Weak-probe absorption and dispersion spectra in a two-level system driven by a strong trichromatic field

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Weak-probe absorption and dispersion spectra in a

two-level system driven by a strong trichromatic field

A.V. Alekseev, A.V. Davydov, N.V. Sushilov, Yu. A. Zinin

To cite this version:

Weak-probe absorption

and

_dispersion

_spectra

in

a

two-level

system

driven

_by

a

_strong

trichromatic field

A. V.

Alekseev,

A. V.

_Davydov,

N. V. Sushilov and Yu. A. Zinin

Pacific

_{Oceanological}

Institute, 7 Radio St., Vladivostok, 690032, U.S.S.R.

(Reçu

le 28 octobre 1988, révisé le 9 décembre 1989,

accepté

sous

_forme

_définitive

le 21 décembre

1989)

Résumé. 2014 On étudie à l’aide d’un

_champ

sonde l’évolution

_temporelle

d’un

_système

à 2 niveaux soumis à un

_{champ quasi}

résonnant à trois

_fréquences

différentes. Les

_équations

de Bloch

correspondantes

sont _{résolues par} une méthode

_{d’exponentielle}

de matrice

_qui

_permet de se

passer de fractions continues. On montre _que

_lorsque

la

_fréquence

de Rabi est suffisamment

grande,

le

_{profil d’absorption}

du

_champ

sonde ressemble à un

_profil

de

_dispersion.

On met en

évidence l’existence d’un nouveau _type de résonances

_{paramétriques}

_qui

ne sont _{pas liées à des}

résonances de Rabi.

Abstract. 2014 With the

help

of a

_probe-field

method the time-behaviour of a two-level _system

subjected

to a _strong trichromatic near-resonant field is studied. The suitable Bloch

_equations

are

solved

_by

a matrix _exponent method which allows us to do without the continued fractions. It is shown that, when the Rabi

_frequency

is

_sufficiently

_large,

the

_profile

of the

_weak-probe

field

absorption

resembles a

_{dispersion profile.}

The existence of a new _type of

_parametric

resonances

not connected with the Rabi resonances is shown. Classification

Physics

Abstracts 42.65G

1. Introduction.

The

_probe-field

method is

_{widely adopted}

in

_studying

the

behaviour of a

_{quantum system}

excited

_by

a

_strong

resonant field. This method consists in

_studying

the

_absorption

or the

dispersion

of the field

_depending

on its

_frequency

_provided

the

_{quantum system}

is excited

_by

a resonant field at the fixed

_frequency.

From the

_absorption

_(dispersion)

line form of the

_probe

field one can draw the conclusions of the nonlinear

_phenomena

in the

_{quantum system.}

Rautian and Sobelman

_[1]

were the first to

_investigate

the

_absorption

of a weak in the presence of a

_strong

wave.

_They

showed that the

_absorption

of the weak

_(probe)

wave

changes

into the

_gain

when the

_strong

field

_amplitude

and the

_frequency

difference between the

strong

and weak field reach certain values. This

_problem

was also considered in

[2-6]

by

the solution of

_equations

for the

_density

matrix of a two-level

_system

which allows us to take into account the relaxation in contrast to the solution of

_equations

for the wave function

amplitude

[1].

It has also been obtained that the

_weak-probe

field

_absorption

_{may be}

_negative

(the

parametric gain

effect)

when the saturation field is

_sufficiently

_strong.

This

_gain

has a maximum when the

_frequency

difference between the

_strong

and weak field is

_equal

f2 = --t f2 RI f2 R is

the nutation

_frequency

in .the

_system

_(the

Rabi

_frequency).

The

_negative

absorption

effect was

_{experimentally}

confirmed in a

_large

_{number of papers both in the} rf-range and in

optics

[7-11]. Subsequently

the

_probe-field

method was

_developed

in two

directions :

₁₎

the case of

_{equal saturating}

and

_probe

field intensities

_(so-called

_strong

modulated

_exciting

_{field) [12-24], 2)}

the case of a number of

_{equal probe}

fields with the

frequencies

which are

_symmetric

relative to the

_saturating

field

_frequency.

The

_amplitudes

of these

probe

fields may be both weak

_[25-28]

and

_equal

to the

_strong

_pump

_{amplitude [29].}

It was shown that there are the resonances at = ±

_{f2 R}

_{and 1 f2 1 = l2R/n, n}

=

1, 2, 3,

...

(the

so-called

sub-radiation structure

_[30])

in their

_absorption

_spectra

when the

_probe

wave

amplitudes

grow. In addition in

[22, 28]

was obtained that the

_dispersion

_profile

resembles an

absorption

one while the

_{absorption profile}

resembles a

_dispersion

one when the

_probe

wave

amplitude

is

_{sufficiently large.}

Now we describe

_briefly

the mathematical methods which have been used

_by

the

above-mentioned authors. It is notorious that the time

_dependence

of the

_density

matrix of a two-level

_system

_interacting

with the resonant radiation is described

_by

the Bloch

_equations

_(see,

for

_example,

_[31]).

When the

_{exciting amplitude}

and

_frequency

are constant the Bloch

equations

are simultaneous linear

_{inhomogeneous equations}

with constant coefficients and

they

can be solved

_[32].

If an

exciting

field is monochromatic but its

_amplitude

is a

_periodic

function of

time,

then the Bloch

_equations

are

_equations

with

_periodic

coefficients. Such a form of the field

_amplitude

modulation is

_equivalent

to the

_presentation

of the

_exciting

field

by

a set of the monochromatic

_components

with the same

_amplitudes

and

_equidistant

frequency

spectrum.

Up

to now these

_equations

have been solved

_by

the

_following

methods :

1)

the

_expansion

on a small

_parameter

_(in

the case of weak

_probe

_{fields) [4,}

5,

25,

26] ; 2)

the

Laplace

(or Fourier)

transformations

_[2,

_5,

_27]

with some

_{simplifying assumptions ;}

₃₎

the introduction of a trial solution

_{[29] ; 4)}

the

_Floquet

theorem method which allows us to

present

the solution

_by

an infinite set of the intermode

_frequency

sidebands. The

_amplitudes

of these sidebands are

_expressed

in terms of continuous fractions which are needed in the numerical calculation. Feneuille et al.

_[12],

_Toptygina

and Fradkin

_[15]

were the first to

_apply

the

_Floquet

theorem and now this method is

_{widely applied}

in

_studying

the Bloch

_equations

with

_periodic

coefficients and constant

_{inhomogeneous}

terms

_[16-24].

However,

when the

_exciting

field modulation is

_{nonperiodical}

_(for

_example,

the

_phase

of a

probe

field is a stochastic function

_[33],

or the field

_spectrum

is

non-equidistant),

the

_Floquet

theorem is

_inapplicable

because the Bloch

_equations

for this case have a

_{nonperiodical}

coefficients or a

_{time-dependent inhomogeneous}

term

_(the

_validity

of the

_application

of the

Floquet

theorem for

_studying

the differential

_equations

have been

_discussed,

for

_example,

in

[34]).

Such

_equations

can be solved with the

_help

of the matrix

_exponent

method which was used in

_[33].

We show below that this method allows us to obtain

_analytic

solutions of the

Bloch

_equations

with both

_periodical

and

_{nonperiodical}

coefficients and without

_using

the continuous fractions.

The

present

paper is devoted to the calculation of the

_{weak-probe absorption}

and

dispersion profiles

in a two-level

_system

saturated

_by

a

_strong

trichromatic field in an

analytic

form,

i.e. we

_apply

the

_probe-field

method to

_study

the nonlinear effects in a two-level

_system

driven

_by

a

_strong

resonant

_{amplitude-modulatéd}

field. In section 2 we obtain

(with

the

help

of the matrix

_exponent

_method)

an

_analytic

solution of the Bloch

_equations

with

_periodic

coefficients and constant

_{inhomogeneous}

terms. These

_equations

describe the behaviour of the two-level

system

density

matrix in the

_strong

trichromatic field with three

_components

of

equal amplitude.

In section 3 we obtain the response of such a two-level

system

to the weak

obtained for the

_absorption

coefficient and

_dispersion

are _very

_complex,

the

_graphs

for these values are calculated

_numerically.

2. Two-level

system

in the resonant trichromatic field.

The trichromatic field with the

_components

of the same

_amplitude

and with the

_equidistant

spectrum

and a

_probe

field can be written in such a form :

where

_Eo

is the

_amplitude

of the

_saturating

field

_component,

the

_frequency

is

_supposed

to be

_equal

to the natural

_{frequency ú)o}

of the two-level

systems, e

is the

_probe

field

_amplitude,

il is the difference between the

_neighbouring

_component

_{frequencies, 8 = ú)w - w,}

wW is the

probe

field

frequency.

The

_equations

for the

_density

matrix elements of a two-level

_system

_interacting

with the field

₍₁₎

in the

_rotating

wave

_{approximation}

(the

_optical

Bloch

equations)

read then :

where

_{a (t) = 2 Re 0’21 (t) ;}

;

f3 (t) = 2 lm (T 21 (t);

;

n (t) = P 22 (t) - P 11 (t) ;

;

0ij (t)

are the

slowly varying

components

of

_density

matrix elements

_{P ij (t) ;}

_{no is the level}

_population

difference per unit volume in the absence of an external

_{field ;}

_f1R

=

dE0/h

is the Rabi

frequency ; d

is the

_dipole

moment of a two-level

_{system ;}

_Fg

1,

Fi

are

the relaxation

_times,

collisional and

radiative,

respectively.

In

_deriving

₍₂₎

we have

_supposed

the

_validity

of the

rotating

wave

_{approximation}

for all

_components

of the trichromatic

_field,

i.e. we have

assumed that the condition f2 « £ô is valid.

Assuming

that the

_probe

field is weak

_{(eleo « 1)}

the solution of

_equations

₍₂₎

is

_sought

as a sum of two solutions - the solutions in the absence of

a weak field

(for

e =

0)

and a small correction to the latter caused

_by

a weak

field,

_namely,

where

u,

v,

_w

₁

_«

_U,

V,

_{W 1.}

. After

substituting

(3)

into

(2)

and

retaining

the terms of the order

_eleo,

we obtain two

_systems

of

_equations

for

U, V,

W and u, v, w instead of

_{(2) :}

The matrix

_{equation (4)}

is the

_equation

with the

_periodical

coefficients and constant

inhomogeneous

term. It can be solved with the

_help

of the

_Floquet

theorem. However we shall show that the matrix

exponent

method allows us to obtain more

_easily

both

_{equation (4)}

and

_equation

₍₅₎

which do not

_satisfy

the

_Floquet

_theorem,

since the

_period

of its

inhomogeneous

term

f(t)

is not

_equal

to the

_period

of

_A(t).

So

_provided

the commutator

_{[A (t ),}

_exp

_{B (t)]}

is

_equal

to zero

_[35]

the solutions of

_{equations (4), (5)}

can be written as

To calculate

_{exp B(t)}

we shall make use of Silvester’s formula

_{[34] :}

where À 1 2 3

are

_eigenvalues

of matrix

B (t ),

I is the

_identity

matrix.

Since we can

_easily

obtain

(provided

where

_{f (t )}

=

f2 R t

₊

(2 nR/n)

sin

f2 t ; p

=

(F2

+

_r1)/2;

y =

(F2 -

r1)/2.

After

sub-stituting

(9)

into

₍₈₎

and

_retaining

the terms of the order

_{y / n R}

we obtain :

The direct calculation shows that

_{exp (B ) exp (- B )}

= 1 within the bounds of our

approxi-mation.

We can now obtain the conditions for the

validity

of solutions

(6), (7).

Using

(10),

one

The commutator is null in four cases

Note that the state of the

_system

at a

_given

time t

_{generally depends}

on its

_previous

_history.

The commutator has then in

_principle

to be null at _any time t’ in _{the range}

_{[0, tJ.}

This is

_only

achieved if y = 0

(case 1).

However the cancellation of the _commutator for t’ --> oo

_{(case 4)}

entails that the

_steady

state

_solution,

which does not

_depend

on the initial

conditions,

can be

obtained

_by

the matrix

_exponent

_method,

whatever y is.

Substituting

(10)

into

_(6),

we obtain the

_analytic

solution for all

_components

of the vector

X(t) :

where

and can be written :

where

_Jn(2p)

are Bessel functions of the first

_{kind, ,p}

=

flR/fl,

Co =

1/2,

_{Cl, c2, ... ,}

cn = 1.

Taking

into account

(14), (15),

solutions

(13)

_can be written _as

(we

write the

steady-state

_parts

_{only) :}

where

The

_{explicit expressions}

for

_{time-independent}

coefficients

_{(A, B, P, Q )f’ f}

are

greatly

these

_expressions

can be written in such a form :

For the sake of

_simplicity

we omitted the

_argument

2 _p for all Bessel functions.

So we obtained the

_analytic

solutions of the Bloch

_equations

in the case of trichromatic resonant excitation without

_using

the

_Floquet

theorem. In contrast to

_[29],

where the trichromatic excitation was also

_considered,

we

_supposed

the

_inequality

of

_{r 2}

and

ri.

The

_following

conclusions can be drawn from

_equations

(16)-(19).

The

steady-state

oscillation

_spectrum

of

_{W(t) (the}

_spectrum

of

_undamped

_nutation)

contains the intermode distance n sidebands. This result

_corresponds

to the results obtained

_by

the

_Floquet

method for the differential

_equations

with the

_periodical

coefficients. The sideband

_amplitudes

are maximum when

_{n R}

= ka

(k

=

1, 2, 3,

... )

i.e. when the Rabi resonance conditions are valid. These

_amplitudes

decrease with

_increasing

k.

3.

_Absorption

and

_dispersion

_spectra

of the

weak-probe

field.

where

_{u (t ), v (t )}

are _{the proper Bloch-vector}

_components.

On the other hand the

_polarization

can be

_expressed

in terms of the medium

_{polarizability}

_{X(w) :}

where eo is the vacuum electric

_{permittivity,}

e is a field

_amplitude.

If the

_exciting

field is

nonmonochromatic,

the total

_polarization

is

the sum of the

_polarization

at both the

frequencies

containing

the

_exciting

field and the combination

_{frequencies :}

When a two-level

_system

is excited

_by

the

_strong

trichromatic field the

polarization

induced

by

the

_probe-field

can be obtained at the first order in e

_by

_substituting

the solutions of

equations (5)

into

₍₂₂₎

and

_{by selecting}

the items at the

_frequency

_lJJq

=

(ow

only.

With the

_help

of

₍₇₎

and

₍₁₀₎

we

obtain :

where

In

_{(23), (24)}

we _may use the

_{only steady-state}

parts

of

_{V (t )}

and

_W(t),

which are written in

(16),

since the substitution of their

_damped

parts

into

_{(23), (24)}

results in

_damped

parts

also.

Consequently,

since one of conditions

(12) t

- oo is

_valid,

we _may

_ignore

the rest conditions.

where are determined in

respectively .

The,

coefficients

_(a,

b,

c,

d,

e,

f,

g,

h ))Î"’

in

₍₂₆₎

are linear combinations of

(A, B, P, Q )7’ f

multiplied by

f24’.

We shall not write these coefficients in an

_explicit

form.

Equations

(25), (26)

enable us to reach the

_following

conclusions.

First,

the weak

_probe

field results in the appearence of another

type

parametric

resonances than the

Rabi-resonances,

namely,

the resonances at 5 = 0 and

st’ f

= 0 for

u (t)

and at

,fZ kem -

0 for

v (t).

These resonances are not the Rabi-resonances

since,

for

_example,

s,t = 0

for

k = f = 0

_{(i = 1),}

for k

= l

_(i

=

2),

i. e. these resonances do not

_depend

on the Rabi

frequency n R.

Second,

the

_weak-probe

field results in the

_undamped

oscillations of the

density

matrix elements both at the intermode distance sidebands and at the combination

sidebands of fi and

_f2R.

This result is a _{consequence of the fact that}

_equation

(5)

does not

Equations

(28), (29)

describe the total linear

_polarization

of a two-level medium induced

by

the

_weak-probe

field.

_Selecting

from these

_equations

the items which are attached to

sin úJ wt

and cos

_{úJ w t,}

we obtain the real and

_imaginary

_parts

of the medium

polarizability

at the

_weak-probe

field

_frequency,

_{respectively.}

Because the coefficients

calculating. Figures

1 and 2 show the

_absorption

and

_{dispersion profiles}

_{(Im Xw(5)}

and

Re Xw (5))

respectively

at the

_following

_{parameters :}

_r2/r1

=

103 ;

II R

=

10 r 2

(a)

and

20

_{l,2 (b) ;}

_{f2 R}

= f2. Summation _on

k,

f,

_{m, n are} carried _on from 0 _to 7. It is

expected

that

Fig.

1. - The

absorption profiles

of the

weak-probe

field for the cases :

_{r 2/ rI}

=

103;

DR

= 1Ii ;

,fl R

= 10

r2 (a)

and 20

_{F2 (b).}

Fig.

2. - The

dispersion profiles

of the

_weak-probe

field for the same choice of _parameters used in

this restriction does not affect

_{significantly}

the results. It is seen that the

absorption

and

dispersion

manifest the extreme behaviour when

_5/QR

is near

_by

_{0; ±1;}

_{± 2 ;}

± 3 ;

± 4 ; ± 5

(note

that

_{ImXw(5)=lmXw(-5); ReXw(5)=-ReXw(-5)).}

In the

neighbourhood

of these

_points

the

_{dispersion profile}

resembles the

_{absorption profile}

of the

monochromatic field. This result was obtained in

[28]

for trichromatic

exciting

field at

f2 R/F2

= 50 and

100,

and also in

_[22]

for

_strong

bichromatic field at

_{f2 R/-r2}

=

4,

6.

So we showed that the

_application

of the matrix

exponent

method to the solution of the Bloch

_equations

allows us to obtain the

_analytic

solutions in most

_general

cases as

_compared

with the

_Floquet

theorem method. Note that there is another method of the solution of the Bloch

_equations.

Prants

_[36]

_{developed recently}

the

_{theoretic-group}

method for

_solving

the Bloch

_equations.

He obtained the

_analytic

solutions for

_arbitrary

time modulation of the field

amplitude

and

_phase

but in the case of

_equal

relaxation times

_Fl

=

r 2.

References

[1]

RAUTIAN S. G., SOBELMAN I. I., Zh.

Eksp.

Teor. Fiz. 41

₍₁₉₆₁₎

456

_{(in Russian).}

[2]

BLOEMBERGEN N., SHEN Y. R.,

Phys.

Rev. 133

₍₁₉₆₄₎

A47.

[3]

CHIAO R. Y., KELLEY P. L., GARMIRE E.,

Phys.

Rev. Lett. 17

₍₁₉₆₆₎

1158.

[4]

APANASEVICH P. A.,

Doklady

Akad. Nauk BSSR 12

₍₁₉₆₈₎

878

_{(in Russian).}

[5]

MOLLOW B. R.,

Phys.

Rev. A 5

₍₁₉₇₂₎

2217.

[6]

HAROCHE S., HARTMANN _F.,

_Phys.

Rev. A 6

₍₁₉₇₂₎

1280.

[7]

KLYSHKO G. N., KONSTANTINOV Yu. S., TUMANOV V. S., Izv. Vuzov

_Radiofizika

8

₍₁₉₆₅₎

513

_(in

Russian).

[8]

CARMEN R. _L., CHIAO R. _Y., KELLEY P. _L.,

_Phys.

Rev. Lett. 17

₍₁₉₆₆₎

1281.

[9]

BONCH-BRUEVICH A. M., KHODOVOI V. A., CHIGIR N. A., Zh.

_Eksp.

Teor. Fiz. 67

₍₁₉₇₄₎

2069

(in Russian).

[10]

WU E. _Y., EZEKIEL _S., DUCLOY _M., MOLLOW B. R.,

Phys.

Rev. Lett. 38

₍₁₉₇₇₎

1077.

[11]

ZEILIKOVICH I. S., PUL’KIN S. A., GAIDA L. S., Zh.

Eksp.

Teor. Fiz. 87

₍₁₉₈₄₎

125

_(in

_{Russian) ;}

Sov.

Phys.

JETP 60

₍₁₉₈₄₎

72.

[12]

FENEUILLE S., SCHWEIGHOFER M.-G., OLIVER G., J.

_Phys.

B Atom. Molec.

_Phys.

9

₍₁₉₇₆₎

2003.

[13]

GORESLAVSKII S. P., KRAINOV V. P., Zh.

_Eksp.

Teor. Fiz. 76

₍₁₉₇₉₎

26

_{(in Russian).}

[14]

BRAUN P. A., MIROSHNICHENKO G. P., Zh.

_Eksp.

Teor.

Fiz.

81

₍₁₉₈₁₎

63

_{(in Russian).}

[15]

TOPTYGINA G. I., FRADKIN E. _E., Zh.

_Eksp.

Teor. Fiz. 82

₍₁₉₈₂₎

422

_{(in Russian) ;}

Sov.

Phys.

JETP 55

₍₁₉₈₂₎

246.

[16]

TSUKADA _N., NAKAMURA T.,

Phys.

Rev. A 25

₍₁₉₈₂₎

964.

[17]

MAK A. _A., PRZHIBELSKII S. G., CHIGIR N. A., Izv. Akad. Nauk SSSR Ser. Fiz. 47

₍₁₉₈₃₎

1976

_(in

Russian).

[18]

AGARWAL G. S., NAYAK _N., JOSA B 1

₍₁₉₈₄₎

164.

[19]

HILLMAN L. W., KRASINSKI J., KOCH K. et al., JOSA 2

₍₁₉₈₅₎

211.

[20]

AGARWAL G. S., NAYAK _N., J.

_Phys.

B : At. Mol.

_Phys.

19

₍₁₉₈₆₎

3385.

[21]

AGARWAL G. S., NAYAK N.,

Phys.

Rev. A 33

₍₁₉₈₆₎

391.

[22]

FRIEDMANN H., WILSON-GORDON A. D.,

Phys.

Rev. A 36

₍₁₉₈₇₎

1333.

[23]

WILSON-GORDON A. _D., FRIEDMANN _H.,

_Phys.

Rev. A 38

₍₁₉₈₈₎

4087.

[24]

CHAKMAKJIAN S., KOCH K., STROUD C. R. Jr., JOSA 5

₍₁₉₈₈₎

2015.

[25]

KUZNETSOVA T. I., RAUTIAN S. G., Zh.

_Eksp.

Teor. Fiz. 49

₍₁₉₆₅₎

1605

_{(in Russian).}

[26]

ARMSTRONG L., FENEUILLE _S., J.

_Phys.

B : At. Mol.

_Phys.

8

₍₁₉₇₅₎

546.

[27]

McCLEAN W. A., SWAIN S., J.

_Phys.

B : At. Mol.

_Phys.

9

₍₁₉₇₆₎

2011.

[28]

BOYD R. _W., RAYMER M. G. , NARUM P. , HARTER D. J.,

Phys.

Rev. A 24

₍₁₉₈₁₎

411.

[29]

THOMANN _P., J.

_Phys.

B : At. Mol.

_Phys.

9

₍₁₉₇₆₎

2411 ; 13

₍₁₉₈₀₎

1111.

[30]

BONCH-BRUEVICH A. M., VARTANYAN T. A., CHIGIR N. _A., Zh.

_Eksp.

Teor. Fiz. 77

_{(1979) (in}

[31]

J

ALLEN L., EBERLY J. H.,

Optical

Resonance and Two-Level Atoms

_(John

_Wiley

& Sons, Inc.,

New

_York)

1975.

[32]

TORREY H. C.,

Phys.

Rev. 76

₍₁₉₄₉₎

1059.

[33]

ALEKSEEV A. V., SUSHILOV N. V., Zh.

_Eksp.

Teor. Fiz. 89

₍₁₉₈₅₎

1951

_{(in Russian) ;}

Sov.

Phys.

JETP 62

₍₁₉₈₅₎

1125.

[34]

YAKUBOVICH V. A., STARZHINSKII V. M., Linear Differential

_Equations

with a Periodic

Coefficients and

they Applications

(Moscow, Nauka)

1972

_{(in Russian).}

[35]

LAPPO-DANILEVSKII I. A.,

Application

of the Matrix Functions to the

_Theory

of a Linear

_Systems

of a Differential

_{Equations (Moscow, Gostehizdat)}

1957

(in Russian).