Lasing without inversion and enhancement of the index of refraction via interference of incoherent pump processes

Optics Communications 87 (1992) 109-114 North-Holland

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Lasing without inversion and enhancement of the index of refraction via interference of incoherent pump processes M. Fleischhauer, C.H. Keitel, M.O. Scully and C. Su Center ~r Advanced Studies and Department of Physics and Astronomy, UniversiO,c~fNew Mexico, Albuquerque, NM 87131, USA and Max-Planck lnstitut ~r Quantenoptik, W-8046 Garching, Germany Received 23 August 1991

For the A quantum beat laser we investigate the generation of coherence between the two lower levels via incoherent pumping of these two levels to a fourth auxiliary level. It will be shown that this way of establishing coherence also leads to lasing without inversion and to an enhancement of the index of refraction at a point of vanishing absorption.

I. Introduction The generation of atomic coherence has been attracting considerable interest in the last few years, since it gives rise to such interesting p h e n o m e n a as lasing without the need of population inversion [ 1-14] and an e n h a n c e m e n t of the refractive index in a transparent m e d i u m [15 ]. it is well known that atomic coherence can be established by the interaction of a multilevel system with strong coherent fields. More recently, it has been shown that u n d e r certain conditions even incoherent processes may generate atomic coherence. It has been pointed out that the radiative decay of two closely lying upper levels with the same J and m j q u a n t u m n u m b e r s to a c o m m o n third level creates coherence between these levels [ 5 7 ]. However, it seems to be almost impossible to meet the necessary conditions in real atomic systems, though the existence of equivalent realizable schemes has been shown [14]. In the present paper we are putting forward an interference effect that occurs in an atomic system, when two lower levels are p u m p e d to a c o m m o n upper level by the same incoherent p u m p field. However, the two lower levels here do not need to have the same J and inj q u a n t u m n u m b e r s any more. The use of magnetic split lower levels with different m j q u a n t u m n u m b e r s does conserve the effect of absorption cancellation if we apply linear polarized light. Moreover, in contrast to the case of interfering radiative decays, the level spacing here only needs to be smaller than the spectral widths of the incoherent p u m p field.

2. Coherence via interference of incoherent pump processes Let us consider an ensemble of atoms with the level configuration sketched in fig. 1. Two closely lying lower levels b and b', for instance with magnetic q u a n t u m n u m b e r s m = 1 and m = - 1 are p u m p e d by a linear polarized incoherent field to a level a with m = 0 . Hence both transitions interact with the same excited p u m p modes establishing a coherence between b and b'. This lower level coherence may then lead to absorption cancellation and lasing without inverted level population. We here describe the optical p u m p process by the semiclassical interaction h a m i l t o n i a n Hpump= - ~ (p,e+ '¢~p [a>" " < b ' [ ) + h . c . ,

(1)

t

0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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where e+ =e, +_ie,., P~.2 are the dipole moments of the atomic transitions, and the propagation direction of the pump field is chosen parallel to an applied magnetic field in the z-dircctiom We further assume x-polarized incoherent pump light :~> = : p e , ,

(2 )

with a broad spectrum or effective/5-like correlation, i.e. < :~,(t) :pft' ) ) =Fp(~(t-t'

(3)

) .

In a first step we calculate the equation of motion for the atomic density matrix according to the free hamiltonian and the pump field interaction H p"mp. We immediately obtain (4a)

/~L, = i[ i v , / h ) :Pp~,,, - c.c. ] + i[ ( p : / ~ ) :,,p ~,., - c.c. ] .

/~,,,= - i [

(p,/l~):,,p~,,- c.c.].

Pi,,, = -i¢o:,,, p},/, q - i ( P l l h

(4b)

):~p'./,

--i(p2/h

(4c)

):pp~ ....

I)',,/, = -- i(o./,p'.:, + i (p~ : p / fi ) (p~,:,--p',,,, ) + i(p2 : p / h )p}, :, .

(4d)

I}~, /, = -- ico.,:,p~, :, -- i(Pt :;p/fi )p~....

(4e)

p. ~,=-Ico.,.p • I

'

.... - i ( p ~ :~,/t~ ) p , , . / , - i ( p ~ : p / f i t

i

)p.

:~1,

¢

:,

.

(4f)

and similar equations for p,,:, and p,,,. We formally integrate these equations and insert the results back into eq. (4). This yield for polarizations p~,:,, and P',, :,: fi~+ ( / ) = -i~o:,, p~,:, + (ith)~p~ i dr :)*(t) :r(r)p~+. exp[-i~o,,:, ( / - r) ] 0

+ (i/h)~P~ i dr :T,(t) :~,(z) P~,:, exp[i~o,,:,(/- r) ] 0

+ ( i / f i ) e p , p2

d r {c~;(t) : ; ( r )

[p},.. ( r ) - /

..(r) l exp[-i~o.~, (t-r)]

0

+4,(t) :;(r)

exp[io).:,(t-r)]I .

[p~+(r)-p'..(r)]

(5)

tS'.:,(t) = - i ( o . / , p ' . : , + (i/,5)~ p~ i dr :p(t) : p ( r ) p 1 , / , ( r ) e x p [ - io.). o ( t - r ) ] 0 t

+ ( i / h ) ' - p , p~ j dr :p(t) :)*,(r) p',,/, (z) e x p [ - i ~ o . , o ( t - ' c ) ] .

(6)

o

We now average over the classical pump field and obtain, using the correlation properties, eq. (3), p •~ +

(t)=-"

N.,,(t)=

I60:,;,PH, , _ ! (2 r,-kr2)p},;,.--½\/7, r2 (p /,t, , + p :, , l,. -- 2p ,,, , ) , " ' " ' / , - 2' V / '' . 1. '.2. . .P a ' t , --I(O,, /,Pu/,-~/tP,~

,

(7) (8)

where r,.: ~ ( p ~ . : / h : ) F ~ , .

110

(9)

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15 January 1992

,

t

/

\

\

Fig. 1. Two closely spaced magnetic sublevels b and b' with ms= + 1, are pumped to a level a with an x-polarized incoherent field. The interaction of both transitions leads to coherence between b and b'.

,

6 b

~I,

-

The equations for the other atomic density matrix elements can be obtained in a similar way. Including the decay processes as indicated in fig. 1 and the interaction with a (x-polarized) laser mode according to the interaction operator H m ' - - ~ [P', ~ l a ' 5""
(10)

i

with p] and p'2 being the dipole matrix elements for the a ' - ~ b and a' ~ b ' transition, respectively, we finally obtain the set o f density matrix equations: p~o- - (F,+F~+yo)p~

' + r2pl,'h" i + x / ~ r 2 ( p i ,~ ,+p~ ,,~ ,) + r, pl,h

(lla)

,

• i J i Pa,~, ' +i(g2,p~,a, - c . c . )+i(~22pi,,~,-c.c.) , = - ( Y ] +72)Pd~. + 7oP.~ •

i

"

:~

(lib) i

p~,,,, - - [ ioJt,,,. + ½(r, + r 2 ) ]p~,,,, - ~ x / ~ r 2 (P~,b +P~,v,. ) - 2 p ~ . + ig2,p.,,,, -i-(22pi,.,

"'

Pa'h

~-

""

"' P~,h. = -

[izl,

+ ~ (,y ,

[iA~+'(y~

, +y'2+r,)]p~,~-½~f~, +

-, 7'2 + r~) ]p.,a, - ~,

r2p..h, + l.(2j (Pt,' b --P,ca' ) + i~22p~,t, -' " ' -, r~zr~po,h + i122(p~.h, --p'~,.. ) + it2,p'bh, ,

( 1 lc) ( 1ld) ( 1 l e)

where ~ = 7 ~ + r j , for j = 1 and 2, and we have introduced slowly varying quantities ~'= do e x p ( - i o g t ) ,

g2j=p~o/h,

p~,,/,=/~'o,t, e x p ( - i o J t ) , P~,/,, =/~,,h, e x p ( - i o J t )

(12) (13)

and A, = ~ , 1 , - o 9 ,

32=o9,,i,,-o9.

(14, 15)

It, moreover, has been assumed that just one x-polarized mode o f frequency ~o and electric field strength d' can develop. These equations are very similar to the corresponding equations for the case o f interfering radiative decays [ 5 ]. Instead o f the interference terms x f ~ 72 in this case we here have ~ r2 indicating that the quantum interference is due to the p u m p process. We notice, in particular, from eq. ( 1 lc) that a steady state population difference between the lowest levels b and b' and the upper level a is always combined with a coherence between these levels• Furthermore, we recognize from eqs. ( 1 l d) and ( 1 l e) that the polarizations p,,~, and P,,,h, which drive the laser field g are correlated. The lower level coherence together with this polarization correlation can lead to inversionless lasing and to an high refractive index without absorption.

3. Lasing without inversion and enhanced refractive index without absorption

The main purpose o f this section is to prove that the system introduced above may lead to inversionless lasing 111

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and to an absorption-free medium with enhanced refractive index. We, therefore, calculate the polarization of the system to the first order in the optical field. Restricting ourselves to the important special cases p~ = P 2 - P and p'~ =P2 =-P', we make the reasonable assumptions rl=r2=--r,

:.,1=72~y,

Ft=F2-=F,

7~=y2=--7',

~QI=-Q2~.

(16)

The existence of just one mode can be guaranteed by assuming a polarizer which allows,just x-polarized light to develop. Because of the symmetry of the system and the imposed conditions, the zeroth order populations in b and b' are equal. Hence the zeroth order solutions can easily be derived from eqs. ( 1 l a - c ) : p,.))

(7o/27,)p,O)

pi~o) ""

=

PI'I°)

=

2v+

Vo

(17)

2r oo~,;, ~ +r2P - 2 r / ( :'~ o T , ' b + r , ) {o~,~,,

i,~P )

(18)

r

. + r - . ( r - i t ° ' " " )(Pi'~°)
- p . o ,o) ) .

(19)

The o p t i m u m lasing condition is given, if yo-*oo, where p,O) tends to zero. The population of the upper level o f the optical transition is thus given by ni<,o?

~-a ,,

- -

r ~o;,~, v--,pi,~? ~" ~07,/,, + r -

)

(20)

"

This indicates that the population is not inverted as long as re +oJ~,,,, -- ( r / y ' )~o~,,, >~0 .

(21)

It is worth noting that in the case 70-+00 the condition 7' >~ I ~om,,I/2 then renders inversion impossible even for arbitrary r values. The complex polarization, P, associated with the atomic transitions a' --+b and a' -~ b' is given by p ( l ) _- ~

n,~¢/'~i(I) ~_n/(I)

u

~t',,'/,

-v.'~,'

) =NP'*(P~,,!l~

(22)

~t-'.'t,',±-{I) ", ,

i

where the N atoms in the sample are assumed equally polarized and the superscript i has thus been dropped. From eqs. ( l I d ) and ( 1 le) we can obtain the first order polarizations p . ~ and p~.!~, yielding p ( i ) _- - i N I p ' I 2~ ~o) ~o) ~D*D [id2(Pl'°)-P"'"'+P'"")+izl'(pl'9'}

_n~O), Y a a

+P},~,))+Y'(Pl,,,

o)

+PI,!5~-

.~.~o) d'Pa'a'

+Pl,°?+Pl,9,?)] D*

"

(23) where D = i ( A t + 3 2 ) (7' + r / 2 ) + y ' E + y ' r - d I A2 .

(24)

Introducing the detuning A = (~o. ~,+ co..~,. ) / 2 - ~ o of the laser field with respect to the midfrequency of the two transitions a' - , b and a' -~b', the real and imaginary part of the susceptibility z=P/~oe~ read as ;( = IP' %hD*D Z,, =

, , [(2d--~°7'l"/2)(l-r/Y')+(27'2+r2)]

[P'~ot~D*DI 2{°),wh/, v" [2A2r2/7, + ( ( O ~ , h . / 4 + y , 2 + y , r + d 2 )

to ~,~, to,q," + r 2 , (r-- 2y' ) ] ~o7,/,.v--~o[,/,,+r~".

(25a) (25b)

In fig. 2 we display the imaginary part of the polarization as a function of d for different values of the p u m p 112

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• ld 16

~10-15

8

6

/ \ /,' ;,

X

X"

4

4

-4

x

- -

x"

- ......

-8

-2

13

' -13 '

'

' :4

'

~

'

0

'

4

13

A/~f ' Fig. 2. The imaginary part of the susceptibility as a function of the detuning Aof the laser field with respect to the midfrequency (oJo.~,. +o~,v,)/2. The susceptibility is normalized to an atomic density of 1 atom per cm3, hence the absolute value has to be multiplied by the number N of atoms per cm3. We have assumed here radiative decay, so that 7' =4[p' [292/3%~23. From top to bottom the pump rates have the values r=0.0, 0.2, 0.5, 1.0, 1.5, 3.0 7'.

-

'

i

,

t

-4

,

r

L

~

,

0

~

,

1

4

,

i

,

J

1

i

8

A/~f' Fig. 3. The real (line) and imaginary (dashed) part of the susceptibility as a function of A. The transition frequency between the two lower levels is I~ob~,J=27' and the pump rate r= 1.27', We see, that at a point a the system is transparent while having a large real part of the susceptibility.

rate. Negative values of Z" indicate gain whereas positive values correspond to effective loss. Here we have considered the case [~o~,j,.I = 2 7 ' , so that according to eq. (21) the population of the level a' is always less than 1/3. Thus, fig. 2 displays lasing without the need of inverted level population. In fig. 3 we display the real and imaginary part of the susceptibility as a function o f d for t0~,,~,=27' and r = 1.27'. We would like to emphasize that at the point a of vanishing imaginary part of Z, m e a n i n g a transparent medium, the real part Z' and hence the index of refraction is still substantial.

4. Conclusion The subject of this letter has been to provide a realistic scheme in which coherence is generated via incoherent processes. Two magnetic field split lower levels are p u m p e d by a linearly polarized incoherent field to one upper level, where the linewidth of the p u m p field is large enough to cover both transitions. The induced coupling between the lower levels then gives rise to lasing without inversion and to an absorption free m e d i u m with an ultra-high refractive index, effects that so far have been realized only via coherent processes. The close relationship between this work and the generation of interference via spontaneous decay coupling has been pointed out. Moreover, we would like to emphasize the m a i n advantages of the incoherent p u m p i n g coupling in comparison with the case of interfering radiative decays: the possibility of generating coherence between magnetic sublevels, and hence the possible realization of the scheme in real atomic systems and the absence of limitations in the spacing between the lower levels. The coupling via spontaneous emission requires more stringent conditions, namely the presence of close lying levels with the same J and m j q u a n t u m numbers, conditions which seem impossible to fulfil in natural systems. The reasons for these strong conditions are the following: the two upper levels need to be closely spaced in order to couple to the same v a c u u m modes at the same strength. The two levels moreover need to have the same q u a n t u m numbers, since otherwise the interference terms have 113

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d i f f e r e n t signs f o r d i f f e r e n t p o l a r i z a t i o n d i r e c t i o n s o f t h e field a n d h e n c e c a n c e l e a c h o t h e r . S u b s t i t u t i n g t h e c o u p l i n g to t h e v a c u u m m o d e s w i t h a r b i t r a r y p o l a r i z a t i o n in t h e case o f s p o n t a n e o u s e m i s s i o n b y c o u p l i n g t o a n i n c o h e r e n t p u m p field w i t h a well d e f i n e d p o l a r i z a t i o n , as d o n e here, r e n d e r s it p o s s i b l e to g e n e r a t e i n t e r f e r e n c e b e t w e e n t r a n s i t i o n s f r o m m a g n e t i c s u b l e v e l s .

Acknowledgments T h e s u p p o r t o f t h e O f f i c e o f N a v a l R e s e a r c h is g r a t e f u l l y a c k n o w l e d g e d .

References [ 1 ] S.E. Harris, Phys. Rev. Len. 62 (1989) 1033. [2] M.O. Scully, S.Y. Zhu and A. Gavrielides, Phys. Rev. Lett. 62 (1989) 2813: M.O. Scully, Noise and chaos in nonlinear dynamical systems, in proc. NATO Advanced Research Workshop, Torino, Italy ( 1989 ). [3] S.E. Harris and J.J. Macklin, Phys. Rev. A 40 (1989) 4135. [4] A. Lyras, X. Tang, P. Lambropoulos and 1. Zhang, Phys. Rev. A 40 (1989) 4131. [5] A. lmamo~lu, Phys. Rev. A 40 (1989) 2835. [6] A. lmamo~lu and S.E. Harris, Optics LeU. 14 (1989) 1344. [7] S.E. Harris, J.E. Field and A. lmamo~,lu, Phys. Rev. Len. 10 (1990) 1107. [8] S. Basile and P. Lambropoulos, Optics Comm. 78 (1990) 163. [9] S.Y. Zhu and E. Fill, Phys. Rev. A 41 (1990) 5684. [ 10] E. Fill, M.O. Scully and S.Y. Zhu, Optics Comm. 77 (1990) 336. [ 11 ] O.A. Kocharovskaya and P. Mandel, Phys. Rev. A 42 (1990) 523. [ 12 ] O.A. Kocharovskaya, R.D. Li and P. Mandel, Optics Comm. 77 (1990) 215; O.A. Kocharovskaya and Ya.l. Khanin, Zh. Eksp. Teor. Fiz. 90 (1986) 1610. [ 13] L.M. Narducci, H.M. Doss, P. Ru, M.O. Scully, S.Y. Zhu and C, Keitel, Optics Comm. 81 ( 1991 ) 379. [ 14] M. Fleischhauer, C.H. Keitel, L.M. Narducci, M.O. Scully, S.Y. Zhu and M.S. Zubairy, in preparation. [ 15 ] M.O. Scully, in preparation.

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