Quantum-fluid dynamics of microcavity polaritons
A. Amo1, D. Sanvitto1, D. Ballarini1, F.P. Laussy2, E. del Valle2, M.D. Martin1,
A. Lemaître3, J. Bloch3, D.N. Krizhanovskii4, M.S. Skolnick4, C. Tejedor2 & L. Viña1
1Dep. Física de Materiales, Univ. Autonóma de Madrid, 28049 Madrid, Spain
2Dep. Física Teórica de la Materia Condensada, Univ. Autonóma de Madrid, 28049
3LPN/CNRS, Route de Nozay, 91460, Marcoussis, France
4Dep. Physics & Astronomy, Univ.. of Sheffield, S3 7RH, Sheffield, U.K.
Semiconductor microcavities offer a unique system to investigate the physics of
weakly interacting bosons. Their elementary excitations, polaritons—a mixture of
excitons and photons—behave, in the low density limit, as bosons that can undergo
a phase transition to a regime characterised by long range coherence1,2.
Condensates of polaritons have been advocated as candidates for superfluidity3;
and the formation of vortices4 as well as elementary excitations with a linear
dispersion5 are actively sought after. In this work, we have created and set in
motion a macroscopically degenerate state of polaritons and let it collide with a
variety of defects present in the sample. Our experiments show striking
manifestations of a coherent light-matter packet that displays features of a
superfluid, although one of a highly unusual character as it involves an out-of-
equilibrium dissipative system where it travels at ultra-fast velocity of the order of
1% the speed of light. Our main results are the observation of i) a linear polariton
dispersion accompanied with diffusion-less motion, ii) flow without resistance
when crossing an obstacle, iii) suppression of Rayleigh scattering and iv) splitting
into two fluids when the size of the obstacle is comparable with the size of the
wavepacket. This work opens the way to the investigation of new phenomenology
of out-of-equilibrium condensates.
Below a critical temperature, a sufficiently high density of bosons undergoes
Bose-Einstein condensation (BEC). Under this condition, the particles collapse into a
macroscopic condensate with a common phase, showing collective quantum behaviour
like superfluidity, quantised vortices, interferences, etc. Up to recently, BEC was only
observed for diluted atomic gases at μK temperatures. Following the recent observations
of non-equilibrium BEC in semiconductor microcavities at temperatures of ~10 K,
using momentum-1 and real-space2 trapping, the quest is now towards the observation of
the superfluid motion of a polariton BEC. For the same reasons that polaritons benefit
from unusually favourable features for condensation, such as very high critical
temperatures, it is expected that their superfluid properties would likewise manifest with
altogether different magnitudes, such as very high critical velocities. Since they have
shown many deviations in their Bose-condensed phase from the cold atoms paradigm, it
is not clear a priori to which extent their superfluid properties would coincide or depart
from those observed with atoms, among which quantised vortices6, frictionless motion7,
linear dispersion for the elementary excitations8, or more recently Čerenkov emission of
a condensate flowing at supersonic velocities9, are among the clearest signatures of
quantum fluid propagation.
Microcavity polaritons are two-dimensional bosons of mixed electronic and
photonic nature, formed by the strong coupling of excitons—confined in semiconductor
quantum wells—with photons trapped in a micron scale resonant cavity. First observed
in 199210, these particles have been profusely studied in the last fifteen years due to
their unique features. Thanks to their photon fraction, polaritons can easily be excited
by an external laser source and detected by light emission in the direction perpendicular
to the cavity plane. However, as opposed to photons, they experience strong inter-
particle interactions owing to their partially electronic fraction. Due to the deep
polariton dispersion, the effective mass of these particles is 104-105 smaller than the free
electron mass, resulting in a very low density of states. This allows for a high state
occupancy even at relatively low excitation intensities. However, polaritons live only a
few 10-12 s in a cavity before escaping and therefore thermal equilibrium is never
achieved. In this respect, a macroscopically degenerate state of polaritons departs
strongly from an atomic Bose-condensed phase. The experimental observations of
spectral and momentum narrowing, spatial coherence and long range order—which
have been used as evidence for polariton Bose-Einstein condensation—are also present
in a pure photonic laser11. The recent observation of long range spatial coherence12,
vortices4 and the loss of coherence with increasing density in the condensed phase13,14,
are in accordance with macroscopic phenomena proper of interacting, coherent
bosons15. But a direct manifestation of superfluidity is still missing.
In this work, using a new combination of temporally and spatially resolved
spectroscopic techniques, we are able to probe directly the dynamics of motion of a
non-equilibrium, coherent polariton quantum fluid, by tracking its space-time evolution.
We report a shape preserving, non-diffusive propagation of matter dressed by the light
field moving at a speed of the order of 106 m s-1. Studying the collision against obstacles
of different sizes, we investigate the analogies of this state of matter with conventional
BEC such as those realised with atomic gases, whose dynamics are known to display
striking properties like superfluidity. We show that the study and manipulation of the
condensate properties can thus be achieved in semiconductor chips of micrometer scale.
To make polaritons flow, we have to address three important issues: 1) the very
short polariton dwell time in the cavity (< 2-5 ps), which hinders detection of their
dynamics; 2) creation of polaritons with well-defined momentum; 3) spatial
inhomogeneities given by sample strain and/or defects. Our experiments are based on
the continuous replenishing of the polariton fluid, at energy ES and momentum kS, from
a higher-lying state, at EP and kP, driven coherently by an external CW laser and ignited
by a short trigger pulse (2 ps) at the idler state, EI and kI, in a configuration of a
triggered optical parametric oscillator (TOPO; see Fig. 1 and the Supplementary
Information)16. The signal polaritons, fed by the pump in resonance with the lower
polariton branch, last for more than 10-9 s after the trigger pulse of 2 ps, thus allowing
the detection of the polariton flow dynamics. This approach solves the first issue listed
above: although the lifetime of a single particle of the polariton droplet remains short
(few picoseconds), the packet itself lasts hundreds of picoseconds. To reduce the effect
of the spatial inhomogeneities present in all semiconductor microcavities17,18, we keep
the pump power at sufficiently high intensity so that most of the potential fluctuations
are smoothed out19. Nonetheless, scattering centres are still present with an approximate
density of 10-2 µm-2 but, as shown below, they can be beneficial to reveal the peculiar
quantum nature of the polariton fluid.
Images of the near- and the far-field of a GaAs based microcavity20 are projected
on to the entrance slit of a 0.5 m imaging spectrometer attached to a streak camera, thus
allowing for simultaneous collection of two dimensional images, at a given energy, and
resolved in time. To avoid problems caused by detection at the pulsed laser energy,
which would bleach the streak camera, the pulsed laser is resonant with the idler rather
than the signal state of the triggered OPO. Furthermore, to retain only the light emission
from the TOPO signal, the OPO-only emission coming from the pump field,
contributing at k=0, is subtracted from all the images. We can thus study the
propagation of the signal polaritons which can have any, specifically selected, in-plane
momentum k-vector given by the phase matching conditions shown in the caption of
The experimental dispersions of the light emitted by the polaritons in different
regimes are depicted in Fig. 2a. The false-colour graph in the upper panel presents the
emission of the polariton fluid generated by the TOPO, after the trigger pulse has
disappeared, and it is compared to the standard parabolic dispersion of the
photoluminescence obtained under non resonant low-power excitation (lower panel).
The most striking feature, under TOPO conditions, is the clear linearization of the
dispersion around the signal state. It is also seen that, for the TOPO, polaritons are blue
shifted (ΔE = 0.6 meV at k=0), due to polariton-polariton interactions, and that the
intensity peaks strongly at kS = +0.6 µm-1, which is determined by the phase-matching
conditions (2kP = kS + kI).
Our system involves macroscopically degenerate states of bosons coupled through
exciton-exciton interactions and so can be accurately described theoretically by a
nonlinear Schrödinger equation for the polariton wavefunction in the presence of a
continuous pump FP and a pulse FI, shining on the microcavity at an angle kP,I, with
frequency ωP,I and localised in a Gaussian spot of spatial extension σP,I:
( , ) x t/2( , )x t ( , )x t
i k xt
t t i k xt
i D i
V F ee
where the operator D is the free-propagation energy of the particles, i.e., in our case,
provides the dispersion relation specific to polariton branches. The lower branch gives
rise to the OPO or TOPO physics of phased-matched scattering. The two last terms are
responsible for maintaining the system out of equilibrium against the decay γ. This
represents a crucial difference with the usual case of atomic condensates. Equation (1)
is integrated numerically, first in the absence of the pulse (FI =0), until a steady state is
reached for |ψ|2. We obtain the energy-k density plot of the system by Fourier-
transforming ψ(x,t). At a given time tI , we release a Gaussian pulse (FI ≠0) that triggers
OPO processes. We track the new evolution of ψ(x,t) until the steady state is restored,
and Fourier-transform again during this time. In order to compare with the experiment,
we subtract the two dispersions obtaining the result shown in Fig. 2b. This reproduces
both the linearization induced by the interactions and a strong signal-idler emission that
proves the triggering of OPO scattering.
In figure 3, real-space images of the signal polaritons are shown along with their
counterpart in k-space. The images show that the polariton droplet undergoes
unperturbed motion without diffusion in either x and y directions, proving that the
dispersion around the signal state has lost its locally parabolic character. At the same
time, the polariton packet also conserves a well defined momentum during all the
propagation time, as seen on Figs. 3b. This confirms that the polariton signal lies on a
linearized dispersion, as predicted theoretically and observed experimentally on Fig. 2.
On the contrary, at low pump powers (without linearization of the dispersion), we have
observed, the very fast appearance of the Rayleigh scattering circle in reciprocal space3,
and the absence of collective motion in real space (See “film_1_k_space” and
“film_1_real_space”, at http://www.uam.es/semicuam/films.html). This observation
demonstrates the suppression of scattering as we cross the threshold into the coherent
This propagation is reproduced by our numerical simulation, as seen on Fig. 4a,
where a snapshot of the signal is shown at regular intervals of times. Clear propagation
within the excitation spot is sustained, after the pulse has triggered the OPO (~2 ps),
until the signal reaches the edge of the excitation spot. Note that the signal movement is
restricted to the pump spot extension. It has a constant speed of 0.95 × 106 m s-1, in
agreement with that obtained experimentally (1.2 × 106 m s-1). To see the motion of the
signal, we take advantage of its separation in energy from both the idler and the pump
by filtering the emission in a window of energy, both in theory and experiments. In Fig.
4b, the spread of the polariton wavepacket is plotted as a function of time both from the
experimental data (black points) and according to calculations based on a parabolic (red
line) or linear (dashed blue) dispersions. The calculation clearly shows that the wave-
packet would expand more than twice as much during the same time of flight without
the linearization due to interactions. Moreover, the excellent agreement with the
experiments using equation (1) implies that polariton interactions are responsible for the
linearization of the signal state.
The quantum character of the polariton fluids created with a TOPO can be
studied by observing their collisions against structural defects, naturally present in the
sample. Figure 5-Ia shows images obtained in the near-field of a polariton fluid
colliding against a defect occurring in the middle of its trajectory. In the course of its
propagation, the signal shows unambiguous signs of interacting with a potential.
However, it clearly maintains its cohesion in this process. This is most strikingly
observed in the k-space counterpart, Fig. 5-Ib, that is left completely unaltered, until the
very end of its trajectory, where the single-state occupancy starts spreading as the signal
dies by moving off the edge of the pump laser region. Note that the images reflect the
addition of two different contributions: i) that of the pump polaritons (extended in an
area of ~ 8×103 µm2) and ii) that of the signal polaritons themselves, which being
constantly feed by the pump reflect its density fluctuations. Figure 1c illustrates how
these two contributions are detected at the signal polariton energy. The fringes observed
around the defect appear due to the local change in density of the pump polaritons,
which is reflected in the spatial structure of the signal. The pump polaritons are injected
in a coherent state, at high energies, high density and with high k-vectors, with a group
velocity higher than the velocity of sound21, vs, (i. e., a Mach number > 1), giving rise,
in the presence of a defect, to very characteristic quantum interferences resembling
Čerenkov waves, observed through the emission of the signal polaritons. Similar
shockwaves have been reported recently for an atomic BEC flowing against a potential
barrier at Mach numbers greater than one9. It is important to note that the visibility of
these waves does not imply that the signal polaritons are also in the “Čerenkov” regime.
On the contrary, for the signal polaritons (which are at lower energy and wavevector),
the group velocity is lower than vs (0.9×106 ms-1, i. e. Mach < 1), and thus the signal
polariton droplet can be expected and is observed to behave as a superfluid whose state
is characterised by a macroscopic wavefunction with a well defined common phase; the
droplet doesn’t change its motion and maintains constant its wavevector while passing
through the obstacle.
On Fig. 5-II, a more striking collision is observed, as the size of the defect is
now comparable with that of the polariton packet. The finite-size travelling polariton
fluid scatters coherently and elastically on the potential and is split in two after the
collision. Note that the process is dissipationless. A normal polariton fluid22,23 would
diffuse both in real and reciprocal space in this configuration, whereas a quantum fluid
whose dispersion has been linearized would pursue coherent and diffusion-less
trajectories as borne out by our experiments and clearly shown in the real-space images
of Fig. 5-IIa, albeit with two new momenta. The linear dispersion is the key element for
this coherent propagation, as any scattered particle from the wave-packet will remain in
phase with the others and at the same group velocity, precluding diffusion both in real
and in reciprocal spaces. Note that this concerns the wave-packet itself, not only its
excitations as is the case in the Landau picture of Helium superfluidity.
Our experimental TOPO configuration differs from the conventional realization
of atomic Bose-Einstein condensates and superfluids formed spontaneously in thermal
equilibrium without the action of any external driving field. In our case, due to the lossy
character of polaritons, an external coherent source needs to drive the system.
Furthermore, the fluid has a finite spatial extension and propagates as a whole, hence
we are investigating the explicit kinematics of a droplet of BEC, rather than the indirect
propagation of a defect inside an infinite size system. Also, thanks to the polariton light
emission, we are able to probe continuously and in real time such features as the motion
of the droplet and its dispersion that, as a manifestation of dominant interactions which
strongly dress the states, is linearized when a signal exists and propagates. This
dispersion is the nonlinear-response equivalent of the Bogoliubov dispersion for
excitations of a superfluid. In the latter case, linearization leads to suppression of weak
scattering and therefore to dissipationless motion. In our case, the dispersion reflects the
dynamics of the whole system, rather than of its excitations only.
The propagation of the signal that keeps its shape unaltered over huge distances
could evoke a soliton surfing on the steady state OPO, what includes correctly the
notion of a non-perturbative excitation (triggered by the pulse). However, elements that
are crucial for the stability of a soliton such as attractive rather than repulsive
interactions, the dependence on the dimensionality of the system or on the particle
densities, are not present in our case, and a plethora of different shapes, width and
heights of the wavepackets can also be sustained on the same dispersion. Therefore,
interactions serve the main purpose of replenishing the signal rather than holding it
together, thus the soliton description is not adequate. Instead, this system provides a
coherent and macroscopic population of bosons isolated in energy on a linear
dispersion. Both ingredients are essential to account for the experimental findings. The
system exhibits the quantum dynamics of BEC. But also in this picture, many novelties
arise from the specifics of polaritons. Up to now, theoretical analysis has only been
made in the perturbative regime24 with regarding to the elementary excitation of the
polariton BEC. These works show that for dissipative systems under incoherent
pumping, the Bogoliubov dispersion becomes diffusive and its characteristic linearity—
which is the key signature of superfluidity in the Landau picture—is lost. However, in
the presence of a coherent source, like in our experiment, a truly linear mode is restored
even with damping3, and superfluid behaviour is observed. Our findings call for further
work, both experimental and theoretical, to reveal new properties of this unusual state of
matter: a coherent, macroscopically-occupied Bose fluid, propagating at ultra high
In conclusion, we have experimentally observed and theoretically reproduced the
motion of a polariton wavepacket travelling at ultrafast velocities in a semiconductor
microcavity. Along with the record breaking speed achieved by these condensed
droplets, their most striking characteristics are to be found in their dynamics,
characteristic of a superfluid with an unperturbed motion while crossing a weak
potential and undergoing coherent scattering against strong defects. This work
demonstrates that microcavity polaritons are ideal candidates to study exotic quantum
bosonic phenomena. Although, as shown by this work, the novel physics associated
with a quantum fluid of polaritons presents both fundamental and subtle deviations from
the atomic case, these might well prove to be assets in their future studies. For instance,
in a dissipative system, the particle-number conservation is lifted and a well defined
phase can be externally imprinted to polaritons, allowing the investigation of symmetry
Supplementary Videos can be found in http://www.uam.es/semicuam/films.html
We thank I. Carusotto and M. Wouters for fruitful discussions and D. Steel for a
critical reading of the manuscript. This work was partially supported by the Spanish
MEC(MAT2005-01388, NAN2004-09109-C04-04 & QOIT-CSD2006-00019), the
CAM (S-0505/ESP-0200). D.B. and E.V. acknowledge a scholarship of the FPU
program of the Spanish ME. D.S and M.D.M. thank the Ramón y Cajal Program.
Correspondence and requests for materials should be addressed to D. S.:
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kP, CW Pump
kP, CW Pump
Pump polaritons Pump polaritons
Signal polaritons Signal polaritons
EE ω= ?
EE ω= ?
-3-3-2 -2-1 -100112233
k (μm k (μm
CW Pump CW Pump
Fig. 1. (a) Experimental microcavity dispersion of the two polariton branches, showing
the condition of excitation of the continuous pump and the formation of a polariton state
at the CW pump energy, which feeds the signal polaritons at a lower energy, fulfilling
the conditions 2kP=kS+kI and 2EP=ES+EI. This process is initialised by a pulse at the
idler state. (b) Schematic drawing of the experimental conditions. Typically, the pump
injects a coherent polariton state at 10o into the lower polariton branch; the pulse arrives
at time t = 0 in resonance with the branch at 16o triggering the OPO and generating a
polariton signal with a finite momentum at 4º. The two laser beams are shown
impinging on the sample surface, the polaritons are created down in the cavity region
(grey circle). The upper distributed Bragg reflector (DBR) of the microcavity has been
depicted as transparent for the sake of visibility. (c) Schematic representation of the
motion of two fluids (pump- and signal-polaritons) against a defect depicted by the red
point. Although the two fluids are spatially in the same plane, they are sketched
separated in the figure to emphasize their energy difference. The lower part depicts the
movement of the signal polariton fluid, at ES=ħωS, represented by the circle running
from left to right on the black plane. It is possible to detect this motion thanks to the
continuous feeding from the pump polaritons at EP= ħωP, which are represented in the
upper part of the figure (in the gray plane).
Fig. 2. (a) Upper panel: false-colour plot of the polariton dispersion. Lower panel:
dispersion obtained under non-resonant low power excitation, for which polaritons are
not in a coherent high density phase. The emission of upper- and lower-panels are
normalised independently and shown with a linear scale. It is important to notice that
the dispersion in which the signal polaritons are moving (bright spot denoted by the
arrow in the upper panel) is different from the bare dispersion and shows a linear
dependence on k vector. The energy blue shift, ΔE (0.6 meV), is due to polariton-
polariton interactions. The white parabola is plotted on top of a set of maxima of the
emission too weak to be visible in the scale of the figure. (b) Computer simulation of
showing the linearization of the signal polariton dispersion due to stimulated
scattering processes occurring between a CW pump (blue circle) and a probe (red circle)
after the pulse has disappeared. The ordinate is referred to the bare exciton energy. A
blow up of the region around the signal is depicted in the inset. The red line describes
the bare dispersion relation of polaritons.