Confinement and precession of vortex pairs in coherently coupled Bose-Einstein condensates

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by  Marek Tylutki

Confinement and precession of vortex pairs in coherently coupled Bose-Einstein
condensates
Marek Tylutki,1,∗ Lev P. Pitaevskii,1,2 Alessio Recati,1,3 and Sandro Stringari1
1INO-CNR BEC Center and Dipartimento di Fisica,
Universita` di Trento, Via Sommarive 14, I-38123 Povo, Italy
2Kapitza Institute for Physical Problems RAS, Kosygina 2, 119334 Moscow, Russia
3Physik Department, Technische Universit¨at Mu¨nchen,
James-Franck-Straße 1, 85748 Garching, Germany
(Dated: May 9, 2016)
6
1 Thedynamicbehaviorofvortexpairsintwo-componentcoherently(Rabi)coupledBose-Einstein
0 condensatesisinvestigatedinthepresenceofharmonictrapping. Wediscusstheroleofthesurface
2 tensionassociatedwiththedomainwallconnectingtwovorticesincondensatesofatomsoccupying
different spin states and its effect on the precession of the vortex pair. The results, based on
y
a the numerical solution of the Gross-Pitaevskii equations, are compared with the predictions of an
M analyticalmacroscopicmodelandarediscussedasafunctionofthesizeofthepair,theRabicoupling
andtheinter-componentinteraction. WeshowthattheincreaseoftheRabicouplingresultsinthe
disintegration of the domain wall into smaller pieces, connecting vortices of new-created vortex
6
pairs. The resulting scenario is the analogue of quark confinement and string breaking in quantum
] chromodynamics.
s
a
PACSnumbers: 03.75.Lm,03.75.Mn,67.85.Fg
g
-
t
n I. INTRODUCTION components, the situation drastically changes. The gas
a
has only a single U(1) symmetry related to the conser-
u
vation of the total particle number, and a gap opens in
q Vortices and solitons are among the most striking fea-
t. tures of superfluids and superconductors. Recent exper- the spin channel, since the coupling locks the relative
a phase of the order parameter of different spin compo-
imental advances in cold atomic gases have made it pos-
m nents. As pointed out in the seminal paper by Son and
sible to study multi-component systems with a vector
- order parameter [1]. The nature of vortices and soli- Stephanov [7] (see also [8]), in coherently coupled uni-
d form two-component superfluids, vortex lines in a single
tons in such systems is far from being a trivial extension
n
component cannot exist as single objects, but only in
o of the single component case. In the context of Bose-
pairs of different spin components (the so called ’half-
c Einstein condensates (BECs) the situation is very rich.
[ Not only it is possible to have condensates with two or vortices’), which are connected by a domain wall. The
more components, but it is also feasible to tune the in- domain wall can be interpreted as a sine-Gordon soli-
2
ton and is characterized by a 2π jump in the relative
v teraction strengths, which allows to explore the nature
phase of the two spin components. Vortices can exist
5 of vortices in different phases (see, for example, [1–3]).
only in pairs because a single vortex would be attached
9 Furthermore, if the components correspond to different
6 spin states of the same atom, it is possible to produce to a domain wall of infinite area, with an infinite en-
3 ergy cost. Stationary configurations of vortex pairs (also
coherent Rabi coupling between the components using
0 knownas’merons’[2])havealreadybeeninvestigatedby
rf transitions, yielding novel topological features in the
1. structure of vortices and solitons. solving the Gross-Pitaevskii (GP) equation in the pres-
0 ence of a rotating trapped two-component gas [9, 10]. In
In this work we consider a mixture of two coher-
6 the same spirit, numerical extensions to more than two
1 ently coupled BECs. Some properties of these systems coherently coupled BECs have been produced in recent
: like coherent Rabi oscillations, internal self-trapping ef-
v years [11, 12]. Pairs of half-vortices connected by a do-
fects,andtheferromagneticclassicalbifurcationhaveal-
i mainwallsharemanyfeatureswithquarkconfinementin
X ready been addressed experimentally [4–6]. In the ab-
quantumchromodynamics(QCD).Inparticular,thesur-
r sence of Rabi coupling, a two-component Bose gas has face energy related to the domain wall is proportional to
a a U(1) U(1) gauge symmetry, which is broken in the
× its length and creates a distance-independent force that
condensed, superfluid phase. In this case vortices be-
attracts vortices to each other. If the surface tension of
have similarly to the single component case as long as
the wall is large enough, it can break, creating new pairs
the inter-component interaction is small enough and the
of vortices similar to string breaking in QCD.
mixture is miscible. In the presence of Rabi coupling,
which produces coherent transitions between the spin In the present work, we provide an explicit investi-
gation of the precession of vortex pairs emphasizing, in
particular,thecompetitiveroleplayedbythetheMagnus
force acting on a moving vortex, and the attractive force
∗Electronicaddress: [email protected] due to the surface tension of the domain wall. Further-
2
more,weshowexplicitlythatthedomainwallconnecting Thisinteractionenergywascalculatedanalyticallyin[16]
the vortex pair breaks for large values of the Rabi cou- in the absence of a coherent coupling (Ω =0).
R
pling. Our analysis is based on the 2D GP equations Starting with the above equations, one can calculate
the precession frequency characterizing the rotation of
1
i(cid:126)∂ ψ = (H +g ψ 2+g ψ 2)ψ (cid:126)Ω ψ , the vortex pair as
t 1 0 | 1| 12| 2| 1− 2 R 2
1 ∂E ∂E ∂d
i(cid:126)∂ ψ = (H +g ψ 2+g ψ 2)ψ (cid:126)Ω ψ , (1) Ω = = , (4)
t 2 0 | 2| 12| 1| 2− 2 R 1 prec ∂L ∂d ∂L
z z
for the order parameters ψ and ψ of the two compo-
1 2 where the angular momentum L of the vortical config-
z
nents respectively. The two equations are coupled not
uration is calculated in terms of the distance d of each
only by the inter-component interaction term propor-
vortex from the center using the relation [14] L =
z
tional to g12, but also by the Rabi coupling ΩR > 0 N(cid:126)(1 d2/R2)2, holding for 2D harmonically t(cid:104)rap(cid:105)ped
between the two spin states. In the above equations − ⊥
condensates.
H0 = −2(cid:126)m2 ∇2 + 12mω⊥2r⊥2 is the single-particle Hamil- For a fixed value of the interaction parameters and
tonian, and r⊥ is the radial coordinate measuring the the Rabi coupling ΩR, the precession angular velocity
distance from the trap’s center (r2 =r2). will depend on the distance 2d. The above formalism
⊥
The presence of the Rabi coupling implies that, at is immediately generalized to the case of a trap rotating
equilibrium,thephasesofthetwocomponentsareequal: withangularvelocityΩ byaddingtheterm Ω L
rot rot z
θ1 = θ2. In the following we will assume g11 = g22 g to the expression for the total energy. − (cid:104) (cid:105)
≡
and consider the miscible (paramagnetic) case, ensured In principle the same formalism would also allow for
by the inequality g12 <g+(cid:126)ΩR/n. Then the Rabi cou- the determination of the size of the pair at equilibrium
pling implies that the equilibrium densities are equal: (vortex molecule). The corresponding condition is fixed
n1 = n2 = n/2, with ni = ψi(r)2, (i = 1,2), and by the equation ∂E/∂d = 0. Notice that this condition
| |
n=n1+n2. implies the absence of precession, as immediately follows
from(4). Theoccurrenceofastablevortexmoleculewith
d = 0 cannot be ensured in the absence of the repulsive
II. THOMAS-FERMI MODEL int(cid:54) eraction term E in Eq.(2). In fact, if E =0, the
int int
stationary solution ∂E/∂d = 0 always corresponds to a
A simple macroscopic description of the dynamics of local maximum of E(d) rather than to a minimum.
these novel configurations is obtained by considering a In order to make the role of Rabi coupling clearer, we
vortex pair located symmetrically with respect to the first consider the case of the absence of inter-component
center of the harmonic potential and by writing the ex- interaction, i.e. we assume g = 0. In this case the
12
cess energy per particle with respect to the ground state precession frequency in the absence of rotation (Ω =
rot
in the form 0) results from the competition between E and E ,
wall v
which act in opposite directions. Assuming d R one
⊥
E(d)=2Ev(d)+Ewall(d)+Eint(d) , (2) finds (cid:28)
where d is the distance of each vortex from the center σ E (cid:114)2(cid:126)√Ω E
(and hence 2d is the distance separating the two vor- Ωprec =−π(cid:126)nd + (cid:126)V =−2 m πdR + (cid:126)V (5)
tices). In the above equation E (d)=NE (1 d2/R2)
v V − ⊥
istheenergyofasinglevortex,forwhichonecanusethe showingthat,ifΩ (andhenceE )issufficientlylarge
R wall
macroscopicestimateEV ≈((cid:126)2/mR⊥2)[ln(1.46R⊥/ξ)−21] theprecessionwillproceedintheoppositedirectionwith
[13–15]; R⊥ is the Thomas-Fermi radius of the trapped respect to the flow generated by the vortex line. In the
(cid:82)
atomic cloud, N = drn(r) is the number of atoms and limit E E , the precession frequency will be pro-
wall v
ξ =(cid:126)/√2mgn is the healing length, while portional to(cid:29)the surface tension of the wall and hence to
√Ω . This result can also be obtained by considering
(cid:90) d R
the equilibrium between the Magnus force and the sur-
E (d) = dr σ(r ,Ω ) , (3a)
wall ⊥ ⊥ R
face tension σ: κ v n/2+σ =0, where v =Ω d is
−d L L prec
×
(cid:126)3/2 (cid:112) the velocity of the vortex line with respect to the fluid.
σ(r⊥,ΩR) = 23/2m1/2n(r⊥) ΩR (3b) The vector κ, with modulus |κ| = 2π(cid:126)/m, is oriented
along the vortex line and indicates the velocity circula-
is the surface energy associated with the domain wall tion. In the calculation of the Magnus force we used the
connecting the two vortices. In the above equations σ unperturbedvaluen/2forthedensityofeachcomponent.
is the surface tension [7] of the wall and n(r ) is the The condition E E is easily achieved exper-
⊥ wall v
(cid:29)
densityofthegas. InauniformmediumtheenergyE imentally. In the following we will consider a gas of
wall
increases linearly with the distance d. Finally, E (d) is sodiumatomsconfinedbyaharmonicpotentialwithtrap
int
the energy of the repulsive interaction between the two frequency ω /2π =10Hz and with a chemical potential
⊥
vortices fixed by the inter-species coupling constant g . µ=gn =50(cid:126)ω , where n is the central density. This
12 0 ⊥ 0
3
corresponds to a Thomas-Fermi radius R = 10ξ =
⊥ ho
66.5µm,whereξho =(cid:112)(cid:126)/mω⊥istheharmonicoscillator ξho
α
length. In the presence of Rabi coupling Ω /2π = 5Hz
R
one then finds that a vortex pair, separated by the dis-
tance 2d = 0.53R , is characterized by the precession
⊥
frequency Ω /2π 2Hz. Setting Ω = 0 one in-
prec R
(cid:39) −
steadwouldfindthevalueΩ /2π +0.45Hzofoppo-
prec
(cid:39)
site sign.
Theabovemodel,includingEq.(3)forthesurfaceten-
sion, is justified as long as the width of the domain wall,
fixed by the relationship dwall = 2(cid:112)(cid:126)/mΩR, is much FIG. 1: (color online) (a) Relative phase θ1−θ2 of the two
larger than the healing length ξ = (cid:126)/√2mgn, but much components near a vortex pair in the presence of Rabi cou-
pling Ω =0.5ω . We can see that the phase jump between
smaller than the size 2d of the vortex pair. If the width R ⊥
the two vortices is confined within the narrow domain wall
d of the domain wall is much larger than the size of
wall stretching between the vortices. The dotted line shows the
the vortex pair (corresponding to very small values of
circlearoundwhichthephaseiscalculatedintherightpanel.
Ω ), the effect of the Rabi coupling should be treated
R (b) Phases θ (red dotted line) and θ (blue solid line) along
1 2
using a perturbative approach [17].
a circle centered in the vortex of the first component. The
phase θ makes a 2π winding around a vortex, with half of
1
the jump concentrated in a short interval of the polar angle
III. NUMERICAL RESULTS α (of the coordinate system centered in the vortex core, as
shown in panel (a)). The phase θ is instead single valued.
2
In Fig. 1 we show the typical behavior of the relative
phase of a vortex pair in the presence of a 2D harmonic 0.5
trap, obtained by solving numerically the coupled GP 0.2
Eqs. (1) with g12 = 0. The structure of the domain 0.0
wall is clearly visible. The result shown in the figure is 0.0
0.2
obtained as follows: first, we symmetrically imprint two − 0.00 0.05 0.10
vortices,oneineachcomponent,farawayfromthetrap’s
⊥0.5
center; then we perform an imaginary time evolution, ω−
/
duringwhichthedomainwallintherelativephaseforms c
e
and the vortices start approaching each other, with the pr 1.0
Ω
energy of the system decreasing. We stop the simulation −
at a certain point, in order to produce a vortex pair of
2d=1.86
the desired size. This configuration serves as the initial 1.5
− 2d=3.17
condition for a subsequent real time evolution, in which
2d=5.27
the pair exhibits precession. The configuration in Fig. 1
2.0
shows the phase after a short precession time, the small − 0 1 2 3 4
asymmetry in the shape of the wall being caused by the ΩR/ω
rotation. ⊥
Fig.1(b)showsexplicitlythebehaviorofthephasesof
FIG. 2: (color online) Dependence of Ω on the Rabi cou-
prec
the two components calculated along the contour shown
plingΩ fordifferentvaluesofthevortexseparation2d. The
R
inFig.1(a),i.e.,aroundthevortexofthespincomponent numerical solution of the GP equations (component 1 - bul-
1. The figure clearly reveals the 2π jump in the relative lets, component 2 - triangles) are in a good agreement with
phase θ θ near the domain wall. theanalyticalexpression,Eq.(5),(solidlines)forlargevalues
1 2
−
By solving the GP equation in real time one can in- of 2d (in units of ξho). The inset is a zoom of the figure for
vestigate the precession of the vortex pair and evaluate small values ΩR, where Ωprec changes sign.
theprecessionfrequencyΩ ,whosedependenceonthe
prec
Rabi coupling Ω is reported in Fig. 2 for different val-
R
ues of the vortex size d. Its dependence on d, for a fixed vortex pairs at the ends of these new fragments. This
value of Ω , is instead shown in Fig. 3. The results of fragmentation is an analogue of string breaking in quan-
R
the numerical calculations are found to agree reasonably tum chromodynamics. The probability of such a frag-
well with the predictions of the macroscopic model (5) mentation becomes larger and larger as Ω grows. In
R
discussed in the first part of the paper (see solid lines Fig. 4 we show the result of the fragmentation of a do-
in the figures). As expected, the discrepancies become mainwallobtainedatΩ =6ω andg =0. Theinitial
R ⊥ 12
smaller if the width of the domain wall becomes much configuration – a domain wall symmetric with respect
smaller compared to the vortex separation d 2d. to the center of the trap – was then allowed to evolve
wall
(cid:28)
For large values of Ω a long domain wall may decay through the time ω t = 1.5. The figure clearly shows
R ⊥
into smaller fragments, resulting in the creation of new three fragments of various sizes, connecting vortices of
4
0.0
0.2
−
⊥0.4
ω−
/
c
e
r
p 0.6
Ω
−
0.8 ΩR=0.5
− ΩR=1.0
ΩR=2.0
1.0
− 0.1 0.2 0.3 0.4 0.5
ξ /2d
ho
FIG. 5: (color online) Relative phase distribution around a
FIG.3: (coloronline)Dependenceoftheprecessionfrequency
singlehalf-vortexinatwocomponentcoherently-coupledsys-
Ω (component 1 - bullets, component 2 - triangles) on
prec tem. Thevortexbuildsadomainwallthatisattachedtothe
1/2d (in units of (cid:112)mω /(cid:126)). The solid lines correspond to
⊥ nearest point of the edge of the cloud. The phase distribu-
the prediction of Eq. (5). As in the Fig. 2, the agreement is
tion corresponds to the evolution time of ω t = 0.2. Then,
⊥
goodaslongasthedistance2dseparatingthetwovorticesis
the vortex starts precessing and induces the appearance of a
sufficiently large.
second vortex in the component 2.
connecting the vortex to the external region outside the
Thomas-Fermi radius, where the density of the atomic
cloud is vanishing (see Fig. 5). When solving the GP
equations (1) in real time, the vortex line in component
1 exhibits the precession according to the macroscopic
prediction (2)-(4) with 2E replaced by E and E
v v wall
calculatedalongthedomainwall. However,wesoonfind
the appearance of a second vortex in component 2, at-
tached to the second end of the wall and emerging from
the border, where its energy cost is vanishingly small.
FIG.4: (coloronline)Fragmentationofthedomainwallafter
theevolutiontimeω t=1.5. Onecanseefragmentsofdiffer- The two vortices then start rotating around each other.
⊥
entsize,connectingvorticesofdifferentcomponents. TheGP Eventuallytheoriginalvortexofthecomponent1reaches
simulation was carried out choosing g = 0 and Ω = 6ω . theborderoftheatomiccloudtodisappearandreappear
12 R ⊥
The initial condition corresponded to a domain wall between again after a while (an analogous behavior of vortices in
the vortex pair of length 2d ≈ 5ξho, symmetric with respect a toroidal trap was observed in Ref. [18]).
tothetrap’scenter. Thefigureisazoomofthecentralregion
of the cloud.
The results presented above were obtained assuming
theinter-componentinteractionparametertobeg =0.
12
different components. Ifg issmallcomparedtotheintra-componentcoupling
12
In the introduction we emphasized the fact that, in a constants,g =g g,wefindthattheinfluenceofg
11 22 12
≡
uniform, infinite system in the presence of the Rabi cou- on the precession is almost negligible, the main role be-
pling, vortices cannot exist alone as single objects but ing played by the long-range surface tension force. This
only in pairs. In a finite system, such as in the pres- behaviorisconsistentwiththefactthatstablemolecules
ence of a confinement, single vortex lines can also exist, have a very small size, even for relatively large values
as the domain wall will cost a finite amount of energy, of g . If g is close to g, one can identify a criti-
12 12
fixed by the size of the atomic cloud. We have explored cal value for the Rabi coupling, given by the expression
single vortex configurations by considering a vortex line Ω = 1(g g )n/(cid:126) [7], above which the domain wall
crit 3 − 12
(correspondingtothe component1) located atsome dis- becomes unstable and the solution of the GP equations
tancefromthecenterofthetrap. Thedomainwallwhich corresponds to a local maximum of energy, rather than
minimizes the energy corresponds to the shortest line to a local minimum.
5
IV. CONCLUSIONS techniques (see, for example, [19]).
We expect that our predictions for the precession of
half-vortex pairs and for the fragmentation of the corre- Acknowledgments
sponding domain wall at large Rabi coupling will stim-
ulate new measurements on coherently coupled BECs. We are grateful to Franco Dalfovo, Alexander Fetter,
Experimentally,pairsofhalf-vortices,connectedbyado- Gabriele Ferrari and Chunlei Qu for stimulating discus-
mainwall,canbecreatedbytheproperimprintingofthe sions. This work was supported by ERC through the
relative phase of the two condensates. The shape of the QGBEgrant,bytheQUICgrantoftheHorizon2020FET
domain wall connecting the two vortical lines is in prin- programandbyProvinciaAutonomadiTrento. A.R.ac-
ciple observable using heterodyne methods giving rise to knowledges support from the Alexander von Humboldt
visible interference in the domain wall region. The pre- foundation. This work was supported in part by the PL-
cessioneffectcouldbemeasuredusingreal-timedetection Grid infrastructure (M.T.).
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