Acoustic waves provide a powerful tool for studying the structure of matter. The speed, attenuation and dispersion of acoustic waves give useful details of the molecular forces and microscopic mechanisms for absorption and scattering of acoustic energy. In solids both compressional and shear waves occur, so-called longitudinal and transverse sound. However, normal liquids do not support shear forces and consequently transverse waves do not propagate in liquids with one notable exception. In 1957 Landau predicted [1] that the quantum liquid phase of ³He might exhibit transverse sound at sufficiently low temperatures where the restoring forces for shear waves are supplied by the collective action of the particles in the fluid. Shear waves in liquid ³He involve displacements of the fluid transverse to the direction of propagation. The displacement defines the polarization direction of the wave similar to electromagnetic waves. We have observed rotation of the polarization of transverse sound waves in superfluid ³He-B in a magnetic field. This magneto-acoustic effect is the direct analogue to the magneto-optical effect discovered by Michael Faraday in 1845, where the polarization of an electromagnetic wave is rotated by a magnetic field along the propagation direction.

Superfluidity in ³He results from the binding of the ³He particles with nuclear spin s = 1/2 into molecules called ``Cooper pairs'' with binding energy, 2D. [2,3,4,5] The pairs undergo a type of Bose-Einstein condensation having a close analogy to the Bardeen-Cooper-Schrieffer condensation [6] phenomenon associated with superconductivity in metals. One important difference is that the pairs that form the condensate in ³He have total spin S = 1 and an orbital wave function with relative angular momentum L = 1 (p-wave). This is in contrast to superconductors which are formed with Cooper pairs of electrons having S = 0 and L = 0 (s-wave) or, as is the case of high temperature superconductors, S = 0 and L = 2 (d-wave). In superfluid ³He, the spin and orbital angular momentum vectors are locked at a fixed angle to one another. This is called broken relative spin-orbit symmetry.[2,3] The equilibrium superfluid state is described as a condenstae of Cooper pairs with a total angular momentum, J = LÅS = 0. In addition, the Cooper pairs can be resonantly excited by sound waves to quantum states with total angular momentum J = 2. [7] This is reminiscent of diatomic molecules which have similar excited states. The above description applies to the B-phase of superfluid ³He, the most stable phase at low pressure. The acoustic Faraday effect occurs in ³He-B as a consequence of spontaneously broken relative spin-orbit symmetry.[8] An applied field magnetically polarizes the spins of the Cooper pairs which, through coupling to their orbital motion, rotates the polarization of transverse sound. The rotational excitations of Cooper pairs are essential [8] to our observation of the propagation of transverse acoustic waves in ³He-B since they significantly increase the sound velocity making the sound mode much easier to detect; the closer the sound energy is to the energy of the Cooper pair excited state, the stronger is this effect. Furthermore the Cooper pair excited states have a linear Zeeman splitting with magnetic field.[9,10] Of the five (2J+1) Zeeman sub-states there is one, m_J = +1, which couples to right circularly polarized transverse sound, and a second, m_J = -1, couples to left circularly polarized sound. Thus the speeds of these two transverse waves are different in a magnetic field. We call this property acoustic birefringence. It leads to the acoustic Faraday effect where the magnetic field rotates the polarization direction of linearly polarized sound. Our measurements show that the rotation angle can be as large as 1.4×10⁷ deg/cm-Tesla, much larger than the ususal magneto-optical Faraday effect. [11]

Excitation and detection of transverse sound is provided by a high Q ( » 3000) AC-cut, quartz transducer with a fundamental resonance frequency of 12 MHz. It generates and detects shear waves with a specific linear polarization. The detection method is based on measurement of the electrical impedance of the transducer using a frequency-modulated cw-bridge spectrometer. [12] All measurements were performed at 82.26 MHz, the 7^th harmonic of the transducer, with frequency modulation at 400 Hz and an amplitude of 3 kHz. The electrical impedance of the transducer is a direct measure of the acoustic impedance of the surrounding liquid ³He in the acoustic cavity that is shown in Fig. 1.

Figure 1: Acoustic cavity for transverse sound. Our short path-length acoustic cavity consists of two quartz transducers 6.3 mm in diameter, labeled (a) and (b). Their ground planes are face-to-face and the spacing between the transducers is established by two parallel gold-coated tungsten wires (c). One of the cavity walls is a 12 MHz, AC-cut transducer (b) used for transverse sound excitation and detection. The other face of the cavity is a 17 MHz, X-cut transducer (a) for excitation and detection of longitudinal sound, which we used to determine the cavity length of 31 mm. The acoustic cavity is immersed in liquid ³He at a pressure near 4.5 bar. The experimental cell is cooled by a copper nuclear demagnetization refrigerator (not shown). A superconducting solenoid (d) is placed outside of the sample liquid enclosed within a superconducting shield. Temperatures were determined by SQUID based lanthanum-diluted cerium-magnesium-nitrate thermometry.

Linearly polarized waves are excited by the transducer, reflected from the opposite surface of the acoustic cell, and detected by the same transducer. Under conditions of high attenuation there is no reflected wave, and the acoustic response is determined by the bulk acoustic impedance, Z_a = rw/q where r is the density of the liquid, w is the sound frequency, q = k+ia is the complex wave number, a is the attenuation, and 2p/k is the wavelength. A change in either the attenuation or the phase velocity, C_f = w/k, produces a change in the impedance, Z_a. On cooling into the superfluid the acoustic response shown in Fig. 2 varies smoothly with temperature in this highly attenuating region. If the attenuation is low, there is interference between the source and reflected waves which modulates the local acoustic impedance detected by the transducer. Consequently the acoustic response oscillates as the phase velocity changes with temperature. The oscillations in Fig. 2 at low temperatures correspond to interference between outgoing and reflected waves, and so they indicate the existence of some form of propagating wave. Each period of the oscillations corresponds to a change in velocity sufficient to increase, or decrease, by unity the number of half wavelengths in the cavity. The amplitude of the oscillations increases as the temperature is reduced indicating that attenuation of the sound mode decreases with decreasing temperature.

Figure 2: Temperature dependence of the acoustic cavity response. Measurements of the transverse acoustic impedance are shown at a pressure of 4.31 bar in zero magnetic field. The feature labeled A corresponds to acoustic pair-breaking, (^h/_2p)w = 2D(T_A). Impedance oscillations develop for temperatures T < 0.6T_c and grow in amplitude at lower temperatures as the attenuation of transverse waves decreases. Near T @ T_B = 0.41 T_c the propagating mode is extinguished by absorption of transverse sound via resonant excitation of Cooper pairs with total angular momentum J = 2 at (^h/_2p)w = 1.5D(T_B). The pair-breaking and resonance conditions of J = 2 Cooper pair excitations for fixed frequency are shown in the inset.

The features labeled A and B in Fig. 2 are identified with known physical processes for sound absorption in superfluid ³He-B. [3,13,14] Feature A corresponds to onset of the dissociation of Cooper pairs by sound where (^h/_2p)w = 2D(T_A). In the temperature range between T_A and the superfluid transition temperature, T_c, the attenuation of the liquid is extremely high owing to this mechanism. The point B corresponds to resonant absorption of sound at (^h/_2p)w = 1.5D(T) by the excited Cooper pairs with angular momentum J = 2. [8] Transverse sound is extinguished below this temperature. Early attempts to observe transverse sound in the normal phase of ³He were inconlusive, [15,16,17] and furthermore, it was originally expected that the transverse mode would be suppressed in the superfluid phase. [18,19] More recent theoretical work [8] clarified the role of the Cooper pair excitations showing that they increase the transverse sound speed which results in a more robust propagating transverse acoustic wave at low temperatures in superfluid ³He-B. The first experimental evidence for this can be found in the acoustic impedance measurements of Kalbfeld, Kucera, and Ketterson. [20]

The proof that the impedance oscillations correspond to a propagating transverse sound mode is given in Fig. 3. In Fig. 3a we show data sets at a pressure of 4.42 bar in magnetic fields of 52 G, 101 G and 152 G. The principal feature is that the magnetic field modulates the zero field oscillations shown in Fig. 2. Our detector is only sensitive to linearly polarized transverse sound having a specific direction. Application of a field of 52 G in the direction of wave propagation suppresses the oscillations near T = 0.465 T_c that were present in zero field. This corresponds to a 90^° rotation of the polarization of the first reflected transverse sound wave making the polarization orthogonal to the detection direction. Doubling the magnetic field restores the transverse sound oscillations at this temperature. The oscillations are suppressed once again by tripling the field to 152 G. Also note that near the points labeled 90^° and 270^°, there are smaller amplitude impedance oscillations with shorter period than the primary oscillations. These come from interference of doubly reflected waves within the cavity. We demonstrate this fact with a simple, but powerful, simulation of the acoustic impedance oscillations shown in Fig. 3b.

Figure 3: Magnetic field dependence of the acoustic cavity response. a: The sound amplitude measured at 4.42 bar. The angles are indicated for rotation of the polarization of transverse sound by 90, 180, and 270^°. b: Simulation of the acoustic impedance for the same path length cell and same temperature, pressure and magnetic fields. The wavelength and attenuation of transverse sound in zero field were used as the input data for the calculation. The Verdet constant for the simulation was determined by the position of the minimum in the impedance oscillations at H = 52 G shown in the left panel.

In zero field, superfluid ³He-B is non-magnetic and non-birefringent. Linearly polarized transverse sound is the superposition of two circularly polarized waves having the same velocity and attenuation. Application of a magnetic field gives rise to acoustic circular birefringence through the Zeeman splitting of the excited states of the Cooper pairs that couple to the transverse sound modes; thus, right- and left-circularly polarized waves propagate with different speeds, C_± = C_f±dC_f. For magnetic fields well below 1 kG the difference in propagation speeds is linear in the magnetic field, dC_fµ H. This implies that a linearly polarized wave generated by the transducer undergoes Faraday rotation of its polarization as it propagates. Upon reflection from the opposite wall of the cavity the linearly polarized wave with q||H reverses direction. The reflected wave propagates with the polarization rotating with the same handedness relative to the direction of the field, i.e. the rotation of the polarization accumulates after reflection from a surface. The spatial period for rotation of the polarization by 360^° is

The Faraday effect produces a sinusoidal modulation of the impedance oscillations as a function of magnetic field with a period that is inversely proportional to the field, i.e. L µ 1/H. The constant of proportionality in magneto-optics is called the Verdet constant, V = 2p/HL.

In Fig. 3b we show the result of our numerical calculation of the sound wave amplitude in the direction detected by the transducer. The oscillations shown in the figure come from interference between the source wave and multiply reflected waves. The calculation uses the attenuation and phase velocity measured in zero field. The Verdet constant is obtained from the measurement at 52 G. The simulation reproduces all the observed features of the impedance as a function of temperature including the maximum in the modulation at T/T_c = 0.415 , H = 101 G and the minimum at T/T_c = 0.415 , H = 152 G, which confirms that the Faraday period is proportional to 1/H. The simulation also produces the fine structure oscillations in the impedance near the points labeled 90^° and 270^°. The fine structure is observed when the polarization rotates by an odd multiple of 90^° upon a single round trip in the cell. Then waves that traverse the cell twice are 180^° out of phase relative to the source wave, and consequently the period of the impedance oscillations is halved. The amplitude of the oscillations is substantially reduced because of attenuation over the longer pathlength. This structure provides proof that impedance oscillations are modulated by the Faraday effect for propagating transverse waves.

The impedance data from our experiments were analyzed to obtain the spatial period for the rotation of the polarization and were found to be in agreement with the theoretical prediction [8] for the Faraday rotation period. The theoretical results for the period can be expressed in the form,

for fields H << 1 kG and temperatures above and near the extinction point B. The temperature, T₊, corresponds to the extinction of transverse sound by resonant excitation of Cooper pairs with J = 2, m_J = +1, at a slightly higher temperature than the B extinction point in zero field as shown in the inset to Fig. 2 (e.g. at H = 100 G, T₊-T_B » 1 mK). The magnitude of the Faraday rotation period depends on accurately known superfluid properties, contained in the parameter K, as well as one parameter that is not well-established, the Landé g-factor, g, for the Zeeman splitting of the J = 2, Cooper pair excited state.

Movshovich,et al.[21] analyzed the splitting of the J = 2 multiplet in the absorption spectrum of longitudinal sound to find a value of g = 0.042. In that experiment it was not possible to resolve the splitting except for fields above 2 kG. At these high fields the non-linear field dependence due to the Paschen-Back effect[22,23] becomes comparable to the linear Zeeman splitting,[24,25] which makes it difficult to determine the Landé g-factor accurately. We have analyzed our measurements of the acoustic Faraday effect to determine the g-factor with high accuracy at low fields, which eliminates the complication of the high-field Paschen-Back effect. We find g = 0.020±0.002. Our significantly smaller value of the Landé g-factor has the interpretation that there are important L = 3 (f-wave) pairing correlations in the superfluid condensate, about 7% of the dominant p-wave interactions.[26]

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