PHYSICAL REVIEW E VOLUME 58, NUMBER 6 DECEMBER 1998
Solitons in highly excited matter: Dissipative-thermodynamic and supersonic effects
Marcus V. Mesquita,* A´ urea R. Vasconcellos, and Roberto Luzzi Instituto de Fı´sica ‘‘Gleb Wataghin,’’ Universidade Estadual de Campinas, Unicamp,† 13083-970 Campinas, Sa˜o Paulo, Brazil ͑Received 6 August 1998͒ Solitary waves — arising out of nonlinearity-induced coherence of optical and acoustical vibrational modes in dissipative open systems ͑polymers and bulk matter͒ — are described in terms of a statistical thermody- namics based on a nonequilibrium ensemble formalism. The undistorted progressive wave is coupled to the normal vibrations, and three relevant phenomena follow in sufficiently away-from-equilibrium conditions: ͑i͒ A large increase in the populations of the normal modes lowest in frequency, ͑ii͒ accompanied by a large increase of the solitary-wave lifetime, and ͑iii͒ emergence of a Cherenkov-like effect, consisting in a large emission of phonons in privileged directions, when the velocity of propagation of the soliton is larger than the group velocity of the normal vibrations. Comparison with experiments is presented, which points out to the corroboration of the theory. ͓S1063-651X͑98͒00412-7͔
PACS number͑s͒: 63.70.ϩh, 05.70.Ln, 63.20.Pw, 87.22.Ϫq
I. INTRODUCTION persion relation qជ ; qជ is a wave vector in reciprocal space running over the Brillouin zone. The vibronic system is Solitary waves are a particular kind of excitation in con- taken to be in contact with a thermal bath, modeled as a densed matter, which nowadays are evidenced as ubiquitous continuum of acousticlike vibrations, with frequency disper- and of large relevance in science and technology. Their role sion relation ⍀ pជ ϭsB͉pជ ͉ and a cutoff Debye frequency ⍀D . as a new concept in applied science was already emphasized System and bath interact via an anharmonic potential, and by Scott et al. in 1973 ͓1͔, who discussed the case of several the whole Hamiltonian is taken as wave systems where the phenomenon may arise. Recently, solitons have been shown to play a very important role in HϭH0ϩHIϭH0SϩH0BϩHI , ͑1͒ three significant areas: conducting polymers ͓2,3͔, fiber op- tics in communication engineering ͓4,5͔, and as conveyors of where energy in biological and organic polymers ͓6–8͔.
We consider here solitary waves arising out of vibronic † 1 H ϭ ជ a a ជ ϩ , 2a modes, both optical and acoustical, when in the presence of 0S ͚ បq͑ qជ q 2 ͒ ͑ ͒ qជ external pumping sources driving the open system arbitrarily away from equilibrium. We evidence the possibility of the † 1 emergence of a particular complex behavior brought about ជ ជ H0Bϭ͚ ប⍀p͑bpជbpϩ 2 ͒, ͑2b͒ by the nonlinearities present in the kinetic equations which pជ govern the evolution of the nonequilibrium ͑dissipative͒ macroscopic state of the system. For that purpose we resort † ͑1͒ † H ϭ Z ជ ជ a ϩ V a ជ b ជ b I ͚ q q qជ ͚ qជ pជ q1 p qជ ϩpជ to the so-called informational statistical thermodynamics ជ ជ ជ 1 1 q q1p ͑IST for short ͓9͔, and see, for example, Refs. ͓10–14͔͒. IST is based on a particular nonequilibrium ensemble formalism, ͑1͒ † ͑1͒ ϩ V a ជ b b ជ ជ ϩ V a ជ b ជ b ជ ជ namely, the nonequilibrium statistical operator method ͚ qជ pជ q1 pជ Ϫq1ϩp ͚ qជ pជ q1 p Ϫq1Ϫp qជ pជ 1 qជ pជ 1 ͑NESOM; see, for example, Refs. ͓15–17͔͒, and Zubarev’s 1 1 approach is by far the most concise, soundly based, and a ͑1͒ † † ͑2͒ † ϩ V a ជ b b ϩ V a ជ a ជ b quite practical one ͓16,17͔. Besides providing microscopic ͚ qជ pជ q1 pជ qជ Ϫpជ ͚ qជ qជ q1 q2 qជ ϩqជ ជ ជ 1 1 ជ ជ 1 2 1 2 foundations to IST, Zubarev’s NESOM yields a nonlinear q1p q1q2 quantum kinetic theory of a large scope ͓16–22͔, the one we ͑2͒ ͑2͒ † used to derive the results we report in what follows. ϩ V a ជ a ជ b ជ ជ ϩ V a a ជ b ជ ជ ͚ qជ qជ q1 q2 Ϫq1Ϫq2 ͚ qជ qជ qជ q2 q1Ϫq2 ជ ជ 1 2 ជ ជ 1 2 1 q1q2 q1q2 ¨ II. FROHLICH CONDENSATION ͑2͒ † † ϩ V a ជ a b ϩH.c. ͑2c͒ ¨ ͚ qជ qជ q1 qជ qជ Ϫqជ AND SCHRODINGER-DAVYDOV SOLITON ជ ជ 1 2 2 1 2 q1q2 Let us consider a system which can sustain longitudinal It consists of the energy of the free system and bath, H0S vibrations, optical and acoustical ͑e.g., polar semiconductors, and H0B , respectively, and in HI are present the interaction polymers, and biopolymers, etc.͒, with, say, a frequency dis- of the system with an external source ͑a mechanism for ex- citation which pumps energy on the system͒, which is the first term on the right, and the anharmonic interaction com- *Electronic address: sousa@ifi.unicamp.br posed of several contributions, namely those associated with †URL: http://www.ifi.unicamp.br/ϳaurea three-particle ͑phonons͒ collisions involving one of the sys-
1063-651X/98/58͑6͒/7913͑11͒/$15.00PRE 58 7913 © 1998 The American Physical Society 7914 MESQUITA, VASCONCELLOS, AND LUZZI PRE 58
(1) † tem and two of the bath ͑we call V the corresponding ជ t , a ជ t , a t ,E , 4 qជ qជ Ј ͕q͑ ͒ ͗ q͉ ͘ ͗ qជ͉ ͘ B͖ ͑ ͒ matrix element͒ and two of the system and one of the bath (2) that is, ͑we call Vqជ qជ the associated matrix element͒. Moreover, aqជ † Ј † (a ជ ) and b pជ (b ជ ) are, as usual, annihilation ͑creation͒ opera- ˆ q p qជ͑t͒ϭTr ͕qជ %͑t͖͒, ͑5͒ tors of, respectively, normal-mode vibrations in the system † and bath and qជ (qជ ) of excitations in the source, with Zqជ ͗aqជ͉t͘ϭTr ͕aqជ %͑t͖͒, ͑6͒ being the coupling strength ͑see also Ref. ͓23͔͒. We recall that the wave vector runs over the system Brillouin zone in EBϭTr ͕H0B%͑t͖͒. ͑7͒ the case of the vibronic modes and between the zero and Moreover, EB ͑the energy of the thermal bath͒ is time Debye cutoff wave vector in the bath. Ϫ1 Next, following NESOM-based IST, we need to define independent as is 0ϭ(kBT0) , because of the assumption the thermodynamic space for the description of the nonequi- that the bath is constantly kept in equilibrium at tempera- librium macroscopic state of the system, in other words the ture T0 . Hence, the whole statistical operator is %(t) ˜ ˜ set of basic variables relevant to the problem at hand: They ϭ%(t)ϫ%B , where now %(t) is Zubarev’s statistical op- are as follows in the present case. First, we take the number erator of the vibronic system and %B is the canonical statis- † tical distribution of the free thermal bath at temperature T of excitations in each mode, i.e., the operator ˆ ជ ϭa a ជ . 0 q qជ q ͑which then plays the role of an ideal reservoir͒. Second, once the formation of a coherent state of vibronic The equations of evolution for the three basic variables modes ͑the solitary wave͒ is expected, we must introduce the † describing the evolution of the vibronic system are derived in amplitudes a ជ and a averaged over the nonequilibrium en- q qជ the NESOM-based kinetic theory ͓15–22͔. Taking into ac- semble. Finally, we take the thermal bath as constantly re- count that the anharmonic interaction is weak, we restrict the maining in equilibrium at a temperature T0 , and then we calculation to the Markovian limit, that is, we consider col- introduce its Hamiltonian H0B as a basic dynamical variable. lision integrals only up to second order in the interaction Therefore the basic set of chosen microdynamical variables strength ͓16,19–21͔. We briefly describe in the second part consists of of Appendix A the fundamentals of these kinetic equations, particularly the origin of the collision operators that are ˆ † ͕qជ ,aqជ ,aqជ ,H0B͖. ͑3a͒ present on the right-hand side of Eq. ͑8͒. After some lengthy calculation, we find that The nonequilibrium statistical operator in NESOM — we 5 recall that we use Zubarev’s approach and call it %(t)—is d a superoperator depending on the above-mentioned basic dy- qជ͑t͒ϭIqជϩ Jqជ ͑t͒ϩqជ͑t͒, ͑8͒ dt ͚ ͑j͒ namical microvariables, and an associated set of Lagrange jϭ1 multipliers ͑which constitute the corresponding set of inten- ជ sive variables in IST, which also completely describes the where Iqជ represents the rate of production of q-mode nonequilibrium macroscopic-thermodynamic state of the sys- phonons generated by the external pumping source, tem͓͒10,13–17͔, which we designate as Ϫ1 ͑0͒ J ជ ͑t͒ϩJ ជ ͑t͒ϭϪ ͓ជ͑t͒Ϫ ͔, ͑9͒ q͑1͒ q͑2͒ qជ q qជ ͕F ជ͑t͒, f ជ͑t͒, f *͑t͒, ͖, ͑3b͒ q q qជ 0 (0) ជ with qជ being the q-mode population in equilibrium, i.e., and in the first part of Appendix A we describe % . Planck distribution at temperature T0 , and qជ is a relaxation The set of basic macrovariables is indicated by time given by
4 1 Ϫ1 ͑1͒ 2 B B ប⍀ជ ϭ V ͓␦͑⍀ជϩ⍀ជ ជϪជ͒ϩ2e p␦͑⍀ជϪ⍀ជ ជϩជ͔͒, ͑10͒ qជ 2 ͑0͚͒ ͉ qpជ͉ pជ qជϪpជ p qϪp q p qϪp q ជ ប qជ p
B where pជ is the population ͑Planck distribution͒ of the phonons in the bath at temperature T0 , and the other terms are
8 ͑2͒ 2 B Jqជ ͑t͒ϭ ͉V ជ ជ ͉ ͓ ជ ជ ͑qជ Ϫqជ ͒Ϫqជ͑1ϩqជ ͔͒␦͑⍀qជ Ϫqជ ϩqជ Ϫqជ ͒, ͑11͒ ͑3͒ 2 ͚ qqЈ qϪqЈ Ј Ј Ј Ј ប qជ Ј
8 ͑2͒ 2 B Jqជ ͑t͒ϭ ͉V ជ ជ ͉ ͓ ជ ជ ͑qជ Ϫqជ ͒ϩqជ ͑1ϩqជ ͔͒␦͑⍀qជ Ϫqជ Ϫqជ ϩqជ ͒, ͑12͒ ͑4͒ 2 ͚ qqЈ qϪqЈ Ј Ј Ј Ј ប qជ Ј
8 ͑2͒ 2 B B Jqជ ͑t͒ϭ ͉V ជ ជ ͉ ͓ ជ ជ ͑1ϩqជ ͒Ϫ͑qជ Ϫ ជ ជ ͒qជ ͔␦͑⍀qជ ϩqជ Ϫqជ Ϫqជ ͒, ͑13͒ ͑5͒ 2 ͚ qqЈ qϩqЈ Ј Ј qϩqЈ Ј Ј ប qជ Ј PRE 58 SOLITONS IN HIGHLY EXCITED MATTER: . . . 7915 and, finally, the term qជ is the one which couples the populations with the amplitudes, namely
2 a ជ t 8 ͦ͗ q͉ ͦ͘ ͑2͒ 2 2 B 2 B ជ͑t͒ϭ ϩ ͉V ͉ ͕ͦ͗a ជ͉tͦ͘ ͑1ϩ ជ ϩ ͒Ϫͦ͗a ជ ͉tͦ͘ ͑ ជ Ϫ ͖͒␦͑⍀ ជ ជ ϩ ជ Ϫ ជ ͒ q 2 ͚ qជ qជ Ј q qЈ qជ Ϫqជ Ј qЈ q qជ Ϫqជ Ј qϪqЈ qЈ q qជ ប qជ Ј 8 ͑2͒ 2 2 B 2 B Ϫ ͉V ͉ ͕ͦ͗a ជ͉tͦ͘ ͑ ជ Ϫ ͒Ϫͦ͗a ជ ͉tͦ͘ ͑1ϩ ជ ϩ ͖͒␦͑⍀ ជ ជ Ϫ ជ ϩ ជ ͒ 2 ͚ qជ qជ Ј q qЈ qជ Ϫqជ Ј qЈ q qជ Ϫqជ Ј qϪqЈ qЈ q ប qជ Ј 8 ͑2͒ 2 2 B 2 B ϩ ͉V ͉ ͕ͦ͗a ជ͉tͦ͘ ͑ ជ Ϫ ͒Ϫͦ͗a ជ ͉tͦ͘ ͑ ជ Ϫ ͖͒␦͑⍀ ជ ជ Ϫ ជ Ϫ ជ ͒. ͑14͒ 2 ͚ qជ qជ Ј q qЈ qជ ϩqជ Ј qЈ q qជ ϩqជ Ј qϩqЈ qЈ q ប qជ Ј In Eqs. ͑11͒–͑14͒, the presence of Dirac’s ␦ function is evident accounting for energy conservation in the anharmonic- interaction-generated collisional processes; momentum conservation is taken care of in the energy operators of Eq. ͑2͒.Inthe case of acoustical vibrational excitations, the matrix elements of the anharmonic interaction are proportional to the square roots of the three wave numbers involved, typically K(1),(2)͓͉qជ͉͉qជЈ͉͉qជϪqជЈ͉͔1/2, with indexes 1 or 2 in K corresponding to the matrix elements V(1) and V(2), respectively; K(1) can be determined via measurements of bandwidths in scattering experiments and K(2) is left an open parameter. The equations of evolution for the amplitudes are
† † † † ץ a ជ t ϭϪi˜ ជ aជ t Ϫ⌫ជ aជ t ϩ⌫ជ a t *ϪiWជ a t *ϩ Rជ ជ aជ t a t ͑ aជ ជ ជ t ϩ a t ͒, ͗ q͉ ͘ q͗ q͉ ͘ q͗ q͉ ͘ q͗ qជ͉ ͘ q͗ qជ͉ ͘ ͚ q1q2͗ q1͉ ͗͘ qជ ͉ ͘ ͗ qϪq1ϩq2͉ ͘ ͗ Ϫqជϩqជ Ϫqជ ͉ ͘ t ជ ជ 2 1 2ץ q1q2 ͑15͒
† ץ ͗a ͉t͘ϭthe complex conjugate of the right-hand side of Eq. ͑15), ͑16͒ t qជץ
˜ where qជ is the frequency renormalized by the anharmonic interaction, with Wqជ being a term of renormalization of frequency, and the lengthy expression for R ជ ជ is given elsewhere ͓24͔͑their detailed expressions are not necessary for our purposes q1q2 here͒. Finally, ⌫qជ (t), which has a relevant role in what follows, is the reciprocal of a relaxation time, given by
4 4 Ϫ1 ͑2͒ 2 B ͑2͒ 2 B ⌫ ជ ͑t͒ϭ ͑t͒ϩ ͉V ͉ ͓1ϩ ជ ϩ ͔␦͑⍀ ជ ជ ϩ ជ Ϫ ជ ͒Ϫ ͉V ͉ ͓ ជ Ϫ ͔␦͑⍀ ជ ជ Ϫ ជ ϩ ជ ͒ q qជ 2 ͚ qជ qជ Ј qЈ qជ Ϫqជ Ј qϪqЈ qЈ q 2 ͚ qជ qជ Ј qЈ qជ Ϫqជ Ј qϪqЈ qЈ q ប qជ Ј ប qជ Ј 4 ͑2͒ 2 B ϩ ͉V ͉ ͓ ជ Ϫ ͔␦͑⍀ ជ ជ Ϫ ជ Ϫ ជ ͒. ͑17͒ 2 ͚ qជ qជ Ј qЈ qជ ϩqជ Ј qϩqЈ qЈ q ប qជ Ј
Equations ͑15͒ and ͑16͒ are coupled together, and contain anilide ͑in which the CO-stretching polar modes are of the linear and trilinear terms. They give rise to two types of same type as those in biopolymers, e.g., the ␣-helix protein͒. solutions: one is a superposition of normal vibrations and the It is shown that Davydov’s soliton-type excitation in the form other is of Davydov’s soliton type ͓6,25,26͔, as we proceed of an undeformed wave packet consisting of a coherent state to show. First, we neglect the coupling of the amplitude of CO stretching ͑or Amide-I͒ vibration is present. However, ͗aqជ͉t͘ and its conjugate, which can be shown to follow when it is damped when propagating in the dissipative medium, a the original Hamiltonian is truncated in the so-called damping dependent on the thermodynamic state of the sys- rotating-wave approximation ͓27͔, which can be used in this tem, as evidenced in the NESOM-IST calculation. Moreover, case. Next, we introduce the averaged ͑over the nonequilib- a calculation in NESOM-based response function theory has rium ensemble͒ field operator allowed us to derive the infrared absorption spectra ͓28͔, characterizing the soliton and obtaining an excellent agree- iqx ment with the experimental data of Careri et al. ͓29͔. For ͑x,t͒ϭ͚ ͗aqជ͉t͘e ͑18͒ q illustration we present in Fig. 1 the infrared spectra in three different conditions, namely at temperatures of 20 K, 50 K, for one-dimensional propagation along the x direction ͑the and 80 K. only one in the case of quasi-one-dimensional polymers or Let us consider next the case of acoustic vibrations, with semiconductor quantum wires͒. At this point we need to de- a frequency dispersion relation qជ ϭs͉qជ͉ (s being the veloc- fine the dispersion relation qជ : we may consider two cases, ity of sound in the system͒. Using this dispersion relation, namely, optical and acoustical vibrations. The first case has and proceeding on the ansatz that a well localized and spa- already been considered ͓28͔ in the particular case of acet- tially undeformed solitary-wave-type solution is expected, 7916 MESQUITA, VASCONCELLOS, AND LUZZI PRE 58
terized by a velocity of propagation v. Resorting to the in- verse scattering method ͓31͔ we obtain that the solution of Eq. ͑19͒ is
M Sv ͑x,t͒ϭ exp i xϪ͑ Ϫi ␥ ͒tϪ A ͭ ͫ ប s s 2ͬͮ 1/2 ͉G͉M S ϫsech ͑xϪvt͒ , ͑20͒ ͩ Aͫ ប ͬ ͪ
where ␥s is the reciprocal lifetime of the excitation. We used Gϭ͉G͉ei and
2 2 ͉G͉ M Sv ϭ A Ϫ , ͑21͒ s 2 4ប
which is an amplitude- and velocity-dependent frequency. We recall that the amplitude and the velocity v are determined by the initial and boundaryA conditions of excita- tion determined by the perturbing source ͑the ‘‘exciting an- tenna array’’͒. Davydov’s soliton of Eq. ͑20͒ can be inter- preted as being that the vibrational acoustic modes are localized by means of the nonlinear coupling with the exter- nal bath; the distortion then reacts — also through anhar- monic coupling — to trap the oscillations while keeping the packet undistorted, in a process also referred to as self- trapping ͓1,7͔. Moreover, as noticed, in conditions of excita- tion in near equilibrium with the bath, the solitary wave is FIG. 1. The infrared absorption spectrum of acetanilide in the Ϫ1 frequency range of the CO-stretching mode, showing the normal damped, relaxing with a lifetime ␥s . However, the situa- band and a redshifted one adjudicated to the soliton. After Ref. tion is substantially modified in sufficiently far-from- ͓28͔: the full line is the calculation in NESOM and the dots are equilibrium conditions, i.e., for high values of the pumping experimental points taken from Ref. ͓29͔. intensity Iqជ in Eq. ͑8͒. In this equation it can be noticed that J ជ and J ជ contain nonlinear contributions in the popula- q(4) q(5) using Eqs. ͑15͒ and ͑18͒,wefind͑see Appendix B͒ that the tions of the modes. These nonlinear contributions have the field amplitude satisfies the local ͑space correlations ne- remarkable characteristic that when qជ Ͻqជ Ј there follows a glected, as noticed͒ equation net transmission of the energy, received from the external source, from the modes higher in frequency to those lower in frequency, in a cascade-down process: This a consequence of the presence of the nonlinear terms ͑containing the product 2ץ ប2 ץ iប ͑x,t͒ϩ ͑x,t͒ϩiប␥ ͑x,t͒ ជ ជ ) in the collision integrals of Eqs. ͑11͒–͑13͒, which are 2 s q qЈ x present in the equation of evolution for the population inץ t 2M Sץ 2 ជ ϭបG͉͑x,t͉͒ ͑x,t͒, ͑19͒ mode q, viz., Eq. ͑8͒. For qជ Ͻqជ Ј , the collision integrals of Eqs. ͑11͒ and ͑13͒ do not contribute, as a consequence of the fact that energy conservation in the collisional events ͑ac- which is formally identical to the one for the optical vibra- counted for the ␦ functions͒ cannot be satisfied. Hence, the 2 collision integral of Eq. 12 survives, giving rise to the al- tions ͓28͔, where ប /2M Sϭបsw, with w being the width of ͑ ͒ the wave packet ͑see below͒ and M S is a pseudomass. This is ready mentioned increase of population in mode qជ , at the a nonlinear Schro¨dinger-type equation with damping ͓1,30͔, expense of all the other modes qជ Ј having higher frequencies and where ␥ and G are the values in the local approximation s than qជ . For qជ Ͼqជ Ј , only the collisional integral of Eq. of the transforms of ⌫ ជ of Eq. ͑17͒ and R ជ ជ in Eq. ͑15͒ to q q1q2 ͑15͒ survives, implying a transmission of energy from mode direct space ͑see Ref. ͓28͔͒. Equation ͑19͒ for the average q to those with lower frequencies, that is, these nonlinear field amplitude admits two types of solutions. One is a terms redistribute energy among the modes. simple plane wave composed of the superposition of the As a consequence, the populations of the modes lowest in normal-mode vibrations ͑corresponding to first-sound-like frequency ͑i.e., those around the zone center͒ are largely in- waves associated with the motion of density͒. The other is a creased. Such a phenomenon was predicted by Fro¨hlich al- Schro¨dinger-Davydov soliton-type excitation: Let us con- most 30 years ago ͓32͔. This so-called Fro¨hlich effect,in sider as an initial and boundary condition an impinged signal sufficiently far-from-equilibrium conditions, has a dramatic with a hyperbolic secant shape, which satisfactorily ap- effect on the propagation of the Davydov soliton described proaches a Gaussian profile. It has an amplitude, say, , above. With increasing population qជ in the modes lowest in which defines its energy content, and a momentum charac-A frequency, the lifetime of these modes of vibration, as given PRE 58 SOLITONS IN HIGHLY EXCITED MATTER: . . . 7917
FIG. 2. Populations of the three relevant modes in the set—as described in the main text—with increasing values of the intensity of the external source pumping modes labeled 2 and 3 in the ultra- sonic region. FIG. 3. The reciprocal of the lifetime of the modes whose popu- lation is shown in Fig. 2. by the reciprocal of the ⌫qជ of Eq. ͑17͒, is largely increased. Therefore, in the field amplitude (x,t), as given by Eq. 5 6 7 Hz, 2ϭ4.5ϫ10 Hz, 3ϭ3.6ϫ10 Hz, 4ϭ2.9ϫ10 ͑18͒, after typically a fraction of a picosecond has elapsed 8 9 10 Hz, 5ϭ2.3ϫ10 Hz, 6ϭ1.8ϫ10 Hz, 7ϭ1.5ϫ10 Hz, after switch-on of the excitation, the amplitudes ͗aqជ͉t͘ for 11 11 modes at intermediate to high frequencies in the dispersion 8ϭ1.2ϫ10 Hz, and 9ϭ9.5ϫ10 Hz. Moreover, for il- relation band die down, but those for the modes lowest in lustration, the open parameter is taken equal to 1, and we frequency ͑in the neighborhood of the zone center͒ survive consider that only the modes 2 and 3 ͑in the ultrasonic re- for long times ͑their lifetime being larger and larger for in- gion͒ are pumped with the same constant intensity SϭI¯, ¯ creasing values of the pump intensity͒. We illustrate this where I2ϭI3ϭI, I1 and In with nϭ4,...,9are null, and is point in Figs. 2 and 3: Consider a sample with the soliton a characteristic time used for scaling purposes ͑as in ͓22͔͒, traveling in a given direction along the extension L of the here equal to 0.17 s. The large enhancement of the popula- sample. Then the permitted vibrational modes are those in tion is evident in the mode lowest in frequency (1), for 19 the interval of wave numbers /LрqрqB , where qB is the S0Ӎ10 , at the expense of the two pumped modes 2 and Brillouin zone-end wave number. We take Lϭ10 cm and 3 , while the modes 4 through 9 have minor modifications the values for the parameters involved in an order of magni- acquiring populations which are very near that in equilibrium tude for typical polymers and thermal bath, namely qB with the thermal bath at temperature T0 ; that is, they are ϭ3.14ϫ107 cmϪ1 ͑hence the lattice parameter has been practically unaltered. The emergence of the Fro¨hlich effect is 5 5 taken as aϭ10 Å͒, sӍ1.8ϫ10 cm/s, sBӍ1.4ϫ10 cm/s, clearly evidenced for this case of acoustical vibrations: In (1) ជ Ӎ10 ps for all qជ , and from the latter we can estimate K fact, pumping of the modes in a restricted ultrasonic band ͑in q 5 in the matrix elements, while we keep as an open parameter the present case in the interval 4.5ϫ10 Hzрр2.8 7 the ratio ϭ͉K(2)͉2/͉K(1)͉2. For these characteristic values it ϫ10 Hz) leads, at sufficiently high intensity of excitation, follows that, because of energy and momentum conservation to the transmission of the pumped energy in these modes to in the scattering events, the set of equations of evolution, those with lower frequencies (Ͻ2), while those with 7 Eqs. ͑8͒, which in principle couple all modes among them- larger frequencies (Ͼ2.8ϫ10 Hz) remain at near equilib- selves, can be grouped into independent sets, each one hav- rium. It may be noticed that for the given value of ¯, S ing nine modes. For example, taking the mode with the low- ϭ1019 corresponds to a flux power, provided by the external est wave number /L, the set to which it belongs contains source in the given interval of ultrasound frequencies being nϪ1 the modes /L, where ϭ(sϩsB)/(sϪsB)ϭ8 in this excited, of the order of milliwatts. case, and nϭ2,3,...,9. Let us call 1 ,...,9 the corre- The dependence of the lifetime with the level of excita- 4 sponding populations, their frequencies being 1ϭ5.6ϫ10 tion is illustrated in Fig. 3: A large increase of the lifetime is 7918 MESQUITA, VASCONCELLOS, AND LUZZI PRE 58
FIG. 5. The population in the steady state for a pumping inten- FIG. 4. The quasichemical potential of the modes labeled 1–3 in sity Sϭ1023 of the modes along the spectrum of frequencies of the Fig. 2, with mode 1 corresponding to the one with the lowest fre- acoustic modes. Dots indicate the modes in the first set ͑the remain- quency in the given set: The emergence of a ‘‘Bose-Einstein-like ing part of the spectrum up to the highest Brillouin frequency B condensation’’ for S approaching a critical value of the order of ϭ9.5ϫ1011 Hz has been omitted͒. 1019 is evident. els of vibronic energy. Also a ‘‘two-fluid-like’’ model may shown for the mode lowest in frequency, that is, the recipro- be considered in a descriptive way, as, in a sense, shown in cal of the lifetime, ⌫1 , largely decreases. Fig. 5.͒ The Fro¨hlich effect can be evidenced in an alternative Otherwise, it can be written way. A straightforward calculation in NESOM leads to the * result that, in terms of the intensive nonequilibrium thermo- Fqជ͑t͒ϭបqជ /kBTqជ ͑t͒, ͑25͒ dynamic variables of Eq. ͑3b͒, the population and the ampli- tude are given by introducing a nonequilibrium pseudotemperature ͑or qua- sitemperature͒ per mode, as used in the physics of the pho- Fqជ͑t͒ Ϫ1 2 toinjected plasma in semiconductors ͑e.g., ͓34–36͔͒; its de- qជ͑t͒ϭ͓e Ϫ1͔ ϩͦ͗aqជ͉tͦ͘ , ͑22͒ pendence on the intensity of the external source is displayed in Fig. 6. ͗aqជ͉t͘ϭϪfqជ͑t͒*/Fqជ͑t͒. ͑23͒ III. FRO¨ HLICH-CHERENKOV EFFECT OR X WAVES Moreover, the intensive thermodynamic variable Fqជ can alternatively be written in either of two forms: One is Moreover, another novel phenomenon may be expected in the out-of-equilibrium nonlinear system we are considering. In both cases of ‘‘optical’’ or ‘‘acoustical’’ Schro¨dinger- Fqជ͑t͒ϭ0 បqជ Ϫqជ͑t͒ , ͑24͒ „ … Davydov solitons that we have described, the amplitude and the velocity of propagation are determined by the initial con- introducing a pseudochemical potential per mode qជ , usu- dition of excitation. Hence, the velocity v can be either ally referred to as a quasichemical potential, as done by smaller or larger than the group velocity of the normal Fro¨hlich ͓32͔ and Landsberg ͓33͔͓we recall that 0 Ϫ1 waves. For the polymer acetanilide in the conditions of the ϭ(kBT0) ͔. The steady-state values of the quasichemical experiment of Careri et al. ͓29͔, v is larger than the group ¯ potential of mode populations j , with jϭ1, 2, and 3, in Fig. velocity of the phonons of the CO-stretching vibrations ͓28͔. 2 versus the intensity of the external source are shown in Fig. In the case of acoustic vibrations in bulk, we may have v 4, where it is evident that 1 approaches 1 for S of the Ͼs, leading to the emergence of a kind of Cherenkov-like 19 ¯ order of 10 , which results in a near singularity in 1 . ͑This effect ͑a so-called superluminal effect in the case of charges phenomenon is sometimes referred to as a kind of nonequi- moving in a dielectric with a velocity larger than the velocity librium ‘‘Bose-Einstein-like condensation’’ because of the of light in the medium ͓37,38͔͒ as we proceed to show. This characteristic of ‘‘piling up’’of excitations in the lowest lev- could be the case in the experiments of Lu and Greenleaf PRE 58 SOLITONS IN HIGHLY EXCITED MATTER: . . . 7919
Consider propagation of a soliton with velocity v (Ͼs) in, say, the x direction in bulk, which introduces a privileged direction in the system. It can be noticed that according to Eq. ͑8͓͒cf. also Eq. ͑22͔͒, the populations of the vibronic modes increase as a result of the direct excitation provided by the source with intensity Iqជ in Eq. ͑8͒, with, as previously shown, such pumped energy being concentrated in the modes lowest in frequency ͑see Figs. 2 and 3͒, and as a consequence of such a so-called Fro¨hlich effect, the lifetime of the soliton is largely increased. Moreover, we notice that for the modes 2 in the Fro¨hlich condensate it can be estimated that ͦ͗aqជ͉tͦ͘ Ӎw2 2/L2, where we recall is the amplitude and we haveA written w for the width ofA the solitary wave packet. On the other hand, for the preferentially populated modes with small qជ , using Eqs. ͑22͒ and ͑24͒ it follows that
kBT0 1 ជ ϭបsq 1Ϫ ln 1ϩ q 2 ͫ បsq ͩ ជ Ϫ͉͗a ជ͉͘ ͪͬ q q
kBT0 ϭបsq 1Ϫ Fជ ϭបvq cos ជ , ͑26͒ ͫ បsq qͬ q
where we have introduced the angle qជ whose cosinus is FIG. 6. The quasitemperature, defined in Eq. ͑25͒, for the modes in Fig. 2. s kBT0 1 cos ជ ϭ 1Ϫ ln 1ϩ ͓39͔; in Fig. 7 we reproduce a related figure ͓40͔ showing on q 2 vͫ បsq ͩ ជ Ϫ͉͗a ជ͉͘ ͪͬ q q the one side the excitation of a normal sound wave, and the other an apparent ͑in our interpretation͒ ‘‘superluminal’’ solitary wave, more aptly called a supersonic solitary wave, s T0 s accompanied by a Cherenkov-like large emission of ϭ 1Ϫ ϵ , ͑27͒ v * ͫ T ជ ͬ vnqជ phonons, as described next. Such excitation has been dubbed q an X wave, and interpreted in terms of an undeformed pro- gressive wave ͓40,41͔, created by the particular excitation after Eq. ͑25͒ is used, and nqជ defines a ‘‘pseudorefraction provided by the pumping transducer. index’’ introduced simply for giving an expression resem-
FIG. 7. Excited normal sound wave ͑upper figure͒ and the undistorted progressive X wave ͑lower figure͓͒40͔. 7920 MESQUITA, VASCONCELLOS, AND LUZZI PRE 58 bling the case of the Cherenkov effect in radiation theory ena a large variety of normal-mode vibrations in matter, such ͑when qជ is the Planck distribution of photons͓͒37,38͔. as, e.g., polaritons, plasmaritons, phonoritons, and all kind of Hence, since excitonic waves propagating in nonlinear media. A particular case that may eventually prove relevant is the case of the Ϫ1 2 so-called ‘‘excitoner,’’ that is, the stimulated amplification of qជ ϭ„exp͕0បsq͓1Ϫ͑v/s͒cosqជ ͔͖Ϫ1… ϩ͉͗aqជ͉͘ ͑28͒ excitons low in energy ͑dubbed a kind of Bose condensation͒ and their propagation in the form of a weakly undamped 2 2 2 packet ͓47,48͔. It is analyzed on the basis of the statistical ͑where ͉͗a ជ͉͘ Ӎw /L ), then it follows that a large emis- q A thermodynamics as described in ͓49͔. sion of phonons follows when cosqជ approaches the value s/v, that is, for T*ជ much larger than T0 ͑cf. Fig. 6͒ and q ACKNOWLEDGMENTS which are emitted in the direction qជ forming an angle qជ with the direction of propagation of the supersonic soliton We acknowledge financial support provided to our group, (vϾs). Forward and backward symmetrical propagations in different forms, by the Sa˜o Paulo State Research Founda- tion ͑FAPESP͒, the National Research Council ͑CNPq͒, the are present because modes Ϯqជ are equivalent (qជ depends Ministry of Planning ͑Finep͒, Unicamp Foundation ͑FAEP͒, on the modulus of qជ ). This is a particular characteristic here IBM Brazil, the National Science Foundation ͑U.S.–Latin of what in radiation theory are the normal and anomalous American Cooperation, Washington͒, and the John Simon Cherenkov effects in a spatially dispersive medium ͓38͔.As Guggenheim Memorial Foundation ͑N.Y.͒. already noticed, the phenomenon, which we call the Fro¨hlich-Cherenkov effect, may provide a microscopic inter- pretation of the X waves in experiments of ultrasonography APPENDIX A: THE STATISTICAL OPERATOR AND THE ͓39͔, shown in the lower part of Fig. 7 ͓40͔. From this figure EQUATIONS OF EVOLUTION we roughly estimate that Ӎ13°, and then v/sӍ1.02, that is, The nonequilibrium statistical operator in Zubarev’s ap- the velocity of propagation of the ultrasonic soliton is 2% proach ͑e.g., ͓15–17͔͒ is larger than the velocity of sound in the medium, once we admit an excitation strong enough to imply that Tq*ӷT0 . t d ¯ ͑tЈϪt͒ ¯ These X waves have been described in terms of a math- %͑t͒ϭexp ln %͑t,0͒Ϫ ͵ dtЈ e ln %͑tЈ,tЈϪt͒ , ematical approach pertaining to the theory of undeformed ͭ Ϫϱ dtЈ ͮ progressive waves ͓41,40͔. This appears to be a particularly ͑A1͒ interesting applied mathematical treatment for a practical ¯ handling of the phenomenon, for example in engineering for where % is the auxiliary ͑sometimes called ‘‘coarse-grained’’ medical imaging ͓39,41͔, as another applied mathematical or ‘‘instantaneous’’ quasiequilibrium͒ statistical operator, in method does for engineering in Refs. ͓42,43͔. The interesting the present case given by case of medical imaging is treated in detail elsewhere ͓44͔, where we use the results presented in this paper. ¯ ˆ %͑t,0͒ϭexp Ϫ͑t͒Ϫ͚ ͓Fqជ͑t͒qជ ϩ f qជ͑t͒aqជ Summarizing, we have described, resorting to a statistical ͭ qជ thermodynamics based on a nonequilibrium ensemble for- malism, the solitary waves which arise out of nonlinearity- † ϩ f *ជ ͑t͒a ជ Ϫ0H0B͔ , ͑A2͒ induced coherence of optical and acoustical vibrations in q q ͮ open systems driven away from equilibrium. The resulting Schro¨dinger-Davydov soliton is coupled to the normal vibra- where (t) ensures its normalization, and tions, and complex behavior is evidenced in the form of three relevant phenomena, namely ͑i͒ a large increase in the popu- 1 %¯ ͑tЈ,tЈϪt͒ϭexp Ϫ ͑tЈϪt͒H %¯ ͑tЈ,0͒ lations of the normal modes lowest in frequency ͑the so- ͭ iប Sͮ called Fro¨hlich condensation͒, ͑ii͒ an accompanying large extension of the solitary-wave lifetime ͑producing a near un- 1 ϫexp Ϫ ͑tЈϪt͒H ͑A3͒ damped soliton͒, and ͑iii͒ large emission of phonons in privi- ͭ iប Sͮ leged directions when the velocity of propagation of the soli- ton is larger than the group velocity of the normal vibrations with HS being the Hamiltonian of Eq. ͑1͒ excluding the in- ͑or Fro¨hlich-Cherenkov effect͒. teraction with the external source ͑i.e., the free system Finally, we call attention to the fact that, in any material Hamiltonian in an interaction representation͒. system, mass and thermal motions are coupled together We recall that is a positive infinitesimal which goes to through thermostriction effects ͑in the case of charged par- zero after the trace operation in the calculation of averages ticles is the thermoelectric effect͒. Thermal motion consists has been performed. Its presence in the exponential intro- of the so-called second sound propagation, for which we duces a so-called fading memory in the formalism, from apply all the considerations we have presented here. Also, which follows irreversible behavior from an initial condition the case of the zero-sound-like excitation in the double pho- of preparation of the nonequilibrated system ͓15–17͔. toinjected plasma in semiconductors ͑the so-called acoustic The equations of evolution for the basic macrovariables, plasmons, with the corresponding first-sound-like excitation Eqs. ͑8͒, ͑15͒, and ͑16͒, consist in the averaging over the being the optical plasmons͒ may be added ͓45,46͔. Similarly, nonequilibrium ensemble of Heisenberg equations of motion, one may consider as candidates for these kinds of phenom- that is, PRE 58 SOLITONS IN HIGHLY EXCITED MATTER: . . . 7921 corresponding to scattering by 2,3,... particles, n is the 1 ץ (n) qជ͑t͒ϭTr ͓ˆqជ ,H͔% ͑t͒ , ͑A4a͒ order of the interaction strength in H present in , and t iប Ј ⍀ץ ͭ ͮ memory effects are included. On the other hand, each one of these collision operators 1 ץ ͗aqជ͉t͘ϭTr ͓aqជ ,H͔%͑t͒ , ͑A4b͒ can be rewritten in the form of a series of partial collision t ͭ iប ͮ operators instantaneous in time, and expressed in the form ofץ correlation functions over the auxiliary ensemble character- † 1 † ץ ͗a ͉t͘ϭTr ͓a ,H͔% ͑t͒ , ͑A4c͒ ized by the coarse-grained operator %¯ (t), that is, ͮ t qជ ͭ i ប qជ ץ ϱ and dE /dtϭ0 because of the assumption that the system of ͑n͒ ͑m͒ B ⍀ ͕qជ͉t͖ϭ ͚ ͑n͒J ͕qជ͉t͖, ͑A7͒ acoustical vibrations remains constantly in equilibrium with mϭn an ideal thermal reservoir at fixed temperature T . 0 and similarly for the case of Eq. ͑A6͒͑for details, see ͓19͔͒. The right sides of Eqs. A4a have a formidable structure ͑ ͒ Introducing Eq. ͑A7͒ in Eq. ͑A5͒, their right-hand sides con- of almost unmanageable proportions. But an appropriate way sist of a double series of partial collision operators. This still of handling them is provided by the NESOM-based kinetic involves extremely complicated calculations, which, how- theory ͓16–22͔. Details are given in these references, where ever, are greatly simplified when the Markovian limit is it is shown that in general we can write, for example for Eqs. taken ͓19,50͔. We recall that the Markovian approach con- ͑A4a͒ and ͑A4b͒, sists of retaining only memoryless-binary-like collisions, an ϱ approximation valid in the weak coupling limit ͓50–52͔, d ͑n͒ which is maintained in the present case of anharmonic inter- qជ͑t͒ϭ͚ ⍀ ͕qជ͑t͉͒t͖, ͑A5͒ dt nϭ0 actions. The corresponding Markov equations retain only the (0) (1) (2) three lowest-order contributions ⍀ , ⍀ , and (2)J in d ϱ (2) ͑n͒ ⍀ , which are the right-hand sides of Eqs. ͑8͒ and ͑15͒.We ͗aqជ͉t͘ϭ ⍀ ͕͗aqជ͉t͉͘t͖, ͑A6͒ (2) dt n͚ϭ0 notice that in the present case, (2)J simply reduces to the golden rule of quantum mechanics averaged over the non- where the ⍀’s for nу2 are interpreted as collision operators equilibrium ensemble. ¨ APPENDIX B: SCHRODINGER-DAVYDOV EQUATION, EQ. „19…
In direct space, after the terms that couple the amplitude ͗aqជ͘ with its conjugate are neglected ͑which, as noticed in the main text, is accomplished using the rotating-wave approximation͒, Eq. ͑15͒ takes the form dxЈ dxЈ ץ iq͑xϪxЈ͒ iq͑xϪxЈ͒ iប ͑x,t͒ϭϪi͚ បqជ ͵ e ͑xЈ,t͒Ϫiប͚ ⌫qជ ͵ e ͑xЈ,t͒ t q L qជ Lץ dx dx Ј Љ iq ͑xϪx ͒ iq ͑xϪx ͒ ϩ Rជ ជ e 1 Ј e 2 Љ ͑xЈ,t͒͑xЉ,t͒*͑x,t͒, ͑B1͒ ͚ q1q2 ͵ ͵ q1q2 L L where we recall qជ ϭsq. Considering the formation of a highly localized packet ͑the soliton͒ centered in point x and with a Gaussian-like profile with a width, say, w ͑fixed by the initial condition of excitation͒ extending along a certain large number of lattice parameter a ͑i.e., wӷa), in Eq. ͑A1͒ we make the expansion
ץ ͑xЈ,t͒Ϸ͑x,t͒Ϫ ͑x,t͒, ͑B2͒ xץ where ϭxϪxЈ is roughly restricted to be smaller or at most of the order of w. The first term on the right-hand side of Eq. ͑A1͒ is
/a LdxЈ ץ dxЈ Ls Ϫi s͉q͉ eiq͑xϪxЈ͒͑xЈ,t͒ϭϪ dq ͓eiq͑xϪxЈ͒ϪeϪiq͑xϪxЈ͔͒͑xЈ,t͒ x͵0 ͵0 Lץ ͚q ͵ L 2
ץ /a xϩw/2 ץ is ϷϪ dq d sin͑q͒ ͑x,t͒Ϫ ͑x,t͒ ͬ xץ ͫ x͵0 ͵xϪw/2ץ
ץ xϩ͑1/2͒w d is xϩ͑1/2͒w d ץ is ϭ 1Ϫcos ͑x,t͒ϩ 1Ϫcos ͑x,t͒ xץͬ xͫ͵xϪ͑1/2͒w ͩ a ͪ ͬ ͫ͵xϪ͑1/2͒w ͩ a ͪ ץ
2ץ is xϩ͑1/2͒w ץ is xϩ͑1/2͒w Ϫ 1Ϫcos d ͑x,t͒Ϫ 1Ϫcos d ͑x,t͒. 2 xץͬ ͪ x ͫ͵xϪ͑1/2͒w ͩ aץͬ ͪ ͫ͵xϪ͑1/2͒w ͩ a ͑B3͒ 7922 MESQUITA, VASCONCELLOS, AND LUZZI PRE 58
But, of the four terms after the last equal sign in Eq. ͑A3͒, the second and third are null, because of the ansatz that a soliton would follow, since the derivative at the center of the packet is null. Consider now the last term, which after the integrations are performed becomes
2ץ isw 2a w Ϫ 1Ϫ sin cos x ͑x,t͒. ͑B4͒ x2ץ ͪ ͩ w a 2 a
But, we notice that the width of the packet is wӷa, and the cosine in Eq. ͑A4a͒ has a period 2a, and then it oscillates many times in w, and with amplitude (2a/w)Ӷ1, and can be neglected. Similarly, the first term becomes proportional to
is ͑x͒ x w x w w 1Ϫcos cos Ϫ2x sin sin , ͑B5͒ x2Ϫ͑w/2͒2ͭ ͫ ͩ a ͪ ͩ 2a ͪͬ ͩ a ͪ ͩ 2a ͪͮ where, on the one hand, the oscillatory terms cancel on average, and, on the other hand, the term decays as xϪ2. Consequently, 2 using these results in Eq. ͑B1͒, after introducing the notation (បsw/)ϵប /(2Ms), and the local approximation in the second and third term on the right-hand side of Eq. ͑B1͒, we find Eq. ͑19͒.
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