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PHYSICAL REVIEW LETTERS

PRL 106, 025302 (2011)

Dynamics of a Tunable Superfluid Junction
L. J. LeBlanc, A. B. Bardon, J. McKeever, M. H. T. Extavour, D. Jervis, and J. H. Thywissen
Department of Physics, University of Toronto, 60 St. George, Toronto ON, Canada, M5S 1A7

F. Piazza and A. Smerzi
INO-CNR, BEC Center, and Dipartimento di Fisica, Via Sommarive 14, 38123 Povo, Trento, Italy(Received 17 June 2010; published 12 January 2011)
We study the population dynamics of a Bose-Einstein condensate in a double-well potential throughout
the crossover from Josephson dynamics to hydrodynamics. At barriers higher than the chemical potential,
we observe slow oscillations well described by a Josephson model. In the limit of low barriers, thefundamental frequency agrees with a simple hydrodynamic model, but we also observe a second, higher
frequency. A full numerical simulation of the Gross-Pitaevskii equation giving the frequencies andamplitudes of the observed modes between these two limits is compared to the data and is used tounderstand the origin of the higher mode. Implications for trapped matter-wave interferometers are
discussed.
DOI: 10.1103/PhysRevLett.106.025302

PACS numbers: 67.85.d, 03.75.Lm, 67.10.Jn, 74.50.+r

Quantum mechanical transport is a consequence of spatial variations in phase. Superfluids behave like perfectinviscid irrotational fluids, whose velocity is the gradientof a local phase, so long as the confining potential issmooth on the scale of the healing length. Where thedensity is small, as it is near surfaces, quantum kinetic
terms must be added to the classical hydrodynamic equations. Macroscopic quantum coherence phenomena, suchas Josephson effects, emerge when superfluids are weakly
linked across such a barrier region.
Josephson effects have been demonstrated with superconductors [1], liquid helium [2,3], and ultracold gases inboth double-well [4,5] and multiple-well optical trapping
potentials [6]. The canonical description of these experiments employs a two-mode model [7–9], in which a sinusoidal current-phase relationship emerges. Hydrodynamics
has also been studied in both liquids and ultracold gases[10]. The relative diluteness of gases makes a satisfying
ab initio description possible [11].
In this Letter, we study the transport of a Bose-Einsteincondensate (BEC) between two wells separated by a tunable barrier and observe the crossover from hydrodynamic
to Josephson transport. As the barrier height Vb is adjusted
from below to above the BEC chemical potential, , thedensity in the link region decreases until it classicallyvanishes when Vb ¼ . The healing length in the linkregion, , increases with Vb and dictates the nature oftransport through this region. Oscillatory dynamics spanning three octaves are observed as we smoothly tune from 0:3d to 2d, where d is the separation between thewells.
Examination of the dynamics of an elongated BEC in a
double well is timely. Recent experiments have createdsqueezed and entangled states by adiabatically splitting a
BEC [12–14]. The degree of squeezing inferred in the0031-9007=11=106(2)=025302(4)elongated case [12,13] seems to exceed what would be
expected in thermal equilibrium [14], raising the possibility that out-of-equilibrium dynamics may be important.
With much remaining to be explored in these systems,
this work represents the first study of the dynamics in the
crossover regime.
Our experiment begins as 87 Rb atoms in the jF ¼ 2;
mF ¼ 2i ground state are trapped on an atom chip andevaporatively cooled in a static magnetic potential BS ðrÞ,as described elsewhere [15]. To prevent gravitational sag
and to compress the trap in the weak direction (withcharacteristic trap frequency !y ¼ 2  95 Hz), we add
an attractive optical potential with a 1064 nm beam. We
dress the static potential with an oscillating radiofrequency (rf) magnetic field [16,17] radiating from twoparallel wires on the atom chip [Fig. 1(a)]. In the rotatingwave approximation, the adiabatic potential created by the
combination of the static chip trap, the rf dressing, and theoptical force is
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃUðrÞ ¼ m0F sgnðgF Þ@ ðrÞ2 þ 2? ðrÞ þ 12m!2y y2 ; (1)where m0F ¼ 2 is the effective magnetic quantum number,ðrÞ ¼ !rf  jb gF BS ðrÞ=@j is the detuning, ? ðrÞ ¼
jb gF Brf;? ðrÞ=2@j is the rf Rabi frequency, Brf;? ðrÞ ¼
jBS ðrÞ  Brf ðrÞj=jBS ðrÞj is the amplitude of the rf field
locally perpendicular to BS ðrÞ, b is the Bohr magneton,gF is the Landé g factor, @ is the reduced Planck’s constant,and m is the atomic mass. By assuming the individual wells
are harmonic near each minimum, calculations show that!z ¼ 2  425 Hz, and !x varies from 2  350 Hz to2  770 Hz as we tune from low to high barriers. For
comparison between theory and experiment, we accountfor small corrections to Eq. (1) beyond the rotating-wave
approximation [18,19].

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PHYSICAL REVIEW LETTERS

FIG. 1 (color online). (a) Schematic of atom chip double-welltrap. Central Z wire [34] carries static trapping current, IS ¼
2 A, which, with uniform external fields Bext ¼
h2:2; 0:11; 0i mT, results in an Ioffe-Pritchard style trapwith harmonic trapping frequencies ð!x0 ;z0 ; !y0 Þ ¼ 2 
ð1300; 10Þ Hz. Side wires are 1.58 mm from trap center andcarry rf currents with amplitude Irf . This rf current produces az-polarized field at the trap location with amplitude Brf ¼
23:6  0:6 T [peak Rabi frequency  ¼ 2  ð82  2 kHzÞ].
A levitation beam [gray (pink)] is positioned to provide a forcecanceling gravity (z direction) while compressing the sample
along y. Atoms are trapped 190 m from the chip surface. (b) A
schematic one-dimensional cut at t ¼ 0:5 ms through trapping
potential along x (solid line) in the presence of linear bias
(dashed line) and (c) balanced potential at t ¼ 0, with Z0 Þ 0.

After turning on the dressing field at a frequency!rf ¼ 2  765 kHz, where the trap is a single well, we
evaporatively cool to produce a BEC with no discernible
thermal fraction. In 20 ms, we adiabatically increase !rf to
a new value characterized by 0  ðr ¼ 0Þ, such that thebarrier Vb rises and the dressed state potential splits along
the x direction into two elongated traps [20].
Using a second 1064 nm beam weakly focused offcenter in x, an approximately linear potential is added
across the double-well junction to bias the population
towards one well [Fig. 1(b)]. By applying the bias beam
before and during the splitting process, we prepare systems
of atoms with a population imbalance Z  ðNR  NL Þ=ðNR þ NL Þ, where NR (NL ) is the number of atoms in theright (left) well. The range of initial population imbalances
Z0 ¼ Zðt ¼ 0Þ we use is 0.05 to 0.10, small enough to
avoid self-trapping [4]. To initiate the dynamics, the powerof the bias beam is ramped off in 0.5 ms (faster than the
population dynamics) and the out-of-equilibrium system isallowed to evolve for a variable time t in the symmetricdouble well [Fig. 1(c)].
To measure the time-dependent population ZðtÞ, we
freeze dynamics by rapidly increasing both Brf and !rf toseparate the wells by 70 m, where Vb =  104 . We
release the clouds from the trap and perform standard
absorption imaging along y after 1.3 ms time-of-flight[Fig. 2(b)]. Analysis of these images allows us to determine
NR and NL to a precision of 50 atoms.
Upon release of the potential bias, we find that the
population ZðtÞ oscillates about Z ¼ 0 [Fig. 2(a)] [21].
To analyze the dynamics, we use a Fourier transform (FT)

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FIG. 2 (color online). (a) Population imbalance, Z, vs time for ¼ 2  ð0:1  0:5Þ kHz, N ¼ 5900  150. The dashed line
is a decaying two-frequency sinusoidal fit to the data, using two
fixed frequencies from the FT (lower inset). Each point is theaverage of six repetitions of the experiment; error bars are
statistical. (b) Averaged absorption image after separation and1.3 ms time of flight, with right and left measurement regions(dashed boxes) indicated. (c) FT amplitude spectrum of data
showing two distinct peaks at 268  6 and 151  13 Hz rising
above the noise floor (grey).

to find the dominant frequency components [Fig. 2(c)]. We
repeat this measurement at many values of Vb =, where is the Thomas-Fermi chemical potential, by varying 0 .
For the purposes of this analysis, we ignore the decay ofthis signal, the 1=e time constant of which is typically two
oscillation periods.
When the barrier is low, ZðtÞ consistently displays two
dominant frequency components. For higher barriers, theamplitude of the higher-frequency mode decreases until
only a single frequency rises above the noise floor. The
white points in Fig. 3 give these frequencies as a functionof the experimental parameter 0 and the calculated ratioof barrier height to chemical potential, Vb =. The ensembles used in Fig. 3 had total atom number N ¼ 6600 
400ð1700Þ, where the error bar is statistical (systematic).
In the low- and high-barrier limits, simple models can beused to understand the dynamics. For low barriers, thehydrodynamic equations of motion can be used to estimate
the frequency of population oscillation. Assuming a harmonic population response for some Z0 , the response
frequency is!2HD ¼ 

2 Z~
n^  rðUþ gÞdS;
mNZ0 S(2)where  is the density of the condensate at t ¼ 0, S is thesurface in the y-z plane bisecting the double well, and n^ isthe vector normal to this surface. Plotting !HD in Fig. 3(dotted line), we find good agreement with the lower frequency mode at low barriers. Since tunneling cannot contribute to hydrodynamic transport, !HD ! 0 as Vb ! .
The breakdown in hydrodynamics also coincides with anincreasing healing length, as shown in the inset of Fig. 3.
In the opposite limit, when tunneling dominates transport, a Josephson model [8] accurately predicts thefrequency of the highest barrier points,

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PHYSICAL REVIEW LETTERS

FIG. 3 (color online). Frequency components of population
imbalance vs rf detuning (measured) and barrier height tochemical potential ratio (calculated). Experimental points (whitecircles) represent the two dominant Fourier components at each
detuning; error bars represent uncertainty contributed by noise in
the FT from a single time series, but do not include shot-to-shot
fluctuations. The spectral weight is represented through the color
map, which has been linearly smoothed between discrete valuesof Vb = and darker colors indicate greater spectral weight. All
calculations use N ¼ 8000 and Z0 ¼ 0:075, and a singleparameter fit of the data to the GPE curves shifts all experimental
points by shift ¼ 2  5:1 kHz [19] to compensate for a systematic unknown in BS ð0Þ. Statistical vertical error bars areshown, while a typical horizontal statistical error bar is shownat Vb =  0:5. Dashed lines represent 3D GPE frequencies, thesolid line the plasma oscillation frequency predicted by the
Josephson model, !p , and the dotted line the hydrodynamic
approximation, !HD =2. White bars at Vb =  0:1 indicate thebounds of the GPE simulation corresponding to the systematic
plus statistical uncertainty in atom number. Inset: ratio of healing
length, , to interwell distance, d, as a function of Vb =.  is
calculated at the center of the barrier.

!2p ¼



1
@loc
;
E
E
þ
N
@N
@2

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The structure and origin of the higher-lying dynamicalmode can be studied within the simulations. If our trap
were smoothly deformed to a spherical harmonic potential,
the two observed modes would connect to odd-parity
modes [11]: the lower mode connects to the lowest
m ¼ 0 mode (coming from the ‘ ¼ 1 mode at sphericalsymmetry, where the quantum numbers ‘ and m label theangular momentum of the excitation and its projection
along the axis of symmetry, y, respectively), while thehigher mode originates from the lowest m ¼ 2 mode(‘ ¼ 3 at spherical symmetry) [24].
With insight from GPE simulations, the observation of asecond dynamical mode, which was not seen in previous
experimental work [4,5], can be explained. In a purely
harmonic trap, a linear bias excites only a dipole mode[25]. By breaking harmonicity along the splitting direction,
x, the barrier allows the linear perturbation (‘ ¼ 1, m ¼ 0,
where x is the azimuthal axis) to excite multiple Bogoliubov
modes [26]. Numerical studies show that two additional
ingredients are required to excite the higher mode. First,
atom-atom interactions couple the x excitation to the transverse (y, z) motion through the nonlinear term in the GPE.
Second, the anisotropy of the trap in the y-z plane mixes the
m ¼ 0 and m ¼ 2 modes such that each of the resulting
modes drives population transfer between wells.
Figure 4 shows the relative strength R1 ¼ a1 =ða1 þ a2 Þof the lower frequency mode as a function of the barrierheight. The amplitude a1 (a2 ) of the lower (higher) frequency mode is extracted from a decaying two-frequency
sinusoidal fit. The modes have comparable strength, evenin the linear perturbation regime, when the barrier is below
the chemical potential. The small spread in the GPE amplitudes shown by the grey band indicates that the higher
mode is excited independently of the initial imbalance, andis not simply due to a high-amplitude nonlinearity.

(3)

where E is the energy difference between the symmetricand antisymmetric ground states of the double-well potential, N is total atom number, and loc is the chemical
potential on one side of the well [19]. The agreement issurprisingly good even for Vb just above , beyond which
the frequency decreases exponentially. To our knowledge,
this constitutes the first direct observation of tunnelingtransport of neutral atoms through a magnetic barrier,
only inferred, for instance, in Refs. [12,22].
To explain the crossover behavior and the existence ofthe higher-frequency mode, we turn to numerical solutionsof a time-dependent three-dimensional Gross-Pitaevskii
equation (GPE) [8,23], which should describe allmean-field dynamics at T ¼ 0. The slope and separationof the measured frequencies are well captured by the GPE,
as shown in Fig. 3, though the decay of population imbalance is not reproduced by these simulations.

FIG. 4. Fraction of low-frequency mode in population dynamics. Dashed line shows the GPE simulation for 8000 atoms withinitial imbalance Zð0Þ ¼ 0:075. The grey shaded area represents
the variation of the GPE calculations over the range of Zð0Þ ¼
0:05 to 0.10. The vertical error bars are statistical; the statisticaluncertainty in  is 2  0:5 kHz (not shown). The GPE calculation gives R1 ¼ 1 when Vb = ’ 1:1.

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PHYSICAL REVIEW LETTERS

The trend in R1 reflects the shape of the trap. When thebarrier is raised from zero, the higher mode is at first moreeasily excited due to an increased anharmonicity along x as
the trap bottom becomes flatter. By further increasing thebarrier, the higher-frequency mode disappears from the
population oscillation spectrum due to the vanishing excitation of transverse modes. As the wave functions in eachindividual well are increasingly localized to the effectively
harmonic minima, the linear bias no longer excites intrawell transverse motion. Furthermore, in the linear perturbation regime, the interwell Josephson plasma oscillation,
like all Bogoliubov modes, cannot itself trigger any othercollective mode.
In conclusion, we have studied the quantum transport ofa BEC in a double-well potential throughout the crossoverfrom hydrodynamic to Josephson regimes. Apart fromfundamental interest, knowing and controlling the natureof superfluid transport is crucial for technological applications of weak-link-based devices, such as double-slit interferometers [12,20,27–29]. The adiabatic transformation of
a BEC from a single- to a double-well trapping potential
has been discussed in recent experimental works[12,14,22,30,31] in the context of the Josephson model,
valid at high barriers [32]. Our work demonstrates that for
Vb < , the lowest mode frequency will lie below that
estimated by the Josephson model. Furthermore, thehigher-lying mode we observe approaches the lowest collective mode as !y  !z [19] and may be important to the
dynamics of splitting in strongly anisotropic double wells[12,33]. Whether using splitting to prepare entangled states[14], or recombination [31] to perform closed-loop interferometry [30], an improved understanding of double-welldynamics provides a foundation for controlling mesoscopic superfluids.
We would like to thank T. Schumm for early experimental work, A. Griffin, P. Krüger, D. McKay, M. Sprague, andE. Zaremba for helpful discussions, and J. Chwedeńczuk
for help with numerical simulations of the GPE. This work
has been generously supported by CIfAR, CFI, CQIQC,
and NSERC.

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