FIG. 1. Plot of the quantityR�1�1 jWm30�z; ��j

2d� as a func-tion of z. Curve a has been obtained in the absence of EIT;curve c has been multiplied by 10. The parameters are the sameas the ones reported in Figure 2 of Ref. [1].

VOLUME 93, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S week ending17 SEPTEMBER 2004

Comment on ‘‘Opening Optical Four-Wave MixingChannels with Giant Enhancement Using UltraslowPump Waves’’

In their recent study [1] on a four-wave mixing (FWM)process, Deng et al. state that ‘‘many orders of magnitudeenhancement in the amplitude of the generated wave’’ isobtained when electromagnetically induced transparency(EIT) is present. Here we will show that their claim isdeeply misleading. In fact, according to the formulasreported in their paper, it is straightforward to showthat the efficiency of the FWM generation process isactually reduced by the presence of EIT, and that themaximum FWM signal is obtained in the absence ofEIT. We are convinced that the approach followed bythe authors of Ref. [1] is adequate to describe the physicalprocess and that the formulas reported in their paper arecorrect. For an easy comparison, we use their notation,where �12 is the field responsible for EIT, �20�z; t�,�m30�z; t� represent the pump and the generated FWMfields which propagate along the z axis, and � is the timeduration of the input pump field �20�0; t�. The main resultof Ref. [1] is contained in Eq. (4) which reports theanalytical expression of the Fourier transformWm30�z; �� (� � !� is the normalized frequency) of thegenerated field �m30�z; t�. In Fig. 1 the quantityR�1�1 jWm30�z; ��j2d�, which is proportional to the energy

of the field generated in the FWM process, is reported forvarious values of �12. The atomic parameters used arethe same as the ones chosen by Deng et al. [1]. Theinterpretation of the behavior shown by Fig. 1 is straight-forward. In the presence of EIT (curves b and c), the pumpfield �20�z; t� propagates with negligible absorption andprovides an almost constant source term for the FWMfield �m30�z; t�. This saturates at a value for which thesource term—proportional to �20�z; t�—compensatesfor the loss term—proportional to �m30�z; t�. The usefulinteraction length is in this case of the order of theabsorption length Lm at the frequency of the generatedfield. In the absence of EIT (curve a), the pump field�20�z; t� propagates with a strong absorption and providesa strongly decreasing source term for the FWM field�m30�z; t�. The optimal interaction length Lmax is inbetween the absorption length Lp at the frequency ofthe pump field and Lm. For interaction lengths much

129401-1 0031-9007=04=93(12)=129401(1)$22.50

longer than Lm the FWM field �m30�z; t� is completelyabsorbed, and, as claimed by Deng et al. [1], the FWMenergy generated in the presence of EIT is many orders ofmagnitude larger than that generated in the absence ofEIT. However, it is a matter of fact that this is only due toa wrong choice of the interaction length and that themaximum FWM energy generated in the absence ofEIT is much larger than the maximum FWM energygenerated in the presence of EIT. It is worth stressingthat, by choosing the right atomic density N, it is alwayspossible to match the value of Lmax with the cell length L.

Roberto Buffa,1 Stefano Cavalieri,2 andMarco V. Tognetti1

1Dipartimento di Fisica, Universita di Siena, and INFM, ViaRoma 56, I-53100 Siena, Italy

2Dipartimento di Fisica and European Laboratory forNonlinear Spectroscopy, Universita di Firenze, and INFM,Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy

Received 29 December 2003; published 16 September 2004DOI: 10.1103/PhysRevLett.93.129401PACS numbers: 42.65.–k

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[1] L. Deng et al., Phys. Rev. Lett. 88, 143902 (2002).

04 The American Physical Society 129401-1