Exciton–polariton laser
We present a review of the investigations realized in the last decades of the phenomenon of the Bose–Einstein condensation (BEC) in the system of two-dimensional cavity polaritons in semiconductor nanostructures. The conditions at which the excitons interacting with cavity photons form new type of q...
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nasplib_isofts_kiev_ua-123456789-1291002025-02-23T17:39:26Z Exciton–polariton laser Moskalenko, S.A. Tiginyanu, I.M. К 100-летию со дня рождения К.Б. Толпыго We present a review of the investigations realized in the last decades of the phenomenon of the Bose–Einstein condensation (BEC) in the system of two-dimensional cavity polaritons in semiconductor nanostructures. The conditions at which the excitons interacting with cavity photons form new type of quasiparticles named as polaritons are described. Since polaritons can form in a microcavity a weakly interacting Bose gas, similarly to the exciton gas in semiconductors, the microcavity exciton–polariton BEC emerged in the last decades as a new direction of the exciton BEC in solids, promising for practical applications. The high interest in BEC of exciton–polaritons in semiconductor microcavities is related to the ultra-low threshold lasing which has been demonstrat-ed, in particular, for an electrically injected polariton laser based on bulk GaN microcavity diode working at room temperature. 2016 Article Exciton–polariton laser / S.A. Moskalenko, I.M. Tiginyanu // Физика низких температур. — 2016. — Т. 42, № 5. — С. 426-437. — Бібліогр.: 64 назв. — англ. 0132-6414 PACS: 71.35.–y, 71.36.+c https://nasplib.isofts.kiev.ua/handle/123456789/129100 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We present a review of the investigations realized in the last decades of the phenomenon of the Bose–Einstein condensation (BEC) in the system of two-dimensional cavity polaritons in semiconductor nanostructures. The conditions at which the excitons interacting with cavity photons form new type of quasiparticles named as polaritons are described. Since polaritons can form in a microcavity a weakly interacting Bose gas, similarly to the exciton gas in semiconductors, the microcavity exciton–polariton BEC emerged in the last decades as a new direction of the exciton BEC in solids, promising for practical applications. The high interest in BEC of exciton–polaritons in semiconductor microcavities is related to the ultra-low threshold lasing which has been demonstrat-ed, in particular, for an electrically injected polariton laser based on bulk GaN microcavity diode working at room temperature. |
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Moskalenko, S.A. Tiginyanu, I.M. |
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Moskalenko, S.A. Tiginyanu, I.M. |
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Moskalenko, S.A. |
| title |
Exciton–polariton laser |
| title_short |
Exciton–polariton laser |
| title_full |
Exciton–polariton laser |
| title_fullStr |
Exciton–polariton laser |
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Exciton–polariton laser |
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exciton–polariton laser |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2016 |
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К 100-летию со дня рождения К.Б. Толпыго |
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Exciton–polariton laser / S.A. Moskalenko, I.M. Tiginyanu // Физика низких температур. — 2016. — Т. 42, № 5. — С. 426-437. — Бібліогр.: 64 назв. — англ. |
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Физика низких температур |
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AT moskalenkosa excitonpolaritonlaser AT tiginyanuim excitonpolaritonlaser |
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2025-11-24T04:28:33Z |
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1849644561816616960 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 5, pp. 426–437
Exciton–polariton laser
S.A. Moskalenko
Institute of Applied Physics, Academy of Sciences of Moldova, Chisinau 2028, Republic of Moldova
E-mail: exciton@phys.asm.md
I.M. Tiginyanu
Institute of Electronic Engineering and Nanotechnologies “D. Ghitu”, Academy of Sciences of Moldova
National Center of Materials Study and Testing, Technical University of Moldova, Chisinau 2004, Republic of Moldova
Received December 14, 2015, published online March 23, 2016
We present a review of the investigations realized in the last decades of the phenomenon of the Bose–Einstein
condensation (BEC) in the system of two-dimensional cavity polaritons in semiconductor nanostructures. The
conditions at which the excitons interacting with cavity photons form new type of quasiparticles named as
polaritons are described. Since polaritons can form in a microcavity a weakly interacting Bose gas, similarly to
the exciton gas in semiconductors, the microcavity exciton–polariton BEC emerged in the last decades as a new
direction of the exciton BEC in solids, promising for practical applications. The high interest in BEC of exciton–
polaritons in semiconductor microcavities is related to the ultra-low threshold lasing which has been demonstrat-
ed, in particular, for an electrically injected polariton laser based on bulk GaN microcavity diode working at
room temperature.
PACS: 71.35.–y Excitons and related phenomena;
71.36.+c Polaritons (including photon–phonon and photon–magnon interactions).
Keywords: excitons, polaritons, microcavity, quantum coherence, Bose–Einstein condensation, inversionless
polariton lasing.
1. Introduction
Our review article is dedicated to the memory of Pro-
fessor Kirill Borisovich Tolpygo commemorating the cen-
tenary of his birthday. S.A. Moskalenko (S.A.M) was for-
tunate to be doctoral student at the Kiev Institute of
Physics (IOP) of the Academy of Science of USSR in
1956–1959 years under the supervision of Professor K.B.
Toplygo. It was a period of time marked by the fundamen-
tal achievements in physics such as the creation of the mi-
croscopic theories of the superfluidity and superconduc-
tivity, the discovery of excitons in semiconductors, and by
the elaboration of the lasers. All these events created an
exceptional scientific atmosphere stimulating the search of
new interconnected physical phenomena. In addition to
these favorable conditions Kirill Borisovich Taplygo hap-
pened to be a scientist with vast and profound knowledge
in physics, democratic and benevolent supervisor provid-
ing an unlimited amount of consultations to his doctoral
students. To complete the characterization of that time in
IOP it is necessary to mention the existence of an excellent
library and the picturesque situation of the Institute near
the Goloseevskii forest. The candidatus scientiarum thesis
fulfilled by S.A.M. under the guidance of K.B. Toplpygo
included some suggestions concerning the collective prop-
erties of high density excitons in semiconductors also. The
suggestions arose under the influence of the microscopic
theory of superfluidity elaborated by N.N. Bogoliubov.
They concerned to the Bose–Einstein condensation of the
excitons in semiconductors. This topic made the contents
of our report at the 3rd International Conference on Nano-
technologies and Biomedical Engineering (ICNBME-
2015) held in Chisinau, Moldova on September 23–26,
2015. Our review article has been written on the base of
this report.
2. Bose–Einstein condensation. Fundamental concepts
Bose introduced the concept of an ideal photon gas and
deduced in this model the Planck’s distribution formula for
the black-body radiation [1]. Einstein generalized these
results applying them to the model of an ideal monoatomic
gas. He came to the conclusion that in the conditions of the
thermodynamic equilibrium the number of atoms in the
© S.A. Moskalenko and I.M. Tiginyanu, 2016
Exciton–polariton laser
given volume and at a given temperature cannot exceed a
well defined density and the surplus of atoms in the system
will accumulate in a macroscopic number in the lowest
state with zero kinetic energy [2]. Due to the contributions
of Bose and Einstein the concepts of the ideal Bose gas,
Bose–Einstein condensation (BEC), and Bose–Einstein
statistics were established in physics. Kapitsa discovered
He II superfluidity phenomenon below the λ-point [3] and
London [4] suggested the idea that the superfluidity and
the superconductivity discovered earlier both are due to the
phenomenon of Bose–Einstein condensation.
In the microscopic theory of superfluidity Bogoliubov
[5] introduced the notion of the amplitude of the conden-
sate considering the Bose operators 0a+ and 0a with wave
vector k = 0 as macroscopic values with the given phases,
the mean values of which 0 0, ,a a+〈 〉 〈 〉 named later quasi-
averaged, remain different from zero if the gauge sym-
metry (1)U of the system is broken. Bogoliubov intro-
duced the unitary canonical transformation for the opera-
tors 0 0,a a+ named as displacement transformation which
is equivalent to the Glauber [6] unitary transformation with
the operator
( ) 0 0 0( )
0 e N a aD N
+ −= , 0 0 0Da D N a+ = + (1)
which was introduced in quantum optics much latter. It de-
termines the coherent states. Because 0N is proportional to
the volume V, it means that Bogoliubov introduced the con-
cept of coherent macroscopic states. The field operators
0 0,a a+ are equivalent to the quantum macroscopic wave
functions *,Ψ Ψ or to the order parameter in the Ginzburg–
Landau [7] theory of superconductivity. In the presence of
the gauge symmetry when the Hamiltonian commutes with
the operators N̂ describing the full number of particles, the
ground state of the system is invariant under the action of
the unitary transformation ˆˆ ( ) eiNU ϕϕ = changing the phase
of the ground state wave function. There are many degener-
ate ground states of the system, the wave functions of which
differ only by their phases. The potential ( )V ϕ represented
in Fig. 1(a) has the minima positioned along the ring. Break-
ing of the gauge symmetry of the system leads to the fixa-
tion of the phase and to the selection of one point in the ring
where the position of the ground state is localized. The exci-
tation of this ground state with changing only its phase does
not need energy in the limit of zero wave vector k. Such
elementary excitations are referred as massless Goldstone
[8] modes or as Nambu–Goldstone gapless modes discussed
by Nambu in Ref. 9. This phenomenon is general and takes
place in any order of the perturbation theory with some ex-
ceptions. The spontaneous breaking of the system continu-
ous symmetry leads to the existence of one Nambu–
Goldstone mode in the spectrum of its elementary excita-
tions. It is true not only in the case of gauge symmetry but
also in the cases of a continuous rotational symmetry. Just
such type of elementary excitations was obtained by
Bogoliubov in the model of a weakly nonideal Bose gas
with the Hamiltonian
( )
, ,2
gT a a a a a a
V
+ + +
+ −= − µ +∑ ∑k k k p q q k p k
k p q k
(2)
where 2 2( )/2T m=k k is the kinetic energy and µ is the
chemical potential. The state of Bose–Einstein condensa-
tion was introduced by the displacement canonical trans-
formation
0 ,0k k ka N= δ + α (3)
which leads to the equation of motion for the condensate
annihilation operator
20
0 0 0
i da
a g a a
dt
= −µ +
(4)
and to the Bogoliubov spectrum of the elementary excitations
( ) 02
0 00 ;2k k
N
E k L g n gT T L
V
= ± = =+ . (5)
In the case of inhomogeneous in real-space and in time
Bose–Einstein condensate the Eq. (4) transforms into the
Gross–Pitaevskii equation of motion of the condensate
wave function ( , )r tψ
( )
2
2( , ) ( , ) ( , )
2
d t gi t t
dt m V
ψ
= − ∆ψ + ψ ψ
r r r r
. (6)
Fig. 1. The potential V(φ) with the minima at |φ| = a and a
local maximum at φ = 0 (a). The free energy density functional
f(Ψ(r, t), T) in dependence on the order parameter 〈Ψ(r, t)〉 (b)
[This drawing is reproduced from the paper by H.T.C. Stoof pub-
lished in Bose–Einstein condensation, A. Griffin, D.W. Snoke, and
S. Stringari (eds.), Cambridge U. Press, Cambridge (1995)].
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 5 427
S.A. Moskalenko and I.M. Tiginyanu
It describes also the vortices formation in a boson sys-
tems discussed by Gross [10] and Pitaevskii [11]. The nu-
cleation and the building of the Bose condensate was in-
vestigated by Stoof [12] who argued that the problem
related to the nucleation of BEC does not lie in the study of
the kinetic stages of the phase transition. The Boltzmann
equation describing the kinetic stages is unable to treat the
buildup of the coherence. It cannot lead to a macroscopic
occupation of the single-particle ground state, since the
production rate of the condensate fraction in the thermody-
namic limit ,N V→ ∞ → ∞ is nonzero only if a conden-
sate already exists. If the gas is quenched sufficiently far
into the critical region of the phase transition, the first
stage of its evolution is then kinetic stage. It leads to the
thermalization of the gas, but does not lead to phase coher-
ence. The first kinetic stage leads to transformation of the
initially highly nonequilibrium distribution into a quasi-
equilibrium one. The dominant time scale of the kinetic
process is determined by the elastic relaxation time el.τ
During this stage in the range of critical temperature the
number of elementary excitations in the system becomes
so high that their interactions cannot be neglected. Taking
them into account and using mean-field approximation
Stoof has obtained another dispersion law different from
the Bogoliubov dispersion relation:
( ) ( ) ( )22
0T gnω = −kk
, (7)
where g plays the role of an effective interaction constant.
The modes for which Tk is less than 0gn are unstable
and their population decreases. It leads to the buildup of
the condensate during the coherent stage with the coher-
ence time cohτ of the same order as el.τ It is demonstated
in the Fig. 1(b).
This coherence stage gives rise to the appearance of the
condensate embryo in a form of a seed which is not in the
thermodynamic quasi-equilibrium with the noncondensed
part of the Bose gas. This goal has to be achieved in the
last stage of the phase transition, which is also a kinetic
stage. Its duration relτ is much longer than the first kinetic
stage and is limited in the case of excitons or polaritons by
their lifetime life.τ In the condition rel life ,τ > τ the com-
plete quasi-thermalization cannot be achieved.
3. BEC of excitons in semiconductors
The fundamental concepts of the Bose gases formed
from the matter particles (helium atoms, Cooper electron
pairs and atomic gases) and photons were investigated in the
conditions of thermodynamic equilibrium. The coherent
macroscopic states arising in the conditions of BEC due to
the thermalization processes were supplemented by the laser
radiation arising due to the photon stimulated emission in
the media with inverse particle distribution instead of ther-
mal equilibrium. The created physical backgrounds of the
microscopic theory of superfluidity elaborated by Bogo-
liubov, as well as of the theory of superconductivity elabo-
rated by Bardeen, Cooper and Schrieffer [13] and refined by
Bogoliubov [5] from the one hand and the creation of the
lasers and coherent sources of radiation in the middle of the
20th century from the other hand stimulated enormously the
search of similar phenomena with the participation of
quasiparticles with finite lifetimes in quasiequilibrium states
but far from the thermodynamic equilibrium. The first at-
tempt in this direction was the investigation of the BEC
phenomenon and the superfluidity of the excitons created by
the bound electron-hole pairs in semiconductors initiated in
the Refs. 14–16. The following step was the study of the
mixed exciton–photon systems composed from the dipole-
active excitons interacting with the photons captured togeth-
er in the resonators, when the full number of excitons and
photons is conserved. In such mixed systems the half-
matter-half-light [17–20] quasiparticles named as polaritons
can be formed. A variant described in [19,20] is the most
effective at present time as will be shown below. Side by
side with the exciton-exciton interaction there is a wide
range of optical nonlinearity sources. In this case instead of
polaritons we deal with the cavity photons interacting be-
tween themselves through the optical nonlinearities and
forming the quantum fluids of light [20].
The present report is related to the mixed exciton–
photon systems enclosed into the resonators when the total
number of enclosed and photons can be conserved due to
high quality mirrors. Their description can be made basing
on a Hamiltonian written in the exciton and photon repre-
sentation with a common chemical potential for the both
kinds of particles in a similar way as in Refs. 17,18, or pas-
sing directly to the polariton representation in special con-
ditions when the polariton creation is not destroyed by the
frequent exciton–exciton scattering processes [19,20]. The
quantum states in the mixed exciton–photon systems large-
ly depend on the ratio between the exciton–exciton scatter-
ing rate and the rate of the exciton–photon conversion pro-
cess. The first process is nonlinear and its rate is
proportional to the exciton density, whereas the last one is
a linear optical process depending on the value of exciton–
photon interaction constant and without any dependence on
the particle densities. The polaritons cannot be formed at
high exciton density when the rate of exciton–exciton scat-
tering is greater than the rate of exciton–photon conversion
as it was considered in the Refs. 17, 18. The BEC of dipole
active excitons takes place firstly in the exciton subsystem
generating posteriori a coherent macroscopic polariton
mode [17] named in Ref. 18 as giant polariton. To avoid
this obstacle and to reach the polariton formation on the
level of individual single-particle formations it was neces-
sary to increase the exciton–photon conversion rate even in
the case of GaAs-type quantum well (QW) embedded into
the resonator, which is characterized by a weak exciton–
photon interaction. It was achieved by an ingenious pro-
posal [19–27] to introduce into the antinodes of the cavity
428 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 5
Exciton–polariton laser
not one but many QWs with a considerable number NQW ≈
≈ 20–30 in the condition of the Dicke [28] superradiant
state, when the rate of the photon emission by QWN QWs
increases by the factor .QWN It means that the Rabi fre-
quency and the rate of the exciton–photon conversion in
such conditions will increase in QWN and QWN re-
spectively and the polariton formation can be possible even
at a high exciton density. The cavity polariton formation is
characterized by a very small effective mass. It is on four
orders of magnitude smaller than the free electron mass
and on nine orders of magnitude less than that of Rb at-
oms. The small effective mass opens the possibility to
reach the conditions necessary for BEC even at the room
temperatures. The small polariton mass leads also to high
polariton–polariton elastic scattering rate and to small
thermalization time of the polaritons on the lower polariton
branch of about same tenth of picoseconds. In spite of a
small lifetime of polaritons in microcavity (tens picose-
conds), due to the smaller thermalization time the BEC of
polaritons in the point with 0=k
on the lower polariton
branch succeeded to be realized. In such a way the BEC of
cavity polaritons emerges in the last decades of our century
as a more perspective variant of the exciton BEC in solids.
The results obtained in last decades by many collectives of
researchers investigating in details the physical processes
related with BEC of cavity exciton–polaritons were repre-
sented in the review articles [20–28] and in some initial
papers [19,30–33]. The investigations of this problem in
the precedent period of time, developing in the second half
of the 20th century, have been reflected in the monograph
[34]. The review article [35] is devoted to the special case
of excitons BEC in Cu2O crystal. Surprisingly, the BEC of
excitons was observed in the conditions of the fractional
quantum Hall effects [33]. The coherent macroscopic state
of dipole-active excitons gives rise to the coherent macro-
scopic beam of photons and their reciprocal transformation
gives rise to the macroscopic coherent mode as it is argued
in the Refs. 17,18. In the same way the coherent macro-
scopic wave created by cavity polaritons condensed on the
lower polariton branch in the point with the in-plane
wavevector 0=k
may emit photons with the same in-
plane wave vector 0,=k
but with the quantized value
/ cLπ of the component zk oriented along the axis of the
cavity and perpendicular to the surface of the QW embed-
ded in it [19,20]. The emitted wave formed by the escaped
photons from the composition of the condensed polaritons
has the same coherent macroscopic property as the con-
densed polariton wave and looks outside the resonator as a
laser radiation. It is named as the exciton–polariton laser
radiation. The system of Bose–Einstein condensed cavity
polaritons arising due to the establishment of the quasi-
equilibrium in the system and emitting outside the cavity
the laser radiation works as a device, which can be named
as polariton laser. Usually the electron–hole pairs in the
QW are excited by the light of the pumping laser. But the
electron-hole pairs can be injected electrically. In this case
we deal with the electrically pumped polariton laser. The
creation and the physics of the room temperature electri-
cally injected exciton-polariton laser is the main topic of
our report. The information concerning this topic is gath-
ered from the Refs. 36–41.
The polariton decay by leakage through the DBR mir-
rors gives rise to the photons carrying the same energy and
momentum as the polaritons in the QWs. The energy can
be measured by spectroscopic method, while the momen-
tum can be measured by the angular direction of the pho-
ton propagation. Below condensation threshold there is a
broad distribution of exciton-polaritons in both energy and
momentum scales of the lower polariton dispersion branch.
As the pumping power is increased, there is a sudden nar-
rowing in both distributions and a large population of
polariton occupies the zero-momentum mode of the system
consistent with the formation of a BEC. Below threshold
the population at k = 0 increases linearly with the pump
power, whereas after condensation it increases nonlinearly.
After the transition region the population returns to linear
increasing and the condensate population dominates the
polariton-population. The macroscopic occupation of the
ground state, the nonlinear threshold behavior and the nar-
rowing of the linewidth are all expected in the case of BEC
transition. All these peculiarities are represented in Fig. 2.
All of them are needed but they are not sufficient fea-
tures to declare unambiguously about the BEC establish-
ing. The supplementary arguments in the favour of BEC
are the measurements of the degree of coherence and of the
correlation functions of the condensate fraction using the
interferometer and the photon counting setup. Some sup-
plementary information concerning the polaritons and the
two-dimensional Bose gas is needed and it is added below.
4. Polaritons
In the Timofeev [28] review article it was fairly men-
tioned once again that the interaction of the electromagnet-
ic field oscillations with the transverse optical phonons in
polar dielectric crystals leading to the formation of the
mixed half-matter-half-electromagnetic field states was
investigated for the first time by Kirill Borisovich Tolpygo
[42] and by Huang Kun [43,44]. Later such mixed states
were named as phonon–polaritons similarly with the
exciton–polariton states described by Hopfield [45]. Hop-
field introduced the concept of polariton as a new
quasiparticle in the solid state physics. It arises in the con-
ditions when the exciton–photon conversion takes place in
the form of multiple, reversible mutual transformations
whereas another irreversible quantum processes such as the
usual emission or the absorption of the light by the exciton
state in crystal, as well as the exciton–phonon or exciton–
exciton scatterings do not. The new exciton–photon states
are representatives of the half-matter-half-light elementary
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 5 429
S.A. Moskalenko and I.M. Tiginyanu
excitations. Their mathematical formulation is written in
the form of the coherent superposition of their wave func-
tions or of the corresponding field operators and will be
presented below. The Hamiltonian describing the quantum
states of the system includes the free-particle terms and the
exciton–photon interaction. Last has the type ,A∇ where
A is the vector–potential of the electromagnetic field. In-
teraction terms of higher order of A and antiresonance type
are excluded from the Hamiltonian for simplicity. Free-
particle terms include the 2D excitons in the quantum well
embedded into the microresonator and the cavity photons
captured in the spacer of the microresonator. Resulting
Hamiltonian has a quadratic form:
_____________________________________________________
|| || || |||| || ||
||
† † † †
ex, || , || , , || / , , ,,, / , , / , ,
,
[ ( ) ( / , ) ( / , )( )],
c cc cj j c c L j j c L j jjj L j L j
j
a a L c c L a c c aπ ππ π= ω + ω π + ϕ π +∑ k k k kkk k k
k
k k k (8)
_______________________________________________
where ex, ||( )jω k is the frequency of the QW exciton with
polarization j and in-plane (in regard to QW plane) wave
vector || ,k ||( / , )c cLω π k is the frequency of the cavity
photons, and ||( / , )j cLϕ π k is the constant of the exciton–
photon interaction. Photon wave vector || 3 zk= +k k a con-
sists of two parts: in-plane ||k and size-quantized normal
component 3 ,zka where 3a is the unit vector oriented
along the axis of the microresonator and perpendicular to
the surface of the QW embedded in it, and / ,z ck L= ±π
( cL is the microresonator length).
The dispersion relations for the both frequencies are:
2 2
ex, || ex, || ex( ) (0) ( )/2 ,j j mω = ω +k k
2
2
|| ||( / , )c c
c c
cL
n L
π
ω π = +
k k . (9)
Here cn is the refractive index of the spacer inside the
microresonator. If length of the in-plane wave vector ||k is
small in comparison with / ,cLπ the cavity photon disper-
sion relation looks as:
2
||
||( / , )
2c c c
c
L
m
ω π = ω +
k
k
, c
c c
c
n L
π
ω = ,
c
c
c
n
m
cL
π
=
, ||| |
cL
π
<k . (10)
Using values 510 cmcL −= and 3cn = we obtain
5
03 10 ,cm m−= ⋅ where 0m is the free electron mass. The
polaritons from the lower branch inherit the excessive
small effective mass of the cavity photons, so their effec-
Fig. 2. (Color online) Polariton occupancy measured at 5 K. (a) Occupancy of the k|| = 0 ground state (solid black diamonds), its energy
blue shift (solid green circles) and linewidth (open red triangles) versus the excitation power. The blue shift is plotted in units of the
Rabi splitting Ω = 26 meV. (b) Polariton occupancy in ground- and excited-state levels is plotted in a semi-logarithmic scale for various
excitation powers. For each excitation power, the zero of the energy scale corresponds to the energy of the k|| = 0 ground state. The fig-
ure adapted from the paper by J. Kasprzak et al., Nature 443, 409 (2006).
430 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 5
Exciton–polariton laser
tive mass becomes small also. One can remember that the
small effective mass of the lower polariton branch plays a
crucial role and facilitates the establishing of the polariton
BEC.
The polariton annihilation operator
|| ,
ˆ jPk can be ex-
pressed as a coherent superposition of the exciton and the
cavity photon annihilation operators:
|| ||
||
2 2
, || , || || ||, ,
ˆ ( ) ( ) , ( ) ( ) 1
c
j j j j j jj
L
P u a v c u vπ= + + =k k k
k k k k .
(11)
Instead of the two bare exciton and cavity photon branch-
es, two other mixed-type branches appear with the disper-
sion laws
,||
ex, || ||( ) ( / , )
2j
j c c
P
Lω + ω π
ω = ±
k
k k
2 2
ex, || || ||
1 ( ( ) ( / , )) 4 | ( / , ) |
2 j c c j cL L± ω − ω π + ϕ πk k k . (12)
They are named as upper and lower polariton branches
with a given j polarization.
5. Two-dimensional Bose-gas
In contrast to the three-dimensional (3D) ideal Bose-gas
and its fundamental concepts mentioned above, in the case
of cavity polaritons new aspects related with their finite
lifetime and with the two-dimensionality of the system
arise. The two-fold dimensionality has an important role,
because as was established in Ref. 46 the BEC of the two-
dimensional (2D) homogeneous Bose gas extended on an
infinite surface area S → ∞ cannot occur at finite temper-
ature. It is possible only at the temperature 0,T = which
cannot be achieved experimentally. An uniform 2D system
of bosons in the thermodynamic limit at finite temperatures
has no phase transition to the highly ordered states, since
the long wavelength thermal fluctuations destroy any long-
range order. In such a way the experimental realization of
the BEC phenomena with the participation of the 2D cavity
polaritons at finite temperatures faced with an obstacle of
principle. Nevertheless, the ingenuity of the experimental
physics succeeded to avoid this obstacle and to reveal the
basic properties of the nature hidden by such restrictions.
The solution was to study the 2D systems with finite area
using the pumping light with finite area spot for excitation.
It should be noted, however, that the confinement of the
Bose gas on a finite area gives rise to the uncertainty of the
in-plane wave vector ||k of the confined polaritons. The
condensate wave vector also exhibits an uncertainty. It
means that the condensate stretches to a finite wave vector
that is disregarded, for example, by the standard Bogo-
liubov theory of superfluidity [5], but can be taken into
account introducing a trap for the 2D polaritons. The spa-
tial extent of the condensed polaritons leads also to the
size-dependent critical density of the condensation. When
the Bose-gas is confined by a spatially varying potential,
the constant density of states well known for the uniform
2D system is strongly modified. As a result the BEC phase
at finite temperatures is recovered [20–32].
Practically any experimental system has a finite size and
a finite number of particles on the single-particle states. In a
2D box with size L the critical conditions for BEC can be
fulfilled at 0.T ≠ If the system is small, the finite size BEC
phase transitions are expected to occur before the
Berezinskii–Kosterlitz–Touless (BKT) phase transition. This
possibility was foreseen and introduced independently by
Berezinskii [47] and Kosterlitz and Thouless [48] in the cit-
ed papers. As was mentioned above, the highly focused spot
for optical excitations can provide a localization of cavity
polaritons. Due to their very small effective mass, the obser-
vation of their 3D quantization is possible for trap diameters
of the order of a few μms. For the trap of 1–2 μm the quan-
tized energy of the polariton equals few meV [20–25]. The
lateral confinement of 2D polaritons creates opportunities
for realization of the future quantum light emitters as
sources of indistinguishable single photons. Here is an op-
portunity following the Refs. 20–28 to remember that the
polariton system has a unique feature compared to the dilute
atomic BEC and dense superfluids. It is related to direct
experimental accessibility to measure the quantum statistical
properties of the condensate. The main decay channel of the
cavity polaritons consists in the photon leakage from the
cavity. During this process two conservation laws are ful-
filled. The energy and the in-plane wave vectors of each
cavity polariton and leaked photon are fully coincident. It is
a new experimental tool to study the non-equilibrium open
system. As was mentioned in the Ref. 22 the photon portion
of the polariton when leaving the microcavity keeps all rele-
vant information concerning not only the polariton popula-
tion, but also its phase that reflects the degree of coherence
of the formed Bose–Einstein condensate. The cavity
polaritons are an unique system for exploring both the cavity
quantum electrodynamics and the many-body physics, open-
ing the possibility to realize experimentally different phase
transitions [20–28]. The phase properties along the spot can
be measured studying interferometrically the interfence
fringes arising from the photons leaked from different points
of the spot. The emitted photons conveyed to the researchers
the information about the polaritons occupancy at the given
wave vectors || ,k about their phase, their coherence, noise,
polarization and polariton dispersion law. The second order
time and space correlation functions can be determined us-
ing the Hanbury–Brown and Twiss setup. The convincing
proofs and evidences of the experimental realization of the
BEC phenomenon do exist. The made measurements testify
that the macroscopic population and the establishing of the
coherent phase of the condensate over whole illuminated
spot take place simultaneously as an unique interdependent
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 5 431
S.A. Moskalenko and I.M. Tiginyanu
process exactly as it was argued by Stoof [12]. It can be re-
peated again following [22] that the ease with which the
internal properties of the condensate can be assessed is due
to the photon portion of the polaritons that leak out the cavi-
ty. The initial demonstrations of the condensation in the
lower polariton branch were made by some research groups
such as the groups of Deng [30] and Kasprzak [31] in Gre-
noble. They used CdTe-based microcavities. The BEC under
the nonresonant excitation of the system relies on many ob-
servations such as: (i) the nonlinear threshold in the accumu-
lation of the polaritons in the point || 0=k with vanishing
chemical potential 0;µ → (ii) the increase of the temporal
coherence with the simultaneous decrease of the linewidth;
(iii) the second order temporal correlation function measured
by the Hanbury–Brown and Twiss photon-counting setup;
(iv) the long-range spatial order measured interfero-
metrically. The latter means the interference between two
shifted one to another images of condensate that allows to
probe the long-range spatial coherence over the entire spot.
The ability to obtain in a very direct way by interferometric
measurements the information about the polariton popula-
tion is a major advantage of polaritons and has served as a
basis for many different observations. Below threshold the
coherence of the polariton could typically extends over mi-
crons, approximately equal to the de Broglie wavelength of
polaritons at the measured temperature 19 K. Above the
condensation threshold, correlations were observed over
most of the excitation spot more than 20 μm.
6. The driven dissipative nature of the polariton
condensate
The polaritons in microcavity have a lifetime comparable
to the thermalization time that gives them an inherently
nonequilibrium nature as was mentioned in the Refs. 20–32.
It raises fundamental questions especially concerning the
possibility to guarantee the effectuation of the relaxation
processes and of establishing thermodynamic quasi-
equilibrium. Unexpectedly, due to an excessive small pola-
riton effective mass and as a result of a very small elastic
polariton–polariton scattering time, many doubts in this di-
rection were removed and the cavity polaritons exhibited
many features that could be expected for the equilibrium
Bose–Einstein condensates. Spontaneous coherence embrac-
ing of a macroscopic number of particles is a fascinating
phenomenon. It is represented by laser radiation, by BEC of
ultra cold atoms, by the formation of the Cooper pairs in the
superconductors as well as by the formation of the superflu-
id fraction in superfluid 4He. Now to this list one can add the
BEC of microcavity polaritons and of the interacting cavity
photons [20,49]. A large number of particles initially pos-
sessing random phases become coherent due to their interac-
tion and the thermalization processes when such parameters
as density and temperature crossed their threshold values.
Exciton polaritons undergoing BEC incorporate new physics
due to their intrinsically non-equilibrium nature. In many
experiments [20–28] the exciton-polariton condensates were
produced in the vicinity of 10 K in GaAs and CdTe semi-
conductors, whereas in GaN, ZnO and organic semiconduc-
tors the polariton condensates were realized at room temper-
atures [20,25]. They stimulated the development of new
physical aspect not only in the fundamental perspectives but
also in quantum technological devices [20,25]. The Bose–
Einstein condensed cavity polaritons with short lifetime,
their evolution in time and in the real space taking into ac-
count the gain and loss terms were described in Ref. 20 by
the generalized Gross–Pitaevskii equation (GPE):
____________________________________________________
( ) ( ) ( )( )( ) ( ) ( ) ( )
2 2, , , 2 , ,
2 2d R R
LP
d rt ii V t R n t g t gn t t
dt m
ψ = − ∆ + + − γ + ψ + ψ
r r r r r
. (13)
_______________________________________________
Here ( ), tψ r is the macroscopic wave function of the con-
densate on the lower polariton branch with effective mass
LPm , ( ),dV tr is a local potential. The gain is effectuated
by the reservoir with the density of the particles ( ),Rn tr
satisfying the kinetic equation and the lost γ is due to the
leakage of photon portion of the polaritons through the
mirrors of the microcavity. In the review article by
Deveaud [22] the experiments were described, when the
pulse duration of the off-resonant excitation was about 100
fsec. The delayed buildup of the population of polaritons in
the lower branch was observed with a very rapid buildup
of the long-range phase coherence in the condensate soon
after the stimulated scattering. The onset of long- range
coherence appears even for points at 10 mµ distance. As
was mentioned by Deveaud [22], this result is amazing
because in some earlier expressed opinions just the phase
relaxation of the condensate was supposed to be in delay
comparable with the relaxation time of the number of the
condensed particles 0.N This result confirms also the theo-
retical statements by Stoof [12] that the relaxation of the
number 0N and of the condensed particle phase are unique
processes taking place simultaneously without delay be-
tween them. When the initial random phase was introduced
in the lower polariton branch, then the set of the vortices in
the excitation spot was observed. The annihilation of the
counter-rotating vortices and the restoration of the phase
coherence take place during only a few picoseconds.
The stability of the elementary excitation spectrum in
non-equilibrium conditions was firstly discussed in Ref. 50
using a Keldysh diagram technique. Another route based
432 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 5
Exciton–polariton laser
on the nonequilibrium GPE was used in Ref. 20 leading to
similar results. The main feature is the diffusive behavior
of the Bogoliubov dispersion at low wave vectors k and
the restoration of the standard Bogoliubov picture at great-
er values of k. The stability of the condensate is guaranteed
by the fact that the imaginary part of the dispersion law is
negative for all values of k [20]. As was reminded in
Ref. 22, the criterion formulated by Penrose and Onsager
[51] for the condensate single-particle density matrix
( , ) ( ) ( )x x x x+ρ = ψ ψ′ ′ does exist. Its limit at | |x x− → ∞′
must be different from zero. It was proved experimental-
ly, that despite the dissipative nature of the polaritons
their distribution follows the criterion proposed by Pen-
rose and Onsager, because above the thresholds the mac-
roscopic occupancy of a single-particle level was clearly
evidenced [22]. In the Ref. 22 it is reminded another out-
standing feature of the Bogoliubov theory of superfluidity,
which consists in the superposition of wave functions of
the corresponding operators of the propagating particle and
of its counter-propagating antiparticle or hole. In the super-
fluid state there are two branches of the Bogoliubov energy
spectrum. One is positive and is named as an energy
branch whereas another is negative and is named as an
quasi-energy branch or even as a ghost branch. Both
branches in the range of small wave vectors have the linear
dispersion relations. The experiments performed in the
Ref. 22 were based on the idea to probe the dispersion law
of the superfluid excitations using the four-wave mixing
(FWM) method. The superfluid fraction of polaritons was
induced in the point || 0=k by the pumping beams. The
properties of the system were probed through the diffraction
of the probe beam sent at an angle to the normal to QW. The
ghost branch appeared only above the threshold density for
the superfluidity of the polariton gas. To describe the be-
havior of the condensate in the case of the FWM it was
used the generalized GPE with the wave function ( , )tψ r
of the form 0( , ) ( , ) ( , )eit t U tψ = ϕ + +krr r r ( , )e iV t∗ − krr
where 0 ( , )tϕ r describes the coherent polariton superfluid
created by the pump beams in the point 0,=k whereas
( , )U tr and ( , )V t∗ r are the wave functions describing
the counter-propagating excitations of the system. At
enough large densities of the polaritons, the ghost branch
shows the same intensity as the normal branch [22].
7. Superfluidity and vortices
In this section we will follow the review articles with
Refs. 20, 22, 25. The criterion of superfluidity was origi-
nally proposed by Landau [52] to explain the phenomenon
of superfluidity of the liquid Helium below the temperature
of the λ-point discovered by Kapitsa [3]. Landau supposed
that the energy spectrum of the elementary excitation in the
superfluid helium has a roton-type behavior with a critical
velocity determined as
( )
mincV
ω
=
k
k
k
. (14)
It concerns to the liquid in a rest state. When the liquid
is moving with respected to the laboratory reference frame,
the energy spectrum due to the Galilean transformation
looks as
( ) ( )Labω = ω +k k Vk . (15)
It can obtain negative values if V > cV and ( ) 0.⋅ <k V In
the flowing liquid with supersonic velocity V, the possibil-
ity to emit unlimitedly elementary excitations and to break
the superfluidity appears. If the moving condensate has the
parabolic dispersion law, its particles encountering the
obstacle will find the resonant states to scatter and to
transmit their energy. In his theory of superfluidity Bogo-
liubov obtains the energy spectrum with a needed linear
dependence in the range of small wave vectors with the
sound velocity
0
c
LP
gn
V
m
= , (16)
where 0 0gn > determines the blue detuning of the exciton
energy level due to the exciton–exciton interaction. In the
case of 0gn ≈ 1 meV and LPm ≈ 10−4
0m the Bogoliubov
sound velocity has the value cV = 108cm/s. The liquid with
flow velocity V smaller than the Bogoliubov sound veloc-
ity cV has no resonant states to scatter on the weak impuri-
ty or obstacle. The superfluidity is due to the suppression
of the scattering processes. But this statement is true only
if the impurities or the obstacles have a weak influence on
the fluid and do not change essentially its density in their
surroundings. In the opposite case the vortices in the liquid
near the surfaces of the impurities and obstacles appear.
The question arises whether the Bogoliubov type disper-
sion law does exist in the conditions of the driven-
dissipative system. It was investigated on the base of gen-
eralized Gross–Pitaevskii equation in the Ref. 53 taking
into account the gain effectuated by an incoherent reservoir
the dynamical variables of which modify strongly the dis-
persion law of the condensate. As a result the instability in
the spectrum appeared in the range of small wave vectors
[53]. Another analysis on the base of the Green’s function
method proposed in Ref. 54 arrived to the conclusion about
the existence of the normal and superfluid components.
The investigations of the quantum vortices in the superflu-
id liquid helium were initiated by Feynman [55–57].
Now the vortices formation in the 2D superfluid
polariton gas will be discussed. As was mentioned above
in the infinite homogeneous 2D system the off-diagonal
long-range order associated with the BEC phenomenon
does not exist. Its existence breaks down being vulnerable
to the thermal fluctuation at nonzero temperatures. How-
ever such systems can exhibit quasi-long-range-order pre-
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 5 433
S.A. Moskalenko and I.M. Tiginyanu
serving superfluidity. Berezinskii [47], Kosterlitz and
Thouless [48] investigated the formation of the vortices in
such system due to the quantum fluctuation. The creation
of one vortex at low temperature is unlikely due to the ne-
cessity of an extensive amount of energy. In spite of it, the
creation of a vortex-anti-vortex (V-AV) pair is possible at
low temperature and does not need an extensive amount of
energy, because the phase disturbances created by the vor-
tex the anti-vortex are canceled out. The pairs V-AV may
exist in free or in bound states. The transition of the V-AV
pairs from the unbound to bound state is named as the
Berezinskii–Kosterlitz-Thouless (BKT) phase transition.
The possible crossover from the BKT and BEC transitions
can be observed changing the density of the Bose gas. We
can expect the existence of the free vortices above the
BKT critical temperature, the presence of the bound V-AV
pairs in the BKT phase and the absence of the vortices in
the BEC phase. At the same time the change of the density
of the electron–hole (e–h) pairs leads to another type BEC–
BCS crossover. It takes place due to the changes of elec-
tron structure of the quasiparticles rather than the vortex or
soliton formations. At low densities of the e–h pairs we
deal with the exciton formation leading to cavity-polariton
existence. At high density of e–h pairs due to the screening
of the e–h Coulomb interaction the excitons as well as the
exciton polaritons do not exist. Instead of them we deal
with the 2D degenerate electron and hole Fermi-gases and
with their Fermi energies. The electrons and holes from the
surroundings of their Fermi energies can take place in the
coherent pairing, forming the Cooper-type e–h pairs as was
established by Keldysh and Kopaev [58] similar with the
electron Cooper pairs in the theory of superconductivity.
At high density of e–h pairs we can expect the formation of
the 2D photon gas instead of 2D polariton gas with a
strong indirect interaction between the photons mediated
by their direct interactions with e–h pairs. Such conclu-
sions were made in the Ref. 59. The photon BEC was in-
vestigated experimentally in the Ref. 49. The direct inter-
action of the photons with the e–h pairs can create not only
the indirect interaction between the photons, but it can cre-
ate also the supplementary effective attraction between the
electrons and holes [25].
8. GaN-based exciton polariton laser
This section contains the information concerning the
Galium Nitride room-temperature electrical-injection
polariton laser elaborated and described in the original
paper [36] and in the review articles [25,27,39]. The inter-
est in this laser is determined by the fact that it represents
the first realization of the ultra-low-energy-coherent light
source. In the previous decades a fast development of the
growth and of the technology of the GaN based devices
took place [60–64]. The industrial production of lasers
emitting in the blue and ultraviolet ranges of spectrum was
realized. These achievements facilitated the creation of the
GaN polariton laser. The previous GaN lasers were based
on the band-to-band quantum transitions and can be named
as band-edge lasers. They take into account the quantum
transitions between the conduction and valence band in the
condition of the inversion population, when the electrons
from the valence band are preliminary excited into the
conduction band so that the bottom of the conduction band
is completely filled by electrons whereas the top of the
valence band is completely filled by the holes. As is fre-
quently mentioned the band-edge laser needs an electron
inversion population with regard to the ground state of the
crystal. The ground state of the crystal means the absence
of the electron-hole pairs at all. Such-type inversion popu-
lation remains the same in the case of high density e–h
pairs, when they being bound in the form of Wannier–Mott
excitons form a weakly non-ideal exciton gas. In this case
a set of kinetic and dynamic processes in the presence of
the pump and loss occur. The quasi-equilibrium therma-
lization and the condensate formation take place without
any supplementary inversion and the lasers constructed in
these conditions are named as inversionless lasers, in spite
of the fact that the initial inversion with regards the ground
state of the crystal persists. The creation of a nonequi-
librium exciton gas, with the density by some orders of
magnitude smaller than the density of e–h pairs in the case
of the band-edge laser, needs smaller excitation energy by
the same order of magnitude. As was underlined in the
Ref. 36 the current density threshold needed for the crea-
tion of the electrically-injected polariton laser is almost
hundred times smaller than for the GaN-based edge-
emitting and surface-emitting lasers. The edge-emitting
laser, as was mentioned by [39], is due to the stimulated
emission of the photons and this process imposed stringent
requirements on the amount of energy needed for its reali-
zations. In contrast to this in the case of exciton–polariton
BEC the condensate emits a coherent-light directly due to
the leakage of the photons from the cavity and without any
supplementary stimulation processes. As a result the pola-
riton laser offers a more effective way to generate coherent
light at very low threshold [39]. Some data concerning the
GaN diode are needed. The GaN diodes consist from a
430 nm thick GaN p–n junction. The energy of the cavity
mode equals to cE =3.57 eV with a linewidth cγ =
= 1.7 meV corresponding to cavity mode lifetime 0.387 ps.
The quality factor of the cavity is Q = 1911. The energy of
the exciton transition at 10 K equals to exE = 3.65 eV
with a linewidth 4.8 meV. The blue shift of the polariton
emission line caused by the polariton–polariton interaction
equals to Eδ = (1.9 ± 0.28) meV when the exciton con-
centration was 3
ex
Dn = (1.53 ± 0.13)⋅1017 cm−3. The Mott
transition density is (1–2)⋅1019 cm−3. In GaN the exciton
has a binding energy of the order of 30 meV and is charac-
terized by extremely high oscillator strength [24]. Wannier
Mott excitons with small binding energy of about 10 meV
434 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 5
Exciton–polariton laser
are unsuitable for many practical applications [24]. Strong
oscillator strength permits to achieve the strong-coupling
regime necessary to obtain the reversible exciton–photon
conversion and the formation of the polaritons even at ele-
vated concentrations of excitons. A dominant radiation in
the case of polariton laser is observed below the exciton res-
onance Ex = = 3.65 eV. The value of the normalized chemi-
cal potential equals to
0
1/ ln 1 0.59Bk T
N
α = −µ = + =
.
The small value of α confirms the establishing of the quan-
tum degeneracy in the system [36]. One can once again
mention that the process of the polariton laser creation is
very different from the stimulated emission of radiation in
the conventional photon laser with inversion distribution.
The novelty and advantages of the polariton devices elabo-
rated in Ref. 36 are related to the successful fabrication of
the low resistance diode and to strong feedback which can
be achieved with high quality DBR mirrors. In those condi-
tions both polariton and photon lasers can be constructed.
The polariton laser, as was underlined in the Ref. 36, origi-
nates from a nonequilibrium BEC obtained by dynamical
balance between injection and loss. A condensate with a size
of about 1.5 µm was achieved. At low current injection level
the emission is depolarized. Above threshold the polariza-
tion of about 30% develops spontaneously. The long-range
spatial coherence along the condensate extent was measured
by the Mach–Zehnder interferometer [36]. As was men-
tioned in [39] electrically pumped polariton lasers are a new
generation of coherent light sources and represent a model
system for the fundamental understanding of strong-coupled
quantum electrodynamics. They promote the development
of nanophotonics.
9. Conclusions
As was underlined in the previous section, the Bose–
Einstein condensate is characterized not only by a macro-
scopic population of the ground state but, that is very im-
portant, by the existence or by the formation of a coherent
macroscopic state described by a macroscopic quantum
wave function Ψ named also as the order parameter, and
this wave function has the properties
†
0 ,NΨ Ψ = 0 e ,iN αΨ = 0 .N V (17)
The fixed or the well defined value of the phase α is a
very important property of the Bose–Einstein condensate.
If the Bose–Einstein condensate is slowly inhomogeneous
in space and in time, in this case we have the functions
0 ( , )N x t and ( , ),x tα which are changing following the
Gross–Pitaevskii equation. But ( , )x tα is well defined and
is not changing randomly as in the case of noncondensed
particles. Their distribution functions ( , )n x tp with wave
vector p different from the condensate wave vector
( )≠k p k are determined by the Boltzmann kinetic equa-
tions. In the homogeneous system and in the case of the
BEC in the single-particle state with wave vector 0≠k
the coherent macroscopic state is characterized by the
wave function 0( , ) e e ,i t ix t N− µ −Ψ = kx where µ is the
chemical potential. They lead to the nondiagonal elements
of the single-particle density matrix
(1) † ( ) ( )0
0
0
1( , ) ( , ) ( , ) e e .i iN
x x x t x t n
V V
− −′ ′ρ = Ψ Ψ = =′ ′ k x x k x x
(18)
Their absolute value in the thermodynamic limit
0N → ∞ and V → ∞ and in the limit − → ∞′x x equals
to 0 0.n ≠ It means that the system has the off-diagonal
long-range order and satisfies to the condition formulated
by Penrose and Onsager [47]. This criterion is a central
concept of the theory of BEC phenomenon [25]. In the
review articles [24,25] and in the original experimental
investigations described in the Refs. 30, 31 a special atten-
tion was paid to the interferometric measurements of the
phase coherence of the cavity polariton condensates. On
this way, in Ref. 31 it was shown that the coherence time
of 0N condensed polaritons is greater than the coherence
time of a single polariton. In such a way it was confirmed
the existence of the macroscopic coherent condensation
fraction. The temporal coherence of the polariton laser
radiation has a collective nature due to the polariton-
polariton interaction in the condition of the pump and of
the loss and, that is very important, it exceeds the lifetime
of a single polariton. Side by side with the measurements
of the population distributions, the interferometric data
testify the establishing of the polariton condensate as a
collective coherent state. Its formation in the condition of
pump and loss is a marvelous event saying us that every
decayed polariton was compensated in the frame of con-
densate by another pumped polariton exactly with the same
phase. Such substitutions are possible due to the bosonic
induced scattering processes, the rate of which is propor-
tional to the number of particles in the final state. In the
investigations represented in the Refs. 30, 31, 24, 25 not
only the temporal coherence of the polariton condensates
but also their spatial coherence were studied. Toward this
end the photons leaked from the microcavity from different
sites of the extended in space condensate were directed to
the interferometer to determine the degree of coherence
along the full surface of the condensate. It was shown ex-
perimentally [31] by the interferometric measurements that
the spatial coherence of the condensate and emitted radia-
tion is much greater than the spatial coherence of a single
photon. Below threshold the spatial coherence of a single
particle is determined by its de Broglie wavelength dBλ
[24]. The dispersion law of the emitted photons escaped
due to the leakage from the cavity coincides with the curve
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 5 435
S.A. Moskalenko and I.M. Tiginyanu
of the lower polariton branch and does not follow the cavi-
ty photon dispersion [24]. Excitons in GaAs-type crystals
have the resulting spin structure with the projections ex
zj
=±1 for the bright dipole-active states and with projections
ex 2zj = ± for the forbidden dark states. The bright states
determine the polarization of the emitted radiation used in
the devices of the spin-optronics. The interaction of the
polaritons gives rise to the shift of the polariton energy
levels, which is enhanced due to the small volume of the
microcavity and to the stronger exciton–photon interaction
in it than in an unlimited 3D space. The self-repulsion of
the polaritons leads to their blue shift and to extremely
high optical nonlinearity. A few numbers of photons in
microcavity with small volume can form a polariton gas
with considerable density. The repulsion between the
polaritons can be used to organize the pushing force and
photon blockade [24]. In such a way the polariton conden-
sates are an excellent platform to study the physics of
quantum gases, quantum phase transitions and quantum
vortices [24]. In another review article in Ref. 25 the dif-
ference between the coherence in the case of the conven-
tional lasers and in the case of polariton condensates is
analyzed. In the first case we deal with the coherence of
the emitted photons in conditions when the media remains
incoherent. In the case of polariton BEC the coherence is
accumulated in the condensed fraction of the polariton gas,
not in the cavity photons [25]. The coherent photon radia-
tion outside the microcavity appears due to the direct trans-
formation of the condensed polaritons into the leaked pho-
tons without any other stimulation processes. The dis-
tinction between two variants concerns the particle species
undergoing the coherence. Laser becomes coherent by
stimulated emission of radiation, whereas the polariton
condensates are coherent owing to the bosonic stimulated
scattering processes. This distinction is clear if the po-
lariton lifetimes are much greater than the thermalization
time. In many cases it is not so and the intermediate re-
gimes do exist [25]. Such regime when the strong-coupling
exciton–photon interaction in the cavity does exist and the
coherent macroscopic polariton condensate with a well
defined fixed phase and with a macroscopic population of
the ground state did appear but without a full therma-
lization process was proposed to be named as polariton
laser. To this proposal it is necessary to add one remark. At
least a partial thermalization is obligatory needed for the
condensate formation. Without many stimulated bosonic
scattering processes the common phase for all condensed
particles could not appear. In fact the Stoof [12] scenario
of the BEC phenomenon is more suitable to describe these
processes. Stoof [12] mentioned that the phase relaxation
and the condensed particle number 0N establishing take
place simultaneously in three stages, namely in two kinetic
stages and in one coherence stage. After the first kinetic
stage with the short characteristic time elτ and the coher-
ence stage with the same duration, during which the con-
densate embryo appears, there is a second kinetic stage
with a longer duration, possibly longer than the polariton
lifetime. During this final stage the condensate embryo has
to grow and to behave in equilibrium with the noncon-
densed fraction of the Bose gas. This last kinetic stage is
not completely realized in the case of the cavity polariton
condensates. Nevertheless the reliable establishing of the
condensate phase coherence in the frame of the polariton
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1. Introduction
2. Bose–Einstein condensation. Fundamental concepts
3. BEC of excitons in semiconductors
4. Polaritons
5. Two-dimensional Bose-gas
6. The driven dissipative nature of the polariton condensate
7. Superfluidity and vortices
8. GaN-based exciton polariton laser
9. Conclusions
|