{'fraction_non_alphanumeric': 0.035243243243243245, 'fraction_numerical': 0.01891891891891892, 'mean_word_length': 5.022786458333333, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Dynamical Phenomena in an Optical-Wavelength Phonon Laser (Phaser): Nonlinear Resonances and Self-Organized Mode Alternation

This is a part of an overview of my early studies on nonlinear spin-phonon dynamics in solid state optical-wavelength phonon lasers (phasers) started in 1984. The main goal of this work is a short description and a qualitative analysis of experimental data on low-frequency nonlinear resonances revealed in a nonautonomous ruby phaser. Under phaser pumping modulation near these resonances, an unusual kind of self-organized motions in the ruby spin-phonon system was observed by me in 1984 for the first time. The original technique of optical-wavelength microwave-frequency acoustic stimulated emission (SE) detection and microwave-frequency power spectra (MFPS) analysis was used in these experiments (description of the technique see: D.N.Makovetskii, Cand. Sci. Diss., Kharkov, 1983). The real time evolution of MFPS was studied using this technique at scales up to several hours. The phenomenon of the self-organized periodic alternation of SE phonon modes was experimentally revealed at hyperlow frequencies from about 0.1 Hz to 0.001 Hz and less (for the phaser pumping modulation at approximately 10 Hz). The nature of this phenomenon was investigated by me later in details (see: arXiv:cond-mat/0303188v1 ; arXiv:cond-mat/0410460v1 ; Tech. Phys. Letters, 2001, Vol.27, No.6, P.511-514 ; Tech. Phys., 2004, Vol.49, No.2, P.224-231).

Stimulated emission (SE) of optical-wavelength microwave-frequency acoustic waves was detected and studied as early as 1960's -1970's in dielectric crystals doped with paramagnetic ions under conditions of inversion of their spin-levels populations [1 -8]. Due to the quantum nature of this laser-like phenomenon, the emitted acoustic flow may be interpreted as an ensemble of quasiparticles -phonons. So, by an analogy with the term laser (light amplification by stimulated emission of radiation), the term phaser (phonon amplification by stimulated emission of radiation) was introduced by U.Kh.Kopvillem for devices based on the phonon SE phenomenon [9]. 1 We will use here this last term.

Autonomous phaser
In an autonomous phaser, the acoustic SE leads to a stationary amplification [1,3,4,6] of an injected optical-wavelength sound, called also microwave ultrasound or hypersound. This simplest experimentally detectable phenomenon takes place in the subthreshold regime of a phaser, when total amplification α of an injected hypersound is less by modulo than its total dissipation d α , i.e. when In the superthreshold regime of an autonomous phaser, the last inequality is reversed, and the phenomenon of the SE causes an instability (self-excitation) of optical-wavelength acoustic waves. A dissipative crystalline system with well-resolved microwave acoustic resonances, e.g. acoustical Fabry-Perot resonator (AFPR), works in this case as mode selector. As the result, the acoustic SE in a crystalline AFPR leads to single-mode or (more typically) multimode generation of phonons, which is an analog of generation of photons in usual electromagnetic lasers.
These two basic regimes (subtreshold amplification and superthreshold generation) are very typical for autonomous phasers as well for autonomous lasers. More complicated an sometimes unexpected regimes for autonomous SEsystems (self-induced collapse of amplification, subthresold generation etc.) are possibly too. These phenomena will be considered in other parts of our work.
Here we will concentrate our attention on unusual dynamics of a nonautonomous phaser.

Nonautonomous phaser
A nonautonomous dynamical dissipative system has at least one timedependent control parameter. Formally, periodically modulated (nonautonomous) dynamical system may be considered as an autonomous one, but in phase space of a higher dimension. So, the simplest 2D Tang-Statz-deMars nonlinear autonomous laser model [10] must be expanded to a 3D model [11], if a single control parameter is periodically modulated.
Third dimension of a dissipative system permits complex nonperiodic motions in the phase space. Moreover, hypersensitivity to initial conditions in many typical laser and laser-like 3D models is the cause of long-time inpredictability of formally deterministic motion (low-dimensional deterministic chaos). These phenomena were observed experimentally by us in ruby phaser modulated at frequency of nonlinear ν -resonance, which is the acoustical analog of the laser nonlinear relaxational resonance [11].
In this work, we report on the first experimental observation of the series of qualitative new nonlinear resonances in a nonautonomous phaser, which, to our knowledge, were not observed in usual, electromagnetic lasers. We call

Conditions of the experiments
Experiments were carried out at liquid-helium temperatures (below 2 K) on artificial pink ruby crystal doped with three-valent chromium ions (at atomic concentration 0.03%). The rod of 2.6 mm diameter and 17.6 mm length with optically plane and parallel faces (acoustical mirrors) was produced from this crystal. It served as AFPR with acoustical quality 6 10 ≈ A Q for longitudinal phonon modes at frequencies about 9 10 9 ⋅ Hz.
The piesoelectric transducer has the form of thin half-acoustic-wavelength textured ZnO film with Al sublayer. The transducer was used both for detection of an optical-wavelength acoustic SE in the AFPR and (optionally) for injection of non-SE optical-wavelength acoustic signal from outside.
The inversion states of chromium active centers were formed by the electromagnetic field pumping at the frequency about 10 10 3 .

⋅
Hz by means of the cylindrical cavity of the 011 H type with quality factor 4 10 ≈ P Q . The AFPR was placed inside of the pumping cavity along its geometrical axis. A modulated klystron generator was used as a source of pumping.
The microwave heterodyne spectrum analyser with the frequency resolution about 1 kHz was used for observation of phaser acoustic power spectra (using electromagnetic signal linearly converted from acoustic one by the transducer).
All the measurements were carried out under direction of static magnetic field H at an angle ) 3 1 ( arccos symm = ϑ to thecrystallographic axis C r of ruby coinsiding with the geometrical axis of the AFPR when the push-pull pumping conditions are realized.

Results of experiments
The real time evolution of MFPS was studied at scales up to several hours. At pumping modulation near 10 = So, in the λ -resonance domains, a large-scale spatially-temporal ordering takes place for our phaser active system under periodic external forcing. This is in contrast to the ν -resonance domain [12,13], where increasing of the external force destroys intermode correlations and leads to deterministic chaos. Naturally, a question about quantitative analysis of the experimental data arises.
A first, but very important step in this direction is made in the following Section.

Why the λ -resonance is at this frequency?
Let us discuss a possible mechanism of λ -resonance formation based on the classical balance SE model [10], taking in account the papers [14,15] on the nonlinear dynamics in multimode lasers. In our work it will be studied the simplest two-mode model with homogeneous transverse SE field. Following to [15] we consider the 5D autonomous flow system and introduce the 1D external periodic perturbation. The 5D vectorial order parameter B for the two-mode phaser reads: where ( ) t J i -normalized dimensionless phonon SE intensity for the − i th mode -Fourier components of the inverse spin-level population difference in active medium ( 2 ; Here -spatially-temporal distribution of inverse spin-level population difference normalized to its thermodynamically equlibrium value; where ( ) where A is pumping parameter; 10 − in our experiments), and using results of work [15], we find that besides of nonlinear resonance at ν ω , which arises even in the supermode approximation, it is once more resonance, caused by the intermode energy exchange. This resonance appears from the complex- In that way the interaction of the phaser modes brings up additional characteristic frequency of the system, which may be identified to the experimentally revealed nonlinear resonance frequency

Final notes
Experimental results in this field were primarily published by me in 1989 in Russian [16]. The nature of the phenomenon described above was experimentally investigated later in details -see [17 -21]. Theoretical interpretation of the self-organized alternation of phonon SE modes (based on antiphase dynamics [14,15]) was proposed by me in [17,19,22]. Some early attempts to understand the nature of the observed mode alternation were published in [23,24]. The original experimental technique (optical-wavelength microwave-frequency acoustic SE detection and the phonon MFPS analysis) used in these experiments was partially described in my dissertation on phaser dynamics [25,26]. More details on the used experimental technique see [19,21].
An alternative theoretical approach concerning phenomena of self-organization in phasers and some other excitable systems was developed in [27 -34].

SE -Stimulated Emission
Literature