Control of Four-Level Quantum Coherence via Discrete Spectral Shaping of an Optical Frequency Comb

0.19 MB·English

by  Matthew C. Stowe

Control of Four-Level Quantum Coherence via Discrete Spectral Shaping of an
Optical Frequency Comb
Matthew C. Stowe, Avi Pe’er, and Jun Ye
JILA, National Institute of Standards and Technology and University of Colorado
Department of Physics, University of Colorado, Boulder, CO 80309-0440
(Dated: February 3, 2008)
We present an experiment demonstrating high-resolution coherent control of a four-level atomic
systeminaclosed(diamond)typeconfiguration. Afemtosecondfrequencycombisusedtoestablish
phase coherence between a pair of two-photon transitions in cold 87Rb atoms. By controlling the
spectral phase of the frequency comb we demonstrate the optical phase sensitive response of the
diamond system. The high-resolution state selectivity of the comb is used to demonstrate the
8 importance of the signs of dipole moment matrix elements in this type of closed-loop excitation.
0 Finally,thepulseshape is optimized resulting in a 256% increase in thetwo-photon transition rate
0 byforcingconstructiveinterferencebetweenthemodepairsdetunedfromanintermediateresonance.
2
PACS numbers: 32.30.-r, 32.80.Qk, 42.50.Gy, 42.62.Eh

n
a
J The field of coherent control of light-matter interac- other resonant modes, this shifts the relative phase be-

0 tions for the purpose of driving a quantum system to a tween the two paths constituting the diamond. By anal-
3 desired state has been drawing increasing interest. Re- ogy with an optical interferometer, the two-photon ex-
search in the field of coherent control generally uses ei- citation is shown to exhibit a sinusoidal variation ver-

] theranultrashortpulsetocreateatomiccoherencesover sus the applied phase, due to the resonant comb modes
h
p a very large bandwidth, or in the other extreme, con- alone. In the second experiment the off-resonant comb
- tinuous wave (cw) lasers to drive transitions with high- modes are of primary importance, in particular we show
t
n resolution. Someofthepioneeringworkoncoherentcon- that the two-photon excitation is enhanced by changing
a trolusedpulseshapingofasinglebroad-bandwidthpulse the phase of the two-photon resonant mode pairs that
u to enhance or diminish two-photonabsorption[1, 2], im- are symmetrically detuned from an intermediate state.
q
provetheresolutionofcoherentanti-StokesRamanscat- The experiments presented in this Letter can be mod-
[
tering [3], and control molecular wavepacket motion [4]. eledastheinteractionofaphasecoherenttrainofshaped

1 Simultaneously, a tremendous amount of research has femtosecond pulses (an optical comb) with a four-level
v been done using narrow-bandwidth cw-lasers to control atomic system in a diamond configuration. A diamond
6
three-levelatomic systems, for example to study electro- configuration has one ground state, two intermediate
9
magnetic inducedtransparency[5]. Particularlyrelevant states, and a single excited state such that there are a
7
4 to this Letter are the many theoretical studies of four- totaloffourpossible resonantdipole allowedtransitions.
. level systems, for example in a double-Lambda [6] or di- There are hundreds of thousands of frequency modes in
1
amond type configuration [7, 8]. The advent of optical the comb spectrum, four of which are tuned to be reso-
0
frequency combs has made a strong impact on the field nantwith the transitionsshownin Fig. 1(a). Due to the
8
0 of high-precision spectroscopy [9], and has recently been equidistantspacingofcombfrequencies,anymodeinthe
: used for coherent control of a three-level transition [10]. spectrumhasanothermodethatformsatwo-photonres-
v
In this work we combine the broad-bandwidthand high- onantpair. Thereforetherearehundredsofthousandsof
i
X resolution of the comb, augmented by discrete spectral mode pairs that are two-photon resonantbut have vary-
r phase shaping, to controlthe phase sensitive response of ing detunings from the intermediate states, these will be
a a four-level atomic system in a diamond configuration. referred to as off-resonant mode pairs. We present our
We use a femtosecond optical frequency comb to ex- experiments in two parts, the first focuses on the dia-
cite a pair of resonant two-photon transitions that form mond configuration excited by resonant modes only, the
acloseddiamondconfigurationincold87Rb. Thebroad- second part utilizes another pulse shape to enhance the
bandwidth of the femtosecond pulses allows for a two- signal from the off-resonantmode pairs.
photon transition to be excited via different intermedi- Two-photonabsorptionviaapairofmodesandasingle
ate states with a separation of 7 THz. Simultaneously intermediatestategivesrisetoanexcitedstateamplitude
the narrowlinewidth ofeachcombmode allowsthe exci- that within second-order perturbation is given by [11],
tationofonlyspecificatomiclevels,anecessarycondition
to isolate the four-level diamond configuration from the E E µ µ
c ∝ n m gi if ×
full setofatomic transitions. The phasesensitiveexcita- gf i(ω −(m+n)2πf −4πf )+πγ
gf r o f
tion of this closed four-level system is demonstrated by
1
discrete spectral shaping of the femtosecond comb. The +
(cid:20)i(ω −2π(nf +f ))+πγ
7 THz separationbetweenintermediate states allowsuse gi r o i
of a pulse shaper to change the phase of the comb mode 1
, (1)
that is resonantwith one intermediate state and not the i(ω −2π(mf +f ))+πγ (cid:21)
gi r o i
2
where E are the electric fields of the nth and mth isthemodeordernumber(oforder106forourlaser). Us-
n,m
modes ofthe comb, γ is the intermediate (final) state ingpriorknowledgeofthe87Rbenergylevelstructurefor
i(f)
decayrate,ω is groundtointermediate(final) state the5S,5P,and5Dhyperfinestates,itispossibletoselect
gi(gf)
transition frequency, f is the repetition frequency of a particular f and f to approximate a diamond config-
r r o
the comb, f is the comb offset frequency, and µ uration with only four resonant levels (see Fig. 1(a)).
o gi(if)
are the dipole moments from the ground to intermedi- We use two values of f , the first is f =100.59660605
r r
ate(intermediatetofinal)states. The totalexcitedstate MHz with f =+16.94 MHz. In this case, the four res-
o
amplitude is then given by the sum of all the possible onant states are: 5S F=2, 5P F=2, 5P F=2, and
1/2 1/2 3/2
two-photon resonant transition pathways resulting from 5D F=1. With this selection of comb frequencies the
3/2
all comb mode pairs in the laser spectrum connecting transitions from 5S F=1, the other ground state, are
1/2
throughvariousintermediate states. The key physics for at least 6 MHz detuned from any intermediate and ex-
the results presented here is that the phase of the ex- cited states. All other possible transitions are further
cited state amplitude is a function of the detuning from detuned. Notethatthe5Sto5Pand5Dlinewidthsare6
the intermediate state, the signs of dipole moment ma- MHz and0.66MHz respectively. Using a slightly shifted
trix elements, and the phase of the two electric fields. f of 100.59660525 MHz and the same f , the comb is
r o
In particular,the phase of the excited state amplitude is resonantwith the sameintermediate andfinalstates but
+90o(-90o) for detunings above(below) the intermediate from 5S F=1. We indirectly measure the 5D popu-
1/2
state,andis0o forzerointermediatestatedetuning. Due lation by counting photons from the 5D-6P-5S cascade
to the phasedifference of180o betweentwoexcitedstate fluorescenceat420nmwithaphotomultipliertube. The
amplitudessymmetricallydetunedaboutanintermediate experimental cycle consists of three parts: first a MOT
resonance, the contribution to the total amplitude from is formed for 6.5 ms, then the atoms are held in opti-
the off-resonant mode pairs cancels to zero for a train of cal molasses for 3 ms while the magnetic field turns off,
transform-limited pulses. finally the atoms are excited for 0.5 ms and the photon
countsarerecordedversustimeonamultichannelscaler.
We use a standard 2f-2f configuration pulse shaper with
5D F=1 a computer controlled spatial light modulator (SLM) to
(a) 3/2 (b) P
5P F7=622 nm 776 n5mP3/2F=2 Power (a.u.) 7627 n7m6 nm 780 7n9m4 nm hase Mask (rad srfrreeeetsdqoutulhcueeetndircoeytnloacotmhfivia∼repx1pim5ho0faizstGeehHtoehfzpetu[hf1lre3sien]c.sogmeTatbvhietsmhisboepidlaMiettsyiOa.wlTiathnlodacatpeteimornppoiaxrraeell
1/2 ia Thefirstexperimentprovidesacleardemonstrationof
n
s) the phase sensitive nonlinear response of the four-level
794 nm 780 nm
5S F={2,1} 725 750 775 800 825 850 875 900
1/2 Wavelength (nm)
250 250
(a) (b)
FIG. 1: (a) Energy level diagram of the diamond configura-
tion in 87Rb, the arrows indicate the resonant comb modes. nts) 200 200
(b) Pulse spectrum and phase mask used for the first exper- cou 150 150
iment. The hatched region in the spectrum and the inset nal ( 100 100
g
energyleveldiagramindicatestheportionofthespectrumto Si
50 50
wrehsoicnhantthewpavhealseengmthasskinisrealpaptiloiend.toTthhee fpohuarsearmroawsks isnpdeicctartael 00Visi4b5ility:9 068%1 3 5 1 8 P0ha2s2e5 Of2f7s0et:3 -1551o360 00Visi4b5ility:9 069%1 3 5 1 8P0has22e5 Of2fs7e0t: 3-11055o360
window. 2000 (c) 2000 (d)
nts)1500 1500
Theexperimentsareconductedonanensembleofcold ou
c
87Rb atoms formed in a magneto-optical trap (MOT). al (1000 1000
n
It is necessary to use cold atoms to ensure that only Sig 500 500

the four intended atomic states are excited, in contrast
Visibility: 26% Phase Offset: -55o Visibility: 23% Phase Offset: -118o
to a Doppler broadened room temperature gas. A Kerr 00 45 90 135 180 225 270 315 360 00 45 90 135 180 225 270 315 360
lensmode-lockedTi:Sapphirelaseroperatingwithanap- Phase Step (degrees) Phase Step (degrees)
proximately 55 nm bandwidth centered at 778 nm with FIG.2: (a)Measuredinterferencefringeswithfitsunderfour
fr ≈ 100 MHz is used to excite all four transitions. fr differentexcitationconditions. Alltheresultsareobtainedby
is phasestabilizedto alowphase-noisecrystaloscillator, scanning the phase Φ of the phase mask shown in Fig. 1(b).
andsteeredviaaCesiumreferencetomaintainthe abso- The top panels correspond to illuminating the atoms from
lute frequency of the comb modes. The offset frequency only one direction (traveling waves). The bottom panels are
f is measured via a f-2f nonlinear interferometer [12] under counter-propagating pulse excitation. The left panels
o
and stabilized to a direct digital synthesizer. Regardless (a)and(c)usethefrforexcitationfromthe5S1/2F=2ground
state; the right panels (b) and (d) use the f for excitation
ofthespectralphaseofthepulses,anycombmodehasan r
absolute frequency givenby, ν =f +N×f , where N from the 5S1/2 F=1 ground state.
N o r
3
discussing the offset of the fringe from Φ=0. Due to the
Intermediate ’gi ’if (MHz) |cgf | large separation of wavelengths used in this experiment
F=2 5P3/2F=2 , mF=1 -0.17 -0.07 0.2 3.0 -3.8o aonthderthoeptsiicgsn,ifiacnayntredsiidspuaerlsciohnirpofotfhtehepupluselseshcaapnercaaunsde
5S1/2 5P1/2F=2 , mF=1 -0.17 -0.09 0.4 3.4 -7.6o an overall phase shift common to all measured fringes.
5P F=1 , m =1 -0.29 -0.26 -11.5 4.3 75.4o However,not allfringes are shifted by the same amount;
1/2 F
in particular, excitation from the two different ground
1 o
F= 5P3/2F=2 , mF=1 0.29 -0.07 -2.6 3.9 -139.1 statesyieldsresultssignificantlyoutofphase. Toexplain
S1/2 5P1/2F=2 , mF=1 -0.29 -0.09 -2.5 4.8 39.8o this relative phase shift we refer to Table 1, in which the
5 o keyparametersto estimate the totalfringevisibility and
5P F=1 , m =1 0.17 -0.26 -14.4 2.1 -101.8
1/2 F phase shift are tabulated. The first and most dominant
TABLE I: Left column is the intermediate state for each effect is due to the sign of the dipole matrix elements;
two-photontransition witharesonantornear-resonantcomb fortransitionsfrom5S F=2allthedipolemomentsare
1/2
mode. Across the top are: the reduced dipole moments, the negative,however,fortransitionsfrom5S F=1thereis
1/2
detuning of the nearest mode from the intermediate state, signdifferencebetweendifferenttwo-photonpaths. Table
therelative magnitudeof thetwo-photonamplitude,and the 1 gives the angular part of the dipole matrix elements,
phase of the amplitude. The top section is for transitions hLm F||ˆr||L′m′ F′i, denoted as µ′ and µ′ . Clearly
from 5S1/2F=2 and thebottom from 5S1/2F=1. the sFign of theFmatrix elements affgeict the iifnterference
in closed loop excitation, an important consideration for
phase-resolvedtwo-dimensional spectroscopy [14].
diamond configuration. We use the spatial light mod- Itisalsonecessarytoconsiderthephaseshiftofapar-
ulator to apply the phase mask indicated in Fig. 1(b), ticular two-photon amplitude due to the detuning from
the effect of this mask is to change the phase of all the therelevantintermediatestate. ForthisweincludeinTa-
combmodepairsthatareresonantfromthegroundstate ble 1 the path through the additional intermediate state
to 5D3/2F=1 and close to the 5P3/2 intermediate states 5P1/2F=1, although the nearest comb mode is detuned,
(denoted by the hatched region in Fig. 1(b)). Specifi- thedipolemomentissufficientlylargetomakeitscontri-
cally, the mask applies a variable phase step of Φ to the butionsignificant. Due tothe detuning ofthis transition
spectralregionfrom772nmto784nm. Duetotheafore- path, there is a large phase shift of the corresponding
mentioned cancellationof the off-resonantamplitudes, it amplitude. The effect of this additional path through
issufficienttoconsideronlythemodepairstunednearest 5P F=1istophaseshiftthetotaltransitionamplitude
1/2
to an intermediate state resonance for this first experi- via 5P relative to the 5P amplitude, and thus the
1/2 3/2
ment. fringe shift. This occurs for two-photon transitions from
Our measured results are shown in Fig. 2, with each both ground states. However, as can be seen from the
fringe fitto a functionof the form,ρ =c +c cos(Φ+ dipole moments and amplitudes in Table 1, the effect
5D 1 2
c )2, where Φ represents the phase applied to the SLM, of the 5P F=1 state is less for the 5S F=1 ground
3 1/2 1/2
c isastaticphaseoffset,andthefringevisibilityisgiven state case. The difference in fringe shift between Fig.
3
by c2 . Due to the fact the phase mask covers both 2(a) and (b), corresponding to excitation from the two
7802nc1m+ca2nd 776nm, the two-photonamplitude fromthe groundstates,is54o. Thisisingoodagreementwiththe
resonant path through 5P F=2 is phase shifted by 2Φ theoretically predicted value of 56o.
3/2
andthereforehasaperiodofπ radians. Thebackground The second feature, the fringe visibility, is a result of
counts due to ambient light at 420 nm and excitation the coherent interference between the 5P and 5P
1/2 3/2
of the hot Rb atoms not trapped in the MOT are mea- paths in the diamond configuration and any additional
sured by repeating the experiment without a MOT and incoherent signal which raises the measured fringe min-
subtracted from the reported data. As mentioned pre- imum. Referring to the results in Fig. 2, the fringe
viously, we choose to conduct this experiment with two visibility is strongly reduced under standing wave exci-
different values of fr. The left panels in Fig. 2 are mea- tation and exhibits little dependence on the choice of
sured with fr set for two-photon transitions from the ground state. In the case of traveling wave excitation,
5S1/2F=2 ground state. The right panels are with the all the atoms in the MOT are excited by the same rela-
secondfr,resonantfromthe5S1/2F=1groundstate. For tive magnitude of electric fields. The visibility predicted
eachfr,wealsomeasuretheinterferencefringeunderex- using the amplitudes presented in Table 1 is 82% for ex-
citationfromasinglepulsepropagationdirection(Fig. 2 citation from the 5S F=2 ground state and 92% for
1/2
top panels) and under counter-propagating pulses (Fig. the 5S F=1 ground state, assuming equal population
1/2
2 bottom panels). distribution among the m sublevels. The best experi-
F
Muchlikeanopticalinterferometer,twoimportantfea- mental results obtained for the visibility are about 70%,
turesofourresultsarethe fringevisibilityandthe phase shownin Fig. 2(a)and (b). Residualfrequency andspa-
offset. Clearly the results presented in Fig. 2 show sig- tialchirpshavelikelyloweredtheobservedvisibilityfrom
nificantdifferencesinboththefringevisibilityandphase the idealcase. Fortravelingwaveexcitationthe interfer-
offset as a function of excitation scheme. We begin by enceeffectisobservedforonlythefirst50µsofexcitation
4
modes near an intermediate resonance. This is due to
Enhancement Factor01122.....50505 (a) Power (a.u.) 762 n7m76 nm780 n79m4 nm (b) 0001....3580Phase Mask (radians) ttttapwhhunhuedodase-sfp.aeamhpcmtIoapntklatyoehstnkahtnhtetofreotansrhpenceecehstosiavtcnasicooedsonntnmtessrmxtiafbprrasuoukejocmrtitirpmoiirvttneyehesnteeoiotnnf,5tttSmeewh1rdeeo/fe2ditruFnwees=snoeFc-c2ipefaghgrn.broceao3tetonu(lwbndnoe)du.aeftnmsoTtaptahfnholtiideesr-
0.0 amplitudes due to the off-resonant mode pairs, increas-
-600 -300 0 300 600 900 1200 725 750 775 800 825 850 875 900
Frequency Offset (GHz) Wavelength (nm) ingthetotalsignal. Recallforeverytwo-photonresonant
FIG. 3: (a) Measured signal enhancement. The ratio of sig- modepairdetunedbelowanintermediatestate,thereisa
nalswith andwithout thephasemask in (b),versusposition pairdetuned approximatelyequallyabovethe state. For
oftheSLMinthespectrum. Thezerooftheoffsetfrequency atransform-limitedpulse train,these twopairsofmodes
is chosen to be the position of the maximum signal increase. give rise to excited state amplitudes that are equal and
(b) The applied phase mask is indicated by the hatched re- opposite, and therefore cancel to zero. By applying the
gion. Exactlyπ radiansofphaseisappliedjustbelow762nm phase mask in Fig. 3(b), those modes that are detuned
and 776 nm to add an extra phase shift to those mode pairs below either the 5P or 5P states (the hatched area
3/2 1/2
that join in the hatched region. in Fig. 3(b)) obtain a 180o phase shift with respect to
thosedetunedabovetheintermediatestate. Thistypeof
spectral phase negates the inherent phase change due to
after which the atoms are Doppler shifted completely off
the detuning arounda resonance(see Eq. 1), andcauses
of resonance due to radiation pressure. It is for this rea-
constructive interference. Figure 3(a) shows we achieve
son the data presented in Fig. 2 is only the first 10 µs
a maximum increaseof 2.56over the normal signal. The
of excitation; using a larger time window significantly
theoretical enhancement is 2.85, however, this estimate
reduced the fringe visibility.
does not include the effects of diffraction at the phase
One method to reduce the effect of radiation pres-
steps in the SLM, which likely reduces the maximum.
sure on the atoms is to balance the average force by
This data is obtained by first coarsely tuning the posi-
probing with counter-propagating pulses [9]. The bot-
tion of the phase mask at the per-pixel resolution. Then
tom panels of Fig. 2 present the fringe measured us-
for finer controlof the location of the phase step applied
ing well overlapped counter-propagating beams of the
to the comb spectrum, the entire SLM is shifted using a
same intensity. Although we measure a constant fringe
micrometerthroughthespectrallydispersedopticalfield.
visibility in this case for an extended time of 300 µs,
In conclusion, we have demonstrated the precise con-
the multi-mode standingwavegeneratedby the counter-
trol of a diamond configuration four-level atomic coher-
propagating pulses reduces the visibility to ∼25%. This
ence over a 32 nm spectral width. The first experiment
effect arisesbecause the four main resonancefrequencies
focuses on the comb modes resonant with intermediate
have different wavelengths and thus different standing-
states, and the second optimizes the two-photon transi-
wave periods, so the magnitudes of the four resonant
tion rate using high-resolutionspectral phase shaping to
electric fields vary spatially. For example, in some re-
force constructive interference between the off-resonant
gions of the atom cloud the 780 nm field is maximum
modes. The demonstratedabilityin high-resolutioncon-
while the 794 nm field is minimum, and there is no in-
trol of the coherence of a four-level system over a very
terference effect. In this case a spatial average over the
broad-bandwidthmayenablefutureresearchinnonlinear
atom cloud, taking into account the different standing
optics of multi-level systems. For example, one proposal
waves, must be conducted. Using the amplitudes given
for a cw-VUV laser utilizes lasing without inversion in a
inTable 1,the visibility under multimode standingwave
four-level diamond configuration [15].
excitationis36%and44%fromtheF=2andF=1ground
states, respectively. We acknowledge funding support from NSF, NIST,
The first experiment focuses entirely on those comb ONR, and DARPA.
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