Nonlinear coherent state of an exciton in a wide quantum dot
aa r X i v : . [ qu a n t - ph ] D ec Nonlinear coherent state of an exciton in a widequantum dot
M. Bagheri Harouni, R. Roknizadeh, M. H. Naderi
Quantum Optics Group, Physics Department, University of IsfahanE-mail: [email protected], [email protected],[email protected]
Abstract.
In this paper, we derive the dynamical algebra of a particle confined inan infinite spherical well by using the f -deformed oscillator approach. We consider anexciton with definite angular momentum in a wide quantum dot interacting with twolaser beams. We show that under the weak confinement condition, and quantizationof the center-of-mass motion of exciton, the stationary state of it can be considered asa special kind of nonlinear coherent states which exhibits the quadrature squeezing.
1. Introduction
The conventional coherent states of the quantum harmonic oscillator, defined by Glauber[1] as the right-hand eigenstates of non-hermitian annihilation operator ˆ a ([ˆ a, ˆ a † ] = 1),have found many interesting applications in different areas of physics such as quantumoptics, condensed matter physics, statistical physics and atomic physics [2]. These statesplay an important role in the quantum theory of coherence, are considered as the mostclassical ones among the pure quantum states, and laser light can be supposed as aphysical realization of them. Due to the vast application of these states, there havebeen many attempts to generalize them [3]. Among the all generalizations, nonlinearcoherent states (NLCS) [4] have been paid attention in recent years because they exhibitnonclassical features such as quadrature squeezing and sub-poissonian statistics [5].These states are defined as the right-hand eigenstates of a deformed operator ˆ A ˆ A = ˆ af (ˆ n ) ˆ A | α, f i = α | α, f i , (1)where the deformation function f (ˆ n ) is an operator-valued function of the numberoperator ˆ n . From (1) one can obtain an explicit form of NLCS in the number staterepresentation | α, f i = N f X n α n √ n ! f ( n )! | n i , N f = "X n | α | n [ f ( n )!] n ! − . (2)A class of NLCS can be realized physically as the stationary state of the center-of-mass motion of a laser driven trapped ion [6, 7]. Furthermore, it has been proposed atheoretical scheme to show the possibility of generating various families of NLCS [8] ofthe radiation field in a lossless coherently pumped micromaser within the frame workof the intensity-dependent Jaynes-Cummings model.Recently, the influences of the spatial confinement [9] and the curvature of physicalspace [10] on the algebraic structure of the coherent states of the quantum harmonicoscillator have been investigated within the frame work of nonlinear coherent statesapproach. It has been shown that if a quantum harmonic oscillator be confined withina small region of order of its characteristic length [9] or its physical space to be a sphere[10], then it can be regarded as a deformed oscillator, i.e., an oscillator that its creationand annihilation operators are deformed operator ˆ A and ˆ A † given by Eq.(1).On the other hand, we can consider nanostructures as systems whose physicalproperties are related to the confinement effects. Thus, we expect that it is possible torealize some natural deformations in these systems [9, 11]. In addition, in nanostructuresdifferent kinds of quantum states can be prepared. One of the most applicable of thesestates is exciton state. Exciton is an elementary excitation in semiconductors interactingwith light, electron in conduction band which is bounded to hole in valance band thatcan easily move through the sample. In one of the nano size systems, quantum dot (QD),due to the confinement in three dimensions, energy bands reduce to quasi energy levels.Therefore, in order to describe the interaction of QD with light we can consider it as afew level atom [12]. These Exciton states can be used in quantum information processes.It has been shown that excitons in coupled QDs are ideal for preparation of entangledstate in solid-state systems [13]. Entanglement of the exciton states in a single QD or in aQD molecule has been demonstrated experimentally [14]. Entanglement of the coherentstates of the excitons in a system of two coupled QDs has been considered [15]. Recently,coherent exciton states of excitonic nano-crystal-molecules has been considered [16].Theoretical approach for generating Dick states of excitons in optically driven QD hasbeen proposed in Ref.[17]. In a QD, the effects of exciton-phonon interaction, exciton-impurity interaction and exciton-exciton interaction play an important role. Theseeffects are the main sources for the decoherence of exciton states [18]. Furthermore,these effects cause the exciton has the spontaneous recombination or scattered to otherexciton modes [19, 20].In this paper we propose a theoretical scheme for generating excitonic NLCS. Wewill show that under certain conditions the quantized motion of wave packet of center-of-mass of exciton can be consider as a special kind of NLCSs. Our scheme is based on theinteraction of a quantum dot with two laser beams. By using the approach consideredin Ref.[6], we propose a theoretical scheme for generation of NLCS of an exciton in awide QD.In section 2, we consider different confinement regimes in a QD, and the explicitforms of the creation and annihilation operators for a particle confined in an infinitewell are derived by using the deformed quantum oscillator approach. In section 3, weconsider an exciton in a wide QD which interacts with two laser beams. We shallshow that under the weak confinement condition, the stationary state of the excitoncenter-of-mass motion can be considered as a NLCS.
2. Algebraic approach for a particle in an infinite spherical well
In nanostructures and confined systems, there are three different confinement regimes.The criteria for this classification is based on the comparison between excitation Bohrradius and the spatial dimensions of the system under consideration. In the case of aQD, these regimes are defined as follows [21].We first introduce three quantities ∆ E c , ∆ E v and V exc which, respectively denote:the electron energy due to the confinement, the hole energy due to the confinement andCoulomb energy between correlated electron-hole (exciton).1) V exc > ∆ E c − ∆ E v : In this case, the exciton energy is much greater than theconfinement energies of electron and hole. If we show the system size by L and theexciton Bohr radius by a , then in this regime L > a . This regime corresponds tothe weak confinement (in some literature the weak confinement is characterized by thesituation in which the electron and the hole are not in the same matter, for example, holebe in QD and excited electron in host matter. In this paper, by the weak confinementregime we mean
L > a and the excitations in the same matter). In this regime due to theconfinement, the center-of-mass motion of the exciton is quantized and the confinementdo not affect electron and hole separately. Hence, the confinement affect the excitonmotion as a whole [22].2) V exc < ∆ E c , ∆ E v : This regime, in contrast to the previous one, is associated withthe cases where L < a . In this regime the exciton is completely localized, and theconfinement affects both the electron and the hole independently and their states becomequantized in conduction and valance bands. This regime is called strong confinement.3) ∆ E c > V exc , ∆ E v : This condition is equivalent to the situation a c < a < a v ,where a c and a v are, respectively, the Bohr radii of electron and hole. Here, due to thedifferent effective masses of electron and hole, the hole which has heavier effective massis localized and the electron motion will be quantized. This regime is called intermediateconfinement.In the first case (weak confinement), in a wide QD, an exciton can move due toits center-of-mass momentum, and because of the presence of the barriers, its center-of-mass motion is quantized. Therefore, it moves as a whole between energy levels of aninfinite well. We consider a wide spherical QD whose energy levels are equivalent to theenergy levels of a spherical well E nl = ~ M α nl R , (3)where α nl is the n’th zero of the first kind Bessel function of order l , j l ( x ). In this energyspectrum according to the azimuthal symmetry around z axis, we have a degeneratespectrum. As mentioned before, in the weak confinement regime, the Coulomb potentialplays an essential role and its spectrum is given by E bk = µe ~ ε k , µ = m e m h m e + m h , (4)where superscript b shows binding energy related to the Coulomb interaction and ε shows dielectric constant of the system. As is usual, we interpret the Coulomb part asan exciton and another degree of freedom (motion between energy levels of the well) asthe exciton center-of-mass motion. Therefore, in a wide QD an exciton has two differentkinds of degrees of freedom: internal degrees of freedom due to the Coulomb potentialand external degrees of freedom related to the quantum confinement. Here we considerthe lowest exciton state, 1 s exciton, because this exciton state has the largest oscillatorstrength among other exciton state. Then the energy of the exciton in a wide QD canbe written as E nlm = E g − E b + ~ M R α nl , (5)where E g is the energy gap of QD, E b = E bk | k =1 is the exciton binding energy, M = m e + m h is the total mass of exciton, and R is the radius of QD. Due to therelation of quantum numbers l and m with the angular momentum and the selectionrules for optical transitions, we can fix l and m (by choosing a certain condition), andhence the energy of exciton depends only on a single quantum number E n = E g − E b + ~ M R α nl . (6)Therefore, we can prepare the conditions under which the exciton center-of-mass motionhas a one-dimensional degree of freedom. Due to the quantization of the exciton center-of-mass motion, we can describe the exciton motion between the energy levels by theaction of a special kind of ladder operators. In order to find these operators we use the f -deformed oscillator approach [4].As mentioned elsewhere [9], if the energy spectrum of the system is equally spaced,such as harmonic oscillator, its creation and annihilation operators satisfy the ordinaryWeyl-Heisenberg algebra, otherwise we can interpret them as the generators of ageneralized Weyl-Heisenberg algebra.The energy spectrum of a particle with mass M confined in an infinite sphericalwell can be written as (3). According to the conservation of angular momentum, weassume that particle has been prepared with definite angular momentum (for exampleby measuring its angular momentum). Then l becomes completely determined, i.e., inthe energy spectrum the number l is a constant. By determining the number l andconsidering the rotational symmetry of the system around the z axis, the angular partof the spectrum becomes completely determined, and the radius part is described by(3). Now we use a factorization method and write the Hamiltonian of the center-of-massmotion of the system as followsˆ H = 12 ( ˆ A ˆ A † + ˆ A † ˆ A ) , (7)where ˆ A and ˆ A † are defined through the relation (1). Therefore the spectrum of ˆ H ,after straightforward calculation, is obtained as E n = 12 [( n + 1) f ( n + 1) + nf ( n )] . (8)By comparing (8) with Eq.(3) we arrive at the following expression for the correspondingdeformation function f (ˆ n ) f ( n ) = vuut ~ M R ( − n n n X i =1 ( − i α i − l . (9)Then, the ladder operators associated with the radial motion of a confined particle in aspherical infinite well is given byˆ A = ˆ a vuut ~ M R ( − n n n X i =1 ( − i α i − l , (10)ˆ A † = vuut ~ M R ( − n n n X i =1 ( − i α i − l ˆ a † . These two deformed operators obey the following commutation relation[ ˆ
A , ˆ A † ] = − nf ( n ) + ~ M R α nl . (11)As is usual in the f -deformation approach, for a particular limit of the correspondingdeformation parameter, the deformed algebra should be reduced to the conventionaloscillator algebra. However, in this treatment we note that there is no thing in commonbetween the harmonic oscillator potential and an infinite spherical well. Only in thelimit R → ∞ , the system reduces to a free particle which has continuous spectrum.As a result, in this section we conclude that the radial motion of a particle confinedin a three-dimensional infinite spherical well can be interpreted by an f -deformed Weyl-Heisenberg algebra.
3. Exciton dynamics in QD
Now we consider the formation of an exciton and its dynamics in a wide QD duringthe exciton lifetime. As mentioned before, in this situation the center-of-mass motionof the exciton is quantized. The exciton is created during the interaction of a QD withlight, and because of the angular momentum conservation, the exciton has a well-definedangular momentum. The exciton is a quasiparticle composed of an electron and a holeand thus the exciton spin state can be in a singlet state or a triplet state. According tothe optical transition selection rules, the triplet state is optically inactive and is calleddark exciton [23]. By adding spin and angular momentum of absorbed photons, theangular momentum of the exciton state can be determined. Hence, the exciton behaveslike a particle in a spherical well with the definite angular momentum. According to theprevious section, the center-of-mass motion of the exciton in the QD and the barriersof QD can be described by an oscillator-like Hamiltonian expressed in terms of the f -deformed annihilation and creation operators given by Eq.(10) H well = 12 ( ˆ A ˆ A † + ˆ A † ˆ A ) , (12)where we interpret the operator ˆ A ( ˆ A † ) as the operator whose action causes the transitionof exciton center-of-mass motion to a lower (an upper) energy state. In fact theHamiltonian (12) is related to the external degree of freedom of exciton. On the otherhand, one can imagine QD as a two-level system with the ground state | g i and theexcited state | e i (associated with the presence of exciton). Thus, for the internal degreeof freedom we can consider the following Hamiltonian H ex = ~ ω ex ˆ S , (13)where ˆ S = | e ih e | − | g ih g | and ~ ω ex = E g − E b is the exciton energy.We consider a single exciton of frequency ω ex confined in a wide QD interacting withtwo laser fields, respectively, tuned to the internal degree of freedom of the frequency ω ex and to the non-equal spaced energy levels of the infinite well. It is necessary thatthe second laser has special conditions, because it should give rise to the transitionsbetween energy levels whose frequencies depend on intensity. The interacting systemcan be described by the Hamiltonianˆ H = ˆ H + ˆ H int , (14)where ˆ H = ˆ H well + ˆ H ex and H int = g [ E e − i ( k ˆ x − ω ex t ) + E e − i ( k ˆ x − ( ω ex − ω n ) t ) ] ˆ S + H.c., (15)in which g is the coupling constant, k and k are the wave vectors of the laser fields,ˆ S = | g ih e | is the exciton annihilation operator, and ω n is the frequency of excitontransition between energy levels of QD due to the spatial confinement. Here, we considertransition between specific side-band levels hence, we show the frequency transition withdefinite dependence to n . We show this by a c-number quantity n .The exciton has a finite lifetime that in systems with small dimension, is increased[24]. The interaction with phonons is the main reason of damping of the exciton [25].Phonons in bulk matter have a continuous spectrum while in a confined system such asQD their spectrum becomes discrete. Hence in a QD, the resonant interaction betweenthe exciton and phonons decreases and in this system the exciton lifetime will increase.Therefore during the lifetime of an exciton, its dynamics is under influence of a bathreservoir, and its damping play an important role. We assume that during the presenceof the exciton in QD, it interacts with the reservoir and hence we can consider its steadystate. We consider an exciton in dark state. Experimental preparation methods of suchexciton has been described in [23]. In this situation lifetime of exciton will increase andexciton has not spontaneously recombination radiation. However, its interaction withphonons causes a finite lifetime for it.The operator of the center-of-mass motion position ˆ x of the exciton in a sphericalQD may be defined asˆ x = κk ex ( ˆ A + ˆ A † ) , (16)where κ being a parameter similar to the Lamb-Dick parameter in ion trapped systemsand is defined as the ratio of QD radius to the wavelength of the driving laser (becauseof the spatial confinement of exciton, its wave function width is determined by thebarriers of QD), and we assume k ≃ k ≃ k ex ( k ex is the wavevector of the exciton).The operators ˆ A and ˆ A † are defined in Eq.(10). The interaction Hamiltonian (15) canbe written as H int = ~ e iω ex t Ω (cid:20) Ω Ω + e − iω n t (cid:21) e iκ ( ˆ A + ˆ A † ) ˆ S + H.c., (17)where Ω = gE ~ and Ω = gE ~ are the Rabi frequencies of the lasers, respectively, tunedto the electronic transition of QD (internal degree of freedom) and the first center-of-mass motion transition of exciton. Since the external degree of freedom is definite, then ω n depends on a special value of n such that it can be consider as a c-number quantity.The frequency ω n is depend on the number of quanta for each transition and hence thelaser tuned to the center-of-mass motion must be so strong that causes transition. Thisallows us to treat the interaction of the confined exciton in a wide QD with two lasersseparately, by using a nonlinear Jaynes-Cummings Hamiltonian [26] for each coupling.The interaction Hamiltonian in the interaction picture can be written as H I = ~ Ω ˆ S (cid:20) Ω Ω + e iω n t (cid:21) exp[ iκ ( e − iω ˆ n t ˆ A + ˆ A † e iω ˆ n t )] + H.c., (18)where ω ˆ n = ~ [(ˆ n + 2) f (ˆ n + 2) − ˆ nf (ˆ n )]. By using the vibrational rotating waveapproximation [6], applying the disentangling formula introduced in [27], and usingthe fact that in the present case the Lamb-Dick parameter is small, the interactionHamiltonian (18) is simplified to H (1) I = ~ Ω ˆ S (cid:20) F (ˆ n, κ ) Ω Ω + iκF (ˆ n, κ )ˆ a (cid:21) + H.c., (19)where the function F i (ˆ n, κ ) ( i = 0 ,
1) is defined by F i (ˆ n, κ ) = e − κ (( n +1+ i ) f ( n +1+ i ) − ( n + i ) f ( n + i )) × (20) n X l =0 ( iκ ) l l !( l + i )! f (ˆ n ) f (ˆ n + i )[ f (ˆ n − l )!] (ˆ a † ) l ˆ a l . It should be noted that this function in the limit f (ˆ n ) → F i (ˆ n, κ ) | f (ˆ n ) → = e − κ ˆ n + i L i ˆ n (cid:0) κ (cid:1) . (21)Now we write the function F i (ˆ n, κ ) (20) F i (ˆ n, κ ) = e − κ (( n +1+ i ) f ( n +1+ i ) − ( n + i ) f ( n + i )) ˆ n + i f (ˆ n )! f (ˆ n + 1)! L if, ˆ n (cid:0) κ (cid:1) , (22)where the function L if, ˆ n ( x ) is defined as L if, ˆ n ( x ) = n X l =0 f (ˆ n − l )!] (ˆ n + i )!(ˆ n − l )! l !( l + i )! ( − x ) l . (23)This function is similar to the associated Laguerre function.The time evolution of the system under consideration is characterized by the masterequation d ˆ ρdt = − i ~ [ ˆ H (1) I , ˆ ρ ] + L ˆ ρ, (24)where L ˆ ρ defines damping of the system due to the different kinds of interactions whichlead to annihilation of exciton. We assume a bosonic reservoir that causes dampingof exciton system. Due to the properties of dark exciton, the rate of spontaneousrecombination and hence spontaneous emission is decrease. On the other hand,interactions of exciton-phonon and exciton-impurities cause the exciton to be damped.In fact in low temperatures it is possible to ignore the phonon effects and by assuminga pure system we neglect the impurity effects. Hence we can write L ˆ ρ = Γ2 (2ˆ b ˆ ρ ˆ b † − ˆ b † ˆ b ˆ ρ − ˆ ρ ˆ b † ˆ b ) , (25)where Γ is the energy relaxation rate, ˆ b and ˆ b † are the annihilation and creation operatorsof the reservoir. Due to the confinement and dark state properties, spontaneousrecombination of exciton decreases and hence the lifetime of exciton becomes so longthat we can consider the stationary solution of Eq.(24). We assume a finite lifetime forexciton, and during this time we neglect damping effects. The stationary solution of themaster equation (24) in the time scales of our interest isˆ ρ = | e i| ψ ih ψ |h e | , (26)where | e i is the electronic excited state correspond to the presence of exciton and | ψ i isthe center-of-mass motion state of the exciton, which can be considered as a right-handeigenstate of the deformed operator ˆ A = F (ˆ n,κ ) F (ˆ n,κ ) ˆ aF (ˆ n, κ ) F (ˆ n, κ ) ˆ a | ψ i = i Ω Ω κ | ψ i . (27)According to Eq.(22) the corresponding deformation function reads as f (ˆ n ) = F (ˆ n − , κ ) F (ˆ n − , κ ) (28)= f (ˆ n ) L f, ˆ n − ( κ ) nL f, ˆ n − ( κ ) e − κ ( ( n +1) f ( n +1) − ( n − f ( n − ) . Hence, we can express the state | ψ i in the Fock space representation as | ψ i = N f X n χ n √ n ! f ( n )! | n i , (29)where χ = i Ω κ Ω . According to the definition (2), it is evident that the state | ψ i can beregarded as a special kind of NLCS. As is seen from equation (27), the eigenvalues of thedeformed operator ˆ A depends on some physical parameters such as the Rabi frequencies,the parameter κ and radius of QD.As is clear from equation (28), the deformation function f (ˆ n ) depends on thequantum number ˆ n and physical parameters such as QD radius and κ which characterizesthe confinement regime. In the limit f (ˆ n ) →
1, (harmonic confinement), whichcorresponds, for example, to a QD in lens shape [28], the function L if, ˆ n reduces tothe ordinary associated Laguerre polynomials, its argument tends to κ and therefore,the deformation function (28) takes the following form f (ˆ n ) = e − κ L n ( κ )[(ˆ n + 1) L n ( κ )] − . (30)This is the deformation function that appears in the center-of-mass motion of a trappedion confined in a harmonic trap [6].In order to investigate the nonclassical behavior of the NLCS | ψ i we consider thequadrature squeezing of the center-of-mass motion. For this purpose, we define thedeformed quadratures operators as followsˆ X = 12 ( ˆ Ae iφ + ˆ A † e − iφ ) , ˆ X = 12 i ( ˆ Ae iφ − ˆ A † e − iφ ) . (31)0In the limiting case f (ˆ n ) →
1, these two operators reduce to the conventional (non-deformed) quadrature operators [29]. The commutation relation of ˆ X and ˆ X is[ ˆ X , ˆ X ] = i n + 1) f (ˆ n + 1) − ˆ nf (ˆ n )] . (32)The variances h (∆ ˆ X i ) i ≡ h ˆ X i i − h ˆ X i i ( i = 1 ,
2) satisfy the uncertainty relation h (∆ ˆ X ) ih (∆ ˆ X ) i ≥
116 ( h (ˆ n + 1) f (ˆ n + 1) − ˆ nf (ˆ n ) i ) (33)A quantum state is said to be squeezed when one of the quadratures components ˆ X and ˆ X satisfies the relation h (∆ ˆ X i ) i < h (ˆ n + 1) f (ˆ n + 1) − ˆ nf (ˆ n ) i i = 1 or s i ( i = 1 ,
2) definedby s i = 4 h (∆ ˆ X i ) i − h (ˆ n + 1) f (ˆ n + 1) − ˆ nf (ˆ n ) i . (35)Then the condition for squeezing in the quadrature component can be simply written as s i <
0. In Fig.(1) we plot the squeezing parameter s versus the parameter Ra B definedas the ratio of the QD radius to the Bohr radius of exciton for two different valuesof ratio Ω Ω . As is clear from Fig.(1) for small values of the parameter Ra B the stateshows quadrature squeezing and by increasing this parameter the quadrature squeezingdisappears.
4. Conclusion
In this paper, we first considered a particle confined in a spherical infinite well andwe found the explicit forms of its creation and annihilation operators by using the f -deformed oscillator approach. Then we considered an exciton in a wide QD interactswith two laser beams. We showed that under the weak confinement condition, theexciton is influenced as a whole and its center-of-mass motion will be quantized. Withinthe framework of the f -deformed oscillator approach, we found that under certaincircumstances of exciton-laser interaction the stationary state of the exciton center-of-mass is a nonlinear coherent state which exhibits the quadrature squeezing. 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 squeezing versus ration ra B . Solid line correspond to Ω Ω = 0 . Ω Ω = 0 .
2. In both plots Lamb-Dick parameter is chosen as κ = 0 ..