Transcript a = l 0

Condensed exciton-polaritons in
microcavity traps
C. Trallero-Giner
Centro Latinoamericano de Fisica, Rio de Janeiro, Brazil
Quito/Encuentro de Fisica/2013
Outline
I. Introduction
II. Mean field description of EPC
III. Bogoliubov excitations
IV. EPC coupled to uncondensed polaritons
V. Conclusions
I. INTRODUCTION
Satyendra Nath Bose y
Albert Einstein
Boson: Statistics
A. Einstein, Sitzungsber. K. Preuss.
Akad. Wiss. Phys. Math., 261, (1924).
Bose-Einstein Condensation
of Rb 87
Phase transition
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Eric Cornell and
Carl Wieman
O. Morsch and M. Oberthaler,
Reviews of Modern Physics, Vol. 78, (2006), 179.
Gross-Pitaievskii equation
μ-the chemical potential
ω-trap frequency
m-the alkaline mass
λ-self-interaction parameter
L.K. Pitaevskii, Sov. Phys. JETP, 13, (1961), 451
5
Bose-Einstein condensation in an optical
lattice
REVIEWS OF MODERN
PHYSICS,
VOLUME 78, JANUARY 2006
6
•Bloch oscillations
Phys. Rev. Lett. 82, 2022 (1999)
•Superfluidity
•Dispersion and effective mass
Phys. Rev. Lett. 86, 4447 (2001)
•Josephson physics in optical lattices
•Mott-insulator transition
7
Superfluidity
S. Burger, et al., Phys. Rev. Lett. 86, 4447 (2001).
Estabilidad de la ecuación de GP y de las
soluciones
X t   center of mass
8
Phys. Rev. Lett. 86, 4447 (2001).
C.Trallero-Giner et al
Eur. Phys. J. D 66, 177
(2012).
9
Exciton-polariton
condensates
Atoms
9
Tc ~ 10 K
polaritons-----m is 0.0001 electron mass Tc ~ 300K
2.- The excitons and
photons interaction
form a new quantum
state= polariton.
Science V. 316
1.- Photons from a laser create electron-hole pairs or
excitons.
Excitations in a Nonequilibrium of Exciton
Polaritons
PRL 99, 140402 (2007)
G. Tosi et. al., Nature
Physics 8, 190 (2012).
Spatially-mapped polariton condensate wavefunctions
Expt. scheme with two 1μm-diameter
pump spots of separation 20μm.
The effective potential V (red)
Tomographic images of polariton emission (repulsive
potential seen as dark circles around pump spots).
G. Tosi et. al., Nature Physics 8, 190 (2012).
Real space spectra along line
between pump spots
Spatially-resolved
polariton
energies on a line
between
pump spots (white
arrows).
Interactions in Confined Polariton Condensates
PRL 106, 126401 (2011)
I. MEAN FIELD DESCRIPTION
OF EPC
The description of the linearly polarized exciton polariton condensate formed i
a lateral trap semiconductor microcavity:
two dimensional time dependent Gross-Pitaievskii equation
g – self-interaction parameter
m – exciton-polariton mass
R – loss
F – generation
Vres – interaction with the reservoir
V(r) - confined
potential
Assuming R, F constant and using the
transformation
we get
Y. Núñez Fernandez et al (to
be published)
Two limit cases
A) Under stationary conditions, R ≈ F and the number
of polaritons in the reservoir is small enough Nr ≪Np
Stationary GPE
Bogoliubov-type elementary excitations
B) Under the conditions
Nr ≫ N p
R < F, t he stability condition of the condensate
t
after a certain time, t ∼ 1/[R-F] .
Text
Linear differential equation
III. BOGOLIUBOV
EXCITATIONS
The collective excitations with frequencies ω
We linearized in terms of
the
amplitudes u and v
Axial symmetry: z-component of the angular momentum, mz and the
principal quantum number, N
s
Perturbation theory approach
-some numbers
Partial conclusions
-The excitation modes are weakly
dependent Λ.
-The total energy of the excited
state, shows almost the same blue-shift
dependence on Λ as the ground state energy.
-The spectrum of the Bogolyubov-type
excitations is nearly equidistant.
IV. EPC COUPLED TO UNCONDENSED
POLARITONS
Vres – interaction with the reservoir
Assuming that the interaction with the reservoir is proportional to the pump profile
vres=
gres is the coupling constant describing the repulsive interaction with uncondensed excitons.
=a/l0
Employing the Bubnov-Galerkin method we solved
the above linear equation.
Normalized energy levels,
EN;0, of the EPC coupled to
uncondensed polaritons
created at the center of the
trap as a function of the laser
excitation power (Λres). Solid
lines are for a = 0.2l0 and
solid lines with full circles for
a = l 0.
Two limiting cases can be distinguished.
-If a = 0 the energy levels tend to the harmonic oscillator
eigenvalues, EN = N + 1
-If a = ∞, EN = N + 1+ Λres.
The level spacings, ∆EN = EN+2-EN, show a strong dependence
on the laser spot size.
Laser pumping setup.
a)The pumping scheme.
For example,
-if a = 0.2l0,
∆EN = EN+2-EN ∼2, as for the 2D harmonic potential.
-if a increases,
∆EN,M=0 ≠ 2 and it depends on the number of polaritons in the
reservoir.
Figure shows the influence of the
uncondensed excitons on the
condensate
-The position of the density maximum is pushed
away from the origin as Λres increases.
-It is linked to the repulsive interactions produced by
the Gaussian density profile of uncondensed polaritons
created in the trap.
-The condensate is repelled from the origin as the
number of uncondensed excitons Nr (proportional
to the pumping beam intensity) increases..
PRL 106, 126401 (2011)
Infuence of the laser spot on the EPC
density.
a=l0
1 2

Veff    vres exp( 2 )
2
a
2
0  0;
1,2
2vres
 a ln 2
a
The dependence of
the condensate
density profile on
Λres for the excited
states with N = 1, 2
and 3.
Acknowledgments
Y. Núñez-Fernández
Havana University,
M. Vasilevskiy
Universidade do Minho,
A. I. Kavokin
University of Southampton
V. CONCLUSIONS
1.- We obtained convenient analytical description of the
Bogolyubov-type elementary excitations. This can be used
to describe the dynamics of the polariton BEC.
I
G. Tosi et. al., Nature
Physics 8, 190 (2012).
2.- The spectrum of these Bogolyubov-type excitations is
almost equidistant even for rather larger values of the
polariton-polariton interaction parameter.
∆EN = EN+2-EN = 2
3.- We obtained a semi-analytical solution for the ground and
excited states of the condensate consider when the
interaction with the reservoir of uncondensed polaritons is the
most important one. It is shown that the states are
"reshaped" by the repulsive interaction with the reservoir. Our
results are in agreement with recent experiments
4.- It is shown that the level spacings between the
condensate states increase with the pump power in
correspondence with the recent experimental observation. We
conclude that the experimentally observed emission patterns
in confined condensates, pumped through polariton reservoir
are not due to Bogolyubov-type elementary excitations in the
condensate itself, rather they are determined by the repulsive
condensate-reservoir interaction reshaping the density profile.
∆EN = EN+2-EN ≠ 2
Nature Physics 8, 190
(2012).
5.- We point out that the spectrum of these Bogolyubov-type
excitations in a condensate whose interaction with
uncondensed polaritons can be neglected, is almost
equidistant even for rather larger values of the polaritonpolariton interaction parameter inside the condensate. This
makes polariton parabolic traps promising candidates for
realization of bosonic cascade lasers. [T.C. H. Liew, et. al., Phys. Rev.
Lett. 110 , 047402 (2013).]
THANKS
Light