Transient excitation and gated or delayed observation provides a narrowing of the inherent line width of the transition. Possible experimental one- and two-photon systems are discussed and the loss of signal with line narrowing is calculated. The general case of lines broadened by dephasing processes is treated. Contact is made with the line narrowing in recent Raman type experiments

Carman

Coleman

Copley

Figger

Giordmaine

Laubereau

M.C. Nuss

Metcalf

Meystre

Shimizu

W. Kaiser

W. Zinth

Walther

Zinth

English

Crossref

Narrowing of spectral lines beyond the natural or dephasing line width

Zinth, Wolfgang

Nuss, M. C.

Kaiser, Wolfgang

Open Access LMU

Volume 44. number 4 OPTICS COMMUNICATIONS 15 January 1983 NARROWING OF SPECTRAL LINES BEYOND THE NATURAL OR DEPHASING LINE WIDTH W. ZINTH, M.C. NUSS and W. KAISER Physik Department der Technischen Universitil~ Mfinchen, M~nchen, Fed. Rep. Germany Received 11 October 1982 Transient excitation and gated or delayed observation provides anarrowing of the inherent line width of the transition. Possible xperimental one- and two-photon systems are discussed and the loss of signal with line narrowing is calculated. The general case of lines broadened by dephasing processes i treated. Contact is made with the line narrowing in recent Raman type experiments. 1. Introduction In spectroscopy one is always faced with the prob- lem of spectral resolution. One wishes to determine the frequency of the quantized transition with highest accuracy and one is interested in detecting fine struc- ture or neighboring transitions. Disregarding experi- mental factors, the ultimate resolution is dictated by the inherent linewidth of the transitions. In optical spectroscopy we know various line-broadening pro- cesses. We point to the Doppler effect or to collision broadening in gases, to dephasing processes in con- densed phases and to the population lifetime which gives rise to the natural linewidth. In recent years different novel techniques have been devised which provide spectral resolution beyond the transition linewidth. For instance, Doppler broadening can be eliminated by saturation spectroscopy or by two counter-propagating beams for two-photon transitions [1 ]. Even measurements beyond the natural ine width have been performed taking biased signals from the fluorescent decay [2 -  5]. Techniques have been proposed where the differ- ence between the decay rates of the two states rather than their sum determines the linewidth [6,7], and narrowing of the natural inewidth by decaying-pulse excitation has been discussed [8]. Very recently we have demonstrated substantial line narrowing of Raman type transitions in con- densed phases [9,10]. The lines were broadened by vibrational dephasing. It is the aim of this article to show new possibilities of line narrowing in coherent transient experiments. In our treatment we include dephasing processes. Without dephasing our results are applicable to line narrowing beyond the natural linewidth. 2. Theory. We note that the spectral resolution in a transient experiment is not limited by the lifetime of the in- vestigated levels, but is determined by the specific experiment. Under favorable conditions the observed line may become substantially narrower than the spontaneous width measured in a steady-state experi- ment. We consider an ensemble of two-level systems with ground state ib) and excited state la) at the fre- quency difference co o . The interacting driving force at the frequency u is denoted by A( t ) .  For the ele- ments of the corresponding density matrix p we ob- tain the equations: Paa = -TaPaa - iA( t )  (Oba -- Pab) , (1)  Pbb = --TbObb + i A(t)  (Oba -- Oab) , (2) Pab = --(it°0 + 3'ab)Pab -- iA ( t )  (Pbb -- Oaa) • (3) 262 0 030-4018/83/0000-0000/$ 3.00 © 1983 North-Holland Volume 44, number 4 OPTICS COMMUNICATIONS 15 January 1983 In the following discussion the driving force A(t) is caused by electro-magn.+etic fields. For one-photon or two-photon processes A(t)  is proportional to E(t) or Ei(t ) L/(t), respectively. The diagonal elements Paa and Pbb describe the population of the two levels with damping constants 7a = l/T1 and 7b, respec- tively. Here we only treat the case of small excitation where Pbb ~ 1 >> Paa" The off-diagonal elements Pab = P~a represent the coherent excitation of the system. Pab relaxes with the dephasing time T 2 = 2/ Tab. The dephasing time is made up of energy relax- ing, T 1 , and pure phase disturbing processes, Tph : I /T 2 = I/2T 1 + 1/Tph. Note, that Tph and T 2 are decay constants of amplitudes, while T l corresponds to the decay of energy (population). The spectral width of a homogeneously broadened transition ob- served in stationary experiments is determined by the dephasing time T 2 , APspon t = 1]TrT 2 . Phase disturbing collisions make the spectral lines broader than the natural inewidth Auna t = 1/21rT 1 . The spectral resolution may be improved beyond the limit imposed by the spontaneous linewidth Auspon t by coherent transient interaction between an electromagnetic f eld and the atomic system. We introduce an observable quantity, the expectation value of the transition operator <r) = rab(Pab + Pba)" It has been shown that (r) obeys the equation of a damped harmonic oscillator with driving force A(t) [111. One obtains from eq. (1) to (3): (h) + ( 2/r2)(r) + co2(r ") cc "A(t) . (4) Introducing plane waves for A'(t) and (r) with slowly varying amplitudes propagating in the x-direction we write: = (1/2)A(t) exp(-iut + ikAX ) + c.c. (5) (r) = (1/2)R(t) exp(-icot + ikRX ) + c.c. (6) and obtain 3R/Ot + [i(co 0 - co) + 1/Tz]R = KA(t).  (7) K stands for a proportionality constant, co is the mo- mentary frequency of the transition amplitude. During the excitation process we have co = ~,, i.e. the system is driven off resonance by Aco = co 0 - u. Eq. (7) is readily integrated to give R(t,  Aco) = K exp(-t /T2)  t × f exp[(iAco + 1/T2)t']A(t' ) dt ' .  (8) - -oc  Depending upon the specific experiment R(t) can be observed irectly or may be interrogated by an addi- tional probing light field Ep(t). For a one-photon resonance,R is proportional to a radiating polarisa- tion and tR 12 is measured, for instance, in a trans- mission experiment using a subsequent pulse. For a non-radiative transition (e.g. a Raman process),R may be monitored by coherent scattering of a probe field Ep. The scattered field E s is deduced from Maxwell's equations containing the nonlinear polarisation pNL ccR(t) X Ep(t) as a source term. It has been shown in ref. [12] that the growth ofE s may be written: OEdOx ~ R(t) X Ep(t). (9) 3. Principle of sub-natural linewidth spectroscopy Up to this point we have treated the atomic system neglecting the question of the spectral resolution. From text books on quantum mechanics we learn that Heisenberg's uncertainty principle limits the ac- curacy of measuring spectral features by the relation: AEX At>~h or AcoAt ~> 1 , (10) where AE, Aoo, At denote the inherent uncertainties of the simultaneous measurement of energy (co fre- quency) and time. In practical experiments he quantities A£" or Aco are directly observed, e.g., at the image plane of a high resolution spectrographic device. The time uncertainty At requires ome considera- tion. It is determined by the experimental observation time and the time constant of the quantum mechani- cal system. In steady-state experiments At is deter- mined by intrinsic time constants: (i) The energy relaxation time T 1 which is obtained from the observation of the exponential decay of the occupation of the upper quantum state. (ii) The phase-relaxation time Tph defined by the phase disturbance of the atomic state, e.g. by colli- sions. 263 Volume 44, number 4 OPTICS COMMUNICATIONS 15 January 1983 Combining the two relaxation rates one obtains the quantum mechanical observation time At ~ T2, where lIT 2 = (1/2T 1 + 1/Tph). Spectral resolution is improved by increasing the observation time beyond T 2 . Two new approaches to the problem using transient coherent experiments are presented in the following sections (4.2 and 4.3). 4. Spectral resolution of different experiments We now wish to discuss the application of eqs. (8) and (9) to several specific experimental situations. 4.1. Steady-state spectroscopy A stationary driving force with amplitude A 0 and frequency uexcites the quantum system. One mea- sures the absolute square of the transition amplitude [R 12 (e.g. the absorption) at some arbitrary time T tuning the frequency v through the resonance w0. Eq. (8) yields with Aco = co 0 - v tR(Aco)I 2 = KA2/[Aco 2 + (l /T2) 2 ] . (11) As expected, in the steady state we obtain a lorentzian line where the line width (FWHM) is equal to the spon- taneous width APspon t = 1/nT 2. 4.2. Transient excitation using a gaussian shaped pulse The quantum system is excited by a pulsed driving force of tunable carrier frequency u and of gaussian temporal shape A(t) = A 0 exp [-(t/tp)22 In 2]. As shown below the pulse duration tp should be longer than the dephasing time T 2 of the excited system. Using eq. (8) we obtain [R(T, Aw)l 2 = ~:A 2 exp [-2T/T 2-(te/tp) 2 4 In 2] I f  • , , 2 (dlt;l 2 X exp(aAwt )exp[--l(t -te)/tp[ 2 In 2] _oo with t e = t2p/(T2 4 In 2). The time T corresponds to the time of observation. For late times T the integral of eq. (12) is proportional to the Fourier transform of the gaussian driving pulse;i.e., the frequency de- pendence of [R 12 has the same gaussian shape as the driving force. Or, in other words, the observed fre- quency bandwidth is determined by the pulse dura- tion tp and not by the dephasing time T 2 (as in the preceding section). Experimentally one observes [R 12 as a function of the frequency detuning/xw of the driving pulse and finds a bandwidth Avco h at a de- layed time T. We introduce the ratio V = AUspont/AVcoh which defines the spectral narrowing found in the transient experiment discussed here. For late times T, AUco h is determined by the gaussian pulse of duration tp and one obtains V = AVspont/AUGauss = tp/1.4 T 2 ; accord- ing to this relation one observes a line narrower than the spontaneous one for tp > 1.4 T 2. The question ow arises concerning the magnitude of JR[ 2 at a late observation time T. In fig. la the ratio of the observed linewidth AUco h to the spectral width of the gaussian pulse APcoh/APGauss i  plotted versus T/tp for different values Av spont /APaauss (i.e. for different T2). As seen from fig. la, APcoh/APGauss = /, o -" ® -g o 3 3 .J z. lo 1 2 3 o Time T/ tp  1~-  I I I I -2 i 164 1 2 3 Spectral  Narrowing ~,Vspont/AVGaus s Fig. 1. (a) Narrowing of observed linewidth versus time of delayed observation. The exciting gaussian pulse has a width auGaus  and a duration tp. The parameter is given y AUspont/~VGau~s where. ~Vspont is the spon)aneous. teaay-state) dnewldtn. (b) Normanzeo peak oi s~gnm ttran- sition amplitude squared IR(T)I 2) versus pectral narrowing. 264 Volume 44, number 4 OPTICS COMMUNICATIONS 15 January 1983 approaches unity for sufficiently large T/tp values. Larger line narrowing V requires later observation times T. The points on the broken line mark the T/tp positions where the observed linewidth Avco h is 10% longer than the linewidth At )Gaus  s of the applied gaussain pulse. As an example we discuss the case of V = 4 and T/tp = 3. With pulses of tp/> 5.6 T 2 and for a delayed observation time of T = 16.8 T 2 one obtains a line narrowing by a factor of 3.6. According to eq. (12) the signal JR(T, ACO)12 de- creases trongly for later observation times T. In fig. lb the peak signal intensity at Aco = 0 is plotted ver- sus the spectral narrowing At)spont/At)Gaus s fora con- stant dephasing time T 2 and for a constant amplitude A 0 of the driving force. Calculations were made for At)coh/AVGaua s = 1.1 (see broken line in fig. la). Larger spectral narrowing ives rise to strongly re- duced signal intensities. In practical cases, the opti- mum spectral resolution is determined by the signal to noise ratio of the experimental system. The accurate determination f the absolute value of the transition frequency coO requires both, a narrow spectral response of the excited system and a high pre- cision of the tunable xcitation frequency v. The ac- curacy of the latter depends upon the band width of the exciting pulse. Gaussian shaped excitation pulses allow to determine the system response and the ex- citation frequency v with equal precision proportional to  At )Gaus  s . Exponentially decaying pulses have a lorentzian Fourier transform (in ref. [8] equal to the spontaneous line width) making the determination ft) less accurate. 4.3. Short excitation and prolonged interrogation (SEPI) spectroscopy Very recently, we have demonstrated a technique to obtain subline width resolution of Raman transi- tions [9,10]. The method is shown schematically in fig. 2. A short driving force A(t) at a frequency v near the resonance coO of the quantum system generates a transition amplitude R(t) (see fig. 2a). The driving force has a short duration or is switched off rapidly at t = 0. After the excitation the exponentially decay- ing transition amplitude R(t), oscillating freely at 6o0, is investigated by a prolonged probing pulse at frequency COp. A pulse with suitably shaped amplitude Ep(t) generates the scattered signal Es(t ) (see eq. (9)) @['•-•T 1 T T~ Shor t  Exc i ta t ion  J 1 ,,, I 0 - ;  - " " " ' ' " "  R .. ._N o Q Pro longed In ter rogat ion  E R r. .~ ES / /~- - :~ .~Ep .~ / ' /  \ \ . \  , ./ -4'  ", J / / "-., \. 162 ] .// / ~ ' .  \ \  _ I "/ / ~"  x - _ I I' 0 2 /. 6 8 T ime t iT  2 Fig. 2. The short excitation and prolonged interrogation (SEPI) technique. (a) A short driving pulse,A(t), excites the exponentially decaying transition amplitude R(t). (b) The transition amplitude R(t) is interrrogated bya long pulse Ep giving rise to the scattered pulse Es(t ). with frequency co s = cop + 60. The frequency depen- dence of the scattered intensity Is(cos)= [Es(cos)l 2has the form +,x, Is(cos )°c J f dt exp(icost)Es(t)] 2 ]_f+~* exp(icost)Ep(t)R(t)exp[-i(co p + c ) t] dt(13J 2 Introducing Aco = cos -- cop + coO and t e = t2/(T2 × 4 in 2) and using a gaussian shape Ep(t) = Epo exp[-((t  - TD)/tp) 2 2 In 2] for the delayed probing pulse we obtain at late delay time T D a gaussian shaped spectrum centered at the frequency cop -+ coo: Is(TD, Aco) cx exp [-2TD/T 2 - (te/tp)24 In 2] × ] . f  exp(iAcot) exp[ - ( t -  te/tp)22 In 2] dt 2 J~T- D (14) 265 Volume 44, number 4 OPTICS COMMUNICATIONS 15 January 1983 For a long delay time T D the width of the observed spectrum is only determined by the duration of the probing pulse. For a sufficiently long pulse, tp > 1.4 T2, tire SEPI lines are narrower than the spontaneous width. The favorable features of the SEPI spectroscopy have been confirmed in Raman type experiments of homogeneously broadened vibrations in liquids [9,10]. We briefly point to another promising application of the SEPI technique, the study of atomic two- photon transitions. Two counter-propagating short pulse of frequencies 0ol = 6°2, close to 6o0/2 , co- herently excite the quantum system. The transition amplitude R(t) subsequently decays exponentially. The excited volume should be monitored by a third delayed long pulse of gaussian shape with center fre- quency 0o3 ~ 6°0/2. A signal pulse of frequency 00 4 = COO - °°3 is generated in opposite direction to the pulse of fiequency 0o 3 and may be analyzed in a spec- trometer. The suggested geometry eliminates Doppler broadening and the SEPI experiment reduces the honrogeneous line width. 4.3. (SEPI): A short pulse excites the quantum mechanical system (tp 1 ~ T2)" At some time T after the excitation (T > tp 1) a long gaussian shaped pulse coherently probes tire transition amplitude R(T) generating a scattered signal. The narrow spectrum of the scattered signal is observed. As a result, short pulses for excitation and long pulses for probing are needed. The technique is particularly useful when di- rect resonant excitation is not possible, e.g. for Raman or two-photon transitions. Both methods provide improved spectral resolu- tion. Gaussian pulses are treated for two reasons: (i) Tenlporally long gaussian input pulses produce sharp gaussian shaped spectra without disturbing ringing and (ii) gaussian shaped pulses are readily realized in a num- ber of laser systems, e.g. mode-locked Nd-lasers. Calcu- lations and experiments show that under practical con- ditions the spectral width may be reduced by a factor of four below the spontaneous linewidth. References 5. Summary and conclusions The salient features of the two techniques discussed in the preceding section are as follows: 4.2. A long gaussian shaped driving force A(t) ex- cites the two-level system (pulse duration tp longer than the dephasing time T2). We wait for some time T > tp and observe the coherent ransition amplitude R. IR t2may be monitored by a gated photodetector or by a transmission experiment using a second short pulse. Varying the frequency of the exciting pulse one obtains an absorption band which is narrower than the one known from spontaneous steady-state measure- mntes. This technique requires exciting pulses of ap- propriate shape with frequency tunability over the resonance of interest. It is applicable to electronic transitions in the UV, the visible and the near IR where dye lasers are available. [ 1 ] For a review see: H. Walther, in: Laser spectroscopy ofatoms and molecules, Topics in Applied Physics, Vol. 2, ed. H. Walther (Springer, Heidelberg, 1976) p. 1. [2] G, Copley, B.P, Kibble and G,W. Series, J. Phys, B1 (1968) 724. 13l H. Figger and H. Walther, Z. Physik 267 (1974) 1. [4] H. Metcalf and W. Phillips, Optics Lett. 5 (1980) 540. [5] F. Shimizu, K. Umezu and H. Takuma, Phys. Rev. Lett. 47 (1981) 825. [6] P. Meystre, M.O. Scully and H. Walther, Optics Comm. 33 (1980) 153. [7] tt.-W. Lee, P. Meystre and M.O. Scully, Phys. Rev. A24 (1981) 1914. [ 8] P.E. Coleman, D. Kagan and P.L. Knight, Optics Comm. 36 (1981) 127. [9] W. Zinth, Optics Comm. 34 (1980) 479. [ l 0] W. Zinth, M.C. Nuss and W. Kaiser, Chem. Phys. Lett. 88 (1982) 257. [ 11 ] J.A. Giordmaine and W. Kaiser, Phys. Rev. 144 (1966) 676. [ 12] R.L. Carman, F. Shinfizu, C.S. Wang and N. Bloembergen, Phys. Rev. A2 (1970) 60; A. Laubereau and W. Kaiser, Rev. Mod. Phys. 50 (1978) 607. 266 

Universität München: Elektronischen Publikationen

Volume 44. number 4 OPTICS COMMUNICATIONS 15 January 1983 
NARROWING OF SPECTRAL LINES BEYOND THE NATURAL OR DEPHASING LINE WIDTH 
W. ZINTH, M.C. NUSS and W. KAISER 
Physik Department der Technischen Universitil~ Mfinchen, 
M~nchen, Fed. Rep. Germany 
Received 11 October 1982 
Transient excitation and gated or delayed observation provides a narrowing of the inherent line width of the transition. 
Possible experimental one- and two-photon systems are discussed and the loss of signal with line narrowing is calculated. 
The general case of lines broadened by dephasing processes is treated. Contact is made with the line narrowing in recent 
Raman type experiments. 
1. Introduction 
In spectroscopy one is always faced with the prob- 
lem of  spectral resolution. One wishes to determine 
the frequency of the quantized transition with highest 
accuracy and one is interested in detecting fine struc- 
ture or neighboring transitions. Disregarding experi- 
mental factors, the ultimate resolution is dictated by 
the inherent linewidth of  the transitions. In optical 
spectroscopy we know various line-broadening pro- 
cesses. We point to the Doppler effect or to collision 
broadening in gases, to dephasing processes in con- 
densed phases and to the population lifetime which 
gives rise to the natural l inewidth. 
In recent years different novel techniques have 
been devised which provide spectral resolution 
beyond the transition linewidth. For  instance, 
Doppler broadening can be eliminated by saturation 
spectroscopy or by two counter-propagating beams 
for two-photon transitions [1 ] .  Even measurements 
beyond the natural line width have been performed 
taking biased signals from the fluorescent decay [ 2 -  
5]. Techniques have been proposed where the differ- 
ence between the decay rates of the two states rather 
than their sum determines the linewidth [6,7], and 
narrowing of the natural linewidth by decaying-pulse 
excitation has been discussed [8]. 
Very recently we have demonstrated substantial 
line narrowing of  Raman type transitions in con- 
densed phases [9,10]. The lines were broadened by 
vibrational dephasing. It is the aim of  this article to 
show new possibilities of line narrowing in coherent 
transient experiments.  In our treatment we include 
dephasing processes. Without dephasing our results 
are applicable to line narrowing beyond the natural 
linewidth. 
2. Theory. 
We note that the spectral resolution in a transient 
experiment is not limited by the lifetime of the in- 
vestigated levels, but is determined by the specific 
experiment.  Under favorable conditions the observed 
line may become substantially narrower than the 
spontaneous width measured in a steady-state experi- 
ment. We consider an ensemble of  two-level systems 
with ground state ib) and excited state la) at the fre- 
quency difference co o . The interacting driving force 
at the frequency u is denoted by A ( t ) .  For the ele- 
ments of  the corresponding density matrix p we ob- 
tain the equations: 
Paa = - T a P a a  - i A ( t )  (Oba -- Pab)  , ( 1 )  
Pbb = --TbObb + i A( t )  (Oba -- Oab) , (2) 
Pab = --(it°0 + 3'ab)Pab -- i A ( t )  (Pbb -- Oaa) • (3) 
262 0 030-4018 /83 /0000-0000 /$  03.00 © 1983 North-Holland 
Volume 44, number 4 OPTICS COMMUNICATIONS 15 January 1983 
In the following discussion the driving force A(t) is 
caused by electro-magn.+etic fields. For one-photon or 
two-photon processes A( t )  is proportional to E(t) or 
Ei(t ) L/(t),  respectively. The diagonal elements Paa 
and Pbb describe the population of the two levels 
with damping constants 7a = l/T1 and 7b, respec- 
tively. Here we only treat the case of small excitation 
where Pbb ~ 1 >> Paa" The off-diagonal elements 
Pab = P~a represent the coherent excitation of the 
system. Pab relaxes with the dephasing time T 2 = 2/ 
Tab. The dephasing time is made up of energy relax- 
ing, T 1 , and pure phase disturbing processes, Tph : 
I /T  2 = I /2T 1 + 1/Tph. Note, that Tph and T 2 are 
decay constants of amplitudes, while T l corresponds 
to the decay of energy (population). The spectral 
width of a homogeneously broadened transition ob- 
served in stationary experiments is determined by the 
dephasing time T 2 , APspon t = 1]TrT 2 . Phase disturbing 
collisions make the spectral lines broader than the 
natural linewidth Auna t = 1/21rT 1 . 
The spectral resolution may be improved beyond 
the limit imposed by the spontaneous linewidth 
Auspon t by coherent transient interaction between 
an electromagnetic field and the atomic system. We 
introduce an observable quantity, the expectation 
value of the transition operator <r) = rab(Pab + Pba)" 
It has been shown that (r) obeys the equation of a 
damped harmonic oscillator with driving force A(t) 
[111. One obtains from eq. (1) to (3): 
(h) + ( 2/r2)(r) + co2(r ") cc "A(t) . (4) 
Introducing plane waves for A'(t) and (r) with slowly 
varying amplitudes propagating in the x-direction we 
write: 
= (1/2)A(t )  exp(-iut  + ikAX ) + c.c. (5) 
(r) = (1/2)R(t) exp(-icot + ikRX ) + c.c. (6) 
and obtain 
3R/Ot + [i(co 0 - co) + 1/Tz]R = KA(t) .  (7) 
K stands for a proportionality constant, co is the mo- 
mentary frequency of the transition amplitude. 
During the excitation process we have co = ~,, i.e. the 
system is driven off resonance by Aco = co 0 - u. Eq. 
(7) is readily integrated to give 
R(t ,  Aco) = K exp ( - t /T2 )  
t 
× f exp[(iAco + 1/T2)t ']A(t '  ) d t ' .  (8) 
- - o c  
Depending upon the specific experiment R(t) can be 
observed directly or may be interrogated by an addi- 
tional probing light field Ep(t). For a one-photon 
resonance,R is proportional to a radiating polarisa- 
tion and tR 12 is measured, for instance, in a trans- 
mission experiment using a subsequent pulse. For a 
non-radiative transition (e.g. a Raman process),R may 
be monitored by coherent scattering of a probe field 
Ep. The scattered field E s is deduced from Maxwell's 
equations containing the nonlinear polarisation 
pNL ccR(t) X Ep(t) as a source term. It has been 
shown in ref. [12] that the growth o fE  s may be 
written: 
OEdOx ~ R( t )  X Ep(t). (9) 
3. Principle of sub-natural linewidth spectroscopy 
Up to this point we have treated the atomic system 
neglecting the question of the spectral resolution. 
From text books on quantum mechanics we learn 
that Heisenberg's uncertainty principle limits the ac- 
curacy of measuring spectral features by the relation: 
A E X  At>~h or AcoAt ~> 1 , (10) 
where AE, Aoo, At denote the inherent uncertainties 
of the simultaneous measurement of energy (co fre- 
quency) and time. 
In practical experiments the quantities A£" or Aco 
are directly observed, e.g., at the image plane of a 
high resolution spectrographic device. 
The time uncertainty At requires some considera- 
tion. It is determined by the experimental observation 
time and the time constant of the quantum mechani- 
cal system. In steady-state experiments At is deter- 
mined by intrinsic time constants: 
(i) The energy relaxation time T 1 which is obtained 
from the observation of the exponential decay of the 
occupation of the upper quantum state. 
(ii) The phase-relaxation time Tph defined by the 
phase disturbance of the atomic state, e.g. by colli- 
sions. 
263 
Volume 44, number 4 OPTICS COMMUNICATIONS 15 January 1983 
Combining the two relaxation rates one obtains 
the quantum mechanical observation time At ~ T2, 
where lIT 2 = (1/2T 1 + 1/Tph). 
Spectral resolution is improved by increasing the 
observation time beyond T 2 . Two new approaches 
to the problem using transient coherent experiments 
are presented in the following sections (4.2 and 4.3). 
4. Spectral resolution of different experiments 
We now wish to discuss the application of eqs. (8) 
and (9) to several specific experimental situations. 
4.1. Steady-state spectroscopy 
A stationary driving force with amplitude A 0 and 
frequency u excites the quantum system. One mea- 
sures the absolute square of  the transition amplitude 
[R 12 (e.g. the absorption) at some arbitrary time T 
tuning the frequency v through the resonance w 0. 
Eq. (8) yields with Aco = co 0 - v 
tR(Aco)I 2 = KA2/[Aco 2 + ( l /T2)  2 ] . (11) 
As expected, in the steady state we obtain a lorentzian 
line where the line width (FWHM) is equal to the spon- 
taneous width APspon t = 1/nT 2. 
4.2. Transient excitation using a gaussian shaped pulse 
The quantum system is excited by a pulsed driving 
force of  tunable carrier frequency u and of  gaussian 
temporal shape A(t) = A 0 exp [-(t/tp)22 In 2]. As 
shown below the pulse duration tp should be longer 
than the dephasing time T 2 of  the excited system. 
Using eq. (8) we obtain 
[R(T, Aw)l 2 = ~:A 2 exp [-2T/T 2-(te/tp) 2 4 In 2] 
I f  • , , 2 (dlt;l 2 X exp(aAwt )exp[--l(t  -te)/tp[ 2 In 2] 
_oo 
with t e = t2p/(T2 4 In 2). The time T corresponds to 
the time of  observation. For late times T the integral 
of eq. (12) is proportional to the Fourier transform 
of the gaussian driving pulse;i.e., the frequency de- 
pendence of  [R 12 has the same gaussian shape as the 
driving force. Or, in other words, the observed fre- 
quency bandwidth is determined by the pulse dura- 
tion tp and not by the dephasing time T 2 (as in the 
preceding section). Experimentally one observes [R 12 
as a function of the frequency detuning/xw of the 
driving pulse and finds a bandwidth Avco h at a de- 
layed time T. 
We introduce the ratio V = AUspont/AVcoh which 
defines the spectral narrowing found in the transient 
experiment discussed here. For late times T, AUco h is 
determined by the gaussian pulse of duration tp and 
one obtains V = AVspont/AUGauss = tp/1.4 T 2 ; accord- 
ing to this relation one observes a line narrower than 
the spontaneous one for tp > 1.4 T 2. 
The question now arises concerning the magnitude 
of  JR[ 2 at a late observation time T. In fig. la the 
ratio of  the observed linewidth AUco h to the spectral 
width of the gaussian pulse APcoh/APGauss is plotted 
versus T/tp for different values Av spont /APaauss (i.e. 
for different T2). As seen from fig. la, APcoh/APGauss 
= /, 
o 
-" ® 
-g 
o 3 
3 
.J 
z. 
lo 1 2 3 
o 
Time T/ tp  
1~ -  I I I I 
-2 
i 
164 
1 2 3 
Spec t r a l  Narrowing ~,Vspont/AVGaus s 
Fig. 1. (a) Narrowing of observed linewidth versus time of  
delayed observation. The exciting gaussian pulse has a 
width auGaus  s and a durat ion tp. The parameter is given 
y AUspont/~VGau~s where. ~Vspont i s t he  spon)aneous .  
teaay-state) dnewldtn.  (b) Normanzeo peak oi  s~gnm ttran- 
sition ampli tude squared IR(T)I 2) versus spectral narrowing. 
264 
Volume 44, number 4 OPTICS COMMUNICATIONS 15 January 1983 
approaches unity for sufficiently large T/tp values. 
Larger line narrowing V requires later observation 
times T. The points on the broken line mark the T/tp 
positions where the observed linewidth Avco h is 10% 
longer than the linewidth A t ) G a u s  s of the applied 
gaussain pulse. As an example we discuss the case of 
V = 4 and T/tp = 3. With pulses of tp/> 5.6 T 2 and 
for a delayed observation time of T = 16.8 T 2 one 
obtains a line narrowing by a factor of 3.6. 
According to eq. (12) the signal JR(T, ACO)12 de- 
creases strongly for later observation times T. In fig. 
lb the peak signal intensity at Aco = 0 is plotted ver- 
sus the spectral narrowing At)spont/At)Gaus s for a con- 
stant dephasing time T 2 and for a constant amplitude 
A 0 of the driving force. Calculations were made for 
At)coh/AVGaua s = 1.1 (see broken line in fig. la). 
Larger spectral narrowing gives rise to strongly re- 
duced signal intensities. In practical cases, the opti- 
mum spectral resolution is determined by the signal 
to noise ratio of the experimental system. 
The accurate determination of the absolute value 
of the transition frequency coO requires both, a narrow 
spectral response of the excited system and a high pre- 
cision of the tunable excitation frequency v. The ac- 
curacy of the latter depends upon the band width of 
the exciting pulse. Gaussian shaped excitation pulses 
allow to determine the system response and the ex- 
citation frequency v with equal precision proportional 
t o  A t ) G a u s  s . Exponentially decaying pulses have a 
lorentzian Fourier transform (in ref. [8] equal to the 
spontaneous line width) making the determination of 
t) less accurate. 
4.3. Short excitation and prolonged interrogation 
(SEPI) spectroscopy 
Very recently, we have demonstrated a technique 
to obtain subline width resolution of Raman transi- 
tions [9,10]. The method is shown schematically in 
fig. 2. A short driving force A(t) at a frequency v near 
the resonance coO of the quantum system generates a 
transition amplitude R(t) (see fig. 2a). The driving 
force has a short duration or is switched off rapidly 
at t = 0. After the excitation the exponentially decay- 
ing transition amplitude R(t), oscillating freely at 
6o0, is investigated by a prolonged probing pulse at 
frequency COp. A pulse with suitably shaped amplitude 
Ep(t) generates the scattered signal Es(t ) (see eq. (9)) 
@['•-•T 
1 T T ~  
S h o r t  E x c i t a t i o n  
J 1 
,,, I 0 - ;  - " " " ' ' " "  R .. 
._N o Q P r o l o n g e d  I n t e r r o g a t i o n  
E 
R r . .~  E S / / ~ - - : ~ . ~ E p  
. ~ / ' /  \ \ . \  
, ./ -4 '  ", J / / "-., \. 
162 ] .// / ~ ' .  \ \  
_ I "/ / ~ "  x - 
_ I I' 
0 2 /. 6 8 
T i m e  t i T  2 
Fig. 2. The short excitation and prolonged interrogation 
(SEPI) technique. (a) A short driving pulse,A(t), excites 
the exponentially decaying transition amplitude R(t). (b) 
The transition amplitude R(t) is interrrogated by a long 
pulse Ep giving rise to the scattered pulse Es(t ). 
with frequency co s = cop + 60. The frequency depen- 
dence of the scattered intensity Is(cos)= [Es(cos)l 2 
has the form 
+,x, 
Is(cos )°c J f dt exp(icost)Es(t)] 2 
]_f+~* exp(icost)Ep(t)R(t)exp[-i(co p + co) t] dt(13J 2 
Introducing Aco = cos -- cop + coO and t e = t2/(T2 × 
4 in 2) and using a gaussian shape Ep(t) = Epo 
exp[ - ( ( t  - TD)/tp) 2 2 In 2] for the delayed probing 
pulse we obtain at late delay time T D a gaussian 
shaped spectrum centered at the frequency cop -+ coo: 
Is(TD, Aco) cx exp [-2TD/T 2 - (te/tp)24 In 2] 
× ] . f  exp(iAcot) e x p [ - ( t -  te/tp)22 In 2] dt 2 
J~T- D 
(14) 
265 
Volume 44, number 4 OPTICS COMMUNICATIONS 15 January 1983 
For a long delay time T D the width of  the observed 
spectrum is only determined by the duration of the 
probing pulse. For a sufficiently long pulse, tp > 1.4 
T2, tire SEPI lines are narrower than the spontaneous 
width. 
The favorable features of the SEPI spectroscopy 
have been confirmed in Raman type experiments of  
homogeneously broadened vibrations in liquids [9,10]. 
We briefly point to another promising application 
of  the SEPI technique, the study of  atomic two- 
photon transitions. Two counter-propagating short 
pulse of  frequencies 0ol = 6°2, close to 6o0/2 , co- 
herently excite the quantum system. The transition 
amplitude R(t) subsequently decays exponentially. 
The excited volume should be monitored by a third 
delayed long pulse of gaussian shape with center fre- 
quency 0o3 ~ 6°0/2. A signal pulse of  frequency 00 4 = 
COO - °°3 is generated in opposite direction to the 
pulse of  fiequency 0o 3 and may be analyzed in a spec- 
trometer. The suggested geometry eliminates Doppler 
broadening and the SEPI experiment reduces the 
honrogeneous line width. 
4.3. (SEPI): A short pulse excites the quantum 
mechanical system (tp 1 ~ T2)" At some time T after 
the excitation (T > tp 1) a long gaussian shaped pulse 
coherently probes tire transition amplitude R(T) 
generating a scattered signal. The narrow spectrum of 
the scattered signal is observed. As a result, short 
pulses for excitation and long pulses for probing are 
needed. The technique is particularly useful when di- 
rect resonant excitation is not possible, e.g. for 
Raman or two-photon transitions. 
Both methods provide improved spectral resolu- 
tion. Gaussian pulses are treated for two reasons: (i) 
Tenlporally long gaussian input pulses produce sharp 
gaussian shaped spectra without disturbing ringing and 
(ii) gaussian shaped pulses are readily realized in a num- 
ber of laser systems, e.g. mode-locked Nd-lasers. Calcu- 
lations and experiments show that under practical con- 
ditions the spectral width may be reduced by a factor 
of  four below the spontaneous linewidth. 
References 
5. Summary and conclusions 
The salient features of the two techniques discussed 
in the preceding section are as follows: 
4.2. A long gaussian shaped driving force A(t) ex- 
cites the two-level system (pulse duration tp longer 
than the dephasing time T2). We wait for some time 
T > tp and observe the coherent transition amplitude 
R. IR t2may be monitored by a gated photodetector 
or by a transmission experiment using a second short 
pulse. Varying the frequency of  the exciting pulse one 
obtains an absorption band which is narrower than the 
one known from spontaneous steady-state measure- 
mntes. This technique requires exciting pulses of  ap- 
propriate shape with frequency tunability over the 
resonance of  interest. It is applicable to electronic 
transitions in the UV, the visible and the near IR where 
dye lasers are available. 
[ 1 ] For a review see: H. Walther, in: Laser spectroscopy of 
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[2] G, Copley, B.P, Kibble and G,W. Series, J. Phys, B1 
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13l H. Figger and H. Walther, Z. Physik 267 (1974) 1. 
[4] H. Metcalf and W. Phillips, Optics Lett. 5 (1980) 540. 
[5] F. Shimizu, K. Umezu and H. Takuma, Phys. Rev. Lett. 
47 (1981) 825. 
[6] P. Meystre, M.O. Scully and H. Walther, Optics Comm. 
33 (1980) 153. 
[7] tt.-W. Lee, P. Meystre and M.O. Scully, Phys. Rev. A24 
(1981) 1914. 
[ 8] P.E. Coleman, D. Kagan and P.L. Knight, Optics Comm. 
36 (1981) 127. 
[9] W. Zinth, Optics Comm. 34 (1980) 479. 
[ l 0] W. Zinth, M.C. Nuss and W. Kaiser, Chem. Phys. Lett. 
88 (1982) 257. 
[ 11 ] J.A. Giordmaine and W. Kaiser, Phys. Rev. 144 (1966) 
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[ 12] R.L. Carman, F. Shinfizu, C.S. Wang and N. Bloembergen, 
Phys. Rev. A2 (1970) 60; 
A. Laubereau and W. Kaiser, Rev. Mod. Phys. 50 (1978) 
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266 


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