Pis'ma v ZhETF, vol. 101, iss. 12, pp. 902-907 © 2015 June 25
Measurements of anisotropic mass of magnons confined in a harmonic
trap in superfluid 3He-B
y. y. Zavjalov1^, S. Autti, V. B. Eltsov, P. J. Heikkinen
Low Temperature Laboratory, Department of Applied Physics, Aalto University, PO Box 15100, FI-00076 AALTO, Finland
Submitted 6 May 2015
We can pump magnons to a nearly harmonic magneto-textural trap in superfluid 3He-B. Using the NMR spectroscopy of levels in the trap we have measured the anisotropic magnon mass and related values of the spin-wave velocities. Based on our measurements we provide values of the Fermi-liquid parameter F".
DOI: 10.7868/S0370274X15120061
Introduction. 3He-B is the topological superfluid with gapless Majorana fermions on the boundary (see recent review [1]). Optical magnons in a magnetotextural trap proved to be a useful and convenient experimental tool for studying various properties of superfluid 3He-B. A number of effects can be observed in this system such as Bose-Einstein condensation of magnons [2], Suhl instability [3] with excitation of other spin-wave modes including a longitudinal Higgs mode [4], self-localization of magnons [5]. It can be used clS cl probe of quantized vortices [6], Andreev bound states and gravity waves on the 3He surface [7], boundary between 3He-A and 3He-B superfluids [8]. It also can be used as a tool for accurate measurements of various 3He parameters and as a thermometer which works below 0.3 mK [8].
For proper interpretation of these measurements, basic properties of magnons should be accurately known. In this paper we report detailed measurements of magnon spectra from which we find the anisotropic magnon mass and spin-wave velocity in 3He-B.
Spin waves in 3He B. The equilibrium state of superfluid 3He-B is described by the order parameter matrix:
A,
0-3
1a
ajj
(1)
where A is the energy gap, y is the phase, and Raj is a rotation matrix which can be written in terms of the rotation axis n and the angle 9 as
: cos 0 Saj + (1 — cos 0) narij — sinö eajknk. (2)
In non-zero magnetic fields the gap becomes anisotropic, but for fields used in this work we can neglect this effect.
Spin waves in 3He-B correspond to oscillations of the rotation matrix Raj. The motion is affected by the
-'-'e-mail: vladislav.zavyalov@aalto.fi
energy of the spin-orbit interaction Fso and the gradient energy Fy.
XbV-1
FSI
1572
-(RjjRkk + RjkRkj)
Fv = -A\KÍGÍ + K2G2 + K3G3),
(3)
(4)
where
G1 = Vji?afcVji?afc, Gl = VjRak^kRaj, G3 = VjRajVkRak,
Xb is the spin susceptibility of the 3He-B, 7 - the gy-romagnetic ratio for the 3He atom, Qb ~ the Leggett frequency, and K\, if2 and are parameters of the gradient energy.
The linear equation of small spin oscillations near the equilibrium value S° = (\b/7) H is [9]
Sr.
[S x 7H],
A 7 [KV2 Sc- K> VjR^R^Vk Sa] -
Xb
ñ • (S — S°)
(5)
where K = 2KX +K2 + K3 and K' = K2 + K3.
In a texture where n is almost parallel to H or in a high magnetic field wl = Qb one can
separate transverse (S — S° _L H) and longitudinal (S — S° || H) oscillations of spin. For a harmonic solution S - S°
= s e .2 „2
one can write
—cj_V - (c| - cjJVj/jZfcVfc + sin2 ßn
= UJ(UJ - LOL) S+, -CiV2 - (Cf - CDVjî/hVk + cos2 ßn
= to2 Sr.,
(6)
902
IlHCbMa b >K3TO tom 101 bun. 11-12 2015
where s+ = + isy), /3n is an angle between n and
H, the orbital anisotropy axis lj = RajSand
Ja2 JA2
2 - 7 (K - K'/2), c\ = 7
C_L =
c2 =
K,
Xb
72A2
Xb
Xb
72A2
Xb
k, a
(7)
■{K-K').
In the case of short wavelengths (when the spin changes on a much shorter distance than the texture) one can write spectra for plane waves with a wave vector k:
ci A-2 + (c¡ -ci)(k-Í)2 + ÍQ|sin2/3n =lü{lü-lül) i,2 i / 2
Cj_k + (Ci - Ci)(k • l)2 + ÍYB eos2 ¡3n
(8)
Here the meaning of all parameters becomes clear: c^ and c|| are velocities of transverse waves, propagating perpendicular and parallel to the 1 direction; C± and C|| are the velocities of longitudinal waves; Qb is a frequency of the uniform longitudinal NMR in a texture with n||H.
The first equation in (8) describes transverse spin waves, which are similar to that in ferromagnets. In the presence of magnetic field it has two solutions oj(k), which are called acoustic (low oj) and optical (high oj) magnons. The second equation for longitudinal waves is unique for 3He. Spin-wave spectrum in a uniform texture is presented on Fig. 1.
k||l k-LÎ
k_LÎ
kill kill k_Ll
Fig. 1. Quasiclassical spin-wave spectrum in a uniform texture (/3n = const). There are two branches of transverse waves and one of longitudinal waves (8). Slopes of the branches at k —s- oc are spin-wave velocities, they depend on the direction of propagation. Values at k = 0 give resonance frequencies in the uniform NMR
Gradient energy coefficients were calculated in Refs. [10, 11]. In particular, they depend on two antisymmetric Fermi-liquid parameters and Fg. Neglecting the high-order parameter Fg, one has:
K i = Ki = —r
h2p
(1 + ±f?)(l -Y0\
A2 40 mm*
1 +
- èw -
(9)
K3 = K i
1 + g-Ff 1 + iFflb '
Here m is the bare mass of 3He, m* is the effective mass of Fermi-liquid quasiparticles, p is the density of 3He, and l'b is the temperature-dependent Yosida function.
Without Fermi-liquid corrections I\\ = K2 = K3 and one has
cl/cii
C±/C\\
(10)
Schrôdinger equation for optical magnons. In
the case of optical magnons with uj « ojl, in the texture where n is almost parallel to H, the first equation in (6) can be rewritten in a form of a Schrôdinger equation for a magnon quasiparticle with an anisotropic mass:
2 m_L
V2
v z
2 m h
U
= Es+. (11)
Here complex value s+ plays role of the magnon wave function, the energy is defined by the precession frequency E = hw, and the values of the magnon mass are
hojL huL
= r, 9 ; TO|| = r, 9 • (12)
2c2
2c2
Potential for magnons U is formed by the order parameter texture /3n and the magnetic field w^:
U=^sm2pn 2 lvl
hujL.
(13)
In our setup we are able to create a harmonic trap for magnons in 3He bulk far from cell walls. Using spectroscopy of levels in the trap we measure the magnon mass.
Experimental setup. We work with a 3He sample confined in a long quartz tube (diameter 5.85 mm, length 15 cm) and cooled in a nuclear demagnetization cryostat. Temperature is measured by two vibrating tuning forks, located in the lower part of the experimental cell. Experiments are performed in the low temperature limit (T = (0.13—0.20) Tc), where such parameters as the gap A, Leggett frequency Qb , spin wave velocities cp ij_, susceptibility xb do not depend on temperature. Pressures 0-29 bar are used.
The experimental volume is located near the upper end of the tube (Fig. 2). The NMR spectrometer includes a transverse pick-up coil made from copper. The coil is a part of the tuned tank circuit. Capacitor of the circuit is installed at the mixing chamber temperature.
QBcosP„
Qfl . 2„
2câ7sin P»
nucbMa B >K3TO TOM 101 Bun. 11-12 2015
904
У. У. Zavjalov, S. Autti, У В. Eltsov, P. J. Heikkinen
4 (A)
-4.0 -3.0 -2.0-1.0-0.5 0.5 1.0 2.0
-4-/-i-\--1-1-ir
Fig. 2. Top part of the experimental cell. Arrows in the cell volume represent the order parameter texture (1 vector)
It can be switched to 8 different values changing the resonance frequency in the range 550-830 kHz, which corresponds to the NMR in 3He at the magnetic field 17.0-25.5 mT. The Q value of the tank circuit is in the range 125-135 depending on the frequency.
In addition to the NMR solenoid, which produces a static magnetic field, a small superconducting longitudinal coil is used to create a minimum of the field at the center of the coil system. For the interpretation of the measurements it is important to know the field profile. We determine the profile using continues-wave (CW) NMR spectra measured in the normal 3He (Fig. 3).
Magnetic field profile. In the simplest model the field of the main solenoid is uniform and proportional to the current I in it. The field of the longitudinal coil can be calculated as that of a current loop with the radius Rm, number of turns iVm, and current Im.
Since both coils are superconducting, they distort the field. We have studied this effect numerically. Distortion of the main solenoid field can be accounted for by introduction of some additional current in the longitudinal coil oc /. Distortion of the longitudinal coil field can be taken into account by introducing an additional uniform field proportional to Im and adjusting the effective radius Rm of the loop.
We also introduce a tiny transverse gradient g = = dH/dx to explain the appearance of the double peak at \Im\ < 0.1 A. This effect is small and not important for most of our measurements (since it does not affect the quadratic terms in ii), we use it only to improve fitting of the normal phase spectra.
25.62 25.64 25.66 25.68 25.70 25.72 25.74 25.76 MI (mT)
Fig. 3. Measured (points) and calculated (lines) NMR signals in normal 3He at different values of /m for the top spectrometer. Upper plot shows /m in the range from —4 to 4 A, amplitude is multiplied by |/m|; lower plot shows Im in the range -0.25 A to 0.25 A
The combined field is
Hz = MI - [Mo + F(r, z) - F(0,0)] I^+gx, (14)
where F(r, z) is the field of a circular loop with Nm turns, radius Rm and 1 A current and parameters have been found by fitting the normal 3He CW NMR spectra:
M = 9.66914 mT/A, M° = 0.22305 ± 0.00005 mT/A, T* — T + 7"° 7"° — —0 02Q2 7"
m m * mi 1m ~ i
Rm = 1.032 ± 0.005 cm, g = 0.02 mT/cm. (15)
Minus sign in front of the second term in (14) shows that the longitudinal coil is directed opposite to the NMR solenoid to provide field minimum along the z-axis for a positive current Im- Measured and calculated spectra in normal 3He are shown in Fig. 3.
In experiments with trapped magnons only quadratic terms in the field distribution near the center of the experimental volume are important. Expansion of the analytical formula for t
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