Table Of ContentSpontaneous creation of non-zero angular momentum modes in tunnel-coupled
two-dimensional degenerate Bose gases
T.W.A. Montgomery, R.G. Scott, I. Lesanovsky, and T.M. Fromhold
Midlands Ultracold Atom Research Centre, School of Physics and Astronomy,
University of Nottingham, Nottingham, NG7 2RD, United Kingdom.
(Dated: 07/10/09)
We investigate the dynamics of two tunnel-coupled two-dimensional degenerate Bose gases. The
0 reduced dimensionality of the clouds enables us to excite specific angular momentum modes by
1 tuning the coupling strength, thereby creating striking patterns in the atom density profile. The
0
extremesensitivityofthesystemtothecouplingandinitialphasedifferenceresultsinarichvariety
2
ofsubsequentdynamics,includingvortexproduction,complex oscillations in relativeatom number
n and chiral symmetry breaking dueto counter-rotation of the two clouds.
a
J
5 I. INTRODUCTION
2
] A major focus of cold-atom research is coupling mul-
s tiple degenerate Bose gases (DBGs) using atom chips
a
and optical lattices [1–14]. The results provide step-
g
- ping stones to future applications, such as interferom-
t
n eters or processors of quantum information. Since atom
a chips and optical lattices typically generate very high FIG.1: (Coloronline)Schematicconstantdensitysurfacesof
u (kHz) trapping frequencies, there is growing interest in theupper(blue/lightgrey)andlower(red/darkgrey)DBGs.
q Arrows show co-ordinate axes.
the dynamics of coupled one- and two-dimensional (1D
.
t and 2D) clouds [15–19]. This interest has also been fu-
a
eled by the radically different physics that has been ob-
m periodically grow and decay exponentially. The growth
servedinlower-dimensionalsystems,suchasthesuppres-
ratesofthesemodesareextremelysensitivetochangesin
-
d sionofequilibration[20],quasicondensation[21],andthe thecouplingandinteractionstrengths,andintherelative
n Kosterlitz-Thouless transition [22–28]. Since this body
phaseofthetwoDBGs. Thegrowthofmodeswithlower
o of work has uncovered such rich dynamics, it is natural
angular momentum becomes unstable due to collisions
c to wonder how coupled lower-dimensional systems will
[ betweenrotatingandnon-rotatingatoms,disruptingthe
behave, and whether they can reveal a crossover to 3D internal structure of the clouds. This leads to a variety
1 phenomena. Some recent work on 1D coupled rings has
of subsequent dynamics such as vortex production, os-
v shown that the reduced dimensionality leads to unex-
cillations in relative atom number, and chiral symmetry
4 pected effects, such as spontaneous population of rotat-
breaking due to counter-rotation of the two clouds.
4
ing excitations and chiral symmetry breaking [16]. Fur-
3
therworkisnowneededtoexploresymmetrybreakingin
4
Josephsonjunctions,theboundariesoflower-dimensional
. II. SYSTEM AND METHODOLOGY
1 physics,andtoestablishhowdouble-wellinterferometers
0
will perform in reduced dimensions. The system consists of two 2D DBGs, referred to as
0
the upper and lower DBGs, in the z = +z and −z
1 In this paper, we investigate the dynamics of coupled 0 0
: 2D disk-shaped DBGs. We find that the reduced di- planes respectively, as shown in Fig 1. The two DBGs
v
arecoupledbyatunneljunctioncreatedbythesymmet-
mensionality of the clouds has profound implications for
i
X the dynamics, because the instabilities of excited states ric double well potential Vdw(z). In the x−y plane, the
DBGs are contained by the harmonic trapping potential
r observed in three dimensions are suppressed [29, 30].
a V(r) = 1mω2r2, where m is the mass of a single atom,
Starting from an irrotational stationary state, and with- 2
out introducing any stirring, we observe spontaneous ω is the trap frequency and r =px2+y2. For |z|≈z0,
occupation of low-lying excitations with non-zero an- Vdw(z) can be approximated by the tight harmonic po-
gular momentum, mediated by the interatomic interac- tential Vα/β(z)= 1mλ2ω2(z∓z )2, where λ≫1 to en-
sw 2 0
tions [16, 31, 32]. We use a linear stability analysis to sure that λ~ω > µ. Consequently, atomic motion in the
predictwhichrotatingBogoliubovmodeswillbeexcited, z direction is frozen into the single-particle groundstate,
and hence identify regimes where we can excite specific and the DBG wavefunction becomes 2D [33]. Hence, in
modes alone by tuning the coupling strength. This tar- the weak coupling limit [34], we may representthe order
geted selection of a single rotating mode creates striking parameter for the two 2D DBGs by the scalar complex
oscillatory patterns in the atomic density profile. As we field
excite different rotating modes, we uncover a rich pa-
rameter space. Modes with a high angular momentum ψ =ζ(z−z )χα(ρ,φ,τ)+ζ(z+z )χβ(ρ,φ,τ) (1)
0 0
2
where the superscripts α and β refer to the upper and
lowerDBGs,ζ isthenormalizedsingle-particleharmonic
α/β
groundstateofV (z),φistheazimuthalangle,andwe
sw
have introduced the dimensionless time τ = ωt and the
dimensionlesslengthρ=r/a ,inwhicha = ~/mω.
ho ho p
Substituting Eq. (1) into the Gross-Pitaevskii equation
results in two coupled equations for χα/β(ρ,φ,τ) [35]
i∂ χα/β =− ∂2+ 1∂ + 1 ∂2−ρ2+µ χα/β
τ h ρ ρ ρ ρ2 φ i (2)
+ γ|χα/β|2χα/β −|κ|χβ/α
where the dimensionless quantities γ = (8πλ)1/2a /a
0 ho
and κ= 1a2 ζ(z+z ) ∂2−2mV (z)/~ ζ(z−z )dz
2 hoR 0 (cid:2) z dw (cid:3) 0
represent the interaction and coupling energy respec-
tively, and a is the s-wave scattering length.
0
At τ = 0, there are an equal number of atoms, N ,
0
in each well, and χα/β is the non-rotating groundstate
of V(r), with chemical potential µ . For this initial
0
configuration, and finite κ, there are two possible sta-
tionary states of Eq. (2). These are the ground state,
defined by χα(r,φ,0) = χβ(r,φ,0), with chemical po-
tential µ − |κ|, and the excited asymmetric station-
0
ary state, henceforth referred to as the π-state, defined
FIG. 2: (Color) Upper plot: Imaginary component of E
by χα(r,φ,0) = −χβ(r,φ,0), with chemical potential κ,l
(scale shown at top) for angular momentum modes |l|=0 to
µ +|κ|[36]. Unsurprisingly,thegroundstateisstablefor
0 4as a function of |κ|. Lower surface plots: three examplesof
allcoupling strengths. However,the π-state shows much
excited modes with (a) l =4 and no radial node (|κ|=1.7),
richer behavior. In three dimensions, the phase discon- (b) l = 1 and no radial node (|κ| = 0.56), (c) l = 1 and a
tinuity may bend, creating vortices via the well known single radial node |κ| = 1.35. The vertical height shows the
snakeinstability [29,30,37,38]. This processcannotoc- atomic density of the mode, and the colors denote its phase
curinoursystem,becausethereduceddimensionalityof (blue= 0, green = π, red = 2π).
the disks precludes the movement of the phase discon-
tinuity. In the following section, we perform a stability
analysis to identify what excitations may occur.
where ǫ=γN and E (≡E in Eq. 3) is the energy of
0 κ,l l
the Bogoliubov mode for a particular κ and l.
The upper panel in Fig. 2 is a gray-scale plot of the
imaginary component of E for |l| = 0 to 4 calculated
III. STABILITY ANALYSIS OF THE π-STATE κ,l
asafunctionof|κ|,forafixedinteractionstrengthǫ=7.
We are interested in complex eigenvalues because they
Weperformastabilityanalysisofthestationarystates indicate unstable modes which grow exponentially at a
by calculating the excitations of the system using the rate proportional to the imaginary component. The re-
Bogoliubov ansatz [39] sults are presented in terms of |l| because each Bogoli-
ubov mode has a degenerate pair with equal and oppo-
χα/β(ρ,φ,τ)=χα/β(ρ,φ,0) siteangularmomentum. Eachshadedregioninthefigure
+ uα/β(ρ,φ)eiElτ +vα/β∗(ρ,φ)eiEl∗τ . corresponds to a single mode with a particular angular
h l l i
momentum and radial excitation. To illustrate this, we
(3)
pick three example couplings of |κ| = {1.7,0.56,1.35},
In the above equation, the u’s, v’s and mode energies
indicated in the upper panel of Fig. 1 by the vertical
E have been explicitly labeled with their angular mo-
l {dotted,solid,dashed}lines,whichpassthroughshaded
mentum quantum number, l, to derive the four coupled
regions labeled {(a), (b), (c)}. The modes correspond-
Bogoliuvbov-de Gennes equations
ing to these shadedregions are shown,also labeled {(a),
E uα/β = − ∂2+ 1∂ − l2 uα/β +2ǫ|χα/β|2uα/β (b), (c)}, in the lower half of Fig. 2 as surface plots,
κ,l l h ρ ρ ρ ρ2i l l whose vertical height denotes the atomic density of the
−ǫ|χα/β|2vα/β −|κ|uβ/α mode, and the colors denote its phase. Mode (a) in the
−l l
−E vα/β = − ∂2+ 1∂ − l2 vα/β +2ǫ|χα/β|2vα/β lowerpanel ofFig. 2 hasfour quanta ofangularmomen-
κ,l −l h ρ ρ ρ ρ2i −l −l tum (l = 4), but no radial node. Mode (b) is similar to
−ǫ|χα/β|2uα/β −|κ|vβ/α mode (a), but has only one quantum of angular momen-
l −l
(4) tum. Aswemovetolargervaluesof|κ|foragiven|l|,we
3
find regionsof instability correspondingto modes with a evolve in time, a macroscopic number of atoms are scat-
greater radial excitation. For example, mode (c) has a tered from the π-state via the inter-atomic interactions
higher radialexcitation than mode (b) because it has an into the l = 4 mode. To preserve angular momen-
additional radial density node [40]. tum, an equal number of atoms transfer into the de-
The positions of the shaded regions in the upper generate l = −4 mode, producing the star-like inter-
panel of Fig. 2 can be understood by considering the ference pattern with eight density antinodes at τ ≈ 95
limiting case as ǫ → 0 (no inter-atomic interactions). shown in Fig. 3(b). To quantify the growth of non-zero
In this limit, the shaded regions shrink to points where angular momentum modes, we introduce the quantity
the coupling strength, κ, matches the single-particle Pα/β = 1/2π| χα/β(e−ilφ+eilφ)ρdρ dφ|2, which is the
energies, n + |l|/2, where n denotes the number of prlojection of tRhe upper or lower DBG onto modes with
radialnodes in the correspondingwavefunction. Clearly, angular momentum |l|. We plot log (Pα) in Fig. 3(d).
10 4
in this case, excitations can be degenerate. However, Thegraphshowsthatthepopulationofthe |l|=4mode
inter-atomic interactions destroy this degeneracy. In increases linearly on the logarithmic plot, indicating ex-
general, such interactions shift the shaded regions to ponentialgrowth,reachingamaximumofapproximately
higher energy. The size of this shift varies, so that there N /3 at τ ≈ 95. At this point, the growth of the l = 4
0
are certain values of |κ| (for example, |κ| = 1.7, shown mode halts, and atoms begin transferring back into the
by the dotted line in the upper panel of Fig. 2) where π-state. The population of the π-state does not reach
only one mode is unstable, and hence we can tune the zerobecausethestabilityanalysispresentedinFig. 2as-
coupling to excite that mode alone. There are also sumes that all atoms are in the π-state, and is therefore
values of κ (for example, |κ|=2.15)where no modes are nolongervalidonceamacroscopicnumberofatomshave
unstable, meaning that the π-state is stable. In addition entered an excited mode. Eventually, the DBG returns
to shifting the positions of the shaded regions to higher to its original configuration at τ ≈ 160 [Fig. 3(c)], at
κ,increasingtheinteractionsfromzeroalsobroadensthe which point the process repeats. This periodic transfer
unstable regionsand increasesthe imaginary component can be maintained because, due to its large angular mo-
of Eκ,l within them. This illustrates that, although the mentum, the excited mode’s peak density is far fromthe
tunnel coupling provides the energy for the growth of origin, meaning that there is little interaction with the
unstable modes, the scattering from the π-state into the remaining atoms in the π-state. Consequently, the sys-
excitation is mediated by the inter-atomic interactions. temisabletosupressesparticletransferbetweenthetwo
We find that ǫ=7 is a suitable value to provide reason- wellsoveralargetime,asshowninFig. 3(e),whichplots
able growth rates of excited modes, without causing the the number imbalance η =(Nα−Nβ)/2N between the
0
unstable regions to become so broad that they overlap, two wells during the simulation [42].
so that specific modes can still be accessed individually. The width of the unstable l = 4 mode in Fig. 2 is
very narrow, indicating that the formation of the star-
like interference pattern is a sharply resonant process.
Hence, the growth rate varies extremely sensitively with
IV. SIMULATIONS OF THE DYNAMICS κ, reaching zero if κ = 1.7 is changed by only ∼ 2%.
Similarly, the system is also highly sensitive to the rela-
To simulate the growth of the unstable modes, we tive phases of the DBGs. For example, when the above
must model inter-atomic scattering events not cap- simulationisrunwithaninitialrelativephaseof0.9πbe-
tured by the Gross-Pitaevskii equation. Hence we add tweenthetwoclouds,thestar-likeinterferencepatternis
non-zero angular momentum fluctuations of the form no longer observed. Moreover, an initial relative phase
C(α/β)r|l|e−ρ2/2eilφ (thesingle-particleexcitations)to of 0.99π causes a 10 % change in the growth rate of the
P l star-likepattern. Since this predictable andmacroscopic
χα/β [41], then evolve Eq. [2] in time. Here, Cl(α/β) is pattern is stable, it could be detected experimentally in
a random complex number such that h|C(α/β)|i=0 and situ using absorption imaging.
l
h|C(α/β)|i2 ≈ 10−8O(N ). These fluctuations are small
0
enough that they do not raise the temperature signifi-
cantly from zero in our model. B. Oscillations in relative atom number and vortex
production (|κ|=0.56)
A. Periodic excitation of a rotating mode (|κ|=1.7) We now tune |κ| to 0.56 to excite the lowest possible
rotating mode, as shown by the vertical solid line in the
Figure3showstheresultsofasimulationfor|κ|=1.7. upper panel of Fig. 2. Comparing the surface plots in
For this coupling,only a single mode with |l|=4 [shown the lower half of Fig. 2, we see that this mode [Fig.
by surface plot (a) in the lower panel of Fig. 2] is un- 2(b)] is similar to the one previously excited [Fig. 2(a)],
stable, as indicated by the vertical dotted line in the except that it has only one quantum of angular momen-
upper panel of Fig. 2. Initially, both clouds have a tum. Consequently, we now expect the formation of an
smooth density profile [Fig. 3(a)]. However, as we interference pattern with two antinodes. Starting from
4
FIG.3: (Color online)Simulation for|κ|=1.7. (a)-(c)Atom FIG. 4: (Color online) Simulation for |κ| = 0.56. (a)-(c)
density plot (black = high) of upper (first row) and lower Atom density plot (black = high) of upper (first row) and
(second row) DBG at (a) τ = 0, (b) τ = 95 and (c) τ = lower (second row) DBG at (a) τ = 0, (b) τ = 25 and (c)
105. (d) Logarithmic plot of the projection P4α of the upper τ = 75. The positions of vortices are indicated by the black
DBG onto angular momentum modes with |l|=4. (e) Atom (minus) plus, which denotes (anti) clockwise rotation. (d)
numberimbalance, η,between upperand lower well. Solid (dashed) line: logarithmic plot of the projection P1α
(P2α)oftheupperDBGontoangularmomentummodeswith
|l| = 1 (|l| = 2). (e) Atom number imbalance, η, between
upperand lower well.
the smooth atom density profile at τ = 0 [Fig. 4(a)],
such an interference pattern begins to form, but the two
antinodes have very different peak densities [Fig. 4(b)].
that, at this low κ, the system has insufficient energy to
The projectionPα shownby the solidcurve in Fig. 4(d)
1 populate other rotating modes and, consequently, it re-
reveals that, as expected from the stability analysis, the
distributes its energy by performing oscillations in rela-
population of the |l|= 1 modes initially grows exponen-
tiveatomnumber. Thisbehaviorcanalternativelybere-
tially. However, as the number of atoms in the |l| = 1
gardedas instability in the symmetric l=0 groundstate
modes becomes macroscopic, there is also a significant
mode,whichoccurswithin the shadedregionlabeled(d)
rise in the projection Pα [dashed curve in Fig. 4(d)],
2 in the upper panel of Fig. 2.
which is not predicted by the stability analysis. We find
theseunexpectedscatteringprocesseswhenweexcitelow Despite the apparent disruption and the macroscopic
energy modes, which are concentrated close to the cen- population of rotating modes in the simulations pre-
ter of the trap, and hence collide more frequently with sentedhitherto,thesystemhasalwaysconservedzeronet
the remaining atoms in the π-state. Despite these unex- angular momentum in each DBG, as illustrated, for ex-
pected scattering events, for 10 . τ . 40 we may still ample,bythepresenceofavortexanti-vortexpairinFig.
observe one oscillation in the population of the |l| = 1 4(c). However the complete system, defined by Eq. (2),
mode in Fig. 4(d). However, at τ ≈ 50 the oscillation only requires that the total angular momentum about
breaksdown,andvortexpairsappearinthe densitypro- the z axis be conserved, meaning that each DBG may
file, as shown in the lower panel of Fig. 4(c), in which break its initial chiral symmetry and obtain a net angu-
the position and(anti)clockwise rotationof the vortex is larmomentum,sothathLαi=−hLβi6=0,wherehLα/βi
z z z
indicated by the (minus) plus sign. At the same time, is the angular momentum of the upper/lower DBG. No
we observe the onset of regular, large amplitude (∼ 0.8) such mode is predicted fom the stability analysis as the
oscillations in the number imbalance η [plotted in Fig. mechanism for spontanous decay of the π-state involves
4(e)] due to inter-well transfer. These dynamics reflect two atoms with zero initial angular momentum gaining
5
2 reveals that an |l| = 2 mode is also unstable for this
coupling, but its imaginary eigenvalue is much smaller
(lighter gray in Fig. 2) than the corresponding value
for the |l| = 1 modes. We would therefore expect the
|l| = 1 modes to grow faster. Our simulation confirms
this: theinitiallysmoothdensityprofile[Fig. 5(a)]trans-
forms into an interference pattern [Fig. 5(b)] with two
antinodes as φ is rotated through 2π, and two antinodes
in the ρ direction. The population of this mode [shown
by the solid curve in Fig. 5(d)] performs one oscillation
for 15 . τ . 45, but by τ ≈ 50 the growing population
of the |l| = 2 modes [shown by dashed line in Fig. 5(d)]
has become significant. The occupation of multiple ro-
tating modes disrupts the DBGs’ internal structure, but
eventually four rough density peaks, encircling the cen-
tral peak, appear in the atomic density profile, revealing
macroscopic population of the |l| = 2 modes [Fig. 5(c)].
Asthisoccurs,weobservetheonsetofoscillationsinrela-
tive atomnumber,indicatedby the periodic fluctuations
inη showninFig. 5(e). Thefrequencyoftheoscillations
ishigherthanthatshowninFig. 4(e),becauseκhasbeen
increased[34]. Unlikethe casepresentedinsectionIVB,
this is accompanied by a transfer of angular momentum
betweenthetwoDBGs,suchthatthetwoDBGscounter-
rotate. We quantify this effect in the enlargement of
the dashed area in Fig. 5(e), which shows not only η
(solid curve, left axis), but also hLα/βi (dashed/dotted
z
curve, right axis). The curves illustrate that, although
hLαi=−hLβi,maintainingthenetzeroangularmomen-
z z
tumofthewhole system,hLαiandhLβiperformperiodic
z z
oscillations, with a maximum angular momentum differ-
enceof1.6~. Numerically,wefindthatthelowestenergy
FIG. 5: (Color online) Simulation for |κ| = 1.35. (a)-(c) counter-rotating mode with hLβi = −hLαi = ~ has an
Atom density plot (black = high) of upper (first row) and z z
energy of 2N (µ +1.3). This explains why macroscopic
lower (second row) DBG at (a) τ = 0, (b) τ = 40 and (c) 0 0
occupationofacounter-rotatingmodeis notobservedin
τ =100. (d) Solid (dashed) line: logarithmic plot of thepro-
jection P1α (P2α) of the upper DBG onto angular momentum the simulation presented in Fig 3, which has an energy
modes with |l|=1 (|l|=2). (e) Atom number imbalance, η, of 2N0(µ0+0.56).
betweenupperandlowerwell. Lowerplotshowsenlargement
ofcurvewithindashedboxin(e)asa(blue)solid curvewith
scaleshownleft,plusLαz/β asa(green)dashed/(black)dotted V. CONCLUSION
curvewith scale shown right.
Insummary,wehaveshownthatthewellknowndecay
mechanismofathree-dimensionalπ-state[29,30,37,38]
equalandoppositeangularmomentumviaanelasticcol-
is suppressed in tunnel-coupled 2D DBGs. This results
lision. Since this process is mediated by the non-linear
inarichvarietyofphysics,suchasspontaneousrotation,
term in Eq. (2), it may only occur between atoms in the
vortex formation, and chiral symmetry breaking due to
same well. Consequently, we conclude that the break-
counter-rotation of the two clouds. Our paper builds on
ing of chiralsymmetry is a secondaryprocess,caused by
previous studies of coupled 1D rings [16], showing that
tunneling of angular momentum between the wells, once
the radial degree of freedom in the 2D disks plays an
rotatingmodeshavebecomepopulated. Wenowpresent
important role. Firstly, excitations in the radial (r) di-
an example of this behavior.
rectionappearinthestabilitydiagram,andmaybeselec-
tively populatedandexperimentallyobserved. Secondly,
the clouds may develop a complex internalstructure, in-
C. Counter-rotation of the two clouds (|κ|=1.35) cluding vortex anti-vortex pairs. Thirdly, the radial po-
sition of the peak density, which is fixed in the case of
We setκ to 1.35(dashedline inFig. 2)in orderto ex- the rings, depends on the angular momentum of the 2D
cite |l| = 1 modes with additional radial energy [shown cloud. As a consequence, disruption arises more readily
in surface plot (c) in the lower panel of Fig. 2]. Figure when we populate modes with low angular momentum.
6
Sincethis workisrestrictedtozerotemperature,itre- Using optical lattices to create this system also offers
mains an open question as to how coupling multiple 2D theintriguingpossibilityofextendingourworktospinor
DBGs will change their dynamics at finite temperature. DBGs or arraysof coupled DBGs [4, 5, 43], which might
Finite temperature will introduce a variation of phase revealacrossoverfrom2Dbehaviortothree-dimensional
acrossthe clouds,whichmightincreasedisruption,caus- band dynamics.
ing the counter-rotationto occur more readily. It is also Finally, we speculate that this system could poten-
possiblethatcoupledsystemsmayprovidealinkbetween tially be developed as a sensor to detect tiny gravi-
2D Kosterlitz-Thouless behavior and three-dimensional tational or electromagnetic forces, in a similar manner
Bose-Einstein condensation. to the production of vortices between merged elongated
We note that these systems can be realized with cur- DBGs[6,29,30]. Inprinciple,oneofthe DBGs couldbe
rent experimental techniques. Similar set ups have al- exposedto a tiny force,causinga phase changeof0.01π,
readybeencreatedbycombiningaharmonictrapwitha which could then be detected as a change in the growth
one-dimensional optical lattice, to investigate quantum rate of rotating modes.
fluctuation-induced localization [4] and the Kosterlitz-
Thouless transition [22]. Using this method, values of λ
upto370havebeenachieved. Asuitablechoiceofparam-
Acknowledgments
eters would be, for example, ω =2π×30Hz, λ=10 and
ǫ=7. In this case, N ≈200 87Rb atoms and the condi-
0
tionsfortwo-dimensionalityaresatisfied. Theseparame- This work is funded by EPSRC-UK. We thank P.
tersarecomparableto thosein previousexperiments [4]. Kru¨ger for helpful discussions.
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