Harmonic salınıcının optimal kontrolü

Abstract

Bu çalışmada, Kuvantum Harmonik Salıma ile modellenmiş moleküler devi nimin dış dipol etkiler altındaki optimal kontrollenmiş devinim denklemlerinin çözümü üzerinde durulmuştur. Bu amaçla, öncelikle, titreşmekte olan cisimleri modellemede kullanılan Kuvantum Harmonik Salmıcı'nın tanımı, özellikleri, dış etkiler altında bulun mayan korunumlu bir sistem için gerekli olan diferansiyel denklemler ve bu denklemlerin çözümü aşamasında kullanılan beklenen değer hesaplamaları ele alınmıştır. Ardından dış dipol etkiler altındaki harmonik salmıcı'nın optimal kontrollenmiş devinim denklemleri ortaya konmuştur. Kuvantum Harmonik Salmıcı'nın dış dipol etkiler altındaki devinim denk lemleri, dalga fonksiyonu olarak adlandırılan xj)(x,t) fonksiyonunun başlangıç biçimi olan /(x)'in bilinmesi durumunda, analitik olarak çözülebilmektedir. Dalga fonksiyonunun bilinmesi ise çeşitli operatörlerin beklenen değerlerinin saptanabilmesi anlamına gelmektedir. Moleküler devinimin optimal kontrolü ise belli ve genellikle gözlenebilen nitelikteki fiziksel özelliklerin, ki bunlar ope ratörler ile temsil edilebilmektedir, devinimin verilen bir T anındaki değerleri nin arzulanan değerlere olabildiğince yakın olabilmesi için alan genlik fonksiyo nunun, yani £(£)'nin, nasıl bir yapıya sahip olması gerektiği ile ilgilenir. Burada izdüşüm, konum ve momentum operatörlerine karşı gelen optimal kontrollen miş denklemlerin açık yapıları ortaya konmuştur. Sonrasında sayısal yöntemler kullanılarak bilgi işlem ortamında bu denklemlerin çözümü yoluna gidilmiştir. Yapılan çalışmalar dış dipol etkiler altındaki Kuvantum Harmonik Salmı cı'nın optimal kontrollenmiş denklemlerinin, ilgilenilen operatörler ve merte belerine bağlı olarak, birden fazla çözümü olabileceği savını doğrulamıştır.

The optimal control of the molecular motion which may be called the alchemy of today is an interesting field for scientists since years. This interest stems from a long-standing desire to actively manipulate dynamical events at the atomic an molecular scale. In general, control is envisoned to be achived by the application of a tailored external optical field. Given that the phenomena are inherently quantum mechanical, the tailored field achive control by deli cately manipulating constructive and destructive quantum wave interferences. In this work, under the assumption of a given initial structure of the molec ular motion modelled by a quantum harmonic oscillator for, at a given time of the oscilation the values of some definite and mostly observable physical properties all of which can be represented by operators, having values as close as possible to their expected, what has to be the structure of the function of field amplitude, is investigated. For this purpose an object functional is con structed and equating its first variation to zero is employed. Thus equations of the optimally controlled oscillations are obtained. Fundamental Equations The equations of the optimally controlled oscillations of an harmonic oscil lator under external dipol effects are as follows,.drj>{x,t) _{ Id2 1 2 dt "1 2dx* + 2 +fi£®X(^Xlt)' ^(*.°) = /(*) (la, 6) x,t) (Id2 1 2 ", * 1 w n - Wp(t) [jdx i>*(x,t)0'tl>(x,t)\ 0VOM), VI A(z, T) = irfdrpÇT) (2a, b) /oo dx^(x,T)ö^(x,T) (36) -oo where î/j(x, <) is the wave function, main variable of the quantum mechanics, A(x,<) is the costate function, ft is a known parameter named as the coefficient of dipol polarizability, E(t) is the unknown amplitude function of the external electromagnetic field and to find this is the main objective. The quantity Wp(t) in (2a) is a weight function and it characterizes the change in time of the weight of the term added to the object functional for the suppression of the value of the operator Ö'. Ö' is an operator that its structure is assumed to be known and the expected value of it wanted to be pulled to zero as close as possible. On the other hand Ö is an operator that is wanted to be as close as possible to O'. The parameter rj in (2b) and (3b) is a quantity that gives the measure of the closeness of the operators Ö and Ö'. Lastly, Ws(t) is the weight function added to the object functional for keeping the field amplitude finite. The parameter s, appeared in the last equation above can only take values 0 or 1 and symbolizes the wish to reach the object exactly or approximately. Equations (3a,b) in principal are in functional nature. That is, they have the property to produce the integral or the differential equations required for the field amplitude function, when equations (la,b) and (2a,b) are solved. For this purpose consider the following quantity. oo */ *(*)=»/ dx \*(x,t)Qxj>(x,t) -or l(f°°dx \*(x,t)Q^(x,t) + f°° dx A(a:,t)QV0M)) (4) Differentiating both sides with respect to t and considering that Q is indepen dent of t, among the terms obtained by differentiation under integral, deriva tives with respect to time of \(x,t) and il>(x,t) and their complex conjugates appear. These derivatives can be replaced by the terms having the hamiltonian of the harmonic oscillator, obtained from (la), (2a) and their conjugates. Hence it is possible to reach an ordinary differential equations for z(t). This differen tial equation will have new quantities. Operator Q is among these quantities vu and instead of it the use of the operators x, - i-ğ^ and X give rise the use of the following functions of time, zi(t) =&( I dx X*(x,t)x^{x,t)j = -!f dx \*(x,t)xij>(x,t) + J dx \(z,t)xi/f*(xtt)) (5) z2(t)=&([ dx\%x,t)l-i-^\i>(x,t)) = \ (J^dx A*(*,i){-^} *(*,*)) + \ {£ J* A(x,0{»^}^(*,0). (6) When in the ordinary differential equation obtained for the z(t), Z\{t) is used instead of z(t) and in the new quantities involved x used instead of Q, then using the obtained equations and considering the definition of zi(t), following differential equation is reached. dzi(t) dt = z2(t) + Wp(t) {I 1/ dx $*{x,t)0'il>(x,t)\ x oo oo dx lj)*(x,İ) (x6' - Ö'ar) i/>(x,t) (7) Similarly, employing Z2(t) instead of z(t), considering the definition of z\(i) and using the equations obtained for the case of Q = J, a differential equation for z2(t) is obtained as dz2{t) dt = -*i(*) + Wp(t) {/: dxi>*{x,t)0'x}){ x,t)\ {/: dx ij)*(x,t) 2\dx dx) ( x,t)\ (8) Integrals appeared in equations (7) and (8) contain wave function only. The important point here is that the wave function depends the unknown field amplitude function £(t). This dependence does not appeares simply on the function S(t) but instead on its integrals given below, zs(t) = f.i I dr sin(i - t)£(t) Jo Z4(t) = n I dr cos(t - t)S(t). Jo (9) (10) Vlll For the soluability of equations (7) and (8), two more equations has to be obtained. For this purpose, when both sides of equations (9) and (10) are differentiated with respect to t, the following equations are obtained. dz3{t) dt dz4(t) dt = z4(t) = -z3(t) + fi£(t) (11) (12) The term £(t) seen in equation (12) however can be written in terms of Zi(t). For this, using equations (3a) £{t) = WJk»® (13) can be obtained. Combining all the equations obtained above, the following system of equa tions are found as the equations of motion of an harmonic oscilator under optimal control, dzi(t) dt dzj(t) dt dz3(t) dt dz4(t) = z2(t) + F1{z3{t),z4{t),t) = -z1{t)+F2(z3(t),z4(t),t) = zA{t) = -*3(*) + 7îr7rv*ı(*) dt ~°w ' w£(ty where Fiiz^t), z4(t),t) and ^2(23 (*)> z4(t),t) are defined as ^2(zz(t\z4(t),t) = Wp(t) J f^ dx 1>*(x,t)6'rl>(x,t) x (14a) (146) (14c) (14d) fi(zz(t),z4(t),t) = Wp(t)< f dx V>*(z,*K?V0M)} x if dx i>*{x,t) I l- (x& - O'x) I xp(x,t)\ (15a) {/: dx il>*(x,t) 2 V ox dx. ( x,t)\ (156) IX Equations (14a,b,c,d) involve four unknown functions and first derivatives and this means that four unknown constants will appear in the solutions. De termination of these constants requires four boundary conditions. Two of these conditions can readily be obtained; in equations (9) and (10) taking t = 0, re sults 23(0) = 0 and 24(0) = 0 are reached. Definitions of the functions z\ (t) and Z2CO show that their values at t = 0 are not easily obtainable. But values at the instant t = T i.e at the last moment of the optimally controlled motion, are more easily computable. Hence, putting t = T in equations (5), (8) and using (2a), equations below are arrived at, z,(T) = Vgi(z3(T),z4(T),T) (16a) z2(T) = rig2(z3(T),z4(T),T) (166) where functions G\(zz(t), z4(t),i) and Q\ (z$ (t),z4 (£),£) are defined as follows, Qx{zz{T),z4{T),T) = r dx f(x,T)U [xö - Ox) X V>(x,T) (17a) G2{z*(T)MT)>T) = J^dx p{*,T) {^ (J^O - O-j^j } iftz,T) (17b) These equations involve an unknown parameter r). Equation (3b) can be uti lized for the determination of this parameter. For this purpose, with the defi nition of, H(z3(T),z4(T),T) = r dx ^(x,T)Ö^(x,T) (18) J -00 corresponding to equation (3b), the following equation is obtained, H(z3{T),z4{T),T)-Ö-sr} = 0, s = 0,1 (19) Solution of the Optimal-Control Equations Solution of the differential equations that produce the field amplitude func tion requires that the function %l)(x,t) called as wave function is known. On the other hand knowledge of the wave function makes it possible to compute the expected values of various operators. In this work it is assumed that if>(x,t) = -A-e-^-'nW*-72^ (20) is given. For the. operator Ö' appeared under the integrals in the right hand side of equations of motion, the operator I - Pi is employed, where I is the unit operator and the projection operator P\ is given as A/(«) = |^°° ¥>i(v)/(v)<*v} ¥>i(*). (21) Again, in the boundary conditions operator P\ is used instead of the operator Ö. Equations of motion obtained for the operators mentioned above display non-linear structures. Consequently it is prefered to linearize these equations and then to solve them numerically. For this purpose a program with the required subroutines is developed in the C programming language. The extensive work done showed that, taking s = 0, the expected value of the operator Ö can not be greater than 0.35 that is, solution of the equations of the optimally controlled motion is not exist. It is also observed that there are more than one solution for the values less than 0.35. This result confirms the claims stating that the equations of an optimally controlled motion has more than one solution.