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INHOMOGENEOUS DEPHASING INHIBITION BY A STRONG STOCHASTIC FIELD
A. Debarre, J.-C. Keller, J.-L. Le Gouet, P. Tchenio
To cite this version:
A. Debarre, J.-C. Keller, J.-L. Le Gouet, P. Tchenio. INHOMOGENEOUS DEPHASING INHI-
BITION BY A STRONG STOCHASTIC FIELD. Journal de Physique Colloques, 1988, 49 (C2),
pp.C2-259-C2-262. �10.1051/jphyscol:1988261�. �jpa-00227678�
JOURNAL DE PHYSIQUE
Colloque C2, Supplbment au n06, Tome 49, juin 1988
INHOMOGENEOUS DEPHASING INHIBITION BY A STRONG STOCHASTIC FIELD
A. DEBARRE, J.-C. KELLER, J.-L. L E GOUET and P. TCHENIO
Laboratoire Aim6 Cotton, CNRS 11, Bdt. 505, Campus d l O r s a y , F-91405 Orsay Cedex, France
R6sum8
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On s1int8resse 3 la formation ae signaux optiques transitoires coherents produits, l o r s de la r e n i s e e n phase de dipoles cr66s sous l t a c t i o n d'une impulsion lumineuse l a r g e bande. Un e f f e t dt inhibition du d6phasage inho m ogene e s t attendu qui repose sur 1' hypothese d' absence de corr6lation 3 long t e r m e dans le champ excitateur. Le rnodele theorique ainsi d6veloppe e s t appliqu6 au c a s particulier des 6chos de photons stimul8s e n configuration non colin6aire. L t e f f e t dtinhibition pr8dit e s t observk dans une vapeur de calcium.Abstract
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We examine t h e formation of coherent transient optical signals produced by t h e rephasing of dipoles which a r e built up by a strong broadband light pulse. The inhibition of inhomogeneous dephasing by a strong stochastic field is predicted with provision t h a t t h e l i g h t field satisfies t h e snort-time correlation condition. The model is applied t o t h e specific situation of stimulated photon echo in a n angled beam configuration. The expected inhibition e f f e c t is demonstrated in a calcium vapor.I
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INTRODUCTIONSome years ago, inhomogeneous broadening inhibition has been evoked t o interpret t h e observed relaxation of optical signals induced by a n intense coherent light /1,2/. The processes under investigation question t h e e f f e c t s of saturation on t h e broadening which results from a specific stochastic perturbation. The f i r s t experimental evidence concerns t h e inexpectedly slow decrease of t h e Free Induction Decay signal produced in a n impurity ion c r y s t a l ~ r : +LaF3 under strong resonant ~ excitation /I/. The flip-flop motion of t h e F nuclear spins originates t h e random fluctuation of t h e resonance frequency of t h e ~ r + ~ i o n s . A similar situation h a s been explored i n vapor phase where t h e relevant fluctuation of t h e atom resonance frequency is produced by collisional velocity changes / 2 / . In both cases, excitation is provided by a monochromatic field. These experiments adtess t h e t h e o r e t i c a l problem of t h e ability of t h e Optical Bloch Equations (O.B.E.) t o describe relaxation phenomena under strong field conditions. This problem is of importance as regard t h e generation of optical coherent t r a n s i e n t since t h e optimum signal intensity is generally expected t o be reached under l a r g e field strength.
In such processes, a sequence of l i g h t pulses resonantly e x c i t e s a n optical transition which connects atomic s t a t e s a and b. The elementary fields r a d i a t e d by t h e oscillating dipoles built o n t h e coherences pab, interfere coherently t o produce t h e transient signal. Depending o n t h e inhomogeneous broadening, e a c h dipole is created with a given initial phase which evolves all along t h e process, and t h e signal intensity is directly r e l a t e d to t h e rephasing of t h e dipoles. Nevertheless, t h e specific situation which is examined in t h e present paper differs ffom t h a t of t h e previously mentioned experiments as regards its stochastic character. Indeed, t h e stochastic perturbation concerns
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988261
C2-260 JOURNAL
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PHYSIQUEt h e fleld itself (since broadband excitation is considered) and t h e broadening which is assumed t o r e s u l t from Doppler effect. I n contrast t o t h e experiment of De Voe and Brewer, all t h e a t o m s undergo simultaneouSLy t h e s a m e fluctuations. The ultimate t e m p o r a l resolution is limited to t h e correlation t i m e T, of t h e source, which may be t o much s h o r t e r t h a n t h e pulse duration TL. T~
identifies with t h e Inverse s p e c t r a l width wL*'. Coherent excitation c a n be produced over a l a r g e inhomogeneous s p e c t r a l width. The work demonstrates t h e ability of a strong broadband Meld t o inhibit inhomogeneous dephasing. I n t h e excitation sequence only one pulse is ? + r o q , with no correlation with o t h e r pulses. This e f f e c t is experimentally demonstrated in a calcium vapor using a stimulated photon echo configuration.
II
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THEORETICAL APPROACHThe studied signal is e m i t t e d by t h e o p t i c a l dipoles which originate from t h e conversion of t h e atomic l e v e l population nab(A,t=O) i n t o coherences pab(A,t) under t h e action of t h e strong probe pulse. The initial velocity dependent phase f a c t o r depends on t h e excitation history before probing. Due t o t h e stochastic c h a r a c t e r of t h e field, t h e O.B.E. which describe t h e evolution of population and coherences contain random coefficients. Only s t a t i s t i c a l averaged data a r e meaningful, which implies many realizations of t h e broadband driving field. The t h e o r e t i c a l approach aims a t expressing t h e two-atom but single t i m e correlation functions 8(A,Q,t)=<pab(A,t)pab(A+Q,t)>
*
which e n t e r t h e expression U(t)> of t h e signal :< ~ ( t ) > a
J I ~ A
~ S Z o(8,n.t) (1 )The broadband field is considered as a stationary gaussian random variable. The procedure uses t h e standard shortrtime correlation techniques /3/. It assumes t h a t t h e field strength which is characterized by t h e averaged Rabi frequency X, is restricted .to t h e range XT < I . An analytical expression of t h e correlation function may be derived for a pulse of finite duration. In t h e strong field l i m i t f o r which T - ' = ~ ~ T ~ > A ~ , it t a k e s t h e following simple form :
1
e ( ~ , a . t ) = l <nab(A,t=O) nab(A+n,t=O)> e -2/3n2tTT < t < r ~ (2)
2 3
where A D is t h e Doppler width. This result expresses t h a t t h e efficiency of t h e strong broadband field t o inhibit t h e dephasing is measured by t h e slow decaying f a c t o r e - 2 / 3 n 2 T ~ T . The phases a r e frozen over frequency domains of t h e order of X ~ < / ~ L f o r which Q2rLTS1. When t h e very strong field condition X ~ < / ~ L > > ~ D is valid, t h e dephasing inhibition is c o m plete and their inho m ogeneous broadening is negligible.
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G E O M E T R I C A L I N T E R P R E T A T I O N O F . D E P H A S I N G I N H I B I T I O N B Y A S T R O N G S T O C H A S T I C FIELDThe relative dephasing of t w o given dipoles c a n be accounted f o r by t h e t e m p o r a l evolution of their angular separation of t h e corresponding Bloch vectors $ ( ~ , t ) and $(A+n,t). This evolution is depicted by a diffusion model of t h e vector headings on t h e Bloch sphere /4/. The angular evolutions of t h e vectors obey random walk laws which express t h e stochastic c h a r a c t e r of t h e field
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In this frame, when X T , < ~ , t h e meaning of t h e parameter T is readily cleared up. After a t i m e interval T has elapsed, a given Bloch vector has l o s t memory of its initial direction but a correlation subsists between the relative evolution of t w o Bloch vectors which make a n angle nT. This angle grows with a random walk of s t e p T. After TL/T steps, t h e final angular separation is S Z T J T L / T
- 2 / 3 P 2 7 ~ T
which demonstrate t h a t t h e decay law of t h e signal is directly related t o t h e magnitude of t h e spread of t h e Bloch vector headings. When t h e angle SZT J ~ is T close t:, unity, t h a t is when Q 2 T ~ L z 1
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t h e correlation between t h e vector evolutions is l o s t and t h e signal drops. On t h econtrary when t h e condition
xJT,/T~>>AD
is fulfilled, a l l t h e Bloch vectors a r e locked t o g e t h e r and relaxation is negligible.This geometrical picture is reexamined to account f o r t h e puzzling case xrc>l which is more appropriate t o comparison with t h e case of a strong monochromatic excitation. In this situation, no analytical solution of t h e O.B.E. is avalaible. A given Bloch vector $(A&) rapidly precesses around t h e direction of t h e field which is assumed t o be constant during t h e elementary time interval T,.
The precession Pequency is 1/x and t h e angular separation which builds up between $(A+o,~) and $(b,t) is at most n / ~ . Following a random walk of t e m p o r a l s t e p T,, a t t i m e TL, t h e final angle between them is w ~ ~ ~The s a m e l a w governs t h e spread of Bloch vectors headings i n both cases x.rc<l / T ~ .
and xrc>l. Thus t h e s a m e relaxation r a t e is expected to be observed f o r t h e signal. To sum up t h e geometrical picture predicts t h a t field inhibition should occur even when t h e s h o r t t i m e correlation condition is ignored.
I V
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EXPERIMENTAL RESULTSThe t w o f i r s t t,, delayed pulses of t h e stimulated photon echo sequence a r e weak. They a r e tuned o n resonance with t h e transition 4s' 'So-4s4p3 P
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of calcium atom. They carve a periodic modulation inside t h e initial velocity distribution of t h e l e v e l population difference.n a b ( A t t = 0 ) = n ~ b ( A ) C ~ ~ ( 2 7 1 t 2 I h 1 )
Under t h e action of a third broadband pulse, t h e memory of t h i s modulation is retained i n t h e initial phase f a c t o r e2"vt12'X1 of t h e dipoles which radiate t h e transient signal a t t i m e t,, h,/Al. The probing field is tuned t o t h e frequency of t h e 4s4p3Pl-4s5s3Sl transition. A s a function of t,,, t h e coherent signal is expected to evolve P o m a dissymetrical shape under weak field conditions t o a s y metrical shape around tl,=O in t h e strong field l i m i t (AD2rLT<<1). In this l a t t e r case t h e expression of t h e signal r e a d s :
and t h e measurement of t h e signal width o f f e r s a n access t o Doppler width
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The intensity of t h e e c h o is recorded a s a function o f t,, by continuously varying, t h e delay between t h e f i r s t t w o pulses.This is done by t h e use of a motorized optical delay line composed of t h r e e corner cube retroreflectors. An angle beam configuration is used and t h e e c h o is Uetected i n t h e direction of t h e second pulse I?, (Figure 1). The illustration of t h e predicted e f f e c t s is depicted on t h e figure 2 where solid lines represent t h e experimental profile, and dotted lines t h e calculated ones. The decay of t h e e c h o intensity is i n good agreement with t h e theoretical model when x-iC<1 and t h e expected field inhibition Is also Observed in t h e case Xr,>l.
The experiments described a t t h e beginning of t h e paper aimed a t demonstrating the abjlity of a strong monochromatic field t o inhibit a n inhomogeneous dephasing in t h e presence of a stochastic perturbation. In t h i s case each atom undergoes a specific fluctuation. The present work put into evidence t h e ability of a stochastic strong field t o inhibit inhomogeneous dephasing. The next s t e p in t h e connection of t h e t w o problems requires t o extend t h e investigations t o t h e situation where both t h e field and the atom parameters fluctuate.
JOURNAL
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PHYSIQUEFigure 1 : -1- l e v e l s c h e m e : (a) t w o-level system, (b) three-level system, -2- t h e excitation pulse sequence : te is t h e t i m e emission of t h e e c h o ; in t h e case (a) te=t3+tl,, in t h e case (b) testr+% t l
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-3- echo direction switching in a n angled-beam configuration : de=d3+d,-dl, d~e=dsA?l-d2. WFigure 2 : Comparison between experimental a n a t h e o r e t i c a l profiles. Curve 1 : weak field l i m i t
~ = 7 . 7 5 10's-I ; Curve 2 : strong field case x ~ ~ < l , ~ = 7 . 7 5 1 0 ~ s - ' , t h e dashed curved is deduced P o m equation 2 ; Curve 3 : very strong field c a s e X T , > ~ . ~ = 7 . 7 5 101Os-l, t h e dotted dashed curve is t h e square of t h e Fourier transform of t h e initial velocity distribution (eq. 3). The dashed a x e s denote t h e t h e o r e t i c a l position f o r t l , = O .
REFERENCES
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1269 (1983) ;52,
1354 (1984)./2/ Yodh A. G., Golub J., Carlson N. W. and Mossberg T. W., Phys. Rev. Lett.
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.659 (1984)./3/ Van Kampen N. G.. Stochastic processes in Physics and Chemistry, Chapter X I V , North Holland ( A msterdam) (1981).
/4/ ~ c h d n i o P., Dibarre A., Keller J. C. and Le Gouet J. L, submitted t o Phys. Rev. A.
