I A Shuda - Phase analysis of superradiance of a quantum-dot ensemble - страница 1

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NANOSYSTEMS

78.67.Но, 78.67.De ©2010

PHASE ANALYSIS OF SUPERRADIANCE OF A QUANTUM-DOT ENSEMBLE

I.A. SHUDA

pacs 73.21.La, 73.21.Fg, Sumy State University

(2, Rymskyi-Korsakov Str., Sumy 40007, Ukraine; e-mail: shudairaSmail. ru)

The phase analysis of the dynamic equations obtained in [13] on the basis of both a microscopic representation of the polarization of a quantum-dot ensemble and the difference of electron-level pop­ulations is carried out. It is shown that, under pumping and real relations between the parameters of a quantum-dot ensemble, the superradiance is realized in the form of a giant pulse regardless of the resonator frequency detuning and the coupling parameters. The obtained results are compared with experimental data.

1. Introduction

The phenomenon of superradiance in atoms and molecules is well known and thoroughly investigated (see, e.g., [1, 2]). Recently, superradiance was also ob­served for Bose-Einstein condensates of atoms [3], nu­clear spins [4], and magnetic molecules [5]. In [6], it was assumed that, in the case where the distances be­tween quantum dots or wells do not exceed the radi­ation length, they get involved in the effective inter­action through the electromagnetic field, due to which nanostructures can pass also to the superradiant mode. The development of up-to-date technologies allows one to form ensembles of nanoobjects with a density suffi­cient to provide such a mode [7]. As a result, superradi­ance was discovered for quantum dots [8], semiconductor heterostructures [9], and photon crystals [10].

From the physical point of view, such a behavior is explained by the fact that the motion of charge carri­ers in quantum wells, wires, and dots is confined to one, two, and three directions, respectively. Due to that, the indicated nanoobjects are characterized by quantum en­ergy levels typical of either isolated atoms or molecules (that is why quantum dots are called "artificial atoms") [11,12]. Taking into account that, with increase in the

dimensions of nanoobjects, the density of their electron states considerably decreases, so that the quantum prop­erties are most pronounced in quantum dots and get much weaker in wires and wells.

A consistent theory of the superradiance of a quantum-dot ensemble based on a microscopic represen­tation was developed in the recent work [13]. It was shown that the evolution of the system can be reduced to the following stages:

fluctuation mode lasting during a short time inter­val of Ю-15 -г 10-14 s, in which quantum dots rep­resenting electric dipoles autonomously emit elec­tromagnetic waves but do not yet interact with one another;

quantum stage lasting till the time moment 10-14^-10-13 s is characterized by the effective photon ex­change between dipoles, but the coherence is still absent;

coherent stage lasts till the time tCOh ~ 10-13 + 10-12 s; quantum dots imitate a superradiance pulse (with a maximum reached at the end of the delay time interval of the order of 5tCOh, while its duration approximately equals 2tCOh);

after the emission of the pulse of electromagnetic radiation, the system relaxes to a non-coherent state during the time T ~ 10_e s;

at t > T, the quantum-dot ensemble not subjected to the external pumping or consisting of weakly interacting dipoles passes to the stationary state corresponding to the attracting node of the phase plane; otherwise, there arises a sequence consisting

of nearly 10 pulses with a period of the order of НГ13 s.

A peculiarity of the approach [13] consists in that the transition from microscopic to macroscopic quantities is performed with the use of two fundamentally different averaging procedures: in the framework of the first one, one searches for statistical states of the quantum sys­tem using the mean-field splitting of all correlators; in the following averaging of dynamic quantities over the stochastic variable, one keeps the even correlators of the amplitudes of their fluctuations. Such an approach al­lows one to successively consider collective effects result­ing in a renormalization of the effective values of param­eters of the system (in contrast to the superradiance of atoms and molecules, these effects play the main role in the ensemble of semiconductor nanoobjects).

The technique proposed in [13] not only describes the developed superradiant mode but also consistently re­produces the mechanism of its reaching due to the am­plification of the electromagnetic radiation. In this case, the use of the single-mode approximation appears insuffi­cient as the noncontradictory picture of the phenomenon requires to consider the total collection of the transverse radiation modes, over which the space averaging is to be performed. The procedure yielded the dynamic equa­tions connecting the rates of change of the polarization of the quantum-dot ensemble P and the differences of the electron-level populations S with the values of P and S. The numerical solution of these equations shows that, at certain relations between the parameters, the system can generate both single pulses of electromagnetic radiation and their sequences. However, the values of parame­ters used in this case are not realized in semiconductor nanoobjects (for example, the nonmonotonous time de­pendences given in Fig. 3 of work [13] were obtained at the anomalously large attenuation parameter of the pop­ulation difference 7i hereinafter denoted by 75). That is why a more detailed investigation of the possible modes of generation of electromagnetic radiation by a quantum-dot ensemble is considered urgent.

As is known from synergetics, the simplest way to perform such an investigation is to employ the phase-plane method that considers a self-consistent change of the quantities P and S (instead of their time depen­dences P(t) and S(t)) expressed by the phase relation P(S) [17-20]. The proposed work is devoted to such a study. Section 2 briefly describes the technique given in [13] that yields dynamic equations for the polarization P and the population difference S. In Section 3, these equations are used for the study of the conditions of su­perradiance. The conclusions about the possible modes of superradiance at real values of the parameters of a quantum-dot ensemble are presented in Section 4.

2. Statement of the Problem

According to [13], the microscopic behavior of a system is determined by the time dependences of the pseudospin operators 0i~(t), and of(t) distributed over nodes

і = 1,... ,N, the vector potential, and the intensities of the electromagnetic radiation field, as well the currents induced by quantum dots and their semiconductor envi­ronment. Moreover, the behavior of the pseudospins is specified by the Heisenberg equation, whereas the elec­tromagnetic field obeys the Maxwell operator equations. Solving the latter, one can express the field potential in terms of the corresponding currents. As a result, it can be presented in the form of a sum of contributions caused by the self-action of dipole momenta of quan­tum dots, their radiation, and a stochastic component related to random changes of the fields of the dipoles and their environment. The use of this potential results in a closed system of equations for the pseudospins in­teracting through the electromagnetic radiation field.

The macroscopic behavior of the system is deter­mined by the quantum averages of the radiation field intensity Ei(t) = 2(aT~(t)), the polarization P,(t) = (at(t)ar(t) + ar(t)a+(t)}, and the pop­ulation difference S,(t) = 2 (af(t)). As the distance be­tween the quantum dots is much less than the radiation wavelength, one can pass from the summation over the nodes to the integration over the coordinate. On the other hand, the size of the medium the radiation passes through considerably exceeds its wavelength, that is why it is suitable to present it in the form of a cylinder with the z-axis parallel to the wave vector к of the radiation field. In this case, one can average all the space de­pendences over the direction normal to the cylinder axis and describe the longitudinal dependence of the radia­tion field by the plane wave e%(kz-ut) with frequency ш. As a result, the effective force acting on a quantum dot takes the form / = Fe^%wt + Ј(t), where the amplitude F determines the deterministic component and the term denotes the stochastic contribution caused by ran­dom changes of the dipole fields and the semiconductor medium.

The equations of motion obtained due to the indicated transformations must be averaged over the stochastic additive In this case, it is worth to consider the

following fundamental fact [13]: whereas the quantum

averaging marked above by the angle brackets supposes the splitting of the pseudospin correlators correspond­ing to different nodes, the averaging over the noise must be performed, by assuming that £*(*)£(*') ф 0 at = 0 (hereinafter, the bar over symbols marks the averaging over £(*)). As a result, the equations of mo­tion that describe the stochastic radiation of quantum dots take the form [131

f = -[іП(5)

TP(S)}E + fS,

dP dt

^2TP(S)P+ (f*E + E*f) S,

(1)

(2)

is specified by the frequency deviation A = w wq-

The substitution of expression (6) into Eqs.(2) and (3) results in the appearance of the terms proportional to the correlator £*(*)£(*') that quickly changes depending on the times t and t'. Averaging over the latter, one obtains the attenuation decrement

7 =

R lim ^ J dt J Є(*)С(*')е_(Ш+Гр)(*"*')Л'. (8) о 0

In this case, Eqs. (2) and (3) take the form dP

dt

^27P(1^ gPS)P+2TS'

(9)

d4 1

— = ^75 (S - Se) - gPlPP - - (f*E + E*f) S. (3)

Here, the collective frequency of radiation П = fl(S) and the effective attenuation decrement Гр = Tp(S) are determined by the expressions

П = w0 + gslpS,      Гр = 7p (1 - gPS),

(4)

where wq stands for the resonator's natural frequency, while 7p, 75 and gp, gs are the attenuation parameters and the coupling constants of the quantities P and S, respectively.

In system (l)-(3), the attention is attracted by the large factor fl(S) on the right-hand side of Eq.(l). That is why assuming that

75

w0

«1, 7p w0 «1, (5)

one can accept that the field E(t) changes much faster than the polarization P(t) and the population difference S(t). This allows one to integrate Eq.(l), by assuming the two last quantities to be constant. As a result, we obtain the time dependence of the radiation field inten­sity in the form [13]

Eq

FS

S + гГр

-(гП+Гр)і _|_

FS

-iujt

S + гГр

dS_ dt

lsSe - (75 + Г) S - gp-ypP,

where the effective attenuation decrement

Г = 7-

\F\2TP Tp + S2

1-е"

-rPt

) ~7 +

(10)

(11)

takes on a constant value at t    Гр1.

Analyzing system (9), (10), it is suitable to measure the time t in units of Wq1 and to relate the frequency ш, the attenuation decrements 7p, 75, 7, and Г and the field amplitude F to the natural frequency wq, and the polarization P to gp2, whereas the population difference S is related to to gp1. After that, Eqs. (9) and (10) take the simplified form

dP

^ = ^27P(l^S)P + 2rS2

dS

— = 75Se - (75 + Г) S - 7pP

containing the effective attenuation decrement 7p|P|2(1^S)

Г~7-

7p (1 ^)2 + [(u;^ 1) ^57pSr

(12)

(13)

(14)

with the ratio of the coupling constants g = gs/gp ~ 1-

+S J С(і')е-(Ш+Гр)(*-*')А'. о

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