Table Of Content’Shut down’ of an atom laser
N.P. Robins, A. K. Morrison, J.J. Hope, and J. D. Close
Australian Centre for Quantum Atom Optics, The Australian National University, Canberra, 0200, Australia.∗
We present experimental results demonstrating that the flux of a continuous atom laser beam
cannot be arbitrarily increased by increasing the strength of output-coupling. We find that a state
5 changing output-coupler for a continuous atom laser switches off as the output-coupling strength,
0 parameterizedbytheRabifrequency,isincreasedaboveacriticalvalueoffourtimestheradialtrap-
0 ping frequency. In addition, we find that the continuous atom laser has large classical fluctuations
2 when the Rabi frequency is greater than the radial trapping frequency, due to the output-coupling
populatingallaccessibleZeemanstates. Thenumericalsolutionofaone-dimensional5-stateGross-
n
Pitaevskii model for theatom laser is in good qualitative agreement with theexperiment.
a
J
PACSnumbers: 03.75.Pp,03.75.Mn
1
3
In precision measurement applications, atom lasers beam in a series of experiments with a 3 ms continu-
]
r have the potential to outperform optical lasers and non- ous atom laser, produced in the F=2 manifold of 87Rb
e
opticaltechniquesbymanyordersofmagnitudeprovided bystatechangingoutputcouplingtoamagneticfieldin-
h
t we canincreasetheir flux,andachieveshotnoise limited sensitivestate,mF =0. Atlowoutput-couplingstrength
o
operationatleastinsomefrequencyband. Anoutstand- the atom laser beam flux increases gradually and homo-
.
t ing goal in atom optics is to produce a high flux, truly geneously until Ω 1 kHz, where Ω = gFµBB is the
a ≈ 2h¯
m continuous atom laser. In this Letter, we make some bare Rabi frequency. At around this value we observe
- early steps along this path and investigate the crossover that the anti-trapped mF =−2,−1 states begin to play
d from weak to strong output-coupling in a continuous apartintheatomlaserdynamics. Thisleadsto increas-
n atom laser based on a radio-frequency (RF) mechanism. ingly severe fluctuations in the density of the atom laser
o
In this work, we define continuous as extended outcou- beam,andalossofatomstotheanti-trappedstates(Fig
c
[ plingwherethewidthinenergyoftheoutcouplingregion 1(a),(b)). A further increase in the coupling strengthin-
is determined by power broadening, rather than pulsed crementally shuts down the output starting around Ω
1 ≈
operation, where it is the duration of the pulse that de- 4 kHz . In the limit of large output coupling strength
v
terminesthe energywidthofthe outcouplingregion. We (to be defined later) we find approximately 70% of the
7
4 donotreplenishthe atomsinthecondensateduringout- atomsremainlocalizedinthecondensate,whiletheother
7 coupling in this work. 30%areemittedshortlyafterthebeginningoftheoutput
1 To produce an atom laser, a Bose-Einstein conden- coupling period.
0
sate (BEC) is used as a quantum degenerate reservoir
5
0 of atoms, from which a coherent output coupling mech-
/ anism convertsatoms from trapped to untrapped states.
t
a Either due to a directed momentum kick [1] or gravi-
m
tational acceleration [2, 3, 4] the output coupled atoms
- form a quasi-collimated beam, with the divergence de-
d
termined by the repulsive interactions between the con-
n
o densate and atomic beam [4]. Our previous experiments
c on a pulsed output-coupler suggested that a continuous
: atom laser would have a stringent limit on peak homo-
v
i geneous flux [5]. Here we show that peak flux into the
X
magneticfieldinsensitive state isindeedsignificantlybe-
r low that which can be provided by the finite reserviorof
a
BEC atoms that we produce. This ’homogeneous flux’
limit is imposed by the interaction between multiple in-
ternalZeemanstatesofthemagneticallyconfinedatoms. FIG. 1: Optical depth plot (35 independent experiments)
showing thespatial structure of a 3 ms atom laser as a func-
Furthermore, we find that a previously predicted effect
tion of output-coupling strength. The system was left to
knownasthe’boundstate’ofanatomlaser[6,7,8]effec-
evolve for a further 4 ms before the trap was switched off
tivelyshutsoffstatechangingoutput-couplingandhence
and 2 ms later the images were acquired. The field of view
the atom laser beam. for each individual image is 0.7×0.3 mm. The final image
The experimental data presented in Figure 1 encapsu- is taken with Ω=16 kHz. Anti-trapped Zeeman states are
late many of the results described herein. The figure clearly visible in (a), (b), and (c) which show an extended
view of the data in themain part figure, (2.7×0.3 mm).
shows the densities of the condensate and atom laser
2
The rest of this paper is organized into three parts.
First, we describe the details and relevant time-scales of
our experiment. In the second part we examine, exper-
imentally, the regime of weak to intermediate output-
coupling. In the third part of the paper we demonstrate
the ’shut down’ or bound state of the atom laser.
Inourexperiment, we produceanF=2,mF =2 87Rb
condensate, consisting of approximately 105 atoms, via
evaporation in a water-cooled QUIC magnetic trap [9]
with a radial trapping frequency of ν⊥ = 260 Hz and
an axial trapping frequency ν = 20 Hz. We operate
z
our trap at a bias field of only B0 = 0.25 G. The trap FIG.3: AtomnumberasafunctionofangularRabifrequency
runs at 12 A (12 V) generated from a low noise power fora10msatomlaser. TheupperplotshowstherawOptical
supply. Thelowpowerdissipationsuppressesheatingre- depth data averaged over 6 identical experiments. The field
lateddriftofthemagnetictrapbiasfield,allowingprecise of view for each image is 1.5mm by 0.5mm.
addressingofthe condensatewithresonantRFradiation
(wemeasureadriftofsignificantlylessthan0.7mGover
8 hours). Unless otherwise stated, in all experiments de- is due to the close proximity of the quadrupole trapping
scribedinthispapertheRFcutisplaced2kHzfromthe coilonwhichthe RFcoilismounted. Usingthis method
upper edge of the condensate. After 50 s of evaporative we estimate the uncertainty in the Rabi frequency to be
cooling, the BEC is left to equilibriate for 100 ms. An around3-5%. The quadraticcontributionof the Zeeman
RF signal generator, set in gated burst mode, produces effect creates resonanceshifts of approximately0.02%at
a weak 3-200 ms pulse which is radiated perpendicular 1 G, and hence our system is modeled accurately by the
to the magnetic bias field of the trap through a 22 mm 5-state GP equation.
radius single wire loop, approximately 18 mm from the To put this work into context, it is critical to under-
BEC. The peak RF magnetic field magnitude, B, and standthetimeandlengthscalesrelevanttotheoperation
hence thebareRabifrequency,Ω, iscalibratedbyfitting ofastatechangingoutput-coupler. IntheThomas-Fermi
thefrequencyaxisofa50µspulsedexperimentwithanu- limit the addressable frequency width of a condensate in
mericalsimulationofpulsedoutputcouplingutilizingthe the F = 2,mF = 2 state is given by W = h¯ωg⊥√2Mµ,
3D Gross-Pitaevskii theory of the atom laser described whereµisthechemicalpotential,Mistheatomicmass,g
in our previous paper [5]. The results of such a calibra- isgravityandω⊥ istheradialtrappingfrequency[3](For
tion are shown in Figure 2. We find that the output our condensates W = 2π 5 kHz). For the continuous
×
coupling magnetic field seen by the atoms is reduced by atom laser the resonance width of the output coupling
around a factor of 5 compared to a simple calculation is determined by the Rabi frequency, Ω, due to power
of the field radiated from a single loop or a free space broadening. A number of definitions exist in the litera-
measurement of the field from the coil. This reduction tureforwhatconstitutesweakoutput-coupling,allbased
onacomparisonofthecouplingstrength,Ωwithvarious
parameters. All require that the resonance width of the
output coupling is significantly less than the frequency
width of a condensate. The most stringent of these def-
initions requires Ω ω⊥ 1.6kHz [10, 11]. A more
≪ ∼
relaxed condition states that Ω 1.6pg/R 3.2kHz,
≪ ∼
where R is the spatial Thomas-Fermi width [12]. The
least stringent condition is Ω 1/τ 12kHz, where τ
≪ ∼
is the time it takes atoms to leave the output coupling
resonance[13]. Strongoutputcouplingcanbeconsidered
to occur when the weak output-coupling conditions are
not met.
In Figure 3 we display experimental data taken under
a similar set of parameters to Bloch et. al [3]. A 10
ms atomlaserbeamis producedandallowedto separate
fromthe condensatepriortoimaging. Thedeclineofthe
FIG.2: OutcoupledfractionintheZeemanstatesoftheF=2
single pulse (50µs) atom laser as a function of angular Rabi trapped condensate atom number clearly follows a sim-
frequency. Solid lines: theoretical calculation with no free ple monotonic decay as found by the previous work [3].
parameters from a 3D GP model of the 5-state F=2 atom However,fromthisdataitcanbeseenthatthedeclineof
laser. the population in the trapped states does not lead to an
3
increaseinthepopulationofthem =0statethatforms
F
the atom laser beam. Our conclusionis that the missing
atoms have been expelled into the anti-trapped states.
We find the same behavior for all output-coupling times
that produce a fully observable atom laser beam (up to
around 25 ms for our imaging field of view). For shorter
output-coupling times, suchas presentedin Figure 1, we
areabletoobservetheanti-trappedstates,whichappear
above an angular Rabi frequency of Ω 800 Hz. For
≈
all our data, we find that the peak output flux occurs
at Ω 1 kHz, and that this value corresponds to the
≈
most homogeneous atom laser beam. As Ω is increased
above1kHztheatomlaserbeamdevelopssteadilywors-
ening density fluctuations, the most prominent feature
of which is a peak in the leading edge of the beam. One
way around this type of flux limiting dynamics is to re-
duce the atom laser system to only two levels: trapped
and un-trapped. Such a system has been demonstrated
byAspectsgroup,utilizingthenonlinearZeemanshiftin
FIG. 5: Theoretical results obtained by solution of the 1D-
the F=1manifoldtocreateaneffective twolevelsystem.
GP equation. Atomic population in the Zeeman states as a
ARamantransitiondirectlyfromtrappedtoun-trapped function of time for a 15 ms atom laser for (a) Ω = 800 Hz
stateswithatwophotonmomentumtransfershouldalso and (b) Ω = 16 kHz. In (c) we show the behaviour of a
increase the homogeneous flux limit as it removes atoms ’pure’twostate atom laser systemfor Ω=16 kHz. (d)Total
more quickly from the output-coupling region. However, atomic population remaining in the magnetic trap after 100
in what follows we show that for all continuous state ms of output coupling plotted as a function of angular Rabi
frequency.
changing output-coupling, including two state systems,
output flux is limited by the underlying physics of the
process.
In Figure 4. we display a measurement of the num- detectionsensitivity. Thecondensateisprogressivelyde-
berofatomsremaininginthe trappedstatesasthe Rabi pleted at increasing, but weak, Rabi frequencies (Figure
frequency is scanned from the weak to strong output- 4,inset)untilweobservenoatomsremaininginthemag-
couplingregimefora100mscontinuousatomlaser(open netic trap. At Ω 4 kHz atoms start to reappear, and
∼
circles in the figure). Note that for this output coupling by Ω 9 kHz the number of atoms remaining in the
∼
time the peak flux in the atom laser beam is below our trap stabilizes to 70% of the initial condensate number.
As described earlier, the other 30% are expelled at the
initial switch on of the output-coupling. Our method of
collectingdataensuresthattherearenosystematicshifts
inthetrapbiasfield. Every10runsoftheexperimentwe
switchtoa10msatomlasertoensureconstantdetuning,
aswellascomparingdetaileddatatoa’coarse’settaken
at the beginning of each run. We emphasize here that
the Rabi frequencies over which this effect occurs corre-
spondtothe cross-overfromthe weaktostrongcoupling
regime. InordertomeasureatomsremainingintheBEC
after 100 ms of output-coupling, the combination of the
magnetictrapandRFfieldinthecross-overregimemust
be creating a trap for the atoms. Equivalently, the atom
laser ceases to operate, or ’shuts down’ in the interme-
diate to strong coupling regime. We have verified that
the ’shut down’ edge is independent of output coupling
duration,as shownin Figure 4 with a 200ms atomlaser
(black dots). For shorter coupling times the condensate
is not fully depleted before this edge is reached (see for
FIG. 4: Trapped fraction as a function of angular Rabi fre- example Figure 1).
quencyfora100msatomlaser(opencircles). Datafora200 Anumberoftheoreticalworkshavesuggestedthatthe
ms output-coupleris shown for comparison (black dots).
atomlasermay havea ’bound’ eigenstate[6, 7, 8], based
4
purelyonthe existence ofcoupling betweena singletrap ejectedatomssimplycorrespondtotheun(anti)-trapped
mode and a continuum of un-trapped states. Further- dressed states. 5(c) shows the same trapping effect, but
more, in the context of producing a two dimensional this time in a simple two-level atom laser system, indi-
BEC,ithasbeenshownrecentlythattrappingofallm cating that the bound mode is inherent to state chang-
F
statesisanaturalconsequenceofcombiningRFcoupling ing output-coupling regardless of the number of states
with a DC magnetic trap [14]. This trapping can be un- involved. Finally, 5(d) shows a prediction of the number
derstoodbyconsideringthe’dressedstate’basisinwhich of atoms remaining in the trap after 100 ms of output
theRFcouplingandDCpotentialsseenbytheatomsare coupling as a function of Rabi frequency. The behaviour
diagonalised. Inthisbasisthedressedeigenstatesarelin- qualitatively matches that of the experiment, indicating
ear combinations of the bare Zeeman states, trapped in that we are indeed observing the ’bound state’ of the
effective potentials created by the avoided crossings. In atom laser in our experiments. The predicted number of
the strong coupling limit, for the F=2 atom laser, diag- atoms remaining in the trap lies within the range of the
onalization yields a prediction of between 31.25% up to strong coupling estimate, but still differs substantially
a maximum of 62.5% of the atoms remaining trapped, from our experimental results. We expect that a full 3D
assuming a non-adiabatic projection onto the dressed model would capture the dynamics of our experiment,
states. Our experimental observation of 70% is incon- but it is beyond the scope of the present work.
sistentwiththesevalues,suggestingsomedegreeofadia-
batictransfer. Inordertogainfurtherqualitativeinsight In conclusion, we have shown that the peak homoge-
we numerically solvethe F=2 Gross Pitaevskiimodel [5] neous output flux of an atom laser beam derived from
of the atom laser in one dimension given by a finite BEC is limited by the requirement to operate
well within the weak coupling regime, in order to avoid
iφ˙2 =( +z2+Gz 2∆)φ2+2Ωφ1 pumping to anti-trapped states and the related density
L −
fluctuations. An obvious short term solution to boost
1
iφ˙1 =( + z2+Gz ∆)φ1+2Ωφ2+√6Ωφ0 the flux would be to chirp the RF detuning across the
L 2 −
condensate, however such a scheme would not be appli-
iφ˙0 =( +Gz)φ0+√6Ωφ1+√6Ωφ−1
L cable to a pumped continuous atom laser. Additionally,
iφ˙−1 =( 1z2+Gz+∆)φ−1+2Ωφ−2+√6Ωφ0 we find that one cannot arbitrarily increase the output
L− 2 flux of a state changing atom laser due to the presence
iφ˙−2 =( z2+Gz+2∆)φ−2+2Ωφ−1, ofbound modes. We are currentlyinvestigating momen-
L− (1) tum kicked continuous Raman coupling and condensate
where φ is the GP function for the ith Zeeman state pumping as a means of boosting atom laser flux.
i
and 1 ∂2 + U(Σ2 φ 2). Here ∆ and Ω
L ≡ −2∂x2 i=−2| i| NPR thanks S.A. Haine, C. M. Savage, C. Figl and
are respectively the detuning of the RF field from
E.AOstrovskayaforthe manydiscussionsrelatedto this
the B0 resonance, and the Rabi frequency, measured
work.
in units of the radial trapping frequency ω⊥ (for the
m = 1 state), U is the interaction coefficient and
F
G = mg ( h¯ )1/2 gravity. The wave functions, time,
h¯ω⊥ mω⊥
spatial coordinates, and interaction strengths are mea-
sured in the units of (h¯/mωz)−1/4, ω⊥−1, (h¯/mω⊥)1/2, ∗ Electronic address: [email protected];
and (h¯ω⊥)−1(h¯/mω⊥)−1/2, respectively. The nonlinear URL:http://www.acqao.org/
interaction strength is derived by requiring that the 1D [1] E. W.Hagley et al.,Science 283, 1706 (1999).
TF chemical potential be equivalent to the 3D case. [2] M.-O. Mewes et al.,Phys.Rev. Lett.78, 582 (1997).
There are no free parameters in this model; we use [3] I. Bloch et al.,Phys. Rev.Lett, 82, 3008 (1999).
−4 [4] Y. Le Coq et al.,Phys. Rev.Lett. 87, 170403 (2001).
U=6.6 10 , G=9.24, Ω=0 13.85, ∆=8.2.
× − [5] N. P. Robinset al., Phys.Rev.A 69, 051602 (2004)
Figure5(a)showsthetimeevolutionofthepopulation [6] J. J. Hope et al.,Phys. Rev.A 61 , 023603 (2000).
numbers in the trapped, un-trapped and anti-trapped [7] J. Jeffers et al.,Phys.Rev.A 62 , 043602 (2000).
Zeeman states for Ω = 800 Hz. The trapped mF = 2,1 [8] G. M. Moy et al.,Phys. Rev.A 59, 667 (1999).
populationstendtowardszero,essentiallymonotonically. [9] T. Esslinger et al., Phys.Rev.A. 58, 2664 (1998).
In 5(b), corresponding to Ω = 16 kHz, we find an en- [10] H. Steck et al.,Phys. Rev.Lett., 80, 1 (1998).
tirely different behavior. After an initial high frequency [11] J. Schneider and A. Schenzle, Appl. Phys. B 69, 353
(1999).
exchangeofthenumberofatomsinthedifferentZeeman
[12] F. Gerbier et al., Phys. Rev. Lett. 86, 4729-4732 (2001)
states, slightly more than 50% of the atoms are ejected and Erratum Phys.Rev.Lett. 93, 059905 (2004).
from the system. The remaining atoms, which populate [13] R. Graham and D. F. Walls, Phys. Rev. A 60, 1429
all Zeeman states, undergo a lossless and slow exchange (1999).
of population. We identify the trapped atoms as being [14] O.ZobayandB.M.Garraway,Phys.Rev.A69,023605
in a superpositionof the dressedstate eigenvectors. The (2004).
