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Voltage-controlled optical bistability of a tunable three-level system in a quantum-dot molecule
ARTICLE IN PRESS Physica E 41 (2008) 70– 73
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Voltage-controlled optical bistability of a tunable three-level system in a quantum-dot molecule J. Li a,, R. Yu b, J. Liu a, P. Huang a, X. Yang a a b
Department of Physics, Huazhong University of Science and Technology (HUST), Wuhan 430074, People’s Republic of China School of Science, Wuhan Institute of Technology, Wuhan 430073, People’s Republic of China
a r t i c l e in f o
a b s t r a c t
Article history: Received 1 March 2008 Received in revised form 4 June 2008 Accepted 4 June 2008 Available online 12 June 2008
We investigate the behavior of optical bistability (OB) in an asymmetry semiconductor double quantum dot (QD) using the tunnel coupling. Such a tunable three-level QD system is driven coherently by a laser field inside the unidirectional ring cavity. The results show that we are able to control efficiently the bistable threshold intensity and the hysteresis loop by tuning the parameters of the system such as the gate voltage and laser frequency. The results obtained can be used for the development of new types of nanoelectronic devices for realizing switching process. & 2008 Elsevier B.V. All rights reserved.
PACS: 78.67.Hc 42.65.Pc 42.50.Ct Keywords: Optical bistability (OB) Interdot tunnel coupling Quantum dots
1. Introduction The research for quantum coherence in atomic systems [1,2] has attracted a lot of interest in the past several decades. Representative examples are electromagnetically induced transparency (EIT) and coherent population trapping (CPT), which not only can modify the linear susceptibility of a medium but also can enhance the nonlinear optical processes in multilevel atomic systems [3]. The nonlinear optical processes related to the atomic coherence have shown their great potential in possible optoelectronic applications, such as optical buffers and modulators based on the slow-light phenomenon [4–7], all-optical switches based on optical bistability (OB) and optical multistability (OM) [8–10], and highly efficient frequency conversion in ultraslow propagation regime [11,12]. In particular, the OB and OM based on quantum coherence in multilevel atomic system have attracted much attention once again. The OB in three-level atomic systems confined in an optical ring cavity has also been extensively studied experimentally [8–10] and theoretically [13,14]. It has been shown that the field-induced transparency and quantum interference effects can significantly decrease the OB threshold [14]. The
Corresponding author. Tel./fax: +86 2787557477.
E-mail address: [email protected] ( J. Li). 1386-9477/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2008.06.002
effects of phase fluctuation [15] and squeezed state fields [16,17] on the OB have subsequently been studied. It has been found that the OB can appear for small cooperation parameters due to the presence of squeezed vacuum field [17]. On the other hand, some similar phenomena involving quantum coherence can also appear in semiconductor quantum dots (QDs) [18–24]. QDs have properties similar to atomic vapours but with the advantage of high nonlinear optical coefficients, large electric-dipole moments of intersubband transitions due to the small effective electron mass as well as ease of integration. Several ideas for quantum coherence in semiconductor QDs have been proposed and analyzed. For instance, Chang-Hasnain et al. proposed a semiconductor optical buffer using EIT in a QD medium and theoretically predicted its performance of slow light [18]. Villas-Boˆas et al. demonstrated coherent control of tunneling in a QD molecule [20]. Recently, the interaction of electrons confined in double QD structures with external electromagnetic driving fields has attracted increasing attention [25–31]. Several interesting phenomena have been recognized when the QD structure contains one or two electrons. For instance, controlled transfer of electrons between the two QDs [25–27] and the creation of maximally entangled states in two-electron QD systems [28–31] have been widely investigated. An optical system which exhibits two steady transmission states for the same input intensity is said to be optically bistable.
ARTICLE IN PRESS J. Li et al. / Physica E 41 (2008) 70–73
To the best of our knowledge, so far no related theoretical or experimental work has been carried out to explore OB properties of a tunable three-level system via tunneling in a coupled QD nanostructure, which motivates the current work. In this paper, we put forward a scheme for electronic OB in a tunable three-level QD molecule formed from an asymmetry double QD system coupled by tunneling. Such a QD molecule can be fabricated using self-assembled dot growth technology [32,33]. The system interacts with a probe laser inside a unidirectional ring cavity. In this scheme, the quantum coherence is created by coupling the two exciton states via tunneling instead of pumping laser. By suitably varying the applied voltage and laser frequency, we can realize the electrical controllability of OB. The remainder of this paper is organized into three parts as follows. In Section 1, the model is presented. The basic dynamics equations of motion are derived. In Section 3, we analyze our results via numerical simulations and further present physical interpretations for the bistable behaviors. Finally, we conclude with a brief summary in Section 4.
2. OB scheme and calculation results In what follows, in dealing with the coherent control of the OB via suitably varying the applied voltage on QD pair in this paper, what we have in our mind is the same physical picture and the same situation as the ones described in Ref. [20]. For simplicity, we also give the schematic plots of band structure and level configuration as shown in Figs. 1(a) and (b). Using this configuration, with the assumption of _ ¼ 1, the resulting 3 3 Hamiltonian for such a double QD system under study can be written in the interaction picture as 0 1 1 0 2d O B C 1 C, Te (1) Hint ¼ B 2d @ O A 0 T e 12d o12 where T e is the electron-tunneling matrix element, and O stands for one-half Rabi frequency for the transition j0i2j1i, i.e., O ¼ m10 Ep =2_, with m10 ¼ ~ m10 ~ eL (~ eL is the unit polarization
1 δ Ep ωp , Ep
ωp
Te
ω12
Te ω10
2 0 ETp
I
Ep
QD Sample
0 M1
R=1 M4
L
M2
R=1 M3
Fig. 1. (a) Band diagram of a QD molecule. It consists of two dots (the left one and the right one) via tunneling. With an external voltage applied to a gate electrode, the conduction-band levels get closer to resonance, greatly increasing their coupling, while the valence-band levels get more off-resonance, resulting in effective decoupling of those levels. (b) Schematic of the energy level arrangement under study. A probe laser with central frequency op and amplitude Ep excites one electron from the valence to the conduction band in the left dot, which can in turn tunnel to the right dot. j0i is the system without excitations, j1i a pair of electron and hole bound in the left dot, and j2i one hole in the left dot with an electron in the right dot. (c) Schematic setup of unidirectional ring cavity containing a QD molecule sample of length L, EIp and ETp are the incident and the transmitted probe fields, respectively.
71
vector of the probe laser) denoting the electric dipole moment for the excitonic transition between states j1i and j0i. It should be noted that another commonly used definition of the Rabi frequency is: O ¼ m10 Ep =_, which differs by a factor of two from the definition used here. d ¼ o10 op is frequency detuning of the probe laser with the exciton resonance (see Fig. 1(b)), and oij ¼ i j with j being the energy of the state j ji ( j ¼ 0; 1; 2). The parameters T e and o12 can be tuned with bias voltage. To obtain the dynamics of the whole system, we use a standard density-matrix approach. The time evolution of the system, expressed using the density operator r, is governed by the Liouville equation which leads to the following equations of motion for the density matrix elements rij :
r_ 00 ¼ g10 r11 þ g20 r22 þ iOðr01 r10 Þ, r_ 11 ¼ g10 r11 þ iT e ðr12 r21 Þ þ iOðr10 r01 Þ, r_ 22 ¼ g20 r22 þ iT e ðr21 r12 Þ, r_ 10 ¼ ðid þ G10 Þr10 iOðr00 r11 Þ iT e r20 , r_ 20 ¼ ½iðd o12 Þ þ G20 r20 þ iOr21 iT e r10 , r_ 12 ¼ ðio12 þ G12 Þr12 iT e ðr22 r11 Þ iOr02 ,
(2a) (2b) (2c) (2d) (2e) (2f)
together with rij ¼ rji and the carrier conservation condition r00 þ r11 þ r22 ¼ 1. gij and Gij are lifetime broadening and dephasing broadening linewidths [18], respectively, and have been added phenomenologically in the above density matrix equations. Usually, Gij is the dominant mechanism. In the following numerical calculations, the choices of the parameters gij and Gij are based on experimental results from Ref. [18]. Now, we put the QD sample in a unidirectional ring cavity (see Fig. 1(c)). For simplicity, we assume that mirrors 3 and 4 have 100% reflectivity, and the intensity reflection and transmission coefficient of mirrors 1 and 2 are R and T (with R þ T ¼ 1), respectively. The total electromagnetic field can be written as E ¼ Ep eiop t þ c:c:, where the probe laser Ep circulates in the ring cavity and the symbol ‘‘c.c.’’ means the complex conjugation. The slowly oscillating term of the induced polarization in the excitonic transition j0i2j1i is determined by Pðop Þ ¼ Nm01 r10 with N being the average density of electrons or QDs (if a QD has one electron) in the QD layer. As a result, under slowly varying envelope approximation, the probe laser field evolves according to the following reduced wave equation:
qEp qEp op NG þc ¼i m01 r10 , qt qz 20
(3)
where c and 0 is the light speed and permittivity in free space, respectively. G is the optical confinement factor of the waveguide [18,19]. We consider the field equation (3) in the steady-state region. Setting the time derivative in Eq. (3) equal to zero for the steady state, we can obtain the field amplitude as follows:
qEp op m01 NG ¼i r10 : qz 2c0
(4)
For a perfectly tuned ring cavity, in the steady state limit, the boundary conditions impose the following conditions between the incident field EIp and the transmitted field ETp pffiffiffi Ep ðLÞ ¼ ETp = T , (5a) pffiffiffi (5b) Ep ð0Þ ¼ T EIp þ REp ðLÞ, where L is the length of the QD molecule sample, and the second term on the right-hand side of Eq. (5b) describes a feedback mechanism due to the mirror, which is essential to give rise to bistability, that is to say, no bistability can occur if R ¼ 0. In the mean-field limit [34], using the boundary conditions pffiffiffi (Eq. (5)) and normalizing the fields by letting y ¼ m10 EIp =_ T and
ARTICLE IN PRESS 72
J. Li et al. / Physica E 41 (2008) 70–73
pffiffiffi x ¼ m10 ETp =_ T , we can get input–output relationship: y ¼ x iC r10 ðxÞ,
3.5 (6)
3
where C ¼ op Ljm01 j2 NG=2_0 cT is the electronic cooperation parameter. It is worthwhile pointing out that the second term on the right-hand side of Eq. (6) is vital for OB or OM to take place.
3.5
Output field |x|
We set the time derivatives qrij =qt ¼ 0 ði; j ¼ 0; 1; 2Þ in the above density matrix equation (2) for the steady state, and solve the corresponding density matrix equation together with the coupled field equation (6) via nice Matlab codes, then we can arrive at the steady-state solutions. In the following we present a few numerical results for the steady state of the output field intensity versus the input field intensity under different parametric conditions in order to demonstrate controllability of the OB, as shown in Figs. 2–4. Fig. 2 shows the dependence of the OB on the voltagecontrolled detuning of the levels j1i and j2i ðo12 Þ, when keeping all other parameters fixed. It is clearly shown that, when the levels of the different dots are in resonance ðo12 ¼ 0Þ, no OB occurs because the effective tunneling is strong [20,21] (see curve A). In contrast, for the case that o12 a0, OB can be generated. It can be easily seen from curves B–G of Fig. 2 that increasing the value of o12 leads to a gradual increasing of the bistable threshold. The reason for the above results can be qualitatively explained as follows. With an increasing o12 between the excitonic states j1i and j2i, the coupling efficiency of interdot tunneling decreases considerably. For high voltage, the two dots are tilted enough such that virtually no coupling occurs, which regresses to isolated twolevel absorption in the left QD. Consequently, when increasing o12 , the absorption for the probe laser at d ¼ 0 can be enhanced apparently as already verified from Figs. 2(c) and (d) of Ref. [21], which makes the cavity field more difficult to reach saturation. This might be useful to manipulate the threshold value and the hysteresis cycle width of the bistable curve via varying o12 . Fig. 3 shows the dependence of the OB on the voltagecontrolled tunneling of the levels j1i and j2i (T e ), when keeping all other parameters fixed. It is found that, when the tunneling T e is
2 1.5 1
F E D
B
A
0 0
0.5
1 Input field |y|
1.5
2
Fig. 3. Output field jxj versus input field jyj for different values of the voltagecontrolled tunneling of the levels j1i and j2i (T e ). Other parameters are C ¼ 50 meV, o12 ¼ 0:2 meV, d ¼ 0 meV, g10 ¼ 0:554 meV, g20 ¼ 0:001 meV, G10 ¼ 5:54 meV, G12 ¼ 2 meV, and G20 ¼ 0:005 meV. Curves A–F are for T e ¼ 0:3; 0:4; 0:6; 0:8; 1:0; and 1.2 meV, respectively.
3 2.5 2 1.5 1 D
0.5
C
B
A
0 0
3
0.5
1
1.5
2
Input field |y| Fig. 4. Output field jxj versus input field jyj for different values of frequency detuning (d). Other parameters are C ¼ 50 meV, T e ¼ 0:6 meV, o12 ¼ 0:2 meV, g10 ¼ 0:554 meV, g20 ¼ 0:001 meV, G10 ¼ 5:54 meV, G12 ¼ 2 meV, and G20 ¼ 0:005 meV. Curves A–D are for d ¼ 0; 0:05; 0:1, and 0.15 meV, respectively.
2.5 Output field |x|
C
0.5
Output field |x|
3. Analysis and discussion
2.5
2 1.5 1 B
A
0.5
C
D
E
F G
0 0
0.5
1 Input field |y|
1.5
2
Fig. 2. Output field jxj versus input field jyj for different values of the voltagecontrolled detuning of the levels j1i and j2i (o12 ). Other parameters are C ¼ 50 meV, T e ¼ 0:6 meV, d ¼ 0 meV, g10 ¼ 0:554 meV, g20 ¼ 0:001 meV, G10 ¼ 5:54 meV, G12 ¼ 2 meV, and G20 ¼ 0:005 meV. Curves A–G are for o12 ¼ 0; 0:05; 0:1; 0:15; 0:2; 0:25, and 0.3 meV, respectively.
adjusted from 0.3 to 1.2 meV, the bistable threshold value decreases obviously and the region of OB becomes narrowed (see curves A–F). When the tunneling T e is strong enough, as expected, OB tends to disappear (see curve F) because the samples are transparent to the probe laser [21]. The electron tunneling between levels j1i and j2i is important for the absorption in such a driven QD molecule, so when we gradually increase the strength of the tunneling, the absorption of the probe field in the medium can be decreased which accounts for the change of the hysteresis cycle as shown in Fig. 3. This means that the threshold value and the hysteresis cycle width of the bistable curve can be controlled by adjusting the tunneling T e . It should be pointed out that, under the condition of allowing the electron to tunnel, the parameter T e can vary with voltage because it changes the potential profile between dots, but that should be a small variation. For a large
ARTICLE IN PRESS J. Li et al. / Physica E 41 (2008) 70–73
variation, we need to vary the distance and material between dots experimentally. In order to have a better idea about how the bistable threshold value changes with the frequency detuning ðdÞ of the probe laser keeping the gate voltage fixed (keeping o12 and T e unchanged), in Fig. 4 we investigate the effects of the detuning of the probe laser with the exciton energy on OB. From this figure, we can observe that with increasing d from 0 to 0.15 mev, the threshold of OB decreases progressively and the area of the hysteresis cycle becomes narrower.
4. Conclusion In summary, we have theoretically explored the OB behaviors in an asymmetric double QD system (a driven QD molecule) using interdot tunnel coupling by means of a unidirectional ring cavity. The tunnel barrier in a QD molecule can be directly controlled by placing a gate electrode between two QDs. By properly tuning the applied voltage and laser frequency, the OB behavior can be controlled efficiently. Our calculations also provide a guideline for optimizing and controlling the optical switching process in the QD solid-state system, which is much more practical than that in atomic system because of its flexible design and the controllable (tunable) interference strength.
Acknowledgments The authors express their gratitude to the referee of the paper for his/her fruitful advice and comment, which significantly improved the paper. The research is supported in part by the National Natural Science Foundation of China (Grant nos. 10634060, 10575040, 60478029 and 10747133) and by National Basic Research Program of China under Contract no. 2005CB724508.
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Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Voltage-controlled optical bistability of a tunable three-level system in a quantum-dot molecule J. Li a,, R. Yu b, J. Liu a, P. Huang a, X. Yang a a b
Department of Physics, Huazhong University of Science and Technology (HUST), Wuhan 430074, People’s Republic of China School of Science, Wuhan Institute of Technology, Wuhan 430073, People’s Republic of China
a r t i c l e in f o
a b s t r a c t
Article history: Received 1 March 2008 Received in revised form 4 June 2008 Accepted 4 June 2008 Available online 12 June 2008
We investigate the behavior of optical bistability (OB) in an asymmetry semiconductor double quantum dot (QD) using the tunnel coupling. Such a tunable three-level QD system is driven coherently by a laser field inside the unidirectional ring cavity. The results show that we are able to control efficiently the bistable threshold intensity and the hysteresis loop by tuning the parameters of the system such as the gate voltage and laser frequency. The results obtained can be used for the development of new types of nanoelectronic devices for realizing switching process. & 2008 Elsevier B.V. All rights reserved.
PACS: 78.67.Hc 42.65.Pc 42.50.Ct Keywords: Optical bistability (OB) Interdot tunnel coupling Quantum dots
1. Introduction The research for quantum coherence in atomic systems [1,2] has attracted a lot of interest in the past several decades. Representative examples are electromagnetically induced transparency (EIT) and coherent population trapping (CPT), which not only can modify the linear susceptibility of a medium but also can enhance the nonlinear optical processes in multilevel atomic systems [3]. The nonlinear optical processes related to the atomic coherence have shown their great potential in possible optoelectronic applications, such as optical buffers and modulators based on the slow-light phenomenon [4–7], all-optical switches based on optical bistability (OB) and optical multistability (OM) [8–10], and highly efficient frequency conversion in ultraslow propagation regime [11,12]. In particular, the OB and OM based on quantum coherence in multilevel atomic system have attracted much attention once again. The OB in three-level atomic systems confined in an optical ring cavity has also been extensively studied experimentally [8–10] and theoretically [13,14]. It has been shown that the field-induced transparency and quantum interference effects can significantly decrease the OB threshold [14]. The
Corresponding author. Tel./fax: +86 2787557477.
E-mail address: [email protected] ( J. Li). 1386-9477/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2008.06.002
effects of phase fluctuation [15] and squeezed state fields [16,17] on the OB have subsequently been studied. It has been found that the OB can appear for small cooperation parameters due to the presence of squeezed vacuum field [17]. On the other hand, some similar phenomena involving quantum coherence can also appear in semiconductor quantum dots (QDs) [18–24]. QDs have properties similar to atomic vapours but with the advantage of high nonlinear optical coefficients, large electric-dipole moments of intersubband transitions due to the small effective electron mass as well as ease of integration. Several ideas for quantum coherence in semiconductor QDs have been proposed and analyzed. For instance, Chang-Hasnain et al. proposed a semiconductor optical buffer using EIT in a QD medium and theoretically predicted its performance of slow light [18]. Villas-Boˆas et al. demonstrated coherent control of tunneling in a QD molecule [20]. Recently, the interaction of electrons confined in double QD structures with external electromagnetic driving fields has attracted increasing attention [25–31]. Several interesting phenomena have been recognized when the QD structure contains one or two electrons. For instance, controlled transfer of electrons between the two QDs [25–27] and the creation of maximally entangled states in two-electron QD systems [28–31] have been widely investigated. An optical system which exhibits two steady transmission states for the same input intensity is said to be optically bistable.
ARTICLE IN PRESS J. Li et al. / Physica E 41 (2008) 70–73
To the best of our knowledge, so far no related theoretical or experimental work has been carried out to explore OB properties of a tunable three-level system via tunneling in a coupled QD nanostructure, which motivates the current work. In this paper, we put forward a scheme for electronic OB in a tunable three-level QD molecule formed from an asymmetry double QD system coupled by tunneling. Such a QD molecule can be fabricated using self-assembled dot growth technology [32,33]. The system interacts with a probe laser inside a unidirectional ring cavity. In this scheme, the quantum coherence is created by coupling the two exciton states via tunneling instead of pumping laser. By suitably varying the applied voltage and laser frequency, we can realize the electrical controllability of OB. The remainder of this paper is organized into three parts as follows. In Section 1, the model is presented. The basic dynamics equations of motion are derived. In Section 3, we analyze our results via numerical simulations and further present physical interpretations for the bistable behaviors. Finally, we conclude with a brief summary in Section 4.
2. OB scheme and calculation results In what follows, in dealing with the coherent control of the OB via suitably varying the applied voltage on QD pair in this paper, what we have in our mind is the same physical picture and the same situation as the ones described in Ref. [20]. For simplicity, we also give the schematic plots of band structure and level configuration as shown in Figs. 1(a) and (b). Using this configuration, with the assumption of _ ¼ 1, the resulting 3 3 Hamiltonian for such a double QD system under study can be written in the interaction picture as 0 1 1 0 2d O B C 1 C, Te (1) Hint ¼ B 2d @ O A 0 T e 12d o12 where T e is the electron-tunneling matrix element, and O stands for one-half Rabi frequency for the transition j0i2j1i, i.e., O ¼ m10 Ep =2_, with m10 ¼ ~ m10 ~ eL (~ eL is the unit polarization
1 δ Ep ωp , Ep
ωp
Te
ω12
Te ω10
2 0 ETp
I
Ep
QD Sample
0 M1
R=1 M4
L
M2
R=1 M3
Fig. 1. (a) Band diagram of a QD molecule. It consists of two dots (the left one and the right one) via tunneling. With an external voltage applied to a gate electrode, the conduction-band levels get closer to resonance, greatly increasing their coupling, while the valence-band levels get more off-resonance, resulting in effective decoupling of those levels. (b) Schematic of the energy level arrangement under study. A probe laser with central frequency op and amplitude Ep excites one electron from the valence to the conduction band in the left dot, which can in turn tunnel to the right dot. j0i is the system without excitations, j1i a pair of electron and hole bound in the left dot, and j2i one hole in the left dot with an electron in the right dot. (c) Schematic setup of unidirectional ring cavity containing a QD molecule sample of length L, EIp and ETp are the incident and the transmitted probe fields, respectively.
71
vector of the probe laser) denoting the electric dipole moment for the excitonic transition between states j1i and j0i. It should be noted that another commonly used definition of the Rabi frequency is: O ¼ m10 Ep =_, which differs by a factor of two from the definition used here. d ¼ o10 op is frequency detuning of the probe laser with the exciton resonance (see Fig. 1(b)), and oij ¼ i j with j being the energy of the state j ji ( j ¼ 0; 1; 2). The parameters T e and o12 can be tuned with bias voltage. To obtain the dynamics of the whole system, we use a standard density-matrix approach. The time evolution of the system, expressed using the density operator r, is governed by the Liouville equation which leads to the following equations of motion for the density matrix elements rij :
r_ 00 ¼ g10 r11 þ g20 r22 þ iOðr01 r10 Þ, r_ 11 ¼ g10 r11 þ iT e ðr12 r21 Þ þ iOðr10 r01 Þ, r_ 22 ¼ g20 r22 þ iT e ðr21 r12 Þ, r_ 10 ¼ ðid þ G10 Þr10 iOðr00 r11 Þ iT e r20 , r_ 20 ¼ ½iðd o12 Þ þ G20 r20 þ iOr21 iT e r10 , r_ 12 ¼ ðio12 þ G12 Þr12 iT e ðr22 r11 Þ iOr02 ,
(2a) (2b) (2c) (2d) (2e) (2f)
together with rij ¼ rji and the carrier conservation condition r00 þ r11 þ r22 ¼ 1. gij and Gij are lifetime broadening and dephasing broadening linewidths [18], respectively, and have been added phenomenologically in the above density matrix equations. Usually, Gij is the dominant mechanism. In the following numerical calculations, the choices of the parameters gij and Gij are based on experimental results from Ref. [18]. Now, we put the QD sample in a unidirectional ring cavity (see Fig. 1(c)). For simplicity, we assume that mirrors 3 and 4 have 100% reflectivity, and the intensity reflection and transmission coefficient of mirrors 1 and 2 are R and T (with R þ T ¼ 1), respectively. The total electromagnetic field can be written as E ¼ Ep eiop t þ c:c:, where the probe laser Ep circulates in the ring cavity and the symbol ‘‘c.c.’’ means the complex conjugation. The slowly oscillating term of the induced polarization in the excitonic transition j0i2j1i is determined by Pðop Þ ¼ Nm01 r10 with N being the average density of electrons or QDs (if a QD has one electron) in the QD layer. As a result, under slowly varying envelope approximation, the probe laser field evolves according to the following reduced wave equation:
qEp qEp op NG þc ¼i m01 r10 , qt qz 20
(3)
where c and 0 is the light speed and permittivity in free space, respectively. G is the optical confinement factor of the waveguide [18,19]. We consider the field equation (3) in the steady-state region. Setting the time derivative in Eq. (3) equal to zero for the steady state, we can obtain the field amplitude as follows:
qEp op m01 NG ¼i r10 : qz 2c0
(4)
For a perfectly tuned ring cavity, in the steady state limit, the boundary conditions impose the following conditions between the incident field EIp and the transmitted field ETp pffiffiffi Ep ðLÞ ¼ ETp = T , (5a) pffiffiffi (5b) Ep ð0Þ ¼ T EIp þ REp ðLÞ, where L is the length of the QD molecule sample, and the second term on the right-hand side of Eq. (5b) describes a feedback mechanism due to the mirror, which is essential to give rise to bistability, that is to say, no bistability can occur if R ¼ 0. In the mean-field limit [34], using the boundary conditions pffiffiffi (Eq. (5)) and normalizing the fields by letting y ¼ m10 EIp =_ T and
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J. Li et al. / Physica E 41 (2008) 70–73
pffiffiffi x ¼ m10 ETp =_ T , we can get input–output relationship: y ¼ x iC r10 ðxÞ,
3.5 (6)
3
where C ¼ op Ljm01 j2 NG=2_0 cT is the electronic cooperation parameter. It is worthwhile pointing out that the second term on the right-hand side of Eq. (6) is vital for OB or OM to take place.
3.5
Output field |x|
We set the time derivatives qrij =qt ¼ 0 ði; j ¼ 0; 1; 2Þ in the above density matrix equation (2) for the steady state, and solve the corresponding density matrix equation together with the coupled field equation (6) via nice Matlab codes, then we can arrive at the steady-state solutions. In the following we present a few numerical results for the steady state of the output field intensity versus the input field intensity under different parametric conditions in order to demonstrate controllability of the OB, as shown in Figs. 2–4. Fig. 2 shows the dependence of the OB on the voltagecontrolled detuning of the levels j1i and j2i ðo12 Þ, when keeping all other parameters fixed. It is clearly shown that, when the levels of the different dots are in resonance ðo12 ¼ 0Þ, no OB occurs because the effective tunneling is strong [20,21] (see curve A). In contrast, for the case that o12 a0, OB can be generated. It can be easily seen from curves B–G of Fig. 2 that increasing the value of o12 leads to a gradual increasing of the bistable threshold. The reason for the above results can be qualitatively explained as follows. With an increasing o12 between the excitonic states j1i and j2i, the coupling efficiency of interdot tunneling decreases considerably. For high voltage, the two dots are tilted enough such that virtually no coupling occurs, which regresses to isolated twolevel absorption in the left QD. Consequently, when increasing o12 , the absorption for the probe laser at d ¼ 0 can be enhanced apparently as already verified from Figs. 2(c) and (d) of Ref. [21], which makes the cavity field more difficult to reach saturation. This might be useful to manipulate the threshold value and the hysteresis cycle width of the bistable curve via varying o12 . Fig. 3 shows the dependence of the OB on the voltagecontrolled tunneling of the levels j1i and j2i (T e ), when keeping all other parameters fixed. It is found that, when the tunneling T e is
2 1.5 1
F E D
B
A
0 0
0.5
1 Input field |y|
1.5
2
Fig. 3. Output field jxj versus input field jyj for different values of the voltagecontrolled tunneling of the levels j1i and j2i (T e ). Other parameters are C ¼ 50 meV, o12 ¼ 0:2 meV, d ¼ 0 meV, g10 ¼ 0:554 meV, g20 ¼ 0:001 meV, G10 ¼ 5:54 meV, G12 ¼ 2 meV, and G20 ¼ 0:005 meV. Curves A–F are for T e ¼ 0:3; 0:4; 0:6; 0:8; 1:0; and 1.2 meV, respectively.
3 2.5 2 1.5 1 D
0.5
C
B
A
0 0
3
0.5
1
1.5
2
Input field |y| Fig. 4. Output field jxj versus input field jyj for different values of frequency detuning (d). Other parameters are C ¼ 50 meV, T e ¼ 0:6 meV, o12 ¼ 0:2 meV, g10 ¼ 0:554 meV, g20 ¼ 0:001 meV, G10 ¼ 5:54 meV, G12 ¼ 2 meV, and G20 ¼ 0:005 meV. Curves A–D are for d ¼ 0; 0:05; 0:1, and 0.15 meV, respectively.
2.5 Output field |x|
C
0.5
Output field |x|
3. Analysis and discussion
2.5
2 1.5 1 B
A
0.5
C
D
E
F G
0 0
0.5
1 Input field |y|
1.5
2
Fig. 2. Output field jxj versus input field jyj for different values of the voltagecontrolled detuning of the levels j1i and j2i (o12 ). Other parameters are C ¼ 50 meV, T e ¼ 0:6 meV, d ¼ 0 meV, g10 ¼ 0:554 meV, g20 ¼ 0:001 meV, G10 ¼ 5:54 meV, G12 ¼ 2 meV, and G20 ¼ 0:005 meV. Curves A–G are for o12 ¼ 0; 0:05; 0:1; 0:15; 0:2; 0:25, and 0.3 meV, respectively.
adjusted from 0.3 to 1.2 meV, the bistable threshold value decreases obviously and the region of OB becomes narrowed (see curves A–F). When the tunneling T e is strong enough, as expected, OB tends to disappear (see curve F) because the samples are transparent to the probe laser [21]. The electron tunneling between levels j1i and j2i is important for the absorption in such a driven QD molecule, so when we gradually increase the strength of the tunneling, the absorption of the probe field in the medium can be decreased which accounts for the change of the hysteresis cycle as shown in Fig. 3. This means that the threshold value and the hysteresis cycle width of the bistable curve can be controlled by adjusting the tunneling T e . It should be pointed out that, under the condition of allowing the electron to tunnel, the parameter T e can vary with voltage because it changes the potential profile between dots, but that should be a small variation. For a large
ARTICLE IN PRESS J. Li et al. / Physica E 41 (2008) 70–73
variation, we need to vary the distance and material between dots experimentally. In order to have a better idea about how the bistable threshold value changes with the frequency detuning ðdÞ of the probe laser keeping the gate voltage fixed (keeping o12 and T e unchanged), in Fig. 4 we investigate the effects of the detuning of the probe laser with the exciton energy on OB. From this figure, we can observe that with increasing d from 0 to 0.15 mev, the threshold of OB decreases progressively and the area of the hysteresis cycle becomes narrower.
4. Conclusion In summary, we have theoretically explored the OB behaviors in an asymmetric double QD system (a driven QD molecule) using interdot tunnel coupling by means of a unidirectional ring cavity. The tunnel barrier in a QD molecule can be directly controlled by placing a gate electrode between two QDs. By properly tuning the applied voltage and laser frequency, the OB behavior can be controlled efficiently. Our calculations also provide a guideline for optimizing and controlling the optical switching process in the QD solid-state system, which is much more practical than that in atomic system because of its flexible design and the controllable (tunable) interference strength.
Acknowledgments The authors express their gratitude to the referee of the paper for his/her fruitful advice and comment, which significantly improved the paper. The research is supported in part by the National Natural Science Foundation of China (Grant nos. 10634060, 10575040, 60478029 and 10747133) and by National Basic Research Program of China under Contract no. 2005CB724508.
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