Theory of quantum nonlinear resonance

PHYSICS LETFERS

Volume 61A, number 5

30 May 1977

THEORY OF QUANTUM NONLINEAR RESONANCE G.P. BERMAN and G.M. ZASLAVSKY Kyrensky Institute of Physics, Siberian Department of the Academy of Sciences, Krasnoyarsk 660036, USSR Received 28 February 1977 The theory of trapping into quantum nonlinear resonance when the coherent field interacts with molecules is developed. The oscillation spectrum structure of probability amplitudes is obtained.

Trapping into nonlinear resonance appears when coherent radiation field interacts with molecules which have high enough anharmonicity of energy spectrum. The trapping condition is y = o~/e>1 (~, are dimensionless parameters of nonlinearity and perturbation respectively). In particular this condition occurs at experiments on dissociation of collisionless molecules [1—3]. [e.g. The classical theory semiclassical of nonlinear analysis resonance is known 4]. Qualitative was proposed in ref. [5]. Quantum theory of nonlinear

the vicinity of n0. If ~

4

‘~

then the expansion for equations (3) in the vicinity of n0 becomes (validity condition will be given further): 2 W,~ W,~ n0) + W, (n n0) 0 + W,~(n 0 = W~ + ~-E,(n n 2 W,~ 0) 0= E~0 n0hv E0 —

~-

—

—

—

resonance is developed below. Let the perturbation dc~cos rt influence the nonlinear oscillator with energy spectrum E~(d is the dipole moment, ~ the external field amplitude). The wave function of the molecule can be written in the form. /i(x, t) = ~C~(t)

~1~n(X) exp[—i(nv +E0/1l)t]

(1)

n

where ~ (x) is the nonperturbed solutionthat of stationary Schrodinger equation. Let us assume the resonance condition is realized for some n 0, then: 1iw,~ = E, °

(2)

1W °

where the prime denotes differentiation with respect to argument. By neglecting the nonresonance terms we receive the equation for en: ih~!~W,~en

+

~

c~~1 +~

W~=E~—Fwn—E0

~ c,~_~

thc~= ~

+

—

V,~0~0÷1 c,~1+ Vnono _lCn_i.

Introducing the amplitude a we have: 1 2ir c~(r)=~— dpa(~p,t)exp[i(n _n0),p—iW~t/fl]. ° (6) Substituting (6) into (5) we receive the equation for a:

f

tha/at = Ha;

2Iap2 +

H=

—

.

~

a

(7 .

U(~’)=V,~0~0~1 exp(i~)+Vno,no_i exp(—i~p). This is the Schrodinger-type equation with complex potential U(~).Operator H is the effective hamiltonian of quantum nonlinear resonance. Let us consider the case n0 ~ 1. Then and U(y,) = 2 V,~0cos p. Equation (7) becomes equivalent to the problem of particle motion in a periodic field with effective mass IE,~0I~. In this case it is the solution of the well known Mathieu equation:

(3)

Hx=Xx where ~ P7~ are the matrix elements of the resonance transition’s. Let ~5n= maxln n0I be an effective number of levels by means of transitions with each other in —

(8)

which we don’t write for shortness. The spectrum X~ has band structure. We need only the solution xq pen295

Volume 61A, number 5

PHYSICS LETTERS

odical in Only those X that lie on the boundaries of the bands and are Mat?iieu functions of first type corresponding with such solutions. Now the solution for a is determined by ~.

xq (~p) exp(_iXqtlfl),

a(p, t) =

(9)

.

q

where the summation is over Xq, ~ as pointed out above. With the help of eqs. (1), (6), (9) we receive finite expressions for probability amplitudes Cn(t) as a function of the initial conditions c~(O): cn(t) = ~~cm(0)Xq(fl

—

0)xq(n0

—

m)

X exp[—i(X~ + Wn)t/hI;

(10)

2ir

,f

d~ exP(in~)xq(~).

From eqs. (5), (7) the validity condition (4) can be written in the form: 112 = y~ 1, ~

sn/n (4Vn/IE; 1)1/2 = (c/a) ,, e = 2V~/P~ 0flWp~~ a = nolEn I~~0 —

—

~

From eq. (11) this condition (4) is equivalent to trapping one (‘~‘> 1). This number of levels trapped in resonance is 6n n /~and can be large enough for n0 ~ 1. According eqs. (7), (8) lower values of A are of order:

296

2. (12) w~0(ac)” So, splitting of energy levels in nonlinear resonance is the order 1/2 and not as in usual perturbance theory. It is useful to note two generalizations of our theory: 1) in eq. (7) we neglected the complex correction to the potential which belongs to especial quantum effects. 2) The resonance transitions n n ±1 were taken in account when obtaining eq. (7). However, the transitions n -÷ n ± m (n > m> 1) are also possible. In this last case the spectrum of A contains not only such A mm Xq

‘~

~Jv~1E~ 0

“

—~

n

q m

Xq(fl) = ~

30 May 1977

that corresponds to band b~undariesbut the values o~ Aq that lie inside the bands. In conclusion we remark that the obtained results can be useful for experimental investigations of the nonlinear molecular spectrum. For example if a l0—~ and condition (4) a> is satisfied then < 5 X 102 CGSE for w 2 x 1014 sec* d = 2 X iO_19 CGSE. References [1] N.R. Isenor, V. Merchant, Halisworth M.C.Richardson, CanadianR.S. Journ. Phys. 51,and 12 (1973)

[2] R.V. Ambartzumian et al., Zh. Eksp. Teor. Fiz. 69 (1975) 72. [3] N.V. Kaniov and A.M. Prochorov, Uspekhi Fiz. Nauk 118 (1976) 583. [4] G.M. Zasiavsky and B.V. Chirikov, Uspekhi Fiz. Nauk 105 (1971)3. [5] E.V. Shuryak, Zh. Eksp. Teor. Fiz. 71(1976) 2039.