transients of the electromagnetically-induced-transparency-enhanced refractive...
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Transients of the electromagnetically-induced-transparency-enhanced refractive Kerr nonlinearity
M. V. Pack, R. M. Camacho, and J. C. Howell*Department of Physics, University of Rochester, Rochester, New York 14627, USA
Received 13 February 2007; published 28 September 2007
We report observations of the dynamics of electromagnetically induced transparency EIT in a systemwhen the ground states are Stark shifted. Interactions of this type exhibit large optical nonlinearities called Kerrnonlinearities, and have numerous applications. The EIT Kerr nonlinearity is relatively slow, which is alimiting factor that may make many potential applications impossible. Using rubidium atoms, we observe thedynamics of the EIT Kerr nonlinearity using a Mach-Zehnder interferometer to measure phase modulation ofthe EIT fields resulting from a pulsed signal beam Stark shifting the ground state energy levels. The rise timesand transients agree well with theory.
DOI: 10.1103/PhysRevA.76.033835 PACS numbers: 42.50.Gy, 32.70.Jz
I. INTRODUCTION
Optical Kerr nonlinearities created via electromagneti-cally induced transparency EIT have received increasingattention recently because these extremely large nonlineari-ties have numerous applications in the fields of low-light-level and quantum optics 1–19. Among the applications forthe EIT Kerr effect are low-light-level switching 2,9,11,quantum nondemolition QND measurements of photonnumber 15, entangling optical wavepackets 7,10 and thesynthesis of optical number states 5. Also, EIT Kerr non-linearities may play an important role in creating single-pho-ton sources for quantum computing. For example, linear op-tics quantum computing requires single-photon sources andhigh efficiency single photon detectors 20, both of whichmay be enabled by EIT enhanced Kerr nonlinearities 15.
For many of the above applications the speed of the Kerrnonlinearity i.e., the inverse of the rise time is as importantas the size of the optical nonlinearity. Pulsed applications,such as photon number resolving QND measurements, re-quire a strong nonlinear response proportional to the pulseenergy. Thus, it is the ratio between the size and the rise timeof the Kerr effect that matters, not just the size. For example,although the EIT Kerr effect can be very large even at verylow light level, the same parameters that make the EIT Kerreffect large also make it slow 21.
The EIT Kerr rise time is inversely proportional to theEIT linewidth, and is directly proportional to the EIT opticaldepth. Similarly, the size of the EIT Kerr effect is inverselyproportional to the EIT linewidth, and is directly propor-tional to the EIT optical depth. Thus, the typical methods forincreasing the size of the EIT Kerr effect leave the ratiobetween size and rise time unaffected.
The dependency of the EIT Kerr rise time on linewidthand optical depth have been shown theoretically, but thesedependencies have not been observed experimentally. In thispaper, we report experimental observations of the EIT Kerrrise time. Note that technically there are two types of EITKerr nonlinearities: Refractive 2 and absorptive 3. In thispaper we discuss exclusively the refractive type of EIT Kerr
nonlinearity. Observations of the dynamics of the absorptiveEIT Kerr nonlinearity have been reported previously12,22.
Coherent transients similar to the EIT Kerr effect havebeen reported in a number of different contexts. Park et al.and Chen et al. have observed absorption transients in -EITmedia induced by rapid changes to the Raman two-photondetuning 23,24. Similarly, Godone et al. have observedtransients due to phase modulation of Raman beams in a system 25. Many others have also studied coherent tran-sients in three level EIT systems in the contexts of coherentRaman beats 26, Zeeman splitting 27–29, and EIT insemiconductors 30,31.
In this paper we begin by reviewing the theory for EITKerr dynamics before presenting the experiment and mea-surement confirming this theory.
II. THEORY
Figure 1 shows a typical system for refractive EIT Kerrnonlinearities. Various parameters such as the Raman detun-ing R, single-photon detuning , and Rabi frequencies iwhere i= P ,C ,S are defined in the figure caption. The en-ergy levels 1, 2, and 3 together with the coupling fieldC and probe field P make up a -EIT system. Coherent
2
CΩ
2γ
γ
Γ
PΩ
SΩ
4
1
3
S∆
∆
2Rδ
2Rδ
2γ
FIG. 1. Simple EIT Kerr system in a four-level N-type system.The detunings are defined as P= 3−2−P, C= 3−1−C, S= 4−2−S, = P+C /2, and R=C−P. Also,the Rabi frequencies are defined as i=iEi / where i= P ,C ,S,i is the dipole moment, and Ei is the electric field. The spontane-ous emission rate out of the excited state is given by and is thedecoherence rate for the ground states.
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population trapping creates a transparency resonance whenthe detunings of the probe and coupling fields are near Ra-man resonance i.e., R=0. This transparency window canbe very narrow with a correspondingly steep dispersioncurve, and it is the steepness of the dispersion that results inlarge phase modulation of the probe field when the signalfield perturbs the EIT system away from Raman resonance.
In Fig. 1 the signal field Stark shifts ground state 1 byS
2 /4S, which changes the Raman detuning to
RS = R0 + S2/4S. 1
Thus the change to the probe phase in radians is approxi-mately
P 32L
c
RenPR
RS − R0 LS2
4vgS, 2
where L is the length of the medium, nP1+P /2 is theprobe index of refraction, S=SES / is the signal Rabifrequency, S is the dipole moment of the signal transition,and vgc / 32nP /R is the group velocity for theprobe we have assumed that ReP−1 1 andnP32nP /R. This cross-phase modulation of the probephase is a third order susceptibility 3 because S2ESES
* i.e., the medium polarization at the probe frequencyis PP=3ESSES
*−SEPP.The steep dispersion which makes possible ultraslow
group velocities in EIT also makes the EIT Kerr nonlinearitypossible. Group velocities as slow as 8 m/s 32 and 17 m/s33 have been measured in EIT experiments and group ve-locities of c 10−5 are routinely achieved in hot atomic va-pors implying that large Kerr nonlinearities are possible atrelatively low light levels. Also, the four-level EIT Kerr sys-tem does not have the large self-phase modulation Kerr non-linearities that are present in other systems with large Kerrnonlinearities such as highly nonlinear fiber.
The theory for the dynamics of the four-level EIT Kerrsystem has been worked out in Ref. 21. Because we useslightly different notation than Ref. 21 we review some ofthe more significant parts of the theory. First, by explicitlyaccounting for the energy shift of state 1 due to the signali.e., replacing R0 by RS we can reduce the four-levelsystem in Fig. 1 to an approximate three-level EIT system inwhich level 4 and the signal field are not included. Thisapproximation ignores the increases of the decoherence rate due to the signal field, which is given by S0+2 S2 /S
2 where /2 is the transverse decoher-ence rate. Ignoring this additional decoherence is a reason-able approximation as long as S. In the measurementsection, Sec. IV, we see that neglecting the additional deco-herence creates some small discrepancies between theoryand experiment, but neglecting it makes the equations muchsimpler.
The master equations for the reduced three-level system1, 2, and 3 are
33 = − 33 + Im2*32 + 1
*31 , 3
22 =
233 − 22 −
1
2 − Im2
*32 , 4
11 =
233 − 11 −
1
2 − Im1
*31 , 5
32 = iS − iRS
2− 32
+ iC
222 − 33 + i
P
221
* , 6
31 = iS + iRS
2− 31
+ iP
211 − 33 + i
C
221, 7
and
21 = iRS − 21 − iP
232
* + iC
*
231, 8
where = + /2+ is the transverse decoherence ratefor the excited state coherences and = + is theground state coherence decay rate. The primed decoherencerates and are dephasing rates i.e., loss of coherencewithout transfer of population from one state to another.
When the excited state coherences 13 and 23 can beadiabatically eliminated this assumption is easily satisfiedfor our choice of experimental parameters, the three-levelmaster equation simplifies to a two-level Bloch vector equa-tion 34. Additionally, the adiabatic elimination of the ex-cited state coherences makes it possible to write the probesusceptibility as a simple function of the coherence betweenthe ground states 21:
p21 N22322
20
1 − 21/21−*
− 2 + i
, 9
where N is the atomic density, 21−=−P
* C /2−P* / is
the ground state coherence GSC of the dark state, and −=P 1−C 2 is the dark state. This result also assumesthat 22C
2 /2, 11P2 /2, and both fields are below
saturation C 21. Finally, the equations simplify evenmore when we assume the fields are real and positive, thesingle-photon detuning is zero, =0, and the Raman detun-ing is much smaller than the homogeneous linewidthR. All of the above assumptions are valid for the ex-perimental parameters discussed in this paper.
Expressing the susceptibility as a function of the GSC, asis done in Eq. 11, has several advantages for discussing thedynamics of the EIT Kerr nonlinearity. The primary advan-tage is that the dynamics and rise times of the GSC have aphysically intuitive interpretation. Also, the GSC interpreta-tion results in simple analytic expressions for the steady-stateand transient solutions of the Bloch vector equation.
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Both of these advantages are demonstrated in Fig. 2. First,Fig. 2a shows that the steady-state GSC as a function ofRaman detuning is a circle solid line in the complex plane;mathematically this is
21ssR =
21−R
R + − iR. 10
The optical pumping rate R2 / is the rate at whichoptical pumping transfers atoms from the bright state to thedark state. Figure 2b shows the real black line and imagi-nary gray line parts of the steady-state probe susceptibility,
pss − i
pc
32
R2 + R + + iRR
R + 2 + R2 , 11
plotted as a function of the Raman detuning, where p=N22 322 /20 is the probe absorption coefficient. Tohelp visualize the mapping of Eq. 11 between GSC andprobe susceptibility, corresponding points on the GSC andsusceptibility plots in Fig. 2 are indicated by various shapese.g., triangle, square, circle, diamond.
The dashed line in Fig. 2a shows a trajectory of thedynamical evolution of the GSC. Mathematically these tra-jectories are given by
21t = 21ss + 210 − 21
ssetiR−R− for t 0,
12
where 210 is the initial GSC and all parameters are heldconstant for times t0 with a steady-state GSC of 21
ss.When the final conditions have RR the trajectories followspiral patterns similar to the trajectory shown in the dashedline in Fig. 2a, and when the final Raman detuning isR R the trajectories are essentially straight lines from210 to 21
ss.
Figure 3 shows the EIT Kerr transients in the time domainthis is the same trajectory as the dashed line in Fig. 2a.Plotted as a function of time the real and imaginary parts ofthe susceptibility exhibit damped simple harmonic oscilla-tions.
From transient curves, like those in Fig. 3, it is possible tofind the 1/e rise times for the refractive and absorptive partsof the EIT Kerr nonlinearity. Figure 4 shows these rise timesas a function of Raman detuning. The refractive and absorp-tive rise times approach two different sets of asymptotes inthe large Raman detuning limit RR and in the small Ra-man detuning limit RR the condition for EIT is Rmaking R+R.
For most refractive EIT Kerr applications it is generallydesirable to work near Raman resonance with changes inRaman detuning that are small relative to the EIT linewidth.That is, RR for all times. This provides a maximal refrac-tive Kerr nonlinearity with minimal change in absorption. Inthis small Raman detuning regime, RR, the 1/e rise andfall times for refraction are given by
Im/ρ
21( -)ρ
21(
)
Re /ρ21(-)
ρ21( )
Ground StateCoherence
δR/2R
ProbeSusceptibility
0 0.5 1-0.5
0
0.5
-2 0 20
0.5
1
Im(χ
)Re
(1-ρ
21/ρ
21(-) )
-4 4-0.5
0
0.5 Re(χ)∝
-Im(ρ
21/ρ//21 (-))
P
P
∝
21
(ss)(δR )ρ21 (t)ρ
(b)(a)
21
(ss)(0 )ρ21
(ss) (1.2R)ρ
FIG. 2. Color online a Steady-state solid line and transientdashed line ground state coherences are plotted in the complexplane, while b the probe susceptibility is plotted as a function ofthe Raman detuning R. All plots are theoretical with R5, andall variables except the Raman detuning are held constant. The tran-sient ground state coherence curve dashed started with steadystate for R=0, then suddenly the Raman detuning changed to 2R.To show the mapping between ground state coherence and probesusceptibility four shapes corresponding to four Raman detuningshave been plotted on both the steady-state ground state coherenceand susceptibility curves.
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Im(χp)
Re(χp)
Time (t /R)
PorbeSusceptibility
(normalized)
FIG. 3. Color online The real and imaginary parts of the probesusceptibility plotted as a function of time. The same parametershave been used as in Fig. 2 i.e., R5, for t0, R=0, and fort0, R=2R.
10-1
100
101
10-2
10-1
100
δ /R
Rise
tim
e(τ
R)
τRef,rise
τAbs,rise
Raman detuning ( )R
FIG. 4. Solid lines show the 1/e rise time for absorption andrefraction resulting from the EIT Kerr nonlinearity. Rise times areplotted logarithmically as a function of the magnitude of the Ramandetuning R. All quantities are dimensionless and normalized by theoptical pumping rate R, and /R=0.2. For small detunings the risetimes asymptotically approach abs=2.15/ R+ and ref=1/ R+. For large two-photon detunings the rise times asymptoticallyapproach abs=1.2/R and ref=0.63R+ /R
2 . These asymptotesare shown as the dashed lines.
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limR→0
ref,rise = limR→0
ref,fall =1 + pz/4
R + 13
and the absorption rise and fall times are
limR→0
abs,rise =1 + 1 + pz/2c1
R + 14
and
limR→0
abs,fall =1 + 3c1pz/8
R + , 15
where c1=ln2+ln2+ln2+ ¯ , and p=pR / R+ isthe depth of the EIT; the atoms occupy the half space z0.The probe and coupling fields are copropagating fromsmaller z to larger z. These rise/fall times can be calculatedanalytically for small optical depths pz1. For large opti-cal depths the dependence of these rise and fall times on pzwas determined by curve fitting numerical simulations of anoptically thick four-level system.
These rise and fall times can be understood intuitively asthe ratio between the total phase change of the GSC and
the rate of GSC phase change we use for the phase ofthe GSC, and for the phase of the probe field. In the limitRR and CP the optically thick equivalent of Eq.10 is
21ssz
PzP0
21−R
R + expi
R
R + , 16
where
Pz P0exp−zp
2R + + iRR
R + , 17
and we have assumed CzC0. The total GSC phasechange is =R1+zp /2 / R+, and from Eq. 12 we
see that =R. Thus, we expect the rise time to be approxi-mately
/ = 1 + zp/2/R + .
In the large Raman detuning limit, i.e., R R+, therise times become more complicated. However, for smalloptical depths the large-Raman-detuning rise times are stillstraightforward to calculate. In this regime the GSC phasechange is −R / R+, and the rise times are
limR→
abs,rise =arccose−1 + R + /R
R18
and
limR→
ref,rise =1 − e−1R +
R2 . 19
For negligible optical thickness i.e., L1, the asymptoticrise times from Eqs. 13, 14, 18, and 19 are shown inFig. 4 as dashed lines to compare with the exact rise times.
This simple four-level model provides a simple intuitivepicture for how the EIT Kerr nonlinearity evolves as a func-
tion of time, and almost all of the ideas presented in thissection generalize to the more complicated experimental sys-tem discussed in the next sections. However, there are dif-ferences between the simplified theory and the experimentalreality, and we discuss these differences as they become im-portant.
For applications of the refractive EIT Kerr nonlinearity,Eq. 13 is the most significant result of this section. Thisequation shows that when RR the rise times are depen-dent on only two parameters: The optical depth pL and theEIT linewidth FWHM=2R+, both of which are straight-forward to measure experimentally. For large Raman detun-ings RR, the rise times are also dependent on the size ofthe Stark shift i.e., Raman detuning which is also an ex-perimentally tunable parameter.
III. EXPERIMENT
Using a Mach-Zehnder interferometer as shown in Fig. 5,we observed the transients of EIT on the D1 line of 85Rb. Thetransients are created by pulsing the signal beam which isabout 2 GHz red detuned from the 85Rb D2 F=2→F= 2,3 transitions. Previous experiments used frequencymodulation spectroscopy to observe similar Stark shifts andrefractive EIT Kerr nonlinearities 11, but such experimentsare unable to observe the transient behavior of the systemsbecause they require lock-in amplification. Thus this is adirect observation of the dynamics and rise times of the re-fractive EIT Kerr nonlinearity.
Both probe and coupling fields are obtained from a singleexternal cavity diode laser ECDL1 with wavelength
AOM
1.5 GHz
PBS
2
Rb CellSpectralfilter
Magnetic Shielding
ECDL1
50/50 BS 50/50 BS
50/50
BS
AOM80 MHz
ECDL2
PBS
PBS
Homodyne
Detection
PZT
Signal
Pump
Probe
Mach-Zehnder
InterferometerComputer
DAQ
AOM
1.5 GHz
PBS
2λ
Rb CellSpectralfilter
Magnetic Shielding
ECDL1
50/50 BS 50/50 BS
50/50
BS
AOM80 MHz
ECDL2
PBS
PBS
Homodyne
Detection
PZT
Signal
PuPumpmp
Probe
Mach-Zehnder
InterferometerComputer
DAQ
FIG. 5. Color online Schematic of the experimental apparatus.The probe beam is sent through a Mach-Zehnder interferometer tomeasure changes in the relative phase between the two arms of theinterferometer. The coupling and signal beams interact with theprobe in the rubidium cell before being separated from the probevia polarization and spectral filtering. A computer controlled dataacquisition system records and averages the difference signal frombalanced homodyne detection.
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=795 nm by splitting the beam at a polarizing beam splitterPBS to obtain a coupling beam and a probe beam. Theprobe beam is double passed through a 1.5 GHz acousto-optic modulator AOM to shift its frequency by approxi-mately 3 GHz the ground state hyperfine splitting of 85Rb.Deriving both EIT fields from a single laser ensures the ex-cellent phase coherence necessary for good EIT. Addition-ally, we are able to control the frequency difference betweenprobe and coupling precisely via the microwave frequencysynthesizer driving the AOM.
A second laser ECDL2 with wavelength =780 nm ispulsed using an 80 MHz AOM with 50 ns rise time to createthe signal pulses. Both ECDL lasers are New Focus Vortexlasers i.e., they are single-mode external-cavity-feedback di-ode lasers with linewidths of less than 300 kHz. The signaland coupling beams are combined on a 50/50 beam splitterBS and are both horizontally polarized so they can becoupled into the Mach-Zehnder interferometer via a PBS.
The probe is split into two paths in the Mach-Zehnderinterferometer: a reference local oscillator path and a probepath that goes through the cell the reference field is about afactor of two larger than the probe field ER2Ep. After theMach-Zehnder interferometer, balanced homodyne detectionmeasures changes in the probe phase. Changes in probe am-plitude can also be measured by blocking the reference armof the interferometer and measuring the current from just oneof the photodiodes. Both polarization and spectral filteringare used after the cell to isolate the probe field from thecoupling and signal fields. When necessary the relative phaseof the two arms of the interferometer scanned and lockedusing the piezomounted mirror PZT after the cell. The ho-modyne signal is given by
Q EREPsinpv + ImpkPL , 20
where ER is the electric field amplitude in the reference legof the interferometer, pv is the relative phase between thetwo arms of the interferometer controlled via the piezovolt-age, L is the length of the cell, and kP=P /c is the wavenumber of the probe field.
The cell apparatus consists of three layers of magneticshielding with a long solenoid inside the innermost layer ofmagnetic shielding. In addition to containing 20 torr of ni-trogen buffer gas, the rubidium cell contains both rubidiumisotopes in their natural abundance 27.8% 87Rb and 72.2%85Rb. The cell is cylindrical in shape with a length of30 mm and a diameter of 25.4 mm. The number density ofrubidium atoms is controlled by heating the cell with electri-cal strip heaters positioned between the outer and middlelayers of magnetic shielding.
At the output of the Mach-Zehnder interferometer the ho-modyne signal is measured using two reverse biasedHamamatsu S2830 photodiodes soldered for differential de-tection. After the photodiodes a 1 MHz low-noise transim-pedance amplifier prepares the signal for computer con-trolled data acquisition DAQ. CW measurements arerecorded by the DAQ, and the fast transient data are recordedand averaged using a 1.5 GHz bandwidth oscilloscope.
Figure 6 shows the hyperfine levels for the 85Rb D1 andD2 lines with laser transitions that are driven in the experi-
ment. The hyperfine splittings for the 5P1/2 361 MHz and5P3/2 220 MHz hyperfine levels are much smaller thanthe Doppler width 600 MHz, such that the individual hy-perfine levels cannot be resolved. Additionally, rubidium-nitrogen collisions create a rapid interchange of atomsamong the velocity classes without decohering the GSC,which means that all EIT parameters such as optical pump-ing R and linewidths must be averaged over all velocityclasses see the Appendix. Finally, the signal field Starkshifts both ground hyperfine levels resulting in a total changein Raman detuning of
RS = R0 +2hf
4SS + hf, 21
where hf=2 3.0357 GHz is the hyperfine splitting of theground states.
The inset in Fig. 6 shows the approximate 1/e2 intensitybeam shapes for the coupling, probe, and signal fields in theexperiment the beams were not perfectly centered. Thecoupling and signal beams were essentially identical withbeam radius of wcws=2.2 mm, while the probe beam wassmaller and slightly elliptical wp,x=1.0 mm and wp,y=1.8 mm.
IV. MEASUREMENTS
To verify the dependence of the EIT Kerr dynamics onoptical depth, EIT linewidth, and Raman detuning, we mea-sured the transient and CW response of EIT to the signalfield while varying the cell temperature optical thicknessL, coupling intensity EIT linewidth R+, and signalintensity Stark shift R. The different conditions underwhich data were taken are summarized in Table I. Each rowin Table I corresponds to a different set of data in which bothCW and transient measurements were taken for several sig-nal powers ranging from 0.2 mW to 4.9 mW. In all measure-ments the probe power was 40 W, and the signal detuningwas S=2 1.9 GHz. In the following sections we prima-rily present the data for the first row of Table I, but the sameanalysis was carried out for each row of Table I.
5S1/2
F=3
F=2
λ = 780 nm
CΩ ↔P
ΩS
Ω↔
↔
5P1/2
5P3/2
λ = 795 nm
-2 -1 0 1 2
-2
-1
0
1
2
x-position (mm)
y-positio
n(m
m)
FIG. 6. Color online Level diagram of 85Rb D1 and D2 lineswith the primary transitions excited by the probe, coupling, andsignal fields. The inset shows the approximate beam sizes andshapes for the probe wp,x=1 mm and wp,y =1.8 mm, couplingwc=2.1 mm, and signal ws=2.2 mm fields.
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A. CW measurements
Figure 7 shows the CW probe absorption as a function ofRaman detuning for several different signal powers, with acell temperature of 58 °C and a coupling power of 0.34 mWfirst row in Table I. Figures 10–15 also use this same dataset. In addition to the signal field Stark shifting the absorp-tion minimum, the signal field also causes the transparencyresonance to become shallower and broader. These effectsare largely due to the additional decoherence created by thesignal field i.e., S=0+S
2 /S2, which is not ex-
plicitly included in the theory of Sec. II. This additional de-coherence is most noticeable when the optical pumping is ofthe same order of magnitude as the decoherence rate i.e.,R, which is the case for the data corresponding to row 1of Table I. For all other sets of parameters in Table I theoptical pumping R is a factor of 3–6 times larger than for
row 1 and the broadening is significantly less noticeable.From the displacement of the probe absorption minimum
we determined the Stark shift as a function of signal powershown in Fig. 8. Figure 8 shows the Stark shifts due to thesignal field only for all rows of Table I.
There is also a Stark shift due to the coupling field inter-acting with the probe transitions and a Stark shift due to theprobe field interacting with the coupling transitions. ThisStark shift is plotted as a function of coupling power in Fig.9. Also, plotted in Fig. 9 is the transparency linewidth versuscoupling power. The linewidth and Stark shift are alsoslightly dependent on the probe power, but probe power issufficiently small that it can be neglected. Figures 8 and 9exhibit the expected linear dependence on intensity.
In addition to measuring the imaginary part of the probesusceptibility via the probe absorption, we also measured therefractive part of the susceptibility using balanced homodynedetection shown in Figs. 10 and 11. The homodyne signalwas measured by setting the phase pv to a desired value andthen stepping the Raman detuning over the desired rangewith a dwell time of 200 ms or greater at each Raman de-tuning.
Figure 10 shows the homodyne signal for the first row ofTable I with no signal field black and a signal power of
TABLE I. Several measured parameters for the EIT line shapeswith no signal present. The beam powers were measured immedi-ately before the PBS in front of the vapor cell. L is the maximumabsorption experienced far from Raman resonance. L is the differ-ence between maximum absorption and the absorption minimumnear Raman resonance, and EIT FWHM is self-explanatory.
TemperaturePowercoupl.
Ltotal
LEIT depth
EITFWHM
58 °C 34020 W 5.23 3.71 11.57 kHz
58 °C 1.01 W 5.32 4.31 31.91.2 kHz
58 °C 1.91 W 5.52 4.61 58.11.8 kHz
48 °C 1.01 W 2.01 1.51 31.97 kHz
52 °C 1.01 W 3.31 2.61 33.27 kHz
72 °C 1.01 W 20.49 16.47 413 kHz
-20 -10 0 10 20 30
2
3
4
5
Raman Detuning (kHz)
Probeabso
rptio
n(αL)
P = 4.5 mWP = 2.5 mWP = 1.5 mWP = 0.4 mWP = 0
s
s
s
s
s
FIG. 7. Color online Probe absorption versus Raman detuningfor different CW signal powers. The signal detuning is S=2 1.9 GHz and the coupling probe power is 340 W 40 W.The fact that with no signal field Raman resonance is alreadyshifted away from the expected zero is due mostly to the 20 torr N2
buffer gas shifting the hyperfine splitting hf=2 240 Hz pertorr N2. The probe and coupling also cause a small about 1 kHzStark shift at these powers.
0 1 2 3 4 5 60
2
4
6
8
10
12
Stark Shift due toSignal Field Only
Signal Power (mW)
Sta
rkS
hift
(kH
z)
theoryαL=1.7, P
C= 1 mW
αL=2.6, PC
= 1 mWαL=3.8, P
C=340µW
αL=4.3, PC
= 1 mWαL=4.6, P
C=1.9 mW
αL=16.4, PC
= 1 mW
FIG. 8. Color online Stark shift due to the signal field forseveral settings of coupling power and EIT optical depth.
0
20
40
60
80EIT Linewidth(FWHM)
FWHM
(kHz)
0 0.5 1 1.5 2 2.50
1
2
3
4
5
Coupling Power (mW)
Stark
Shift(kHz)
theory
αL≈1.5
αL≈2.6
αL≈4.5
αL≈16
theory
αL≈1.5
αL≈2.6
αL≈4.5
αL≈16
Stark shift due to Couplingand Probe fields only
FIG. 9. Color online Stark shift due to the coupling and probefields as a function of coupling power. Also shown is the FWHM ofthe EIT resonance versus coupling power. The signal was off for allof these measurements.
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4.5 mW nonblack. The dashed curves show an envelope ofmaximum and minimum homodyne signal i.e.,±Er EpR, which is determined for each Raman detuningR by measuring amplitude of the homodyne sinusoid whenthe phase pv is scanned over several cycles. In Fig. 11 wehave used the data from Fig. 10 to calculate the real andimaginary parts of the probe susceptibility. The homodyneenvelope gives the absorptive imaginary part of the probesusceptibility, and the refractive real part of the probe sus-ceptibility is extracted using Eq. 20, the homodyne enve-lope, and the homodyne signal.
B. Transients measurements
Homodyne detection was also used to measure the tran-sient response of the probe field when the signal field isturned on and off with a fast rise time. Figure 12 shows thetransient homodyne signal and homodyne envelope when the
peak signal power was 4.5 mW and all other parameters aregiven by the first row in Table I. From these measurements,the absorptive and refractive transients of the probe field canbe extracted. Figure 13 shows the extracted change in probeabsorption and probe phase for different choices of peak sig-nal power. The inset shows the measured absorption and dis-persion curves from Fig. 11 when PS=0. The vertical lines inthe Fig. 11 inset show the Stark shifts corresponding to dif-ferent signal powers. The square dashed line in Fig. 11shows the time during which the signal was on.
There are several characteristics of these transients worthnoting. First, the steady-state changes in absorption and re-fraction can be estimated by looking at the Fig. 11 inset. Thereal and imaginary parts of the susceptibility are Starkshifted from their Raman resonance values to approximatelythose values indicated by the vertical lines in the Fig. 11inset. For example, for small signal powers e.g., PS=0.2 mW and PS=0.4 mW the change in refraction is largerthan the change in absorption, because near Raman reso-
-30 -20 -10 0 10 20 40 50
-3
-2
-1
0
1
2
Raman detuning (kHz)
Homodyn
esignal(arb.units
)
30
FIG. 10. Color online Raw CW homodyne data for couplingpower of 340 W. Two sets of data are shown: one for Ps=0 andthe other for Ps=4.5 mW. The dotted lines show the homodyneenvelope i.e., the maximum and minimum values which the homo-dyne signal can obtain.
Re(χp)
Im(χp)
-20 -10 0 10 20 30 40 50
Raman Detuning (kHz)
Probesu
sceptib
ility(∆φin
rad)
Probesu
sceptib
ility
(αL/2)
1
2
1
3
2
0
3
2
0
1
-1
1
0
0
-1
-30
δ = 8 kHzR
δ = 0R
FIG. 11. Color online Measured absorption L and phase shift derived from the homodyne data in Fig. 10. In addition toobtaining refraction information we are able to confirm the absorp-tion measurements shown in Fig. 7.
0 100 600 700 800
-1
0
1
µTime ( s)200 500
HomodyneSignal(arb.units)
Homodyne
Envelope
Signal
FIG. 12. Color online Homodyne measurements as a functionof time Ps=4.5 mW, Pc=340 W. The signal was turned on att=0 and then turned off again at t=500 s. Similar to Fig. 10, theenvelope shows the maximum and minimum values obtainable bythe homodyne signal.
Phaseshift(rad)
Absorption(α
L/2)
0
0.2
0.4
0.6
0.8
1
0 50 100 1500
0.5
1
1.5
µ
500 550 600 650 700
P =4.5 mW
P =2.5 mW
P =1.5 mW
P =0.8 mW
P =0.4 mW
P =0.2 mW
Time ( s)
0 10 20Raman Detuning (kHz)
30
Re(χ
)andIm
(χ)
pp
s
s
s
s
s
s
FIG. 13. Color online Change in refraction and absorption as afunction of time for different values of peak signal power. Bothrefraction and absorption are measured relative to their respectivevalues when the signal is off.
TRANSIENTS OF THE ELECTROMAGNETICALLY-… PHYSICAL REVIEW A 76, 033835 2007
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nance the slope of the dispersion curve is greater than theslope of the absorption curve. According to the theory fromSec. II the change in absorption is quadratic for small Starkshifts around Raman resonance. However, the theory ne-glects the small linear change in the absorption due to theincrease in decoherence S=0+2S
2 /S2 arising
from the signal field. For small signal powers the Stark shiftis a constant.
For large signal fields the rise time decreases and the os-cillatory nature of the transients becomes more noticeable.The fall time is independent of the signal field and is neveroscillatory because R0 when the signal field is off.
Additional insight comes from considering the dynamicsof the GSC. In Fig. 14 we have used the inverse mapping ofEq. 11 to obtain the measured GSC transients. The real partof the GSC is plotted versus the imaginary part of the GSCparametrically as a function of time. The rise dynamics lookvery similar to the transient trajectory shown in the theoryplot in Fig. 2. The fall dynamics are essentially straight linesback toward the dark state GSC.
Figure 15 shows measured 1/e rise and fall times for thetransient measurements presented in Fig. 13. The solid linesare theoretical rise times in which optical thickness is ac-counted for by multiplying the theory curves from Fig. 4 bythe factor 1+ L /4 for the refractive curves and the absorp-tive theory curves have been multiplied by 1+c1L / 2+2c1. The fall times are mostly independent of the size ofthe Stark shift as expected the theory curves are only for therise times, not the fall times. The theory curves agree verywell with the measured rise times, except for the absorptiverise and fall times for small Stark shifts. For small Starkshifts the change in absorption is very small, as can be seenin Fig. 13, making this discrepancy of little practical impor-tance. Also, there is no simple analytic theory for the risetime when the Stark shift is large i.e., RR, so it is some-what surprising that the theory curves agree as well as theydo in this region.
Figure 16 shows the rise time plots for the data sets cor-responding to the second through fifth rows of Table I. Foreach data set in Fig. 15 we include the EIT half width at halfmaximum, HWHM=R+, and EIT depth which were usedto calculate the theory curves.
Finally, Fig. 17 shows the measured and theoretical risetimes versus optical thickness for small Stark shifts. Themeasured refractive rise and fall times fit the theory curvewell. For small optical depths L6 the absorptive rise timeagrees with theory. However, the absorptive fall times differfrom the theory, and both absorptive rise and fall times dis-agree with theory for large optical depths. It is not clear whythese differences exist, and they are not significant for appli-
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.1
0
0.1
0.2
0.3
0.4
Re /ρ21(-)
ρ21( )
Im/ρ
21( -)ρ
21(
)0.5
P =4.5 mW
P =2.5 mW
P =1.5 mW
P =0.4 mW
s
s
s
s
Fall Dynamics
Rise Dynamics
Dynamics of Ground State Coherence
FIG. 14. Color online Trajectories of the ground state coher-ence GSC in the complex plane. GSC was derived using Eq. 11and the refraction and absorption transients in Fig. 13. Steady stateGSC values black line and transient data nonblack textured linesare both shown. The curved or spiraling lines show the rise dynam-ics R0, while the straighter lines show the fall dynamics R
=0.
1 10
10
100
Stark shift (kHz)
Risetim
e(
s)µ
meas. τrise
(Amp.)
theory τrise
(Amp.)
meas. τrise
(Ref.)
theory τrise
(Ref..)
meas. τfall
(Ref.)
meas. τfall
(Amp.)
20.5 5 202
5
20
50
200
FIG. 15. Color online Rise times of the refraction and absorp-tion resulting from the EIT Kerr nonlinearity Pc=340. Squaresand triangles show the measured refractive rise and fall times. Dia-monds and circles show the measured absorptive rise and fall times,and solid curves are theory for the rise times. The values L=3.75 and R+=2 kHz are used to obtain the theory curve fit.
1 10
10
100
10
100
1 10
αL = 4.3
R+Γ = 12 kHz
αL = 1.7
R+Γ = 12 kHz
αL = 2.6
R+Γ = 12 kHz
αL = 4.6
R+Γ = 20 kHz
Raman detuning (kHz)
Rise&falltim
es(µs)
2
2
(a)
(d)(c)
(b)
~~
~~
FIG. 16. Color online Rise- and fall-time measurements forthe conditions a T=48 °C and Pc=1 mW, b T=52 °C and Pc
=1 mW, c T=58 °C and Pc=1 mW, and d T=58 °C and Pc
=1.9 mW. The definitions for measured data and curve fits are thesame as in Fig. 15, and the parameters used in the curve fits aregiven in each figure.
PACK, CAMACHO, AND HOWELL PHYSICAL REVIEW A 76, 033835 2007
033835-8
cations based on the refractive EIT Kerr nonlinearity.
V. DISCUSSION
Although the simple theory from Sec. II shows generalagreement with the measured results, there are several differ-ences which should be explained. These differences mostlyarise from two sources: Approximations made in the theory,and the experimental system is more complicated than thetheoretical system note that the theoretical system assumesplane waves, a single velocity class, and a four-level atom.
We have already mentioned that in addition to creating theStark shift the signal field also adds to the decoherence be-tween ground states; i.e.,
S 0 + S2/S
2. 22
By leaving this additional decoherence out of the theory seeSec. II the equations become simpler, but this also leads tosome inaccuracy. Figure 18 illustrates the results of neglect-ing the signal dependent decoherence, by comparing themeasured and theoretical GSC dynamics. First, the measured
steady-state GSC is not a circle as predicted by the theory. Inthe theory near R=0 the change in refraction is linear insignal intensity Rep Im21S
2 whereas the ab-sorption is quadratic in signal intensity ImpRe21S
4 resulting in a smooth quadratic steady-state GSC curvenear Raman resonance. However, in reality both the absorp-tion and refraction have a linear dependence on the signal,and the measured steady-state GSC curve comes to a cuspnear Raman resonance. The signal dependent decoherence isalso the source for the differences between the transient mea-sured and theoretical GSC curves. When the signal depen-dent decoherence is used in calculations simple analytic re-sults are no longer possible but the theory agrees much moreclosely with measurements. Even with its inadequacies thetheory does a good job of predicting the measurements asshown in Fig. 18.
Other differences arise because the experimental system isactually a collection of many systems i.e., velocity classes,etc. that must be averaged. This averaging takes severalforms. First, the 85Rb level structure for the D1 and D2 linesis rather complicated eight hyperfine levels and 48 magneticsublevels resulting in five dark states and seven brightstates. Second, at room temperature 85Rb experiences signifi-cant Doppler broadening, requiring us to average over atomsfrom different velocity classes. Finally, the experiment usesGaussian-like beams with spatially varying intensities, re-quiring spatial averaging.
The experimental EIT Kerr system with a total of 48 lev-els will obviously be more complicated than the four-levelsystem assumed in Sec. II. Even so, the theory of Sec. IIaccurately predicts most experimental observations. One sig-nificant difference resulting from the experimental levelstructures is that four of the five dark states are only semi-dark. These semidark states are much darker than brightstates, but they are still weakly coupled with the excitedstates resulting in nonzero transparency even when =0. Forthe parameters in the experiment these semidark states limitour best transparency to about L /7. The Appendix has thedetails about how we average over dark states and velocityclasses to obtain the effective optical pumping rate Rave andhomogeneous linewidth eff.
0 2 4 6 8 10 12 14 160
20
40
60
80
100
Optical Depth (αL)
RiseTim
e(µs)
Ref. rise (meas.)Ref. rise (theory)Ref. fall (meas.)Ref. fall (theory)Abs. rise (meas.)Abs. rise (theory)Abs. fall (meas.)Abs. fall (theory)
FIG. 17. Color online Rise and fall times for the refractive andabsorptive parts of the EIT Kerr nonlinearity versus optical depth.The Stark shift is small R R+ for all measurements andtheory.
Measured GSC Dynamics
0 0.2 0.4 0.6 0.8
0
0.2
0.4
Re /ρ21(-)
ρ21( )
Im/ρ
21( -)ρ
21(
)
0.6
0 0.2 0.4 0.6 0.8
0.2
0.4
0.6
0
Re /ρ21(-)
ρ21( )
Im/ρ
21( -)ρ
21(
)
Theoretical GSC Dynamics
(b)(a)
FIG. 18. a Measured transient solid line with dots and steady-state solid line GSC and b transient dashed line and steady-statesolid line GSC calculated using the theory from Sec. II. Both plots show the normalized real part of the GSC plotted versus the imaginarypart of the GSC. The GSCs are plotted parametrically as a function of time transient data, or as a function of Raman detuning steady-statedata. The transient data show GSC starting out in steady state for zero Raman detuning, and evolving toward steady state when the Ramandetuning is R1.2R+, after which the Raman detuning is sharply switched back to zero, and the GSC again evolves toward steady state.
TRANSIENTS OF THE ELECTROMAGNETICALLY-… PHYSICAL REVIEW A 76, 033835 2007
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The spatial averaging over variations in field intensitieshas the primary result that the simple relation between therise times and the EIT HWHM must be slightly modified tobecome
ref = f1 + L/4
HWHM, 23
where HWHM=R+ and f is a integration factor accountingfor spatial averaging. The factor f arises because the effect ofspatial averaging is slightly different for the 1/e rise timeand the HWHM. Diffusion is also important in spatial aver-aging because it tends to coarse-grain the averaging. Thecharacteristic diffusion length is lDvl0TR, where v is themean velocity, l0 is the mean free path length between87Rb-N2 collisions, and TR1/R is the optical pumping pe-riod. Atoms that are within the characteristic diffusion lengthwill on average have encountered the same average field in-tensity over the last optical pumping period TR regardless ofthe difference between the field intensities at their currentlocations. This diffusion length sets the characteristic lengthfor coarse graining the spatial average.
In the experiment transit-time broadening 35 is thedominant source of GSC decoherence, which means that thediffusion length is related to the beam waist by R /wp / lD2 the probe beam waist wp is the limiting beamwaist. Thus, when R the diffusion length is similar tothe beam diameter and f 1 and spatial averaging does noth-ing. When R f approaches its maximum value, whichdepends on the relative probe and coupling beam diameters.Experimentally, the changes in f can be calculated for differ-ent ratios R / by taking the ratio of the values for R+ inFigs. 15 and 16 and the measured EIT FWHM from Fig. 9.For coupling powers of Pc=0.34 mW, Pc=1 mW, and Pc=1.9 mW we find that f 1, f 1.3, and f 1.4, respec-tively. This agrees with the fact that as the optical pumpingrate increases the diffusion length decreases and the value off monotonically approaches its maximum value. The maxi-mum value of f 1.5 can be calculated by spatial averagingwhile assuming an infinitesimal diffusion length.
VI. CONCLUSIONS
The transients and rise times of the EIT Kerr nonlinearityare well modeled by the theory of Ref. 21 this theory isalso reviewed in Sec. II. This agreement between theory andexperiment is especially good for the refractive Kerr nonlin-earity when intensity variations in the fields e.g., Gaussianbeam shapes are accounted for. Also, by thinking in terms ofthe GSC it is possible to develop a more intuitive under-standing of the physical mechanisms behind the EIT Kerrnonlinearity and its dynamics.
Our primary result is that the rise and fall times of therefractive Kerr nonlinearity are accurately predicted by Eq.13 for the small Raman detunings. This means that in thelimit of greatest interest for most EIT Kerr applications i.e.,L1 and RR, the EIT Kerr rise time is inversely pro-portional to the optical thickness and linearly proportional tothe EIT linewidth.
Both of these proportionalities may prove problematic forpulsed applications of the refractive EIT Kerr nonlinearitysuch as QND measurements of photon number. Pulsed EITKerr applications require a strong nonlinear response at lowpulse energies, which implies that the EIT Kerr nonlinearityshould be both large and fast. However, one advantage of theEIT Kerr nonlinearity is that it can theoretically be madearbitrarily large because it is linearly proportional to the op-tical thickness and inversely proportional to the EIT line-width. Thus, making the EIT Kerr nonlinearity large alsomakes it slow and vice versa. Applications that are limited bythe slow rise time of the EIT Kerr nonlinearity include: QNDmeasurement of photon number, entangling optical wavepackets, synthesis of number states, and single photonsources and detectors. All of these applications requirepulsed few-photon operation.
Although the slow rise time of EIT Kerr nonlinearity lim-its its applicability, the EIT Kerr nonlinearity is still com-paratively large relative to other Kerr nonlinearities, and ithas other advantages such as weak self-phase modulation. Inmost parametric nonlinear processes the rise time is so fastthat it is ignored. For EIT enhanced nonlinear processes therise time is sufficiently slow that it cannot be ignored formost applications.
ACKNOWLEDGMENTS
This work was supported by DARPA Slow Light, the Na-tional Science Foundation, and Research Corporation.
APPENDIX: CALCULATIONS OF AVERAGEOPTICAL PUMPING RATE AND EFFECTIVE
HOMOGENEOUS LINEWIDTH
In this appendix we discuss how the complicated 85Rbsystem can be distilled into a couple of parameters: The av-erage optical pumping rate Rave, and the effective homoge-neous linewidth eff. These parameters allow us to use thesimple four-level theory of Sec. II to predict the EIT dynam-ics in 85Rb. In the experiment EIT is created on the D1 line of85Rb using orthogonal linearly polarized probe and couplingbeams. Choosing the quantization axis along coupling beampolarization, Fig. 19 shows the probe and coupling transi-tions with their transition coefficients.
In order to achieve coherent population trapping and EIT,the dark states must be simultaneously dark for both the F=2 and F=3 excited hyperfine levels. There is only onesuperposition of ground states that satisfies this condition.
There are also four other semidark states which are notcompletely decoupled from the excited states, but are moreweakly coupled to the excited than the bright states. To cal-culate the coupling between the ground and excited states wesolve for the eigenvalues of the time independent Hamil-tonian. The Hamiltonian accounts for only the D1 line dipoleallowed transitions explicitly shown in Fig. 19 and includesall field and detuning parameters while assuming a rotatingreference frame and the rotating wave approximation. Theeigenvalues and eigenvectors are calculated for the casewhen R=0 and +, where
PACK, CAMACHO, AND HOWELL PHYSICAL REVIEW A 76, 033835 2007
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+ = 2D1EP
2 + EC2 /3, A1
is the back-of-the-envelope “bright” Rabi frequency for theD1 line and where D1
=2.54 10−29 C m is the dipole mo-ment for the D1 line D1
D2/2. The calculated eigen-
values are the Stark shifted excited and ground state ener-gies, and the amount of the Stark shift gives the strength ofthe coupling. The 12 eigenvectors with eigenvalues closest tothe original ground state energy can be projected onto theground states and normalized to create a new orthonormalbasis for the ground states. The Stark shifts of the energy forthese new basis ground states are used to determine the Rabifrequency i using
Ui = − i2/4, i = 1,2, . . . ,12 , A2
where Ui is the Stark shift for each of the ground states.Figure 20 shows each of the 12 ground state Rabi frequen-cies i normalized by the back-of-the-envelope “bright”Rabi frequency + plotted as a function of the ratio betweenprobe and coupling field amplitudes EP /EC for the experi-ment discussed in this paper 0.2P /C0.6. A new or-thonormal basis for the excited states can be calculated byprojecting the 12 excited state eigenvectors onto the excitedstates and normalizing. Each of the new basis ground stateswill couple to one of the new basis excited states with the
exception of the completely dark states which are completelydecoupled from all other states.
The effective optical pumping rate can be approximatedas the average of the optical pumping rates out of each brightstate into one of the five dark states. The optical pumpingrate out of bright state m is given by
Rm = n=1
5
l=1
12
Pel+mRel→− n, A3
where Pel +m is the steady-state conditional probabilitythat if the atom is in either bright state + m or one of theexcited states, then it is in excited state el, and Rel→−n
isthe spontaneous decay rate from excited state el to darkstate −n. The effective optical pumping rate is then
Rave =1
7 m=1
7
Rm, A4
where we have assumed that the bright states are equallypopulated in steady state in reality there will be some varia-
0 0.2 0.4 0.6 0.8 10.36
0.38
0.40
0.42
B=R
γ/Ω
aveeff
+
E /E
2
P C
5/12
FIG. 21. Color online Plot of B=Raveeff /+2 versus the ratio
EP /EC. Also plot is the value 5/120.417, which is the ratio of thenumber of dark states to ground states. Over the range 0.2EP /EC0.6 B is approximately 3/8.
F’=3
F=2
F=3
F’=2
mF=-2 m
F=-1 m
F=0 m
F=1 m
F=2m
F=-3 m
F=3
Cωh
Pωh
27
10−
27
15−
27
6−
27
1−
3
1−
27
1−
3
1−
27
6−
27
10−
27
15−
27
4−
3
1−
27
1−0
27
1
27
4
3
1
=
27
10−
27
15−
27
6−
27
1−
3
1−
27
1−
3
1−
27
6−
27
10−
27
15−
27
4−
3
1−
27
1−0
27
1
27
4
3
1
27
4−
3
1−
27
1−0
27
1
27
4
3
1
F’=3
F=2
F=3
F’=2
mF=-2 m
F=-1 m
F=0 m
F=1 m
F=2
(a)
mF=-3 m
F=3
27
5
27
8
27
5
3
1
27
8
27
5
27
8
27
5
3
1
27
8
27
2−
3
1−
27
2−
3
1−
27
2−
3
1−
27
2−
3
1−
27
2
3
1
3
1
27
2
27
2
3
1
3
1
27
2
Cωh
Pωh
(b)
FIG. 19. Color online Transitions excited by the vertically po-larized probe laser gray lines and horizontally polarized couplinglaser black lines. The quantization axis is chosen along the hori-zontal polarization. Also shown are the hyperfine matrix elementsfor each excited transition, which are related to the Clebsch-Gordoncoefficients by the factor −1F+J+1+I2F+12J+1 J J 1
F F I,
where I=3/2 is the nuclear spin, J=J=1/2 are the orbital plus spinangular momenta for the ground and excited state, respectively, andwe have used the Wigner 6-j symbol.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
EP/EC
RabiF
requency
(Ω/Ω
)
Completely Dark State
Quasi-dark States
Bright States
i+
FIG. 20. Color online Normalized Rabi frequencies i /+
plotted versus the ratio EP /EC. There are 12 Rabi frequencies:seven bright states, four quasidark states, and one completely darkstate.
TRANSIENTS OF THE ELECTROMAGNETICALLY-… PHYSICAL REVIEW A 76, 033835 2007
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tion in the bright state populations but it should be small forthe parameters considered in the experiment. Evaluating Eq.A4 we find that in the range 0.2Cp0.6C that
Rave 3+
2
8eff, A5
where eff is the effective homogeneous linewidth that arisesdue to a combination of Doppler broadening and buffer gaseffects. Figure 21 shows the value of Raveeff /+
2 as a func-tion of P /C. Naively one might expect this value to be5/12, which is the ratio between the number of bright statesand the total number of ground states, but calculations shownin 21 reveal that it is slightly smaller than 5/12 and is weaklydependent on the ratio P /C.
The optical pumping rate also has a spatial dependencedue to the inhomogeneity of the field intensities i.e., thebeams are approximately Gaussian. Thus quantities that aredependent on the optical pumping rate such as the rise timesand transparency FWHM must be averaged spatially.
There are two primary contributions to the effective ho-mogeneous linewidth: homogeneous broadening due to thebuffer gas and Doppler broadening. The buffer gas actual hastwo distinct effects. First, collisions between 85Rb and N2
atoms change the velocities of the atoms. Second, the colli-sions randomize the phase of the 5S1/2→5P1/2 transitionswhich broadens the homogeneous linewidth. For 85Rb with aN2 buffer gas the measured increase in homogeneous line-width is 16.3 MHz per Torr N2 36,37. In our experimentwe have 20 Torr N2 resulting in =330 MHz before con-sidering Doppler broadening. Although buffer gas collisionsrandomize the phase of the 5S1/2→5P1/2 coherences, theground state coherences GSC are essentially unaffected bythe 85Rb-N2 collisions. Because the 85Rb-N2 collision rate,which is about 2 50 MHz, is much faster than the opticalpumping rate in the experiment R30 kHz, the atoms willon average “wander” through every velocity class severaltimes in one optical pumping period T=1/R without loss ofground state coherence GSC. Thus, processes such as EITwhich are determined by the GSC must be averaged over thecontributions from all velocity classes. Thus the effectivehomogeneous linewidth is
1
eff=
2D
−
exp−2
2D2
42 + 2 d , A6
and eff=660 MHz when D=530 MHz and =330 MHz.
1 P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb,M. S. Shahriar, and P. Kumar, Opt. Lett. 20, 982 1995.
2 H. Schmidt and A. Imamoglu, Opt. Lett. 21, 1936 1996.3 S. E. Harris and Y. Yamamoto, Phys. Rev. Lett. 81, 3611
1998.4 S. E. Harris and L. V. Hau, Phys. Rev. Lett. 82, 4611 1999.5 G. M. D’Ariano, L. Maccone, M. G. A. Paris, and M. F. Sac-
chi, Phys. Rev. A 61, 053817 2000.6 M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett. 84, 5094
2000.7 M. D. Lukin and A. Imamoglu, Phys. Rev. Lett. 84, 1419
2000.8 M. Yan, E. G. Rickey, and Y. Zhu, Phys. Rev. A 64,
041801R 2001.9 E. Paspalakis and P. L. Knight, J. Mod. Opt. 49, 87 2002.
10 D. Petrosyan and G. Kurizki, Phys. Rev. A 65, 033833 2002.11 H. Kang and Y. Zhu, Phys. Rev. Lett. 91, 093601 2003.12 D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, Phys. Rev. A
68, 041801R 2003.13 A. B. Matsko, I. Novikova, G. R. Welch, and M. S. Zubairy,
Opt. Lett. 28, 96 2003.14 D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris,
Phys. Rev. Lett. 93, 183601 2004.15 R. G. Beausoleil, W. J. Munro, and T. P. Spiller, J. Mod. Opt.
51, 1559 2004.16 Y.-F. Chen, G.-C. Pan, and I. A. Yu, Phys. Rev. A 69, 063801
2004.17 M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod.
Phys. 77, 633 2005.18 V. Balić, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris,
Phys. Rev. Lett. 94, 183601 2005.19 C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, Phys. Rev. A
73, 010301R 2006.20 E. Knill, R. Laflamme, and G. Milburn, Nature London 409,
46 2001.21 M. V. Pack, R. M. Camacho, and J. C. Howell, Phys. Rev. A
74, 013812 2006.22 H. Wang, D. Goorskey, and M. Xiao, Opt. Lett. 27, 1354
2002.23 Y.-F. Chen, G.-C. Pan, and I. A. Yu, J. Opt. Soc. Am. B 21,
1647 2004.24 S. J. Park, H. Cho, T. Y. Kwon, and H. S. Lee, Phys. Rev. A
69, 023806 2004.25 A. Godone, S. Micalizio, and F. Levi, Phys. Rev. A 66,
063807 2002.26 K. Harada, K. Motomura, T. Koshimizu, H. Ueno, and M.
Mitsunaga, J. Opt. Soc. Am. B 22, 1105 2005.27 A. Mair, J. Hager, D. F. Phillips, R. L. Walsworth, and M. D.
Lukin, Phys. Rev. A 65, 031802R 2002.28 P. Valente, H. Failache, and A. Lezama, Phys. Rev. A 65,
023814 2001.29 A. Belyanin, C. Bentley, F. Capasso, O. Kocharovskaya, and
M. O. Scully, Phys. Rev. A 64, 013814 2001.30 M. Lindberg and R. Binder, Phys. Rev. Lett. 75, 1403 1995.31 Y. Wu and X. Yang, Phys. Rev. A 71, 053806 2005.32 D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yash-
chuk, Phys. Rev. Lett. 83, 1767 1999.33 L. Hau, S. Harris, Z. Dutton, and C. Behroozi, Nature Lon-
don 397, 594 1999.34 R. S. Bennink, Ph.D. thesis, University of Rochester, Institute
of Optics, 2004.
PACK, CAMACHO, AND HOWELL PHYSICAL REVIEW A 76, 033835 2007
033835-12
35 Transit-time broadening derives from atomic diffusion result-ing in optically prepared atoms in the laser beam being re-placed by thermal atoms from outside the beam.
36 J. Vanier and A. Audoin, The Quantum Physics of Atomic Fre-
quency Standards Adam Hilger, Philadelphia, 1989.37 M. D. Rotondaro and G. P. Perram, J. Quant. Spectrosc. Ra-
diat. Transf. 57, 497 1997.
TRANSIENTS OF THE ELECTROMAGNETICALLY-… PHYSICAL REVIEW A 76, 033835 2007
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