 × 105 s−1 ,  = 0.5 1 ,  2 =  1 ,  3 = 1.5 1 , c = 4 1 . The group index ng versus p is plotted in Fig. 3. It is shown that, on the whole, the group index is changed with increasing p . It is changed from positive to negative value, that is, from subluminal to superluminal. So, the probe detuning can be used as a knob to adjust the group velocity of the probe field. This result may be helpful for the experiment studies of measuring vg . According to Eqs. (3) and (4), the control field ˝c has effects on the absorptive–dispersive line. Thus, in Fig. 4, the dependence of ng on the Rabi-c field is plotted. Here, p =  1 . It is seen that ng is decreased with the enhancement of ˝c . That is, increasing the coupling field ˝c leads to the group index minimal. Correspondingly,

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