The evolution of longitudinal and transverse acoustic waves in a medium with paramagnetic impurities

We study the bipartial interaction of longitudinal and transverse acoustic pulses with a system of paramagnetic impurities with an effective spin S=1/2 in a crystalline layer or on a surface in the presence of an arbitrarily directed external constant magnetic field. We derive a new system of evolution equations that describes this interaction and show that, in the absence of losses, for equal phase velocities of these acoustic components, and under the condition of their unidirectional propagation, the original system reduces to a new integrable system of equations. The derived integrable system describes the pulse dynamics outside the scope of the slow-envelope approximation. For one of the reductions of the general model that corresponds to the new integrable model, we give the corresponding equations of the inverse scattering transform method and find soliton solutions. We investigate the dynamics and formation conditions of the phonon avalanche that arises when the initial completely or incompletely inverted state of the spin system decays. We discuss the application of our results to describing the interaction dynamics of spins and acoustic pulses in various systems with an external magnetic field.     

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 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.     

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.     

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.     

Plasmon-exciton self-induced transparency

The possibility of forming stable bound plasmon-polariton states in an extended metallic cylinder surrounded by a two-level medium has been investigated. The dynamics of plasmons is described in the hydrodynamic approximation. It has been shown that the equations of motion of charge-density bunches and the Bloch equations for the two-level medium are reduced in certain approximations to integrable equations for both transverse and longitudinal plasmons. In the former case, the initial system of equations after the application of the slow-envelope approximation is reduced to equations equivalent to the Maxwell-Bloch equations. In the latter case, the equations describe wave dynamics beyond the slow-envelope approximation. In the approximation of unidirectional wave propagation, the initial system of equations is reduced to equations related to the reduced Maxwell-Bloch equations. Soliton and breather-like solutions of the derived equations describe plasmon-exciton self-induced transparency.     

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.     

On the soliton-like dynamics of ultrashort acoustic pulses in a paramagnetic crystal

A system of nonlinear equations describing the dynamics of the longitudinal and transverse components of an acoustic pulse in a cubic crystal containing paramagnetic impurities has been obtained. On the basis of this system, the dynamics of a two-component ultrashort acoustic pulse propagating under the Zakharov-Benney resonance conditions has been analytically investigated.     

Polarization dynamics of unidirectional optical pulse evolution in a laser amplifier

Polarization dynamics of optical pulses in an isotropic two-level medium is analyzed by solving an integrable system of evolution equations without using the slowly varying envelope approximation. The analysis is focused on the regime of unidirectional pulse generation in an initially inverted medium. Qualitative difference in polarization dynamics is revealed between few-cycle and quasi-monochromatic pulse propagation     

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)     

Optical filaments and optical bullets in dispersive nonlinear media

We have investigated the evolution of picosecond and femtosecond optical pulses governed by the amplitude vector equation in the optical and UV domains. We have written this equation in different coordinate frames, namely, in the laboratory frame, the Galilean frame, and the moving-in-time frame and have normalized it for the cases of different and equal transverse and longitudinal sizes of optical pulses or modulated optical waves. For optical pulses with a small transverse size and a large longitudinal size (optical filaments), we obtain the well-known paraxial approximation in all the coordinate frames, while for optical pulses with relatively equal transverse and longitudinal sizes (so-called light bullets), we obtain new non-paraxial nonlinear amplitude equations. In the case of optical fields with low intensity, we have reduced the nonlinear amplitude vector equations governing the light-bullet evolution to the linear amplitude equations. We have solved the linear equations using the method of Fourier transform. An unexpected new result is the relative stability of light bullets and the significant decrease in the diffraction enlargement of light bullets with respect to the case of long pulses in the linear propagation regime.     

Amplification of extremely short pulses in optical media

The dynamics of pulses with durations comparable to the inverse transition frequency that propagate in an optical medium is studied in terms of two integrable systems of Maxwell-Bloch equations. The first model describes the field interaction with a nondegenerate medium with a permanent dipole moment and permanent external pumping. A general formula is derived for the N-soliton solution. Particular solutions are used as examples to investigate the effect of permanent dipole moment and pumping on the soliton dynamics. The second model describes the interaction between two-component electric-field pulses and a two-level degenerate medium with permanent upper-level pumping. For different initial magnetic-sublevel populations, soliton solutions are used as examples to show that pumping causes a change in polarization dynamics. A two-soliton solution is used to analyze the interaction of solitons in a two-level medium with external pumping.     

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.     

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 self-induced transparency for transverse waves in a system with resonant and quasi-resonant transitions

A theoretical analysis of acoustic self-induced transparency is presented for transverse elastic waves propagating perpendicular to an applied magnetic field through a crystal with spin-3/2 paramagnetic impurities. The interaction between an acoustic pulse and magnetic field is described by Maxwell-Bloch-type equations for a system with transitions inhomogeneously broadened because of a quadrupole Stark shift. If the pulse carrier frequency is resonant with one transition and quasi-resonant with another transition, then the evolution of a one-dimensional pulse is described by an integrable Konno-Kameyama-Sanuki (KKS) equation. The underlying physics of its soliton solution and the corresponding behavior of the medium are analyzed. Self-focusing and self-trapping conditions are found for a pulse of finite transverse size. In the latter regime, the pulse stretches along the propagation direction, transforming into a “hollow bullet,” while its transverse size remains constant.     

Generation of acoustic solitons by a single-mode electromagnetic field

Mechanisms of acoustic pulse generation by a single-mode electromagnetic field propagating in a photoelastic material are analyzed. The anisotropy induced by acoustic excitations in an isotropic medium leads to nonlinear coupling between the polarization components of a single-mode electromagnetic field. For different conditions, it is shown that the acoustic-electromagnetic wave interaction due to mixing of the polarization components of light and acoustic waves can give rise to soliton-like coherent acoustic excitations in a thin crystal plate. When spatial dispersion is ignored, the governing system of equations for unidirectional acoustic solitons can be reduced to an integrable model. It is shown that qualitatively different scenarios of formation of acoustic solitons are possible, depending on the directions of deformation and field polarization.     

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.     

Four-wave mixing of quasi-monochromatic waves and few-cycle pulses

Four-wave coherent mixing models for two quasi-monochromatic pumping fields and pulses of a two-component Stokes field with an elliptic polarization and a duration on the order of the period of oscillations have been derived for a two-level medium with a forbidden dipole transition. It is shown that, under the unidirectional wave propagation conditions and in the absence of depletion of pumping, the system of Maxwell-Bloch equations can be reduced to a new completely integrable system of equations. Nonsoliton radiation dynamics of generation of Stokes field pulses is studied in the framework of the integrable reduction of this model. The apparatus for the inverse problem algorithm corresponding to the solvable problem is developed. An approximate asymptotic expression for the leading front of the pulse packet being generated is obtained for various initial and boundary conditions. The application of these results for describing parametric processes involving various types of waves is discussed.     

Propagation of surface elastic waves in the Cosserat medium

The problem of surface elastic wave propagation in the Cosserat medium (half-space) is considered. The strained state is characterized by two independent vectors: displacement and rotation. Solutions to the equations of motion are sought in the form of wave packets specified by an arbitrary Fourier spectrum. It is shown that, if the solution is sought in the form of a three-component displacement vector and a three-component rotation vector dependent on time, depth, and longitudinal coordinate, the initial system splits into two systems, one of which describes the Rayleigh wave and the other corresponds to a transverse wave decaying with depth. For both waves, analytical solutions in terms of displacements are obtained. It should be particularly noted that, unlike the Rayleigh wave, the solution for the transverse surface wave has no analogues in the classical elasticity theory. The transverse wave solution is numerically compared with the Rayleigh wave solution.