Soliton-like propagation modes of picosecond acoustic pulses in a paramagnetic crystal

The nonlinear propagation of picosecond acoustic pulses at an arbitrary angle to an external magnetic field is studied in an elastically isotropic paramagnetic crystal at low temperatures. Various soliton-like propagation modes arising due to spin-phonon interaction and acoustic anharmonicity are revealed, and the stability of these modes with respect to transverse perturbations is analyzed. In the case of defocusing cubic nonlinearity, the crystal can support the propagation of compression pulses, which undergo defocusing, and rarefaction pulses can propagate in the self-channeling mode. In the case of focusing cubic nonlinearity, only compression pulses can propagate if the conditions of stability with respect to self-focusing are satisfied.     

Acoustic activity effect for picosecond soliton-like pulses

A theoretical analysis is made of the acoustic activity for interfering picosecond acoustic soliton-like pulses of down to a single oscillation period. An analysis is made of the case where these pulses propagate parallel to an external magnetic field and one of the acoustic axes in a cubic crystal containing paramagnetic impurities having effective spin S = 1. Allowance is made for natural, magnetic (Faraday), and cross acoustic activity. This cross activity is caused by the significant spatial nonlocality of the spin-phonon interaction for such short pulses in crystals having no center of inversion in the presence of paramagnetic impurities. A system of nonlinear equations is obtained for the transverse and longitudinal components of the strain in the form of a coupling between the “differentiated” nonlinear Schrödinger equation (with nonlinearity after the derivative sign) and the Korteweg-de Vries equation which generalizes the known systems of long-short-wavelength resonance to the case where the slowly varying envelope approximation is not valid. An approximate solution of this system is used to study the structure of an elastic soliton-like pulse whose transverse component has a rotating plane of polarization, which propagates under conditions of nonlinear coupling with the longitudinal strain.     

The influence of transverse perturbations on the propagation of picosecond acoustic pulses in a paramagnetic crystal

The influence of transverse perturbations on the dynamics of a picosecond soliton-like acoustic pulse in a paramagnetic crystal in an external magnetic field is investigated. The nonlinear and dispersion effects are governed by the intrinsic properties of the crystal and the spin-phonon interaction. The effect of different nonlinear mechanisms and an external magnetic field on the stability against transverse perturbations is analyzed. It is shown that, in the absence of paramagnetic impurities, there can exist only a compression pulse that propagates in a defocusing regime. In the presence of paramagnetic ions in the crystal, there can arise rarefaction pulses that, under specific conditions, can propagate in self-focusing and self-channeling regimes.     

Soliton generation via continuous stokes acoustic self-scattering of hypersonic waves in a paramagnetic crystal

A new mechanism is proposed for continuous frequency down-conversion of acoustic waves propagating in a paramagnetic crystal at a low temperature in an applied magnetic field. A transverse hypersonic pulse generating a carrier-free longitudinal strain pulse via nonlinear effects is scattered by the generated pulse. This leads to a Stokes shift in the transverse hypersonic wave proportional to its intensity, and both pulses continue to propagate in the form of a mode-locked soliton. As the transverse-pulse frequency is Stokes shifted, its spectrum becomes narrower. This process can be effectively implemented only if the linear group velocity of the transverse hypersonic pulse equals the phase velocity of the longitudinal strain wave. These velocities are renormalized by spin-phonon coupling and can be made equal by adjusting the magnitude of the applied magnetic field. The transverse structure of the soliton depends on the sign of the group velocity dispersion of the transverse component. When the dispersion is positive, planar solitons can develop whose transverse component has a topological defect of dark vortex type and longitudinal component has a hole. In the opposite case, the formation of two-component acoustic “bullets” or vortices localized in all directions is possible.     

Primary acoustic echo during excitation of a paramagnetic crystal by picosecond elastic video pulses

The primary acoustic echo formed during excitation of a paramagnetic crystal with effective spin S=1 by two transverse picosecond elastic video pulses is investigated theoretically. Both exciting video pulses are applied perpendicular to the external magnetic field. It is shown that the primary acoustic echo in the general case consists of six longitudinal and transverse signals at the frequencies of the transitions within a Zeeman triplet. The optimal parameters of the exciting video pulses for the appearance of different echo signals are determined. Fiz. Tverd. Tela (St. Petersburg) 41, 623–628 (April 1999)     

Dynamics of a longitudinal-and-transverse acoustic wave in a crystal with paramagnetic impurities

The evolution of longitudinal-and-transverse acoustic pulses propagating along an external magnetic field through a system of resonant paramagnetic impurities with effective spin S=1/2 is studied theoretically. It is shown that, when the group velocities of longitudinal and transverse waves are equal and the impurity concentration is sufficiently small, the initial system of equations is reduced to new evolution equations, which are integrable within the framework of the inverse scattering problem approach. These equations qualitatively describe the new coherent dynamics of acoustic pulses.     

Nonlinear acoustic transparency for longitudinal-transverse picosecond pulses in a paramagnetic crystal at low temperatures

Nonlinear propagation of longitudinal-transverse acoustic pulses of duration shorter than one oscillation period (video pulses) is studied theoretically in a system of paramagnetic centers with effective spin S=1. It is shown that, depending on the relationship between the magnitudes of the longitudinal and transverse strain components and on the detuning of their linear velocities, various regimes of propagation corresponding to different dynamics of the field and the medium can occur. In the case where the velocities of longitudinal and transverse hypersonic waves differ only slightly, an effect similar to self-induced transparency is analyzed. For substantial velocity detuning, propagation in the form of rational solitons is possible. If the transverse component is dominant, these solitons can produce full population inversion of Zeeman sublevels. In the opposite limit, the populations remain practically unchanged.     

Self-induced acoustic transparency in the long-short-wave resonance mode

The effect of self-induced acoustic transparency for transverse-longitudinal pulses propagating along an external magnetic field in a system of resonance paramagnetic impurities with the effective spin S=1/2 is theoretically investigated. In this case, the short-wave transverse component of the pulse causes quantum transitions, and the longitudinal long-wave component dynamically shifts the frequency of those transitions. When the speeds of the longitudinal and transverse acoustic waves in the crystal matrix are close to each other, both components interact in the mode of the long-short-wave resonance, which is described by a system of nonlinear integro-differential equations. It is shown that this interaction results, in particular, in the modulation of the carrier frequency of the circular-polarized component of the pulse. More precisely, the frequency in the neighborhood of the signal’s maximum is less than in the vicinity of its edges. Solutions in the form of traveling 2π-pulses are analyzed analytically and numerically. It is shown that there exist solutions that include a longitudinal component and cannot be reduced to well-known transverse solitons of the sinus-Gordon equation.     

Nonlinear effects of resonance transparency for two-frequency acoustic pulses in a paramagnetic crystal

Two modes of nonlinear propagation of two-frequency acoustic pulses in a low-temperature crystal containing paramagnetic resonance impurities with an effective spin S = 1 in an external magnetic field and a field of the static strain are considered. It is shown that the spin-phonon transitions occurring within spin triplets according to the V scheme are responsible for two-frequency self-induced acoustic transparency. When the spin-phonon transitions follow the Λ scheme, there can arise an acoustic effect similar to electromagnetically induced transparency in a pulsed mode, which is accompanied by trapping of the population of the spin sublevels.     

Nonlinear picosecond acoustics of low-temperature paramagnetic crystals

The interaction of transverse picosecond deformation pulses with a system of S=1/2 spins, forming a paramagnetic crystal, is investigated theoretically. The influence of acoustic dispersion, owing to the lattice structure, and the anharmonicity of atomic vibrations on the formation of stationary soliton-like video pulses, interacting with the spins of paramagnetic atoms, is studied for the example of a cubic crystal. The phenomenon of total reflection of deformation pulses from the paramagnetic crystal is predicted. The amplification and self-compression of nonresonant pulses in nonequilibrium paramagnetics is studied, and it is shown that generation of higher-order (up to tenth-order) harmonics of the frequency of the initial signal is possible. It is concluded that for appropriate crystal parameters, a limiting solition with a duration of one period of the oscillations, corresponding to spatial size of the order of the lattice constant of the crystal, is possible at the final stage of amplification.V. V. Kuibyshev Tomsk State University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 94–113, July, 1993.     

Self-induced acoustic transparency of three-component longitudinal-transverse pulses

The features of the nonlinear dynamics of three-component elastic pulses in a low-temperature crystal containing paramagnetic impurities of electron and nuclear spins have been analyzed in the slowly varying envelope approximation. The presence of the electron spin subsystem makes it possible to equate the velocities of longitudinal sound and transverse acoustic waves; as a result, all components of the strain field efficiently interact with each other through the nuclear spin subsystem. The system of equations for envelopes of harmonics of the components of the strain field and the spin variables has been derived. The relations between the amplitudes and phases of the components have been obtained, the spectral composition has been analyzed, and the regimes of acoustic transparency of three-component longitudinal-transverse pulses have been discussed.     

Propagation of picosecond transverse acoustic pulses in a kramers doublet system

Propagation of extremely short transverse-strain pulses is studied in a low-temperature crystal containing paramagnetic impurities with an effective spin S = 1/2 in Voigt geometry. It is shown that various types of acoustic solitons can form due to the spin-acoustic interaction between elastic-field components.     

Integrable models for the dynamics of a longitudinal-transverse acoustic wave in a crystal with paramagnetic impurities

We theoretically study the evolution of longitudinal-transverse acoustic pulses propagating parallel to an external magnetic field in a system of resonant paramagnetic impurities with an effective spin S=1/2. For equal group velocities of the longitudinal and transverse waves, the pulse dynamics is shown to be described by evolution equations. In limiting cases, these equations reduce to equations integrable in terms of the inverse scattering transform method (ISTM). For the most general integrable system of equations that describes the dynamics of acoustic pulses outside the scope of the slow-envelope approximation, we derive the corresponding ISTM equations. These equations are used to find a soliton solution and a self-similar solution. The latter describes the leading edge of the packet of acoustic pulses generated when the initial unstable state of a spin system decays. Analysis of our solutions and models indicates that the presence of a longitudinal acoustic wave leads not only to a change in the amplitude and phase of the transverse wave but also to a qualitatively new dynamics of sound in such a medium.     

On the propagation of hypersonic solitons in a strained paramagnetic crystal

Nonlinear dynamics of a subnanosecond transverse elastic pulse in a low-temperature paramagnetic crystal placed into a magnetic field and statically strained in the same direction is investigated. Paramagnetic impurities implanted into the crystal have an effective spin of 3/2, and the pulse propagates at right angles to the magnetic field. In the general case, the structure of the pulse is such that the approximation of slowly varying envelopes, which is standard for quasi-monochromatic signals, is inapplicable. Under certain conditions, the pulse propagation in the 1D case is described by the Konno-Kameyama-Sanuki integrable wave equation for strain, which is transformed into the Hirota equation for the envelope of the given strain in the quasi-monochromatic limit. The effect of transverse perturbations on extremely short and quasi-monochromatic solitons is studied in detail. The conditions and features of self-focusing and defocusing of acoustic solitons in the form of extremely short pulses and envelope solitons are revealed. The propagation of an extremely short “half-wave” hypersonic pulse in the “acoustic bullet” regime in the medium with a quasiequilibrium population of quantum sublevels of effective spins is predicted.     

Effects of soliton dynamics of acoustic pulses in a paramagnetic crystal

Nonlinear dynamics of longitudinal-transverse acoustic pulses in the deformed cubic crystal containing a paramagnetic impurity with effective spin S = 1 is theoretically investigated. Soliton solutions of systems of equations describing the propagation of extremely short and ultrashort pulses at an arbitrary angle to the direction of external deformation parallel to the crystal symmetry axis are obtained. Classification of longitudinal-transverse soliton types and resonant acoustic transparency regimes is given. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 31–36, August, 2007.     

Vector solitons in the dynamics of anharmonic monatomic lattices

It is shown that three types of solitary acoustic waves can develop in anharmonic crystal lattices corresponding to the three branches of acoustic phonons. A system of three nonlinear Schrödinger equations is derived to describe this situation. For greatly different group velocities, the interaction between solitons reduces collisions between them. When the group velocities of the different acoustic modes in a lattice are close to one another, bound states of the corresponding types of solitary waves occur. Bound states of this sort are vector solitons, whose polarization varies along the pulse. If the transverse acoustic modes are degenerate in velocity, the situation is extremely similar to the propagation of pulses in optical fibers.     

Soliton mechanism of acoustic pulse frequency conversion to the red region

Self-scattering of a transverse acoustic pulse from a longitudinal strain video pulse (induced by an acoustic pulse) in a paramagnetic crystal in a magnetic field is predicted. This effect is accompanied by a continuous frequency shift of the hypersonic pulse to the red region; this shift is proportional to the pulse intensity.     

Nonlinear acoustic transparency phenomena in strained paramagnetic crystals

Nonlinear dynamics of longitudinal acoustic pulses propagating in a strained paramagnetic cubic crystal at low temperature is analyzed. The direction of constant uniform strain is parallel to one of the fourth-order symmetry axes. Effective-spin 1 ions are considered as paramagnetic impurities with the strongest spin-lattice interaction. In this medium, normally degenerate magnetic sublevels are dynamically shifted by the quadrupole Stark effect, and the frequencies of the transitions induced by an acoustic pulse change accordingly. The self-consistent system of equations derived in this study without using the slowly varying envelope approximation describes pulse propagation at an arbitrary angle to the direction of the static strain. Exponentially and rationally decaying monopolar and breather-like solutions to the system are obtained. An analysis of the solutions reveals an asymmetry of the pulse polarity depending on the type of strain (extension or compression) and the pulse propagation direction. In particular, it is shown that acoustic transparency associated with monopolar strain pulses exhibits threshold behavior. The sign of the time area (zeroth harmonic) of breather-like strain pulses is such that the transition frequency averaged over an oscillation period dynamically decreases. This behavior determines the efficiency of generation of high-order harmonics of acoustic pulses in strained paramagnetic crystals.     

Acoustic study of the ferroelastic phase transition in LiCsSO4 crystal

The paper reports on an acoustic study of the temperature dependences of the ultrasonic-wave velocity and attenuation in a LiCsSO₄ crystal within the 190–295 K temperature region, which includes the interval of the pseudoproper second-order ferroelastic phase transition (202 K). The velocity of the transverse xy acoustic mode is found to decrease by more than six times at the phase transition. The possibility of performing ultrasonic studies both in the region of the ferroelastic phase transition temperature and below it is demonstrated. The results are treated in terms of Landau’s theory. Waves not associated with the soft mode are shown to exhibit anomalies which are supposedly due to an intermediate phase, whose existence was suggested in a number of publications.     

Theoretical investigation of an acoustic wave in molecular magnets via electromagnetically induced transparency

Using the Schrödinger-Maxwell equations, we theoretically investigate the propagation properties of a transverse acoustic wave in a crystal of molecular magnets in the presence of two strong ac resonant magnetic fields and a weak acoustic wave. The acoustic wave can freely propagate in the magnetic molecule medium (under appropriate conditions) due to quantum interference. Furthermore, using the slowly varying envelope approximation, we discuss the propagation equation of the acoustic wave, which includes the high order nonlinear term. The results show that a crystal of molecular magnets can support the propagation of acoustic wave solitons via electromagnetically induced transparency. We also obtain the analytical expressions for the phase shift and absorption coefficient of the acoustic wave within certain parameters.