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Phase control of optical steady-state behaviors from Fano-type interference in triple-semiconductor quantum wells Wen-Xing Yang a,b,∗ , Ai-Xi Chen c , Yanfeng Bai a , Ray-Kuang Lee b a b c

Department of Physics, Southeast University, Nanjing 210096, China Institute of Photonics Technologies, National Tsing-Hua University, Hsinchu 300, Taiwan Department of Applied Physics, School of Basic Science, East China Jiaotong University, Nanchang 330013, China

a r t i c l e

i n f o

Article history: Received 27 March 2014 Accepted 13 May 2015 Available online xxx Keywords: Optical steady-state behaviors Optical bistability and multistability Quantum interference Semiconductor quantum well

a b s t r a c t We analyze the optical steady-state behaviors based on intersubband transitions in a triple quantum well structure driven coherently by a probe laser ﬁeld and two control laser ﬁelds embedded in a unidirectional cavity. It is shown that the optical bistability behavior can be observed and controlled efﬁciently through the frequency detuning of the probe ﬁeld, the intensities of two control ﬁelds, the electronic cooperation parameter, and the Fano-type interference. More interestingly, with Fano-type interference, switch from optical bistability to optical multistability or vice versa can be also controlled by adjusting the relative phase between two coherent control ﬁelds. © 2015 Elsevier GmbH. All rights reserved.

1. Introduction In the last decade, coherent interaction between electromagnetic ﬁelds with intersubband transitions (ISBT) in semiconductor quantum wells (QW) intrigues many interesting quantum phenomena [1–7], such as lasing without inversion [8,9], slow light [10–13], enhancement of four-wave mixing [14,15], optical switches [16], controlling steady-state behaviors including optical bistability (OB) and optical multistability (OM) [17–20]. In particular, controllable OB and OM in QW have attracted an increasing number of interests for their potentially important applications in optoelectronics and solid-state quantum information science. Even though, population decay processes from longitudinal optical phonon emissions in intersubbands may destroy the coherent property; quantum interference between discrete states and continuum component in QW systems, the Fano interference, can lead to nonreciprocal absorptive and dispersive proﬁles [21,22]. Furthermore, the relative phase of applied laser ﬁelds has been widely used for the coherent control of ISBT in QW systems, coined as the phase control technology [23]. Phase control has already been applied for the coherent manipulation of population dynamics and absorption-dispersive properties in QW systems [24].

∗ Corresponding author at: Department of Physics, Southeast University, Nanjing 210096, China. Tel.: +86 15996365158. E-mail address: [email protected] (W.-X. Yang).

Motivated by the phase control and Fano-interference, in this paper, we investigate the optical steady-state behaviors in a triple QW structure, where two excited states are coupled by tunneling to the same electronic continuum. The proposed QW structure embedded in a unidirectional ring cavity is driven by a probe and two control laser ﬁelds. We reveal that the OB phenomenon can be observed and the OB behavior can be modiﬁed by the intensity of control ﬁelds, the frequency detuning of the probe ﬁeld, the electronic cooperation parameter, and the Fano-type interference. More interestingly, in presence of the Fano-type interference, one can easily realize the switch between OB and OM as well as a controllable OB just by adjusting the relative phase between two control ﬁelds. 2. Model and motion equations The proposed quantum well structure includes a deep well and two shallow wells coupled by tunneling to a common continuum of energies through a thin barrier [25]. This triple QW band structure and related level conﬁgurations are shown in Fig. 1, respectively. In this model, a deep 7.1-nm-think GaAs well is coupled to two shallow 6.8-nm-thick Al0.2 Ga0.8 As wells by a 2.5-nm-thick Al0.4 Ga0.6 As barrier. The two shallow wells are separated by a 2.0-nm-thick Al0.4 Ga0.8 As barrier. Both sides of quantum well contact with 36 nm Al0.4 Ga0.8 As. The electronic wave functions of the ground state of deep well and three excited states are denoted by |3, |2, |1, and |0, respectively. A probe ﬁeld with Rabi frequency p (amplitude Ep and angular frequency ωp ) is applied to the transition |3 ↔ |2;

http://dx.doi.org/10.1016/j.ijleo.2015.05.048 0030-4026/© 2015 Elsevier GmbH. All rights reserved.

Please cite this article in press as: W.-X. Yang, et al., Phase control of optical steady-state behaviors from Fano-type interference in triple-semiconductor quantum wells, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.048

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where the dots denote derivative with respect to t, together 3 with ij = ji∗ and the carrier conservation condition = 1. j=0 jj The population decay rates and dephasing rates are added phenomenologically in the above equations. The population decay rates for state |j, denoted by j , come primarily from the longitudinal optical phonon emission events at low temperature. The total dph dph decay rates ij are given by 12 = 1 + 12 , 02 = 0 + 02 , and dph

Fig. 1. Schematic diagram our proposed triple QW structure. It consists of a deep well and two shallow wells, separated by thin tunneling barriers. Related energy level and the corresponding wave functions of the ground state of deep well and three excited states are labeled as |3, |2, |0, and |1, respectively. p is the Rabi frequency of probe ﬁeld which is applied to the transition |3 ↔ |2, b and c are the Rabi frequencies of two control ﬁelds which mediate transitions |2 ↔ |1 and |2 ↔ |0, respectively.

while the transitions |2 ↔ |1 and |2 ↔ |0 are mediated by coherent control ﬁelds Eb (Rabi frequency b and angular frequency ωb ) and Ec (Rabi frequency c and angular frequency ωc ). Under the rotating-wave and electro-dipole approximation, in the interaction picture, the semiclassical Hamiltonian describing the system under study can be written as (taking =1 and the ground state |3 is the energy origin):

⎛ + ⎞ 0 −c 0 p c ⎜ 0 p + b −b 0 ⎟ ⎜ ⎟ ⎜ ⎟ ∗ ∗ ⎟ Hint = ⎜ − − − p p ⎟, c b ⎜ ⎜ ⎟ ⎝ 0 0 −∗p 0 ⎠

dph

23 = 3 + 23 , where ij , determined by carrier–carrier scattering, interface roughness, and phonon scattering processes, is the dephasing decay rates of quantum coherence of the |i ↔ |j transition. The population decay rates can be calculated by solving the effective mass Schrödinger equation. It is known that the initially nonthermal carrier distribution should be quickly broadened due to the inelastic carrier-carrier scattering [33]. For the temperatures up to 10 K, the carrier density smaller than 1012 cm−2 , dph the dephasing decay rates ij can be estimated according to Ref. √ [33]. = 12 02 /|12 02 | and the term /2 1 0 denotes a cross coupling term between the excited states |2 and |3, due to the Fano-type interference in the electronic continuum [1]. It is noted that represents the strength of Fano-type interference, where the values = 0 and = 1 correspond, respectively, to no interference and perfect interference. In order to study the related phase effect on the system, we use c and b to denote the relevant phases of two control ﬁelds, respectively. We express c = Gc ei c and b = Gb ei b , where Gc and Gb are the real parameters, other density elements are jj = jj (i = 0 −3), 02 = 02 e−i c , 12 = 12 e−i b , 13 = 13 e−i b , 03 = 03 e−i c and 10 = 10 ei( c − b ) . Then, Eq. (2) can be rewritten as follows: ˙ 00 = −0 00 + iGc 20 − iGc 02 − ˙ 11 = −1 11 + iGb (21 − 12 ) −

(1)

1 0 (e−i 10 + ei 01 ) 2

1 0 (e−i 10 + ei 01 ) 2

˙ 33 = 2 22 + iGp (23 − 32 ) ˙ 02 = −(02 + 23 + ic )02 + iGc (22 −00 )−iGb 01 −iGp 03 − ˙ 03 = −[12 + i(p + c )]03 + iGc 23 − iGp 02 −

where p = (ε2 − ε3 )/ − ωp , b = (ε1 − ε2 )/ − ωb and c = (ε0 − ε2 )/ − ωc are the detunings of probe ﬁeld and two coherent control ﬁelds, respectively, and εj (j = 0 −3) is the energy of state |j. p = 23 Ep /2, b = 12 Eb /2, and c = 02 Ec /2 are the one-half Rabi frequencies for the respective transitions. Relevant dipole moments are denoted by 23 , 12 , and 02 , respectively. Using the Weisskopf-Wigner theory, we can obtain the master equation for this triple QW system [26–32]:

1 0 e−i 13 2

˙ 10 = −[12 + 02 +i(b − c )]10 +iGb 20 − iGc 12 −

1 0 ei (11 + 00 ) 2

˙ 12 =−(12 +23 + ib )12 +iGb (22 −11 )−iGc 10 − iGp 13 − ˙ 13 = −[02 + i(b + p )]13 + iGb 23 − iGp 12 −

1 0 e−i 12 2

1 0 ei 02 2

1 0 ei 03 2

˙ 23 = −(23 + ip )23 + iGp (33 − 22 ) + iGb 13 + iGc 03

(3)

1 0 (10 + 01 ) 2 = −1 11 + ib 21 − i∗b 12 − 1 0 (10 + 01 ) 2

˙ 00 = −0 00 + ic 20 − i∗c 02 − ˙ 11

˙ 33 = 2 22 + i∗p 23 − ip 32 ˙ 02 = −(02 + 23 + ic )02 + ic (22 − 00 ) − ib 01 − i∗p 03 −

1 0 13 2 = −[12 + 02 + i(b − c )]10 + ib 20 − i∗c 12 − 1 0 (11 + 00 ) 2 = −(12 + 23 + ib )12 + ib (22 − 11 ) − ic 10 − i∗p 13 − 1 0 02 2 = −[02 + i(b + p )]13 + ib 23 − ip 12 − 1 0 03 2

˙ 03 = −[12 + i(p + c )]03 + ic 23 − ip 02 − ˙ 10 ˙ 12 ˙ 13

1 0 12 2

(2)

˙ 23 = −(23 + ip )23 + ip (33 − 22 ) + i∗b 13 + i∗c 03 Please cite this article in press as: W.-X. Yang, et al., Phase control of optical steady-state behaviors from Fano-type interference in triple-semiconductor quantum wells, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.048

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Fig. 2. Schematic setup of a unidirectional ring cavity containing the triple semiconductor quantum well (SQW) sample with the length L. EpI and EpT are the incident and transmitted ﬁelds, respectively. Ec and Eb represent the two control ﬁelds that are not circulating inside the cavity.

where = c − b is the relative phase difference of two control

3

= 1. ﬁelds. Similarly, Eq. (3) are restricted by ij = ij∗ and j=0 jj Now, we consider a medium of length L composed by the QW system described above, which is embedded in a unidirectional ring cavity [34–37], as shown in Fig. 2. For simplicity, we assume both the mirrors 3 and 4 have a 100% reﬂectivity, and the intensity reﬂection and transmission coefﬁcients for mirrors 1 and 2 are R and T (with R + T = 1), respectively. The total electromagnetic ﬁeld can be written as E = Ep e−iωp t + Ec e−iωc t + Eb e−iωb t + c . c ., where only the probe ﬁeld Ep circulates in the ring cavity. The propagation of the pulsed probe ﬁeld in such a medium is governed by Maxwell’s wave equation: 2

∇ 2 Ep −

2

1 ∂ Ep 1 ∂ P = , c 2 ∂t 2 ε0 c 2 ∂t 2

(4)

where P = N32 23 exp [− iωp t)] with N, c, and ε0 being the concentration, velocity of light in vacuum, and vacuum dielectric constant, respectively. Under the slowly varying envelope approximation [38], the Maxwell equation can be reduced to the ﬁrst-order equation. Thus we can obtain the slowly varying envelope equation for describing the probe ﬁeld evolution:

∂Ep ∂Ep ωp P(ωp ), +c =i 2ε0 ∂t ∂z

(5)

where P(ωp ) = N32 23 is the slowly oscillating term for the induce polarization in the transition |3 ↔ |2. The input probe ﬁeld EpI , entering the cavity under the assumption for a perfectly tuned ring cavity, satisﬁes the following boundary conditions together with the output ﬁeld EpT : Ep (L) =

EpT

, L √ Ep (0) = T EpI + REp (L),

(6) (7)

with Ep (0) and Ep (L) electric ﬁeld at the entry and the end of the sample. We should note that the second term on the right-hand side of Eq. (7) describes a feedback mechanism due to the mirror, which is essential to give rise to OB or OM. That is to say, there is no OB or OM when R = 0. Under the steady-state approximation, the time derivative in Eq. (5) ∂Ep /∂t can be neglected. Then the amplitude of the probe ﬁeld can be given as follows:

∂Ep ωp P(ωp ). =i 2cε0 ∂z

(8)

In the mean-ﬁeld limit [39], using the boundary conditions√in Eqs. (6) and (7) and normalizing the ﬁeld by letting x = 23 EpT /2 T √ and y = 23 EpI / T , we can get the input-output relation as follows y = x − iC23 ,

(9)

3

Fig. 3. Output intensity |x| versus input intensity |y| (a) for different values of the probe ﬁeld detuning p with b = c = 12 meV; (b) for different intensities of control ﬁelds c,b with p = 20 meV. Other parameter values are chosen as c = b = 0, = 0, dph

dph

dph

C = 300, = 0, 0 = 1.72 meV, 1 = 2.24 meV, 2 = 1.32 meV, and 21 = 32 = 02 = 1.65 meV, respectively.

where C = Nωp L232 /2cε0 T deﬁnes the electronic cooperation parameter. The second term on the right-hand side of Eq. (8) is very important for OB or OM to occur. The ﬁrst-order steady-state solution 23 can be found by the set of equations in Eq. (3) with the (0) (0) assumption of 00,11,22 = 0 and 33 = 1: id0 ei (d1 b − id0 e−i c )p − d1 (id0 ei b − d2 c )p

(1)

23 =

(d1 b − id0 e−i c )(id0 d3 ei − b c ) + (id0 ei b − d2 c )(d1 d3 − |c |2 )

,

(10) √ where d0 = 1 0 /2, d1 = i 12 − (c + p ), d2 = i 2 − (b + p ), and d3 = i 23 − p . From the above Eq. (10), we can ﬁnd the expressions 23 is a complicated fractional form of two polynomials. Thus, it is difﬁcult to get a straightforward relationship between the cavity input and output intensities. Therefore, we will solve the density matrix Eq. (3) together with the input–output Eq. (9) in the following section. 3. Numerical results and physical analysis In this section, we present typical numerical results for the steady state solution of output ﬁeld, in terms of ﬁeld intensities for different corresponding parameters. As mentioned in the above section, the QW sample for current study is based on the one used in Ref. [25] so that we can keep the same parametric conditions here. To give a clear illustration, we choose b = c = 0, c,b , p = 0.1 meV, 0 = 1.72 meV, 1 = 2.24 meV, 2 = 1.32 meV, and dph

dph

dph

21 = 32 = 02 = 1.65 meV. In the absence of the Fano-type interference = 0, we analyze how the frequency detuning of probe ﬁeld p and the intensities of two control ﬁelds modify the bistable behavior. In Fig. 3a we plot the input–output ﬁeld characteristics of the unidirectional cavity for several values of the detuning frequency p , which demonstrates the threshold intensity and the width of OB can be manipulated by the frequency detuning of probe ﬁeld. As shown in Fig. 3a, one can ﬁnd that the threshold intensity of OB increases and the area of hysteresis loop becomes narrower as the detuning p goes from p = 5 meV to 15 meV. Interestingly, it is shown that no OB occurs when the probe ﬁeld is on resonance (p = 0) with the corresponding transition |2 ↔ |3. In Fig. 3b, we analyze the effect of intensities of two control ﬁelds on the bistable behavior in this triple QW system. From Fig. 3b, one can ﬁnd that the width and related threshold of bistable region can be effectively controlled by the intensities of the control ﬁelds. Moreover, the threshold of OB decreases as the intensities of the control ﬁelds increases; meanwhile, the area of hysteresis loop become narrower. In order to ﬁnd the physical reason of these phenomena illustrated in Fig. 3, we plot in Fig. 4 the dependence of the probe

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6 C=50 C=100 C=200 C=300

Output field |x|

5 4 3 2 1 0

0

1

2

3

4

5

6

Input field |y| Fig. 4. The dependence of probe absorption on the frequency detuning p and the intensities of control ﬁelds c,b . Other parameter values are chosen as p = 0.1 meV, c = b = 0, = 0, C = 300, = 0, 0 = 1.72 meV, 1 = 2.24 meV, 2 = 1.32 meV, dph dph dph and 21 = 32 = 02 = 1.65 meV, respectively.

absorption on the frequency detuning p and the intensities of the control ﬁelds c,b . As shown in Fig. 4, with a gradual increment in the frequency detuning, that is p from 5 to 15 meV, the absorption for probe ﬁeld increases dramatically, which makes the cavity ﬁeld difﬁcult to reach saturation. Accordingly, the threshold intensity of OB increases and the area of hysteresis loop becomes narrower. From Fig. 4 we can also ﬁnd that there is transparency window around the probe resonant position p = 0. In other words, the probe ﬁeld can propagate through the medium without absorption. Thus the OB behavior does not appear as illustrated in Fig. 3a. As a result, we can have a desired bistable curve and even switch the OB behavior for disappearance or appearance through a adjusting the frequency detuning of probe ﬁeld. With the unique tunability in our proposed QW structure, one can switch on and off for OB by varying the corresponding resonance conditions of input probe ﬁeld by a given time-dependent function for detuning frequencies [11]. In addition, with a certain frequency detuning of the probe ﬁeld, Fig. 4 also demonstrates that that the absorption of probe ﬁeld on the transition |3 ↔ |2 and the Kerr nonlinearity of QW medium modiﬁed when one increases the control ﬁeld between the transitions |2↔1 or |2 ↔ |0. As we can see in Fig. 3b, a smaller threshold intensity comes from the reduction in the effective saturation intensity due to the increasing of the intensity of the control ﬁelds. It is worth noting that the present triple QW system has been studied in several previous works [18,25]. The electron sheet density takes values between 109 and 1011 . These values ensure that the system is initially in the lowest subband. The cooperation parameter C in Eq. (9) is linearly proportional to the electron sheet density N. One can easily ﬁnd from Eq. (9) that the cooperation parameter will affect the input–output curves. Thus we should consider the effect of the cooperation parameter C on the OB behavior. In Fig. 5, we analyze the effect of the electronic cooperation parameter on the bistable behavior in this triple QW system. The results of Fig. 5 show that the OB threshold increases when the electronic cooperation parameter C becomes larger. Since the increment in cooperation parameter enhances the absorption of QW sample, which makes the cavity ﬁeld harder to reach saturation. Then, the required OB threshold goes higher [6,41,33]. Furthermore, we can optimize the optical switching process for controlling the OB behavior by choosing the electron sheet density of the corresponding system properly. As shown in the above results, we have demonstrated the realization of OB in the proposed triple QW system and discussed the inﬂuences of some system parameters without including the Fano-type interference effect, i.e., = 0. When the Fano-type

Fig. 5. Output intensity |x| versus input intensity |y| for different values of electronic cooperation parameter C. Other parameters values are chosen as b,c = 12 meV, dph

c = b = 0, p = 5 meV 0 = 1.72 meV, 1 = 2.24 meV, 2 = 1.32 meV, and 21 = dph 32

=

dph 02

= 1.65 meV, respectively.

Fig. 6. (a) Output intensity |x| versus input intensity |y| for different values of the Fano-type interference and the relative phase with p = 0. (b) Output intensity |x| versus input intensity |y| for different relative phase with p = 30 meV and = 0.9. Other parameters values are chosen as b,c = 20 meV, c = b = 0, 0 = 1.72 meV, dph

dph

dph

1 = 2.24 meV, 2 = 1.32 meV, and 21 = 32 = 02 = 1.65 meV, respectively.

interference is included, from Eq. (10), one can ﬁnd that the Fanotype interference makes probe absorption quite sensitive to the relative phase between two control ﬁelds. These -dependent terms are always accompanied with a phase dependent term exp(± i ). In Fig. 6, we present numerical results for analyzing the Fano-type interference and the phase on the steady state of output ﬁeld amplitude |x| as a function of input ﬁeld amplitude |y| for different cases. From Fig. 6a, one can ﬁnd that OB output can be achieved in presence of the Fano-type interference (i.e., = 0.9) even if the two-photon resonance condition is satisﬁed p = c = b = 0. However, the OB phenomenon disappears again when the relative phase is tuned from = 0 to = . When the frequency detuning is changed from p = 0 to p = 30 meV, c = b = 0, = 0.9, and c,b = 20 meV are ﬁxed, it is found from Fig. 6b that the relative phase plays an important role on the steady behavior. Compared with the curve ( = 0) in Fig. 6a and b illustrates that the OB phenomenon can be still observed and the corresponding threshold increases slightly. When the relative phase is changed from 0 to /2, the threshold of OB becomes more pronounced. More interestingly, OM phenomenon appears when the relative phase is changed from /2 to . In other words, we can easily realize a switch from OB to OM or vice versa just by adjusting the relative phase between two control ﬁelds. These phenomena also demonstrate a phase-sensitive property in the proposed triple QW system. Speciﬁcally, the relative phase between two control ﬁelds can effectively control the optical response in a triple QW system, which leads to the appearance of OM. For a better insight into the effects of phase and frequency detuning on global behavior of input–output curves, we plot in Fig. 7 the contour map of the probe absorption as the function of

Please cite this article in press as: W.-X. Yang, et al., Phase control of optical steady-state behaviors from Fano-type interference in triple-semiconductor quantum wells, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.048

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References

Fig. 7. The dependence of probe absorption on the frequency detuning p and the relative phase . Other parameter values are chosen as p = 0.1 meV, b,c = 20 meV, dph

c = b = 0, C = 300, = 0.9, 0 = 1.72 meV, 1 = 2.24 meV, 2 = 1.32 meV, and 21 = dph 32

=

dph 02

= 1.65 meV, respectively.

both the phase and detuning . The reason of these interesting phenomena shown in Fig. 6 can be found from the changes of absorption properties illustrated in Fig. 7. In the case of = 0, the probe absorption is closed to zero at the probing resonance position (p = 0) when the Fano-type interference is neglected ( = 0). Therefore, the input ﬁeld EpI can be approximately proportional to the output ﬁeld EpT , resulting in the case without appearance of OB phenomenon. When the Fano-type interference is strong enough ( = 0.9), the large absorption of the medium leads to the appearance of the OB phenomenon with = 0. However, even with a strong Fano-type interference, the probe absorption is shown to be closed to zero when the relative phase between the two control ﬁelds is tuned to . Thus the OB phenomenon disappears again as illustrated in Fig. 6a. When the pulsed probe ﬁeld is far from the resonance i.e., p = 30 meV, from Fig. 7, one can also ﬁnd that the absorption of the medium increases obviously. Especially, the corresponding absorption enhances dramatically, which leads to the parameter y in Eq. (9) is not a cubic polynomial of the variable x, and thus the OM phenomenon can be observed. 4. Conclusion In this work, we proposed a model system to investigate optical bistability (OB) and optical multistability (OM) behaviors in a four-subband triple QW system inside a unidirectional ring cavity. Driven coherently by two control and one probe ﬁelds, we demonstrate numerically that a controllable OB can be achieved by adjusting different parameters in the QW system. Under the steadystate condition, the threshold intensity and related hysteresis cycle area for the OB can also be modiﬁed by modulating the frequency detuning of probe ﬁeld, electronic cooperation parameter, intensity of driving ﬁelds, Fano-type interference strength, as well as the relative phase between two control ﬁelds. More interestingly, we demonstrate the switch from OB to OM with the existence of an optimal Fano-type interference. With these results, we believe that phase controlled OB and OM provide another feasible approach for applications in nanoelectronics and quantum information science. Acknowledgments The research is supported in part by National Natural Science Foundation of China under Grant Nos. 11374050 and 61372102, by Qing Lan project of Jiangsu, and by the Fundamental Research Funds for the Central Universities under Grant No. 2242012R30011.

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