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On Bose-Einstein condensation and superп¬‚uidity of trapped photons with

coordinate-dependent mass and interactions

Oleg L. Berman1,2, Roman Ya. Kezerashvili1,2, and Yurii E. Lozovik3,4

1Physics Department, New York City College of Technology, The City University of New York,

2The Graduate School and University Center, The City University of New York,

Brooklyn, NY 11201, USA

3Institute of Spectroscopy, Russian Academy of Sciences, 142190 Troitsk, Moscow, Russia

4National Research University Higher School of Economics, Moscow, Russia

New York, NY 10016, USA

(Dated: June 29, 2017)

The condensate density proп¬Ѓle of trapped two-dimensional gas of photons in an optical micro-

cavity, п¬Ѓlled by a dye solution, is analyzed taking into account a coordinate-dependent eп¬Ђective

mass of cavity photons and photon-photon coupling parameter. The proп¬Ѓles for the densities of the

superп¬‚uid and normal phases of trapped photons in the diп¬Ђerent regions of the system at the п¬Ѓxed

temperature are analyzed. The radial dependencies of local mean-п¬Ѓeld phase transition temperature

T 0

c ^{®} and local Kosterlitz-Thouless transition temperature Tc^{®} for trapped microcavity photons

are obtained. The coordinate dependence of cavity photon eп¬Ђective mass and photon-photon cou-

pling parameter is important for the mirrors of smaller radius with the high trapping frequency,

which provides BEC and superп¬‚uidity for smaller critical number of photons at the same temper-

ature. We discuss a possibility of an experimental study of the density proп¬Ѓles for the normal and

superп¬‚uid components in the system under consideration.

Key words: Photons in a microcavity; Bose-Einstein condensation of photons; superп¬‚uidity of

photons.

PACS numbers: 03.75.Hh, 42.55.Mv, 67.85.Bc, 67.85.Hj

I.

INTRODUCTION

When a system of bosons is cooled to low temperatures, a substantial fraction of the particles spontaneously occupy

the single lowest energy quantum state. This phenomenon is known as Bose-Einstein condensation (BEC) and its

occurs in many-particle systems of bosons with masses m and temperature T when the de Broglie wavelength of the

Bose particle exceeds the mean interparticle distance [1]. The most remarkable consequence of BEC is that there

should be a temperature below which a п¬Ѓnite fraction of all the bosons вЂњcondenseвЂќ into the same one-particle state

with macroscopic properties described by a single condensate wavefunction, promoting quantum physics to classical

time- and length scales.

Most recently, the observations at room temperature of the BEC of two-dimensional photon gas conп¬Ѓned in an optical

microcavity, formed by spherical mirrors and п¬Ѓlled by a dye solution, were reported [2вЂ»5]. The interaction between

microcavity photons is achieved through the interaction of the photons with the non-linear media of a microcavity,

п¬Ѓlled by a dye solution. While the main contribution to the interaction in the experiment, reported in Ref. 2, is

thermooptic, it is not a contact interaction.

It is known that BEC for bosons can exist without particle-particle

interactions [6] (see Ref. 1 for the details), but at least the interactions with the surrounding media are necessary to

achieve thermodynamical equilibrium. For photon BEC it can be achieved by interaction with incoherent phonons [7].

The inп¬‚uence of interactions on condensate-number п¬‚uctuations in a BEC of microcavity photons was studied in Ref. 8.

The kinetics of photon thermalization and condensation was analyzed in Refs. 9вЂ»11. The kinetics of trapped photon

gas in a microcavity, п¬Ѓlled by a dye solution, was studied, and, a crossover between driven-dissipative system laser

dynamics and a thermalized Bose-Einstein condensation of photons was observed [12].

In previous theoretical studies the equation of motion for a BEC of photons conп¬Ѓned by the axially symmetrical

trap in a microcavity was obtained.

It was assumed that the changes of the cavity width are much smaller than

the width of the trap [13]. This assumption results in the coordinate-independent eп¬Ђective photon mass mph and

photon-photon coupling parameter g. In this Paper, we study the local superп¬‚uid and normal density proп¬Ѓles for

trapped two-dimensional gas of photons with the coordinate-dependent eп¬Ђective mass and photon-photon coupling

parameter in a an optical microcavity, п¬Ѓlled by a dye solution. We propose the approach to study the local BEC

and local superп¬‚uidity of cavity photon gas in the framework of local density approximation (LDA) in the traps of

larger size without the assumption, that total changes of the cavity width are much smaller than the size of the trap.

In this case, we study the eп¬Ђects of coordinate-dependent eп¬Ђective mass and photon-photon coupling parameter on

the superп¬‚uid and normal density proп¬Ѓles as well as the proп¬Ѓles of the local temperature of the phase transition for

trapped cavity photons. Such approach is useful for the mirrors of smaller radius with the high trapping frequency,

2

which provide BEC and superп¬‚uidity for smaller critical number of photons at the same temperature.

The paper is organized in the following way.

In Sec. II, we obtain the condensate density proп¬Ѓle for trapped

microcavity photon BEC with locally variable mass and interactions. The expression for the number of particles in a

condensate is analyzed in Sec. III. In Sec. IV, the dependence of the condensate parameters on the geometry of the

trap is discussed. In Sec. V, we study the collective excitation spectrum and superп¬‚uidity of 2D weakly-interacting

Bose gas of cavity photons. The results of our calculations are discussed in Sec. VI. The proposed experiment for

measuring the distribution of the local density of a photon BEC is described in Sec. VII. The conclusions follow in

Sec. VIII.

II. THE CONDENSATE DENSITY PROFILE

While at п¬Ѓnite temperatures there is no true BEC in any inп¬Ѓnite untrapped two-dimensional (2D) system, a true

2D BEC quantum phase transition can be obtained in the presence of a conп¬Ѓning potential [14, 15]. In an inп¬Ѓnite

translationally invariant two-dimensional system, without a trap, superп¬‚uidity occurs via a Kosterlitzв€«Thouless

superп¬‚uid (KTS) phase transition [16]. While KTS phase transition occurs in systems, characterized by thermal

equilibrium, it survives in a dissipative highly nonequilibrium system driven into a steady state [17].

The trap for the cavity photons can be formed by the concave spherical mirrors of the microcavity, that provide

the axial symmetry for a trapped gas of photons. Thus the transverse (along xy plane of the cavity) conп¬Ѓnement

of photons can be achieved by using an optical microcavity with a variable width. Let us introduce the frame of

reference, where zв€«axis is directed along the axis of cavity mirrors, and (x, y) plane is perpendicular to this axis. The

energy spectrum E (k) for small wave vectors k of photons, conп¬Ѓned in z direction in an ideal microcavity with the

coordinate-dependent width L^{®}, is given by [2]

E (k) =

ВЇhПЂcЛњn

…

[23] L. Onsager, вЂњStatistical Hydrodynamics, вЂќ Nuovo Cimento Suppl. 6, 279 (1949).

[24] R.P. Feynman, вЂњApplication of Quantum Mechanics to Liquid Helium, вЂќ Prog. Low Temp. Phys. 1, 17 (1955).

[25] P.C. Hohenberg and P.C. Martin, вЂњMicroscopic Theory of Superп¬‚uid Helium, вЂќ Ann. Phys. 34, 291 (1965).

[26] G. Blatter, M.Y. FeigelвЂman, Y.B. Geshkenbein, A.I. Larkin, and V.M. Vinokur, вЂњVortices in high-temperature super-

conductors, вЂќ Rev. Mod. Phys. 66, 1125 (1994).

[27] N.S. Voronova and Yu. E. Lozovik, вЂњExcitons in cores of exciton-polariton vortices, вЂќ Phys. Rev. B 86, 195305 (2012);

N.S. Voronova, A.A. Elistratov, and Yu. E. Lozovik, вЂњDetuning-Controlled Internal Oscillations in an Exciton-Polariton

Condensate, вЂќ Phys. Rev. Lett. 115, 186402 (2015) .

[28] A. Griп¬ѓn, вЂњConserving and gapless approximations for an inhomogeneous Bose gas at п¬Ѓnite temperatures, вЂќ Phys. Rev. B

53, 9341 (1996).

[29] A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Prentice-

Hall, Englewood Cliп¬Ђs. N.J., 1963).

[30] O.L. Berman, Yu. E. Lozovik, and D.W. Snoke, вЂњTheory of Bose-Einstein condensation and superп¬‚uidity of two-

dimensional polaritons in an in-plane harmonic potential, вЂќ Phys. Rev. B 77, 155317 (2008).

[31] O.L. Berman, R. Ya. Kezerashvili, and K. Ziegler, вЂњSuperп¬‚uidity and collective properties of excitonic polaritons in gapped

graphene in a microcavityвЂќ, Phys. Rev. B 86, 235404 (2012).

[32] A. Amo, J. Lefr`ere, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. HoudrВґe, E. Giacobino, and A. Bramati, вЂњSuperп¬‚uidity

of polaritons in semiconductor microcavities, вЂќ Nature Physics 5, 805 (2009).

[33] J.P. FernВґandez and W.J. Mullin, вЂњThe Two-Dimensional Boseв€«Einstein Condensate, вЂќ J. Low. Temp. Phys. 128, 233