Master equation and conversion of environmental heat into coherent electromagnetic energy

Progress in Quantum Electronics 34 (2010) 349–408

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Review

Master equation and conversion of environmentalheat into coherent electromagnetic energy

Eliade Stefanescu�

Center of Advanced Studies in Physics at the Institute of Mathematics Simion Stoilow of the Romanian Academy,

13 Calea 13 Septembrie, 050711 Bucharest S5, Romania

Abstract

We derive a non-Markovian master equation for the long-time dynamics of a system of Fermions

interacting with a coherent electromagnetic field, in an environment of other Fermions, Bosons, and

free electromagnetic field. This equation is applied to a superradiant p–i–n semiconductor

heterostructure with quantum dots in a Fabry–Perot cavity, we recently proposed for converting

environmental heat into coherent electromagnetic energy. While a current is injected in the device,

a superradiant field is generated by quantum transitions in quantum dots, through the very thin

i-layers. Dissipation is described by correlated transitions of the system and environment particles,

transitions of the system particles induced by the thermal fluctuations of the self-consistent field of

the environment particles, and non-local in time effects of these fluctuations. We show that, for a

finite spectrum of states and a sufficiently weak dissipative coupling, this equation preserves the

positivity of the density matrix during the whole evolution of the system. The preservation of the

positivity is also guaranteed in the rotating-wave approximation. For a rather short fluctuation time

on the scale of the system dynamics, these fluctuations tend to wash out the non-Markovian integral

in a long-time evolution, this integral remaining significant only during a rather short memory time.

We derive explicit expressions of the superradiant power for two possible configurations of the

superradiant device: (1) a longitudinal device, with the superradiant mode propagating in the

direction of the injected current, i.e. perpendicularly to the semiconductor structure, and (2) a

transversal device, with the superradiant mode propagating perpendicularly to the injected current,

i.e. in the plane of the semiconductor structure. The active electrons, tunneling through the i-zone

between the two quantum dot arrays, are coupled to a coherent superradiant mode, and to a

dissipative environment including four components, namely: (1) the quasi-free electrons of the

conduction n-region, (2) the quasi-free holes of the conduction p-region, (3) the vibrations of the

crystal lattice, and (4) the free electromagnetic field. To diminish the coupling of the active electrons

to the quasi-free conduction electrons and holes, the quantum dot arrays are separated from the two

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E. Stefanescu / Progress in Quantum Electronics 34 (2010) 349–408350

n and p conduction regions by potential barriers, which bound the two-well potential corresponding

to these arrays. We obtain analytical expressions of the dissipation coefficients, which include simple

dependences on the parameters of the semiconductor device, and are transparent to physical

interpretations. We describe the dynamics of the system by non-Markovian optical equations with

additional terms for the current injection, the radiation of the field, and the dissipative processes. We

study the dependence of the dissipative coefficients on the physical parameters of the system, and the

operation performances as functions of these parameters. We show that the decay rate of

the superradiant electrons due to the coupling to the conduction electrons and holes is lower than the

decay rate due to the coupling to the crystal vibrations, while the decay due to the coupling to the free

electromagnetic field is quite negligible. According to the non-Markovian term arising in the optical

equations, the system dynamics is significantly influenced by the thermal fluctuations of the self-

consistent field of the quasi-free electrons and holes in the conduction regions n and p, respectively.

We study the dependence of the superradiant power on the injected current, and the effects of the

non-Markovian fluctuations. In comparison with a longitudinal device, a transversal device has a

lower increase of the superradiant power with the injected current, but also a lower threshold current

and a lesser sensitivity to thermal fluctuations.

& 2010 Elsevier Ltd. All rights reserved.

Keywords: Non-Markovian quantum master equation; Superradiance; Correlated transitions; Semiconductor

heterostructure; Photon; Phonon

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

2. Quantum master equation for a matter-field system and the positivity of the

density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

3. Long-time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

4. Superradiant dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

5. Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

6. System structure and microscopic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

7. Wave functions and dipole moments of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

8. Coupling to the conduction electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

9. Coupling to the crystal vibrations and the free electromagnetic field. . . . . . . . . . . . . . . 386

10. Superradiant semiconductor device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

11. Operation conditions for the device parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

12. Operation conditions for the separation barriers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

13. Dissipative coefficients and stationary regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

14. Non-Markovian fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

15. Discussion and concluding remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

1. Introduction

The dissipative dynamics, as a characteristic of any realistic system, is an interesting fieldof research [1–4], especially due to the difficulties raised by a correct description of thedissipative coupling [5]. Important applications where dissipation cannot be neglected aregiven in [6–10]. The dissipative coupling is essential for a device we recently proposed for

E. Stefanescu / Progress in Quantum Electronics 34 (2010) 349–408 351

converting the environmental heat into usable energy, called quantum heat converter[11–14]. In the most general form, such a coupling can be described by an additional,dissipation term, in the dynamic equation of the density matrix that, with this, becomes amaster equation [15–17].

A generalization of the Schrodinger equation of a closed system, to a master equationdescribing an openness of this system, has been performed by taking into account variousarguments [18]. In [15], the dynamic equation of a harmonic oscillator is generalized byconsidering the canonical coordinates as being complex quantities, with imaginary partsdepending on noise operators. When an equation for the real coordinate and momentum isderived from the dynamic equation with a noisy effective Hamiltonian, a quantum masterequation is obtained. In [16], the total system composed of a harmonic oscillator of interestand an environment of other harmonic oscillators is quantized, and the reduced dynamicsis obtained in the framework of the path integral theory.

In [17], Lindblad adopts a quite radical approach of the problem, by using amathematical generalization of the dynamic group of the quantum states of a system to atime-dependent semigroup, having in view only the positivity preservation of the densitymatrix. Ten years later, this equation has been put into a form with friction and diffusioncoefficients, for a harmonic oscillator [19]. For the dissipative coefficients, althoughunspecified in this framework, fundamental constraints are obtained from theirdependence on Lindblad’s axiomatic coefficients. These relations are in agreement withthe well-known theorem of dissipation and fluctuations. The essential merit of thisequation in comparison with the previous ones is the full agreement with the quantum-mechanical principles: the positivity of the density matrix during the whole evolution of thesystem and the uncertainty relation. When the matrix elements of the openness operatorssatisfy certain conditions depending on temperature, Lindblad’s master equation is inagreement also with the detailed balance principle [20]. For a harmonic oscillator at quasi-equilibrium, the dissipative parameters reduce to only two, the two diffusion coefficients ofcoordinate and momentum becoming functions of the friction coefficient and temperature.Important efforts continued to improving existing models, or developing new physicalmodels [21–38].

By a procedure previously used in [39], in this paper we calculate the long-time reduceddynamics of a system of Fermions, interacting with a coherent electromagnetic field, in adissipative environment of other Fermions, Bosons, and a free electromagnetic field, bytaking into account correlated transitions of the system and environment particles, andrandom fluctuations of the self-consistent field of the environmental Fermions. We derive amaster equation including a phase-operator fðtÞ, which describes fluctuations of thedensity matrix due to the thermal fluctuations of the self-consistent field of theenvironment particles, and a memory time t, much longer than the fluctuation time ofthis field, but much shorter than the decay time. This equation also includes a fluctuationHamiltonian ‘ zijc

þi cj, which is similar to the hopping Hamiltonian (3) in [40]. Besides this

fluctuation Hamiltonian, a non-Markovian term of the second-order in the fluctuationmatrix elements zij arises [39], from the dissipative quantum dynamics in theapproximation of a weak dissipative coupling [28]. We derive conditions for thepreservation of the positivity of the density matrix.

We apply this master equation to a superradiant n–i–p semiconductor heterostructurerepresented in Fig. 1, as a basic element of a quantum heat converter [11–14], whichworks on the principle of the photon-assisted tunneling [41–45]. This device consists in a

AD

LD

n

ND Ne

i na

pa

Ne

p n na i p

a p

NA NAND Ne Ne

n−i−p junction 1 n−i−p junction 2 n−i−p junction Nt

nb pb

nb

pb

N3 N4 N3 N4

•I I

x

y

z

LongitudinalRadiation Field

TransversalRadiation Field

◦ • ◦

• ◦• ◦

Fig. 1. Superradiant semiconductor heterostructure with a thickness LD and an area AD, as a packet of a number

N t of superradiant junctions connected in series. By quantum transitions (red arrows), of electrons in quantum

dots (red ovals) of a concentration Ne, from states of donor atoms (small disks) in the na very thin layer, to states

of acceptor atoms (small circles) in the pa very thin layer, a superradiant field is generated. This field propagates in

the x-direction for a longitudinal structure, or in a y-direction, perpendicular to the direction of the electron

transitions, for a transversal structure. (For interpretation of the references to color in this figure legend, the

reader is referred to the web version of this article.)


Fabry–Perot cavity including a packet of n–i–p superradiant junctions with quantum dotsas pairs of donor–acceptor atoms on the two sides of the i-layer. While a current is injectedin the device, a superradiant field is generated by quantum transitions in this quantumsystem. The donor array na, of a concentration Ne, is separated from the n-region of adonor concentration ND by a very thin layer nb of a larger forbidden band and a lowerdonor concentration N3oNe, while the acceptor array pa, of the concentration Ne, isseparated from the p-region of a concentration NA by a very thin layer pb of a largerforbidden band and a lower acceptor concentration N4oNe. While a current I is injected inthis device, supposed as a circuit of diodes connected in series, the electrons of thequantum dot donors undertake superradiant quantum transitions to the hole states of thequantum dot acceptors, thus generating a coherent electromagnetic field. The superradiantmode is selected by a Fabry–Perot cavity, as a system of two metal layers with controlledtransparency (mirrors), made on two opposite sides of the semiconductor chip. Such acavity could be for a field mode propagating in the x-direction of the electron transitions(longitudinal structure), or for a mode propagating in a y-direction that is perpendicular tothe direction of the electron transitions. Here, we choose realistic values for the physical


parameters, and study the dissipation coefficients and the superradiant power as functionsof these parameters.

The superradiation process [46–48] is usually described by using optical equations for a two-level atom interacting with a single mode of the electromagnetic field [49–51]. These equationsdescribe the time-evolution of a Bloch vector ~s in the space ðsx;sy;szÞ of the pseudo-spin, as a

geometrical representation of the population difference sz, and polarization sx þ isy [49].

When the system is closed, this vector rapidly rotates around the sz-axes with the resonancefrequency o0. It is remarkable that, when the system is open, the Bloch vector has a quitedifferent evolution, which essentially depends on the perturbing coupling to the environment[52–57]. Really, since in a closed system the atom continuously changes energy with the field,

the population difference sz and the amplitude of the polarizationffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2x þ s2y

qslowly oscillate

between �1 and 1 with the Rabi frequency, while the polarization components sx;sy rapidly

oscillate with the frequency o0. When the system is open, as long as no electromagnetic fieldexists, the component sz takes a constant value sz0 according to the detailed balance principle,while the polarization components sx, sy vanish. While a quasi-resonant electromagnetic field

arises, after a transitory time, the amplitudes of the Bloch vector components sx, sy, sz take

constant values, depending on the coupling coefficients of the system to the field and to thedissipative environment. In this case, the energy absorbed by the atom from the field isdissipated by the environment. Thus, we notice that dissipation plays a central role in theatom-field interaction. Important efforts have been devoted to the microscopic description ofthe dissipative processes in the atom-field interaction [40,58–60]. In [40] a many-electronmicroscopic model, including a hopping Hamiltonian, an electron–electron Coulombinteraction, and an electron–phonon interaction, is developed.

The atom-field dissipative dynamics gets a more systematic approach in the frameworkof the quantum theory of open systems, where the diagonal elements of the density matrixare the populations of the states, while the non-diagonal ones represent polarizations infield equations [7–10,61–65]. In [66], we found that, besides the well-known decay anddephasing rates of the optical equations, Lindblad’s master equation [17] produces anadditional term, describing a coupling of the polarization to the population through theenvironment. It is remarkable that this coupling is somehow similar to the couplingproduced by an electromagnetic field. For certain values of the atomic detuning, thiscoupling leads to negative values of the absorption coefficient, which means a slightamplification of the electromagnetic field on the account of the environment energy.

In this paper we present a physical principle and a semiconductor superradiant deviceconverting the environmental heat in coherent electromagnetic energy with a highefficiency [11–14]. In Section 2, we obtain a non-Markovian quantum master equationwith explicit microscopic coefficients, describing the long-time dynamics of this device. Weshow that for a finite spectrum of states and a sufficiently weak dissipative coupling, thepositivity of the density matrix is preserved, the tendency of the non-Markovian term tocarry the density out from its positivity domain being canceled by the Markovian term,bringing it back into this domain. In Section 3, we derive equations for the matrixelements. In the rotating-wave approximation [58,67,68], for the diagonal matrix elementswe obtain equations preserving the positivity. We describe the long-time dynamicsof the system, by introducing a memory time in the non-Markovian integral, and aphase fluctuation operator in the exponentials under this integral. In Section 4, forthe usual model of an assembly of two-level systems in a Fabry–Perot cavity, we derive


non-Markovian equations for polarization, population, and field, with additional termsfor the injected current and the field radiation and dissipation. In Section 5, we find thatthe time non-local equations do not have stationary solutions. However, since thecontributions of the non-Markovian terms consist only in oscillations round theMarkovian solution, we consider this solution as describing the stationary regime of thisdevice. In the Markovian approximation, we derive expressions of the superradiant poweras functions of the coupling, dissipation, and radiation characteristics.In Section 6, we present the microscopic model of the device superradiant junction, and

calculate the energy levels as functions of impurity concentrations. We obtain explicitexpressions of the electric dissipative coefficients, as functions of electric materialcharacteristics and matrix elements of the two-body potentials between the electrons ofthe superradiant system and the conduction electrons and holes. In Section 7, we derivewave-functions and transition dipole moments. These moments mainly depend on theoverlap in the i-layer of the wave-functions of the active electrons in the two states ofthe superradiant system. In Section 8, we calculate the matrix elements for the coupling tothe quasi-free electrons and holes, and derive the electric dissipative coefficients as functionsof physical characteristics of the semiconductor structure. These coefficients depend on thetransition dipole moments, the transition energy, the donor and acceptor concentrations,and the thicknesses of the separation barriers. In Section 9, we derive the phonon dissipativecoefficients, which depend on the transition energy and the sound velocity in the crystal. Wealso derive the decay rates due to the coupling to the free electromagnetic field, whichdepend only on physical constants and the transition dipole moment.In Section 10, we present the basic idea of a quantum heat converter, and the physical

models for two possible versions, which depend on the position of the semiconductor structurein the Fabry–Perot cavity, i.e. on the surfaces of the structure which the mirror metalizationsare made on: (1) a longitudinal device with the superradiant field propagating in the directionof the injection current, which is perpendicular to the semiconductor layers, and (2) atransversal device with the superradiant field propagating perpendicularly to the direction ofthe injection current, which is in the plane of the semiconductor layers. In Section 11, we getexplicit expressions for the coefficients of the optical equations, as depending only on physicalcharacteristics of the system. We discuss the dependences on these characteristics of thesuperradiant power, threshold currents, and operation conditions. In Section 12, we deriveconditions for the separation barriers to provide the necessary injection current. In Section 13,we give a numerical example, and study the dependence on the i-zone thickness of the deviceparameters that mainly depend on this thickness: dissipation rates, coupling coefficients,threshold currents, and the quantum dot density necessary for entirely including the internalfield in the quantum dot region. We find that, due to the separation barriers, the electriccoupling to the conduction electrons gets weaker than the phonon coupling to the crystallattice. We study the dependence of the dissipation coefficients and superradiant power onphysical characteristics of the system. In Section 14, we study the effects of the thermalfluctuations on the superradiation process for the two versions of the device. In Section 15 wediscuss the results and give some conclusions.

2. Quantum master equation for a matter-field system and the positivity of the density matrix

The dissipative dynamics of a system of Fermions interacting with an electromagneticfield, in an environment of other Fermions, Bosons, and a free electromagnetic field, has


been previously described by a quantum master equation with a Markovian term for thecorrelated transitions of the system and environment particles [29–31]. Recently, we havetaken into account the presence of a self-consistent field of the environment Fermions,described in the master equation by a non-Markovian term [39]. We notice that all thesecouplings are essential for the superradiant dynamics of a semiconductor structure, which,besides the active quantum system, includes a complex environment of quasi-free electronsand holes, and vibrations of the crystal lattice. For the total system, we consider thegeneral quantum dynamical equation:

d

dt~wðtÞ ¼ �

i

‘e ~V ðtÞ þ e ~V

EðtÞ; ~wðtÞ

h i; ð1Þ

where VE is the potential of interaction of the system of interest with the environment,while V is the potential of interaction from the Hamiltonian of the system of interest

H ¼ HS0 þHF þ V ; ð2Þ

which includes the terms

HS0 ¼

Xk

ekcþk ck ð3Þ

for the system of Fermions, and HF for the electromagnetic field. In this equation, tildedenotes operators in the interaction picture, e.g.

~wðtÞ ¼ ei=‘ ðHEþHS0þHF ÞtwðtÞe�i=‘ ðHFþHS

0þHE Þt; ð4Þ

where HE is the Hamiltonian of the environment. According to a general proceduredisclosed in [28], we take a total density of the form

~wðtÞ ¼ R ~rðtÞ þ e ~wð1ÞðtÞ þ e2 ~wð2ÞðtÞ þ � � � ; ð5Þ

where rðtÞ is the reduced density matrix of the system of Fermions and electromagneticfield, while R is the density of the dissipative environment at the initial momentof time, t=0, the time-evolution of environment being taken into account by the higher-order terms ~wð1ÞðtÞ; ~wð2ÞðtÞ; � � �. The parameter e is introduced to handle the ordersof the terms in this expansion, and is set to 1 in the final results. The reduced density of thesystem is

~rðtÞ ¼ TrEf ~wðtÞg; ð6Þ

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

TrEf ~wð1Þg ¼ TrEf ~wð2Þg ¼ � � � ¼ 0: ð7Þ

If initially the environment is in the equilibrium state R, the initial density matrix wðtÞ of thetotal system is of the form wð0Þ ¼ Rrð0Þ. We suppose that at the moment t=0, due to theinteraction V of the system of Fermions with the electromagnetic field, or due to a non-equilibrium initial state rð0ÞarT , a time-evolution begins, while the reduced densitysatisfies a quantum dynamical equation of the form

d

dt~rðtÞ ¼ e ~B

ð1Þð ~rðtÞ;tÞ þ e2 ~B

ð2Þð ~rðtÞ;tÞ þ � � � : ð8Þ

We take the equilibrium density matrix of the environment

R ¼ RF � RB � RFE ð9Þ


as including the density matrix

RF ¼Xab

RFabcþa cb ð10Þ

of the Fermi component, and similar expressions for the Bose and the free electromagneticfield components. In this case, the perturbation term

V E ¼ ‘X

ij

Gijcþi cj ð11Þ

includes operators

Gij ¼ GFij þ GB

ij þ GFEij ; ð12Þ

with terms for the Fermion part of the environment

GFij ¼

1

‘

Xab

/aijVF jbjScþa cb; ð13Þ

and similar expressions for the Boson part and for the free electromagnetic field. Here, wetake into account only the dissipative coupling of the system of Fermions, since thedissipation of the field is a well-known problem. That means that the interaction with theelectromagnetic field is described only by the Hamiltonian term, while the field dissipationcan be taken into account merely introducing the dissipative terms of the correspondingharmonic oscillator [39]. In the density matrix (5) of the total system, we distinguish thefirst term, which describes the evolution of the system of interest while the state of theenvironment remains unchanged (Markov approximation). The higher-order terms of thisseries expansion describe the evolution of the environment correlated with the evolution ofthe system. In the second-order approximation, these terms are

~wð1ÞðtÞ ¼ �i

‘R

Z t

0

½ ~V ðt0Þ; ~rðt0Þ� dt0�R

Z t

0

~Bð1Þ½ ~rðt0Þ;t0� dt0�i

Xij

Z t

0

½ ~Gijðt0Þ~cþi ðt

0Þ~cjðt0Þ;R ~rðt0Þ� dt0;

ð14aÞ

~wð2ÞðtÞ ¼ �i

‘

Z t

0

½ ~V ðt0Þ; ~wð1Þðt0Þ� dt0�R

Z t

0

~Bð2Þ½ ~rðt0Þ;t0� dt0�i

Xij

Z t

0

½ ~Gijðt0Þ~cþi ðt

0Þ~cjðt0Þ; ~wð1Þðt0Þ� dt0;

ð14bÞ

while for the reduced dynamics of the system we get the terms:

~Bð1Þ½ ~rðtÞ;t� ¼ �

i

‘½ ~V ðtÞ; ~rðtÞ��i

Xij

TrE ½ ~G ijðtÞ~cþi ðtÞ~cjðtÞ;R ~rðtÞ�; ð15aÞ

~Bð2Þ½ ~rðtÞ;t� ¼ �i

Xij

TrE ½ ~GijðtÞ~cþi ðtÞ~cjðtÞ; ~wð1ÞðtÞ�: ð15bÞ

In these equations we consider the reduced density matrix in the interaction picture ~rðtÞ asslowly varying in time. We assume the time-symmetry, which means that the time-integralsdo not depend on the origin of time, but only on the relative intervals t�t0 between the


actual time t and the past time t0. With the transition operators in the interaction picture,

~cþi ðtÞ~cjðtÞ ¼ eioij tcþi cj ;oij ¼ei�ej

‘; ð16Þ

we obtain:

~Bð1Þ½ ~rðtÞ;t� ¼ �

i

‘½ ~V ðtÞ; ~rðtÞ��i

Xij

zij0½~cþi ðtÞ~cjðtÞ; ~rðtÞ�; ð17aÞ

~Bð2Þ½ ~rðtÞ;t� ¼

Xij

lijf½cþi cj ~rðtÞ;cþj ci� þ ½c

þi cj ; ~rðtÞcþj ci�g

þXijkl

zij0zkl0

Z t

0

½~cþi ðtÞ~cjðtÞ;½~cþk ðt0Þ~clðt

0Þ; ~rðt0Þ�� dt0: ð17bÞ

These equations include two families of dissipative coefficients: the matrix elements of theself-consistent field of the environment particles

zij0 ¼

1

‘Y F

ZðaÞ/aijVF jajSf F

a ðeaÞgFa ðeaÞ dea ð18Þ

and the decay rates

lFij ¼

p‘Y F

ZðbÞj/aijV F jbjSj2½1�f F

a ðeaÞ�fFb ðebÞg

Fa ðeaÞg

Fb ðebÞjoab¼oji

deb; ð19aÞ

lBij ¼

p‘Y B

ZðbÞj/aijV BjbjSj2½1þ f B

a ðeaÞ�fBb ðebÞg

Ba ðeaÞg

Bb ðebÞjoab¼oji

deb ð19bÞ

for the coupling with the system of Fermions and with the system of Bosons, respectively.In these expressions, YF, YB represent the total numbers of the environment Fermions andBosons, in the quantization volumes of these particles. Although, by diagonalizing thepotential VF in (18) the transition elements zij

0 vanish, such transitions continue to begenerated by the thermal fluctuations of the environment particles over the states jaS. Wetake into account these fluctuations by considering in (18) the variances of the potential VF:

zij0�!zij ¼

1

‘

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

Y F

ZðaÞ/aijðV F Þ

2jajSf F

a ðeaÞgFa ðeaÞ dea

s: ð20Þ

Using the transition operators (16) with the resonance condition oab ¼ oji, and the seriesexpansion (8) with the terms (17), in the second-order approximation of the dissipativecoupling we obtain the non-Markovian quantum master equation [39]:

d

dtrðtÞ ¼ �

i

‘½H;rðtÞ��i

Xij

zij½cþi cj ;rðtÞ� þ

Xij

lijð½cþi cjrðtÞ;cþj ci� þ ½c

þi cj ;rðtÞcþj ci�Þ

þXijkl

zijzkl

Z t

0

½cþi cj ;e�i=‘HS

0ðt�t0Þ½cþk cl ;rðt0Þ�ei=‘HS

0ðt�t0Þ� dt0: ð21Þ

The dissipative generator of this equation is composed of a Hamiltonian part with thematrix elements zij, which describe transitions stimulated by the fluctuations of the self-consistent field of the environment particles, a Markovian part, of Lindblad’s form, withthe decay rates lij, which describe correlated transitions of the system and environment


particles, and a non-Markovian part, as a time-integral of the system operators, describingmemory effects, which are proportional to the fluctuations of the self-consistent field of the

environment particles. We do not diagonalize the dissipative HamiltonianP

ijzijcþi cj, since

the matrix elements zij describe fluctuations that arise in any basis of states. The dissipative

coefficients of the Markovian part

lij ¼ lFij þ lB

ij þ gij ð22Þ

include explicit terms for the coupling to an environment of Fermions, Bosons and the freeelectromagnetic field. These terms depend on the dissipative two-body potentials VF, VB,

the densities of the environment states gFa ðeaÞ; g

Ba ðeaÞ, the occupation probabilities of these

states f Fa ðeaÞ; f

Ba ðeaÞ; and temperature T. For a rather low temperature, T5eji; j4i, these

terms become

lFij ¼

p‘j/aijV F jbjSj2½1�f F

a ðejiÞ�gFa ðejiÞ; ð23aÞ

lFji ¼

p‘j/aijV F jbjSj2f F

a ðejiÞgFa ðejiÞ ð23bÞ

for the Fermi environment,

lBij ¼

p‘j/aijV BjbjSj2½1þ f B

a ðejiÞ�gBa ðejiÞ; ð24aÞ

lBji ¼

p‘j/aijV BjbjSj2f B

a ðejiÞgBa ðejiÞ ð24bÞ

for the Bose environment, and

gij ¼2a

c2‘ 3~r2ije

3ji 1þ

1

eeji=T�1

� �ð25Þ

for the Bose environment of the free electromagnetic field, where~rij are the transition dipole

moments. The terms of the master Eq. (21), with the dissipative coefficients (22)–(25),describe single-particle transitions of the system and environment, with energy conserva-tion, eji ¼ eab, in agreement with the quantum-mechanical principles, and with the detailed

balance principle [32]. The non-Markovian part of this equation takes into account thefluctuations of the self-consistent field of the environment Fermions, with the coefficients

(20), where Y F is the total number of these particles occupying the states jaS in aquantization volume.Since the master Eq. (21) is derived as a second-order approximation of the total

dynamics, we investigate the preservation of the positivity of the density matrix generatedby this equation. We consider that at the initial moment of time t=0, the density matrix ispositive and, consequently, can be diagonalized:

rð0Þ ¼X

i

licþi ci; riið0Þ ¼ li40: ð26Þ

Introducing this expression in the master equation (21), and using the commutation relations

eð�i=‘ ÞHS0tcþk ¼ eð�i=‘ Þektcþk eð�i=‘ ÞH

S0t; ð27aÞ

ckeði=‘ ÞHS0t ¼ eði=‘ Þekteði=‘ ÞH

S0tck; ð27bÞ


we get the evolution of the diagonal matrix elements riiðtÞ in a very short interval of time t, asdepending on the whole energy spectrum ej:

d

dtriiðtÞ

��t¼t¼ 2X

j

lij�‘ jzijj

2

ej�ei

sin½ðej�eiÞt�

!lj� lji�

‘ jzijj2

ej�ei

sin½ðej�eiÞt�

!li

" #:

ð28Þ

According to (22)–(25), for a finite energy spectrum, the decay/excitation coefficients takenon-zero, positive values, lij ; lji40, which means that, for a rather small but non-zerointerval of time t, the coefficients of Eq. (28) also remain positive:

lij�jzijj2t40; lji�jzijj

2t40: ð29Þ

When a diagonal element riiðtÞ 2 ½0; 1�, of the density matrix, approaches one of its limits, thevariation of this element gets a sign bringing it back to the inner of its definition domain:

riiðtÞ-0)d

dtriiðtÞ

��t¼t¼ 2X

j

ðlij�jzijj2tÞlj40; ð30aÞ

riiðtÞ-1)d

dtriiðtÞ

��t¼t¼ �2

Xj

ðlji�jzijj2tÞlio0: ð30bÞ

However, if we consider a time interval t satisfying the uncertainty relation ðej�eiÞt4‘ ,for a finite spectrum of states we get the positivity conditions

‘ jzijj2

jej�eijolij ;

‘ jzijj2

jej�eijolji; ð31Þ

depending on the decay rates lij and the excitation rates lji. From (30) and (31), we noticethat, although the non-Markovian fluctuation rates zij of this equation have the tendency toalter the positivity of the density matrix, for a finite spectrum of states and a sufficiently weakdissipative coupling, the positivity is still preserved, this tendency being canceled by the decay/excitation rates lij ; lji. In the following, we derive explicit expressions for these coefficients, asfunctions of the physical parameters of the system.

We notice that Eq. (28) of the diagonal matrix elements includes only non-diagonalmatrix elements zij . From (20) and (23b), we notice that the excitation matrix elements lji

and the non-diagonal fluctuation matrix elements zij could be considered approximately ofthe same order of magnitude, which means that the conditions (31) reduce to the conditionthat these elements are much smaller than the distance between the energy levels of thesystem:

‘ jzijj5jej�eij: ð32Þ

These conditions are always satisfied in the assumption of a weak dissipative coupling,when the density matrix elements have slowly varying amplitudes:

rjiðtÞ ¼ ~rjiðtÞe�ioji t: ð33Þ


3. Long-time evolution

From the master Eq. (21) we derive equations for the density matrix elements that, withthe matrix elements of the transition operators /kjcþi cjjlS ¼ dkidjl , take the form

d

dtrjiðtÞ ¼ �iojirjiðtÞ�

i

‘

Xk

½VjkðtÞrkiðtÞ�rjkðtÞVkiðtÞ�

�iX

k

½zjkrkiðtÞ�rjkðtÞzki� þX

k

½2dijljkrkkðtÞ�ðlki þ lkjÞrjiðtÞ�

þX

kl

Z t

0

fzjk½zklrliðt0Þ�rklðt

0Þzli�eioikðt�t0Þ�½zjkrklðt

0Þ�rjkðt0Þzkl �zlie

iolj ðt�t0Þg dt0:

ð34Þ

In these equations, we introduce density matrix elements of the form (33), and neglect therapidly varying terms in the amplitudes ~rjiðtÞ (rotating-wave approximation). Thus, for theslowly varying amplitudes ~rjiðtÞ, we get the equations

d

dt~rjiðtÞ ¼ �

i

‘

Xk

½VjkðtÞrkiðtÞ�rjkðtÞVkiðtÞ�eioji t þ

Xk

½2dijljkrkkðtÞeioji t�ðlki þ lkjÞ ~rjiðtÞ�

�iX

k

½zjk ~rkiðtÞe�iokit� ~rjkðtÞe

�iojktzki�eioji t

þX

kl

Z t

0

fzjk½zkl ~rliðt0Þe�ioli t

0

� ~rklðt0Þe�iokl t

0

zli�eioji teioikðt�t0Þ

�½zjk ~rklðt0Þe�iokl t

0

� ~rjkðt0Þe�iojkt0zkl �zlie

ioji teiolj ðt�t0Þg dt0; ð35Þ

where we neglect the rapidly oscillating terms. From these equations we obtain non-Markovian equations for the non-diagonal matrix elements

d

dtrjiðtÞ ¼ �i½oji þ ðzjj�ziiÞ�rjiðtÞ�

i

‘

Xk

½VjkðtÞrkiðtÞ�rjkðtÞVkiðtÞ�

�X

k

ðlki þ lkjÞrjiðtÞ þ ðzjj�ziiÞ2e�ioji t

Z t

0

~rjiðt0Þ dt0; ð36Þ

and Markovian equations for the diagonal ones:

d

dtrjjðtÞ ¼ �

i

‘

Xk

½VjkðtÞrkjðtÞ�rjkðtÞVkjðtÞ��2X

k

½lkjrjjðtÞ�ljkrkkðtÞ�: ð37Þ

That means that in the rotating-wave approximation the non-Markovian component ofthe dissipative dynamics does not alter the quantum and statistical properties of the densitymatrix, as the normalization, positivity, and detailed balance. We notice that the non-diagonal elements rjiðtÞ of the density matrix does not depend on the non-diagonalelements of the field fluctuations, but only on the diagonal elements zjj and zii of thesefluctuations, which means that, in this case, the positivity conditions (31) and (32) arealways satisfied in the rotating-wave approximation.These equations take a simpler form for a two-level system with a transition frequency

o0 � o10, interacting with an electromagnetic field with an amplitude EðtÞ and a frequencyo, which is not very far from resonance (o � o0). Using the notations SðtÞ for the


polarization of the system as a slowly varying amplitude of the non-diagonal element ofthe density matrix,

r10ðtÞ ¼12SðtÞe�iot; ð38Þ

and

wðtÞ ¼ r11ðtÞ�r00ðtÞ with the condition ð39aÞ

1 ¼ r11ðtÞ þ r00ðtÞ ð39bÞ

for the population difference, we obtain

d

dtSðtÞ ¼ �g?ð1�idoÞSðtÞ þ igEðtÞwðtÞ þ g2n

Z t

0

Sðt0Þeiðo�o0Þðt�t0Þ dt0; ð40aÞ

d

dtwðtÞ ¼ �gJ½wðtÞ�wT ��i

g

2½EðtÞS�ðtÞ�E�ðtÞSðtÞ�: ð40bÞ

The coefficients of these equations are the damping rate of the polarization (dephasingrate)

g? ¼ l01 þ l10 þ l00 þ l11; ð41Þ

the decay rate of the population difference

gJ ¼ 2ðl01 þ l10Þ; ð42Þ

the fluctuation rate of the self-consistent field of the environment particles (non-Markovian coefficient)

gn � jz11�z00j; ð43Þ

the relative atomic detuning

do ¼o�o0�gn

g?; ð44Þ

the equilibrium population difference

wT ¼ �l01�l10l01 þ l10

; ð45Þ

and the coupling coefficient of the system with the coherent electromagnetic field

g ¼e

‘~r01~1E ð46Þ

as a function of the transition dipole moment~r01 and the polarization vector of the field ~1E .The non-Markovian term of Eq. (40a) describes the effects of the fluctuations of the self-consistent field of the environment particles on the system polarization SðtÞ, cumulated duringthe whole history of the system from t0 ¼ 0 to t0 ¼ t. This description is certainly valid for ashort pulse of the field EðtÞ interacting with the system. For a longer time, we have to take intoaccount that the term gn in the exponential factor ðo�o0Þðt�t0Þ ¼ ðg?doþ gnÞðt�t0Þ of thenon-Markovian integral is the mean-value of a fluctuation. Thus, in system (40) we introduce a


random phase fnðt0Þ 2 ½0; 2p� with a fluctuation time tn ¼ 1=gn:

d

dtSðtÞ ¼ �g?ð1�idoÞSðtÞ þ igEðtÞwðtÞ þ g2n

Z t

0

Sðt0Þei½fnðt0Þþðo�o0Þðt�t0Þ� dt0; ð47aÞ

d

dtwðtÞ ¼ �gJ½wðtÞ�wT ��i

g

2½EðtÞS�ðtÞ�E�ðtÞSðtÞ�: ð47bÞ

In the following, by numerical calculations it will be found that the fluctuation time 1=gn

is very short on the scale of the time-evolution of the system. Thus, during a long-timeevolution t�t0bt4tn, these fluctuations have the tendency to washing out the term under theintegral, the non-Markovian term remaining significant only during a rather short memorytime t. That means that, for a long-time evolution, the quantum master Eq. (21) takes thegeneral form:

d

dtrðtÞ ¼ �

i

‘½H;rðtÞ��i

Xij

zij½cþi cj ;rðtÞ� þ

Xij

lijð½cþi cjrðtÞ;cþj ci� þ ½c

þi cj ;rðtÞcþj ci�Þ

þXijkl

zijzkl

Z t

t�t½cþi cj ;e

�i½fðt0Þþð1=‘ ÞHS0ðt�t0Þ�½cþk cl ;rðt0Þ�ei½fðt

0Þþð1=‘ ÞHS0ðt�t0Þ�� dt0;

ð48Þ

where fðt0Þ is a phase fluctuation operator. With this operator, in Eqs. (34) of the matrixelements, any memory phase oikðt�t0Þ is altered by a fluctuation phase fikðt

0Þ: oikðt�t0Þ-fikðt

0Þ þ oikðt�t0Þ. In this equation, fðt0Þ describes thermal fluctuations induced by the self-consistent field of the environment particles, while t is a memory time, which is much longerthan the fluctuation time of this field. In the following, we find that, for realistic values of thephysical parameters, the memory time t is much shorter than the decay/excitation times andthe period of the Rabi oscillation that characterizes the Hamiltonian dynamics. We notice thatthe fluctuation Hamiltonian ‘ zijc

þi cj in (48) is similar to the hopping Hamiltonian (3) in [40].

Besides this fluctuation Hamiltonian, a non-Markovian term of the second-order in thefluctuation matrix elements zij arises from the dissipative dynamics (1)–(8) in the approximationof a weak dissipative coupling.

4. Superradiant dynamics

We apply the master Eq. (48) to a superradiant n–i–p semiconductor heterostructure,working on the principle of the photon-assisted tunneling [41–45], as is represented inFig. 1. We consider this system as an assembly of AD Ne N t two-level quantumsystems, with an area AD and a length LD in a Fabry–Perot cavity, coupled to the twocounter-propagating modes of the electromagnetic field in this cavity, as is represented inFig. 2. This matter-field system is described by the Hamiltonian (2), including theHamiltonian (3) for the system of electrons, the Hamiltonian

HF ¼ ‘oðaþþaþ þ aþ�a� þ 1Þ ð49Þ

for the two counter-propagating waves of the electromagnetic field in the Fabry–Perotcavity, and the interaction potential

V ¼e

M~p~A: ð50Þ

√ AD

√ AD

LD

1

2

√ NeAD

1

2

√ NeAD

1 Nt

√

√ 1−

0=0

2

Fig. 2. Model of an assembly of N t of superradiant junctions, of area AD ¼ffiffiffiffiffiffiffiAD

p

ffiffiffiffiffiffiffiAD

pand length LD. These

junctions are conceived as square arrays with Ne ¼ffiffiffiffiffiffiNe

p

ffiffiffiffiffiffiNe

ptwo-level quantum dots per square meter,

coupled to an electromagnetic field with two counter propagating waves G,ffiffiffiffiffiffiffiffiffiffi1�Tp

G, existing in a Fabry–Perot

cavity with the transmission coefficient T of the output mirror.


This potential depends on the momentum of the system

~p ¼ iMX

ij

oij~rijcþi cj ð51Þ

and the potential vector

~A ¼‘e~K ðaþeikx þ aþþe�ikx þ a�e�ikx þ aþ�eikxÞ ð52Þ

for the electric field

~E ¼ i‘oe~K ðaþeikx�aþþe�ikx þ a�e�ikx�aþ�eikxÞ ð53Þ

propagating in the x-direction. In these expressions, M is the mass of the electron, ‘oij ¼

ei�ej is the energy of a transition j jS-jiS,~rij is the dipole moment of this transition, and~K ¼ ~1y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðl=VÞ

pis a vector in the y-direction of the field, depending on the fine-structure


constant a ¼ e2=4pe‘ c � 1137

, the field wavelength l ¼ 2p=k, and the quantization volumeof the electromagnetic field V.From the master Eq. (48) for a two-level system, we derive optical equations in the

approximation of the slowly varying amplitudes. We consider the non-diagonal matrixelements in the mean-field approximation

r10ðtÞ ¼ r�01ðtÞ ¼12½SþðtÞeikx þ S�ðtÞe�ikx�e�iot; ð54Þ

and of the population difference

wðtÞ ¼ r11ðtÞ�r00ðtÞ with the normalization condition; ð55aÞ

1 ¼ r11ðtÞ þ r00ðtÞ: ð55bÞ

Calculating the matrix elements of the two-level system, and averaging over the field states,from Eq. (48) we get

d

dtr10ðtÞ ¼ �½l01 þ l10 þ iðo0 þ z11�z00Þ�r10ðtÞ

þ~K ½ð/aþSþ/aþ�SÞeikx þ ð/aþþSþ/a�SÞe�ikx�o0~r10½r00ðtÞ�r11ðtÞ�

þðz11�z00Þ2

Z t

t�tr10ðt

0Þe�i½f10ðt0Þþo0ðt�t0Þ�dt0; ð56aÞ

d

dtr11ðtÞ ¼ �

d

dtr00ðtÞ ¼ 2½l10r00�l01r11�

þ~K ½ð/aþSþ/aþ�SÞeikx þ ð/aþþSþ/a�SÞe�ikx�o0~r10½r10ðtÞ þ r01ðtÞ�:

ð56bÞ

Since the mean-values of the annihilation operators of the superradiant field are of theform

/aþS ¼ ~aþðtÞe�iot; ð57aÞ

/a�S ¼ ~a�ðtÞe�iot; ð57bÞ

the mean-value of the electric field of the superradiant mode is

/~ES ¼ 12½~E ðtÞe�iot þ ~E�ðtÞeiot�; ð58Þ

with the slowly varying in time amplitudes

~E ðtÞ ¼ ~EþðtÞeikx þ ~E�ðtÞe�ikx; ð59Þ

and the slowly varying in space amplitudes

~EþðtÞ ¼ 2i‘oe~K ~aþðtÞ; ð60aÞ

~E�ðtÞ ¼ 2i‘oe~K ~a�ðtÞ: ð60bÞ

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

~E�ðtÞ ¼ �ffiffiffiffiffiffiffiffiffiffi1�Tp

~EþðtÞ: ð61Þ


With the notations

~g ¼e

‘~r10 ð62Þ

for the coupling coefficient,

g? ¼ l01 þ l10 ð63Þ

for the dephasing rate,

gJ ¼ 2ðl01 þ l10Þ ð64Þ

for the decay rate,

gn ¼ jz11�z00j ð65Þ

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

wT ¼ �l01�l10l01 þ l10

; ð66Þ

from (54) to (61) we obtain equations for the slowly varying amplitudes

d

dtSþðtÞ ¼ �½g? þ iðo0 þ gn�oÞ�SþðtÞ þ i~g~EþðtÞwðtÞ

þ g2n

Z t

t�tSþðt0Þei½�f10ðt

0Þþðo�o0Þðt�t0Þ� dt0; ð67aÞ

d

dtwðtÞ ¼ �gJ½wðtÞ�wT � þ ð2�T Þi~g

1

2½~E�þðtÞSþðtÞ�~EþðtÞS�þðtÞ�: ð67bÞ

In Eq. (67b) we have taken into account that the term

FþðtÞ ¼ i~g12½~E�þðtÞSþðtÞ�~EþðtÞS�þðtÞ� ð68Þ

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

F�ðtÞ ¼ i~g12½~E��ðtÞS�ðtÞ�~E�ðtÞS��ðtÞ� ð69Þ

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

F�ðtÞ ¼ ð1�T ÞFþðtÞ: ð70Þ

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

/cþi cjS ¼ rjiðtÞ; ð71Þ

from Eq. (48) we get the field equation

d

dt/aþS ¼ �io/aþSþ ~Ko0~r10½r10ðtÞ�r01ðtÞ�e

�ikx: ð72Þ

Thus, with (54), (57) and (60), we get a field equation for slowly varying amplitudes

d

dt~EþðtÞ ¼ �io0

‘oe~K ð~K~r10ÞSþðtÞ: ð73Þ


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

SþðtÞ ¼ uðtÞ�ivðtÞ; ð74Þ

and F ðtÞ and GðtÞ of the electromagnetic field

EþðtÞ ¼ F ðtÞ þ iGðtÞ; ð75Þ

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

d

dtF ðtÞ ¼ �gFF ðtÞ�g

‘o0

2eV vðtÞ; ð76aÞ

d

dtGðtÞ ¼ �gFGðtÞ�g

‘o0

2eV uðtÞ: ð76bÞ

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:

d

dtV 12eF 2ðtÞ

� �¼ �T c

1

2eF 2ðtÞA�gFVeF 2ðtÞ�g

‘o0

2FvðtÞ; ð77aÞ

d

dtV 12eG2ðtÞ

� �¼ �T c

1

2eG2ðtÞA�gFVeG2ðtÞ�g

‘o0

2GuðtÞ: ð77bÞ

At the same time, from (67b) with (74) and (75), we derive the equation for the populationdifference (55a), and introduce the particle flow I in a two-level system, due to the electriccurrent I ¼ eADNeI injected in the device:

d

dtwðtÞ ¼ �gF ½wðtÞ�wT � þ 2I þ ð2�T Þg½F ðtÞvðtÞ þ GðtÞuðtÞ�: ð78Þ

From (77) and (78) with (55), we get an equation of energy conservation:

‘o0I ¼d

dt‘o0r11ðtÞ þ ð2�T ÞV

1

2e½F 2ðtÞ þ G2ðtÞ�

� þ gJ r11ðtÞ�

1þ wT

2

� �‘o0

þð2�T Þ T cAV þ 2gF

� �V 12e½F 2ðtÞ þ G2ðtÞ�: ð79Þ

This equation describes the transition power ‘o0I of the active system as providing thetransfers of energy involved in the dissipative superradiant decay: (1) the energy variation ofthe electron-field system, (2) the dissipative decay of the electron energy, proportional to gJ,(3) the radiation of the field energy, proportional to the light velocity c and the transmissioncoefficient T of the output mirror, and (4) the dissipation of the field energy, proportionalto gF . In this equation, both waves leaving the quantum system and propagating in thecavity, the forward wave with an amplitude coefficient 1 and the backward wave with anamplitude coefficient R ¼ 1�T , are taken into account with the coefficient 1þR ¼ 2�T .From the polarization Eq. (67a) with (74) and (75), the population Eq. (78), and the field

Eqs. (77), we obtain the equations of the slowly varying amplitudes of the system:

d

dtuðtÞ ¼ �g?½uðtÞ�dovðtÞ��gGðtÞwðtÞ

þg2n

Z t

t�tfuðt0Þcos½fnðt

0Þ þ ðo�o0Þðt�t0Þ� þ vðt0Þsin½fnðt0Þ þ ðo�o0Þðt�t0Þ�g dt0;

ð80aÞ


d

dtvðtÞ ¼ �g?½vðtÞ þ douðtÞ��gF ðtÞwðtÞ

þg2n

Z t

t�tfvðt0Þcos½fnðt

0Þ þ ðo�o0Þðt�t0Þ��uðt0Þsin½fnðt0Þ þ ðo�o0Þðt�t0Þ�g dt0; ð80bÞ

d

dtwðtÞ ¼ �gJ½wðtÞ�wT � þ 2I þ ð2�T Þg½GðtÞuðtÞ þ F ðtÞvðtÞ�; ð80cÞ

d

dtF ðtÞ ¼ � 1

2T cAV F ðtÞ�gFF ðtÞ�g

‘o0

2eV vðtÞ; ð80dÞ

d

dtGðtÞ ¼ � 1

2T cAV GðtÞ�gFGðtÞ�g

‘o0

2eV uðtÞ; ð80eÞ

where fnðt0Þ � f01ðt

0Þ � �f10ðt0Þ is the phase fluctuation with a fluctuation time tn ¼ 1=gn,

and

do ¼o�o0�gn

g?ð81Þ

is the relative detuning. In these equations, the coupling of the electron system to theelectromagnetic field is described by a coupling coefficient for the electric dipole interaction(46). These equations also describe a dissipative decay of the electron system by thecoefficients gJ, g?, non-Markovian effects by time-integrals in the polarization Eqs. (80a),(80b), a decrease of the electron-field coupling due to the field radiation by the termproportional to the coefficient ð2�T Þ in (80c), and a decrease of field by the radiationterms proportional to the product cT in (80d), (80e), and by the terms proportional to thedecay rate gF .

5. Steady state

The dynamic Eqs. (80) take a simpler form in a stationary regime when the timederivatives become zero and the polarizations can be taken out from the integrals.Considering an integration over a fluctuation time tn ¼ 1=gn, we get

� g?�g2n

sin½ðo�o0Þ=gn�

o�o0

� �uþ g?doþ g2n

sin2½ðo�o0Þ=ð2gnÞ�

ðo�o0Þ=2

� �v�gGw ¼ 0; ð82aÞ

� g?doþ g2nsin2½ðo�o0Þ=ð2gnÞ�

ðo�o0Þ=2

� �u� g?�g

2n

sin½ðo�o0Þ=gn�

o�o0

� �v�gFw ¼ 0; ð82bÞ

�gJðw�wT Þ þ 2I þ ð2�T ÞgðGuþ FvÞ ¼ 0; ð82cÞ

�1

2T cAV þ gF

� �F�g

‘o0

2eV v ¼ 0; ð82dÞ

�1

2T cAV þ gF

� �G�g

‘o0

2eV u ¼ 0: ð82eÞ


From Eqs. (82a)–(82b) and (82d)–(82e), we get the existence condition of a quasi-stationary solution ðF ;GÞ of the field:

g?�gG

GF

w�g2nsin½ðo�o0Þ=gn�

o�o0

� �2

þ g?doþ g2nsin2½ðo�o0Þ=ð2gnÞ�

ðo�o0Þ=2

� �2

¼ 0; ð83Þ

where we used the notations:

G ¼ g‘o0

2eV ; ð84aÞ

GF ¼1

2T cAV þ gF : ð84bÞ

Generally, this condition cannot be fulfilled, that means that such a solution in fact doesnot exist, i.e. the system continuously oscillates under the influence of the fluctuations ofthe environment particles.In the following we solve Eqs. (82) in the Markovian approximation. Thus, instead of

Eqs. (82a) and (82b), we obtain the Markovian polarization equations:

�g?½u�dov��gGw ¼ 0; ð85aÞ

�g?½vþ dou��gFw ¼ 0: ð85bÞ

From the system of Eqs. (85a)–(85b) for the resonance case ðdo ¼ 0Þ and (82c)–(82e), wecalculate the flow density of the electromagnetic energy radiated by the device:

S ¼ T c12eðF 2 þ G2Þ: ð86Þ

We get

S ¼

‘o0

ð2�T ÞA

1þ2gFVT cA

I� �wT

gJ2þ

1

2T cAV þ gF

g2‘o0

g?gJeV

0BB@

1CCA

2664

3775: ð87Þ

We notice that this expression of the flow density S has a nice physical interpretation beingproportional to the product of the transition energy ‘o0, divided to the radiation areaof a quantum dot A, with the difference between the particle flow I and a thresholdvalue depending on the coupling, radiation, and dissipation coefficients. This expressionis valid when the quantization volume V of the field corresponds to the electromagneticenergy delivered by the whole system of NeNt quantum dots to a volume unit, whichmeans

V½m3� ¼1

Ne½m�2�Nt½m�1�; ð88Þ

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

�1] is the number ofsuperradiant junctions per length unit. For a longitudinal device, the N t (dimensionless)quantum dots in the x-direction, radiate through an area 1=Ne½m

�2�, which means

AL½m2� ¼

1

Ne½m�2�N t

; ð89Þ


while for a transversal device, theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNe½m�2�AD½m2�

pquantum dots in the y-direction,

radiate through an area ðLD½m�=N tÞð1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNe½m�2�

pÞ, which means

AT ½m2� ¼

LD½m�

Ne½m�2�N t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAD½m2�

p : ð90Þ

With the radiation area AL (AT ) of a quantum dot, from (87) we derive the flow density SL

(ST), and the total flow of the electromagnetic field radiated by the device in the twoversions:

FL ¼ ADSL; ð91aÞ

FT ¼ LD

ffiffiffiffiffiffiffiAD

pST : ð91bÞ

We obtain

FL ¼N t

ð2�T Þ 1þ 21LgF

T c

� � � ‘o0

eðI�I0LÞ; ð92aÞ

FT ¼N t

ð2�T Þ 1þ 21LgF

T c

A1=2D

LD

! � ‘o0

eðI�I0T Þ ð92bÞ

as a function of the injected current I and the threshold currents

I0L ¼1

2eNeADgJ �wT þ

eg?g2

L‘o0NeN t

ðT cþ 2 � 1LgF Þ

� �; ð93aÞ

I0T ¼1

2eNeADgJ �wT þ

eg?g2

T‘o0NeN t

T cLD

A1=2D

þ 2 � 1LgF

!" #; ð93bÞ

where we used the notation 1L ¼ N t=Nt for the length unit. The threshold current isproportional to the threshold population, which includes three terms for the threedissipative processes that must be balanced by current injection for creating a coherentelectromagnetic field: (1) the threshold value �wT, necessary to reach an inversion state ofpopulation, (2) the population inversion proportional to the light velocity c and thetransmission coefficient T , necessary to balance the radiation of the field, and (3) thepopulation inversion proportional to decay rate gF , necessary to balance the dissipation ofthe field in the cavity. The second term arises only due to the openness of the cavity, whilefor a closed cavity, when T ¼ 0 and no energy is lost by radiation, this term vanishes.

From (82c) and (92), we notice that when the injection current I ¼ eNeADI is under thethreshold value I0L (I0T), the radiation field is F þ iG ¼ 0, while the population differencew increases with this current. When the injection current I reaches the threshold current I0L

(I0T), the population difference w reaches the radiation value

wR ¼

T cAV þ 2gF

g2‘o0

g?eV

: ð94Þ


Increasing the injection current I beyond the threshold value I0L (I0T), the populationdifference keeps this value, while the superradiant field and the polarization variables

u ¼ �g

g?wRG; ð95aÞ

v ¼ �g

g?wRF ð95bÞ

increase with this current according to (92) with (86) and (91). However, the polarization(u, v) cannot increase indefinitely, being constrained by the condition of the Bloch vectorlength:

ð2�T Þðu2 þ v2Þ þ w2rw2T : ð96Þ

For the maximum value (uM, vM) of the polarization, while u2M þ v2M ¼ ðw

2T�w2

RÞ=ð2�T Þ,the superradiant field reaches its maximum flow density

SM ¼T ce

2ð2�T Þ w2T

g2‘2o2

0

e2V2

T cAV þ 2gF

� �2�g2?g2

26664

37775: ð97Þ

From this equation with Eq. (87) for S=SM, we get the value IM ¼ eNeADIM of theinjection current producing the maximum flow of the electromagnetic energy. Increasingthe injection current beyond this value, the polarization (u, v) will not increase any more,but the population will increase, leading to a rapid decrease of the polarization due to thecondition (96). Neglecting the current increase from IM to the value I 0M when thepolarization vanishes, from Eq. (82c) with w=�wT and u=v=0, we get a simple,approximate expression

IM � I 0M ¼12eNeADgJð�wT�wT Þ; ð98Þ

which can be compared with (93). Thus, from the operation condition I0L, I0ToIM , we getconditions for the coupling, dissipation, and radiation coefficients. These conditions will bederived as functions of the physical characteristics of the system, and will be studied bynumerical calculations.

6. System structure and microscopic model

We consider a superradiant semiconductor heterostructure, as an assembly of n–i–psemiconductor junctions as represented in Fig. 1. Such a junction contains four GaAs-layers, with a narrower forbidden band, for the two conduction zones n and p, and for thetwo narrow quantum wells determining the quantum zone with two energy levels E1 and E0

(Fig. 3). The potentials Uc and Uv of the two conduction regions n and p depend on theconcentrations of donors ND and acceptors NA of these regions. Between these twoconduction regions there is an active quantum region separated by potential barriersformed by two very thin slightly doped layers of Alx Ga1�xAs. The active quantum regionhas two states separated by a very thin i-layer of Alx Ga1�xAs with a larger forbiddenband. The active quantum region is composed of pairs of donors and acceptors (quantumdots) of concentration Ne, placed in two GaAs layers na and pa neighboring the i-layer. For

Uc

U3

i n

E1

E0

U1

U2

U0

U00

U4

Uv

p

Ie

Ih

U

0

y z

− √1−

√

x3 x1 x3 x2 x4 x5 x

0=0 T

GaAs

GaAs GaAs

AlxGa1−xAs

AlxGa1−xAs

AlxGa1−xAs

GaAs

Fig. 3. Dissipative superradiant n–i–p structure with quantum dots with the energy levels E1 and E0. By quantum

transitions between these levels, a superradiant electromagnetic field with two counter-propagating waves with the

amplitudes G andffiffiffiffiffiffiffiffiffiffi1�Tp

G is generated in a Fabry–Perot resonator with transmission coefficients of the mirrors

T 0 ¼ 0 and T40.


the potential wells/barriers, we use a rectangular model describing the dependences on themain physical characteristics: the widths and the heights of these wells/barriers. While thewidths of the wells/barriers correspond to the thicknesses of the GaAs/AlxGa1�x As layers,their heights are functions of two-dimensional doping arrays of donors/acceptorsembedded in these layers. When a current of electrons Ie is injected into the n-zone ofthe device, by tunneling through the barrier with the height U3 and width x1�x3, theseelectrons decay from the energy level E1 to the energy level E0, and tunnel through thebarrier with the height U4 and width x5�x4, into the hole states of the p-zone. Byrecombination, a hole current Ih ¼ Ie � I is extracted from the p-zone, closing the circuit.A quantum transition between the energy levels E1 and E0 of an active electron iscorrelated with the superradiant field of amplitude G, and with single-particle transitions ofthe dissipative systems: (1) the quasi-free electrons of the n-region, (2) the quasi-free holesof the p-region, (3) the vibrations of the crystal lattice, and (4) a free electromagnetic fieldof temperature T.

We consider the two-level system with the energies E1, E0 interacting with the twocounter-propagating waves of the electromagnetic field, which are reflected between thetwo mirrors of transmission coefficients T 0 ¼ 0 and T40 with the amplitudes G and�

ffiffiffiffiffiffiffiffiffiffi1�Tp

G. With the creation–annihilation operators aþþ�aþ and a�

þ�a� of the two

counter-propagating waves, we describe the dissipative quantum dynamics of the systemby the quantum master Eq. (48) with the Hamiltonian (2) including the terms

HS0 þ V ¼

ð~p þ e~AÞ2

2MþUð~rÞ ð99Þ


and (49), which depend on the potential Uð~rÞ, the potential vector ~A of the superradiantfield, and the potential of interaction V of an active electron with this field. In aweak-field approximation, when the term in ~A

2can be neglected, we get the interaction

potential (50) as a product of the two operators (51) and (52), for a field propagatingin the x-direction, where ‘oij ¼ ei�ej , a ¼ e2=4pe‘ c � 1=137, l ¼ 2p=k is the radiationwavelength, and V is the quantization volume of the electromagnetic field. We notice thatthe momentum (51) is non-diagonal, while the square of this momentum with thematrix elements

/mj~p2jnS ¼

M2

‘ 2

Xj

ðej�emÞðej�enÞ~rmj~rnj ; m;n;j ¼ 0;1 ð100Þ

is diagonal for a two-level system. Thus, we obtain non-diagonal two-body potentials that,due to the momentum conservation, are proportional to ~p, and diagonal fluctuations ofthese potentials, which are proportional to ~p2.We calculate the decay/excitation rates by using the general expressions (23)–(25) for

non-diagonal matrix elements of the dissipative potential, and the diagonal elements zii byusing the general expression (20). From Eqs. (36) to (37), we notice that these elementsdetermine entirely the dynamics of the system. With the assumption that in a quantizationvolume Vn (Vp), of a conduction region n (p), any electron (hole) is a quasi-free particlewith a kinetic energy Ea, we get the density of states [69]

gðnÞa ðEaÞ ¼ Vn

ffiffiffi2p

M3=2n

p2‘ 3

ffiffiffiffiffiffiEa

p; ð101aÞ

gðpÞa ðEaÞ ¼ Vp

ffiffiffi2p

M3=2p

p2‘ 3

ffiffiffiffiffiffiEa

p; ð101bÞ

where Mn (Mp) is the effective mass. For the two conduction regions n and p, we considerthe non-degenerate case, when the donor and acceptor concentrations ND and NA aresufficiently low to approximate the Fermi–Dirac distributions of electrons and holes withBoltzmann distributions:

f ðnÞa ðEaÞ ¼1

eðUcþEaÞ=T þ 1� e�ðUcþEaÞ=T ; ð102aÞ

f ðpÞa ðEaÞ ¼1

eð�UvþEaÞ=T þ 1� e�ð�UvþEaÞ=T ; ð102bÞ

where Ea is the kinetic energy in the conduction or valence band. Integrating the particlenumbers (102) over the states with the densities (101), we obtain the number of particles inthe quantization volume:Z 1

0

f ðnÞa ðEaÞgðnÞa ðEaÞ dEa ¼ VnND; ð103aÞ

Z 10

f ðpÞa ðEaÞgðpÞa ðEaÞdEa ¼ VpNA; ð103bÞ

where ND is the concentration of donors in the n-region and NA is the concentration ofacceptors in the p-region. With the approximate expressions (102), in Eqs. (103) we get an


analytically integrable formZ 10

e�Ea=TffiffiffiffiffiffiEa

pdEa ¼

ffiffiffipp

2T3=2: ð104Þ

Thus, we obtain the margins of the two conduction bands:

UcðTÞ ¼ T lnNcðTÞ

ND

; NcðTÞ ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMnT=2p

p‘

!3

; ð105aÞ

UvðTÞ ¼ �T lnNvðTÞ

NA

; NvðTÞ ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMpT=2p

p‘

!3

: ð105bÞ

For the two very thin GaAs layers of the quantum dot region na and nb, with margins U1

and U2 of the conduction bands determined by monoatomic layers of impurities of aconcentration Ne (see Fig. 3), we use a two-dimensional model. We obtain constantdensities of states:

gð1Þ ¼ Ae

Mn

p‘ 2; ð106aÞ

gð2Þ ¼ Ae

Mp

p‘ 2; ð106bÞ

where Ae is a quantization area. Integrating the particle numbers (102) over the states withthe densities (106),Z 1

0

gð1ÞdE

eðU1þEÞ=T þ 1¼ AeNe; ð107aÞ

Z 10

gð2ÞdE

eð�U2þEÞ=T þ 1¼ AeNe; ð107bÞ

we obtain the two potentials of these layers

U1ðTÞ ¼ �T ln ep‘2Ne=MnT�1

�; ð108aÞ

U2ðTÞ ¼ T ln ep‘2Ne=MpT�1

�: ð108bÞ

Similar expressions are obtained for the very thin AlxGa1�xAs-layers of the separationbarriers, with the margins U3 and U4 (see Fig. 3) as functions of the donor and acceptorarrays with concentrations N3 and N4 embedded in these layers:

U3ðTÞ ¼ �T ln ep‘2N3=MnT�1

�; ð109aÞ

U4ðTÞ ¼ T ln ep‘2N4=MpT�1

�: ð109bÞ

Since we choose the potentials U3, U4 of the separation barriers from the condition thatthese barriers provide the necessary current for the device operation by quantum


tunneling, we use the inverse relations to calculate the corresponding concentrations:

N3 ¼MnT

p‘ 2lnð1þ e�U3=T Þ; ð110aÞ

N4 ¼MpT

p‘ 2lnð1þ eU4=T Þ: ð110bÞ

From (23) and (20) with the densities of states (101), and the occupation probabilities ofthese state (102), we calculate the dissipation coefficients of the coupling to an n-clusterwith a quantization volume Vn

lðVnÞ

01 ð~R0Þ ¼

M3=2n

ffiffiffiffiffiffiffiffi2e10p

p‘ 4½e�ðUcþe10Þ=T þ 1�

j/a0jV ðnÞð~R0Þjb1Sj2Vn; ð111aÞ

lðVnÞ

10 ð~R0Þ ¼

M3=2n


p‘ 4½eðUcþe10Þ=T þ 1�

j/a0jV ðnÞð~R0Þjb1Sj2Vn; ð111bÞ

½zðVnÞ

ii ð~R0Þ�

2 ¼M

3=2n T3=2

pffiffiffiffiffiffi2pp

‘ 5e�Uc=T/ij½V ðnÞð~R0Þ�

2jiSVn; i ¼ 0;1; ð111cÞ

and to a p-cluster with a quantization volume Vp

lðVpÞ

01 ð~R0Þ ¼

M3=2p


p‘ 4½e�ð�Uvþe10Þ=T þ 1�

j/a0jV ðpÞð~R0Þjb1Sj2Vp; ð112aÞ

lðVpÞ

10 ð~R0Þ ¼

M3=2p


p‘ 4½eð�Uvþe10Þ=T þ 1�

j/a0jV ðpÞð~R0Þjb1Sj2Vp; ð112bÞ

½zðVpÞ

ii ð~R0Þ�

2 ¼M

3=2p T3=2


‘ 5eUv=T/ij½V ðpÞð~R0Þ�

2jiSVp; i ¼ 0;1; ð112cÞ

where e10 ¼ E1�E0 � ‘o0 is the transition energy of the system, ~R0 is the coordinate of adissipative cluster with the volume Vn (Vp), and V ðnÞ (~R0) ðV

ðpÞ ð~R0ÞÞ is the two-bodypotential of interaction with an electron of the n-conduction region (a hole of thep-conduction region) (Fig. 4). In (111a), (111b), (112a), and (112b), the initial, lower quasi-free state jbS corresponds to an energy that we approximate as Eb ¼ T=2, while thefinal, higher quasi-free state jaS corresponds to the excitation energy Ea ¼ Eb þ e10. In(111c) and (112c) we neglected the dimensions of a dissipative cluster in comparisonwith the distance ~R0, which means /aij½V ðnÞð~R0Þ�

2jaiS ¼ /ajaS/ij½V ðnÞð~R0Þ�2jiS ¼

/ij½V ðnÞð~R0Þ�2jiS, and a similar relation for V(p).

We take into account that the dissipation coefficients lij represent transition

probabilities due to the coupling to the environment particles, and zii fluctuations ofthe self-consistent field of these particles. That means that the total probability of atransition due to the coupling to the environment is a sum of transitions probabilitiesdue to the couplings to the components of this environment. At the same time, a

n-region-quasi-free electrons

p-region-quasi-free holes

quantumdots

R0

kP

kFEF

α0|V (n) (R0)|β1 α0|V (p) (R0)|β1

Eβ

Eα

Eα

Eβ

E1

E0

n p

−Lp / 2 Lp / 20−Lp / 2 Lp / 20

n p

Fig. 4. Dissipative couplings of an active electron to the environment. A decay j1S-j0S of the electron of a

quantum dot is correlated with: (1) a transition jbS-jaS of a quasi-free electron in a quantization volume Vn, (2)

a transition jbS-jaS of a quasi-free hole in a quantization volume Vp, (3) a phonon creation with a wave vector~kP, and (4) a photon creation with a wave vector ~kFEF .


fluctuation of the total environment is the quadratic sum of the fluctuations ofthe components of this environment. Thus, we take the dissipative coefficients of thecoupling to the entire system of the conduction electrons and holes as integrals of the

components (111) and (112) with the quantization volumes as differentials d3~R0 ¼ Vn, Vp.

We obtain

lE01 ¼ lðnÞ01 þ lðpÞ01 ; ð113aÞ

lE10 ¼ lðnÞ10 þ lðpÞ10 ; ð113bÞ

z2ii ¼ ½zðnÞii �

2 þ ½zðpÞii �2; i ¼ 0;1; ð113cÞ

with the components

lðnÞ01 ¼M

3=2n


p‘ 4½e�ðUcþe10Þ=T þ 1�

ZðnÞ

j/a0jV ðnÞð~R0Þjb1Sj2 d3~R0; ð114aÞ

lðnÞ10 ¼M

3=2n


p‘ 4½eðUcþe10Þ=T þ 1�

ZðnÞ

j/a0jV ðnÞð~R0Þjb1Sj2 d3~R0; ð114bÞ

½zðnÞii �2 ¼

M3=2n T3=2


‘ 5e�Uc=T

ZðnÞ

/ij½V ðnÞð~R0Þ�2jiSSnð~R0Þ d

3~R0; i ¼ 0;1 ð114cÞ


for the coupling to the conduction electrons in the n-zone, and

lðpÞ01 ¼M

3=2p


p‘ 4½e�ð�Uvþe10Þ=T þ 1�

ZðpÞ

j/a0jV ðpÞð~R0Þjb1Sj2 d3~R0 ð115aÞ

lðpÞ10 ¼M

3=2p


p‘ 4½eð�Uvþe10Þ=T þ 1�

ZðpÞ

j/a0jV ðpÞð~R0Þjb1Sj2 d3~R0 ð115bÞ

½zðpÞii �2 ¼

M3=2p T3=2


‘ 5eUv=T

ZðpÞ

/ij½V ðpÞð~R0Þ�2jiSSpð~R0Þ d

3~R0; i ¼ 0;1 ð115cÞ

for the coupling to the holes in the p-zone. In (114c) and (115c) we introduced the

screening functions Snð~R0Þ and Spð~R0Þ, which take into account that a field fluctuation

generated inside the n (p) layer is mostly absorbed in its propagation toward the outside.To calculate the matrix elements in these expressions, in the next section we derive thewave-functions.

7. Wave functions and dipole moments of the system

A wave-function c1ðxÞ (c0ðxÞ), with the energy E1 (E0), is considered extended only inthe barriers bounding the corresponding potential well, i.e. the tails of the wave-functionbeyond these barriers are neglected. Thus, we get

c1ðxÞ ¼ A1 cos k1ðx0�xÞ�arctana1k1

� �; x1rxrx0; ð116aÞ

c1ðxÞ ¼ A1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE1�U1

U0�U1

re�a1ðx�x0Þ; x0rxrx2; ð116bÞ

c1ðxÞ ¼ A1


U3�U1

re�a3ðx1�xÞ; x3rxrx1 ð116cÞ

for the firs well, and

c0ðxÞ ¼ A0 cos k0ðx�x2Þ�arctana0k0

� �; x2rxrx4; ð117aÞ

c0ðxÞ ¼ A0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU2�E0

U2�U00

re�a0ðx2�xÞ; x0rxrx2; ð117bÞ

c0ðxÞ ¼ A0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU2�E0

U2�U4

re�a4ðx�x4Þ; x4rxrx5 ð117cÞ

for the second well. These wave-functions depend on the wave-vectors in the two wells

k1 ¼1

‘

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MnðE1�U1Þ

p; ð118aÞ

k0 ¼1

‘

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MpðU2�E0Þ

p; ð118bÞ


on the attenuation coefficients in the corresponding barriers

a1 ¼1

‘

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MnðU0�E1Þ

p; ð119aÞ

a0 ¼1

‘

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MpðE0�U00Þ

p; ð119bÞ

a3 ¼1

‘


p; ð119cÞ

a4 ¼1

‘

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MpðE0�U4Þ

p; ð119dÞ

and on the normalization factors

A1 ¼ffiffiffi2p

x0�x1 þ‘ffiffiffiffiffiffiffiffiffiffi2Mn

p1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

U0�E1

p þ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

U3�E1

p

� �� 1=2; ð120aÞ

A0 ¼ffiffiffi2p

x4�x2 þ‘ffiffiffiffiffiffiffiffiffiffi2Mp

p 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE0�U00

p þ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

E0�U4

p

� �" #�1=2: ð120bÞ

We obtain the energy levels E1 and E0 as solutions of the equations:

E1�U1 ¼‘ 2

2Mnðx0�x1Þ2

arctan


E1�U1

rþ arctan


E1�U1

r� �2

; ð121aÞ

U2�E0 ¼‘ 2

2Mpðx4�x2Þ2

arctan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE0�U00

U2�E0

rþ arctan


U2�E0

r� �2

: ð121bÞ

Thus, the wave functions c1ðxÞ and c0ðxÞ are defined in the domains (x3,x2) and (x0,x5),respectively, their tails outside these domains being neglected. Due to the overlap of thesewave-functions in the domain (x0,x2) of the i-layer, a tunneling through this layer takesplace. The overlap function depends on the wave numbers (118), the attenuationcoefficients (119), and the normalization factors (120), as functions of the energy levelsgiven by (121). This tunneling is not a simple quantum process, due only to an overlap, buta complex process, including other processes of superradiance and dissipation in thesemiconductor structure and the Fabry–Perot cavity.

Two superradiant modes are possible: (1) a longitudinal mode, propagating in thetunneling direction, i.e. perpendicularly to the quantum dot layers, and (2) a transversalmode, propagating perpendicularly to the tunneling direction, i.e. in the plan of thequantum dot layers (Fig. 1). At this point, we have to take into account that while in thex-direction an electron is confined in the potential represented in Fig. 3, in the y, z-directionsthe electron moves quasi-freely with wave-numbers knðTÞ ¼ ð1=‘ Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MnEnðTÞ

p�

ð1=‘ ÞffiffiffiffiffiffiffiffiffiffiffiMnTp

in the n-zone and kpðTÞ ¼ ð1=‘ Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2MpEpðTÞ

p� ð1=‘ Þ

ffiffiffiffiffiffiffiffiffiffiffiMpT

pin the p-zone,

corresponding to temperature T of the system (Fig. 5). Thus, the total wave-functions of thesystem are of the form

C1ðx;y;zÞ ¼ c1ðxÞf1ðyÞw1ðzÞ; ð122aÞ

C0ðx;y;zÞ ¼ c0ðxÞf0ðyÞw0ðzÞ: ð122bÞ

x

y

z

1/√N e

1/√N

e

Electron Hole

kn

kp

na pai

E1 E0

Fig. 5. Quantum dot with an electron confined in the x-direction by the n-type active layer na, and a hole confined

in this direction by the p-type active layer pa. While an electron decays from the higher energy E1 in the na-layer to

the lower energy E0 in the pa-layer, recombining with a hole, these particles move quasi-freely in the plane (y, z) of

the two layers, with the wave vectors ~knðTÞ, ~kpðTÞ corresponding to temperature T of these layers.


With the overlap functions

cðxÞ01 ¼

Z x2

x0

c0ðxÞc1ðxÞ dx

��; ð123aÞ

cðyÞ01 ¼

Z y2

y1

f0ðyÞf1ðyÞ dy

��; ð123bÞ

cðzÞ01 ¼

Z z2

z1

w0ðzÞw1ðzÞ dz

��; ð123cÞ

and the dipole moments

x01 ¼

Z x2

x0

c0ðxÞxc1ðxÞ dx

��; ð124aÞ

y01 ¼

Z y2

y1

f0ðyÞyf1ðyÞ dy

��; ð124bÞ

z01 ¼

Z z2

z1

w0ðzÞzw1ðzÞ dz

��; ð124cÞ

we obtain the total dipole moments corresponding to a transition through the i-layer:

xðCÞ01 ¼ x01c

ðyÞ01 cðzÞ01 ; ð125aÞ

yðCÞ01 ¼ c

ðxÞ01 y01c

ðzÞ01 ; ð125bÞ

zðCÞ01 ¼ c

ðxÞ01 cðyÞ01 z01: ð125cÞ


Using (116)–(120), we obtain the overlap functions

cðxÞ01 ¼

A1A0

a0�a1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðE1�U1ÞðU2�E0Þ

ðU0�U1ÞðU2�U00Þ

sðe�a1ðx2�x0Þ�e�a0ðx2�x0ÞÞ; ð126aÞ

cðyÞ01 ¼ c

ðzÞ01 ¼ 1; ð126bÞ

corresponding to transitions with momentum conservation kn ¼ kp � k, which is observedin a weak-field approximation of the superradiant tunneling. In such a transition, the totalenergy, including the crystal lattice motion that is responsible for the effective mass variationfrom Mn to Mp, is conserved, while, due this mass variation, the particle energy is notconserved. In this case, a part of the initial energy ‘ 2k2=ð2MnÞ ¼ T is taken by the crystallattice, while the final energy of the particle takes a smaller value ‘ 2k2=ð2MpÞ ¼ ðMn=MpÞT .Only by a subsequent relaxation process, the particle takes its thermal value‘ 2k2

pT=ð2MpÞ ¼ T . We notice that the overlap function in the x-direction has a simpledependence on the ratios of the two zero-point vibration energies to the correspondingbarrier heights (E1�U1)/(U0�U1) and (U2�E0)/(U2�U00), and on the i-layer thicknessx2�x0. For transitions between the states (116) and (117) in the x-direction, and thermalstates of the form

f1ðyÞ ¼ffiffiffi2pA�1=4 cos½knðTÞy�; ð127aÞ

f0ðyÞ ¼ �ffiffiffi2pA�1=4 sin½kpðTÞy� ð127bÞ

in a quantization area A, we obtain the dipole moments

y01 ¼ z01 ¼‘

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MnEn

p ‘

2ffiffiffiffiffiffiffiffiffiffiffiMnTp ; ð128aÞ

y10 ¼ z10 ¼‘

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2MpEp

p ‘

2ffiffiffiffiffiffiffiffiffiffiffiMpT

p : ð128bÞ

With these expressions, we get the total dipole moments

xðCÞ01 ¼ c

ðxÞ01

x2�x0

2�

1

a0�a1

� �; ð129aÞ

yðCÞ01 ¼ z

ðCÞ01 ¼ c

ðxÞ01

‘

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MnEn

p cðxÞ01

‘

2ffiffiffiffiffiffiffiffiffiffiffiMnTp ; ð129bÞ

yðCÞ10 ¼ z

ðCÞ10 ¼ c

ðxÞ01

‘

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2MpEp

p cðxÞ01

‘

2ffiffiffiffiffiffiffiffiffiffiffiMpT

p : ð129cÞ

We notice that the longitudinal x-dipole moment is given by the half-thickness of thei-zone x2�x0, diminished with the attenuation width 1=ða0�a1Þ, and multiplied by theoverlap function. The transversal y- and z-dipole moments are given by the half-widthsdefined as ‘ c=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMnc2T

pand ‘ c=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMpc2T

p, respectively, also multiplied by the overlap

function. Due to the variation of the effective mass, the transversal dipole moments of thetransition from the n-region to the p-region are higher than those of the inverse transitions,from the p-region to the n-region. This is because at the same energy, a lighter particle has alarger wavelength.


For the second-order moments, by taking into account the contributions of the wave-function only inside the two potential wells of the two states, we get the simple,approximate expressions:

ðx2Þ11 �16

A21ðx

30�x3

1Þ ðx2Þ00 �

16A2

0ðx34�x3

2Þ; ð130aÞ

ðy2Þ11 ¼ ðz2Þ11 ¼ ðy

2Þ00 ¼ ðz2Þ00 �

1

12Ne

: ð130bÞ

In the next section, we show that, while the dipole moments (129) determine transitionsbetween the two energy levels E1, E0 induced by the environmental particles, the diagonalsecond-order moments (130) determine fluctuations of these levels, due the thermal motionof the environmental particles.

8. Coupling to the conduction electrons

A significant component of the dissipative dynamics comes from the Coulombinteraction between the active electrons, mainly located in the interval (x3, x5), and theenvironment of the conduction electrons and holes in the conduction regions ð�1;x3Þ andðx5;þ1Þ, respectively (Fig. 3). We use the notations~r for the position vector of an activeelectron, and ~R0 þ ~R for the position vector of a dissipative electron (hole), where ~R0 is theposition vector of an arbitrary n (p) cluster, and ~R ¼ ~1xX þ~1yY þ~1zZ is the position ofan electron (hole) in this cluster (Fig. 6). In this case, the Coulomb potential in a first-order

Quantum dot

Dissipative cluster

−Ln

2

Ln

2

x

y

z

X

Y

Z

R0

r

R

Fig. 6. An active electron with a position vector~r ¼ ~1xxþ~1yyþ~1zz in a quantum dot is coupled to a quasi-free

electron with a position vector ~R ¼ ~1xX þ~1yY þ~1zZ in a dissipative cluster as a cube with a position vector~R0 ¼ ~1xX0 þ~1yY0 þ~1zZ0 and a side Ln ¼ N

�1=3D .


approximation of the two-body term ~R~r ¼ Xxþ Yyþ Zz is

VCð~R;~rÞ ¼a‘ c

j~R0 þ ~R�~rj¼

a‘ c

j~R0j

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

2~R0ð~R�~rÞ

j~R0j2þ~R

2�2~R~r þ~r2

j~R0j2

s

�a‘ c

j~R0j1þ

Xxþ Yyþ Zz

~R2

0

!: ð131Þ

From this expression, only the second term, bilinear in the coordinates of an activeelectron and of an electron (hole) of the environment, yields contributions in the transitionmatrix elements of the decay/excitation rates (23):

VF ð~R;~rÞ ¼a‘ c

j~R0j3ðXxþ Yyþ ZzÞ: ð132Þ

To obtain the dissipative coefficients for the coupling to the entire system of conductionelectrons and holes, we consider (114a)–(114b) and (115a)–(115b) with potentials of theform (132), V ðnÞ ¼ VF ;V ðpÞ ¼ �V F , and integrate with ~R0 over the n- and p-regions from aradius equal to the distance to the nearest cluster, RD=ND

�1/3/2-x3 for the n-region,and RA=NA

�1/3/2þx5 for the p-region, up to infinity (Fig. 7).With the notations

~rðCÞ01 ¼~1xx

ðCÞ01 þ

~1yyðCÞ01 þ

~1zzðCÞ01 ; ð133aÞ

~RðnÞ

ab ¼~1xX

ðnÞab þ

~1yYðnÞab þ

~1zZðnÞab ; ð133bÞ

~RðpÞ

ab ¼~1xX

ðpÞab þ

~1yYðpÞab þ

~1zZðpÞab ; ð133cÞ

we get the dissipative coefficients:

lðnÞ01 ¼a2c2ð2MnÞ

3=2e1=210

3‘ 2 N�1=3D

2�x3

!3�ð~RðnÞ

ab~rðCÞ01 Þð

~RðnÞ

ba~rðCÞ10 Þ

e�ðUcþe10Þ=T þ 1; ð134aÞ

lðnÞ10 ¼a2c2ð2MnÞ

3=2e1=210

3‘ 2 N-1=3D

2-x3

!3�ð~RðnÞ

ab~rðCÞ01 Þð

~RðnÞ

ba~rðCÞ10 Þ

eðUcþe10Þ=T þ 1ð134bÞ

for the coupling to the sea of electrons in the n-region, and

lðpÞ01 ¼a2c2ð2MpÞ

3=2e1=210

3‘ 2 N�1=3A

2þ x5

!3�ð~RðpÞ

ab~rðCÞ01 Þð

~RðpÞ

ba~rðCÞ10 Þ

e�ð�Uvþe10Þ=T þ 1; ð135aÞ

Quantum dot

n-dissipative clusters p-dissipative clusters

0x3−RD x5 RA

NA−1/3ND

−1/3

R0

Fig. 7. The electron of a quantum 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).


lðpÞ10 ¼a2c2ð2MpÞ

3=2e1=210

3‘ 2 N-1=3A

2þ x5

!3�ð~RðpÞ

ab~rðCÞ01 Þð

~RðpÞ

ba~rðCÞ10 Þ

eð-Uvþe10Þ=T þ 1ð135bÞ

for the coupling to the sea of holes in the p-region, where ND�1/3/2 and NA

�1/3/2 are thehalf-sides of the dissipative clusters of the n-region with the donor concentration ND andof the p-region with the acceptor concentration NA, respectively (see Fig. 7). Accordingto (129), and the discussion that follows regarding the effects of the different effectivemasses Mn and Mp of the particles in thermal motion, in (134) and (135) we have takeninto account different matrix elements for the transitions jb1S-ja0S and for thereverse ones. These expressions describe single-particle transitions of the systembetween the two states C1ðx; y; zÞ and C0ðx; y; zÞ, with dipole moments x

ðCÞ01 ; y

ðCÞ01 ; z

ðCÞ01 ,

correlated to single-particle transitions of an environmental cluster between two quasi-freestates

Cað~RÞ ¼ caðX ÞfaðY ÞwaðZÞ; ð136aÞ

Cbð~RÞ ¼ cbðX ÞfbðY ÞwbðZÞ; ð136bÞ


with the dipole moments

Xab ¼

Z X2

X1

caðX ÞXcbðX Þ dX ; ð137aÞ

Yab ¼

Z Y2

Y1

faðY ÞYfbðY Þ dY ; ð137bÞ

Zab ¼

Z Z2

Z1

waðZÞZwbðZÞ dZ: ð137cÞ

That means that, while an active electron decays from C1ðx; y; zÞ to C0ðx; y; zÞ with acertain probability rate lðnÞ01 ðl

ðpÞ01 Þ, an electron (hole) in a n (p) dissipative cluster is excited

from a state CbðX ;Y ;ZÞ to a state CaðX ;Y ;ZÞ. We consider quasi-free states of differentparities in a quantization volume Vn ¼ L3

n:

cðnÞa ðX Þ ¼ �ffiffiffi2pV�1=6n sinðkðnÞa X Þ; kðnÞa ¼

1

‘

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MnEa

p; ð138aÞ

cðnÞb ðX Þ ¼ffiffiffi2pV�1=6n cosðk

ðnÞb X Þ; k

ðnÞb ¼

1

‘

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MnEb

pð138bÞ

for an initial energy Eb corresponding to the thermal motion, and the final energy Ea ¼

Eb þ e10 for an excitation with e10. With these states, we derive the dipole moments

XðnÞab � �2V

�1=3n

Z Ln=2

�Ln=2sinðkðnÞa X ÞXcosðk

ðnÞb X Þ dX

¼ L�1n Xcos½ðkðnÞa þ k

ðnÞb ÞX �

kðnÞa þ k

ðnÞb

þcos½ðkðnÞa �k

ðnÞb ÞX �

kðnÞa �k

ðnÞb

( )��Ln=2

�Ln=2

�L�1n

Z Ln=2

�Ln=2

cos½ðkðnÞa þ kðnÞb ÞX �

kðnÞa þ k

ðnÞb

þcos½ðkðnÞa �k

ðnÞb ÞX �

kðnÞa �k

ðnÞb

( )dX : ð139Þ

We notice that the second term of this expression is an integral of a rapidlyoscillating function that can be neglected. This integral vanishes exactly with theboundary conditions

cos ðkðnÞa þ kðnÞb Þ

Ln

2

� �¼ 1; cos ðkðnÞa þ k

ðnÞb Þ�Ln

2

� �¼ 1; ð140aÞ

cos ðkðnÞa �kðnÞb Þ

Ln

2

� �¼ 1; cos ðkðnÞa �k

ðnÞb Þ�Ln

2

� �¼ 1 ð140bÞ

in the quantization volume Vn, while the first term yields

XðnÞab ¼

2kðnÞa

ðkðnÞa Þ

2�ðk

ðnÞb Þ

2: ð141Þ


With the initial and the final energies Eb ¼ T=2;Ea ¼ Eb þ e10 in the wave-numbers (138),we obtain the dipole moments for the n-zone

XðnÞab ¼ Y

ðnÞab ¼ Z

ðnÞab ¼

‘e10

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2e10 þ T

Mn

s; ð142Þ

and similar expressions for the p-zone:

XðpÞab ¼ Y

ðpÞab ¼ Z

ðpÞab ¼

‘e10

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2e10 þ T

Mp

s: ð143Þ

With the dipole moments (129) of the system and the dipole moments (142) and (143) ofthe conduction electrons and holes, the dissipative coefficients (134) and (135) take theexplicit form:

lðnÞ01 ¼4a2c2

ffiffiffiffiffiffiffiffiffiffi2Mn

pe10 þ

T

2

� �jcðxÞ01 j

2m201

3N�1=3D

2�x3

!3

e3=210 ðe�ðUcþe10Þ=T þ 1Þ

; ð144aÞ

lðnÞ10 ¼4a2c2


pe10 þ

T

2

� �jcðxÞ01 j

2m201

3N

-1=3D

2-x3

!3

e3=210 ðeðUcþe10Þ=T þ 1Þ

; ð144bÞ

lðpÞ01 ¼4a2c2

ffiffiffiffiffiffiffiffiffiffi2Mp

pe10 þ

T

2

� �jcðxÞ01 j

2m201

3N�1=3A

2þ x5

!3

e3=210 ðe�ð�Uvþe10Þ=T þ 1Þ

; ð144cÞ

lðpÞ10 ¼4a2c2


pe10 þ

T

2

� �jcðxÞ01 j

2m201

3N

-1=3A

2þ x5

!3

e3=210 ðeð-Uvþe10Þ=T þ 1Þ

; ð144dÞ

where

m201 ¼x2�x0

2�

1

a0�a1þ

‘ffiffiffiffiffiffiffiffiffiffiffiMnTp

� �x2�x0

2�

1

a0�a1þ

‘ffiffiffiffiffiffiffiffiffiffiffiMpT

p !

: ð145Þ

The dissipation coefficients of the coupling to the conduction electrons (holes) get largerwith the overlap function c01

(x), and smaller with the distance �x3 (x5) of separation fromthe conduction region. With these coefficients, from (113a) to (113b) we obtain the termslE01; l

E10 of the total decay rates that, according to the general Eqs. (22) applied to our

system, include terms for the couplings to the crystal lattice vibrations lP01, l

P10, and to the

Fluctuations of theself-consistent field

(Cn) (Cp)x

−RD RA0

x0 + x12

x2 + x42

ND−1/3

Fig. 8. Quantum dot in the self-consistent field fluctuations of the quasi-free electrons in the n-region and the

quasi-free holes in the p-region.


free electromagnetic field g01, g10:

l01 ¼ lE01 þ lP

01 þ g01; ð146aÞ

l10 ¼ lE10 þ lP

10 þ g10: ð146bÞ

The electric decay/excitation processes, between the two quantum states j1S and j0S, ofthe coupling to the conduction electrons and holes, enter in competition with the processesof phonon creation/annihilation of the coupling to the crystal lattice, and with thesuperradiant processes. From (144), we notice that, for rather short separation distances�x3, x5, as is the case of a conventional light-emitting or laser diode, where the activetransitions are between conduction electrons and holes, the electric decay/excitation ratescan take very large values.

With (138), by neglecting the rapidly varying terms under the integrals of the second-order moments, we get

ðX 2Þn ¼ ðY2Þn ¼ ðZ

2Þn ¼ 2V�1=3n

Z Ln=2

�Ln=2X 2 cos2ðkðnÞa X Þ dX ¼

1

12N�2=3D ; ð147aÞ

ðX 2Þp ¼ ðY2Þp ¼ ðZ

2Þp ¼ 2V�1=3p

Z Lp=2

�Lp=2X 2 cos2ðkðpÞa X Þ dX ¼

1

12N�2=3A : ð147bÞ

We calculate the field fluctuations (114c) and (115c) considering that, due the planegeometry, the field fluctuations generated inside a conducting layer are emittedperpendicularly to this layer. We take into account this effect integrating not in thewhole n (p) region, but only in the cylinder Cn (Cp) of the neighboring clusters with anarea of the basis 1/Ne corresponding to the quantum dot area in the plane (y,z),and extended in the n (p) region from the nearest cluster up to infinity (Fig. 8). Due to thevery strong decrease with the distance j~R0j of the square of the potential (132), with power6 of this distance, in the integrals (114c) and (115c) we neglect the screening effect, i.e. we


take Snð~R0Þ ¼ Spð~R0Þ ¼ 1:

½zðnÞii �2 ¼

M3=2n T3=2


‘ 5e�Uc=T

ZðCnÞ

/ij½V ðnÞð~R0Þ�2jiSd3~R0; i ¼ 0;1; ð148aÞ

½zðpÞii �2 ¼

M3=2p T3=2


‘ 5eUv=T

ZðCpÞ

/ij½V ðpÞð~R0Þ�2jiSd3~R0; i ¼ 0;1: ð148bÞ

Thus, these integrals become one-dimensional, with the variable j~R0j ¼ ðx0 þ x1Þ=2�x fori=1 and j~R0j ¼ x�ðx2 þ x4Þ=2 for i=0, the differential element is d3~R0 ¼ dx=Ne, and thelimits of integration from �1 to �RD ¼ x3�ðN

�1=3D Þ=2 for the cylinder (Cn), and from

RA ¼ x5 þ ðN�1=3A Þ=2 to 1 for the cylinder (Cp) (see Fig. 8). By including under the

integrals the potential (132), which leads to the second-order moments (130), (147), andusing (105), we get the fluctuation coefficients zðnÞ11 , z

ðpÞ11 , z

ðnÞ00 , z

ðpÞ00 :

½zðnÞ11 �2 ¼

a2c2M3=2n T3=2

360pffiffiffiffiffiffi2pp

‘ 3�

N1=3D A2

1ðx30�x3

1Þ þ1

Ne

� �

NeNc

N�1=3D

2�x3 þ

x0 þ x1

2

!5; ð149aÞ

½zðpÞ11 �2 ¼

a2c2M3=2p T3=2


‘ 3�

N1=3A A2

1ðx30�x3

1Þ þ1

Ne

� �

NeNv

N�1=3A

2þ x5�

x0 þ x1

2

!5; ð149bÞ

½zðnÞ00 �2 ¼

a2c2M3=2n T3=2


‘ 3�

N1=3D A2

0ðx34�x3

2Þ þ1

Ne

� �

NeNc

N�1=3D

2�x3 þ

x4 þ x2

2

!5; ð149cÞ

½zðpÞ00 �2 ¼

a2c2M3=2p T3=2


‘ 3�

N1=3A A2

0ðx34�x3

2Þ þ1

Ne

� �

NeNv

N�1=3A

2þ x5�

x4 þ x2

2

!5: ð149dÞ

We notice that, from the physical point of view, these coefficients have understandabledependences on the parameters of the system. Thus, they get larger with the concentrationsof donors/acceptors ND/NA in the conduction regions and with temperature, get smallerwith the concentration of quantum dots in the active layer Ne, and rapidly decrease withthe separation distances �x3, x5.

9. Coupling to the crystal vibrations and the free electromagnetic field

The quantum transitions of the active electrons are correlated with transitions of theBose environment of the crystal vibrations according to (24). To obtain the matrix


elements in these expressions, we consider the electron system with the Hamiltonian (3)interacting with the phonon system with the Hamiltonian

HP ¼Xna

‘on aþnaana þ1

2

� �; ð150Þ

where a designates the polarization of a phonon mode. The process of a phonon creationby an electron decay, and an electron excitation by a phonon absorption, is described byan electron–phonon potential of the form:

VEP ¼X

ioj;n;a

ðV EPijnac

þi cja

þna þ V EP

jinacþj cianaÞ: ð151Þ

We consider the electron–phonon system with the Hamiltonian

HEPT ¼ HS

0 þHP þ V EP: ð152Þ

From the energy conservation of this system, according to Heisenberg’s equation ofmotion, we obtain

½V EP;HS0 � þ ½V

EP;HP� ¼ 0: ð153Þ

Of course, this equation does not describe the dynamics of our system of interest that hasmany other couplings, but it is used here only to derive the matrix elements of the electron–phonon potential (151). For a two-level system with the transition frequency o0, thisequation takes a formX

na

ðon�o0ÞðVEP01nac

þ0 c1aþna�VEP

10nacþ1 c0anaÞ ¼ 0; ð154Þ

which gives the resonant conditions:

on ¼ o0: ð155Þ

We take the momentum of the electron system (51), and the momentum of the phonon system

~P ¼ iXna

~1a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM‘on

2

rðaþna�anaÞ; ð156Þ

whereM is the mass of a quantization volume VP of the phonon field in the crystal. FromHeisenberg’s equation of motion for the conservation of the total momentum~p þ ~P, we obtain

½HS0 þ V EP;~p� þ ½HP þ V EP;~P� ¼ 0: ð157Þ

With the resonance condition (155), this equation takes the explicit form:

M‘o20~r01

~1a þ VEP01a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM‘o0

2

r !cþ0 c1 þ M‘o2

0~r01~1a þ V EP

10a


2

r !cþ1 c0

þ‘o0


2

r�

1

‘V EP

01aM~r01~1aðcþ1 c1�cþ0 c0Þ

" #aþa

(

�


2

r�

1

‘VEP

10aM~r01~1aðcþ1 c1�cþ0 c0Þ

" #aa

)¼ 0: ð158Þ

Since the phonon creation–annihilation operators are non-diagonal in a particle numberrepresentation, from the mean-value of this equation over the phonon states we obtain the


matrix elements of the electron–phonon potential:

V EP01a ¼ VEP

10a ¼ �‘o0Mo0~r01~1affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM‘o0=2

p : ð159Þ

At the same time, for the density of phonon states of energy ‘o0 in the quantization volume VP

of the phonon field in the crystal, we obtain an expression similar to (101):

gPð‘o0Þ ¼ VP

ffiffiffi2pM3=2

p2‘ 3

ffiffiffiffiffiffiffiffiffi‘o0

p: ð160Þ

We consider the sound velocity v from the phonon wavelength expressions

lP �2pkP

¼2p‘ v

e10¼

2p‘ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Me10p ; ð161Þ

and the crystal density

D �MVP

¼2p2‘ 2

VPl2Pe10

: ð162Þ

With (159)–(162), from (24) we obtain the decay/excitation rates:

lP01 ¼

Xa

E2e e

510

p‘ 6c4v3D�ð~r01~1aÞð~r10~1aÞ

1�e�e10=T; ð163aÞ

lP10 ¼

Xa

E2e e

510

p‘ 6c4v3D�ð~r01~1aÞð~r10~1aÞ

ee10=T�1; ð163bÞ

where Ee=M c2 is the rest energy of the electron, and v is the sound velocity, which can becalculated from the Young elasticity coefficient E and of the crystal density D:

v �

ffiffiffiffiE

D

r: ð164Þ

In the following, we show that these rates are the dominant terms in the total decay/excitationrates (146).In comparison with the Mosbauer effect, where a large nuclear transition energy e10

generally does not correspond to a phonon wavelength, our transition of rather smallenergy will always find a resonant phonon mode, in the quasi-continuous spectrum of thevibrational states, with wave-lengths much larger than the crystal constant.Similar relations, but with different matrix elements, are obtained from Eq. (25) for a

two-level system coupled to the free electromagnetic field:

g01 ¼2ae310‘ 3c2

�~r01~r10

1�e�e10=T; ð165aÞ

g10 ¼2ae310‘ 3c2

�~r01~r10

ee10=T�1: ð165bÞ


Due to the Boson nature of the two free fields, of photons and phonons, we obtainproportionality relations between the corresponding decay/excitation rates:

lP01 ¼

E2e e

210

Pað~r01

~1aÞð~r10~1aÞ

2paDc2ð‘ vÞ3~r01~r10g01; ð166aÞ

lp10 ¼

E2e e

210

Pað~r01

~1aÞð~r10~1aÞ

2paDc2ð‘ vÞ3~r01~r10g10: ð166bÞ

We notice that, while the electromagnetic decay/excitation rates (165) are proportional tothe transition energy e10 with power 3, the phonon decay/excitation rates (163) areproportional to this energy with a higher power, 5. These dependences are differentbecause the coupling to the phonon field is proportional to the momentumP ¼ ‘kP ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Me10p

, while the coupling to an electromagnetic field is proportional tothe potential vector A1=

ffiffiffiffiffiffie10p

[10]. This strong dependence of the phonon decay rate lP01,

on the transition frequency e510, means that, in our structure with a transition frequencye100:2 eV, the coupling to the phonon field is four orders of magnitude weaker than thatin a conventional light-emitting or laser diode, which has a transition frequencye101:4 eV. At the same time, we notice that the phonon decay/excitation rates (163) areinverse proportional to the sound velocity in the crystal with power 3, and to the crystaldensity.

U1

U2

I

U1

U2

U0

I

E1

E2

E0

Quantumtransition

Classicalpath

Superradiant

field I

Φ

Fig. 9. (a) Water flow of a hydroelectric plant with an initial energy level U1, and a final energy level U2;

(b) similar system with an intermediate level U0, much lower than the initial and the final levels U1 and U2; and

(c) superradiant quantum system, with a radiation power F ¼ ðE1�E0�ELÞI much larger than the electric power

ðE1�E2ÞI injected in the system by an electron flow I .


10. Superradiant semiconductor device

We consider the semiconductor structure of a certain number N t of n–i–p superradiantjunctions we described in Section 1, as a basic element of a quantum heat converter [12,14].The physical principle of this device can be understood in simple terms. Really, if we have a

Transistor 1 Transistor 2

n i p i n n i p i n

Hea

t abs

orbe

nt

Front electrode Rear electrodeoutput mirror

I − +

EI

ER

ET

Uc

0Deep LevelPath

Thermal Junction Superradiant Junction

Emitter Base Collector

Uv

Heat Absorption

E1

E0

Uc1

Uv1

SuperradiantTransition

I

I

I

n nb na pa pb p i i n Internal field

ER

EI

Fig. 10. (a) Quantum heat converter and (b) superradiant transistor.


classical system as the water flow of a hydroelectric plant (Fig. 9a), we obtain a power

F ¼ ðU1�U2�ELÞI ; ð167Þ

as the product of the difference between the initial potential energy U1 and the finalpotential energy U2 of the mass unit with the mass flow I, while EL is an energy loss. If wetake a similar system with a much lower intermediate energy level U0oU1;U2 (Fig. 9b),nothing happens, the power being the same as is given by Eq. (167). However, if instead ofthe classical system in Fig. 9b, we consider a quantum system of electrons as is representedin Fig. 9c, something extraordinary might happen: by a quantum transition from theinitial energy level E1 to the much lower intermediate level E0, the large energyE1�E0 might be given to a photon of a superradiant field, which means an electromagneticpower

F ¼ ðE1�E0�ELÞI ð168Þ

much larger than the power ðE1�E2ÞI injected in the device to obtain the electron flow I ,while EL is a term taking into account that some transitions are lost by dissipation.

The actual device [12], represented in Fig. 10a, can be regarded as a succession of n–i–p–i–n superradiant transistors (Fig. 10b), with an electron transfer in two steps: (1) asuperradiant quantum transition from emitter to base, and (2) a classical transfer, withthermal excitation of electrons from base to collector. This process involves a quasi-ohmiccontact between base and collector, that means a deep-level path crossing the energy gap ofthe base-collector junction. On this deep level path, the electrons are carried up by theinternal field of the p–i–n base-collector junction, while the energy of this field is recoveredby heat absorption, when a kinetic energy is provided to the electrons building up the fieldby diffusion. This electron transfer by the internal field of a junction is similar to thetransistor effect [70], with the difference that in an ordinary bipolar transistor the electronscross the base-collector junction decaying through the conduction band, with energydissipation (Fig. 11), while in a superradiant transistor (Fig. 10b), the electrons go up,absorbing energy from the internal field, which is from the environment. This phenomenoncan be understood from statistical reasons. From Fig. 10b, we notice that the injectedcurrent I increases the population of the lower states and decreases the population of thehigher states of the deep level path. That means that this region becomes colder, absorbing

n n

Uc

Uv

IE

IB

ICUc1

Uv1

0

Internalfield

Emitter Collector Base

p

Fig. 11. Ordinary bipolar transistor: while a small current IB is injected in base, an important part IC of the

emitter current IE is carried through the base-collector junction by the internal field.


heat from the environment. This heat absorption has the tendency to remake the initialstatistical distribution, modified by current injection. In the following calculations weneglect the temperature variation due to the heat transfer throughout the semiconductorstructure. To take into account this temperature variation, one has to make corrections ofthe parameters, to obtain the same transition frequency on the whole chain of superradiantjunctions.

Uc

U3

i n

E1

E0

U1

U2

U0

U00

U4

Uv

p Ie

Ih

x

z U y

x1 x0 x2x3 x4 x50

− √1−√

I

M1( 0=0) M2( )

1 2

na

nb

AlxGa1−xAs

GaAs

GaAs GaAs

pa pbGaAs

AlxGa1−xAs

AlxGa1−xAs

Uc

U3

i n

E1

E0

U1

U2

U0

U00

U4

Uv

p

Ie

Ih

x

z U y

x1 x0 x2x3 x4 x50

− √1−TG

√

I

M1(T0=0)

M2( )

1 2

na

nb

AlxGa1−xAs

GaAs

GaAsGaAspa

pbGaAs

AlxGa1−xAs

AlxGa1−xAs

Fig. 12. Dissipative superradiant n–i–p device with two injection electrodes E1 and E2 and a Fabry–Perot cavity

with the mirrors M1 and M2 of transmission coefficients T 0 ¼ 0 and T , respectively, in two possible versions (a)

and (b). (a) Longitudinal superradiant device with the Fabry–Perot cavity oriented in the x-direction of the

injected current I=Ie=Ih, i.e. perpendicular to the semiconductor layers. (b) Transversal superradiant device with

the Fabry–Perot cavity oriented in the y-direction, perpendicular to the injected current I=Ie=Ih, i.e. in the plane

of the semiconductor layers.


Such a device can be realized in two versions schematically represented in Fig. 12: (a) alongitudinal device with the two mirror metalizations M1 and M2 made on the two surfacesin the plane of the chip, of transmission coefficients T 0 ¼ 0 and T40, which form aFabry–Perot cavity coupling a superradiant mode that propagates in the x-direction of theinjection current; (b) a transversal device with the two mirror metalizations M1 and M2

made on two lateral surfaces of the chip, of transmission coefficients T 0 ¼ 0 and T40,which form a Fabry–Perot cavity coupling a superradiant mode that propagates in they-direction, perpendicular to the injection current. While in version (a) the roles of themirrors M1 and M2, and of the injection electrodes E1 and E2, are played by the samemetalizations, made on the two surfaces in the plane of the chip, in version (b) the mirrormetalizations M1 and M2, which are made on two lateral surfaces, are different from theelectrode metalizations E1 and E2.

The two devices have the same semiconductor structure, including layers of GaAs, with anarrower forbidden band and a heavier doping, for the quantum wells, and layers ofAlxGa1�xAs, with a larger forbidden band and a lighter doping, for the potentialbarriers. The margins of these bands are determined by the concentrations of the donors/acceptors embedded in the semiconductor layers. For the potential distribution, weconsider a simple rectangular model taking into account the essential characteristics ofthe system.

11. Operation conditions for the device parameters

We describe the superradiant dynamics of the systems represented in Fig. 12 by theoptical equations (80) with coefficients that take simple forms as functions of the physicalparameters of the system and temperature. These equations describe the coupling of asystem of electrons to the electromagnetic field, by a coupling coefficient for the electricdipole interaction, which according to (62) and (129), is

g ¼ gL ¼e

‘xðCÞ01 ¼

e

‘cðxÞ01

x2�x0

2�

1

a0�a1

� �ð169Þ

for a transversal device, while for a longitudinal device we take

g ¼ gT ¼e

‘

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiyðCÞ01 y

ðCÞ10

q¼

e

‘cðxÞ01

‘

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

MnMp

pT

q : ð170Þ

In Sections 8 and 9, we obtained explicit expressions of the decay and excitation rates l01and l10 of the electron system for three dissipative couplings. With these terms, from (42)and (146), we obtain the decay and dephasing rates

gJ ¼ 2g? ¼ gEJ þ gP

J þ gEMJ ; ð171Þ

where gEJ stands for the electric coupling to the conduction electrons, gP

J for the phononcoupling to the crystal lattice, and gEM

J for the coupling to the free electromagnetic field.The electric decay rate has two components for the interaction with the two conductionregions n and p:

gEJ ¼ gðnÞJ þ gðpÞJ : ð172Þ


From (42), for the two components n and p of the environment of conduction electronsand holes, with (144), we get

gðnÞJ ¼8a2c2


pe10 þ

T

2

� �jcðxÞ01 j

2m201

3N�1=3D

2�x3

!3

e3=210

; ð173Þ

and

gðpÞJ ¼8a2c2


pe10 þ

T

2

� �jcðxÞ01 j

2m201

3N�1=3A

2þ x5

!3

e3=210

: ð174Þ

These two expressions describe dipole-dipole couplings of an active electron to the quasi-free electrons and holes in the two conduction regions n and p, which are inverseproportional to the cubes of the separation distances N

�1=3D =2�x3 and N

�1=3A =2þ x5,

respectively. The electric decay rates (173) and (174) get lower with the transition energye10 ¼ E1�E0, an increase of this energy leading to transitions of the environment electrons/holes into states with wave-functions more rapidly varying in space, which form smallerdipole moments. From Eq. (42) for a phonon environment, with (165) and (166), we obtainthe phonon decay rate

gPJ ¼

Xa

2E2e e

510ð~r01

~1aÞð~r10~1aÞ

p‘ 6c4v3D�

ee10=T þ 1

ee10=T�1; ð175Þ

where Ee=M c2 is the rest energy of the electron, and ~1a are the polarization vectors of thephonon modes. We notice that the phonon decay rate gets higher with the transitionenergy e10 with power 5, with the dipole moments~r01,~r10, and with temperature T. At thesame time, it gets lower with the sound velocity with power 3, which, according to (164) is afunction of the Young elasticity coefficient E and the crystal density D. With (165), we getthe electromagnetic decay rate

gEMJ ¼

4ae310~r01~r10‘ 3c2

�ee10=T þ 1

ee10=T�1; ð176Þ

which also gets higher with the dipole moments ~r01, ~r10, and temperature T, but with thetransition energy e10 only with power 3. In the numerical case in Section 5, the phonondecay rate (175) dominates the electric decay rate (172)–(145), while the electromagneticdecay (176) is negligible. The non-Markovian coefficient (43), with the components (149),depends on the relative fluctuations between the two levels, induced by the twocomponents n and p of the environment that we add quadratically:

g2n ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½zðnÞ11 �

2 þ ½zðpÞ11 �2

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½zðnÞ00 �

2 þ ½zðpÞ00 �2

q� �2

: ð177Þ

The non-Markovian coefficient (177) with the components (149) arises due to the distancedifference of the two states from the two conduction regions: the field fluctuations of aconduction region have a stronger influence on the closer state than on the farer one. It is


interesting to notice that these fluctuations occur like a near-field effect, stronglydecreasing with the distance and with the quantum dot density Ne.

For the two versions of the device, we obtain the superradiant powers (92) as functionsof the injected current I and the threshold currents (93). A superradiant power is obtainedwhile the current I injected in the device is higher than the corresponding threshold currentI0L or I0T, and lower than the maximum current (98). From the necessary conditionIM4I0L; I0T , we find conditions of operation for the physical parameters of thesuperradiant system:

wIL ¼e0g?

g2L‘o0NeN t

ðT cþ 21LgF Þo�wT ð178aÞ

wIT ¼e0g?

g2T‘o0NeN t

T cLD

A1=2D

þ 21LgF

!o�wT : ð178bÞ

For instance, having in view that �wT � 1, we could chose physical parameters leading toinversion populations of the order wIL;wIT0:1. Inequalities (178) mean rather largevalues of the coupling coefficients gL, gT, of the transition energy ‘o0, of the quantum dotdensity Ne, and of the number of superradiant transistors N t, in comparison with theradiation characteristics c and T , polarization decay rate g?, and decay rate of the field gF .In this case, we can choose an injection current I, satisfying the condition

I0L;I0ToIoIM : ð179Þ

For an injection current I4IM , the active zone is invaded by electrons occupying bothenergy levels, which means that the normalization of the density matrix is no morepreserved, the charge accumulation leading to a detuning that inhibits the superradiationprocess. From (92), we notice that the radiation energy flow FT of the transversal mode, incomparison with the radiation energy flow FL of the longitudinal mode, is attenuated withthe factor A

1=2D =LD, due to the field propagation along the active layers. From Eqs. (93), we

notice that due to this propagation, the current threshold I0T of the transversal mode islower than the current threshold I0L of the longitudinal mode.

12. Operation conditions for the separation barriers

We notice that the power of the device essentially depends on the coupling coefficient ofthe active electron system to the superradiant field gL (gT), to the conduction electrons gE

J ,to the phonon system gP

J , and on the dissipation coefficient of the superradiant field gF . Inprinciple, these coefficients are in a competition mainly depending on the thickness x2�x0

of the i-zone that strongly influence the overlap of the two wave-functions of the activesystem, and on the distances x3 and x5 of separation from the two conduction zones thatdetermine the coupling to the quasi-free electrons and holes of these zones, respectively.However, the separation barriers must be not too large, to enable the electron transfer bytunneling. These barriers must have a higher penetrability P than the necessary value toprovide the injected current I, which means that this current must be smaller than thethermal current 1

6eNDvT P emergent from a unit volume and crossing the barrier with the

thermal velocity vT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiT=Mn

p. From the penetrabilities of the two barriers in the regions


n and p, P ¼ e�Bn ; e�Bp , we find the conditions:

BnoIn; ð180aÞ

BpoIp; ð180bÞ

for the barrier coefficients

Bn ¼1

‘

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Mn U3�Uc�

T

2

� �sðx1�x3Þ;

Bp ¼1

‘

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Mp Uv�U4�

T

2

� �sðx5�x4Þ; ð181Þ

and the injection coefficients

In ¼1

2ln

eNDAD

6I

ffiffiffiffiffiffiffiT

Mn

r� �;

Ip ¼1

2ln

eNAAD

6I


Mp

s !: ð182Þ

These coefficients depend on the current ratios eNDAD c/I and eNAADc/I, and the energyratios T/(Mnc2) and T/(Mpc2) for the n and p regions, respectively. With (119c) and (119d),we get conditions for the attenuation coefficients of the wave-functions in the separationbarriers a3 ¼ ð1=‘ Þ


p, a4 ¼ ð1=‘ Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MpðE0�U4Þ

p, and the thicknesses of

these barriers:

a3ðx1�x3Þo1

2ln

eNDAD

6I


Mn

r� �; ð183aÞ

a4ðx5�x4Þo1

2ln

eNAAD

6I


Mp

s !: ð183bÞ

The thickness x2�x0 of the i-zone must be sufficiently small to obtain the necessarycoupling coefficients (169) and (170) with the overlap function (126a). However,diminishing this thickness, the dissipative coupling to the conduction electrons also gethigher. This dissipative coupling can be diminished by increasing the separation distancesx3, x5, but only in the limits imposed by inequalities (183). In the following section, weshow that, even with these limitations, a large radiation power can be obtained.

13. Dissipative coefficients and stationary regime

We consider a chip of a GaAs–Al0.37Ga0.63As semiconductor heterostructure, as asquare with a side of 2 cm and a thickness of 2mm (see Fig. 1), which comprise 1045superradiant n–i–p junctions with a thickness equal to the wavelength of the superradiantfield, which is l ¼ 1:914mm for the chosen parameters of the system (Fig. 12). In thefurther calculations, we use the relative effective masses Mnr=0.05 of the electron andMpr= 0.5 of the hole, the relative electric permittivity er ¼ 12, the mobilities mn ¼

8000 cm2=Vs of the electron and mp ¼ 400 cm2=Vs of the hole, a Young elasticity


coefficient E ¼ C11 ¼ 5:38 1011 dyn=cm2, the thermal resistivity k ¼ 0:37 cal=ðcm s0 KÞ,and the crystal density D=5.317 g/cm3.

For the two GaAs conduction zones with the forbidden band Eg(GaAs)=1.426 eV wetake equal donor and acceptor concentrations ND ¼ NA ¼ 3:16 1016 cm�3. We considera temperature T ¼ 10ˆC, while from Eqs. (105), we get the margins of the conduction andvalence bands Uc=51.1meV and Uv=�135.4meV, respectively, which correspond toNc=2.57 1017 cm�3 and Nv=8.14 1018 cm�3. The quantum zone includes three layersof Al0.37Ga0.63As with the forbidden band Eg(Al0.37Ga0.63As)=1.92 eV: (1) the i-layer witha thickness that here is taken in the interval x2�x0 2 ð5; 10Þnm, in fact a weakly dopedlayer with the barrier height U0=0.5 eV, corresponding to an array of N0=6.4243 106

m�2 donors, (2) the nb barrier with a thickness x1�x3=10 nm and a heightU3=Ucþ0.05=0.1011 eV corresponding to an array of N3=8.01 1013m�2 donors, and(3) the pb barrier with a thickness x5�x4=3nm and a height U4=Uv�0.05=�0.1854 eV,corresponding to an array of N4=2.552 1013m�2 acceptors.

For a thickness of the i-layer x2�x0=5.5 nm and a density of quantum dots Ne ¼

1:476 1016 m�2 donors in the na–GaAs layer, and acceptors in the pa–GaAs layer, from(108), we get the potentials U1=�69.3meV and U2=�26.6meV. For energy levels of thetwo quantum dot wells equal to the margins of the two conduction and valence bands(E1=Uc and E0=Uv), from (121), we calculate the widths x0�x1=4.189 nm andx4�x2=1.576 nm of these wells, while, for the considered Ne-value, we get the internalfield entirely confined between the two quantum dot arrays. At the same time, thetransition energy is e10 � E1�E0 ¼ 0:1866 eV, the wavelength of the radiation in GaAstakes the value l ¼ 1:9158mm, while the dipole moments of coupling to the superradiant

field are xðCÞ01 ¼ 5:171 10�4 nm and


ðCÞ10

q¼ 4:689 10�4 nm, for the transversal and

the longitudinal device, respectively. From Eqs. (144)–(145) and (163)–(164), we get the

dissipative coefficients of the electric interaction lðnÞ01 ¼ 5:4517 105 s�1, lðnÞ10 ¼ 32:0194 s�1,

lðpÞ01 ¼ 4:8894 106 s�1, lðpÞ10 ¼ 9:0810 s�1, and of the phonon interaction lP01 ¼ 1:8947

107 s�1, lP10 ¼ 9:0572 103 s�1. We notice that the electric decay rate is an order of

magnitude lower than the phonon decay rate.

For a transmission coefficient of the output mirror T ¼ 0:1, the threshold currents takerather large values I0L=24.1149A and I0T=23.4528A with a maximum currentIM=46.0995A. For the chosen separation barriers, even with a rather large injectioncurrent as, for instance I=45A, the necessary relations between the barrier and theinjection coefficients are satisfied: In=Bn ¼ 1:727641, Ip=Bn ¼ 1:548741. At the sametime, for the non-Markovian coefficient we get the value gn ¼ 3:8015 1011 s�1. Taking alength of the p-zone of l=4, the total series resistance is Rs ¼ 7:1mO, while the electricallydissipated power is Pel=14.3643W. For the longitudinal device, we obtain a totalsuperradiant power FL ¼ 1:2843 kW, which means a power density PL ¼ 3:2107MW=m2.For the transversal device, we obtain a total superradiant power FT ¼ 0:288 kW, i.e. apower density PT ¼ 0:720MW=m2.

In Fig. 13, we represent the components gEJ , g

PJ , g

EMJ of the decay rate gJ, and the non-

Markovian coefficient gn, as functions of the thickness x2�x0 of the i-zone that essentiallydetermines the overlap of the two wave functions of the initial and final states and, by this,the transition dipole moment (see Eqs. (126a)–(129)). First of all, we notice that theelectromagnetic decay rate gEM

J is negligible in comparison with the electric and phonon

5 6 7 8 9 10101

102

103

104

x2−x0[nm]

[m V

−1 s

−1]

gL

gT

Fig. 14. The dependence of the coupling coefficients on the thickness of the i-zone.

5 6 7 8 9 1010−2

10−1

100

101

102

103

x2−x0 [nm] x2−x0 [nm]

[A] IM

I0LI0T

5 6 7 8 9 1010−2

10−1

100

101

102

103

[A] IM

I0LI0T

Fig. 15. The dependence of the threshold currents on the thicknesses of the i-zone for two values of the

transmission coefficient of the output mirror: (a) T ¼ 0:1 and (b) T ¼ 0:5.

5 6 7 8 9 10

100

105

1010

x2−x0[nm]

[s−1

]

γn

γ||

γ||

γ||

P

E

EM

Fig. 13. The dependence of the dissipative coefficients on the thickness of the i-zone.


5 6 7 8 9 100.9

1

1.1

1.2

1.3

1.4

1.5

1.6 x 1016

x2−x0 [nm]

Ne

[m−2

]

Fig. 16. The dependence of the quantum dot density on the thickness of the i-zone.


decay rates gEJ ; g

PJ , and that the phonon decay dominates the other decay processes. We

also notice that the variation of the decay rates with the thickness of the i-zone is verystrong, while the dependence of the non-Markovian coefficient gn is weak, this coefficientbeing essentially determined by the distance between the active electrons and the quasi-freeelectrons and holes in the conduction regions. The non-Markovian coefficient gn is muchlarger than the decay rates, this coefficient describing fluctuations of a mean-time 1=gn

much shorter than the decay time 1=gJ.In Fig. 14, we represent the dependence on the thickness x2�x0 of the coupling

coefficients gL, gT. As we can see from (169) to (170), since these coefficients areproportional to the dipole-moment, while the decay rates (173)–(175) are proportional tothe square of this moment, the inversion populations (178) do not depend strongly on these

coefficients and on the decay rates, but only on other parameters as Ne, N t, T and gF .From (169) and (170), it is interesting to notice that, although the two coefficients gL andgT have quite different expressions, depending on different parameters, they are of thesame order of magnitude. However, the dependence on x2�x0 of gT, which is proportional

to xðCÞ01 , is weaker than that gL, which is proportional to


ðCÞ10

q.

In Fig. 15, we represent the dependence on the thickness x2�x0 of the threshold currents.According to (93), the decrease of these currents with the thickness x2�x0 is mainlydetermined by the decrease of the decay rate gJ, the inversion populations wIL, wIT in thesquare parentheses depending only weakly on this thickness. It is interesting that, as onecan notice from (92), (98), and (179), by increasing the dissipation rate gJ, one obtains ahigher radiation power, given by a higher current IoIM . The threshold current of atransversal device is lower than that of a longitudinal one, due to the factor LD=A

1=2D in the

radiation term of the inversion population (178b). From Fig. 15b, in comparison withFig. 15a, we notice that, although according to (93) the threshold current gets larger withthe transmission coefficient T , the operation conditions (178) are satisfied for large valuesof T .

In Fig. 16, we represent the dependence on the thickness x2�x0 of the quantum dotdensity Ne, obtained from the condition that the entire internal field between the two


regions n and p is contained between the two na and pa arrays of the quantum dot system.If this density is lower, spatial charge zones arise at the boundaries of the two conductionregions to complete the deficit of charge in the quantum dot region. If this density ishigher, a mobile charge is attracted at the boundaries of the conduction regions to cancelthe excess of charge in the quantum dot region. We notice that this dependence is ratherweak.In Fig. 17a we represent the electric power and the radiation powers for the longitudinal

and transversal configurations of the device, as functions of the injected current. Aradiation power arises only when the injection current exceeds a threshold value. From(93b), we notice that, due to the factor LD=A

1=2D in the radiation term of the population

inversion, the threshold current of a transversal device is lower than the threshold current(93a) of a longitudinal one. However, due the same factor at the denominator of theradiation power (92b) of a transversal device, the increase with the injection current of thispower is lower than that of the radiation power (92a) of a longitudinal one. In Fig. 17b thetotal temperature variation in the semiconductor structure is represented. We notice that arather high power of 200W, that means 0.500MW from an active area of 1m2, can beobtained at a rather low temperature difference of about 7 1C.The radiation power of a transversal device becomes much higher by increasing the

transmission coefficient from T ¼ 0:1 to 0.5 and the transition dipole moment bydiminishing the thickness of the i-zone from x2�x0=6.5 nm to x2�x0=6 nm as isrepresented in Fig. 18. In this case, the threshold current of the transversal device becomessignificantly lower than that of the longitudinal one. The threshold current of thelongitudinal device is significantly lowered by decreasing the transmission coefficient fromT ¼ 0:5 to 0.2 as is represented in Fig. 19. It is remarkable that in the three casesrepresented in Figs. 17–19 the electric power dissipated in the device by the injectioncurrent I is much lower than the superradiant power. This is because, as one can noticealso from (92), the superradiant power produced by the injected current corresponds to the

4 4.5 5 5.5 6 6.5 7 7.5 80

50

100

150

200

250

I [A]

[W] ΦL

ΦT

PE

4 4.5 5 5.5 6 6.5 7 7.5 80

1

2

3

4

5

6

7

I [A]

[°C

] ∆TL

∆TT

Fig. 17. (a) The radiation powers FL and FT and the electric power PE as functions of the injection current I,

for x2�x0=6.5 nm, T ¼ 0:1, and gF ¼ 107 s�1 and (b) the temperature variations DTL, DTT as functions of the

injection current I.

8 10 12 14 16 18 200

100

200

300

400

500

600

700

I [A]

[W]

PE

ΦT

ΦL

8 10 12 14 16 18 200

5

10

15

20

25

I [A]

[°C

]

ΔTT

ΔTL

Fig. 19. The radiation powers FL and FT , the electric power PE, and the temperature variations DTL, DTT as

functions of the injection current, for x2�x0=6nm, T ¼ 0:2, and gF ¼ 107 s�1.

0100200300400500600700800900

I [A]

[W]

PE

ΦT

ΦL

8 10 12 14 16 18 20 8 10 12 14 16 18 200

5

10

15

20

25

30

I [A]

[°C

] ΔTT

ΔTL


functions of the injection current, for x2�x0=6nm, T ¼ 0:5, and gF ¼ 107 s�1.


high transition energy ‘o0 between the two zones n and p, while the power electricallydissipated by this current corresponds to a very low potential difference Uc�Uc1, necessaryfor carrying this current through the two rather thin highly conducting zones n and p(Fig. 10b). The difference between these two powers is obtained by heat absorption, whenthe electrons are excited from the lower potential of the p-zone to the higher potential ofthe n-zone of the base-collector junction. In Fig. 20 we consider a much larger decay rateof the electromagnetic field, gF ¼ 108 s�1 instead of gF ¼ 107 s�1, when the operationcondition (178) is also satisfied. In this case, we also obtain a high radiation power, butwith a higher injection current, which, however, does not produce an important electricalpower dissipated in the device.

0 0.5 1 1.5 2 2.5 3x 10−7

0

500

1000

1500

2000

2500

t [s]

ΦL

[W]

Non−Markovianfluctuation

Markovian evolution

0 0.5 1 1.5 2 2.5 3x 10−7

−0.5

0

0.5

1

t [s]

w(t)

u(t)

v(t)

Fluctuaion

1.9176 1.9177 1.9177 1.9178 1.9178 1.9179 1.9179

x 10−7

−0.5

0

0.5

1

t [s]

Fluctuation

u(t)

w(t)

v(t)

Fig. 21. Dynamics of a longitudinal superradiant device with x2�x0=5.5 nm and T ¼ 0:1 when a step current of

I=45A is injected in the device: (a) superradiant power; (b) polarization and population; and (c) polarization

fluctuation in a short timescale.

25 30 35 40 450

100200300400500600700800900

I [A]

[W]

ΦL

PE

ΦT

25 30 35 40 450

5

10

15

20

25

30

I [A]

[°C

]

ΔTT

ΔTL


functions of the injection current, for x2�x0=5.5 nm, T ¼ 0:5, and gF ¼ 108 s�1.



14. Non-Markovian fluctuations

Non-Markovian fluctuations are time-evolutions of polarization, population andfield due to the self-consistent field of the environment particles that, in our case, arethe quasi-free electrons and holes in the conduction regions of the device. In Fig. 21, werepresent the dynamics of a longitudinal device with a thickness of the i-zonex2�x0=5.5 nm and a transmission coefficient of the output mirror T ¼ 0:1, while thethreshold current is I0L=24.1149A and the maximum current is IM=46.0995A. Weconsider a step current of amplitude I=45A injected at time t=0. In the Markovianapproximation, a superradiant power FLðtÞ is generated as in Fig. 21a, while thepopulation wðtÞ and polarization variables u(t), v(t) have the time-evolutions represented inFig. 21b. At t=0, the population increases from the equilibrium value wT for thetemperature T, to wð0Þ ¼ wT þ 2I=ðeNeADgJÞ and, after that, while the radiation fieldincreases, the population decreases tending to an asymptotic value. With an appropriatechoice of the phase of the initial polarization, v(0)=0, while u(0) takes a valuecorresponding to the maximum value �wT of the Bloch vector, which is

uð0Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½w2

T�w2ð0Þ�=ð2�T Þp

. In the Markovian approximation, the electromagnetic

power is growing to a certain value, and after a short oscillation tends to the asymptotic

value that according to (92a) is FL ¼ 1:2843 103W.However, in the non-Markovian approximation, random fluctuations of the polariza-

tion, population, and field arise. In Fig. 21, we consider such a fluctuation arising at acertain moment of time. In Eqs. (80a) and (80b), we take a positive fluctuation with aduration tn ¼ 1=gn ¼ 2:6305 10�12 s, followed by a negative one with the same duration.From Fig. 21b, we notice that the polarization variables u(t), v(t) undergo very rapidvariations, which start a much longer evolution of these variables and of the radiation field(Fig. 21a). In Fig. 21c, these rapid variations are represented in a short timescale.

In Fig. 22, we represent the dynamics of the transversal device with the samesemiconductor structure and injected current, while the threshold current takes alower value I0T=23.4528A. We notice that, while the radiation power is lower, this

0 0.5 1 1.5 2 2.5x 10−7

0

50

100

150

200

250

300

350

t [s]

ΦT

[w]

Markovian evolution

Non−Markovian fluctuation

0 0.5 1 1.5 2 2.5x 10−7

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t [s]

Fluctuation

w(t)

u(t)

v(t)

Fig. 22. Dynamics of a transversal superradiant device with x2�x0=5.5 nm and T ¼ 0:1 when a step current of

I=45A is injected: (a) superradiant power and (b) population and polarization.

0 0.5 1 1.5 2 2.5x 10−7

0

50

100

150

200

250

300

350

t [s]

ΦL

[W]

Non−Markovianfluctuation

Markovian evolution

0 0.5 1 1.5 2 2.5x 10−7

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t [s]

w(t)

u(t)

v(t)

Fluctuation

Fig. 24. Dynamics of a transversal superradiant device with a negative fluctuation ðfn ¼ pÞ, followed by a

positive one ðfn ¼ 0Þ.

x 10−7

0

500

1000

1500

2000

2500

t [s]

ΦL

[W]

Markovian evolution

Non−Markovian fluctuation

0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3x 10−7

−0.5

0

0.5

1

t [s]

w(t)

u(t)

v(t)

Fluctuation

Fig. 23. Dynamics of a longitudinal superradiant device with a negative fluctuation ðfn ¼ pÞ, followed by a

positive one ðfn ¼ 0Þ.


device is much less sensitive to the thermal fluctuations described by the non-Markovianterm.In Figs. 21 and 22, we considered a positive fluctuation followed by a negative one, which

means an integration over a first interval of time tn ¼ 1=gn with a phase fn ¼ 0 followed byan integration over a second interval of time tn with a phase fn ¼ p in Eqs. (80a) and (80b).Changing the phases of the fluctuations, i.e. taking a negative fluctuation followed by apositive one (Figs. 23 and 24), we get similar evolutions but with opposite signs.

15. Discussion and concluding remarks

We obtained a non-Markovian master equation for a system of Fermions interactingwith an electromagnetic field, for the long-time dynamics of this system. By a Markovian


term, this equation describes transitions of the system particles correlated with single-particle transitions of a complex environment of other Fermions, Bosons and freeelectromagnetic field. This equation also includes a hopping potential for the fluctuationsof the self-consistent field of the environmental Fermions and a time-integral for time non-local effects of these fluctuations. In comparison with the similar master equation wepreviously derived for short pulses, the new equation includes a memory time, and arandom phase in the non-Markovian integral. We found that the fluctuation time isabout three orders of magnitude shorter than the characteristic times of the Hamiltonianand Markovian evolutions. Taking into account such fluctuations during a long-timeevolution, we conceive this integral as being split into terms with short integration-timeintervals, and random phases of the harmonic functions under the integral. In along-time evolution, these terms cancel one-another, only the terms in a rather shortmemory-time remaining significant. We showed that, for a finite spectrum of states and asufficiently weak dissipative coupling, this equation preserves the positivity of the densitymatrix. In the rotating-wave approximation, the time non-local term affects only the non-diagonal elements of the density matrix, the diagonal ones remaining of Lindblad’s form.Thus, the fundamental physical properties guaranteed by the Lindbladian time local term,as positivity and detailed balance, remain untouched.

As an application, we considered a superradiant semiconductor n–i–p heterostructure,as a basic element of a device we recently proposed for the conversion of the environmentalheat into usable energy. We derived polarization equations with additional terms for thethermal fluctuations of the environment particles, a population equation with anadditional term for a current injected in the semiconductor structure, and field equationswith additional terms for the radiation of coherent field from the Fabry–Perot cavityincluding this structure. On this basis, one can study the competition of the superradiantprocess, which is useful in this application, against dissipative effects that could deterioratethe performances of the device.

In the framework of a two-level model, we calculated the dissipative coefficients,where we have taken into account the coupling of the active electrons to the conductionelectrons and holes, lattice vibrations, and the free electromagnetic field. We obtainedthe dependence of the dissipative coefficients on the physical parameters of the device asthe density of quantum dots, the thicknesses and the impurity concentrations of thesemiconductor zones, and temperature. Thus, for the coupling with the conductionelectrons and holes, we got a strong decrease of the of the corresponding decay rates andfluctuation coefficients with the thicknesses of the separation barriers, namely with thesethicknesses with powers 3, and 5

2, respectively. In comparison with a light-emitting or laserdiode, based on recombination processes between conduction flows of carriers that comevery close to one another, and with a transition energy of the order of the forbidden band,our device, with an active region separated from the conduction regions by potentialbarriers, and with a much smaller transition energy, is much less dissipative. At the sametime, for the coupling to the crystal lattice vibrations, we obtained a strong increase of thecorresponding decay rate with the transition energy, namely with this energy with power 5.

We performed numerical calculations for a realistic semiconductor device including anumber of n–i–p superradiant junctions connected in series. A superradiant power that issignificant for applications is obtained for quite feasible values of the device parameters.While a current is injected in the semiconductor structure, the most part of the transitionenergy is converted into coherent electromagnetic energy, a smaller part is transferred to


the crystal vibrations, a still smaller part is dissipated in the conduction regions, and aquite negligible part is emitted as thermal radiation. The electron transfer through thequasi-ohmic contacts between the superradiant junctions is provided with energy byheat absorption from the environment. The operation characteristics of the device asfunctions of its physical parameters can be understood by simple physical interpretations.Thus, the decrease of the electric decay rate with the transition energy is an effect of thedecrease of the dipole moment of the conduction electrons that, being excited at higherenergies in the conduction band, get wave functions more rapidly oscillating in space. Thisdecrease of the dipole moment with the transition energy dominates the increase of thedensity of states.The increase of the phonon decay rate with the transition energy can be understood

by the increase of the density of states and of the interaction potential. However, thedensity of these phonon states increases with the transition energy as long as theirwavelength is much longer than the distance between atoms. When the phonon wavelengthapproaches the distance between atoms, the density of phonon states can no morebe considered quasi-continuous. For energies that are not in resonance with the vibrationalmodes, the coupling begins to decrease, finally vanishing as in the Mosbauer effect. In ourcase of rather low transition energies, when the phonon spectrum of states can beconsidered quasi-continuous, an optimum value of the transition energy exists, whenthe decay rate and, consequently, the threshold injection current take the minimumvalue. However, this minimum value of the threshold current could not be veryadvantageous for a high superradiant power. A low value of the threshold current means asmall current injected in the device, which cannot be much larger than the threshold value.This current must be smaller than a maximum value IM, otherwise altering thenormalization of the active electron distribution in the two quantum states and, by this,dramatically altering the difference between the corresponding energy levels. A largerdecay rate enables a larger injected current and, consequently, a larger superradiant poweras long as the threshold current I0L (I0T) remains significantly lower than the maximumcurrent IM.We studied two versions of this superradiant device: (1) a longitudinal device, with the

superradiant field propagating in the direction of the injected current, i.e. perpendicularlyto the semiconductor structure, and (2) a transversal device, with the superradiant fieldpropagating perpendicularly to the direction of the injected current, i.e. in the plane of thesemiconductor structure. We derived analytical expressions of the superradiant power forthe two versions of the device working in stationary regime. The time-dependent equationsof population, polarization, and field, have been numerically solved in the Markovianapproximation. In these equations, the non-Markovian dynamics is described by a time-integral of the polarization variables multiplied by harmonic functions with a frequencyequal to the fluctuation rate gn, and a random phase with the fluctuation time tn ¼ 1=gn.For the physical system considered here, the fluctuation time is much shorter than thedecay time: tn51=gJ. When such a fluctuation arises at a certain moment of time, a long-standing non-Markovian evolution of the radiation power, population, and polarization isstarted. The amplitude of such a fluctuation of the superradiant power is far from beingnegligible, but consists only in an oscillation around the Markovian value. Thus, wedescribe the device operation as a Markovian evolution of the superradiant system ofelectrons, with a noise generated as a time non-local effect of the thermal fluctuations ofthe self-consistent field of the conduction electrons and holes.


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Master equation and conversion of environmental heat into coherent electromagnetic energy

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