Observation of Rabi oscillations between Bloch bands in an optical ...
Observation of Rabi oscillations between Bloch bands in an optical ...
Observation of Rabi oscillations between Bloch bands in an optical ...
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RAPID COMMUNICATIONS<br />
PHYSICAL REVIEW A VOLUME 58, NUMBER 4<br />
OCTOBER 1998<br />
<strong>Observation</strong> <strong>of</strong> <strong>Rabi</strong> <strong>oscillations</strong> <strong>between</strong> <strong>Bloch</strong> <strong>b<strong>an</strong>ds</strong> <strong>in</strong> <strong>an</strong> <strong>optical</strong> potential<br />
M. C. Fischer, K. W. Madison, Qi<strong>an</strong> Niu, <strong>an</strong>d M. G. Raizen<br />
Department <strong>of</strong> Physics, The University <strong>of</strong> Texas at Aust<strong>in</strong>, Aust<strong>in</strong>, Texas 78712-1081<br />
Received 8 June 1998<br />
We report <strong>an</strong> experimental study <strong>of</strong> atomic motion <strong>in</strong> the <strong>Bloch</strong> b<strong>an</strong>d <strong>of</strong> a periodic potential. Our system<br />
consists <strong>of</strong> cold sodium atoms <strong>in</strong> a far detuned st<strong>an</strong>d<strong>in</strong>g wave <strong>of</strong> near reson<strong>an</strong>t light. We prepare the atoms <strong>in</strong><br />
the lowest motional b<strong>an</strong>d, <strong>an</strong>d then impose phase modulation <strong>in</strong> order to drive <strong>in</strong>terb<strong>an</strong>d tr<strong>an</strong>sitions. We<br />
observe <strong>Rabi</strong> <strong>oscillations</strong> <strong>between</strong> the first <strong>an</strong>d second b<strong>an</strong>d as a function <strong>of</strong> modulation <strong>in</strong>tensity <strong>an</strong>d frequency.<br />
We also observe damped <strong>oscillations</strong> <strong>of</strong> the population as a function <strong>of</strong> modulation frequency, which<br />
are primarily due to a spread <strong>in</strong> <strong>Rabi</strong> frequencies over the <strong>Bloch</strong> b<strong>an</strong>d. S1050-29479850110-2<br />
PACS numbers: 42.50.Vk, 32.80.Pj<br />
The qu<strong>an</strong>tum behavior <strong>of</strong> particles <strong>in</strong> a periodic potential<br />
was <strong>an</strong>alyzed by <strong>Bloch</strong> <strong>an</strong>d Zener almost 70 years ago <strong>in</strong><br />
order to underst<strong>an</strong>d electron conduction. They showed that<br />
stationary states <strong>of</strong> electrons <strong>in</strong> a lattice are pl<strong>an</strong>e waves<br />
modulated by periodic functions <strong>of</strong> position 1. A wave<br />
packet that is <strong>in</strong>itially localized <strong>in</strong> a potential well will<br />
spread via reson<strong>an</strong>t ‘‘<strong>Bloch</strong> tunnel<strong>in</strong>g,’’ <strong>an</strong>d will eventually<br />
become delocalized. The qu<strong>an</strong>tized energy levels are broadened<br />
<strong>in</strong>to energy ‘‘<strong>b<strong>an</strong>ds</strong>’’ due to this tunnel<strong>in</strong>g process. The<br />
picture becomes considerably more complex when external<br />
fields time vary<strong>in</strong>g or static are imposed, <strong>an</strong>d theoretical<br />
studies <strong>of</strong> such problems are still ongo<strong>in</strong>g.<br />
Qu<strong>an</strong>tum tr<strong>an</strong>sport <strong>of</strong> particles <strong>in</strong> lattices has grown <strong>in</strong>to<br />
<strong>an</strong> active area <strong>of</strong> research <strong>in</strong> recent years with the development<br />
<strong>of</strong> semiconductor superlattices. In a parallel development,<br />
the cool<strong>in</strong>g <strong>of</strong> atoms to ultralow temperatures has<br />
opened up <strong>an</strong>other field <strong>of</strong> study <strong>in</strong> which their center-<strong>of</strong>mass<br />
motion exhibits m<strong>an</strong>ifestly qu<strong>an</strong>tum mech<strong>an</strong>ical behavior,<br />
<strong>an</strong>d experiments on qu<strong>an</strong>tum tr<strong>an</strong>sport <strong>in</strong> <strong>optical</strong> lattices<br />
have probed a wide r<strong>an</strong>ge <strong>of</strong> phenomena that are closely<br />
related to the condensed matter counterparts 2. In this<br />
Rapid Communication, we report the observation <strong>of</strong> <strong>Rabi</strong><br />
<strong>oscillations</strong> <strong>between</strong> <strong>Bloch</strong> <strong>b<strong>an</strong>ds</strong> <strong>of</strong> cold sodium atoms <strong>in</strong><br />
the periodic potential <strong>of</strong> a st<strong>an</strong>d<strong>in</strong>g wave <strong>of</strong> light 3. This<br />
work relies on the long coherence times possible <strong>in</strong> atom<br />
optics, <strong>an</strong>d opens up possibilities for coherent control <strong>of</strong> motional<br />
states <strong>in</strong> <strong>Bloch</strong> <strong>b<strong>an</strong>ds</strong>.<br />
To beg<strong>in</strong> our discussion we consider the effect <strong>of</strong> a st<strong>an</strong>d<strong>in</strong>g<br />
wave <strong>of</strong> light on cold atoms. For sufficiently large detun<strong>in</strong>g<br />
from reson<strong>an</strong>ce the effect <strong>of</strong> spont<strong>an</strong>eous emission c<strong>an</strong><br />
be neglected. Adiabatic elim<strong>in</strong>ation <strong>of</strong> the excited state leads<br />
to <strong>an</strong> effective one-dimensional potential <strong>of</strong> the form V(x)<br />
V 0cos(2k Lx), where k L2/ L is the light wave number.<br />
The amplitude V 0 <strong>of</strong> the <strong>optical</strong> dipole potential is proportional<br />
to the laser <strong>in</strong>tensity <strong>an</strong>d <strong>in</strong>versely proportional to the<br />
detun<strong>in</strong>g from reson<strong>an</strong>ce 4. We add a time-dependent<br />
modulation <strong>of</strong> the st<strong>an</strong>d<strong>in</strong>g wave position <strong>in</strong> the form<br />
Vx,tV 0cos2k Lx cos2t, 1<br />
where is the modulation frequency <strong>an</strong>d is the dimensionless<br />
modulation amplitude. To elucidate the effect <strong>of</strong> this<br />
modulation we c<strong>an</strong> exp<strong>an</strong>d the potential for small modulation<br />
amplitude . To first order, one obta<strong>in</strong>s<br />
Vx,tV 0cos2k LxV 0 s<strong>in</strong>2k Lxcos2t. 2<br />
The first term <strong>in</strong> Eq. 2 is the stationary periodic potential<br />
that determ<strong>in</strong>es the eigenstates <strong>an</strong>d eigenenergies <strong>of</strong> the system.<br />
The result<strong>in</strong>g set <strong>of</strong> qu<strong>an</strong>tized eigenenergies is periodic<br />
<strong>in</strong> momentum space <strong>an</strong>d produces the well-known b<strong>an</strong>d<br />
structure. Figure 1 shows the energy <strong>b<strong>an</strong>ds</strong> <strong>in</strong> the reduced<br />
zone scheme with<strong>in</strong> the first Brillou<strong>in</strong> zone. The graph corresponds<br />
to a typical well depth <strong>of</strong> V 0 /h70 kHz, where h<br />
is Pl<strong>an</strong>ck’s const<strong>an</strong>t. When no perturbation <strong>of</strong> the potential is<br />
present, the quasimomentum q <strong>an</strong>d the b<strong>an</strong>d <strong>in</strong>dex are conserved.<br />
The time-dependent second term <strong>in</strong> Eq. 2 has a<br />
small amplitude <strong>an</strong>d c<strong>an</strong> be treated perturbatively. It acts as a<br />
harmonic driv<strong>in</strong>g term that c<strong>an</strong> <strong>in</strong>duce tr<strong>an</strong>sitions <strong>between</strong><br />
the <strong>b<strong>an</strong>ds</strong>, leav<strong>in</strong>g q unch<strong>an</strong>ged. As <strong>in</strong>dicated <strong>in</strong> Fig. 1 the<br />
drive frequency c<strong>an</strong> be chosen to be reson<strong>an</strong>t with the<br />
tr<strong>an</strong>sition <strong>between</strong> the first two <strong>b<strong>an</strong>ds</strong> (1↔2), but far from<br />
reson<strong>an</strong>ce for tr<strong>an</strong>sitions <strong>between</strong> successive <strong>b<strong>an</strong>ds</strong> arrows<br />
a <strong>an</strong>d b. In this case only two eigenstates <strong>of</strong> the atoms<br />
need to be considered 5. The population evolution <strong>in</strong> such a<br />
two-level system exhibits <strong>Rabi</strong> <strong>oscillations</strong>. If the drive frequency<br />
is close to reson<strong>an</strong>ce for both 1↔2 <strong>an</strong>d 2↔3 arrow<br />
FIG. 1. B<strong>an</strong>d structure for <strong>an</strong> atom <strong>in</strong> a far-detuned st<strong>an</strong>d<strong>in</strong>g<br />
wave. The energies are calculated relative to the bottom <strong>of</strong> the<br />
potential for a well depth <strong>of</strong> V 0 /h70 kHz. In the reduced zone<br />
scheme the quasimomentum q is limited to the first Brillou<strong>in</strong> zone<br />
k L ,k L. The arrows correspond to a modulation frequency <strong>of</strong><br />
a 70 kHz, b 75 kHz, <strong>an</strong>d c 85 kHz. The dashed l<strong>in</strong>e <strong>in</strong>dicates<br />
the top edge <strong>of</strong> the potential (2V 0).<br />
1050-2947/98/584/26484/$15.00 PRA 58 R2648 © 1998 The Americ<strong>an</strong> Physical Society
c <strong>in</strong> Fig. 1 more th<strong>an</strong> two levels participate <strong>in</strong> the <strong>in</strong>teraction<br />
<strong>an</strong>d more complicated dynamics are to be expected.<br />
The experimental setup for measur<strong>in</strong>g the tr<strong>an</strong>sition <strong>between</strong><br />
<strong>Bloch</strong> <strong>b<strong>an</strong>ds</strong> is based on the system previously used to<br />
study W<strong>an</strong>nier-Stark ladders <strong>an</strong>d tunnel<strong>in</strong>g <strong>in</strong> <strong>optical</strong> lattices<br />
6,7. Several steps were necessary to prepare the atoms <strong>in</strong><br />
the lowest <strong>Bloch</strong> b<strong>an</strong>d. We beg<strong>an</strong> by cool<strong>in</strong>g <strong>an</strong>d trapp<strong>in</strong>g a<br />
cloud <strong>of</strong> approximately 10 5 sodium atoms <strong>in</strong> a magneto<strong>optical</strong><br />
trap MOT <strong>in</strong> a configuration 8. The atom<br />
distribution <strong>in</strong> the MOT had a typical Gaussi<strong>an</strong> width <strong>of</strong><br />
x(0.200.06) mm <strong>in</strong> position <strong>an</strong>d p(4.50.5)k L <strong>in</strong><br />
momentum. After the cool<strong>in</strong>g <strong>an</strong>d trapp<strong>in</strong>g sequence the<br />
magnetic field coils <strong>an</strong>d the MOT beams were switched <strong>of</strong>f<br />
<strong>an</strong>d the <strong>in</strong>teraction beam was turned on.<br />
The <strong>in</strong>teraction potential was a st<strong>an</strong>d<strong>in</strong>g wave created by<br />
two l<strong>in</strong>early polarized counterpropagat<strong>in</strong>g laser beams with<br />
parallel polarization vectors. The light was far detuned from<br />
the (3S 1/2 ,F2)↔(3P 3/2 ,F3) tr<strong>an</strong>sition, <strong>an</strong>d detun<strong>in</strong>gs<br />
typically r<strong>an</strong>ged from 30 to 40 GHz. The power <strong>in</strong> each <strong>of</strong><br />
the beams was adjusted up to 100 mW <strong>an</strong>d was monitored<br />
with photodiodes dur<strong>in</strong>g the <strong>in</strong>teraction. The beams were<br />
spatially filtered to form a beam waist <strong>of</strong> 2.1 mm at the<br />
position <strong>of</strong> the atomic cloud. Due to the large momentum<br />
spread <strong>in</strong> the MOT, switch<strong>in</strong>g on the <strong>in</strong>teraction potential<br />
<strong>in</strong>itially populated several <strong>of</strong> the lower energy <strong>b<strong>an</strong>ds</strong>. As<br />
<strong>in</strong>dicated <strong>in</strong> Fig. 1, the atoms projected <strong>in</strong>to the lowest b<strong>an</strong>d<br />
are trapped with<strong>in</strong> the potential wells, whereas atoms <strong>in</strong> the<br />
second b<strong>an</strong>d are only partially trapped. Atoms <strong>in</strong> even higher<br />
<strong>b<strong>an</strong>ds</strong> have energies well above the potential <strong>an</strong>d, hence, are<br />
effectively free. To empty all but the lowest b<strong>an</strong>d, the st<strong>an</strong>d<strong>in</strong>g<br />
wave was then accelerated to a velocity <strong>of</strong> v 040 v rec ,<br />
where v rec3 cm/s is the s<strong>in</strong>gle-photon recoil velocity. L<strong>in</strong>early<br />
chirp<strong>in</strong>g the frequency <strong>of</strong> one <strong>of</strong> the counterpropagat<strong>in</strong>g<br />
beams, while keep<strong>in</strong>g the frequency <strong>of</strong> the other beam<br />
fixed, resulted <strong>in</strong> a const<strong>an</strong>t acceleration. As discussed <strong>in</strong> our<br />
previous work, accelerat<strong>in</strong>g the potential leads to a loss <strong>of</strong><br />
population <strong>in</strong> the lower <strong>b<strong>an</strong>ds</strong> due to the tunnel<strong>in</strong>g <strong>of</strong> atoms<br />
<strong>in</strong>to higher untrapped <strong>b<strong>an</strong>ds</strong> 7. The tr<strong>an</strong>sport acceleration<br />
a tr was chosen to maximize tunnel<strong>in</strong>g out <strong>of</strong> the second b<strong>an</strong>d<br />
while m<strong>in</strong>imiz<strong>in</strong>g losses from the first trapped b<strong>an</strong>d. For<br />
typical experimental parameters <strong>of</strong> V 0 /h70 kHz <strong>an</strong>d a tr<br />
2000 m/s 2 , the L<strong>an</strong>dau-Zener expression for the lifetime<br />
<strong>of</strong> the first <strong>an</strong>d second <strong>b<strong>an</strong>ds</strong> yields 24 ms <strong>an</strong>d 40 s, respectively<br />
9. This ensured that, after 600 s <strong>of</strong> acceleration,<br />
only the first b<strong>an</strong>d still conta<strong>in</strong>ed a signific<strong>an</strong>t number <strong>of</strong><br />
atoms. After reach<strong>in</strong>g the velocity v 0 the chirp was stopped<br />
<strong>an</strong>d the frequency difference was held const<strong>an</strong>t. At that<br />
po<strong>in</strong>t, phase modulation was added to one <strong>of</strong> the two counterpropagat<strong>in</strong>g<br />
beams form<strong>in</strong>g the st<strong>an</strong>d<strong>in</strong>g wave. The modulation<br />
was switched on <strong>an</strong>d <strong>of</strong>f smoothly over 16 s to avoid<br />
<strong>an</strong>y discont<strong>in</strong>uous phase ch<strong>an</strong>ges <strong>in</strong> the potential that could<br />
<strong>in</strong>duce tr<strong>an</strong>sition to higher <strong>b<strong>an</strong>ds</strong>. After a fixed time <strong>in</strong>terval<br />
<strong>of</strong> zero acceleration the frequency chirp<strong>in</strong>g resumed at a rate<br />
correspond<strong>in</strong>g to a tr . This separated the rema<strong>in</strong><strong>in</strong>g trapped<br />
atoms <strong>in</strong> the lowest b<strong>an</strong>d from those <strong>in</strong> higher <strong>b<strong>an</strong>ds</strong> <strong>in</strong> momentum<br />
space. After reach<strong>in</strong>g a f<strong>in</strong>al velocity <strong>of</strong> 80 v rec the<br />
<strong>in</strong>teraction beams were switched <strong>of</strong>f suddenly.<br />
In the detection phase we needed to dist<strong>in</strong>guish three<br />
classes <strong>of</strong> atoms: 1 atoms that were not <strong>in</strong>itially trapped <strong>in</strong><br />
the lowest b<strong>an</strong>d <strong>an</strong>d immediately tunneled out <strong>of</strong> the well<br />
dur<strong>in</strong>g the <strong>in</strong>itial acceleration, 2 atoms that were trapped <strong>in</strong><br />
RAPID COMMUNICATIONS<br />
PRA 58 OBSERVATION OF RABI OSCILLATIONS BETWEEN ...<br />
R2649<br />
FIG. 2. Measured survival probability <strong>in</strong> the lowest b<strong>an</strong>d as a<br />
function <strong>of</strong> modulation duration. The modulation amplitude was set<br />
to be a 0.1, b 0.2, <strong>an</strong>d c 0.3. The data were taken at<br />
a well depth <strong>of</strong> V 0 /h71 kHz <strong>an</strong>d a modulation frequency <strong>of</strong> <br />
70 kHz, correspond<strong>in</strong>g to a drive near the b<strong>an</strong>d edge as <strong>in</strong>dicated<br />
<strong>in</strong> Fig. 1, arrow a. Each run was repeated several times <strong>an</strong>d the<br />
error bar denotes the one-sigma error <strong>of</strong> the me<strong>an</strong>. The solid l<strong>in</strong>e<br />
displays the fit <strong>of</strong> <strong>an</strong> exponentially damped cos function to the data.<br />
the first b<strong>an</strong>d at the beg<strong>in</strong>n<strong>in</strong>g <strong>of</strong> the <strong>in</strong>teraction, but were<br />
driven out by the modulation, <strong>an</strong>d 3 atoms that rema<strong>in</strong>ed <strong>in</strong><br />
the first b<strong>an</strong>d dur<strong>in</strong>g the entire sequence. S<strong>in</strong>ce the atoms <strong>in</strong><br />
different classes had left the trapp<strong>in</strong>g potential at a different<br />
stage <strong>of</strong> the experimental sequence, they were accelerated to<br />
different velocities. Therefore, after drift<strong>in</strong>g <strong>in</strong> the dark for<br />
2.5 ms, these classes separated <strong>in</strong> space <strong>an</strong>d could be dist<strong>in</strong>guished<br />
by record<strong>in</strong>g their position. For this purpose the<br />
MOT beams were turned back on with no magnetic-field<br />
gradient present. This temporarily restricted movement <strong>of</strong><br />
the atoms <strong>in</strong> a ‘‘freez<strong>in</strong>g molasses’’ stage, while the fluorescence<br />
was imaged onto a charged coupled device camera.<br />
The two-dimensional image was then <strong>in</strong>tegrated <strong>in</strong> the direction<br />
that was perpendicular to the axis <strong>of</strong> the <strong>in</strong>teraction<br />
beams to obta<strong>in</strong> a one-dimensional distribution along the<br />
beam direction, which conta<strong>in</strong>ed all three classes <strong>of</strong> atoms.<br />
In order to reduce the sensitivity to fluctuations <strong>in</strong> the number<br />
<strong>of</strong> atoms <strong>in</strong> the MOT, the number <strong>of</strong> survivors atoms <strong>in</strong><br />
class 3 was normalized by the total number <strong>of</strong> atoms <strong>in</strong>itially<br />
trapped <strong>in</strong> the first b<strong>an</strong>d, which was obta<strong>in</strong>ed by summ<strong>in</strong>g<br />
up the contributions <strong>of</strong> classes 2 <strong>an</strong>d 3. To observe<br />
the temporal evolution <strong>of</strong> the fundamental b<strong>an</strong>d population<br />
we repeated the experiment for various modulation durations,<br />
hold<strong>in</strong>g the modulation frequency <strong>an</strong>d amplitude <br />
fixed.
RAPID COMMUNICATIONS<br />
R2650 FISCHER, MADISON, NIU, AND RAIZEN<br />
PRA 58<br />
FIG. 3. The solid squares show the measured <strong>Rabi</strong> frequency<br />
versus the modulation amplitude for a well depth <strong>of</strong> V 0 /h<br />
71 kHz <strong>an</strong>d a drive frequency <strong>of</strong> 70 kHz. Uncerta<strong>in</strong>ties <strong>in</strong> the<br />
least-squares fit <strong>of</strong> the frequency are <strong>in</strong>dicated as error bars. The<br />
l<strong>in</strong>e depicts the result <strong>of</strong> a l<strong>in</strong>ear least-squares fit through the experimental<br />
data. The hollow dots are the calculated <strong>Rabi</strong> frequencies<br />
for a drive at the b<strong>an</strong>d edge, correspond<strong>in</strong>g to the experimental<br />
parameters.<br />
Figure 2 compares the evolution <strong>of</strong> the first b<strong>an</strong>d survival<br />
probability for <strong>in</strong>creas<strong>in</strong>g modulation amplitude. The data<br />
were recorded for V 0 /h71 kHz <strong>an</strong>d 70 kHz, which<br />
corresponds to a drive reson<strong>an</strong>t with states near the b<strong>an</strong>d<br />
edge as <strong>in</strong>dicated by arrow a <strong>in</strong> Fig. 1. All graphs clearly<br />
show damped <strong>Rabi</strong> <strong>oscillations</strong> <strong>of</strong> the population <strong>in</strong> the first<br />
b<strong>an</strong>d. The damp<strong>in</strong>g c<strong>an</strong> be expla<strong>in</strong>ed by tak<strong>in</strong>g <strong>of</strong>f-reson<strong>an</strong>t<br />
tr<strong>an</strong>sitions <strong>in</strong>to account. Atoms with a quasimomentum close<br />
to the value for which the reson<strong>an</strong>ce occurs c<strong>an</strong> undergo<br />
<strong>Rabi</strong> <strong>oscillations</strong> with different frequencies <strong>an</strong>d different amplitudes<br />
10. Summ<strong>in</strong>g over the distribution <strong>of</strong> quasimomenta<br />
leads to a dephas<strong>in</strong>g <strong>of</strong> the <strong>oscillations</strong> <strong>an</strong>d, therefore,<br />
to a decrease <strong>in</strong> the average oscillation. It is import<strong>an</strong>t to<br />
note that this damp<strong>in</strong>g effect is not caused by level decay,<br />
s<strong>in</strong>ce the <strong>Bloch</strong> states <strong>in</strong>volved are stable. The plots <strong>of</strong> the<br />
survival probability <strong>in</strong> the first b<strong>an</strong>d <strong>in</strong> Fig. 2 show <strong>an</strong> overall<br />
<strong>of</strong>fset from unity at zero modulation duration. We attribute<br />
this to the residual phase modulation <strong>of</strong> our st<strong>an</strong>d<strong>in</strong>g wave,<br />
caused by <strong>in</strong>complete ext<strong>in</strong>ction <strong>of</strong> the modulation signal,<br />
which drives tr<strong>an</strong>sitions to higher <strong>b<strong>an</strong>ds</strong>. This <strong>in</strong>troduces a<br />
const<strong>an</strong>t loss <strong>in</strong>dependent <strong>of</strong> the chosen modulation duration<br />
<strong>an</strong>d does not affect the curve shape. Because the strength <strong>of</strong><br />
the residual modulation depends on the set modulation amplitude<br />
, the curve <strong>of</strong>fset ch<strong>an</strong>ges with <strong>in</strong>creas<strong>in</strong>g .<br />
As is evident from Fig. 2, the frequency <strong>of</strong> the <strong>Rabi</strong> oscillation<br />
<strong>in</strong>creases with the modulation amplitude. By fitt<strong>in</strong>g<br />
<strong>an</strong> exponentially damped cos<strong>in</strong>e function to the experimental<br />
data, the value for the oscillation frequency c<strong>an</strong> be extracted.<br />
The solid squares <strong>in</strong> Fig. 3 show the result <strong>of</strong> the leastsquares<br />
fits. The error bars denote the uncerta<strong>in</strong>ty <strong>in</strong> the frequency<br />
fitt<strong>in</strong>g parameter. The plot <strong>in</strong> Fig. 3 shows a <strong>Rabi</strong><br />
frequency that varies l<strong>in</strong>early with modulation amplitude .<br />
From the <strong>an</strong>alytic solution <strong>of</strong> a driven two-level system one<br />
expects this l<strong>in</strong>ear relationship for the case <strong>of</strong> exact reson<strong>an</strong>ce.<br />
To compare this solution to the experimental data we<br />
calculated the <strong>Rabi</strong> oscillation frequency for a modulation<br />
driv<strong>in</strong>g tr<strong>an</strong>sition only at the b<strong>an</strong>d edge. Accord<strong>in</strong>g to firstorder<br />
perturbation theory, the expression for the reson<strong>an</strong>t<br />
<strong>Rabi</strong> frequency is 1/h 1V 0 s<strong>in</strong>(2k Lx) 2, where<br />
1, 2 are the unperturbed <strong>Bloch</strong> states <strong>in</strong> the first <strong>an</strong>d<br />
FIG. 4. Measured survival probability <strong>in</strong> the lowest b<strong>an</strong>d as a<br />
function <strong>of</strong> the modulation duration for three different frequencies.<br />
All <strong>of</strong> the data were recorded at a well depth <strong>of</strong> V 0 /h71 kHz <strong>an</strong>d<br />
a modulation amplitude <strong>of</strong> 0.3. The po<strong>in</strong>ts are connected by<br />
solid l<strong>in</strong>es for clarity. The <strong>in</strong>sets show the results <strong>of</strong> numerical<br />
simulations at V 0 /h72 kHz <strong>an</strong>d 0.3. Note that the modulation<br />
duration does not <strong>in</strong>clude the 16-s turn-on <strong>an</strong>d turn-<strong>of</strong>f times. The<br />
modulation frequency was set to be a 70 kHz, b <br />
75 kHz, <strong>an</strong>d c 85 kHz, correspond<strong>in</strong>g to drives that are <strong>in</strong>dicated<br />
by the arrows <strong>in</strong> Fig. 1.<br />
second <strong>b<strong>an</strong>ds</strong> respectively. The result<strong>in</strong>g frequencies are displayed<br />
as hollow dots <strong>in</strong> Fig. 3, <strong>in</strong> which a slight detun<strong>in</strong>g <strong>of</strong><br />
the drive from the tr<strong>an</strong>sition 1↔2 has been taken <strong>in</strong>to account.<br />
The calculation results are <strong>in</strong> good overall agreement<br />
with the experimental data. The slight deviation <strong>in</strong> frequency<br />
<strong>of</strong> the measured versus the calculated data c<strong>an</strong> be attributed<br />
to the experimental uncerta<strong>in</strong>ty <strong>in</strong> the determ<strong>in</strong>ation <strong>of</strong> the<br />
well depth V 0 . We c<strong>an</strong> measure the <strong>in</strong>tensity <strong>of</strong> the <strong>in</strong>teraction<br />
beams to with<strong>in</strong> 10%, which leads to the same relative<br />
error <strong>in</strong> the value for the well depth. To check the validity<br />
<strong>of</strong> restrict<strong>in</strong>g the r<strong>an</strong>ge <strong>of</strong> quasimomenta to the b<strong>an</strong>d<br />
edge, we performed a numerical <strong>in</strong>tegration <strong>of</strong> Schröd<strong>in</strong>ger’s<br />
equation, <strong>in</strong>clud<strong>in</strong>g the full potential <strong>in</strong> Eq. 1, with <strong>an</strong> <strong>in</strong>itial<br />
condition that was taken to be a uniform distribution <strong>of</strong><br />
atoms <strong>in</strong> the first b<strong>an</strong>d. The frequencies <strong>of</strong> the result<strong>in</strong>g<br />
population <strong>oscillations</strong> are not plotted <strong>in</strong> Fig. 3 because, on<br />
the scale used, they are <strong>in</strong>dist<strong>in</strong>guishable from the calculated<br />
values. The calculation <strong>an</strong>d the numerical <strong>in</strong>tegrations were<br />
performed us<strong>in</strong>g experimental values, for the well depth <strong>an</strong>d
drive frequency, with no adjustable parameters.<br />
Driv<strong>in</strong>g tr<strong>an</strong>sitions at the b<strong>an</strong>d edge has several adv<strong>an</strong>tages,<br />
one <strong>of</strong> which is the high density <strong>of</strong> states <strong>in</strong> that region.<br />
This results <strong>in</strong> a large number <strong>of</strong> atoms that c<strong>an</strong> participate<br />
<strong>in</strong> the population tr<strong>an</strong>sfer, therefore yield<strong>in</strong>g a large<br />
detection signal. Another adv<strong>an</strong>tage is the slow damp<strong>in</strong>g rate<br />
<strong>of</strong> the <strong>Rabi</strong> <strong>oscillations</strong>. S<strong>in</strong>ce there is a large fraction <strong>of</strong><br />
atoms contribut<strong>in</strong>g to the oscillation with the same frequency,<br />
the small number <strong>of</strong> <strong>of</strong>f-reson<strong>an</strong>tly driven atoms will<br />
not signific<strong>an</strong>tly decrease the amplitude <strong>of</strong> the averaged oscillation.<br />
Away from the b<strong>an</strong>d edge, however, the relative<br />
weight <strong>of</strong> the reson<strong>an</strong>t oscillation frequency becomes less<br />
dom<strong>in</strong><strong>an</strong>t <strong>an</strong>d the <strong>of</strong>f-reson<strong>an</strong>t drives lead to <strong>an</strong> <strong>in</strong>creased<br />
damp<strong>in</strong>g rate. The evolution <strong>of</strong> the first b<strong>an</strong>d population for<br />
three different drive frequencies is depicted <strong>in</strong> Fig. 4. The<br />
data were recorded at a well depth <strong>of</strong> V 0 /h71 kHz <strong>an</strong>d a<br />
modulation amplitude <strong>of</strong> 0.3. The <strong>in</strong>sets show the result<br />
<strong>of</strong> the numerical <strong>in</strong>tegrations, for which the well depth was<br />
adjusted to V 0 /h72 kHz <strong>in</strong> order to produce match<strong>in</strong>g<br />
damp<strong>in</strong>g rates. The modulation frequencies <strong>in</strong> Figs. 4a<br />
1 F. <strong>Bloch</strong>, Z. Phys. 52, 555 1929; C. Zener, Proc. R. Soc.<br />
London, Ser. A 145, 523 1934.<br />
2 M. G. Raizen, C. Salomon, <strong>an</strong>d Q. Niu, Phys. Today 50 7, 30<br />
1997.<br />
3 X.-G. Zhao, G. A. Georgakis, <strong>an</strong>d Q. Niu, Phys. Rev. B 54,<br />
R5235 1996.<br />
4 R. Graham, M. Schlautm<strong>an</strong>n, <strong>an</strong>d P. Zoller, Phys. Rev. A 45,<br />
R19 1992.<br />
5 Assum<strong>in</strong>g that all the atoms are <strong>in</strong>itially <strong>in</strong> the first b<strong>an</strong>d, the<br />
reson<strong>an</strong>t coupl<strong>in</strong>g for arrow a <strong>between</strong> the second <strong>an</strong>d third<br />
<strong>b<strong>an</strong>ds</strong> for a different quasimomentum c<strong>an</strong> be neglected, s<strong>in</strong>ce<br />
those states are not populated.<br />
RAPID COMMUNICATIONS<br />
PRA 58 OBSERVATION OF RABI OSCILLATIONS BETWEEN ...<br />
R2651<br />
through 4c match the correspond<strong>in</strong>g arrows <strong>in</strong> Fig. 1 a<br />
70 kHz, b 75 kHz, <strong>an</strong>d c 85 kHz. Larger damp<strong>in</strong>g rates<br />
for <strong>in</strong>creas<strong>in</strong>g modulation frequencies are clearly visible. In<br />
addition, we observed a decrease <strong>in</strong> oscillation amplitudes<br />
due to a smaller density <strong>of</strong> states at the center <strong>of</strong> the b<strong>an</strong>d. At<br />
the opposite side <strong>of</strong> the energy b<strong>an</strong>d lower damp<strong>in</strong>g rates are<br />
recovered. Modulat<strong>in</strong>g the potential with frequencies beyond<br />
the b<strong>an</strong>d edges leads to <strong>Rabi</strong> <strong>oscillations</strong> with a higher frequency<br />
<strong>an</strong>d a lower amplitude, as expected for <strong>of</strong>f-reson<strong>an</strong>tly<br />
driven systems not shown.<br />
In summary, we have observed <strong>Rabi</strong> <strong>oscillations</strong> <strong>of</strong><br />
atomic population <strong>between</strong> <strong>Bloch</strong> <strong>b<strong>an</strong>ds</strong> <strong>in</strong> <strong>an</strong> <strong>optical</strong> potential.<br />
Future directions <strong>in</strong>clude spectroscopic studies <strong>of</strong> the<br />
b<strong>an</strong>d structure modified by external fields, <strong>an</strong>d the preparation<br />
<strong>of</strong> novel motional states <strong>in</strong> <strong>Bloch</strong> <strong>b<strong>an</strong>ds</strong>.<br />
We th<strong>an</strong>k Roberto Diener for help with the calculations<br />
<strong>an</strong>d numerical simulations. This work was supported by the<br />
Robert A. Welch Foundation <strong>an</strong>d the National Science Foundation.<br />
6 S. R. Wilk<strong>in</strong>son, C. F. Bharucha, K. W. Madison, Q. Niu, <strong>an</strong>d<br />
M. G. Raizen, Phys. Rev. Lett. 76, 4512 1996.<br />
7 C. F. Bharucha, K. W. Madison, P. R. Morrow, S. R. Wilk<strong>in</strong>son,<br />
B. Sundaram, <strong>an</strong>d M. G. Raizen, Phys. Rev. A 55, R857<br />
1997; S. R. Wilk<strong>in</strong>son, C. F. Bharucha, M. C. Fischer, K. W.<br />
Madison, P. R. Morrow, Q. Niu, B. Sundaram, <strong>an</strong>d M. G.<br />
Raizen, Nature London 387, 575 1997.<br />
8 For a recent review, see S. Chu, Science 253, 861 1991.<br />
9 Q. Niu, X.-G. Zhao, G. A. Georgakis, <strong>an</strong>d M. G. Raizen, Phys.<br />
Rev. Lett. 76, 4504 1996.<br />
10 The <strong>Rabi</strong> frequency also depends on the tr<strong>an</strong>sition matrix element<br />
for the given quasimomentum. However, for the tr<strong>an</strong>sition<br />
1↔2, the matrix elements are const<strong>an</strong>t with<strong>in</strong> about 10%.