arXiv:physics/9707012v4 [math-ph] 4 May 1998 2.Ntc httephysical the that Notice [2]. and uesmercprnr.Tefe apn etneuto rea equation Newton damping s free the we The construct note, and this partners. mechanics In classical supersymmetric in [1]. case decades two damping last free the critical in areas research many to tively. rtcldmig n a bani h nedmigcs hrigo chirping a case underdamping frequency the Startin physical in taneous obtain mechanics. can one quantum damping, supersymmetric critical in construction ner’s pnigi unu ehnc oaptnilwihi eoadrelate fundame and c a zero damping has is critical which which of domain at case potential time quency the the a with in to Schr¨odinger-type equation start mechanics a we quantum Thus, in sponding [3]. Rosner to similar non to fact in corresponding regimes. symbols mathematical convenient only qiaeto on tt nteuulqatmmcais noth In mechanics. quantum equation usual Riccati the “fermionic” in the solve state we bound a of equivalent aete“hrig so h upyia)type (unphysical) the of is “chirping” the case a epti h olwn croigrfr ntetm domain time the in Schr¨odinger form following the in put be can rpdnnsiltn eaain,adoedmig(lwnonoscillatin critica (slow i.e., overdamping relaxation), and ation), (oscillating relaxation), underdamping nonoscillating of (rapid cases classical the or sn h ag hneo eedn variable dependent of change gauge the Using b a eatmnod ´sc el nvria eGaaaaa Corregid Guadalajara, de Universidad la F´ısica de de Departamento 2 1 y h ehiuso uesmercqatmmcaisgv e im new a gave mechanics quantum supersymmetric of techniques The Supersymmetry. - 11.30.Pb PACS Summary enwpoedwt h uesmercshm htw pl nama a in apply we that scheme supersymmetric the with proceed now We -al [email protected] E-mail: [email protected] E-mail: nttt eFıiad aUiesddd unjao poPsa E Postal Apdo Guanajuato, de Universidad la F´ısica de de Instituto ′′ uesmercprnrcipn fNwoinfe damping free Newtonian of chirping partner Supersymmetric ω − o ω r h nedme n vrapdfeunyprmtr,r parameters, frequency overdamped and underdamped the are d 2 y − ,where 0, = ω − u 2 ω ecnettecasclfe apn ae ymaso Ros- of means by cases damping free classical the connect We - . .. igeoclaigrlxto oeta ecnie sthe as consider we that mode relaxation oscillating single a i.e., , u 2 .Rosu H. = ω d 2 ω 40 udljr,Jl M´exico Jal, Guadalajara, 44100 < d 2 a y ( = 75 eo,Go M´exico Le´on, Gto, 37150 1 ′′ m 0, ω .L Romero L. J. , − W ω 2 d dt γ/ ( ω 2 d 2 h t 1 2 x 2 c 2 ) 2 rqece r eaie h oiieoe are ones positive the negative, are frequencies − m 2 ∝ + = γ m W ) γ 2 ω ω 1 dx − u ′ dt 2 d 2 2 1 sech + k/m − and 0 = + ω b m d 2 k 2 kx n .Socorro J. and ( hs n a ics separately, discuss can one Thus, . 0 = i ω y 0 = u x t ω 0 = ,weesi h overdamped the in whereas ), , = 2 ( . ω t y ) , o 2 exp( ∝ = ω ω − o 2 sec a d 2 2 2 γ m > 2 t physics/9707012 ( ,ti equation this ), ω ,respectively 0, corresponding o ds t atwt the with tart ,where ), ihthe with g damping l oscillating rwords, er r 500, ora instan- f tlfre- ntal relax- g -143, espec- petus orre- tto it nner (2) (3) (1) ω u i.e., 2 − ′ − 2 W1 W1 ωu =0 , (4) to find Witten’s superpotential W1(t) = −ωu tanh[ωut] and next go to the “bosonic” Riccati equation 2 ′ 2 − 2 W1 + W1 + ω1(t) ωu =0 , (5) 2 − 2 2 in order to get ω1(t)= 2ωusech [ωut]. Moreover, one can write the Schr¨odinger equation corresponding to the “bosonic” Riccati equation as follows − ′′ 2 − 2 y˜ + ω1(t)˜y = ωuy˜ , (6) with the localized solutiony ˜ ∝ ωusech(ωut). The physical picture is that of a chirping sech soliton profile containing a single oscillating relaxation mode − 2 self-trapped at ωu within the frequency pulse. One can employ the scheme recursively to get several oscillating relaxation modes embedded in the chirping frequency profile. Indeed, suppose we would like to introduce N oscillating 2 − 2 2 relaxation modes of the type ωn = n ωu, n =1, ...N in the sech chirp. Then, one has to solve the sequence of equations 2 − ′ 2 2 2 Wn Wn = ωn−1 + n ωu (7a)
2 ′ 2 2 2 Wn + Wn = ωn + n ωu (7b) inductively for n = 1...N [3]. The chirp frequency containing N underdamped 2 2 2 − 2 2 frequencies n ωu, n = 1...N is of the form ωN (t) = N(N + 1)ωusech (ωut). The relaxation modes can be written in a compact form as follows
† † † † N−n+1 y˜n(t; N) ≈ A (t; N)A (t; N − 1)A (t; N − 2)...A (t; N − n + 2)sech ωut , (8) † − d − i.e., by applying the first-order operators A (t; an) = dt anωu tanh(ωut), where an = N − n, onto the “ground state” underdamped mode. This − 2 scheme can be easily generalized to embedding frequencies of the type ωu,i = 2 (γi/2m) − k/m and moreover, to the construction of chirp profiles having a given continuous spectrum of relaxational modes but we shall not pursue this task here. On the other hand, in the case of overdamping the “fermionic” Riccati equa- tion 2 − ′ 2 W1 W1 + ωo = 0 (9) leads to the solution W1 = ωotan(ωot) and from the “bosonic” Riccati equation
2 ′ 2 2 W1 + W1 + ω1 + ωo =0 , (10)
2 2 2 one will find ω1(t)=2ωosec (ωot). Consequently, the Schr¨odinger equation − ′′ 2 2 y˜ + ω1(t)˜y = ωoy˜ (11)
has solutions of the typey ˜ ∝ ωosec(ωot), and therefore the approach leads to unphysical results. Referring again to the underdamped case, we also remark that an interesting, polar analysis of the chirp frequency profile can be performed by means of the θ change of variable t = ln(tan 2 ) leading to an associated Legendre equation in the spherical polar coordinate θ
d2y˜ dy˜ n2 + cot θ + N(N + 1) − y˜ =0 . (12) dθ2 dθ h sin2 θ i
2 Finally, we mention that there may be potential applications of supersym- metric approaches to chirping phenomena in many areas, such as semiconductor laser physics [4], the propagation of chirped optical solitons in fibers [5], and optimal control of quantum systems by chirped pulses [6].
Acknowledgment The work was supported in part by the CONACyT Projects No. 4868- E9406, No. 3898-E9608, and a “Scientific Summer” grant from the University of Guanajuato.
References
[1] For a recent review see, Cooper F., Khare A. and Sukhatme U., Phys. Rep. 251 (1995) 267. [2] See for example, Fowles G.R., Analytical Mechanics (CBS College Publish- ing) pp. 64-68, 1986. [3] Rosner J.L., Ann. Phys. 200 (1990) 101; Kwong W. and Rosner J.L., Prog. Theor. Phys. Suppl. 86 (1986) 366. [4] Chow W.W., Koch S.W. and Sargent M., Semiconductor Laser Physics (Berlin: Springer) 1994. [5] Malomed B.A., Parker D.F. and Smith N.F., Phys. Rev. E 48 (1993) 1418 [6] Amstrup B. et al., Phys. Rev. A 48 (1993) 3830.
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