Q -deformed description of excitons and associated physical results
aa r X i v : . [ qu a n t - ph ] D ec Q -deformed description of excitons and associatedphysical results M. Bagheri Harouni, R. Roknizadeh, M. H. Naderi
Quantum Optics Group, Physics Department, University of Isfahan, IranE-mail: [email protected], [email protected],[email protected]
Abstract.
We consider excitons in a quantum dot as q -deformed systems.Interaction of some excitonic systems with one cavity mode is considered. Dynamicsof the system is obtained by diagonalizing total Hamiltonian and emission spectrumof quantum dot is derived. Physical consequences of q -deformed exciton on emissionspectrum of quantum dot is given. It is shown that when the exciton system deviatesfrom Bose statistics, emission spectra will become multi peak. With our investigationwe try to find the origin of the q -deformation of exciton. The optical response ofexcitons, which affected by the nonlinear nature of q -deformed systems, up to thesecond order of approximation is calculated and absorption spectra of the system isgiven.PACS numbers: 73.20.La, 71.35.Cc, 03.65.Fd Submitted to:
J. Phys. B: At. Mol. Phys.
1. Introduction
Exciton is an elementary excitation of a semiconductor which consists of a pair of twocorrelated fermions, the electron and the hole. Analogous to the Hydrogen atom, itis characterized by a binding energy E b and a Bohr radius a B . Because an exciton iscomposed of two fermions, it is a composite boson. Particularly, in a bulk semiconductor,when the excitation of system is dilute, i.e. n ex a B ≪
1, where n ex is the exciton density,a bosonic description of system is convenient [1]. Also Bose-Einstein condensation ofexcitons, which is an essential characteristic of boson systems has been consideredtheoretically [2]. When density of excitons increase, the above condition is violated.In this situation the statistics of excitons deviates from Bose statistics.In low dimensional semiconductor systems such as quantum well (QW), quantumwire and quantum dot (QD), due to the small dimensions and loss of translationalsymmetry, exciton excitation differs from exciton in bulk materials. In semiconductornanostructures the size of the system strongly affects exciton properties. For example,in the case of quantum well it is shown [3], if the well width is larger (smaller) thanthe Bohr radius of exciton, the spectrum of quantum well has properties similar to thesituation in which excitons are boson (fermion). Hence, the size of the system directlyaffects the quantum statistics of excitons in that system. Recently similar results hasbeen obtained for QD [4]. In Ref.[4] the effects of different statistics of excitons onemission spectra of a QD is investigated, and the origin of different statistics of excitonsis considered. The same results have also been obtained for the quantum well. If thesize of QD is smaller (larger) than the exciton Bohr radius, excitons behave like fermion(boson). Real statistics of excitons in the interaction is considered in [5] and referencestherein. As mentioned before in high density regime, exciton statistics deviates fromBose statistics. This is due to the increase of mutual forces between the excitations ofthe system and then the Pauli exclusion principle plays a dominant role [6]. Appearanceof Bose statistics of exciton-biexciton system and Pauli exclusion effects in superlatticehas been considered experimentally [7].Bosons and fermions are the only two kinds of particles realized in nature. Theconditions mentioned for excitons (in one regime they are like bosons and in another onelike fermions) are property of a special kind of statistics called intermediate statistics[8]. Bose and Fermi statistics are two limiting cases of this statistics. Properties ofthis statistics have been considered by many authors [9] − [11]. Operator realizationof intermediate statistics is similar to q -deformed operators [12]. Bosonic q -deformedoperators [13] are a generalization of the Heisenberg algebra obtained by introducing adeformation parameter q . Deviation of this parameter from 1 shows deviation of algebrafrom the Heisenberg algebra. It is shown that it is possible to describe correlated fermionpairs with q -deformed bosons [14]. Therefore it is reasonable to consider an excitonsystem as a q -deformed system. We assume the creation and annihilation operators ofexcitons obey a q -deformed algebra. A q -deformed description of Frenkel exciton hasbeen considered recently [15].The algebra generated by q -deformed operators are given by[ˆ b q , ˆ b † q ] q = ˆ b q ˆ b † q − q − ˆ b † q ˆ b q = q ˆ n , (1)[ˆ n, ˆ b † q ] = ˆ b † q , [ˆ n, ˆ b q ] = − ˆ b q . where ˆ n = ˆ b † ˆ b is the usual particle number operator. Representation of this algebra isgiven in [16]. In the case of excitons, q -parameter can depend on excitation number andphysical size of system.In this paper we consider the interaction of light with a QD embedded in amicrocavity. By considering excitons in QD as q -deformed bosons (case of q -deformedfermion is straightforward) we study the emission spectrum of the system. As is clear,the commutator (1) explicitly depends on the number of excitons. Hence, this is asystem in which light interacts with a nonlinear active medium. Therefore, we shallobtain the linear and nonlinear response of a q -deformed exciton system. Knowledgeof interaction of light with a nonlinear medium ( q -deformed excitons) and its opticalresponse is important for the interpretation of experimental results such as [30]. Onthe other hand, we compared the obtained results with some experimental ones and inthis manner we investigate the physical origin of q -deformation of excitons. In section2 we derive the spectrum of QD when one exciton mode interacts with a single modecavity-field. In section 3 we consider the interaction of two exciton modes with a singlecavity mode. In section 4 the nonlinear response of QD is derived up to second orderof approximation. Finally we summarize our conclusions in section 5.
2. Model Hamiltonian
We consider a QD embedded in a microcavity which interacts with a single mode cavity-field. We assume the excitations in QD have an intermediate statistics [4], and theircreation and annihilation operators obey q -deformed algebra. We can express the q -deformed operators in terms of ordinary boson operators by the following mapsˆ b q = ˆ b s q ˆ n − q − ˆ n ˆ n ( q − q − ) , ˆ b † q = s q ˆ n − q − ˆ n ˆ n ( q − q − )ˆ b † , (2)where ˆ b and ˆ b † are the ordinary boson operators and ˆ n = ˆ b † ˆ b . Ordinary commutator of q -deformed exciton operators is[ˆ b q , ˆ b † q ] = qq + 1 [ q n + q − ( n +1) ] ≡ k (ˆ n ) . (3)Deviation of this commutator from ordinary boson algebra ([ˆ b, ˆ b † ] = 1) relates todeviation of q -parameter from 1. It is clear, this generalized commutator depends onthe number of excitations. It seems that by using this algebra we can consider somenonlinear phenomena in the system related to the population of excitons. For example,biexciton effects can be considered in this manner as an effective approach. So that thedeformation parameter q can represent some physical parameters such as the ratio of sizeof system to the Bohr radius of exciton. Interaction of QD with single mode cavity-fieldin rotating wave approximation can be described by the following Hamiltonianˆ H = ~ ω ˆ a † ˆ a + ~ ω ex ˆ b † q ˆ b q + ~ g (ˆ a ˆ b † q + ˆ a † ˆ b q ) , (4)where ˆ a and ˆ a † are creation and annihilation operators of cavity field and [ˆ a i , ˆ a † j ] = δ ij .We shall consider a phenomenological damping for the system which relates to bothsubsystems: photon and exciton. As is clear from the Hamiltonian (4), the excitonnumber is not a constant of motion. Because of the dependence of exciton operator,ˆ b q , on the exciton number, resulting equations of motion become a nontrivial set ofcoupled equations. On the other hand, since the total number of excitation (excitonand photon) is conserved we can diagonalize the Hamiltonian in the subspace of a definiteexcitation. To consider this dynamics we propose an approach based on diagonalizationof the Hamiltonian by using the polariton transformation [17]. This procedure dependson some unitary transformations which diagonalize the model Hamiltonian. As isusual in this procedure [18], new operators have the same commutation relation asthe original operators (free operators). Here, there are two distinct sets of operators,the cavity mode operators which obey the usual boson commutation relation and excitonoperators that are q -deformed boson. Therefore, with the presence of these two differentstatistics, mixed operators (polariton operators) do not have specific statistics. Theycan be considered as ordinary boson operators or q -deformed operators. We considerboth situations and we study the physical results associated with each situation in theresonance fluorescence spectrum of QD. In order to solve the dynamical system, we perform the following transformationˆ p k = u k ˆ b q + v k ˆ a. (5)Due to the presence of q -deformed operator ˆ b q , we call this transformation a polariton-like transformation. As mentioned before, ˆ b q depends on the number of excitonsexplicitly and this causes the Hopfield coefficients u k and v k will depend on the numberof excitons. Hence, the transformation (5) can be considered as a nonlinear polaritontransformation. This kind of transformation has been considered recently for the case ofBogoliubov transformation [19, 20]. We assume polariton-like operators obey the usualboson commutation relations[ˆ p k , ˆ p † k ′ ] = δ kk ′ ⇒ [ˆ p k , ˆ p † k ] = | u k | k (ˆ n ) + | v k | = 1 , (6)where the operator valued function k (ˆ n ) was introduced by Eq.(3). We choose unknowncoefficients u k and v k so that the Hamiltonian (4) becomes diagonal in terms of thepolariton-like operatorsˆ H = ~ X k Ω k ˆ p † k ˆ p k , (7)where Ω k is the polariton spectrum and k refers to different polariton branches. Bytaking into account a phenomenological damping for exciton and photon systemsseparately, the unknown parameters satisfy following set of equations[ ω ex k (ˆ n ) − Ω k − iγ ex ] u k + v k g = 0 , u k gk (ˆ n )+( ω − Ω k − iγ ph ) v k = 0 . (8)In this set of equations, γ ex and γ ph are the exciton and photon damping constants,respectively. From these equations the polariton spectrum can be obtained asΩ k = ω ex k (ˆ n ) + ω − i ( γ ex + γ ph )2 ± q [ ω ex k (ˆ n ) − ω − i ( γ ex − γ ph )] + 4 g k (ˆ n )) . (9)It is apparent that q -deformed description of excitons causes the splitting between theseenergy eigenvalues be increased in compare to the case of bosonic description of exciton.Using the set of equations (8) and the polariton spectrum (9) we find the coefficientsfor two polariton branches u k = s ω − iγ ph − Ω k k (ˆ n )[ ω − k + ω ex k (ˆ n ) − i ( γ ex + γ ph )] , (10) v k = − s ω ex k (ˆ n ) − iγ ex − Ω k ω − k + ω ex k (ˆ n ) − i ( γ ex + γ ph ) . By employing these coefficients all necessary parameters for the polariton Hamiltonianare determined.Now we can consider the dynamics of polariton operators. The time evolution ofpolariton operators is governed by the polariton Hamiltonian (7)ˆ˙ p k = − i ~ [ˆ p k , ˆ H ] = − i Ω k ˆ p k . (11)Let us consider damping effects by taking into account a phenomenological dampingterm and noise operator in the dynamical equations of polariton operators. Hence, thetime evolution of polariton operator is given byˆ˙ p k = − i Ω k ˆ p k − Γ k ˆ p k + ˆ F ˆ p k ( t ) , (12)where ˆ F ˆ p k ( t ) is the Langevin noise operator which depends on the reservoir variables andΓ k is the damping constant of k th polariton branch given by Γ k = γ ex + γ ph . Correlationfunctions of the noise operators determine physical properties of the system. TheLangevin noise operator are such that their expectation values h ˆ F x i vanishes, but theirsecond order moments do not [21]. They are intimately linked up with the globaldissipation and in a Markovian environment they take the form h ˆ F † ˆ p k ( t ) ˆ F ˆ p k ( t ′ ) i = 2Γ k δ ( t − t ′ ) . (13)With neglecting the phonon effects by decreasing the temperature, other sources ofdamping like spontaneous recombination of exciton and photon loss are considered asMarkovian procedures. It follows, on solving Eq.(12), thatˆ p k ( t ) = ˆ p k (0) e ( − i Ω k − Γ k ) t + Z t e ( − i Ω k − Γ k )( t − t ′ ) ˆ F ˆ p k ( t ′ ) dt ′ . (14)In this equation we set initial time equal zero.The power spectrum of the scattered light for statistical stationary fields is givenby [22] S ( r, ω ) = 1 π Re Z ∞ h ˆ E − ( r, t ) ˆ E + ( r, t + τ ) i e iωτ dτ, (15)where ˆ E ± are the positive and negative frequency parts of the electric field operator.Expressing field operators in terms of creation and annihilation operators we have S ( r, ω ) = A ( r ) π Re Z ∞ h ˆ a † (0)ˆ a ( τ ) i e iωτ dτ. (16)Here, we set t = 0, and A ( r ) depends on mode function of the cavity-field.Now we can express, the field and exciton creation and annihilation operators interms of polariton ones:ˆ a = v ∗ ˆ p + v ∗ ˆ p , ˆ b q = k (ˆ n )( u ∗ ˆ p + u ∗ ˆ p ) , (17)and at the time t we haveˆ a ( t ) = v ∗ ˆ p ( t ) + v ∗ ˆ p ( t ) . (18)Now to calculate the resonance fluorescence spectrum we have to determine the initialstate of system. we assume at t = 0, the cavity-field is in a coherent state | α i , andthe exciton subsystem in its vacuum state. Under this condition, by using Eq.(14) theresonance fluorescence spectrum is obtained as S ( r, ω ) = A ( r ) | α | π (cid:20) | v | Γ ( ω − Ω ) + Γ + | v | Γ ( ω − Ω ) + Γ (cid:21) . (19)In deriving this result we implicitly assume that at t = 0 the noise operator and polaritonoperators are uncorrelated. Fig.(1) shows the plot of S ( r, ω ) versus ω for differentvalues of deformation parameter q. Material parameters are chosen as ω = 1 . eV , ω ex = 1 . eV , g = 200 µeV , γ ex = 20 µeV , γ ph = 40 µeV [23], n = 100 and | α | = 9 .As is clear when q = 1, spectrum has similar variation as experimental results [23]. Thisfigure shows that when q = 1 (nondeformed case) the power spectrum of the fluorescencelight is a double peak centered at ω = Ω and ω = Ω . By increasing deviation of qfrom 1, it is apparent from the different plots in this figure that splitting between twopeaks increases and the height of one of peaks decreases. This result has been reported inresonance fluorescence of excitons when the biexcitonic interaction is taken into account.It has been shown [3] that biexcitonic effects are a red shift of the transition frequencies,emergence of sidebands due to the switch-on forbidden transitions and asymmetry of theemission spectrum. The binding energy of biexciton in QD causes a shift in the spectrumof the system. In the present model the splitting of spectrum (Rabi splitting) dependson the q -parameter. Hence, changing this parameter affects the spectrum. Then as aone reason of deviation of excitons from ideal Bose system we can consider Coulombinteraction between them. On the other hand, q -deformed exciton operators depend onthe total number of exciton, and biexciton interaction occurs when there are more thanone exciton. This similarity makes this clue that the q -deformation can be consider asan effective approach to take into account the biexciton effects. As mentioned before,the q -parameter can depend on the size of sample. The plotted resonance fluorescencespectrum in Fig.(1) makes clear some differences of optical properties of different sizeQD. For large values of q, compare with 1, spectrum will reduce to one peak. This caseis a characteristic of the weak coupling regime. Q -deformed polaritons In this subsection we assume that the polariton operators are q -deformed operators.According to the q -deformed nature of the exciton system we assume the followingalgebra for polariton operators[ˆ p k , ˆ p † k ] s = ˆ p k ˆ p † k − s − ˆ p † k ˆ p k = s ˆ n k , (20)where s denotes the deformation parameter corresponding to the polariton system andˆ n k shows the number operator for k th polariton branch. Ordinary commutator for theseoperators is [ˆ p k , ˆ p † k ] = | u k | k (ˆ n ) + | v k | = ss + 1 [ s ˆ n k + s − (ˆ n k +1) ] = M (ˆ n k ) . (21)Using the same approach of the previous subsection we obtain the following set ofequations for the coefficients of transformation[( ω ex k (ˆ n ) − iγ ex − Ω ′ k M ( n k )] u k + v k g = 0 ,u k gk (ˆ n ) + [ ω − iγ ph − Ω ′ k M ( n k )] v k = 0 . (22)From this set of equations we derive the deformed polariton spectrum asΩ ′ k = ω ex k (ˆ n ) + ω − i ( γ ex + γ ph )2 M ( n k ) ± p [ ω ex k (ˆ n ) − ω − i ( γ ex − γ ph )] + 4 g k (ˆ n ))2 M ( n k ) . (23)and the transformation coefficients read as u k = − s M ( n k )[ ω − iγ ph − Ω ′ k M ( n k )] k (ˆ n )[ ω − ′ k M ( n k ) + ω ex k (ˆ n ) − i ( γ ex + γ ph )] , (24) v k = s M ( n k )[ ω ex k (ˆ n ) − iγ ex − Ω ′ k M ( n k )] ω − ′ k M ( n k ) + ω ex k (ˆ n ) − i ( γ ex + γ ph ) . By determining all the variables, polariton Hamiltonian (diagonal Hamiltonian) willbe determined. By applying the same procedure as before we derive the resonancefluorescence spectrum in this case as follows S ( r, ω ) = A ( r ) | α | ( | v | + | v | ) π X i =1 , | v i | Γ i ( ω − Ω ′ i M ( n k )) + Γ i . (25)Fig. (2) shows the plot of S ( r, ω ) versus ω for different values of polariton deformationparameter s . This figure shows that changes of s-parameter (deformation parameterof polariton) does not cause any shift in transition frequencies, but causes strengths ofpeaks increase.
3. Interaction of light with two exciton modes
We now consider the interaction of one cavity mode with QD when two exciton modesare coupled to the field mode. As before, we assume exciton system is expressed by the q -deformed operators. The total Hamiltonian of the system under consideration can bewritten as followsˆ H = ~ ω ˆ a † ˆ a + ~ X i =1 , ω ex i ˆ b † q i ˆ b q i + ~ g X i =1 , (ˆ a ˆ b † q i + ˆ a † ˆ b q i ) . (26)We assume both excitons have the same coupling constant with the cavity mode. Wesolve this system as before by diagonalizing the Hamiltonian. For this purpose weperform the following transformationˆ p k = u k ˆ b q + x k ˆ b q + v k ˆ a. (27)We consider the situation in which the polariton operators obey the nondeformed Bosestatistics [ˆ p k , ˆ p † k ] = | u k | k (ˆ n ) + | x k | k (ˆ n ) + | v | = 1 , (28)where ˆ n i ( i = 1 ,
2) represents the number operator for each excitonic mode. As isclear in this case there are three polariton branches. Assuming the transformation (27)diagonalizes the Hamiltonian (26), this polariton Hamiltonian takes the following formˆ H = ~ X k Ω k ˆ p † k ˆ p k , (29)where summation is over all polariton branches. The following equation determines thepolariton spectrum( c − Ω k )[( d − Ω k )( ω − iγ ph − Ω k ) − g k (ˆ n )] − g k (ˆ n )( d − Ω k ) = 0 , (30)where c = ω ex k (ˆ n ) − iγ ex and d = ω ex k (ˆ n ) − iγ ex . By deriving the polaritonspectrum the transformation parameters are obtained as u k = g [( d − Ω k )( ω − iγ ph − Ω k ) − g k (ˆ n )] A , (31) x k = g k (ˆ n ) Av k = − ( c − Ω k )[( d − Ω k )( ω − iγ ph − Ω k ) − g k (ˆ n )] A , where the parameter A is given by A = ([ g k (ˆ n ) + ( c − Ω k ) ][( d − Ω k )( ω − iγ ph − Ω k ) − g k (ˆ n )] + g k (ˆ n ) k (ˆ n )) . (32)In this manner, all the parameters which appear in the polariton Hamiltonian aredetermined. By repeating the approach of previous section the resonance fluorescencespectrum of system with different initial conditions can be determined. If we assumeat t = 0 the cavity mode is in the coherent state | α i and QD in vacuum state | i , theresonance fluorescence spectrum is given by S ( r, ω ) = | α | A ( r ) π X k | v k | Γ k Γ k + ( ω − Ω k ) . (33)To show complex structure (multi-peak structure) of this spectrum Fig. (3) presentsthe spectra on a logarithmic scale. For the sake of clarity, we have powered some peakscompare to other ones in this figure. In the case of q = 1 (nondeformed exciton) thespectrum has two peaks. Increasing the q -parameter causes that splitting between peaksbe increased and spectrum becomes multi-peaks. Multi-peaks structure in emission ofexciton such as Mollow triplet was predicted when excitons obey statistics differentfrom Bose statistics [3, 4]. When, q -parameter is changed, the energy and intensities ofemission change. Effects of exciton number on absorption spectrum of QD is considered. Due to the relation of absorption spectrum and resonance fluorescence, similar resultis obtain in [24].
4. Nonlinear response of excitons in q -deformed regime In previous sections we considered some physical results of q -deformed descriptionof excitons. The q -deformed description can be served as a nonlinear description ofexcitons. It is well-known that different kinds of nonlinearity in an exciton systemlead to different orders of nonlinear response of the system [25, 26]. Therefore, we tryto obtain optical response of a driven quantum dot, which its optical excitations areconsidered as q -deformed systems. For this purpose we will calculate the coefficientabsorption of a QD in this regime. In this section we neglect all damping effects and weconsider the Hamiltonian of the system as followsˆ H = ~ ω ˆ a † ˆ a + ~ ω ex ˆ b † q ˆ b q + ~ g (ˆ a ˆ b † q + ˆ a † ˆ b q ) . (34)In the electron picture, the induced dipole moment by transition of an electron isdescribed by ˆ µ = ˆ a † v ˆ a c + ˆ a † c ˆ a v [27]. The operator ˆ a † v (ˆ a v ) is the creation (annihilation)operator for an electron in the valance band (level in the case of QD), and ˆ a † c (ˆ a c ) isthe creation (annihilation) operator for an electron in the conduction band. Hence,creation of an exciton is denoted by ˆ a † c ˆ a v = ˆ b † q . Therefore we can write the dipoleoperator of QD as ˆ µ = ˆ b † q + ˆ b q . The macroscopic polarization is expectation valueof polarization operator. The optical response function represents the reaction of thesystem to an external classic field E ( t ) coupled to the variables of system [28], i.g., thedipole operator. Hence, we consider an external field as a pump source and we treatthe reaction of QD to it. Then the total Hamiltonian of system is then given byˆ H = ~ ω ˆ a † ˆ a + ~ ω ex ˆ b † q ˆ b q + ~ g (ˆ a ˆ b † q + ˆ a † ˆ b q ) − [ ~d vc · ~E ( t )ˆ b q + ~d cv · ~E ( t )ˆ b † q ] , (35)where ~d vc denotes the dipole matrix element. The Hamiltonian in the interaction picturehas the form ˆ H int = ~ g h ˆ a ˆ b † q e − i [ ω − ω ex k (ˆ n )] t + ˆ a † e i [ ω − ω ex k (ˆ n )] t ˆ b q i (36)0 − h ~d cv · ~E ( t ) e − iω ex k (ˆ n ) t ˆ b q + ~d vc · ~E ( t )ˆ b † q e iω ex k (ˆ n ) t i . The observable of interest for the optical response is the time-dependent dipole density µ ( t ) = h ˆ b q ( t ) i + h.c. = T r ex (ˆ b q ρ ex ( t )) + h.c. , where T r ex means trace over the excitonsystem and ρ ex ( t ) = T r f ρ ( t ), which ρ ( t ) is the total time dependent density matrix ofthe system and ρ ex ( t ) is the time dependent density matrix of exciton system. The totaltime dependent density matrix is given byˆ ρ ( t ) = ˆ U ( t, t ) ˆ ρ ( t ) ˆ U − ( t, t ) , (37)where U ( t, t ) = ˆ T exp[ − i ~ R tt ˆ H int ( t ′ ) dt ′ ] is the time ordered evolution operator and ˆ ρ ( t )is the total density matrix of system at initial time. We assume that the quantum fieldand exciton system are both in vacuum state. Therefore, the time dependent densitymatrix of excitons is given byˆ ρ ex ( t ) = X n h n | ˆ U ( t, t )( | i f | i ex )( ex h | f h | ) ˆ U − ( t, t ) | n i , (38)where summation is carried on field state and the matrix elements of the time evolutionoperator are in the basis of field states. By using the Feynman disentanglement theorem[29] the matrix elements of the time evolution operator ˆ U ( t, t ) can be evaluated. Wecan write Hamiltonian in (36) as ˆ H int = ˆ H ( t ) + ˆ H ( t ), whereˆ H ( t ) = ~ g h ˆ a ˆ b † q e − i [ ω − ω ex k (ˆ n )] t + ˆ a † e i [ ω − ω ex k (ˆ n )] t ˆ b q i (39)ˆ H ( t ) = − h ~d cv · ~E ( t ) e − iω ex k (ˆ n ) t ˆ b q + ~d vc · ~E ( t )ˆ b † q e iω ex k (ˆ n ) t i . As is clear ˆ H ( t ) depends only on exciton operators. The time evolution operator canbe written as ˆ U ( t, t ) = ˆ T exp[ − i ~ Z tt ( ˆ H ( t ′ ) + ˆ H ( t )) dt ′ ]= ˆ T exp[ − i ~ Z tt ˆ H ( t ′ ) dt ′ ] exp[ − i ~ Z tt ˆ H ( s ) ds ] . (40)In this equation we use Feynman notation [29]. These two exponential terms are notdisentangle from each other. They are correlated and in doing integration, we have totake into account ordering of operators. In calculation of matrix element of this operatorin the basis of field states, second exponential can be considered as a ordinary c-numberfunction of t ′ , because it is independent of field operators: h i | ˆ U ( t, t ) | j i = h i | ˆ T exp[ − i ~ Z tt ˆ H ( t ′ ) dt ′ ] | j i exp[ − i ~ Z tt ˆ H ( s ) ds ] . (41)On the other hand we consider all the exciton operators in ˆ H ( t ) as ordinary c-numberfunctions, and we can write h i | ˆ U ( t, t ) | j i = h i | ˆ T exp[ − i ~ Z tt (ˆ a † t ′ e iωt ′ B ( t ′ ) + ˆ a t ′ e − iωt ′ B ∗ ( t ′ )) dt ′ ] | j i× exp[ − i ~ Z tt ˆ H ( s ) ds ] , (42)1where B ( t ) is a ordinary function corresponding to exciton operators. As is clear thismatrix element is a function of exciton operators. The influence of exciton system iscompletely contained in this operator functional and factored term in (42). By usingFeynman theorem, above matrix element can written as h i | exp[ − ig Z tt ˆ a † t ′ e iωt ′ B ( t ′ ) dt ′ ] exp[ − ig Z tt ˆ a ′ t ′ e − iωt ′ B ∗ ( t ′ ) dt ′ ] | j i , (43)where in this equation ˆ a ′ t ′ = ˆ V − ( t )ˆ a t ˆ V ( t ), andˆ V ( t ) = exp[ − ig ˆ a † Z tt B ( t ′ ) e iωt ′ ] . (44)In this manner the density matrix of exciton system takes the following formˆ ρ ex ( t ) = X n n ! S (ˆ n, ˆ b q , ˆ b † q ) | i ex h | S (ˆ n, ˆ b q , ˆ b † q ) , (45)where S (ˆ n, ˆ b q , ˆ b † q ) = (cid:20) − g ˆ b † q e iω ex k (ˆ n ex )( t − t ) ω ex k (ˆ n ex ) (cid:21) n e − g L (ˆ n ex ) f (ˆ b q , ˆ b † q ) ,S (ˆ n, ˆ b q , ˆ b † q ) = (cid:20) − g ˆ b q e − iω ex k (ˆ n ex +1)( t − t ) ω ex k (ˆ n ex + 1) (cid:21) n e − g L (ˆ n ex ) f − (ˆ b q , ˆ b † q ) ,L (ˆ n ex ) = ˆ b † q ˆ b q [ e − iω ex [ k (ˆ n ex +1) − k (ˆ n ex − t − t ) [ ω − ω ex k (ˆ n ex − ω − ω ex k (ˆ n ex + 1)] (46)+ e − iω ex [ k (ˆ n ex +2) − k (ˆ n ex ))]( t − t ) [ ω − ω ex k (ˆ n ex + 2)][ ω − ω ex k (ˆ n ex )] ] , (47)and f (ˆ b q , ˆ b † q ) = ˆ T exp [ i ~ Z tt dt ′ ( ~d cv · ~E ( t ′ )ˆ b q e − iω ex k (ˆ n ex +1) t ′ + ~d vc · ~E ( t ′ )ˆ b † q e iω ex k (ˆ n ex ) t ′ )] , (48)By expanding the function f (ˆ b q , ˆ b † q ) up to second order in E ( t ) and using (45) we obtainthe time-dependent dipole density as follows p ( t ) = X n g n n ! h ( n )! q f q ( n )! e − g L ( n ) × [ i ~ h ( n + 1)! e − g L (1) q f q ( n + 1)! f q ( n + 1) Z tt ~d cv · ~E ( t ′ ) e iω ex t ′ dt ′ − i ~ q f q ( n ) h ( n )! e − g L (0) q f q ( n )! f q ( n ) Z tt ~d cv · ~E ( t ′ ) e iω ex k ( n − t ′ dt ′ + i ~ q f q ( n + 1) h ( n + 2)! e − g L (0) q f q ( n + 2)! f q ( n + 2) × Z tt Z tt Z tt ~d vc · ~E ( t ′ ) ~d cv · ~E ( r ) ~d cv · ~E ( s ) e iω ex [ s + r − k ( n +1) t ′ ] dt ′ drds + i ~ q f q ( n ) h ( n )! e − g L (0) q f q ( n )! f q ( n ) × Z tt Z tt Z tt ~d cv · ~E ( t ′ ) ~d vc · ~E ( r ) ~d cv · ~E ( s ) e iω ex [ k ( n − t ′ − ( r − s )] dt ′ drds − i ~ q f q ( n + 1) h ( n + 1)! e − g L (1) q f q ( n + 1)! f q ( n + 1) Z tt Z tt Z tt × ~d vc · ~E ( r ) ~d cv · ~E ( s ) ~d cv · ~E ( t ′ ) e − iω ex [ k ( n +1)( r − s ) − t ′ ] dt ′ drds − i ~ q f q ( n ) h ( n + 1)! e − g L (1) q f q ( n + 1)! f q ( n + 1) Z tt Z tt Z tt × ~d cv · ~E ( r ) ~d vc · ~E ( s ) ~d cv · ~E ( t ′ ) e iω ex [ k ( n +1)( r − s )+ t ′ ] dt ′ drds ] , (49)where h i ( n ) = e ( − iiωexk ( n + i )( t − t ω ex k ( n + i ) and f q ( n ) = q q n − q − n q − q − . These equation shows that inthis conditions second order response function is equal zero. Now we can calculatelinear and nonlinear electric susceptibility of this exciton system from this equation.Generalized linear and nonlinear absorption spectra of this system is shown in figures(4)-(6) for different values of q -parameter. In these plots, 1 s -exciton is considered. Inthese figures we choose ~ = e = 1, g = 200 µev and ω ex = 1574 mev . Fig.(4) shows plotsof linear absorption spectra and Fig.(6) shows plots of nonlinear spectra. On the otherhand, 3-dimension plot of linear absorption coefficients is given in figure (5). It is clearthat changes of q -parameter strongly affects absorption spectra of the system. Thesefigures show in the presence of q -values absorption of probe beam shows a complexstructure: a multiple-like absorption pattern appears with one strong peak and someside bands. Presence of these side bands is a signature of the optical generation of annonlinear exciton (an exciton which expresses with q -deformed operator). Negative partof the absorption spectrum demonstrates gain of the probe beam. Due to the resonanceinteraction of pump with exciton transition, the gain effect comes from the coherentenergy exchange between the pump and probe beams through the QD nonlinearity.The obtained absorption spectra are very similar to experimental results [30]. In Ref.[30] absorption spectra of a driven charged QD is derived experimentally. ChargedQD is a nonlinear medium and is similar to our model. Then It can be consider as aexperimental test of our model.
5. Conclusion Q -deformed description of excitons in a QD and its physical consequences wasconsidered. We showed that increasing the q -parameter will lead to increase of splittingbetween peaks in the spectrum and asymmetry of spectrum. Similar effects were observewhen biexciton effects taken into account. In experiments of QD it is shown [23] thesame results are obtained in different temperatures. Then we can associate this physicalparameter as source of q -deformation. The temperature dependence of emission energyof system can be attributed to the change in the refractive index of its active mediumwith temperature. We have derived the optical response of QD with q -deformed exciton.As mentioned before q -deformed description of excitons will lead to dependence of opticalresponse on q parameter. Hence, due to the wide range of q parameter and its effects3on optical response we can consider some parameters like temperature and interactionbetween the excitons which affects the optical response of QD as sources of q -deformationof excitons. As mentioned, the relation of quantum statistics of excitons in the QDand the size of QD has been considered. Then we can consider the ratio of excitonBohr radius to dimension of system and exciton population as two main sources of q -deformation. Q -deformed operator depends on total number of associated particles ofsystem. Therefore we can interpret q -deformed operator as an operator which consistsof effects of other excitations of system implicitly. Then it is reasonable to considerthis description as an effective description which takes into account some nonlinearityin exciton system. As we saw, in the case of interaction of light with two excitons,when q = 1 this system showed a two peaks spectrum. While by increasing deviation ofexciton from Bose statistics, spectrum becomes multi peak. Due to the nonlinear natureof q -deformed exciton we showed that different orders of nonlinear response function ofthis system can be calculated. From coincidence of obtained results and experimentalresults, we can conclude that q -deformed description of excitons can be a considerablemodel for excitons. With comparing the obtained results in this paper with experimentalones we can investigate the origin of this description of excitons. As pointed out theratio of system dimension to the Bohr radius of exciton is one of the sources of deviationof excitons from usual boson. The obtained results are very similar to the effects of theexciton-exciton interaction [3],[31] which is relates to exciton population and biexcitonbinding energy. On the other hand, it is shown that [1] exciton density is anothersource of their deviation from ordinary bosons. To sum up we attribute the origin of q -deformation of the excitons to their density, their mutual interactions, confinementsize and other parameters which cause fluctuation of optical response of the system. Q -deformed description of an active medium causes that the optical properties of systemdepend on the q -parameter. Then, it is seem that parameters which can affect opticalproperties of the active medium (like refractive index) their effects can be considered bythis formulation. The q -parameter can be considered as a variation parameter which itsvalues can be obtained from comparison of theoretical and experimental results. Acknowledgment
The authors wish to thank the Office of Graduate Studies ofthe University of Isfahan and Iranian Nanotechnology initiative for their support.
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Plots of S ( ω ) versus ω . Parameter are choose as ω = 1 . eV , ω ex = 1 . eV , g = 200 µeV , γ ex = 20 µeV , γ ph = 40 µeV , n = 1 and | α | = 9. Solidplot corresponds to q = 1, nondeformed case. Dotted one corresponds to q = 1 .
01, andfor dash line q is equal 1 . PSfrag replacements ω ( ev ) s ( ω ) Figure 2.
Plots of S ( ω ) versus ω . Parameter are choose as ω = 1 . eV , ω ex = 1 . eV , g = 200 µeV , γ ex = 20 µeV , γ ph = 40 µeV , n = 1 and | α | = 9.In all figures we have q = 1. Solid line corresponds to case s = 1. In dotted one wehave s = 1 .
007 and for dash line s = 1 . PSfrag replacements ω ( ev ) s ( ω ) Figure 3.
Plots of S ( ω ) versus ω . Parameter are choose as ω = 1 . eV , ω ex = 1 . eV , ω ex = 1 . eV , g = 200 µeV , γ ex = γ ex = 200 µeV , γ ph = 45 µeV , n = 1, n = 1 and | α | = 9. Dotted line corresponds to nondeformed case q , q = 1.For solid line q , q = 1 .
04. In the case of dashed line q , q = 1 . - - PSfrag replacements ω ( mev ) α ( ω ) Figure 4.
Plots of spectrum absorption versus ω . We consider 1s-exciton andParameter are choose as ~ = e = 1, g = 200 µev and ω ex = 1574 mev . Solidplot corresponds to nondeformed case q = 1. For dotted one q = 1 .
01 and in dash one q = 0 . PSfrag replacements ω ( mev ) α ( ω ) qq Figure 5. ω and deformation parameter q .Physical parameter are the same as Fig.(4). - - - · - · - PSfrag replacements ω ( mev ) α (3) (3 ω ) Figure 6.
Plots of nonlinear spectrum absorption versus ω . We consider 1s-excitonand Parameters are choose as ~ = e = 1, g = 200 µev and ω ex = 1574 mev Solid plotcorresponds to nondeformed case q = 1. In dotted plot q = 1 .
01. In dash plot q = 0 ..