Basics of Physical Lasing

We present schematically the basic scheme of the functioning of physical laser and its basic components (Fig. 3.1). For the moment, we do not consider the functioning of the optical cavity (see Chapters 7 and 8).

Laser: History of Invention

The invention of laser was one of the main technological outputs of quantum physics.

• • The theoretical basis for future creation of laser was implicitly present in the works of Einstein on stimulated absorption and emission (1917).
• • In 1939, the Soviet physicist Fabrikant predicted the use of stimulated emission to amplify "short” waves.
• • In 1953, Townes, with Gordon and Zeiger, produced the first microwave amplifier, a device operating on principles similar to the laser, but amplifying microwave radiation rather than infrared or visible radiation. Townes's maser was incapable of continuous output.

Figure 3.1 Laser’s structure: ф output beam; © pump source (e.g., flashlamp); © gain medium; 0 high reflectivity mirrow; © partially reflective mirrow, © optical cavity (the block of two mirrows 4 + 6) -laser resonator.

• In 1955, Prokhorov and Basov suggested optical pumping of a multilevel system as a method for obtaining population inversion. Later, the Prokhorov-Basov approach became the main method of laser pumping.

Spontaneous and Stimulated Emission

We start with a description of the processes of spontaneous and stimulated emission for excited two-level atoms with energy levels E = Ei, E₂. Denote the resonance energy of an atom by E_A = E₂ - E. Typically in the physical literature one proceeds with frequencies and speaks about the resonance frequency a>_A = where h is the Planck constant. To proceed to the social modeling, we tiy to eliminate the space-time picture from the model. In particular, we want to operate solely with energies.

We consider the quantum electromagnetic field interacting with atoms and describe the physical processes generated by the interaction. ^[1]

Figure 3.2 Stimulated emission.

it is impossible to predict neither the instance of time nor the direction of the emitted photon.

• (2) Atoms in the ground state can absorb from this field only photons having the resonance energy E_A = E₂ — E.
• (3) An excited atom interacting with photons with energy E_A emits a photon of the same energy.
• (4) This output flow of photons is coherent. All photons produced from the "seed photon" have the same features: direction of flow, polarization, and energy.

Thus one photon produces two. These two interact with two atoms and produce four photons. After n steps this process generates N = 2" photons. We note that this description and its illustration by Fig. 3.2, although typical for physics textbooks, are too straightforward. In fact, an atom interacts not with a single photon, but with all photons in the field (see section 4.8 and equality (4.4)).

Population Inversion

The cascade process described above plays a crucial role in generating laser beams carrying huge energy. The main problem is spontaneous emission.¹ Pumping photons into a gain medium (an ensemble of atoms in the ground state) transfers ground-state atoms into excited atoms. But they can spontaneously fall back to the ground level. The basic step in generating lasing is approaching population inversion. This is transition from an ensemble of atoms in the ground state to an ensemble in which more than 50% of atoms are in the excited state.

We remark that, for a gain medium composed of two level atoms, population inversion cannot be approached (at least straightforwardly), because the transition probabilities are equal: p(Ei -> E₂) = p[E₂ -*■ £1). These probabilities are known as the Einstein coefficients. Their equality for the electromagnetic field can be proven by using thermodynamics for indistinguishable systems following Bose-Einstein statistics, (see section 8.2.1 for the standard physical presentation; see Chapter 6 for a social counterpart of Einstein’s theory.) For the electromagnetic field, one needs a gain medium composed of atoms with at least three energy levels, Ei < E₂ < E₂.

The above considerations are about the physical laser based on the quantum electromagnetic field. But, in general, information thermodynamics [168] does not imply the coincidence of the Einstein coefficients for a two-level system. In principle, the social laser can be based on the simplest s-atoms having two levels of social energy. ^[2]

Basics of Social Lasing

We start considerations with a general foundational discussion. We remark that in quantum theory value a of the observable A obtained in its measurement cannot be interpreted as the property of system S on which the measurement is performed. By the Copenhagen interpretation this value is generated in the complex process of interaction of S and a device used for the measurement of A.

We recall that in the mathematical formalism a (pure) state is represented by the normalized vector H endowed with the scalar product ( I-), i.e., (i/r|^) = 1. So-called mixed states, probabilistic mixtures of pure states, are also considered. They are represented by density operators. The latter state representation plays the basic role in the theory of open quantum systems. A quantum observable is mathematically represented by a Hermitian operator. We shall denote an observable and the corresponding Hermitian operator by the same symbol.

Only if state t/f = |a) of S is an eigenstate of Hermitian operator A representing the observable measured on S, then eigenvalue a can be considered as the property of 5 (in this eigenstate). For this state, the A-observable takes the value a with probability 1.

Here we use the Dirac notation |a) for the eigenstate corresponding to the eigenvalue a, i.e.,

In fact, the label "Copenhagen interpretation” is assigned to a bunch of interpretations of quantum mechanics which can differ essentially [205]. As was proposed by A. Plotnitsky, it is better to speak about interpretations in the spirit of Copenhagen [see, e.g., our joint paper [209] for the extended discussion). Generally, quantum theory is characterized by a huge diversity of interpretations. From my viewpoint, this diversity is a sign of the deep crisis in quantum foundations. At the same time, some experts in foundations do not consider the diversity of interpretations as a problem: various interpretations can peacefully coexist. One can choose an interpretation for his own convenience. Other experts would consider this position as a form of scientific opportunism and they badly want to create a single interpretation of quantum mechanics. One of the most successful recent attempts to resolve this foundational crisis was the creation of quantum Bayesianism (QBism), the subjective probability interpretation of quantum mechanics [Chapter 11). For physicists, it is difficult (if possible at all) to accept that quantum theory describes the process of decision making about possible outputs of quantum measurements. Moreover, QBism claims that these decisions are made by individual agents, e.g., experimenters. However, QBism provides sufficiently consistent interpretation of quantum mechanics. In particular, in this framework it is possible to proceed without such notions as "quantum nonlocality" and action at a distance.^[3]

Once again we emphasize the role of QBism as a bridge between quantum physics and applications of quantum formalism and methodology outside physics, e.g., in sociophysics.

We presented this overview of the present-day situation in quantum foundations to support our interpretational analysis of the notion of social energy and other emotion-based observables in sociophysics. A few times I was confronted with veiy sharp opposition for the quantum-like modeling of cognition and social processes. In particular, reviewers can be very critical of operating with the notion of social energy. At the same time, such reviewers are (surprisingly) sure that in quantum mechanics the notion of physical energy can be straightforwardly interpreted. I think that this is not the case. The interpretation of observables, such as energy, spin, polarization, or temperature, for quantum systems is not crystal clear.

Coming back to the interpretations in the spirit of Copenhagen, we point out that this book is based on, so to say, the genuine Bohr interpretation. Quantum mechanics is a theory of observations, and the whole experimental arrangement (context) has to be taken into account.

Social Energy

In physics, the Copenhagen interpretation has to be applied to energy observable E. Although one often says, e.g., "the energy of the electron,” the correct meaning of this statement is about the output of the E measurement. The Copenhagen interpretation is well accommodated to the notion of superposition of states. Quantum system S can be in state ^ of superposition of two states |£i>, E₂) corresponding to two different energy levels E, E₂

where c, are complex numbers, probability amplitudes, such that kil² + кг |² = 1. If system S is in this state, then the probability to get the value £, of the energy observable is equal to p, = |c, |². Only, for states Ei), | E₂), the energy can be considered as the property of 5.

Supported by the Copenhagen interpretation of the notion of energy in physics, we are ready to consider the very complex notion of social (mental) energy. This notion has been actively used in cognition, psychology, social and political science (since the works of James [118-120] and Freud [84, 85], later by Jung [121], see also [166]), and recently in economics and finances, multi-agent modeling, evolution theory, and industrial dynamics ([55,101,117], see also [168, 169, 176] for details). And, of course, these previous studies are supporting for our model. However, we emphasize once again that the application of the Copenhagen methodology simplifies and clarifies essentially the issue of the social energy.

Energy of Social Atoms

The simplest quantification of social energy (mental energy) can be done by the question (observable) E = "Are you in the state of relaxation or excitement?" This observable takes two values, say Ei = 0, E₂ = 1. The Copenhagen interpretation is strongly involved. Before being asked the E question, s-atom can be in superposition of these two states. Only by confronting the E question the s-atom determines his or her state. The social energy observable can be represented in different forms; for example, in the form of the question E = "Will you go to the demonstration against Trump or Brexit?" Of course, we need not be restricted to the simplest yes-no, to be or not to be, quantification.

Finer quantifications of social energy can be useful as well. Different types of s-atoms can emit and absorb portions of social energy of different magnitudes. Here a type of s-atom is determined by her or his psyche and social environment. In the operational formalism, we can proceed with some grading of the possible social energy levels for s-atoms. Since we restrict consideration to two- level s-atoms, they are characterized by the social energy levels E = Ei, E2, the ground and excited states, and the resonance energy E_a = E₂ — EWe remark that in principle the ground-level energy for one type of s-atoms can be higher than the excitation level energy for another type. Social lasing is possible only for a gain medium with the homogeneous structure of energy levels.

The Copenhagen methodology demystifies the notion of social (mental) energy. (Of course, the same methodology can be applied to any social [mental] observable.) One should not be surprised that the methodology of quantum physics is applicable outside it. The

Copenhagen interpretation presents the veiy general methodology which is applicable to any kind of measurement. We remark that the use of this measurement methodology does not imply that the whole apparatus of quantum theory can be applicable. One should be careful by checking constraints on the class of systems and observables leading to applicability of one or another part of quantum theory. The quantum-like approach does not mean to copy straightforwardly the complete quantum theory to, say, social science. For example, to derive the Bose-Einstein statistics, we have to assume the indistinguishability of information quanta and excitations of the quantum information field (see section 4.2).

Energy of the Quantum Information Field

In physics, energy can be assigned not only to atoms, material systems, but also to carriers of interactions, e.g., photons or neutrinos, which are excitations of corresponding quantum fields. (Here "assigned” has the operational meaning: "can be measured.”) In social lasing, the interaction processes are formally modeled with the aid of a quantum information field generated by a variety of information sources (see section 4.2). Communications "emitted" by them carry portions of social energy. These quanta of social energy are interpreted as the excitations of the quantum information field.

Again, as in the case of s-atoms, we can proceed with "to be or not to be” quantification of the social energy carried by communications. If an s-atom in the ground state absorbs energy from communication C (and becomes excited), then C carries social energy E₂ = 1. If C cannot excite a social atom, then C's energy Ei = 0.

This social energy quantification depends on the concrete ensemble of s-atoms, a social group. It is easy to give examples of social and political communications which would excite the average Englishman or American, but not Russian or Chinese, and vice versa. Thus the definition of an information field’s energy is purely operational. In some sense, it is even "more operational” than the definition of the energy for a quantum physical field. It is meaningless to speak about the social energy of the information field without describing the class of "detectors”; in our case, these are s-atoms. As in the case of s-atoms, it is possible to proceed with models based on finer scales of the social energy assigned to excitations of the information field. Each communication C is characterized by social energy Ec. It can be absorbed by an s-atom with the resonance energy E_A = Ec.

A variety of communications can carry the same portion of social energy. All communications with Ec = E, where E is the fixed portion of the social energy, are considered equivalent from the viewpoint of energy delivering. They can be represented by the same field’s state | E). We say that E determines a mode of the quantum information field: E is the analogue of the characteristic energy E_w = hw of the electromagnetic mode with frequency со.

How many elementary excitations of energy E are carried by the E mode of the quantum information field? This number is determined by the power of information sources emitting communications belonging to the E mode.

4.2 Quantum Field Representation of the

• [1] An atom in the excited state can spontaneously emit a photon.This process is irreducibly random, i.e., even for a single atom
• [2] 4n fact, the primary problem is that the thermodynamic heat-bath Boltzmanndistribution leads to higher population in the lower levels than in the upper levels.
• [3] 4 remember the stormy debates after the talk of T. Hansh at one of theVaxjo conferences on quantum foundations. (He is physicist who got the NobelPrize for experimental laser physics.) When Hansh was aggressively attacked byrepresentatives of the realist interpretations of quantum mechanics (especially byL. Vaidman) and accused of using QBism, which is totally foreign to physics, he saidthat he was not able to find another consistent interpretation. He prefers to live withQBism than with a bunch of quantum paradoxes and mysteries.