Slow and stored light and optical depth
M. Klein, Y. Xiao, A.V. Gorshkov, M. Hohensee, C.D. Leung, M.R. Browning, D.F. Phillips, I. Novikova, R.L. Walsworth
SSlow and stored light and optical depth
M. Klein a,b , Y. Xiao a , A. V. Gorshkov b , M. Hohensee a,b , C. D. Leung a,b , M. R. Browning a,b ,D. F. Phillips a , I. Novikova c , and R. L. Walsworth a,ba Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, 02138 USA b Department of Physics, Harvard University, Cambridge, MA, 02138 USA c Department of Physics, College of William & Mary, Williamsburg, VA 23185, USANovember 9, 2018
ABSTRACT
We present a preliminary experimental study of the dependence on optical depth of slow and stored light pulsesin Rb vapor. In particular, we characterize the efficiency of slow and stored light as a function of Rb density;pulse duration, delay and storage time; and control field intensity. Experimental results are in good qualitativeagreement with theoretical calculations based on a simplified three-level model at moderate densities.
Keywords:
Electromagnetically-induced transparency, slow light, stored light, vapor cell, buffer gas, opticaldepth
1. INTRODUCTION
Applications of slow and stored light based on electromagnetically-induced transparency (EIT), in both quantuminformation processing and optical communication, will benefit from improved efficiency and delay-bandwidthproduct. In quantum information, stored light has emerged as a promising technique for applications suchas single-photon generation on demand and quantum memories and repeaters.
However, practicalapplications will require significant improvements in the efficiency of writing, storing, and retrieving an inputphoton state beyond values achieved to date.
In classical communications, optical buffers require adjustabledelay time (i.e., group index) for input signal pulses with minimal pulse distortion and loss and high compressionof the input pulse for high data density inside the delay medium. However, high efficiency and large delay-bandwidth product have not fully been realized in all-optical systems. Here we present a preliminary study ofthe optimization of slow and stored light using EIT in warm Rb vapor as a function of optical depth, laser power,and input pulse bandwidth.In EIT, a weak signal pulse propagates with very slow group velocity through an otherwise absorbingmedium, which is established by a strong control field. The reduced group velocity with which a signalpropagates under EIT is the result of the transfer of photonic excitation in the pulse to the collective spin co-herence of the atoms. The group velocity is proportional to the control field intensity, and can be reduced tozero by shutting off the control light. In this case the photonic information of the pulse is completely transferredto the atoms. This coherent transfer is reversible, and thus after some storage time the pulse may be retrieved,converting it back into photonic excitation when the control light is turned back on.Optimization of slow and stored light requires reducing the absorption of signal light and spin decoherence, aswell as increasing optical depth to achieve a large delay-bandwidth product. In warm atom vapor cells, signallight absorption is typically driven by incomplete polarization of the atomic medium due to such mechanismsas radiation trapping and competing nonlinear processes. Atomic decoherence arises from mechanisms suchas collisions with buffer gas atoms and cell walls, diffusion out of the laser beam, and residual magnetic fieldgradients.Modeling the atomic medium as a collection of stationary three-level Λ-systems (Fig. 1a), we find that slowand stored light properties improve with increasing rubidium density or optical depth. The EIT bandwidth, γ EIT , for moderate to large optical depths is found to be
19, 20 γ EIT = | Ω C | (cid:112) γg N L/c (1)1 a r X i v : . [ qu a n t - ph ] J u l pectrum analyzer Rb cell,oven,solenoid,shieldsEOM AOM ! /2 PBSLASER MMNPBSfiber, PD beam stop beam stop ! Fp ! Fs ! Fs C " probe (a) (b) Figure 1. (a) The relevant Rb atomic energy levels in a Λ-scheme used for EIT, slow, and stored light measurements.(b) Schematic of the apparatus. Output of laser is split with a half-wave plate ( λ/
2) and polarizing beam splitter (PBS)into a reference beam and main beam; electro-optic (EOM) and acousto-optic (AOM) modulators shape the signal andcontrol fields; the vapor cell is housed inside an oven, solenoid, and magnetic shielding; a fast photodetector measures thebeat between the unmodulated reference beam and the output from the vapor cell.
See text for details. where Ω C is the Rabi frequency of the control field, γ the excited-state line width, N the total number of three-level atoms, L the length of the medium (the optical depth, d = g N L/γc ), g the light-atom coupling coefficient,and c the speed of light. Similarly, the group velocity v g resulting from the steep resonant dispersion for thesignal pulse in the slow light medium is
19, 20 v g = c g N/ | Ω C | ≈ c | Ω C | g N . (2)The absolute pulse delay, ∆ T abs , is then given by the length of the medium divided by the group velocity andwe find that ∆ T abs ∝ d , that is the delay is proportional to the optical depth. A careful theoretical study ofstorage efficiency η (defined as the ratio of the number of output photons to the number of input photons),optimized for a given optical depth, finds that the optimal temporal pulse width T opt ∼ d , and finds that η scales as 1 − /d at high optical depth, and increases more rapidly at lower d . This treatment impliescontinued efficiency improvement with increasing optical depth. In practice, experiments have found that slowand stored light efficiency improves with increasing optical depth for modest optical depths. However, at largeroptical depths, slow and stored light efficiency reach a maximum and then degrade. Note that a procedure foroptimizing stored light efficiency for a given d , using the time-reversed output pulse to determine the shape ofthe input pulse, has been developed theoretically and directly implemented experimentally. In the present study we report preliminary experimental investigations of the behavior of slow and storedlight over a wide range of optical depths, by varying the temperature of Rb in a vapor cell. Measuring coherenceloss rates and including them in the efficiency optimization model yields good agreement between theory andexperiment as η falls off with increasing atomic density. We focus on the role played by optical depth in coherenceloss, in particular the effect of radiation trapping, where incoherent photons are repeatedly re-absorbed by themedium before exiting. We also report promising preliminary efficiency results for a new cell geometry designedto reduce coherence loss due to radiation trapping.
2. EXPERIMENTAL SETUP
We measured EIT and slow and stored light, using the D transition in a warm Rb vapor. The relevant energylevels and coupling fields are shown in Fig. 1a: a strong control field is resonant with the F = 2 → F (cid:48) = 2transition and a weak signal field is resonant with F = 1 → F (cid:48) = 2. The optical fields were generated by anexternal cavity diode laser tuned to λ = 795 nm, amplified by a tapered amplifier, and spatially filtered by apinhole (Fig. 1b). An electro-optic modulator (EOM) phase modulated the laser at the ground state hyperfinesplitting ( ∼ .
835 GHz); the +1 sideband acted as the signal field, with a maximum signal to control intensity2atio of 2 . δ of the EIT transition for measuring its line shape. The overallintensity was regulated with an acousto-optic modulator (AOM), which also shifted the frequency by +80 MHz.A quarter-wave plate ( λ/
4) before the cell converted the beam to circular polarization. The collimated beamentering the cell had a diameter of approximately 7 mm. The output fields were measured by sending the laserinto an optical fiber, then into a fast photodetector (PD) along with a reference beam picked off prior to theEOM and AOM. The control field, signal field, or the unused − .
835 GHz + 80 MHz, and 6 .
835 GHz −
80 MHz respectively.The transmission of the − ∼ T abs was the difference betweenpulse peak times and efficiency η = (signal pulse area)/( − ◦ C, or atomic number density between 4 × and 1 × cm − . Three layers ofcylindrical high-permeability shielding surrounded the oven to screen out stray laboratory magnetic fields, anda solenoid inside was used to cancel any small constant background field. Three Rb vapor cells were used in ourexperiments; all three contained isotopically enriched Rb and were filled with buffer gas to confine atoms andextend their coherence life times. The cell used for all measurements except where specifically noted had length L = 7 . D = 2 .
3. SLOW AND STORED LIGHT EFFICIENCY MEASUREMENTS
We studied slow and stored light as a function of optical depth by measuring the delay-bandwidth product andefficiency of “optimized” pulses as the vapor cell temperature was varied. Such pulses were constructed throughan iterative optimization procedure in which a signal pulse was stored and retrieved, its output shape measuredand a new pulse stored with the shape of the output pulse. This process is repeated until convergence. Thisprocedure has been shown both theoretically and experimentally to produce slow-light pulse shapes thatoptimize storage efficiency for the given optical depth and laser power. In effect this procedure selects a pulsebandwidth that balances fractional delay and absorption to yield the largest output signal pulse.Measured EIT bandwidths and slow light delays scaled as expected with optical depth. Figure 2a showsEIT linewidth (FWHM, as extracted from a Lorentzian fit) vs. atomic density, which follows the γ EIT ∝ / √ d trending expected from Eq. 1. Figure 2b shows that absolute pulse delay ∆ T abs ∝ d ; and Figure 2c showsthat the optimal pulse width T opt ∝ d . We measured line widths and delays at two control field powers,3 . . and Rabi frequencies of6 . T opt is the temporal pulse width) to themeasured EIT bandwidth (Fig. 2d), we find that 1 /T ∼ γ EIT /
3, over a large range of fractional delays (from 0 . > .
20, 24 where γ EIT
T >> >>
10. Note that based on the scalings of ∆ T abs ∼ d and T opt ∼ d , we would expect the optimal frequencybandwidth to be quadratic in the EIT line width which is not clearly observed in Fig. 2d.Stored light efficiency was measured for a range of rubidium densities between 4 × cm − (40 ◦ C) and1 × cm − (80 ◦ C) and storage intervals from τ = 50 µ s to τ = 1 . τ = 400 µ s inFig. 3. The efficiencies peak at temperatures between 60 and 65 ◦ C, falling at higher optical depth. The1 /e coherence lifetimes during storage (Fig. 4) also fall at high density. This points to possible high-densityor high-temperature phenomena adversely affecting slow and stored light. Increased loss during storage andreadout at high optical depth agrees with predictions from theoretical simulations when the density dependenceof decoherence is included (Fig. 3). The simulations, described fully in earlier work,
18, 21, 22 consist of iteratedsolutions to the stored light dynamics equations for an ensemble Λ-system. The dashed curve in Fig. 3 usesthe measured rates of decoherence during storage and readout taken from Fig. 4 as an input parameter for thespin-wave decay rate, and most closely matches the experimental results.3 igure 2.
Measured EIT line width, slow light delay, and optimized pulse width at various optical depths and fortwo control field powers (3.8 mW and 8.8 mW). (a) EIT line widths are consistent with power broadening and densitynarrowing. (b) Absolute pulse delay (= time elapsed between the peaks of the reference pulse and the slowed pulse) and(c) optimized temporal pulse widths are approximately linear in atomic density. (d) Optimized slow light bandwidth vs.EIT line width. Error bars for all measurements are derived from a systematic uncertainty in the Rb temperature of ∼ ◦ C and uncertainty in the laser frequency of ∼
150 MHz. Dashed lines are fits of data to the simplified three-levelmodel described in text. igure 3. Measured and calculated storage efficiency at fixed storage time τ = 400 µ s vs. atomic density for control fieldlaser powers of (a) 3 . . µ swith no density-dependent decoherence (dotted curves) and for the measured coherence lifetimes shown in Fig. 4 (dashedcurves). Comparison of the two calculated results shows that density or temperature-dependent decoherence during thestorage interval dominates high-density losses. The remaining discrepancy between measurement and calculation is likelydue to radiation trapping. Error bars are from systematic uncertainty in Rb temperature of ∼ ◦ C and of the laserfrequency of ∼
150 MHz.
Curves added to guide the eye.
Figure 4.
Lifetimes of stored coherence as a function of atomic density, derived from measurements described in text.The coherence lifetime drops at high atomic density, likely due to the effect of temperature-dependent losses such asRb-Rb spin exchange. Error bars are derived from Rb temperature and laser frequency uncertainty as well as statisticaluncertainty in fits to coherence lifetimes.
Dashed line added to guide the eye. igure 5. (a) Slow light efficiency (output pulse area divided by input pulse area) as a function of atomic density, forinput pulse shapes optimized for maximum stored light efficiency (solid triangles) and for input Gaussian pulses withthe same FWHM (hollow triangles). Also shown are measurements relevant to stored light efficiency: the fraction ofthe optimized input pulse that escapes the medium before the pulse is stored (open squares). The dashed line combineslight storage data at τ = 400 µ s and coherence lifetime measurements to predict slow light efficiency ( see text ) in goodagreement with measured efficiency. Error bars are from systematic uncertainty in Rb temperature of ∼ ◦ C and in thelaser frequency of ∼
150 MHz. Error bars on other data sets are of equal magnitude. (b) Sample data for density ofapprox. 8 × cm − showing the retrieved and unstored pulses (dark, solid line), the slow light pulse with optimizedshape (dotted line), and the reference pulse (gray solid line). In Figure 5 we verify that the loss mechanisms for stored light are all self-consistent. A slow light pulse thatis not trapped in the medium (the control field is left on) has loss from imperfect EIT transmission and frompulse bandwidth lying outside the region of maximum transmission (after normalization to a reference pulse, thisis the “slow-light efficiency”). A stored and retrieved pulse will experience that same loss, and additional lossfrom two sources: (i) part of the pulse escapes the medium before the pulse is stored (fractional delay less thanunity implies significant losses from this mechanism); and (ii) atomic coherence decay during the storage andreadout periods. Figure 5a shows the directly measured slow light efficiency η slow , the area ratio of the outputpulse to the reference pulse, and an inferred η slow,predict from the storage measurement. Using the measureddecoherence rate from Fig. 4 and the storage readout areas at τ = 400 µ s, we can back out the storage readoutat τ = 0; combining this with the area of the not-stored pulse leakage we infer η slow (see Fig. 5b): η slow,predict = η leakage + η × exp(400 µs/τ coherence ) , (3)where η leakage is the leaked pulse area divided by the reference pulse area, and τ coherence is the atomic coherencelifetime from Fig. 4. The measured and inferred slow light efficiencies match very well, only deviating significantlyat the highest atomic density, where the term with exp(400 µ s /τ coherence ) becomes very sensitive to the shortcoherence lifetime measurement, which is more error prone due to the low SNR in the signal transmission.As indicated from both the storage efficiency and the comparison between slow and stored light, at high Rbdensities the effectiveness of the medium is reduced. We believe that several mechanisms are responsible for this.First, at higher densities, temperature or density-dependent losses such as Rb-Rb spin exchange become moresignificant, contributing to decoherence even during the storage interval (see Fig. 4). Second, in the presence oflight at high Rb densities, radiation trapping becomes significant: light absorbed and then incoherently radiatedby one rubidium atom is reabsorbed by another rubidium atom, and this process leads to decoherence and lossof efficiency. In the next sections, we present preliminary investigations of high-density losses.6 igure 6. (a) Measured and calculated optical line widths for the signal field when detuned from two-photon resonance asa function of Rb density, consistent with an effectively reduced optical depth at high densities due to radiation trapping.(b) Fluorescence signal rise time increases as the atomic density is increased, whereas transmission signal rise time hasminimal density dependence, indicating that absorbed and re-emitted photons are trapped in the medium for a longertime due to the large transverse optical depth. ( See text. ) Error bars are smaller than data points except where shown(derived from uncertainty in Rb temperature).
4. OPTICAL DEPTH AND RADIATION TRAPPING MEASUREMENTS
Direct measurements of large optical depths are difficult due to the large corresponding absorption of the opticalfield. To characterize the optical depth for comparison with the optimization models described above, we usedoptical absorption line width as a proxy for the optical depth. The signal field was detuned from two-photonresonance by changing the frequency driving the EOM and the signal intensity monitored as the one-photon(laser) detuning was swept through resonance. The line width of the signal field absorption serves as a proxyfor the optical depth because the logarithm of the measured absorption is proportional to the product of thedetuning and the optical depth.Figure 6a shows a comparison of the measured one-photon linewidth and the expected linewidth from asimple numerical model based on an optically thick sample of three-level Λ-systems. At high densities (andoptical depths), the measured linewidth stops growing while the model shows continued line broadening. Weattribute this to radiation trapping: photons scattered inside the medium depolarize the rubidium ground state.Fewer atoms are polarized in the signal field channel, leading to bothdecoherence and absorption of the signalfield.We also directly measured the presence of radiation trapping in our vapor cells, with a simplified apparatus.Light from an extended cavity diode laser was focused through an AOM with a small spot size to allow forrapid rise times in the AOM. This AOM output was turned on and off at a rate of 1 MHz with a rise time of ∼
75 ns. The beam diameter was then expanded to approximately 2 mm and sent into a Rb vapor cell housedinside a small plastic oven, unshielded from stray laboratory magnetic fields, and warmed with resistive heatersto temperatures between 54 and 90 ◦ C. Output light from the cell was measured with two photodetectors: onefor transmission, and the other for fluorescence mounted to the side of the cell oven where a lens close to the cellfocused scattered light onto the detector. When light was turned on via AOM modulation, the rise time of thesignal was observed in both detectors.Evidence for radiation trapping is shown in Fig. 6b. At high atomic densities, the rise time in the fluorescencedetection signal increases significantly. The greater the transverse optical depth, the greater the number ofabsorption and re-emission cycles that occur before absorbed photons exit the medium, so the rise time seen influorescence detection should grow with the transverse optical depth. In Fig. 6b we see rise times up to 350 ns,whereas the lifetime for the excited state of the 5 p / state of Rb is ∼
28 ns, indicating many scattering eventsfor each absorbed photon. 7 . NEW VAPOR CELL GEOMETRY
In order to minimize the decoherence resulting from radiation trapping in our vapor cells, we recently developeda cell designed to minimize its effects. A 15 cm long, 1.2 cm diameter cell with N buffer gas was designed toreduce radiation trapping in three ways: (i) the aspect ratio is four times larger than previous cells, allowingfluorescence to escape the cell in the transverse direction with fewer depolarizing interactions with Rb atoms;(ii) the absolute length of the cell is twice that of previous vapor cells allowing the equivalent optical depth to bereached at lower Rb density, minimizing density-dependent effects, including spin exchange; (iii) N buffer gasacts to quench radiation trapping by collisionally de-exciting Rb atoms before the atoms fluoresce, preventingthe emission of unpolarized photons that could destroy the coherence of nearby atoms.Initial results of stored light efficiency in this cell appear promising. For a laser power of P = 4 . ◦ C (the same longitudinal optical depth as the density = 40 × cm − data in Fig. 3),efficiency of η = 40% was achieved. This is three times greater than the η for P = 3 . P = 8 .
6. CONCLUSIONS
We reported a preliminary experimental study of slow and stored light at a variety of Rb densities (i.e., opticaldepths), and as a function of pulse duration, delay and storage time, and control field intensity. Experimentalresults are in good qualitative agreement with theoretical calculations based on a simplified three-level model atmoderate densities; and indicate that radiation trapping is an important limitation at high atomic density. Wehave developed a new cell geometry which we expect to reduce losses and improve efficiency through reducedradiation trapping and density-dependent decoherence.We are grateful to A. Glenday for useful discussions. This work was supported by ONR, DARPA, NSF, andthe Smithsonian Institution.
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