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High-Temperature Bose-Einstein Condensation of Polaritons: Realization Under the Intracavity Laser Pumping of Matter Condition

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High-Temperature Bose-Einstein Condensation of Polaritons: Realization Under the Intracavity Laser Pumping of Matter Condition View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server High-temperature Bose-Einstein condensation of polaritons: realization under the intracavity laser pumping of matter condition V.A. Averchenko1), A.P. Alodjants2*), S.M. Arakelian2), S.N. Bagayev3), E.A. Vinogradov4), V.S. Egorov1), A.I. Stolyarov1), I.A. Chekhonin1) 1) St. Petersburg State University, Ul'yanovskaya ul. 1, 198504 St. Petersburg, Staryi Peterhof, Russia 2) Vladimir State University, ul. Gor'kogo 87, 600000 Vladimir, Russia 3) Insitute of Laser Physics, Russian Academy of Sciences, prosp. akad. Lavrent'eva 13/3, 630090 Novosibirsk, Russia 4) Institute of Spectroscopy, Russian Academy of Sciences, 142190 Troitsk, Moscow region, Russia Abstract. A quantum model of Bose-Einstein condensation based on processes involving polaritons excited in an intracavity absorbing cell with resonance atoms, which is manifested in the spectral characteristics of the system, is considered. It is shown that the spectral 'condensation' appears which is directly related to the degeneracy of a weakly interacting gas of polaritons resulting in quasi-condensation at room temperature. The possibility of obtaining polariton condensation as a new phase state by using the confinement of polaritons in an atomic optical harmonic trap is discussed. Keywords: polaritons, quasi-condensation, Bose-Einstein condensation, polariton laser. 1. Introduction Experiments on the Bose-Einstein condensation (BEC) of macroscopic numbers of atoms (N ≥ 106 ) is one of the most spectacular recent advances, which have made a great influence on the development of various directions in modern quantum and laser physics and newest technologies (see, for example, [1]). In the case of BEC, when under conditions of the temperature phase transition a macroscopic number of atoms are in the ground (lower) quantum level, a new coherent state of matter is formed. This is manifested in the fact that, for example, at the limiting temperature T=0 an ensemble of condensate atoms, as each individual atom, is described by the common wave function corresponding to a coherent state. In this aspect, the BEC phenomenon is similar to lasing, for example, when strict phase locking of laser modes occurs in laser cavities [2, 3]. In addition, in the case of BEC, we can say about the realisation of a Bose laser (boser) emitting coherent ensembles of atoms [3, 4]. A remarkable feature of such macroscopic quantum states of matter is the possibility to use them for the development of new physical principles of quantum information processing and communication [5, 6]. However, despite spectacular achievements in this direction, there exist a number of practical difficulties imposing principal restrictions on the possibility of real applications of the atomic BEC for these purposes. Thus, one of the basic problems is the necessity of maintaining extremely low temperatures (tens of nK) to realise such devices. In this connection the problem of obtaining macroscopic coherent (quantum) states of matter at high (room) temperatures becomes very important. One of the most attractive approaches to the solution of this problem is the preparation of a quasi- condensate of the two-dimensional Bose gas of weakly interacting polaritons (in atomic physics [7]) and _______________________________________________________ *) Email: [email protected] excitons (in solid state physics [8-10] *)). Such collective states of the medium (quasi-particles) represent a superposition of photons and spin waves in the atomic medium and can be obtained, for example, within the framework of the Dicke model used to describe superradiance [12]. Although these states cannot be treated as a condensate in a strict thermodynamic sense due to the nonequilibrium state of the system as a whole, under certain conditions imposed on the type of atomic optical interactions in the system, polaritons do form a condensate, their distribution being described by the Bose-Einstein distribution function for an ideal gas of bosons [13]. In this paper, we considered the interaction of a system of two-level atoms with an electromagnetic field in the cavity in the case of the so-called strong coupling, when the inequality 2 1/2 ⎛⎞2πωdn0 1 ωc = ⎜⎟ , (1) ⎝⎠ 2τ coh is fulfilled, where ωc is the cooperative frequency determining the collective interaction of atoms with the field; ω0 is the atomic transition frequency; d is the transition dipole moment; τ coh is the characteristic coherence time of the atomic medium; n is the atomic gas density; and is Planck's constant. In this case, the field itself is weak (in the number of photons). The so-called condensation of the spectrum occurs when inequality (1) is fulfilled [14, 15]. This effect consists in the fact that under some threshold conditions imposed on the concentration of absorbing atoms and pump intensity, radiation of a broadband laser with a narrowband absorbing intracavity cell is concentrated ('condensed') near the strongest absorption lines of matter. This phenomenon was observed experimentally and interpreted within the framework of a clear classical model of parametric excitation of two coupled oscillators (electromagnetic field and atoms of matter) upon coherent energy transfer between them. However, this model is one-dimensional and is not quantum one, which obviously restricts the field of its applications. In this paper, we propose a detailed quantum model of spectral condensation realised for polaritons excited in an intracavity absorbing cell [16]. We show that spectral condensation can be directly related to the condensation (quasi-condensation) of polaritons in the cavity if a strong coupling between the electromagnetic field and medium is provided. The latter statement is in itself of interest, and in this paper we substantiate for the first time the possibility of obtaining the true BEC in the polariton system at high (room) temperatures in the case of spectral condensation. In this respect, of interest are the experimental data [8, 9] obtained in semiconductor microcavities, which confirm the above assumption. 2. Basic relations Consider the interaction of two-level atoms (with levels a and b) with a quantum electromagnetic field, + which is described by the photon annihilation (creation) operators ffkk() for the k-th mode. Within the framework of dipole approximation, such a system can be described by the Hamiltonian [13] ++++++1 H ()kEkffEbbaagfabbaf=+−+−∑ phkk ()∑∑2 atkkkk ( ) ( kkkkkk ), (2) kk k where a (a+ ) and b ( b+ ) are the Bose operators of annihilation (creation) of atoms at the lower and upper levels, respectively; Ekph ()= ck is the dispersion relation for the photons in the resonator; EEat ≡ 0 is the transition energy in a two-level atom (here we neglect the motion of atoms in the resonator); g is the coefficient determining the strength of coupling between the field and an atom; and c is speed of light in vacuum. In the limiting case of small perturbations of the atomic system, atoms occupy mainly the lower energy level a, and the inequality _______________________________________________________ *) Here, we are dealing with the so-called Kosterlitz-Thouless phase transition to the superfluid state of two-dimensional Bose systems in which the true Bose-Einstein condensation (in the absence of confinement of gas particles in a trap) is impossible [11]. ++ bbkk aa kk (3) is fulfilled. In this approximation, Hamiltonian (2) can be diagonalised by using the unitary transformation + + Ф1,kkkkkk=−µνfab, Ф2,kkkkkk=+νµfab, (4) where the introduced annihilation operators Фjk, (j=1,2) characterise quasi-particles (polaritons) in the atomic medium, corresponding to two types of elementary perturbations, which in approximation (3) satisfy the boson commutation relations ⎡⎤+ ⎣⎦ФФik,,; jk= δ ij, ij,1,2= . (5) The transformation parameters µk and vk in expression (4) are real Hopfield coefficients satisfying the condition 22 µkk+=v 1 , which determine the contributions of the photon and atomic (excited) components to a polariton, respectively: 4g 2 δδ++(4)221/2g µ 2 = , ν 2 = kk , (6a,b) k 221/2221/2⎡⎤k 2(δ 221/2+ 4g ) 2(δδδkkk+++ 4gg )⎣⎦ ( 4 ) k where δkatph=−EEk()is the phase mismatch determining the contributions of the photon and atomic components to expression (4) for polaritons. In particular, in the limiting case, when δ k −2 g , we have 2 2 µk →1 ( vk → 0 ), which corresponds to the negligible contribution of the photon part to the polariton Ф2,k . In 2 2 opposite limit, when 2 g δ k , we have µk → 0 ( vk →1), which means that the photon contribution to the coherence of polaritons of this type increases. Expression (6) shows that the polariton is a half-matter and half- 22 photon ( µν==1/2 ) quasi-particle under the resonance condition δ k = 0 . Taking expressions (4) and (6) into account, Hamiltonian (2) takes the form ++ Hk()=+∑ E1 () kФФ 1,kk 1,∑ Ek 2 ()ФФ 2, k 2, k, (7) kk where Ek1,2 () determine the dispersion dependence of polaritons: 1 ⎧⎫2 1/2 ⎡⎤2 Ek1,2 ()=+±−⎨⎬ Eat Ek ph ()⎣⎦ E at Ek ph () + 4 g . (8) 2 ⎩⎭{ } Figure la presents dispersion dependences Ek1,2 () (8) of polaritons for the interaction of atoms with the quantum field in free space. One can see that the two allowed energy states, polaritons of the upper [()]Ek1 and lower [()]Ek2 branches, correspond to each value of the wave vector k . When the medium is placed into the resonator, the wave-vector component k⊥ orthogonal to the mirror surface is quantised. At the same time, a continuum of modes exists in the direction parallel to the mirror surface due to the absence of boundary conditions. This means that in the single-mode (single-frequency for each value of k⊥ ) regime, the dispersion of polaritons is determined only by the wave-vector component k parallel to the mirror surface. Then, under the condition kk ⊥ which corresponds physically to the paraxial approximation in optics (see, for example, [17]), the dispersion relation for photons in the resonator has the form ⎡ kk23⎛⎞⎤ Ek()=+= ckk (221/2 ) ck⎢ ++ O⎜⎟⎥ .
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