Self-Localized Solutions of the Nonlinear Quantum Harmonic Oscillator
aa r X i v : . [ qu a n t - ph ] M a r Self-Localized Solutions of the Nonlinear Quantum Harmonic Oscillator
Cihan Bayındır ∗ Associate Professor, Engineering Faculty, ˙Istanbul Technical University, 34467 Maslak, ˙Istanbul, Turkey.Adjunct Professor, Engineering Faculty, Bo˘gazi¸ci University, 34342 Bebek, ˙Istanbul, Turkey.International Collaboration Board Member, CERN, CH-1211 Geneva 23, Switzerland.
We analyze the existences, properties and stabilities of the self-localized solutions of the nonlinearquantum harmonic oscillator (NQHO) using spectral renormalization method (SRM). We show thatself-localized single, dual and triple soliton solutions of the NQHO do exists, however only single anddual soliton solutions satisfy the necessary Vakhitov and Kolokolov slope condition, therefore triplesoliton solution is found to be unstable, at least for the parameter ranges considered. Additionally,we investigate the stability characteristics of the single and dual soliton solutions using a split-stepFourier scheme. We show that single and dual soliton solutions are pulsating during time stepping.We discuss our findings and comment on our results.
PACS numbers: 03.65.w, 05.45.-a, 03.75.b
I. INTRODUCTION
Quantum harmonic oscillator (QHO) is one of the fun-damental models in quantum mechanics [1–5]. It can beviewed as the quantum mechanical analog of the simpleharmonic oscillator of the classical vibration theory. Thevibrations of atomic particles and molecules under theeffect of restoring spring like potential due to molecularbonding are modeled within its frame [1–5]. QHO admitsexact solutions in terms of Hermite polynomials and canbe extended to N-dimensions to model multidimensionalatomic and molecular vibrations [1–5].Nonlinear quantum mechanics studies, on the otherhand, are becoming increasingly popular recently [6–11].Majority of the studies on nonlinear quantum phenom-ena are modeled in the frame of dynamic equations, i.e.the nonlinear Schr¨odinger equation (NLSE). Comparedto the linear Schr¨odinger equation, NLSE can adequatelymodel the cubic nonlinearity effects on the wavefunction.Such nonlinearities give rise to many interesting quantummechanical phenomena including but are not limited tosolitons, rogue waves, nonlinear quantum entanglementand quantum chaos. Analogs of these phenomena mayalso appear in the macroscopic physical environment.Various studies which investigate the effects of nonlin-earity on quantum oscillations do also exist in the lit-erature [12–18]. As discussed in the relevant literature,nonlinearity can arise in different ways. One possible waythat gives rise to nonlinearity is the nonlinear behavior ofbonding spring like stiffness and its corresponding poten-tial. Another source of nonlinearity, which is investigatedin this paper, arises due to strong electric and magneticfields, which eventually leads to the cubic nonlinear termin the NQHO models.In this paper, we consider the NQHO model first pro-posed in [12] and recently generalized by us [13]. Thismodel equation can only be solved numerically, but for ∗ Electronic address: [email protected] some limiting cases exact analytical solutions do exist[12]. We study the self-localized solutions of this NQHOmodel using the spectral renormalization method (SRM).More specifically, we obtain the self-localized single, dualand triple soliton solutions of the NQHO using the SRMand discuss their properties. Additionally, we investigatethe stability characteristics of those self-localized solu-tions using a split-step Fourier scheme which is used toperform the time stepping. We show that self-localizedsingle and dual soliton solutions of the NQHO are stableand have pulsating behavior in time, at least for some ofthe parameter ranges considered in this paper. We alsoshow that the self-localized triple soliton solution of theNQHO is not stable since it violates the necessary Vakhi-tov and Kolokolov slope condition for stability, at leastfor the parameter ranges considered.
II. A NONLINEAR QUANTUM HARMONICOSCILLATOR MODEL
Linear quantum harmonic oscillator’s (LQHO) Hamil-tonian (energy) can be given by [1–5] b H = b p m + 12 kx = b p m + 12 mω x (1)In this formula b H denotes the Hamiltonian of the LQHO, m denotes the particle mass and k is the bonding stiff-ness of the atomic particle, which is analogous to springconstant in a classical mass-spring-dashpot system. Inthis formulation, the momentum operator can be givenby b p = − i ~ ∂/∂x where ~ is the reduced Planck’s con-stant. Thus, the unsteady Schr¨odinger equation can bederived using the Hamiltonian formalism as i ~ ∂ψ∂t + ~ m ∂ ψ∂x − mω x ψ = 0 (2)where i denotes the imaginary unity, t is time variable, x is the position and ψ ( x, t ) is the wavefunction. This formof the LQHO is commonly studied in the literature [1–5]. While majority of the studies on quantum harmonicoscillations utilizes linear models, few different forms ofNQHO models are proposed in [12–18]. Generally, twoforms of nonlinearity are considered in these studies dueto different nonlinear behaviors. One of them is the non-linearly behaving molecular bond, which is commonlymodeled using spring constant analogy, which is repre-sented by different forms of the potential function thanthe commonly used trapping-well potential. However, inorder to account for the effects of high-order electric andmagnetic fields on wavefunction, a NQHO model is pro-posed in [12]. The form of the NQHO proposed in [12]can be given as iψ t + ∂ ψ∂x − x ψ + σ | ψ | ψ = 0 (3)In here σ is a constant which controls the strength ofthe nonlinearity. This equation can be derived by apply-ing the non-dimensional parameters given in [12] to theLQHO and including the nonlinear term of the NLSE.Setting, σ = 0, NQHO can be linearized and reduced tothe LQHO, which admits solutions in the form of ψ ( x, t ) = U ( x ) e − iµt (4)The analytical solution in this form can only be de-rived for the discrete spectrum of µ n = 1 + 2 n where n = 0 , , , ... [12]. For this discrete spec-trum, the amplitude functions can be derived as U n =(2 n n ! √ π ) − / e − x / H n ( x ) where H n ( x ) are the Hermitepolynomials. These polynomials are described by H n ( x ) = ( − n e x / d n ( e − x / ) dx n (5)giving H = 1, H = 2 x , H = 4 x − H = 8 x − x ,... etc. [12, 19, 20]. Eq.(3) requires numerical solutionexcept for some limiting cases [12], and a continuous fre-quency spectrum is considered in [12] for its numericalsolution.In this paper, we study a slightly more general versionof the NQHO given by Eq.(3) which was first proposedby Kivshar [12] and later extended by us recently [13].To model the effects of variable potential due to varyingbonding (spring) stiffness, we use a potential well con-stant, α . Thus, the form of the non-dimensional NQHOequation studied in this paper can be written as iψ t + ∂ ψ∂x − αx ψ + σ | ψ | ψ = 0 (6)As before, the t denotes the non-dimensional temporalparameter and x denotes the non-dimensional spatial pa-rameter. In the next sections of this paper, we study theexistences and properties of the self-localized solutions ofthe NQHO given by Eq.(6) using the spectral renormal-ization method (SRM). Additionally, using a split-stepFourier scheme, we study the stability characteristics ofsuch solutions of the NQHO. III. SPECTRAL RENORMALIZATIONMETHOD FOR FINDING THESELF-LOCALIZED SOLUTIONS OF THENONLINEAR QUANTUM HARMONICOSCILLATOR
There are few different techniques used to find the self-localized solutions of nonlinear systems. Some of thesetechniques are the shooting, self-consistency and relax-ation techniques [21–25]. One of the most commonly usedmethods for this purpose is the Petviashvili’s method, inwhich the governing nonlinear equation is transformedinto Fourier space similar to the other Fourier spectralmethods using FFT routines [26–40]. Then, a conver-gence factor is used in accordance with the degree ofthe nonlinear term [21, 22]. Petviashvili was the per-son who proposed this approach and he also applied thismethod to the Kadomtsev-Petviashvili equation [21]. Inorder to treat various forms of the homogeneities differentthan the fixed ones, Petviashvili’s method is extended tothe spectral renormalization method (SRM), which canbe used to find the self-localized solutions of more gen-eral nonlinear equations [22–24]. Later, another exten-sion of the Petviashvili’s method is proposed by us [25],which is capable of finding the self-localized solutions innonlinear waveguides under missing spectral information.This method is named as the compressive spectral renor-malization method (CSRM) [25]. The SRM transformsthe governing equation into wavenumber space using theFourier transforms and then couples it to a nonlinearintegral equation. The iterations are performed in thewavenumber space. Due to the coupling of these equa-tions, the energy is conserved and the initial conditionsconverge to the self-localized solutions of the system stud-ied [22]. Details and other possible uses of the SRM canbe seen in [22].In this section we apply the SRM to the NQHO modelgiven by Eq.(6) in order to study the self-localized solu-tions of the NQHO. We start with rewriting the NQHOmodel given by Eq.(6) in the form of iψ t + ψ xx − V ( x ) ψ + σ | ψ | ψ = 0 (7)where V ( x ) = αx is the trapping-well potential func-tion. Eq. (7) can be rewritten as iψ t + ψ xx − V ( x ) ψ + N ( | ψ | ) ψ = 0 (8)where N ( | ψ | ) = σ | ψ | is the nonlinear term of theNQHO. Using the ansatz, ψ ( x, t ) = η ( x, µ )exp( iµt ),Eq. (8) becomes − µη + η xx − V ( x ) η + N ( | η | ) η = 0 (9)where µ is the soliton eigenvalue. Iterations performedusing the spectral representation of Eq. (9) may becomesingular [22]. In order to avoid the singularity of thescheme, a pη term with p > b η ( k ) = F [ η ( x )] = Z + ∞−∞ η ( x ) exp[ i ( kx )] dx (10)thus becomes b η ( k ) = ( p + | µ | ) b ηp + | k | − F [ V η ] − F h N ( | η | ) η i p + | k | (11)The iteration formula given in Eq. (11) can be usedto find the self-localized solutions of the NQHO modelequation, however the iterations may diverge or it maytend to zero [22]. This problem can be solved by defininga new variable as η ( x ) = βξ ( x ), which has the Fouriertransform given by b η ( k ) = β b ξ ( k ). Using these substitu-tions, Eq. (11) can be rewritten as b ξ j +1 ( k ) = ( p + | µ | ) p + | k | b ξ j − F [ V ξ j ] p + | k | + 1 p + | k | F h σ | β j | | ξ j | ξ j i = R β j [ b ξ j ( k )] (12)which is the iteration equation of the SRM in wavenum-ber domain. The algebraic condition on the parameter β ,which prevents the scheme from diverging or tending tozero, can be derived using the energy conservation prin-ciple. Multiplying Eq. (12) with the b ξ ∗ ( k ) term, where ∗ shows the complex conjugation, and integrating the re-sulting equation to evaluate the total energy of the sys-tem, it is possible to derive the algebraic condition as Z + ∞−∞ (cid:12)(cid:12)(cid:12)b ξ ( k ) (cid:12)(cid:12)(cid:12) dk = Z + ∞−∞ b ξ ∗ ( k ) R β [ b ξ ( k )] dk (13)This becomes the normalization constraint, which pre-vents the scheme from diverging or tending to zero. Themethod summarized above is the SRM [22] and appliedto NQHO in this paper. Starting the simulations usinga single or multi-Gaussians as initial conditions, Eq. (11)and Eq. (13) are applied iteratively to find the self-localized solutions of the NQHO. The iterations are con-tinued until the parameter β convergences, for which thecut-off criteria is given for different simulations in thenext part of this paper. IV. RESULTS AND DISCUSSION
In this section we apply the SRM summarized aboveto find the self-localized solutions of the NQHO modelequation. In our simulations throughout this paper weuse N = 1024 spectral components. Starting with a sin-gle humped Gaussian in the form of exp ( − ( x − x ) ),where x is taken as 0, SRM converges to single humpedself-localized soliton solutions of the NQHO. We plot these self-localized solutions of the NQHO model equa-tion in Fig. 1 for various values of, α , the trapping-wellpotential coefficient. Other parameters are selected as p = 30 , σ = 1 , µ = 10 . β to be less than 10 − . The singlehumped Gaussian converges to self-localized solution ofthe NQHO rapidly. The exact form of the analytical so-lution is unknown, however the profile resembles solitarywaves in shape, however they have an asymmetric struc-ture. This result can also be verified with the findingspresented in [12]. As one can realize from the figure,the increase in trapping-well potential strength results inbigger waves. One possible explanation for this result isthat, bigger α values represents the trapping of the soli-tons in a more confined well, thus such bigger solitonsare formed. -20 -15 -10 -5 0 5 10 15 20 x -50510152025 =0=0.2=0.4=0.6=0.8=1.0=1.2 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 x Increasing
FIG. 1: Self-localized single solitons for different trapping wellpotential strength, α . Next we turn our attention to investigate the effects ofnonlinearity coefficient, σ , on the existence and proper-ties of the self-localized solitons of the NQHO. With thismotivation, we depict Fig. 2. As before, the simulationparameters are selected as p = 30 , µ = 10 . α = 1.Various values of σ are used as indicated in figure andagain, the cut-off criteria for the SRM iterations are se-lected as normalized change of β being less than 10 − .As Fig. 2 confirms, self-localized solutions of the NQHOexists for various values of σ as well and they tend to besmaller as σ grows larger.It is important to discuss the stability characteristics ofthe self-localized solutions of the NQHO model equationfound using the SRM. For a soliton to be stable two con-ditions must be satisfied. The first conditions is knownas the slope condition, dP ( µ ) /dµ <
0, as first derived byVakhitov and Kolokolov [41, 42]. In here P = R | ψ | dx denotes the soliton power. Therefore the soliton poweras a function of soliton eigenvalue must be examined toanalyze the stability characteristics of the self-localizedsolutions of the NQHO. With this aim, we plot the soli-ton power as a function of soliton eigenvalue in Fig. 3 forthe single humped self-localized solution of the NQHOusing parameters of p = 30 , α = 1 , σ = 1. We investi- -20 -15 -10 -5 0 5 10 15 20 x =0.5=1.0=1.5=2.0=2.5 Increasing
FIG. 2: Self-localized single solitons for different nonlinearitystrength, σ . gate the stability characteristics of such solitons withinthe soliton eigenvalue interval of µ = 0 −
10 15 20 25 30 35 40 45 50050001000015000
P( )
11 12 13 14 15 16 17 182000400060008000100001200014000
P( )
P( )
FIG. 3: Self-localized single soliton power as a function ofsoliton eigenvalue, µ . The figure clearly indicates that the Vakhitov andKolokolov slope condition, dP ( µ ) /dµ <
0, is satisfiedpiecewise. This results suggests that self-localized soli-tons of the NQHO may be stable for some ranges ofthe soliton eigenvalue, µ , such as µ ≈ [10 . − .
8] forthe parameters considered. Being a necessary condi-tion for the soliton stability, the Vakhitov and Kolokolovslope condition is not a sufficient one. The secondcondition for the soliton stability is the spectral condi-tion. The spectral conditions states that the operator L + = − ∆ + V − N ( η ) − η N ′ ( η )] − µ for the NQHOproblem that we analyze [42, 43], should have at mostone eigenvalue which should be nonzero [42]. In here, ∆shows the diffraction term, V is the trapping-well poten-tial and N ( η ) is the nonlinear term given above. Spectralcondition can be analyzed analytically or numerically.For various nonlinear models studied in the literature,a numerical approach is the more commonly used one.With this aim, we investigate the temporal stabil-ity of self-localized solutions of the NQHO model equa-tion given by Eq.(6) using a split-step Fourier method(SSFM). This SSFM is recently proposed by us [13] and itis validated using the analytical solutions of the some lim- iting cases of the NQHO model equation given by Eq.(6).It is also tested against a 4 th order Runge-Kutta inte-grator. The SSFM, splits the governing NQHO modelequation into linear and nonlinear parts. As a possiblesplitting, the nonlinear part can be written as iψ t = − ( − αx + σ | ψ | ) ψ (14)which can be exactly solved. This integrations gives˜ ψ ( x, t + ∆ t ) = e i ( − αx + σ | ψ | )∆ t ψ (15)In here, ψ = ψ ( x, t ) denotes the initial condition. ∆ t indicates the time step, which is selected as ∆ t = 5 × − throughout this study, which does not cause stabilityproblems. The remaining linear part of the NQHO modelequation is iψ t = − ψ xx (16)One can compute the linear part of the NQHO equationin periodic domain using spectral techniques. Using themost commonly utilized Fourier spectral technique, thelinear part can be evaluated as ψ ( x, t + ∆ t ) = F − h e − ik ∆ t F [ ˜ ψ ( x, t + ∆ t )] i (17)In here k is the wavenumber parameter [25]. InsertingEq.(15) into Eq.(17), the complete form of the SSFMiteration formula for the numerical solution of the NQHOmodel equation can be derived as ψ ( x, t + ∆ t ) = F − h e − ik ∆ t F [ e i ( − αx + σ | ψ | )∆ t ψ ] i (18)Starting from the initial conditions, we perform the in-tegration of the NQHO model equation using this pro-cedure. In order to study the temporal stabilities of theself-localized solutions of the NQHO model equations,the initial conditions are taken as the self-localized so-lutions found by the SRM, such as the ones depicted inFig. 1 and Fig. 2. Using the normalized self-localized soli-ton obtained by SRM for p = 30 , α = 1 , σ = 1 , µ = 10 . t = 500 is given.In our simulations we observe the pulsating recurrencetype behavior between these two profiles, that is theseforms are interchanging to each other gradually duringtime stepping in a recursive way.Next, we turn our attention to dual humped self-localized solitons of the NQHO model equation. Using N = 1024 spectral components as before, selecting thecomputation parameters are p = 150 , σ = 1 , µ = 10 . t P ( t ) FIG. 4: Self-localized single soliton power as a function oftime, t . -20 -15 -10 -5 0 5 10 15 20 x | | t=0t=500 FIG. 5: Self-localized single soliton at two different times, t = 0 and t = 500. -20 -15 -10 -5 0 5 10 15 20 x =0=0.2=0.4=0.6=0.8=1.0=1.2 -11 -10.5 -10 -9.5 -9 x Increasing Increasing
FIG. 6: Self-localized dual solitons for different trapping wellpotential strength, α . of β to be less than 10 − as before, we depict thedual humped self-localizes solutions of the NQHO modelequation in Fig. 6 for various values of α . The initialcondition for the SRM is selected as exp ( − ( x − x ) ) +exp ( − ( x − x ) ) for which the locations of the Gaussiansare selected as − x = x = 10.Setting α = 1 and keeping the other parameters as be-fore, the dual humped self-localized solitons are obtainedby the SRM for various values of the nonlinearity coeffi- -20 -15 -10 -5 0 5 10 15 20 x =0.5=1.0=1.5=2.0=2.5 Increasing
FIG. 7: Self-localized dual solitons for different nonlinearitystrength, σ . cient, σ , and they are depicted in Fig. 7. As Fig. 6 andFig. 7 confirms, the dual humped self-localized solutionsof the NQHO model equation to also exist. As in thecase of single humped solitons, the dual humped solitonsare also asymmetric about the vertical axis, similar tothe soliton profiles given in [12]. P( )
P( )
FIG. 8: Self-localized dual soliton power as a function of soli-ton eigenvalue, µ . In order to check the stability characteristics of dualhumped self-localized solitons, we depict soliton power asa function of soliton eigenvalue, µ , in Fig. 8. The param-eters of computation are selected as p = 150 , α = 1 , σ = 1and the soliton eigenvalue interval of µ = 0 −
50 isscanned. As indicated in Fig. 8, the Vakhitov andKolokolov slope condition is only satisfied in a small in-terval of µ ≈ . − .
25, thus stable solitons can be foundin this range for the parameters considered.In order to check the temporal stability of the dualhumped self-localized soliton we again use the SSFMsummarized above. Starting the time stepping using thenormalized dual humped self-localized soliton obtainedfor p = 150 , α = 1 , σ = 1 , µ = 1, as the initial condi-tion, the soliton power as a function of time is depictedin Fig. 9, and the dual humped soliton at two differenttimes of t = 0 and t = 200 is depicted in Fig. 10. Simi-lar to the single humped self-localized solitons, the dualhumped soliton also exhibits a pulsating behavior, the t P ( t ) FIG. 9: Self-localized dual soliton power as a function of time, t . -20 -15 -10 -5 0 5 10 15 20 x | | t=0t=200 FIG. 10: Self-localized dual soliton at two different times, t = 0 and t = 200. form is gradually and recursively interchanging from theform given at t = 0 to the one given at t = 200. -50 -40 -30 -20 -10 0 10 20 30 40 50 x =0=0.2=0.4=0.6=0.8=1.0=1.2 -29 -28 -27 -26 -25 -24 -23 -22 -21 x Increasing Increasing Increasing Increasing
FIG. 11: Self-localized triple solitons for different trappingwell potential strength, α . Lastly, we investigate the properties of triple humpedself-localized solutions of the NQHO model equation ob-tained by the SRM. For the SRM to converge to such soli-ton solutions, parameters needed to be relaxed and areselected as p = 1000 , α = 1 , σ = 1 , µ = 1. The conver-gence criteria for the SRM is also relaxed to be the nor-malized change of β being less than 10 − . The initial con- dition used in SRM to obtain the triple humped solitonsis selected as superimposed three Gaussians in the formof exp ( − ( x − x ) )+exp ( − ( x − x ) )+exp ( − ( x − x ) )where x = − , x = 0 , x = 10. The triple humpedself-localized solutions of the NQHO model equation aredepicted in Fig. 11 and Fig. 12, for various values of α and σ , respectively. -50 -40 -30 -20 -10 0 10 20 30 40 50 x =0.5=1.0=1.5=2.0=2.5 -0.5 0 0.5 x Increasing
FIG. 12: Self-localized triple solitons for different nonlinearitystrength, σ . P( )
FIG. 13: Self-localized dual soliton power as a function ofsoliton eigenvalue, µ . Contrary to results presented for single and dualhumped self-localized solitons of the NQHO model equa-tion earlier in this paper, the soliton power vs solitoneigenvalue graph is depicted in Fig. 13 indicates thatVakhitov-Kolokolov slope condition is not satisfied forthe triple humped self-localized solutions of the NQHOmodel equation. We should note that scanned range ofthe soliton eigenvalue is µ = 0 −
50, however only somepart of it is depicted for illustrative purposes. Since thegraph increases monotonically, it is possible to concludethat triple humped self-localized solitons of the NQHOmodel equation are unstable, at least for the parameterranges considered in this study.Findings and the computational approach based onthe SRM and the SSFM we have proposed in this paperfor the investigations of the self-localized solitons of theNQHO model equation can be used to analyze nonlinearquantum harmonic oscillations. Additionally atomic vi-bration resonances, bonding and bond breaking strengthof molecules under the effect of nonlinear electric andmagnetic fields and trapping well potentials having vari-able strengths can also be studied within this frame. Theprocedure proposed in this paper and our findings canalso find many possible applications in the macroscopiclevel, such as Bose-Einstein condensation.
V. CONCLUSION AND FUTURE WORK
In this paper we analyzed the existences and proper-ties of the self-localized solitons of a nonlinear quantumharmonic oscillator model. In order to do so, we im-plemented a numerical approach based on the spectralrenormalization scheme and showed that single, dual andtriple self-localized soliton solutions of NQHO do exists.We compared our findings with the results existing inthe literature. We also studied stability characteristics of such soliton solutions of the nonlinear quantum har-monic oscillator model using a split-step Fourier scheme.We showed that single and dual humped self-localized so-lutions of nonlinear quantum harmonic oscillator modelequation are stable within some subintervals of the soli-ton eigenvalue, which may be considered as a piecewisecontinuous spectrum. Time stepping analysis showedthat single and dual humped self-localized solitons of thenonlinear quantum harmonic oscillator model equationare pulsating in time. However, the triple humped self-localized soliton solution of the nonlinear quantum har-monic oscillator turned out to be unstable since it vio-lates the necessary Vakhitov-Kolokolov slope condition.Our findings can be used to analyze nonlinear quantumharmonic oscillations under varying molecular bond stiff-ness and/or nonlinear field effects. Some other similarphenomena observed at the macroscopic level, such asin the Bose-Einstein condensation, can also be investi-gated using our results and the framework proposed inthis paper. [1] E. Schrdinger, Annalen d. Physik (4), 79, 489 (1926).[2] D. J. Griffiths,
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