Vortex waves in trapped Bose-Einstein condensates

1.1 MB·English

by  T. P. Simula

Vortex waves in trapped Bose-Einstein condensates
T. P. Simula, T. Mizushima, and K. Machida
Department of Physics, Okayama University, Okayama 700-8530, Japan
We have theoretically studied vortex waves of Bose-Einstein condensates in elongated harmonic
traps. OurfocusisontheaxisymmetricvaricosewavesandhelicalKelvinwavesofsinglyquantized
vortex lines. Growth and decay dynamics of both types of vortex waves are discussed. We propose
a method to experimentally create these vortex waves on demand.
PACSnumbers: 03.75.Lm,67.85.De
I. INTRODUCTION icsofsuperfluidKelvinwavesiscrucialtounderstanding
9
thedetailsofquantumturbulence[23,24]. Itisalsocon-
0
ceivable that the quantum gases would support varicose
0 Substancewhirlingarounda line inspaceforms avor-
2 waves in which an axisymmetric pulsating perturbation
tex. Eddies in a flowing water and whirlwinds in the
propagates along the vortex line causing periodic modu-
n atmosphere are perhaps the most common examples of
lation of the vortex core diameter.
a vortices encountered in nature, although such structures
J In this paper, we mainly focus on two types of vor-
exist in nearly every imaginable medium including laser
6 light and superconductors. Vortices are objects of fun- tex waves in trapped Bose-Einstein condensed gases. In
theSec. IIwedescribeournumericalmethodsandinthe
damental importance in fluid dynamics and they are in-
] Sec. IIIwefirstextendourpreviousresults[21]onKelvin
r herent to turbulent flows featuring normal mode excita-
e waves followed by an introduction to varicose wave exci-
tions analogous to vibrating classical strings [1]. Lord
h tations. Finally, the Sec. IV is devoted to discussion.
Kelvin (Sir W. Thomson) derived a dispersion relation
t
o for a class of normal modes of classical hydrodynamic
.
t vortices [2]. A particular type of vortex excitations in
a II. METHODS
which the vortex twists around its equilibrium position
m
forming a helical structure have since become known as
- We consider a Bose-Einstein condensate composed of
Kelvinwaves[3]. Suchchiralvortexvibrationsareanele-
d
87Rb particles trapped in a harmonic potential whose
n mental property of vortices ranging from tiny whirlpools
transverse and axial frequencies are ω = ω = ω =
o in creeks to tornadoes of atmospheric scales. Another x y ⊥
c type of axially propagating excitation mode—coined a 2π×98.5Hzandωz =2π×11.8Hz,respectively,asinthe
[ experimentofBretinet al. [12]. Thenumberofparticles
varicose wave—may exist in fluids confined to cylindri-

inthecondensateis1.3×105unlessotherwisestated. The
2 cal geometry. These radially pulsating modes have the
ground state and its dynamics are accurately described
v varicose shape, well known in the context of blood flow
3 through veins in the cardiovascular systems [4]. bythetime-dependentGross-Pitaevskiiequation[25,26]
9
9 Superfluids are distinguished from classical fluids by i(cid:126)∂ ψ(r,t)=(cid:0)H +V (r,t)+g|ψ(r,t)|2(cid:1)ψ(r,t), (1)
t 0 pert
their ability of frictionless flow [5]. Rotational mo-
0
. tion of superfluids is supported by quantized vortices where the single-particle quantum harmonic oscillator
9
0 [3]. Suchquantumvorticespossessmanypropertiesakin operator is given by
to their classical counterparts—including Kelvin waves
8
0 [3, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Superfluid Kelvin waves (cid:126)2∇2 1
H =− + m(ω2x2+ω2y2+ω2z2), (2)
: were first experimentally introduced in the context of 0 2m 2 x y z
v
liquid helium [3, 6, 7, 9]. Later the discovery of Bose-
i
X Einstein condensates of dilute atomic gases [15, 16, 17] and the last term in Eq.(1) accounts for the contact
r facilitated the creation and direct imaging of quantized interaction of strength g between particles. The time-
a vorticesinhighlycontrollablesuperfluidsystems[18,19]. dependent potential Vpert(r,t) may be engineered to res-
Inparticular,Kelvinwavesonanisolatedquantizedvor- onantly excite various normal modes by perturbing the
tex line were explicitly excited and imaged for the first ground state. Such collective excitation modes are also
time by Bretin et al. [12]. Their excitation method was obtained within the linear response theory by solving
described in terms of Beliaev decay of the quadrupole the coupled Bogoliubov-de Gennes eigenvalue equations
mode into a pair of Kelvin waves [12, 20]. Further in- [25, 26]
sight for this process was recently obtained by direct dy-
namical simulations [21]. Hydrodynamically it may be Luq(r)+gψ(r)2vq(r) = Equq(r)
understood in terms of a parametric resonance between Lv (r)+gψ(r)∗2u (r) = −E v (r), (3)
q q q q
thequadrupolemodeandapairofKelvinmodesofoppo-
siteaxialmomenta,whichinducesanellipticalinstability whereu andv aretheusualquasiparticlewavefunctions
q q
to the initially straight vortex line [21, 22]. The dynam- corresponding to the excitations with eigenenergies E ,
q
2
and the single-particle operator local induction approximation the self-induced fluid ve-
locityinthevicinityofthevortexcoreisgivenby[3,29],
L=H +2g|ψ(r,t)|2−µ (4)
0 v (r)=Cs(cid:48)×s(cid:48)(cid:48), (7)
loc
contains the condensate chemical potential µ.
where C is a system dependent constant and a prime
We solve the time-dependent Gross-Pitaevskii equa-
denotes a derivative with respect to the curve parameter
tion by utilizing a parallel MPI implementation of a sothats(cid:48)(cid:48)isthelocalcurvaturevectorofthevortexands(cid:48)
finite-element discrete variable representation method in
is tangent to the vortex line, pointing to the direction of
a fully three dimensional cartesian coordinate system
the local vorticity vector. The Kelvin-Helmholz theorem
[27, 28]. For finding solutions to the Bogoliubov-de
guarantees the conservation of circulation which implies
Gennes equations, we deploy the cylindrical symmetry
thatavortextendstomovewiththefluiditisimmersed
of the axisymmetric single vortex stationary state. As a
in. Hence according to Eq.(7) the local motion of the
resultwehaveagoodazimuthalquantumnumber(cid:96)thus
vortex line element is in the direction determined by the
reducingtheeffectivecomputationaleffortbyonespatial
binormal of the vorticity and the curvature vectors. In
dimension. To diagonalize the matrix form of Eq.(3), we
a uniform system this then implies that the Kelvin wave
use a hybrid basis consisting of radial Laguerre polyno-
vortex helix must rotate in the direction opposite to the
mials and finite-difference discretation in the axial direc-
fluid flow around the unperturbed vortex.
tion. This yields a matrix which has a computationally
However, in the presence of external fields such as a
beneficial band shape. harmonic trapping field, the local curvature s(cid:48)(cid:48) may be-
comesignificantlymodified. Aparticularlyelusiveexam-
pleisasinglyquantizedvortexprecessingaboutthetrap
III. RESULTS axisinaharmonicallytrappedBose-Einsteincondensate
[30]. In such system the vortex is bent even in the pan-
By solving the eigenvalue problem for the axisymmet- cake limit and the effective curvature vector points away
ric single vortex initial state we obtain full information from the trap centre and therefore the local self-induced
of the small-amplitude normal modes of our superfluid velocity,whichdeterminesthedirectionofvortexpreces-
vortex state. Using dynamical simulations we can study sion, is in the same direction as the unperturbed con-
the excitation and subsequent decay dynamics of vortex densate flow. The Kelvin modes correspond to a small
waves beyond linear stability analysis. Combining these amplitudemotionofthevortexcoreandthecorrespond-
two resources we are able to analyze excitation mecha- ing mode densities are highly localized within the core
nisms, dispersion relations, and decay processes of the volume of the vortex.
collective vortex excitations. In the following we sepa- Dispersionrelation.—ThedrivingforceofKelvinwave
rately focus on helical (cid:96)=−1 Kelvin waves and axisym- motion involves velocity instead of acceleration and the
metric (cid:96)=0 varicose waves. resulting semiclassical long-wavelength, |a0kz| (cid:28) 1, dis-
persion relation ω(k ) = (cid:126)k2ln(1/|a k |)/2m depends
z z 0 z
logarithmically on the wave vector k . In the case of
z
A. Helical (cid:96)=−1 Kelvin waves an axisymmetric vortex line in a harmonically trapped
Bose-Einstein condensate the kelvon dispersion relation
is better described by [21]
Physical characterization.—Kelvin waves or their
quanta—kelvons—are helical vortex waves propagating (cid:126)k2 (cid:18) 1 (cid:19)
along the vortex line with an axial wave vector kz. The ω(k0+kz)=ω0+ 2mz ln |r k | , |rckz|(cid:28)1, (8)
c z
path s (ζ,t) traced by the core of a vortex executing an
0
idealizedclassicalKelvinwavemotionofanamplitudeR where rc is a core parameter which is of the order of the
may be curve parametrized as vortex core size. The “continuum” branch described by
the above formula is cut off at low frequency ω <0 and
0
s (ζ,t)=Rcos(k ζ+ωt)ˆe −Rsin(k ζ+ωt)ˆe +ζˆe , long-wavelength k ∼ 2π/D , where D is the Thomas-
0 z x z y z 0 z z
(5) Fermi length of the condensate. In addition, the spec-
where t denotes time and ω is the rotation frequency of trumcontainslow-energybendingmodeswhichliebelow
the line element about the z-axis. A particle following the continuum branch. From a qualitative point of view
such trajectory has a negative helicity. One may use the modes with ω < 0 rotate in the same direction as the
Biot-Savart law to obtain the velocity field [3] unperturbedcondensateflow,whereasmodeswithω >0
rotate in the opposite direction. Modes with zero fre-
(cid:126) (cid:90) (s −r)×ds quency remain stationary in the laboratory frame. Fig-
v(r)= 0 0. (6)
2m |s −r|3 ure 1 shows the computed kelvon dispersion relations
0
for three different system sizes N = 1×104 (squares),
The divergence in the integral may be removed by intro- N = 5 × 104 (bullets), and N = 5 × 105 (triangles).
ducing a cutoff a = |r−s |, which causes the internal The solid curves are plotted using the dispersion rela-
0 0
structure of the vortex core to be neglected. Within the tion, Eq.(8), where ω and k are determined from the
0 0
3
FIG.2: Kelvinwavesofdifferentwavenumbersexcitedusing
V /(cid:126)ω = 6.0 (a) 4.0 (b), and -4.0 (c). The duration of the
0 ⊥
quadrupole perturbation is 35 ms for all cases. The frames
(a)-(c) are for times 75, 86, and 112 ms, respectively.
possible to control the energies of the kelvons with re-
spect to the condensate ground state energy. Since this
FIG. 1: Kelvon dispersion relation for different system sizes,
may be performed in a fashion which does not dramat-
N =1×104(squares),5×104(bullets),and5×105(triangles).
ically affect the resonant quadrupole mode frequency, it
Thesolidcurvesareplottedusingthedispersionrelation,Eq.
is possible to modify and choose the kelvon wave vector
(8), with core parameter r = 0.13 µm for all cases. The
c
dashedlineillustratesthesensitivityofthedispersionrelation which resonates with the quadrupole mode. Quantita-
to the change of the core parameter as specified in the text. tively such controlled kelvon excitation process may be
achieved by using the time-dependent perturbation
numerical results and the core parameter rc = 0.13 µm V (r,t) = V σ02 exp(−2r2/σ(z)2) (9)
for all cases. In contrast, the value of the healing length pert 0σ(z)2
variesbetween0.2and0.4µm. Thedashedcurveisplot- + (cid:15)mω2(cos(2Ωt)(x2−y2)+2sin(2Ωt)xy)/2,
ted for N = 1×104 using r = ξ = 0.4 µm highlighting ⊥
c
thefailureofapproximatingthecoreparameterr bythe wherethefirstlinemodelsaGaussianlaserfieldandpro-
c
healing length ξ. However, all cases shown in Fig. 1 are vides the frequency shift to the kelvon dispersion. It is
still in the Thomas-Fermi regime and for very small sys- also worth noting that at finite temperatures the kelvon
tems the dispersion relation shows much stronger finite- frequencies may experience an additional frequency shift
size effects. The modes below the continuum dispersion [31, 32]. The weak potential, (cid:15) = 0.025, in the second
are the bending modes [21]. As the number of particles line of Eq.(9) may be realized by rotating an elliptically
inthecondensateisincreaseditbecomescorrespondingly deformed trap and is responsible for the resonant exci-
longer and hence k approaches zero. tation of the quadrupole mode. In Eq.(9), Ω = −0.6ω
0 ⊥
Excitation method.— An elegant way to excite Kelvin is the rotation frequency of the elliptical perturbation,
(cid:112)
waves has been realized in the experiment of Bretin et σ(z) = σ 1+(z/z )2 and we have chosen σ = 2µm
0 R 0
al. [12]. In this method, an elliptical rotating per- with a Rayleigh range of z = 20µm. It may be experi-
R
turbation first populates the quadrupole (cid:96) = −2 col- mentallychallengingtoachievefocusingtosuchanarrow
lective surface oscillation mode of the condensate thus beam waist, however, the method also works for wider
reducing the orbital angular momentum of the system. beams although the accuracy in determining the disper-
Subsequently, the quadrupole mode population sponta- sion relation may then suffer. Furthermore, it would
neouslydecayspairwiseinto(cid:96)=−1Kelvinmodesviathe probablybewisertoswitchonthepinningbeamalready
energy-momentumconservingBeliaevprocessinwhicha before creating the condensate in order to avoid super-
quadrupole quasiparticle interacts with the condensate, fluous excitations.
scattering to a pair of oppositely traveling Kelvin modes Frames (a)-(c) of figure 2 show a gallery of kelvons
[12, 20, 21]. Hydrodynamically one may explain such with different wave vectors which have been gener-
process in terms of a parametric (subharmonic) reso- ated by applying the above prescription with V =
0
nance between a quadrupole mode and a pair of Kelvin {6.0,4.0,−4.0} (cid:126)ω , respectively. The duration of
⊥
modes, leading to an elliptical instability of the initially the perturbation was set to 35 ms and the plots are
straight vortex line [21, 22]. for times 75, 86, and 112 ms after the beginning
The shortcoming of this resonant excitation method of the perturbation. The initial in-plane sinusoidal
is that it is constrained to a specific kelvon. However, shape of the vortex s (ζ,t) = 2Rcos(k ζ)cos(ωt)ˆe −
± z x
it is possible to circumvent this constraint by deploy- 2Rcos(k ζ)sin(ωt)ˆe +ζˆe ,seenineachframeofFig.(2),
z y z
ing the core localized nature of the Kelvin modes [21]. formsasasuperpositionoftwohelicalwavestravelingin
By applying an external pinning potential, such as pro- opposite axial directions. It might also be possible to
duced by a suitably detuned laser beam, at the vortex directlyexcitekelvonsbyusingsuitablytunedLaguerre-
core thereby changing the local effective potential it is Gaussian beams or by applying an axially propagating
4
velocity field such as a frequency matched traveling op-
tical lattice with an appropriately chosen lattice period
whichcouldprovidearesonantcouplingtoKelvinmodes.
Decay dynamics.— Thedecayof Kelvin wavesmay be
studied by measuring the length of the vortex line
(cid:90) Z
(cid:112)
S(t)= 1+[dX(z,t)/dz]2+[dY(z,t)/dz]2dz
−Z
(10)
as a function of time t, where X(z,t) and Y(z,t) are
the parametrized x and y positions of the vortex core,
respectively. We define a reduced line length density
L(t)=(S(t)/S(0)−1)mω /(cid:126),whereS(0)isthereference
⊥
length of a straight vortex. The quantity L(t)/L(t(cid:48)) is
shown in Fig. (3), where S(t(cid:48))/S(0)=1.06 and the data
is obtained for V /(cid:126)ω =0. At t≈0.1 s the vortex line
0 ⊥
FIG. 3: Reduced length of the vortex as a function of time.
shape rapidly changes from being straight to oscillating
Initially the vortex line remains straight until at 0.1 s its
sinusoidallyinplane. Theincreaseinthelinelengthmay
length rapidly increases due to the exponentially growing in-
be estimated by an exponential function, Aexp(γω t)
⊥ stability. Thelinelengthdecayismodeledusingtheequation
plottedinFig. (3)usingA=0.001andγ =0.3. Between
(12).
t ≈ 0.1 s and t ≈ 0.3 s the line length diminishes as the
kelvons decay to kelvons of smaller wave numbers. The
final state is a bent vortex which causes the data in Fig.
(3) to saturate at a finite value. For our system, energy
isconservedbothduringthegrowthandthedecayofthe
line length. In the context of superfluid turbulence and
Kelvin wave cascade, the decay of the vortex line length
densityisfrequentlymodeledusingarateequationofthe
form [24, 33, 34, 35, 36, 37]
dL(t)
=−νL(t)α, (11)
dt FIG. 4: Fourier transformed dipole moment k1 (a) and
z
monopolemomentk0 (b)ofthesystemasafunctionoftime.
where,forα=2,ν hastheunitsofaneffectivekinematic z
viscosity. This may be integrated to yield
L(t) 1 moments
= . (12)
L(t(cid:48)) 1+νL(t(cid:48))(t−t(cid:48))
(cid:90) ∞(cid:90) ∞
ThedecayingcurveinFig. (3)isplottedusingν/κ=0.2, Q(cid:96)(z,t)= y(cid:96)|ψ(x,y,z)|2dxdy (13)
where κ=2π(cid:126)/m is the quantum of circulation. It is in- 0 0
teresting to compare this result to, e.g., those in Refs.
[34, 36, 37]. However, more detailed investigation is re- wemaygainfurtherinformationonthecollectiveexcita-
quiredifonedesirestodrawadirectconnectionbetween tions present in the system. Figure 4(a) and (b) display
thedecayofKelvinwavesonasinglevortexinaharmon- the respective spatially Fourier transformed dipole (cid:96)=1
icallytrappedBose-Einsteincondensateandthedecayof and monopole (cid:96)=0 signals
vortex tangle in turbulent superfluid helium systems.
Further insight to the excitation and decay of Kelvin
k(cid:96)(t)=F{Q(cid:96)(z,t)−Q(cid:96)(z,0)}, (14)
waves may be obtained in terms of the multipole mo- z
ments of the system. Initially, the condensate wavefunc-
tion attains strong quadrupole (cid:96)=−2 moment since the whereF denotesaspatialFouriertransform,obtainedfor
elliptical rotating drive builds up population in the ap- V /(cid:126)ω = 0. At t ≈ 0.1 s a clear dipole moment signal
0 ⊥
propriatequadrupolemode. Whenthequadrupolemode emergesatk1 =0.8and0.9µm−1 signifyingtheonsetof
z
population decays to (cid:96) = −1 kelvons, the wavefunction the kelvon excitation, see Fig. 4(a). Soon afterwards, a
gains a corresponding dipole moment and finally, as the strongmonopolesignalemergesatk0 =0.2and0.4µm−1
z
kelvons further decay to phonons and kelvons of longer due to the phonon emission from the decaying kelvons,
wavelength, they reveal themselves in the monopole mo- see Fig. 4(b). Since the cylindrical symmetry is broken
ment of the wavefunction. Thus by analyzing the z- whenthekelvonsareexcited,theirpresenceinthesystem
dependence of the real monopole (cid:96)=0 and dipole (cid:96)=1 is manifest in both the monopole and dipole signals.
5
FIG. 6: Varicose waves of different wave vectors k =0.2 (a)
z
0.4 (b), and 0.6 µm−1(c).
FIG.5: Dispersionrelationforvaricosewaves(boxes),sound
waves (circles) and Kelvin waves (triangles). Solid (open)
markers are plotted for even (odd) z-parity modes. The
solidcurvesaretherespectiveanalyticalvaricon,phononand
kelvon dispersion relations discussed in the text.
B. Axisymmetric (cid:96)=0 varicose waves
Physical characterization.— We assign the term vari-
cose wave—or varicon to denote their quanta—to de-
FIG.7: Wavevectorasmeasuredfromthemonopolemoment
scribe a class of axisymmetric, (cid:96)=0, collective modes of
according to Eq.(14) as a function of time. The varicon with
the condensate. Here we focus on varicons of lowest fre-
k = 0.4 µm−1 is first resonantly excited by the external
quencies which involve transverse motion and hence our z
perturbation. For later times, these varicons couple to the
varicose wave having zero axial momentum corresponds phononbranch,inparticulartotheonewithk =1.1µm−1.
z
to the usual radial breathing (or monopole) mode [38]. The inset shows the condensate at t=48 ms.
The modes with finite axial wave number exhibit sim-
ilar sinusoidal breathing oscillation with z−dependent
phase. In these systems the collective modes are den- Excitation method.— Varicose waves may be reso-
sity waves and therefore also the axial phonons produce nantly excited using a perturbing potential of the form
similar density modulation along the vortex core due to
V (r,t)=(cid:15)mω2(x2+y2)cos(Ωt)cos(k z), (16)
the local density variations caused by the axially propa- pert ⊥ z
gatingsoundwaves[39]. Incontrasttothekelvonswhich
where Ω estimates the varicon frequency and k is the
only exist in the presence of a vortex and are localized z
corresponding wave vector. In the limit k → 0 the
in the vortex core, varicons correspond to the transverse z
perturbation corresponds to a simple modulation of the
swell-squeeze motion of the condensate and exist also in
strength of the radial trapping potential resulting in the
the absence of the vortex.
excitationofthefundamentalmonopolebreathingmode.
Dispersion relation.— Figure 5 shows the computed
Figure 6 shows a collection of varicose waves of differ-
varicose mode frequencies (boxes) as a function of their
ent wave numbers excited using the potential given by
wavevector. Forcomparison,wehavealsoincludedboth
Eq.(16), where the frequency and wavevector are chosen
the phonon (circles) and kelvon (triangles) modes. The
according to Fig. 5. The duration of the perturbation is
upper-most solid curve follows the frequency shifted free
24msandtheframes(a)-(c)areplottedfortimes48,26,
particle dispersion relation
and 27 ms, respectively.
(cid:126)k2 Decay dynamics.— Figure 7 displays the wave vectors
ω(kz)=ω0+ 2mz (15) of the varicose waves as measured from the monopole
moment Q0 according to Eq.(14), as a function of time.
describing a varicose mode in the presence of a vortex Theresonantexternalperturbationfirstexcitesthevari-
line. Here ω corresponds to the fundamental monopole con with k = 0.4 µm−1 which then couples to phonons
0 z
frequencyofthecondensate[38]. Thelowestcurveisthe and in particular to the one with k = 1.1 µm−1, see
z
kelvon dispersion relation, Eq.(8), and the straight line also Fig.7. The varicose wave perturbations propagate
isthelinearphonondispersionω =c|k |, wherecisthe in both directions along the vortex axis. On arriving
p z
density averaged speed of sound. at the low-density ends of the condensate, the varicose
6
waves produce ring currents which become pinched off kelvons.
fromthemothercondensateasseenintheinsetofFig.7. Theonlydissipativemechanismoperationalinthezero
temperature limit which may return a turbulent super-
fluid system to a laminar flow state is a reduction of
IV. DISCUSSION the total length of the vortex lines. It is believed that
in an isolated vortex the Kelvin wave cascade process
Inconclusion,wehavestudiedaxiallypropagatingcol- is responsible for this transfer of the excess energy from
lective excitations of quantized vortex lines in harmon- small to large wave numbers and eventually phonon ra-
ically trapped elongated Bose-Einstein condensates. In diation is expected to absorb the surplus energy of the
particular, we have focused on two types of axial vortex turbulence [23, 24, 40, 41, 42, 43]. In contrast to this
waves: helical Kelvin waves and axisymmetric varicose scenario—if applicable to our freely decaying single vor-
waves. We have studied the excitation mechanisms, dis- tex case—here the cascade proceeds predominantly to-
persion relations and decay processes of both types of ward smaller wave numbers, i.e., toward longer wave-
collective excitation modes. We have employed efficient lengths and smaller kelvon frequencies.
computational methods to solve: (i) the time-dependent In addition, we have studied axisymmetric excitations
Gross-Pitaevskii equation for the full three-dimensional which exhibit a varicose shape. The long wavelength
quantum dynamics of the vortex motion and (ii) the dispersion relation of the varicose waves studied here is
Bogoliubov-de Gennes equations for the stationary ax- quadratic in the wave vector. The varicose waves are
isymmetric single vortex eigenstates. The combination readily visualized as periodic modulations of the vortex
of these two tools provides us with a comprehensive de- core diameter along the vortex axis, although the modes
scription for this zero temperature system. themselvesexistindependentofthepresenceofavortex.
The Kelvin wave dispersion relation at the long wave The varicons are found to decay via coupling to phonon
lengthlimithasbothpositiveandnegativeenergyeigen- modes of higher wave vector. Detailed understanding
modes with respect to the condensate energy and these of quantized vortex waves and their dynamics may shed
have distinct physical manifestations: positive modes lightonopenquestionsregardingvariousvorticalsystems
counter-rotatewithrespecttotheunperturbedsuperflow including turbulent superfluids.
whilethenegativemodesareco-rotating. Thekelvonfre-
quenciesmaybeengineeredusingexternalpinningpoten-
tials. In particular, it is possible to choose the sense of
Acknowledgments
rotationofparticularkelvonandtoexcitekelvonsofspe-
cific wave numbers. Experimentally, Kelvin waves may
be created indirectly by exciting the quadrupole mode This work was financially supported by the Japan So-
of the system and allowing it to resonate with a pair of ciety for the Promotion of Science (JSPS).
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