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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
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
PACS numbers: 03.75.Hh, 42.55.Mv, 67.85.Bc, 67.85.Hj
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 . 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  (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 .
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 .
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 . 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,
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
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 . While KTS phase transition occurs in systems, characterized by thermal
equilibrium, it survives in a dissipative highly nonequilibrium system driven into a steady state .
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 
E (k) =
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