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PHYSICAL REVIEW A 82, 015801 (2010)

Analytical solutions for a two-level system driven by a class of chirped pulses

Pankaj K. Jhal,* and Yuri V. Rostovtsev1,2

'Institute for Quantum Science and Engineering and Department of Physics, Texas A&M University, College Station, Texas 77843, USA

2Department of Physics, University of North Texas, 1155 Union Circle 311427, Denton, Texas 76203, USA

(Received 14 April 2010; published 6 July 2010)

We present analytical solutions for the problem of a two-level atom driven by a class of chirped pulses. The

solutions are given in terms of Heun functions. By use of the appropriate chirping parameters, an enhancement

of four orders of magnitude in the population transfer is obtained.DOI: 10.1103/PhysRevA.82.015801

Interaction of coherent optical pulses with quantum systems

is a fundamental problem [1] that is closely related to

important applications [2,3]. Nowadays, laser systems produce

controlled, intense, ultrashort optical pulses [4]. Various

technologies have been used for pulse shaping [5], which

allows researchers to provide coherent optical control [6]

of excitation in quantum systems; this has a broad range of

applications from nonlinear laser spectroscopy to generation

of coherent radiation. For example, chirped pulses [2,5] are

used to produce maximal coherence in atomic and molecular

systems. Maximal coherence can be used for generation

of short-wavelength radiation, which has been a focus of

research recently in an atomic system under the action of a

far-off-resonance strong pulse of laser radiation; it has been

shown that such pulses can excite remarkable coherence in

high-frequency far-detuned transitions; and this coherence can

be used for efficient generation of soft x-ray and ultraviolet

radiation [3,7,8]. Maximal coherence can also be used for

molecular spectroscopy, for example, time-resolved coherent

Raman spectroscopy, to obtain molecule-specific signals from

molecules that can serve as marker molecules for bacterial

spores [2].

In this Brief Report, we investigate two classes of chirped

pulses for which the problem can be solved exactly in

analytical form. By use of the appropriate chirping parameters,

the population transfer, after the the pulse is gone, can be

optimized, and for the pulse considered here an enhancement

by four orders of magnitude was obtained. An unchirped pulse

corresponding to the Heun and confluent Heun equations has

recently been investigated extensively in Ref. [8] where we

included an estimate of the energy of emission of soft x-ray

and ultraviolet radiation via excited quantum coherence in the

atomic system. The estimate shows good potential for a source

of coherent radiation based on the discussed mechanism.

The equation of motion for the probability amplitudes for

the states la) and Ib) [see Fig. 1(a)] of a two-level atom

[9,10] interacting with a classical field (under the rotating-

wave approximation) with nonzero chirping [12] is given as(la)

(lb)Ca = iQ(t)eit(t)Cb,

Cb = i 2*(t)e-it(t)Ca,

pkjha@physics.tamu.eduPACS number(s): 42.50.-p

where O(t) = At + q(t). Here A = co - v [11] and Q(t)

pS(t)/2h. Let us define the dimensionless parameters asA f20

r= at, , =-, y= o.

a a(2)

To solve for Ca, we can get a second-order linear differential

equation for Ca from Eq. (1), which in terms of the dimen-

sionless parameters of Eq. (2) is given asCa - i+f + i (r) Ca + 2 (t)Ca

\ (t) /0. (3)

A. Class I: Heun equation

To find an analytical solution for Eq. (3), we introduce a

new variable 9p = qp(r) defined by(4)

and make an ansatz for the pulse envelope Q(r) and the

chirping function q(r) as(5a)

(5b)2g(1 -g9) 1/2 Y

S(c - ) + v

( -2c + 2[(< + ) + c( + r)l q)

() - c)( X)In terms of the variable qp(r) and the definition of Q(r) and

q(r) from Eq. (5), Eq. (3) takes the form,, a( v )C, abo - q

Ca + -+ +- Ca + Ca

-1 (g - c) t ( - ) -cwhere c > 1 and

( ), /

=0,

(6)1 ((I + )

2F i 21 y2

v 2 i, q 2, a =0,

2 21 i/3X

b=2 2,c(7)

8+1

2The parameters of a Heun equation [13,14] are constrained,

by the general theory of Fuschsian equations, as p + a + v =

a + b + 1, which provides us the first constraint relation for

the chirping parameters <, r9, asS++ =0.

(8)

2010 The American Physical Society

r = (1/2)ln[=p/(1 - ))+ ],

1

21050-2947/2010/82(1)/015801(4)

015801-1

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Jha, Pankaj K. & Rostovtsev, Yuri V. Analytical solutions for a two-level system driven by a class of chirped pulses, article, July 6, 2010; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc103257/m1/1/: accessed April 23, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.