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electromagnetically induced transparency ( eit ) @xcite , in which a `` pump '' field of laser light can allow a weaker `` signal '' field to propagate through an otherwise opaque atomic gas , has been inspiring a number of applications based on the underlying coherent interaction of laser light with atomic media .
these include nonlinear optics at low light levels @xcite and ultra - sensitive magnetic field measurements @xcite .    of particular interest
has been the recent observation of ultra - slow light ( usl ) @xcite in atomic gases , at group velocities on the order of 10  m / s , due to a steep linear dispersion in the index of refraction associated with the narrow eit feature .
this could allow for controllable true - time delay devices for classical light pulses , with applications in fiber - optic telecommunications @xcite and radar signal processing @xcite .
later extensions of the technique to stored light @xcite ( for several milliseconds ) has also raised the possibility of quantum memory devices @xcite .
while the narrow frequency feature ( below the natural linewidth and doppler width ) of eit is one of its attractive features for precision applications @xcite , this has drawbacks in regards to delay and storage applications .
optical communications and radar processing typically desire @xmath4  ghz bandwidth .
similarly , single photon sources and other tools of potential quantum information technologies may emit photons over a broad band . in usl experiments to date
, the width of eit transparency window is much narrower .
eit and usl work best when the atom can be well described with a @xmath5 energy level structure .
the signal field is near - resonant with a stable state ( which we label @xmath6 ) and a radiatively decaying excited state ( @xmath7 ) .
the pump field is resonant with another stable state @xmath8 and the common excited level @xmath7 .
we consider two energy level schemes in @xmath0rb , shown in fig.[fig : diagram](a ) .
the schemes are labeled `` a '' ( dashed , blue arrows ) and `` b '' ( solid , red arrows ) .
the transparency and slow , distortion - free propagation of the signal pulse that we desire occur only when the frequency difference of the two lasers @xmath9 matches the energy level difference between levels @xmath6 and @xmath8 to within the narrow eit width .
frequency components of the signal outside this width are strongly absorbed and distorted .
this width is directly proportional to the pump power and practical limits on the pump power ( @xmath10  mw/@xmath11 ) limit it to @xmath1  mhz .
[ cols="^ " , ]     because of the extremely strong dependence of @xmath12 with power we found numerically that one always benefits from using smaller @xmath13 , so this parameter should be chosen to be the smallest reasonable value over which the beam can be easily focused .
though too small a value @xmath13 would lead to a higher loss from @xmath14 this problem is almost always dominated by the diffusion in the magnetic field gradient and so is not a big consideration .
similarly , in choosing @xmath15 we found that the gain in transparency from higher intensities tended to outweigh the gain one got from lower slopes .
figure  [ fig : parameterdrift](b ) shows the maximum delay - bandwidth product versus @xmath16 , but keeping the pump power constant and adjusting the interaction with @xmath15 such that the bandwidth @xmath17  mhz was also kept constant . indeed one sees that one gains by using steeper gradients over smaller areas .
ultimately , the slope @xmath16 that can be used in practice will be determined by the manner in which the magnetic field gradient and signal dispersion can be generated .    in the calculations in figs .
[ fig : wideband ] and [ fig : parameterdrift](a - b ) we chose pressures @xmath18  torr and 25  torr for schemes a and b , respectively . numerically these were found to be about optimal . in fig .
[ fig : parameterdrift ] we plot the dependence of the of the maximum delay - bandwidth on the pressure for several cases . in scheme
a , the optimal pressures are larger than in the homogenous case due to increased importance of reducing diffusion in the magnetic field gradient . interestingly , even in scheme b , higher pressures eventually reduce the performance .
this can be understood from the factor @xmath19 in the analytic estimate for @xmath12 above .
the physical origin of this factor is the fact that the eit width @xmath20 decreases with @xmath21 and therefore makes the resonance more sensitive to the averaging over nearby magnetic fields eq .
( [ eq : rhodiffint ] ) .
we plot the dependence on pressure for two different gradients in each scheme . while the optimal pressures are slightly different , the dependence on pressure is rather weak and so a sensitive parameter search versus @xmath22 should not be required .
we have performed a comprehensive and systematic analysis of eit resonances , and the resulting pulse propagation characteristics , in @xmath0rb vapors , including effects of couplings to additional levels in the hyperfine structure and a buffer gas .
we then calculated the delays , transmissions , and bandwidths for propagation of light tuned to these resonances .
we analyzed two particular @xmath5 level schemes ( diagrammed in fig .
[ fig : diagram ] ) and found that scheme b was far superior , in terms of achievable delays and delay - bandwidth products , due to the lack of coupling to additional levels . despite its poorer performance , scheme a still provides reasonable performance and may be desirable since it is much easier to initialize , simply with optical pumping .
importantly , we found the eit resonance could be shifted over a wide range of frequencies by applying a homogeneous magnetic field , and that the resonance characteristics were quite insensitive to this field over range of about 500  mhz .
this analysis serves as a useful model to study eit in conventional slow light , and also as a basis for study of our channelization architecture .
we then presented a model to analyze the effect of an inhomogenous magnetic field , which causes a strong variation of the eit resonance frequency in the transverse direction .
this was then applied to analyze the performance of our proposed channelization architecture for wide - band slow light , where a signal pulse is spatially dispersed according to frequency and an inhomogeneous magnetic field is applied in a such a way that the eit resonance frequency matches this dispersion .
we found that by choosing the magnetic field gradient so the change in the two - photon resonance is slightly mismatched from the transverse dispersion of the signal , one could achieve eit and slow light conditions over a much larger bandwidth than with conventional slow light .
this is essential for applications in many signal processing applications .
we found that the diffusion of atoms in the field tended to reduce the delay - bandwidth products with bandwidth . in scheme b ,
this architecture should allow a delay - bandwidth greater than unity up to bandwidths of about @xmath2  mhz , where delays are @xmath3  ns ( see fig .  [
fig : pulsedrift ] ) .
furthermore , we note either the pump field power or the magnetic field gradient can be used to control @xmath23 and thus the delay , making it a controllable time delay system .
the buffer gas is important in reducing the diffusion of atoms from into regions of widely varying magnetic field and so higher buffer gas pressures are generally desirable for higher magnetic field gradients .
however , we also found that higher pressures narrow the eit feature and can therefore increase the sensitivity of the dispersive slope and absorption profile to magnetic field gradients .
balancing these two considerations leads to an optimal pressure , which we found this optimum to be near @xmath2410  torr for scheme a and @xmath25  torr for scheme b , for reasonable parameters .
this optimal pressure was not very sensitive to the exact value of the gradient and other parameters .
we also found that one generally benefited from tight focusing and high magnetic gradients .    in future work , it will be useful to consider the effects of atomic diffusion at a more microscopic level .
in particular , it has been found that the model used here for diffusion out of the interaction region may overestimate the loss in real systems due to the fact that atoms can diffuse back into the interaction region @xcite . additionally ,
dynamical jumps of velocity of individual rubidium atoms upon collisions with the buffer gas has also been found to be an important consideration @xcite .
finally , for implementation of this system , work is also needed to develop optimal methods for transversely dispersing the signal field and producing large linear magnetic field gradients .
the role played by the differential phase shift in this system is interesting in its own right and merits further investigation .
it is not entirely clear that the pulse will not be significantly more slowed than our analysis here shows , due to subtleties with the transverse dispersion of the signal .
perhaps the signal dispersion or magnetic field could be engineered in such a way that the group velocity is governed by the _ local _ ( and much larger ) @xmath26 , rather than @xmath23 , allowing much larger delays .
furthermore , it may be possible to combine this method with aspects of previous light storage experiments @xcite to significantly increase the delay times .
the authors wish to thank irina novikova , mikhail lukin , and fredrik fatemi for helpful discussions .
this work was supported by the office of naval research and the defense advanced research projects agency ( darpa ) slow light program .
rotondaro and g. p. perram , `` collisional broadening and shift of the rubidium @xmath28 and @xmath29 lines ( @xmath30 ) by rare gases , @xmath31 , ''
_ j. quant .
radiat . transfer _ * 57 * pp .
497507 , 1997 .
we propose a `` channelization '' architecture to achieve wide - band electromagnetically induced transparency ( eit ) and ultra - slow light propagation in atomic @xmath0rb vapors . 
 eit and slow light are achieved by shining a strong , resonant `` pump '' laser on the atomic medium , which allows slow and unattenuated propagation of a weaker `` signal '' beam , but only when a two - photon resonance condition is satisfied . 
 our wideband architecture is accomplished by dispersing a wideband signal spatially , transverse to the propagation direction , prior to entering the atomic cell . 
 when particular zeeman sub - levels are used in the eit system , then one can introduce a magnetic field with a linear gradient such that the two - photon resonance condition is satisfied for each individual frequency component . because slow light is a _ group _ velocity effect , utilizing differential phase shifts across the spectrum of a light pulse , one 
 must then introduce a slight mismatch from perfect resonance to induce a delay . 
 we present a model which accounts for diffusion of the atoms in the varying magnetic field as well as interaction with levels outside the ideal three - level system on which eit is based . 
 we find the maximum delay - bandwidth product decreases with bandwidth , and that delay - bandwidth product @xmath1 should be achievable with bandwidth @xmath2  mhz ( @xmath3  ns delay ) . 
 this is a large improvement over the @xmath1  mhz bandwidths in conventional slow light systems and could be of use in signal processing applications .