Laser Resonator

In physics, the laser resonator in the form of an optical cavity plays a crucial role in the process of amplifying the output beam and making it coherent. We model the functioning of the social laser resonator by "distilling" the physical scheme to exclude the straightforward connection with optics. We proceed with the quantum master equation for the density operator describing excitations of the quantum information field inside the resonator of the social laser (see section 7.3). The main aim is establishing proper social interpretations of the basic quantities and parameters of this dynamical system.

The Internet-based echo chamber is considered as an important example of social resonators. Its functioning is mathematically represented by the field of social excitations in the form of posts and comments interacting with the human gain medium.

One of the basic consequences of quantum-like dynamics is the existence of the threshold for the pump parameter (section 7.3.5). If this parameter exceeds the threshold, then practically all social energy pumped into the human gain medium is transferred into the output beam of social actions. This is a good place to emphasize once again the role of the huge power of the information flows generated by the modern mass media.

In this chapter, we again proceed with gain media composed of two-level s-atoms. Generalization in the case of multilevel s-atoms is straightforward (cf. section 6.3).

Resonators of Physical Lasers

In laser physics, one of the main problems in creating laser is approaching population inversion. However, population inversion is not enough to generate a laser effect. Stimulated and spontaneous emissions are compete with each other. Thus, before becoming an amplifying device, a gain medium pumped by an external energy source is first radiated as a usual electric "lamp.” Here, spontaneous emission is dominating. The light power is distributed over a variety of frequencies and directions of propagation, generally uniformly distributed. It is the optical cavity, the laser resonator, that creates the conditions necessary for stimulated emission to become predominant over spontaneous emission. The cavity or resonator is composed of two mirrors (Fig. 3.1) that bounce the beam back and forth through the gain medium. One of the mirrors is only partially reflecting (the left-hand side mirror) and another is totally reflecting (the right-hand side mirror).

In further considerations it is assumed that the gain medium has approached population inversion.

Spontaneous Initiation of Physical Lasing

When functioning of the laser begins, the gain medium emits spontaneously (see Fig. 7.1) in all directions. However, a part of such spontaneous radiation goes along the axis of the optical cavity. These spontaneously emitted photons can travel backwards and forwards between two mirrors (see Fig. 3.1). Thus, over time, as a result of stimulated emission from the gain medium, the energy of the electromagnetic field along the cavity axis, say x, increases very rapidly. At the same time, radiation emitted in directions deviating from the x axis leaves the cavity after a few iterations due to reflection from mirrors with angles different from тг/2.

Another crucial feature of the electromagnetic field propagating along the resonator axis is its coherence in time. Because the

Figure 7.1 Spontaneous emission initiating lasing.

periodic boundary conditions in cavity (with the period 2nc/L, where c is the velocity of light) the electromagnetic x waves interfere constructively. The waves propagating in other directions exhibit destructive interference of some degree.

In the photonic picture we can speak about increasing the number n of photons propagating along the x axis. The rays typically used in the light representation give the directions of the photons’ momentum vectors. The concentration of the field inside the cavity increases the probability of stimulated emission rather than spontaneous emission occurring. This is a basic feature of bosons (see section 4.8). We point out that the behavior of bosons is similar to human behavior, and this is known as the bandwagon effect (see again section 4.8).¹ In a laser cavity, for each atom, periodically the density of photons surrounding this atom and oriented along the cavity axis becomes very high, resulting in temporal coherence. This makes the probability of stimulated emission of a photon from this atom very high as well. And the emitted photon is directed along the x axis.

Stimulated Initiation of Physical Lasing

Instead of spontaneous emission initiating lasing and filtering improper directions through interference, it is possible to initiate lasing by stimulated emission (see Fig. 3.2). An electromagnetic 'This effect is very natural. This form of human behavior was created evolutionally as an important survival factor. At the same time, quantum theory cannot explain the origin of the "bandwagon effect for physical particles," e.g., photons. It only provides a formal mathematical description.

pulse oriented along the x axis and having a frequency matching the energy levels of atoms in the gain medium is injected in the optical cavity. The photons in this pulse interact with atoms and generate stimulated emission along the same axis. It is crucial that these photons have the same phase. One can imagine them as a cloud of increasing size moving between mirrors. Synchronism in movement produces the bosonic stimulated emission effect described above, since each atom is periodically surrounded by an electromagnetic field of high density which is x-oriented.

Of course, spontaneous emission takes place and it can produce photons with momentum vectors different from the x direction. And each of such spontaneously emitted photons can generate a cascade of photons. However, in contrast to the process described in section 7.1.1, there is no need to eliminate such spontaneously emitted photons. They can stay in the cavity, but cascades stimulated by them are veiy weak compared with the basic cascade generated by the initially injected pulse of photons with momentum x. It is important that photons generated by this pulse propagate coherently and generate the bosonic effect by increasing of the probability of stimulated emission. Photons spontaneously emitted by different atoms as well as the cascades stimulated by these photons are not coherent. And the probability that at the same instant of time a few atoms would spontaneously emit photons in the same direction, say different from x, is practically zero.

It seems that the initializing pulse need not be powerful. One can even find statements such as, to generate a cascade strongly dominating over cascades induced by possible spontaneous emissions, it is enough to coherently inject just two photons. We also remark that, for some gain media, just one or two cascading iterations are sufficient to approach a veiy high density of photons in the cavity.

Resonators of Social Lasers

The same competition between spontaneous and stimulated emission plays a crucial role in social processes. People in the excited state may "radiate” social energy spontaneously, say in debates with relatives and friends about political and social problems. Social colors of excitations in such spontaneous radiation are typically randomly distributed, often uniformly distributed. Such emission of social energy cannot lead to coherent social actions.

Structure and Functioning of the Social Resonator

A social resonator consists of a gain medium composed of s-atoms that have already approached population inversion And, say, an Internet-based communication system, e.g., some social network. We call such a system echo chamber. We restrict modeling to the Internet-based echo chambers (see also Barbera et al. [20]). Consider the following model.

Each s-atom in the excited state can emit a quantum of social energy in the form of a post or a comment on some post. We call the posts and comments excitations of the social resonator. By posting or commenting, i.e., emitting an excitation, an s-atom falls to the ground state. The resonator's excitations play the role of photons in the optical cavity, the resonator of the physical laser. Moreover, to simplify the model, we assume that the social resonator under consideration accepts only excitations of a concrete color, a. This is a strong constraint that is in visible contradiction to the functioning of typical Internet-based social networks. We shall relax it in later modeling. The social color of an excitation plays the role of the direction of propagation of the output beam of photons emitted by the laser resonator, the x axis of the optical cavity.

Suppose that at a fixed instance of time in the social resonator there are n excitations. Each member of the gain medium interacts with all these excitations—with the information field. The boson behavior of excitations implies that the probability that the concrete agent would fall to the ground state and emit an excitation increases with n (section 4.8). It is crucial that if all excitations of the social resonator have a fixed color a, the color of excitations emitted by this agent is also a. This dynamics leads to an exponential increase in the number n of excitations having the a color inside this social resonator (see section 7.3 for the modeling of temporal dynamics). Excitations of colors different from a also can be spontaneously emitted by the gain medium. But they cannot generate the cascade process, since in the present model they are simply blocked.

Output beam from the echo chamber

When n becomes sufficiently large (see section 7.3), it is possible to open the output channel of the echo chamber and generate a stable flow of high-intensity excitations of a fixed color a. In "outer space," this flow is realized in the form of meetings, demonstrations, and brutal protest actions.

On a spatial picture of quantum physical processes

We remark that in a physical laser resonator, an optical cavity, the light beam bounces back and forth through the amplifying medium. Its temporal dynamics can be represented in the spatial picture, as waves propagating back and forth and reflecting from mirrors. However, this picture of waves propagating in the physical space is merely an artifact of the classical physical model of light. The latter is very useful for visualization of physical processes and it is widely used in laser physics. However, the real theory of lasers is quantum theory.^[1] Here, one operates not with classical waves propagating in physical space-time, but with photons, excitations of the quantum electromagnetic field. Although it might be illustratively convenient to treat individual photons as systems moving in space, this picture does not match with quantum theory. A photon's wave function i/r(t.x), x s R^[2], is not well defined. Consequently, by modeling the laser resonator, we have to use particle (quanta) representation and to operate not with spatial waves, but with excitations carrying energy quanta. Such processes are modeled with the aid of creation and annihilation operators (section 7.3.1). The latter mathematical model matches our social applications well. To proceed to the social laser, we have to eliminate the spatial picture from the laser theory.

Waves in the physical space are good for visualization, but not waves in the information space.

Stimulated Initiation of Social Lasing

As was stressed, the straightforward blocking of excitations with colors different from one fixed color a is a strong assumption. Of course, moderators of social networks block some posts and comments, e.g., having extremist, racist, or sexist content. However, the proportion of blocked excitations seems not to be so high; in any event it is far from 100%. Therefore we have to improve the above model.

As was presented in section 7.1, there are two possible scenarios for initiating lasing:

• (1) spontaneous emission and eliminating photons with momentum vectors deviating from the x axis by using the optical cavity
• (2) stimulated emission generated by a coherently injected ensemble of photons with the x momentum vector.

In social lasing, the second scenario is preferable, because the social mechanism of elimination of "wrongly directed and spontaneously emitted social excitations” is not so straightforward as in optics (see section 7.2.3).

Thus generation of the beam of social excitations having the same color a is started by injecting a block of «-colored posts into the Internet echo chamber. They are injected at the same moment of time.^[3] This initializing block generates a cascade of stimulated emissions. After a few interactions the propagating wave of excitations is so big that the probability of stimulated emission becomes very close to 1. This is a good time to open the output channel of the echo chamber and to transform information excitations into physical social actions.

Of course, spontaneously posted excitations of colors different from a can also be generated in this echo chamber. But they are generated in different moments and have a variety of colors. Even if such a post starts to generate its own cascade, its power is negligible compared with the dominating cascade started with injection of a posts.

7.2.3 Spontaneous Initiation of Social Lasing and

Spontaneous Initiation of Social Lasing and Elimination of "Wrongly Colored'' Information Excitations

As we have seen in section 7.1, in an optical cavity, direction filtering is done through reflection from mirrors. If the direction of a field's propagation, say u, differs from the x axis of the cavity, then the ray optics implies that soon or later the field radiation will leave the cavity. A bigger difference between directions и and x implies a shorter time needed for escaping from the cavity, or a smaller number of reflections.

We remark again that in the quantum framework, instead of waves propagating in the physical space, we can consider probability waves represented by complex probability amplitudes, normalized vectors of the complex Hilbert space. The inner product determines the angle between two states, two waves of probability.

To be concrete, let us assume that colors of social excitations are labeled by real numbers. Thus a is the real parameter. Now, instead of a fixed color a characterizing the echo chamber, we consider the Gaussian distribution of colors centered at color a with the standard deviation a. In principle, this distribution of colors in the echo chamber can be very sharp mimicking the fixed color a.

We now consider an information analogue of optical reflection. What does "information excitation V leaves the echo chamber” mean? It means that the chamber’s gain medium does not interact any longer with this post or comment. How can the interaction be terminated? There are two possible ways:

• (1) aforementioned straightforward blocking
• (2) implicit blocking with the aid of a reflection-like process

So, let V have color different from the chamber's color a. If the difference is veiy large, sayfi—a > 4ст, then the «-colored chamber does not interact with V. In particular, such V can be blocked by the moderator. Suppose that the difference is not so large, say

Р — a <2a. Then the echo chamber should start to direct this excitation out of it, i.e., the chamber’s gain medium should stop interacting with V. This is done through elimination of p color from the echo chamber through posting comments to V having colors deviating more strongly from a than color p. So, reacting to V appearance, somebody posts comment V with color pi, which differs even more from a, Pi — a| > p — a|. Then V can generate posts and comments which differ from a even more. After a few iterations the difference becomes so big, say p_k — a > 4a, that the chamber stops interacting with such posts.

This strategy is actively used in Internet echo chambers. For example, consider the Russian news portals, say, vz.ru or gazeta.ru. Suppose some news starts to generate antigovernment comments. One can see that such comments are often interrupted by injections of posts on Russian-Ukrainian relations. Suddenly and without any relation to the news, one writes that all "Moscals” are slaves by nature and Ukranians are free people. Immediately in a few comments it is claimed that Ukranians prostitute for America and so on and so forth. The original topic, say the increase in retirement age, is completely forgotten.

Energy Spectrum of the Output Beam: Physical versus Social Lasing

The problem of filtration of the energy of excitations is not less important than the problem of filtration of "wrongly colored” excitations. It is impossible to create a gain medium (either physical or social) with a fixed difference between energy levels.

In physics, the cavity acts as a frequency (energy) filter owing to the numerous round trips: only certain frequencies will be favored, the resonance frequencies of the cavity. In this way, the cavity produces a specific beam of radiation which is composed of a comb of frequencies v_ku ..., v_kn matching with the basic frequencies of the cavity given by the formula

where c is the velocity of light and L is the optical length of the cavity, к = 0, 1, 2,____This comb is centered on the spontaneous emission spectrum. The spectral bandwidth of a laser is given by the width of the spontaneous emission.

Typically, a laser is considered the emitter of monochromatic radiation. This terminology may make the impression that the output radiation is concentrated just at one fixed frequency, i.e., it emits only photons of fixed energy. However, one has to be careful while operating with the notion of "monochromatic laser." The spectrum of a laser, e.g., helium-neon laser, is monochromatic in the sense that only one color is visible to the naked eye as the line is very narrow. But this is not just one fixed frequency. In reality the physical laser produces a beam with the Gaussian distribution of frequencies (energies) centered on the basic frequency (energy) of the cavity.

In modeling of social processes, we are not interested in the generation of a discrete comb of energies. It is fine to generate a narrow Gaussian distribution of social energies. Echo chambers act as social energy filters: only posts and comments matching the (Gaussian) distribution of energy in the gain medium interact with excited people in this medium and generate new posts and comments. For example, in an anti-globalism echo chamber, posts carrying too high social energy, e.g., to stop the process of globalization of economics by a revolutionary uprising against the capitalist system, would be ignored. They would not generate stimulated emission in the anti-globalism gain medium. Such a gain medium interacts with posts and comments carrying essentially lower portions of social energy. Such a portion is sufficient for stimulating, for example, flight from California to Sweden and participation in an anti-globalism manifestation, but not more. In the same way, a low-energy post proposing to start with yourself and save electric energy at home would be ignored as well. It would not generate new posts and comments.

• [1] We remark that in the laser physics community it is sufficiently common to the
• [2] claim that the functioning of laser can be modeled by means of the semiclassicalmodel [111]: quantized matter and the classical electromagnetic field. However,the semiclassical model is not useful for our social applications, because its rigidcoupling to the physical space representation.
• [3] By speaking about "the same moment of time," one has to take into account thetemporal scale of the echo chamber’s functioning. "The same moment" is not thesame moment of physical time. It can cover a few days or even weeks.