Spatial confinement effects on quantum harmonic oscillator I: Nonlinear coherent state approach
aa r X i v : . [ qu a n t - ph ] D ec Spatial confinement effects on quantum harmonicoscillator I: Nonlinear coherent state approach
M. Bagheri Harouni, R. Roknizadeh, M. H. Naderi
Quantum Optics Group, Department of physics, University of Isfahan, Isfahan, IranE-mail: [email protected], [email protected],[email protected]
Abstract.
In this paper we study some basic quantum confinement effects throughinvestigation of a deformed harmonic oscillator algebra. We show that spatialconfinement effects on a quantum harmonic oscillator can be represented by adeformation function within the framework of nonlinear coherent states theory. Weconstruct the coherent states associated with the spatially confined quantum harmonicoscillator in a one-dimensional infinite well and examine some of their quantumstatistical properties, including sub-poissonian statistics and quadrature squeezing.
1. Introduction
The harmonic oscillator is one of the models most extensively used in both classical andquantum mechanics. The usefulness and simplicity make this model a subject of lots ofstudies. One of the most important aspects of quantum harmonic oscillator (QHO) is itsdynamic algebra i.e. Weyl-Heisenberg algebra. This algebra appears in many areas ofmodern theoretical physics, as an example we notice that the one-dimensional quantumharmonic oscillator was successfully used in second quantization formalism [1].Due to the relevance of Weyl-Heisenberg algebra, some efforts have been devotedto studying possible deformations of the QHO algebra [2]. A deformed algebra isa nontrivial generalization of a given algebra through the introduction of one ormore deformation parameters, such that, in a certain limit of parameters the non-deformed algebra is recovered. A particular deformation of Heisenberg algebra has ledto the notion of f -oscillator [3]. An f -oscillator is a non-harmonic system, that frommathematical point of view its dynamical variables (creation and annihilation operators)are constructed from a non canonical transformation throughˆ A = ˆ af (ˆ n ) , ˆ A † = f (ˆ n )ˆ a † , (1)where ˆ a and ˆ a † are the usual (non-deformed) harmonic oscillator operators with[ˆ a, ˆ a † ] = 1 and ˆ n = ˆ a † ˆ a . The function f (ˆ n ) is called deformation function which dependson the number of excitation quanta and some physical parameters. The presence ofthe operator-valued deformation function causes the Heisenberg algebra of the standardQHO to transform into a deformed Heisenberg algebra. The nonlinearity in f -oscillatorsmeans dependence of the oscillation frequency on the intensity [4]. On the other hand,in contrast to the standard QHO, f -oscillators have not equal spaced energy spectrum.For example, if we confine a simple QHO inside an infinite well, due to the spatialconfinement, the energy levels constitute a spectrum that is not equal spaced. Therefore,in this case it is reasonable to investigate the corresponding f -oscillator.The confined QHO can be used to describe confinement effects on physicalproperties of confined systems. Physical size and shape of the materials strongly affectthe nature, dynamics of the electronic excitations, lattice vibrations, and dynamics ofcarriers. For example, in the mesoscopic systems, the dimension of system is comparablewith the coherence length of carriers and this leads to some new phenomena that theydo not appear in a bulk semiconductor, such as quantum interference between carrier’smotion [5]. Recent progress in growth techniques and development of micromachinigtechnology in designing mesoscopic systems and nanostructures, have led to intensivetheoretical [6] and experimental investigations [7] on electronic and optical propertiesof those systems. The most important point about the nanoscale structures is that thequantum confinement effects play the center-stone role. One can even say, in general,that recent success in nanofabrication technique have resulted in great interest in variousartificial physical systems (quantum dots, quantum wires and quantum wells) with newphenomena driven by the quantum confinement. A number of recent experiments havedemonstrated that isolated semiconductor quantum dots are capable of emitting light[8]. It becomes possible to combine high-Q optical microcavities with quantum dotemitters as the active medium [9]. Furthermore, there are many theoretical attemptsfor understanding the optical and electronic properties of nanostructures especiallysemiconductor quantum dots [10]. On the other hand, a nanostructure such as quantumdot, is a system that carrier’s motion is confined inside a small region, and during theinteraction with other systems, the generated excitations such as phonons, excitons andplasmons are confined in small region. In order to describe the physical properties ofthese excitations one can consider them as harmonic oscillator.As another application of deformed algebra we can refer to the notion ofparastatistics [11]. The concept of parastatistics has found many application in fractalstatistics and anyon theory [12]. In addition to the anyon theory, the parastatistics hasfound many interesting application in supersymmetry and non-commutative quantummechanics [13].The construction of generalized deformed oscillators corresponding to well-knownpotentials and study of the correspondence between the properties of the conventionalpotential picture and the algebraic one has been done [14]. Recently, the generalizeddeformed algebra and its associated generalized operators have been considered [15].By looking at the classical correspondence of the Hamiltonian, the potential energy andthe effective mass function is obtained. In this contribution we derive the generalizedoperators associated with a definite potential by comparing the physical properties ofsystem and physical results of generalized algebra.One of the most interesting features of the QHO is the construction of its coherentstates as the eigenfunctions of the annihilation operator. As is well known [3], onecan introduce nonlinear coherent states (NLCSs) or f -coherent states as the right-handeigenstates of the deformed annihilation operator ˆ A . It has been shown [16] that thesefamilies of generalized coherent states exhibit various non-classical properties. Due tothese properties and their applications, generation of these states is a very importantissue in the context of quantum optics. The f -coherent states may appear as stationarystates of the center-of-mass motion of a trapped ion [17]. Furthermore, a theoreticalscheme for generation of these states in a coherently pumped micromaser within theframe-work of intensity-dependent Jaynes-Cummings model has been proposed [18].One of the most important questions is the physical meaning of the deformationin the NLCSs theory. It has been shown [19] that there is a close connection betweenthe deformation function appeared in the algebraic structure of NLCSs and the non-commutative geometry of the configuration space. Furthermore, it has been shownrecently [20], that a two-mode QHO confined on the surface of a sphere, can beinterpreted as a single mode deformed oscillator, whose quantum statistics dependson the curvature of sphere.Motivated by the above-mentioned studies, in the present contribution we areintended to investigate the spatial confinement effects on physical properties of astandard QHO. It will be shown that the spatial confinement leads to deformation ofstandard QHO. We consider a QHO confined in a one-dimensional infinite well withoutperiodic boundary conditions, and we find its energy levels, as well as associated ladderoperators. We show that the ladder operators can be interpreted as a special kind ofthe so-called f -deformed creation and annihilation operators [3].This paper is organized as follows: In section 2, we review some physical propertiesof f -oscillator and its coherent states. In section 3 we consider the spatially confinedQHO in a one-dimensional infinite well and construct its associated coherent states. Weshall also examine some of their quantum statistical properties, including sub-Poissonianstatistics and quadrature squeezing. Finally, we summarize our conclusions in section4. f -oscillator and nonlinear coherent states In this section, we review the basics of the f -deformed quantum oscillator and theassociated coherent states known in the literature as nonlinear coherent states. In thefirst step, to investigate one of the sources of deformation we consider an eigenvalueproblem for a given quantum physical system and we focus our attention on theproperties of creation and annihilation operators, that allow to make transition betweenthe states of discrete spectrum of the system Hamiltonian [21]. As usual, we expandthe Hamiltonian in its eigenvectorsˆ H = ∞ X i =0 E i | i ih i | , (2)where we have choosed E = 0. We introduce the creation (raising) and annihilation(lowering) operators as followsˆ a † = ∞ X i =0 p E i +1 | i + 1 ih i | , ˆ a = ∞ X i =0 p E i | i − ih i | , (3)so that ˆ a | i = 0. These ladder operators satisfy the following commutation relation[ˆ a, ˆ a † ] = ∞ X i =1 ( E i +1 − E i ) | i ih i | . (4)Obviously if the energy spectrum is equally spaced that is, it should be linear in quantumnumbers, as in the case of ordinary QHO, then E i +1 − E i = c , where c is a constantand in this condition the commutator of ˆ a and ˆ a † becomes a constant (a rescaled Weyl-Heisenberg algebra). On the other hand, if the energy spectrum is not equally spaced,the ladder operators of the system satisfy a deformed Heisenberg algebra, i.e. theircommutator depends on the quantum numbers that appear in the energy spectrum.This is one of the most important properties of the quantum f -oscillators [3].An f -oscillator is a non-harmonic system characterized by a Hamiltonian of theharmonic oscillator formˆ H D = Ω2 ( ˆ A ˆ A † + ˆ A † ˆ A ) ( ~ = 1) , (5)( ˆ A = ˆ af (ˆ n )) with a specific frequency Ω and deformed boson creation and annihilationoperators defined in (1). The deformed operators obey the commutation relation[ ˆ A , ˆ A † ] = (ˆ n + 1) f (ˆ n + 1) − ˆ nf (ˆ n ) . (6)The f -deformed Hamiltonian ˆ H D is diagonal on the eingenstates | n i in the Fock spaceand its eigenvalues are E n = Ω2 [( n + 1) f ( n + 1) + nf ( n )] . (7)In the limit f →
1, the ordinary expression E n = Ω( n + ) and the usual (non-deformed)commutation relation [ˆ a , ˆ a † ] = 1 are recovered.Furthermore, by using the Heisenberg equation of motion with Hamiltonian (5) i d ˆ Adt = [ ˆ
A , ˆ H D ] , (8)we obtain the following solution for the f -deformed operators ˆ A and ˆ A † ˆ A ( t ) = e − i Ω G (ˆ n ) t ˆ A (0) , ˆ A † ( t ) = ˆ A † (0) e i Ω G (ˆ n ) t , (9)where G (ˆ n ) = 12 (cid:0) (ˆ n + 2) f (ˆ n + 2) − ˆ nf (ˆ n ) (cid:1) . (10)In this sense, the f -deformed oscillator can be interpreted as a nonlinear oscillator whosefrequency of vibrations depends explicitly on its number of excitation quanta [4]. It isinteresting to point out that recent studies have revealed strictly physical relationshipbetween the nonlinearity concept resulting from the f -deformation and some nonlinearoptical effects, e.g., Kerr nonlinearity, in the context of atom-field interaction [22].The nonlinear transformation of the creation and annihilation operators leadsnaturally to the notion of nonlinear coherent states or f -coherent states. The nonlinearcoherent states | α i f are defined as the right-hand eigenstates of the deformed operatorˆ A | α i f = α | α i f . (11)From Eq.(11) one can obtain an explicit form of the nonlinear coherent states in anumber state representation | α i f = C ∞ X n =0 α n d n | n i , (12)where the coefficients d n ’s and normalization constant C are, respectively, given by d = 1 , d n = (cid:16) √ n ![ f ( n )]! (cid:17) − , [ f ( n )]! = n Y j =1 f ( j ) ,C = ∞ X n =0 d n | α | n ! − . (13)In recent years the nonlinear coherent states have been paid much attentions becausethey exhibit nonclassical features [16] and many quantum optical states, such as squeezedstates, phase states, negative binomial states and photon-added coherent states can beviewed as a sort of nonlinear coherent states [23].
3. Quantum harmonic oscillator in a one dimensional infinite well f -deformed oscillator description of confined QHO In this section we consider a quantum harmonic oscillator confined in a one dimensionalinfinite well. Many attempts have been done for solving this problem (see [24],[25],and references therein). In most of those works, authors tried to solve the problemnumerically. But in our consideration we try to solve the problem analytically, to revealthe relationship between the confinement effect and given deformation function. Westart from the Schr¨odinger equation ( ~ = 1) (cid:20) − m d dx + 12 kx + V ( x ) (cid:21) ψ ( x ) = Eψ ( x ) , (14)where V ( x ) = ( − a ≤ x ≤ a ∞ elsewhere . According to the approach introduced in previous section, we can obtain raising andlowering operators from the spectrum of Schr¨odinger operator. On the other hand, bycomparing the energy spectrum of particular system with energy spectrum of general f -deformed oscillator (7), one could obtain deformed raising and lowering operators.Hence, we need an analytical expression for energy spectrum of the system whichexplicitly shows dependence on special quantum numbers. The original problem,confined QHO (14), can be solved only by using the approximation methods. Whenapplying perturbation theory, one is usually concern with a small perturbation of anexactly solvable Hamiltonian system. In the case of confined QHO we deal with threelimits. Inside the well, for small values of position we have harmonic oscillator, forlarge values we have an infinite well and at the positions of the boundaries the twopotentials have the same power. Hence the approximation method can not lead toacceptable results. Therefore, we model the original problem by a model potential thathas mathematical behavior such as confined QHO. Instead of solving the Schr¨odingerequation for the QHO confined between infinite rectangular walls in positions ± a , wepropose to solve the eigenvalue problem for the potential V ( x ) = 12 k (cid:18) tan( δx ) δ (cid:19) , (15)where δ = π a , is a scaling factor depending on the width of the well. This potentialmodels a QHO placed in the center of the rectangular infinite well [26]. The potential V ( x ) (15) fulfills two asymptotic requirements: 1) V ( x ) → kx when a → ∞ (freeharmonic oscillator limit). 2) V ( x ) at equilibrium position has the same curvatureas a free QHO, h d Vdx i x =0 = k . This model potential belongs to the exactly solvabletrigonometric P¨oschl-Teller potentials family [27]. Stationary coherent states for specialkind of this potential have been considered [28].Now we consider the following equation " − m d dx + 12 k (cid:18) tan( δx ) δ (cid:19) − E ψ ( x ) = 0 . (16)To solve analytically this equation, we use the factorization method [29]. By changingthe variable and some mathematical manipulation, the corresponding energy eigenvaluesare found as E n = γ ( n + 12 ) + p γ + ω ( n + 12 ) + γ , (17)where γ = π a m ,and ω = q km is the frequency of the QHO. The first term in theenergy spectrum can be interpreted as the energy of a free particle in a well, the secondterm denotes the energy spectrum of the QHO, and the last term shifts the energyspectrum by a constant amount. It is evident that if a → ∞ then γ → γ as thecorresponding deformation parameter. In Table 1 the numerical results associated withthe original potential, given in Ref. [24], are compared with the generated results fromthe model potential under consideration. As is seen, the results are in a good agreementwhen boundary size is of order of characteristic length of the harmonic oscillator. Theoriginal oscillator potential when approaches to the boundaries of the well becomesinfinite suddenly, while the model potential is smooth and approaches to the infinityasymptotically. Therefore, the model potential (15) is more appropriate for the physicalsystems.If we normalize Eq.(17) to energy quanta of the simple harmonic oscillator andintroduce the new variables n + = h , q γ ω + 1 = η , and γ ′ = γω then it takes thefollowing form E l = γ ′ h + ηh + γ ′ . (18)By comparing this spectrum with the energy spectrum of an f -deformed oscillator, givenby (7), we find the corresponding deformation function as f (ˆ n ) = p γ ′ ˆ n + η. (19)Furthermore, the ladder operators associated with the confined oscillator underconsideration can be written in terms of the conventional (non-deformed) operatorsˆ a , ˆ a † as followsˆ A = ˆ a p γ ′ ˆ n + η , ˆ A † = p γ ′ ˆ n + η ˆ a † . (20)These two operators satisfy the following commutation relation[ ˆ A, ˆ A † ] = γ ′ (2ˆ n + 1) + η. (21)It is obvious that in the limiting case a → ∞ ( γ ′ → η → n , and the deformed algebrareduces to a the conventional Weyl-Heisenberg algebra for a free QHO.Classically, harmonic oscillator is a particle that attached to an ideal spring, andcan oscillate with specific amplitude. When that particle be confined, boundaries canaffect particle’s motion if the boundaries position be in a smaller distance in comparisonwith a characteristic length that particle oscillates within it. This characteristic lengthfor the QHO is given by ~ mω ( ~ = 1) , and if 2 a ≤ mω , then the presence of the boundariesaffects the behavior of QHO, otherwise it behaves like a free QHO. Therefore, one caninterpret l = mω as a scale length where the deformation effects become relevant. Now, we focus our attention on the coherent states associated with the QHO underconsideration. As usual, we define coherent states as the right-hand eigenstates of thedeformed annihilation operatorˆ A | β i f = β | β i f . (22)From (22) and using the NLCS formalism introduced in (11)-(13) the explicit form ofthe corresponding NLCS of the confined QHO is written as | β i f = N X n β n p n !( γ ′ n + η )! | n i , (23)where N = (cid:16)P n | β | n [ f ( n )!] n ! (cid:17) − is the normalization factor, β is a complex number, and thedeformation function f ( n ) is given by Eq.(19). The ensemble of states | β i f labeled bythe single complex number β is called a set of coherent states if the following conditionsare satisfied [30]: • normalizability f h β | β i f = 1 , (24) • continuity in the label β | β − β ′ | → ⇒ k | β i f − | β ′ i f k → , (25) • resolution of the identity Z c d β | β i f f h β | w ( | β | ) = ˆ I, (26)where w ( | β | ) is a proper measure that ensures the completeness and the integrationis restricted to the part of the complex plane where normalization converges.The first two conditions can be proved easily. For the third condition, we choose thenormalization constant as N = | β | η I γ ′ η (2 | β | ) , (27)where I γ ′ η ( x ) = ∞ X s =0 s !( γ ′ s + η )! ( x s + η , (28)is similar to the modified Bessel function of the first kind of the order η with theseries expansion I η ( x ) = P ∞ s =0 1 s !( s + η )! ( x ) s + η . Resolution of the identity of the deformedcoherent states | β i f can be written as Z d β | β i f h β | w ( | β | ) = π X n | n ih n | n !( γ ′ n + η )! Z ∞ d | β || β || β | n (29) × | β | η I γ ′ η (2 | β | ) w ( | β | ) . Now we introduce the new variable | β | = x and the measure w ( √ x ) = 8 π I γ ′ η (2 √ x ) K m (2 √ x ) x l , (30)where K m ( x ) is the modified Bessel function of the second kind of the order m , m = ( γ ′ − n + α , and l = ( γ ′ − n + 1. Using the integral relation R ∞ K ν ( t ) t µ − dt =2 µ − Γ (cid:0) µ − ν (cid:1) Γ (cid:0) µ + ν (cid:1) [31], we obtain Z d β | β i f f h β | w ( | β | ) = X n | n ih n | = ˆ I. (31)We therefore conclude that the states | β i f qualify as coherent states in the sensedescribed by the conditions (24)-(26).We now proceed to examine some nonclassical properties of the nonlinear coherentstates | β i f . As an important quantity, we consider the variance of the number operatorˆ n . Since for the coherent states the variance of number operator is equal to its average,deviation from the Poissonian statistics can be measured with the Mandel parameter[32] M = (∆ n ) − h ˆ n ih ˆ n i . (32)This parameter vanishes for the Poisson distribution, is positive for the super-Poissoniandistribution (bunching effect), and is negative for the sub-Poissonian distribution(antibunchig effect). Fig. 1 shows the size dependence of the Mandel parameter fordifferent values of | β | . As is seen, the Mandel parameter exhibits the sub-Poissonianstatistics and with further increasing values of a it is finally stabilized at an asymptoticalzero value corresponding to the Poissonian statistics. In addition, the smaller theparameter | β | is, the more rapidly the Mandel parameter tends to the Poissonianstatistics.As another important nonclassical property we examine the quadrature squeezing.For this purpose we first consider the conventional quadrature operators ˆ X a and ˆ Y a defined in terms of nondeformed operators ˆ a and ˆ a † as [33]ˆ X a = 12 (ˆ ae iφ + ˆ a † e − iφ ) ˆ Y a = 12 i (ˆ ae iφ − ˆ a † e − iφ ) . (33)0In this equation, φ is the phase of quadrature operators which can effectivly affect thesqueezing properties. The commutation relation for ˆ a and ˆ a † leads to the followinguncertainty relation(∆ ˆ X a ) (∆ ˆ Y a ) ≥ |h [ ˆ X a , ˆ Y a ] i| = 116 . (34)For the vacuum state | i , we have (∆ ˆ X a ) = (∆ ˆ Y a ) = and hence (∆ ˆ X a ) (∆ ˆ Y a ) = .A given quantum state of the QHO is said to be squeezed when the variance of one ofthe quadrature components ˆ X a and ˆ Y a satisfies the relation(∆ ˆ O a ) < (∆ ˆ O a ) vacuum = 14 ( ˆ O a = ˆ X a or ˆ Y a ) . (35)The degree of quadrature squeezing can be measured by the squeezing parameter s ˆ O defined by s ˆ O = 4(∆ ˆ O a ) − . (36)Then, the condition for squeezing in the quadrature component can be simply writtenas s ˆ O <
0. In Fig. 2 we have plotted the parameter s ˆ X a corresponding to the squeezingof ˆ X a with respect to the phase angle φ for three different values of a . As is seen,the state | β i f exhibits squeezing for different values of the confinement size, and when a l = al = 2 .
5, the quadrature ˆ X a exhibits squeezing for all values of the phase angle φ . Fig. 3 shows the plot of s ˆ X a versus the dimensionless parameter a l = al fordifferent values of the phase φ . As is seen, with the increasing value of a l ( al ), thequadrature component tends to the zero according to the vacuum fluctuation. Let usalso consider the deformed quadrature operators ˆ X A and ˆ Y A defined in terms of thedeformed operators ˆ A and ˆ A † asˆ X A = 12 ( ˆ Ae iφ + ˆ A † e − iφ ) , ˆ Y A = 12 i ( ˆ Ae iφ − ˆ A † e − iφ ) . (37)By considering the commutation relation (6) for the deformed operators ˆ A and ˆ A † ,the squeezing condition for the deformed quadrature operators ˆ O A (= ˆ X A , ˆ Y A )can bewritten as S = 4(∆ ˆ O A ) − h (ˆ n + 1) f (ˆ n + 1) i + h ˆ nf (ˆ n ) i < . (38)In Fig. 4 we have plotted the parameter S ˆ X A versus the dimensionless parameter al for three different values of | β | . As is seen, the deformed quadrature operator exhibitssqueezing for all values of a . Furthermore, with the increasing value of | β | the squeezingof the quadrature ˆ X A is enhanced.
4. Conclusion
In this paper, we have considered the relation between the spatial confinement effects anda special kind of f -deformed algebra. We have found that the confined simple harmonicoscillator can be interpreted as an f -oscillator, and we have obtained the correspondingdeformation function. By constructing the associated NLCSs, we have examined the1 Table 1.
Calculated energy levels of the confined QHO in a one dimensional infinitewell by using our model potential in comparison with the numerical results given inRef.[24] state boundary size model potential numerical results0 a=0.5 4.98495312 4.951129320 1 1.41089325 1.298459830 2 0.67745392 0.537461200 3 0.57321464 0.500391080 4 0.54003728 0.500000491 a=0.5 19.88966157 19.774534171 1 5.46638033 5.075582011 2 2.34078691 1.764816431 3 1.85672176 1.506081521 4 1.69721813 1.500014612 a=0.5 44.66397441 44.452073822 1 11.98926850 11.258825782 2 4.62097017 3.399788242 3 3.41438455 2.541127252 4 3.00861155 2.500201173 a=0.5 79.30789166 78.996921153 1 20.97955777 19.899696493 2 7.51800371 5.584639073 3 5.24620303 3.664219643 4 4.47421754 3.501691534 a=0.5 123.82141330 123.410710504 1 32.43724814 31.005254504 2 11.03188752 8.368874424 3 7.35217718 4.954180474 4 6.09403610 4.50964099effects of confinement size on non-classical statistical properties of those states. Theresult show that the stronger confinement leads to the strengthening of non-classicalproperties. We hope that our approach may be used in description of phonons in thestrong excitation regimes, photons in a microcavity and different elementary excitationsin confined systems. The work on this direction is in progress.
Acknowledgment
The authors wish to thank the Office of Graduate Studies ofthe University of Isfahan and Iranian Nanotechnology initiative for their support.2
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Quantum Optics (Cambridge University Press, 1997). l - - - - - M Figure 1.
Plots of the Mandel parameter versus the dimensionless parameter a l = al .For | β | = 0 . | β | = 1 (longdashed curve), for | β | = 1 . | β | = 4 . - - - s Figure 2.
Plot of s ˆ X a versus φ for | β | = 4. The dashed, longdashed and solid curvesrespectively relate to a = 2 . a = 1, a = 0 . a are renormalized to l ). l - - - - s Figure 3.
Plots of s ˆ X a versus the dimensionless parameter a l = al for different phasesand | β | = 1. Dashed curve, solid curve and bold curve ,respectively, correspond to φ = 100, φ = 110 and φ = 90. l - - - - - - S l - - - - S Figure 4.
Plots of deformed squeezing parameter S X A versus the dimensionlessparameter a l = al . The dashed curve, longdashed curv, solid curve and bold curve arerespectively, correspond to | β | = 1, | β | = 1 . | β | = 2 . | β |2