Incoherent processes: master equation and Lindblad terms

The correct description of the system dynamics must include the decoherence processes, as we explained in Section 2.1. The first element to take into account is dissipation. The excitations (photons, excitons or polaritons) will eventually leak out of the system (the cavity in our case). Apart from the decoherence induced in the system, these excitations can be detected from the outside and provide valuable information of the light-matter coupling inside the cavity. Therefore, at zero temperature--as we have seen when studying the thermal equilibrium states--photons and excitons in the cavity have a finite lifetime ( $1/\kappa$ ). At $T\neq0$ , not only the dissipation rate is now given by Eq. (2.32), but also there is an intrinsic particle income from the environment with a rate given by Eq. (2.31). In order to take this into account, the model is upgraded from a Hamiltonian [Eq. 2.53)] to a Liouvillian description with a quantum dissipative master equation for the density matrix of the system $\rho$ :

$\displaystyle \frac{d\rho}{dt}=i[\rho,H]+\mathcal{B}^a_T\rho+\mathcal{B}^b_T\rho\,.$

(2.69)

Each superoperator $\mathcal{B}^c_T$ (

) is composed of two so-called Lindblad terms, after the study of equations of motion for density matrices by Lindblad (1976). They represent the outcoming and incoming particles, respectively:

$\displaystyle \mathcal{B}^c_T\rho= \frac{\kappa_c(1+\bar n_T)}2(2c\rho\ud{c}-\u... ...ho\ud{c}c) +\frac{\kappa_c\bar n_T}2(2\ud{c}\rho c-c\ud{c}\rho-\rho c\ud{c})\,.$

(2.70)

Mainly, there are two possible (and complementary) derivations of these terms in the literature. The first one is the microscopic approach described by Carmichael (2002) or Gardiner (1991). It consists in considering dissipation as a coupling to a bath of oscillators and taking the following steps:

In the interaction picture, we apply second order perturbation theory on the coupling constants.
Born approximation: we assume that the coupling is so weak that system and reservoir are separable at all times and that the bath is too large to be affected by the system dynamics.
We consider a Markovian environment: the interactions with the system take place at a longer timescale than the environment's internal dynamics and, therefore, any correlations induced in the environment by such interactions are quickly lost.
We trace out the reservoir degrees of freedom making use of their bosonic statistics in thermal equilibrium.
We change back to the Schrödinger picture.

In this frame, the escape of the cavity photons is accounted for by the coupling to a reservoir of exterior photons in thermal equilibrium. The $\kappa_a$ parameter is inversely proportional to the cavity quality factor : $\kappa_a=\omega_a/Q_a$ .

There is also the spontaneous decay of the QD into other modes than that of the cavity, following the mechanism of Weisskopf and Wigner, even if they are kept in the vacuum state (at zero temperature). Although their density of states has been greatly reduced, it will be nonzero in a realistic cavity. Nonradiative de-excitation also takes place due to the coupling to phonons and other particles of the solid state environment. The total QD decay rate, $\kappa_b$ , is typically much smaller than the cavity emission rate $\kappa_a$ but still they can induce significant deviations from the ideal case and should be included in an accurate description. The environment also induces pure dephasing on the light-matter coupling, causing the off-diagonal terms of the density matrix, linked to light-matter coupling, to decay. I will not consider this effect for simplicity and because in general it only contributes to destroy the coherence, and its role is well understood.

The last essential ingredient is the excitation of the system. In semiconductor experiments, one usually excites--optically or with electrical injection--the electronic levels far above resonance. Then, a reservoir of electron-hole pairs is created in the wetting layer with further relaxation to the exciton level. A detailed microscopic analysis of carrier capture in QDs has been developed by Nielsen et al. (2004) taking into account semiconductor many-body physics. It showed that the Coulomb scattering of electrons and holes, in delocalized states of the wetting layer, can provide efficient transitions into the discrete localized QD states. Also LO-phonons can be an important mechanism responsible for such a relaxation. Another approach with a microscopic derivation of the pumping mechanism has been recently investigated by Averkiev et al. (2009).

In this work, the pumping terms will represent only carrier capture due to phonons, processes where a fully correlated electron-hole pair is created in the QD. Our aim, therefore, is not to make a systematic analysis of all the relaxation processes which are taking place in the system. Rather, it is to develop an heuristic model where one can investigate the impact of the pumping mechanism at a fundamental level. The pumping is modelled by a coupling to a reservoir of electron-hole pairs and phonons. However, some conceptual changes are needed in the microscopic derivation of these terms that we described above. The case of electronic pumping, for instance, is similar to the process of laser gain: the medium requires an inversion of electron-hole population, something that cannot be achieved by means of a simple HO heat bath. The actual process of gaining an exciton in the QD involves the annihilation of an electron-hole pair in an external reservoir out of equilibrium and the emission of a phonon, that carries the excess of energy, to another external reservoir (which can be in thermal equilibrium). A simple effective description of this nonequilibrium process can be made by an inverted HO with levels $E_p=- \omega_\mathrm{p}(p+1/2)$ maintained at a negative temperature, as explained by Gardiner (1991). Since the raising operator for the energy decreases the number of quanta of this oscillator, the role of creation and destruction operators is indeed reversed with respect to the usual case of damping. Effectively, this results in new Lindblad terms for the incoming particles, like those in the last term of Eq. (2.70), but that can be controlled externally and independently. This mechanism of direct excitation of the excitonic degree of freedom is sketched in the right side of Fig. 2.2, where is represented the QD under study, represented by its two levels and interacting with the single-photonic mode with coupling strength .

**Figure 2.2:** Schema of our system for the SS case: self-assembled QDs in a semiconductor microcavity. The QD sketched on the right is in strong coupling with the cavity mode with coupling strength , while the one of the left is in weak coupling. The electron-hole pairs created by the incoherent pumping of the structure provide an effective electronic pumping, of the dot of interest, while the pumping of the assembly of nearby dot results in an effective cavity pumping through rapid conversion of the excitons into cavity photons.
$\includegraphics[width=.6\linewidth]{chap2/fig2-schema.pdf.eps}$

I also consider another type of pumping, that offers a counterpart for the cavity by injecting photons incoherently. The major factor to account for such a term is the presence of many other QDs, that have been grown along with the one of interest. Those only interact weakly with the cavity. In most experimental situations so far, it is indeed difficult to find one dot with a sufficient coupling to enter the nonperturbative regime. When this is the case, all the other dots that remain in weak coupling (WC) become ``spectators'' of the strong coupling (SC) physics between the interesting dot and the cavity, and their presence is noticed by weak emission lines in the luminescence spectrum and an increased cavity emission. They are also excited by the electronic pumping that is imposed by the experimentalist, but instead of undergoing SC, they relax their energy into the cavity by Purcell enhancement or inhibition, depending on their proximity with the cavity mode. This, in turn, results in an effective pumping of the cavity as was also pointed out by Keldysh et al. (2006).

The second possible procedure to derive the Lindblad terms is based on Monte Carlo methods and quantum jumps. In the books of Gerry & Knight (2005) and Haroche & Raimond (2006), this approach is preferred as it is closer to quantum information and measurement theories. The time evolution of a system is conceived as a succession of coherent periods of the Hamiltonian dynamics (inside a manifold) and incoherent events (between manifolds), taking place with some probability, which force the collapse of the wavefunction into a given realization. In this image, the microscopic origin of the incoherent processes is overlooked and they are just assumed to exist with a given probability and give rise to random flows of incoming and outcoming particles. Once we have analyzed the most relevant processes leading to incoherent dissipation and pumping, we adopt this point of view, as it goes better with the spirit of our study. We can define the Liouvillian $\mathcal{L}^c$ that acts in the density matrix through the jump operator as $\mathcal{L}^c=2c\rho\ud{c}-\ud{c}c\rho-\rho\ud{c}c$ and consider the general master equation

$\displaystyle \frac{d\rho}{dt}=i[\rho,H]+\frac{\gamma_a}{2}\mathcal{L}^a\rho+\f... ...+\frac{P_a}{2}\mathcal{L}^{\ud{a}}\rho+\frac{P_b}{2}\mathcal{L}^{\ud{b}}\rho\,,$

(2.71)

which includes the total rates for decay $\gamma$ and pump

for both modes

. If

is an annihilation operator, the Lindblad term leads to decay (loss of particles

) and if it is creation operator, it leads to pumping (injection of particles

). All together, these elements can be put in the form the total superoperator $\mathcal{L}$ and constitute the master equation as $d\rho/dt=\mathcal{L}\rho$ . In what follows, we consider that these parameters can take all possible values as long as they drive the system to some SS (with nondivergent populations). We will not be concerned with the exact experimental situation that leads to such parameters. They could well correspond to the action of one or more thermal baths, with positive of negative temperatures.

It is interesting to look at the difference between the bosonic, , and fermionic, $\sigma$ , density matrices for SS of the free particles under pump and decay. For bosons, solving the master equation for the general term $\rho^b_{n,p}=\bra{n}\rho^b\ket{p}$ ,

$\displaystyle \frac{d}{dt}\rho^b_{n,p}$	$\displaystyle =$	$\displaystyle -[i \omega_b(n-p)+(\gamma+P)\frac{n+p}{2}+P]\,\rho^b_{n,p}$
		$\displaystyle +P\sqrt{(n+1)(p+1)}\,\rho^b_{n+1,p+1}+\gamma\sqrt{np}\,\rho^b_{n-1,p-1}\,,$	(2.72)

leads, as we know, to the thermal mixture with average number give by Eq. (2.34): $\rho^b_{n,n}=(\frac{P}{\gamma})^n(1-\frac{P}{\gamma})$ . The fermionic master equation,

$\displaystyle \begin{eqnarray}\frac{d}{dt}\rho^\sigma_{0,0}&=&-P\rho^\sigma_{0,... ...mega \, \rho^\sigma_{0,1}-\frac{\gamma+P}{2}\rho^\sigma_{0,1}\,, \end{eqnarray}$

(2.73a)

leads to the counterpart, $\rho^\sigma_{n,n}=(\frac{P}{\gamma})^n(1+\frac{P}{\gamma})^{-1}$ (with only

). They converge at low pump $P/\gamma\ll 1$ when the population grows linearly, $\rho_{1,1}\approx P/\gamma$ , in both cases. At the limit $P\rightarrow \gamma$ , the excitation is equally shared between all the manifolds. In the fermionic system, this simply means $\rho^\sigma_{n,n}=1/2$ , but in the bosonic case where there are an infinite number of available manifolds, it implies $\rho^b_{n,n}\approx 1 -P/\gamma \rightarrow 0$ , and the divergence of the average $\langle n\rangle$ . At $P=\gamma/2$ there is an ``inversion of population'' in both cases. For the 2LS, this means that $\rho^\sigma_{1,1}>\rho^\sigma_{0,0}$ or, equivalently, that $\langle n_\sigma\rangle >0.5$ . However, for the HO, a thermal state imposes that vacuum is the most populated state and $\rho^b_{n,n}>\rho^b_{n+1,n+1}$ . Therefore, the inversion consists rather in a probability to have particles larger than that of the vacuum: $\sum_{n>0}\rho^b_{n,n}>\rho^b_{0,0}$ .

In the master equation of the 2LS, we note again the equivalence between pump and decay from the symmetry under exchange $\gamma\leftrightarrow P$ and $0\leftrightarrow 1$ . Mathematically, this it is inherited from the simplicity of the 2LS operators, $\ud{\sigma}=\ket{E}\bra{G}$ , that makes equivalent the Lindblad terms for pump and decay.

Many other Lindblad terms have been considered in the literature. Cross terms in which both modes appear together, $a\rho\ud{b}$ for instance, lead to entanglement of the modes. Such terms can be originated from mechanisms of pump or decay that apply to linear combinations of and particles rather to the bare modes. We will come back to this point in Chapter 3, to discuss polariton pumping, and 6 to discuss the joint indistinguishable pump of two identical QDs.

Other interesting Lindblad terms are those to describe incoherent transfer of population between levels. Holland et al. (1996) used them to describe the evaporative cooling and later Porras & Tejedor (2003) and Laussy et al. (2004) included them in their models of polariton condensation, in order to account for polariton-polariton and photon-polariton scattering. It represents the scattering process of two particles from level into another two levels 0 and (assuming energy and momentum conservation) when level is adiabatically eliminated. The jump operator describing this event is $\ud{a_0}a_1^2$ . In the same way, the operator for polariton relaxation down its dispersion, from level to 0, by phonon emission (with energy $\omega_1-\omega_0$ ), is $\ud{a_0}a_1$ . These few-body Lindblad terms couple the modes and generate entanglement between their populations. This results in Poissonian (coherent) statistics for the fields even though the dynamics have a incoherent character (see Ref. 15 in the list of my publications, in page ).