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Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at SciVerse ScienceDirect

Optics Communications

0030-40http://d

n CorrE-m

Pleas

journal homepage: www.elsevier.com/locate/optcom

Controllable optical bistability via tunneling induced transparencyin quantum dot molecules

Zhiping Wang n, Shenglai Zhen, Xuqiang Wu, Jun Zhu, Zhigang Cao, Benli YuKey Laboratory of Opto-Electronic Information Acquisition and Manipulation of Ministry of Education, Anhui University, Hefei 230601, China

67

a r t i c l e i n f o

Article history:Received 9 January 2013Received in revised form10 April 2013Accepted 11 April 2013

Keywords:Optical bistabilityTunneling induced transparencyQuantum dot molecules

18/$ - see front matter & 2013 Published by Ex.doi.org/10.1016/j.optcom.2013.04.024

esponding author.ail address: wzping@mail.ustc.edu.cn (Z. Wan

e cite this article as: Z. Wang, et al.,

a b s t r a c t

We investigated the optical bistability of a quantum dot molecule under coherent excitation andconsidering the spontaneous exciton decay and pure dephasing as two decoherence channels. Because ofthe efficient destructive quantum interference established by the tunneling coupling, it is found that theappearance and disappearance of OB can easily be controlled by adjusting the position of the tunnelinginduced transparency window. Our scheme opens the possibility to control OB with electric gates.

& 2013 Published by Elsevier B.V.

686970717273747576777879808182838485868788899091929394959697

1. Introduction

Over the last few years, quantum optical phenomena based onquantum coherence have attracted attention of many researchersin coherent media [1–18]. One of the interesting phenomena, theoptical bistability (OB) in multilevel atoms confined in an opticalring cavity, has been the subject of many recent studies because ofits potential wide applications in all-optical switches, memories,transistors, and logic circuits [19]. Studies show that there aremany different mechanisms [20–32] to control OB such as thephase fluctuation, the squeezed state field, the spontaneouslygenerated coherence, and so on.

On the other hand, the OB behaviors in semiconductor quan-tum wells and quantum dots have also been extensively studiedrecently. For instance, Joshi and Xiao [33] reported the bistablebehavior in a three-level semiconductor quantum well that inter-acts with a strong driving electromagnetic field under two-photonresonant condition, and the results show that the threshold forswitching to upper branch of the bistable curve can be reduceddue to the presence of quantum interference. Li [34] analyzed theOB behavior in a four-subband quantum well system drivencoherently by the control and probe fields inside the unidirec-tional ring cavity. He found that the energy splitting of the twoupper excited states the coupling strength of the tunneling, theFano-type interference, the driving field intensity as well as thefrequency detuning can affect the OB behavior dramatically, whichcan be used to manipulate efficiently the bistable threshold

lsevier B.V.

g).

Optics Communications (20

intensity and the hysteresis loop. More recently, two schemes[35,36] for realizing the OB in quantum dots have also beenproposed.

In this work, we investigate the optical bistability in a quantumdot molecule system inside an optical ring cavity. It is found thatthe optical bistability can be easily controlled via adjustingproperly the corresponding parameters of the system. Our workis mainly based on the Refs. [20–36], however, our scheme is verydifferent from those works. First, we are interested in showing thecontrollability of the OB behavior via tunneling induced transpar-ency (TIT) in weak tunneling regime. A few works have discussedOB behaviors in quantum dots [35,36] focusing on the strongtunneling regime. Unlike those works, we will mainly discuss theOB behavior in weak tunneling regime. Second, the appearanceand disappearance of OB can easily be controlled by adjusting theposition of the TIT window. Third, we consider two decoherencemechanisms here: the spontaneous decay of excitonic states andthe pure dephasing. Our paper is organized as follows. In Section 2,we present the theoretical model and establish the correspondingequations. Our numerical results and physical analysis are shownin Section 3. In Section 4, the some simple conclusions are given.

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2. The model and dynamic equations

We consider a three-level quantum dot molecule system asshown in Fig. 1. Such a quantum dot molecule can be fabricatedusing self-assembled dot growth technology [16]. With appliedgate voltage, conductionband levels get into resonance, increasingtheir coupling, while valence-band levels become even more off-resonance, resulting in effective decoupling of those levels.

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EI ET

M1M2

M3M41R1R

L0sample

1 Te 12

0

1

10 20

10

22

1

Fig. 1. (a) Scheme of energy levels, decoherence channels, detunings, and relevantHamiltonian parameters. With applied gate voltage, conductionband levels get intoresonance, increasing their coupling, while valence-band levels become even moreoff-resonance, resulting in effective decoupling of those levels. A probe laser fieldexcites one electron from the valence band that can tunnel to the other dot. Weassume that the hole cannot tunnel in the time scale we are considering here [16].(b) A unidirectional ring cavity containing four mirrors and quantum dot moleculessample of length L. M3 and M4 are perfect mirrors, and the intensity reflection andtransmission coefficient of mirrors M1 and M2 are R and T (with R+T¼1),respectively. EI and ET are the incident and transmitted fields, respectively.

Z. Wang et al. / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎2

A probe laser field E (frequency ω) is introduced by the usualdipole interaction, which couples the ground state j0⟩ (the systemwithout excitations) with the exciton state j1⟩ (a pair of electronand hole bound in the first dot). The electron tunneling, on theother hand, couples the exciton j1⟩ with the indirect exciton statej2⟩ (one hole in the first dot with an electron in the second dot).Under the rotating-wave approximation, the Hamiltonian of thissystem is given as follow

H¼ ∑2

j ¼ 0ℏωjjj⟩⟨jj þ Tej1⟩⟨2j þ ℏΩ expðiωtÞj0⟩⟨1j þ H:c:Þ; ð1Þ

where H.c. means Hermitian conjugation, ℏωj is the energy of statejj⟩, Te is the tunneling strength, Ω¼ μ01E=2ℏ is the one-half Rabifrequency for the probe laser field, here μ01 is the dipole momen-tum matrix element and E is the electric field amplitude.

The system dynamics is described by Liouville-von Neumman-Lindblad equation [37]

∂ρ∂t

¼−iℏ½H; ρðtÞ� þ LðρÞ; ð2Þ

being ρðtÞ the density matrix operator and LðρÞ represents theLiouville operator that describes the decoherence process.

Assuming the Markovian approximation, the Liouville operatorthat describes both dissipative processes is given by

LðρÞ ¼ 12∑iΓjið2 j⟩⟨i ρ i⟩⟨j −ρ i⟩⟨i − i⟩⟨i ρÞ

����������������þγið2jj⟩⟨ijρji⟩⟨jj−ρji⟩⟨ij−ji⟩⟨ijρÞ; ð3Þ

where the first term describes the spontaneous decay processfrom the state ji⟩ to the state jj⟩ with rate Γj

i and the second term isthe pure dephasing with rate γi. Here, we consider two decoher-ence mechanisms: the spontaneous decay of excitonic states andthe pure dephasing. The main source of exciton pure dephasing inquantum dots is the electronacoustic phonon interaction [38]. Inquantum dot molecules, the effects of pure dephasing were

Please cite this article as: Z. Wang, et al., Optics Communications (20

experimentally explored by Borri et al. [39], who showed thatfor quantum dot molecules there is an enhancement of the puredephasing channels, in contrast to quantum dots, where at lowtemperatures no pure dephasing occurs.

Using the Eqs. (2) and (3), the density-matrix equations ofmotion in dipole and rotating-wave approximations for thissystem can be written as follows

_ρ11 ¼ i½Ωðρ10−ρ01Þ þ Teðρ12−ρ21Þ�−Γ10ρ11;

_ρ22 ¼ iTeðρ21−ρ12Þ−Γ20ρ22;

_ρ01 ¼ i½Ωðρ00−ρ11Þ þ Teρ02 þ δ1ρ01�−Γ10

2þ γ1

2

� �ρ01;

_ρ02 ¼ i12ðδ1 þ δ2Þρ02 þ Teρ01−Ωρ12

� �−

Γ20

2þ γ2

2

� �ρ02;

_ρ12 ¼ i12ðδ2−δ1Þρ12 þ Teðρ11−ρ22Þ−Ωρ02

� �−

Γ10 þ Γ2

0

2þ γ1 þ γ2

2

� �ρ12;

ð4Þconstrained by ρmn ¼ ρnnm (m,n¼0, 1, 2) and ρ00 þ ρ11 þ ρ22 ¼ 1.Here, the detunings are defined as δ1 ¼ ω10−ω and δ2 ¼ δ1 þ 2ω12

with ωij the transition frequency between ji⟩ and jj⟩ states.The behavior of the above-described quantum dot molecule

system will be investigated in the unidirectional cavity (see Fig. 1(b)). As we know, the probe field circulates in the ring cavity. So,the dynamics of the probe field in the optical cavity is governed byMaxwell's equation, under slowly varying envelope approxima-tion, which is given by

∂Ep∂t

þ c∂Ep∂z

¼ iωpΓopt

2ε0PðωpÞ; ð5Þ

where PðωpÞ is the slowly oscillating term of the induce polariza-tion in the transition j0⟩-j1⟩, which is given by PðωpÞ ¼Nμ01ρ10,and N is the average density of electrons. Γopt is the opticalconfinement factor. ε0 and c are the permittivity of free spaceand the light speed, respectively.

In the steady state, the term ∂Ep=∂t in the Eq. (5) is equal tozero. Then the Eq. (5) can easily be given as follows

∂Ep∂z

¼ iNωpμ01Γopt

2cε0ρ10: ð6Þ

For a perfectly tuned ring cavity, in the steady state limit, theboundary conditions impose the following conditions between theincident field EI and the transmitted field ET

EpðLÞ ¼ ET=ffiffiffiT

p; ð7Þ

Epð0Þ ¼ffiffiffiT

pEI þ REpðLÞ; ð8Þ

where L is the length of the quantum dot molecule sample, andthe second term on the right-hand side of Eq. (8) describes afeedback mechanism due to the mirror, which is essential to giverise to OB, that is to say, there is no OB when R¼0.

In the mean-field limit [40], normalizing the fields by lettingy¼ μEI=ℏ

ffiffiffiT

pand x¼ μET=2ℏ

ffiffiffiT

pand using the boundary condi-

tions, we can obtain the input-output relationship

y¼ 2x−iCρ10; ð9Þwhere C ¼ LNωpjμ01j2Γopt=2ℏcε0T is the usual cooperation para-meter. The second term on the right hand side of Eq. (9) is veryimportant to achieve the OB.

3. Numerical results

In the following, we set the time derivatives ∂ρij=∂t ¼ 0ði; j¼ 0;1;2Þ in the above density matrix Eq. (4) for the steadystate, and solve the corresponding density matrix equationtogether with the coupled field Eq. (9) via simple Matlab codes,

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Z. Wang et al. / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3

then we can arrive at the steady-state solutions. Now, we present afew numerical results for the steady state of the output fieldintensity versus the input field intensity under various parametricconditions in order to demonstrate controllability of the OB, asshown in Figs. 2–4.

In Fig. 2(a), we show the dependence of the optical bistabilitybehavior on the detuning δ1. As can be seen, on the condition ofδ1 ¼ −10 μeV, the OB does not occur (the solid line in Fig. 2(a)). Whenthe detuning is tuned to δ1 ¼ 0 μeV, the OB appears (the dotted linein Fig. 2(a)). The reason for the above results can be qualitativelyinterpreted as follows. When the tunneling strength Te and the decayrate of direct exciton Γ1 obey the condition Te=Γ1 ≤0:5, the tunnelingcoupling can establish an efficient destructive quantum interferencepath that creates a transparency window in the absorption spectra inthe limit of low optical excitation [18]. The detuning δ1 is animportant system parameter that can modify the position of theTIT window, which makes the appearance and disappearance of theOB. In order to gain deeper insight into the phenomenon, we plot theimaginary ½Imðρ10Þ� and real ½Reðρ10Þ� parts of ρ10 versus thefrequency detuning δ1 in the Fig. 2(b). We can see that the TITwindow localizes at δ1 ¼ −ω12 ¼ −10 μeV. Thus, OB does not takeplace in this case. However, if δ1 ¼ 0 μeV, the imaginary ½Imðρ10Þ� andreal ½Reðρ10Þ� parts of ρ10 are not again equal to zero at this point. So,the OB occurs. Therefore, by controlling the position of the TITwindow with the detuning δ1,we are able to control bistablebehavior efficiently.

We display the influence of the tunneling strength Te on theoptical bistability behavior in Fig. 3(a). Clearly, when the tunneling

0 20 40 60 80 100 120 1400

10

20

30

40

50

60

Out

put

field

|x|

Input field |y|

1=0

1=-10 eV

-50 -40 -30 -20 -10 0 10 20 30 40 50-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1( eV)

Re( )

Im( 10

10

)

Fig. 2. (a) Output field jxj versus input field jyj for different values of δ1, the otherparameters are C ¼ 100 eV, Γ1 ¼ 6:6 μeV, Γ2 ¼ 10−3Γ1, Te ¼ 3 μeV, and ω12 ¼ 10 μeV.(b) Imðρ10Þ and Reðρ10Þ as a function of laser detuning δ1, the other parameters arethe same as in panel (a) except for Ωp ¼ 2 μeV.

Please cite this article as: Z. Wang, et al., Optics Communications (20

strength is adjusted to Te ¼ 3 μeV, we can realize the OB (thedashed line in Fig. 3(a)). However, when the tunneling strength istuned to Te ¼ 30 μeV, the OB disappears (the solid line in Fig. 3(a)).The reason for this is that the variational tunneling strength candramatically modify the absorption and the Kerr nonlinearity ofthe medium, which leads to the appearance and disappearance ofthe OB. We also plot the imaginary ½Imðρ10Þ� and real ½Reðρ10Þ� partsof ρ10 as a function of the tunneling strength Te in Fig. 3(b). It canbe easily seen that imaginary ½Imðρ10Þ� and real ½Reðρ10Þ� parts arealmost equal to zero when Te ¼ 30 μeV. As a result, the abovedescription may provide a clue for optimizing the optical switch-ing process via adjusting properly the tunneling strength Te.

Finally, the effects of the effective dephasing parameter Γ1 andcooperation parameter C on the OB can be clearly seen from Fig. 4(a) and (b), respectively. By increasing Γ1 in Fig. 4(a) from 3 μeV to6 μeV, and then to 9 μeV, one can see that the bistability thresholdintensity goes up when the effective dephasing parameter Γ1

becomes large. For this scenario, the physical reason can bequalitatively explained as follows. By increasing the effectivedephasing parameter Γ1, the absorption for laser field on thetransition j0⟩↔j1⟩ will be enhanced, which makes the cavity fieldharder to reach saturation. The dependence of the OB behavior onthe cooperation parameter C is given in Fig. 4(b). From this figure,we can observe that the cooperation parameter leads to anincrease in the threshold of OB. This phenomenon can also bedue to the varying absorption of the medium. From the termC ¼ LNωpjμ01j2Γopt=2ℏcε0T , we can find that the cooperation para-meter is directly proportional to the average density of electrons,

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10

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40

50

60

Out

put f

ield

|x|

Input field |y|

Te=3 eV

Te=30 eV

0 5 10 15 20 25 30 35 40 45 50-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Te ( eV)

Im(10

)

Re( 10)

Fig. 3. (a) Output field jxj versus input field jyj for different values of Te, the otherparameters are C ¼ 100 μeV, Γ1 ¼ 6:6 μeV, Γ2 ¼ 10−3Γ1, δ1 ¼ 0 μeV, andω12 ¼ 10 μeV. (b) Im ðρ10Þ and Re ðρ10Þ as a function of the tunneling strength Te,the other parameters are the same as in panel (a) except for Ωp ¼ 2 μeV.

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100

0 20 40 60 80 100 120 1400

10

20

30

40

50

60O

utpu

t fie

ld |x

|

Input field |y|

1= 2 eV

1= 5 eV

1= 10 eV

0 20 40 60 80 100 120 1400

10

20

30

40

50

60

Out

put f

ield

|x|

Input field |y|

C= 80 eVC= 100 eVC= 120 eV

Fig. 4. (a) Output field jxj versus input field jyj for different values of Γ1, the otherparameters are C ¼ 50 μeV, Γ2 ¼ 10−3Γ1, Te ¼ 3 μeV, δ1 ¼ 0 μeV, and ω12 ¼ 10 μeV.(b) Output field jxj versus input field jyj for different values of C, the otherparameters are Γ1 ¼ 6:6 μeV, Γ2 ¼ 10−3Γ1, Te ¼ 3 μeV, δ1 ¼ 0 μeV, and ω12 ¼ 10 μeV.

Z. Wang et al. / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎4

and the increase of the number density of electrons will enhancethe absorption of the sample, which accounts for the raise of thebistable threshold. Thus, the threshold value and the hysteresiscycle width of the bistable curve can be controlled simply byadjusting the density of electrons.

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4. Conclusions

In summary, we have shown a scheme for realizing the opticalbistability in a quantum dot molecule system inside an optical ringcavity. A three-level physical model is presented to explain theobserved OB phenomena. By manipulating the absorption andnonlinear optical properties of this optical system via the quantuminterference created by an external electric field, the appearanceand disappearance of OB can easily be controlled. Quantum dotshave properties similar to atomic vapours but with the advantagesof high nonlinear optical coefficients, large electric-dipolemoments of intersubband transitions due to the small effectiveelectron mass as well as ease of integration, which are very useful

Please cite this article as: Z. Wang, et al., Optics Communications (20

in building all-optical switches and logic-gate devices for opticalcomputing and quantum information processing [27].

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no. 11205001), the National BasicResearch Program of China (Grant no. 2010CB234607), and thePostdoctoral Science Foundation of Anhui University.

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