June 1, 2000 / Vol. 25, No. 11 / OPTICS LETTERS 847

Enhanced nonlinearity and transparencyvia autoionizing resonance

Takashi Nakajima

Institute of Advanced Energy, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan

Received January 3, 2000

The nonlinear response of an autoionizing medium in which coherence is established essentially by nonradiativeinteractions is analyzed. It is found that, by proper tuning of two radiation fields, the third-ordernonlinearities can be enhanced while the absorption is canceled. The proposed scheme is particularly usefulfor enhancing nonlinearities in the VUV region. 2000 Optical Society of America

OCIS codes: 190.3270, 270.1670, 020.1670.

The use of the coherent property of radiation hasled to several interesting findings in recent years,which are represented, for example, by the subjectsof lasing without inversion,1,2 electromagnetically in-duced transparency,3 enhanced linear and nonlinearrefractive indices,4 – 6 and coherent population trans-fer.7,8 In all these studies, laser-induced coherencebetween bound states plays an essential role in modify-ing light–matter interactions. As for related studiesinvolving autoionizing states, we note that only a fewhave been reported (e.g., Refs. 9–12).

In this Letter the nonlinear response of an autoioniz-ing medium is analyzed. The proposed scheme wouldbe particularly useful for enhancing nonlinearities inthe short-wavelength region, since the transition wave-length between the ground and the autoionizing statesusually lies in the UV–VUV region. The essential dif-ference between the scheme proposed here and otherbound-state schemes4 – 6 is that, for the former, owingto the nonradiative configuration interaction, coher-ence already exists even without laser fields, whereasfor the latter coherence is induced only when exter-nal laser fields resonantly couple the bound states.As a result, the use of an autoionizing state consid-erably simplif ies the scheme in terms of the numberof states and lasers involved. For example, a bound-autoionizing system could be viewed as a quasi-two-level system in which the absorption profile exhibitsa well-known asymmetry and the medium becomestransparent at its absorption minimum near an au-toionizing resonance. Note that the transparency inthis case is due to the nonradiative coherence betweendiscrete and continuum components of an autoionizingstate. In contrast, if a transparency is to be createdin a bound-state system, at least three states are re-quired. Such a peculiar property of an autoionizingsystem has been used to observe interference effects inthird-harmonic generation,10 pulse propagation,11 andlinear susceptibilities and ionization yields in the pres-ence of strong dressing fields.12 Another advantage ofusing an autoionizing medium is that various broad-ening mechanisms, such as Doppler broadening, colli-sional broadening, and laser bandwidth, which causea serious degradation of laser-induced coherence forbound-state systems, can be neglected. That this is

0146-9592/00/110847-03$15.00/0

so can be easily understood by comparison of a widthof autoionization with that of Rabi broadening: Anautoionization width typically ranges from a few to afew hundred cm21, which roughly corresponds to in-tensities of tens of MW�cm2 centimeter to hundredsof GW�cm2, if such a coupling strength is to be cre-ated by a laser field. Considering the fact that thetypical Doppler and collisional widths are at mostone tenth of a cm21 or less for a gaseous medium,the neglect of these effects is perfectly justif ied foran autoionizing medium. A similar argument holdsfor the neglect of laser bandwidth. Closely related tothe scheme presented here is a four-level scheme, asshown in Fig. 1, that was recently proposed by Schmidtand Imamoglu for enhancement of Kerr-type non-linearity.6 In that scheme, electromagnetically in-duced transparency again plays an essential role.

The quasi-three-level scheme that we propose isdepicted in Fig. 2(a). It consists of two bound states,j1� and j3�, and an autoionizing state j2� embedded ina continuum jc�, where the bar implies that diagonal-ization of discrete state j2� and continuum jc�, whichare coupled by configuration interaction V , has al-ready been made. For the following analysis it is moreconvenient to recast this scheme into the one shownin Fig. 2(b), in which j2� and jc� represent the dis-crete and continuum components, respectively, of theautoionizing state j2� before diagonalization. Statesj1� and j2� and j2� and j3� are coupled by laser fieldswith frequencies va and vb, respectively, with the

Fig. 1. Four-level scheme proposed by Schmidt andImamoglu to enhance the Kerr-type nonlinearity.6 j2� andj20� are strongly coupled by the laser field.

2000 Optical Society of America

848 OPTICS LETTERS / Vol. 25, No. 11 / June 1, 2000

Fig. 2. Proposed quasi-three-level scheme: An autoion-izing quasi-three-level system (a) can be decomposed intotwo bound states, j1� and j3�, and a discrete state j2� cou-pled to continuum jc� by nonradiative configuration inter-action V (b).

corresponding Rabi frequencies given by V1 and V2.After diagonalization of states j2� and jc�, these Rabifrequencies become complex quantities defined by V1and V2, where the tilde implies a complex quantity.Using asymmetry parameters q1 and q2, we give therelations between the Rabi frequencies as V1 � V1�1 2

i�q1� and V2 � V2�1 2 i�q2�, where q1 � 2V1�p

g1G2

and q2 � 2V2�p

G2g3 , with g1, g3, and G2 being thedirect photoionization rates from j1� and j3� and anautoionizing rate from j2�, respectively. By compari-son of Figs. 1 and 2(b), the similarity as well asthe difference between the two schemes become evi-dent. The two schemes are similar in that both con-sist of four states or components, j1�, j2�, j3�, and j20� orjc�. There is an essential difference, however, arisingfrom the different characters of the states involved, i.e.,a bound state j20� and a continuum state jc�. As indi-cated by thin lines in Fig. 2(b), there exist additionalbound-continuum couplings in our system, since bothj2� and jc� belong to the same parity, whereas no suchcoupling exists in Fig. 1 between j1� and j20� and be-tween j2� and j3�. More details of the dynamics of asimilar system involving an autoionizing state were re-ported in Ref. 12.

For the description of our system [Fig. 2(b)] weemploy the following set of density matrix equations12:

�s11 � 2 g1s11 1 2 Im�V1s21� 1 l2s22 1 l33s33

1 lcscc , (1)

�s22 � 2�l2 1 G2�s22 2 2 Im�V1�s21� 1 2 Im�V2s32� ,

(2)

�s33 � 2�l3 1 g3�s33 2 2 Im�V2�s32� , (3)

�s21 �

∑id1 2

12

�g1 1 l2 1 G2�∏s21 2 iV1s11

1 iV1�s22 2 iV2s31 , (4)

�s31 �

∑i�d1 1 d2� 2

12

�g1 1 l3 1 g3�∏s31 2 iV2s21

1 iV1�s32 , (5)

�s32 �

∑id2 2

12

�l2 1 G2 1 l3 1 g3�∏s32 2 iV2s22

1 iV2�s33 1 iV1s31 , (6)

where the decay rates from j2�, j3�, and continuum jc�back to lower state j1� are defined by l2, l3, and lc,respectively. d1 and d2 are the detunings of the firstand the second lasers.

To obtain the linear and the nonlinear susceptibili-ties we solve Eqs. (1)–(6) to the third order, startingfrom s

�0�00 � 1, under the steady-state condition. For

example,

s�3�21 �

V1s�2�11 2 V1

�s�2�22 1 V2s

�2�31 2 ig1s

�1�21 �2

d1 1 i�l2 1 G2��2, (7)

s�3�32 �

V2s�2�22 2 V1

�s�2�31

d2 1 i�l2 1 l3 1 G2��2. (8)

The polarization of the medium with frequency va isgiven by

Pva � m12s21 1Xc

m1sc1

� x �1�ea 1 �x �3�SPMea

3 1 x�3�XPMeaeb

2� 1 . . . , (9)

where

x �1�ea � m12s�1�21 1

Xc

m1cs�1�c1 , (10)

x�3�SPMea

3 1 x�3�XPMeaeb

2 � m12s�3�21 1

Xc

m1cs�3�c1 . (11)

The subscripts SPM and XPM refer to self- and cross-phase modulation, respectively. s

�1�c1 and s

�3�c1 are the

first- and the third-order matrix elements between j1�and jc�, respectively, where jc� has already been elimi-nated in Eqs. (1)–(6). These matrix elements can becomputed from

s�1�c1 �

1dc

�Dc1s�0�11 1 Vc2s

�1�21 � , (12)

s�3�c1 �

1dc

�Dc1s�2�11 1 Vc2s

�3�21 1 Dc3s

�2�31 � . (13)

Using Eqs. (12) and (13), we can obtain the generalforms for x

�3�SPM and x

�3�XPM, after some algebra:

x�3�SPMea

3 �2m12

2eas�2�11 1 jm12j

2eas�2�22 2 ig1

0ea2s

�1�21 �2

d1 1 i�l2 1 G2��2

2

µs10 2 i

g10

2

∂eas

�2�11 , (14)

June 1, 2000 / Vol. 25, No. 11 / OPTICS LETTERS 849

x�3�XPMeaeb

2 �2m12m23ebs

�2�31

d1 1 i�l2 1 G2��22 m

�2�13ebs

�2�31 , (15)

where mjk denotes the complex dipole matrix elementbetween jj� and jk�, after the adiabatic elimination ofthe continuum, i.e., m12 � V1�ea, m32 � V2�eb, andm

�2�13 �

RdEc�m1cmc3���v1 1 va 2 vc�. g1

0jeaj2��g1�

and s1 0jeaj2 stand for the photoionization rate from j1�

and an associated shift, respectively. Note that thecontrollable parameters for x

�3�SPM and x

�3�XPM are d1 and

d1 and d2, respectively.It is well known that there is an absorption minimum

for an autoionizing resonance at the detuning d1 �2q1G2�2, which we define as d1

0, and we furthersimplify x

�3�SPM and x

�3�XPM at d1

0. x�3�SPM then becomes

x�3�SPM�d1 � d1

0� �28m12

4

�q1G2�3

∑µ2 2

l2

lc

∂2

iq1

µ1 2

l2

lc

∂∏

1

µs10 2 i

g10

2

∂4m12

2

�q1G2�2

µ1 2

l2

lc

∂.

(16)

Recall the relation g10 � 4m12

2�q12G2; it can be shown

that Im x�3�SPM�d1 � d1

0� � 0. This result indicatesthat the third-order nonlinearity x

�3�SPM can be en-

hanced without absorption, even without the aid ofthe second laser. A similar result for the linear sus-ceptibility for an autoionizing system was reportedrecently in Ref. 12. This is quite in contrast to thewell-known result for the electromagnetically inducedtransparency scheme, in which Re x

�3�SPM � Im x

�3�SPM �

0 when Im x �1� � 0.6 Similarly, an expression forx

�3�XPM�d1 � d1

0,d2� can be derived as

x�3�XPM�d1 � d1

0,d2� �m12m23�m12m23 2 �q1G2�2�m�2�

13 ��q1G2�2�2��d2 2 q1G2�2� 1 il3�2�

.

(17)

Note that both the numerator and the denominator ofthe right-hand side of Eq. (17) are complex quantities.Although the value of the numerator is uncontrollableonce the system is chosen, the denominator is control-lable through detuning d2. If d2 is chosen in such away that

m12m23�m12m23 2 �q1G2�2�m�2�13 �

~ �q1G2�2�2��d2 2 q2G2�2� 1 il3�2� , (18)

it is obvious that x�3�XPM becomes pure real. In

other words, by tuning the first laser to the ab-sorption minimum of the autoionizing resonance,i.e., d1 � 2q1G2�2, we obtain Im x �1��d1 � d1

0� �Im x

�3�SPM�d1 � d1

0� � 0 and Re x�3�SPM�d1 � d1

0� fi 0, sug-

gesting that the third-order nonlinearity associatedwith the self-phase modulation can be enhanced with-out absorption. Furthermore, by applying the secondlaser and varying the detuning d2, we can control theratio of Re x

�3�XPM�d1 � d1

0,d2��Im x�3�XPM�d1 � d1

0,d2� atwill. Note that the origin of these interesting proper-ties can be traced to the fact that the matrix elements,or equivalently the Rabi frequencies, are complex foran autoionizing system, as can be seen from Eqs. (14)and (15). Thus the use of an autoionizing systemwould be particularly useful for enhancing nonlin-earity in the VUV region with the help of the secondlaser, since the intensity of the first laser, whosewavelength is most likely in the UV–VUV region, maynot be sufficient to induce significant nonlinearitythrough x

�3�SPM. Another important remark is that,

although Im x�3�XPM cannot be made completely zero for

the bound four-level scheme,6 this is possible in ourquasi-three-level scheme. This is another advantageof our scheme over the bound four-level scheme forenhancing nonlinearity.

In conclusion, we have studied the nonlinear re-sponse of a quasi-three-level autoionizing system andfound that proper choice of the laser detunings couldlead to enhanced nonlinearities through two kinds ofthird-order process associated with self- and cross-phase modulations, while canceling the linear and thenonlinear absorptions. The proposed scheme would beparticularly attractive in the VUV region, since nei-ther of the two laser fields needs to be highly coher-ent, which, again, is due to the fact that the coherencein this system is established by strong nonradiativeinteractions. For the experimental realization of thisscheme an alkali-earth atom would be a good candi-date because of the availability of the laser wavelengthin the UV–VUV region.

T. Nakajima’s e-mail address is t-nakajima@iae.kyoto-u.ac.jp.

References

1. S. E. Harris, Phys. Rev. Lett. 62, 1033 (1989).2. S.-Y. Zhu and E. E. Fill, Phys. Rev. A 42, 5684 (1990).3. K.-J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev.

Lett. 66, 2593 (1991).4. M. O. Scully, Phys. Rev. Lett. 67, 1855 (1991).5. U. Rathe, M. Fleischhauer, S.-Y. Zhu, T. W. Hansch,

and M. O. Scully, Phys. Rev. A 47, 4994 (1993).6. H. Schmidt and A. Imamoglu, Opt. Lett. 21, 1936

(1996).7. K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod.

Phys. 70, 1003 (1998).8. T. Nakajima, Phys. Rev. A 59, 559 (1999).9. P. Lambropoulos and P. Zoller, Phys. Rev. A 24, 379

(1981).10. S. J. van Enk, J. Zhang, and P. Lambropoulos, Phys.

Rev. A 50, 3362 (1994).11. E. Paspalakis, N. J. Kylstra, and P. L. Knight, Phys.

Rev. A 60, 642 (1999).12. T. Nakajima, Phys. Rev. A 60, 4805 (1999).