Download Exciton-Polariton Bose-Einstein Condensation: Advances and

Int. J. Nanotechnology, Vol. x, No. x, xxxx
Exciton-Polariton Bose-Einstein Condensation:
Advances and Issues
Maxime Richard1,*, Jacek Kasprzak2, Augustin
Baas3, Konstantinos Lagoudakis3, Michiel
Wouters4, Iacopo Carusotto5, Régis André1,
Benoit Deveaud-Plédran3, Le Si Dang1
1
Institut Néel, CNRS/UJF
25 rue des Martyrs, BP 166 - 38042 Grenoble cedex 9
2
School of Physics and Astronomy, Cardiff University, Cardiff, CF24
3AA, United Kingdom
3
IPEQ, Ecole Polytechnique Fédérale de Lausanne(EPFL), Station 3,
1015 Lausanne, Switzerland.
4
ITP, Ecole Polytechnique Fédérale de Lausanne(EPFL), Station 3,
1015 Lausanne, Switzerland.
5
INFM-CNR BEC and Dipartimento di Fisica, Universita di Trento,
via Sommarive 14 38050 Povo (Trento) ITALY
*Corresponding author: [email protected]
Fax: +33476881191
Abstract: In this review, we present a comprehensive set of experimental
results on microcavity-polariton Bose-Einstein Condensation (BEC), obtained
in a close collaboration between Institut Néel, Grenoble and EPFL, Lausanne.
First, the main observations are presented, i.e. massive occupation of ground
state and onset of long range order, which led to the conclusion that polariton
BEC indeed occurs. Then, the highly disordered environment in which the
condensation takes place is analysed: we show how interactions are a necessary
ingredient for polariton BEC. Finally we discuss quantized vortices evidenced
for the first time in polariton condensates. Their unusual features are shown to
be inherited from the disordered environment and the dissipative-driven
character of the polariton BEC.
Keywords: Bose-Einstein condensation, polariton, superfluidity, quantized
vortices, microcavity, strong coupling
Biographical notes: Maxime Richard was born in France in 1977. He joined
the Laboratoire de Spectrométrie Physique, Grenoble, France, in 2001. After
his PhD degree in 2004, he stayed at the Institute of Photonics and Quantum
Electronics at EPFL, Lausanne, Switzerland, for 2.5 years as a postdoctoral
fellow. Since the end of 2007, he is a research associate (CNRS) at Institut
Néel, Grenoble. His current research is mainly focused on spontaneous
formation and behavior of coherence, in semiconductor nanostructures (BoseEinstein condensation, superfluidity, superradiance).
Copyright © 200x Inderscience Enterprises Ltd.
1
Exciton-Polariton Bose-Einstein Condensation
2
Le Si Dang was born in Viet Nam. After his doctorate in Grenoble and his
post-doctoral stay at Lehigh University (USA), he joined the Laboratoire de
Spectrométrie Physique (Université Joseph Fourier) in 1980 and the Institut
Néel (CNRS, Grenoble) in 2007. He is a CNRS Directeur de recherche.
Presently he is coordinating the Associated International Laboratory “Center
for Photonics and Nanostructures” of CNRS and KIST (Korea Institute of
Science and Technlogy). His current research interests include macroscopic
quantum coherence in low dimensional semiconductor heterostructures (BoseEinstein condensation, superradiance) and optics and transport in single
nanowires.
1
Introduction
Particles in nature can be divided into two categories, depending on their spin: those
with integer spins are bosons, and those with half-integer spins are fermions. By contrast
to fermions, which cannot simultaneously occupy the same quantum state as a result of
the Pauli exclusion principle, bosons can undergo a Bose-Einstein condensation (BEC)
when their thermal de Broglie wavelength becomes comparable to their average
separation [1]. Then a large fraction of bosons in the system massively occupy the lowest
energy quantum state, resulting in the formation of a macroscopic matter wave. Since the
de Broglie wavelength scales as the inverse of the square root of the particle mass and
temperature, the BEC criterion is most easily satisfied at high temperature for bosons
with a light mass. BEC was demonstrated for the first time in a dilute gas of rubidium
atoms in 1995 [2, 3]. Because of the heavy atom mass (about five orders of magnitude
larger than the electron mass), sophisticated cooling techniques were needed to lower the
atomic gas temperature below the micro-Kelvin range to achieve condensation. For solid
state systems, excitons in semiconductors have long been considered as a promising
candidate for BEC. The idea was first proposed in 1962 [4, 5]. Indeed, their very light
mass as compared to atoms, of the order of the electron mass, would permit BEC at
temperatures of a few Kelvin, reachable by standard cryogenic techniques. However, in
spite of intense research efforts over the past three decades, no convincing evidence of
exciton condensation has ever been firmly established [6].
On the other hand, in 1992, semiconductor epitaxy has become sufficiently wellcontrolled to permit the growth of advanced heterostructures, such as high-quality
monolithic Fabry-Perot microcavities. When quantum wells are inserted in such a highfinesse microcavity, a strong coupling takes place between the cavity photon field and the
quantum well excitons. The resulting eigenstates are no longer exciton or photon modes
but a mixture of both called microcavity-polaritons [7, 8]. Figure 1 shows typical
dispersions of polaritons in the high and low energy branches (solid lines) and those of
bare exciton and photon modes (broken lines) in the (x, y) plane, z being the confinement
axis of the microcavity and quantum wells. In the small momentum range of the figure,
the exciton dispersion appears completely flat, by contrast to the strong dipersion of
polaritons which comes from their photonic component. Thus with a mass lighter than
that of excitons by about four orders of magnitude, microcavity-polaritons are excellent
candidates for solid-state BEC in their lower polariton branch [8, 9]. The drawback is
their very short lifetime (~1 ps), inherited also from their photonic component, so that
polariton thermalization could be problematic. In fact the polariton gas can become fully
thermalized, as a result of strong polariton-polariton interaction caused by their excitonic
Exciton-Polariton Bose-Einstein Condensation
3
component [10, 11]. In this paper, we will review the main results obtained on polariton
BEC, in collaboration between the Institut Néel and EPFL. First we will recall how BEC
has been demonstrated for a polariton temperature of 20K in CdTe based microcavities
[12]. Important signatures of the phase transition, such as Bose-Einstein-like distribution,
bosonic stimulation, and spectral narrowing have been first reported. The defining proof
is brought by a study of the order parameter across the transition. In the second part of
this paper, the mechanism of the condensation is studied and discussed: on one hand, and
contrarily to the case of cold atom condensation, interactions between polaritons are
shown to be necessary to overcome the environment disorder in which polariton BEC
takes place [13]. On the other hand, and more importantly, polaritons are a dissipativedriven system. This non-equilibrium situation is a major difference with conventional
atomic-like BEC, and directly affects the superfluid behaviour of the polariton
condensate [14].
2
Demonstration of polariton BEC
Figure 1 polaritons optical excitation
scheme represented in the (Energy E,
momentum k) plane. The dashed lines show
the pure cavity photon and quantum-well
exciton dispersion. The solid lines represent
the lower polariton branch (low energy
side) and upper polariton branch (high
energy side).
To demonstrate polariton BEC, a monolithic CdTe-based microcavity embedding 16
quantum wells has been grown by Molecular Beam Epitaxy. Energy dispersions of
excitons, photons, and polaritons in this microcavity are displayed in Fig. 1, where k// is
the in-plane momentum. The polariton state of zero momentum at the bottom of the
lower polariton branch constitutes the ground state of the system, in which BEC can take
place. To generate the polariton gas, hot electron-hole pairs are injected in the
microcavity by continuous wave laser pumping (Fig.1). Excitons are formed and relax
their excess energy through exciton-phonon and exciton-exciton scatterings. When
entering the light cone, they strongly couple with the cavity field to form polaritons. To
probe polaritons inside the microcavity, we take advantage of a very important property
of the strong photon-exciton coupling: the polariton field and its photonic part, which
leaks out of the microcavity as emitted photons, behave identically. In other words,
intensity, momentum, polarization and correlations at any order of the microcavity
emission are linearly connected with those of the polariton field [15]. In particular, the
Exciton-Polariton Bose-Einstein Condensation
4
distribution of the polariton gas in momentum space is fully reproduced in the angular
distribution of the microcavity far-field emission [16].
The large amount of works carried out to demonstrate Bose Einstein condensation of
excitons in semiconductors, e.g. in large gap semiconductors as Cu2O [17-19] and CuCl
[20, 21] or coupled quantum wells [22-24], as well as the huge experimental and
theoretical works developed for cold atoms BEC, show that the claim of Bose Einstein
condensation requires the observation of (at least) two defining signatures.
Of course, the transition from a Boltzmann distribution of the particles towards a BoseEinstein distribution is a prerequisite to claim BEC. The unique quantum statistics of
bosons should result in a very strong nonlinearity of the ground state population versus
increasing boson number: any bosons newly added to the system have an increasing
probability to end up in the ground state, instead of choosing randomly among the
available quantum states. This is in fact the observation first reported in 1995 [2, 3] that
convinced the scientific community of BEC in ultracold atom gases. It is important to
note that the temperature of the atom system was precisely assessed, and that the phase
transition was observed at thermal equilibrium by properly lowering the system
temperature. In semiconductors, the temperature of excitons is in general not precisely
measured, and thermal equilibrium is not usually clearly assessed. For polaritons, their
very steep energy dispersion around k// ~ 0 allows to measure precisely their distribution
in momentum space and thus derive their temperature when they are in thermal
equilibrium.
BEC should also result in the onset of spontaneous coherence. Thus a strong increase of
coherence time is expected across the transition. In the special case of dissipative-driven
condensates, such as polariton BEC, and in strong analogy with laser properties, the
coherence time in the condensed regime is not limited anymore by the lifetime of the
single particle in the system. Similarly, it is interesting to have a look at the second order
time correlation of the polariton system, i.e. fluctuations in time of the polariton emission
intensity, which allows distinguishing very clearly a mixed thermal state from a coherent
state. However, these studies are necessary but not sufficient to demonstrate polariton
BEC since the second order time correlation is only an indirect measurement of the order
parameter of the phase transition.
In the framework of phase transition phenomena, the most defining evidence of BEC is
the appearance of long range order [25, 26]. The proper and most natural order parameter
is the condensate wavefunction, which, for free and non interacting bosons, corresponds
to the single particle state of zero momentum. This idea of long range coherence and
macroscopic wavefunction unifies the understanding of BEC, superfluidity,
superconductivity and lasing [25]. However, this last one remains very distinct from
polariton BEC since it describes coherence of a pure photon field. Experimentally, the
order parameter can be monitored by measuring the spatial correlations in polarization
and phase within the polariton gas. Below the condensation density, these correlations
should be short range and characterized by the thermal de Broglie wavelength and then
extend as far as the edges of the polariton system when the condensate is formed.
Exciton-Polariton Bose-Einstein Condensation
5
Polaritons distribution
Figure 2 shows 2D images of the angular distribution of the polariton emission intensity
(coded in false color) measured at T = 5 K (lattice temperature), for three excitation
powers. As stated before, and within small corrections that will be discussed below, this
angular distribution of the emission intensity reflects the distribution of the polariton
population in the momentum space k// [16]. Below but close to the critical density (left
panel), polaritons are smoothly distributed around k// ~ 0 on the lower polariton branch.
Above some critical excitation P0 (middle and right panels), a massive build-up of the
population in the zero momentum ground state occurs, whereas the excited state
occupation remains almost unchanged. This bimodal distribution is a unique feature of
the Bose-Einstein distribution and typical of BEC at finite temperature. For this
experiment, it is important to note that, unlike cold atom BEC, the transition is driven by
the polariton density at fixed lattice temperature T = 5 K. However, although it is less
convenient from the experimental point of view, the transition can also be driven by the
temperature while keeping the density constant (not shown). The highest critical lattice
temperature for polariton BEC is found to be around 50 K.
Figure 2: (Energy E, momentum k) images displaying population distribution in the lower
polariton branch measured at T = 5 K, for three laser pumping powers: 0.84, 1.12, and
1.21 P0, where P0 is the condensation threshold power. The color scale is identical for
the three images. The bimodal distribution, consisting of massively occupied ground
state on one side and thermal cloud at higher energies on the other side, clearly emerges
above threshold.
Data such as those shown in Figure 2 can be used to extract precisely the occupation
of polariton states as a function of their energy. To do so, one needs to correct the
emission intensity by the photonic Hopfield coefficients, and by the polariton density of
states [8]. This correction leads to the result of Fig. 3.a: under very weak excitation
(white open circles), the polariton distribution is not thermalized. This is the so-called
“bottleneck effect” due to poor acoustic phonon relaxation on the lower polariton branch,
as a result of the very steep energy dispersion around k// ~ 0 [27, 28]. Upon increasing the
excitation power, a large exciton reservoir is formed at large k//, thus allowing exciton-
Exciton-Polariton Bose-Einstein Condensation
6
exciton scatterings to take place to populate the lower polariton branch [29]. In this
regime, polariton-polariton interaction becomes also significant, and the dynamics is fast
enough to achieve thermal equilibrium within the polariton population during the very
short polariton lifetime (~ 1 ps). This is shown by the red circle plot of Fig. 3.a: a
Boltzmann distribution fits the data accurately for a temperature of 19 K, significantly
higher than the lattice temperature of 5 K. At first sight, the observation of a polariton gas
in internal thermal equilibrium could appear puzzling considering the short polariton
radiative lifetime. In fact, numerical simulations of polariton relaxation using the
semiclassical Boltzmann equation show that equilibration can be achieved in the
polariton gas (but not necessarily at the lattice temperature), provided the excitonic part
in the polariton wavefunction is significant (> 50%, as is the case in this work) [11].
As shown by the solid diamond plot in Fig. 3b, the ground state occupation changes from
a linear to a quadratic dependence with respect to the excitation power just below the
condensation threshold. Increasing further the excitation, the occupation increases
exponentially. This is a direct consequence of the bosonic stimulation of the relaxation:
the relaxation rate toward the ground state increases with increasing occupation. The
polariton distribution becomes now very peaked on the low-energy side, and deviates
strongly from a Boltzmann distribution and resembles much more a BE distribution (Fig.
3a). It is important to note that the BE distribution cannot fit properly these data; in
particular the condensate linewidth is too large. This is a well understood BEC feature in
systems of strongly interacting bosons: strong scatterings are expected for bosons in the
condensate towards the low energy side of the thermal cloud [26, 30]. In the case of
polaritons, this effect limits the condensate fraction to about 50%, as compared to more
than 90 % in atom condensates with much weaker interactions.
Figure 3 (a) Polariton occupancy versus energy for several laser pumping powers. The power is
normalized to the threshold power of BEC. (b) Spectroscopy of the ground state across
the BEC transition: black filled diamonds plot is the ground state occupancy, hollow red
triangles plot is the ground state linewidth, filled green circles plot is the ground state
spectral blueshift normalized to the Rabi splitting.
7
Exciton-Polariton Bose-Einstein Condensation
Time coherence
BEC of cold atoms is achieved in a closed system. On the contrary, the polariton
system continuously leaks towards the outside world at a very quick rate. In that sense it
resembles very much a laser, and similar behaviours are expected, such as enhancement
of time coherence beyond the single particle radiative lifetime. This is shown in Fig.3.b
(open triangle plot): at the condensation threshold (~0.5 kW.cm-2), the ground state
linewidth changes abruptly from 0.7 meV to 0.3 meV, as a result of increased time
coherence. Using a low noise excitation laser, coherence times as long as 150 ps (~10μeV
linewidth) have been reported at threshold [31]. Far above threshold, the linewidth gets
much broader due to increasing interactions within the condensate and with the
environment (hot excitons, free carriers...). This loss of coherence, also observed by
measuring the second order time correlations, is a direct consequence of the mixed nature
of half excitons and half photons of polaritons.
Second order time correlation
Second order time correlation provides access to the particle number fluctuations in
an ensemble of particles. The properly normalized fluctuations are given by the quantity
(1)
where n(t) is the particle number at time t, T the time delay characterizing the fluctuation
correlation, and the averaging is performed over time t. Note that this definition of g(2)(T)
is only valid for fluctuations above the standard quantum noise. It can be shown that this
quantity is intimately connected with the coherence degree of the system: for instance,
photoluminescence from a thermal gas (following Boltzmann distribution) features
important fluctuations corresponding to g(2)(0)=2 (“photon bunching”), on a time scale
which is that of the (short) coherence time of the emission. On the other hand, a purely
coherent state (for example, at thermal equilibrium: a BEC, or out of thermal equilibrium:
a laser) should show up the smallest possible classical fluctuations, also denoted as the
standard quantum noise, g(2)(0)=1.
Making use of a Hanbury-Brown and Twiss interferometer, we have performed a
detailed study of the second order time correlation of polaritons in the same 16 quantum
well CdTe-based microcavity, non-resonantly pumped by either pulsed or continuous
laser [32]. The main results obtained in this measurement are the following: (1)
fluctuations measured just above the condensation threshold are clearly those of a
coherent state, as expected for BEC. (2) Upon increasing further the pumping density,
fluctuations increase again, as a result of increasing polariton interaction with increasing
density. This last result is in full agreement with the linewidth broadening observed at
high density in Fig. 3b. It also implies that the condensate should contain a significant
excitonic fraction [33].
8
Exciton-Polariton Bose-Einstein Condensation
First order spatial correlation
Any BEC claim should include a study of the first order spatial correlation across the
transition. As previously mentioned, the polariton field inside the microcavity is
transmitted to the outside world by the emitted photon field of the microcavity. Thus the
first order spatial correlation of the polariton system can be measured by analyzing the
microcavity emission spot with an imaging Michelson interferometer, which was
appropriately modified in order to extract a spatial correlation mapping of the following
form:
(2)
where E(x,y) is the polariton emission field at the specified position, and the averaging is
performed over time. A typical result is shown Fig.4 for a polariton gas of 25 μm
diameter. The point (x=0,y=0) of the autocorrelation is chosen in the middle of the
polariton gas. Below threshold, a very short range correlation peak of about 1 μm
diameter is observed. This is expected for any classical gas: at thermal equilibrium the
mean correlation length is given by the de Broglie wavelength
(3)
where h and k are the Planck and Boltzmann constant, T the temperature and m the
particle mass. For polaritons at thermal equilibrium at 20 K (see Fig. 3a), the de Broglie
wavelength is λdB ~ 2 μm, slightly larger than the spatial correlation observed in Fig.4a.
In fact, polaritons are an out of equilibrium system, in which correlation length is
expected to be shorter than the de Broglie wavelength. Above threshold (Fig.4.b) very
long range correlations appear, limited only by the size of the condensate. This
demonstrates unambiguously the occurrence of BEC.
Figure 4 Correlation mapping of the polariton gas below (a) and above (b) BEC threshold. The
color scale represents the normalized first order space correlation g(1) between the
points (x,y) and (-x,-y). The central point (0,0) of the image has been chosen
experimentally in the middle of the condensate.
Exciton-Polariton Bose-Einstein Condensation
9
Another important feature of the correlation mapping of Fig.4.b is its rather
inhomogeneous character. This is a consequence of the disordered potential experienced
by the polariton condensate. This point will be addressed in the next section.
To conclude this section, it should be emphasized that we are dealing with polaritons
and not with pure photons, although all measurements are performed with the photon
field emitted by the microcavity. This crucial point relies on two unambiguous evidences:
- At and above threshold, the energy blue shift of the polariton emission energy remains
very small as compared to the Rabi splitting (~4% at threshold, cf. Fig.3.b green circles
plot). This shows that the microcavity is still operating with polaritons in the strong
coupling regime.
- Above threshold, the condensate fluctuations increase with increasing density. This is
an unambiguous proof that we are not dealing with pure photons but rather with
polaritons, which contain a significant excitonic fraction. Indeed for lasers (involving
only pure photons), it is expected and observed that fluctuations decrease steadily upon
increasing density [34].
Figure 5 First row displays real space images of polaritons clouds uncondensed (a) and condensed
(b) and (c). (b) and (c) were obtained at two different places with two different disorder
landscape. Second row represents a measurement where a vertical slice of real space
image of polaritons have been dispersed in a monochromator to obtain spatial and
spectral resolution at the same time. (d) is the result obtained below threshold and (e) at
the same place but above threshold.
Exciton-Polariton Bose-Einstein Condensation
10
Influence of disorder on the polariton BEC*
Nowadays, the growth of semiconductor heterostructures such as microcavities can be
controlled at the atomic monolayer scale in thickness. This ultimate limitation in a single
or two wavelength thick microcavity can induce significant fluctuations of the cavity
mode energy, around 0.8 meV in average in our microcavity sample. Such disorder
amplitude is comparable with the polariton linewidth, and thus could affect the polariton
system.
Below the condensation threshold, this disorder has little effect on the polariton cloud
in real space: as shown in Fig.5a, the polariton gas looks rather homogeneously
distributed over its whole diameter, with no visible disorder effect. This can be explained
by looking at the spectrally resolved image of the polariton distribution in Fig.5.d. The
existence of disorder is evidenced by the fact that the lowest energy polaritons (low
energy edge of the distribution) feature a position dependent energy. However, this
energy fluctuation is smaller than the average kinetic energy of polaritons in the cloud,
and therefore hardly visible in the polariton distribution. On the other hand, above the
condensation threshold, the effect of disorder in real space is dramatic, provoking a very
inhomogeneous density distribution (Fig.5.b and Fig.5.c). What seems to be a paradox at
first sight is that the spatial fluctuation of the polariton ground state energy has virtually
vanished (Fig.5e), as compared to the non-condensed situation (Fig. 5d).
This is a quite puzzling observation: polaritons being localized in a strongly
disordered environment below condensation, how a spatially extended condensate with a
well defined energy can be formed? There are two possible responses to this question:
either, condensation takes place in an extended state delocalized over several potential
minima, and not visible in a photoluminescence measurement like those of Fig.5d and
Fig5.e, or a new extended ground state is formed at high density as a result of nonlinear
interactions. The first hypothesis can be checked by measuring the first order spatial
correlation with spectral resolution for excitation power below (Fig.6.a) and above
(Fig.6.b) the condensation threshold [13]. Above threshold, the very long correlation
length (length over which interference fringes are visible along the red line of Fig.6.a) is
observed for the condensate, in complete agreement with data in Fig.4b. Below threshold,
it is strongly reduced (red line of Fig.6.b), ruling out the existence of any intrinsically
extended ground state. This suggests that a nonlinear effect is required to build up the
extended ground state in which condensation takes place.
Figure 6: Spatially (vertical axis) and spectrally (horizontal axis) resolved spatial correlations.
Unlike Fig. 4, the correlations are visible here as interference fringes: along the vertical
axis. The more contrasted the fringes the higher the correlations. The more extended the
fringes the longer the correlations. (c) was obtained for uncondensed polaritons and (d)
for a polariton BEC. The red dashed line shows the ground state energy.
11
Exciton-Polariton Bose-Einstein Condensation
The scenario of condensation in disordered environment could be as follows. At low
excitation pumping, polaritons are localized in different potential minima with different
densities. With increasing pumping, polaritons in deeper minima experience stronger
blueshifts due to higher densities, thereby reducing the initial spatial detuning between
the local minima. This could eventually lead to linear coupling and frequency
synchronization between the different localized sites, allowing BEC to occur in an
extended state of well defined energy. This vision has been confirmed by simulation
using a mean-field based calculation [35].
By contrast to the textbook condensation, the critical density of polariton BEC
depends very much on the local details of the disorder, and requires nonlinear interaction.
3
Vortices in a polariton superfluid
One of the most striking quantum effects that may occur in an interacting Bose gas at
low temperature is superfluidity. First observed in liquid helium, this macroscopic
coherence phenomenon has been studied quite extensively in a variety of systems. It
shows remarkable features such as the persistence of superflows and the emergence of
quantized vortices [36]. The achievement of BEC in dilute atomic gases provided the
opportunity to observe and study superfluidity in an extremely clean and well-controlled
environment [37, 38]. For polariton condensates, the situation is not so clear due to their
intrinsic dissipative-driven character and the disordered environment.
Figure 7 Left: Inset: local spatial phase mapping φ(x,y) within the condensate. The image is
centred on a phase singularity due to a vortex. The main plot shows the spatial phase
along circles centred on the vortex. Right: red filled circles: Density profile along a
vertical axis intersecting the vortex in (y=0). Blue filled circles: Density profile along a
horizontal axis intersecting the vortex in (x=0).
3
10x10
profile along x
profile along y
intensity(arb. un.)
9
8
7
6
5
4
-1.0
-0.5
0.0
0.5
relative position (μm)
1.0
Exciton-Polariton Bose-Einstein Condensation
12
Vortices in superfluids carry quantized phase winding and circulation of the superfluid
particles around their cores, characterized by:
- A rotation of the phase around the vortex by an integer multiple of 2π, commonly
known as the topological charge of the vortex.
- A vanishing superfluid density at the vortex core.
To look for these possible vortex signatures in polariton condensates, we have used
the same imaging Michelson interferometer as discussed before [14]. A locally
homogeneous area of the condensate is selected to set the phase reference of
interferogram patterns generated with other areas, from which both the phase and the
density of these areas can be mapped out. In the left panel of Fig.7, the inset presents
such a phase mapping in a vortex area. The vortex core is clearly identified by the
peculiar distribution of the phase, which increases steadily when following a closed loop
around the core. The condensate phase along closed circulation loops (shown in inset)
centred at various distances from the vortex core is plotted versus the angular coordinate
in the left panel of Fig.7. It is found to increase steadily from 0 to exactly 2π providing
the first signature of a vortex of single topological charge. Note that the azimuthal
dependence of the phase is not a straight line due to the local disorder.
The other vortex signature is obtained by mapping the condensate density around the
vortex. The position of the vortex core is at the center of the closed phase loops, and at
this very position, in spite of the poor spatial resolution of our setup, we indeed observe a
local minimum of the condensate density (Fig.7 right panel).
These two findings constitute clear evidence that single quantized vortices can be
found in polariton condensates. However, their features are rather unusual compared to
other systems: they are present without the need to stir the condensate; moreover their
position, topological charge and sign (sense of rotation) are permanently pinned in space
and time.
Vortices have been reported in several other kinds of superfluids, like supercurrent of
Cooper pairs [39], superfluid Helium [36], and atomic BECs. In this latter case, vortex
lattices have been generated upon stirring the superfluid [37]. Then the position, charge
and sign of quantized vortices are determined by the size of the condensate and the
externally applied stirring motion. For superfluids at rest, vortices can also be thermally
activated. But in this case, their lifetime is limited and their characteristics randomly
fluctuate in time and space. Such a situation could be found in a quasi-2D condensate:
between the pure BEC phase and the normal phase, the system undergoes the
Berezinskii–Kosterlitz–Thouless (BKT) transition. It corresponds to the formation of a
condensate containing thermally activated vortices, which cause the spatial correlations
to drop exponentially. This phase has been nicely realized in cold atom condensates [38].
However, its features are in clear contradiction with our observations, since polariton
condensates display a fully deterministic vortex behaviour and spatial correlations that do
not decay exponentially but remain constant versus distance (not shown).
To get a better understanding of polariton vortices, we have performed a mean field
calculation of the system, which included driving and loss terms as well as spatially
inhomogeneous potential [14]. With this model, we found out that permanent vortices of
fixed location, charge and sign could be generated by the combined effect of the
Exciton-Polariton Bose-Einstein Condensation
13
dissipative-driven character of polaritons and the disorder potential. Actually, the
condensate polaritons can be pictured as flowing constantly down the hills of the disorder
potential landscape, along permanent streamlines. In some places of the disorder, this
motion generates permanent vortices.
Optical vortices have also been observed in the near field of VCSEL laser beams, as a
result of the nonlinear interaction between the light and the amplifying medium [40].
However the physical situation is totally different as compared to microcavity polaritons
since VCSELs are operated in the weak coupling regime of the light-matter interaction,
and since the nonlinearity is of completely different nature in the two systems.
4
Conclusion
Although polariton BEC has already shown very rich physics, many questions are still
open. Let us mention a few interesting issues. The first one is the condensate excitation
spectrum, which determines the superfluid character of the condensate, using the Landau
criterion. For interacting bosons, this spectrum follows the Bogoliubov linear dispersion
[41], as reported for condensates of Rb atoms [42]. In the polariton case, the Bogoliubov
dispersion is expected to be modified on the low energy side due to the strong
dissipative-driven character [43]. Another interesting issue is to clarify the role of
disorder effect. Presently it is not clear whether disorder is favorable or not to polariton
condensation, as we have no way to remove or modify it. On one side, disorder breaks
down the 2D character of the polaritons (condensation of purely 2D bosons is forbidden
at the thermodynamic limit), but on the other side, it prevents the formation of long range
order until interactions come in.
To address these points and study many other fascinating effects, like Josephson
oscillation and self-trapping [44], dissipationless flow [45], Bloch oscillation [46] or
room temperature polariton BEC and lasing [47], it is worth exploring the strong
coupling regime with other photonic structures (photonic crystal resonators [48],
microdisks [49]) and materials with stronger excitonic effects [50, 51].
The part of this work done at Institut Néel has been carried out within the joint group
CNRS-CEA-UJF NanoPhysique et SemiConducteurs. The part of this work done at
EPFL has been carried out within the Quantum OptoElectronics group of the institute of
Photonics and Quantum Electronics.
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