I A Shuda - Phase analysis of superradiance of a quantum-dot ensemble - страница 1
78.67.Но, 78.67.De ©2010
PHASE ANALYSIS OF SUPERRADIANCE OF A QUANTUM-DOT ENSEMBLE
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  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.
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 , nuclear spins , and magnetic molecules . In , 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 . As a result, superradiance was discovered for quantum dots , semiconductor heterostructures , and photon crystals .
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 . 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  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  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  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  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 , 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 : 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,
^2TP(S)P+ (f*E + E*f) S,
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
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
— = ^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),
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
«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 
S + гГр
S + гГр
lsSe - (75 + Г) S - gp-ypP,
where the effective attenuation decrement
Г = 7-
\F\2TP Tp + S2
) ~7 +
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
^ = ^27P(l^S)P + 2rS2
— = 75Se - (75 + Г) S - 7pP
containing the effective attenuation decrement 7p|P|2(1^S)
7p (1 ^)2 + [(u;^ 1) ^57pSr
with the ratio of the coupling constants g = gs/gp ~ 1-
+S J С(і')е-(Ш+Гр)(*-*')А'. о