Negative temperature. Temperature All year round the temperature is negative at

Absolute temperature in molecular kinetic theory is defined as a value proportional to the average kinetic energy of particles (see section 2.3). Since kinetic energy is always positive, absolute temperature cannot be negative. The situation will be different if we use a more general definition of absolute temperature as a quantity characterizing the equilibrium distribution of particles of a system over energy values ​​(see Section 3.2). Then, using the Boltzmann formula (3.9), we will have

Where N 1 – number of particles with energy 𝜀 1 , N 2 – number of particles with energy 𝜀 2 .

Taking logarithms of this formula, we get

In the equilibrium state of the system N 2 is always less N 1 if 𝜀 2 > 𝜀 1 . This means that the number of particles with a higher energy value is less than the number of particles with a lower energy value. In this case always T > 0.

If we apply this formula to such a nonequilibrium state, when N 2 > N 1 at 𝜀 2 > 𝜀 1, then T < 0, т.е. состоянию с таким соотношением числа частиц можно формально по аналогии с предыдущим случаем приписать определенную отрицательную абсолютную температуру. Поскольку при этом формула Больцмана применена к неравновесному распределению частиц системы по энергии, то отрицательная температура является величиной, характеризующей неравновесные системы. Поэтому отрицательная температура имеет иной физический смысл, чем понятие обычной температуры, определение которой неразрывно связано с равновесием.

Negative temperature is only achievable in systems that have a finite maximum energy value, or in systems that have a finite number of discrete energy values ​​that particles can accept, i.e. with a finite number energy levels. Since the existence of such systems is associated with the quantization of energy states, in this sense the possibility of the existence of systems with a negative absolute temperature is a quantum effect.

Let's consider a system with negative absolute temperature, which has, for example, only two energy levels (Fig. 6.5). At absolute zero temperature, all particles are at the lowest energy level, and N 2 = 0. If you increase the temperature of the system by supplying energy to it, then the particles will begin to move from the lower level to the upper one. In the limiting case, one can imagine a state in which there are the same number of particles at both levels. Applying formula (6.27) to this state, we obtain that T = at N 1 = N 2, i.e. a uniform energy distribution of particles in a system corresponds to an infinitely high temperature. If you somehow inform the system extra energy, then the transition of particles from the lower level to the upper one will continue, and N 2 will be greater than N 1 . Obviously, in this case the temperature, in accordance with formula (6.27), will take a negative value. The more energy is supplied to the system, the more particles will be at the upper level and the more negative the temperature will be. In the limiting case, one can imagine a state in which all particles are collected at the top level; wherein N 1 = 0. Therefore, this state will correspond to a temperature of 0 K or, as they say, the temperature of negative absolute zero. However, the energy of the system in this case will be infinitely large.

As for entropy, which, as is known, is a measure of the disorder of a system, depending on the energy in ordinary systems it will increase monotonically (curve 1, Fig. 6.6), so

as in conventional systems there is no upper limit to the energy value.

Unlike conventional systems, in systems with a finite number of energy levels, the dependence of entropy on energy has the form shown by curve 2. The section shown by the dotted line corresponds to negative values ​​of absolute temperature.

To more clearly explain this behavior of entropy, let us turn again to the example of a two-level system discussed above. At absolute zero temperature (+0K), when N 2 = 0, i.e. all particles are at the lower level, the system is maximally ordered and its entropy is zero. As the temperature rises, the particles will begin to move to the upper level, causing a corresponding increase in entropy. At N 1 = N 2 particles will be evenly distributed across energy levels. Since this state of the system can be represented in the greatest number of ways, it will correspond to the maximum entropy value. The further transition of particles to the upper level leads to some ordering of the system in comparison with what took place when the particles were unevenly distributed over energies. Consequently, despite the increase in the energy of the system, its entropy will begin to decrease. At N 1 = 0, when all the particles gather at the upper level, the system will again have maximum order and therefore its entropy will become zero. The temperature at which this happens will be the temperature of negative absolute zero (–0K).

Thus, it turns out that the point T= – 0K corresponds to the state furthest from the usual absolute zero (+0K). This is due to the fact that on the temperature scale the region of negative absolute temperatures is located above the infinitely large positive temperature. Moreover, the point corresponding to an infinitely large positive temperature coincides with the point corresponding to an infinitely large negative temperature. In other words, the sequence of temperatures in ascending order (from left to right) should be like this:

0, +1, +2, … , +

It should be noted that a negative temperature state cannot be achieved by heating a conventional system in a positive temperature state.

The state of negative absolute zero is unattainable for the same reason that the state of positive absolute zero temperature is also unattainable.

Despite the fact that the states with temperatures +0K and –0K have the same entropy, equal to zero, and correspond to the maximum order of the system, they are two completely different states. At +0K the system has a maximum energy value and if it could be achieved, it would be a state of stable equilibrium of the system. An isolated system could not come out of such a state on its own. At –0K the system has a maximum energy value and if it could be achieved, it would be a metastable state, i.e. a state of unstable equilibrium. It could only be maintained with a continuous supply of energy to the system, since otherwise the system, left to itself, would immediately come out of this state. All states with negative temperatures are equally unstable.

If a body with a negative temperature is brought into contact with a body with a positive temperature, then energy will transfer from the first body to the second, and not vice versa (as with bodies with ordinary positive absolute temperature). Therefore, we can assume that a body with any finite negative temperature is “warmer” than a body with any positive temperature. In this case, the inequality expressing the second law of thermodynamics (second particular formulation)

can be written in the form

where is the amount by which the heat of a body with a positive temperature changes over a short period of time, is the amount by which the amount of heat of a body with a negative temperature changes over the same time.

Obviously, this inequality can be satisfied if and only if the value = is negative.

Since the states of a system with a negative temperature are unstable, in real cases it is possible to obtain such states only if the system is well isolated from surrounding bodies with a positive temperature and provided that such states are maintained by external influences. One of the first methods for obtaining negative temperatures was the method of sorting ammonia molecules in a molecular generator created by domestic physicists N.G. Basov and A.M. Prokhorov. Negative temperatures can be obtained using a gas discharge in semiconductors exposed to a pulsed electric field, and in a number of other cases.

It is interesting to note that since systems with negative temperatures are unstable, when radiation of a certain frequency passes through them, as a result of the transition of particles to lower energy levels, additional radiation will appear, and the intensity of the radiation passing through them will increase, i.e. systems have negative absorption. This effect is used in the operation of quantum generators and quantum amplifiers (in masers and lasers).

Note that the difference between the usual absolute zero temperature and negative is that we approach the first from the side of negative temperatures, and the second from the side of positive ones.

If we proceed from the definition of temperature that was given at the beginning of this book, i.e., that temperature is proportional to the average kinetic energy of particles, then the title of this paragraph seems to be devoid of meaning: after all, kinetic energy cannot be negative! And for those atomic systems in which energy contains only kinetic energy movement of particles, negative temperature actually has no physical meaning.

But let us remember that in addition to the molecular kinetic determination of temperature, in Chap. I also noted the role of temperature as a quantity that determines the energy distribution of particles (see page 55). If you use this more general concept temperature, then we will come to the possibility of the existence (at least in principle) of negative temperatures.

It is easy to see that Boltzmann’s formula (9.2)

formally “allows” the temperature to take not only positive, but also negative values.

In fact, in this formula, this is the proportion of particles in a state with energy, and this is the number of particles in a state with some initial energy, from which the energy is measured. From the formula it is clear that the higher, the lower the proportion of particles possessing this energy. So, for example, when times are less than the base of natural logarithms). And a significantly smaller fraction of particles already have energy: in this case, times less. It is clear that in an equilibrium state, to which, as we know, Boltzmann’s law applies, it is always less than

Taking the logarithm of equality (9.2), we obtain: whence

From this expression for it is clear that if then

If, however, it turned out that there is an atomic system in which there can be more than this, this would mean that the temperature can also take negative values, since at becomes negative.

It will be easier for us to understand under what circumstances this is possible if we consider not a classical system (in which negative temperatures cannot be realized), but a quantum one, and also use the concept of entropy, which,

as we have just seen, is a quantity that determines the degree of disorder in the system.

Let the system be represented by a diagram of its energy levels (see, for example, Fig. 1, p. 17). At absolute zero temperature, all particles of our system are at their lowest energy levels, and all other levels are empty. The system under such conditions is maximally ordered and its entropy is zero (its heat capacity is also zero).

If we now increase the temperature of the system by supplying energy to it, then the particles will move to higher energy levels, which, thus, also turn out to be partially populated, and the higher the temperature, the greater the “population” of higher energy levels. The distribution of particles over energy levels is determined by the Boltzmann formula. This means that it will be such that there will be fewer particles at higher levels than at lower ones. The “dispersal” of particles across many levels, of course, increases the disorder in the system and its entropy increases with increasing temperature. The greatest disorder, and therefore the maximum entropy, would be achieved with such a distribution of particles by energy in which they are evenly distributed over all energy levels. Such a distribution would mean that in the formula it means, Therefore, a uniform distribution of particles over energies corresponds to an infinitely high temperature and maximum entropy.

However, in the quantum system we are talking about here, such a distribution is impossible, because the number of levels is infinitely large, and the number of particles is finite. Therefore, the entropy in such a system does not pass through a maximum, but increases monotonically with temperature. At an infinitely high temperature, the entropy will also be infinitely high.

Let us now imagine a system (quantum) in which there is an upper limit to its internal energy, and the number of energy levels is finite. This, of course, is only possible in a system in which the energy does not include the kinetic energy of particle motion.

In such a system, at absolute zero temperature, particles will also occupy only the lowest energy levels, and entropy will be equal to zero. As the temperature rises, the particles “settle” at higher levels, causing a corresponding increase in entropy. In Fig. 99, and a system with two energy levels is presented. But, since the number of energy levels of the system, like the number of particles in it, is now finite, a state can ultimately be achieved in which the particles are evenly distributed among the energy levels. As we have just seen, this state corresponds to infinitely high temperature and maximum entropy.

In this case, the energy of the system will also be at some maximum, but not infinitely large, so our old definition of temperature as the average energy of particles becomes inapplicable.

If we now somehow impart additional energy to the system, which is already at an infinitely high temperature, then the particles will continue to move to a higher energy level, and this will lead to the fact that the “population” of this high energy level will become greater than that of the lower one (Fig. 99, b). It is clear that such a predominant accumulation of particles at high levels already means some ordering in comparison with the complete disorder that existed when, i.e., with a uniform distribution of particles over energies. Entropy, which has reached a maximum at, therefore begins to decrease with further energy supply. But if with increasing energy entropy does not increase, but decreases, then this means that the temperature is not positive, but negative.

The more energy supplied to the system, the more particles will be at the highest energy levels. At the limit, one can imagine a state in which all particles are collected at the highest levels. This state is obviously also quite ordered. It is no “worse” than the state when all particles occupy the lowest levels: in both cases, complete order prevails in the system, and entropy is zero. We can therefore designate the temperature at which this second completely ordered state is established by -0, in contrast to the “ordinary” absolute zero. The difference between these two “zeros” is that we come to the first of them from the negative, and to the second - from the side of positive temperatures.

Thus, the conceivable temperatures of the system are not limited to the interval from absolute zero to infinity, but extend from through to , and coincide with each other. In Fig. Figure 100 shows a curve of entropy versus energy of the system. The part of the curve to the left of the maximum corresponds to positive temperatures, to the right of it - to negative ones. At the maximum point the temperature value is

From the point of view of orderliness, and therefore entropy, the following three extreme states are possible:

1. Complete ordering - particles are concentrated at the lowest energy levels. This state corresponds to "normal" absolute zero

2. Complete disorder - particles are evenly distributed across all energy levels. This state corresponds to temperature

3. Complete ordering again - particles occupy only the highest energy levels. The temperature corresponding to this state is assigned the value -0.

We are dealing here, therefore, with a paradoxical situation: in order to reach negative temperatures, we did not have to cool the system below absolute zero, which is impossible, but, on the contrary, increase its energy; negative temperature turns out to be higher than infinitely high temperature!

There is a very important difference between the two completely ordered states that we just mentioned - states with temperatures .

The state of “ordinary” absolute zero, if it could be created in the system, would remain in it indefinitely, provided that it is reliably isolated from environment, is isolated in the sense that no energy is supplied to the system from this medium. This state is a state of stable equilibrium, from which the system by itself, without outside intervention, cannot exit. This is due to the fact that the energy of the system in this state has a minimum value.

On the other hand, the state of negative absolute zero is an extremely nonequilibrium state, since. the energy of the system is maximum. If it were possible to bring the system to this state, and then leave it to its own devices, then it would immediately come out of this nonequilibrium, unstable state. It could only be maintained by continuously supplying energy to the system. Without this, particles located at higher energy levels will certainly “fall” to lower levels.

The common property of both “zeros” is their unattainability: to achieve them requires the expenditure of infinitely large amounts of energy.

However, not only the state corresponding to a temperature of -0 is unstable, nonequilibrium, but also all states with negative temperatures. All of them correspond to values, and for equilibrium the inverse relation is necessary

We have already noted that negative temperatures are more high temperatures than positive ones. Therefore, if we bring

a body heated (one cannot say cooled) to negative temperatures comes into contact with a body whose temperature is positive, then energy will transfer from the first to the second, and not vice versa, and this means that its temperature is higher, although it is negative. When two bodies with a negative temperature come into contact, energy will transfer from the body with a lower absolute temperature to a body with a higher numerical temperature.

Being in an extremely nonequilibrium state, a body heated to a negative temperature very willingly gives up energy. Therefore, in order for such a state to be created, the system must be reliably isolated from other bodies (in any case, from systems that are not similar to it, that is, not having a finite number of energy levels).

However, a state with a negative temperature is so unbalanced that even if a system in this state is isolated and there is no one to transfer energy to, it can still give off energy in the form of radiation until it passes into a state (equilibrium) with a positive temperature .

It remains to add that atomic systems with a limited set of energy levels, in which, as we have seen, a state with a negative temperature can be achieved, is not only a conceivable theoretical construction. Such systems actually exist and negative temperatures can actually be obtained in them. Radiation arising during the transition from a state with a negative temperature to a state with a normal temperature is practically used in special devices: molecular generators and amplifiers - masers and lasers. But we cannot dwell on this issue in more detail here.

Negative temperature

negative absolute temperature, a quantity introduced to describe nonequilibrium states of a quantum system in which higher energy levels are more populated than lower ones. At equilibrium, the probability of having energy E n is determined by the formula:

Here E i - system energy levels, k- Boltzmann constant, T- absolute temperature characterizing the average energy of the equilibrium system U = Σ (W n E n), From (1) it is clear that when T> 0 lower energy levels are more populated by particles than upper ones. If a system, under the influence of external influences, goes into a nonequilibrium state, characterized by a greater population of the upper levels compared to the lower ones, then formally we can use formula (1), putting in it T

In thermodynamics, absolute temperature T determined through the reciprocal value 1/ T, equal to the derivative of entropy (See Entropy) S based on the average energy of the system with other parameters constant X:

From (2) it follows that O. t. means a decrease in entropy with increasing average energy. However, thermodynamics is introduced to describe nonequilibrium states to which the application of the laws of equilibrium thermodynamics is conditional.

An example of a system with O. t. is a system of nuclear spins in a crystal located in a magnetic field, very weakly interacting with thermal vibrations of the crystal lattice (See Vibrations of the crystal lattice), that is, practically isolated from thermal motion. The time required to establish thermal equilibrium between the spins and the lattice is measured in tens of minutes. During this time, the system of nuclear spins can be in a state with O. t., into which it passed under external influence.

In a narrower sense, OPT is a characteristic of the degree of population inversion of two selected energy levels of a quantum system. In the case of thermodynamic equilibrium of the population N 1 And N 2 levels E 1 And E 2 (E 1 E 2), i.e. the average numbers of particles in these states are related by the Boltzmann formula:

Where T - absolute temperature of a substance. From (3) it follows that N 2 N 1. If you disturb the equilibrium of the system, for example, influence the system with monochromatic electromagnetic radiation, the frequency of which is close to the frequency of transition between levels: ω 21 = ( E 2 - E 1)/ħ and differs from the frequencies of other transitions, then it is possible to obtain a state in which the population of the upper level is higher than the lower one N 2 > N 1. If we conditionally apply the Boltzmann formula to the case of such a nonequilibrium state, then with respect to a pair of energy levels E 1 And E 2 You can enter O.t. using the formula:

Despite the formal nature of this definition, it turns out to be convenient in a number of cases; for example, it allows one to describe fluctuations in equilibrium and nonequilibrium systems with O. t. by similar formulas. The concept of optical energy is used in quantum electronics (see Quantum electronics) for the convenience of describing the processes of amplification and generation in media with population inversion.

D. N. Zubarev.

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what “Negative temperature” is in other dictionaries:

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negative temperature- Characteristic of the inverse state, meaning the transition temperature. [Collection of recommended terms. Issue 75. Quantum electronics. Academy of Sciences of the USSR. Committee of Scientific and Technical Terminology. 1984] Topics: quantum electronics EN... ... Technical Translator's Guide

negative temperature- neigiamoji temperatūra statusas T sritis Standartizacija ir metrologija apibrėžtis Temperatūra, žemesnė už 0 ºC. atitikmenys: engl. negative temperature vok. negative Temperature, f rus. negative temperature, f pranc. temperature au dessous… … Penkiakalbis aiškinamasis metrologijos terminų žodynas

negative temperature- neigiamoji temperatūra statusas T sritis fizika atitikmenys: engl. negative temperature vok. negative Temperature, f rus. negative temperature, f pranc. temperature negative, f … Fizikos terminų žodynas

negative temperature- Characteristics of the inverse state, meaning the transition temperature... Polytechnic terminological explanatory dictionary

Temperature characterizing the equilibrium states of a thermodynamic system in which the probability of finding the system in a microstate with a higher energy is higher than in a microstate with a lower one. In quantum statistics this means that... ... Wikipedia

Temperature (from the Latin temperatura, proper mixing, proportionality, normal state), a physical quantity that characterizes the state of thermodynamic equilibrium of a macroscopic system. T. is the same for all parts of the isolated system...

I Temperature (from the Latin temperatura, proper mixing, proportionality, normal state) is a physical quantity that characterizes the state of thermodynamic equilibrium of a macroscopic system. T. is the same for all parts of the insulated... Great Soviet Encyclopedia

Dimension Θ Units of measurement SI K ... Wikipedia

A negative quantity having the dimension of temperature, characterizing the degree of population inversion of energy levels of systems (atoms, ions, molecules) ... Big Encyclopedic Dictionary

Thermodynamic systems in which the probability of finding the system in a microstate with a higher energy is higher than in a microstate with a lower one.

In quantum statistics, this means that it is more likely to find a system at one energy level with a higher energy than at one level with a lower energy. An n-fold degenerate level is considered to be n levels.

In classical statistics, this corresponds to a higher probability density for points in phase space with higher energy compared to points with lower energy. At a positive temperature, the ratio of probabilities or their densities is the opposite.

For the existence of equilibrium states with negative temperatures, the statistical sum must converge at this temperature. Sufficient conditions for this are: in quantum statistics - the finiteness of the number of energy levels of the system, in classical statistical physics - that the phase space accessible to the system has a limited volume, and all points in this accessible space correspond to energies from a certain finite interval.

In these cases, there is the possibility that the energy of the system will be higher than the energy of the same system in an equilibrium distribution with any positive or infinite temperature. An infinite temperature will correspond to a uniform distribution and a final energy below the maximum possible. If such a system has an energy higher than the energy at infinite temperature, then the equilibrium state at such energy can only be described using a negative absolute temperature.

The negative temperature of the system remains long enough if this system is sufficiently well insulated from bodies with a positive temperature. In practice, negative temperature can be realized, for example, in a system of nuclear spins.

At negative temperatures, equilibrium processes are possible. When there is thermal contact between two systems with different temperature signs, the system with a positive temperature begins to heat up, and the one with a negative temperature begins to cool. For the temperatures to become equal, one of the systems must pass through an infinite temperature (in a particular case, the equilibrium temperature of the combined system will remain infinite).

Absolute temperature + ∞ (\displaystyle +\infty ) And − ∞ (\displaystyle -\infty )- this is the same temperature (corresponding to a uniform distribution), but the temperatures T=+0 and T=-0 differ. Thus, a quantum system with a finite number of levels will be concentrated at the lowest level at T=+0, and at the highest at T=-0. Passing through a series of equilibrium states, the system can enter a temperature region with a different sign only through infinite temperature.

In a level system with population inversion, the absolute temperature is negative if it is determined, that is, if the system is sufficiently close to equilibrium.

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Absolute temperature ➽ Physics grade 10 ➽ Video lesson

First, we note that the idea of ​​states with negative absolute temperature does not contradict Nerst’s theorem about the impossibility of reaching absolute zero.

Consider a system with negative absolute temperature that has only two energy levels. At absolute zero temperatures, all particles are at the lowest level. As the temperature increases, some particles begin to move from the lower level to the upper level. The relationship between the number of particles at the first and second levels at different temperatures will satisfy the energy distribution in the form:

As the temperature increases, the number of particles at the second level will approach the number of particles at the first level. In the limiting case of infinitely high temperatures, there will be the same number of particles at both levels.

Thus, for any ratio of the number of particles in the interval

our system can be assigned a certain statistical temperature in the interval determined by equality (12.44). However, in special conditions it is possible to ensure that in the system under consideration the number of particles at the second level is greater than the number of particles at the first level. A state with such a ratio of the number of particles can, by analogy with the first case considered, also be assigned a certain statistical temperature or distribution modulus. But, as follows from (12.44), this module of the statistical distribution must be negative. Thus, the considered state can be assigned a negative absolute temperature.

From the example considered, it is clear that the negative absolute temperature introduced in this way is in no way a temperature below absolute zero. Indeed, if at absolute zero the system has minimal internal energy, then with increasing temperature the internal energy of the system increases. However, if we consider a system of particles with only two energy levels, then its internal energy will change as follows. When all particles are at the lower level with energy, therefore, internal energy. At an infinitely high temperature, particles are evenly distributed between levels (Fig. 71) and internal energy:

that is, it has a finite value.

If we now calculate the energy of the system in the state to which we have assigned a negative temperature, it turns out that the internal energy in this state will be greater than the energy in the case of an infinitely large positive temperature. Really,

Thus, negative temperatures correspond to higher internal energies than positive ones. During thermal contact of bodies with negative and positive temperatures, energy will transfer from bodies with negative absolute temperature to bodies with positive temperature. Therefore, bodies at negative temperatures can be considered “hotter” than at positive ones.

Rice. 71. Towards an explanation of the concept of negative absolute temperatures

The above considerations about the internal energy with a negative distribution modulus allow us to consider the negative absolute temperature as if higher than an infinitely large positive temperature. It turns out that on the temperature scale the region of negative absolute temperatures is not “below absolute zero”, but “above infinite temperature”. In this case, an infinitely large positive temperature “is next to” an infinitely large negative temperature, i.e.

A decrease in negative temperature in magnitude will lead to a further increase in the internal energy of the system. When the energy of the system will be maximum, since all particles will gather at the second level:

The entropy of the system turns out to be symmetrical with respect to the sign of the absolute temperature in equilibrium states.

The physical meaning of negative absolute temperature comes down to the idea of ​​a negative module of the statistical distribution.

Whenever the state of a system is described using a statistical distribution with a negative absolute value, the concept of negative temperature can be introduced.

It turns out that such states for some systems can be realized under different physical conditions. The simplest of them are the finiteness of the energy of the system with weak interaction with surrounding systems with positive temperatures and the ability to maintain this state by external forces.

Indeed, if you create a state with a negative temperature, i.e., do more, then thanks to spontaneous transitions, particles will be able to move from a state with to a state with lower energy. Thus, a state with a negative temperature will be unstable. To maintain it for a long time, it is necessary to replenish the number of particles at the level by reducing the number of particles at the level

It turned out that systems of nuclear magnetic moments satisfy the requirement of finite energy. Indeed, spin magnetic moments have certain number orientation and therefore energy levels in a magnetic field. On the other side; in a system of nuclear spins, with the help of nuclear magnetic resonance, most spins can be transferred to the state with the highest energy, i.e., to the highest level. To return to the lower level, nuclear spins will have to exchange energy with the crystal lattice, which will take quite a long time. During periods of time shorter than the spin-lattice relaxation time, the system can be in states with negative temperatures.

The considered example is not the only way to obtain systems with negative temperatures.

Systems with negative temperatures have one interesting feature. If radiation with a frequency corresponding to the difference in energy levels is passed through such a system, then the transmitted radiation

will stimulate particle transitions to the lower level, accompanied by additional radiation. This effect is used in the operation of quantum generators and quantum amplifiers (masers and lasers).