Real Squashing Mode Research Articles

Zero sound spectroscopy of3He-B below 250 μK

Using non-resonant, broadband transducers fabricated from PVDF films, frequency sweeps have been performed in3He-B from nominally 10 to 60 MHz at nearly constant temperatures below 250 μK at about 1 bar. By studying the changes in the group velocity and attenuation as a function of frequency, the real squashing (J=2+) mode has been identified in theT→0 limit.

Experiments on nonlinear acoustics in3He-B

Observations and applications of nonlinear acoustic phenomena in superfluid3He-B are reported. Two-phonon absorption (TPA) by the real squashing (rsq) mode has been detected under several experimental conditions below p = 3.5 bar, using two coincident sound pulses. The attenuation peak height has been investigated as a function of the energy densities of the two sound waves. We discovered the five-fold Zeeman splitting of TPA by parallel sound pulses in an applied magnetic field and the two-fold dispersion splitting due to the finite wave vector of the mode when the two sound pulses are mutually perpendicular. The dispersion relation of the real squashing mode has been investigated at zero pressure and in zero magnetic field by exciting the mode with two parallel, perpendicular, or antiparallel sound waves. Experimental values for the parameters that determine the collective-mode velocities have been extracted from the positions of the observed attenuation maxima. An anomalous structure has been observed in the attenuation and phase velocity spectra of a single high-intensity sound wave near the threshold for pair breaking by two phonons; in an applied magnetic field the phase velocity anomaly splits into a triplet.

Observation of dispersion in the J=2(+) collective modes of 3He-B by nonlinear acoustic spectroscopy.

Parametric excitation yields new results, which cannot be obtained by conventional linear spectroscopy, on the collective modes in superfluid $^{3}\mathit{B}$. In our ultrasonic experiments, performed in a chamber with four quartz crystals, we have excited the real squashing (J=${2}^{+}$) modes by two simultaneous sound waves which are mutually either parallel, orthogonal, or antiparallel. Qualitatively these results are in agreement with the theoretical dispersion relation of the J=${2}^{+}$ modes, and quantitatively we have been able to extract the collective-mode velocities from the line splittings in zero magnetic field.

Experimental investigation of two-phonon absorption by the real squashing mode in superfluid 3He-B

We report on investigations of a nonlinear parametric excitation of the real squashing (RSQ) collective mode in superfluid 3He-B. Two-phonon absorption (TPA) at fs + fp = frsq has been observed by two coincident zero sound pulses, with the signal and the pump wave frequencies at fs and fp, respectively. We have also discovered and studied the Zeeman splitting of TPA in a magnetic field, which is a characteristic feature of the J = 2+ mode in the linear response regime. The dependence of the mode frequency on the magnitude of the wave vector has been investigated; this can be done easily in parametric measurements only. In our earlier, preliminary observations, the coupling of the RSQ mode by TPA was weaker than theory predicted; the discrepancy has now been systematically studied and clarified.

Nonlinear acoustic effects in superfluid 3He-B

We consider the nonlinear interaction of zero sound with the collective modes of the order-parameter in superfluid 3 He-B. The approximate particle-hole symmetry of the 3 He-Fermi liquid determines selection rules for the linear and nonlinear coupling of zero sound to the collective modes. Starting from the quasiclassical theory of superfluid 3 He, we have shown that the coupling strenghts have a simple representation in terms of Feynman diagrams. We predict measurable two-photon absorption and nonlinear-Raman scattering by the J = 2 + (real squashing) modes at low pressures. Recent observations of two-phonon absorption by a group in Helsinki are compared to the theoretical predictions. Two-phonon absorption can be used to determine the dispersion of the J = 2 + modes.

Excitation of collective modes in superfluid 3He B by magnetic resonance techniques.

Using a phenomenological set of equations we argue that the real squashing and other collective modes of superfluid {sup 3}He {ital B} may be excited by a radiofrequency excitation which acts on the {ital J}=1, {ital J}{sub 2}=0 static, field-induced nuclear polarization of the liquid.

Splitting of real squashing mode in acoustic-impedance experiments on 3He-B.

A general analysis is presented for various time-reversal-preserving symmetry-breaking fields that split the collective modes of $^{3}\mathit{B}$. In the absence of these fields, the excited states of the order-parameter factor into pure-real and pure-imaginary components and these two categories have different frequencies (i.e., are nondegenerate). The symmetry analysis made in this paper avoids the usual analogies with the quantum-mechanical theory of angular momentum (which involve complex wave functions and hence do not preserve the real-imaginary separation of the order-parameter components). Our analysis is based on the interplay of real representations of the rotation groups SO(3) and SO(2). For the J=${2}_{+}$ mode we use the fact that the real, irreducible, five-dimensional representation of SO(3) can be resolved, due to the time-reversal-preserving symmetry breaking mentioned above, into the direct sum of two two-dimensional, and one one-dimensional (trivial) real representations of the two-dimensional rotation group SO(2): ${5}_{\mathrm{SO}(3)}$=${2}_{\mathrm{SO}(2)}$+${2}_{\mathrm{SO}(2)}$+${1}_{\mathrm{SO}(2)}$. A similar decomposition is possible for the J=${1}_{+}$ mode; corresponding statements can be made for the pure-imaginary (-) modes.

Two-phonon absorption by the real squashing mode in superfluid 3He-B.

We have observed in superfluid {sup 3}He-{ital B} two-phonon absorption, which is one of the nonlinear effects recently predicted by McKenzie and Sauls. The {ital J}=2{sup +} real squashing mode was excited with two coincident sound pulses at the temperature {ital T}{sup +}, f{sub signal}+{ital f}{sub pump}={ital f}{sub rsq}({ital T}{sup +}), under 0, 2.7, and 3.5 bars pressure. The heights of the attenuation peaks scale almost linearly with the energy of the higher-intensity wave, in qualitative agreement with the theory. Furthermore, a satellite peak appears even in zero magnetic field, which originates from the reflected wave.

The influence of electric fields on collective excitations in helium-3-B

The collective excitation spectrum of He-B is investigated in the presence of electric fields (EF). We have taken the gap distortion caused by EF into account and considered the case without dipole interaction as well as with it. It has been shown that a 3-fold splitting of the real squashing, squashing and pair breaking modes took place. To observe this splitting experimentally one needs to have the fields of order from 5∗105 up to 5∗106 V/cm (in dependence on pressure). Thus this splitting could be observed at part of phase diagram (at not too high pressure). EF does not change the goldstone mode spectrum (it changes only sound velocity).

Investigation of the rsq mode of helium-3-B using non-resonant transducers

We present systematic measurements of the frequency of the real squashing mode (RSQ) in 3.5. Superfluid 3 He-B. The use of ultrasonic transducers with broadhand response enabled us to track RSQ frequency down to its zero temperature limit with the sample held at constant pressure.

The flare-in texture in rotating superfluid 3He-B

A new first-order phase transition is proposed to manifest itself in rotating 3He-B for the vortex-free state, owing to counterflow. The 'flare-out' texture is found to become spontaneously unstable towards a new 'flare-in' texture, which is suggested to explain the transition, first observed with the use of NMR by Hakonen and Nummila in 1987 and confirmed by Korhonen and co-workers in 1989, to a previously unidentified state characterised by a signal corresponding to beta =90 degrees , i.e., with a vector n perpendicular to H// Omega //z. This transition is also expected to be observable in other experiments, such as ones based on ultrasonic measurements, on the vortex-free state of rotating superfluid 3He-B, a new tenfold splitting of the real squashing mode is predicted to occur for the flare-in texture.

Collective-mode spectroscopy of textures in superfluid3He-B

A phenomenological theory for the propagation of the real squashing modes in superfluid3He-B is presented. This allows one to calculate the splitting of the real squashing (rsq) mode spectrum caused by the combined effects of magnetic field, dispersion, and texture in the experimentally important range of magnetic fields from 0 to 103 Gauss. This serves to provide a tool for the rsq-mode spectroscopy of the $$\hat n$$ -textures in3He-B. In particular, a new “gyrosonic effect” is suggested: the intensity of the rsq modes generated in textures depends on the sense of rotation—even when the axis of rotation $$\hat \Omega $$ coincides with the direction of the ultrasound propagation $$\hat q$$ .

Coupling of zero sound to the real squashing mode in rotating 3He-B.

Rotation of superfluid $^{3}\mathit{B}$ in an magnetic field enhances the coupling of the nonzero ${\mathrm{m}}_{\mathrm{J}}$ substrates of the real squashing collective mode to the zero sound, and the fivefold line splitting becomes observable even when H is parallel to \ensuremath{\Omega} and to the direction of sound propagation. Equilibrium vortex lattices and vortex-free states can be distinguished by their characteristic absorption spectra. The dependence of the sound attenuation on the angular velocity in magnetic fields up to 32 mT is reported; the data are qualitatively compared with theory.

Pressure dependence of the zero temperature real squashing mode in 3He-B.

We present measurements of the real squashing mode (RSQ) frequency at temperatures approaching T=0. The data are obtained through repeated frequency sweeps that yield the frequency of the RSQ attenuation maximum over a broad range oaf temperatures, at fixed pressure. We are thus able to record the value to which the RSQ frequency tends as T\ensuremath{\rightarrow}0, a number that is largely independent of temperature scale. Essential to these measurements is the use of nonresonant ultrasonic transducers.

Doublet splitting and the low-field evolution of the real squashing modes in superfluid 3He-B.

We consider the effects of textures on the collective-mode spectrum of /sup 3/He. Our result for the frequency splitting of the real-squashing-mode doublet agrees well with the measurements of Shivaram et al. The smooth disappearance of the doublet and the nonlinear evolution of the real-squashing-mode frequencies at low magnetic fields are explained in terms of a crossover to the dispersion-dominated regime when (qv/sub F/)/sup 2//..delta..>..gamma..H.

Eigenfrequencies of the Collective Modes of Superfluid 3He–B

The eigenfrequencies of both the squashing and real squashing modes are determined independently of the temperature scale. This is accomplished by the identification of 2Δ(T) at each sound frequency. Measurements have been made at 34.2, 44.2 and 54.0 MHz and pressures of 3.90, 2.52, 1.57 bar where strong coupling effects are believed small. The eigenfrequencies of the two modes are close to the theoretical values, with no interaction effects, of (12/5)½Δ and (8.5)½Δ. This indicates that ℓ=3 pairing fluctuations are much weaker than had previously been suggested.

A Microscopic Approach to Nonlinear Sound Propagation in Superfluid 3He–B

A microscopic approach to nonlinear sound propagation observed in 3He–B will be given. Using the eigen-solution of the real squashing modes we define the quasi-spin operators from the triplet-pairing order-parameters and study the coupling between the quasi-spin and the zero-sounds. The resulted nonlinear equations are shown equivalent to those encountered in optical two-level systems.

F-wave correlations in superfluid 3He.

Recent analyses of the collective mode data in both the $A$ and $B$ phases of superfluid $^{3}\mathrm{He}$ suggest that the $f$-wave pairing interaction is attractive and large enough to lead to observable shifts in the frequencies of the real squashing mode of $^{3}\mathrm{He}\ensuremath{-}B$ and the clapping mode of $^{3}\mathrm{He}\ensuremath{-}A$. Here I consider the possibility that $f$-wave pairing correlations exist in the ordered phases of $^{3}\mathrm{He}$, with or without further reduction in symmetry, as a result of an attractive $f$-wave pairing interaction.

Response functions and collective modes of superfluid 3He-B in strong magnetic fields.

The field dependence of the magnetic susceptibility and collective mode frequencies of superfluid $^{3}\mathit{B}$ are calculated perturbatively in (\ensuremath{\gamma}H/\ensuremath{\Delta}) using the quasiclassical theory. With the aid of this perturbation theory, we interpret the gap distortion, nonlinear susceptibility, and collective excitations of the order parameter in terms of additional correlations between Cooper pairs that are induced by a magnetic field. We also calculate the dispersion splittings of the real squashing modes in a magnetic field. In addition to the nonlinear Zeeman shifts of these modes, which occur in strong magnetic fields (\ensuremath{\gamma}H\ensuremath{\sim}0.10\ensuremath{\Delta}), there are low-field (\ensuremath{\gamma}H\ensuremath{\sim}${q}^{2}$${v}_{f}^{2}$/\ensuremath{\Delta}) nonlinearities of the collective mode frequencies resulting from the field dependence of the quantization axis. Our results for all of these response properties of $^{3}\mathit{B}$ depend upon a small number of material parameters; thus measurements of these properties can provide detailed information on the quasiparticle interactions.

Collective modes and sound propagation in a magnetic field in superfluid3He-B

A high-resolution, ultrasonic (12–89 MHz) acoustic impedance technique has been used to investigate the order parameter collective modes in superfluid3He-B over a pressure range of 0–15 bar and in magnetic fields up to 180 mT. In agreement with earlier experiments, theJ=2 real squashing mode has been observed to split into five components in small magnetic fields. However, contrary to earlier theoretical estimates, the Zeeman shifts have been found to become extremely nonlinear as the magnetic field is increased. The extent of this nonlinearity is largest at low pressures and at temperatures close toTc. In comparison with recent theoretical work, the nonlinear Zeeman shifts may be explained as a result of two effects. First, there is a significant distortion of the B-phase energy gap in large magnetic fields. Second, there is an important coupling between the sameJzsubstates of the differentJ modes. In this sense the nonlinear evolution of the real squashing mode constitutes the observation of the Paschen-Back effect in3He-B. A comparison of the observed Zeeman shifts with theoretical expressions has yielded information about particle-particle and particle-hole interaction effects in the superfluid. In the limitT → 0 and above a threshold field, the real squashing mode has been found to possess additional structure. TheJz=0 substate has been observed to split into a doublet. The separation between the two components of the doublet is of the order of 100–200 kHz and remains independent of the magnetic field. The origin of the doublet may be understood in terms of a recent theory which postulates a texture-dependent collective mode frequency. Further, at extremely small fields the effects due to dispersion of the real squashing modes have been found to be important. The magnitude of the dispersion-induced mode splitting in zero field is found to be consistent with theoretical predictions. TheJ=2 squashing mode has also been studied in the presence of a magnetic field. TheJz=0 state of the squashing mode is observed to shift to lower temperatures in a magnetic field. An additional field dependence of the observed acoustic impedance is interpreted as the evolution of theJz=−1, −2 states, but appears to be inconsistent with theoretical predictions.