Optical hole burning with finite excitation time

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Optical hole burning with finite excitation time

Szabo, A.; Muramoto, T.

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PHYSICAL REVIE%' A VOLUME 37, NUMBER 10 MAY 15, 1988

Optical

hole burning with

finite

excitation

time

A.

Szabo and

T.

Muramoto'

Division

of

Physics,

¹tiona/

Research Council

of

Canada, Ottawa, Ontano, Canada KIAOR6

(Received 23 November 1987)

This paper extends earlier calculations [M.Yamanoi and

J.

H. Eberly, Phys. Rev. A 34, 1609

(1986)]ofhole burning using continuous excitation tothe finite excitation time used in experiments on Pr:LaF3. Both Gaussian-Markov (GM) and random-telegraph (RT)dephasing models are stud-ied. FortheGM model, anew feature isthe appearance ofnutation structure near the center ofthe hole for a400™@secexcitation time. Incontrast forRT,asimilar, although much weaker, structure appears in the wings ofthe hole. Itisnoted that forSnite excitation times, adetuning aswell as the

usual intensity dependence ofthe dephasing time appears.

INTRODUCTION

The failure

of

the conventional optical Bloch equa-tions' (OBE) todescribe measurements

of

saturation in a low-temperature solid has led to many theoretical stud-ies. ' These studies have used various models

of

atom-icfrequency fiuctuations that lead todephasing and have concentrated on atime-domain description

of

saturation,

i.e.

,free-induction decay

(FID).

So far, however, there is

no clear unanimity concerning which model,

if

any, best describes the data (see,

e.

g., discussion by Herman' ).

It

has recently been suggested" that studies in the frequen-cy domain may be useful in clarifying the correspondence between the various theories and experiment, and some hole-shape calculations have been presented. In these calculations, as well asin earlier

FID

work, an infinite ex-citation period was assumed. Experimentally, however, a

finite excitation period is necessary because

of

optical-pumping effects and the limited time during which a nar-row laser linewidth (&1kHz}can be maintained.

In this paper we present calculations on the effects

of

a

finite excitation time on hole burning for both Gaussian-Markov (GM) and random-telegraph (RT)dephasing models. A principal new result is that in addition tothe usual intensity dependence often noted for the various models, a detuning dependence appears for finite excita-tion times. In particular, for

GM,

a nutation structure appears near the center

of

the hole and persists for excita-tion times as long as400@secin spite

of

a much shorter (21.7 @sec) dephasing time.

THEORY

The general form

of

the reduced

OBE

isgiven by the matrix equation

T& isthe upper-state lifetime. The 3

g

3matrix

M

can be written as y

0

0

0

M=

—

oy

0

0 0

2y

0

I

~2

I

i3

I

22

I

23

0

0

0

0

0

0

—

₀

0

where

y'=(5to)

r„Q'=(b,

+Q

)'~,

Q/2n. is the Rabi frequency, b, /2m is the detuning frequency, and 5to and

~,

are parameters

of

the frequency fluctuation model de-scribed in

Ref. 4

and equivalently in

Ref.

8; and (3} random-telegraph dephasing model where

i (1/~,

+y)(1/v, +2y)

I"_ll

_—

—

_a

,

(1/~,

+y)(1/~, +2y)+Q'

I"22——_a

P

where the first matrix accounts for lifetime damping, the second is the coherent driving matrix, and the third is a generalized damping matrix.

In the calculation we consider three cases: (1)the con-ventional

OBE

for which

I

],2

—

—

I

2,

—

—

I

)3

—

—

I"23

—

—

0

and

I

_»

—

—

I

2z——T2 '

—

y;

(2) the Gaussian-Markov dephasing model, where for

y'r,

«1,

I"„=I"2i=y'[1+(b~,

)

]/[1+(Q'r,

)

],

I

_i3——

_y'Qb

_r,

_/[1+

_{(Q'~, )}

],

I"

23

—

——

y'Q~,

/[1+

(

Q'r,

)~],

~12 ~21

where

_p

is the Bloch vector expressed as a three-component column vector,

p(1)=u,

p(2)=v, p(3)=w

using the usual notation, and

L(1)

=L(2) =0,

L

(3)

=2ym,

q. %'euse the equilibrium value m,

= —

l and as-sume a closed two-level system so that 2y

=1/T„where

and

5(1/7, +2y)

I

„=

—

r„=

—

a2

I'

I

~3=I 23=0

BRIEF

REPORTS

T2—1

=f+Q

2 (5)

The form

of

the solution to Eq. (1) depends on the roots

of

the characteristic equation (CE) for M;

det(M+XI)=0,

where

I

is the unit matrix. There are two cases to consider.

Case

I:

CErootsuand b%is

In the case

of

one real and two complex-conjugate roots, the solution for

P

is

of

the form

P(h,

t)=

A exp(

—

at)+exp(

—

bt)

8

cos(st)+C

I'

_=[(I/i,

_{+y)'+5'](I/r,}

_+2@)+Q'(I/r,

_+y)

.

If

we limit our consideration only to the values

of

param-eters which lead to nearly exponential decay (i.e.,

ar,

&~1),

then the parameter

a

isrelated tothe

dephas-ing time T2 obtained from photon echoes by

Figure 1 shows plots

of

the normalized population

&(&)=[I+i'(b,

)]/[I+w(0)]

—

0.

5 for the GM model

using the relations

1/T,

=2@

and

I/T2

—

—

y+y'.

All

of

the plots show four curves,

C,

„

isthe conventional

OBE

with cw excitation,

C

is the conventional

OSE

with 400-psec pulse excitation, and similary

_M,

„and

M

for the modified

OBE. For

Rabi frequencies up

to

10kHZ, it is evident that, aside from an expected slight decrease

of

the holewidth, the pulsed and cw results are similar. However, at 0/2m

=30

kHz a new feature is the appear-ance

of

nutation structure near the center

of

the hole (curve M~). This structure becomes more prominent as 0/2m increases, as shown in

Fig.

2, where, for clarity, we

plot directly the Bloch vector ic(b, ). At first sight this is

a surprising result since the 400-p,sec pulse is about 20 times longer than the dephasing time

of

21.

7 @sec.

Q/27r = _{3 kHz}

Q

psec

where A,

8,

Q, and

P„are

three-component column vec-tors given by

A

=[(a

b)

+s

]

—

'[(b

+s

)E+2bF+G],

400

C=aA+bB+F,

(7)

-O.

e-

Mcw where

_P„

is the value

of

the Bloch vector with cw

excita-tion and

_g

_=Po

—P„,

_{where Po is the initial} value

of P.

Also,

F=MPO+L

and

@=M

F.

Written in this form, the coefficients may then be easily evaluated by a

comput-er.

0.

6-(b)

0/2'

= ₁₀kHz

Qp.

sec

Case2: CKroots a,b,and

c

In the case

of

three real roots,

P(h,

t)

=

A exp(

—

at)+8

exp(

bt)+

C

exp(

—

c—

t)+P„,

where

A

=[(b

c)/D][bcE+(b—

+c)F+g],

8

=[(c

a)/D][caE+(c

—

+a)F+g],

C=E

—

A

—

8,

D

=ab

(a

b)+

bc(b

—

c—

)+ca

(c

—

a)

.

It

may be readily veriffed that the vectors A,

8,

and

C

for cases 1and 2become equivalent as

s

~0

and

c

~b.

(c)

Q/2 7r= 30 _kHz

Tc "- 9

+sec

RESULTSAND DISCUSSIQN

%e

present calculations using parameters appropriate

to

the DeVoe-Brewer experiment

(Pr:I.

aF3,

T

i

—

—

500

p,sec, T2

—

—

21.

7psec). Also, for comparison with

earlier"

cw hole-burning calculations (GM model), we choose a correlation time v,

=9

psec.

b,

(krad

I

sec

)

FIG.

1. Normalized inversion

h(h)

vs detuning

6

for the Gaussian-Markov dephasing model. A correlation time ~,

=9

BRIEF

REPORTS G,/2~= 50kHz Tc = 9 _P.sec Q/2~ = 50kHz TC = _{9 fLSec} 0.

5-k 4 f I I l l I I I I l 1000

-0.

5-

-0.5-A4g (b) 0/2m. = ₁₀₀kHz Tc= _9+88C Q/2~ = ₁₀₀kHz Tc 9Ps8 00 —

0.

5-

_-0.

5-G/2m= 200kHz 9p.sec {c) g/2~= 200 kHz ~c= _{9 p.sec} 0.

5-

0.

5-00

-0.

5-

—

_0.

5-6

(kradjsec)

b,

_(krad/sec)

FIG.

2. Unnormalized inversion m(h) vs detuning

5

for the Gaussian-Markov dephasing model at high Rabi frequencies. Correlation time v;

=9

psec.

FIG.

3. Unnormalized inversion m(h) vs detuning

6

for the random-telegraph dephasing model at high Rabi frequencies. Correlation time ~,

=9

psec.

Indeed, as expected, no such structure appears for the conventional

OBE

holes. The reason is that, from

Eq.

(3),the efkctive dephasing time

I,

_,' depends both on the detuning

5

as well as the usual light intensity. In partic-ular,

I",

_,' is longest near the center

of

the hole.

For

the cw case (curves

_M,

„),

the nutation structure is absent, since the nutation must decay eventually, given enough time.

It

is

of

interest to compare these results with those us-ing other dephasing models. %'e show in

Fig.

3 calcula-tions for the

RT

model similar

to

those

of

Fig. 2.

It

is evident that the pulsed hole shapes are quite diferent from those

of

the GM model. In particular, nutation structure is now absent near the center

of

the hole, and a very weak fine-structured nutation appears in the wings, asexpected from

Eq.

(4).

CONCLUSIONS

While various theoretical treatments appear to

success-fully describe the optical saturation behavior in the DeVoe-Brewer experiment (using a range

of

correlation times

_r,

=5

—

37 p,sec), Berman '0 as well as Javanainen have noted that these descriptions are invalid either be-cause

of

a violation

of

the range

of ~,

for which the theory is valid and/or because the theories predict nonex-ponential decay, aresult that disagrees with the observed exponential decay

of

both

FID

and photon echoes.

It

ap-pears that the random-telegraph theory (with

r,

=

8

psec&T2

—

—

21.

7 psec) comes closest to explaining all the data in a consistent manner. However, more complex formulations'

of

dephasing which cannot be simply

ex-37 BRIEFREPORTS 4043

pressed asmodifjted Blochequations remain tobe numeri-cally investigated.

It

is evident that additional experiments using both fre-quency and time-domain techniques would be useful to help identify which

of

the various dephasing models (if

any) consistent1y describe optical saturation behavior in low-temperature solids. In particular, this paper demon-strates that there are striking differences in hole-burning line shapes for cw versus 5nite excitation times. In the

latter case, the holes display a DETUNING as well as the

usual intensity dependence

of

the effective dephasing time, the nature

of

which is strongly model dependent.

Finally, as noted by Schenzle et

al.

, studies

of

nuta-tion or rotary echoes'

'

should provide a further

power-ful test

of

the various models which in general predict an intensity-dependent echo decay time. Experiments using an ultranarrow (

—

1-kHz linewidth) dye laser are

present-ly in progress and will be reported on later.

'Permanent address: Shiga University„Otsu 520,Japan.

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P. Feynman,

F.

L. Vernon,

Jr.,

and

R.

W. Hellwarth,

J.

Appl. Phys. 28,49 (1957).

2R. G. DeVoe and

R.

G. Brewer, Phys. Rev. Lett. 50, 1269

{1983}.

3E.Hanamura,

J.

Phys. Soc. Jpn. 52, 2258(1983).

4M. Yamanoi and

J.

H.Eberly, Phys. Rev.Lett. 52, 1353(1984);

J.

Opt. Soc.Am. B1,751(1984).

5J.Javanainen, Opt. Commun. 50,26(1984).

P. A. Apansevich, S. Ya. Kilin, A. P. Nizovtsev, and N. S.

Onishchenko, Opt.Commun. 52, 279(1984).

7A. Schenzle, M.Mitsunaga,

R.

G.DeVoe, and

R. G.

Brewer, Phys, Rev.A30,325(1984).

SP.

_R.

_Herman and

R. G.

Brewer, Phys. Rev.A32, 2784 (1985). 9K.%'odkiewicz and

J.

H.Eberly, Phys. Rev.A32, 992 (1985).

oP.R.Berman,

J.

Opt. Soc.Am B 3,564(1986);3,572(1986). M.Yamanoi and

J.

H.Eberly, Phys. Rev.A34, 1609 (1986). N.C.%'ong, S.S.Kano, and

R.

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J.