Table Of ContentObervation of non-linear stationary spin waves in superfluid 3He-B
A. S. Chen, Yu. M. Bunkov, H. Godfrin, R. Schanen, F. Scheffler.
Centre de Recherches sur les Tr`es Basses Temp´eratures,
Centre National de la Recherche Scientifique,
BP 166, 38042 Grenoble, Cedex 9, France.
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(February 7, 2008)
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1 Duetoitsbrokenspinandorbitrotationsymmetries,superfluid3Heplaysauniquerolefortesting
n rotational quantumproperties on a macroscopic scale. In thissystem theorbital momentum forms
a texturesthatprovideaneffectivepotentialwellforthecreationofstationaryspinwaves. Inthelimit
J
of thelowest temperatures presently attainable, we observebyNMR techniquesa profound change
5 in the spin dynamics. The NMR line shape becomes asymmetric, strongly hysteretic and displays
1 substantial frequency shifts. This behavior, quantitatively described by an anharmonic oscillator
model,indicatesthattheparametersofthepotentialwelldependonthespinwavesamplitude,and
]
n thereforethattheorbitalmotion isnotdampedinthisnewregime,notconsidered bythestandard
o Leggett-Takagi theory. This regime of non-linear stationary spin waves is shown to give rise to the
c pulsed NMR ”Persistent Signals” reported recently.
-
r
p PACS number: 67.57.Np, 68.45.Gd
u
s Thegroundstateofsuperfluid3HeisformedbyCooper frequency). The equations for the spin precession de-
.
at pairs of 3He atoms in a triplet state with Spin and Or- rived by Leggett and Takagi (L-T) [1], written down for
m bital momentum quantum numbers equal to unity. In hydrodynamicconditions,describewellthehightemper-
- consequence superfluid 3He shows not only the usual su- ature experimental results. The superfluid component is
d perfluid properties arising from the broken gauge sym- responsible for spin supercurrents, the order parameter
n
metry, but also displays macroscopic quantum rotation stiffness and the dipole frequency shift, while the nor-
o
phenomena relatedto the brokenspinandorbitrotation malcomponentdetermines magneticrelaxationandspin
c
[ symmetries. The spin concerned is that of the nuclear diffusion (a review is given in ref. [2]).
1 rotation and is thus associated with a nuclear magnetic Itisusuallybelievedthattheseequationsremainvalid
v moment. The orbital motion consists of the mutual or- innon-hydrodynamicconditions,withsomesortofrenor-
9 biting of two neutral 3He atoms and is thus not associ- malisation of the relaxation parameters. This renormal-
5 ated with a magnetic moment. These two motions are isation was tested in NMR experiments in the tempera-
1
not independent, since the dipole-dipole interaction of ture regionabove about 0.4 T and found to be in good
1 C
0 two nuclearmagnetic moments depends ontheir relative agreement with the theoretical predictions [3]. The ef-
8 orientation. fective value of the magnetic interaction decreases very
9 In the 3He-B phase, at zero field, the magnetization rapidly in the non-hydrodynamic region, as 1/(ωτ).
/
t and the average orbital momentum are equal to zero. In the present work, performed at much lower tem-
a
m Applying an external magnetic field induces a magneti- peratures, we are able to reach a new condition where
zation M~ and, via the spin-orbit symmetry of the wave thespinandorbitdynamicsofthesuperfluidcomponent
d- function, an orbital momentum L~. This symmetry is arenolongerdampedbytheinteractionwiththenormal
n usually characterized by the rotation matrix Rˆ of angle component, where we can study true superfluid proper-
o Θ around an axis~n. The orbital momentum is therefore ties of rotation quantum dynamics. We describe in this
c L~ =Rˆ(~n,Θ)M~. article the observation of a new non-linear phenomenon
:
v The early investigations of superfluid 3He have been at temperatures of about 0.2 TC, now accessible to ex-
i
X performedatrelativelyhightemperaturesoforder0.4T perimental investigation.
C
r and above. In these conditions, first, the orbital motion Let us first remind the properties of spin dynamics at
a
is blocked due to an effective interaction with quasipar- temperatures above 0.4 T . First, the equilibrium lo-
C
ticles. Consequently, the NMR of the spin system con- cal orientation of the orbital momentum L~ with respect
sidered up to now only involves quasi-stationary orbital to the external magnetic field supplies an additional po-
dynamics. Second, there is an effective magnetic inter- tential for NMR and leads to a NMR frequency shift.
actionbetweenthe superfluidandnormalcomponentsof Usuallythe orbitalmomentumorientationis determined
3He, which is characterized by an effective quasiparticle byboundaryconditionsatthe wallsofthe cell,magnetic
lifetime τ. One has to distinguish two different regimes field anisotropy and bending energy. Since the angle Θ
◦
ofspindynamics,hydrodynamicandnonhydrodynamic, is fixed at about 104 by dipole-dipole interactions, the
for ωτ smaller or larger than unity (here ω is the NMR relative orientation of M~ is then determined by ~n which
1
canformacomplicatetexture. ThelocalNMRfrequency a)weobserveanenormousnumberofSSWnewmodes,
at relatively high field is: b) the SSW modes exhibit strong non-linearbehavior.
ω2 =(γH)2+n2⊥Ω2D (1)
whereΩ characterizesthedipole-dipoleinteractionand
D
n⊥ is the component of ~n normal to the magnetic field.
The distribution of the orientation of L~ gives rise to a
Ω2
broadening of the NMR line by about D, which can be
γH
on the order of tens of kHz.
Second, owing to the stiffness of the order parameter,
the spin systemexhibits non-localresonantmodes - Sta-
tionary Spin-Waves modes (SSW) trapped by a texture.
These modes were observed in the pioneering work by
Osheroff[4]. Later SSW were used for studying spin dif-
fusion[5],texturesandvortices[6]. TheSSWcanbecre-
atedinapotentialwelldeterminedbythewallsofthecell
Amplitude (% of HPD signal)
andthefieldgradient[5],bythewallandatexture[4]or
only by a texture, for e-xample the ”Flared-out” texture
[6]. The equation of m0otion for the transverse compo- 0 1
nent of t4he magnetizat.5ion M+ in the tex0ture potential l.5evFeIl:GH.11.=1T0h−e5wTheosllea1,CmWagNneMtiRc glrinaediaentt0:.5.2521T0−C4. TEexsclait.amt−io1n.
wellβL(r7)hastheformoftheSchr¨odingerequation[4,6]: Nearly 100 SSW modes are observed.
0
Ω2 2 24
(ω−ω(z))M+ = D ( sin2β (r)− ξ2∇2)M+. (2) As shown in Fig.1 we observe at this temperature an
2ω(z) 5 L 65 d
enormous number of NMR peaks that cover practically
whereβ isthe localdeflectionofthe orbitalmomentum all the NMR line. By changing the magnetic field gradi-
L
from the direction of magnetic field and ξ the dipole ent, we first determine the positions within the cell that
d
coherent length. The SSW modes have been studied at generate these SSW modes. We performed a calibration
relatively4hightemperatures,above0.4TC,andfoundto in normal 3He and then, at 0.22 TC, we studied the fre-
be in a g7ood agreement with theory. quency vs. magnetic field gradient dependence of each
Thene2wGrenoblerefrigeratorallowsustocool3Heto peakinthe limitofsmallexcitationlevel. This allowsus
significantly lowertemperatures. Owing to the exponen- toshowthatthetwolargestpeakscorrespondtothe top
tial dependence of the number of quasiparticles on tem- and to the bottom of the cell (as was expected in this
perature, we are able to decrease their density by many texturalconfiguration,and confirmedby the same study
orders of magnitude, thus suppressing the magnetic and onSSW at0.44 TC), with dipole shifts from the Larmor
orbitfriction. Inthis articlewereporttheobservationof frequency on the order of 150 Hz. Finally all the peaks
a dramatic change in the spin and orbit dynamics, ob- correspond either to the fundamental state at different
4
served in a CW-NMR experiment in this ultra-low tem- places within the cell, or to harmonics.
7
perature4region. We then fixed the magnetic field gradient at a value
The experiments were performed in a cylindrical ex- of 5 10−4 Tesla.m−1 and studied the behavior of sev-
perimental cell of diameter 6 mm and height 5 mm, at eral low-frequency-shift SSW modes, when changing the
F
pressures ranging from 0 to 6 bars. The magnetic field, excitation level and the frequency sweep direction. In
r
correesponds to a NMR frequency of about 500 kHz and Fig.2 we show the signals of a few low-frequency-shift
q
was directed parallel to the axis of the cell. In addition, SSW modes (absorption and modulus) at 0.44 T and
C
u
weewereable to changethe magnetic fieldgradientin or- 0.22 TC, for different excitation levels H1 (note that our
der to de4termine the localisation of NMR signals within sweepisgiveninunitsoffrequency,whichisrecalculated
n
7
thecell. Intheseconditionsa”flaredout”textureshould from our real field sweep. Sweeping the field up corre-
c
6
be yformed. This is shown schematically in the inset of spondsthereforetosweepingthefrequencydown.). Even
Fig .2: at the top and the bottom of the cell, the orbital if these signals are radiated by the same SSW modes
mom(entumisorientedparalleltothefield,whilethever- at both temperatures, their properties are very differ-
k
ticaHl walls orient L~ perpendicular to the field. This tex- ent: at low temperatures, the broadening of the signals
turazlconfigurationgivesaregionofminimaofenergyfor does not correspond to a relaxation process, but to a
NM)R of SSW near the top and the bottom of the cell. non-linear frequency shift with excitation. Let us look
We ob4served SSW signals at temperatures down to at signals for a relatively large rf field (H1=1.8 10−6
0.22 TC,7i.e. significantly lower than in previous investi- Tesla). If we sweep the field up (frequency down), the
gations.8We report here two new features: low-temperature-signalsappearatthesamefrequencyas
2
A
D
4 b
athightemperature,buttheygrowrapidlyevenwellbe- zero until point 3 and jumps to point 4, thus showing
low the high temperature resonance frequency. At some hysteretical behavior. We find that λ=0.001 kHz is the
criticalpoint,thesignaldisappears. Inthesweepbackwe damping factor giving the width of our resonant curve,
observe a strong hysteresis since the signal appears only while the anharmonic coefficient (κ=-1700 kHz.A−2) re-
around the high temperature resonance frequency. This lates the frequency to the amplitude of excitation. Here
behavior is typical of a non-linear oscillator, where the the amplitude A was measured in units of the full HPD
frequency strongly depends on the excitation level. The signal (Homogeneously Precessing Domain: in all the
◦
biggerthe excitation,the biggerfrequencyshift that can sample, the magnetization is deflected by 104 and pre-
be achieved. cesses homogeneously). For 0.44 T we find λ=0.007
C
kHz, while the anharmonic coefficient is negligible.
B
B A B' HPD) 0.6 2
%
e ( 0.5
A ud
H1=5.2 10-6 T mplit 0.4
0.22 Tc n a
o
ati 0.3
z
eti
n
H1=1.8 10-6 T mag 0.2 4
0.22 Tc e
s
er 0.1
v
1 %am opfl iHtuPdDe H0.12=21 .8T c10-6 T Trans 0 3 1
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06
H1=0.9 10-6 T w-w 0 (kHZ)
0.22 Tc
H=1.8 10-6 T
1
0.44 Tc
469.9 470 470.1 470.2 470.3 470.4 FIG.3. Anon-linearSSWsignal(points)anditsfitbythe
Frequency (kHz)
non-linear oscillator equation (line).
FIG.2. CWNMRabsorption(negativesignals)andNMR
amplitudes(that correspond toM+,positive signals) for dif-
This anharmonic oscillator model corresponds to the
ferenttemperaturesandamplitudesofrffield,H1. Thearrows fact that the potential well radiating the SSW becomes
indicatethedirectionofthefrequencysweep. Theinsetshows
theL~ textureandtheregions(labeledAandB)radiatingthe anharmonicwhenthetransversemagnetizationincreases.
This effect can be quantitavely described by incorporat-
SSW signals A, B, and the higherharmonics, like B’.
ingafeedbackcorrectionintothepotentialterm,sin2β ,
L
of the Schr¨odinger equation (2): β should no longer
Following Landau and Lifchitz [7], we consider an an- L
depends only on r, but also on M+. Indeed, the non-
harmonic oscillator with a third order non-linearity:
linearSSWareobtainedbypumping the energyinto the
M¨++2λM˙++ω2M+ =Fcos(ωt)−αM+2−βM+3 (3) spinwavemodeslocalizedinthepotentialwellformedby
0
the orbital texture. When the intensity of a given spin
The amplitude of the transverse component of the mag- wave increases, it influences the orbital degrees of free-
netization,M+ =Acos(ωt),isfoundbysolvingtheequa- dom. As a result, the potential well should be modified
tion and change the eigen-frequency of the spin-wave mode,
which in turn affects the orbital texture. The frequency
4ω02A2((ω−ω0−κA2)2+λ2)=F2, (4) of this self-consistent precessing state thus depends on
theexcitationlevelanddecreaseswiththemagnetization
where κ describes the dynamic frequency shift. Equa-
deflection. Qualitatively,whentheprecessingmagnetiza-
tion(4)isofordersixinA,andweareonlyinterestedin
tion is deflected, the texture becomes shallower, causing
real and positive solutions. The coefficients of equation
a decrease of the spin waves frequency shift.
(4) are obtained by fitting our experimental data. Fig.3
shows the comparisonbetween the theoretical curve and The signal from the bottom of the cell, labeled A in
theexperimentalsignalamplitudefromthetopofthecell Fig.2, has a more complicate behavior. Its amplitude
for a rf field level of 1.8 10−6 Tesla. When sweeping the increases faster than in the model considered here. Nev-
frequency ω down, the amplitude of the transversemag- ertheless,themainfeaturesaresimilar. Possibly,thedif-
netization follows the part 1−2 and then drops down ference originates from the spatial extension of the SSW
to zero. When the frequency increases, it stays equal to mode at high levels of excitation.
3
extremely helpful for guiding future experimental work.
In conclusion, we have shown that the spin dynamics
15.5 in superfluid 3He are profondly modified in the limit of
ultra-low temperatures. In this new regime the station-
15.4
ary spin waves become extremely non-linear, displaying
15.3
the typical behavior of a third order anharmonic oscil-
Hz) 15.2 lator. This effect can be ascribed to a softening of the
k texture potential for the spin waves, and modeled by a
y, ( 15.1
c non-linearSchr¨odingerequationwhichgovernsthetrans-
n
que 15 verse magnetization dynamics. We also show that the
e
Fr 14.9 origin of the Persistent Signals observed in pulsed NMR
experiments is directly related to the non-linear dynam-
14.8 Dispersion
ics of the system. Our new qualitative and quantitative
14.7 Absorption results call for an extension of the ”Standard Model of
14.6 Frequency, (kHz) 3He” given by Leggett-Takagi in order to describe the
14.814.9 15 15.115.215.315.415.5 ultra-low temperature regime, where the orbital motion
14.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 plays an important role in the spin dynamics. Finally,
Time, (s)
we would like to mention that spin waves in superfluid
3He-B can serve as an interesting experimental model
FIG. 4. The time dependent spectrum of Persistent Sig-
for Particle Physics systems described by the non-linear
nals observed after a NMR pulse. Inset: the two main CW
Schr¨odinger equation.
NMR signals in the same conditions. (the base frequency of
We are grateful to I. A. Fomin, A. J. Leggett,
thepulse NMR spectrometer (467 kHz) was subtracted from
G. R. Pickett, O. D. Timofeevskaya and G. E. Volovik
both signals)
for many stimulating discussions.
Finally, we investigated the relation between the ex-
citations reported here and Persistent Signals (PS). A
Persistent Signal is a small but extremely long lived in-
ductiondecaysignalthatwasobservedinpreviouspulsed
NMR experiments [8]. Its frequency increases with time
[9](whichisoppositetoconventionalHPD!),anditisra-
[1] A. J. Leggett, S. Takagi, Annals of Physics, 106, 79
diatedbyatexture[10]. WemadepulsedandCWNMR, (1977).
in the same low temperature conditions. In CW NMR, [2] Yu.M.Bunkov,”SpinSupercurrentandNovelProperties
we observed the non-linear SSW described before, while ofNMRin3He”,in”ProgressinLowTemp.Physics”14,
inpulsedNMR,weobservedPersistentSignals. Weshow 69, ed. W. Halperin, Elsevier (1995).
in Fig.4 the time dependent spectrum of the signal after [3] Yu. M. Bunkov, V. V. Dmitriev, A. M. Markelov,
an NMR pulse. We find two PS whose frequencies in- Yu. M. Mukharsky, D. Einzel, Phys. Rev. Lett, 65, 867
(1990).
crease with time, while their amplitudes decrease. The
[4] D. D. Osheroff, Physica B, 90, 20 (1977).
twoPSappearatthesamefrequencies,andtheyhavethe
[5] D. Candela, D. O. Edwards, A. Heft, N. Masuhara,
same width and frequency vs. amplitude dependence as
D. S. Sherrill, R. Combescot, Can. J. Phys., 65, 1330
thetwomainnon-linearCWNMRsignals. So,theyseem
(1987).
to be radiatedfrom the same texture by the same mech-
[6] P.J.Hakonen,O.T.Ikkala,S.T.Islander,O.V.Lounas-
anism. This identification is particularly informative on maa, G. E. Volovik, 1982, J. Low Temp. Phys., 53, 425
the natureof the PSandsuggestsa physicalscenariofor (1983).
their creation. At low temperatures, the magnetization [7] L. D. Landau, E. M. Lifchitz, Physique Theorique-
precession with high deflection angles is very unstable Mecanique, Editions de Moscou, Mir, 1966.
[11]. So, after the NMR pulse, the deflected magnetiza- [8] Yu.M.Bunkov,S.N.Fisher,A.M.Gu´enault,G.R.Pick-
tion decays quickly to the spin waves modes. However, ett, Phys. Rev. Lett., 69, 3092 (1992).
[9] Yu.M.Bunkov,S.N.Fisher,A.M.Gu´enault,G.R.Pick-
since the orbital momentum has also some kind of flex-
ett, S. R.Zakazov, Physica B,194, 827 (1994).
ibility, the PS signal arises as a combined solution for
[10] Yu.M.Bunkov,D.J.Cousins,M.P.Enrico,S.N.Fisher,
the magnetizationandtexture motion. Thisexplanation
G. R. Pickett, N. S. Shaw, W. Tych, Czechoslovak J.
is consistent with the feedback mechanism for SSW in a
Phys., 46, S1, 233 (1996).
”flared out” texture proposed earlier to explain some of
[11] Yu. M. Bunkov, V. L. Golo, O. D. Timofeevskaya,
the PSproperties[12]. Inprinciple,itshouldbe possible Czechoslovak J. Phys. 46, S1, 213 (1996).
to develop a quantitative theory for PS as well as non- [12] Yu. M. Bunkov, Czechoslovak J. Phys. 46, S1, 231
linear SSW by numerical simulations. These would be (1996).
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