Coexistence of photonic and atomic Bose-Einstein condensates in ideal atomic gases

We have studied conditions of photon Bose-Einstein condensate formation that is in thermodynamic equilibrium with ideal gas of two-level Bose atoms below the degeneracy temperature. Equations describing thermodynamic equilibrium in the system were formulated; critical temperatures and densities of photonic and atomic gas subsystems were obtained analytically. Coexistence conditions of these photonic and atomic Bose-Einstein condensates were found. There was predicted the possibility of an abrupt type of photon condensation in the presence of Bose condensate of ground-state atoms: it was shown that the slightest decrease of the temperature could cause a significant gathering of photons in the condensate. This case could be treated as a simple model of the situation known as"stopped light"in cold atomic gas. We also showed how population inversion of atomic levels can be created by lowering the temperature. The latter situation looks promising for light accumulation in atomic vapor at very low temperatures.


I. INTRODUCTION
Bose-Einstein condensation is a vivid manifestation of quantum nature of macroscopic scale matter physics.This phenomenon is basic for many physical effects as, for example, superfluidity and superconductivity which have been known for a long time and used in practical applications.This is the fact that caused unabated interest to different Bose-Einstein condensate (BEC) related phenomena and effects study.Obtaining of BEC direct experimental performance in alkali metal vapors (in this regard see [1][2][3]) opened the prospect of experimental and theoretical predictions of new effects, which are possible in systems with BEC [4,5].For instance, such projections include the phenomenon of slow light [6][7][8] or even stopped light [9] in a BEC, storage of light in atomic vapor at extremely low temperatures [10,11].Studies [7,8] predicted the ability to control group velocity of light in gases with BEC using an external magnetic field [12] as well as the possibility of using BEC for filtering optical electromagnetic signals [13].Interesting effects associated with the passage of charged particles through systems with BEC were predicted in [14].
To complete the overall academic research experiments on BEC of photons study were required.For several reasons there is a small number of theoretical works devoted to this phenomenon (see, for example [15][16][17]).Indeed, when it comes to Bose-Einstein condensation the possibility of such phenomena in gases as the simplest physical systems was studied first.As we have already mentioned, in first experiments within such systems the conditions of condensation were achieved (see [1][2][3]).To get such a state in bosonic gas it is required for particles to have a mass and to conserve their total number in the system.It is known that mass of photon is zero in vacuum and to observe Bose condensation we need to reduce the temperature.It is difficult to find a method for lowering the temperature in gas consisting only of photons and creating such a system is even more difficult.One may see that photons could behave like particles with non-zero mass and lowering the temperature of that gas is possible when photons interact with the matter.However, in this case one should find the way how to compensate the loss of photons, because decreasing the environment temperature causes their absorption.Just recently all of these obstacles have been overcome [18,19]: during the simple and elegant experiments BEC of free photons was obtained in dye-filled optical microcavity.Photons appeared in the system when pumping the dye solution with an external laser.Thermal equilibrium of the photon gas was received as a result of absorption and re-emission.This fact made it possible to observe the condensation: due to cut-off frequency the effective photon mass became nonzero.Note that by cut-off frequency we mean the finite value of the photons wave vector at zero frequency.Scientific community admitted this experiment to be a real breakthrough: since it has been expected for a long time to receive photonic condensate and during the experiment it was observed at room temperature.Also, this phenomenon may be used in practical applications.For example, it could help with gathering and focusing sunlight in solar panels when the weather is cloudy; in creating new sources of short-wavelength laser radiation; in reducing the size of electronic microchips, etc.
It is obvious that since there are still a lot of issues to be discussed, the research needs to be continued in a number of ways.For example, is it possible to achieve BEC of photons in other systems?Which temperatures are required?Is it possible to observe atomic and photonic Bose-Einstein condensates at the same time?In this ar-ticle authors tried to answer the last question.We've studied the thermodynamic equilibrium of photonic and atomic gases at ultra-low temperatures, when BEC occurs in atomic gases.Note that the possibility of photons Bose-Einstein condensate formation in atomic nondegenerated gas was described in details [20].

II. EQUATIONS OF THERMODYNAMIC EQUILIBRIUM OF PHOTONS AND TWO-LEVEL
IDEAL BOSE-GAS Actually, in this section we will receive even more general task compared with the one announced in the Introduction.We will study the possibility of Bose-Einstein condensation of photons that are in thermodynamic equilibrium with a two-level gas.Let us consider a gas below the degeneracy temperature that can consist of bosons as well of fermions.
As it was mentioned above, there were photons that are in thermodynamic equilibrium with two-level atom ideal gas at ultralow temperatures.This model implies that the atom has only two possible states -the ground state and the exited state.It means that when the atom absorbs a photon it becomes excited and, when atom emits a photon, it changes the state from excited to the ground one.Thus, an excited atom can be considered as a bound state consisting of photon and non-excited atom.All these three components -photons, excited and non-excited atoms -are in thermodynamic equilibrium.Let us assign subindex "1" to ground state physical characteristics and subindex "2" to excited physical characteristics correspondingly (see also [20]).Distribution functions for two sorts of atoms -"1" and "2"are: Sign"+" in the function given above corresponds to the case of fermionic atoms, and sign"−" corresponds to case of bosonic atoms.The quantum numbers are specified by parameters α 1 , α 2 for all sorts of atoms.The values ε αi are the energy levels of stationary atoms; they are negative ε αi < 0, because an atom can be regarded as a bound state of some particles.Chemical potentials corresponding to both quantum-mechanical states are denoted as µ i , i = 1, 2. We imply that atomic number in this system is conserved.
The photon distribution function looks as follows: where ω (k) is photon dispersion law, and µ * is photon chemical potential.Existence of non-zero chemical po-tential µ * points to the fact that total number of photons N ph is conserved.It is important that the total number N ph of the photons consists of free photons and those ones absorbed by atoms (note that the number of excited atoms is equal to the number of absorbed photons).Current paper does not cover reasons for the total number N ph conserving, but it could be supposed that some system of mirrors with high reflectivity provides that it is possible.However mirrors should be located far enough apart from each other to diminish influence of boundary conditions so that they could be neglected.
From formulas (1), ( 2) we obtain these equations of the balance below: that follow from the fact of the number of atom N and photon number N ph conservation in the system.The parameter g α1 (g α2 ) corresponds to degeneracy of atomic levels with the set of quantum numbers α 1 ( α 2 ).For instance, parameters g α1 , g α2 can take into account spin state degeneracy, and g * -the degeneracy of a photon with wave vector k -may be caused by it's polarization.Further we wont take into account that characteristics of particles can depend of the spin in future consideration.
To get the complete description of the system consisting of atoms and photons we need to add the phase equilibrium condition to the system of equations (3) (condition of chemical reaction in this regard see in [21], for example): To calculate the sum over k in each equation in (3) photon dispersion law needs to be specified.Subsequently the dispersion relation is assumed to be quadratic in the wave vector and it is given by the equation: where ω 0 -is the cut-off frequency of photon spectrum and m * -its effective mass.Let's remark that we used quadratic photon dispersion relation as well as it was done in [18,19] with the difference that in these articles it had two-dimensional wave vector.This paper doesn't explain the reasons why we use this exact law and the dependence between the values ω 0 and m * .For example, the dispersion relation ( 5) can be obtained in case when formula gives photon energy for the photon in some medium.The expression ( 6) is similar to the expression for the energy of a relativistic object where v -is the phase velocity of light in matter.If the wave vectork satisfies the inequality v 2 k 2 /ω 2 0 ≪ 1, formula (6) can be represented as: and then it becomes possible to introduce the effective photon mass m * which can be addressed as photon rest energy in the matter, defined by: To denote the photon dispersion relation in certain matter and to get the cut-off frequency ω 0 and speed v of electromagnetic waves propagation in the matter one needs to formulate and to solve the dispersion relation of electromagnetic waves in the matter.Some basics of solving such problems in the case of electromagnetic waves propagation in ultracold atomic Bose gases can be found in [22] and in [7,8].It is interesting that the electromagnetic waves in plasma (see , eg, [23]) provide us with the dispersion laws that fully satisfy the formulas ( 6) - (8).
In particular, dispersion relation for longitudinal waves in plasma is like formula (5): and the one for the transverse electromagnetic waves is given by a formula similar to (6): where ω 0 is Langmuir oscillation frequency, r D is Debye radius, c is the speed of light in vacuum.In eq. ( 3) when replacing sums over the momentum by the integrals and introducing the system volume V, atomic density n ≡ N/V and photon density n ph ≡ N ph /V and when taking into account (4) we get the following system of equations: Here it was assumed that the photon dispersion relation is given by (5).The system of equations ( 9) can be used as a starting point when studying the thermodynamic equilibrium of photons with an ideal gas of twolevel atoms in wide temperature range.The system gets rather simplified in some specific cases and conditions of Bose-Einstein condensation can be obtained analytically.For instance, one of such possible situations was considered in [20], where the conditions of Bose-Einstein condensation of free photons theoretically were received at the high temperature.The term high temperature means that atomic gas is nondegenerate thus bosons and fermions behave similarly and there is no need to distinguish the difference between them.As it was mentioned above, this case was specified in detail in [20]; for this reason, in the next section we will use equations (9) with the temperature close to the degeneracy temperature of atomic gas components to study the conditions of photon Bose-Einstein condensation.In this temperature range the difference between Fermi-Dirac and Bose-Einstein statistics is very significant.Specifically, bosons can form a BEC below a certain temperature.In the next section we will find out when two Bose condensates -atomic and photonic -can coexist in the system.
To summarize this section, we will define the densities of atomic components n αi (T ) where i = 1, 2, the density of free photons n ph (T ) in the system as follows: and the atomic distribution function n αi (p) and photon distribution function n ph (p):

III. THE COEXISTENCE CONDITIONS OF BOSE EINSTEIN CONDENSATES OF NONEXCITED ATOMS AND PHOTONS
Three essentially different cases are possible when Bose-Einstein condensation of photons appears in system at low temperatures.Each case is determined by the type of atoms present in the condensate.In the first case, ground-state atoms form Bose-Einstein condensate; here for simplicity we will consider the gas of excited atoms to be nondegenerate.In the second case, excited atoms form Bose-Einstein condensate and (again for simplicity) the gas of ground-state atoms is nondegenerate.In the third case all atomic components (gases of excited and non-excited atoms) form Bose-Einstein condensate.
When studying the listed cases let's stick to the general system of equations rearranged by taking into account the fact that atomic subsystem consists of two-level boseatoms: Let's remind that equation ( 9) assumes that the system is in thermodynamic equilibrium.Note that according to (10) and ( 11) two first equations ( 12) can be written as follows: First we study the case of Bose-Einstein condensates formed by photons and ground-state atoms, whereas the gas of excited atoms is non-degenerate.Since the gas is considered to be non-degenerate, the chemical potential of excited atoms µ 2 satisfies the condition: The previous inequality makes it easy to calculate integrals in the last expressions of the first and second equations of system (12) and to get the next equations set: Note that all components of the studied system are ideal gases.Consequently, photonic component and component of atoms in the ground state are required to satisfy equalities (see in this regard, for example, [24]) for Bose condensate to appear in the system: There were introduced T c the condensation temperature of the ground state atomic gas and T * c -the one of the photon gas correspondingly.As a result of expression ( 16) and the last equation in ( 15) transformation we get chemical potential µ 2 formula: Taking into account ( 16), ( 17), ( 13) first two equations of ( 15) can also be rearranged as: where n α1 (p) , n ph (p) were defined in ( 10), ( 11) and symbol ∆ means: In the case being studied the value of ∆ should be greater than zero; the inequality ∆ > 0 is required because excited atoms are non-degenerated , see (14).Equations ( 18), (19) are the ones to be studied as the initial equations to define characteristics of conditions under which photons and non-excited atoms BEC coexist.The density of photons n ph (p) and atoms n α1 (p) distribution function below the transition temperatures T c and T * c over the momentum can be represented in the form below (in this regard see, for example [24]): where n 0 α1 (T ) -BEC density of atoms, n 0 ph (T ) -BEC density of free photons.Let's put (20) into (18) to obtain expressions for such densities: where ζ (x) -Riemann zeta function.We emphasize that BEC disappear in the transition point; this fact infers the following Thus, equations (22) together with ( 21) should be used to define transition temperatures T c and T * c .To analyze the cases possible for the first equation in ( 21) lets regard the temperature to be equal to T c -ground-state atoms condensation temperature; for the second equation in (21) let's regard the temperature to be equal to T * c -photons condensation temperature.As a result we get: It is easy to see that equations (23) are transcendental; they do not have analytical solution -only the numerical one.Nevertheless, in some cases analytical solution can be obtained.Lets analyze the first equation in (23) when temperatures are supposed to be low: it means that inequality exp (−∆/T c ) ≪ 1 (or (T c /∆) ≪ 1) is valid.When temperatures are low the first equation in ( 23) can be solved by means of perturbation theory using parameter exp (−∆/T c ) ≪ 1 and we get: The first order of perturbation theory gives us the following equation using method of successive approximation: This equation gives low temperature approximation tool adaptability criteria: We have a quite different case when defining free photons condensation temperature.Let us write the second equation of the system (23) as follows: It should be pointed out that the m * /m value -the relation of photon mass to the mass of atom -is very small.For instance, in [18] authors estimated photon mass as 6.7 • 10 −33 g. (see also [20]).Even for lithium that relation equals order-of-magnitude (m * /m) ∼ 10 −10 .This circumstance gives us two possible cases determined by two inequalities: If there are lithium atoms in the system the values (m * /m) 3/2 and exp (−∆/T * c ) will be of the same order of magnitude at To analyze the situation to which the validity of the first or the second equality in (28) may lead let's perform as follows: we divide the photon density (when system temperature is T * c ) by atoms density (when system temperature is T c ).As a result we have: All important characteristics of the system -condensation temperatures T * c , T c and the ratio of the masses m * /m and of the densities n ph n -are included in (30).As it was mentioned above, photons condensation temperature was assumed to be far below degeneration temperature of atomic gas in [20].Here quite different case is studied: atomic and photonic condensates were considered as coexisting ones.For definiteness, we've attributed photons condensation temperature to T * c and assigned it to be below atomic condensation temperature T * c T c .
If the first inequality in (28) is valid we can neglect its compounds with exponents in (30) and in the main approximation we will get: When atomic gas is helium the latter ratio is estimated to be (n ph /n) ∼ 10 −15 .In other words the fulfillment of the first case in (28) implied that photons amount is negligibly small in comparison with the number of atoms in the system.This particular value -the density of photons -defines maximum possible photon condensate density in the system (according to (21)).This circumstance makes the case (defined by first formula in (28)) not worth studying in this paper; since its not within the scope of priorities -to obtain photonic BEC -stated at the beginning of our research.
We will get definitely more interesting case, if the second condition in (28) is valid where the obtaining of BEC became more promising.From ( 12) one can get: According to second condition in (32) the inequality n ph n ≪ is maintained.Despite that fact the ratio (32) gives us the ability to observe BEC coexistence of atoms and photons at ultralow temperatures at higher values of photons density than formula (31) provides and taking into account (29), ( 27), (32) we can find the following equation for the critical temperature or write it as follows: It is easy to see in this case the critical temperature dependence on the density n ph is not power-behaved and essentially non-typical for transition points with BEC in ideal gases ( see [24] for example); this behavior was first noticed in [20].The condition T * c /T c 1 taking into consideration ( 25), (33) can be represented as: and gives us the condition under which the situation being studied is possible : and this do not contradict with (32).Equations (33) (or (34)) are both transcendental and do not have any analytical solution; it is easy to verify that the solution exists only when densities ratio satisfies n ph /n ≪ 1 (condition (32) is also fulfils), because the temperature belongs to the next 1/30<(T * c /∆) ≪ 1 temperature range.For example, when one supposes ) ≈ 4.5 × 10 −5 and consequently takes into consideration (33), (34) and if the fact that (T c /T * c ∼ 1) is true finally, we get the following: In other words, if particle density satisfies n ∼ 10 12 -10 14 cm −3 (these densities are typical for experiments performed at ultralow temperatures, see [1][2][3]) our studied case may be expected to occur when total density of photons satisfies n ph ∼ 10 7 -10 9 cm −3 .Such photon transition temperature under which BEC appears may be estimated from Fig. 1 where the numerical solution of (34) has been shown.The dimensionless quantities have been introduced as: and equation (38) has been transformed as Approximated formulas for densities of photonic n 0 ph (T ) and atomic n 0 α1 (T ) components in system being studied can be obtained from ( 21) using ( 25), (28), (31) -(33); for the first order of smallness by exp(−∆/T ) one can obtain: According to formula (40) photon BEC occurs rapidly (we call it "avalanche" condensate mode) together with temperature decrease, that can be easily seen when calculating the derivative ∂n 0 ph /∂T value at the following temperatures T T * c : The number of excited atoms decreased in the same manner: one may obtain it from formulas (18), (19) and when taking into account the equation ( 13) for such atoms density n α2 (T ) we get the following: The opposite statement is also correct: when temperature of the system is decreasing the state with BEC disappears and photons "captured" by atoms are being emitted.The statement about "avalanche" character of condensation is fully illustrated in Fig. 2: it expresses such normalized density n 0 ph /n ph behavior that depends on dimensionless parameters: T /T * c and ∆/T * c that is responsible for low-temperature approximation.There we used precise expression for photons condensate density n 0 ph : that could be easily obtained from the second inequality (21) when using the second equation in (23).On getting the value of n 0 ph /n ph we suggest that condensation temperatures are comparable T ∼ T * c and T T * c .Fig. 2 n ph 0 n ph FIG. 2. The demonstration of "avalanche" photons condensation possibility when being in equilibrium with ultracold atomic gas.
was performed in the same way as, for example, rock massif mapping in cartography, but here the higher altitude is indicated with lighter color.Solid line curves in the figure are contour lines that join the dots with the same value of n 0 ph /n ph (like isopleths in map making).One may see in this contour graph that in certain range of T /T * c and ∆/T * c values the avalanche photons condensation is possible: when nondimentional temperature T /T * c decreases slightly, the value of n 0 ph /n ph almost immediately grows up to its maximum (in mapping terms it looks like vast "plateau").This "sharp" character of photons condensation corresponds to "sharp" decrease of non-excited atoms population according to (42).
The area of the contour graph with n 0 ph /n ph > 0.9 is the lightest (where normalized density n 0 ph /n ph is the highest).For instance, if ∆/T * c ≈ 20 the latter inequality is valid for T /T * c 0.9, and if ∆/T * c ≈ 10 the nondimentional temperature needs to be decreased to T /T * c 0.85 to satisfy that inequality.
One may notice from approximated formulas (40), (41) that illustrated avalanche character of condensation does not occur for all parameters ∆ that satisfy ∆ values the condensation law becomes "standard" as for ideal gas when photon BEC density obtains power-law behaved dependency on a temperature The power-law mode of condensation is mapped with dashed lines in Fig. 2; it is evident that in upper area of graph (when ∆/T * c > 30) the "avalanche" type of condensation switches to rather different one -powerbehaved type of condensation.Obtained expression (40), (43) allows us to reach some other intriguing conclusions.It is easy to see that when temperatures are subjected to the condition: almost all photons are in BEC.Besides, almost all atoms remain unexcited with the set of quantum numbers α 1 and form BEC. Such situation could be interpreted as stopped light in BEC (let us remind that that appearance of such phenomenon was first shown in [9][10][11]).Indeed, this particular case does not only concern the light itself but also electromagnetic waves in general.Let's clarify the statement.It is known that group velocity v g of electromagnetic waves propagation through some matter can be defined as: When using "relativistic" dispersion law ( 6) for small wave vectors (see ( 7)), one can get the following: where v -is phase velocity of light in the matter.The expression similar to (45) that was also supposed to apear in this paper will be also valid in case of quadratic photons dispersion law.Zero momentum photons form BEC. It means that their wave vector is zero and correspondingly to (45) their group velocity is zero.This is the situation that was foreseen when we stated a possibility of stopped light phenomena in studied system.Thus, according to formulas (40)-( 45), photons could be captured in BEC of atomic gas with their possible following transition into coherent state, because each photon in the condensate has ω (k)| k=0 = ω 0

IV. ON POSSIBILITY OF BEC COEXISTENCE IN PHOTON COMPONENT AND EXCITED ATOMS SUBSYSTEM.
Let us now study the case when BEC is formed not only by photons but also by excited atoms; for simplicity, ground state atoms are assumed to be nondegenerate (it means they are far away from possibility of BEC forming).Consequently, when densities of photons n ph (p) and atoms n α2 (p) are below the transition temperatures T c and T * c of excited atoms and photons correspondingly the distribution functions densities can take the form (analogous to (20)): Here n 0 α2 (T ) is excited atoms BEC density and n 0 ph (T ) is photonic BEC density.The condition for excited atomic gas to be considered as nondegenerated one is formulated in analogous to (14) way and is it is given by the following: Equations ( 46) were obtained in correspondence with the expressions below that are required for BEC of the system components appearance (see in this regard [24]] and ( 16)).Formulas ( 46), (47) allow to rewrite (12) as follows: where ∆ is still defined by (19).We emphasize that in the case currently being studied this value must be less than zero ∆ < 0 to satisfy the condition (47) that makes possible to regard nonexcited atomic gas as nondegenerated one.Equations (49) give us condensate densities n 0 α2 (T ) and n 0 ph (T ) at T ≤ T c and at T ≤ T * c correspondingly: It can also provide us with transition temperatures T c and T * c , if we take into account that the densities of condensate become zero in transition points.
Let us remark here that because of photon condensate density should have positive value (including the case when T → 0) the second equation in (50) implies the following inequality Later on in this article we will get back to question of circumstances when n 0 ph (T ) has positive value in all allowed temperatures range.
For definiteness it is assumed that T * c < T c (as we've done it in the previous section).The expressions to define critical temperatures look as follows: When temperatures are supposed to be low one may obtain the result from the first equation in ( 53) It is easy to see that if we replace coefficient g α1 with g α2 expression (54) coincides with the similar one for transition temperature (24) in the main approximation of the first order of smallness.It is clear that in the case being studied the approximation tool adaptability criteria is defined by relation (26) where one needs to replace g α1 with g α2 , and ∆ value needs to be replaced with its absolute value |∆|.Let's analyze the second equation in (53); when using (54) it can be rearranged into the form: For simplicity in further calculations we'll assume that T * c < T c and the temperatures T c and T * c are of the same order of magnitude , from which we get the following formula: that evidently implies that inequalities are valid: The first one in (57) is trivial; the second one results from inequality exp − |∆| Tc ≪ 1 when considering Tc ∼ 1.In addition, the expression n ph ∼ n to regard atoms as two level ones needs to be valid.As it can be seen from ( 57), (52) the value of n ph −n n has upper limit: Such limitation means that the value of is negligibly small.Actually, as we mentioned above m * /m ratio is very small (see formulas (28)-( 32)); for lithium atom this value is (m * /m) ∼ 10 −10 , and consequently for this case we have 56) allows us to find the analytical expression for the transition temperature T * c that has to be of the same order of magnitude as T c .
Let us get back to the question about positivity of photon condensate density (see (50)) at any temperature below the condensation one.It is easy to analyze that condition (52) does not provide this value positivity at any temperature, because n 0 ph (T ) is nonmonotonic function and, consequently, not all values of |∆| T * c ≫ 1 assure the photonic condensate density to be positive.The evidence of that can be seen in Fig. 3: here n 0 ph (T ) dependences were plotted according to expressions (50), (53) for some values of |∆| T * c when it was still supposed that |∆| T * c ≫ 1 (cases when density n 0 ph (T is negative in some temperature range were drawn with dashed lines).From Fig. 3 one may see that for some |∆| T * c and became positively defined when |∆| T * c 36.According to the "hint" given in Fig. 3 we can find more precise limitation for studied system parameters which provides positivity of n 0 ph (T ) along the whole range of allowed temperatures.To do so, let's suppose that when T = T * c the derivative As we can see from Fig. 3 such a claim certainly provides us with n 0 ph (T ) value monotonic increase when temperature is lowering within 0 < T ≤ T * c range.By calculating This is the inequality (61) that defines (in addition (57), (58)) another one coexistence condition of photonic BEC and BEC of excited atoms when atoms in the ground state are nondegenerated and all components are in thermodynamic equilibrium.Besides this relation (61) allows us to find approximated formula of photonic condensate density in studied system: The density of excited atoms BEC accordingly to (50), (54) looks as follows: and density of atoms n α1 (T ) in the ground state will be exponentially decreased by lowering the temperature and density takes the form (see the similar approach in (49)): almost all photons in the system are basically absorbed by atoms in the ground state within the possibility of their transfer to excited state; almost all excited atoms form atomic BEC in the system because of this the number of atoms in the ground state becomes exponentially small depending on the temperature.In other words, when the temperature of the studied system is lowering the population inversion of atomic levels takes place with formation of BEC of photons and excited atoms.Moreover, such pumping occurs in "avalanche" -type mode of condensation according to (64) and taking into account expression (65).Note, that such "avalanche" pumping looks unexpected enough: the number of excited atoms increases when temperature decreases.This situation can be also treated as "stopped light" one (actually electromagnetic waves are meant here), because group velocity of photons in BEC equals to zero (see ( 45)).Nevertheless, in this current section such a statement is less significant than it was within the previous section: now we can have a few photons in the system and consequently, a few one can be seen in the condensate.The described situation is likely to be considered as the one having relevance to the storage of light in atomic vapor at ultralow temperatures [10,11].
As the next step to complete the overall picture of our research well study the possibility of three BEC simultaneous coexistence at the same temperature of system components (photonic and two atomic components).

V. ON POSSIBILITY OF BEC CO-EXISTENCE IN ALL SYSTEM COMPONENTS
The equations (( 12), ( 13) can be considered as the initial ones to study this case.To obtain Bose Einstein condensates in three subsystems the following is needed (see.[24] and ( 16), (48)): Equations (66) change the third one in (9) as: where ∆ is still defined by (19).As it follows from two previous sections, if (67) is valid then two atomic components can form BEC simultaneously at the same temperature T c .We showed that if the inequality ∆ > 0 fulfills it makes possible to achieve BEC in photonic and ground state atomic subsystem, whilst the validity of ∆ < 0 provides coexistence of photons and excited atoms in BEC.
From this particular fact we come to conclusion that BEC formation of two atomic gas components is only possible at the same temperature for both.
When temperature is below T c and T * c the latter statement gives us the following distribution functions densities of atoms n α1 (p), n α2 (p) and photons n ph (p) (see (10), ( 11)); these functions can take the form (see also (20), (49)): where n 0 α1 (T ), n 0 α2 (T ), n 0 ph (T ) are condensates densities of atomic components and free photons correspondingly.By inserting (68) into (13) one may obtain the next system of equations at T < T c , T < T * c temperatures: This system of equations is not complete: it includes two equations for the three unknown (n 0 α1 (T ), n 0 α2 (T ) and n 0 ph (T )).We can easy add the third equation to the system (69) if we notice that the first equation in (13) and the definitions (68) when µ 1 | T ≤Tc = ε 1 and µ 2 | T ≤Tc = ε 2 give us the following: Performing easy calculations using (68) the latter equation can be rearranged into the form This expression can be regarded as the one we lacked to make (69) a closed system of equations.The solutions of (69), (70) are given by: One may obtain the transition temperatures of system components from previous formulas by taking into account that the Bose condensates disappear in the transition point: The condition ∆ = 0 (see (67)) leads to the similar temperature T c of transition to BEC states for both atomic components as and it is easy to see that this condition follows from last two equations in (48).In the studied system photon condensation temperature can be calculated using (72) and taking into account (71): where n ef f ph is some effective density of photons and it is defined by formula Since photon condensation density has to be positive at any temperature including T → 0 from (71) we get density limit of photons and atoms in the system to make the coexistence of BEC in all three system components possible: Besides, if condensastion temperatures are supposed to satisfy inequality T * c < T c but to have the same order of magnitude T * c ∼ T c one can get more precise limitation of n ef f ph rather than (76) (see (75)).From (73), (64) we can obtain As we mentioned several times, the masses ratio m * /m is extremely small (for lithium (m * /m) ∼ 10 −10 , see above) and, consequently, m/m * is rather high; for that reason to satisfy T * c ∼ T c one needs the densities ratio n ef f ph /n in (77) to be small (like in (36), (37)): In terms of the critical temperatures T c , T * c (see ( 73), (74)) expressions (71) for condensate densities of all three system components may be rewritten in more common way (see for example [24] ) We can get from (79), (78) that photons condensate density in the system under the conditions being studied is negligibly small compared with atomic condensate density.This is the reason why in this section the studied case is not of great interest in the context of atomic and photonic BEC coexistence.

VI. CONCLUSION
By this means, weve studied all three possible variants when photonic Bose condensate coexists in thermodynamic equilibrium with Bose condensate of ideal twolevel atomic gas.That is to say, it was supposed that photonic BEC can always be created in the system, although photons critical temperature was considered to be below the atomic one.In context of atomic Bose-Einstein condensation three cases were assumed to be possible for implementation: 1) BEC can be formed by ground state atoms with nondegenerate gas component of excited atoms.
2) BEC can be formed by excited atoms with nondegenerate gas component of ground state atoms.
For all these situations weve found critical temperatures, condensates densities for atomic and photon components and conditions of their coexistence.The first case (when condensates of ground state atoms and photons coexist) was shown to be the most effective one from experimental implementation point of view, because avalanche photon condensation when temperature is lowering was predicted for this case.This situation in authors opinion is the closest to the one which can be treated as stopped light in BEC.Also, it was shown that other cases may as well concern the storage of light in atomic vapors at ultralow temperatures.Some features of our system model need to be improved.For example, in this article weve used the approach common enough in theoretical physics, optics and photonics -atoms of the gas were treated as are two-level ones.Besides, from the standpoint of current article such an assumption is not crucial: it only simplifies the calculations greatly and even allows to get some analytical results of calculations.Equations of such type as ( 9) or ( 12) may be formulated for arbitrary large number of components, but in this case a complicated problem appears -to trace such numerical calculations for these equations.
There are some atoms interactions and some scattering processes that can influence on studied condensates coexistence conditions.However, it is rather challenging task to take such interactions into account, which is in our opinion beyond this paper framework.Moreover, such task must be based on microscopic approach that may be provided by quantum electrodynamics (at low temperatures).In our opinion, some perspective approach was developed and presented in [25]; we are currently studying such issue.
It should be noted that one may also set a problem on forming photonic BEC in thermodynamic equilibrium with ideal Fermi gas.Let us recall that equations (9) were written for both Fermi and Bose gases being in thermodynamic equilibrium with photons.These are the initial equations that were provided by formulas (9) in the case of photon Bose condensation in thermodynamic equilibrium with degenerated Fermi gas.Preliminary estimates show that in the latter case photon condensate densities are negligibly small compared to the atomic ones; also, photon transition temperatures may be significantly lower than common temperatures of Fermi gas degeneration.However, such estimates were made by authors in the area of physical parameters that were suitable for analytical calculations.This fact makes such a study a separate task, which claims more detailed research using special numerical methods.

FIG. 1 .
FIG. 1.The transition temperature dependence on and overall photonic density.

FIG. 3 .
FIG. 3.Illustration of nonmonotonic dependence of condensate density n 0 ph (T ) on the temperature for different ∆ values.

)
Formulas (62)-(64) generally solve the problem announced in this section as for coexistence of BEC of excited atoms and photonic BEC when ground state atoms are nondegenerated.However, we have to mention that due to limitation (58) photons BEC density in this case is negligibly small compared with atoms in the excited state BEC density.For this reason considering the problem stated in this article (BEC coexistence conditions study) the case appears to be of little interest for us.It can have some interest for the other reason: as it is easy to see from (62)-(64) within temperatures range