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Photon switch in a quantum well by quantum interference in interband transitions
Optics Communications 249 (2005) 231–237 www.elsevier.com/locate/optcom
Photon switch in a quantum well by quantum interference in interband transitions Yan Xue
b,c,*
, Xue-Mei Su
b,c
, Gang Wang
b,c
, Yi Chen
b,c
, Jin-Yue Gao
a,b,c,*
a CCAST (World Laboratory), P.O. Box 8370, Beijing 100080, PR China College of Physics, Jilin University, Jiefang Road No.119, Changchun 130023, PR China Key Laboratory of Coherent Light, Atomic and Molecular Spectroscopy, Educational Ministry of China, Changchun, PR China b
c
Received 24 November 2004; received in revised form 20 December 2004; accepted 21 December 2004
Abstract We propose a scheme for a photon switch based on quantum interference in interband transitions in an asymmetric semiconductor coupled double quantum well structure. It is shown that, due to the existence of tunneling induced quantum interference, a weak resonant probe field, in the ultraviolet or visible spectral region, can pass through this structure with little absorption. A second coupling field can switch off the probe laser by destroying the destructive interference at the resonant probe frequency. The photon switch can be achieved easily with relatively small pulse intensity (104 W/cm2) because of a large dipole moment length (108 m) in the interband transition. 2004 Elsevier B.V. All rights reserved. PACS: 42.50.Gy Keywords: Quantum interference; Optical switch
The generation of atomic coherence has been attracting considerable interests in last few years, since it gives rise to such interesting phenomena as electromagnetically induced transparency (EIT) [1] , light amplification without population inversion [2,3] and enhanced nonlinearity with reduced absorption [4]. After having been successful in atomic physics, the studies in quantum wells, such as EIT [5–7] and Fano interference [8], have drawn more and more attentions in recent years, for their inherent advantages like large electric dipole moment due to the small effective electron mass, high nonlinear optical coefficients [9] and a great flexibility in device design by choosing the materials and structure dimensions. Xue-Mei Su and Jin-Yue Gao [10] presented a photon switch in a quantum well by quantum interference in intersubband transitions, where one *
Corresponding authors. Tel./fax: +86 431 5650478. E-mail addresses: [email protected] (Y. Xue), [email protected] (J.-Y. Gao).
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.12.045
232
Y. Xue et al. / Optics Communications 249 (2005) 231–237
light controled another in infrared region and the photon switch was realized with small pulse intensity (106 W/cm2). In this letter we predict, on the basis of a model calculation, that a photon switch may be realized in an asymmetric semiconductor coupled double quantum well by quantum interference in interband transitions. Different from the intersubband transition in infrared region, the interband transition in the ultraviolet or visible region [11,12] has very large dipole moment length (108 m) [13], so a photon switch may be realized with relatively smaller pulse intensity (104 W/cm2). The asymmetric semiconductor coupled double quantum well structure, considered here (Fig. 1), consists ˚ ) thick wide well and a 35-monolayer (100 A ˚ ) thick narrow well, sepaof 10 pairs of a 51-monolayer (145 A rated by a Al0.2Ga0.8As buffer layer [14,15]. In this quantum well structure, the first (n = 1) electron levels in the wide well and the narrow well can be energetically aligned with each other by applying a static electric field, while the corresponding n = 1 hole levels are never aligned for this polarity of the field [14]. The electrons then delocalize over both wells while the holes remain localized. For transitions from the narrow well and wide well, the Coulomb interaction between the electron and the hole downshifts the value of the electric field where the resonance condition is fulfilled to that corresponding to the built-in field. Level |3æ and level |4æ, respectively corresponding to bonding state |+æ and anti-bonding state |æ, between which there exists a tun3 C4 ffi [16], are tunneling coupled new states of the first (n = 1) elecneling induced cross-coupling term Xc ¼ pCffiffiffiffiffiffiffiffi c31 c42 tron level in the wide well and the narrow well. Level |1æ in the narrow well and level |2æ in the wide well are localized hole states. For more details on this system we refer the reader to [11]. The coherent coupling by tunneling makes the probe laser wp transparent while the switch laser ws is absent. However, with the presence of the switch laser ws, the quantum interference is destroyed and the probe laser wp is switched off. Working in the interaction picture, utilizing the rotating-wave approximation and the electric-dipole approximation, we derive the Hamiltonian for our quantum well structure as HI ¼ h dp j3ih3j þ ds j2ih2j þ dc j4ih4j h Xp ðj3ih1j þ j1ih3jÞ þ Xc ðj3ih4j þ j4ih3jÞ þXs ðj4ih2j þ j2ih4jÞ; lEp 2 h
ð1Þ
lEs 2 h
where Xp ¼ and Xs ¼ are Rabi frequency corresponding to respective laser fields. Here dp = (x3 x1) xp, dc = (x4 x1) xp xc and ds = (x2 x1) xp xc + xs are detunings of the probe, the coherent perturbation by tunneling and the switch, respectively, and Ci is the natural linewidth of level |iæ. For simplicity, we assume that Rabi frequency of external fields Xp, Xs and coherent coupling by
Fig. 1. Schematic band diagram of the ASCDQW structure. Levels |3æ and |4æ are the bonding and antibonding electron state of the coupled system and levels |1æ and |2æ are the localized hole state. Xp and Xs are Rabi frequencies of probe field and switch field, respectively.
Y. Xue et al. / Optics Communications 249 (2005) 231–237
233
tunneling Xc are real. The master equation of motion for density operator in the interaction picture can be written as oq 1 ¼ ½H I ; q þ Kq: ot i h
ð2Þ
By expanding Eq. (2), we can easily carry out the density matrix equations of motion oq34 ¼ ½c34 þ idc q34 iXs q32 þ iXp q14 þ iXc ðq44 q33 Þ; ot oq23 ¼ c23 þ i dp ds q23 iXc q24 þ iXs q43 iXp q21 ; ot oq24 ¼ ½c24 þ iðdc ds Þq24 þ iXs ðq44 q22 Þ iXc q23 ; ot oq14 ¼ ðc14 þ idc Þq14 þ iXp q34 iXs q12 iXc q13 ; ot oq13 ¼ c13 þ idp q13 þ iXp ðq33 q11 Þ iXc q14 ; ot oq11 ¼ iXp ðq31 q13 Þ þ C31 q33 þ C21 q22 þ C41 q44 ; ot oq22 ¼ iXs ðq42 q24 Þ C21 q22 þ C42 q44 þ C32 q33 ; ot oq33 ¼ iXp ðq13 q31 Þ þ iXc ðq43 q34 Þ C3 q33 ; ot oq12 ¼ ðc12 þ ids Þq12 þ iXp q32 iXs q14 ; ot q11 þ q22 þ q33 þ q44 ¼ 1;
ð3Þ
qij ¼ qji ; where Cij represents the damping of population from state |iæ to state |jæ and cij(i6¼j) is the total coherence deph deph C3 C4 C2þC3 relaxation rate, given by c12 ¼ ðC22 þ cdeph þ cdeph 12 Þ; c13 ¼ ð 2 þ c13 Þ; c14 ¼ ð 2 þ c14 Þ; c23 ¼ ð 2 23 Þ; deph deph C2þC4 C4þC3 c24 ¼ ð 2 þ c24 Þ and c34 ¼ ð 2 þ c34 Þ. Natural linewidths of the excited levels in quantum well structure are largely affected by temperature, well width [17], intensity of applied dc field [18] and so on. The dephasing linewidth cdeph corresponding to respective transition, determined by electron–electron scatij tering, interface roughness, phonon scattering and alloy scattering contribution [19], largely affected by temperature and electron population [20], should be reduced as low as possible to obtain a better Fano interference [8]. To start, we take all fields as monochramatic and assume that probe field is weak enough. Since the intensity of the probe laser is very weak, we can neglect the second and other higher order solutions in Xp and solve Eq. (3) in the steady state to get iXp ðc12 ids Þðc14 idc Þ þ X2s ð1Þ : ð4Þ q13 ¼ ðc12 ids ÞX2c þ ðc12 ids Þðc14 idc Þ þ X2s c13 idp Defining the polarization P ðwp Þ ¼ 12e0 Ep ½vp ðwp Þeiwp t þ H c, we can get the expression of the susceptibility at the probe frequency as ik ðc12 ids Þðc14 idc Þ þ X2s ; vp ¼ ð5Þ ðc12 ids ÞX2c þ ðc12 ids Þðc14 idc Þ c13 idp þ X2s c13 idp
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Y. Xue et al. / Optics Communications 249 (2005) 231–237 N l213
where k ¼ e0 h is a constant, and lij is the effective dipole moment with respect to the transition |iæ ! |jæ. We neglect ds and dp as compared to their respective linewidths. Inserting dp = dc(x3 = x4), and the golden rule 2 2 X2 transition rates W p ¼ 2c13p ; W c ¼ 2cX34c and W s ¼ 2cX24s [10] into Eq. (5), we can get vp ¼ k
2c212 c34 W c dp ð2c12 c34 þ c12 c13 c14 þ 2c13 c24 W s Þ2 þ c212 c213 d2p
ik
ð2c24 W s þ c12 c14 Þð2c12 c34 W c þ c12 c13 c14 þ 2c13 c24 W s Þ þ c212 c13 d2p ð2c12 c34 þ c12 c13 c14 þ 2c13 c24 W s Þ2 þ c212 c213 d2p
:
ð6Þ
Combining Eq. (6) with the MaxwellÕs equation, we can obtain expressions for the phase shift bl, the power loss 2al, and the group velocity delay time for the probe pulse of frequency xp. They are bl ¼ Nlr13
2c212 c13 c34 W c dp
;
ð7Þ
c13 ð2c24 W s þ c12 c14 Þð2c12 c34 W c þ c12 c13 c14 þ 2c13 c24 W s Þ þ c212 c213 d2p ; 2 2c12 c34 þ c12 c13 c14 þ 2c13 c24 W 2s þ c212 c213 d2p
ð8Þ
h i 2 2 2 2 2 2 2c c W 2c c þ c c c þ 2c c W c c d c 12 34 12 13 14 13 24 s 12 34 12 13 p oðblÞ ¼ ¼ Nlr13 ; h i2 odp 2 ð2c12 c34 þ c12 c13 c14 þ 2c13 c24 W s Þ þ c212 c213 d2p
ð9Þ
2al ¼ Nlr13
TD
2
ð2c12 c34 þ c12 c13 c14 þ 2c13 c24 W s Þ þ c212 c213 d2p
where r13 is the absorption cross-section. We can get power transmission exp(2al) as a function of dp. The critical switch Ws is obtained when the power loss 2al = 1. Next, we assume that probe laser Xp and switch laser Xs are pulse lasers and ask for conditions for which Eqs. (7)–(9) are still valid. The state vector in this quantum well structure can be written as jWðtÞi ¼ b1 j1i þ b2 j2i þ b3 j3i þ b4 j4i;
ð10Þ
where bi stands for the slowly varying probability amplitude corresponding to the state |iæ. By inserting Eqs. (1) and (10) into the schrodinger equation, we can obtain the probability amplitude equation of the model in study. Neglecting the frequency detunings as compared to the corresponding linewidths and droping the derivatives of the probability amplitudes of state |3æ and state |4æ because they vary slowly as compared to their linewidths, the probability amplitude equation can be rewritten as ob1 ¼ iXp b3 ; ot ob2 ¼ iXs b4 c12 b2 ; ot
ð11Þ
c13 b3 ¼ iXp b1 þ iXc b4 ; c14 b4 ¼ iXs b2 þ iXc b3 : With the same approach mentioned by Harris [21], we can obtain the condition that Eqs. (7)–(9) are still valid for the pulse fields 1 oXp 1 : Xp ot Wc
ð12Þ
Y. Xue et al. / Optics Communications 249 (2005) 231–237
235
That is to say, the maximum variation rate of Xp(t) and therefore the switch bandwidth are set by the (adjustable) golden rule transition rate Wc [22]. Now, we estimate the possibility for the photon switch studied here in virtual use. We let dephasing time cdeph ¼ cdeph ¼ cdeph ¼ cdeph ¼ cdeph ¼ 18 THz [23] and optical depth NLr13 = 2. Neogi pointed out that the 14 13 12 24 34 magnitude of lifetime of an excited state is about 1 ps [24], so we assume C4 ¼ 56 THz and C3 = 1 THz. We take C2 = 0 for the lifetime of level |2æ is 3 ns [25], which is very large compared to that of the excitation state. Fig. 2(a)–(c) show the power transmission exp(2al), phase shift bl and group velocity delay time TD as a function of the detuning dp. The results tell us that the design for optical switch in this asymmetric semiconductor quantum well structure is feasible. We should emphasize three points about the results shown in Fig. 2(a). One is that the power transmission at open state for probe laser is only 75.3% because of the larger dephasing rate cdeph as compared to 12 that in an intersubband transition of a quantum well or in an atomic system. We can get a better power transmission with a smaller cdeph 12 , which have been done by Grandjean [26]. The other point we should emphasize is that the power transmission at closed state for probe laser in Fig. 2(a) is about 33.5%. We may reduce the power transmission at a closed sate to 25% by a weak optical switch Ws = 0.8Wc (shown in Fig. 3). When the value of Ws is increased, we can get a lower power transmission up to 15% in a closed state by a strong optical switch Ws = 6Wc (shown in Fig. 3). Hereafter, the power transmission in a closed state varies slowly with Ws. Finally, we should accentuate the critical value Ws chosen for switch at a closed state. When Ws = 0.4Wc, the Rabi frequency of the switch field Xs 0.5Xc is in the order of
Fig. 2. Power transmission (a), phase shift (b) and group velocity delay time (c) as a function of the detuning dp. The switch is shown open (Ws = 0) and closed (critical Ws = 0.4Wc). The parameters are cdeph ¼ cdeph ¼ cdeph ¼ cdeph ¼ cdeph ¼ 18 THz; C4 ¼ 56 THz, 14 13 12 24 34 C3 = 1 THz, C1 = C2 = 0, Wc = 0.98 THz and NLr13 = 2.
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Y. Xue et al. / Optics Communications 249 (2005) 231–237
Fig. 3. Power transmission with different Ws when the switch is closed.The other parameters are similar with the values in Fig. 2.
7 · 1011 Hz. In this quantum well structure, a very large electric dipole moment length (108 m) [13] requires the intensity of the switch pulse for critical weak to be in the order of 104 W/cm2. In summary, we have described a scheme for a photon switch, in ultraviolet or visible spectra region, in an asymmetric semiconductor quantum well structure. It is much more feasible than in atomic system for its flexible design and the controllable interference strength. It is a method to realize one light controls another, in ultraviolet or visible region, in semiconductor.
Acknowledgement The authors thank the support from the National Natural Science Foundation of China under the number of 10334010.
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[21] S.E. Harris, Y. Yamamoto, Phys. Rev. Lett. 81 (1998) 3611. [22] J.H. Eberly, Quantum Semicalss. Opt. 71 (1995) 373. [23] Gia-Wei Chern, Chi-Kuang Sun, Yue-Kai Huang, Kung-Hsuan Lin, SPIE Photonics WEST 2003, Ultrafast Phenomena in Semiconductors VII, San Jose, California 4992–28. [24] A. Neogi, Opt. Commun. 133 (1979) 479. [25] A. Neogi, H. Yoshida, T. Mozume, O. Wada, Opt. Commun. 159 (1999) 225. [26] N. Grandjean, B. Damilano, J. Massies, G. Neu, M. Teissere, J. Appl. Phys. (2000) 183.
Photon switch in a quantum well by quantum interference in interband transitions Yan Xue
b,c,*
, Xue-Mei Su
b,c
, Gang Wang
b,c
, Yi Chen
b,c
, Jin-Yue Gao
a,b,c,*
a CCAST (World Laboratory), P.O. Box 8370, Beijing 100080, PR China College of Physics, Jilin University, Jiefang Road No.119, Changchun 130023, PR China Key Laboratory of Coherent Light, Atomic and Molecular Spectroscopy, Educational Ministry of China, Changchun, PR China b
c
Received 24 November 2004; received in revised form 20 December 2004; accepted 21 December 2004
Abstract We propose a scheme for a photon switch based on quantum interference in interband transitions in an asymmetric semiconductor coupled double quantum well structure. It is shown that, due to the existence of tunneling induced quantum interference, a weak resonant probe field, in the ultraviolet or visible spectral region, can pass through this structure with little absorption. A second coupling field can switch off the probe laser by destroying the destructive interference at the resonant probe frequency. The photon switch can be achieved easily with relatively small pulse intensity (104 W/cm2) because of a large dipole moment length (108 m) in the interband transition. 2004 Elsevier B.V. All rights reserved. PACS: 42.50.Gy Keywords: Quantum interference; Optical switch
The generation of atomic coherence has been attracting considerable interests in last few years, since it gives rise to such interesting phenomena as electromagnetically induced transparency (EIT) [1] , light amplification without population inversion [2,3] and enhanced nonlinearity with reduced absorption [4]. After having been successful in atomic physics, the studies in quantum wells, such as EIT [5–7] and Fano interference [8], have drawn more and more attentions in recent years, for their inherent advantages like large electric dipole moment due to the small effective electron mass, high nonlinear optical coefficients [9] and a great flexibility in device design by choosing the materials and structure dimensions. Xue-Mei Su and Jin-Yue Gao [10] presented a photon switch in a quantum well by quantum interference in intersubband transitions, where one *
Corresponding authors. Tel./fax: +86 431 5650478. E-mail addresses: [email protected] (Y. Xue), [email protected] (J.-Y. Gao).
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.12.045
232
Y. Xue et al. / Optics Communications 249 (2005) 231–237
light controled another in infrared region and the photon switch was realized with small pulse intensity (106 W/cm2). In this letter we predict, on the basis of a model calculation, that a photon switch may be realized in an asymmetric semiconductor coupled double quantum well by quantum interference in interband transitions. Different from the intersubband transition in infrared region, the interband transition in the ultraviolet or visible region [11,12] has very large dipole moment length (108 m) [13], so a photon switch may be realized with relatively smaller pulse intensity (104 W/cm2). The asymmetric semiconductor coupled double quantum well structure, considered here (Fig. 1), consists ˚ ) thick wide well and a 35-monolayer (100 A ˚ ) thick narrow well, sepaof 10 pairs of a 51-monolayer (145 A rated by a Al0.2Ga0.8As buffer layer [14,15]. In this quantum well structure, the first (n = 1) electron levels in the wide well and the narrow well can be energetically aligned with each other by applying a static electric field, while the corresponding n = 1 hole levels are never aligned for this polarity of the field [14]. The electrons then delocalize over both wells while the holes remain localized. For transitions from the narrow well and wide well, the Coulomb interaction between the electron and the hole downshifts the value of the electric field where the resonance condition is fulfilled to that corresponding to the built-in field. Level |3æ and level |4æ, respectively corresponding to bonding state |+æ and anti-bonding state |æ, between which there exists a tun3 C4 ffi [16], are tunneling coupled new states of the first (n = 1) elecneling induced cross-coupling term Xc ¼ pCffiffiffiffiffiffiffiffi c31 c42 tron level in the wide well and the narrow well. Level |1æ in the narrow well and level |2æ in the wide well are localized hole states. For more details on this system we refer the reader to [11]. The coherent coupling by tunneling makes the probe laser wp transparent while the switch laser ws is absent. However, with the presence of the switch laser ws, the quantum interference is destroyed and the probe laser wp is switched off. Working in the interaction picture, utilizing the rotating-wave approximation and the electric-dipole approximation, we derive the Hamiltonian for our quantum well structure as HI ¼ h dp j3ih3j þ ds j2ih2j þ dc j4ih4j h Xp ðj3ih1j þ j1ih3jÞ þ Xc ðj3ih4j þ j4ih3jÞ þXs ðj4ih2j þ j2ih4jÞ; lEp 2 h
ð1Þ
lEs 2 h
where Xp ¼ and Xs ¼ are Rabi frequency corresponding to respective laser fields. Here dp = (x3 x1) xp, dc = (x4 x1) xp xc and ds = (x2 x1) xp xc + xs are detunings of the probe, the coherent perturbation by tunneling and the switch, respectively, and Ci is the natural linewidth of level |iæ. For simplicity, we assume that Rabi frequency of external fields Xp, Xs and coherent coupling by
Fig. 1. Schematic band diagram of the ASCDQW structure. Levels |3æ and |4æ are the bonding and antibonding electron state of the coupled system and levels |1æ and |2æ are the localized hole state. Xp and Xs are Rabi frequencies of probe field and switch field, respectively.
Y. Xue et al. / Optics Communications 249 (2005) 231–237
233
tunneling Xc are real. The master equation of motion for density operator in the interaction picture can be written as oq 1 ¼ ½H I ; q þ Kq: ot i h
ð2Þ
By expanding Eq. (2), we can easily carry out the density matrix equations of motion oq34 ¼ ½c34 þ idc q34 iXs q32 þ iXp q14 þ iXc ðq44 q33 Þ; ot oq23 ¼ c23 þ i dp ds q23 iXc q24 þ iXs q43 iXp q21 ; ot oq24 ¼ ½c24 þ iðdc ds Þq24 þ iXs ðq44 q22 Þ iXc q23 ; ot oq14 ¼ ðc14 þ idc Þq14 þ iXp q34 iXs q12 iXc q13 ; ot oq13 ¼ c13 þ idp q13 þ iXp ðq33 q11 Þ iXc q14 ; ot oq11 ¼ iXp ðq31 q13 Þ þ C31 q33 þ C21 q22 þ C41 q44 ; ot oq22 ¼ iXs ðq42 q24 Þ C21 q22 þ C42 q44 þ C32 q33 ; ot oq33 ¼ iXp ðq13 q31 Þ þ iXc ðq43 q34 Þ C3 q33 ; ot oq12 ¼ ðc12 þ ids Þq12 þ iXp q32 iXs q14 ; ot q11 þ q22 þ q33 þ q44 ¼ 1;
ð3Þ
qij ¼ qji ; where Cij represents the damping of population from state |iæ to state |jæ and cij(i6¼j) is the total coherence deph deph C3 C4 C2þC3 relaxation rate, given by c12 ¼ ðC22 þ cdeph þ cdeph 12 Þ; c13 ¼ ð 2 þ c13 Þ; c14 ¼ ð 2 þ c14 Þ; c23 ¼ ð 2 23 Þ; deph deph C2þC4 C4þC3 c24 ¼ ð 2 þ c24 Þ and c34 ¼ ð 2 þ c34 Þ. Natural linewidths of the excited levels in quantum well structure are largely affected by temperature, well width [17], intensity of applied dc field [18] and so on. The dephasing linewidth cdeph corresponding to respective transition, determined by electron–electron scatij tering, interface roughness, phonon scattering and alloy scattering contribution [19], largely affected by temperature and electron population [20], should be reduced as low as possible to obtain a better Fano interference [8]. To start, we take all fields as monochramatic and assume that probe field is weak enough. Since the intensity of the probe laser is very weak, we can neglect the second and other higher order solutions in Xp and solve Eq. (3) in the steady state to get iXp ðc12 ids Þðc14 idc Þ þ X2s ð1Þ : ð4Þ q13 ¼ ðc12 ids ÞX2c þ ðc12 ids Þðc14 idc Þ þ X2s c13 idp Defining the polarization P ðwp Þ ¼ 12e0 Ep ½vp ðwp Þeiwp t þ H c, we can get the expression of the susceptibility at the probe frequency as ik ðc12 ids Þðc14 idc Þ þ X2s ; vp ¼ ð5Þ ðc12 ids ÞX2c þ ðc12 ids Þðc14 idc Þ c13 idp þ X2s c13 idp
234
Y. Xue et al. / Optics Communications 249 (2005) 231–237 N l213
where k ¼ e0 h is a constant, and lij is the effective dipole moment with respect to the transition |iæ ! |jæ. We neglect ds and dp as compared to their respective linewidths. Inserting dp = dc(x3 = x4), and the golden rule 2 2 X2 transition rates W p ¼ 2c13p ; W c ¼ 2cX34c and W s ¼ 2cX24s [10] into Eq. (5), we can get vp ¼ k
2c212 c34 W c dp ð2c12 c34 þ c12 c13 c14 þ 2c13 c24 W s Þ2 þ c212 c213 d2p
ik
ð2c24 W s þ c12 c14 Þð2c12 c34 W c þ c12 c13 c14 þ 2c13 c24 W s Þ þ c212 c13 d2p ð2c12 c34 þ c12 c13 c14 þ 2c13 c24 W s Þ2 þ c212 c213 d2p
:
ð6Þ
Combining Eq. (6) with the MaxwellÕs equation, we can obtain expressions for the phase shift bl, the power loss 2al, and the group velocity delay time for the probe pulse of frequency xp. They are bl ¼ Nlr13
2c212 c13 c34 W c dp
;
ð7Þ
c13 ð2c24 W s þ c12 c14 Þð2c12 c34 W c þ c12 c13 c14 þ 2c13 c24 W s Þ þ c212 c213 d2p ; 2 2c12 c34 þ c12 c13 c14 þ 2c13 c24 W 2s þ c212 c213 d2p
ð8Þ
h i 2 2 2 2 2 2 2c c W 2c c þ c c c þ 2c c W c c d c 12 34 12 13 14 13 24 s 12 34 12 13 p oðblÞ ¼ ¼ Nlr13 ; h i2 odp 2 ð2c12 c34 þ c12 c13 c14 þ 2c13 c24 W s Þ þ c212 c213 d2p
ð9Þ
2al ¼ Nlr13
TD
2
ð2c12 c34 þ c12 c13 c14 þ 2c13 c24 W s Þ þ c212 c213 d2p
where r13 is the absorption cross-section. We can get power transmission exp(2al) as a function of dp. The critical switch Ws is obtained when the power loss 2al = 1. Next, we assume that probe laser Xp and switch laser Xs are pulse lasers and ask for conditions for which Eqs. (7)–(9) are still valid. The state vector in this quantum well structure can be written as jWðtÞi ¼ b1 j1i þ b2 j2i þ b3 j3i þ b4 j4i;
ð10Þ
where bi stands for the slowly varying probability amplitude corresponding to the state |iæ. By inserting Eqs. (1) and (10) into the schrodinger equation, we can obtain the probability amplitude equation of the model in study. Neglecting the frequency detunings as compared to the corresponding linewidths and droping the derivatives of the probability amplitudes of state |3æ and state |4æ because they vary slowly as compared to their linewidths, the probability amplitude equation can be rewritten as ob1 ¼ iXp b3 ; ot ob2 ¼ iXs b4 c12 b2 ; ot
ð11Þ
c13 b3 ¼ iXp b1 þ iXc b4 ; c14 b4 ¼ iXs b2 þ iXc b3 : With the same approach mentioned by Harris [21], we can obtain the condition that Eqs. (7)–(9) are still valid for the pulse fields 1 oXp 1 : Xp ot Wc
ð12Þ
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That is to say, the maximum variation rate of Xp(t) and therefore the switch bandwidth are set by the (adjustable) golden rule transition rate Wc [22]. Now, we estimate the possibility for the photon switch studied here in virtual use. We let dephasing time cdeph ¼ cdeph ¼ cdeph ¼ cdeph ¼ cdeph ¼ 18 THz [23] and optical depth NLr13 = 2. Neogi pointed out that the 14 13 12 24 34 magnitude of lifetime of an excited state is about 1 ps [24], so we assume C4 ¼ 56 THz and C3 = 1 THz. We take C2 = 0 for the lifetime of level |2æ is 3 ns [25], which is very large compared to that of the excitation state. Fig. 2(a)–(c) show the power transmission exp(2al), phase shift bl and group velocity delay time TD as a function of the detuning dp. The results tell us that the design for optical switch in this asymmetric semiconductor quantum well structure is feasible. We should emphasize three points about the results shown in Fig. 2(a). One is that the power transmission at open state for probe laser is only 75.3% because of the larger dephasing rate cdeph as compared to 12 that in an intersubband transition of a quantum well or in an atomic system. We can get a better power transmission with a smaller cdeph 12 , which have been done by Grandjean [26]. The other point we should emphasize is that the power transmission at closed state for probe laser in Fig. 2(a) is about 33.5%. We may reduce the power transmission at a closed sate to 25% by a weak optical switch Ws = 0.8Wc (shown in Fig. 3). When the value of Ws is increased, we can get a lower power transmission up to 15% in a closed state by a strong optical switch Ws = 6Wc (shown in Fig. 3). Hereafter, the power transmission in a closed state varies slowly with Ws. Finally, we should accentuate the critical value Ws chosen for switch at a closed state. When Ws = 0.4Wc, the Rabi frequency of the switch field Xs 0.5Xc is in the order of
Fig. 2. Power transmission (a), phase shift (b) and group velocity delay time (c) as a function of the detuning dp. The switch is shown open (Ws = 0) and closed (critical Ws = 0.4Wc). The parameters are cdeph ¼ cdeph ¼ cdeph ¼ cdeph ¼ cdeph ¼ 18 THz; C4 ¼ 56 THz, 14 13 12 24 34 C3 = 1 THz, C1 = C2 = 0, Wc = 0.98 THz and NLr13 = 2.
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Fig. 3. Power transmission with different Ws when the switch is closed.The other parameters are similar with the values in Fig. 2.
7 · 1011 Hz. In this quantum well structure, a very large electric dipole moment length (108 m) [13] requires the intensity of the switch pulse for critical weak to be in the order of 104 W/cm2. In summary, we have described a scheme for a photon switch, in ultraviolet or visible spectra region, in an asymmetric semiconductor quantum well structure. It is much more feasible than in atomic system for its flexible design and the controllable interference strength. It is a method to realize one light controls another, in ultraviolet or visible region, in semiconductor.
Acknowledgement The authors thank the support from the National Natural Science Foundation of China under the number of 10334010.
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