Realizing and optimizing an atomtronic SQUID

2.9 MB·

by  Amy C. Mathey

Realizing and optimizing an atomtronic SQUID
Amy C. Mathey1, and L. Mathey1,2
1Zentrum für Optische Quantentechnologien and Institut für Laserphysik, Universität Hamburg, 22761 Hamburg, Germany
2The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, Hamburg 22761, Germany
We demonstrate how a toroidal Bose-Einstein condensate with a movable barrier can be used
to realize an atomtronic SQUID. The magnitude of the barrier height, which creates the analogue
6 of an SNS junction, is of crucial importance, as well as its ramp-up and -down protocol. For too
1 low of a barrier, the relaxation of the system is dynamically suppressed, due to the small rate of
0 phaseslipsatthebarrier. Forahigherbarrier,thephasecoherenceacrossthebarrierissuppressed
2 due to thermal fluctuations, which are included in our Truncated Wigner approach. Furthermore,
we show that the ramp-up protocol of the barrier can be improved by ramping up its height first,
n
and its velocity after that. This protocol can be further improved by optimizing the ramp-up and
u
ramp-downtimescales,whichisofdirectpracticalrelevanceforon-goingexperimentalrealizations.
J

4
1 I. INTRODUCTION Density, n(ρ,θ,z=0) θ-current, jθ(ρ,θ,z=0)

(a) (b)
] 20
s
a The advancement of cold atom technology, and the
g levelofcontrolthatcanbeachievedinsuchsystems,has y/l 0
- motivated the question if it can be used to emulate elec-
t
n tronic circuitry, and possibly move beyond its features, -20
a Ref. [1]. Whilearealizationof,say,theequivalentofelec- t=300 ∆t
u tronsmovinginasemiconductingmaterialisaninterest- (c) (d)
q 20
ing directionin itself, it is particularlyintriguing to cap-
.
t italizeonthespecificfeaturesofcoldatomsystems,such
a y/l 0
m as long-range phase coherence in Bose-Einstein conden-
- sates. This motivates to realize systems inspired by su- -20
d perconducting circuitry. Experimentally, an interesting t=410 ∆t
n starting point, and a remarkable achievementof its own, (e) (f)
o istherealizationofBose-Einsteincondensates(BECs)in 20
c
toroidal geometries, Refs. [2–5].
[ y/l 0
In this paper, we study the equivalent of an electronic
2
SQUID. The condensate wave function is the equivalent -20
v
of the superconducting wave function, and a potential t=1000 ∆t
1
3 barrier, at which the condensate density is suppressed, (g) (h)
20
4 replaces the SNS interface. This barrier is then moved
5 at a constant speed, which imitates a non-zero magnetic
0 flux through the ring. This setup can also be seen as a y/l 0
.
1 stirring experiment, testing the superfluid properties of -20
0 condensates,seeRefs. [6,7]. Othertheoreticalstudiesof t=2000 ∆t
6 the ring geometry were reported in Refs. [8]. -20 0 20 -20 0 20
x/l x/l
1
We demonstrate that a regime of a controlled and ef-
:
v fective realization of an atomtronic SQUID exists for 0 10 20 30 40 50-5 0 5 10 15 20
i
X sufficiently large potential barrier heights, for realistic
Figure 1: The density, n(ρ,θ) and the θ component of the
temperatures. For small barrier heights, the dynamical
r current, j (ρ,θ) at z = 0 before (t = 300∆t = 217.2 ms),
a relaxation of the condensate to the ground state phase θ
during (t = 410∆t = 296.8 ms and t = 1000∆t = 724 ms)
winding is suppressed, because phase slips at the bar-
andafter(t=2000∆t=1448ms)stirring. Thebarrierheight
rier occur only at a very small rate. For larger barrier
is Vb/J =1.8 and the stirring frequency is ωs/ω0 =3.8.
heights, the phase coherence across the barrier is sup-
pressed because of thermal fluctuations in the bulk of
the ring. This results in a smoothed-out response of the
order. Interestingly,we find thatthe latter resultsin no-
rotation,approachingalineardependenceonthestirring
ticeably less heating, and that the new ground state is
velocity,ratherthanaquantized,step-likeresponse,that
reached more quickly. We then discuss how this proto-
is characteristic for a SQUID. Furthermore, we consider
col can be further improved by different choices for the
two seemingly similar ramp-up processes for the SQUID
ramp-up and -down time scales.
operation: toeitherfirstrampupthebarriertofullspeed
and then ramp up the barrier height, or use the reverse This paper is organized as follows: In Sect. II we de-
2
scribe our simulation method, in Sect. III we discuss Inthisrepresentation,theequationsofmotiontakethe
the phase slip dynamics. In Sect. IV we compare the form
different barrier ramp-up scenarios, and in Sect. V we
conclude. d l2
i~dtψ˜ijk = −J ψ˜i+1jk+ψ˜i−1jk −2ψ˜ijk + 4ρ2ψ˜ijk
" i
(cid:16) (cid:17)
II. SIMULATION METHOD +ρρ202 ψ˜ij+1k +ψ˜ij−1k −2ψ˜ijk
i
(cid:16) (cid:17)
BecausetheSQUIDdynamicsisdominatedbythedy-
namics at the barrier, and because of the low density at + ψ˜ijk+1+ψ˜ijk−1−2ψ˜ijk
the barrier,andtherefore the low mean-fieldenergy,it is #
(cid:16) (cid:17)
imperativetoincludethermalfluctuationsofthesystem. + V +U(ρ )|ψ˜ |2 ψ˜ , (2)
ijk i ijk ijk
We include boththermalfluctuations, andthe lowestor-
h i
der of quantumfluctuations, within a TruncatedWigner
where J = ~2/(2ml2) is the tunneling energy, and
approximation (TWA), see e.g. [9]. The approach that
we use is closely related to the one of Ref. [10]. the interaction term is given by U(ρi) = U0ρ0/ρi,
We describe the system with the Hamiltonian where U0 = gl−3. On the lattice, the external po-
~2∇2 tential is given by Vijk = 4~J2l2 ωρ2(ρi−ρ0)2+ωz2zk2 +
Hˆ = dr Ψˆ†(r) − +V(r,t) Ψˆ(r) α(t)V exp −0.5ρ2(θ −θ (t))2/hl2 . i
2m b i j b b
Z g (cid:16) (cid:20) (cid:21) As ment(cid:2)ioned above, we initia(cid:3)lize the dynamics by
+ Ψˆ†(r)Ψˆ†(r)Ψˆ(r)Ψˆ(r) (1) sampling from the initial Wigner distribution. For the
2
initial state, we choose a non-interacting Bose gas in the
(cid:17)
where m is the atomic mass and g is the interac- toroidal trap, of zero temperature. After the initializa-
tion strength, for which we use the approximation tion, the interactions are turned on slowly to generate
g = 4πa ~2/m, with a being the s-wave scattering the desired interacting ensemble in the trap [12]. This
s s
length. The external potential, V(r,t) = V (r) + process of turning on the interaction results in a non-
tr
V (r,t) consists of the trapping potential, V (r) = zero temperature that is comparable or larger than the
bar tr
12mωρ2(ρ − ρ0)2 + 21mωz2z2, with ρ = x2+y2, and mean-fieldenergyofthesystem[13]. Thistemperatureis
the time-dependent stirring potential, V (r,t) = measuredbyweaklycouplingharmonicoscillatorsto the
bar
α (t)V exp −0.5ρ2(θ−θ (t))2/l2 , withpθ being the az- currentasdescribedinRef. [10]. Forexamplesdiscussed
b b b b
imuthal angle. ρ0 is the radius to the ring. Within the in this paper, the temperature of the atomic cloud after
TWA, the o(cid:2)perators, Ψˆ are repla(cid:3)ced by classical fields, initialization is approximately 4.0 J = 43 nK [14].
which are propagated according to the equations of mo- Throughout this paper, we use a lattice with the di-
tion of this Hamiltonian. The initial condition of these mensions Nρ = 17, Nθ = 126, and Nz = 5, and N =
classical fields are generated from the Wigner distribu- 50000 atoms. The trapping frequencies are ωρ/J = 0.5
tion of the initial state. and ωz/J = 2.5. We propagate and average over 72 ini-
In order to carry out the calculations, we discretize tial states and the length scale of the barrier is lb/l =3.
the real-space description by introducing a lattice ap- We setU0/J =0.07,whichcorrespondsto a lengthscale
proximation and work in cylindrical coordinates. We re- l = 0.99µm, a time scale ∆t = ~/J = 0.71 ms and
place the continuous wave function, ψ(ρ,θ,z) by a dis- an energy scale J = 10.8 nK for sodium atoms. The
crete wavefunction, ψ˜ = ψ˜(ρ ,θ ,z ), with the map- healing length in the bulk of the system is ξ ≈ 0.9µm.
ijk i j k
This length is small compared to system size in the ra-
ping
dial direction and comparable to the system size in the
ψ(ρ,θ,z)→ ρρ0l3 1/2ψ˜ijk zcr-odsirse-ocvtieornr,egwihmicehbeptuwtesenthtewosyastnedmthirneethdeimdeinmseionnssio.nal
(cid:18) i (cid:19) We now consider the following experiment: Starting
where ρi = ρ0 + l[i−(Nρ−1)/2], i ∈ [0,...,Nρ − 1], att1 =100∆t,we rampthe barriermagnitude from0 to
θj =lj/ρ0,j ∈[0,...,Nθ−1],andzk =[k−(Nz −1)/2]l, the final barrier height Vb over a time period of 200∆t,
k ∈ [0,...,Nz −1], and ρ0 = lNθ/(2π) is the radius of whilekeepingthebarrierstationary. Thenatt2 =300∆t,
the ring at the trap minimum. We emphasize that the thebarrierisacceleratedtoitsfinalstirringfrequency,ω
s
discretization length is not constant, but increases lin- over200∆t. Att3 =500∆t,theatomiccloudisstirredat
earlywithincreasingdistancefromthe centeraxisofthe constant frequency for a time period of 1200∆t at max-
ring. Therefore, the curvature of the ring geometry is imum barrier height. At t4 = 1700∆t the barrier height
fully taken into account. In this representation, |ψ˜ |2 is ramped down over the time 200∆t while continuing
ijk
corresponds to the number of atoms per unit cell, where to stir the atoms at the frequency ω . We refer to this
s
the volume ofthe unit cell is l3ρi/ρ0. For the lattice size procedure as protocol 1.
chosenhere, the volume of the unit cell varies from 0.6l3 Additionally, we compare this protocol to the process
for i=0 to 1.4l3 for i=16. of turning on the barrier height to V , while stirring
b
3
time [s] the current in Fig. 1 (h), the stirring of this ring shaped
0 0.362 0.724 1.09 1.45 condensate has imparted a finite current circulating in
(a) 4 the ring.
ding 3 ωs/ω0=2.4 For a system with a complex orderparameter,such as
win a condensate of atoms or a superconductor, this current
se 2 is related to the well-defined phase. In the condensed
a
Ph 1 phase,thisresultsinaquantizationofthephasewinding
0 around a ring geometry. We determine the phase along
(b) 4 the centralline alongthe ring,whichtracksthe maximal
ding 3 ωs/ω0=3.4 density for a given azimuthal angle, and in the z = 0
n plane. We calculate the phase at the angle θ via
wi
e 2
Phas 1 φ(θ)=φ(ρ0,θ,0) = tan−1(Im ψ(θ)/Re ψ(θ)). (3)
0 The phase winding is the sum of the phase differences
g(c) 4 ωs/ω0=3.6 aroundthe ring,n∆φ =( θδφ(θ))/2π,wherethe phase
din 3 difference between two points, δφ(θ) = φ(θ+θ0)−φ(θ)
n P
wi is between −π and π. The phase winding is calculated
se 2 Vb/J=1.0 ineachindividualrealizationandthenaveragedoverthe
Pha 1 VVbb//JJ==21..48 realizations to generate the average phase winding.
0
0 500 1000 1500 2000
time, t/∆t
III. PHASE SLIP DYNAMICS
Figure 2: We depict the time evolution of the average phase
In Fig. 2 we show the time evolution of the average
winding around the center of the toroid for stirring fre-
quencies, ωs/ω0 = 2.6,3.4,3.6, and barrier heights Vb/J = phase winding along the center of the toroidal trap for
1.0,1.8,2.4. threedifferentbarrierheightsandthree differentstirring
frequencies. For the smallest barrier height of V /J =
b
1, and for all three rotation frequencies ω , the phase
s
at constant frequency ωs, starting at t1 and reaching windingdoesnotrelaxtothenewgroundstate. Instead,
the maximum barrier height at t2. The atomic cloud onthetimescalesofatypicalexperiment,itremainsina
is stirred for 1400∆t before ramping down the barrier, long-lived,metastablestate. Ontheotherhand,forlarge
while continuing to stir at constant frequency. This sec- barrier heights such as Vb/J = 2.4, the coupling across
ond protocol is reminiscent of the experiments reported thebarrierissmallcomparedtothetemperature,sothat
in Ref. [2]. As we discuss below, the first protocol is the dynamics are oscillatory and noisy for the length of
the preferable protocol for large stirring frequencies, be- the experiment. Here, the dynamics of the experiment
causeitavoidsexcitingphaseslipsbeforethe barrierhas results in a distribution of phase windings, rather than
reached its maximum height. We elaborate on this pro- - essentially - a single phase winding number. However,
tocolfurtherinSect. IVbyconsideringdifferentramping we observe that for intermediate barrier heights such as
time scales. Vb/J =1.8, the oscillations damp out in a realistic time.
In Fig. 1, we illustrate the dynamics of the system by We note that the magnitude the phase slip rate, and
depicting the density n(ρ,θ) in the z = 0 layer, and the its dependence on the system properties war discussed
currentj (ρ,θ)alongtheazimuthaldirectionofthering, in Ref. [10]. In particular, it was discussed that the
θ
at four different times, for a barrier heightof V /J =1.8 phase slip rate is consistent with the scaling τ−1 ∼
b ph
and stirring frequency ωs/ω0 =3.8 for the first protocol, exp(−Eb/kBT), i.e. an Arrhenius law. The energy scale
with ω0 = ~/(mρ20). At the first time, t = 300∆t, the Eb is controlled by the barrier height and width. This
trappotentialisfullyrampedup,asisvisibleintheden- strong,exponentialdependence isreflectedinthe behav-
sity depletion of the ring shaped condensate, but is still ior we observe here, where the relaxation dynamics of
stationary. Here,thephasewindingisstillzero,andboth the system is nearly suppressed at V /J = 1.0, while at
b
positiveandnegativecurrentfluctuationsarevisible. We around V /J ≈2.4 the barrier is so high that any coher-
b
note again that these fluctuations are predominantly of ence across the barrier is suppressed.
thermalorigin. Atthesecondtime,t=410∆t,onephase To understand the origin of the oscillatory behavior
slip has occurred. Now the current has acquired a pre- we show the time evolution of the density along the
ferred direction, as is immediately visible in Fig. 1 (d). ring. We define radially projected density n1D(θ) =
At the third time, t = 1000∆t, a second phase slip has dρ dzn(ρ,θ,z). We emphasize that while this quan-
occurred,andthemagnitudeofthecurrenthasincreased, tity has the dimensions of a one-dimensional density we
R R
Fig. 1 (f). Finally, at time t = 2000∆t, the barrier has do not imply that the dynamics can be reduced to that
beenrampeddown,sothedensityofthecondensatedoes ofaone-dimensionalsystem. Infact,aswehaddiscussed
notdisplayadepletedregion. However,asisvisiblefrom in Ref. [10], the phase slips that occur in the system are
4
due to a non-trivial process of a vortex traversing the
barrier region. We merely use this integrated density as
a convenient way to depict the system evolution.
In Figure 3 (a) and (c ) the time evolution of the ra-
dially projected density is shown for V /J = 1.8 and
b
Vb/J = 2.4, respectively, and for ωs/ω0 = 3.4. In (b)
and (d), the same evolution is shown in the reference
frame of the barrier. As is visible in this figure, the os-
cillations in the average phase winding are due phonon
pulses generated during the initialization and accelera-
tion of the barrier. The density waves observed in the
density travel at the speed of sound, which is approx-
imately v = 1.52l/∆t or 2.1mm/s. With this velocity,
s
theperiodoftheoscillationsoftheaveragephasewinding
is of the order of 2T = 166∆t, corresponding to travel-
ing back and forth along the circumference of the ring.
This matches the period of the oscillations observed in
Fig. 2. We observe that the oscillations for V /J = 1.8
b
damp out during the time of the simulation, whereas for
V /J = 2.4 they do not. This is due to the reflection of
b
thephononpulseatthebarrier. Forthehighbarrierthis
reflection is essentially complete, and the pulse travels
back and forth with only little damping. For the inter-
mediate barrier height, the reflection is partial, which
results in dephasing and damping.
Motivatedbythisobservation,wesuggestwaystomin-
imize the detrimental effect of these phonons pulses and
the resulting oscillations in the phase winding in Sect.
IV,asa steptowardsimprovingthe SQUID operationof
the condensate ring.
The phase winding that emerges from this dynamical
evolution is shown in Fig. 4. We depict the final, av-
erage phase winding after the barrier has been ramped
down, as a function of the stirring frequency for several
barrierheights,inthe mainpanel. As indicated, the ide-
alized, fully quantized, and fully relaxed phase winding
shows steps at ωs/ω0 = n+1/2, with n being an inte-
ger. This is shown as a black line. For too small of a
barrierheight,suchasV /J =1,the systemremains dy-
b
namically trapped at a smaller phase winding than for
the fully relaxed system. As the barrier height is in-
creased, the behavior of the phase winding approaches
a smoothed-out step-like behavior, for V /J = 1.8 and
b
V /J =2.4. Wenotethatthissmoothedoutstep-likere-
b
sponse in the phase winding is comparableto the results
that were obtained experimentally in Refs. [2]. In the
inset of Fig. 4, we plot the averagephase winding, time-
averaged over four phonon periods, [t4 −4T,t4], before
the barrier is ramped down. This is the phase winding
thatthesystemrelaxestoatlongertimes,aftertheoscil-
latory behavior has damped down, with the stirring on.
We note that for V /J = 1.0 and V /J = 1.8 the time-
b b
averagedphasewindingisveryclosetothephasewinding
Figure 3: We depict the time evolution of the radially pro- shown in the main panel, because the oscillatory behav-
jected density, n1D(θ), for ωs/ω0 = 3.4. In panels (a) and ior has damped out on the time scale of the experiment.
(b), we use the barrier height Vb/J = 1.8, and (c ) and (d) However,forVb/J =2.4,thephasewindingthatisshown
Vb/J = 2.4. In (b) and (d) the density is plotted in the ref- intheinsetissmoothedouttoanalmostlinearbehavior.
erence frame of the barrier. The phonons travel as straight
This indicates that the classical limit of this response is
lines in this depiction.
5
tion along the ring falls off exponentially, with a length
e winding 2334....5050 Time-avg. phase winding01234.....000000.0 1.0 2.0 3.0 4.0 fFdtsohceoraenrlsTeNceidlr0=φcau≈=t5mo0Nm~fen2,srNK.ean0,nTc/dwe(hπeiNosmhflaρebtvn0heekgienBtlhgrφTinst)=chg.a,e3NleLt30o6chtµiaas=mslta.th2oteπoTbmρnhe0uismnc≈uorbemmes1rupb2laeo4trrfs.,e4cdiµanonmtndoa-.
has ωs/ω0 ratio Lc/lφ that is smaller than 1. However, this does
al p 2.0 suggest, that the phase coherence across the barrier can
e fin be further stabilized by increasing this ratio. This ratio
ag 1.5 can also be written as
er
v
A 1.0 Lc 4π2
= (4)
0.5 Vb/J=1.0 lφ n2Dλ2T
Vb/J=1.8
0.0 Vb/J=2.4 where λT is the thermal de Broglie wavelength, λT =
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (2π~2/(mkBT))1/2, and n2D =N/(πρ20) is a the density
Barrier sweep frequency, ωs/ω0 of a hypothetical system of area πρ20 with N atoms. An-
other way of stating the meaning of this ratio is that it
Figure 4: We depict the average final phase winding as a describes the magnitude of the phase difference across
function of the barrier sweep frequency for barrier heights, the barrier, (∆φ)2 ∼ L /l . Therefore, the figure of
c φ
Vb/J =1.0,1.8,2.4. For Vb/J =1, the system is remains dy- merit that determines if the elongated 3D condensate
namicallytrappedbelowtheequilibriumphasewindingnum-
is phase coherent along its extended 1D axis, is of the
ber. For V /J = 1.8 and V /J = 2.4 a step-like response is
b b form of an inverse 2D phase space density. The optimal
achieved. The black line indicates steps at ωs/ω0 =n+1/2, regime is that of low temperatures, and ring condensate
where n is an integer. The inset shows the time averaged
with a small radius and high density. If, on the other
phase winding before the barrier is ramped back down. For
largebarrierheightsthesystemapproachesalinearresponse. hand, one wants to explore the effective 1D regime of a
ThistendencyisalreadyvisibleforV /J =2.4,andcontinues phase-fluctuating condensate, this figure of merit has to
b
for higher values. be increased above 1.
nearlyreached,asexpectedfora fully disconnectedring. IV. OPTIMIZING THE BARRIER PROTOCOL
We thereforeconclude thatthis limit shouldbe observed
in experiment for high barrier potentials, at long stir- As a last point, we compare the two ramp-up proto-
ring times and instantaneous ramp-down of the barrier. cols,mentionedabove,aswellasdifferentrampingtimes
Furthermore, we conclude that the re-emergence of the for the barrier. In Fig. 5 (b) we show the case in which
step-like behavior after the ramp-down of the barrier, is thebarrierheightisrampedupfirst,whileremainingsta-
duetothenon-zerotimeofthisramp-down,duringwhich tionary,and then its velocity. This is the case which has
the system develops a well-defined phase winding. This been discussed thoughout this paper. In Fig. 5 (c) we
motivates our proposal to increase this ramp-down time show the case in which the velocity is always at its final
in Sect. IV. magnitude, and the height is ramped up from zero. We
In Fig. 5 (a) and (b) we elaborate on the behavior seethatinthelattercasetheresponseisvisiblymoreos-
shown in Fig. 4. We show the distribution of the phase cillatoryandnoisy. Here,the phase slips begin to occurs
winding, as a function of time, which goes beyond the around t = 200∆t, when the barrier is still well below
expectation value of this distribution that was shown in itsmaximumheight. Eachphaseslipgeneratesphononic
Fig. 4. In Fig. 5 (a) we show the case of the larger excitations which are released into the bulk of the con-
value of the barrierheight, V /J =2.4,and in Fig. 5 (b) densate, as visible in Fig. 5 (c). Similar processes were
b
we show the case with V /J =1.8. For the intermediate observed in Ref. [10]. Additionally, we observe that it
b
barrier height of V /J = 1.8, the system converges to a takes longer for the system to relax to a single phase
b
single value of the phase winding over the time of the winding. This suggests that the density at the barrier
experiment. The transient oscillations are damped out should be minimized when the phase slips occur to re-
onthis time scale. Forthe largervalue ofV /J =2.4the duce undesirable excitations.
b
distribution of phase windings is wider and stays oscilla- This is illustrated in Fig. 6. Here, we show the total
tory throughout the experiment time. It does not settle energyperparticleduringtheevolutionofthesetwopro-
to a single value, but rather a distribution that mostly tocols. A larger magnitude of this quantity will result in
includes n∆θ =3 and 4, in this example. a higher temperature of the system after it has thermal-
We note that the width of this distribution is con- ized,andthereforecanbeusedasameasureforhowwell
trolled by the long range phase fluctuations along the the SQUID is implemented. As visible, in this protocol
quasi-1D geometry of the ring-shaped condensate. As additional undesired excitations are created, along with
discussedinRef. [11],thesingleparticlecorrelationfunc- the desired phase slips, which leads to longer relaxation
6
time [s] time [s]
0 0.362 0.724 1.09 1.45 0 0.362 0.724 1.09 1.45
(a) 5 ωs/ω0=3.8 10000 t1 t2 t3 2.10
Phase winding 1234 E - E [units of J]0 68000000 11..2668 cle, (E - E)/N [nK]0
-01 Vb/J=2.40 nergy, 4000 0.84 er parti
E p
(b)5 Total 2000 0.42 nergy
ng 4 pprroottooccooll 12 E
di 3 0 0.00
e win 2 0 500 tim 1e0,0 t0/∆t 1500 2000
s
ha 1
P
0 Figure 6: We depict the dynamics of the energy for sweep
-1 Vb/J=1.80 frequency ωs/ω0 = 3.8 and barrier height, Vb/J = 1.8. In
protocol 1 the barrier is stationary while it is ramped up,
(c)
5 from t1 =100∆t tot2 =300∆t andthenaccelerated at max-
g 4 imum height, from t2 =300∆t to t3 =500∆t. In protocol 2,
n thebarrierstirsatconstantfrequencywhilethemagnitudeis
di 3
se win 2 reanmerpgeydautpt,=f0r.om t1 = 100∆t to t2 = 300∆t. E0 is the total
ha 1
P
0
-1 Vb/J=1.80 sequences with slower ramp-up and ramp-down. Fur-
0 500 1000 1500 2000 thermore, in panel (g) we show the resulting increase of
time, t/∆t the energy of the system, which is again improved for
the slower ramp times. For these comparatively small
0 0.2 0.4 0.6 0.8 1
stirringfrequencies,the improvementis primarily due to
Figure 5: We plot the occupation of each phase winding for the slower ramp-down. For higher stirring frequencies,
sweep frequency ωs/ω0 = 3.8 and barrier heights Vb/J = the slower ramp-up time leads to a further improvement
1.8,2.4. In (a) and (b), the barrier is stationary while it is oftheoperation,becausethephononpulsethatiscreated
ramped up, from t1 = 100∆t to t2 = 300∆t and then accel- by the barrier acceleration is reduced. We give an indi-
erated at maximum height, from t2 = 300∆t to t3 = 500∆t cation for this behavior in panel (b). Among the three
(protocol 1). In (c), the barrier stirs with a constant fre- phase winding evolutions that are shown, the protocol
quency while the magnitude is ramped up, from t1 = 100∆t that is shown in panel (e) is the least oscillatory.
to t2=300∆t (protocol 2).
Furthermore, we point out that the slow ramp-down
time also allows for the sloshing motion of the system to
damp down. The detrimental feature of this motion is
timesandadditionalheatingofthesystem. Theseeffects
not due to the magnitude of the phase fluctuations, but
are more pronounced at higher stirring frequencies than
because it can the lead the system to arrive at a phase
at lower stirring frequencies and are expected to play a
winding other than the ground state one, if the barrier
bigger role as the stirring frequency is further increased.
is ramped down fast. As mentioned above, this phonon
Again, we observe that the preferable operation of the
motion damps out for intermediate values of the barrier,
SQUID consists of first ramping up the barrier height,
butnotforhighbarriers. Ifthe ramp-downis donesuffi-
and then the barrier velocity.
cientlyslow,thisprovidesenoughtimefortheoscillations
Nextweaddresstheinfluenceoftheramptimesonthe
to damp out, while the barrier is at intermediate values.
SQUIDoperation. Asakeyexample,wefocusonthefirst
stepofthephasewinding,aroundωs/ω0 ≈1/2. InFig.7,
weconsiderthreedifferenttimesequences. Thesequence
V. CONCLUSIONS
that we discussedup to here,is shownin panel(c ). Ad-
ditionally,weconsidertwoothersequences,shownin(d)
and(e). The sequencein(d) featuresa slowrampdown, In conclusion, we have demonstrated the atomtronic
and the sequence in (e) features both a slow ramp-up implementation of a SQUID in a toroidal Bose-Einstein
and a slow ramp down. For stirring frequencies near the condensate, with particular emphasis on the effect of
first step, these two sequences both lead to an improve- thermal fluctuations. These are included in our Trun-
ment: In panel (f) we show the resulting phase winding. cated Wigner approximation of a realistic system, as re-
The phase winding increases more steeply for the time alized in the experiments of Refs. [2, 3]. We show that
7
the regime of SQUID operation is viable for sufficiently
time [s]
0 0.362 0.724 1.09 1.45 large barrier heights, which imitates an SNS junction of
1.2
g (a) a solid state SQUID. For too low of a barrierheight, the
se windin 00 ..168 Vωbs=/J0=.25.54ω0 rthateeeqoufiplihbarisuemslpiphsasies wtoinodlionwg nfourmtbheer fsoyrstaemgivteonrmeaocvh-
pha 0.4 ing barrierspeed. Forlargerbarrierheights,the thermal
Average - 00 ..022 τττbbb000===τττωωω===242000000∆∆∆ttt;;; τττbbbfff===1822000000∆∆∆ttt prwiheinar.dseinTflghueicstcuahacathriioaencvtseedsruifspotpircr,eeisststheetphr-eliinkctoeehrbemerheendavciaeitoaercvoraoflsutsheteshoepfhbtaahsree-
ng 2 (b)
di barrier height, or during the ramp-down of the barrier,
n
e wi 1.5 ωs=1.60ω0 if this occurs on a sufficiently long time scale. Further-
s
a 1 more,weinvestigatetworamp-upprotocolsofthebarrier
h
p
ge 0.5 height and the barrier velocity, and show that ramping
a
er upthebarrierheightfirst,beforesettingitinmotion,re-
Avα(t)b 01 (c) ω=0 ω= ωs sauchltisevinedlebsysihnecarteiansgin.gAbnotahddthiteiornaamlpim-upprotivmemeeonftthceanbabre-
αα(t)(t)bb 0011 ((de)) ω=0 ωω== ω0s ω=ωs ritnhieeraabrnaeddrrutiehcrteiorisnamsoepft-tdihnoewmpnhototiinmoonen.. pTTuhhleseeslstolhowawetreirrsaremammpi-ptut-epddorwewsnuhleotnsf
0 the barrierimprovesthe reemergenceofanintegerphase
0 500 1000 1500 2000
time, t/∆t winding after the stirring. We emphasize that these re-
1.2 400 sults and considerations will similarly apply to all atom-
g
ndin 1 (f) (g) troniccircuits,thatarebasedoncondensatedynamicsin
ge final phase wi 000...468 123000000E-E [units of J]f0 ntceiorrnceu-sttirtitrvoyiatwlhitetrhaempBgeoresgoei-mnEgeitnfirsietelesdi,noafncodimnadirteeanttsihanetgreessf.uopreerocfobnrdouacdtiinng-
era 0.2
v
A 0 0
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1
Barrier sweep frequency, ωs/ω0 Barrier sweep frequency, ωs/ω0
Figure 7: (a) Time dependenceof theaverage phasewinding
for three different barrier sequences. Starting at t1 =100∆t,
thebarrierheightisrampedupoverτb0,thenacceleratedover
τω. The cloud is stirred at constant angular velocity and the
barrierisrampeddownoverτ ,startingat1900∆t−τ . The Acknowledgments
bf bf
normalizedbarrierheightisdepictedin(c)τb0 =200∆t,τω =
200∆t, τbf = 200∆t, (d) τb0 = 200∆t, τω = 200∆t, τbf =
1200∆t, and (e) τb0 = 400∆t, τω = 400∆t, τbf = 800∆t.
Average final phase winding (f) and final energy per particle We acknowledge support from the Deutsche
(g) as a function of the stirring frequency. The maximum Forschungsgemeinschaft through the SFB 925 and
barrier height is Vb/J = 2.4 and the stirring frequency is the Hamburg Centre for Ultrafast Imaging, and from
ωs = 0.55ω0 for (a) and (c )–(g). In (b) we show the phase the Landesexzellenzinitiative Hamburg, supported by
winding obtained for ωs/ω0 =1.6. the Joachim Herz Stiftung.
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