Bose-Einstein condensates in toroidal traps: instabilities, swallow-tail loops, and self-trapping

1.2 MB·English

by  Soheil Baharian

Bose-Einstein condensates in toroidal traps: instabilities, swallow-tail loops, and
self-trapping
Soheil Baharian and Gordon Baym
Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green St., Urbana, IL 61801, USA
(Dated: January 29, 2013)
Westudythestabilityanddynamicsofanultra-coldbosonicgastrappedinatoroidalgeometry
and driven by rotation, in the absence of dissipation. We first delineate, via the Bogoliubov mode
expansion,theregionsofstabilityandthenatureofinstabilitiesofthesystemforbothrepulsiveand
attractive interaction strengths. To study the response of the system to variations in the rotation
rate, we introduce a “disorder” potential, breaking the rotational symmetry. We demonstrate the
3
1 breakdown of adiabaticity as the rotation rate is slowly varied and find forced tunneling between
0 the system’s eigenstates. The non-adiabaticity is signaled by the appearance of a swallow-tail loop
2 in the lowest-energy level, a general sign of hysteresis. Then, we show that this system is in one-
to-one correspondence with a trapped gas in a double-well potential and thus exhibits macroscopic
n
quantumself-trapping. Finally,weshowthatself-trappingisadirectmanifestationofthebehavior
a
of the lowest-energy level.
J

5
2 I. INTRODUCTION where m is the particle mass, V = 2π2r2R is the vol-
0
ume of the annulus, g is the strength of contact in-
]
s Superfluidflowinatoroidaltrapisstabilizedbyalarge teractions, and N0 and N1 are the number of particles
a
in |0(cid:105) and |1(cid:105) respectively with N = N + N the to-
g energy barrier between the current-carrying state and a 0 1
tal number of particles. The state with N = N is a
- state with lower angular momentum [1, 2]. However, in 1
t single-vortex state and that with N = N is the ground
n mesoscopic systems, such as atomic Bose-Einstein con- 0
state. Figure 1, which shows the energy per particle as
a densates, the barrier can be sufficiently small that the
u system can tunnel quantum-mechanically to a state of a function of N1 for different values of the interparticle
q interaction strength, illustrates the energy barrier that
lowerangularmomentum[3,4];furthermore,iftheshort-
. appears between the single-vortex state and the non-
t range interparticle interactions are attractive or very
a rotating ground state when gN/V > (cid:126)2/2mR2. With
m weakly repulsive, such a barrier does not exist, and the
weakening interaction strength, the barrier decreases,
systemcantransitionsmoothlyfromthecurrent-carrying
- and for gN/V ≤ (cid:126)2/2mR2, it disappears, leading to in-
d state, to, e.g., the non-rotating ground state. Recent ex-
stability of the single-vortex state.
n perimentsinultra-coldbosonicsystemsintoroidaltraps,
o stimulated by the possibility of shining new light on the We first delineate the regions of stability and the na-
c stabilityanddecayofsupercurrents[5]aswellasbypos- ture of instabilities of the full system as functions of the
[
sible applications in other areas, e.g., interferometry [6] external rotation frequency of the trap, for both posi-

2 andatomtronics[7],haveseensuchcurrentdecays[8–10]. tive and negative interaction strengths. In general, the
v The stability of superflow depends on the interparti-
6
cle interactions, the rotation rate of the trap, disorder
7
in the trapping potential, and temperature. We con-
2
sideragasofinteractingbosonsatzerotemperatureina
2
. toroidal trap rotating at angular velocity Ω and address
1
the question of how the single-vortex condensate with a
1
metastable or unstable superflow evolves in the absence
2
1 of dissipation, driven either by varying the rotation rate
: ofthetraporbyvaryingtheinterparticleinteractionvia
v
a Feshbach resonance.
i
X We consider, throughout this paper, a quasi-one-
r dimensional gas of N bosons in a thin annulus of radius 0 1
a Randcross-sectionalradiusr (cid:28)Ratzerotemperature.
0
Thebasicphysicsofthestabilitycanbemostsimplyun-
derstood by considering just two single-particle levels of
FIG. 1: (Color online) Energy landscape of the two-level
the annulus, the non-rotating state, |0(cid:105), and the state model as a function of the number of particles in state
with azimuthal angular momentum (cid:126) per particle, |1(cid:105). |1(cid:105). Note the energy barrier between the single-vortex and
TheHamiltonianofthissystemhasthefamiliarNozi`eres groundstatesforgN/V >(cid:126)2/2mR2 (solidline,inblack); for
form [11] gN/V = (cid:126)2/2mR2 (dashed line, in red), the slope vanishes
at N = N, while for gN/V < (cid:126)2/2mR2 (dot-dashed line,
1
(cid:126)2 g in blue) no barrier exists, indicating instability of the single-
H2 = 2mR2N1+ 2V (N02+N12+4N0N1) (1) vortex state.
2
stability of the flow is manifest in the small-amplitude We briefly note related theoretical studies in simi-
Bogoliubov fluctuations about the current-carrying con- lar toroidally trapped systems: Bose condensates with
densate. Starting from a mean-field condensate, we in- dipolarinterparticleinteractions[21]whichinduceanef-
clude Bogoliubov fluctuations [12] and find the eigenen- fective double-well Josephson junction, leading to self-
ergies of the quasiparticle excitations. With decreasing trapping [22]; Bose-Einstein condensates with a modu-
repulsion or trap rotational frequency, an energetic in- lated,spatially-dependentscatteringlength[23];andhol-
stability[13,14]canappearinthesystemviaexcitations low pipe optical waveguides with an azimuthally modu-
that decrease the angular momentum of the system by latedrefractiveindexwhichgeneratesaneffectivedouble-
one unit; the system can lower its energy by exciting well potential configuration [24].
these quasiparticles. Moreover, we find a dynamical in- In Sec. II, we discuss the stability regime of the con-
stabilityforsufficientlyattractiveinteractions,wherethe densate by studying the energies of the Bogoliubov ex-
quasiparticle eigenenergy becomes complex [13, 14] and citations. We then analyze the energy landscape of the
the system is driven exponentially rapidly in time away two-mode system in Sec. III and demonstrate a swallow-
from the initial state. For a system to evolve due to an tailloopintheenergyofthegroundstate. Weconstruct
energeticoradynamicalinstability,thepresenceofdissi- a mean-field description of this system in the presence
pationisnecessaryinordertoremoveenergyandangular of the disorder potential in Sec. IV. The appearance of
momentum; in this paper, we do not include dissipative swallow-tailloopsandcuspsintheenergylevelsandtheir
effects, but will in a future publication. With a knowl- relation to extrema in the energy landscape are studied
edge of the instabilities, we then study simple ground inSec.IVA.Finally,inSec.IVB,wediscusstheconnec-
states that encompass the underlying physics, consisting tionofthissystemtoatrappedcondensatetunnelingina
of the two lowest-lying single-particle states. (Another double-well potential, and the corresponding connection
example of how an instability indicates the presence of a ofself-trappinginthedouble-wellsystemtothebehavior
lower-energymetastablegroundstateisgiveninRef.[15] of the adiabatic energy levels discussed in the previous
where we studied a rapidly rotating trapped Bose gas in subsection.
thelowestLandaulevelwithavortexatthecenterofthe
trap.)
For the system to feel the presence of the trap, the II. STABILITY OF THE GROUND STATE
trapping potential must break the rotational symmetry.
We describe the coupling of the system to the container
Forasufficientlythinannulus, theradialandaxialex-
by an asymmetric “disorder” potential stationary in the
citations are frozen out, and the angle around the ring
frame rotating at angular velocity Ω. Within mean-field
becomes the only effective degree of freedom. The nor-
theory, we determine the stationary states of the con-
malizednon-interactingsingle-particleeigenstatesofthis
√
densateformedfromthesingle-particlestates|0(cid:105)and|1(cid:105) system are ϕ (θ)=(cid:104)r|l(cid:105)=eilθ/ 2πR with eigenenergies
l
and find that for sufficiently large interaction strengths, (cid:15) =((cid:126)l)2/2mR2 where(cid:126)l istheangularmomentumand
l
dependent on the disorder potential, the system exhibits
r=(R,θ)isthepositionvector. TheHamiltonianinthe
a non-adiabatic response [17, 18] to variations of the ro-
laboratory frame is
tation frequency, even if Ω is changed arbitrarily slowly.
This behavior arises from the presence of multiple min- (cid:88) ((cid:126)j)2 1 g (cid:88)
ima in the energy landscape (separated by a maximum H= 2mR2 a†jaj + 2 V a†j−ma†k+makaj (2)
or saddle-point which represents an unstable mode) and j j,k,m
is characterized by the appearance of a swallow-tail loop
in the lowest-lying adiabatic energy level and a fold-over whereaj istheannihilationoperatorforaparticleofan-
in the occupation probability of the corresponding state gular momentum (cid:126)j, and g = 4π(cid:126)2a/m is the two-body
as functions of the rotation frequency (see Fig. 5 below). contact interaction strength with a the s-wave scatter-
The swallow-tail loop implies that the response of the ing length. The Hamiltonian in the frame rotating at Ω
system to external rotation exhibits hysteresis [17]. (denoted by a prime) can be written as [25]
Moreover, we show that the quasi-one-dimensional
(cid:126)Ω2 (cid:88)1(cid:16) (cid:17)2
Bose-Einstein condensate in a rotating annulus can be H(cid:48) =−N +(cid:126)Ω j− Ω a†a
mapped onto the problem of a condensate trapped in a 2Ω0 0 2 Ω0 j j
j
double-well potential with Josephson tunneling between
1 g (cid:88)
the two wells. Therefore, macroscopic quantum phe- + 2 V a†j−ma†k+makaj (3)
nomenon of self-trapping in double wells [19, 20] also j,k,m
appears in such rotating Bose gases, where the system
acquires a non-zero time-averaged population difference where Ω =(cid:126)/mR2 is the characteristic scale of rotation
0
betweenthetwocomponents. Theonsetofself-trapping, inthesystem. Thenon-interactingsingle-particleenergy
which is a steady-state population imbalance, exactly levelsofH(cid:48),depictedinFig.2,areperiodicinΩ. Atthis
correspondstothebehavioroftheenergylevelsdiscussed stage we do not include the disorder potential.
above. In the laboratory frame, the condensate ψ (θ,t) obeys
c
3
12 j(cid:45)(cid:87)(cid:87)02 Note that for 14ν2 +ηNc < 0, these energies are com-
2.0 plex, indicating that the condensate is dynamically un-
(cid:72) (cid:144) (cid:76) stable [13, 14].
Theoscillationsofthecondensatecanalsobepictured
1.5
insecond-quantizationasquasiparticleexcitationsofthe
condensate. In the usual second-quantized Bogoliubov
formalism, the coherence factors are given in terms of
1.0
j = -2 j = 2 (cid:15) [13, 28] by
ν
 
0.5 1 1ν2+ηN
j = -1 j = 1 |uν|2 = 2|ν|2(cid:113)1ν2+ηcN +1 (9)
j=0 4 c
(cid:87)(cid:87)
(cid:45)2 (cid:45)1 0 1 2 0  
1 1ν2+ηN
FmIeGas.u2r:edSiinngulen-iptsarotfic(cid:126)leΩ0e,naesrgfyunlcetvieolnssinoftΩh.e rotati(cid:144)ng frame, |vν|2 = 2|ν|2(cid:113)14ν2+ηcNc −1, (10)
andtheexcitationenergyintherotatingframeofaquasi-
the time-dependent Gross-Pitaevskii (GP) equation particle carrying ν units of angular momentum relative
to the condensate becomes
(cid:20) (cid:126)2 ∂2 g (cid:21)
i(cid:126)∂ ψ (θ,t)= − + |ψ (θ,t)|2 ψ (θ,t). (cid:113)
t c 2mR2 ∂θ2 πr2 c c (cid:15)(cid:48)(Ω¯)=ν(1−Ω¯)+|ν| 1ν2+ηN . (11)
0 ν 4 c
(4)
To determine the stability of the system, we construct Self-consistency dictates that N +(cid:80) |v |2 =N.
the normal modes of the condensate by perturbing c ν(cid:54)=0 ν
We now analyze the stability of the ground state in
the system around the stationary solution ψ (θ,t) =
c terms of the normal modes. For weak interactions,
e−iµt/(cid:126)ψc(θ), where µ is the chemical potential. We |ηN | (cid:28) 1, expansion to first order leads to (cid:15)(cid:48)(Ω¯) (cid:39)
expand the condensate wave function in terms of time- 1ν2c+ν(14−Ω¯)+ηN . Thus, at Ω¯ = 0 and forν repul-
dependent modes with angular momentum ν measured 2 c
sive interactions, only (cid:15) (cid:39) −1 +ηN is negative; the
relative to the condensate by writing −1 2 c
ψ(θ,t)=e−iµt/(cid:126)(cid:2)ψ (θ)+δψ(θ,t)(cid:3) (5)
c
2.0
where
δψ(θ,t)=eiS(θ)(cid:88)(cid:2)u ϕ (θ)e−i(cid:15)vt/(cid:126)−v∗ϕ∗(θ)ei(cid:15)vt/(cid:126)(cid:3) 1.5 stable
ν ν ν ν
ν(cid:54)=0
(6)
with (cid:15) the eigenenergies, S(θ) the phase of ψ (θ), and 1.0
v c
u and v complex numbers to be determined. (Follow-
ν ν
ing Fetter’s notation [13], we explicitly take the phase
of the condensate out of the sum, whereas other authors Nc 0.5
Η
include the exponential factor in the definition of excita-
tion wave functions [26].) From now on, for brevity, we
measure angular momentum in units of (cid:126), time in units 0.0
energetically unstable
of Ω−1, energy in units of (cid:126)Ω , and define the dimen-
0 0
sionless parameters η = mRg/2π2(cid:126)2r2 = 2aR/πr2 and
0 0 (cid:45)0.5
Ω¯ =Ω/Ω0. dynamically unstable
We focus, in particular, on the lowest-energy single-
vortex state, with a condensate of N atoms in the state
c (cid:45)1.0
|1(cid:105),forwhichtheGPequationimpliesthatµ= 21+ηNc. 0.0 0.2 0.4 0.6 0.8 1.0
The modes are described by the two coupled equa- (cid:87)(cid:87)
0
tions [12]
(cid:18)ν+(cid:2)1ν2+ηN (cid:3) −ηN (cid:19)(cid:18)u (cid:19) (cid:18)u (cid:19) FIG. 3: (Color online) Stability (cid:144)phase diagram in the rota-
2ηN c ν−(cid:2)1ν2+cηN (cid:3) vν =(cid:15)ν vν tionrate–interactionstrengthplane,fortheν =−1normal
c 2 c ν ν mode of a condensate with one unit of angular momentum
(7) per particle. Energetic instabilities are caused by excitations
from which we find the eigenenergy with negative energy, whereas those with complex energies
lead to dynamical instabilities. The current experiments in
(cid:113)
(cid:15) =ν+|ν| 1ν2+ηN . (8) Refs. [9, 10] lie in the stable region.
ν 4 c
4
ν = −1 mode is energetically unstable and anomalous,
indicating that the correct ground state has lower angu-
lar momentum than the original single-vortex state. For
attractive interactions, (cid:15)−2 (cid:39) ηNc is also negative, and 1 0
the ν =−2 mode is anomalous as well.
The general stability phase diagram of the ν = −1
mode is shown in Fig. 3 in the interaction strength –
external rotation frequency plane. In the hashed region
where ηNc < −41, the quasiparticle energy is complex. Lb
Note that the regions of dynamical instability and en- L L
1 0
ergetic instability are in agreement with the arguments
in the appendix of Ref. [14]. In a dynamically unsta-
ble mode, where the eigenenergy is complex, one of the
two components in Eq. (6) grows exponentially in time
whiletheotherdecaysexponentially. Anunstablemode,
living around a maximum or a saddle-point in the en- FIG. 4: (Color online) The energy per particle, in units of
ergy landscape, hints at the existence of a stable lower- (cid:126)Ω , of the states |N,0(cid:105) (labeled 0) and |0,N(cid:105) (labeled 1)
0
energy state, corresponding to a modified condensate. and that of the barrier state |b(cid:105) (dashed line) as functions
However, a small-amplitude analysis does not, in gen- of the rotation frequency, for ηN = 1/4. An energy loop
eral, reveal the nature of the new stable state (see, e.g., (labeled by Lb, L0, and L1) emerges due to the existence of
a maximum in the energy landscape.
Refs. [15, 16] where such modified condensates are ex-
plicitly discussed). The solid black line, the solution of
(cid:15)(cid:48) = 0, shows the critical values of interaction strength
ν interactingsystem;itsregionsofstabilityinthepresence
and rotation frequency needed for stability. For a non-
of interactions, for j−1 < Ω¯ < j, are identical to those
interacting system, the ν = −1 mode becomes stable at
Ω¯ = 1. Interestingly, faster rotations shrink the ener- shown in Fig. 3.
2
geticallyunstableregionandstabilizethismodeevenfor
weakly attractive interactions. At Ω¯ = 1, the energet-
III. TWO-MODE APPROXIMATION
ically unstable region completely vanishes, and the gas
becomes stable for ηN >−1. As mentioned before, due
to the absence of dissicpation4in our model, the energy is As discussed above, when the energy of the ν = −1
conserved, andtheinstabilities(althoughpresent)failto mode becomes negative, the system, condensed in |1(cid:105),
changethestateofthesystemintoonewithlowerenergy prefers a ground state with smaller angular momentum
and lower angular momentum. than (cid:126) per particle. As shown in Fig. 2, over the entire
range 0 < Ω¯ < 1, the lowest-lying single-particle states
IntheexperimentinRef.[10],whereN ∼8×104atoms
are |0(cid:105) and |1(cid:105). The states |−1(cid:105) and |2(cid:105) have, in gen-
of 87Rb with unit circulation were held in a ring trap of
eral, muchhigherenergiesandarenotmixedinwiththe
radius R ∼ 9µm, we find ηN ∼ 1.8×103. Also, the ex-
ground state by weak interactions; only for ηN (cid:38) 1 is
perimentinRef.[9],withN ∼8×104 atomsof23Naina
the mixing significant at Ω¯ (cid:39) 0 and Ω¯ (cid:39) 1. Hence, for
ring trap of radius R∼20µm, has ηN ∼2.9×103. The
sufficientlyweakinteractions(ηN (cid:46)1),wecankeeponly
initialstatesincurrentexperiments[9,10]arewithinthe
|0(cid:105) and |1(cid:105) in the description of the system and work in
stable regime discussed here, far from encountering any
a two-mode approximation with the truncated Hamilto-
energetical or dynamical instabilities [37]. However, by
nian in the frame rotating at Ω¯,
suddenlychangingthestrengthoftheinterparticleinter-
action from repulsive to sufficiently attractive via a Fes-
H(cid:48) =(1 −Ω¯)N + 1η(N2+N2+4N N ) (12)
hbach resonance, thereby bringing the system from the 2 2 1 2 0 1 0 1
stable region into the dynamically unstable region, one
where N = a†a . The eigenstates of H(cid:48) are the Fock
wouldbeabletoinvestigateexperimentallytheevolution j j j 2
states
of the system in the presence of a dynamical instability.
The above stability analysis was done for an initial |N ,N (cid:105)= √ 1 (cid:0)a†(cid:1)N0(cid:0)a†(cid:1)N1|vac(cid:105) (13)
condensate in |1(cid:105). Due to the periodicity of the single- 0 1 N !N ! 0 1
0 1
particle energy levels with respect to the external rota-
tionfrequency(seeFig.2),wecanextendthesameargu- where |vac(cid:105) is the vacuum.
mentseasilytocondensatesinhigherangularmomentum This system has been studied extensively in Ref. [17];
states. For a condensate in |j(cid:105), the chemical potential is we briefly recap the results here. Similar to the non-
µ = 1j2 +ηN , and the quasiparticle energies become interacting case, in the ground state for Ω¯ < 1, all the
(cid:15)(cid:48)(Ω¯)2=ν(j−Ω¯c)+|ν|(cid:0)1ν2+ηN )1/2. Thus,theanoma- particles are condensed into |0(cid:105), while for Ω¯ >2 1, they
ν 4 c 2
lous ν =−1 mode (which connects |j−1(cid:105) and |j+1(cid:105) to are condensed into |1(cid:105). As one finds by extremizing the
the condensate) becomes stable at Ω¯ = j− 1 for a non- energy, the spectrum also acquires an energy maximum
2
5
when (cid:12)(cid:12)Ω¯ − 1(cid:12)(cid:12)<ηN, corresponding to the state The angular momentum per particle of the system
2
changes according to
(cid:12) (cid:69)
|b(cid:105)=(cid:12)(cid:12)12N +(Ω¯ − 12)/2η,12N −(Ω¯ − 12)/2η . (14) ∂ (cid:104)(cid:96)(cid:105)=−i∂ (cid:90)dθψ∗(θ,t) ∂ ψ (θ,t)=2vIm(cid:2)c∗c (cid:3) (18)
t t c ∂θ c 1 0
This state can be seen as the maximum in Fig. 1 for
Ω¯ =0 and ηN > 1; as ηN decreases below 1, the region whichvanishes,asitshould,intheabsenceofv andalso
2 2
in Ω¯ for which such a maximal state exists shrinks, as when the phase of c∗c equals 0 or π.
1 0
depicted by the dashed line in Fig. 4 for ηN = 1. This
4
state acts as a barrier between the two ground states,
makingitenergeticallyexpensivefordensitymodulations A. Swallow-tail loops
(vortex-inducedphaseslips)todrivethesystemfromone
minimumtotheother,evenasΩ¯ isvaried. Notetheloop We now ask how the system responds dynamically as
structure in Fig. 4 indicated by the lines labeled L , L , the external rotation rate is varied. As we show, the
b 0
and L , whose effect on the response of the system to non-linearityinherentintheGPequationleadstoforced
1
changes in Ω¯ will be discussed in the next section. tunneling between the energy levels of the system in the
presence of v. To see this behavior, we first construct
thesteady-statesolutionsoftheGPequationintheform
IV. SYMMETRY BREAKING IN MEAN-FIELD |ψ (t)(cid:105)=e−iµt|ψ (0)(cid:105) where µ is the chemical potential,
c c
THEORY a function of N; then, Eq. (17) gives
Wenowturntothequestionofhowthecondensatere- µc =(cid:104)ηN(cid:0)2−|c |2(cid:1)(cid:105)c +vc ,
0 0 0 1
spondsdynamicallytochangesinitsrotationrate. Inre- (19)
alisticexperiments,thepotentialsfeltbytheparticlesare µc =(cid:104)(cid:0)1 −Ω¯(cid:1)+ηN(cid:0)2−|c |2(cid:1)(cid:105)c +vc .
1 2 1 1 0
neverfullyrotationallyinvariant(see,e.g.,Ref.[10]);the
breaking of rotational invariance changes the way single- The occupation probabilities of |0(cid:105) and |1(cid:105) as functions
particle states of given angular momentum are mixed. of µ are
In order to couple the system to rotations of the trap,
we include in the Hamiltonian a small “disorder” poten- (cid:0)1 −Ω¯(cid:1)+ηN −µ
|c |2 = 2 =1−|c |2. (20)
tial, v(θ−Ωt), which is stationary in the frame rotating 0 (cid:0)1 −Ω¯(cid:1)+2(ηN −µ) 1
2
at Ω. Such a potential favors a coherent superposition
of states (a single condensate) over a fragmented (Fock) The eigenstates, then, have the form
state [25, 29, 30]; we assume the following variational
(cid:18) (cid:19) (cid:18) (cid:19)
|c | |c |
form for the time-dependent two-mode condensate wave |I(cid:105)= 0 , |II(cid:105)= 0 (21)
−|c | |c |
function 1 1
√
(cid:2) (cid:3) where|I(cid:105)denotesthegroundstatebranchand|II(cid:105)theex-
|ψ (t)(cid:105)= N c (t)|0(cid:105)+c (t)|1(cid:105) (15)
c 0 1
cited state branch, since a phase difference of π between
where |c (t)|2 +|c (t)|2 = 1. The time-evolution of the c0 and c1 minimizes the coupling energy 2NvRe(cid:2)c∗1c0(cid:3)
0 1 whereas a phase difference of 0 maximizes it.
condensate wave function in the rotating frame is gov-
Thechemicalpotential(the“adiabaticenergylevel”in
erned by the GP equation
thesenseofRefs.[14,18])isfoundfromthedeterminant
(cid:20) 1 ∂2 ∂ (cid:21)
i∂tψc(θ,t)= −2∂θ2+iΩ¯∂θ+η|ψc(θ,t)|2+v(θ) ψc(θ,t). (cid:12)(cid:12)(cid:12)(cid:12)ηN(cid:0)2−v|c0|2(cid:1)−µ (cid:0)1 −Ω¯(cid:1)+ηNv(cid:0)2−|c |2(cid:1)−µ(cid:12)(cid:12)(cid:12)(cid:12)=0
2 1
In the two-mode model, we can, with no loss of gener- (22)
ality, take v(θ)=2vcosθ with v real and positive, cor- together with Eqs. (20). The result is a fourth-order
responding to a coupling between the states |0(cid:105) and |1(cid:105) equationforµ, withtwotofourrealsolutionsdepending
of the form 2NvRe(cid:2)c∗c (cid:3), so that the mean-field Hamil- on the values of ηN/2v ≡ Λ and Ω¯. The chemical po-
1 0
tonian becomes tential is simply related to the energy per particle in the
rotating frame by
H2(cid:48) =(1−Ω¯)|c |2+1ηN(cid:0)1+2|c |2|c |2(cid:1)+2vRe(cid:2)c∗c (cid:3). (cid:16) (cid:17)
N 2 1 2 0 1 1 0 E(cid:48) =µ− 1ηN 1+2|c |2|c |2 . (23)
(16) 2 0 1
Withthiscoupling,theamplitudesobeythetwocoupled
The energy levels corresponding to states |I(cid:105) and |II(cid:105),
differential equations
in general, exhibit an avoided crossing as a function of
i∂ c =ηN(cid:2)2−|c |2(cid:3)c +vc , Ω due to the presence of disorder. Since µ and E(cid:48) are
t 0 0 0 1 (17) simply related by Eq. (23), the physical content of their
i∂ c =ηN(cid:2)2−|c |2(cid:3)c +vc +(cid:0)1 −Ω¯(cid:1)c . correspondingplotsisidentical. Toillustratethephysics,
t 1 1 1 0 2 1
6
II
II
II
B
I
A
I I
I I I
B
A
II
II
II
FIG. 5: (Color online) Adiabatic energy levels (top panels, in black), measured in units of (cid:126)Ω , and occupation probabilities
0
of |0(cid:105) (bottom panels, in blue) as functions of Ω¯ for ηN = v (left column), ηN = 2v (middle column), and ηN = 3v (right
column), with v =1/5. In all graphs, the lowest energy branch and the corresponding population are indicated by I, and the
top energy level and its corresponding population by II. The arrows A and B in the right column indicate the discontinuous
change in the population of |0(cid:105) and the forced tunneling of particles to |1(cid:105) as Ω¯ is changed past the folding point.
we plot the behavior in terms of µ since it is graphically I (indicated by the arrow A in Fig. 5) or to the upper
clearer. TheupperpanelsofFig.5showtherealsolutions branchII(indicatedbythearrowB).Similarly,theoccu-
forµasfunctionsofΩ¯ forselectedvaluesofηN andv. As pation probability of |0(cid:105) adiabatically follows the change
shown in the figure, and as we will prove at the end, for in Ω¯ until the branch starts to fold over itself, at which
Λ<1, the two energy levels have an avoided crossing at point a sudden change in the population of that state
thecriticalvalueoftherotationrateΩ¯ = 1; atΛ=1, a becomes inevitable with further increase of the rotation
c 2
cusp appears in the lower branch at this frequency; and frequency, as indicated by the arrows in Fig. 5. In other
for Λ > 1, the cusp gives birth to a loop in the lower words, a fraction of the particles in |0(cid:105) are forced to tun-
branch. The loop discussed earlier in Sec. III in the ab- nel to |1(cid:105). Sweeping Ω¯ in the other direction forces a
senceofthedisorderpotential(seeFig.4)evolvesintothe similar behavior on the system as well.
present loop as the disorder is turned on. Note that at a The folded-over section of branch I of the occupation
givenrotationfrequency,Fig.4showseithertwoorthree probability and the respective top part of the swallow-
states, while Fig. 5 shows two or four states; the extra tail loop of the lowest-energy level (in the right column
state arises from mixing of the upper maximum-energy of Fig. 5) are inaccessible through a sweep of Ω and cor-
state with lower-energy states (not shown in Fig. 4). respond to unstable states. In direct analogy with the
As seen in the lower panels of Fig. 5, the derivative of barrierstatediscussedearlier, theyindicatethepresence
the occupation, |c |2, of |0(cid:105) with respect to Ω¯ diverges ofmorethanoneminimumintheenergylandscape,sep-
0
as the cusp appears in the lowest energy band, and the arated by a maximum or a saddle-point [17, 35]. As we
occupation folds over itself (the characteristic S shape show in the next subsection, the appearance of the cusp
seeninfoldcatastrophes[32,34,35])astheloopemerges (along with the corresponding divergence of ∂|c0|2/∂Ω¯
for Λ>1. The swallow-tail loop indicates hysteresis [17] at Ω¯c) and the swallow-tail loop (along with the corre-
and a lack of adiabatic evolution with Ω¯ [14, 18, 27, 31, sponding fold-over in the level population) are related to
33] in a condensate in an annulus. the macroscopic quantum phenomenon of self-trapping
or self-locked population imbalance.
To see the physics of the swallow-tail loop, imagine
that we prepare the system, with Λ > 1, on the lower
branch at Ω¯ =0 and very slowly increase Ω¯ to avoid any
tunneling to the other branch. The system will, then, B. Self-trapping
followthisbranchadiabaticallyuntilthepointwherethe
branch terminates and folds back on itself (at Ω¯ > 1). Interesting quantum phenomena, including Josephson
2
Upon further increase of Ω¯, the system is forced to make effects analogous to those in superconductors and also
a discontinuous jump either to the lower part of branch chaotic dynamical behaviors, arise from the dynamical
7
(a) (b) (c) (d)
FIG. 6: (Color online) Population difference z (solid line, in red) and phase difference φ/π (dashed line, in blue) as functions
of τ for ∆E =1/2 and (a) Λ=1 below the critical value, (b) 7.75, (c) 8.2365 just below the critical value, and (d) 8.2374 the
critical value. The initial conditions are z(τ =0)=0.6 and φ(τ =0)=0. Note how the oscillatory behavior of z(τ) changes
from purely harmonic to anharmonic as Λ increases; finally, z(τ) and φ(τ) both become time-independent as Λ reaches the
critical value.
behaviorofthemacroscopicphasedifferencebetweenthe partofthecurveinFig.6(c)]. AtacriticalvalueofΛ,de-
two components of the time-dependent condensate (15). pendent on z(0), the oscillation period becomes infinite,
Toseethisconnection,werecastthetime-evolutionequa- andthepopulationimbalancebecomestime-independent
tions (17) in terms of the population difference z = atz(τ)=z [seeFig.6(d)]. ForΛlargerthanthiscritical
s
|c0|2 −|c1|2 (where −1 ≤ z ≤ 1) and the phase differ- value,z(τ)oscillatesintimeentirelyabovezs (notshown
ence φ = α −α between the two constituent states, in Fig. 6). Similarly, for fixed ∆E and Λ, there exists a
0 1
where αj =arg[cj]. Then, we find critical value zc of the initial population difference for
which the oscillations cease and z(τ) becomes constant
(cid:112)
∂ z =− 1−z2 sinφ (24) after a finite time (the plateau continues indefinitely).
τ
z This evolution to a state with a non-zero time-averaged
∂ φ=∆E+Λz+ √ cosφ (25)
τ valueofz(τ)(independentof∆E andforΛgreaterthan
1−z2
or equalto the criticalvaluediscussed above) is theana-
where we rescale the time to τ =2vt and set logofthephenomenonofself-trappinginthedouble-well
∆E =(cid:0)1 −Ω¯(cid:1)/2v. (26) system [19, 20].
2 The condition to develop self-trapping is that the two
time derivatives, ∂ z and ∂ φ, vanish simultaneously.
These equations are identical to Eqs. (3a) and (3b) of τ τ
From Eq. (24), this requires φ = 0 or π (although it
Ref. [19] which describes coherent tunneling between
appears that the singular point z = 1 also makes z(τ)
two Bose-Einstein condensates in a double-well poten-
time-independent, the correct solution is actually time-
tial; therefore,alltheresultsofthatpaperdirectlyapply
dependent [20], as can be deduced from Eq. (17), and
to the present system. The two levels, |0(cid:105) and |1(cid:105), cor-
is thus unacceptable). From Eq. (25), the steady-state
respond to the two wells. In the double-well system, an
value z is given in terms of ∆E and Λ by
applied DC voltage between the two wells induces tun- s
neling and, therefore, an AC particle current (cid:112)
∆E+Λz ±z / 1−z2 =0. (29)
s s s
I =2Nv∂ z (27)
τ
Once the system reaches this plateau, it must stay there
between the two condensates [20]. Analogously, in a forever, since the equations of motion are first-order in
toroidal trap, an external rotation induces a population time. The critical initial population difference z that
c
transfer between the two levels. leads to self-trapping can be found from energy conser-
The Hamiltonian (16) written in terms of φ and z is vation. For an initial phase difference, φ(0), we find
given, to within a constant term, by
(cid:112)
∆Ez + 1Λz2− 1−z2 cosφ(0)
H(cid:48) (cid:112) c 2 c c
2 =−∆Ez− 1Λz2+ 1−z2 cosφ (28) =∆Ez + 1Λz2∓(cid:112)1−z2. (30)
Nv 2 s 2 s s
and is a conserved quantity. Given that the dynamics is As illustrated in Fig. 6, the stationary self-trapped pop-
Hamiltonian, the quantum analog of the Poincar´e recur- ulation imbalance z is, in general, non-zero and only
s
rence theorem holds [36], and, therefore, the system is vanishes for ∆E =0.
inherently periodic in time. The steady-state self-trapped solution is, in fact, re-
The time evolution of the system, calculated numeri- latedtotheadiabaticenergylevelsdiscussedabove. The
cally, is shown in Fig. 6 for different Λ. With increase stationaryvalueφ=π leadstothegroundstate(branch
of Λ for a given initial population imbalance, z(0), the IinFig.5),whereasφ=0givestheexcitedstate(branch
oscillations in z(τ) change from purely harmonic to an- IIinFig.5). WefocusfirstonthepointΩ¯ =Ω¯ atwhich
c
harmonic and a plateau appears in z(τ) [the nearly flat the cusp and the tip of the swallow-tail loop appear and
8
for which ∆E = 0; choosing the minus sign in Eq. (29) rotation rate of the trap. Describing the coupling of
(corresponding to the ground state), we find three solu- the system to the rotation of the trap by a symmetry-
tions breaking disorder perturbation to the original Hamilto-
nian, we investigate the steady-state and the general dy-
(cid:112)
zs =0,± 1−Λ−2. (31) namics of this system in a two-mode mean-field model.
We find that in the presence of these disorders, swallow-
For Λ < 1, two solutions are complex and unphysical; tail loops appear in the lowest-lying energy band for suf-
however, when Λ = 1, all three solutions become degen- ficientlystronginteractionstrengthsand,astherotation
erate at zs = 0; and for Λ > 1, we have three distinct rate is varied, force a sudden, non-adiabatic change in
solutions. This change in the number of solutions at the the state of the system and the population of the two
criticalpointΛ=1isanotherindicationoftheoccurence constituentstates. Finally,weinvestigatetheconnection
of the catastrophe [34, 35] when a swallow-tail loop ap- of this system with a system of two condensates tunnel-
pears. Moreover, Eq. (29) also yields inginadouble-wellpotentialandfindthesetwosystems
to have identical dynamics; therefore, the analog of the
∂zs =− 1 =2∂|c0|2 (32) phenomenon of self-trapping also appears in the system
∂Ω¯ Λ(cid:0)1−Λ2(cid:1) ∂Ω¯ studied in this paper. We calculate the onset and the
properties of self-trapping, and show how the steady-
for ∆E = 0. For Λ = 1, this quantity diverges; for state self-trapped states are described in terms of the
Λ < 1, it is negative; and for Λ > 1, it is positive. To- energy eigenstates.
gether, these two quantities exhibit the exact behavior, The next step, which we discuss in a future publica-
with varying Λ, seen in the bottom panels of Fig. 5 at tion, is to include the effects of non-zero temperature on
Ω¯ = Ω¯c. In general, for non-zero ∆E, finding zs as a thedetailedtime-dependenceofaBosegasrotatinginan
function of Ω by solving Eq. (29), we indeed see the be- annulus. At finite temperature, the angular momentum
havior for |c |2 depicted in Fig. 5. Hence, the ceasing and energy of the condensate need not be conserved. As
0
of the coherent oscillation between the two components aresult,thermal,aswellasquantum,fluctuationscanin-
of the condensate (as the system becomes self-trapped) duce transitions between metastable states, as has been
and the appearance of a cusp or a swallow-tail loop in observed experimentally [10]. Furthermore, the coupling
the lowest-lying adiabatic energy level are in one-to-one of an unstable condensate to the environment would al-
correspondence. Thisprovesourpreviousstatementthat low it to evolve to a stable configuration.
the critical disorder strength for which a cusp or a loop
first appears is Λ=1 or, in other words, v =ηN/2.
Acknowledgments
V. SUMMARY
In this paper, we analyze the stability of a Bose- This work was supported in part by National Science
Einsteincondensateinarotatingtoroidaltrapintermsof FoundationGrantsNo.PHY07-01611andPHY09-69790.
the normal modes of small-amplitude deviations from a Author S.B. would like to thank Alexander Fetter for an
metastablecurrent-carryingstate. Weidentifyregionsof illuminating discussion and Brian DeMarco for critical
energetic and dynamical instabilities in the phase space comments on the manuscript and further useful sugges-
as functions of the interparticle interaction strength and tions.
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