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Electrical and optical control of optical gain in a coupled triple quantum dot system operating

in telecommunication window

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2014 Laser Phys. 24 125201

(http://iopscience.iop.org/1555-6611/24/12/125201)

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1. Introduction

The study of optical properties of nanostructures, especially quantum dots (QDs), has an important role in optoelectronic devices. The optical properties of QDs and quantum wells can be substantially modified by the external field and inter-dot tunnel coupling [1–9]. Quantum dot molecules (QDMs) have found special attention because of their similar properties to atomic vapors, but with the advantage of flexible design and controllable interference strength. In addition, long dephas-ing times [10, 11], atomic-like properties (e.g. discrete energy levels) at high temperatures and large bandwidth due to fast carrier dynamics have given a potential application of such molecules. Quantum coherence in a QD structure can be induced by applying the laser field or by electron tunneling. It is important because controlling light by light or by electron tunneling are essential for the next generation of all optical communication and optoelectronic devices. An important advantage of a QD lies in the fact that carriers are confined in all directions, while in a quantum well carriers are only confined in one direction. Confinement of carriers reduces

the density of states (DOS). Reducing the DOS makes the QDs a more effective laser by offering lucrative potential benefits such as a lower and temperature-insensitive thresh-old current compared with a quantum well. Also, because of the 3D confinement of carriers in QDs, for incident light with any direction carriers exhibit quantum optical behavior. These properties of QDs open up a new route for designing ultra-narrowband switches and filters by applying quantum optical ideas for a new class of optical integrated circuits. The most important idea in lasing without inversion (LWI) of QDs is the absorption cancellation by tunneling coher-ence. Also study of light amplification, gain without inver-sion (GWI) or LWI has been considerable [12–18]. Many schemes for LWI have been proposed and the dependence of optical gain on various system parameters has been examined [19–22]. Experimental observations of inversionless gain and lasing have been reported by several groups [23–27]. Lasing without population inversion may be useful in achieving laser actions in the spectral regions where lasing with population inversion is impractical with conventional pumping schemes. Among the proposed schemes of lasing without population

Laser Physics

Electrical and optical control of optical gain in a coupled triple quantum dot system operating in telecommunication window

Mohammad Reza Mehmannavaz1 and Hamed Sattari2

1 Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran2 Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz 51665–163, Iran

E-mail: b.mehmannavaz@gmail.com

Received 13 July 2014Accepted for publication 1 September 2014Published 7 October 2014

AbstractWe investigate the light amplification and gain without inversion (GWI) in triple quantum dot molecules in both steady-state and transient state. We demonstrate that the light amplification and GWI of a light pulse can be controlled through the rates of the incoherent pumping and tunneling between electronic levels. The required switching times for switching of a light pulse from absorption to gain and vice versa is then discussed. We obtain switching time at about 40 ps, which resembles a high-speed optical switch in nanostructure. The proposed approach in QDMs may provide some new possibilities for technological applications in optoelectronics and solid-state quantum information science.

Keywords: amplification of light, GWI, optical switching, absorption-gain, quantum dot molecules

(Some figures may appear in colour only in the online journal)

M R Mehmannavaz and H Sattari

Printed in the UK

125201

lP

© 2014 Astro ltd

2014

24

laser Phys.

lP

1054-660X

10.1088/1054-660X/24/12/125201

Papers

12

laser Physics

Astro Ltd

KB

1054-660X/14/125201+8$33.00

doi:10.1088/1054-660X/24/12/125201Laser Phys. 24 (2014) 125201 (8pp)

M R Mehmannavaz and H Sattari

2

inversion, most are based on the utilization of external coher-ent fields that induce atomic coherence and interference lead-ing to optical gain in the absence of population inversion. The implementation of light amplification, LWI and tunneling induced transparency (TIT) [28, 29] in semiconductor-based devices is very appealing from a more applied point of view.

In this paper, we investigate light amplification in a QDM composed of three QDs. We find that the absorption spectra of the light pulse can be changed to gain via the effect of inco-herent pumping field and inter-dot tunnel couplings of QDs. Also, we investigate the dynamical response of the system to inspect the required switching time when the absorption of the light pulse changes from positive to negative and vice versa. This is important due to its novel application in fast optical switching, which is an important technique in quantum com-puting and quantum information [30, 31].

The paper is organized as follows: in section 2, we present the model and obtain the density matrix equations of motion for the system. In section 3, we discuss the effects of inco-herent pumping field and inter-dot tunnel couplings of QDs on probe absorption and dispersion. We finally conclude in section 4.

2. Model and equations of motion

InAs is chosen as a material for quantum dot structure and imbedded into a GaAs/Al0.45GA0.55As because of its lattice mismatch with GaAs/Al0.45GA0.55As layer. The InAs quan-tum dots are grown by molecular epitaxy on [0 0 1] GaAs substrate, that have a lens-shaped geometry with a low aspect ratio of ~ 0.1 (height over diameter) as seen in figure 1(a). Typical heights span the 1.5–2.5 nanometer range. The dot core is constituted of pure InAs with a sharp interface between InAs and the GaAs barrier with a roughness below 0.5 nm. In the calculation, the InAs volume is thus modeled as fol-lows: a portion of a sphere, 2.5 nm high and 25 nm in diameter, lying on the bottom of a 0.5 nm thick wetting layer. Also, we have for center dot (QD1), 3.5 nm high and 35 nm in diam-eter. From structural measurements, the dot density is around 4  ×  1010 cm−2. The nanostructure system under consideration is illustrated schematically in figure  1. Panel (a) shows the cross section of the planned triple QDM sample structure, and panel (b) the detailed band structure and energy level of the QDM, which consists of three dots that are coupled by the electron tunneling and four atomic levels in QDM as depicted in figure 1. Two levels 0 and 1 are the lower and upper con-ducting band levels of the left QD, respectively. Levels 2 and 3 are the excited conducting level of the second and tertiary QDs, respectively. It is assumed that the energy difference of the three excited levels and the lower level is large, so their tunneling couplings can be ignored. By applying a gate volt-age, levels 2 and 3 get closer to level 1 . An incoherent pumping field and a weak tunable probe field of the frequency

ω with Rabi frequency Ω = ℘ ℏ⃗ ⃗E . ∕ 2 are applied to the transition →0 1 . Here, E and ℘ ⃗are amplitude and dipole moment of the probe field, respectively. Under the resonant coupling of a probe field with the QD1, an electron is excited

from the 0 band to the 1 band of the QD1. This electron can be transferred via tunneling to QD2 and QD3 (the levels 2 and 3 ). The tunnel barrier, in a triple QD molecule, can be controlled through placing a gate electrode between the neigh-boring QDs. We take a sheet density of 3.7  ×  1011cm−2 for QDs. For a practical case, proposed QDMs may be realized on a platform of the self-assembled QD system [32–36].

We write the Hamiltonian for the system shown in figure 1: with neglect from the hole tunneling. We follow the method developed in [37, 38] in the rotating wave approximation (RWA). So, the total Hamiltonian describing the interaction of the probe and incoherent pumping fields with the triple QD system can be expressed in the form of

∑ Ω ε= + +

+ + +

ω

=

−⎡⎣

⎤⎦

(

)

H E j j e P

T T H C

0 1 0 1

1 2 2 3 . .

jj p

i t

A B

0

3

1p

(1)

where ω= ℏEj j denotes the energy of state i . P1 is the dipole moment of the atomic transition corresponding to the pump-ing from 0 to 1 and the electric field ε implies the elec-trical amplitude of the incoherent pumping field. The dipole moment P1 corresponds to the transitions from 0 to 1 and in principle has different directions. However, the electric field can be chosen in a polarization mode such that its coupling to the transition is a maximum or minimum. TA and TB are the non-zero electron tunneling matrix elements and correspond to inter-dot tunneling between ‘QD1’ and ‘QD2’ and between ‘QD2’ and ‘QD3’, respectively. The electron tunneling in a barrier can be described by perturbation theory which can be given by Bardeen’s approach [39]. According to Bardeen’s approach, the probability of tunneling an electron in state Ψ

Figure 1. Cross section of InAs triple coupled QDs imbedded into a GaAs sample structure (a), schematic of triple coupled QDs (QD1, QD2, and QD3), which shows the detailed band structure, quantized energy level and coupling scheme for the three tunnel-coupled QDs (b).

(a)

(b)

Laser Phys. 24 (2014) 125201

M R Mehmannavaz and H Sattari

3

with energy EΨ from the first QD to state Φ with energy EΦ in the second QD is given by Fermi’s golden rule [40]:

π δ=

ℏ−Ψ ΦW T E E

2( ) .e

2 (2)

The tunneling matrix elements can be obtained by an integral over a surface in the barrier region lying between the QDs:

∫ Φ Ψ Ψ Φ= ℏ ∂∂

− ∂∂=

⎛⎝⎜

⎞⎠⎟T

m z zS

2*

*d ,e

z z0

(3)

where m is the effective mass of the electron and z0 lies in the barrier. We note also that TA,B are related to the applied bias of the molecule. Applying a bias voltage V, the current is

∫π ρ ε ρ ε ε=ℏ

− + +Ie

E eV E T4

( ) ( ) d ,eV

F F e0

1 22 (4)

this means that the current is proportional to the local den-sity of states of each QD ρ ρ( , )0 1 at the Fermi energy (EF). Therefore, it is possible to tune the magnitude of coupling between two QDs, by modifying the bias applied to the mol-ecule. For Te = TA,B ≠  0, some interaction terms should be applied in total Hamiltonian as described in equation (1).

The density matrix approach used for obtaining the den-sity operator in an arbitrary multilevel QD system can be written as

ρ ρ∂

∂= −

ℏt

iH[ , ]. (5)

Substituting equation (1) in equation (5), the density matrix equations of motion can be obtained as

ρ γ ρ γ ρ γ ρ Ω ρ Ω ρ ρ

ρ γ ρ Ω ρ Ω ρ ρ ρ ρ

ρ γ ρ ρ ρ ρ ρ

ρ γ ρ ρ ρ

ρ δ Γ ρ Ω ρ ρ ρ

ρ δ ω Γ ρ Ω ρ ρ ρ

ρ δ ω ω Γ ρ Ω ρ ρ

ρ ω Γ ρ ρ ρ Ω ρ ρρ ω ω Γ ρ ρ ρ Ω ρρ ω Γ ρ ρ ρ ρρ ρ ρ ρ

= + + + + − −

= − + − − + − +

= − + − − −

= − + −

= − − − − −

= + − − + − +

= + + − − + −= − + + − − − += − + + + − − −= − + − − −+ + + =

( )( )

( )( )

( )( )( )

( )( )

( )( )

( )

( )

( )( )

R i R

R i i T T R

i T T i T T

i T T

i R i iT

i R i i T T

i R i iT

i R iT i iT

i R i T T i

i iT iT

˙ * ,

˙ * * ,

˙ * * ,

˙ * ,

˙ ( ) ,

˙ / 2 * ,

˙ / 2 * ,

˙ ( / 2) ( ) * ,

˙ ( ( ) / 2) ,

˙ ( ) ( ) * ,

1,

p p

p p A A

A A B B

B B

p p A

p p A B

p p B

A p B

A B p

B A

00 10 11 20 22 30 33 10 01 00

11 10 11 10 01 21 12 00

22 20 22 12 21 23 32

33 30 33 23 32

01 10 01 00 11 02

02 12 20 02 12 01 03

03 12 23 30 03 13 02

21 12 21 21 22 11 20 31

31 12 23 31 31 32 21 30

32 23 32 32 33 22 31

00 11 22 33 (6)

where ρ = =m n m n( , 0, 1, 2, 3)mn for ρmn = ∣m⟩ ⟨m∣ (m = 0, 1, 2, 3) represent the population operators for the QDs and for m  ≠ n, the electron and exciton transition operators between levels m and n are in total form. We get ω12 = ω10 − ω20 and ω23 = ω20 − ω30. The laser field detuning with respect to the QD transition frequencies is δ ω ω= − ,p p10 and appropri-ate wavelengths with that are ω π λ ω π λ= =c c2 / , 2 / .10 0 We take λ = 1550 μm in our investigation because that is the most important wavelength for communication applications. The term Γ= ℏ( )( )R P2 / P1

2 2 is the incoherent pumping rate. Note that the incoherent pumping process also can take place via some unspecified auxiliary levels. So, we assume that

the electric field has a broad frequency spectrum or effec-tively δ-like correlation, i.e. ε ε Γ δ′ = − ′t t t t* ( ) ( ) ( ) .P The effect can be summarized through the pumping param-eter Γ= ℏ( )R P2 / P1

2 2 [32]. TA and TB are the first and second coherent tunneling rates (tunneling matrix elements) between neighboring dots. These tunneling rates can be modulated by the gate voltage [41]. The spontaneous emission rates and the dephasing rates of the QDs are added phenomenologically in the above density matrix equations  (6). The spontaneous emission rates for sub-band i , denoted by γ10, are due pri-marily to longitudinal optical (LO) phonon emission events at low temperature. The total decay rates Γ ≠i j( )ij are given

by Γ γ γ Γ γ γ γ= + = + +/ 2 , ( ) / 2 ,n n ndph

mn n m mndph

0 0 0 0 0 m, n = 1, 2, 3 and m  ≠  n, here γ ,mn

dph determined by electron–electron, interface roughness and phonon scattering processes, are the dephasing rates of the quantum coherence of the ↔n m pathway. Usually, γmn

dph is the dominant mechanism in a semi-conductor solid-state system. Equation  (6) can be solved to obtain the steady-state response of the medium. The main observable is the susceptibility of the QD molecule to the probe field. The susceptibility of the system for a weak probe field is determined by coherence term ρ01

χε

ρ= ℘N

E

2,

001 (7)

where N is the QD number density in the medium. Note that the real and imaginary parts of χ χ χ χ= ′ + ′′i( ) correspond to the dispersion and the absorption, respectively.

3. Results and discussion

In this section by using the numerical result from the density matrix equation of motions ρij, we investigate the absorption-gain and transient evolution for light amplification and estimate of switching time in a coupled triple QD system for response from different respects on the incoherent pumping and inter-dot tunneling rates. As is well known, gain-absorption of the probe field on transition →0 1 are proportional to the imag-inary part of ρ01which can be obtained from equation  (6). If

ρ >Im ( ) 001 the system exhibits absorption for the probe field, while for ρ <Im ( ) 0,01 the probe laser will be amplified. We now present the numerical results of equation (6) through figures 2–6. It is assumed that the system is initially in the ground state, i.e. ρ00 (0) =1 and ρij (0) =0, (i, j = 0, 1, 2, 3). We take typically slow dephasing rates γ10 = γ = 1 GHz and Γ10 = 1.6 GHz and introduce the other dephasing rates by the factor of these rates, Γ10 = 1.6 GHz, Γ20 = 0.1 Γ10, Γ30 = 0.01 Γ10, Γ21 = 0.05 Γ10, Γ31 = 0.025 Γ10, Γ32 = 0.05 Γ10, Ω = 0.11 Γ10, γ20 = 0.6 γ10, γ30 = 0.01 γ10. Introduced rates are equivalent to dephasing times in the order of several nanoseconds. Figure  2 shows absorp-tion of linear susceptibility for values of incoherent pumping and inter-dot tunneling rates versus wavelength. In figure 2(a), we show the absorption of the probe field in the absence of tunneling effect and incoherent pumping field. We observe a large absorption peak at the medium. We are interested in the effect of inter-dot tunneling TA and TB and incoherent pump-ing field in the absorption of pulse propagation. In this case

Laser Phys. 24 (2014) 125201

M R Mehmannavaz and H Sattari

4

the coherence in the system is created by the coupling of two excited states via tunneling [42]. Note that in an atomic sys-tem the coherence can be created by the coupling laser field; here such coherence is created by tunneling effects. In fig-ures 2(b, c), we show the tunneling effects on the absorption for

γ γ= =T T2 , 0A B and γ γ= =T T2 , 1.7A B respectively, in the absence of incoherent pumping field. In the first case (figure 2(b)), the probe absorption at λ = 1.550 μm is negligible, so TIT is established through the inter-dot tunnel coupling of QD1 and QD2 instead of the conventional coupling field induced coher-ence. An interesting frequency region for light propagation is a region in which the system does not show absorption. This is due to the fact that the large absorption in the system does

not permit the pulse to propagate inside the medium. We have a TIT at telecommunication wavelength namely λ = 1.550 μm. In the case of figure 2(c), by establishing a second tunneling TB the close to zero probe field absorption converts to an absorp-tion peak and two points close to zero in λ = 1.552 μm and λ = 1.548 μm are created. In figures 2(d), (e) we investigate the impact of incoherent pumping on the absorption of the pro-posed QDM in the presence of tunneling effect. We have plot-ted the probe absorption in the presence of both the incoherent pumping field and tunneling for γ γ γ= = =T T r2 , 0 , 4.3A B (d) and γ γ γ= = =T T r2 , 1.7 , 0.75A B (e). In comparison with figure 2(b), we have a nonzero incoherent pumping rate in figure  2(d). By comparing figure  2(b) with figure  2(d), it

Figure 2. Absorption spectrum of susceptibility as a function of normalized probe wavelength detuning for (a) γ γ γ= = =T T r0 , 0 , 0 ,A B (b) γ γ γ= = =T T r2 , 0 , 0 ,A B (c) γ γ γ= = =T T r2 , 1.7 , 0 ,A B (d) γ γ γ= = =T T r2 , 0 , 4.3 ,A B and (e) γ γ γ= = =T T r2 , 1.7 , 0.75 .A B Other parameters are Γ Γ Γ Γ Γ Γ Γ Γ Γ Ω Γ= = = = = =THz1.6 , 0.1 , 0.01 , 0.05 , 0.025 , 0.11 ,10 20 10 30 10 21 10 31 10 10 Γ32 = 0.05Γ10, γ10 = γ = 1 THz, γ20 = 0.6γ10, γ30 = 0.01γ10, ω12 = ω23 = 0.

1.54 1.542 1.544 1.546 1.548 1.55 1.552 1.554 1.556 1.558 1.56x 10−6

0

0.02

0.04

0.06

0.08

0.1

0.12

Wavelength

Im[ ρ

01]

1.54 1.542 1.544 1.546 1.548 1.55 1.552 1.554 1.556 1.558 1.56

x 10−6

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Wavelength

Im[ ρ

01]

1.54 1.542 1.544 1.546 1.548 1.55 1.552 1.554 1.556 1.558 1.56x 10−6

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Wavelength

Im[ ρ

01]

1.54 1.542 1.544 1.546 1.548 1.55 1.552 1.554 1.556 1.558 1.56

x 10−6

1

1.5

2

2.5

3

3.5

4x 10−3

Wavelength

Im[ ρ

01]

1.54 1.542 1.544 1.546 1.548 1.55 1.552 1.554 1.556 1.558 1.56x 10−6

0

0.005

0.01

0.015

0.02

0.025

0.03

Wavelength

Im[ ρ

01]

(a) (b)

(c) (d)

(e)

Laser Phys. 24 (2014) 125201

M R Mehmannavaz and H Sattari

5

appears that, despite two absorption peaks for the case of zero pumping rate convert to one absorption peak in figure  2(d) (with nonzero pumping rate), the magnitudes of the absorption through the spectrum is lowered about 10 times). In figure 2(e), we show reduced coherence by the incoherent pumping field for figure 2(c), with small absorption. In figures 3(a), (b), we show convert absorption to gain with an appropriate choice of the intensity of tunneling and incoherent pumping field. In figure 3(a) we show the absorption-gain spectrum for dif-ferent values of second tunneling, TB = 0γ (solid), TB = 0.45 γ (dashed) TB = 1.2 γ (dash-dot). It is realized that the probe absorption is changed to the probe amplification by increas-ing the second tunneling rate. Note that a negative absorption expresses the amplification of a propagating wave, while a positive value corresponds to the attenuation of the probe laser. In figure 3(b) we show the absorption-gain spectrum for dif-ferent values of incoherent pumping rate, r = 0 γ (solid), r = 0.75 γ (dashed), r = 2.5 γ (dash-dot). It is realized that the probe absorption is changed to the probe amplification by increas-ing the incoherent pumping rate. Physically, the incoherent pump field pumps the population from the lower level to the upper level. Strong incoherent pumping rates bring more of the population from the lower level to the upper level leading to absorption reduction and even gain. Another key point is that second tunneling rate and incoherent pumping process can be implemented for amplification of propagating light through the novel semiconductor systems. These results show amplification at a wavelength λ = 1.550 μeV, which is a very practical wave-length and might be interesting for laser applications. However, there is a net gain in the medium and light propagation with less attenuation is desirable in optical devices.

Figure 4 illustrates the time evolution of the population distribution for levels 0 , 1 , 2 and 3 that correspond to the ρ00 ρ11 ρ22 and ρ33 respectively. In figure  4 (column a) we investigate the population distribution for two different values of second tunneling rate TB = 0 γ (solid), TB = 1.2 γ (dashed). With increasing second tunneling rate, population levels ρ00 ρ11 and ρ22 begin to decrease and population level ρ33 begins to increase. In this case, the population inversion does not occur and according to figure 3(a) in this situation gain is generated. So, we have GWI by the second tunneling effect. In figure 4 (column b) we investigate the population distribution for two different values of the incoherent pump-ing field r = 0 γ (solid), r = 2.5 γ (dashed). With increasing incoherent pumping field the population inversion does not occur and according to figure 3 (b) in this situation gain is generated. So, we have GWI by the incoherent pumping field. Also, we have GWI at a wavelength λ = 1.550 μeV that is a very practical wavelength. These results are useful for laser applications.

Figure 5 shows the time evolution of the gain-absorption coef-ficient ρ01 for two different values of second tunneling rate and incoherent pumping field rate. In figure 5 (a), we investigate the time evolution of the gain-absorption coefficient ρ01 for two values of second tunneling rate TB = 0 γ (solid) and γ=T 1.2 (dashed) .B With second tunneling, levels 2 and 3 are coupled, and thus a level is added to the excited levels. So this makes the electrons spend more time in the excited levels, resulting in decreased

absorption of the system. With increasing intensity of second tunneling rate the absorption is reduced and then converted to gain, as shown in figure 5(a). For γ=T 0 (solid) ,B the absorption exhibits an oscillatory behavior in a short time and finally reaches a positive steady-state value. When we have a nonzero tunnel-ling rate, namely γ=T 1.2 (dashed) ,B after some initial evolution the transient absorption gradually reaches a stable negative value that resembles the gain in the system. Thus, a non-periodic gain-absorption can be converted to a periodic one, just by increas-ing the intensity of second tunneling rate TB. In figure 5 (b), we investigate the time evolution of the gain-absorption coefficient ρ01 for two values of incoherent pumping field rate γ=r 0 (solid) and γ=r 2.5 (dashed) . In the absence of the incoherent pump-ing field, the time evolution of the absorption curve has a posi-tive value. When the incoherent pumping field is applied, more QDMs go to the excited state and thus reduce the absorption of the system. With further increasing of the incoherent pumping field intensity, absorption decreases further, and is converted to gain. As shown in figure 5(b); for γ=r 2.5 (dashed) , the tran-sient absorption after a short time with an oscillatory behavior finally disappears before reaching a negative (gain) steady-state value. Thus, a non-periodic gain-absorption can be converted to a periodic one, just by increasing the intensity of the incoherent pumping field r.

Figure 3. Absorption and gain spectrum of susceptibility as a function of normalized probe wavelength detuning, (a) for different values of the intensity of the second tunneling (TB),

γ γ γ γ= = = =T r T T2 , 1.5 , 0 (solid) , 0.45 (dashed) ,A B B TB = 1.2 γ (dash-dot) and (b) for different values of the intensity of the incoherent pumping field (r),

γ γ γ γ= = = =T T r r2 , 1.7 , 0 (solid) , 0.75 (dashed) ,A B  r = 2.5γ (dash-dot). Other parameters are as in figure 2.

1.54 1.542 1.544 1.546 1.548 1.55 1.552 1.554 1.556 1.558 1.56x 10−6

−0.01

−0.005

0

0.005

0.01

0.015

Wavelength

Im[ ρ 01

]

1.54 1.542 1.544 1.546 1.548 1.55 1.552 1.554 1.556 1.558 1.56x 10−6

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Wavelength

Im[ ρ 01

]

(a)

(b)

Laser Phys. 24 (2014) 125201

M R Mehmannavaz and H Sattari

6

Figure 4. Time evolution of the population distribution ρ ρ ρ, ,00 11 22 and ρ33. Column (a) γ γ γ γ= = = =T r T T2 , 1.5 , 0 (solid) , 1.2 (dashed)A B B . Column (b) γ γ γ γ= = = =T T r r2 , 1.7 , 0 (solid) , 2.5 (dashed)A B . Other parameters are as in figure 2.

0 5 10 15 20 25 30 35 40 45 500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

tγ

ρ 00

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

tγ

ρ 11

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

tγ

ρ 22

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

tγ

ρ 33column (a) column (b)

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

tγ

ρ 00

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

tγ

ρ 11

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

tγ

ρ 22

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

tγ

ρ 33

Laser Phys. 24 (2014) 125201

M R Mehmannavaz and H Sattari

7

According to the possibility of optical switching of absorp-tion to gain and vice versa, it would be worthwhile to discuss the suitable switching time of such an optical controllable switch in the proposed QD system. In figure  6, we investi-gated the optical switching of the absorption for various values of second tunneling and incoherent pumping field, in accordance with what is shown in figure 5. In figure 6(a), we investigate the effect of the second tunneling rate on the optical switching of the absorption-gain for two values of

γ γ=T 0 and 1.2 .B Here, in the first case γ=T 0 ,B the absorp-tion is positive and when we turn it to γ=T 1.2B the absorp-tion will be negative. Switching time in this case is equal to 60 γ and for case γ=T 1.2B to γ=T 0B it is equal to 10 γ . In figure 6(b), we investigate the effect of the incoherent pumping field on the optical switching of the absorption-gain for two values of γ γ=r 0 and 2.5 . Here also, in the first case γ=r 0 , the absorption is positive and when we turn it to γ=r 2.5 the absorption will be negative. Switching time in this case is equal to 40 γ and for case γ=r 2.5 to γ=r 0 it is equal to 40 γ. This approach can be utilized to produce a switch operating only by controlling the second tunneling rate and incoher-ent pumping field. Currently, control of the tunneling rates between two QDs by applying an electrical bias is a develop-ment technique. An optically controllable switch for the wave propagation between absorption and gain propagation can be proposed based on the presented results in figures 6(a) and (b). It should be noted that the proposed scheme is significantly different compared with the gaseous atomic system, where the background permittivity is 1. In this case, the density of the

medium is not homogeneous and the background permittivity of the medium is not 1. Now we are interested in the required switching time for changing the absorption to gain and vice versa. We do this for figure 6(b). It is apparent that the switch-ing time from absorption to gain and vice versa is about 40 γ. For a realistic case if we take γ γ= = −s10 ,10

12 1 we reach a switching time equal to 40 ps (4  ×  10−11 s), that is an appro-priate time for such a QDM-based switch. Such a switch has the advantage of having control parameters TB and r, which makes it controllable electrically (by applying bias) or even optically. So, we found out the switching time is sensitive to the intensity of the incoherent pumping field and tunneling rates. Therefore, by proper choice of the physical parameters, the applied fields’ intensities and the geometry of the QDs, one can decrease the switching rise/fall time and consequently the total switching time.

4. Conclusion

We investigated the electrical and the optical control of light amplification in a coupled triple QD system in both steady-state and transient state. It is shown that the light amplification or GWI of the medium can be controlled by the intensity of the incoherent pumping field and tunneling between neighboring QDs. It has also been shown that the medium can be used as an optical switch in which the propagation of the laser pulse can be controlled with tunneling between neighboring QDs and the incoherent pumping field. Switching time is obtained in about 40 ps (4   ×  10−11 s), which is a high-speed optical switch and may be useful for understanding the switching

Figure 5. Time evolution of the gain-absorption coefficient ρIm 01 for (a) γ γ γ γ= = = =T r T T2 , 1.5 , 0 (solid) , 1.2 (dashed) ,A B B and (b) γ γ γ γ= = = =T T r r2 , 1.7 , 0 (solid) , 2.5 (dashed) .A B Other parameters are as in figure 2.

0 10 20 30 40 50 60 70 80 90 100−0.005

0

0.005

0.01

0.015

0.02

0.025

tγ

Im[ρ

01]

TB = 0 γ

TB = 1.2 γ

0 10 20 30 40 50 60 70 80 90 100−0.02

0

0.02

0.04

0.06

0.08

0.1

tγ

Im[ρ

01]

r = 0 γr = 2.5 γ

(a)

(b)

Figure 6. Switching process for (a) γ γ= =T r2 , 1.5 ,A γ= ↔T 0 1.2 ,B (b) γ γ= =T T2 , 1.7 ,A B γ= ↔r 0 2.5 . Other

parameters are as in figure 2.

0 50 100 150 200 250 300 350 400 450 500−0.005

0

0.005

0.01

0.015

0.02

0.025

tγ

Im[ ρ

01]

TB = 0 γ

40γ60γ

90γ

10γ

TB = 1.2 γTB = 0 γ

0 50 100 150 200 250 300 350 400 450 500−0.02

0

0.02

0.04

0.06

0.08

0.1

tγ

Im[ ρ

01]

60γ40γ

60γ40γ

r = 0 γ r = 2.5 γ r = 0 γ

(a)

(b)

Laser Phys. 24 (2014) 125201

M R Mehmannavaz and H Sattari

8

feature of the nanostructure system and have potential appli-cation in optical information processing, quantum information networks, communication and transmission. The proposed approach can be used for optimizing and controlling the opti-cal switching process in the QD solid-state system because of its flexible design and several easily controllable parameters.

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