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Modern Physics Letters B

Theory of Spontaneous and Stimulated Emission in Two Band Intrinsic Semiconductor

Vladislav Cheltsov

Published May 10, 1997• 0 citations
Publisher

Abstract

The behavior of a single mode of radiation field coupled to an excited two band intrinsic semiconductor has been investigated with the help of the commutator version of evolution operator method. As dependent on the excitation level three states of the field has been shown to exist: the equilibrium with the number of photons determined by the Bose–Einstein distribution with nonzero chemical potential; the quasi-equilibrium with the average number of photons equal to unity and accompanied by fluctuations; the state of photon avalanche.

DOI
10.1142/s0217984997000608

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Journal of the Optical Society of America•Sep 1, 1983•18 citations
The behavior of a single mode of radiation field coupled to an excited two band intrinsic semiconductor has been investigated with the help of the commutator version of evolution operator method. As dependent on the excitation level three states of the field has been shown to exist: the equilibrium with the number of photons determined by the Bose–Einstein distribution with nonzero chemical potential; the quasi-equilibrium with the average number of photons equal to unity and accompanied by fluctuations; the state of photon avalanche.
The basic results of theoretical nonperturbative analysis of spontaneous and stimulated emission from two-band excited intrinsic semiconductor interacting with a single mode of radiation field in a lossless micro-cavity are presented. The self-consistent dynamics of the coupled system ''semiconductor + single mode radiation field", accompanied with spontaneous and stimulated emission was investigated as dependent of the initial level of the excitation of semiconductor in the presence initially of the physical field vacuum. Calculations have been carried out with the help of the commutation version of the evolution method without introducing explicitly the phonon mechanisms. The initial non-equilibrium electron and hole populations in the bands have been supposed to be created by ultra-short laser pulses and described by the Fermi-Dirac quasi-equilibrium distributions with the given chemical potentials. The interaction between the electron-hole subsystem and the mode field has been adopted to be switched on suddenly. The calculations have shown that depending on the initial excitation level of electron-hole subsystem three states of the mode field are realized: the equilibrium Bose-Einstein photon distribution with nonzero chemical potential; the quasi-equilibrium distribution with the average number of photons equal to unit and accompanied with their fluctuations, and finally, the state of photon avalanche. The first regime takes the place when the coefficient of the light amplification is negative. Then, the equilibrium between the electron-hole subsystem and arising photons exists until the average number reaches the unit. For negative and =1 the equilibrium still exists but is accompanied with field fluctuations. Finally, when the steady growth arises the lower but stable values are established
Thebasic results oftheoretical nonperturbative analysis ofspontaneous andstimulated emission from two-band excited intrinsic interacting with asingle modeofradiation field inalossless micro-cavity arepresented. Theself-consistent dynamics ofthecoupled system semiconductor + single moderadiation field, accompanied withspontaneous andstimulated emission wasinvestigated asdependent oftheinitial level ofthe excitation ofsemiconductor inthepresence initially ofthephysical field vacuum. Calculations havebeen carried out withthehelpofthecommutation version oftheevolution method without introducing explicitly thephonon mechanisms. Theinitial non-equilibrium electron andholepopulations inthebands havebeensupposed tobe created byultra-short laser pulses anddescribed bytheFermi-Dirac quasi-equilibrium distributions withthegiven chemical potentials. Theinteraction between theelectron-hole subsystem andthemodefield hasbeenadopted tobe switched onsuddenly. Thecalculations haveshownthat depending ontheinitial excitation level ofelectron-hole subsystem three states ofthemodefield arerealized: theequilibrium Bose-Einstein photon distribution with nonzero chemical potential; thequasi-equilibrium distribution withtheaverage number ofphotons equal tounitand accompanied with their fluctuations, andfinally, thestate ofphoton avalanche. Thefirst regime takes theplace when thecoefficient ofthelight amplification isnegative. Then, theequilibrium between theelectron-hole subsystem and arising photons exists until theaverage number reaches theunit. Fornegative and=1theequilibrium still exists but isaccompanied withfield fluctuations. Finally, whenthesteady growth arises thelower butstable values are established. Theself-consistent spontaneous andstimulated emission fromanexcited intrinsic two-band confined inamicro-cavity andcoupled toasingle resonance field modehave beeninvestigated with thehelp ofthecommutation version ofevolution operator (1,2) method in theinteraction representation. Phonons werenotintroduced directly into theHamiltonian but through theinitial Fermi-Dirac distributions forelectrons andholes inthebands. Ithasbeen suggested that theelectron-hole subsystem isexcited byanultra-short laser pulses. Sothedynamics ofphoton emission hasbeentraced through times shorter thenthe characteristic relaxation time caused bythephonon mechanism. Theinitial state ofelectron-hole andmodefield subsystems hasbeenadopted tobepresented with asuperposition It>= ({ne,h,m,},t=0) {nh}> .m>
Photon statistics are obtained from exact closed-form solutions for both spontaneous and stimulated coherent emissions by a large number of two-level atoms in a Dicke state. Depending on the initial excitation level of the atoms, three types of photon statistics are obtained for the emitted radiation field. Moments of the photon number of the emitted field are calculated in the case of initial atomic coherent states as well as Dicke states, and in the case of n-photon stimulating pulses as well as coherent stimulating pulses. In the particular case of an initial "superradiant" Dicke state stimulated by a coherent-radiation field, the emitted radiation pulse is also fully coherent. In other cases the photon distributions undergo large fluctuations.
Spontaneous emission from a two-level atom embedded in photonic crystals was investigated in this paper. Due to the photon localization and the quantum interference between the two dressed states, some changes in the radiation field take place: when the excited level moves from the gap to the band, the energy of the localized field is transferred to the propagating field through the diffusion field. If the localized field and the propagating field could coexist, the population evolution in the excited level would exhibit quasi-oscillation behavior. If they could not coexist, there would be no quasi-oscillation. The steady-state trapped atom population in the excited level is changed also. All these characteristics depend not only on the relative position δ of the excited level ω _1 with respect to the band-edge ω _c but also on the photon density of states ρ(ω_k) (or the band-edge smoothing parameter e ) near the band edge.
We investigate spontaneous emission from a two-level atom embedded in photonic crystals. Due to the photon localization and the quantum interference between the two dressed states, some changes of the radiation field happen, and the energy is transferred among the excited atom, the localized field, the propagating field and the diffusion field. If there is a localized field, it appears quasi-oscillation behaves of the population evolution in the excited level. If there isn't a localized field, it appears no quasi-oscillation. Moreover, there have some steady-state trapped atom populations in the excited level, and the spontaneous emission spectrum near the forbidden gap edge is different from that in the vacuum. All these characteristics depend not only on the relative position of the excited level from the band-edge but also on the photon density of states (or the band-edge smoothing parameter) near the band edge.
Rate equations for photons and for electrons in systems of two groups of energy levels (energy separation kTηG) are set up allowing for absorption, stimulated and spontaneous emission, nonradiative transitions, photon loss or dissipation as well as photon gain from the ambient, and an externally applied pumping rate. The rate equations are discussed qualitatively, and the stability of the solutions is obtained by the use of phase portraits. The quantitative discussion includes steady-state photon distributions. Earlier work showing that these distributions can obey the Bose–Einstein law with a nonzero chemical potential kTF is taken into the laser regime, which was not covered by recent relevant work. The second-order phase transition that occurs at threshold is established explicitly, and it is shown that the transition requires the neglect of certain generation terms in the rate equations. Some time-dependent solutions of the rate equations are also discussed, using the slaving principle of adiabatic elimination for the electrons. The application of this work to semiconductors is discussed.