In these expressions, *M* is the electron mass, *c*+ *i* − *cj* are Fermion operators, *ωij* are transition frequencies of the active electron,*rij* are dipole moments, *ω* is the frequency of the field, and

$$
\vec{K} = \vec{1}\_{\mathcal{Y}} \sqrt{\alpha \frac{\lambda}{\mathcal{V}}} \tag{9}
$$

is a vector in the *y*-direction of this field, depending on the wavelength *λ*, the fine structure constant *α* = *e*2 4*πε*0*hc*¯ ≈ 1 137 , and the quantization volume V.

From (5) and (6), we notice that the electron-field interaction of the system depends on the transition dipole moment

$$
\vec{r}\_{01} = \vec{r}\_{10} = \int\_{\mathcal{V}\_s} \Psi\_0 \vec{r} \Psi\_1 d^3 \vec{r}\_{\prime} \tag{10}
$$

where

$$\Psi\_1(\mathbf{x}, y, z) = \psi\_1(\mathbf{x})\phi\_1(y)\chi\_1(z) \tag{11a}$$

$$\Psi\_0(\mathbf{x}, y, z) = \psi\_0(\mathbf{x})\phi\_0(y)\chi\_0(z) \tag{11b}$$

are eigenfunctions of the Hamiltonian (4), for the ground states of the two quantum wells. For the potential *U*(*r*), we distinguish seven regions, of four *GaAs* quantum wells, and three *AlxGa*1−*xAs* potential barriers, determined by the impurity concentrations of these regions (see figure 1). For the two thick conduction regions with the potentials *Uc* and *Uv*, we use a three-dimensional model, with a quantization volume V*n* = 1/*ND* for the n-region of donor concentration *ND*, and V*p* = 1/*NA* for the p-region of acceptor concentration *NA*. In these quantization volumes, for an electron with the effective mass *Mn*, and an electron hole with the effective mass *Mp*, we consider the densities of states (8)

$$g^{(n)}(E\_{\mathfrak{a}}) = \mathcal{V}\_n \frac{\sqrt{2}M\_n^{3/2}}{\pi^2 \hbar^3} \sqrt{E\_{\mathfrak{a}}} \tag{12a}$$

$$g^{(p)}(\mathbf{E}\_a) = \mathcal{V}\_p \frac{\sqrt{2}M\_p^{3/2}}{\pi^2 \hbar^3} \sqrt{\mathbf{E}\_{a\nu}}\tag{12b}$$

4 Will-be-set-by-IN-TECH

while the term in *A* 2 is negligible in the non-relativistic approximation. This potential depends

*ωijrijc*+

+*e*−i*kx* + *a*−*e*−i*kx* + *a*+

+*e*−i*kx* + *a*−*e*−i*kx* − *a*+

*MpA* , (5)

−*e* i*kx*

> −*e* i*kx*

*i* − *cj* are Fermion operators, *ωij* are transition

V (9)

*r*, (10)

√*Eα* (12a)

√*Eα*, (12b)

*i cj*, (6)

, (7)

. (8)

*V* = *e*

*p* = i*M* ∑ *ij*

i*kx* + *a*+

i*kx* − *a*+

 *K* =1*y*

137 , and the quantization volume V.

*r*01 =*r*10 =

the effective mass *Mp*, we consider the densities of states (8)

*g*(*n*)

*g*(*p*)

(*Eα*) = V*n*

(*Eα*) = V*p*

√2*M*3/2 *n π*2*h*¯ 3

√2*M*3/2 *p π*2*h*¯ 3

frequencies of the active electron,*rij* are dipole moments, *ω* is the frequency of the field, and

is a vector in the *y*-direction of this field, depending on the wavelength *λ*, the fine structure

From (5) and (6), we notice that the electron-field interaction of the system depends on the

 V*s*

are eigenfunctions of the Hamiltonian (4), for the ground states of the two quantum wells. For the potential *U*(*r*), we distinguish seven regions, of four *GaAs* quantum wells, and three *AlxGa*1−*xAs* potential barriers, determined by the impurity concentrations of these regions (see figure 1). For the two thick conduction regions with the potentials *Uc* and *Uv*, we use a three-dimensional model, with a quantization volume V*n* = 1/*ND* for the n-region of donor concentration *ND*, and V*p* = 1/*NA* for the p-region of acceptor concentration *NA*. In these quantization volumes, for an electron with the effective mass *Mn*, and an electron hole with

Ψ0*r*Ψ1*d*3

Ψ1(*x*, *y*, *z*) = *ψ*1(*x*)*φ*1(*y*)*χ*1(*z*) (11a) Ψ0(*x*, *y*, *z*) = *ψ*0(*x*)*φ*0(*y*)*χ*0(*z*) (11b)

 *α λ*

and the interaction potential

on the electron momentum

and the vector potential

of the electric field

constant *α* = *e*2

where

transition dipole moment

*A* = *h*¯ *e K a*+*e*

 *E* = i *h*¯ *ω e K a*+*e*

4*πε*0*hc*¯ ≈ 1

In these expressions, *M* is the electron mass, *c*+

and the occupation probabilities of these states with the kinetic energies *Eα*:

$$f^{(n)}(E\_a) = \frac{1}{e^{(\mathcal{U}\_c + E\_a)/T} + 1} \approx e^{-(\mathcal{U}\_c + E\_a)/T} \tag{13a}$$

$$f^{(p)}(E\_a) = \frac{1}{e^{(-\mathcal{U}\_v + E\_a)/T} + 1} \approx e^{-(-\mathcal{U}\_v + E\_a)/T},\tag{13b}$$

where approximate expressions are taken into account for the usual case of a non-degenerate semiconductor. Considering the integral of the number of particles occupying the states of a quantization volume, one gets the two potentials

$$\mathcal{U}\_{\rm c}(T) = T \ln \frac{N\_{\rm c}(T)}{N\_{\rm D}}, \qquad N\_{\rm c}(T) = 2 \left( \frac{\sqrt{M\_{\rm tr} T / 2 \pi}}{\hbar} \right)^{3} \tag{14a}$$

$$M\_{\overline{v}}(T) = -T \ln \frac{N\_{\overline{v}}(T)}{N\_A}, \qquad N\_{\overline{v}}(T) = 2 \left( \frac{\sqrt{M\_p T / 2 \pi}}{\hbar} \right)^3. \tag{14b}$$

For the very thin layers of the quantum wells and potential barriers, we use a two-dimensional model. For a quantization area *Ae*, in n and p regions, one gets the densities of states

$$\mathcal{g}^{(1)} = \mathcal{A}\_{\mathcal{C}} \frac{M\_{\text{ll}}}{\pi \hbar^2} \tag{15a}$$

$$g^{(2)} = \mathcal{A}\_{\mathfrak{e}} \frac{M\_p}{\pi \hbar^2}.\tag{15b}$$

By using the Fermi-Dirac distribution in the particle number integral, we obtain expressions similar to (14) for the potentials of the two *GaAs* quantum wells, as a function of the surface quantum dot density *Ne*

$$\mathcal{U}I\_1(T) = -T \ln \left( e^{\frac{\pi \hbar^2 N\_\text{fr}}{M\_\text{tr} T}} - 1 \right) \tag{16a}$$

$$\mathcal{U}l\_2(T) = T \ln \left( e^{\frac{\pi \hbar^2 N\_\ell}{M\_p T}} - 1 \right). \tag{16b}$$

Similar expressions are obtained for the separation barriers, as functions of the donor and acceptor arrays with concentrations *N*3, *N*4 embedded in the very thin *AlxGa*1−*xAs*-layers of these barriers,

$$\mathcal{U}I\_3(T) = -T \ln \left( e^{\frac{\pi \hbar^2 N\_3}{M\_3 T}} - 1 \right) \tag{17a}$$

$$\mathcal{U}L\_4(T) = T \ln \left( e^{\frac{\pi \hbar^2 N\_4}{M\_p T}} - 1 \right),\tag{17b}$$

and for the potential barrier *U*0 between the two quantum wells,

$$\mathcal{U}l\_0(T) = -T \ln \left( e^{\frac{\pi \hbar^2 N\_0}{M\_0 T}} - 1 \right),\tag{18}$$

while the energy eigenvalues are given by the equations:

2*Mn*(*x*0 − *x*1)2

2*Mp*(*x*4 − *x*2)2

the quantum dot region, which means a quantum dot surface density

*d* = 1 2

thermal velocity *vT* = √*T*/*Mn*, and crossing the barrier. We get the conditions

1 2 ln

1 2 ln

Thus, a quantum injection dot is a two-level system, with the energy levels *E*0 and *E*1, in a

*x*5 − *x*3. In this volume, we consider the two wave-functions (19)-(20) for the coordinate *x*, while, for the coordinates *y* and *z* in the plane of the quantum dot array, we consider the wave-functions *φ*1(*y*), *φ*0(*y*) and *χ*1(*y*), *χ*0(*y*), describing a thermal motion with an energy mean-value *T*. For a longitudinal device, when the electromagnetic field propagates in the direction *x* of the injected current, the electric component *Ey* of this field is coupled with the component *y*01 of the transition dipole moment of the system between thermal states. For a transversal device, when the electromagnetic field propagates in a direction *y* perpendicular to the direction *x* of the injected current, i.e. in the plane of the quantum dot array, the electric component *Ex* of this field is coupled with the transition dipole moment *x*01 between the two states (19)-(20) of the system. These dipole moments also determine the dissipative couplings of the quantum dot. They essentially depend on the width *x*2 − *x*0 of the quantum dot barrier

*eND AD* 6*I*

*eNA AD* 6*I*

 *T Mn*

 *T Mp*

*α*3(*x*1 − *x*3) <

*α*4(*x*5 − *x*4) <

quantization volume shaped as a parallelepiped with the basis area A = 1

arctan

Quantum Injection Dots 305

arctan

We take the two energy eigenvalues as *E*1 = *Uc*, *E*0 = *Uv*. In this case, the whole internal potential of the n-i-p junction is included on the distance *d* between the two charge layers of

The energy levels *E*1 = *Uc* and *E*0 = *Uv* can be obtained from (14), as functions of the donor and acceptor concentrations *ND* and *NA* of the conduction regions. By choosing appropriate values for the separation and quantum dot barriers, from (17) and (18) one gets the surface concentrations *N*3, *N*4 and *N*0 of these barriers. With the widths *x*1 − *x*3 and *x*5 − *x*4, the separation barrier must have a higher penetrability *P* than the necessary value to provide the injected current *I* through the device area *AD*, which means that the density of this current

*E*1 − *E*0

*Ne* = *ε*

*U*0 − *E*1 *E*1 − *U*1

*E*0 − *U*00 *U*2 − *E*0

+ arctan

+ arctan

*U*3 − *E*1 *E*1 − *U*1

> *E*0 − *U*4 *U*2 − *E*0

*e*2*d* , (25)

(*x*2 − *x*0 + *x*4 − *x*1). (26)

6 *eNDvTP* emergent from a unit volume with the

2

2

(24a)

(27a)

. (27b)

*Ne* and the height

. (24b)

*E*1 − *U*1 = *h*¯ 2

*U*2 − *E*0 = *h*¯ 2

while this distance can be approximated as

must be smaller than the thermal current 1

with a slight donor concentration *N*0, controlling this potential. For these wells, we consider harmonic wave-functions with exponential tails in the neighboring barriers

$$\psi\_1(\mathbf{x}) = A\_1 \cos \left[ k\_1 (\mathbf{x}\_0 - \mathbf{x}) - \arctan \frac{\alpha\_1}{k\_1} \right], \quad \mathbf{x}\_1 \le \mathbf{x} \le \mathbf{x}\_0 \tag{19a}$$

$$\psi\_1(\mathbf{x}) = A\_1 \sqrt{\frac{\mathbf{E}\_1 - \mathbf{U}\_1}{\mathbf{U}\_0 - \mathbf{U}\_1}} e^{-\mathbf{a}\_1(\mathbf{x} - \mathbf{x}\_0)}, \qquad \qquad \mathbf{x}\_0 \le \mathbf{x} \le \mathbf{x}\_2 \tag{19b}$$

$$\psi\_1(\mathbf{x}) = A\_1 \sqrt{\frac{E\_1 - \mathcal{U}\_1}{\mathcal{U}\_3 - \mathcal{U}\_1}} e^{-\mathfrak{a}\_3(\mathbf{x}\_1 - \mathbf{x})} , \qquad \qquad \mathfrak{x}\_3 \le \mathbf{x} \le \mathfrak{x}\_1 \tag{19c}$$

and

$$\psi\_0(\mathbf{x}) = A\_0 \cos \left[ k\_0 (\mathbf{x} - \mathbf{x}\_2) - \arctan \frac{\mathbf{x}\_0}{k\_0} \right], \qquad \mathbf{x}\_2 \le \mathbf{x} \le \mathbf{x}\_4 \tag{20a}$$

$$\psi\_0(\mathbf{x}) = A\_0 \sqrt{\frac{\mathcal{U}\_2 - E\_0}{\mathcal{U}\_2 - \mathcal{U}\_{00}}} e^{-a\_0(\mathbf{x}\_2 - \mathbf{x})}, \qquad \qquad \mathbf{x}\_0 \le \mathbf{x} \le \mathbf{x}\_2 \tag{20b}$$

$$\psi\_0(\mathbf{x}) = A\_0 \sqrt{\frac{\mathcal{U}\_2 - E\_0}{\mathcal{U}\_2 - \mathcal{U}\_4}} e^{-\mathfrak{a}\_4(\mathbf{x} - \mathbf{x}\_4)}, \qquad \qquad \mathfrak{x}\_4 \le \mathfrak{x} \le \mathfrak{x}\_5. \tag{20c}$$

while the tails beyond these barriers are neglected. These wave-functions depend on the wave-numbers

$$k\_1 = \frac{1}{\hbar} \sqrt{2M\_n(E\_1 - \mathcal{U}\_1)}\tag{21a}$$

$$k\_0 = \frac{1}{\hbar} \sqrt{2M\_p (\mathcal{U}\_2 - E\_0)}\,\tag{21b}$$

attenuation coefficients

$$\alpha\_1 = \frac{1}{\hbar} \sqrt{2M\_n(\mathcal{U}\_0 - E\_1)}\tag{22a}$$

$$\alpha\_0 = \frac{1}{\hbar} \sqrt{2M\_P(E\_0 - \mathcal{U}\_{00})} \tag{22b}$$

$$\alpha\_3 = \frac{1}{\hbar} \sqrt{2M\_n(\mathcal{U}\_3 - E\_1)}\tag{22c}$$

$$\alpha\_4 = \frac{1}{\hbar} \sqrt{2M\_p (E\_0 - \mathcal{U}\_4)},\tag{22d}$$

and normalization coefficients

$$A\_1 = \sqrt{2} \left[ \mathbf{x}\_0 - \mathbf{x}\_1 + \frac{\hbar}{\sqrt{2M\_n}} \left( \frac{1}{\sqrt{\mathcal{U}\_0 - E\_1}} + \frac{1}{\sqrt{\mathcal{U}\_3 - E\_1}} \right) \right]^{-1/2} \tag{23a}$$

$$A\_0 = \sqrt{2} \left[ \mathbf{x}\_4 - \mathbf{x}\_2 + \frac{\hbar}{\sqrt{2M\_p}} \left( \frac{1}{\sqrt{E\_0 - \mathbf{U}\_{00}}} + \frac{1}{\sqrt{E\_0 - \mathbf{U}\_4}} \right) \right]^{-1/2},\tag{23b}$$

6 Will-be-set-by-IN-TECH

with a slight donor concentration *N*0, controlling this potential. For these wells, we consider

*k*1(*x*0 − *x*) − arctan

*e*−*α*1(*x*−*x*0)

−*α*3(*x*1−*x*)

*k*0(*x* − *x*2) − arctan

*e*−*α*0(*x*2−*x*)

*e*−*α*4(*x*−*x*4)

while the tails beyond these barriers are neglected. These wave-functions depend on the

*α*1 *k*1

*α*0 *k*0  , *x*1 ≤ *x* ≤ *x*0 (19a)

, *x*2 ≤ *x* ≤ *x*4 (20a)

, *x*0 ≤ *x* ≤ *x*2 (19b)

, *x*3 ≤ *x* ≤ *x*1 (19c)

, *x*0 ≤ *x* ≤ *x*2 (20b)

, *x*4 ≤ *x* ≤ *x*5, (20c)

2*Mn*(*E*1 − *U*1) (21a)

2*Mp*(*U*2 − *E*0), (21b)

2*Mn*(*U*0 − *E*1) (22a)

2*Mp*(*E*0 − *U*00) (22b)

2*Mn*(*U*3 − *E*1) (22c)

2*Mp*(*E*0 − *U*4), (22d)

−1/2

−1/2

(23a)

, (23b)

1 √*U*3 − *E*1

> 1 √*E*0 − *U*4

harmonic wave-functions with exponential tails in the neighboring barriers

*E*1 − *U*1 *U*0 − *U*1

*E*1 − *U*1 *U*3 − *U*1

*U*2 − *E*0 *U*2 − *U*00

*U*2 − *E*0 *U*2 − *U*4

> *k*1 = 1 *h*¯

> *k*0 = 1 *h*¯

*α*1 = 1 *h*¯

*α*0 = 1 *h*¯

*α*3 = 1 *h*¯

*α*4 = 1 *h*¯

*h*¯ √2*Mn*

*h*¯ 2*Mp*  1 √*U*0 − *E*1

 1 √*E*0 − *U*00

+

+

*x*0 − *x*1 +

*x*4 − *x*2 +

*e*

*ψ*1(*x*) = *A*1 cos

*ψ*1(*x*) = *A*1

*ψ*1(*x*) = *A*1

*ψ*0(*x*) = *A*0 cos

*ψ*0(*x*) = *A*0

*ψ*0(*x*) = *A*0

and

wave-numbers

attenuation coefficients

and normalization coefficients

*A*1 = √ 2

*A*0 = √ 2  while the energy eigenvalues are given by the equations:

$$E\_1 - U\_1 = \frac{\hbar^2}{2M\_{\text{fl}}(\mathbf{x}\_0 - \mathbf{x}\_1)^2} \left( \arctan \sqrt{\frac{\mathcal{U}\_0 - E\_1}{E\_1 - \mathcal{U}\_1}} + \arctan \sqrt{\frac{\mathcal{U}\_3 - E\_1}{E\_1 - \mathcal{U}\_1}} \right)^2 \tag{24a}$$

$$\mathcal{U}\_2 - \mathcal{E}\_0 = \frac{\hbar^2}{2M\_p(\mathbf{x}\_4 - \mathbf{x}\_2)^2} \left( \arctan \sqrt{\frac{\mathcal{E}\_0 - \mathcal{U}\_{00}}{\mathcal{U}\_2 - \mathcal{E}\_0}} + \arctan \sqrt{\frac{\mathcal{E}\_0 - \mathcal{U}\_4}{\mathcal{U}\_2 - \mathcal{E}\_0}} \right)^2. \tag{24b}$$

We take the two energy eigenvalues as *E*1 = *Uc*, *E*0 = *Uv*. In this case, the whole internal potential of the n-i-p junction is included on the distance *d* between the two charge layers of the quantum dot region, which means a quantum dot surface density

$$N\_{\varepsilon} = \varepsilon \frac{E\_1 - E\_0}{c^2 d},\tag{25}$$

while this distance can be approximated as

$$d = \frac{1}{2}(\mathbf{x}\_2 - \mathbf{x}\_0 + \mathbf{x}\_4 - \mathbf{x}\_1). \tag{26}$$

The energy levels *E*1 = *Uc* and *E*0 = *Uv* can be obtained from (14), as functions of the donor and acceptor concentrations *ND* and *NA* of the conduction regions. By choosing appropriate values for the separation and quantum dot barriers, from (17) and (18) one gets the surface concentrations *N*3, *N*4 and *N*0 of these barriers. With the widths *x*1 − *x*3 and *x*5 − *x*4, the separation barrier must have a higher penetrability *P* than the necessary value to provide the injected current *I* through the device area *AD*, which means that the density of this current must be smaller than the thermal current 1 6 *eNDvTP* emergent from a unit volume with the thermal velocity *vT* = √*T*/*Mn*, and crossing the barrier. We get the conditions

$$a\_3(\mathbf{x}\_1 - \mathbf{x}\_3) < \frac{1}{2} \ln \left( \frac{eN\_D A\_D}{6I} \sqrt{\frac{T}{M\_n}} \right) \tag{27a}$$

$$
\alpha\_4(\mathbf{x}\_5 - \mathbf{x}\_4) < \frac{1}{2} \ln \left( \frac{e N\_A A\_D}{6I} \sqrt{\frac{T}{M\_p}} \right). \tag{27b}
$$

Thus, a quantum injection dot is a two-level system, with the energy levels *E*0 and *E*1, in a quantization volume shaped as a parallelepiped with the basis area A = 1 *Ne* and the height *x*5 − *x*3. In this volume, we consider the two wave-functions (19)-(20) for the coordinate *x*, while, for the coordinates *y* and *z* in the plane of the quantum dot array, we consider the wave-functions *φ*1(*y*), *φ*0(*y*) and *χ*1(*y*), *χ*0(*y*), describing a thermal motion with an energy mean-value *T*. For a longitudinal device, when the electromagnetic field propagates in the direction *x* of the injected current, the electric component *Ey* of this field is coupled with the component *y*01 of the transition dipole moment of the system between thermal states. For a transversal device, when the electromagnetic field propagates in a direction *y* perpendicular to the direction *x* of the injected current, i.e. in the plane of the quantum dot array, the electric component *Ex* of this field is coupled with the transition dipole moment *x*01 between the two states (19)-(20) of the system. These dipole moments also determine the dissipative couplings of the quantum dot. They essentially depend on the width *x*2 − *x*0 of the quantum dot barrier

We consider a system of interest, including a system of Fermions *S* with the Hamiltonian *HS*

Quantum Injection Dots 307

and an electromagnetic field *F* with the Hamiltonian *HF*, in a dissipative environment with the Hamiltonian *HE*. Taking into account a potential of interaction *V* between the Fermion system *S* and the field *F*, and a system-environment potential *VE*, the dynamics of the total

*εV*˜(*t*) + *εV*˜ *E*(*t*), *χ*˜(*t*)

*h*¯ (*HF*+*HS*

2*χ*˜(2)

*χ*(*t*)*e*− *i*

(*t*) + *ε*

where *ρ*(*t*) is the reduced density matrix of the system of interest, while *R* is the density matrix of the dissipative environment at the initial moment of time, *t* = 0, the time-evolution of the environment being described by the higher-order terms *χ*˜(1)(*t*), *χ*˜(2)(*t*), .... The parameter *ε* is introduced to handle the orders of the terms in this series expansion, and is set to 1 in the final

} = *TrE*{*χ*˜

(*ρ*˜(*t*), *t*) + *ε*

*i cj*, *ρ*(*t*)]

From the dynamic equation (28), with expressions (30)-(33), we obtain the quantum master

*i cj*, *ρ*(*t*)*c*+

�*αi*|(*VF*)2|*αj*�*f F*

If initially the environment is in the equilibrium state *R*, the density matrix of the total system is of the form *χ*(0) = *Rρ*(0). We suppose that at time *t* = 0, due to the interaction *V* of the system of Fermions with the electromagnetic field, or due to a non-equilibrium initial state *ρ*(0) �= *ρT*, a time-evolution begins, while the reduced density satisfies a quantum dynamical

(2)

2*B*˜(2)

*j ci*])

)][*c*+ *k cl*, *ρ*(*t* � )]*e* i[*φ*(*t* � )+ 1 *h*¯ *HS* 0 (*t*−*t*�

*α* (*εα*)*gF*

0 +*HE*)*t*

*ρ*˜(*t*) = *TrE*{*χ*˜(*t*)}, (31)

} = ... = 0. (32)

(*ρ*˜(*t*), *t*) + ... . (33)

)]]*dt*� ,

*α* (*εα*)*dεα* (35)

(34)

. (28)

. (29)

(*t*) + ... , (30)

system is described by an equation of motion of the form: *d*

*χ*˜(*t*) = *e*

results. The reduced density of the system is

*d*

[*H*, *ρ*(*t*)] − i∑

*t*−*τ* [*c*+

*ζij* = 1 *h*¯

*dt ρ*˜(*t*) = *εB*˜(1)

*ij*

 1 *YF* (*α*)

*i cjρ*(*t*), *c*+

*ζij*[*c*+

*i cj*,*e*−i[*φ*(*t*

*j ci*]+[*c*+

� )+ 1 *h*¯ *HS* 0 (*t*−*t*�

equation of the form

equation

*d*

*dt ρ*(*t*) = − i

where the coefficients

*h*¯

*λij*([*c*+

*ζijζkl t*

+ ∑ *ij*

+ ∑ *ijkl* *dt χ*˜(*t*) = − *i*

In this equation, tilde denotes operators in the interaction picture, e.g.

*χ*˜(*t*) = *R* ⊗ *ρ*˜(*t*) + *εχ*˜(1)

while the higher-order terms of the total density have the property:

*TrE*{*χ*˜(1)

*i h*¯ (*HE*+*HS*

*h*¯

0 +*HF*)*t*

According to a general procedure disclosed in (9), we take a total density of the form

0

of height *U*0, which determines the overlap of the two wave-functions *ψ*1(*x*) and *ψ*0(*x*). When this width is chosen for reasonable values of the electron-field and dissipative couplings, and *N*3, *N*4 for reasonable values of the separation barriers satisfying the conditions (27), from equations (16) and (24)-(26) the geometrical characteristics *x*0*x*1, *x*2, *x*3, *x*4 are obtained.

As an example, for a concentration *ND* = *NA* = 3.16 × 1016 *cm*−3 of a super radiant junction, working at temperature *T* = 10 0*C*, we get a transition frequency *E*1 − *E*0 = 0.1866 *eV*. A quantum dot *Al*0.37*Ga*0.63*As*-barrier of *U*0 = 0.5 *eV* is obtained for a surface concentration of *N*0 = 6.4243 × 106 *m*−2 donors, and separation barriers of *U*3 − *Uc* = *Uv* − *U*4 = 0.05 *eV* are obtained for the surface concentrations of these barriers *N*3 = 8.01 × 1013 *m*−2 and *N*4 = 2.552 × 1013 *m*−2. For a width *x*2 − *x*0 = 5.5 *nm* of the quantum dot barrier, we get a quantum dot surface concentration *Ne* = 1.476 × 1016 *m*−2 and the widths of the two quantum dot wells *x*0 − *x*1 = 4.189 *nm* and *x*4 − *x*2 = 1.576 *nm*, while separation barriers with widths *x*1 − *x*3 = 10 *nm* and *x*5 − *x*4 = 3 *nm* satisfy the conditions (27) for an injected current *I* = 45 *A* in a device with an area *AD* = 4 *cm*2.

#### **3. Dissipative dynamics of quantum injection dots**

We consider a quantum injection dot with two energy levels *E*1, *E*0, coupled to a super radiant field and a complex dissipative environment of a semiconductor structure as it is represented in figure 1: (1) the quasi-free electrons of the n-conduction region *x* < *x*3, (2) the quasi-free holes of the p-conduction region *x* > *x*5, (3) the phonons of the crystal at temperature *T*, and (4) the photons of the free electromagnetic field existing at temperature *T* (see figure 2). In (3)

Fig. 2. Dissipative couplings of a quantum injection dot to the environment. A decay |1�→|0� of the active electron is correlated with: (1) a transition |*β*�→|*α*� of a quasi-free electron in a quantization volume V*n*, (2) a transition |*β*�→|*α*� of a quasi-free hole in a quantization volume V*p*, (3) a phonon creation with a wave vector�*kP*, and (4) a photon creation with a wave vector�*kFEF*.

we showed that, for a semiconductor structure with the characteristics mentioned at the end of the preceding section, the decay rate corresponding to the phonon environment is dominant, the decay rate due to the conduction electrons and holes is smaller, while the decay rate given by the free electromagnetic field is negligible.

8 Will-be-set-by-IN-TECH

of height *U*0, which determines the overlap of the two wave-functions *ψ*1(*x*) and *ψ*0(*x*). When this width is chosen for reasonable values of the electron-field and dissipative couplings, and *N*3, *N*4 for reasonable values of the separation barriers satisfying the conditions (27), from equations (16) and (24)-(26) the geometrical characteristics *x*0*x*1, *x*2, *x*3, *x*4 are obtained.

As an example, for a concentration *ND* = *NA* = 3.16 × 1016 *cm*−3 of a super radiant junction, working at temperature *T* = 10 0*C*, we get a transition frequency *E*1 − *E*0 = 0.1866 *eV*. A quantum dot *Al*0.37*Ga*0.63*As*-barrier of *U*0 = 0.5 *eV* is obtained for a surface concentration of *N*0 = 6.4243 × 106 *m*−2 donors, and separation barriers of *U*3 − *Uc* = *Uv* − *U*4 = 0.05 *eV* are obtained for the surface concentrations of these barriers *N*3 = 8.01 × 1013 *m*−2 and *N*4 = 2.552 × 1013 *m*−2. For a width *x*2 − *x*0 = 5.5 *nm* of the quantum dot barrier, we get a quantum dot surface concentration *Ne* = 1.476 × 1016 *m*−2 and the widths of the two quantum dot wells *x*0 − *x*1 = 4.189 *nm* and *x*4 − *x*2 = 1.576 *nm*, while separation barriers with widths *x*1 − *x*3 = 10 *nm* and *x*5 − *x*4 = 3 *nm* satisfy the conditions (27) for an injected current *I* = 45 *A*

We consider a quantum injection dot with two energy levels *E*1, *E*0, coupled to a super radiant field and a complex dissipative environment of a semiconductor structure as it is represented in figure 1: (1) the quasi-free electrons of the n-conduction region *x* < *x*3, (2) the quasi-free holes of the p-conduction region *x* > *x*5, (3) the phonons of the crystal at temperature *T*, and (4) the photons of the free electromagnetic field existing at temperature *T* (see figure 2). In (3)

n-region - quasi-free electrons p-region - quasi-free holes quantum dots

> ✶ �*kP*

�*kFEF*

*R*0)|*β*1�

*Eα Eβ*

V*p*

V*p*

−*Ln*/2 0 *Ln*/2 −*Lp*/2 0 *Lp*/2

we showed that, for a semiconductor structure with the characteristics mentioned at the end of the preceding section, the decay rate corresponding to the phonon environment is dominant, the decay rate due to the conduction electrons and holes is smaller, while the decay rate given

Fig. 2. Dissipative couplings of a quantum injection dot to the environment. A decay |1�→|0� of the active electron is correlated with: (1) a transition |*β*�→|*α*� of a quasi-free electron in a quantization volume V*n*, (2) a transition |*β*�→|*α*� of a quasi-free hole in a quantization volume V*p*, (3) a phonon creation with a wave vector�*kP*, and (4) a photon

*R*0)|*β*1� �*α*0|*V*(*p*)(�

*E*1 *E*0

in a device with an area *AD* = 4 *cm*2.

✾

creation with a wave vector�*kFEF*.

by the free electromagnetic field is negligible.

*Eβ Eα*

V*n*

V*n*

**3. Dissipative dynamics of quantum injection dots**

� *R*0

�*α*0|*V*(*n*) (� We consider a system of interest, including a system of Fermions *S* with the Hamiltonian *HS* 0 and an electromagnetic field *F* with the Hamiltonian *HF*, in a dissipative environment with the Hamiltonian *HE*. Taking into account a potential of interaction *V* between the Fermion system *S* and the field *F*, and a system-environment potential *VE*, the dynamics of the total system is described by an equation of motion of the form:

$$\frac{d}{dt}\tilde{\chi}(t) = -\frac{i}{\hbar} \left[ \varepsilon \tilde{V}(t) + \varepsilon \tilde{V}^E(t) \, \_\prime \tilde{\chi}(t) \right]. \tag{28}$$

In this equation, tilde denotes operators in the interaction picture, e.g.

$$\tilde{\chi}(t) = e^{\frac{i}{\hbar}(H^E + H\_0^S + H^F)t} \chi(t) e^{-\frac{i}{\hbar}(H^F + H\_0^S + H^E)t}. \tag{29}$$

According to a general procedure disclosed in (9), we take a total density of the form

$$\tilde{\chi}(t) = \mathbb{R} \otimes \tilde{\rho}(t) + \varepsilon \tilde{\chi}^{(1)}(t) + \varepsilon^2 \tilde{\chi}^{(2)}(t) + \dots \tag{30}$$

where *ρ*(*t*) is the reduced density matrix of the system of interest, while *R* is the density matrix of the dissipative environment at the initial moment of time, *t* = 0, the time-evolution of the environment being described by the higher-order terms *χ*˜(1)(*t*), *χ*˜(2)(*t*), .... The parameter *ε* is introduced to handle the orders of the terms in this series expansion, and is set to 1 in the final results. The reduced density of the system is

$$
\vec{\rho}(t) = \text{Tr}\_E\{\vec{\chi}(t)\}\_\prime \tag{31}
$$

while the higher-order terms of the total density have the property:

$$\operatorname{Tr}\_E\{\tilde{\chi}^{(1)}\} = \operatorname{Tr}\_E\{\tilde{\chi}^{(2)}\} = \dots \quad = 0. \tag{32}$$

If initially the environment is in the equilibrium state *R*, the density matrix of the total system is of the form *χ*(0) = *Rρ*(0). We suppose that at time *t* = 0, due to the interaction *V* of the system of Fermions with the electromagnetic field, or due to a non-equilibrium initial state *ρ*(0) �= *ρT*, a time-evolution begins, while the reduced density satisfies a quantum dynamical equation of the form

$$\frac{d}{dt}\tilde{\rho}(t) = \varepsilon \tilde{B}^{(1)}(\tilde{\rho}(t), t) + \varepsilon^2 \tilde{B}^{(2)}(\tilde{\rho}(t), t) + \dots \tag{33}$$

From the dynamic equation (28), with expressions (30)-(33), we obtain the quantum master equation

$$\begin{split} \frac{d}{dt}\rho(t) &= -\frac{\mathbf{i}}{\hbar}[H\_{\prime}\rho(t)] - \mathbf{i}\sum\_{ij}\xi\_{ij}[c\_{i}^{+}c\_{j\prime}\rho(t)] \\ &+ \sum\_{ij}\lambda\_{ij}([c\_{i}^{+}c\_{j}\rho(t),c\_{j}^{+}c\_{i}] + [c\_{i}^{+}c\_{j\prime}\rho(t)c\_{j}^{+}c\_{i}]) \\ &+ \sum\_{ijkl}\xi\_{ij}\xi\_{kl}\int\_{t-\tau}^{t}[c\_{i}^{+}c\_{j\prime}e^{-\mathbf{i}[\phi(t') + \frac{1}{\hbar}H\_{0}^{S}(t-t')]}[c\_{k}^{+}c\_{l\prime}\rho(t')]e^{\mathbf{i}[\phi(t') + \frac{1}{\hbar}H\_{0}^{S}(t-t')]}]dt', \end{split} \tag{34}$$

where the coefficients

$$\mathcal{L}\_{ij} = \frac{1}{\hbar} \sqrt{\frac{1}{Y^F} \int\_{(\alpha)} \langle \alpha i | (V^F)^2 | \alpha j \rangle f\_a^F(\varepsilon\_a) g\_a^F(\varepsilon\_a) d\varepsilon\_a} \tag{35}$$

describe transitions stimulated by the thermal fluctuations of the self-consistent field of the *YF* environmental Fermions in a certain quantization volume - hopping potential (10), *φ*(*t* � ) is a phase fluctuation operator, while the coefficients

$$
\lambda\_{i\dot{j}} = \lambda\_{i\dot{j}}^F + \lambda\_{i\dot{j}}^B + \gamma\_{i\dot{j}} \tag{36}
$$

**Quantum dot**

*r* ✲

−*RDx*3 0 *x*5 *RA*

 *R*0

*D N*−1/3

Fig. 3. The electron of a quantum injection dot is coupled to the quasi-free electrons of the n-dissipative clusters (n-region) and quasi-free holes of the p-dissipative clusters (p-region)

From the wave-functions derived in the preceding section, we obtain the dipole moment of a

01 *x*2 − *x*0

(*x*) 01

(*x*) 01

and a quantity that we call the state separation distance. At the same time, with the initial and the final energies *Eβ* = *T*/2, *Eα* = *Eβ* + *ε*10, we obtain the dipole moments for the n-zone

*ε*10

*ε*10

2

2

(*x*)

(Ψ) 01 = *c*

(Ψ) 10 = *c*

(*E*1 − *U*1)(*U*2 − *E*0) (*U*0 − *U*1)(*U*2 − *U*00)

*αβ* = *h*¯

*αβ* = *h*¯

*αβ* = *Z*(*n*)

*αβ* = *Z*(*p*)

*A*

*R*,*r*) = *αhc*¯ | *R*0|3 *Rr*.

2 − 1

*h*¯

*h*¯

 *e*

2*ε*10 + *T Mn*

2*ε*10 + *T Mp*

*α*0 − *α*1

−*α*1(*x*2−*x*0) − *e*

≈ 2¯*h Mnω*0

≈ 2¯*h Mpω*0

√*MnT* (42b)

*MpT* , (42c)

−*α*0(*x*2−*x*0)

, (44)

, (45)

. (43)

*N*−1/3

*x* (Ψ) 01 = *c*

*y* (Ψ) 01 = *z*

*y* (Ψ) 10 = *z*

by an electric dipole-dipole interaction: *VF*(*R*0,

as a product of the overlap function

01 = *A*1*A*0 *α*0 − *α*1

> *X*(*n*) *αβ* = *Y*(*n*)

> *X*(*p*) *αβ* = *Y*(*p*)

*c* (*x*)

and for the p-zone,

quantum dot:

✛

✛

 *R* ✲

(42a)

**n-dissipative clusters p-dissipative clusters**

Quantum Injection Dots 309

describe correlated transitions of the system Fermions with environment particles, including explicit terms for Fermions, Bosons, and photons of the free electromagnetic field. These terms depend on the dissipative two-body potentials *VF*, *VB*, the densities of the environment states *gF*(*εα*), *gB*(*εα*), the occupation probabilities of these states *f F*(*εα*), *f B*(*εα*), and temperature *T*. For a rather low temperature, *T* � *εji*, *j* > *i*, these terms become

$$
\lambda\_{ij}^F = \frac{\pi}{\hbar} |\langle ai | V^F | \beta j \rangle|^2 [1 - f^F(\varepsilon\_{ji})] g^F(\varepsilon\_{ji}) \tag{37a}
$$

$$
\lambda\_{ji}^F = \frac{\pi}{\hbar} |\langle ai \vert V^F \vert \beta j \rangle|^2 f^F(\varepsilon\_{ji}) g^F(\varepsilon\_{ji}) \,\tag{37b}
$$

for the Fermion environment,

$$
\lambda\_{ij}^{B} = \frac{\pi}{\hbar} |\langle ai | V^{B} | \beta j \rangle|^{2} [1 + f^{B}(\varepsilon\_{ji})] g\_{a}^{B}(\varepsilon\_{ji}) \tag{38a}
$$

$$
\lambda\_{ji}^{B} = \frac{\pi}{\hbar} |\langle ai | V^{B} | \beta j \rangle|^{2} f^{B}(\varepsilon\_{ji}) g\_{a}^{B}(\varepsilon\_{ji}) \tag{38b}
$$

for the Boson environment, and

$$\gamma\_{\vec{i}\vec{j}} = \frac{2\alpha}{c^2 \hbar^3} \left| \vec{r}\_{\vec{i}\vec{j}} \right|^2 \varepsilon\_{\vec{j}\vec{i}}^3 (1 + \frac{1}{e^{\varepsilon\_{\vec{j}\vec{i}}/T} - 1}) \tag{39}$$

for the Boson environment of the free electromagnetic field, where�*rij* are the transition dipole moments. The dissipative terms of the master equation (34) with coefficients (36)-(39) describe correlated transitions of the system and the environment particles, with energy conservation, *εji* = *εαβ*, in agreement with the quantum-mechanical principles and the detailed balance principle (11). The non-Markovian part of this equation takes into account the fluctuations of the self-consistent field of the environment Fermions, with the coefficients (35).

A significant component of the dissipative dynamics comes from the Coulomb interaction of the active electrons, mainly located in the interval (*x*3, *x*5), with the conduction electrons and holes in the conduction regions (−∞, *x*3) and (*x*5, +∞), respectively (figure 1). We use the notations �*r* for the position vector of an active electron, and � *R*0 + � *R* for the position vector of a dissipative electron (hole), where � *R*0 is the position vector of an arbitrary n (p) cluster, and � *R* = �1*xX* +�1*yY* +�1*zZ* is the position of an electron (hole) in this cluster (figure 3). In this case, the Coulomb potential in a first-order approximation of the two-body term � *R*�*r* = *Xx* + *Yy* + *Zz* is

$$V^{\mathbb{C}}(\vec{R}, \vec{r}) = \frac{a\hbar c}{|\vec{R}\_0 + \vec{R} - \vec{r}|} \approx \frac{a\hbar c}{|\vec{R}\_0|} \left( 1 + \frac{\mathbf{X}\mathbf{x} + \mathbf{Y}\mathbf{y} + \mathbf{Z}z}{\vec{R}\_0^2} \right). \tag{40}$$

From this expression, only the second term, bilinear in the coordinates of an active electron and of an electron (hole) of the environment, yields contributions in the two-body transition matrix elements of the decay (excitation) rates (37):

$$V^{\vec{r}}(\vec{R}\_0, \vec{R}, \vec{r}) \dot{=} V^{(\eta)}(\vec{R}\_0, \vec{R}, \vec{r}) = -V^{(p)}(\vec{R}\_0, \vec{R}, \vec{r}) = \frac{a\hbar c}{|\vec{R}\_0|^3} (\mathbf{X}\mathbf{x} + \mathbf{Y}\mathbf{y} + \mathbf{Z}z). \tag{41}$$

10 Will-be-set-by-IN-TECH

describe transitions stimulated by the thermal fluctuations of the self-consistent field of the *YF* environmental Fermions in a certain quantization volume - hopping potential (10), *φ*(*t*

*ij* + *λB*

describe correlated transitions of the system Fermions with environment particles, including explicit terms for Fermions, Bosons, and photons of the free electromagnetic field. These terms depend on the dissipative two-body potentials *VF*, *VB*, the densities of the environment states *gF*(*εα*), *gB*(*εα*), the occupation probabilities of these states *f F*(*εα*), *f B*(*εα*), and temperature *T*.

*h*¯ |�*αi*|*VB*|*βj*�|2[1 + *f B*(*εji*)]*gB*

*h*¯ |�*αi*|*VB*|*βj*�|2 *f B*(*εji*)*gB*

for the Boson environment of the free electromagnetic field, where�*rij* are the transition dipole moments. The dissipative terms of the master equation (34) with coefficients (36)-(39) describe correlated transitions of the system and the environment particles, with energy conservation, *εji* = *εαβ*, in agreement with the quantum-mechanical principles and the detailed balance principle (11). The non-Markovian part of this equation takes into account the fluctuations of

A significant component of the dissipative dynamics comes from the Coulomb interaction of the active electrons, mainly located in the interval (*x*3, *x*5), with the conduction electrons and holes in the conduction regions (−∞, *x*3) and (*x*5, +∞), respectively (figure 1). We use the

*R* = �1*xX* +�1*yY* +�1*zZ* is the position of an electron (hole) in this cluster (figure 3). In

 1 +

≈ *αhc*¯ |� *R*0|

From this expression, only the second term, bilinear in the coordinates of an active electron and of an electron (hole) of the environment, yields contributions in the two-body transition

> (� *R*0, �

this case, the Coulomb potential in a first-order approximation of the two-body term �

*R* −�*r*|

*R*,�*r*) = −*V*(*p*)

*λij* = *λF*

a phase fluctuation operator, while the coefficients

For a rather low temperature, *T* � *εji*, *j* > *i*, these terms become

*λF ij* = *π*

*λF ji* = *π*

*λB ij* = *π*

*λB ji* = *π*

> *γij* = 2*α c*2*h*¯ 3 �*rij* 2 *ε* 3 *ji*(1 +

notations �*r* for the position vector of an active electron, and �

*R*,�*r*) = *αhc*¯ |� *R*0 + �

> (� *R*0, �

of a dissipative electron (hole), where �

*VC*(�

matrix elements of the decay (excitation) rates (37):

*R*,�*r*)=˙ *V*(*n*)

and �

*Xx* + *Yy* + *Zz* is

*VF*(� *R*0, �

the self-consistent field of the environment Fermions, with the coefficients (35).

for the Fermion environment,

for the Boson environment, and

� ) is

*ij* + *γij* (36)

*α* (*εji*) (38a)

) (39)

*R* for the position vector

. (40)

*R*�*r* =

*α* (*εji*) (38b)

*h*¯ |�*αi*|*VF*|*βj*�|2[1 − *f F*(*εji*)]*gF*(*εji*) (37a)

*h*¯ |�*αi*|*VF*|*βj*�|2 *f F*(*εji*)*gF*(*εji*), (37b)

*R*0 + �

*R*0 is the position vector of an arbitrary n (p) cluster,

*Xx* + *Yy* + *Zz* � *R*2 0

*R*,�*r*) = *αhc*¯ |�

*R*0|3 (*Xx* + *Yy* + *Zz*). (41)

1 *eεji*/*T* − 1

Fig. 3. The electron of a quantum injection dot is coupled to the quasi-free electrons of the n-dissipative clusters (n-region) and quasi-free holes of the p-dissipative clusters (p-region) by an electric dipole-dipole interaction: *VF*(*R*0, *R*,*r*) = *αhc*¯ | *R*0|3 *Rr*.

From the wave-functions derived in the preceding section, we obtain the dipole moment of a quantum dot:

$$\mathbf{x}\_{01}^{(\mathbf{y})} = \mathbf{c}\_{01}^{(\mathbf{x})} \left( \frac{\mathbf{x}\_2 - \mathbf{x}\_0}{2} - \frac{1}{\mathbf{a}\_0 - \mathbf{a}\_1} \right) \tag{42a}$$

$$y\_{01}^{(\mathbf{y})} = z\_{01}^{(\mathbf{y})} = c\_{01}^{(x)} \frac{\hbar}{2\sqrt{M\_{\text{ll}}T}} \tag{42b}$$

$$y\_{10}^{(\mathbf{y})} = z\_{10}^{(\mathbf{y})} = c\_{01}^{(\mathbf{x})} \frac{\hbar}{2\sqrt{M\_{\mathbf{y}}T}} \tag{42c}$$

as a product of the overlap function

$$c\_{01}^{(\mathbf{x})} = \frac{A\_1 A\_0}{a\_0 - a\_1} \sqrt{\frac{(E\_1 - U\_1)(U\_2 - E\_0)}{(U\_0 - U\_1)(U\_2 - U\_{00})}} \left( e^{-a\_1(\mathbf{x}\_2 - \mathbf{x}\_0)} - e^{-a\_0(\mathbf{x}\_2 - \mathbf{x}\_0)} \right). \tag{43}$$

and a quantity that we call the state separation distance. At the same time, with the initial and the final energies *Eβ* = *T*/2, *Eα* = *Eβ* + *ε*10, we obtain the dipole moments for the n-zone

$$X\_{a\notin}^{(n)} = Y\_{a\notin}^{(n)} = Z\_{a\notin}^{(n)} = \frac{\hbar}{\varepsilon\_{10}} \sqrt{\frac{2\varepsilon\_{10} + T}{M\_{\text{lt}}}} \approx \sqrt{\frac{2\hbar}{M\_{\text{lt}}\omega\_0}}\tag{44}$$

and for the p-zone,

$$X\_{a\circledast}^{(p)} = Y\_{a\circledast}^{(p)} = Z\_{a\circledast}^{(p)} = \frac{\hbar}{\varepsilon\_{10}} \sqrt{\frac{2\varepsilon\_{10} + T}{M\_p}} \approx \sqrt{\frac{2\hbar}{M\_p \omega\_0}}\tag{45}$$

The Markovian dissipative coefficients (46) describe a very strong, exponential decrease of the decay rate with the width *x*2 − *x*0 of the quantum dot barrier, given by the square of the overlap function (43), and a strong decrease with the distances *x*3 and *x*5 of the quantum dot separation from the two conduction regions, which enter at the denominators with power 3. The non-Markovian coefficients (48) describe a very strong decrease of the fluctuation rate with the separation distances *x*3 and *x*5, which enter at the denominators with power 5.

Quantum Injection Dots 311

The coupling of a quantum dot electron with a vibrational mode *α* is described by the potential

10*α* = −*h*¯ *ω*3/2 0

where M is the mass of the phonon quantization volume VP. For the density of phonon states

√2M3/2 *π*2*h*¯ 3

, *λP* ≡ 2*π*

= 2*π*2*h*¯ 2 VP*λ*2 *Pε*10 *kP*

(*x*) 01 | 2*μ*2 01

= 2*πh*¯ √2M*ε*10

*Mr*011*α* √M*h*¯ /2 , (49)

*h*¯ *ω*0. (50)

. (52)

, (53b)

1 − *e*−*ε*10/*T* (53a)

*D* . (54)

01 = 2 × 107 *s*−1. As we found by direct calculations (3), for

, (51)

*VEP* 01*α* = *VEP*

*ν* = 2*πv*

With (49)-(52), from (38) we obtain the decay (excitation) rates

*λP*

*λP*

*gP*(*h*¯ *ω*0) = VP

We consider the sound velocity *v* from the phonon wavelength expressions

*ω* = 2*πhv*¯ *ε*10

> *D* ≡ M VP

01 = *E*2

10 = *E*2

calculated from the Young elasticity coefficient *E* and the crystal density *D*:

*e ε*5 10 *πh*¯ 6*c*4*v*3*D* · |*c*

*e ε*5 10 *πh*¯ 6*c*4*v*3*D* ·

*v* ≈

where *Ee* = *Mc*2 is the rest energy of the electron, and *v* is the sound velocity, which can be

We notice that both systems of dissipation coefficients (46) and (53) are proportional to the squares of the state separation distance and overlap function. Expressions (53) describe a very strong dependence of the decay rates on the transition energy *ε*10, being proportional to this energy with power 5. However, they are valid for phonon wavelengths much larger than the distance between the atoms of the crystal lattice. Otherwise, the number of the density modes can no more be treated as a quasi-continuous function of frequency, and the probability of any non-resonant interaction vanishes (Mösbauer effect). We also found that for the rather low transition energies specific to the quantum injection dots, the decay rate due to the phonon coupling is rather low, e.g. for the structure presented at the end of the preceding section

the quantum injection dots, which are separated by potential barriers from the conduction electrons, the decay rate due to these electrons is much lower than the decay rate due to the

*E*

of energy ¯*hω*0, we obtain an expression similar to (12):

*λP* = *v*

matrix element (3):

and the crystal density

with *ε*10 = 0.1866 *eV*, we got *λP*

coupling to the phonons of the crystal lattice vibrations.

where we used the notations *ε*10=˙ *h*¯ *ω*0=˙ *E*1 − *E*0. With the quantum dot dipole moments (42), and the environment dipole moments (44)-(45), we calculate the matrix elements of the two-body potential (41). With these matrix elements, from (37) with the densities of states (12) and the occupation probabilities of these states (13), one obtains the dissipative coefficients for the coupling of a quantum dot with a dissipative cluster. By integration over all clusters of both hemispheres of the n and p conduction regions, with the quantization volumes V*n* = 1 *ND* and V*p* = 1 *NA* as differential volumes in these integrals, we obtain the dissipation coefficients (3):

$$\lambda\_{01}^{(\boldsymbol{n})} = \frac{4a^2c^2\sqrt{2M\_n}(\varepsilon\_{10} + \frac{T}{2})|c\_{01}^{(\boldsymbol{x})}|^2\mu\_{01}^2}{3\left(\frac{N\_D^{-1/3}}{2} - \chi\_3\right)^3\varepsilon\_{10}^{3/2}(e^{-(\mathcal{U}\_c + \varepsilon\_{10})/T} + 1)}\tag{46a}$$

$$\lambda\_{10}^{(\eta)} = \frac{4\alpha^2 c^2 \sqrt{2M\_n} (\varepsilon\_{10} + \frac{T}{2}) |c\_{01}^{(\chi)}|^2 \mu\_{01}^2}{3\left(\frac{N\_D^{-1/3}}{2} - \chi\_3\right)^3 \varepsilon\_{10}^{3/2} (e^{(\mathcal{U}\_\ell + \varepsilon\_{10})/T} + 1)} \tag{46b}$$

$$\lambda\_{01}^{(p)} = \frac{4\alpha^2 c^2 \sqrt{2M\_p} (\varepsilon\_{10} + \frac{T}{2}) |c\_{01}^{(x)}|^2 \mu\_{01}^2}{3\left(\frac{N\_A^{-1/3}}{2} + \varkappa\_5\right)^3 \varepsilon\_{10}^{3/2} (\varepsilon^{-(-L\_v + \varepsilon\_{10})/T} + 1)}\tag{46c}$$

$$\lambda\_{10}^{(p)} = \frac{4a^2 c^2 \sqrt{2M\_p} (\varepsilon\_{10} + \frac{T}{2}) |c\_{01}^{(\chi)}|^2 \mu\_{01}^2}{3 \left(\frac{N\_A^{-1/3}}{2} + \varkappa\_5\right)^3 \varepsilon\_{10}^{3/2} (e^{(-\mathcal{U}\_0 + \varepsilon\_{10})/T} + 1)},\tag{46d}$$

where

$$
\mu\_{01}^2 = \left(\frac{\mathbf{x}\_2 - \mathbf{x}\_0}{2} - \frac{1}{\mathbf{a}\_0 - \mathbf{a}\_1} + \frac{\hbar}{\sqrt{M\_n T}}\right) \left(\frac{\mathbf{x}\_2 - \mathbf{x}\_0}{2} - \frac{1}{\mathbf{a}\_0 - \mathbf{a}\_1} + \frac{\hbar}{\sqrt{M\_p T}}\right). \tag{47}
$$

is the square of the separation distance of the two states Ψ0(*r*) and Ψ1(*r*). From (35), by similar calculations we obtain the fluctuation coefficients of a quantum dot in the self-consistent field of dissipative clusters:

$$\left[\zeta\_{11}^{(n)}\right]^2 = \frac{a^2 c^2 M\_n^{3/2} T^{3/2}}{360\pi \sqrt{2\pi} \hbar^3} \cdot \frac{N\_D^{1/3} [A\_1^2 (\mathbf{x}\_0^3 - \mathbf{x}\_1^3) + \frac{1}{N\epsilon}]}{N\_\ell N\_\ell \left(\frac{N\_D^{-1/3}}{2} - \mathbf{x}\_3 + \frac{\mathbf{x}\_0 + \mathbf{x}\_1}{2}\right)^5} \tag{48a}$$

$$\left[\zeta\_{11}^{(p)}\right]^2 = \frac{a^2 c^2 M\_p^{3/2} T^{3/2}}{360\pi \sqrt{2\pi} \hbar^3} \cdot \frac{N\_A^{1/3} [A\_1^2 (\mathbf{x}\_0^3 - \mathbf{x}\_1^3) + \frac{1}{\mathcal{N}\epsilon}]}{N\_\ell N\_\mathbb{v} \left(\frac{N\_A^{-1/3}}{2} + \mathbf{x}\_5 - \frac{\mathbf{x}\_0 + \mathbf{x}\_1}{2}\right)^5} \tag{48b}$$

$$\left[\zeta\_{00}^{(n)}\right]^2 = \frac{a^2 c^2 M\_n^{3/2} T^{3/2}}{360\pi \sqrt{2\pi} \hbar^3} \cdot \frac{N\_D^{1/3} [A\_0^2 (\mathbf{x}\_4^3 - \mathbf{x}\_2^3) + \frac{1}{\text{Ne}}]}{N\_\ell N\_\ell \left(\frac{N\_D^{-1/3}}{2} - \mathbf{x}\_3 + \frac{\mathbf{x}\_4 + \mathbf{x}\_2}{2}\right)^5} \tag{48c}$$

$$\left[\zeta\_{00}^{(p)}\right]^2 = \frac{a^2 c^2 M\_p^{3/2} T^{3/2}}{360\pi \sqrt{2\pi} \hbar^3} \cdot \frac{N\_A^{1/3} \left[A\_0^2 (\mathbf{x}\_4^3 - \mathbf{x}\_2^3) + \frac{1}{N\epsilon}\right]}{N\_\ell N\_\upsilon \left(\frac{N\_A^{-1/3}}{2} + \mathbf{x}\_5 - \frac{\mathbf{x}\_4 + \mathbf{x}\_2}{2}\right)^5}. \tag{48d}$$

*ND*

(46a)

(46b)

(46c)

, (46d)

5 (48a)

5 (48b)

5 (48c)

5 . (48d)

. (47)

12 Will-be-set-by-IN-TECH

where we used the notations *ε*10=˙ *h*¯ *ω*0=˙ *E*1 − *E*0. With the quantum dot dipole moments (42), and the environment dipole moments (44)-(45), we calculate the matrix elements of the two-body potential (41). With these matrix elements, from (37) with the densities of states (12) and the occupation probabilities of these states (13), one obtains the dissipative coefficients for the coupling of a quantum dot with a dissipative cluster. By integration over all clusters of both hemispheres of the n and p conduction regions, with the quantization volumes V*n* = 1

> 3 *ε* 3/2

> 3 *ε* 3/2

> 3 *ε* 3/2

> 3 *ε* 3/2

is the square of the separation distance of the two states Ψ0(*r*) and Ψ1(*r*). From (35), by similar calculations we obtain the fluctuation coefficients of a quantum dot in the self-consistent field

√2*πh*¯ 3 · *N*1/3

√2*πh*¯ 3 · *N*1/3

√2*πh*¯ 3 · *N*1/3

√2*πh*¯ 3 · *N*1/3

*NeNc*

*NeNv*

*NeNc*

*NeNv*

01 = 4*α*2*c*2√2*Mn*(*ε*10 + *T*

10 = 4*α*2*c*2√2*Mn*(*ε*10 + *T*

01 = 4*α*2*c*22*Mp*(*ε*10 + *T*

10 = 4*α*2*c*22*Mp*(*ε*10 + *T*

+

*n T*3/2

*p T*3/2

*n T*3/2

*p T*3/2

*h*¯ √*MnT*

*NA* as differential volumes in these integrals, we obtain the dissipation coefficients

2 )|*c* (*x*) 01 | 2*μ*2 01

2 )|*c* (*x*) 01 | 2*μ*2 01

10 (*e*−(*Uc*+*ε*10)/*T* + 1)

10 (*e*(*Uc*+*ε*10)/*T* + 1)

10 (*e*−(−*Uv*+*ε*10)/*T* + 1)

10 (*e*(−*Uv*+*ε*10)/*T* + 1)

2 − 1

*α*0 − *α*1

1) + 1 *Ne* ]

1) + 1 *Ne*]

2) + 1 *Ne* ]

2) + 1 *Ne*]

2

2

2

2

2 − *x*3 + *x*0+*x*1

2 + *x*5 − *x*0+*x*1

2 − *x*3 + *x*4+*x*2

2 + *x*5 − *x*4+*x*2

+

*h*¯ *MpT*

2 )|*c* (*x*) 01 | 2*μ*2 01

2 )|*c* (*x*) 01 | 2*μ*2 01

*x*2 − *x*0

*D* [*A*2 1(*x*3 0 − *x*3

*A* [*A*2 1(*x*3 0 − *x*3

*D* [*A*2 0(*x*3 4 − *x*3

*A* [*A*2 0(*x*3 4 − *x*3

> *N*−1/3 *A*

 *N*−1/3 *D*

 *N*−1/3 *A*

 *N*−1/3 *D*

and V*p* = 1

*λ*(*n*)

*λ*(*n*)

*λ*(*p*)

*λ*(*p*)

2 − 1

*x*2 − *x*0

 *ζ* (*n*) 11 2

 *ζ* (*p*) 11 2

 *ζ* (*n*) 00 2

 *ζ* (*p*) 00 2 3 *N*−1/3 *D* 2 − *x*3

3 *N*−1/3 *D* 2 − *x*3

3 *N*−1/3 *A* 2 + *x*5

3 *N*−1/3 *A* 2 + *x*5

*α*0 − *α*1

= *α*2*c*2*M*3/2

360*π*

= *α*2*c*2*M*3/2

360*π*

= *α*2*c*2*M*3/2

360*π*

= *α*2*c*2*M*3/2

360*π*

(3):

where

*μ*2 01 =

of dissipative clusters:

The Markovian dissipative coefficients (46) describe a very strong, exponential decrease of the decay rate with the width *x*2 − *x*0 of the quantum dot barrier, given by the square of the overlap function (43), and a strong decrease with the distances *x*3 and *x*5 of the quantum dot separation from the two conduction regions, which enter at the denominators with power 3. The non-Markovian coefficients (48) describe a very strong decrease of the fluctuation rate with the separation distances *x*3 and *x*5, which enter at the denominators with power 5.

The coupling of a quantum dot electron with a vibrational mode *α* is described by the potential matrix element (3):

$$V\_{01a}^{EP} = V\_{10a}^{EP} = -\hbar\omega\_0^{3/2} \frac{M\vec{r}\_{01}\vec{1}\_a}{\sqrt{M\hbar\hbar/2}} \,\prime \,\tag{49}$$

where M is the mass of the phonon quantization volume VP. For the density of phonon states of energy ¯*hω*0, we obtain an expression similar to (12):

$$\mathbf{g}\_P(\hbar\omega\_0) = \mathcal{V}\_{\mathcal{P}} \frac{\sqrt{2}\mathcal{M}^{3/2}}{\pi^2 \hbar^3} \sqrt{\hbar\omega\_0}.\tag{50}$$

We consider the sound velocity *v* from the phonon wavelength expressions

$$
\lambda\_P = \frac{v}{\nu} = \frac{2\pi v}{\omega} = \frac{2\pi \hbar v}{\varepsilon\_{10}}, \qquad \lambda\_P \equiv \frac{2\pi}{k\_P} = \frac{2\pi \hbar}{\sqrt{\mathcal{D} \mathcal{M} \varepsilon\_{10}}}, \tag{51}
$$

and the crystal density

$$D \equiv \frac{\mathcal{M}}{\mathcal{V}\_{\mathcal{P}}} = \frac{2\pi^2 \hbar^2}{\mathcal{V}\_{\mathcal{P}} \lambda\_{\mathcal{P}}^2 \varepsilon\_{10}}.\tag{52}$$

With (49)-(52), from (38) we obtain the decay (excitation) rates

$$
\lambda\_{01}^P = \frac{E\_e^2 \varepsilon\_{10}^5}{\pi \hbar^6 c^4 v^3 D} \cdot \frac{|c\_{01}^{(\chi)}|^2 \mu\_{01}^2}{1 - e^{-\varepsilon\_{10}/T}} \tag{53a}
$$

$$
\lambda\_{10}^P = \frac{E\_\varepsilon^2 \varepsilon\_{10}^5}{\pi \hbar^6 c^4 v^3 D} \cdot \frac{|c\_{01}^{(\chi)}|^2 \mu\_{01}^2}{e^{\varepsilon\_{10}/T} - 1} \tag{53b}
$$

where *Ee* = *Mc*2 is the rest energy of the electron, and *v* is the sound velocity, which can be calculated from the Young elasticity coefficient *E* and the crystal density *D*:

$$
v \approx \sqrt{\frac{E}{D}}.\tag{54}$$

We notice that both systems of dissipation coefficients (46) and (53) are proportional to the squares of the state separation distance and overlap function. Expressions (53) describe a very strong dependence of the decay rates on the transition energy *ε*10, being proportional to this energy with power 5. However, they are valid for phonon wavelengths much larger than the distance between the atoms of the crystal lattice. Otherwise, the number of the density modes can no more be treated as a quasi-continuous function of frequency, and the probability of any non-resonant interaction vanishes (Mösbauer effect). We also found that for the rather low transition energies specific to the quantum injection dots, the decay rate due to the phonon coupling is rather low, e.g. for the structure presented at the end of the preceding section with *ε*10 = 0.1866 *eV*, we got *λP* 01 = 2 × 107 *s*−1. As we found by direct calculations (3), for the quantum injection dots, which are separated by potential barriers from the conduction electrons, the decay rate due to these electrons is much lower than the decay rate due to the coupling to the phonons of the crystal lattice vibrations.

varying terms are neglected. We consider the summations over the environmental states as integrals over a quasi-continuum of states, with the densities *g*(*εα*) and *g*(*εβ*) and occupation probabilities *f*(*εα*) and *f*(*εβ*), and neglect the thermal energies *εβ* ∼ *T* in comparison with the

Quantum Injection Dots 313

i*kx* + *a*−*e*−i*kx*) − �*c*+

Λ = *πh*¯ *ω*2*g*(*h*¯ *ω*)(�

2¯*h*

Unlike the master equation for an electromagnetic field mode derived in (12), we considered explicit expressions of the electron-field potential of interaction, and neglected the thermal energy in comparison with the transition energy. With (65)-(67), the dissipation coefficient

Λ = Ω V*S*

and the two quantization volumes, V of the electromagnetic field and V*S* of the dissipative

of the electromagnetic field V means a measuring process corresponding to a confinement in this volume. The electromagnetic field can not be quantized in a volume V*S*, but in a much larger one, with much larger dimensions than the field wavelength *λ*. For an electromagnetic

the radiation density is equal to the electromagnetic field density times the light velocity, *S* =

We notice that the master equation (63) describes an electromagnetic field quantized in a unit volume V = 1*m*3 in interaction with an electron system occupying a much smaller volume V*S* = *V*(*n*), *V*(*p*). A system of *N* dissipative electrons can be taken into account by multiplying

2*Mc*2*ω*

*NA* for a p-type region. From physical point of view, a quantization volume

2 [*J*/*m*3] is calculated with this quantization volume in the

Ω = 4*α*

electron system. We consider a quantization volume V*S* = *Vn* = 1

field, we consider a unit quantization volume V = 1*V* = 13

√2*M*3/2 *π*2*h*¯ 3

√

+]+[*a*−*ρ*(*t*), *a*+

*K*� *Rαβ*)

*i cj*�(*a*+

+*e*−i*kx* + *a*+

V (65)

*Mω* (66)

V (68)

*h*¯ , (69)

*ND* for an n-type region,

*L* = 1*m*3, because, in this case,

−]+[*a*−, *ρ*(*t*)*a*+

−*e*

−] ,

2, (64)

*h*¯ *ω*. (67)

i*kx*), *ρ*(*t*)

(63)

transition energy ¯*hω*. We obtain the quantum master equation

−*a*−, *ρ*(*t*)]

+]+[*a*+, *ρ*(*t*)*a*+

depending on quantities which, according to (9), (44), and (12), are of the form

*K* = *α λ*

*Rαβ* =

*g*(*h*¯ *ω*) = V*S*

+*a*+ + *a*+

[*a*+*ρ*(*t*), *a*+

*d*

(64) takes a form

or V*S* = *Vp* = 1

depending on the quantity

*wEc* [*W*/*m*2], where *wE* = *ε*0*E*2

expression (8)-(9) of the field.

*dt ρ*(*t*) = − i*ω*[*a*+

+ ∑ *j*>*i ωji*� *K*�*rij* �*c*+ *j ci*�(*a*+*e*

+ Λ

with the dissipation coefficient

#### **4. Dissipative dynamics of electromagnetic field**

The operation of a semiconductor device with quantum dots interacting with a quasi-resonant electromagnetic field is based on the transparency of the host semiconductor structure, with a band to band transition frequency much higher than the quantum dot transition frequency. That means that this electromagnetic field is absorbed by the host semiconductor structure mainly by intra-band transitions, essentially depending on overlap functions of thermal states with excited states populated (depopulated) by these transitions.

The dynamics of this electromagnetic field, with a potential *V*˜(*t*) of interaction with a quantum dot and a potential *V*˜ *E*(*t*) of interaction with a conduction electron (hole), is described by a system of equations of the form (28), (30), (31) (33), which, in the second-order approximation, provides

$$\frac{d}{dt}\vec{\rho}(t) = \mathcal{B}^{(1)}[\vec{\rho}(t), t] + \mathcal{B}^{(2)}[\vec{\rho}(t), t] \tag{55a}$$

$$\tilde{\boldsymbol{B}}^{(1)}[\tilde{\boldsymbol{\rho}}(t), t] = -\frac{\mathbf{i}}{\hbar} \text{Tr}\_{E}[\tilde{\boldsymbol{V}}(t) + \tilde{\boldsymbol{V}}^{E}(t), \boldsymbol{R} \otimes \tilde{\boldsymbol{\rho}}(t)] \tag{55b}$$

$$\tilde{\mathcal{K}}^{(1)}(t) = \int\_0^t \left\{ -\frac{\mathbf{i}}{\hbar} [\tilde{V}(t') + \tilde{V}^E(t'), \mathbb{R} \otimes \tilde{\rho}(t')] - \mathbb{R} \otimes \tilde{\mathcal{B}}^{(1)}[\tilde{\rho}(t'), t'] \right\} dt' \tag{55c}$$

$$\tilde{\mathcal{B}}^{(2)}[\tilde{\rho}(t),t] = -\frac{\mathrm{i}}{\hbar}Tr\_E[\tilde{V}(t) + \tilde{V}^E(t), \tilde{\chi}^{(1)}(t)].\tag{55d}$$

The interaction potential *V*˜(*t*) is obtained from (5)-(6), while the dissipative potential *V*˜ *E*(*t*) is given by the similar expressions

$$V^E = \frac{e}{M}\vec{P}\vec{A}\tag{56}$$

and

$$\vec{P} = \mathrm{i}M \sum\_{\alpha\beta} \omega\_{\alpha\beta} \vec{R}\_{\alpha\beta} c\_{\alpha}^+ c\_{\beta}. \tag{57}$$

From (5)-(7) and (56)-(57), we derive expressions of these potentials depending only on the positive transition frequencies *ωji*(*j* > *i*) and *ωαβ*(*α* > *β*), and take into account the so called "rotating-wave approximation", which includes only conservative processes, when an electron excitation is correlated only with a photon annihilation, while an electron decay is correlated only with a photon creation:

$$\nabla V = \mathbf{i} \sum\_{j>i} \hbar \omega\_{j\bar{i}l} \vec{\mathbf{K}} \vec{r}\_{l\bar{j}} \left[ c\_{\bar{j}}^{+} c\_{i} \left( a\_{+} e^{i\mathbf{k}\cdot\mathbf{x}} + a\_{-} e^{-i\mathbf{k}\cdot\mathbf{x}} \right) - c\_{i}^{+} c\_{\bar{j}} \left( a\_{+}^{+} e^{-i\mathbf{k}\cdot\mathbf{x}} + a\_{-}^{+} e^{i\mathbf{k}\cdot\mathbf{x}} \right) \right] \tag{58}$$

$$V^{E} = \mathbf{i} \sum\_{a>\beta} \hbar \omega\_{a\beta} \vec{\mathcal{K}} \vec{\mathcal{R}}\_{a\beta} \left[ c\_{a}^{+} c\_{\beta} \left( a\_{+} e^{i\mathbf{k}\cdot\mathbf{x}} + a\_{-} e^{-i\mathbf{k}\cdot\mathbf{x}} \right) - c\_{\beta}^{+} c\_{a} \left( a\_{+}^{+} e^{-i\mathbf{k}\cdot\mathbf{x}} + a\_{-}^{+} e^{i\mathbf{k}\cdot\mathbf{x}} \right) \right]. \tag{59}$$

We consider the time-dependent expressions of these operators in the interaction picture,

$$\vec{a}(t) = a e^{-i\omega t}, \tag{60}$$

$$\vec{a}^+(t) = a^+ e^{i\omega t} \tag{60}$$

$$
\mathfrak{c}\_{i}^{+}(t)\mathfrak{c}\_{j}(t) = \mathfrak{c}\_{i}^{+}\mathfrak{c}\_{j}e^{-\mathrm{i}\omega\_{\parallel}t}, \qquad \qquad \mathfrak{c}\_{j}^{+}(t)\mathfrak{c}\_{i}(t) = \mathfrak{c}\_{j}^{+}\mathfrak{c}\_{i}e^{\mathrm{i}\omega\_{\parallel}t} \tag{61}
$$

$$
\tilde{c}\_{\beta}^{+}(t)\tilde{c}\_{a}(t) = c\_{\beta}^{+}c\_{a}e^{-i\omega\_{a\beta}t}, \qquad \tilde{c}\_{a}^{+}(t)\tilde{c}\_{\beta}(t) = c\_{a}^{+}c\_{\beta}e^{i\omega\_{a\beta}t}.\tag{62}
$$

and take equations (55) for the mean-values of the electron operators. We retain only the slowly time varying terms, obtained from the resonance condition *ωαβ* = *ω*, while the rapidly 14 Will-be-set-by-IN-TECH

The operation of a semiconductor device with quantum dots interacting with a quasi-resonant electromagnetic field is based on the transparency of the host semiconductor structure, with a band to band transition frequency much higher than the quantum dot transition frequency. That means that this electromagnetic field is absorbed by the host semiconductor structure mainly by intra-band transitions, essentially depending on overlap functions of thermal states

The dynamics of this electromagnetic field, with a potential *V*˜(*t*) of interaction with a quantum dot and a potential *V*˜ *E*(*t*) of interaction with a conduction electron (hole), is described by a system of equations of the form (28), (30), (31) (33), which, in the second-order approximation,

), *R* ⊗ *ρ*˜(*t*

*TrE*[*V*˜(*t*) + *V*˜ *E*(*t*), *χ*˜(1)

The interaction potential *V*˜(*t*) is obtained from (5)-(6), while the dissipative potential *V*˜ *E*(*t*) is

*ωαβ*� *Rαβc*+

From (5)-(7) and (56)-(57), we derive expressions of these potentials depending only on the positive transition frequencies *ωji*(*j* > *i*) and *ωαβ*(*α* > *β*), and take into account the so called "rotating-wave approximation", which includes only conservative processes, when an electron excitation is correlated only with a photon annihilation, while an electron decay is correlated

i*kx* + *a*−*e*−i*kx*

We consider the time-dependent expressions of these operators in the interaction picture,

, *a*˜

*i cje*−i*ωjit*

*β cαe*−i*ωαβt*

i*kx* + *a*−*e*−i*kx*

, *c*˜

, *c*˜

and take equations (55) for the mean-values of the electron operators. We retain only the slowly time varying terms, obtained from the resonance condition *ωαβ* = *ω*, while the rapidly

− *c*+ *i cj a*+ +*e*

> − *c*+ *β cα a*+

+(*t*) = *a*+*e*

*j* (*t*)*c*˜*i*(*t*) = *c*+

*α* (*t*)*c*˜*β*(*t*) = *c*+

+

+

*VE* = *e M*� �

[*ρ*˜(*t*), *t*] (55a)

[*ρ*˜(*t* � ), *t* � ]

(*t*)]. (55d)

*PA*� (56)

−i*kx* + *a*+ −*e* i*kx*

+*e*−i*kx* + *a*+

*j cie*

*α cβe* i*ωαβt* −*e* i*kx*

i*ωt* (60)

i*ωjit* (61)

, (62)

*α cβ*. (57)

*dt*� (55c)

(58)

. (59)

*TrE*[*V*˜(*t*) + *V*˜ *E*(*t*), *R* ⊗ *ρ*˜(*t*)] (55b)

)] − *R* ⊗ *B*˜(1)

**4. Dissipative dynamics of electromagnetic field**

provides

and

*d*

*B*˜(1)

*χ*˜(1) (*t*) =

*B*˜(2)

given by the similar expressions

only with a photon creation:

*VE* = i ∑ *α*>*β*

*V* = i∑ *j*>*i* *h*¯ *ωji*� *K*�*rij c*+ *j ci a*+*e*

*h*¯ *ωαβ*� *K*� *Rαβ c*+ *α cβ a*+*e*

*c*˜ +

*c*˜ +

*a*˜(*t*) = *ae*−i*ωt*

*i* (*t*)*c*˜*j*(*t*) = *c*+

*β* (*t*)*c*˜*α*(*t*) = *c*+

*dt ρ*˜(*t*) = *B*˜(1)

[*ρ*˜(*t*), *t*] = − i

 *t* 0 − i *h*¯ [*V*˜(*t* � ) + *V*˜ *E*(*t* �

[*ρ*˜(*t*), *t*] = − i

with excited states populated (depopulated) by these transitions.

[*ρ*˜(*t*), *t*] + *B*˜(2)

*h*¯

*h*¯

� *P* = i*M* ∑ *αβ*

varying terms are neglected. We consider the summations over the environmental states as integrals over a quasi-continuum of states, with the densities *g*(*εα*) and *g*(*εβ*) and occupation probabilities *f*(*εα*) and *f*(*εβ*), and neglect the thermal energies *εβ* ∼ *T* in comparison with the transition energy ¯*hω*. We obtain the quantum master equation

$$\begin{split} \frac{d}{dt}\rho(t) &= -\mathbf{i}\omega \left[a\_{+}^{+}a\_{+} + a\_{-}^{+}a\_{-}\rho(t)\right] \\ &+ \sum\_{j>i} \omega \gamma\_{ji} \vec{\mathcal{K}} \vec{r}\_{ij} \left[\langle c\_{j}^{+}c\_{i}\rangle (a\_{+}e^{\mathrm{i}\mathbf{x}} + a\_{-}e^{-\mathrm{i}\mathbf{x}}) - \langle c\_{i}^{+}c\_{j}\rangle (a\_{+}^{+}e^{-\mathrm{i}\mathbf{x}} + a\_{-}^{+}e^{\mathrm{i}\mathbf{x}}), \rho(t)\right] \\ &+ \Lambda\left\{[a\_{+}\rho(t),a\_{+}^{+}] + [a\_{+},\rho(t)a\_{+}^{+}] + [a\_{-}\rho(t),a\_{-}^{+}] + [a\_{-},\rho(t)a\_{-}^{+}]\right\}, \end{split} \tag{63}$$

with the dissipation coefficient

$$
\Lambda = \pi \hbar \omega^2 g(\hbar \omega) (\vec{\mathcal{K}} \vec{\mathcal{R}}\_{\text{eff}})^2,\tag{64}
$$

depending on quantities which, according to (9), (44), and (12), are of the form

$$K = \sqrt{\alpha \frac{\lambda}{\mathcal{V}}}\tag{65}$$

$$R\_{a\beta} = \sqrt{\frac{2\hbar}{M\omega}}\tag{66}$$

$$g(\hbar\omega) = \mathcal{V}^S \frac{\sqrt{2}M^{3/2}}{\pi^2 \hbar^3} \sqrt{\hbar\omega}.\tag{67}$$

Unlike the master equation for an electromagnetic field mode derived in (12), we considered explicit expressions of the electron-field potential of interaction, and neglected the thermal energy in comparison with the transition energy. With (65)-(67), the dissipation coefficient (64) takes a form

$$
\Lambda = \Omega \frac{\mathcal{V}^S}{\mathcal{V}} \tag{68}
$$

depending on the quantity

$$
\Omega = 4\pi \sqrt{\frac{2Mc^2 \omega}{\hbar}},
\tag{69}
$$

and the two quantization volumes, V of the electromagnetic field and V*S* of the dissipative electron system. We consider a quantization volume V*S* = *Vn* = 1 *ND* for an n-type region, or V*S* = *Vp* = 1 *NA* for a p-type region. From physical point of view, a quantization volume of the electromagnetic field V means a measuring process corresponding to a confinement in this volume. The electromagnetic field can not be quantized in a volume V*S*, but in a much larger one, with much larger dimensions than the field wavelength *λ*. For an electromagnetic field, we consider a unit quantization volume V = 1*V* = 13 *L* = 1*m*3, because, in this case, the radiation density is equal to the electromagnetic field density times the light velocity, *S* = *wEc* [*W*/*m*2], where *wE* = *ε*0*E*2 2 [*J*/*m*3] is calculated with this quantization volume in the expression (8)-(9) of the field.

We notice that the master equation (63) describes an electromagnetic field quantized in a unit volume V = 1*m*3 in interaction with an electron system occupying a much smaller volume V*S* = *V*(*n*), *V*(*p*). A system of *N* dissipative electrons can be taken into account by multiplying

we notice that, in equations (70), this relation selects only the close terms, with *x*� ≈ *x*, while the farer terms in *x* − *x*� are washed up by the uncertainty Δ*k* in the oscillating functions under

Quantum Injection Dots 315

1*L*

1*L*

We notice that these are non-local in space equations including retarded contributions of the

the coordinate *x*. We consider the quantities under these integrals as spectral lines integrated

By integrating the first integral of the first equation two times by parts, for a large distance

i*k* A+(*x*�

*d dx*� A+(*x*�

2

2*k* A+(*x*, *t*) +

A+(*x*, *t*) sin *k*Δ*x* − *ω*Δ*t*

√3

cos *k*Δ*x* − *ω*Δ*t*

)]*dx*� = 1

 *x*−Δ*x*/2 0

= −1 + i

while, taking into account that on a vary short distance Δ*x* the field amplitude is practically

i*k* A+(*x*� , *t* �

These terms describe a slight variation of the wave-vector *k*, *k*� = *k* + *κ*, which means that the

1+ 2*ck*21 √ *L* 3Ω*D*

A+(*x*, *t*)

 = −1 2 *ω*0�

A+(*x*, *t*) = A˜+(*x*, *t*)*e*

)]*dx*� = 1

= 2 *k*

− Ω*D* 1*L*

uncertainty Δ*x*, in the vicinity of *x*, by extending these *x*�

*K*�*r*01S+(*x*, *t*) − Ω*D*

*K*�*r*01S−(*x*, *t*) + Ω*D*

+ Ω*D* 1*L*

over half-widths, *k*Δ*x* − *ω*Δ*t* = (*k* + Δ*k*)Δ*x* − (*ω* + Δ*ω*)Δ*t* = Δ*k*Δ*x* − Δ*ω*Δ*t* = *π*

− 1 i*k*

)−*ω*(*t*−*t* �

*dx* A+(*x*, *t*)

constant, the second integral of this equation takes a simple form

amplitude of the mean-value of the field operator takes a form

 *∂ ∂x*

)−*ω*(*t*−*t*�

*ck*21*L* � 1, while *κ* = *k*

A+(*x*, *t*) + *α*�

2 *ω*0�

> 2 *ω*0�

dissipation processes along the distance |*x* − *x*�

A+(*x*, *t*) + 1

)*e*−i[*k*(*x*−*x*�

*k*2 *d*

)*e*−i[*k*(*x*−*x*�

> *x*−Δ*x*/2 0

> *x*+Δ*x*/2 *x*−Δ*x*/2

> > *LD x*+Δ*x*/2

 *x*+Δ*x*/2 *x*−Δ*x*/2

A+(*x*� , *t* �

A+(*x*� , *t* � )*e*

A−(*x*� , *t* � )*e*−i[*k*(*x*�

A−(*x*� , *t* � )*e*−i[*k*(*x*�

> , *t* �

)*e*−i[*k*(*x*−*x*�

, *t* �

)*e*−i[*k*(*x*−*x*�

2

� *k*, we get a field equation

)−*ω*(*t*−*t* �

)−*ω*(*t*−*t* � )]*dx*�

−*x*)−*ω*(*t*−*t*�

−*x*)−*ω*(*t*−*t*�

)]*dx*� (76a)

)]*dx*� (76b)

)]*dx*� .

6 + *π* 6 > 1 2 + 1 2 .

*x*−Δ*x*/2 0

)]*dx*�

(77)

(78)

)*e*−i[*k*(*x*−*x*�

−i[*k*(*x*−*x*�

)−*ω*(*t*−*t*� )]

− i sin *k*Δ*x* − *ω*Δ*t*

)−*ω*(*t*−*t*� )]

> = 1 *k*

i*κx*. (79)

√3 − i 2*k*2

)−*ω*(*t*−*t*�

2

*dx* A+(*x*, *t*),

*x*+Δ*x*/2

*x*−Δ*x*/2

A+(*x*, *t*).

*K*�*r*01S+(*x*, *t*), (80)

*d*

)*e*−i[*k*(*x*−*x*�

the *x*�

*d*

*d*

*x* − *x*� we obtain *x*−Δ*x*/2 0

> = − i *k*

 *x*+Δ*x*/2 *x*−Δ*x*/2

We obtain the field equations

*dt*A+(*x*, *t*) = −1

*dt*A−(*x*, *t*) = −1

A+(*x*� , *t* �

A+(*x*� , *t* �

Taking into account that Ω*D*

*∂ ∂t*

A+(*x*, *t*) + *c*

the dissipative coefficient Λ with *N*. However, in such a description, the electromagnetic field is considered of a constant amplitude inside the quantization volume V = 1*V*, while, in fact, this amplitude undertakes a spatial variation due to the interaction with the system quantized in a volume V*S* � V, i.e. propagation characteristics as the absorption coefficient and the refractive index inside the field quantization volume are not taken into account. We take into account the spatial variation of the electromagnetic field by considering this field as being given by an *x* dependent density matrix, as product of density matrices for the two counter-propagating waves, *ρ*(*x*, *t*) = *ρ*+(*x*, *t*)*ρ*−(*x*, *t*), and taking the dissipative terms as integrals over the paths traveled by these waves. Considering a distribution of *ND* dissipative clusters over the the thickness *LD* of the device, from the master equation (63), we get

*d dt ρ*+(*x*, *t*)=−i*ω*[*a*+ +*a*+, *ρ*+(*x*, *t*)]+∑ *j*>*i ωji*� *K*�*rij* �*c*+ *j ci*�*a*+*e* i*kx*−�*c*+ *i cj*�*a*+ +*e*−i*kx*, *ρ*+(*x*, *t*) (70a) + Ω*D* 1*L x* 0 [*a*+*ρ*+(*x*� , *t* � ), *a*+ +]+[*a*+, *ρ*+(*x*� , *t* � )*a*+ +] *e* −i*k*(*x*−*x*� ) *dx*� *d dt ρ*−(*x*, *t*)=−i*ω*[*a*+ −*a*−, *ρ*−(*x*, *t*)]+∑ *j*>*i ωji*� *K*�*rij* �*c*+ *j ci*�*a*−*e*−i*kx*−�*c*+ *i cj*�(*a*+ −*e*i*kx*, *ρ*−(*x*, *t*) (70b) + Ω*D* 1*L LD x* [*a*−*ρ*−(*x*� , *t* � ), *a*+ −]+[*a*−, *ρ*−(*x*� , *t* � )*a*+ −] *e*−i*k*(*x*� −*x*) *dx*� ,

depending on the dissipative coefficient

$$
\Omega\_D = \Omega \frac{\mathbf{1}\_L^2 L\_D}{\mathbf{1}\_L^3} = \Omega \frac{L\_D}{\mathbf{1}\_L} \,\tag{71}
$$

obtained by summation over the dissipative clusters with the volume 12 *LLD* in the quantization volume 13 *L*. The exponential factors in the integrals describe the delay of the field propagating from the coordinate *x*� of a dissipative element to the coordinate *x* of the density matrix of this field at this coordinate. These equations describe the dissipative dynamics of a forward electromagnetic wave, propagating from *x*� = 0 to *x*� = *x*, and of a backward electromagnetic wave, propagating from *x*� = *LD* to *x*� = *x* < *LD*. With these equations, we calculate the mean-values of the field operators

$$a\_{+}(\mathbf{x},t) = \langle a\_{+}\rangle = \text{Tr}\left\{a\_{+}\rho\_{+}(\mathbf{x},t)\right\} = \mathcal{A}\_{+}(\mathbf{x},t)e^{-\dot{\mathbf{k}}\omega t} \tag{72a}$$

$$a\_{-}(\mathbf{x},t) = \langle a\_{-}\rangle = \text{Tr}\left\{a\_{-}\rho\_{-}(\mathbf{x},t)\right\} = \mathcal{A}\_{-}(\mathbf{x},t)e^{-i\omega t}.\tag{72b}$$

For an array of two-level systems with the coordinate *x*, interacting with the electromagnetic field with the frequency *ω* ≈ *ω*0, we define the time slowly-varying amplitude of the polarization S(*x*, *t*) by the relations

$$
\rho \langle \mathbf{c}\_i^+ \mathbf{c}\_j \rangle = \rho\_{jl}(\mathbf{x}, t) = \frac{1}{2} \mathcal{S}(\mathbf{x}, t) e^{-i\omega t},\tag{73}
$$

$$\mathcal{S}(\mathbf{x},t) = \mathcal{S}\_{+}(\mathbf{x},t)e^{i\mathbf{k}\mathbf{x}} + \mathcal{S}\_{-}(\mathbf{x},t)e^{-i\mathbf{k}\mathbf{x}},\tag{74}$$

where S+(*x*, *t*), S−(*x*, *t*) are slowly-varying in space and time amplitudes of the polarization induced by the two counter-propagating waves of the field. Having in view Heisenberg's uncertainty principle

$$
\Delta k \Delta x \geq \frac{1}{2} \, \qquad \qquad \Delta \omega \Delta t \geq \frac{1}{2} \, \tag{75}
$$

16 Will-be-set-by-IN-TECH

the dissipative coefficient Λ with *N*. However, in such a description, the electromagnetic field is considered of a constant amplitude inside the quantization volume V = 1*V*, while, in fact, this amplitude undertakes a spatial variation due to the interaction with the system quantized in a volume V*S* � V, i.e. propagation characteristics as the absorption coefficient and the refractive index inside the field quantization volume are not taken into account. We take into account the spatial variation of the electromagnetic field by considering this field as being given by an *x* dependent density matrix, as product of density matrices for the two counter-propagating waves, *ρ*(*x*, *t*) = *ρ*+(*x*, *t*)*ρ*−(*x*, *t*), and taking the dissipative terms as integrals over the paths traveled by these waves. Considering a distribution of *ND* dissipative clusters over the the thickness *LD* of the device, from the master equation (63), we get

i*kx*−�*c*+

+]+[*a*+, *ρ*+(*x*�

*j ci*�*a*−*e*−i*kx*−�*c*+

−]+[*a*−, *ρ*−(*x*�

= Ω *LD* 1*L*

*L*. The exponential factors in the integrals describe the delay of the field propagating

*i cj*�*a*+

*i cj*�(*a*+

−i*ωt*

, *t* � )*a*+ +] *e* −i*k*(*x*−*x*� ) *dx*�

, *t* � )*a*+ −] *e*−i*k*(*x*� −*x*) *dx*� ,

+*e*−i*kx*, *ρ*+(*x*, *t*)

−*e*i*kx*, *ρ*−(*x*, *t*)

*LLD* in the quantization

−i*ωt* (72a)

, (73)

−i*kx*, (74)

, (75)

. (72b)

, (71)

 (70a)

 (70b)

*d*

*d*

volume 13

*dt ρ*+(*x*, *t*)=−i*ω*[*a*+

*dt ρ*−(*x*, *t*)=−i*ω*[*a*+

+*a*+, *ρ*+(*x*, *t*)]+∑

−*a*−, *ρ*−(*x*, *t*)]+∑

 *LD x*

 *x* 0

+ Ω*D* 1*L*

+ Ω*D* 1*L*

depending on the dissipative coefficient

mean-values of the field operators

polarization S(*x*, *t*) by the relations

uncertainty principle

*j*>*i ωji*� *K*�*rij* �*c*+ *j ci*�*a*+*e*

*j*>*i ωji*� *K*�*rij* �*c*+

Ω*D* = Ω

obtained by summation over the dissipative clusters with the volume 12

�*c*+

[*a*−*ρ*−(*x*�

[*a*+*ρ*+(*x*�

, *t* � ), *a*+

, *t* � ), *a*+

12 *LLD* 13 *L*

from the coordinate *x*� of a dissipative element to the coordinate *x* of the density matrix of this field at this coordinate. These equations describe the dissipative dynamics of a forward electromagnetic wave, propagating from *x*� = 0 to *x*� = *x*, and of a backward electromagnetic wave, propagating from *x*� = *LD* to *x*� = *x* < *LD*. With these equations, we calculate the

*a*+(*x*, *t*) = �*a*+� = *Tr* {*a*+*ρ*+(*x*, *t*)} = A+(*x*, *t*)*e*

*a*−(*x*, *t*) = �*a*−� = *Tr* {*a*−*ρ*−(*x*, *t*)} = A−(*x*, *t*)*e*

*i cj*� = *ρji*(*x*, *t*) = 1

1 2

S(*x*, *t*) = S+(*x*, *t*)*e*

Δ*k*Δ*x* ≥

For an array of two-level systems with the coordinate *x*, interacting with the electromagnetic field with the frequency *ω* ≈ *ω*0, we define the time slowly-varying amplitude of the

where S+(*x*, *t*), S−(*x*, *t*) are slowly-varying in space and time amplitudes of the polarization induced by the two counter-propagating waves of the field. Having in view Heisenberg's

2

, Δ*ω*Δ*t* ≥

S(*x*, *t*)*e*

i*kx* + S−(*x*, *t*)*e*

−i*ωt*

1 2 we notice that, in equations (70), this relation selects only the close terms, with *x*� ≈ *x*, while the farer terms in *x* − *x*� are washed up by the uncertainty Δ*k* in the oscillating functions under the *x*� -dependent integrals. By definition, these *x*-dependent integrals describe the attenuation of an electromagnetic wave squeezed in the *x*-domain, Δ*x* = 0. We take into account a finite uncertainty Δ*x*, in the vicinity of *x*, by extending these *x*� -integrals with the half-width Δ*x*/2. We obtain the field equations

$$\begin{split} \frac{d}{dt}\mathcal{A}\_{+}(\mathbf{x},t) &= -\frac{1}{2}\omega\_{0}\vec{\mathcal{K}}\vec{r}\_{01}\mathcal{S}\_{+}(\mathbf{x},t) - \frac{\Omega\_{D}}{1\_{L}}\int\_{0}^{\mathbf{x}-\Delta\mathbf{x}/2} \mathcal{A}\_{+}(\mathbf{x}',t')e^{-i[k(\mathbf{x}-\mathbf{x}')-\omega(t-t')]}d\mathbf{x}' \\ &- \frac{\Omega\_{D}}{1\_{L}}\int\_{\mathbf{x}-\Delta\mathbf{x}/2}^{\mathbf{x}+\Delta\mathbf{x}/2} \mathcal{A}\_{+}(\mathbf{x}',t')e^{-i[k(\mathbf{x}-\mathbf{x}')-\omega(t-t')]}d\mathbf{x}' \\ \frac{d}{dt}\mathcal{A}\_{-}(\mathbf{x},t) &= -\frac{1}{2}\omega\_{0}\vec{\mathcal{K}}\vec{r}\_{01}\mathcal{S}\_{-}(\mathbf{x},t) + \frac{\Omega\_{D}}{1\_{L}}\int\_{\mathbf{x}+\Delta\mathbf{x}/2}^{L\_{D}} \mathcal{A}\_{-}(\mathbf{x}',t')e^{-i[k(\mathbf{x}'-\mathbf{x})-\omega(t-t')]}d\mathbf{x}' \\ &+ \frac{\Omega\_{D}}{1\_{L}}\int\_{\mathbf{x}-\Delta\mathbf{x}/2}^{\mathbf{x}+\Delta\mathbf{x}/2} \mathcal{A}\_{-}(\mathbf{x}',t')e^{-i[k(\mathbf{x}'-\mathbf{x})-\omega(t-t')]}d\mathbf{x}'. \end{split} \tag{76b}$$

We notice that these are non-local in space equations including retarded contributions of the dissipation processes along the distance |*x* − *x*� |, and absorption processes in the vicinity Δ*x* of the coordinate *x*. We consider the quantities under these integrals as spectral lines integrated over half-widths, *k*Δ*x* − *ω*Δ*t* = (*k* + Δ*k*)Δ*x* − (*ω* + Δ*ω*)Δ*t* = Δ*k*Δ*x* − Δ*ω*Δ*t* = *π* 6 + *π* 6 > 1 2 + 1 2 . By integrating the first integral of the first equation two times by parts, for a large distance *x* − *x*� we obtain

$$\begin{split} \int\_{0}^{\mathbf{x}-\Delta\mathbf{x}/2} \mathcal{A}\_{+}(\mathbf{x}',t') e^{-i[\mathbf{k}(\mathbf{x}-\mathbf{x}')-\omega(t-t')]} d\mathbf{x}' &= \frac{1}{i\mathbf{k}} \mathcal{A}\_{+}(\mathbf{x}',t') e^{-i[\mathbf{k}(\mathbf{x}-\mathbf{x}')-\omega(t-t')]} \Big|\_{0}^{\mathbf{x}-\Delta\mathbf{x}/2} \\ &- \frac{1}{i\mathbf{k}} \int\_{0}^{\mathbf{x}-\Delta\mathbf{x}/2} \frac{d}{d\mathbf{x}'} \mathcal{A}\_{+}(\mathbf{x}',t') e^{-i[\mathbf{k}(\mathbf{x}-\mathbf{x}')-\omega(t-t')]} d\mathbf{x}' \\ &= \left[ -\frac{\mathbf{i}}{k} \mathcal{A}\_{+}(\mathbf{x},t) + \frac{1}{k^{2}} \frac{d}{d\mathbf{x}} \mathcal{A}\_{+}(\mathbf{x},t) \right] \left[ \cos\left(\frac{k\Delta\mathbf{x}-\omega\Delta t}{2}\right) - \mathbf{i} \sin\left(\frac{k\Delta\mathbf{x}-\omega\Delta t}{2}\right) \right] \\ &= -\frac{1+\mathbf{i}\sqrt{3}}{2k} \mathcal{A}\_{+}(\mathbf{x},t) + \frac{\sqrt{3}-\mathbf{i}}{2k^{2}} \frac{d}{d\mathbf{x}} \mathcal{A}\_{+}(\mathbf{x},t), \end{split} \tag{77}$$

while, taking into account that on a vary short distance Δ*x* the field amplitude is practically constant, the second integral of this equation takes a simple form

$$\begin{split} \int\_{\mathbf{x}-\Delta\mathbf{x}/2}^{\mathbf{x}+\Delta\mathbf{x}/2} \mathcal{A}\_{+}(\mathbf{x}',t')e^{-\mathbf{i}[\mathbf{k}(\mathbf{x}-\mathbf{x}')-\omega(t-t')]}d\mathbf{x}' &= \frac{1}{i\mathbf{k}}\mathcal{A}\_{+}(\mathbf{x}',t')e^{-\mathbf{i}[\mathbf{k}(\mathbf{x}-\mathbf{x}')-\omega(t-t')]} \Big|\_{\mathbf{x}-\Delta\mathbf{x}/2}^{\mathbf{x}+\Delta\mathbf{x}/2} \\ &= \frac{2}{\mathbf{k}}\mathcal{A}\_{+}(\mathbf{x},t)\sin\left(\frac{\mathbf{k}\Delta\mathbf{x}-\omega\Delta t}{2}\right) = \frac{1}{\mathbf{k}}\mathcal{A}\_{+}(\mathbf{x},t). \end{split} \tag{78}$$

These terms describe a slight variation of the wave-vector *k*, *k*� = *k* + *κ*, which means that the amplitude of the mean-value of the field operator takes a form

$$\mathcal{A}\_{+}(\mathbf{x},t) = \mathcal{A}\_{+}(\mathbf{x},t)e^{i\mathbf{x}\cdot\mathbf{x}}.\tag{79}$$

Taking into account that Ω*D ck*21*L* � 1, while *κ* = *k* 1+ 2*ck*21 √ *L* 3Ω*D* � *k*, we get a field equation

$$\frac{\partial}{\partial t} \mathcal{A}\_{+}(\mathbf{x}, t) + c \left[ \frac{\partial}{\partial \mathbf{x}} \mathcal{A}\_{+}(\mathbf{x}, t) + \mathbf{a}^{\prime} \mathcal{A}\_{+}(\mathbf{x}, t) \right] = -\frac{1}{2} \omega\_{0} \vec{\mathcal{K}} \vec{r}\_{01} \mathcal{S}\_{+}(\mathbf{x}, t), \tag{80}$$

with an absorption coefficient

$$
\alpha' = \frac{2\alpha c}{1\_L} \sqrt{\frac{2M}{\hbar \omega}} \frac{L\_D}{1\_L}. \tag{81}
$$

From the expression (8) of the quantized electric field

mean-values of the annihilation operators of the form

� *E*� = 1 2

while the amplitudes of the two counter-propagating waves are

E (*t*) = E+(*t*)*e*

boundary condition for the output mirror of transmission coefficient T :

for the fluctuation rate of the self-consistent field, and

+*γ*2 *n t t*−*τ*

from (84)-(91) we obtain equations for the slowly-varying amplitudes

S+(*t* � )*e* −i[*φ*10(*t* � )+*ω*(*t*−*t*�

*dt w*(*t*) = −*γ*�[*w*(*t*) − *wT*]+(2 − T )i*g*

E−(*t*) = −

E+(*t*) = 2i *h*¯ *ω*

E−(*t*) = 2i *h*¯ *ω*

In this description we neglect the variation of the amplitudes inside the cavity, by taking into account these two amplitudes only as mean-values over the space coordinate, related by the

√

*g* = *e h*¯

*wT* = − *λ*01 − *λ*10 *λ*01 + *λ*10

*dt* S+(*t*) = −[*γ*⊥ + i(*ω*0 + *γn* − *ω*)]S+(*t*) + i*g*E+(*t*)*w*(*t*) (97a)

1 2 E ∗

)]*dt*�

we get the mean-value of this field,

With the notations

for the coupling coefficient,

for the dephasing rate,

*d*

*d*

for the decay rate,

with the time slowly-varying amplitude

�*a*−� = *a*¯−(*t*)*e*

Quantum Injection Dots 317

E (*t*)*e*−i*ωt* + E ∗(*t*)*e*

*e*

*e*

−i*ωt*

i*ωt*

*E* in the plane-wave approximation, and

, (87b)

, (88)

�*a*+� = *a*¯+(*t*)*e*−i*ωt* (87a)

i*kx* + E−(*t*)*e*−i*kx*, (89)

*Ka*¯+(*t*) (90a)

*Ka*¯−(*t*). (90b)

1 − T E+(*t*). (91)

*r*10 (92)

, (96)

+(*t*)

. (97b)

*γ*⊥ = *λ*01 + *λ*10 (93)

*γ*� = 2(*λ*01 + *λ*10) (94)

*γn* = |*ζ*11 − *ζ*00| (95)

+(*t*)S+(*t*) − E+(*t*)S∗

In the following calculations, we are interested in a form of this equation in a cavity with the length *LD*,

$$\frac{d}{dt}\mathcal{A}\_{+}(\mathbf{x},t) = -\gamma\_{F}\mathcal{A}\_{+}(\mathbf{x},t) - \frac{1}{2}\omega\_{0}\vec{K}\_{01}^{\sharp}\mathcal{S}\_{+}(\mathbf{x},t),\tag{82}$$

with a field decay rate

$$\gamma\_F = c a' = \frac{2ac^2}{1\_L^2} \sqrt{\frac{2M}{\hbar \omega}} L\_{D\prime} \text{ or } 1\_L \gamma\_F = \frac{2ac^2}{1\_L} \sqrt{\frac{2M}{\hbar \omega}} L\_{D\prime} \tag{83}$$

which we call field decay velocity. In section 6, it will be shown that the field decay velocity 1*LγF* describes the field loss by dissipation, as the quantity T *c* describes the electromagnetic energy loss by radiation through the output mirror with the transparency T . We notice that the decay rate of an electromagnetic field in a cavity is proportional to the length of this cavity. For a semiconductor chip with the thickness *LD* = 2 *mm*, we considered in our calculations in (3), from (83) we get a field decay rate *γF* = 2.05 × 107 *s*−1, which is in agreement with the empirical values *γF* = 107, 108 *s*−1, we considered in these calculations. It is interesting that, in this model, the decay rate does not depend on the concentration of the dissipative clusters, since an increase of this concentration means a decrease of the density of states in every cluster, which, in this way, becomes smaller. These two variations cancel exactly one another in the final result. By taking into account the spreading of a dissipative electron wave-function beyond the the boundaries of its cluster due to the thermal motion, one obtains a lower value of the decay rate, but with an increase with the concentration of these clusters.

#### **5. Optical equations for a system of quantum injection dots**

From the quantum master equation (34), we derive optical equations for a two-level system. In the approximation of the slowly varying amplitudes, we consider the non-diagonal matrix elements

$$\rho\_{10}(t) = \rho\_{01}^\*(t) = \frac{1}{2} \left[ \mathcal{S}\_+(t) e^{i\mathbf{k}\mathbf{x}} + \mathcal{S}\_-(t) e^{-i\mathbf{k}\mathbf{x}} \right] e^{-i\omega t} \,, \tag{84}$$

and the population difference

$$w(t) = \rho\_{11}(t) - \rho\_{00}(t), \text{ with the normalization condition} \tag{85a}$$

$$1 = \rho\_{11}(t) + \rho\_{00}(t). \tag{85b}$$

Calculating the matrix elements of the two-level system, and averaging over the field states, from the master equation (34) we get:

$$\frac{d}{dt}\rho\_{10}(t) = -[\lambda\_{01} + \lambda\_{10} + \mathrm{i}(\omega\_{0} + \zeta\_{11} - \zeta\_{00})]\rho\_{10}(t) \tag{86a}$$

$$+ \vec{\mathcal{K}} \left[ (\langle a\_{+} \rangle + \langle a\_{-}^{+} \rangle)e^{i\mathbf{k}\cdot\mathbf{r}} + (\langle a\_{+}^{+} \rangle + \langle a\_{-} \rangle)e^{-i\mathbf{k}\cdot\mathbf{r}} \right] \omega\_{0}\vec{r}\_{10}[\rho\_{00}(t) - \rho\_{11}(t)]$$

$$+ (\zeta\_{11} - \zeta\_{00})^{2} \int\_{t-\tau}^{t} \rho\_{10}(t')e^{-i[\phi\_{0}(t') + \omega(t-t')]}dt'$$

$$\frac{d}{dt}\rho\_{11}(t) = -\frac{d}{dt}\rho\_{00}(t) = 2[\lambda\_{10}\rho\_{00} - \lambda\_{01}\rho\_{11}] \tag{86b}$$

$$+ \vec{\mathcal{K}} \left[ (\langle a\_{+} \rangle + \langle a\_{-}^{+} \rangle)e^{i\mathbf{k}\cdot\mathbf{r}} + (\langle a\_{+}^{+} \rangle + \langle a\_{-} \rangle)e^{-i\mathbf{k}\cdot\mathbf{r}} \right] \omega\_{0}\vec{r}\_{10}[\rho\_{10}(t) + \rho\_{01}(t)].$$

18 Will-be-set-by-IN-TECH

In the following calculations, we are interested in a form of this equation in a cavity with the

which we call field decay velocity. In section 6, it will be shown that the field decay velocity 1*LγF* describes the field loss by dissipation, as the quantity T *c* describes the electromagnetic energy loss by radiation through the output mirror with the transparency T . We notice that the decay rate of an electromagnetic field in a cavity is proportional to the length of this cavity. For a semiconductor chip with the thickness *LD* = 2 *mm*, we considered in our calculations in (3), from (83) we get a field decay rate *γF* = 2.05 × 107 *s*−1, which is in agreement with the empirical values *γF* = 107, 108 *s*−1, we considered in these calculations. It is interesting that, in this model, the decay rate does not depend on the concentration of the dissipative clusters, since an increase of this concentration means a decrease of the density of states in every cluster, which, in this way, becomes smaller. These two variations cancel exactly one another in the final result. By taking into account the spreading of a dissipative electron wave-function beyond the the boundaries of its cluster due to the thermal motion, one obtains a lower value of the decay rate, but with an increase with the concentration of these clusters.

From the quantum master equation (34), we derive optical equations for a two-level system. In the approximation of the slowly varying amplitudes, we consider the non-diagonal matrix

Calculating the matrix elements of the two-level system, and averaging over the field states,

i*kx* + (�*a*+

i*kx* + (�*a*+

*ρ*10(*t* � )*e*−i[*φ*10(*t* � )+*ω*(*t*−*t* � )]*dt*�

*dt ρ*10(*t*) = −[*λ*01 + *λ*10 + i(*ω*0 + *ζ*11 − *ζ*00)]*ρ*10(*t*) (86a)

S+(*t*)*e*i*kx* + S−(*t*)*e*−i*kx*

*w*(*t*) = *ρ*11(*t*) − *ρ*00(*t*), with the normalization condition (85a) 1 = *ρ*11(*t*) + *ρ*00(*t*). (85b)

+� + �*a*−�)*e*−i*kx*

*dt ρ*00(*t*) = 2[*λ*10*ρ*00 − *λ*01*ρ*11] (86b)

+� + �*a*−�)*e*−i*kx*

2*M h*¯ *ω*

*LD* 1*L*

> 2 *ω*0

*LD*, or 1*LγF* = 2*αc*2

1*L*

2*M h*¯ *ω*

*e*−i*ωt*

*ω*0*r*10[*ρ*00(*t*) − *ρ*11(*t*)]

*ω*0*r*10[*ρ*10(*t*) + *ρ*01(*t*)].

. (81)

*Kr*01S+(*x*, *t*), (82)

*LD*, (83)

, (84)

*α*� = 2*αc* 1*L*

*dt*A+(*x*, *t*) = −*γF*A+(*x*, *t*) − 1

2*M h*¯ *ω*

with an absorption coefficient

*d*

*γF* = *cα*� = 2*αc*2

12 *L*

**5. Optical equations for a system of quantum injection dots**

01(*t*) = 1 2

*ρ*10(*t*) = *ρ*∗

(�*a*+� + �*a*+

(�*a*+� + �*a*+

+(*ζ*11 − *ζ*00)2

−�)*e*

−�)*e*

 *t t*−*τ*

length *LD*,

elements

*d*

*d*

*dt ρ*11(*t*) = − *d*

and the population difference

from the master equation (34) we get:

+ *K*

+ *K*

with a field decay rate

From the expression (8) of the quantized electric field *E* in the plane-wave approximation, and mean-values of the annihilation operators of the form

$$
\langle a\_{+} \rangle = \overline{a}\_{+}(t)e^{-\mathrm{i}\omega t} \tag{87a}
$$

$$
\langle a\_{-}\rangle = \vec{a}\_{-}(t)e^{-\mathrm{i}\omega t},\tag{87b}
$$

we get the mean-value of this field,

$$
\langle \vec{E} \rangle = \frac{1}{2} \left[ \vec{\mathcal{E}}(t) e^{-\mathrm{i}\omega t} + \vec{\mathcal{E}}^\*(t) e^{\mathrm{i}\omega t} \right], \tag{88}
$$

with the time slowly-varying amplitude

$$
\vec{\mathcal{E}}(t) = \vec{\mathcal{E}}\_{+}(t)e^{\mathrm{i}\mathbf{k}\mathbf{x}} + \vec{\mathcal{E}}\_{-}(t)e^{-\mathrm{i}\mathbf{k}\mathbf{x}},\tag{89}
$$

while the amplitudes of the two counter-propagating waves are

$$
\vec{\mathcal{E}}\_{+}(t) = 2\mathbf{i}\frac{\hbar\omega}{e}\vec{\mathcal{K}}\vec{a}\_{+}(t) \tag{90a}
$$

$$
\vec{\mathcal{E}}\_{-}(t) = 2\mathbf{i}\frac{\hbar\omega}{\varepsilon}\vec{\mathcal{K}}\vec{a}\_{-}(t). \tag{90b}
$$

In this description we neglect the variation of the amplitudes inside the cavity, by taking into account these two amplitudes only as mean-values over the space coordinate, related by the boundary condition for the output mirror of transmission coefficient T :

$$
\vec{\mathcal{E}}\_{-}(t) = -\sqrt{1-\mathcal{T}}\vec{\mathcal{E}}\_{+}(t). \tag{91}
$$

With the notations

$$
\vec{g} = \frac{e}{\hbar} \vec{r}\_{10} \tag{92}
$$

for the coupling coefficient,

$$
\gamma\_\perp = \lambda\_{01} + \lambda\_{10} \tag{93}
$$

for the dephasing rate,

$$
\gamma\_{\parallel} = \mathcal{Z}(\lambda\_{01} + \lambda\_{10}) \tag{94}
$$

for the decay rate,

$$\gamma\_n = |\mathbb{Z}\_{11} - \mathbb{Z}\_{00}| \tag{95}$$

for the fluctuation rate of the self-consistent field, and

$$w\_T = -\frac{\lambda\_{01} - \lambda\_{10}}{\lambda\_{01} + \lambda\_{10}},\tag{96}$$

from (84)-(91) we obtain equations for the slowly-varying amplitudes

$$\frac{d}{dt}\mathcal{S}\_{+}(t) = -[\gamma\_{\perp} + \mathrm{i}(\omega\_{0} + \gamma\_{n} - \omega)]\mathcal{S}\_{+}(t) + \mathrm{i}\vec{\mathcal{S}}\vec{\mathcal{E}}\_{+}(t)w(t) \tag{97a}$$

$$+ \gamma\_{n}^{2} \int\_{t-\tau}^{t} \mathcal{S}\_{+}(t')e^{-\mathrm{i}[\phi u(t') + \omega(t - t')]}dt'$$

$$\frac{d}{dt}w(t) = -\gamma\_{\parallel}[w(t) - w\_{T}] + (2 - \mathcal{T})\mathrm{i}\vec{\mathcal{S}}\frac{1}{2}\left[\vec{\mathcal{E}}\_{+}^{\*}(t)\mathcal{S}\_{+}(t) - \vec{\mathcal{E}}\_{+}(t)\mathcal{S}\_{+}^{\*}(t)\right]. \tag{97b}$$

At the same time, from (97b) with (104) and (105), we derive the equation for the population difference (85a), and introduce the particle flow I in a two-level system, due to the electric

Quantum Injection Dots 319

*ε*[F2(*t*) + G2(*t*)]

+(2 − T )(T *c*

This equation describes the transition power ¯*hω*0I of the active system providing the energy transfer processes involved in the dissipative super radiant decay: (1) the energy variation of the electron-field system, (2) the dissipative decay of the electron energy, proportional to *γ*�, (3) the radiation of the field energy, proportional to the light velocity *c* and the transmission coefficient T of the output mirror, and (4) the dissipation of the field energy, proportional to *γF*. In this equation, both waves leaving the quantum system and propagating in the cavity, the forward wave with an amplitude coefficient 1 and the backward wave with an amplitude

From the polarization equation (97a) with (104) and (105), the population equation (108), and the field equations (107), we obtain the equations of the slowly varying amplitudes of the

*dt u*(*t*) = −*γ*⊥[*u*(*t*) − *δωv*(*t*)] − *g*G(*t*)*w*(*t*) (110a)

*dt v*(*t*) = −*γ*⊥[*v*(*t*) + *δωu*(*t*)] − *g*F(*t*)*w*(*t*) (110b)

*dt w*(*t*) = −*γ*�[*w*(*t*) − *wT*] + 2I + (2 − T )*g*[G(*t*)*u*(*t*) + F(*t*)*v*(*t*)] (110c)

*h*¯ *ω*0

*h*¯ *ω*0

*δω* = *ω* − *ω*0 − *γn γ*⊥

is the relative atomic detuning. In these equations, the coupling of the electron system to the electromagnetic field is described by a coupling coefficient for the dipole interaction *g* = *g*1*E*. These equations also describe a dissipative decay of the electron system by the coefficients *γ*�

� )] − *u*(*t* �

� )] + *v*(*t* �

From (107) and (108) with (85), we get an equation of energy conservation:

2

coefficient R = 1 − T , are taken into account with the coefficient 1 + R = 2 − T .

)+(*ω* − *ω*0)(*t* − *t*

)+(*ω* − *ω*0)(*t* − *t*

*dt w*(*t*) = −*γF*[*w*(*t*) − *wT*] + 2I + (2 − T )*g*[F(*t*)*v*(*t*) + G(*t*)*u*(*t*)] (108)

+ *γ*�

) sin[*φn*(*t*

) sin[*φn*(*t*

�

�

2*ε*V *v*(*t*) (110d)

2*ε*V *u*(*t*), (110e)

) is the phase fluctuation with a fluctuation time *τn* = 1/*γn*,

)+(*ω* − *ω*0)(*t* − *t*

)+(*ω* − *ω*0)(*t* − *t*

� )] *dt*�

� )] *dt*�

(111)

V + 2*γF*)V 1

A

*ρ*11(*t*) − 1 + *wT*

2

2

*ε*[F2(*t*) + G2(*t*)].

 *h*¯ *ω*0

(109)

current *I* = *eADNe*I injected in the device:

*h*¯ *ω*0*ρ*11(*t*)+(2 − T )V 1

) cos[*φn*(*t*

) cos[*φn*(*t*

�

�

V F(*t*) − *γF*F(*t*) − *g*

V G(*t*) − *γF*G(*t*) − *g*

�

) ≡ −*φ*10(*t*

*d*

*dt*

*h*¯ *ω*0I = *d*

system:

*d*

+*γ*2 *n t t*−*τ u*(*t* �

*d*

+*γ*2 *n t t*−*τ v*(*t* �

*d*

*d*

*d*

where *φn*(*t*

and

*dt* F(*t*) = −1

*dt* G(*t*) = −1

�

2 T *c* A

2 T *c* A

) ≡ *φ*01(*t*

�

In equation (97b) we have taken into account that the term

$$\Phi\_{+}(t) = \text{i}\vec{\text{g}}\frac{1}{2}\left[\vec{\mathcal{E}}\_{+}^{\*}(t)\mathcal{S}\_{+}(t) - \vec{\mathcal{E}}\_{+}(t)\mathcal{S}\_{+}^{\*}(t)\right] \tag{98}$$

is a particle flow due to the forward electromagnetic wave propagating in the cavity, while

$$\Phi\_{-}(t) = \text{i}\vec{\text{g}}\frac{1}{2}\left[\vec{\mathcal{E}}\_{-}^{\*}(t)\mathcal{S}\_{-}(t) - \vec{\mathcal{E}}\_{-}(t)\mathcal{S}\_{-}^{\*}(t)\right] \tag{99}$$

is a particle flow due to the backward electromagnetic wave, which means that the two flows satisfy the boundary condition for the energy flow of the electromagnetic field

$$
\Phi\_{-}(t) = (1 - \mathcal{T})\Phi\_{+}(t). \tag{100}
$$

At the same time, calculating the mean-value of the field operator *a*, averaging over the states of the two-level system, and taking into account the relation

$$
\langle \mathbf{c}\_{\mathbf{i}}^{+} \mathbf{c}\_{\mathbf{j}} \rangle = \rho\_{\overline{j}\overline{i}}(\mathbf{t}) , \tag{101}
$$

from equation (34) we get the field equation

$$\frac{d}{dt}\langle a\_{+}\rangle = -\mathrm{i}\omega \langle a\_{+}\rangle + \mathrm{\vec{K}}\omega\_{0}\vec{r}\_{10}[\rho\_{10}(t) - \rho\_{01}(t)]e^{-\mathrm{i}\mathbf{k}\cdot\mathbf{r}}.\tag{102}$$

Thus, with (84), (87) and (90), we get a field equation for slowly-varying amplitudes

$$\frac{d}{dt}\vec{\mathcal{E}}\_{+}(t) = -\mathrm{i}\omega\_{0}\frac{\hbar\omega}{e}\vec{\mathcal{K}}(\vec{K}\vec{r}\_{10})\mathcal{S}\_{+}(t). \tag{103}$$

We consider this equation for the components *u*(*t*) and *v*(*t*) of the polarization amplitude

$$\mathcal{S}\_{+}(t) = \mathfrak{u}(t) - \mathrm{i}\upsilon(t),\tag{104}$$

and F(*t*) and G(*t*) of the electromagnetic field

$$\mathcal{E}\_{+}(t) = \mathcal{F}(t) + \mathrm{i}\mathcal{G}(t),\tag{105}$$

and take into account the field dissipation described by the dissipation rate *γF*. We get

$$\frac{d}{dt}\mathcal{F}(t) = -\gamma\_\mathcal{F}\mathcal{F}(t) - g\frac{\hbar\omega\_0}{2\varepsilon\mathcal{V}}v(t) \tag{106a}$$

$$\frac{d}{dt}\mathcal{G}(t) = -\gamma\_F \mathcal{G}(t) - g \frac{\hbar \omega\_0}{2\varepsilon \mathcal{V}} \mu(t). \tag{106b}$$

We consider these equations for the electromagnetic energy in the quantization volume V, and introduce the energy flow through the surface A of this volume:

$$\frac{d}{dt}\left[\mathcal{V}\frac{1}{2}\varepsilon\mathcal{F}^2(t)\right] = -\mathcal{T}c\frac{1}{2}\varepsilon\mathcal{F}^2(t)\mathcal{A} - \gamma\_\mathcal{F}\mathcal{V}\varepsilon\mathcal{F}^2(t) - g\frac{\hbar\omega\_0}{2}\mathcal{F}v(t) \tag{107a}$$

$$\frac{d}{dt}\left[\mathcal{V}\frac{1}{2}\epsilon\mathcal{G}^2(t)\right] = -\mathcal{T}c\frac{1}{2}\epsilon\mathcal{G}^2(t)\mathcal{A} - \gamma\_\mathcal{F}\mathcal{V}\epsilon\mathcal{G}^2(t) - g\frac{\hbar\omega\_0}{2}\mathcal{G}u(t). \tag{107b}$$

20 Will-be-set-by-IN-TECH

is a particle flow due to the forward electromagnetic wave propagating in the cavity, while

is a particle flow due to the backward electromagnetic wave, which means that the two flows

At the same time, calculating the mean-value of the field operator *a*, averaging over the states

*dt*�*a*+� = −i*ω*�*a*+� + *K ω*0*r*10[*ρ*10(*t*) − *ρ*01(*t*)]*e*

We consider this equation for the components *u*(*t*) and *v*(*t*) of the polarization amplitude

and take into account the field dissipation described by the dissipation rate *γF*. We get

*dt* F(*t*) = −*γF*F(*t*) − *g*

*dt* G(*t*) = −*γF*G(*t*) − *g*

1 2

1 2

We consider these equations for the electromagnetic energy in the quantization volume V, and

*h*¯ *ω e K*(

Thus, with (84), (87) and (90), we get a field equation for slowly-varying amplitudes

*dt* E+(*t*) = −i*ω*0

+(*t*)S+(*t*) − E+(*t*)S∗

−(*t*)S−(*t*) − E−(*t*)S∗

+(*t*)

−(*t*)

Φ−(*t*)=(1 − T )Φ+(*t*). (100)

*i cj*� = *ρji*(*t*), (101)

S+(*t*) = *u*(*t*) − i*v*(*t*), (104)

E+(*t*) = F(*t*) + iG(*t*), (105)

*h*¯ *ω*0

*h*¯ *ω*0

*ε*F2(*t*)A − *γF*V*ε*F2(*t*) − *g*

*ε*G2(*t*)A − *γF*V*ε*G2(*t*) − *g*

−i*kx*. (102)

*Kr*10)S+(*t*). (103)

2*ε*V *v*(*t*) (106a)

2*ε*V *u*(*t*). (106b)

2 F*v*(*t*) (107a)

2 G*u*(*t*). (107b)

*h*¯ *ω*0

*h*¯ *ω*0

(98)

(99)

In equation (97b) we have taken into account that the term

Φ+(*t*) = i*g*

Φ−(*t*) = i*g*

of the two-level system, and taking into account the relation

*d*

*d*

*d*

introduce the energy flow through the surface A of this volume:

= −T *c*

= −T *c*

from equation (34) we get the field equation

*d*

and F(*t*) and G(*t*) of the electromagnetic field

*d dt* V 1 2 *ε*F2(*t*)

*d dt* V 1 2 *ε*G2(*t*)  1 2 E∗

1 2 E∗

satisfy the boundary condition for the energy flow of the electromagnetic field

�*c*+

At the same time, from (97b) with (104) and (105), we derive the equation for the population difference (85a), and introduce the particle flow I in a two-level system, due to the electric current *I* = *eADNe*I injected in the device:

$$\frac{d}{dt}w(t) = -\gamma\_F[w(t) - w\_T] + 2\mathcal{I} + (2 - \mathcal{T})g[\mathcal{F}(t)v(t) + \mathcal{G}(t)u(t)] \tag{108}$$

From (107) and (108) with (85), we get an equation of energy conservation:

$$\begin{split} \hbar\omega\_{0}\mathcal{T} = \frac{d}{dt}\left\{\hbar\omega\_{0}\rho\_{11}(t) + (2-\mathcal{T})\mathcal{V}\frac{1}{2}\varepsilon[\mathcal{F}^{2}(t)+\mathcal{G}^{2}(t)]\right\} + \gamma\_{\parallel}\left[\rho\_{11}(t) - \frac{1+w\_{T}}{2}\right]\hbar\omega\_{0} \\ &+ (2-\mathcal{T})(\mathcal{T}c\frac{\mathcal{A}}{\mathcal{V}}+2\gamma\_{\parallel})\mathcal{V}\frac{1}{2}\varepsilon[\mathcal{F}^{2}(t)+\mathcal{G}^{2}(t)]. \end{split} \tag{109}$$

This equation describes the transition power ¯*hω*0I of the active system providing the energy transfer processes involved in the dissipative super radiant decay: (1) the energy variation of the electron-field system, (2) the dissipative decay of the electron energy, proportional to *γ*�, (3) the radiation of the field energy, proportional to the light velocity *c* and the transmission coefficient T of the output mirror, and (4) the dissipation of the field energy, proportional to *γF*. In this equation, both waves leaving the quantum system and propagating in the cavity, the forward wave with an amplitude coefficient 1 and the backward wave with an amplitude coefficient R = 1 − T , are taken into account with the coefficient 1 + R = 2 − T .

From the polarization equation (97a) with (104) and (105), the population equation (108), and the field equations (107), we obtain the equations of the slowly varying amplitudes of the system:

$$\frac{d}{dt}u(t) = \frac{-\gamma\_{\perp}[u(t) - \delta\omega v(t)] - g\mathcal{G}(t)w(t)}{\int\_{t}^{t} \left[\int\_{t-\tau}^{t} \omega(t)\omega(t-\tau)d\tau + \int\_{t-\tau}^{t} \omega(t-\tau)d\tau + \int\_{t}^{t} \omega(t-\tau)d\tau + \int\_{t-\tau}^{t} \omega(t-\tau)d\tau\right]dt} = \frac{1}{2}\frac{\mathcal{G}(t)}{\mathcal{G}(\tau)}\tag{110a}$$

$$\begin{aligned} \left[ +\gamma\_n^2 \int\_{t-\tau} \left\{ u(t') \cos[\phi\_n(t') + (\omega - \omega\_0)(t - t')] + v(t') \sin[\phi\_n(t') + (\omega - \omega\_0)(t - t')] \right\} dt' \right] \\ \int\_{\omega\_0(t)} \left[ -\gamma\_n(t) + \chi\_{\text{vac}}(t) \right] \quad \text{or } \mathfrak{T}(t)^{\omega\_0} \end{aligned} \tag{110b}$$

$$\frac{d}{dt}v(t) = \begin{cases} -\gamma\_\perp \left[ v(t) + \delta \omega u(t) \right] - g\mathcal{F}(t)w(t) \\\\ +\gamma\_n^2 \int\_{t-\tau}^t \left\{ v(t') \cos[\phi\_n(t') + (\omega - \omega\_0)(t - t')] - u(t') \sin[\phi\_n(t') + (\omega - \omega\_0)(t - t')] \right\} dt' \\ \end{cases}$$

$$\frac{d}{dt}w(t) = -\gamma\_{\parallel}[w(t) - w\_T] + 2\mathcal{I} + (2 - \mathcal{T})g[\mathcal{G}(t)u(t) + \mathcal{F}(t)v(t)]\tag{110c}$$

$$\frac{d}{dt}\mathcal{F}(t) = -\frac{1}{2}\mathcal{T}c\frac{\mathcal{A}}{\mathcal{V}}\mathcal{F}(t) - \gamma\_F \mathcal{F}(t) - g\frac{\hbar\omega\_0}{2\varepsilon\mathcal{V}}v(t) \tag{110d}$$

$$\frac{d}{dt}\mathcal{G}(t) = -\frac{1}{2}\mathcal{T}\varepsilon\frac{\mathcal{A}}{\mathcal{V}}\mathcal{G}(t) - \gamma\_F \mathcal{G}(t) - g\frac{\hbar\omega\_0}{2\varepsilon\mathcal{V}}u(t),\tag{110e}$$

where *φn*(*t* � ) ≡ *φ*01(*t* � ) ≡ −*φ*10(*t* � ) is the phase fluctuation with a fluctuation time *τn* = 1/*γn*, and

$$
\delta\omega = \frac{\omega - \omega\_0 - \gamma\_n}{\gamma\_\perp} \tag{111}
$$

is the relative atomic detuning. In these equations, the coupling of the electron system to the electromagnetic field is described by a coupling coefficient for the dipole interaction *g* = *g*1*E*. These equations also describe a dissipative decay of the electron system by the coefficients *γ*�

In a first approximation, we neglect the temperature variation due to the heat transfer through the semiconductor structure. To take into account this temperature variation, one has to make corrections of the parameters, to obtain the same transition frequency on the whole chain of

Quantum Injection Dots 321

Such a device can be realized in two versions schematically represented in figure 4: (a) a longitudinal device, with the two mirrors *M*1 and *M*2 made on the two surfaces in the plane of the chip, of transmission coefficients T0 = 0 and T > 0, which form a Fabry-Perot cavity coupling a super radiant mode that propagates in the *x*-direction of the injection current; (b) a transversal device, with the two mirrors *M*1 and *M*2 made on two lateral surfaces of the chip, of transmission coefficients T0 = 0 and T > 0, which form a Fabry-Perot cavity coupling a super radiant mode that propagates in the *y*-direction, perpendicular to the injection current. While in version (a) the roles of the mirrors *M*1 and *M*2, and of the injection electrodes E1 and E2, are played by the same metalizations, made on the two surfaces in the plane of the chip, in version (b) the mirror metalizations *M*1 and *M*2, which are made on two lateral surfaces, are

The two devices have the same structure, including layers of *GaAs*, with a narrower forbidden band and a heavier doping, for the quantum wells, and layers of *AlxGa*1−*xAs*, with a larger forbidden band and a lighter doping, for the potential barriers. The margins of these bands are determined by the concentrations of the donors (acceptors) embedded in the semiconductor layers. For a longitudinal device (figure 4a), the *N*¯*t* (dimensionless) quantum dots in the

, which means

*Ne*[*m*−2]*N*¯*t*

, which means

Φ*L* = *ADSL* (119a)

*ADST*. (119b)

*e* (*I* − *I*0*L*) (120a)

*e* (*I* − *I*0*T*), (120b)

*AD*[*m*2]

, (117)

. (118)

*Ne*[*m*−2]

*N*¯*t*

A*T*[*m*2] = *LD*[*m*] *Ne*[*m*−2]*N*¯*t*

Φ*T* = *LD*

1 + 2 1*LγF* T *c* · *h*¯ *ω*0

1 + 2 1*LγF* T *c*

*A*1/2 *D LD* · *h*¯ *ω*0

Φ*L* = *N*¯*t* (2 − T )

Φ*T* = *N*¯*t* (2 − T )

A*L*[*m*2] = 1

while for a transversal device (see figure 4b), the *Ne*[*m*−2]*AD*[*m*2] quantum dots in the

With the radiation area A*L* (A*T*) of a quantum dot, from (115) we derive the flow density *SL* (*SD*), and the total flow of the electromagnetic field radiated by the device in the two versions:

1 *Ne*[*m*−2]

super radiant junctions.

different from the electrode metalizations E1 and E2.

*x*-direction, radiate through an area 1

*y*-direction, radiate through an area *LD*[*m*]

We obtain

and *γ*⊥, non-Markovian effects by time-integrals proportional to the fluctuation coefficient *γ*2 *n* in the polarization equations (110a) and (110b), a decrease of the electron-field coupling due to the field radiation by the term proportional to the coefficient (2 − T ) in (110c), and a decrease of field by the radiation terms proportional to the product *c*T in (110d) and (110e), and by the terms proportional to the decay rate *γF*.

#### **6. Superradiant quantum injection dots and heat conversion**

The dynamic equations (110) take a simpler form in a stationary regime when the derivatives with time become zero and the polarization variables can be taken out from the integrals. Considering an integration over a fluctuation time *τn* = 1/*γn*, we get a time oscillating term, generated by the fluctuations of the environment particles. In the Markovian approximation, when these oscillations are neglected, we get the steady-state equations:

$$-\gamma\_\perp \left[ \mu - \delta \omega v \right] - g\mathcal{G}w = 0\tag{112a}$$

$$-\gamma\_{\perp}[v+\delta\omega u]-g\mathcal{F}w=0\tag{112b}$$

$$-\gamma\_{\parallel}(w - w\_{T}) + 2\mathcal{T} + (2 - \mathcal{T})\mathbf{g}(\mathcal{G}u + \mathcal{F}v) = 0\tag{112c}$$

$$-\Gamma\_F \mathcal{F} - G\upsilon = 0\tag{112d}$$

$$-\Gamma\_F \mathcal{G} - \mathcal{G}u = 0,\tag{112e}$$

where

$$G = g \frac{\hbar \omega\_0}{2\varepsilon \mathcal{V}} \tag{113a}$$

$$
\Gamma\_F = \frac{1}{2}\mathcal{T}c\frac{\mathcal{A}}{\mathcal{V}} + \gamma\_F. \tag{113b}
$$

From the system of equations (112), for the resonance case (*δω* = 0), we calculate the flow density of the electromagnetic energy radiated by the device:

$$S = \mathcal{T}c\frac{1}{2}\varepsilon(\mathcal{F}^2 + \mathcal{G}^2). \tag{114}$$

We get

$$S = \frac{\frac{\hbar\omega\_0}{(2-\mathcal{T})\mathcal{A}}}{1 + \frac{2\gamma\_F\mathcal{V}}{\mathcal{T}c\mathcal{A}}} \left[ \mathcal{T} - \left( -w\_T \frac{\gamma\_\parallel}{2} + \frac{\frac{1}{2}\mathcal{T}c\frac{\mathcal{A}}{\mathcal{V}} + \gamma\_F}{\frac{g^2\hbar\omega\_0}{\gamma\_\perp\gamma\_\parallel c\mathcal{V}}} \right) \right].\tag{115}$$

This expression of the flow density *S* has a nice physical interpretation, being proportional to the product of the transition energy ¯*hω*0, divided to the radiation area of a quantum dot A, with the difference between the particle flow I and a threshold value depending on coupling, radiation, and dissipation coefficients. This expression is valid when the quantization volume V of the field corresponds to the electromagnetic energy delivered by the whole system of *NeNt* quantum dots to a volume unit, which means

$$\mathcal{V}[m^3] = \frac{1}{N\_\varepsilon[m^{-2}]N\_t[m^{-1}]},\tag{116}$$

where *Ne*[*m*−2] is the number of quantum dots per area unit, and *Nt*[*m*−1] is the number of super radiant junctions per length unit.

*n*

22 Will-be-set-by-IN-TECH

and *γ*⊥, non-Markovian effects by time-integrals proportional to the fluctuation coefficient *γ*2

in the polarization equations (110a) and (110b), a decrease of the electron-field coupling due to the field radiation by the term proportional to the coefficient (2 − T ) in (110c), and a decrease of field by the radiation terms proportional to the product *c*T in (110d) and (110e), and by the

The dynamic equations (110) take a simpler form in a stationary regime when the derivatives with time become zero and the polarization variables can be taken out from the integrals. Considering an integration over a fluctuation time *τn* = 1/*γn*, we get a time oscillating term, generated by the fluctuations of the environment particles. In the Markovian approximation,

> −*γ*⊥[*u* − *δωv*] − *g*G*w* = 0 (112a) −*γ*⊥[*v* + *δωu*] − *g*F*w* = 0 (112b) −*γ*�(*w* − *wT*) + 2I + (2 − T )*g*(G*u* + F*v*) = 0 (112c) −Γ*F*F − *Gv* = 0 (112d) −Γ*F*G − *Gu* = 0, (112e)

> > 2*ε*V (113a)

*ε*(F2 + G2). (114)

⎞ ⎠ ⎤

⎦ . (115)

, (116)

V + *γF*. (113b)

terms proportional to the decay rate *γF*.

where

We get

**6. Superradiant quantum injection dots and heat conversion**

when these oscillations are neglected, we get the steady-state equations:

*G* = *g*

Γ*F* = 1 2 T *c* A

*S* = T *c*

⎡ ⎣I −

density of the electromagnetic energy radiated by the device:

*h*¯ *ω*0 (2−T )A 1 + 2*γF*V T *c*A

*S* =

*NeNt* quantum dots to a volume unit, which means

super radiant junctions per length unit.

*h*¯ *ω*0

From the system of equations (112), for the resonance case (*δω* = 0), we calculate the flow

*γ*� 2 +

*Ne*[*m*−2]*Nt*[*m*−1]

where *Ne*[*m*−2] is the number of quantum dots per area unit, and *Nt*[*m*−1] is the number of

This expression of the flow density *S* has a nice physical interpretation, being proportional to the product of the transition energy ¯*hω*0, divided to the radiation area of a quantum dot A, with the difference between the particle flow I and a threshold value depending on coupling, radiation, and dissipation coefficients. This expression is valid when the quantization volume V of the field corresponds to the electromagnetic energy delivered by the whole system of

1 2 T *c* A

V + *γF g*2 *h*¯ *ω*0 *γ*⊥*γ*�*ε*V

1 2

⎛ ⎝−*wT*

V[*m*3] = 1

In a first approximation, we neglect the temperature variation due to the heat transfer through the semiconductor structure. To take into account this temperature variation, one has to make corrections of the parameters, to obtain the same transition frequency on the whole chain of super radiant junctions.

Such a device can be realized in two versions schematically represented in figure 4: (a) a longitudinal device, with the two mirrors *M*1 and *M*2 made on the two surfaces in the plane of the chip, of transmission coefficients T0 = 0 and T > 0, which form a Fabry-Perot cavity coupling a super radiant mode that propagates in the *x*-direction of the injection current; (b) a transversal device, with the two mirrors *M*1 and *M*2 made on two lateral surfaces of the chip, of transmission coefficients T0 = 0 and T > 0, which form a Fabry-Perot cavity coupling a super radiant mode that propagates in the *y*-direction, perpendicular to the injection current. While in version (a) the roles of the mirrors *M*1 and *M*2, and of the injection electrodes E1 and E2, are played by the same metalizations, made on the two surfaces in the plane of the chip, in version (b) the mirror metalizations *M*1 and *M*2, which are made on two lateral surfaces, are different from the electrode metalizations E1 and E2.

The two devices have the same structure, including layers of *GaAs*, with a narrower forbidden band and a heavier doping, for the quantum wells, and layers of *AlxGa*1−*xAs*, with a larger forbidden band and a lighter doping, for the potential barriers. The margins of these bands are determined by the concentrations of the donors (acceptors) embedded in the semiconductor layers. For a longitudinal device (figure 4a), the *N*¯*t* (dimensionless) quantum dots in the *x*-direction, radiate through an area 1 *Ne*[*m*−2] , which means

$$\mathcal{A}\_L[m^2] = \frac{1}{N\_\varepsilon[m^{-2}]\bar{N}\_t},\tag{117}$$

while for a transversal device (see figure 4b), the *Ne*[*m*−2]*AD*[*m*2] quantum dots in the *y*-direction, radiate through an area *LD*[*m*] *N*¯*t* 1 *Ne*[*m*−2] , which means

$$\mathcal{A}\_T[m^2] = \frac{L\_D[m]}{N\_\varepsilon[m^{-2}]\bar{N}\_t\sqrt{A\_D[m^2]}}.\tag{118}$$

With the radiation area A*L* (A*T*) of a quantum dot, from (115) we derive the flow density *SL* (*SD*), and the total flow of the electromagnetic field radiated by the device in the two versions:

$$
\Phi\_L = A\_D \mathcal{S}\_L \tag{119a}
$$

$$
\Phi\_T = L\_D \sqrt{A\_D} S\_T. \tag{119b}
$$

We obtain

$$\Phi\_{L} = \frac{\bar{N}\_{l}}{(2 - \mathcal{T})\left(1 + 2\frac{\mathbf{1}\_{L}\gamma\_{\rm F}}{\mathcal{T}c}\right)} \cdot \frac{\hbar\omega\_{0}}{e}(I - I\_{0L})\tag{120a}$$

$$\Phi\_T = \frac{\ddot{N}\_l}{(2-\mathcal{T})\left(1+2\frac{\mathbf{1}\_L\gamma\_F}{\mathcal{T}c}\frac{A\_D^{1/2}}{L\_D}\right)} \cdot \frac{\hbar\omega\_0}{e}(I-I\_{0T})\_\prime \tag{120b}$$