- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 347674, 12 pages

http://dx.doi.org/10.1155/2012/347674

## Mathematical Models of Dissipative Systems in Quantum Engineering

Academic Center of Optical Engineering and Photonics, Polytechnic University of Bucharest, 313 Spl. Independentei, 060042 Bucharest, Romania

Received 9 February 2012; Accepted 18 March 2012

Academic Editor: Ezzat G. Bakhoum

Copyright © 2012 Andreea Sterian and Paul Sterian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper shows the results of theoretical research concerning the modeling and characterization of the dissipative structures generally, the dissipation being an essential property of the system with self-organization which include the laser-type systems also. The most important results presented are new formulae which relate the coupling parameters from Lindblad equation with environment operators ; microscopic quantitative expressions for the dissipative coefficients of the master equations; explicit expressions which describe the changes of the environment density operator during the system evolution for fermion systems coupled with free electromagnetic field; the generalized Bloch-Feynman equations for -level systems with microscopic coefficients in agreement with generally accepted physical interpretations. Based on Maxwell-Bloch equations with consideration of the interactions between nearing atomic dipoles, for the dense optical media we have shown that in the presence of the short optical pulses, the population inversion oscillates between two extreme values, depending on the strength of the interaction and the optical pulse energy.

#### 1. Introduction

An essential problem of the quantum information systems is the controllability and observability of the quantum systems. In this context, Fermi systems are essential for several important physical effects in quantum engineering as the dynamics of semiconductor nanostructures and high temperature superconductivity, nuclear resonances, fusion-fission reactions, and analysis of optical quantum systems. These effects are essentially determined by the dissipative coupling of the system.

Dissipation in quantum systems is a complex phenomenon which raises important theoretical investigations. A dissipative system is a system of interest, coupled with another system usually considered as being of much larger-environment. Fundamental and difficult problem of dissipative quantum theory is to design the total system (system of interest + environment) on the space system of interest. In this way obtain a quantum master equation describing the evolution of the system using two terms: (1) a hamiltonian term for processes with energy conservation and (2) a nonhamiltonian term with coefficients that depend on the dissipative coupling. A master equation is based on approximations that consist in mediating rapid oscillations of reduced density matrix describing the interaction.

Such an approximation is the assumption that the evolution operators of a dissipative system forms a semigroup, not a group like for isolated systems. In this framework was derived a quantum master equation with dissipative terms which is consistent with all principles of quantum mechanics. Considering two operators, coordinate and momentum , master equation was used to describe the harmonic oscillator. In this theoretical framework, dissipation is described by the friction and diffusion coefficients that satisfy certain conditions called basic restrictions and Heisenberg’s uncertainty relations are observed during the whole evolution of the system.

A rigorous method for deducting the master equation with microscopic expressions of the dissipative coefficients is developed in the literature.

For a weak dissipative coupling one obtains a master equation of Lindblad form [1], but with the microscopic expressions of the dissipative coefficients.

In the development of quantum theory of dissipative systems an important step was the connection between Lindblad’s generator and the previous phenomenological descriptions, realized by Săndulescu and Scutaru [2]. Besides, we must mention Isar et al.’s contributions [3]. This school developed by the above-mentioned researchers in the field are well recognized in the scientific world [4–8].

Firstly, in the paper general expressions which relate the coupling parameters in Lindblad equation with environment operators have been established [9–11]. In this way, became possible deeper causality understanding of processes of friction and diffusion and of related quantum effects: broadening and shift of spectral lines, tunneling rates, bifurcations and instability [12, 13].

Secondly, for a system of fermions, coupled with a dissipative environment quantitative microscopic expressions for the coefficients of the dissipative master equation depending on the potential matrix elements, the densities of states of the environment and the occupation probabilities of these states are presented [14–19].

The study continue with the systems of fermions coupled by electric dipole interactions of free electromagnetic field for which has established general explicit expressions which describe the changes of the environment density operator during the system evolution. This description is not restricted to the Born approximation, taking into account the environment time evolution as a function of the system evolution. The results of the dissipative dynamics of the system of fermions in the presence of laser field are applicable to the dissipative structures [14, 20–29].

Next, generalized Bloch-Feynman equations for -level systems with microscopic coefficients in agreement with generally accepted physical interpretations are presented.

In the last part, we study the dynamics of dense media under the action of ultrafast optical pulses using Maxwell-Bloch formalism to include interaction between close atomic dipoles [30–34]. It is shown that, in a system initially without inversion, in the presence of optical pulses, the final population has two extreme values, results which contribute to understanding the specific mechanisms of switching for applications, with specific examples concerning the coherent radiation generation and amplification [35–43]. A computational specific software, to verify the experimental and numerical existing models and in the same time to discover new important situations for operative systems design and implementation, was developed [44–48].

#### 2. Relationship between Coupling Coefficients in Lindblad Master Equation and Environment Observables

Research on dissipative processes has led to evidence for the first time concerning the relationship between coupling coefficients in the Lindblad equation:

depending on the system Hamiltonian and *the operators of opening *:

where are system operators, are complex *coupling coefficients* or *amplitudes* and operators of environment defined using the interaction Hamiltonian as

These relationships have been established under the form [10]

and allow an understanding of the physical causes of quantum processes of friction and diffusion, with their known effects: broadening and shift of spectral lines [11], increased rates of tunneling, nonlinear characteristics, leading to bifurcation, instability, and chaos.

#### 3. Microscopic Quantitative Expressions of the Dissipative Coefficients in Master Equations

A general quantum master equation for a many-level many-particle system, with microscopic coefficients, that preserves the quantum-mechanical properties of the density matrix was obtained [12]:

with dissipative coefficients:

including a component for a dissipative environment of fermions and a component for a dissipative environment of bosons.

Equation (3.1) is of Lindblad’s form, with dissipative operators depending on the transition/population operators . For a system with levels, the total number of these operators is the number of the independent operators defined.

If we denote by and the interaction dissipative potentials of the environment containing fermions and bosons, respectively, it is possible to write the expressions of the coefficients şi for the resonant transition of the system coupled with environmental transition, with fermionic state having densitiesaly and populations , , and bosonic states with densities , and population şi . The probability that the final state of the environment to be free is while the probability the initial state of the environment to be occupied is .

General expressions of the dissipative coefficients are written for this type of interaction in the form:

#### 4. The Environment Dynamics Correlated with that of a Fermion Systems Coupled with Free Electromagnetic Field

We consider a system of charged fermions with the coordinates and momenta in a single-particle potential , while represents the two-particle residual potential. This system is coupled to the modes of the free electromagnetic field. In order to describe the dynamics of this system, for simplicity, we neglect the particle spin and its dimensions with respect to the electromagnetic field wavelength (the electric dipole approximation). In this case, the total hamiltonian is of the form [14]

In the total hamiltonian (4.1),

is the system-field interaction potential, while

is the fermion system hamiltonian, and

is the field hamiltonian, where

is the field mode hamiltonian.

Let us take the density operator of the total system with hamitonian (4.1) and the reduced density matrix

over the environment states.

The total density operator satisfies the equation of motion:

where the sign above designs operators within the framework of interaction picture of the system and environment

while is an intensity parameter used to show the orders of the series expansion of this density. Considering the radiation field of the black body in the initial state , the total density operator of the system can be taken under the form:

where , represent modifications of the field during the system evolution. The first term of this expression corresponds to the Born approximation when the environment state is a constant state , while the higher-order terms, which satisfy the normalization relations

describe the environment dynamics that is correlated to the system dynamics. For an equation of motion of the form

From (4.7), (4.9), and (4.11) we get a system of coupled equations:

By calculating the partial traces over the environment states and using the normalization conditions (4.10), from these equations we get successively the terms of the equation of motion (4.11):

while, integrating by time, we get "excitation" terms of the total density operator (4.9):

The first-order equation (4.13) represents the system evolution when the environment is considered as being in a constant state , while for the higher-order term (28), we take into consideration some changes of the environment matrix (4.14). Further on, we will show that the first-order terms (4.13) describe the hamiltonian dynamics of the system, while the second-order term (28) describes system one-particle transitions related to environment.

#### 5. The Generalized Bloch-Feynman Equations

An alternative description of dissipative system dynamics is given by Bloch-Feynman equations for systems of fermions obtained by defining the pseudo-spin operators [14].

In particular, for a system with two-level known Bloch-Feynman, equations are obtained, where, is the field operator, is the polarization operator, and is population operator:

with microscopic coefficients şi expressed by dissipative coefficients of the master equation:

The condition is a confirmation of master equation (4.7) which led to the establishment of Bloch equations-Feynman, because this condition is verified experimentally.

#### 6. Dynamics of Dense Media under the Action of Short Optical Pulses

Maxwell-Bloch equations of a two-level atomic medium generalized to include interactions between the dipoles approach [15, 16, 32] have been used to describe the system dynamics under the action of ultrafast optical pulses. These equations, for systems with homogeneous broadening of spectral lines in about semiclassical treating, were established using the density matrix formalism as

In the above equations, is the inversion of population, nondiagonal elements of density matrix slow variable, indices and refer to lower and higher energy states, with the gap , is slowly varying local field is the frequency deviation in relation to the center frequency of the field resonance frequency, is the transition matrix element of the electric dipole, and, are longitudinal and transverse relaxation rates.

Contributions of the dipole-dipole interactions occur in (6.2) by term , where is the strength parameter of dipole-dipole interactions having a dimension of a frequency. Equations (6.1) and (6.2) for the atomic variables and for field variables realise the description Maxwell-Bloch of optically dense environment. These equations were generalized and used to study intrinsic optical bistability, propagation effects in nonlinear media, and so forth.

For numerical simulation, we considered the case resonant , a characteristic distance between dipoles much smaller than the wavelength of the central field (propagation effects are negligible) and ultrafast pulses (pulses much shorter than ; this enables us to neglect dissipation processes). In these conditions, the matrix element is decomposed into its real and imaginary parts , resulting in system

whose outcome is possible only numerically.

In the above equations is the normalized time, is the measured width pulse, is the instantaneous Rabi frequency, and is the intensty of electrical pulse.

In Figure 1, we present the final population inversion function of maximum Rabi frequency for hyperbolic secant pulses . As long as the Rabi frequency has a value so that , the final population inversion is . In the region , the final population inversion has an oscillatory behavior, almost rectangular wave. As the parameter value is greater, the oscillation period decreases, the transitions become abrupt, and the first half cycle of the rectangular wave becomes more centered to .

In Figure 2, temporal evolution of the system is presented for a hyperbolic secant pulse with a peak higher than one (when , the population inversion performs a number of oscillations before reaching a value 1; under certain conditions when , after a number of oscillations, the system remains in the ground state).

#### 7. Conclusions

General expressions which relate the coupling parameters in Lindblad equation with environment operators have been established. These expressions allow deeper understanding of causal processes of friction- and diffusion-related quantum effects: broadening and shift of spectral lines, tunneling rates, bifurcations, and instability.

For a system of fermions coupled with a dissipative environment quantitative microscopic expressions for the coefficients of the dissipative master equation are presented.

These coefficients depend on the potential matrix elements, the densities of states of the environment, and the occupation probabilities of these states.

Expressions of the dependence of the particle distributions on temperature are taken into account. It can be shown that a system of fermions located in a dissipative environment of bosons tends to a Bose-Einstein distribution.

Studying the systems of fermions coupled by electric dipole interactions of free electromagnetic field, has established general explicit expressions which describe the changes of the environment density operator during the system evolution for fermion systems coupled with free electromagnetic field. This description is not restricted to the Born approximation, taking into account the environment time evolution as a function of the system evolution. The study can be continued with the calculation of the higher-order term of the reduced matrix equation in order to describe the correlated transition of the system particles. The results of the dissipative dynamics of the system of fermions in the presence of laser field are applicable to the dissipative structures.

Generalized Bloch-Feynman equations for -level systems with microscopic coefficients in agreement with generally accepted physical interpretations are presented. On this basis, the problem of a quantum system control is explicitly formulated in terms of microscopic quantities: matrix elements of the dissipative two-body potential, densities of the environment states, and occupation probabilities of these states.

Studying the dynamics of dense media under the action of ultrafast optical pulses using Maxwell-Bloch formalism to include interaction between close atomic dipoles showed that, in a system initially without inversion, in the presence of optical pulses, the final population has two extreme values, the ratio of Rabi frequency and the parameter that describes the interactions between close dipoles, which contribute to understanding the specific mechanisms of switching.

#### References

- G. Lindblad, “On the generators of quantum dynamical semigroups,”
*Communications in Mathematical Physics*, vol. 48, no. 2, pp. 119–130, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Săndulescu and H. Scutaru, “Open quantum systems and the damping of collective modes in deep inelastic collisions,”
*Annals of Physics*, vol. 173, no. 2, pp. 277–317, 1987. View at Publisher · View at Google Scholar - A. Isar, A. Sandulescu, H. Scutaru, E. Stefanescu, and W. Scheid, “Open quantum systems,”
*International Journal of Modern Physics E*, vol. 3, no. 2, pp. 635–714, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. Stefanescu and A. Sandulescu, “Microscopic coefficients for the quantum master equation of a Fermi system,”
*International Journal of Modern Physics E*, vol. 11, no. 2, pp. 119–130, 2002. View at Publisher · View at Google Scholar · View at Scopus - A. Sandulescu and E. Stefanescu, “New optical equations for the interaction of a two-level atom with a single mode of the electromagnetic field,”
*Physica A*, vol. 161, no. 3, pp. 525–538, 1989. View at Publisher · View at Google Scholar · View at Scopus - E. Stefanescu, R. J. Liotta, and A. Sandulescu, “Giant resonances as collective states with dissipative coupling,”
*Physical Review C*, vol. 57, no. 2, pp. 798–805, 1998. View at Publisher · View at Google Scholar · View at Scopus - G. W. Ford, J. T. Lewis, and R. F. Connell, “Master equation for an oscillator coupled to the electromagnetic field,”
*Annals of Physics*, vol. 252, no. 2, pp. 362–385, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - E. Stefanescu, A. Sandulescu, and W. Scheid, “The collisional decay of a fermi system interacting with a many-mode electromagnetic field,”
*International Journal of Modern Physics E*, vol. 9, no. 1, pp. 17–50, 2000. View at Google Scholar · View at Scopus - E. N. Stefanescu, P. E. Sterian, and A. R. Sterian, “Fundamental interactions in dissipative quantum systems,”
*The Hyperion Scientific Journal*, vol. 1, no. 1, pp. 87–92, 2000. View at Google Scholar - E. Ştefǎnescu and P. Sterian, “Exact quantum master equations for Markoffian systems,”
*Optical Engineering*, vol. 35, no. 6, pp. 1573–1575, 1996. View at Publisher · View at Google Scholar · View at Scopus - E. Stefanescu, A. Sandulescu, and P. Sterian, “Open systems of fermions interacting with the electromagnetic field,”
*Romanian Journal of Optoelectronics*, vol. 9, no. 2, pp. 35–54, 2001. View at Google Scholar - E. Ştefănescu, P. Sterian, and A. Gearbă, “Spectral line broadening in the two level systems with dissipative coupling,”
*Romanian Journal of Optoelectronics*, vol. 6, no. 2, pp. 17–22, 1998. View at Google Scholar - E. Stefanescu, P. Sterian, and A. Gearba, “Absorption spectra in dissipative two level system media,” in
*Proceedings of the EPS 10 Trends in Physics 10th General Conference of the European Physical Society*, September 1996. - E. N. Stefanescu, A. R. Sterian, and P. E. Sterian, “Study of the fermion system coupled by electric dipol interaction with the free electromagnetic field,” in
*Proceedings of the Advanced Laser Tehnologies*, A. Giardini, V. I. Konov, and V. I. Pustavoy, Eds., vol. 5850 of*Proceeding of the SPIE*, pp. 160–165, 2005. - E. Stefanescu and P. Sterian, “Dynamics of a Fermi system in a complex dissipative environment,” in
*Proceedings of the International Conference on Advanced Laser Technologies (ALT '02)*, vol. 4762 of*Proceeding of SPIE*, pp. 247–259, Constanta, Romania, 2002. - E. Stefanescu and P. Sterian, “Non-Markovian effects in dissipative systems,” in
*Proceedings of the 5th Conference on Optics (Romopto '97)*, vol. 3405 of*Proceeding of SPIE*, pp. 877–882, Bucharest, Romania, 1998. - E. Stefanescu, A. Sandulescu, and P. Sterian, “Quantum tunneling through a dissipative barrier,” in
*Proceedings of the 4th Conference In Optics (Romopto '94)*, vol. 2461, pp. 218–225, 1995. - E. Stefanescu, P. Sterian, and A. R. Sterian, “The lindblad dynamics of a Fermi system in a particle dissipative environment,” in
*Proceedings of the International Conference on Advanced Laser Technologies (ALT '03)*, vol. 5147 of*Proceeding of SPIE*, pp. 160–168, Adelboden, Switzerland, 2003. - E. Stefanescu, P. Sterian, and A. Sandulescu, “Quantum tunneling in open systems,” in
*Proceedings of the 1st General Conference of the Balkan Physical Union*, K. M. Paraskevopoulos, Ed., Hellenic Physical Society, 1991. - E. Stefanescu and A. Sandulescu, “Dynamics of a superradiant dissipative system of tunneling electrons,”
*Romanian Journal of Physics*, vol. 49, pp. 199–207, 2004. View at Google Scholar - E. Stefanescu, “Quantum master equation for a system of charged fermions interacting with the electromagnetic field,”
*Romanian Journal of Physics*, vol. 48, pp. 763–770, 2003. View at Google Scholar - E. Stefanescu and A. Sandulescu, “The decay of a Fermi system through particle-particle dissipation,”
*Romanian Journal of Physics*, vol. 47, pp. 199–219, 2002. View at Google Scholar - E. Stefanescu and A. Sandulescu, “Dissipative processes through Lindblad's master equation,”
*Romanian Journal of Optoelectronics*, vol. 7, p. 59, 2000. View at Google Scholar - E. Stefanescu and P. Sterian, “Open systems of fermions interacting with the electromagnetic field,”
*Romanian Journal of Optics*, vol. 9, pp. 35–55, 2001. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - E. Stefanescu, P. Sterian, I. M. Popescu et al., “Time-dependent problem algorithm of the optical bistability,”
*Revue Roumaine de Physique*, vol. 31, pp. 345–350, 1986. View at Google Scholar - . E. Stefanescu and P. Sterian, “Optical equations for an open system of fermions,” in
*Proceedings of the 11th General Conference of the European Physical Society, Trends in Physics*, London, UK, Septmber 1999. - E. Stefanescu, I. M. Popescu, and P. Sterian, “The Semi-classical Approach of the Optical Bistability,”
*Revue Roumaine de Physique*, vol. 29, no. 2, pp. 183–188, 1984. View at Google Scholar · View at Scopus - E. Stefanescu and P. Sterian, “Dynamics of a fermi system in a complex dissipative environment,” in
*Proceedings of the Advanced Laser Technologies*, vol. 4762 of*Proceeding of SPIE*, pp. 247–259, Constanţa, Romania, Septmber 2001. - E. Stefanescu and A. Sandulescu, “Dynamics of a superradiant dissipative system of electrons tunneling in a micro-cavity,”
*Romanian Journal of Physics*, vol. 50, pp. 629–638, 2005. View at Google Scholar - C. M. Bowden and J. P. Dowling, “Near-dipole-dipole effects in dense media: generalized Maxwell-Bloch equations,”
*Physical Review A*, vol. 47, no. 2, pp. 1247–1251, 1993. View at Publisher · View at Google Scholar · View at Scopus - V. Ninulescu and A. R. Sterian, “Dynamics of a two-level medium under the action of short optical pulses,” in
*Proceedings of the Computational Science and Its Applications (ICCSA '05)*, O. Gervaşi, M. L. Gavrilova, V. Kumar et al., Eds., vol. 3482 of*Lecture Notes in Computer Science-LNCS*, pp. 635–642, Springer, 2005. View at Publisher · View at Google Scholar - C. Vasile, A. R. Sterian, and V. Ninulescu, “Optically dense media described by help of Maxwell- Bloch equations,”
*Romanian Journal of Optoelectronics*, vol. 10, no. 4, pp. 81–86, 2002. View at Google Scholar - M. E. Crenshaw, M. Scalora, and C. M. Bowden, “Ultrafast intrinsic optical switching in a dense medium of two-level atoms,”
*Physical Review Letters*, vol. 68, no. 7, pp. 911–914, 1992. View at Publisher · View at Google Scholar · View at Scopus - V. Ninulescu, I. Sterian, and A. R. Sterian, “Switching effect in dense medie,” in
*Proceedings of the International Conference, Advanced Laser Technologies (ALT '01)*, vol. 4762 of*Proceeding of SPIE*, pp. 227–234, 2002. - A. R. Sterian, “Coherent radiation generation and amplification in erbium doped systems,” in
*Advances in Optical Amplifiers*, P. Urquhart, Ed., InTech, Vienna, Austria, 2011. View at Google Scholar - A. R. Sterian and V. Ninulescu, “Nonlinear phenomena in erbium doped lasers,” in
*Proceedings of the Computational Science and Its Applications (ICCSA '05)*, O. Gervasi et al., Ed., vol. 3482 of*Lecture Notes in Computer Science, LNCS*, pp. 643–650, Springer, 2005. - A. R. Sterian,
*Laserii în Ingineria Electrică*, Printech Publishing House, Bucharest, Romania, 2003. - A. R. Sterian and V. Ninulescu, “The nonlinear dynamics of erbium-doped fiber laser,” in
*Proceedings of the 13th International School on Quantum Electronics: Laser Physics and Applications*, Burgas, Bulgaria, September 2004. - M. Ghelmez, C. Toma, M. Piscureanu, and A. R. Sterian, “Laser signals' nonlinear change in fatty acids,”
*Chaos, Solitons and Fractals*, vol. 17, no. 2-3, pp. 405–409, 2003. View at Publisher · View at Google Scholar · View at Scopus - A. R. Sterian,
*Amplificatoare Optice*, Printech Publishing House, Bucharest, Romania, 2006. - C. Rosu, D. Manaila-Maximean, D. Donescu, S. Frunza, and A. R. Sterian, “Influence of polarizing electric fields on the electrical and optical properties of polymer-clay composite system,”
*Modern Physics Letters B*, vol. 24, no. 1, pp. 65–73, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. G. Bakhoum and C. Toma, “Specific mathematical aspects of dynamics generated by coherence functions,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 436198, 10 pages, 2011. View at Publisher · View at Google Scholar - E. G. Bakhoum and C. Toma, “Dynamical aspects of macroscopic and quantum transitions due to coherence function and time series events,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 428903, 13 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - F. C. Maciuc, C. I. Stere, A. R. Sterian et al., “Time evolution and multiple parameters variations in a time dependent numerical model applied for Er+3 laser system,” in
*Proceedings of the 1th International School on Quantum Electronics: Laser Physics and Applications*, P. A. Atanasov and S. Cartaleva, Eds., vol. 4397 of*Proceeding of SPIE*, pp. 84–89, 2001. - A. R. Sterian and F. C. Maciuc, “Numerical model of an EDFA based on rate equations,” in
*Proceedings of the 12th International School an Quantum Electronics, Laser Physics and Application*, vol. 5226 of*Proceeding of SPIE*, pp. 74–78, 2003. - C. Cattani and I. Bochicchio, “Wavelet analysis of bifurcation in a competition model,” in
*Proceedings of the Computational Science (ICCS '07)*, vol. 4488, part 2 of*Lecture Notes in Computer Science*, pp. 990–996, 2007. - F. C. Maciuc, C. I. Stere, and A. R. Sterian, “Rate equations for an Erbium laser system, a numerical approach,” in
*Proceedings of the 6th Conference on Optics (ROMOPTO '01)*, vol. 4430 of*Proceeding of SPIE*, pp. 136–146, 2001. - D. A. Iordache, P. Sterian, F. Pop, and A. R. Sterian, “Complex computer simulations, numerical artifacts, and numerical phenomena,”
*International Journal of Computers, Communications and Control*, vol. 5, no. 5, pp. 744–754, 2010. View at Google Scholar · View at Scopus