Asymmetric double well system as effective model for the kicked one
aa r X i v : . [ n li n . C D ] J u l Asymmetric double well system as effective model for thekicked one
V.I. Kuvshinov ∗ , A.V. Kuzmin † and V.A. Piatrou ‡ Joint Institute for Power and Nuclear Research,Krasina str. 99, Minsk, 220109, Belarus
Abstract
Effective Hamiltonian for the kicked double well system was derived using the Campbell-Baker-Hausdorff expansion formula. Asymmetric model for the kicked system was con-structed. Analytical description of the quasienergy levels splittings for the low layingdoublets was given in the framework of the model. Numerical calculations confirm appli-cability of the proposed effective asymmetric approach for the double well system withthe kick-type perturbation.
The connection between the semiclassical properties of chaotic systems and purely quantumprocesses such as tunneling is a reach rapidly developing field of research nowadays. Our insightin some novel phenomena in this field was extended in the last decades. The most intriguingamong them are the chaos assisted tunneling (CAT) and the closely related coherent destructionof tunneling (CDT).The first one is an enhancement of tunneling in perturbed low-dimensional systems atrelatively high external field strengths and high driving frequencies (in order the singlet-doubletcrossings to occur) [1, 2, 3]. This phenomenon takes place when levels of the regular doubletundergo an avoided crossing with the chaotic state [4, 5]. At the semiclassical level of descriptionone considers tunneling between KAM-tori embedded into the ”chaotic sea”. The region ofchaotic motion affects tunneling rate because compared to direct tunneling between tori it iseasier for the system to penetrate primarily into the chaotic region, to travel then along someclassically allowed path and finally to tunnel onto another KAM-torus [6, 7].CDT phenomenon is a suppression of tunneling when values of amplitude and frequency ofdriving force belong to some one-dimensional manifold in the perturbation parameters’ space [8].This phenomenon occurs due to the exact crossing of two states with different symmetries fromthe tunneling doublet. In this parameter region tunneling time diverges which means the totallocalization of quantum state on the initial torus. ∗ E-mail:[email protected] † E-mail:[email protected] ‡ E-mail:[email protected] n to kick n + 1 andgreatly facilitates theoretical analysis.The main idea of this paper is to study the possibility to construct an effective autonomousmodel for the non-autonomous perturbed system using Campbell-Baker-Hausdorff expansionformula and to test it in numerical calculations of the quasienergy spectrum. Both CAT andCDT are connected with the behavior of the quasienergy spectrum (avoided or exact crossingof the levels). Thus the development of the methods for calculation of this spectrum is impor-tant for extending one’s knowledge in CAT and CDT. We regard the kick-type perturbationwhich is proportional to x . This perturbation, in contrast to perturbation proportional to x ,destroys the spacial symmetry in the system which is important for presence of the CAT phe-nomenon [28]. The main role in quantum dynamics of our system is played by the classicalasymmetry. There is no chaos induced processes in it but in our future work we will use thisapproach to system with CAT and CDT.In this paper we propose the effective model for the kicked double well system which givesa possibility to simplify the numerical calculations of the quasienergy spectrum and allows todetermine both analytically and numerically the quasienergy splitting dependence on both theperturbation strength and frequency. Now lets construct the effective Hamiltonian for the double well system with the perturbationof the kick-type. Hamiltonian of the particle in the double-well potential can be written in thefollowing form: H = p m + a x − a x , (1)where m - mass of the particle, a , a - parameters of the potential.2e consider the perturbation of the kick-type which is proportional to xV per = ǫ x + ∞ X n = −∞ δ ( t − nT ) , (2)where ǫ - perturbation strength, T - perturbation period, t - time.Full Hamiltonian of the system is the following: H = H + V per . (3)Now we will construct an effective Hamiltonian for the system under investigation using thefollowing definition: exp ( − iH eff T ) = exp ( − iǫx ) exp ( − iH T ) , (4)where RHS is a one-period evolution operator. We restrict our consideration by sufficientlysmall values of both the perturbation strength and period. Using the Campbell-Baker-Hausdorffexpansion formula for the kicked dynamical systems we can rewrite the last expression (4) inthe following way [29]: H eff = H + ǫ ν π Z ds g h exp ( − iǫs ˆ x ) exp ( − i ˆ H T ) i x, (5)where g ( z ) = ln zz − ∞ X n =0 ( − n n + 1 ( z − n and ν = 2 πT . With the definition of g ( z ) formula (5) can be expanded in the following form: H eff = H + ǫ ν π ∞ X n =0 n + 1 n X k =0 ( − k n ! k !( n − k )! Z ds h exp ( − iǫs ˆ x ) exp ( − i ˆ H T ) i k x, (6)where the expression under the integral is a map in the power k which for sufficiently smallvalues of both the perturbation strength and period can be rewritten as follows: h exp ( − iǫs ˆ x ) exp ( − i ˆ H T ) i k x = x − kTm p + O ( ǫ , ǫT, T ) . (7)Substituting (7) in expression (6) one obtains the following form of the effective Hamiltonian: H eff = p m + a x − a x + ǫ ν π x + ǫ m p + O ( ǫ , ǫT, T ) . The fouth term in RHS of the last expression has the same order in perturbation parameters asfirst three main terms. The fifth term is proportional to small parameter, namely perturbationstrength. We restrict our consideration by terms without small parameters and neglect allterms with order higher than zero in the perturbations parameters. As a result we have thefollowing effective Hamiltonian for the kicked double well system: H eff = p m + a x − a x + ǫ ν π x. (8)This is the Hamiltonian for the asymmetric double well potential without perturbation. Incontrast to the kicked system it is autonomous. In the next section we will consider theproperties for this system and construct the effective asymmetric model for the kicked system.In section 4 the correspondence between kicked and asymmetric effective double well systemswill be tested numerically. 3 Effective asymmetric model for the kicked system
Hamiltonian of the classical particle in the asymmetric double-well potential is the following: H as = p m + a x − a x + σ r a a x, (9)where a and a are parameters of the potential, σ - asymmetric parameter. The asymmetricpotential with parameters m = 1 , a = 1 / , a = 1 / σ = 0 .
15 is shown in the figure 1(thick solid line). Eight lowest energy levels are shown in the same figure with thin solid lines.The form of the last term in the RHS of the Hamiltonian (9) is more handy due to specialchoice of the asymmetric parameter σ . Parameter σ for this form of the Hamiltonian is equalto shift between bottoms of the wells (dashed lines in the figure 1). -2-1.5-1-0.5 0 0.5 1 1.5 -6 -4 -2 0 2 4 6 P o t en t i a l , V Coordinate, x σ ∆ ∆ -2-1.5-1-0.5 0 0.5 1 1.5 -6 -4 -2 0 2 4 6 P o t en t i a l , V Coordinate, x σ ∆ ∆ -2-1.5-1-0.5 0 0.5 1 1.5 -6 -4 -2 0 2 4 6 P o t en t i a l , V Coordinate, x σ ∆ ∆ -2-1.5-1-0.5 0 0.5 1 1.5 -6 -4 -2 0 2 4 6 P o t en t i a l , V Coordinate, x σ ∆ ∆ -2-1.5-1-0.5 0 0.5 1 1.5 -6 -4 -2 0 2 4 6 P o t en t i a l , V Coordinate, x σ ∆ ∆ -2-1.5-1-0.5 0 0.5 1 1.5 -6 -4 -2 0 2 4 6 P o t en t i a l , V Coordinate, x σ ∆ ∆ -2-1.5-1-0.5 0 0.5 1 1.5 -6 -4 -2 0 2 4 6 P o t en t i a l , V Coordinate, x σ ∆ ∆ -2-1.5-1-0.5 0 0.5 1 1.5 -6 -4 -2 0 2 4 6 P o t en t i a l , V Coordinate, x σ ∆ ∆ -2-1.5-1-0.5 0 0.5 1 1.5 -6 -4 -2 0 2 4 6 P o t en t i a l , V Coordinate, x σ ∆ ∆ -2-1.5-1-0.5 0 0.5 1 1.5 -6 -4 -2 0 2 4 6 P o t en t i a l , V Coordinate, x σ ∆ ∆ -2-1.5-1-0.5 0 0.5 1 1.5 -6 -4 -2 0 2 4 6 P o t en t i a l , V Coordinate, x σ ∆ ∆ Figure 1: Asymmetric double well potential (thick solid line) with eight lowest energy levels(thin solid lines). Minima shift ( σ = 0 .
15) is shown by dashed lines. The model parameters are m = 1 , a = 1 / , a = 1 / σ eff = r a a ǫ νπ . (10)Now we can give definition of the proposed effective model: asymmetric double well systemwith the effective parameter σ eff defining by expression (10) is an effective model for the kickedsystem with the perturbation parameters ǫ and ν . The parameters a , a and m are the samefor both systems.The definition of the effective asymmetric parameter (10) shows that perturbation strengthand frequency appears in it as a product ǫν . This is the first advantage of the effective approach.We have effectively only one asymmetric parameter ( σ ) instead of two perturbation parameters( ǫ and ν ). The second advantage is the more simple way of the numerical calculations whichwill be discussed in the next section. 4he third advantage of the proposed approach is that splitting for the doublets layingbelow the barrier hump in the asymmetric double well can be described analytically. Theasymmetric model can be considered as a pair of shifted harmonic oscillators. The shift isequal to asymmetric parameter σ . It is obvious that in this constructed system non-degenerateenergy doublets have splitting ∆ = σ . In asymmetric system splitting between levels remainsclose to σ for all doublets lying below the barrier top as for case shown in the figure 1. Thiscorrespondence can be used in order to give analytical description of the quasienergy spectrumin the kicked system. Using expression (10) we can write down the formula for low layingquasienergy doublets’ splittings of the time-dependent system (3)∆ = r a a ǫ νπ . (11)It worth to mention the linear dependence of the levels splitting on both the perturbationstrength and frequency. The applicability of this analytical description will be tested in thenumerical calculations in the next section. For the computational purposes it is convenient to choose the eigenvectors of harmonic oscil-lator as basis vectors. In this representation matrix elements of the Hamiltonian (1) and theperturbation (2) are real and symmetric. They have the following forms ( n ≥ m ): H m n = δ m n (cid:20) ~ ω (cid:18) n + 12 (cid:19) + g (cid:18) g a (2 m + 2 m + 1) − a ′ (2 m + 1) (cid:19)(cid:21) + δ m +2 n g g a (2 m + 3) − a ′ ) p ( m + 1)( m + 2)+ δ m +4 n a g p ( m + 1)( m + 2)( m + 3)( m + 4) ,x m n = δ m +1 n r g √ m + 1 , where g = ~ /mω and a ′ = a + m ω / ~ is Planck constant which we put equal to 1, ω -frequency of harmonic oscillator which is arbitrary, and so may be adjusted to optimize thecomputation. We use the value ω = 0 . m = 1 , a = 1 / , a = 1 /
4. Thematrix size is chosen to be equal to 200 × η k ) in the kicked double well system directly we cal-culate eigenvalues ( λ k ) of the one-period evolution operator e − iHT e − iV and express quasienergylevels through the definition η k = i ln λ k /T . Then we get ten levels with the lowest one-periodaverage energy which is calculated using the formula h v i | H + V /T | v i i ( | v i i are the eigenvectorsof the one-period evolution operator). The dependence of the quasienergies of these ten levelson the strength of the perturbation is shown in the figure 2. Quasienergies of the two doubletswith the minimal average energy (thick lines in the figure 2) has a linear dependence on thestrength of the perturbation in the considered parameter region. They are strongly influencedby the perturbation while some of the quasienergy states are not.5 Q ua s i ene r g y , η Perturbation strength, ε -0.2-0.1 0 0.1 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 Q ua s i ene r g y , η Perturbation strength, ε -0.2-0.1 0 0.1 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 Q ua s i ene r g y , η Perturbation strength, ε -0.2-0.1 0 0.1 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 Q ua s i ene r g y , η Perturbation strength, ε -0.2-0.1 0 0.1 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 Q ua s i ene r g y , η Perturbation strength, ε -0.2-0.1 0 0.1 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 Q ua s i ene r g y , η Perturbation strength, ε -0.2-0.1 0 0.1 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 Q ua s i ene r g y , η Perturbation strength, ε Figure 2: Quasienergy spectrum for the ten lowest average energy levels. All levels are numberedin order of the average energy values. Solid lines - quasienergy levels for the kicked system.Thick lines - two doublets with the minimal average energy. Empty squares - shifted energylevels of the asymmetric model. The model parameters are m = 1 , a = 1 / , a = 1 / ν = 0 . Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation strength, ε (a) ν = 0.4 ν = 0.6 ν = 0.8 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation strength, ε (a) ν = 0.4 ν = 0.6 ν = 0.8 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation strength, ε (a) ν = 0.4 ν = 0.6 ν = 0.8 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation strength, ε (a) ν = 0.4 ν = 0.6 ν = 0.8 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation strength, ε (a) ν = 0.4 ν = 0.6 ν = 0.8 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation strength, ε (a) ν = 0.4 ν = 0.6 ν = 0.8 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation strength, ε (a) ν = 0.4 ν = 0.6 ν = 0.8 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation strength, ε (a) ν = 0.4 ν = 0.6 ν = 0.8 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation strength, ε (a) ν = 0.4 ν = 0.6 ν = 0.8 0 0.005 0.01 0.015 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation frequency, ν (b) ε = 0.01 ε = 0.02 0 0.005 0.01 0.015 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation frequency, ν (b) ε = 0.01 ε = 0.02 0 0.005 0.01 0.015 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation frequency, ν (b) ε = 0.01 ε = 0.02 0 0.005 0.01 0.015 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation frequency, ν (b) ε = 0.01 ε = 0.02 0 0.005 0.01 0.015 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation frequency, ν (b) ε = 0.01 ε = 0.02 0 0.005 0.01 0.015 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Q ua s i ene r g y s p li tt i ng , ∆ η Perturbation frequency, ν (b) ε = 0.01 ε = 0.02 Figure 3: Quasienergy splitting as a function of the strength ( a ) and frequency ( b ) of theperturbation. Filled circles - results of the numerical calculations for the kicked system. Emptysquares - shifted energies for the asymmetric model. The model parameters are m = 1 ,a = 1 / , a = 1 / − ν , ν ) in order to compare results with ones in the kicked system. The result ofcalculations on the base of this procedure is shown in the figure 2 by empty squares for twolowest doublets. Comparing results of direct and model calculations we make the conclusionthat the levels of the doublets laying below the potential hump are correctly described by theeffective model.Performed numerical calculations for the kicked and the effective system give the depen-6ence of the ground quasienergy splitting both on the strength (fig.3(a)) and the frequency(fig.3(b)) of the perturbation. Filled circles in figures correspond to kicked double well system,empty squares to numerical results obtained in the framework of the effective model. There isgood agreement between splitting’s dependencies for these two systems. They are linear as itwas predicted by expression (11). It should be mentioned that all dependencies for the effectivemodel was obtained from one series of the numerical calculations. We fix model parameters a , a , m and calculate numerically one set of numerical points for the dependence on the asym-metric parameter. In kicked system we should to perform one series of numerical calculationsfor every dependency. This is the first advantage which was discussed in the previous section.The second advantage which we should discuss after description of the used numericalmethods is a more simple algorithm of the calculations. In the kicked system we should tocalculate eigenvalues of the matrix exponents. This is more difficult task than in asymmetricmodel where we calculate the eigenvalues of the system Hamiltonian.Analytical result (11) which was put forward as third advantage of the method is plotted inthe figures 3 (a) and 3 (b) by straight solid lines. Numerical points lie close to these lines. Theagreement between numerical calculations and analytical expression (11) is good (near 6%) inthe parametric region considered. Effective Hamiltonian for the kicked double well system was obtained using the Campbell-Baker-Hausdorff expansion formula. Effective autonomous asymmetric model for this systemwas constructed. This model is more convenient in numerical calculations than kicked one.Results of numerical calculations show that model correctly describes quasienergy spectrum ofthe kicked system for low laying levels.The analytical formula for the ground quasienergy splitting dependence on both the per-turbation strength and frequency was obtained in the framework of the effective asymmetricmodel. This formula predicts linear dependence of the ground quasienergy splitting on theseparameters for the small perturbation strength and period values. Numerical results for thequasienergy splitting as a function of the perturbation frequency and strength demonstratelinear dependence as well. They are in a good agreement with the formula (11). Proposedapproach will be used in future for investigation of the CAT and CDT phenomena.
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