1. Introduction

Experimental observations of ultraslow and stored light in coherent media [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] have naturally led to suggestions for application of these phenomena to fabrication of miniature delay lines and all-optical buffers [11, 12, 13]. The advantages of this kind of buffers include a potentially small size, as well as the feature of tunability of the delay/storage times, and the possibility of externally controllable release of the optical information. Such optical devices are among the components that are critically important in photonics, as well as in optical computing and communications.

To be practical and useful, an optical buffer must delay an optical pulse for a period of time that exceeds the pulse duration, while the absorption and distortion of the delayed pulse should be reasonably small. This, generally, is not the case in the slow light experiments, where the measured time delays are smaller than the pulse duration (see, e.g., [2, 3, 8, 9, 10]). The only exceptions to this, known to us, are the slow light delays measured in hot isotopically pure lead vapors [14] and in cold alkali atomic vapors [1]. In the first experiment a pulse was delayed by approximately four times its full width at a half maximum, and was absorbed by half. In the second experiment a Gaussian pulse was delayed by less than three times its full width at a half maximum and was absorbed by more than seven times. Practical usefulness of those delays in photonics is questionable. In the first case, this is due to the short wavelength coherent radiation (406 nm and 283 nm) and ultra-hot atomic vapor cell and also due to very large residual Doppler broadening of the EIT resonance limiting its width to hundreds of MHz. In the second case this is because of comparably short lifetime of the cold atomic clouds in the traps and the relatively bulky equipment involved. A large number of experiments performed by various scientific groups, including ours, in various coherent media gives us ground to think that there exist some specific fundamental problems preventing the realization of efficient slow light delays at room temperature.

To study the fundamental properties of slow light buffers, we focus on the particular examples of transitions in alkali atoms, although the problems outlined in the paper can be shown to have a general character. We further restrict our consideration to the class of interaction schemes where coherent population trapping is possible. We compare the performance of the atomic slow light delay lines with that of fiber and optical resonators. We show that while a cold, naturally broadened alkali atomic vapor could be used as an efficient delay line, a hot atomic vapor cannot. In some cases the reason is in the incomplete compensation of the probe absorption by the atomic coherence as well as by the high order dispersion of the electromagnetically induced transparency (EIT) resonance. In other cases the performance of the optical buffer is limited by the high nonlinearity of the atomic media and the consequent wave mixing processes. Fiber and optical resonator buffers operated at room temperature do not suffer from these effects, and are are more preferable for the practical applications than the slow light optical buffers.

A recent study [15] has also addressed nearly the same question of whether there are any fundamental limits to the maximum time delay that can be achieved for a pulse propagating through a slow-light medium. In that work, however, the authors studied only naturally broadened, ideal Λ media, omitting inhomogeneous broadening, residual absorption, and wave mixing effects present in real systems. Thus their main conclusion, stating that through optimization it should be possible to delay a pulse by very many pulse lengths in a slow light medium [15], appears to be opposite to the conclusion of the present paper. However, our present study, which accounts for the “parasitic” effects mentioned above, also naturally leads to the same answer in the limit where these effects are ignored.

Unlike the classical “slow light”, the so called “stored light” can be considered as a candidate for optical buffering. It was shown theoretically that a probe light can be stored and released from a coherent three-level EIT medium by manipulating parameters of the drive light [4, 5]. This method seems to be even more preferable for optical buffering than the slow light because the maximum delay of a “stopped” pulse of light could be much larger than the delay of the “slow” light pulse observed under the same conditions. In the first case the delay is determined by the atomic coherence decay rate, while in the second case, the delay is determined by the power broadening induced by the optical drive field. However, to our knowledge no experimental evidence of storage (and release) of an entire light pulse in a room temperature atomic cell has as yet been reported; only a segment of a probe pulse has been stored and released. Moreover, there are significant controversy in the literature regarding the explanation of the results of some of the light storage experiments [16]. We believe, that the theoretical model presented in [17] is flawless in the sense that the light can really be stored in the coherent vapor of three level atoms. The storage is unlikely to be either possible or practically useful in the majority of real systems because of the same reasons we already mentioned, nonlinear wave mixing and residual absorption, that diminish performance of the slow light buffers. We will clarify this statement in the following sections of this paper. Finally, we mention that the use-fulness of the “stored light” system can be partially rehabilitated if specific external pulse shape restoring/reshaping measures are applied [18].

We would like to stress that in the present paper we do not introduce any new formalism, and instead refer to results of previous studies. The main novelty of our study is in the attempt to create a complete picture of the slow and stored light phenomena, where not only the desirable, but also unavoidable, undesirable features are taken into account. Our work takes advantage of earlier research addressing the possible limitations of slow light media used for information processing [11, 13]. With this picture we estimate the dynamic range of the slow and stored light optical buffers and show their advantages and their deficiencies.

The paper is organized as follows. In Section 2 we introduce several criteria for performance to characterize the optical buffer. We apply those criteria to slow and stored light based optical buffers in sections 3 and 4, respectively. Comparison of the slow light, optical fiber, and resonator optical buffers is presented in Section 5.

2. Basic properties and characteristics of a delay line

An optical buffer can be characterized with three parameters: the maximum rate of bits B that can be processed by the buffer, the maximum number of bits simultaneously stored in the buffer N_B , and the buffer geometrical size L [11, 13]. B depends on the time slot T_B for an optical pulse carrying the information. In general, the larger B is, the larger is the information flow that can be processed with the buffer; the larger N_B /B is, the longer is the achievable time delay. And finally, the smaller L is, the smaller is the geometrical dimensions of the buffer.

By definition,

where V_g is the group velocity of light in the buffer. Besides its direct meaning, the factor N_B also indicates the relative time delay τ _g of the pulse center with respect to the pulse duration

In the case of the return-to-zero modulation format where the pulses are well separated in time, the width of the pulse spectrum, Δν_B ≃1/τ _B , exceeds the bit rate B, where B≃1/T_B , because T_B >τ _B . Hence, the group delay exceeds the pulse duration if N_B ≥1. For simplicity, in the following we assume that Δν_B ≃B and T_B ≃τ _B , which means that we are considering a pulse train with 50% duty cycle. This simplification only relaxes the condition of usefulness of the optical buffer.

The dynamic range of change of L, B, and N_B can be found from the following evaluation criteria: i) the spectrum of the optical pulse must fit into the transparency window of the buffer; ii) the buffer dispersion and/or nonlinearity broadened pulse should remain confined in its own bit slot; iii) the number of bits stored should exceed unity, i.e. N_B ≥1; iv) the buffer absorption should not be too large; v) wave mixing should be suppressed; and vi) the noise added by the buffer to the signal should be reasonably small.

To quantify criterion (ii) we introduce the modified wave number β=ω(n(ω)-1)/c for the electromagnetic wave propagating in the buffer, where ω is the wave frequency, and n(ω) is the index of refraction. The dispersion of the medium at frequency ω ₀ is characterized with derivatives ${\beta }_n=d^n\beta ⁄d{\omega }^n{\mid }_{\omega ={\omega }_0}$. It is obvious that β ₁=1/V_g . Criterion (ii) is satisfied for a Gaussian pulse if (see [19])

To quantify criterion (iv) we introduce the power absorption coefficient exp(-αL) of the medium of the buffer, and require αL≤1. Some EIT media are unstable with respect to four-wave mixing and hyper-parametric oscillations. Criterion (v) assumes that the buffer operates below the threshold of such processes. Finally, the smallness of the noise, criterion (vi), implies that the noise added by the buffer should not significantly exceed the shot noise of the signal at the buffer exit.

3. Atomic coherent medium as an optical delay

3.1. General remarks

Various restrictions placed on atomic coherent media used for an optical delay have been studied earlier. For example, spectral reshaping of the incident pulse due to the frequency dependence of the material absorption as well as dispersion was recognized in [20]. The influence of the atomic off-resonance energy levels on the EIT absorption was studied in [21]. It was shown that those levels not only contribute significantly to the absorption of the probe radiation, but also result in additional broadening of the probe pulses for a weak enough drive field. In what follows we generalize those studies, take into account instabilities of the system, and substitute the results of our calculations into the criteria indicated above.

We consider a unified EIT model. The drive and probe light interact with four level atomic double-Λ system (see in Fig. 1a). The drive and probe are nearly resonant with transitions |c〉→|α〉 and |b〉→|α〉 respectively, and off-resonant with transitions |c〉→|α ₂〉and |b〉→α ₂〉 respectively.

 

Fig. 1. Double-lambda scheme. (a) Usual level configuration for EIT observation. The atomic coherence is created through two-photon transition |c〉⃡|α〉⃡|b〉 while off resonant level |α ₂〉 results in residual absorption of the light. (b) Four-wave mixing process that could mask EIT if ω_bc >γ, where γ is the natural decay rate of state |a〉. Presence of the off resonant level shown by dashed line is not necessary to observe the wave mixing process.

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It is worth emphasizing two of the usually underestimated and unwanted phenomena among the multiple effects that the slow light buffers suffer from: absorption by residual off-resonant levels and nonlinear wave mixing. The absorption may occur because the drive and probe fields do not create a dark state while interacting with far detuned transitions |c〉→|α ₂〉 and |b〉→|α ₂〉, where |α ₂〉 is an energy level or a family of levels [21]. The dark state is not created, for example, because the matrix elements of those transitions may have different signs and(or) amplitudes compared with transitions |b〉→|α〉 and |c〉→|α〉. This picture is valid for Zeeman sublevel based EIT schemes [3, 4].

The residual absorption is naturally suppressed if the off resonant level is located far from the excited state of the Λ system compared with the spectral width of the excited state, determined by either natural, homogeneous, or inhomogeneous broadening. In room temperature alkali vapors, however, the Doppler broadening is comparable with the splitting between the excited hyperfine sublevels, so the absorption should always be considered. For instance, this kind of absorption is important for EIT observed on D2 line of any alkali atomic vapor [22].

There are schemes where the probe and the drive create the same dark state interacting either with |c〉→|α〉 and |b〉→|α ₂〉 transitions, or with |c〉→|α ₂〉 and |b〉→|α ₂〉 transitions. If the drive laser power is large enough there is no residual absorption of light in this scheme. However, such systems could be unstable because of four-wave mixing processes (see in Fig. 1b). This picture is valid for hyperfine sublevel based EIT [2, 27].

Four-wave mixing is a common effect in EIT experiments. It does not require presence of a fourth level (|α ₂〉) in the Λ configuration. The process is always possible when the polarizations of the drive and the probe have the same components so the powerful drive could interact with both |c〉→|α〉 and |b〉→|α2〉 transitions. From the symmetry arguements, the optimum EIT scheme formed on alkali hyperfine transitions involves the drive and probe having the same polarization (see, e.g., [26, 27, 29]).

The four-wave mixing effect is suppressed if the drive and the probe are either both resonant, almost degenerate, and have the opposite circular polarizations, as in Zeeman sublevel based EIT schemes, or are strongly nondegenerate. Both of these cases are not practical. In the first case, it is difficult to distinguish between the drive and probe photons. In the second case, the scheme suffers from residual Doppler broadening of the atomic two photon transition. The ideal case could be realized when the drive and probe are degenerate and have the opposite circular polarizations. However, such a field configuration does not form an efficient Λ scheme in alkalis. Moreover, polarization separation is not efficient in the case of atomic cells because of polarization self-rotation due to non-linear Faraday effect. We will return to a discussion of this effect in the experimental section below.

We discuss the above mentioned restrictions of the slow light systems in more details below. To do this we introduce the drive Ω and the probe Ω _p Rabi frequencies as Ω=E℘/h̄ and Ω _p =E_p℘_p /h̄, where E and E_p are the electric fields, assuming the probe and drive atomic transitions have the same matrix elements ℘_p =℘. The Rabi frequency of the probe is much less than the Rabi frequency of the drive, |Ω|≫Ω _p |. We introduce radiative decay γ for level |α〉 and γ_{α 2} for level |α ₂〉. The levels are separated by detuning Δ, where Δ≫γ, γ_{α 2}. In the case of a Doppler broadened system, the Doppler distribution frequency width Δ _D is comparable with Δ. The probe and drive fields have (two-photon) detuning δ from the corresponding two-photon resonance. The coherence has a decay rate equal to γ_bc ≪γ, γ_{a 2}.

3.2. Dispersion and residual absorption of EIT resonances

Let us consider the scheme in Fig. 1a and assume that the splitting between ground state levels ω_bc is smaller than γ and γ_{α 2}. The system is stable with respect to the four wave mixing process.

The dispersion of the naturally broadened system can be described by

where κ=3Nλ ²/(8π), the term proportional to γ_{α 2} results from the light interaction with off resonant level |α ₂〉. Approximation (5) describes dispersion of the double-Λ level configuration well enough under conditions |Δ|²≫|Ω|²≫γ_bcγ (see Fig. 2). It is worth mentioning that for |Ω|≥|Δ| the role of the atomic coherence is not significant and the properties of the four-level system can be described using several far detuned two-level systems. We do not consider this case here.

Using expression (5) we find for |δ|²≪γγ_bc (c.f. [13])

The absorption of the light in the atomic vapor of the naturally broadened double-Λ atoms is where the term proportional to γ ² _α2 results from light interaction with the off resonant level |α ₂〉. Again, Eq. 9 describes absorption of the double-Λ level configuration well enough under conditions |Δ|²≫|Ω|²≫γ_bcγ (see Fig. 2).

 

Fig. 2. Normalized absorption and refractivity of the double-Λ system shown in Fig. 1a. Red (solid) curves correspond to the exact solution of the Maxwell-Bloch equations for the system, while blue (dashed) lines stand for the analytical approximations (9) and (5). The parameters used in the calculations are |Ω|=0.1γ, |Ω _p |=0.01γ, γ_bc =0.001γ, and Δ=200γ

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determined by

To find the width of the transparency window of the slow light optical buffer we should take into account density narrowing of the spectral window originating from the finite optical thickness of the medium in the atomic cell. For small δ the power absorption is

Eq. (10) allows us to estimate the transparency width of the EIT resonance (see [26, 27] for details):

Let us turn now to the inhomogeneously broadened atomic vapor. We consider a quasi Doppler-free configuration which is characterized by a co-propagating drive and probe. The configuration can be considered as Doppler-free if ω_αbγbc /Δ _D ≫ω_bc , where Δ _D the single-photon frequency Doppler width.

This problemwas extensively studied in [23, 24, 25]. It was shown there that, under condition |Ω|²≫${\mathrm{\Delta }}_D^2$ γ_bc /γ, i) the width of the EIT resonance in optically thin media is equal to |Ω|²/Δ _D ; ii) the group velocity is the same as for the naturally broadened system, and iii) the maximum absorptionα~2κγ/Δ _D is achieved at δ~±|Ω|. We phenomenologically introduce an expression for the refractive susceptibility of the Doppler broadened system, shown in Fig. 1a, using those approximations:

This expression reasonably well describes the dispersion of the Doppler broadened system in the vicinity of the EIT resonance (see Fig. 3). Using expression (12) we derive

 

Fig. 3. Normalized absorption and refractivity of the Doppler-broadened double-Λ system shown in Fig. 1a. Red (solid) curves correspond to the exact solution of the Maxwell-Bloch equations for the system, while blue (dashed) lines stand for the analytical approximations (16) and (12). The parameters used in the calculations are |Ω|=10γ, |Ω _p |=0.3γ, γ_bc =0.004γ, Δ _D =100γ, and Δ=300γ.

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To take into account the influence of the light propagation on the width of the EIT resonance we write, in the same way as it was done for the naturally broadened system, the input-output power ratio

Eq. (17) allows us to estimate the transparency width of the EIT resonance:

3.3. Limitations on the parameters of the slow light buffer arising from the high order dispersion as well as from the residual absorption

We note at this point that the bit rate characterizing the optical buffer is restricted from above by two values resulting from both the third order dispersion, B≤B _β3, and EIT window width, B≤B_δEIT . To satisfy both conditions we introduce B≤[$B_{\beta 3}^{-1}$+$B_{\delta \mathit{\text{EIT}}}^{-1}$ ]^-1. It is also useful to introduce the number of absorption lengths for the atomic cell. It is ξ=κL for a naturally broadened medium, and ξ _D =κLγ/Δ _D for the Doppler broadened medium.

We can now use the optical buffer criteria and derive the following conditions for L, B, and N_B . For the bit rate we have:

It might seem from the above expressions that the bit rate could be arbitrary because it is proportional to the externally adjustable power of the drive field. In reality, under our assumption of the 50% duty cycle, the bit rate B is restricted from below by the minimum value of the drive Rabi frequency due to the finite coherence decay rate; it cannot be smaller than γ ₀. The maximum bit rate is determined by the hyperfine splitting in the atom. It cannot exceed Δ, in our model.

The number of bits stored in the optical buffer is

It is clear from (21,22) that the higher is the optical thickness, the larger in N_B .

The main point of the present study is that there are factors that restrict the maximum feasible optical thickness. We derive the condition on the maximum optical thickness of the EIT medium using the inequality αL≤1 and the expressions for the unavoidable “incoherent” absorption due to the off-resonance level. This “incoherent” absorption can prevail over the “coherent” absorption, which is proportional to a small coherence delay rate γ_bc .

Let us find the maximum optical thickness and corresponding number of stored bits for the slow light scheme realized on Zeeman sublevels of D1 line of ⁸⁷Rb, for example. We use Δ=812 MHz, Δ _D =500 MHz, γ_{α 2}≃γ=5 MHz and find ξ _max ≃1.3×10⁴ and ξ _Dmax ≃1.8, which means N_Bmax ≃10² and N_{BD max} ≃0.2. Our estimates imply that a light pulse can be delayed in a naturally broadened rubidium vapor for a period up to a hundred times its duration. On the other hand, the pulse can barely be delayed for its duration in an atomic vapor cell, which makes the cell in Zeeman configuration unsuitable as an optical delay line. The same can be derived for D ₂ line based both Zeeman and hyperfine EIT schemes.

3.4. Slow light on Zeeman sublevels: the experiment revisited

We repeated the slow light experiment by [3] using a 1 cm-long cell with pure ⁸⁷ Rb buffered with 4 torr of Argon and 4.75 torr of Neon. The best EIT resonance can be observed in the cell when the laser interacts with 5S _1/2,F=2 to 5P _1/2,F ^′=1 transition. The residual absorption results from the transition 5S _1/2,F=2→5P _1/2,F ^′=2 the probe and the drive light interact with.

The cell temperature was 99°C, the optical beams diameter was 9 mm. A Pockels cell rotated the initially linear polarization of the drive by a small angle, therefore producing the desired sequence of the probe pulses polarized orthogonally to the drive. The first quarter-wave plate converted both polarizations to circular with eccentricity of less than 1 per cent, and the second similar plate converted them back to linear. The polarizer was set to suppress the drive polarization, which was achieved with the extinction ratio of better than 0.7%. This resulted in the high-contrast probe optical pulses detected at the output of our system. We verified that the shape of these pulses is rectangular, as well as that of the voltage pulses applied to the Pockels cell. There is no detectable time lag between these two sets of pulses at the exit of the Pockels cell.

 

Fig. 4. Left: Transmission of a bi-rectangular probe pulse. Relative power of the probe pulse at the entrance (blue) and exit (red) of the atomic cell. The probe power transmission is 20%. Right: Transmission of a comb of short rectangular probe pulses. The color indicates here simultaneous reduction of the both pump and probe power (from red to blue). The corresponding relative probe power transmission is 40, 30, 25, 20, 15 and 10 % (from red to blue). All output probe pulses are normalized to unity for convenience of perception.

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We carried out a delay measurement for a pair of square pulses, shown in Fig. 4(Left). In this measurement, the drive and probe powers at the cell input were approximately 0.5mW and 0.1 mW, and the absorption in the cell was over 50% and 80% respectively. The EIT resonance was power-broadened to approximately 15 kHz, which is larger than the inverse duration of each individual pulse. As a result, each pulse experienced an average delay of 12 µs, but was considerably distorted. This distortion occurs because the rectangular pulse spectrum is broader than the EIT width, and hence strong absorption and phase distortion apply to the outer parts of this spectrum. A characteristic feature of this distortion is the increase of the second pulse amplitude at the expense of the first pulse: the build-up of the atomic coherence responsible for the EIT is not complete during the first pulse and is continued by the second pulse. Also notice that, for the same reason, the probe light is not completely quenched between the pulses.

A further increase of the cell temperature (optical thickness) resulted in significant absorption of the drive which led to an even more significant absorption as well as distortion of the probe pulse. We were unable to measure a group delay comparable with the pulse duration. The experiment confirms our conclusion that the optical thickness is restricted by the residual absorption.

A more dramatic demonstration of the probe pulse absorption and distortion effect can be done with pulse trains consisting of a large number of shorter probe pulses, e.g. five, see Fig. 4(Right). In this measurement, we reduced the pulse duration by a factor of two, and also reduced the drive and probe powers. As a result, the pulses did not experience any appreciable delay, but their distortion was severe. The distortion is greater for lower powers, which corresponds to a narrower EIT width. The effect of pulse-to-pulse coherence build-up is also very clear. This process saturates by the end of the pulse train, whose duration roughly corresponds to the inverse of the EIT width. It is also interesting to notice that if we consider the entire train as a single pulse, then the delay of this pulse does take place. This delay depends on the optical power and is consistent with the estimated EIT width.

In addition to the problems connected with residual absorption of the probe and drive, we observed another unwanted effect. The polarization rotation in the atomic medium resulted in unbalancing the measurement system in the followingway. The mutual orientation alignment of the polarization elements involved in our experiment was done far from the optical resonance. When, after that balancing, the laser was tuned at the F=2 to F ^′=1 manifold, this alignment no longer yielded the maximum probe polarization suppression. Both the input and the output quarter wave plates needed to be rotated again in order to achieve the highest contrast of the probe pulses. During the probe pulse, when the polarization state of light inside the atomic cell is elliptical, this effect is a natural consequence of Faraday polarization rotation. Remarkably, it is this effect that was highlighted in the critique of the interpretation of light storage experiments [16] as unjustly omitted. It is interesting, however, that a considerable polarization self-rotation is observed, in the vicinity of the atomic resonance, even when the probe is off. The reason is probably that the drive itself is not perfectly circularly polarized, e.g. due to the quarter-wave plate imperfection or the cell windows that are slanted at 15 degrees with respect to the beam direction to avoid back-reflection. The circular polarization at the cell output may mean elliptical polarization at the cell entrance as well as inside the cell. Residual magnetic fields could also result in the rotation. Hence, it is difficult to separate the frequency degenerate drive and probe based only on their polarization. Nondegenerate EIT schemes seem to be more promising. However, as is shown in the following section, those schemes could be unstable with respect to the nonlinear wave mixing.

3.5. Four-wave mixing

We have not yet taken into account a variety of wave mixing processes occurring in realistic coherent atomic systems, which can restrict the application as delay lines. Wave mixing results in the generation of new optical fields as well as in nonlinear power exchange between the drive and the probe. The wave mixing generally occurs when the drive field is able to interact with the probe transition, i.e. has the same polarization component as the probe. Atomic coherence leads to a significant reduction of the threshold for those processes and wave mixing becomes significant when optical thickness ξ is large enough [26].

Wave mixing was observed in the slow light experiments (see, e.g., [2]) when the probe and drive of the same circular polarization, but different frequency, were used. Such a choice of the polarizations leads to the realization of an almost ideal three-level Λ system out of the actual multi-level atomic transition. This is true for the circularly polarized light interacting width D1 line of ⁸⁷Rb, for instance. There is no off-resonance sublevels like |α ²〉 that result in residual absorption in the system. However, the four-wave mixing process is strong, especially in the optically thick medium [27].

Let us consider the scheme in Fig. 1b and assume that the splitting between the ground state levels ω_bc is larger than γ as well as γ_{a 2}, and the drive Rabi frequency is large enough: |Ω|²>ω_bc γ_bc . This four-level system is unstable with respect to the four wave mixing process. Moreover, level |α ₂〉 does not change significantly the instability condition if ω_bc ≫Δ, which is generally the case. Even three-level Λ system can be unstable.

The instability results in generation of light with frequencies ωd±ω_bc if only the drive ω_d is present in the system. One of those frequencies corresponds to the frequency of the probe light. Therefore, the four-wave mixing could result in appearance of “1” instead of “0” signals in the probe channel.

The threshold for the four-wave mixing process does not depend on the drive power and could be estimated from [27]. The four wave mixing is small if

There exists another four-wave mixing process in the system. This process results in generation of the field shown by dashed line in Fig. 1b. The new field is generated at the expense of the probe field. This process does not have a threshold (similar effect is described in [28]), however the efficiency of the process is small if conditions (25) and (26) are fulfilled.

For D1 line of ⁸⁷Rb, we substitute ω_bc ≃6.8 GHz. Hence, ξ _max ≃1.5×10³ and ξ _Dmax ≃15, which means N_Bmax ≃15 and N_{BD max} ≃1. Our estimates imply that a light pulse can be delayed in a naturally broadened rubidium vapor up to a period a dozen times its durations. On the other hand, again, the pulse can only be delayed for only its duration in an atomic vapor cell.

Finally, the wave mixing processes could be reduced by increasing ω _bc . From this point of view, Cs vapor is more suitable for light storage than Rb vapor. This explains why strongly nondegenerate systems like Pb allow achieving large relative group delays [14].

4. Light storage

Our calculations are related to active coherent delay lines (light storage experiments) [4, 5, 29]. In this section we qualitatively discuss the basic principles of the coherent light storage and show why the light storage in hot atomic vapors is unlikely to be suitable for optical buffering.

In light storage experiments a continuous wave drive (“writing drive pulse”) is switched off when a probe pulse has completely entered the coherent medium. The information about coherence as well as quantum properties of the probe is then transferred to the coherent medium via the stimulated Raman process. The atomic coherence has a long lifetime, much longer than the inverse width of the radiatively broadened EIT resonance at the moment when the probe pulse enters the medium. Theoretically, the probe pulse should be completely regenerated after switching the drive field (“reading drive pulse”) on. The drive field scatters off the atomic coherence and generates the probe. This makes “light storage” very attractive as a delay line because it results in long delays for comparably short pulses.

The problem with light storage is the distortion of the probe pulse shape during the writing-reading process. Numerous experiments demonstrate that usually only a segment of the probe pulse is stored in the medium. Indeed, in order to have the entire probe pulse confined in the atomic cell at the first stage of a light storage experiment, the parameter N_B must exceed unity, because, by definition, N_B ≃L/V_g τ _B , where τ _B is the duration of the probe pulse and V_g is the group velocity of the probe when it enters the atomic cell. We have shown that N_B can exceed unity in an atomic cell only in the case of strong distortion or absorption of the pulse. Therefore, it is unlikely that the entire probe pulse could be restored in a “stopped light” experiment with a hot atomic cell. It is worth mentioning, however, that truncation of the pulse shape in the writing-reading process still allows saving and restoring the quantum properties of the probe light, such as photon-number squeezing, which is not related significantly to the pulse shape.

On the other hand, “light storage” delays are much more tolerant to the values of N_B compared with slow light delays. Indeed, one has to realize N_B ≈1 to obtain an efficient delay for a single pulse. Our estimations show that experiments with hot rubidium vapor could give N_B of the order of 1. Using cesium vapor, for example, could increase this value a little, because of reduced wave mixing efficiency, but enough to make a delay line. Our calculations show why all light-storage experiments in room temperature vapor cells, performed so far, could not store the entire pulse. They do not show however that this is impossible.

5. Discussion

We have considered a train of pulses with a 50% duty cycle. The case of a single pulse can be easily derived from this treatment by formally redefining the bit rate B as the pulse spectral width Δν_B . Then N_B loses its meaning as the number of simultaneously stored pulses, but remains equal to τ _g /τ _B . Its value should still be greater than unity for the pulse to be delayed by more than its width, so the results we have derived above for the pulse train also apply to a single pulse case.

Let us compare the EIT-based delay line with a fiber delay line. For a conventional optical fiber delay Eq.(1) transforms to

where n _f is the refractive index of fiber, typically in the range of 1.4-1.5. The typical for fibers parameters are α≥10^-6 cm^-1, β ₂ 50 fs2/cm, and β ₃~500 fs³/cm, though the dispersion could be smaller with dispersion shifted fibers. Unlike the EIT systems, fiber delay lines are efficient for short optical pulses and high bit rates. Hence, EIT delay lines are complimentary to fiber delay lines in the sense that the EIT systems are more efficient for long optical pulses and low bit rates. Advantages of the EIT delays include their small size and tunability by controlling the power of the drive field. On the other hand, small achievable values of N_B in EIT systems limit them in many practical applications where fibers are involved.

The first question that arises with respect to the results of our study is related to their generality. We cannot prove rigorously, that the entire concepts of slow and stored light are in-principle useless with respect to their practical applications as optical delays. However, the examples discussed in the paper clearly show that the vast majority of the dark state based delay lines suffer from strong residual absorption and/or large nonlinear wave mixing. A lot of other, rather technical, difficulties of applications of optically thick coherent media for optical buffering, like radiation trapping [30], are still left untouched in the paper. Therefore, it is worthwhile to consider alternative ways of “slowing light” and to study possibilities of their practical implementation.

We conclude this section by mentioning one such alternative way, that is chains of optical resonators as delay lines. This approach has also attracted a lot of attention recently [31, 32]. An advantage of these delay lines is the arbitrariness of their operational wavelength, as in optical fiber delay lines, as well as small size and tunability, as in the EIT delay lines. An advantage of the resonator based delays over EIT based delays is the possibility for making N_B large. Generally, N_B ≤1/2 for a resonator. It means that several resonators should be placed in a chain to get a large N_B . A disadvantage of a resonator delay line is the narrow bandwidth of the resonator systems. The bit rate is restricted from the above in those systems, as in the EIT delay lines. Thus the bit rate must be smaller than the linewidth of the resonators’ modes and, hence, less than the free spectral range of the resonators.

6. Conclusion

It is shown that while a vapor of ideal three-level Λ atoms can always be used as an efficient optical buffer for long light pulses, the behavior of realistic incarnations of the Λ system is more complex. We studied both theoretically and experimentally the performance of an optical delay line based on the effect of electromagnetically induced transparency in several realistic systems. We demonstrated that while the naturally broadened coherent atomic media, whose behavior can be described with the models we studied, can be utilized as an optical delay, the hot atomic media can not. Inhomogeneous broadening results in the relative enhancement of the residual absorption, dispersion, and nonlinearity that substantially deteriorates the delay line performance. In this respect, optical resonator based delay lines look more promising compared with the EIT systems.