Quasi-condensate growth on an atom chip

0.44 MB·English

by  N. P. Proukakis

Quasi-condensate Growth on an Atom Chip
N.P. Proukakis1∗, J. Schmiedmayer2 and H.T.C. Stoof1
1Institute for Theoretical Physics, Utrecht University,
Leuvenlaan 4, 3584 CE Utrecht, The Netherlands and
2Physikalisches Institut, Universit¨at Heidelberg, 69120 Heidelberg, Germany
6
0
We discuss and model an experiment to study quasi-condensate growth on an atom chip. In
0
particular, we consider the addition of a deep dimple to the weak harmonic trap confining an
2
ultracoldone-dimensionalatomicBosegas,beloworclosetothecharacteristictemperatureforquasi-

n condensate formation. The subsequent dynamics depends critically on both the initial conditions,
a and the form of the perturbing potential. In general, the dynamics features a combination of
J
shock-wavepropagationinthequasi-condensate,andquasi-condensategrowthfromthesurrounding

0 thermal cloud.
2
PACSnumbers: 03.75.Kk,05.30.Jp,03.75.-b

]
h
c
I. INTRODUCTION the usual condensate growth. The dimensionality of our
e
m envisaged 1D experiment ensures the system remains in
the regime of large phase fluctuations throughout its en-
- The recent achievement of effectively one-dimensional
t tireevolution. Moreover,underconditionsofslowcooling
a ultracold atomic samples on microfabricated surfaces,
fromabovethetransitionpoint,ourresultspointtowards
t known as ‘atom chips’ [1] opens up the way for a num-
s a slow initial linear-like growth which is consistent with
. ber of applications, such as precision interferometric
t bosonicenhancement. However,adetailedcomparisonto
a measurements [2, 3, 4, 5] and quantum computation
theaboveexperimentwouldrequireafull3Dcalculation,
m [6, 7, 8]. One-dimensional (1D) geometries, in which
and therefore lies beyond the scope of this paper.
- the transverseconfinement exceeds allother relevanten-
d ergy scales in the system, offer improved atomic guid- The experimental scenario considered here is the fol-
n lowing: An atom chip is first loaded with a certain
ance and device miniaturization [9, 10]. This comes
o number of atoms, at densities low enough to be in the
at the expense of increased phase fluctuations, which
c 1D regime, and cooled to a prescribed low temperature.
[ tend to destroy the coherence of the ensemble. Var-
Theapproximatelyharmoniclongitudinalconfinementof
ious equilibrium studies have already been performed
2 the atoms is then perturbed by the addition of a deep
both experimentally [11, 12, 13, 14] and theoretically
v dimple microtrap, of variable width, as shown in Fig.
[15, 16, 17, 18, 19, 20, 21, 22, 23, 24] to understand
4
1. Such a technique of local phase-space compression
5 in particular this latter effect. However,a deeper under-
[31, 32] has been used to force a system of 23Na atoms
1 standingofthe intrinsicdynamicsofsuchsystemsisstill
intothe quantum-degenerateregimeinareversibleman-
9 lacking, including the issue of quasi-condensate forma-
0 tion, andthe role ofphase fluctuations onits growthdy- ner in 3D [26]. A related approach has been used to
5 create quantum-degenerate 133Cs in 3D [33, 34], and in
namics. Although condensate formation has been stud-
0 2D [35]. In 1D, such a dimple can be created by optical
ied in 3D geometries[25, 26, 27, 28, 29, 30], suchstudies
/ andmagnetic traps, althoughthe optimum technique on
t are only now possible in 1D. In this paper, we discuss
a an atom chip appears to be the application of electric
m a realistic experiment which captures the dynamics of
fields to a separate electrode, as discussed in Appendix
quasi-condensate formation and relaxation into a per-
-
d turbed potential, by adding a dimple microtrap on an
n atom chip containing a gas of 1D ultracold Bose atoms.
2000
o
The issue of quasi-condensate formation may be re- (a) (b)
:c lated to a recent 3D condensate growth experiment [27], ωh)z 1000
iv which to date remains largely not understood. In par- n(z) ( 0
rX tsilcouwlacro,otlihnigs,eaxnpeurnimexepnetctreedvesalloewd,liunnedarerinciotinadlitcioonndseno-f V(z), g −1000
a sategrowth. Asthesefeatureswereobservedclosetothe
−2000
transitionpoint,theauthorssuggestedthismightbedue −30 −15 0 15 30 −30 −15 0 15 30
z (l) z (l)
to the enhanced phase fluctuations in this region, lead- z z
ingto the formationofaquasi-condensatethatpreceded
FIG. 1: Initial (brown lines) and final (black) longitudinal
harmonicpotentials (solid) andtotal densities (dashed)for a
system of 2500 87Rb atoms at a temperature T = 50nK for
s∗iPtyreosfenNteAwcdadsrtelses,:MSecrhzoColooufrtM,NatehwecmasattliecsNaEn1d7SRtUat,isUtincist,edUnKiivnegr-- (wai)thazwDid=e d0im, Vp0le=w−=240l0z0,h¯aωnzd.(bH)erae,tiglzht=dimp¯hl/emwω=z i0s.5tlhze,
longitudinal harmonic oscillator length.
dom
p
2
A. We study the resulting growth dynamics focusing on at a rate [37, 38, 39]
theregimeofaverydeepdimple,whosepotentialgreatly
exceedsallotherenergiesinthesystem[36],asthisleads iR(z,t) = β¯hΣK(z)
to the most interesting non-equilibrium regime. If there −4
isalreadyaquasi-condensatepresentintheoriginaltrap, ¯h2 2
theadditionofthedimple isfoundtoleadtoshock-wave ∇ +Vext(z) µ+g Φ(z,t)2 . (2)
× − 2m − | |
formation and subsequent relaxation of the perturbed (cid:18) (cid:19)
quasi-condensate in the dimple. In the opposite case of In accordance with the fluctuation-dissipation theorem,
an initially incoherent ultracold atomic sample, we ob- there is an associated gaussian noise contribution η(z,t)
servedirect quasi-condensategrowthfrom the surround- obeying η∗(z,t)η(z′,t′) =(i¯h2/2)ΣK(z)δ(z z′)δ(t t′),
ing thermal cloud. The nature of the dynamics addi- wherethhebracketsdenoiteaveragingoverthe−realiza−tions
tionallydependsonthewidthofthedimplecomparedto of the noise. The dependence of these quantities on the
the size of the atomic cloud, and on the dimple location. Keldyshself-energyh¯ΣK(z)ensuresthatthetrappedgas
The distinct regimes of quasi-condensate dynamics can relaxes to the correct thermal equilibrium, as addition-
be probed by controlling the initial temperature, atom ally verified by direct comparisonto the modified Popov
number and perturbing potential. theory discussed elsewhere [16, 17, 18]. For simplic-
This paper is structured as follows: Sec. II discusses ity, we choose in this work both the longitudinal and
the methodology used to model the envisaged experi- the transverse atom chip confinement to be harmonic,
ment. Thedynamicsfollowingtheadditionofthedimple with respective trap frequencies ω = 2π 5 Hz and
z
×
in the limit of low temperature is then discussed in Sec. ω⊥ = 2π 5000 Hz, and present explicitly results for
III, paying particular attention to shock-wave formation 87Rb, with×a = 5.32nm. Unless otherwise specified, re-
(Sec. III A), its effect on the dynamics of the density at sults will be given in dimensionless harmonic oscillator
the trap centre (Sec. III B), and the effect of the width units of the originaltrap, such that lengths are scaledto
of the dimple compared to the size of the atomic cloud thelongitudinalharmonicoscillatorlengthl = ¯h/mω
z z
(Sec. III C). The effect of temperature is discussed in and energies to h¯ω .
z
p
Sec. IV for both wide and tight dimples, with the latter Once the system has fully relaxed to the desired equi-
case highlighting the interplay between spontaneous and librium,wesuddenlyturnonagaussiandimplepotential
stimulatedgrowthinthe dimple. Sec. Vsummarizesthe andsimultaneouslyremovethecouplingtotheheatbath.
examinedregimesbymeansofasuitablegraph,andSec. Switching off driving and dissipation terms, i.e., taking
VI features a brief summary. The discussion of experi- R(z,t)=η(z,t)=0,ensuresthatthetotalatomnumber
mentaltechniqueswhichcanbeusedtocreatethedimple ontheatomchipremainsfixed,asintheenvisagedexper-
trap on an atom chip has been deferred to Appendix A. iment,andthusonlyintrinsicdynamicsofthesystemare
taken into account. The dimple potential added is given
by V = V e−(z−zD)2/2w2 where V is the dimple depth,
D 0 0
w its width, and z its location. A sudden introduction
D
II. METHODOLOGY of the dimple potential is chosen, since this generates a
highly non-equilibrium situation, which gives rise to the
To model the proposed experiment, we first prepare most interesting dynamics. For the chosen parameters,
the desired initial state on the atom chip stochastically. thiscorrespondstoatrapturn-ontimescaleof6µs,which
This is achieved by means of the Langevin equation is an experimentally realistic timescale. The subsequent
analysisis basedonaveragingovermany different initial
realizations, which ensures that both density and phase
∂Φ(z,t) ¯h2 2
fluctuationsareaccuratelyincluded. Thisisequivalentto
i¯h = ∇ +V (z) µ iR(z,t)
ext
∂t "− 2m − − averagingoveralargenumberofindependentexperimen-
tal realizations with a variable initial phase, as typically
+g Φ(z,t)2 Φ(z,t)+η(z,t), (1) done when investigating growth dynamics [25, 26, 27].
| | # In our discussion, we assume the dimple is turned on at
t=0.
describing the dynamics of the order parameter
Φ(z,t) [37, 38, 39]. Here V (z)=mω2z2/2 denotes the
ext z
longitudinal harmonic confinement, and g = 2h¯aω⊥ is III. LOW TEMPERATURE LIMIT
the one-dimensionalcoupling constantobtainedby aver-
aging over transverse gaussianwavefunctions,where a is The sudden addition of a deep gaussian dimple on a
the three-dimensional scattering length and ω⊥ the trap harmonically confined pure condensate leads to a large
frequencyin the transversedirections[40]. The chemical atomic flux towards the center of the dimple, and thus
potential,µ,determines,foragiveninitialtrappotential to the development of a large local density gradient at
and temperature, the total atom number in the system. symmetric points about the dimple centre, which, in
The atom chip trap is pumped from a thermal reservoir turn, leads to the formation of two counter-propagating
3
shock wavefronts. The dynamics of the initial shock (a) (b) (c)
waveformationstagefor apure coherentcondensatehas
been discussed under various related conditions in Refs.
[41, 42, 43]. In addition, shock waves have already been
observed experimentally in the context of rapid poten-
(d) (e) (f)
tial perturbations related to the formation of dark soli-
tons [44] and vortex lattices [45]. In this paper, we dis-
cuss both short and long term dynamics in the dimple,
and identify the distinct growth dynamics induced by
1000
the addition of the dimple. In particular, we highlight ω)z 1000 (g) 500 (h) (i)
the competing effects of temperature, chemicalpotential 0) (h 500 0 −5 5
and dimple width on these processes. n(
g
We start our analysis by considering the most general 0
−20 0 20
scenario of experimental relevance. The dimple of Fig. z (l)
z
1(a) is added onto a quasi-condensate with about 3,400
atoms. This particulardimple hasbeen chosen,as itcan (j)
be easily generated with existing atom chips. A detailed 1000
discussion of how this can be achieved can be found in ω)z
h 1500
ApIpnenbdriiexf,Aw.e find the subsequent dynamics to be es- n(0) ( 500 1000
g
sentially controlled by two parameters: (i) Firstly, the 500
dynamics depends critically on the amount of quasi- 0 0 0.2 0.4 0.6
0
condensation present in the initial system. This is de- 0 2 4 6 8 10
t (ω−1)
termined by the ratio of T/Tc, where Tc is the effective z
1D ‘transition’ temperature. (ii) Secondly, the dynam-
ics is sensitive to the relationof the spatial extent of the FIG.2: (coloronline)Averagedsnapshotsoftotaldensityin
atomic cloud in the original trap compared to the effec- the initial harmonic (a) and perturbed ((b)-(i)) trap, taken
at equal time intervals, ∆t = 0.073ω−1, for a gas of 3400
tive width of the perturbing dimple potential. z
87Rb atoms. Profiles are shown at T = 50nK ≪ Tc (black
lines) and T = 200nK ≈ Tc (brown). (i) Indistinguishable
equilibrium profiles due to large dimple depth. Insets com-
A. Non-equilibrium Dynamics in the Dimple paresingle-runresultsbasedontheGross-Pitaevskiiequation
(green) to averaged stochastic profiles at T =50nK. (j) Cor-
respondingoscillationsinaverageddensityatthetrapcentre.
Forsimplicity,wefocusinitiallyontheregimeT T ,
c
≪ Inset highlights the first two oscillations, showing explicitly
for whichmostofthe atoms arein the quasi-condensate.
thetimes of theplotted snapshots (a)-(h),and comparing to
Typicaldensitysnapshotsofthegrowthdynamicsinthis
results of Gross-Pitaevskii (green). Dimple parameters as in
limit are shown by the black lines in Figs. 2 (a)-(i). In
Fig. 1(a).
our analysis,the ‘transition’ temperature T in the pres-
c
ence of interactions is determined numerically from our
modified Popov theory [16, 17, 18, 19].
Once the dimple is turned on, the central density in- into counter-propagating lower peaks, and so forth, un-
creases rapidly, until the instability towards shock-wave til the gas equilibrates to the profile shown in Fig. 2(i).
formation is reached [41, 42, 43]. The ensuing shock Such shock-wave dynamics would be largely suppressed
waves propagate towards the dimple edge and subse- ifthedimplewasmuchshallower,orifitwereintroduced
quently reflect back into the dimple centre. Individual on a much slower timescale.
profiles canbe simulated by solving the Gross-Pitaevskii The long-term system dynamics is portrayed in the
equation for a fully coherent condensate, yielding den- evolution of the central density of the gas shown by the
sity profiles with large density variations, as shown by black lines in Fig. 2(j). It features oscillations, on top
the green curves in the insets of Figs. 2(d)-(h). How- of a growth curve. This complicated dynamics is a re-
ever,adetailedunderstandingofgrowthdynamicswhich sult of a number of competing processes occuring simul-
enables a straightforwardcomparisonto experiments re- taneously. Firstly, the observed oscillations are a direct
quires the addition of random initial phases and subse- consequence of averaged shock-wave propagation in the
quent averaging over such initial configurations. Such quasi-condensate. Infact,theinitialoscillatorydynamics
averaged profiles, obtained within our analysis, reveal in the low temperature limit resembles a suitably aver-
the initial formation of a large central peak, followed aged single-run of the Gross-Pitaevskii equation, shown
by its break-up into two smaller peaks, moving in oppo- by the green lines in the inset to Fig. 2(j). The ‘growth’
sitedirectionstowardsthedimpleedges,withthecentral part of the dynamics arises as a result of spatial com-
peak re-emerging after a characteristic time determined pression of the trap, and features various contributions:
by the dimple potential. The latter peak further splits In addition to quasi-condensate compression, there is an
4
additional process whereby all thermal atoms located in ω)z100
the dimple area fall into the perturbed trap by joining (a) 0) (h 50
the quasi-condensate. This is ensured by the depth of gn( 0
the applied dimple, which largely exceeds all other rel-
evant energies of the system. Moreover, thermal atoms (b) ω)z 0
located in the tails of the initial atomic cloud (as well as h
anyquasi-condensateatomslocatedoutsideoftheregion V (D
−1000
of the dimple) are pulled into the trap centre, continu-
ously interacting with the propagating quasi-condensate
shock wave, and eventually leading to additional quasi-
−2000
condensategrowth. Intheremainingpartofthepaperwe −20 −10 0 10 20 −20 −10 0 10 20
z (l) z (l)
z z
discussthesecompetingprocessesinmoredetailbyiden-
tifying suitable experimentally realistic limiting regimes.
FIG.3: (coloronline)(a)InitialdensityforagasofN ≈960
atoms at T = 25nK and µ = 100¯hωz, such that µ/kBT ≈
1 (solid blue lines), versus corresponding zero-temperature
Thomas-Fermi profiles (dashed green lines). The harmonic
potential is also shown for easier visualization (solid black).
B. Oscillation Frequencies
(b)Initial(solidblack)versusfinal(solidbrown)potentialsfor
the two dimples of Fig. 1. Corresponding dimple potentials
Firstly, we comment on the observedoscillations. The VD are shown bythe dashed red lines.
addition of the dimple corresponds essentially to an in-
stantaneouslocalcompressionofthecentralregionofthe
trap. For an easy visualization which captures the main C. Effect of Dimple Width
dynamics discusssed here, let us consider the simplified
case of a harmonic dimple of width comparable to the The above example featured a combination of shock-
system size, such that essentially all of the dynamics is wave propagation and growth. These effects can be iso-
contained within a suitably defined ‘dimple region’. In lated by changing the form of the perturbation. Shock
this case, the effect induced by the addition of the dim- wavesariseinthequasi-condensatealreadypresentprior
ple is similar to that of an instantaneous increase in the to the addition of the dimple. On the other hand, the
harmonic frequency. This leads to excitation of the low- observed increase in the central density is the result of
est compressionalmode, which, for a pure condensate in both local compression of the atoms in the dimple re-
the 1D mean field regime, has been predicted to occur gion,andof directgrowtharisingfrom (mainly thermal)
at a frequency of √3ωD [46, 47], as already observedex- atomsmovingintothecentralregionfromthetrapedge.
perimentally [48], and additionally verified numerically Asaresult,oscillationsdominatewhenthedimplewidth
in our simulations. is comparable to the quasi-condensate system size, and
In the experimentally more realistic gaussian dimples for relatively small atom numbers, such that growth is
considered here, we can define an effective harmonic minimized. On the other hand, the growth features are
frequency ω = √V /w at the central dimple region. dominant in very tight traps for which the dimple width
D 0
Although the initial oscillation in the gaussian dimple is much smaller than the system size.
occurs very close to the predicted frequency, the fre- Thesetwocontrastingregimesforthetwooppositelim-
quency shifts towards lower values, as soon as the quasi- itsofwideandtightdimples,withrespecttotheeffective
condensate no longer feels a purely harmonic confine- system size, are shown in Fig. 3 for a reduced number
ment,i.e.,forsystemsizeslargerthantheeffectivedimple of N 960 atoms for computational convenience. To
≈
width. This shift becomes more pronounced in time as facilitate an easy comparison between these two limits
the quasi-condensate grows in the dimple. In addition, we keep the dimple depth fixed. To give a simple vi-
the observed oscillations are damped due to various ef- sualization of this distinction, we note that, for the low
fects. Firstly, unlike for a purely coherent system, the temperature case considered here, the zero-temperature
averaging performed over random initial phases in the Thomas-Fermi profile (dashed green line in Fig. 3(a)) is
presence of fluctuations leads to dissipation, precisely as a reasonable first approximation to the system density
would be the case in averaged condensate growth ex- prior to the addition of the dimple (solid blue line). In
periments in the presence of a non-negligible thermal the chosen harmonic oscillator units, the zero temper-
cloud. Furthermore, the relative motion between the ature Thomas-Fermi radius is defined by RTF = √2µ,
shock waves and the (mostly thermal) atoms entering where µ is the chemical potential of the system.
the dimple region from the edges is expected to create An effective dimple width, R can also be approxi-
D
an additional channel for dissipation. Finally, the addi- mately defined as the point at which the dimple depth
tion of a gaussiandimple leads to excitations of multiple fallsto0.01ofitsmaximumvalueV ,implyingR 3w.
0 D
≈
frequencies,whichleadto beating effects andanacceler- Fig. 3 plots both initial densities and potentials (top)
ated decay of the oscillation amplitude. and perturbed potentials (bottom) for the two opposite
5
values. Although equilibration timescales appear to be
comparable, the initial dynamics is slower in the tight
trap,presumablyduetothereducedoverlapbetweenthe
states in the initial and final traps.
The initial dynamics, including the shock-wave-
induced oscillations are more appropriatelyinvestigated,
whenthesameresultsareplottedintermsoftheeffective
dimple timescale ω−1, as in Fig. 4(b). This is because
D
the oscillations are actually fixed by the frequency ω
D
of the perturbed trap (multiplied by the factor of √3),
which varies with dimple width w. When plotted in this
manner, the initial dynamics feature a striking similar-
(b) 600 ity in both amplitude and frequency of the respective
600 central density oscillations, as shown in the inset. This
ω)z 300 is due to the fact that the initial dynamics is set by the
h quasi-condensatewhichis,inbothcases,presentoverthe
n(0) ( 300 w = 0.5 lz 0 0 10 20 30 entire extent of the dimple region. However, the subse-
g quentgrowthisnoticeablydifferent,duetotheenhanced
compression in the tight dimple, leading to a much en-
w = 4 l
z
hanced growth in the central density.
0
0 50 100 150 200
t (ω −1)
D
IV. EFFECT OF TEMPERATURE
FIG. 4: (color online) (a)-(b) Dynamics of dimple central
densityfor N ≈960atoms uponaddingabroad dimplewith
w = 4lz (black) or a narrow one with w = 0.5lz (brown), as All earlier discussion focused on the regime T ≪ Tc.
inFig. 3. Timeplottedintermsof(a)originaltraptimescale We now discussthe effect ofchangingthe initialtemper-
ωz−1and(b)dimpletimescaleωD−1,withωD =11.2ωz (black), ature of the system, with respect to Tc.
or 89.4ωz (brown). Here T = 25nK and µ = 100¯hωz, with
µ/kBT ≈ 1. Insets: (a) Same plot with densities approxi-
matelyscaledtofinalequilibriumvaluesnEQ(0),and(b)Ini- A. Wide Dimple
tial regime highlighting theoverlapping density oscillations.
The density snapshots shown in Fig. 2 correspond to
fairly ‘typical’ experimental regimes, for which R (0)
TF
regimes of R R (left) and R R (right).
D TF D TF is approximately equal to a few R , such that the ad-
≈ ≪ D
Note that the equilibrium densities in both dimple traps
dition of the dimple is accompanied by a combination
are essentially given by the Thomas-Fermi profiles in
of oscillations and growth. However, as the temperature
these traps. This is true for all atom numbers consid-
of the initial sample increases towards T at fixed atom
c
eredinthispaper(N <3500),forwhichthechosenlarge
number, the propagating secondary density peaks in the
dimpledepthensuresthatallatomscanbeaccomodated
averaged profiles become washed out, as shown by the
withinthecentralapproximatelyharmonicdimpleregion
brownlinesinFig. 2(a)-(h). Thisisduetotheenhanced
(see also the density profiles in Fig. 1).
fluctuations in the initial state prior to the addition of
The system dynamics in these two opposite regimes the dimple. Nonetheless, the final averaged profile is al-
is portrayed by the evolution of the central density in most independent of temperature, due to the large dim-
Fig. 4. This evolution is plotted both in terms of the ple depth, which overshadows all other energies of the
oofrigthinealefftreacptivteimdeismcapllee ωtiz−m1esincalFeigω.−14(ain),Faingd. i4n(bt)e,rmass sysBteymc.omparing these two cases, we note that the am-
D
these revealdifferent dynamicalfeatures that we wish to plitude of the initial oscillation is controlled by the ra-
comment on. tio of the chemical potential µ to the thermal energy
Theactualrelaxationtimescale,andlong-termdynam- k T. Underconditionsofstrongcondensation,suchthat
B
ics in these opposing regimes are best compared when µ/k T >1, the initial oscillation amplitude is large and
B
both curves are plotted in terms of the characteristic can even exceed the equilibrium value. In the opposite
timescale ω−1 of the initial harmonic trap. Fig. 4(a) regime, its amplitude is largely suppressed. In the limit
z
thus shows that the central density in the tight dimple ofasufficientlysmallatomnumberandarelativelybroad
increases much more significantly, due to the enhanced dimple, the addition of the dimple perturbs the central
spatial compression. A comparison of the growth rate density only mildly. Thus, the additional constraint of
in the different dimple traps is facilitated in the inset to extremely low temperature (k T µ), leads essentially
B
≪
Fig. 4(a), by plotting the dynamics of the peak densi- onlytotheappearanceofoscillationswithoutsubstantial
ties,approximatelyscaledtotheirrespectiveequilibrium growth.
6
B. Tight Dimple
The tight dimple, whose width is much smaller than
theeffectivesystemsize,createsanaturaltemporalsepa-
rationbetweentheinitialshock-wave-inducedoscillations
and the subsequent growth dynamics. These competing
effects exhibit differentcharacteristics,depending onthe
ratio T/T , as shown in Fig. 5(a). This ratio can be
c
controlled either by changing the temperature at con-
stant total atom number, as discussed for Fig. 2 above, 200
or by varying the atom number at fixed temperature. 600
In this section, we choose to discuss the latter, as this (b) (c)
enables a further distinction between spontaneous and
400
stimulatedbosonicgrowth. Notethat,forthesmallatom QCD100
N
numbers 400<N <1000 considered here, the condition
200
R (0) R is always satisfied for the dimple with
TF ≫ D T ~ T T << T
w =0.5l . C C
z
0 0
Regardingtheinitialoscillationsinthecentraldensity, 0 20 40 60 80 0 20 40 60 80
t (ω−1) t (ω−1)
wenotethat,astheatomnumberdecreases,sodoesalso z z
theratioofµ/k T determiningtheamplitudeofthefirst
B
oscillation. This leads to very strong suppresion when FIG. 5: (color online) (a) Central density dynamics in tight
µ/k T 1,asshownbythebottomcurveintheinsetto dimple (w = 0.5lz) with increasing atom number (from bot-
Fig.B5(a≪). Afterquenchingoftheseinitialoscillations,we tom to top) N ≈ 420, 490, 620 and 960, with RD ≪RTF in
all cases. The temperature is fixed to T =25nK, and corre-
observea‘secondarygrowth’dynamicalphase. Forsmall
sponding chemical potentials are µ/¯hwz = 5,30,60,100. In-
atom numbers, corresponding to large T/T , we observe
c set: Comparisonofinitialoscillatorydynamics. (b)-(c)Quasi-
a very slow initial growth in the central density. This condensate growth in the dimple for (b) N ≈ 420 (T ≈ Tc),
will be shown to be consistent with spontaneous quasi- and (c) N ≈ 960 (T ≪ Tc). Each computed curve (black)
condensate growth from the thermal cloud. Increasing is fitted with the growth curve of Eq. (3) (dashed brown),
the atom number leads to a lower value of T/T , and to and therelaxation curveof Eq. (4) (solid green). The initial
c
asignificantquasi-condensatefractionintheinitialtrap, number of condensate atoms in the dimple is bigger in (c),
which yields a faster initial growth rate in the dimple. dueto thelarger chemical potential.
To understand this in more detail, Fig. 5(b)-(c) plots
theintegratedquasi-condensateatomnumberinthedim-
ple as a function of time for the cases (b) T T , and
c
≈
(c)T T . Thequasi-condensateatomnumberisdeter-
c
≪
mined from a combination of local quadratic and quar-
tic correlations of the order parameter Φ(z,t), as will
and subsequent stimulated quasi-condensate growth in
be discussed in more detail elsewhere. The evolution is
the dimple, analogous to the first condensate growth
in both cases fitted by two distinct growth models, as
studies in 3D systems. Although the exponential re-
in [25]. The first model includes both spontaneous and
laxation model fails to describe the entire dynamics in
stimulatedbosonicgrowth,andisfittedbydashedbrown
this case, it nonetheless works well if fitted after a cer-
lines. It obeys
tain ‘initiation’ time [28]. In the opposite regime of
−1/δ T T shown in Fig. 5(c), the initial dimple dynamics
δ c
N0(t)=N0(i)eγt 1+ N0(i)/N0(f) eδγt−1 , is ≪mainly governed by re-equilibration of the perturbed
(cid:20) (cid:16) (cid:17) (cid:0) (cid:1)(cid:21) (3) quasi-condensate in the combined trap. This obeys the
exponential relaxation model well, indicating that the
whereN thetotalquasi-condensateatomnumberinthe
0 dynamics is dominated by stimulated growth.
dimple, the superscripts (i) and (f) denote respectively
initial and final values, and γ corresponds to the initial
growth rate. The value δ = 2/3 assumes a growth rate
The intermediate temperature regime features inter-
linear in the difference of chemical potential of the ther-
esting, but complicated dynamics, as the effects of spon-
malcloudandthequasi-condensate,forwhichµ N2/3.
∝ 0 taneousandstimulatedgrowthcompetewitheachother.
The second model, fitted by the solid green lines, de-
The above distinction between spontaneous and stimu-
scribes simple exponential relaxation to equilibrium via
lated growth becomes less pronounced with increasing
N (t)=N(f) 1 e−γt . (4) atom number, with the initial growth dynamics being
0 0 − well-modeledbyanexponentialrelaxationcurveevenfor
The limit T Tc shown in(cid:0)Fig. 5(b(cid:1)) is well described the tight dimple considered here, when N is a few 1000
≈
by the growth model of Eq. (3), indicating spontaneous atoms.
7
V. SUMMARY OF DISTINCT REGIMES
The effect of adding a dimple trap to the centre of a
weaker harmonic trap containing a 1D quasi-condensate
wasconsideredinthelimitwhenthedimpledepthgreatly
exceeds all other relevant energies in the system, i.e.,
V µ,k T,¯hω . Thepresentedanalysiswasrestricted
D B z
≫
to the typical experimental regime satisfying the follow-
ing length scale separation
l⊥ <ξ <(λdB,lD) lz (RD,RTF(0)) (5)
≪ ≪
where l⊥ = ¯h/mω⊥ is the transverse harmonic oscil-
lator length, ξ = h¯/√4µm is the healing length of the
p
system, λ = 2π¯h2/mk T is the thermal de Broglie
dB B
wavelength, l q= ¯h/mω is the effective harmonic
D D
dimple oscillator length, R (0) is the zero-temperature
TF
p
Thomas-Fermi radius in the original harmonic trap, and
R is the effective dimple width. Within this regime,
D
the dynamics in the dimple displays aninteresting inter-
FIG. 6: Visualization of distinct regimes accessible in ul-
play between shock-wave propagation in the perturbed
tracold 1D Bose gases, upon addition of a deep dimple
quasi-condensate, and direct quasi-condensate growth. (V0 = −2000h¯ωz) at the centre of a weaker harmonic trap
The following important conclusions were reached (i.e. zD = 0), as manifested by the dynamics in the central
about the accessible dynamical regimes in such systems: density oscillations. From top to bottom, N ≈ (a) 3400, (b)-
(c) 540, and w = (a)-(b) 4lz, and (c) 0.5lz. Left images (i)
correspond to T ≪ Tc, with right images (ii) showing the
The initial dynamics for T < Tc is dominated by oppositeregimeT ≈Tc. Eachfiguredisplaysthecorrespond-
• shock-wave propagation, leading to large oscilla- ing approximatevalues of α=µ/kBT, and β=RTF(0)/RD,
determining thesystem dynamics.
tions in the central density. The initial amplitude
of such oscillations is controlled by the ratio of
(µ/k T), yielding large oscillations when this is of
B
(c) AverytightdimpleR R (0),featuring‘long-
order unity or larger. D TF
≪
term’pronouncedgrowth,afterdampingofanyini-
The long-term system dynamics is dominated by tial oscillations occuring. Note that the sponta-
• quasi-condensate compression and growth in the neous bosonic growth contribution is pronounced
dimple, with this effect largely pronounced when here due to the small number of atoms.
the system size greatly exceeds the dimple width.
Having identified the importance of the relation of the
size of the atomic cloud compared to the dimple width,
The different accessible regimes, along with the asso-
and of the chemical potential compared to the tempera-
ciatedrelevantparametersfor (µ/k T)andR (0)/R
B TF D
ture of the system, we can now ‘tailor’ our experimental
are summarized by the evolution of the central density
conditions to produce the desired dynamics.
showninFig. 6. Inthisfigure,eachhorizontalline(a)-(c)
Notethatalldimpleswereconsideredtohavethesame
corresponds to a particular configuration of fixed initial
depth, which exceeds all other relevant energy scales in
atom number and dimple width. Left images (i) show
thesystem. Thus,fortheatomnumbersconsideredhere,
thecaseT T ,whilerightimages(ii)displaythehigh-
c
≪ thefinaldensitiesarealldescribedbyThomas-Fermipro-
temperature limit T T . The images shown portray,
c
≈ files in the dimple, and the dimple width is the relevant
from top to bottom, the following cases:
parameter determining the ensuing 1D dynamics.
(a) A relatively large atom number N = 3400, such The case ofan off-centereddimple, with z0 =0,yields
6
that R (0) is slightly larger than R . This qualitatively similar behaviour, that will be discussed in
TF D
leads to noticeable growth, and additionally fea- more detail elsewhere. When the dimple falls within
turespronouncedand‘long-lived’oscillationswhen the initial quasi-condensate size, shock wave formation
µ/k T 1. plays a key role in the subsequent dynamics. However,
B
≈
in the opposite limit of a dimple located outside of the
(b) A sufficiently small atom number N = 540 and quasi-condensate,the oscillationsare largelysuppressed,
comparatively wide dimple, R (0) R , for and quasi-condensate relaxation competes with quasi-
TF D
≈
whichgrowthissuppressedandoscillationsarethe condensate growth. The whole process is further modi-
dominant feature in the low temperature limit. fied by the asymmetry in the initial density distribution
8
with respect to the dimple centre, an effect which be- Starting from a 1D potential we can create a dimple
comes particularly pronounced for shallow dimples. on an atomchip by using (i) magnetic fields, (ii) electric
fields and (iii) dipole potentials. In the following dis-
cussion of creating a dimple we neglect the longitudinal
VI. CONCLUSIONS confinement of the original trap.
In conclusion, we studied one-dimensional quasi-
1. Creating the dimple with magnetic fields
condensate growth dynamics in a dimple microtrap cre-
atedontopoftheharmonicconfinementofanatomchip.
Onan atomchip, the 1D trapis formedin a magnetic
Rich novel dynamics were observedin the case of a deep
minimumcreatedbysuperimposingthemagneticfieldof
dimple,whosedepthexceedsallotherenergiesofthesys-
a current in a straight wire by a homogeneous bias field
tem, displaying an interesting interplay between shock-
[49,51,54,55]. Weassumethetrappingwiretobeparal-
wavepropagationintheperturbedquasi-condensate,and
lel to the z-direction (Fig. 7) and the homogeneous bias
direct quasi-condensate growth. The optimum observa-
tion of shock-wave dynamics, without the added signifi- field Bbias = (Bx,By,BIoffe) to be nearly orthogonal,
cant growth of the central density, requires low temper- i.e., | BIoffe | ≪ | B⊥ | with B⊥ = (Bx,By,0). The
component of the field along the wire direction, the Ioffe
atures T T , a comparatively large chemical potential
c
≪ field B , defines the minimum of the trapping field.
µ>k T, and a dimple of width comparable to the sys- Ioffe
B A dimple can be created by a second wire, located
temsize. Understandingtheseprocessesinmoredetailis
alongthey-direction,andthuscrossingthetrappingwire
expected to contribute to fundamental issues in the dy-
(Fig. 7)[56]. Asmallcurrentj inthis dimple-wire cre-
namics of degenerate one-dimensional Bose gases, with D
atesamagneticfieldwhicheithersubtracts from,oradds
potential applications in atom lasers and atom interfer-
to, B , thus creating a dimple or a barrier respec-
ometers. Ioffe
tively. For a linear current crossing the trapping wire at
z =0,withthe 1Dtrapatheighthaboveit(Fig. 7(a)),
the dimple potential is given by
Acknowledgments
1
V (z)=V . (A1)
We acknowledge discussions with P. Kru¨ger, S. Wil- D 01+(z/h)2
dermuth, and in particular E. Haller. This work was
The depth of the dimple potential V =
supported by the Nederlandse Organisatie voor Weten- 0
g µ m (µ /2π)(j /h) is determined by the mag-
schaplijk Onderzoek (NWO), by the European Union, F B F 0 D
nitude ofthe magneticfieldcreatedbythe currentj at
contract numbers IST-2001-38863 (ACQP), MRTN- D
the position of the trap. Here µ is the Bohr magneton,
CT-2003-505032 (AtomChips), HPRN-CT-2002-00304 B
g the Lande factor of the trapped atomic state F,m
(FASTNet), and the Deutsche Forschungsgemeinschaft, F | Fi
and µ = 4π Gmm/A is the vacuum permeability. The
contract number SCHM 1599/1-1. 0
longitudinal size (i.e. width) of the dimple is set by the
distance h.
The geometry of the dimple depends on the ratio be-
APPENDIX A: EXPERIMENTAL CREATION OF tween h, the height of the trap above the dimple wire,
1D DIMPLE POTENTIALS
anddtrap,thedistanceoftheatomstothetrappingwire:
This Appendix discusses various experimental tech- ω⊥ h B⊥
= | | (A2)
niqueswhichcanbeusedtocreateadimple potential,in ωD dtrap 2BIoffeBD
a quasi-1Datomic gas on an atom chip [1, 49, 50], in or-
dertoobservethedynamicsdiscussedinthispaper. The The parameters h and dtrappcan be controlled indepen-
basic idea is to modify the 1D trapping potential with a dently, by either mounting the dimple wire above the
small but localized attractive potential, which creates a trappingwire(Fig. 7(a))orbyrotatingthetraptowards
tight potential minimum along the weak confining axis the surface by rotating the bias field (Fig. 7(b)), thus
of the 1D trap. yielding complete control of the dimple trap anisotropy.
To create the situation described in this paper one When the 1D gas is trapped relatively far from the
needs exceptionally smooth trapping potentials as ob- trappingwire, the remainingresidualdisorderpotentials
tained for example with atom chips nano-fabricated in [9, 10, 36] are reduced to a point where they are not
gold layers on semiconductor substrates [10, 36, 51, 52]. visible even for tight transverse confinement of typical
On these atom chips, we can routinely create 1D traps frequency ω⊥ 2π(5 kHz) [53, 57].
∼
with an aspect ratio of larger than 1000 [53]. The lon- Inarealexperimentalsetup,thewidthofthewiresand
gitudinal confinement in the traps can be harmonic or the exact form of the current distribution in the wires
box-like on the energy scale of the chemical potential, determines the potential. By measuring the magnetic
depending on the actual setup. dimple potential, as described in [9], we found that it is
9
a) b) c)
by suitably choosing the position of the trap, as shown
elongated
trap in Fig. 7(c).
dimple dimple dimple Such a 1D electric dimple can be easily createdon the
wire wire electrode atom chip with ω⊥ 2π(5 kHz) described above, once
∼
dimple trap the trap is rotated towards the chip surface and posi-
tioned above an orthogonal wire. In the electric case, a
Bbias
z few Volts of electric potential on the dimple wire create
y an adjustable dimple, whose parameters are similar to
trapping wire trapping wire trapping wire
the ones described in this paper for typical trap heights
of order h (5 25)µm [57].
∼ −
x In a real setup one must again take into account the
detailedgeometry,theexactboundaryconditionsandthe
y
dielectricpropertiesofthechipsubstrate. Measuringthe
potentials we found that they can be described, to good
FIG.7: (coloronline)Wireconfigurationsforcreatingadim-
accuracy, by a gaussian shape [57].
ple in a quasi-1D gas on an atom chip. (a) The dimple is
Creating the dimple with electric fields has the ad-
created by the crossing wire placed above the trapping wire.
Themagneticfieldcreatedbythecurrentflowinginthecross- vantage that a switchable dimple can be created even
ing dimple wire subtracts from the minimum of the 1D trap without the need for crossing the wires on the chip, as
(IoffefieldB ). (b)By rotatingthebiasfieldthe1Dtrap shown in figure 7(c). In addition, high resistivity ma-
Ioffe
isdisplacedandthedimpleiscreatedonthesideofthetrap- terials can be used for charged structures, and this will
ping wire. To keep the trap symmetric around the dimple, dramatically reduce the thermal noise induced by John-
a wire crossing is again required. (c) In the rotated geome- sonnoisecurrents[59,60,61,62,63,64,65,66,67],such
trycharginganelectrodecreatesastronglyconfiningdimple,
that distances of 1 µm to the charged dimple wire are
without theneed for wire crossing.
feasible.
actuallyverywelldescribedbyagaussianshape,asused
3. Creating a dimple by superposing a dipole
in the calculations presented in this paper [57].
potential
On anatom chip, a 1D trapwith ω⊥ 2π(5 kHz) can
∼
be typically created by a 1000 mA current and a B⊥
28G. If the trapis then rotatedtowardsthe chipsurfac∼e A third way of creating a dimple on a atom chip is
and positioned above an orthogonal wire at heights h following[26]andsuperimposingadipolepotentialonthe
(5 25)µm, a current j (10 100)µA in the dimp∼le atom chip trap [68]. The dimple is created by a tightly
D
wir−ecreatespotentialslike∼theon−esdescribedinthemain focused red-detuned laser beam. The size of the dimple
part of this paper. The larger values of h correspond to is given by the focus size of the beam. The form of the
wider dimples and require larger jD. dimple potential is gaussian,VD =V0 e−(z−z0)2/2σ2, and
the size of the dimple is set by the numericalaperture of
the imaging lens, and will typically be of order of 3-10
µm.
2. Creating the dimple by adding electric fields
Thedipoledimplecanbecreatedatarbitrarydistance
from the surface of the chip. It is the simplest way to
A second way of implementing a controllable dimple
create a tight dimple in cold atom experiments, and has
on an atom chip is by using electric fields (Fig. 7(c)).
beenused inthe condensategrowthexperiments atMIT
An atomwith electric polarizability α feels an attractive
[26], andthe 133Cs experiments at Innsbruck[33, 35]. In
potential Vel = 1αE2. Charging up a wire positioned
−2 an atom chip environment, one has to take care of the
orthogonalunderneathatrapwillcreateadimplepoten-
stray light from the chip surface and the resulting inter-
tial [58].
ference structure inside the dimple. Scattered light will
In a simple model of a 1D trap at height h above the
create speckles and form a additional disorder potential.
line charge Q, the electric dimple potential
el 1 Q
VD(z)=−2αz2+h2 (A3) 4. Switching the dimple
has the same form as the magnetic dimple discussed All three examples of dimple traps discussed above
above(Eq. (A1)). Theratioofthelongitudinaltotrans- can be switched very fast. In the electric and magnetic
verse confinement in the dimple depends again on the cases,onlyverysmallcurrentsorchargesareneeded,and
distance dtrap between the trapping wire and the atom rapid switching times (τ 1µs), much faster than the
≪
trap, and on the height h above the line charge. This timescale of the transverse trapping frequency ω⊥, can
ratio can thus be adjusted over a broad range of values, easily be obtained. The same can be said for the dipole
10
dimple,forwhichtheswitchingdependsonlyonthelight switching time, which can be again τ 1µs.
≪
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