Table Of ContentResonant control of spin dynamics in ultracold quantum gases by microwave dressing
Fabrice Gerbier,∗ Artur Widera, Simon F¨olling, Olaf Mandel, and Immanuel Bloch
Institut fu¨r Physik, Johannes Gutenberg-Universit¨at, 55099 Mainz, Germany.
(Dated: February 6, 2008)
We study experimentally interaction-driven spin oscillations in optical lattices in thepresence of
an off-resonant microwave field. We show that the energy shift induced by this microwave field
6 can be used to control the spin oscillations by tuning the system either into resonance to achieve
0 near-unity contrast or far away from resonance to suppress the oscillations. Finally, we propose a
0 schemebased on thistechniqueto createa flat sample with eithersingly- or doubly-occupiedsites,
2 starting from an inhomogeneous Mott insulator, where singly- and doubly-occupied sites coexist.
n
PACSnumbers: 03.75.Mn,03.75.Lm,03.75.Qk
a
J
8
The fact that coupling an atomic system to a strong m=-1 m=0 m=+1
] radiation field can modify its level structure is well- a b
er known,leadingtosucheffectsastheAutler-Townesdou- F=2
h blet or the Bloch-Siegert shift (see, e.g., [1]). Today, (1)
z
t this phenomenon is often explained in a dressed-atom H
o G
t. picture, where the bare atomic states are replaced by 6.8 wµ
a eigenstates of the total atom-plus-field system. For cold ~ ω (2)
m atoms, this picture helps to understand how magnetic F=1
m=-1
- trapping potentials can be engineered with off-resonant m=0 m=+1 (3)
d
radio-frequency or microwave radiation [2], allowing for
n
example to create nearly two-dimensional [3] or double-
o
c well trapping potentials [4, 5].
[ Here, we show how an off-resonant microwave field FIG. 1: Anoff-resonant microwave field with approximately
1 (hereafter called dressing field) can be used for coher- π-polarizationandfrequencyωµw ((a))isusedtocontrolthe
detuningforcoherentspinoscillationsinopticallattices. Case
v ent control of the dynamics of a spinor quantum gas.
(b1) corresponds to zero or weak microwave intensity. With
1 Ultracold gases with internal (spin) degrees of freedom
properly chosen microwave parameters, one can compensate
5
[6] are novel quantum systems where atom-atom inter- thisdetuningandachievefullresonance(b2)orovercompen-
1
actions play a key role. In an optically trapped gas of sate to suppress thespin-changing collisions (b3).)
1
0 spin-f = 1 atoms, spin-dependent interactions can pro-
6 mote anatompair fromaninitial two-atomstate ,where
0 both atoms are initially in the m=0 magnetic sublevel, the topic of spin oscillations in optical lattices. Control-
/
t to a final state with one atom in m = +1 and the other ling the spin oscillations enables one to use them as a
a
inm= 1[7,8,9,10,11]. Thecoherenceofthisprocess tool for quantum state detection, preparation or manip-
m −
was demonstrated in [12, 13] for atom pairs in optical ulation. Ina separatepaper,we have describedhow this
-
d lattices and in [14, 15] for a seeded Bose-Einstein con- technique can be used to probe atom number statistics
n densate. In the former case, the dynamics is controlled across the superfluid to Mott insulator (MI) transition
o by the energy mismatch, or detuning, between the ini- [16]. We propose here to use it in order to create an
c
tial and final spin states of the colliding atom pair. This homogeneous MI with singly-occupied sites out of a in-
:
v detuningoriginatesnotonlyfromdifferentquadraticZee- homogeneous system where singly- and doubly-occupied
Xi manshiftsintheinitialorfinaltwo-atomstates,butalso sites coexist.
fromdifferentinteractionenergies. Ifpositive,asin87Rb In our case, the microwave is relatively weak, so that
r
a [10,13,15],this“residual”interactiondetuningprevents the dressed states almost coincide with the bare atomic
to reach the resonance for spin oscillations by magnetic states and the level shifts can be calculated perturba-
means. tively [1]. We consider a 87Rb atom in a constant and
Inthispaper,weshowhowtotunethespinoscillations homogeneous magnetic field B, plus an arbitrary po-
inandoutofresonanceusingtheenergyshiftinducedby larized microwave field with amplitudes B(µw) , with
q q
an off-resonant dressing field (see Fig.1). We apply this q = 0, 1 for π and σ± polarizations, re{spectiv}ely. Far
±
technique to an ensemble of atom pairs in a deep optical from any hyperfine resonance, we find the level shift for
lattice and initially polarized in the m=0 sublevel. We the state F =1,m as
| i
showthattheconversiontoanensembleofm= 1pairs
canbe achievedwith nearunit efficiency, evenin±a finite ~Ω2 µ B
∆E πf x= B . (1)
magneticfield. Thesignificanceofthisworkgoesbeyond m ≈ 4∆ m(cid:18) ~∆ (cid:19)
2
1 . After these pulses, the magnetic field is eventu-
y − i
sit0,15 ally ramped down to a final value (typically B = 0.4G
n
a bde0,10 for the experiments described here). Subsequently, colli-
al0,05 sional spin oscillations take place at each individual lat-
c
pti0,00 tice site where an atom pair is present. As discussed in
O
-600 -200 0 200 600 [12,13],theatompairpreparedwithbothatomsinitially
Position(µm)
in m = 0 behaves as a two-level system, and undergoes
Rabi-likeoscillationsat aneffective oscillationfrequency
FIG. 2: (a) Absorption image of an expanded cloud after
tosc = 0 ms (upper image) and tosc = 15 ms (lower image).
Thedashedlineindicatestheboxused tofitthem=0com- ~Ωeff = (~Ω0)2+(∆ǫ+Us)2. (3)
ponent. (b) Horizontal cut through the lower image in (a) p
(black line). Subtracting a Gaussian fit (red dashed line) to Here the coupling strength ~Ω0 = 2√2Us and the in-
the m=0 component yields a new image (blue line) used to teraction part of the detuning are both fixed by a spin-
count separately thepopulation in m=±1by integration. dependent interaction energy U = U˜(a a )/3, with
s 2 0
U˜ an on-site interaction energy per Bo−hr radius a
B
[13] and a the scattering length characterizing scatter-
In Eq. (1), ∆ = ω ω denotes the microwave de- F
µw − hf ing in the total spin-F channel [6, 12]. The quantity
tuning,whereω isthemicrowavefrequencyandwhere
ωhf isthehyperfiµwnefrequencysplittinginthe5S1/2 man- ∆enǫce=inǫ+Z1ee+maǫ−n1e−ner2gǫi0es∝ǫ Bb2ectowrereensptohnedisnittoialthaenddififfnear-l
ifold of Rubidium 87. We have introduced the Rabi m
states. Without dressing field, ∆ǫ = (µ B)2/2~ω is
frequency Ω = µ B(µw)/~ for the π-polarized transi- B hf
π B π determined by the magnetic field. In the presence of the
tion, with µ the Bohr magneton, and assumed that
B dressing field, the Zeeman sublevels shift according to
x=µ B/~∆ 1. The function
B ≪ Eq. (1), so that one obtains
3C I 1
fm(x)=Xq 4m,q Iπq 1−(m+ q2)x, (2) ∆ǫ+Us =~×α(Ω2π −Ω2res), (4)
accounts for different microwave polarizations and the whereα=(f+1(x)+f−1(x)−2f0(x))/4∆andwherethe
resonance Rabi frequency Ω solves
multi-level structure. Here C = F = 1,m;1,q F = res
m,q
|h |
2,m + q 2 are square moduli of Clebsch-Gordan coef- (µ B)2
cfircoiewnatvse, cia|onmdpIoqneisntt.hWe einctaenlisbirtayteofthtehemiqc−ropwoalavreizinedtenmsii-- ~αΩ2res+ 2~Bωhf +Us =0. (5)
tiesforeachpolarizationq bydrivingfast,single-particle When normalized to the total atom number, the oscilla-
Rabi flopping between the hyperfine states F = 1,m = tion amplitude for the lattice as a whole is reduced by a
|
1 and F = 2,m = 1+q . From this, we obtain factor n , corresponding to the fraction of atom pairs in
− i | − i 2
Iσ−/Iπ ≈ 0.33 and Iσ+/Iπ ≈ 0.02 for our experimental the lattice [12, 13]. Forourtrappingparametersandour
parameters. The peak Rabi frequency observed in our atom number, approximately half of the atoms are in a
experiment is Ωπ =2π×82(1) kHz. central region with two atoms per site (n2 ≈0.5) [12].
To observe the microwave-induced shift, a MI of We have investigated the resonantly dressed spin dy-
around 2 105 Rb atoms is first created in the F = namics at a magnetic field of 0.4 G, by varying the mi-
× |
1,m= 1 Zeemansublevel. This MI is heldina purely crowave power (or equivalently the dressing frequency),
− i
optical lattice potential at a magnetic field of 1.2 G. In andrecordingseveralperiodsofspinoscillationsforeach
this work, we use a typical lattice depth of 40Er, with power. The microwavefieldwas detuned by severalhun-
Er = h2/2Mλ2 and λ = 840nm the lattice laser wave- dred MHz to the red of the hyperfine resonances to sup-
length. Afteranexperimentalphasedescribedbelowand press population transfer to F = 2. Three representa-
a 12-ms-long free expansion including a magnetic gradi- tive examples are shown in Fig. 3a, corresponding to
ent pulse, we detect the population in each Zeeman sub- the cases illustrated in Fig. 1: the unperturbed case
level as shown in Fig. 2 [12, 13]. We find the population (Ω Ω ,squares),thenear-resonantcase(Ω Ω ,
π res π res
in the m=0 sublevelby fitting a Gaussianfunction to a circle≪s) where the microwave dressing compensa≈tes ex-
restricted area, by substracting it from the data, and by actlythe“bare”detuning,andtheovercompensatedcase
integrating over the cloud areas to find the populations (Ω Ω , triangles), where the dynamics is blocked
π res
in m= 1 (see Fig. 2). due ≫to the large detuning induced by the microwave
±
The main topic of this paper is the study of spin dy- dressing. We have found empirically that the data were
namics in a dressing field. In our experiments, the spin well described by a fitting function of the form
dynamics is initialized as described in [12, 13] by first
transferring the sample to F = 1,m = 0 by apply- N t
| i 0 =A 1 e−γofft +Bcos 2π e−γosct. (6)
ing two microwave π pulses starting from F = 1,m = N − (cid:18) T (cid:19)
− | (cid:0) (cid:1) osc
3
1
0.5
0.8
s
N 0.4 a ne
N)/-1 0.3 2 1 ationsianstat 0.6
+ 0.2 ulm 0.4
1 pe
+ oe
N PZ 0.2
( 0.1
0 3 0
0 100 200 300
Time(ms)
0 10 20 30 40 50 60 70 80
Time(ms)
FIG. 4: Spin oscillations recorded for a magnetic field B =
30
) 1.2G.FilledcirclesdenotethepopulationintheZeemansub-
ms 25 b statem=0,andhollowcirclesthesumofthepopulationsin
2
( substates m=±1.
d 20 3
o
eri 15 1
P
10 expected behavior. The fit parameters include the same
0 5 10 15 20 25 30 35 40 parameters α,Ω as in the previous case, plus an ad-
RabiFrequencyΩπ (kHz) ditional parametreesr n corresponding to the fraction of
2
atoms participating in the oscillation. The values for α
0.4
ne andΩ areconsistentwiththosededucedfromthefitto
od 0.3 c res
cillatimplitu 0.2 itshselimghetalsyusrmedaplleerritohda.nTthheeofrvaecrtailolnamn2pl=itu0d.5eonf2a≈tom0.4p0a(i3rs)
s
Oa 0.1 expectedintheMottphase. Wehaveperformedanother
measurementwhere the atomic sample is held at a mag-
0
0 5 10 15 20 25 30 35 40 netic field of 1.2G using the dressing field on resonance
RabiFrequencyΩπ (kHz) accordingtoEq.(5). Thespinoscillationcurveshownin
Fig. 4 yields an amplitude of n 0.47(2), much closer
2
≈
to the expectation than for the B = 0.4G case. We at-
FIG. 3: Spin oscillations at a magnetic field B = 0.4 G. (a) tribute part of this 15 % reduction in contrast to a
Spin oscillations without microwave dressing (squares), for a finite ramp time of t∼he magnetic field in the latter case,
resonantdressing(circles)andforanovercompensateddress-
inducing a small amount of adiabatic population trans-
ing (triangles). (b) and (c), period and amplitude of the
fer at the beginning of the oscillation [13]. From the
oscillation (circles), together with a fit to the expected Rabi
datashowninFig.4,wehavealsodeterminedthedamp-
dependance.
ing times γ ,γ and the offset value A, as defined in
osc off
Eq. 6. Within our experimental accuracy, we find that
The second term describes the spin oscillations with pe- they are independent of microwave power and magnetic
riod T , including a damping term with rate γ . The field. The average offset value is A 0.6, and the av-
osc osc
first term describes a slowly varying offset approaching erage damping times γ 140s−1 a≈nd γ 160s−1.
osc off
the value A with a time constant γ−1. Although the origin of the≈damping is not enti≈rely clear,
off
The measured period of the oscillation is plotted in the fact that γ γ indicates a correlation between
osc off
≈
Fig. 3b as a function of the Rabi frequency Ω of the the damping of the oscillations and the rise of the offset.
π
dressing field. The period is fitted to the expected Possible damping mechanisms can be either local (i.e.
form of the Rabi frequency given in Eq. (3,4) treat- involving only on-site processes) or non-local (involving
ing U , Ω and α as free parameters. We find good tunneling to a neighboring site, particularly effective if
s res
agreement between the expected 2πα = 0.069s, Ω = atoms are transferredto excited Bloch bands by heating
res
2π 23.4kHzandthe measuredvalues 2πα=0.066(1)s, processes). On-site damping mechanisms could possibly
×
Ω = 2π 22.8(1)kHz. The “bare” Rabi frequency be off-resonant Raman transitions between the m = 1
res
× ±
U /h = 10.5(1)Hz is in excellent agreement with an sublevels induced by the lattice beams. We believe the
s
independent measurement of the spin-dependent colli- limitations of the present experiment to be of technical
sionalcoupling,whichweredonewithoutadressingfield origin. In principle, even longer coherence times could
[12, 13]. be achieved.
In Fig. 3c, we show the oscillation amplitude versus Because of these long coherence times, the technique
microwave Rabi frequency Ω , together with a fit to the presented here could find applications beyond the study
π
4
singly- doubly- tem with only singly-occupied sites in the ground state.
occupied occupied This would help to remove local defects with two atoms
F=2
onasite,thusprovidinganovelcoolingtechniqueunique
F=1
to the MI state.
We acknowledgesupportfromthe DFG,fromAFOSR
and from the EU under the OLAQUI program.
∗
email: [email protected]
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