Stability of dynamic squeezed state in four-wave mixing

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Stability of dynamic squeezed state in four-wave mixing

A.A. Bakasov, B.M. Narbaev

To cite this version:

1367

LE

JOURNAL

DE

PHYSIQUE

Classification

Physics Abstracts

42-50T--05.45 Short Communication

Stability

of

_{dynamic squeezed}

state

in

four-wave

_mixing

A.A. Bakasov

_(1,*)

and B.M. Narbaev

₍₂₎

(1)

Joint Institute for Nuclear

Research,

Laboratory

of Nuclear

_Problems,

Head Post

_Office,

P.O. Box

79, 101000, Moscow,

U.S.S.R.

(2)

Moscow

_{Engineering Physics}

_Institute,

Kashirskoe

_{sh., 31,115409 Moscow,}

U.S.S.R.

(Reçu

le 29 mars

_1990,

_accepté

le 23 avis

₁₉₉₀₎

Abstract. 2014 One of the realistic models of the

squeezed

state

_generation

in the four-wave

_mixing

has been discussed. An

_{asymptotic stability}

of the

_generation

_regime

of the

_{light dynamic squeezing}

is

strictly

shown. The absence of the

_periodic

bifurcation for the

_experimental

_parameter values reveals that the

_{multifrequency generation}

of the

_squeezed

_light

has not taken

_place.

J

_Phys.

France 51

₍₁₉₉₀₎

1367-1371 ler JUILLET

_1990,

A considerable number of the

_papers

_[1-4]

has been devoted to the theoretical

_{investigation}

of the

_squeezed

state

_generation

in wave

_mixing

_processes.

Holm et al.

_[4]

with the

_help

of the method

_{developed by}

them

_[5,6]

have described the

_squeezed

state

_génération

in the

_process

of the

four-wave

_mixing

and have obtained a

_{good degree}

of

_agreement

with the

_experiment

of Slusher et

aL

_[7].

_However,

the

_{important problem}

of the

_stability

of the solutions

_describing

the

_squeezing

dynamic

still remains

_open.

In

_reality

the recurrence of the effect in the

_experiments

testifies to

the

_stability

of those solutions of the

_dynamic

model,

which describes this effect.

_Therefore,

the aim of the

_present

letter is the

_{investigation}

of

_dynamic

_squeezing

in the model of Holm et al.

_[4]

to clear

_up

the

_qualitative

concordance of the model and

_experiment.

The model Hamiltonian in the interaction

_{representation}

and in the

_{rotating-wave}

approxi-mation has the form

_[4]

In this

_expression

_aj

and

_a+j

are the annihilation and creation

_operators

_respectively

for the field

_j-mode;

vl, v2 and v3 are the

_frequencies

of the

_{electromagnetic}

field

_arbitrary

tuned from

1368

atomic resonance w.

Frequencies

vl and v3 are

_{symmetrically}

located relative to v2 .

Further,

_{Uj =}

Uj (r) -

corresponding spatial

mode

factor, A

= _{v2 - vl, g - constant} of the atom-field

coupling

and V2

=

-pC212h,

where p -

matrix element of the atomic

dipole

moment,

C2 -

amplitude

of mode 2 which is considered to be classic.

_{Operators uz}

and

u:%

describe the atoms:

Following

[4]

let us consider the

_equation

of motion for the field

_density

_operator

where coefficients

_{A1, Bl,}

_Cl,

_Dl

are determined

_by

the

_expressions:

where is the

dimen-sionless

_intensity

of the classical

field,

r is

thé

_{upper-to-lower-level decay}

_constants

is the

_dipole

decay

constant, F

=

r/(r’

₊

ià)

and N - total number of the _atoms. Consider the linear

_{superposition}

of side-band

_components

1 and 3

where 0 -

_phase

shift of the local oscillator.

The

_operators

of the field

_quadratures

and the variances have the forms:

As seen, variance Ad) > is the linear function of the averages afai > , a#a3 >, afa#

>

and aiâ3 > .

_Using

_(3),

we shall obtain the

_équation

of motion for thèse

_quantities

Equations

for

_{a3 a3}

> and

_{a3 al}

> are obtained

_by

the

_change

1-->3 in

_equation

₍₉₎

and with

_complex

_conjugation

of

_{equation (10).}

Here,

as in

_[4],

_v/Q

is the rate of

_cavity

losses and Ai =

n2 - v2

is the

detuning

of the

cavity

mode from that of field 2.

It is convenient to

_pass

to the real values for a further

_{investigation}

of this

_system

of the linear

equations.

Determine new variables and coefhcients:

Then the

_investigated

_system

of the

_equations

will take the form:

where

The

_{corresponding}

characteristic

_equation

of the linear

_system

matrix

₍₁₂₎

has the form:

where

The solutions of

_{equation (12)}

are

_{asymptotically}

stable if

_{polynomial (13)}

with the real coeffi-cients has the roots

_only

with the

_negative

real

_parts.

For this

_purpose

it is sufficient that the

1370

where

_Ti

_{(i =1, 4) -}

Hurwitz

_minors,

The

_generation

_process

in the

_experiment

of Slusher et al.

_[7]

is carried out far from the

saturation due to a

_great

_tuning

of

_pumped

mode 2 from the atomic resonance in

_spite

of the

_high

intensity

of

_pumping

_12.

It means that the

_expressions

for the coefficients in formulae

₍₄₎

can be

replaced by

the

_{corresponding}

12

-linear terms:

where

Ai

= W - v;

0,

expressions

for coefficients

ReX3, ImX3, Rea3, Ima3

are obtained from

(21) by

the

_change

1->3.

It is seen that

_{Rexl, ImXI, ImX3, Rea3,}

_{Im (al}

+

_a3)

are

_positive;

as to

_ReX3

and

_Real

they

can become

_negative

at some values of

_dl

and

A3.

We shall restrict our consideration to

the area of the

_parameters

determined

_by

the

_inequality

_{Real >}

0. It is also

_possible

to make

sure that 6 > 0.

_{Consequently,}

all the Hurwitz minors of

_{equation (13)}

are

_positive

at

_b4

=

4Rea1 Rea3

E - 16 >

0. Then the trivial solution of

_system

₍₁₂₎

is

_{asymptotically}

stable in the area of the

_parameters

_which,

as considered

[4, 8, ],

_corresponds

to the

_experiment.

It follows from the

_stability

theorem for the linear

_system

solutions

_[11]

that all the solutions of

_dynamic

_system

(12)

are

_{asymptotically}

stable and limited.

Hence,

the solution

_{corresponding}

to the

_dynamic

squeezing

within the framework of the model considered is

_{asymptomatically}

stable and limited

as

_well,

as therefore the

_squeezing

cannot reach 100% .

Let us now consider the

_problem

of

_periodic

bifurcation from the

_stationary

solution in the

system,

i.e. whether the

_squeezing

_generation

is

_possible

under the

_{multifrequency}

evolution of variances. A

_necessary

condition for this is the

_passage

of the real

_part

of _any

_complex

root of

equation

(13) through

zero value. The number of the roots with the

_positive

real

_parts

is

_equal

to

the number of the

_{sign changes}

in the

_following

_sequence

_(Hurwitz

theorem

_[10])

Any

complex

root of

_{equation (13)}

has the

_conjugate

one, their real

parts

are

_equal,

i.e. if

the real

_part

of _any

_complex

root

_passes

_through

_zero, then the number of the

_{sign changes,}

which

we

_designate

as

_1,

should exceed or be

_equal

to two. If K 2 the

_necessary

condition of the

K is less than 2.

It is seen from formulae

_{(14), (16)}

and

₍₁₈₎

that

_only

the

_following

_{inequalities determining}

the

_signs

in

_sequence

₍₁₈₎

and, hence,

K as

_well,

can be carried out in the area of the

_parameters

(19):

Then it follows from

₍₁₈₎

and

₍₁₉₎

that five

_sign

_sequences

and

_{corresponding}

numbers of

their

_changes

are

_possible:

Thus,

it has been found that the conditions of the

_squeezed

state

_{multifrequency generation}

is not

_performed

in the

_system

for the

_parameter

values

_(19).

For the similar

_experimental

situation the

_study

of

_stability

has been also carried out in terms

of the field

_{amplitude, polarization}

and inversion variables

_[12].

But,

evidently,

the

_stability

of these variables doesn’t mean the

_stability

of the variables we used and vice versa.

_Finally,

we can

state that

_general

_{investigation}

of

_system

_{(12) requires symbolic}

and numeric

_computer

calcula-tions. This is the reason

_why

we consider

_only

the

_experimental

_parameter

_range

[7, 4].

Références

[1]

YUEN H.P. and SHAPIRO

J.H.,

Opt.

Lett. 4

₍₁₉₇₉₎

_334;

BONDURANT

_R.S.,

KUMAR

_P.,

SHAPIRO J.H. and MAEDA

_M.,

_Phys.

Rev. A 30

₍₁₉₈₄₎

343.

[2]

REID M.D. and WALLS

_D.F.,

_Phys.

Rev. A 31

₍₁₉₈₅₎

1622.

[3]

KLAUDER

_J.R.,

McCALL S.L. and YURKE

_B.,

_Phys.

Rev. A 33

₍₁₉₈₆₎

3204.

[4]

HOLM D.A. and SARGENT III

_M.,

_Phys.

Rev. A 35

₍₁₉₈₇₎

2150.

[5]

SARGENT III

_M.,

HOLM D.A. and ZUBAIRY

_M.S.,

_Phys.

Rev. A 31

₍₁₉₈₅₎

3112.

[6]

STENHOLM

S.,

HOLM D.A. and SARGENT III

M.,

Phys.

Rev. A 31

₍₁₉₈₅₎

3124.

[7]

SLUSHER

_R.E.,

HOLLBERG

_L.W.,

YURKE

_B.,

MERTZ J.C. and VALLEY

J.F.,

Phys.

Rev. Lett. 55

₍₁₉₈₆₎

2409.

[8] J. Opt.

Soc. Am. B 4

₍₁₉₈₇₎

4

_(special

_issue).

[9] J.

Mod.

_Opt.

34

₍₁₉₈₇₎

6/7

(special

issue).

[10]

GANTMAKHER

F.R.,

Theory

of

Matrix,

in Russian

_{(Nauka, Moscow)1988.}

[11]

PONTRYAGIN

L.S.,

Ordinary

Differential

_Equations,

in Russian

_{(Nauka, Moscow)}

1982.