Direct Imaging of Slow, Stored, and Stationary EIT Polaritons
Geoff T Campbell, Young-Wook Cho, Jian Su, Jesse Everett, Nicholas Robins, Ping Koy Lam, Ben Buchler
DDirect Imaging of Slow, Stored, and Stationary EITPolaritons
Geoff T Campbell , , Young-Wook Cho , Jian Su , JesseEverett , Nicholas Robins , Ping Koy Lam , Ben Buchler Centre for Quantum Computation and Communication Technology,Department of Quantum Science, The Australian National University, Canberra,ACT 2601, Australia Department of Quantum Science, The Australian National University, Canberra,ACT 2601, AustraliaE-mail: [email protected] Abstract.
Stationary and slow light effects are of great interest for quantuminformation applications. Using laser-cooled Rb87 atoms we have performed sideimaging of our atomic ensemble under slow and stationary light conditions, whichallows direct comparison with numerical models. The polaritions were generated usingelectromagnetically induced transparency (EIT), with stationary light generated usingcounter-propagating control fields. By controlling the power ratio of the two controlfields we show fine control of the group velocity of the stationary light. We alsocompare the dynamics of stationary light using monochromatic and bichromatic controlfields. Our results show negligible difference between the two situations, in contrast toprevious work in EIT based systems. a r X i v : . [ qu a n t - ph ] J un irect Imaging of Slow, Stored, and Stationary EIT Polaritons Introduction
Electromagnetically induced transparency (EIT) [1, 2, 3] is a technique that enables finecontrol of the propagation of light fields. It is ordinarily achieved using a probe beamthat copropagates with a control beam through an atomic ensemble. In the absenceof the control light, the probe is absorbed into the atomic ensemble. The additionof control light induces transparency for the probe, which can then pass through theensemble with, potentially, very little absorption. Modulation of the control field allowsthe slowing and even storage of light within the atomic medium. Accordingly, EIThas been used to demonstrate the slowing, storage and retrieval of quantum states oflight [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and has been proposed to have uses in sensingapplications [14, 15] where slow light can be advantageous. It has also been extendedto demonstrate a wide range of coherent control techniques to manipulate light. Oneinteresting example is that of stationary light, where two counter-propagating controlfields are used to stop the propagation of an EIT polariton even though a portion ofthe excitation remains optical [16, 17, 18, 19, 20, 21]. The optical part can be used togenerate a nonlinear interaction, necessary for optical quantum gates [22, 23, 24, 25].In most stationary light experiments to date, only the optical field at the outputend of the ensemble has been detected and compared to theoretical predictions. It ispossible, however, to use side-imaging of a cold atomic ensemble to directly observethe dynamics of EIT polaritons. The technique has been applied in Bose Einsteincondensates [26, 27] and warm atomic vapours [28] to observe the dynamics of EITslow light. Recently, we have also used side-imaging to directly observe the dynamics ofpolaritons in an off-resonant stationary light scheme [29] that was performed in a coldatomic cloud.Here, we use the side-imaging technique to observe the propagation of EIT-basedpolaritons in a laser-cooled atomic ensemble under various conditions. The results arein good agreement with simulations of a simple three-level model and illustrate a rangeof effects such as slow light, storage, backward retrieval, and stationary light. Theexperiments provide strong evidence in support of the existence of stationary light underEIT conditions. We further demonstrate that the group velocity of propagating lightcan be precisely controlled by changing the ratio of forward and backward propagatingcontrol fields.One aspect of this system, that is the subject of ongoing investigations, is the impactof atomic motion on the dynamics of EIT-based stationary light. When the counter-propagating control fields form a standing wave there is potential for rapid decay of thestationary light, but this depends on the temperature of the atoms [21, 30, 31, 32, 33, 34].In our experiments we investigate the impact of the standing wave control field andcompare this situation to a bichromatic control field without a standing wave. Ourexperiments show no significant difference between these two cases, in contrast toprevious work. We speculate on the reasons for this and offer suggestions for furtherwork that may resolve this issue. irect Imaging of Slow, Stored, and Stationary EIT Polaritons Slow-Light with Counter-Propagating Control Fields
The dynamics of EIT slow-light can be understood via the Maxwell-Bloch equations ofmotion. In the pure-state and paraxial approximations, these equations can be written[35] as ∂ t P = − Γ P + ig √ N E + i Ω S (1) ∂ t S = − γS + i Ω ∗ P (2)( ∂ t + c∂ z ) E = ig √ N P . (3)Referring to the atomic level scheme shown in Fig. 1b, P ( z, t ) is the envelope of theexcited state coherence | (cid:105) ↔ | (cid:105) , S ( z, t ) is the envelope of the spin coherence | (cid:105) ↔ | (cid:105) and E ( z, t ) is the envelope of the probe field. Decay rates Γ and γ are imposed for P and S , respectively, g is the probe field coupling rate, N is the number of atoms and Ω isthe control field Rabi frequency. If the envelope E ( z, t ) varies slowly enough, P ≈ E ≈ − Ω / ( g √ N ) S . The light and the atomic spin coherence then propagate together asa polariton defined as ψ = ( E sin θ − S cos θ ) where tan θ = Ω / ( g √ N ) [4]. The equationof motion for the polariton is (cid:0) ∂ t + sin θc∂ z + cos θγ (cid:1) S = 0 which, in a transformed aset of coordinates τ = t − z/c , ξ = z/L , can be written (cid:16) ∂ τ + cL tan θ∂ ξ + γ (cid:17) ψ = 0 (4)where L is the length of the ensemble.To determine the dynamics of slow light with counter-propagating control fields, wetreat the forward and backward traveling components of P independently by defining P = P + e ikz + P − e − ikz , where k is the wavenumber of the excited state coherence [19, 21].These are coupled to the atomic ground state and meta-stable state through E ± andΩ ± respectively, as illustrated in Fig. 1(b). Standing waves formed by the counter-propagating control and probe fields will create higher spatial frequencies in S of theform S = S + (cid:80) ∞ n =1 S n + e i nkz + S n − e − i nkz . We expect a rapid decay of the higherspatial frequencies of the coherence due to atomic motion and truncate to n = 1.Because E ± are counter-propagating, there is no coordinate transformation thatwill remove the time derivative from the propagation equations for both the forwardand backward traveling waves. We can, however, omit the time derivatives by assumingthat the speed of light is sufficiently large that L/c , which on the order of 10 − s, ismuch shorter than any other timescale in the system dynamics. We then define a spatialcoordinate ξ ( z ) = (cid:90) z dz (cid:48) η ( z (cid:48) ) /N (5)that is scaled by the relative optical density along the ensemble, η ( z ). Introducing theoptical depth d = g N L/ (Γ c ) and scaling E by a dimensionless factor of (cid:112) c/ (Γ L ) allowsus to write the Maxwell-Bloch equations in a compact form P ± = i √ d E ± + i (Ω ± / Γ) S + i (Ω ∓ / Γ) S ± (6) ∂ t S = − γS + i Ω ∗ + P + + i Ω ∗− P − (7) irect Imaging of Slow, Stored, and Stationary EIT Polaritons ∂ t S ± = − ˜ γS ± + i Ω ∗∓ P ± (8) ∂ ξ E ± = i √ dP ± (9)where we have made the adiabatic approximation Γ P ± (cid:29) ∂ t P ± . The higher spatialfrequencies are assumed to decay at a rate ˜ γ as a result of averaging due to atomicmotion. These higher spatial frequencies have been shown to result in additionaldiffusion, decay and pulse splitting of the stationary polariton [30, 31, 32, 33] forsufficiently cold atoms.To find a compact analytic solution, we assume that the higher spatial frequencies S ± dissipate sufficiently quickly to be negligible, although we include them for numericalsolutions [21]. The equations can then be solved in the Fourier domain of the normalisedspatial coordinate X ( ξ, t ) = (cid:82) dκe − iκξ X ( κ, t ) [36]. Combining equations (6) and (9) andexpanding to the first order in κ/d we obtain E ± (cid:39) − Ω ± √ d Γ (1 ± iκ/d ) S. (10)The quantity κ/d is the spatial variation of the envelope of S relative to the absorptiondepth of the ensemble. Substituting this into Eq. 7 and transforming out of the Fourierdomain yields an equation of motion (cid:20) ∂ t + Γ tan θ (cid:18) cos 2 φ∂ ξ − d ∂ ξξ (cid:19) + γ (cid:21) S = 0 (11)where tan θ ≡ | Ω | d Γ ; tan φ ≡ | Ω − | | Ω + | ; | Ω | ≡ | Ω + | + | Ω − | . (12)This is a shape-preserving advection equation with a velocity v = Γ tan θ cos 2 φ and adiffusion term (Γ /d ) tan θ∂ ξξ S . The diffusion is due to finite optical depth and arises inthe equation of motion due to taking the more relaxed adiabatic approximation ∂ t P (cid:28) Γ P ± instead of P ± ≈
0. In the limit of large optical depth or slow pulses, k/d (cid:28) ψ = sin θ ( E + cos φ + E − sin φ ) − S cos θ can bedefined for the system. The mixing angles θ and φ are governed by the total controlfield power and the ratio between the power of the forward and backward control fields,respectively. Either mixing angle can be used to reduce the polariton velocity to zeroand φ can be used to reverse the direction of propagation. Experimental methods
We experimentally analysed the dynamics of slow-light by sending a probe pulse into anelongated cloud of cold atoms [37, 38] that was uniformly illuminated by a bright controlfield. The magnitude of the spin coherence as the pulse propagated along the length ofthe cloud could then be observed by absorption imaging from the side of the cloud. Wefirst prepared the atoms in an elongated magneto-optical trap (MOT), which providedan ensemble of atoms with a temperature of approximately 100 µ K and an amplitudeoptical depth of d = 190 along the trap axis. The atom cloud was approximately 4 cm irect Imaging of Slow, Stored, and Stationary EIT Polaritons µ m. The probe field was aligned along the cloud axisand focused with a beam diameter of 110 µ m. The control fields were collimated to alarger diameter to ensure uniform coverage of the atomic ensemble.Figure 1 shows the layout of the experiment (a) and the atomic transitions usedfor EIT and imaging (b). Phase-matching between the forward and backwards EITprocesses was achieved by placing the control fields on the transition with the shorter c) Distance (cm)0 4 i) I ii) I Out iii) d
Image b) P P S a) Propagation direction Phase-matchingCCD Camera T i m e ( µ s ) Ω + Ω + Ω - Ω - ++ -- D2 (Imaging) D1(EIT) d I m a g e d I m a g e Figure 1. (a) Schematic of the experimental layout. (b) Energy level diagram showingthe EIT tripod and imaging beam. (c) Characterization of the atom distribution in theMOT showing the imaging beam intensity I , transverse absorption from the MOT I z and calculated transverse optical depth d image . (d) Absorption images taken every 3 µ s of an EIT polariton traveling through the ensemble. Slow light, storage and releasecan be seen. irect Imaging of Slow, Stored, and Stationary EIT Polaritons transition. The shadow of the atoms was imaged onto a CCD camerausing a large aperture lens.The magnitude of | S | is proportional to the number of atoms, which in turn isproportional to the optical depth as seen by the imaging beam. This can be foundaccording to d image = − log ( I out /I ) / I is the intensity of the imaging beam inthe absence of any atoms. Figure 1(c) shows images of I , I out , and d image for the entireatom cloud. This was obtained by optically pumping the ensemble into | (cid:105) which is thetransition used for imaging.For a slow light experiment, the cloud is initially prepared in the | (cid:105) state and aprobe pulse is sent into the cloud while the forward control field is on to build somecoherence between the | (cid:105) and | (cid:105) . The CCD camera is exposed for 300 µ s, spanningthe entire duration of a slow-light or storage experiment, but the imaging beam is gatedto illuminate the ensemble for only 1 µ s. This allows a stroboscopic measurement of thelocation of the spin coherence as it travels through the cloud. Repeating the experimentand shifting the exposure time in 1 µs intervals allowed composition of a space-timeimage of the spinwave propagation. Figure 1(d) shows samples of the images taken ofpulse propagation during slow-light, storage, and release.To map | S | into the normalised spatial coordinate ξ , the images can be binnedalong the propagation dimension according to the optical density of the atom cloud,as in Eq. 5. This binning reduced the length of each image from 1384 to 200 bins,each of which contains a roughly equal number of atoms. The transverse region of theimage that contains the pulse was then integrated to obtain a one-dimensional arrayproportional to | S | . In the following results we will present data showing scaled valuesof | S | as a function of time and ξ to allow comparison with numerical models. Results
Storage and Retrieval
Figure 2(a) shows the propagation of the coherence | S | in the normalised coordinate asa probe pulse enters the ensemble under the conditions of EIT slow-light. As is shownin the timing diagram (i), a pulse enters the ensemble while the control field is on andcan be seen in the imaging data propagating slowly through the ensemble (ii). Thepropagation is stopped by turning off the control field, and is then resumed by restoringthe control field. The optical pulse emerges from the ensemble and is recorded ona photo-detector (iii). The shape-preserving nature of the slow-light pulse propagationcan be seen to be in good agreement with numerical simulations of Eqs. 6-9 (iv), and thesimple advection Eq. 11 (v). All of the parameters used in the simulations correspond irect Imaging of Slow, Stored, and Stationary EIT Polaritons Stationary Light
In addition to using either the forward or backward control fields, both control fieldscan be used simultaneously to modify the propagation of the polariton by changingthe mixing angle φ in Eq. 11. Figure 3 shows a number of experiments demonstratingdifferent slow-light effects with both control fields. For each experiment, the observedpropagation data is shown along with the solutions to numerical simulations of Eqs. 6-9, and to the corresponding advection Eq. 11. Column (a) shows a reduced backwardpropagation velocity due to the addition of a forward control field with an amplitudethat is half that of the backward control field. Column (b) shows a complex sequenceof forward slow light, backward propagation with some forward control as well, storedlight and then forward recall. The sequence shows that the use of counter-propagatingcontrol fields can be used to controllably push the coherence in either direction withinthe ensemble.In columns (c,d), stationary light is demonstrated by illuminating the ensemblewith both control fields at equal amplitude simultaneously, once the polariton haspropagated to the centre of the cloud. In (c), the stationary light is formed directly Three-level modelAdvection equation ξ ( a . u . ) ξ ( a . u . ) ξ ( a . u . )
01 Time (µs) 500500 Time (µs) b)Experiment100.10 10 ξ ( a . u . )
01 Experiment0.10 Time (µs) 500 c)a) |s|
Propagation Direction: Input: Forward Output: Backward Output: x10 ξ ( a . u . ) ξ ( a . u . )
01 Time (µs) 500 Three-level modelAdvection equationiv)v) + -
Figure 2.
Propagation of an EIT polariton as it experiences slow-light, storage, andeither forward retrieval (a) or backward retrieval (b). The experimentally measuredpropagation (ii) is compared to the results of a numerical simulation (Eqs. 6-9) (iv)and to the solution of a shape-preserving advection equation (Eq. 11) (v). The opticalfield at the input of the ensemble (i) and output (iii) is measured by photo-detectors.Some transmission is visible on the output photo-detector; this is a spurious frequencycomponent of the probe field that is far-detuned from resonance and is an artifact fromhow we generate the probe field. The values of | S | are scaled to the maximum valueof | S | . irect Imaging of Slow, Stored, and Stationary EIT Polaritons ξ ( a . u . ) ξ ( a . u . ) ξ ( a . u . )
01 Time (µs) 500 Time (µs) 400Time (µs) 500 Time (µs) 400 E xp e r i m e n t - L e v e l S i m A dv ec ti on T i m i ng b) c) d)a) Figure 3.
Experimental polariton propagation compared to simulations and solutionsof an advection equation for various control field timings and amplitudes. (a) Forwardslow light (FSL), storage (S) and retrieval using quasi-stationary light with imbalancedcontrol fields (QSL). (b) A sequence of FSL, QSL, S, FSL. (c) FSL followed bystationary light. (d) FSL, S, stationary light with half-intensity control fields. Thevalues of | S | are scaled to the maximum value of | S | . from slow light by turning on the counterpropagating control field. In (d) The polaritonis stopped by turning off the control field, and stationary light is formed by turning onboth control fields at half of the initial amplitude. In both cases, the polariton is heldnearly stationary while both control fields are on. From this, and from the agreementbetween the observed propagation dynamics and those that are predicted, we infer thata stationary optical field is present in the ensemble.The diffusion of the polariton arises from limited optical depth, the standing wavepattern formed by the control fields, and thermal motion of the atoms in the cloud.The temperature of the atoms in our system was measured to be 100 µ K in a previousexperiment [37], giving a mean atomic velocity of 10 cm/s. This is slow enough thatwe can neglect diffusion due to atomic motion. The solution of the advection equation,Eq. 11, includes only the effect of limited optical depth while the numerical solutionsof Eqns. 6-9 also takes into account the standing wave. The decay rate of the higherspatial frequencies that we use in the simulation is estimated from the atomic thermalmotion according to ˜ γ = 4 π (cid:112) k B T /m/λ = 2 π × .
25 MHz. We note that diffusion isactually reduced with increasing temperature because atomic motion becomes significantcompared to the length scale of the standing wave but not compared to the length scaleof the polariton. This eliminates the diffusion term resulting from the standing wavebut diffusion that arises directly from thermal motion remains negligible. Our results,however, show less diffusion than would be expected for the measured temperature.To further investigate the diffusion we are able to manipulate the effective decay rate irect Imaging of Slow, Stored, and Stationary EIT Polaritons
400 Time (µs) 500Time (µs) Time (µs) 500 ξ ( a . u . ) ξ ( a . u . )
01 Single-Colour Single-ColourSingle-ColourTwo-ColourTwo-Colour Two-Colour a) b) c)d) e) f) T i m i ng S i m E xp . T i m i ng S i m E xp . ξ ( a . u . ) ξ ( a . u . ) Figure 4.
A comparison of stationary light using control fields with equal frequencies(a,c,e) and with frequencies that are symmetrically detuned from resonance by 4 MHz(b,d,f). The stationary light is formed directly from slow-light (a,b) and from stoppedlight (c,d). Numerical simulations predict reduced dispersion when the control fields aredetuned, however, a rapid decay of the polariton in the experimental case obscures theeffect. Quasi-stationary light is observed (e,f) with imbalanced control field powers.Again, differences between the single- and two-color cases are not resolvable in theexperiment. The values of | S | are scaled to the maximum value of | S | . for the standing wave terms by introducing a frequency difference between the forwardand backward propagating components [21]. In this case, the interference betweencounter-propagating control fields forms a travelling wave that averages out the finespatial structure. Running the experiment in this regime may allow one to distinguishthe diffusion of the stationary light due to finite optical depth and diffusion due to thestanding wave of the control field. We therefore performed stationary light experimentswith the control fields symmetrically detuned from the excited state transition by ± γ eff = ˜ γ + 2 π × irect Imaging of Slow, Stored, and Stationary EIT Polaritons | (cid:105) state is performed by an axially aligned beam. Reflection
All results presented thus far were obtained by first allowing the polariton to propagateinto the ensemble as a slow-light polariton with only the forward control field beforeturning on the backward control to create stationary light. If both control fields arepresent when the probe pulse is incident on the ensemble, reflection can be observed.Figure 5 (top-row) shows the observed (a) and simulated (b,c) reflection from theensemble when it is illuminated with equal or unequal forward and backward controlfields. The two simulations show the results that include the standing wave terms (b)and that neglect them (c). For the case where the forward control field is stronger thanthe backward control field, a polariton can be observed propagating into the ensemble(bottom-row). For the cases where the forward control field intensity is equal or lessthan that of the backward control field, no polarition can be seen entering the ensemblein the imaging data (not shown), although the simulations show a small region spincoherence at the start of the ensemble (not shown).
100 Time (µs) 20 I n t e n s it y ( a . u . ) ξ ( a . u . ) a)d) 0 Time (µs) 20 Input:Ω + = Ω - Ω + = 2 Ω - Ω + = - Reflection:Experiment Simple Simulation c)0 Time (µs) 20
Full Simulation b)10 Ω2 e) f) Figure 5.
Reflection off of the ensemble is observed for equal and imbalanced controlfield powers. Experimental data showing the power reflected from the atomic ensembleis shown in (a) and theoretical simulations with (b) and without (c) the standing waveterms. The bottom row shows data for the case of Ω + = 2Ω − . The experimentaldata (d) and models (e,f) demonstrate that a polariton propagates some way into theensemble before being reemitted in the backward direction. The values of | S | are scaledto the maximum value of | S | . irect Imaging of Slow, Stored, and Stationary EIT Polaritons Group velocity control
The group velocity as a function of the mixing angles θ and φ can be directly determinedfrom the imaging data. Figure 6 shows the measured group velocities along withthose calculated for some of the combinations of control field powers used for the datapresented in figs. 2-5. The measured group velocities are compared to the expectedvalues for each point based on the measured control field powers. There are slightdifferences in the values for control field powers used here and those used in the numericalsimulations. Differences in the transverse size of the control fields resulted in a differentcalibration between the measured control power and the control Rabi frequency for theforward and backward controls. The numerical simulations were run with the intendedratio between forward and backward fields while the data in Fig. 6 uses a calibrationthat is based on the measured group velocities of forward and backward slow light. -5x10 -2x10 v g , no r m a li s e d ( s - ) φ v g ( m / s ) Г Tan θ Cos 2φ MeasuredГ Tan θ n Cos 2φ (1) (2) (3) (4) (5) (θ =3.9 mrad)(θ =4.5 mrad)(θ =3.3 mrad)(θ =5.0 mrad)(θ =4.5 mrad) (6) (θ =5.9 mrad)(θ =4.2 mrad) Figure 6.
Measured group velocities (red dots) for different control field powerscompared to the expected values (blue squares). The mixing angle θ differed slightlybetween measurements and the values for each measurement are shown in the lowerleft quadrant. The solid curve shows the expected behavior if the total control fieldpower had been held constant across all the measurements. Discussion
While our results are in generally good agreement with the theoretical models, thereis stronger attenuation and less dispersion of the stationary polariton than predicted.To quantify both the attenuation and dispersion for stationary light, we fit Gaussianenvelopes to spatial profiles of the spinwaves under stationary light conditions as shownin Fig. 7 (a) for single-color stationary light and (b) for two-color stationary light.These cross sections are made from the data presented earlier in Fig. 4 (a,b). Plottingthe decay of the spinwave amplitude and the Gaussian full-width-half-maximum as afunction of time we arrive at Figs. 7 (c) and (d) respectively. Figure 7(c) shows that the irect Imaging of Slow, Stored, and Stationary EIT Polaritons P u l s e F W H M ( a . u . ) Time µs ξ (a.u.) | S | ( a . u . ) ξ (a.u.) | S | ( a . u . ) a) b)c) d) | S | A m p lit ud e ( a . u . ) Time µs
Single-colour τ = 7.1 µsTwo-colour τ = 7.6 µst = 1 µst = 4 µst = 7 µst = 10 µs t = 1 µst = 4 µst = 7 µst = 10 µsSingle-colour Two-colour Single-colour Two-colour
Figure 7. a) Gaussian fits to spatial profile of the spinwave for single-colour stationarylight. b) Gaussian fits to spatial profile of the spinwave for two-colour stationary light.c) Decay of the spinwave amplitude for one- and two-colour stationary light. d) Spatialwidth of the spinwave as the stationary light decays for the one- and two-colour data. time-constants for the decay of the polaritons are 7 . ± µ s for the single-color case and7 . ± . µ s for the two-color case respectively. Figure 7(d) shows that no diffusion of thepulses is apparent, although the pulse width fits have a large uncertainty due to noisein the images. From these images we conclude that there is no observable differencebetween the monochromatic and bichromatic controls fields in our experiment. Thisis in contrast to other observations and theoretical predictions [21, 30, 31, 32, 33, 34]which predict a large diffusion due to the standing wave terms in the coherence. Asnoted above, a possible cause of the discrepancy may be some net longitudinal motionof the ensemble. The temperature measurement is insensitive to any motion of thecloud that is common to all atoms in the ensemble. Such motion, however, wouldstill average out the high spatial frequencies associated with the standing wave of thecounter-propagating control fields.The observed decay time, shown in Fig. 7 is significantly shorter than the estimateddecoherence rate γ ≈
500 Hz for our ensemble. A possible cause of this attenuation isatomic population present in the | F = 1 , m F = {− , }(cid:105) states. Transitions from thesestates will absorb the probe field since there is no corresponding control field to provideelectromagnetically induced transparency. This residual Beer’s law absorption can beeasily included in the equations of motion, however, the initial slow light propagationseen in figures 2 and 3 is consistent with negligible additional absorption. irect Imaging of Slow, Stored, and Stationary EIT Polaritons Conclusion
We have applied the technique of side absorption imaging to visualize the dynamicsof stationary and non-stationary electromagnetically induced transparency polaritonswhen driven by counter-propagating control fields. Our results demonstrate that EITstationary light can be modelled with a simple equation of motion. We have also shownhow tuning the power ratio of the counterpropagating control fields allows fine controlof the group velocity of the stationary light. Absent from the results are signaturesthat arise from high spatial frequencies due to standing wave control field. Furthercooling our ensemble may reveal the modification of dynamics that is expected from thestanding wave terms.
Acknowledgments
We thank J. R. Ott, A. S. Sørensen and their team for very helpful discussions on therole of higher-order spatial frequencies in the atomic coherence. Our work was fundedby the Australian Research Council (ARC) (CE110001027, FL150100019).
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