University of Ljubljana Faculty of Mathematics and Physics. Photons in a double quantum resonator

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1 University of Ljubljana Faculty of Mathematics and Physics Photons in a double quantum resonator Author : Jernej Györköš Mentor: Norma Mankoč Borštnik Date : Abstract The interaction of two-level atoms with two-mode electromagnetic field inside a resonator, formed by two weakly coupled cavities, is represented. The first ecited atom passes through the first cavity and ecites the two field modes. The second unecited atom then passes through the other cavity and tests the ecitation. Since the photon generated by the first atom may find itself in the first or the second cavity, quantum resonance phenomena are observed. 1

2 Contents 1 Introduction 3 2 Brief history of quantum resonators 3 3 Basic realization of interaction of atom with field 4 4 Theoretical background Electromagnetic field inside resonator Electromagnetic field inside the cubic resonator with dielectric barrier Interaction of atom inside single cavity resonator 11 6 Interaction of two-level atom with double cavity resonator 12 7 Summary 16 2

3 1 Introduction A quantum theory predicts a number of interesting phenomena, which indicate a crucial difference between classical and quantum systems. Our area of interest in this seminar is quantum electrodynamics in a cavity. In particular the interaction of an atom with a photon inside a weakly coupled double resonator. This resonator was proposed by the authors and was also the PhD. work of M. Škarja from the University of Ljubljana [1]. The resonator is formed by two weakly coupled cavities. It has to have a high quality factor ( defined as number of oscillations during a lifetime ) in order to be able to observe quantum effects. The field inside a resonator is maintained by passing an ecited atom through the first cavity. The temperature must be lower then a few Kelvins, otherwise the thermal photons prevail over the photon created by interaction of the atoms with the field and the phenomena vanish. The field inside the resonator is controlled by a second atom passing in the ground state through the second cavity. The second atom interacts with the field and if the atom is tested outside the resonator, the information about the state of the field inside the resonator can be deduced. In such a way the life of a single photon within the resonator can be followed. 2 Brief history of quantum resonators First resonators based on quantum mechanical principals were called masers. A maser is a device that produces coherent electromagnetic waves through amplification of stimulated emissions. When optical coherent oscillators were developed, they were called lasers. The principal of the maser was described by N. Basov and A. Prokhorov from Lebedev Institute of Physics in 1952 and their results were published in Independently, C.H. Townes, J.P. Gordon and H.J. Zeiger built the first maser at Columbia University in The device used a stimulated emission of ammonia molecules to produce amplification of microwaves at a frequency of 24 GHz. Townes later worked with A.L. Schawlow to describe the principle of the optical maser or laser, which T.H. Maiman first demonstrated in For their research in this field Townes, Basov and Prokhorov were awarded the Nobel Prize in Physics in Masers serve as high precision frequency references. These atomic frequency standards are one form of atomic clock. They are also used as electronic amplifiers in radio telescopes. 3

4 3 Basic realization of interaction of atom with field Eperiments, where this kind of phenomena can be observed, are usually performed in resonators with super conductive walls and they must be cooled down to several Kelvins. The quality factor Q of such resonators is around Q 10 11, frequencies ν are up to few 10 GHz which means that a lifetime of a field is around τ lifetime = Q ν 1s. Figure 1: Basic eperimental setting [3] In figure 1 a typical setting of equipment is presented. At the bottom of the picture an atom beam source is placed. Then atoms travel through a collimator. Before entering into the resonator the atom flies through a laser which has been previously frequency stabilized. The laser ecites the atom into desired energy level. The atom usually interacts only with one state of a resonator s field. After the atom eits the cavity it is tested by two different detectors which work like ionization cells. They test the atom to find out whether it is in the ground state or still in the ecited state. Actually the atom does not find itself in the ground state but in a lower ecited state so the following statement is being satisfied : n = n e n g << n e, n g. 4

5 The n g is found to be around 50 to 60, so it is better to label it as the upper and the lower state. Velocities of atoms, that are coming out of source, are distributed by the Mawell s distribution : dp dv e ma v 2 2 k B T. The speed determines the interaction time with the resonator s field. When a fied velocity is required, it is achieved by a laser which points under a suitable angle on a beam of atoms. Under the Doppler s effect only atoms with the desired velocity are ecited in an upper state with around 2% accuracy [4]. The ideal chemical elements appear to be atoms of the alkali metals. They are ecited to energy levels with a high quantum number n ( between 50 and 60 ). Atoms with so high quantum number n also have large electrical matri elements, which could be classically interpretable as a large radius and a large dipole moment of the atom. If an approimation of the hydrogen atom is made, the following epression can be written for an average radius: r = r b 2 (3n2 l(l + 1)), where r b is the Bohr s radius and l is the quantum number representing an angular moment. If values n = 50 and l = 0 are put into the upper epression, an average radius is around 100nm. If n n << n is to be valid, a matri element is represented by epression : nl e 0ˆr n l = 3 2 e 0 r b n 2 f(n n ), where f(n n ) represents a value between 0 and 1. The energy of ecited states is proportional to n 2 and energy difference between two ecited states is proportional to n 3, if the condition n n << n is satisfied for large n. For n = 51 and n = 50 the energy difference is E 51, ev. The corespondent frequency is 50 GHz, which is in the region of the centimeter wave length. 4 Theoretical background 4.1 Electromagnetic field inside resonator This is not strictly the subject of the this seminar but it will help us to understand the basic theoretical background for a weakly coupled resonator. But for now a basic cubic resonator with super-conductive walls will be introduced. 5

6 The electric field in a space with no free currents and charges satisfies the wave equation: 2 E ɛµ 2 E c 0 t 2 = 0 where ɛ and µ are the dielectric constant and the magnetic permeability of the space respectively, c 0 is the velocity of the light in a vacuum. The spacial part of the electric field applies the Helmholtz s equation: 2 E + k 2 E = 0 with k 2 = ɛµ ω2 c 2 0 The magnetic field is correlated to the electric field trough the Mawell s equation: E( r, t) = B( r, t) t If the resonator is empty, both field constants are set to 1 (ɛ = 1, µ = 1). From the boundary conditions on super-conductive walls : E (on walls) = 0, B (on walls) = 0 the epressions for k can be gained : k L = n π, k y L y = n y π, k z L z = n z π, n, n y, n z = 0, 1, 2, 3... where L, L y, L z are the dimensions of the resonator. And the Coulomb gauge E = 0. gives us the solution a electric field inside a cavity: ɛ k cos(k ) sin(k y y) sin(k z z) E k ( r) = E k ɛ ky sin(k ) cos(k y y) sin(k z z) ɛ kz sin(k ) sin(k y y) cos(k z z) = E k ɛ k u k ɛ ky u ky ɛ kz u kz (1) where ɛ = (ɛ k, ɛ k, ɛ k ) is so called a polarization vector of the electric field. The magnetic field is correlated to the electric field as shown above (through the Mawell s equation): B k ( r) = B k ɛ k sin(k ) cos(k y y) cos(k z z) ɛ ky cos(k ) sin(k y y) cos(k z z) ɛ kz cos(k ) cos(k y y) sin(k z z) = B k ɛ k v k ɛ ky v ky ɛ kz v kz, B k = k ω k E k (2) k is known as a wave vector and k = k 2 + k 2 y + k 2 z is a size of the wave vector. For convenience, the functions u k ( r) and v k ( r) are introduced into 6

7 epressions for electric and magnetic field and are known as the eigenfunctions of electric and magnetic field, respectively. They are perpendicular to each other for every k, but they are not normalized: uk ( r) u k ( r)d3 r vk ( r) v k ( r)d3 r } = Ṽkδ k,k where the Ṽk has been introduced as the volume of an oscillation mode of the field. In a cubic resonator all oscillation modes have same Ṽk : Ṽ k = L L y L z 8 Time dependence for each oscillation mode of the electric field is selected to be cos(ω k t) and for the magnetic field sin(ω k t). The total electrical and magnetic field inside the resonator are: ( ) ( E( r, t) = B( r, k E ) k u k ( r) cos(ω k t) t) k k E (3) k ω k v k ( r) sin(ω k t) E k = 1 E( r) u k ( r)d 3 r (4) Ṽ k Epression for electric polarization field comes from the Coulomb s gauge and shows perpendicularity between the polarization vector and the wave vector: ɛ k1 k = 0 Because for a single E, two k can be found, it means that the electric field is degenerated two times and one more polarization vector has to be defined. The second polarization vector is perpendicular on the first one and on the wave vector, so it can be written as: ɛ k2 = k k ɛ k1 The magnetic polarization is also twice degenerated and is perpendicular to the electric polarization vectors and, of course, on the wave vector: ɛ k1 = k k ɛ k1 = ɛ k2 ɛ k2 = k k ɛ k2 = ɛ k1 7

8 4.2 Electromagnetic field inside the cubic resonator with dielectric barrier A resonator with super conductive walls and barrier was invented by professor N. Mankoč Boštnik from University of Ljubljana. The calculations and simulations were made by M. Škarja as his PhD. work [1]. The barrier is put inside the resonator to divide it into two cavities. The low transmission of the barrier is selected to gain weakly coupled cavities. A very low transmission can be achieved by using a very high dielectric constant; or using a lower dielectric constant then it is the one of the cavities but the condition of the total reflection has to be met ( Figure 2 ). Figure 2: The resonator [2]. By solving the wave equation with the belonging boundary conditions the eigen frequencies and the oscillation modes are gained. The double degeneracy from the previous eample of the single resonator is lost and only one mode of oscillations is found (only the TE or the TM polarization). If the transmission of the barrier is low enough and the cavities are the same size, the oscillation modes with the close enough frequencies are coupled in to pairs as shown on Fig. 3. For a plain wave the frequency difference ( ω T E, ω T M ) is approimately proportional to an amplitude of the transmission (C T E = C T E (k, K (b), ɛ (b) ), C T M = C T M (k, K (b), ɛ (b) ): ω T E = ωc T E e d/δ ω T M = ωc T M ɛ (b) e d/δ (5) The shape of the eigenfunctions are, inside of the double cavity resonator, symmetrical or anti symmetrical regarding the middle of the barrier. If one mode from the pair is symmetrical then the other is anti symmetrical, as shown on figures 4 7. The resonator described above is used to find the quality properties of the system of the two weakly coupled cavities. The main advantage is that only numerical calculations that are used, are used to calculate the eigen frequencies: 8

9 For L 1 = L 2 and ɛ 1 = ɛ 2 and for the TE polarization: sin(k (1) sin(k (1) L 1 ) sin(k (b) d 2 ) k(1) k (b) L 1 ) cos(k (b) d 2 ) + k(1) And for the TM polarization: cos(k (1) cos(k (1) L 1 ) cos(k (b) d L 1 ) sin(k (b) d k (b) 2 ) ɛ bk (1) ɛ 1 k (b) 2 ) + ɛ bk (1) ɛ 1 k (b) cos(k (1) cos(k (1) sin(k (1) sin(k (1) L 1 ) cos(k (b) d 2 ) = 0 L 1 ) sin(k (b) d 2 ) = 0 L 1 ) sin(k (b) d 2 ) = 0 L 1 ) cos(k (b) d 2 ) = 0 The field inside the resonator is ecited by two-level atoms. A transformation of the field between the two cavities will be observed if the atoms ecite the two oscillation modes of the field. A high efficiency can be achieved by selecting the same class size of a frequency difference of the atom and the field, and the constant of the interaction between the atoms and the field (g). The constant of the interaction g in a microwave range is estimated to be s 1 so the frequency difference has to be in range of 10 5 s 1. If such a resonator is used, the dielectric constant of the barrier has to be around ɛ b or the ɛ b should be lower then the dielectric constant of the Figure 3: A first few eigen frequencies of the TE polarization ( n y = 1 and n z = 1 of the wave vector; the thickness of the barrier d is chosen to be for the lower pairs at ɛ = 100 to be k b d = 3π 2.) 9

10 Figure 4: TE polarization, ɛ 1,2 = 1 of the cavities, ɛ (b) = 1, an amplitude of the transmission E(2) 0 = 1. E (1) 0 Figure 5: TE polarization, ɛ 1,2 = 1 of the cavities, ɛ (b) = 0, 92, an amplitude of the transmission 0 = 0, 13. E(2) E (1) 0 Figure 6: TE polarization, ɛ 1,2 = 1 of the cavities, ɛ (b) = 0, 6, an amplitude of the transmission 0 = E(2) E (1) 0 Figure 7: TE polarization, ɛ 1,2 = 1 of the cavities, ɛ (b) = 0, 001, an amplitude of the transmission E(2) 0 = E (1) 0 10

11 cavities and the total reflection has to be met. Since a matter with so high ɛ could not be found (the highest dielectric constant have ferro-electrics with ɛ 10 5 ) it is clearly that the second choice is selected. The results of such a choice are the quality properties of a resonator with two cavities connected by a wave guide with the critical frequency higher then the frequency of the modes inside the cavities. 5 Interaction of atom inside single cavity resonator This chapter is about an interaction of the two-level atom and the single mode field inside a single cavity resonator. The dimensions of the resonator is chosen so that only one oscillation mode of the field interacts with the atomic transition. The frequency of the interacting atomic transition has to be close enough to the frequency of the chosen oscillation mode. The resonator is consider to be an ideal so the energy losses are negligible and the differences in an energy are considered to be only due to an interaction with the atom. In practice this can be achieved by using the super conductive resonators with the quality factor up to Q The life time of the field can last up to 2s and the time the atoms need to pass through the resonator is around 50µs. The decay of the atoms during the flight is neglected because of the highly ecited states of the atoms, that have a longer life time then low ecited ones. Since the main quantum number of the ecited atoms is around 50, the approimation is valid and for a low quantum number of an angular momentum the life time epends to 0, 5ms. But for the maimal size and projection of the angular momentum the life time increases to 0, 2s. So the degree of the atom s ecitation can be chosen regarding to the needs of the eperiment. The special phenomena of the interaction of the two-level atoms with single mode field are known as Rabbi oscillation. They appear, vanish and Figure 8: The Rabbi oscillation of the initial field state with 20 photons inside the cavity where the Poisson s statistics is taken. w b is the probability that the atom find itself in upper state. 11

12 reappear when the single atom interacts as shown in figure 8. But when the sequence of atoms interact with the field, where only a single atom can be in the cavity at the time, another phenomenon is observed: the trapping states. If the dumped oscillation of the field and a great number of passing atoms is taken into a consideration, the static field inside the resonator can be established. In the literature the resonator, where the static field is maintained with the sequence of atoms, is known as micromaser. 6 Interaction of two-level atom with double cavity resonator This section is the main theme of this seminar, where a basic principal of the interaction between a two-level atom and a two-mode field is represented. Parameters of the resonator are chosen so that the pair of frequencies between two eigen oscillation modes are close together. The same goes for the frequency difference of atom levels and the mode of the field; and frequency difference of levels in the atom itself. Consecutively the oscillation modes are strongly jointed with the chosen atomic translation and so this differences of frequencies can not overcome the constant of interaction between the atom and the field. If both of the field s oscillation modes are ecited, the field starts to pass over the barrier into the net cavity and returns back. The time period of such oscillation is 2π/ ω 1 ω 2, where ω 1 and ω 2 are frequencies of the field oscillation modes. How much the field is ecited, is determined by the oscillation modes amplitudes of each cavity and the amplitude of ecitation of these modes. Complete field is passed over only when cavities are the same size and amplitudes of the modes are equal. To ecite the two modes of such a resonator, we let a two-level atom with atomic frequency ω 0 close to the two frequencies of pair of modes ω 1 and ω 2 to pass trough one of the two cavities. Time of the interaction is labeled as τ 1. The second, testing atom enters the second cavity after t 1 time has passed. De-ecitation of the field lasts t 1 τ 1 which is equal to the period of field s oscillation mentioned earlier. The field can be calculated using annihilation operator a for each of the field mode, a 1 and a 2, respectively. Ground and the ecited state of the field are chosen to be g > and e >, respectively. The atom, when passing through one of the cavities (i.e α th cavity ) interacts with the resonator s field. Hamilton operator of a dipole using the rotating-wave approimation is: H = H a + H f + H int ; (6) 12

13 where H a, H f and H int represent the operator for a free atom, a free field and an interaction between an atom and a field, respectively. Hamilton operator for a free atom is: H a = j=g,e hω j j >< j ; (7) where j runs over the atomic states. An energy difference between both of states is hω 0 = hω e hω g. Hamilton operator for a free field is: H f = i=1,2 hω i a i a i ; (8) where inde i labels the two oscillation modes of the field. And the last of Hamiltonians in the epression (6), Hamiltonian of an interaction: H (α) int = ˆ d ˆ E (9) ˆ d = < g d e > ( g >< e + e >< g ) ˆ E = i H (α) int = i h 2 k=1 2 k=1 hωk ( ) u k ( r) â k â k 2ɛ 0 Ṽ k g (α) k ( r) ( g >< e a k a k e >< g ) (10) where g (α) k is a coupling constant of a k th mode in the α th cavity, u (α) k is a kth mode function in the α th cavity, d is a matri element for the atomic electric dipole transition, and Ṽk is the mode s volume. Since one of the oscillation modes is anti-symmetric across the barrier it changes sign in one of the cavities while the other is symmetrical and does not change the sign. As shown in figures 4-7 the two coupling constants have the same sign in one cavity and the opposite sign in other cavity. General solution of the Schrödinger equation H ψ(t) >= i h t ψ > can be written as: ψ(t) > = m,n(a m,n (t) e i(ωe+n ω 1+m ω 2 )t e > m, n > + + b m,n (t) e i(ωg+n ω 1+m ω 2 )t g > m, n >) (11) The relevant subspace of states we have to consider is e > 0, 0 >, g > 1, 0 > and g > 0, 1 > so an analytical solution can be found. To simplify situation, we consider the cavities to be identical with the atomfield interaction constants: g (1) 1 = g (1) 2 = g (2) 1 = g (2) 2 = g. We choose atomic transition frequency ω 0 to be in the middle of the frequencies of field modes ( ω 1, ω 2 ): ω 1 ω 0 = (ω 2 ω 0 ) = ω, as shown in figure 9. 13

14 The probability w g (1) (τ 1 ) for the first atom to be found in a lower state when eiting the first cavity at time t = τ 1 ( the time needed to pass the cavity 1 ) can be written: w (1) g (τ 1 ) = < ψ(τ 1 ) g > 1 2 = 1 1 g 4 ( ω 2 + 2g 2 cos( gτ 1 )) 2 (12) with the g = ω 2 + 2g 2,the initial conditions are a (1) 0,0 (0) = 1, b(1) 1,0 (0) = 0 and b (1) 0,1 (0) = 0. Whole energy is transfered, from an ecited atom to a field, if the epression in brackets is equal to zero. That happens when 2g 2 / ω 2 1.The figure 10 shows eamples when 2g 2 / ω 2 1 and 2g 2 / ω 2 1. Figure 9: The atomic transition frequency is in the middle of the frequencies modes. Figure 10: The time dependence of the probability that the first atom leaves the first cavity in the lower state for ω = 3g/4 (the plot that shows a value maimums at 1) and ω = 3g (the plot that shows a value maimums at 0, 6). The period is equal to 2π/ g, which is different from the echange period of the field between two cavities and, as mentioned in previous sections, is equal to 2π/ ω 2 ω 1 when an atom is not present. Second atom enters the second cavity at time t 1 > τ 1 and interferes with the field. Wave function of the interaction of the two atoms and two field modes now has the form: ψ(t) > = a (1) 0,0 (τ 1) e > 1 g > 2 0, 0 > +b (2) 1,0 (t) g > 1 g > 2 1, 0 > + + b (2) 0,1 (t) g > 1 g > 2 0, 1 > +a (2) 0,0 (t) g > 1 e > 2 0, 0 > (13) where subscript and superscript refer to a first and second atom. The initial conditions are a (2) 0,0 (t 1) = 0, b (2) 1,0 (t 1) = b (1) 1,0 (τ 1), b (2) 0,1 (t 1) = b (1) 0,1 (τ 1) The probability that the second atom leaves the cavity in ecited state is represented by w (2) e (τ 1, τ 2, t 1 ) = < ψ(t 1 + τ 2 ) e > 2 2 (14) 14

15 where τ 2 represents the time needed to pass through a cavity and interacts with a field. The result is represented in figure 11 and it is not depended on whether the first atom is detected before, during or after the interaction of the second atom with the cavity field. A plot, shown in figure 11, also Figure 11: The probability that the second atom leaves the second cavity in eited state w (2) 2. The plot is generated ω = g. Figure 12: The time τ 2 is chosen to be such that maimum of the entangled states is generated. Other parameters are ω = g and t 1 = π 2g reveals the periodic behavior with periods 2π/ ω 2 ω 1 on (t 1 τ 1 )g ais and 2π/ g on τ 2 g ais. The oscillation in t 1 dependence of the equation (14) is a quantum interference effect and is a consequence of indistinguishable paths that lead from initial state to the final state of a system. The final state of the system with the second atom in the ecited state, g > 1 e > 2 0, 0 >, can be achieved via two intermediate states, g > 1 g > 2 1, 0 > and g > 1 g > 2 0, 1 >, from the initial state e > 1 g > 2 0, 0 >. The quantum interference can be shown to be a function of delay time between the first and the second atom. If the information of which mode was used to transfer a system from initial to the final state is gained by testing, the interference effects would no longer be observed. Quantum correlations can be regulated by adjusting the interaction times τ 1 and τ 2, time between the atoms t 1 and the mode splitting ω. The probability that the first atom ends up the state k > and the second atom leaves the cavity in l > state, is represented by P k,l (τ 1, τ 2, t 1 ): P e,g (τ 1, τ 2, t 1 ) = a (1) 0,0 (τ 1) 2 = 1 w (1) g (τ 1 ) P g,e (τ 1, τ 2, t 1 ) = a (2) 0,0 (τ 2) 2 = w (2) e (τ 2 ) P g,g (τ 1, τ 2, t 1 ) = b (2) 1,0 (τ 2) 2 + b (2) 0,1 (τ 2) 2 = w g (1) (τ 1 ) w e (2) (τ 2 ) P e,e (τ 1, τ 2, t 1 ) = 0 (15) 15

16 Because of the non classical nature of an atom - atom correlation is present, maimum entangled states can be generated. That can be done by insuring the equal probability that the first atom leaves the cavity in the upper or the lower state, i.e w (1) g (τ 1 ) = 1 2. τ 1 is adjusted to : τ 1 = 1 g ( g 2 arccos / 2 ω 2 ) 2g 2 (16) A second thing that has to be done, is to choose the parameters in a way that w e (1) (τ 1, τ 2, t 1 ) = 1 2 and there for b(2) 1,0 (τ 2) 2 = b (2) 0,1 (τ 2) 2 = 0. The wave function (13) looks like : ψ(t) > = 1 2 ( g > 1 e > 2 +e iφ e > 1 g > 2 ) (17) where the phase factor is e iφ = a(1) w (1) e (τ 1, τ 2, t 1 ). 7 Summary 0,0 (τ 1) a (2) 0,0 (τ 2). Figure 12 shows the result of the In this seminar the double-cavity resonator is represented. The two cavities are separated by a dielectric barrier where its constant is selected to be low relatively to a cavity constant (ɛ barrier < ɛ cavity ). A low amplitude of the transmission( 10 6 ) is needed to observe passages of the electromagnetic field from one cavity to another. The eigenfrequencies couple into pairs with small frequency differences ( ω 1 ω s 1 ) of oscillation modes. The field is passing through a barrier if at least one pair is ecited. That can be done if we let an ecited atom pass one of the cavities. Its transition frequency has to be close to the frequencies of oscillation modes. If a photon passes a barrier, it is tested by a second unecited atom, which passes through the second cavity. That kind of resonator is suitable for studying the interaction of a field with atoms. The probability that the second atom is ecited, is found out to be periodic function of the entering time between the two atoms. Because a symmetrical resonator is chosen, the first atom can transfer its ecited energy completely to the field, and the second atom can then, regarding to the entering time, collect the energy from a passing field or can stay unecited with a probability 1. The periodical dependence is a consequence of a different phase dependence of time of the modes which can be observed as an energy oscillation between the two cavities. In a case where the interacting constant is much 16

17 larger then the frequency difference between oscillation modes and the transition frequency, an atom echanges energy only with the cavity it is in, where the second cavity receives a small amount of atoms energy. A realization of the double resonator was conferred with the coauthors of the article [2] with professor H. Walther and phd. M. Löffer from the Institute of Ma Planck, from Germany, but I haven t found the results of the eperiments. So, with the double resonator we can count almost individual photon that comes in the second cavity and from information of the photon we can deduce through a quantum mechanical approach whether the source of photon, that is atom, was in an ecited state or not. That kind of advantage could be used in quantum computers (may be as a read and write unit). References [1] M.Škarja, PhD. Thesis, University of Ljubljana, 1999 [2] M.Škarja, N. Manloč Borštnik, M.Löffler and H. Walther, Quantum interference and atom-atom entanglement in a two-mode, two-cavity micromaser, Phys. Rev. A 60, 4 (1999) [3] D.Meschede, H. Walther, G.Müller, One-Atom maser, Phys. Rev. Lett. 54, 6 (1985) [4] G. Rampe, F.Schmidt-Karel, H. Walther, Sub-poissonian atomic statistics in a micromasar, Phsys. Rev. A 42, (1990) 17