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Journal of Low Temperature Physics manuscript No.(will be inserted by the editor)

C.A. Collett · J. Pollanen · J.I.A. Li · W.J.Gannon · W.P. Halperin

Zeeman splitting and nonlinearfield-dependence in superfluid3He

January 7, 2014

Abstract We have studied the acoustic Faraday effect in superfluid3He up to sig-nificantly larger magnetic fields than in previous experiments achieving rotationsof the polarization of transverse sound as large as 1710. We report nonlinear fieldeffects, and use the linear results to determine the Zeeman splitting of the imagi-nary squashing mode (ISQ) frequency in3He-B.

PACS numbers: 43.35.Lq, 67.30.H-, 74.20.Rp, 74.25.Ld

1 Introduction

Superfluid3He is ap-wave, spin-triplet superfluid of great interest due to its un-conventional pairing symmetry. While numerous experimental probes exist, trans-verse zero sound has recently become an excellent tool to explore the order pa-rameter structure, after its prediction by Landau1 and Moores and Sauls2, anddiscovery by Leeet al.3 The coupling between transverse sound and the Imagi-nary Squashing Mode (ISQ), an order parameter collective mode, allows for spec-troscopy of the mode and its behavior in a magnetic field. The ISQ has total an-gular momentumJ = 2 with five Zeeman sub-states. Transverse sound couplesto themJ = ±1 states, causing a rotation of the sound polarization in a magneticfield, called the Acoustic Faraday Effect (AFE). It is this effect that both provesthe existence of transverse zero sound in the superfluid3 and provides a sensitiveprobe into the magnetic field dependence of the ISQ.4

Transverse sound interacts with the ISQ for acoustic frequencies near the modeaccording to the dispersion relation,2

ω2

q2v2F

= Λ0+Λ2−ω2

ω2−Ω 2(T,P,H)− 25q2v2

F

, (1)

Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA.E-mail: ccollett@northwestern.edu

2

2.0

1.5

1.0

0.5

0.0

Fre

quency/∆

(0)

1.00.80.60.40.20.0T/Tc

Acoustic

Response

(µV

)

Pressure (bar)

Fig. 1 (Color online) ISQ (blue line) and pair-breaking (green dashed line) frequencies plottedagainst reduced temperature. A typical pressure sweep alters the sound frequency normalizedto the zero temperature gap to follow the red line, and transverse sound only propagates in thegrey shaded area. Inset: An acoustic trace versus pressure at H = 0.04 T, illustrating the cavityoscillations from changes in the sound velocity and the envelope from the AFE. The black arrowsindicate AFE rotations of 90, 270, and 450, from right to left.

whereω is the sound frequency,q the wavevector,vF the Fermi velocity, andΩ theISQ frequency, with a zero-field value5,4 of ∼

√

12/5 times the weak-coupling-

plus gap.6 Λ0 =FS

115(1−λ)(1+ Fs

25 )/(1+λ Fs

25 ) is the quasiparticle restoring force,

andΛ2− =2Fs

175 λ(1+ Fs

25 )2/(1+ λ Fs

25 ) is the superfluid coupling strength, where

theFsn are Landau parameters, andλ is the Tsuneto function.7 We model the field

dependence of the dispersion relation over a wide range of frequency by simplyreplacing the denominator on the right-hand side of Eq. 1 with

D2 = ω2−Ω 20 −

25

q2v2F −mJAγeffH −Bγ2

effH2−mJCγ3

effH3. (2)

HereΩ0 is the zero-field ISQ frequency,γeff is the effective gyromagnetic ratioof 3He,H is the external magnetic field, and the terms containingA, B, andC de-scribe the linear, quadratic, and cubic magnetic field dependence of the dispersion,respectively. ThemJ dependence of each term is simplified to reflect the fact thatmJ =±1, making the linear and cubic terms either positive or negative but havingno effect on the quadratic term.

2 Experiments

In these experiments we use a dilution refrigerator and demagnetization stagesetup described previously4 to reach∼ 600 µK with a 3He cavity formed by atransducer and a reflecting plate. To probe the Faraday rotation, we lower thepressure of the helium from∼ 6 to 3 bar, changing the frequencies of the ISQand pair-breaking, which transform relative to the acoustic frequency of 88 MHzas shown by the trajectory of the red arrow in Fig. 1, taking into account heatinginherent in the experiment which raises the temperature from∼ 0.5→ 0.6Tc. This

3

1500

1000

500

0

θ

0.300.250.200.150.100.050.00

(ω2−Ω0

2)/ω

2

300

200

100

0

ct(m

/s)

0.100.080.060.040.02

Magnetic Field (T)

a

b

Fig. 2 (Color online) Data taken from acoustic traces at all experimental fields, plotted againstthe shift: (a) Sound polarization rotation angle data,θ . Open circles are taken from the envelopeamplitude, while solid circles correspond to minima in the acoustic traces. (b) Sound velocitydata,ct . The data points at each shift mostly overlap.

causes changes inD2 and thus the sound velocity,ct , leading to oscillations in thetransducer response, or cavity oscillations, such as thoseseen in the inset to Fig.1. With a magnetic field present, the polarization rotation is seen as the overallenvelope in the inset with minima indicated by the black arrows. These effects aregoverned by

VZ ∝ cosθ sin(2dω

ct

)

, (3)

whereVZ is the transducer voltage,θ is the AFE rotation angle, andd = 31.6±0.1µm is the cavity spacing.4

Using Eq. 3, we can get bothθ andct from our acoustic response data. Bymeasuring the envelope we calculateθ based on its amplitude, using the nodes asfixed anglesθ = n×90; n = 1,3,5. . .. We also convert the period of the cavityoscillations to changes in the sound velocity using

1 Period= 2dω2π

∣

∣

∣

1ct f

−1cti

∣

∣

∣. (4)

After calculating an initial value ofct near the ISQ, we use the sound velocitychanges calculated from Eq. 4 to extend that initial sound velocity over the entiredata range.

4

10

8

6

4

2

0

χ2

red

0.20.10.0

(ω2−Ω0

2)/ω

2

Fig. 3 (Color online) Values ofχ2redcalculated using Eq. 7 for fittingθ to the dispersion using

A andC as free parameters, withB as the upper bound from fittingct .

3 Results and Discussion

We show our data in Fig. 2, plotted against the normalized difference in the squareof the frequencies,(ω2−Ω 2

0)/ω2. This scaling, which we refer to as the frequencyshift, or just the shift, better reflects the changes in the dispersion caused by thechanges in temperature and pressure during an experimentalrun. θ shows strongfield dependence, butct appears to be nearly field independent at these fields.We relate this data to the dispersion by taking Eq. 1, with theright-hand sidedenominator given by Eq. 2, and using the following definitions:7,8

θ = 2d δq, (5)

ct = 2ω/(q++q−), (6)

whereδq = |q+−q−|/2, andq± is obtained by solving Eq. 1 forq, settingmJ =±1. Using Eqs. 5 and 6 we find thatct depends most strongly on the quadraticfield term, whileθ depends on the linear and cubic terms, due to the cancellingeffects ofmJ =±1.

As mentioned above,ct shows very little field dependence. In order to quantifythis we group the data in frequency shift bins spaced by(ω2−Ω 2

0)/ω2 = 0.005,and fit each bin to Eq. 1 as a function of field, using Eq. 2 withB as the onlyfree parameter and settingA = C = 0. All fits are done using the Levenberg-Marquardt algorithm9,10 for nonlinear least-squares minimization. We evaluatethese fits quantitatively by calculatingχ2

red, a measure of how well a model fitsdata with a normal sample distribution, given by

χ2red=

1ν

N

∑i

(yi − fi)2

σ2i

, (7)

whereν = N −M is the number of degrees of freedom,N is the number of datapoints,M is the number of fit parameters,yi is the measured value,fi is the cal-culated value, andσi is the standard deviation of the measurement. Forct we

5

-4

-3

-2

-1

0

1

B

0.250.200.150.100.050.00

(ω2−Ω0

2)/ω

2

0.12

0.08

0.04

0.00

A/ω

12

8

4

0

C•ω

a

b

c

Fig. 4 (Color online)A, B, andC parameters for Eq. 2, made dimensionless by normalizing toappropriate orders ofω . (a) The linear field dependenceA/ω , red circles. (b) An upper boundon the quadratic field dependenceB, green triangles. (c) The cubic field dependence,C ·ω , solidorange diamonds. The upturn nearω = Ω0 is in a region of greater uncertainty due to the lackof the highest field data, as described in the text.

calculateσi directly from the data, assuming zero field dependence, resulting invalues ofσ ∼ 2 m/s, while forθ we estimateσ = 10 from our measurements.For a fit eitherσ or 2σ from our data,χ2

red∼ 1.25 or 5.5, respectively.

The results of ourct fits only give us an upper bound on its quadratic fielddependence, which constrains theB term such that it has little to no effect onθ .Consequently, to account for the non-linear dependence ofθ when we fitθ toEq. 1 as above, we must include the cubicC term in Eq. 2 as well as theA termas free parameters, using theB from ct as an input, giving an accurate fit to thedata resulting in theχ2

red values shown in Fig. 3. At the lowest and highest shifts

χ2red increases, which is reflected in larger error bars forA andC at these shifts.

We attribute these larger error bars to a larger uncertaintyin θ due to a greaterdifficulty in precisely identifying the rotation angle in these regions. OurA, B, andC results are shown in Fig. 4, given in a dimensionless form by normalizing toappropriate orders ofω.

The appearance of an upturn at low shifts inC can be attributed to the lack ofhigh fieldθ data at those shifts, due to the rotation angle changing too fast to bedetermined. At higher shifts,C is a fairly constant value. Our data thus shows adecreasing linear term, an increasing quadratic bound, anda constant cubic termwith increasing shift.

Although there are predictions11 for the quadratic field dependence of the ISQ,ourB values cannot be directly compared with them due to the insensitivity of ct to

6

302520151050Pressure (bar)

0.08

0.06

0.04

0.02

0.00

g

Fig. 5 (Color online) Landeg-factor values as a function of pressure, for our data (red circle,error bars inside data point) and reanalyzed data from Daviset al. (blue diamonds).4,14

the magnetic field. As our data provides only an upper bound onthe magnitude ofthe quadratic term in the dispersion we can only say that previous measurements12

seem to fall within that bound at the mode. There are currently no theoreticalpredictions for the cubic field dependence of the ISQ.

By extrapolating ourA data toω = Ω0, we can apply the theory of Sauls andSerene13 to extract the Landeg-factor of the ISQ. Theg-factor is a measure ofthe magnitude of the linear Zeeman splitting of the ISQ, and is predicted13 todepend on the magnitude off -wave pairing interactions in3He. As the theoryapplies only in the region aroundω ∼ Ω , we cannot extract more than a singledata point, which, using the relationshipA0 = 2gω, whereA0 is the extrapolatedvalue ofA to ω =Ω0, is shown as a red circle in Fig. 5. We also present reanalyzeddata from Daviset al. as blue diamonds.4 The reanalysis, which focuses on theirextrapolation of their data toT = 0 rather than toω = Ω0, as well as a discussionof the applicability of the theory, will be discussed in a forthcoming publication.14

4 Conclusion

We have measured the splitting of the ISQ using transverse sound, finding thatnonlinear field effects play a significant role at fields up toH = 0.11 T. Using asimple model for the effect of splitting on the dispersion wehave quantified thefield dependence up to cubic order. We have set an upper bound on the quadraticsplitting, which is small at these fields, and have determined the linear and cubicsplitting parameters, as well as the Landeg-factor of the ISQ.

5 Acknowledgments

We would like to thank J.A. Sauls for his help with this work, and acknowledgethe support of the National Science Foundation, DMR-1103625.

7

References

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