Polariton Condensation in Dynamic Acoustic Lattices
E. A. Cerda-Me
´
ndez,
1
D. N. Krizhanovskii,
2
M. Wouters,
3
R. Bradley,
2
K. Biermann,
1
K. Guda,
2
R. Hey,
1
P. V. Santos,
1
D. Sarkar,
2
and M. S. Skolnick
2
1
Paul-Drude-Institut fu
¨
r Festko
¨
rperelektronik, Berlin, Germany
2
Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom
3
ITP, Ecole Polytechnique Fe
´
de
´
rale de Lausanne, 1015 Lausanne, Switzerland
(Received 7 June 2010; revised manuscript received 6 August 2010; published 9 September 2010)
We demonstrate that the tunable potential introduced by a surface acoustic wave on a homogeneous
polariton condensate leads to fragmentation of the condensate into an array of wires which move with the
acoustic velocity. Reduction of the spatial coherence of the condensate emission along the surface
acoustic wave direction is attributed to the suppression of coupling between the spatially modulated
condensates. Interparticle interactions observed at high polariton densities screen the acoustic potential,
partially reversing its effect on spatial coherence.
DOI: 10.1103/PhysRevLett.105.116402 PACS numbers: 71.36.+c, 42.50.p, 63.20.kk, 73.21.Cd
Microcavity polaritons are hybrid light-matter quasi-
particles arising from the strong coupling between excitons
and photons confined in semiconductor microcavities.
Their photonic character lends them very low masses (on
the order of 10
5
of the electron mass), which favors
condensation at low injected polariton densities [1]. The
properties of this solid-state system can be controlled by
structure design and manipulated and probed by external
light beams. Unlike better known Bose-Einstein conden-
sates (BECs) [2] such as cold atoms or liquid helium,
polariton condensates are intrinsically out-of-equilibrium
[3]. They exhibit characteristic condensate properties such
as vorticity [4], extended spatial [1], and temporal coher-
ence [5] as well as collective fluid behavior [6].
Cold atoms and atomic BECs in optical lattices have
attracted significant interest, providing access to the co-
herent transport properties associated with matter field
coherence or decoherence [7]. In contrast, investigation
of a spatially modulated polariton condensate, a nonequi-
librium and strongly interacting light-matter system, pro-
vides a highly complementary insight into this field.
Polariton condensates in a static lateral periodic potential
have been addressed [8]. However, polariton condensation
in a strong tunable periodic potential, which is essential to
study coherence or decoherence of this system, remains
unexplored. Recently localization or delocalization of
noncondensed excitons in coupled quantum wells was
reported [9].
In this Letter, we achieve polariton condensation in
m-sized periodic tunable potentials created by coher-
ent surface acoustic waves (SAWs). We show that the
application of the SAW to a uniform extended condensate
creates an array of condensate wires trapped at the minima
of the SAW potential, which moves with the acoustic
velocity. The observed reduction of the spatial coherence
of the array along the SAW propagation direction with
increasing acoustic intensity is attributed to the suppressed
coupling between the wires, which leads to independent
phases between adjacent condensates. We also observe
coherence reduction along the SAW wires, which we sug-
gest arises from increased phase fluctuations as the dimen-
sionality is reduced. Finally, nonlinear interparticle
interactions at high quasiparticle densities screen the ap-
plied acoustic potential, thus partially reversing the effect
of the SAW.
We study GaAs-based microcavities with six 15 nm-
thick quantum wells grown by molecular beam epitaxy.
The microcavities have a Rabi splitting of 6 meV and a
quality factor Q 2000 with nearly zero detuning between
the cavity and the exciton modes at k ¼ 0. The condensates
were generated at 5 K by pumping with a continuous
wave laser with in-plane wave vector k
ðpÞ
¼½k
ðpÞ
x
;k
ðpÞ
y
¼
ð1:8; 0Þ m
1
[Fig. 1(a)] in resonance with the lower
polariton (LP) branch. In this optical parametric oscillator
(OPO) configuration [Fig. 1(a)][6,10], two pump polar-
itons scatter into states fulfilling energy and momentum
conservation. As a result, three coupled macroscopically
occupied states (pump, signal, and idler) are formed. The
phase of the signal is independent of that of the pump laser
[11], implying the spontaneous breakdown of the U(1)
symmetry at the phase transition as in a standard BEC [12].
The spatial modulation of the polariton potential by
acoustic fields was introduced by a SAW of wavelength
SAW
¼ 8 m propagating along a nonpiezoelectric h100i
surface direction [the y direction in Fig. 1(b)]. The use of
this type of SAW, which does not carry a longitudinal
piezoelectric field, is essential to prevent field-induced
exciton ionization [13] and to observe condensation. The
SAWs were generated by interdigital transducers placed
on a piezoelectric ZnO island on the sample surface
[Fig. 1(b)][13]. Below we state the amplitude u of the
acoustic field in terms of square root of the nominal rf
power (u
ffiffiffiffiffiffiffi
P
rf
p
) applied to the interdigital transducer.
The SAW induces a spatial modulation of the polariton
energy along the y direction [cf. Figure 1(b)] through two
mechanisms: (i) a type I deformation potential modulation
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of the quantum wells band gap [13] and (ii) an in-phase
modulation of the microcavity resonance energy [14]. This
spatial and temporal modulation creates a one-dimensional
(1D) lattice for polaritons, where the energy (E) vs in-plane
wave vector [k ¼ðk
x
;k
y
Þ] dispersion is folded along k
y
within a mini-Brillouin zone (MBZ) of dimension k
SAW
¼
2=
SAW
. Figure 1(c) compares the unperturbed polariton
dispersion (i.e., calculated without acoustic excitation,
dashed lines) with the corresponding dispersion with a
SAW displayed in the extended (thick solid line) and
reduced (thin line) zone schemes [15]. The anticrossing
of the folded branches induces energy gaps in the center
and at the edges of the MBZ. For optical excitation den-
sities I
p
below the OPO condensation threshold I
OPO
th
¼
600 W=cm
2
, the far-field photoluminescence (PL) reveals
the one-particle polariton dispersion, as illustrated in
Figs. 1(d) and 1(e) in the absence and presence of a
SAW, respectively. The well-resolved energy gaps in the
center and at the edges of the first and second MBZs (i.e., at
nk
SAW
=2 ¼n=
SAW
, where n is an integer) indicate
the formation of a polariton lattice [14]. The superimposed
lines display the dispersion calculated using a perturbative
approach [14]. For the small acoustic modulation ampli-
tudes used in the present studies (up to 0.36 meV), the
width E of the gap at the edges of the first MBZ [see
Fig. 1(e)] becomes essentially equal to the peak-to-peak
amplitude modulation V
SAW
(V
SAW
u) of the LP branch
[15].
Above the OPO threshold, self-organization sets in [16],
forming a signal condensate at k 0 containing 10
3
par-
ticles as well as a weaker idler at k 2k
ðpÞ
. Figures 2(a)–
2(c) display E vs k
y
images of the signal emission recorded
under increasing SAW powers for a fixed optical excitation
power density of I
p
¼ 2I
OPO
th
¼ 1:2 10
3
W=cm
2
distrib-
uted over a 50 m diameter Gaussian spot. In the absence
of a SAW [Fig. 2(a)], a condensate is formed at k ¼ 0 and
energy of 1.5345 eV, which is approximately 0.4 meV
above the bottom of the LP branch below threshold. The
blueshift arises from interparticle interactions between
polaritons in the OPO modes [17].
When a SAW is applied the signal emission broadens in
momentum space [Figs. 2(b) and 2(c)]. We note that the
linewidth of the total stimulated polariton emission (reso-
lution limited to a FWHM of 60 eV, which is one third
of the subthreshold value), remains constant and much
smaller than the SAW modulation amplitude over the
whole range of acoustic powers, thus indicating persistence
of temporal coherence. In addition, the condensate emis-
sion intensity does not change with SAW power, indicating
negligible effect of the SAW on the pump rate and losses in
the system.
As will be further justified below, the SAW potential
fragments the original extended condensate into an array of
identical, weakly interacting condensates confined at the
minima of the acoustic potential, thus forming the conden-
sate wires oriented along the SAW wave fronts illustrated
FIG. 2 (color online). (a),(c) Energy vs k
y
images of conden-
sates for different SAW amplitudes (u) at fixed pump power I
p
¼
2I
OPO
th
¼ 1:2 10
3
W=cm
2
. (d) Correlation setup to measure the
g
ð2Þ
ðy; Þ function of moving condensates using two detectors
(A and B) separated by a distance y. (e) g
ð2Þ
ðy; Þhistograms
measured at SAW amplitude u ¼ 3:5mW
1=2
at different y ¼
m
SAW
=2 (m ¼ 0, 0.5, and 1). The curves are shifted down
along the vertical axis for clarity.
FIG. 1 (color online). (a) Schematic diagram of a polariton
optical parametric oscillator. (b) Sample structure with SAWs
propagating along the y ¼½010 direction of the (001) micro-
cavity surface. The polaritons are excited by a laser beam
incident in the (x , z) plane. (c) Folded dispersion of the lower
polariton (LP) (d),(e) LP dispersion recorded under weak optical
excitation (I
p
¼ 20:8W=cm
2
) in the absence and presence of a
SAW (u ¼ 5:65 mW
1=2
), respectively. The lines display the
calculated dispersion (the dotted lines indicate optically inactive
states).
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in Fig. 2(d). Each condensate wire moves with the acoustic
velocity v
SAW
¼ 3 10
3
m=s and emits independently
from the others at high acoustic powers. The flattening of
the lowest polariton branch [Fig. 1(e)] and the k
y
spreading
of the emission [Figs. 2(b) and 2(c)] are attributed to the
small width y
SAW
=2 of the wires in accordance with
Heisenberg’s uncertainty principle (k
y
=y
2=
SAW
). Their spectral emission energies, however,
overlap within the measured linewidths, thus attesting to
the high homogeneity of the modulation potential.
Interestingly, at intermediate SAW amplitudes [u ¼
2:3mW
1=2
, cf. Fig. 2(b)] the emission peaks not around
k
y
¼ 0 but at k
y
¼k
SAW
=2, corresponding to the s
0
states
in Fig. 1(c) at the edge of the lowest folded branch. A likely
explanation for the s
0
-state emission arises from the spatial
modulation of the pump polariton density by SAW. The
excitation of an OPO condensate requires a laser energy
slightly blueshifted with respect to the LP branch [17].
Above the OPO threshold pump-pump polariton interac-
tions shift the LP branch into resonance with the laser [17].
Under a SAW, this blueshift is larger at the minima of the
SAW potential, thereby leading to a higher pump polariton
density at these positions. This spatial modulation leads to
diffraction replicas of the pump beam with k
y
¼k
SAW
and a measured intensity of 0.5% of I
p
, from which we
estimate a pump density modulation amplitude of about
20%. The diffraction replicas may undergo a parametric
process with the unperturbed pump state at k
y
¼ 0,
namely, k
ðpÞ
y
ð¼ 0Þþk
ðpÞ
y
ð¼ k
SAW
Þ!k
ðsÞ
y
ð¼ k
SAW
=2Þþ
k
ðiÞ
y
ð¼ k
SAW
=2Þ thus providing the observed parametric
gain at k
y
¼k
SAW
=2. The coupling (and coherence)
between adjacent wires reduces at higher SAW amplitudes
(u 5:7mW
1=2
) due to the stronger confinement. The
momentum of the condensate is thus no longer well defined
and the emission pattern in momentum space spreads over
the whole MBZ branch [Fig. 2(c)].
The condensate wires were directly detected using a
correlation setup to measure the classical second order
intensity autocorrelation function g
ð2Þ
ðy; tÞ¼
hI
PL
ð0; 0Þ;I
PL
ðy; tÞi of the PL intensity (I
PL
) from two
small spots (2 2 m
2
) separated by a distance y
[Fig. 2(d)]. As expected, the g
ð2Þ
function exhibits maxima
at time intervals t multiples of the SAW period
SAW
,
when condensates cross the collection regions of both
detectors. Figure 2(e) displays g
ð2Þ
functions recorded for
different y. The phase of the oscillations changes by
when the spots are displaced by y ¼
SAW
=2: the fact that
the rate of change y= ¼ v
SAW
proves that the confined
condensate regions are spatially separated by
SAW
and
move at v
SAW
. Quantitative information about the modu-
lation amplitude can be obtained by assuming that
I
PL
ðy; tÞ¼I
0
þ I sinðk
SAW
y þ !
SAW
tÞ, yielding
the g
ð2Þ
ðy; tÞ¼I
2
0
þ I
2
=2 sinðk
SAW
y þ !
SAW
tÞ.
From the maxima to minima ratio of about 1.25 in
Fig. 2(e), we determine a signal population modulation
(/I=I
0
)of45%. While these results demonstrate the
confinement and motion of the condensate wires, they do
not imply the long-range transport of a coherent state, since
the measured coherence time of unperturbed condensates
of 200–500 ps [5,11] is significantly shorter than
SAW
¼
2:65 ns.
The coherence among the wire condensates was further
investigated by recording first-order spatial correlation
functions g
ð1Þ
ðr; rÞ for the wire arrays using the proce-
dure described in Ref. [1]. The full width at 1=2:7 of the
maxima of g
ð1Þ
ðr; rÞ gives the corresponding coherence
lengths L
x
and L
y
along the x and y directions, respec-
tively. In the absence of a SAW, the emission of the
condensate occurs at k ¼ 0 and is spatially homogeneous
with equal L
y
L
x
25 m [cf. Figure 3(a)] limited by
the size of the excitation spot. The spreading of the disper-
sion in k space in Figs. 2(b) and 2(c) is accompanied by a
reduction of both L
y
and L
x
[Fig. 3(b)].
The SAW modulation acting directly on the signal re-
duces the tunneling coupling between adjacent conden-
sates, thus allowing their phases to evolve independently
with the consequent collapse of L
y
to values close to
SAW
=2 ¼ 4 m [ 18]. Using the measured polariton
mass m
pol
¼ 4:5 10
5
m
e
(m
e
is the free electron mass)
and assuming a potential barrier V
SAW
¼ 0:16 meV [cf.
Fig. 1(e)] we estimate a tunneling time
t
¼ h=J 200 ps
[J ¼ 4V
SAW
expð
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f2m
pol
V
SAW
g
SAW
q
=@Þ is the tunneling
energy]. This time is comparable to the condensate coher-
ence time, hence preventing the full establishment of spa-
tial coherence between adjacent condensates.
The 20% spatial modulation of the pump (and hence
modulation of the gain [19]) deduced from the diffraction
replicas is also likely to contribute to the signal spatial
modulation and hence to the reduction of its coherence. A
20% modulation is also expected for the signal density,
given the measured linear dependence of the signal inten-
sity on the pump power above threshold. We note, how-
ever, that we have observed a similar reduction of
FIG. 3 (color online). First-order spatial correlation functions
g
ð1Þ
of condensates for (a) u ¼ 0 and (b) u ¼ 2:3mW
1=2
.
(c) Dependence on SAW amplitude u of the spatial coherence
lengths L
x
and L
y
along the x and y directions, respectively.
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coherence length with SAW power for condensates excited
resonantly with the excitonic LP branch at a large inci-
dence angle of 30
[8]. In this case the direct modulation of
the condensate by the SAW potential plays the major role
in the reduction of its coherence, because the modulation
of the pump states by the SAW is negligible due to the
broad exciton resonance (FWHM 1 meV).
The spatial coherence in Figs. 3(b) and 3(c) also reduces
along the wires (x direction). We suggest that this is
associated with phase fluctuations in the 1D polariton
system. This phenomenon is well known at thermody-
namic equilibrium [2,20] and has also been predicted for
nonequilibrium systems such as the 1D OPO [21]. For the
extended OPO the phase fluctuations are negligible and the
signal coherence is predicted to be limited only by the size
of the excitation spot [22]. For a nonequilibrium conden-
sate the phase fluctuations are predicted to determine the
coherence length, which becomes proportional to 1=,
where is the decay rate of excitations (diffusive
Goldstone mode) in the condensate [21]. is estimated
to be of the order of the inverse polariton lifetime and plays
a role analogous to the temperature in equilibrium systems.
We note that coherence lengths of the order of 100 m
have recently been reported in nonresonantly pumped 1D-
condensates in high-Q microresonators (Q 15 000)[23].
The lower L
x
values in our wires can be attributed to the
lower Q (and, consequently, the shorter lifetimes and
higher ) in our microcavities. Also, the small excitation
spot in Ref. [23](3 m) is expected to reduce noise due
to spatial decoupling of propagating polaritons from the
pump reservoir, thus potentially increasing the coherence
length.
We show finally that the degree of lateral confinement of
the wires can be reversed by increasing the polariton
density. As presented in Fig. 4, for a fixed SAW amplitude
both coherence lengths L
y
and L
x
increase with I
p
.
Repulsive (signal-signal and signal-pump) interactions,
which are stronger in the minima than in the maxima of
the SAW potential, raise the energy of the trapped con-
densates above the percolation limit. As a result, the con-
finement effects induced by the SAW are partially reversed
and the long-range spatial coherence of the signal is rees-
tablished, as observed in Fig. 4. In agreement with the
screening mechanism, the effects are more pronounced
for low acoustic powers, where the energy renormalization
due to polariton-polariton interactions more easily over-
comes the energies barriers imposed by the SAW potential.
In conclusion, we have demonstrated that the coherent
spatial modulation of polariton condensates by a SAW
creates a condensate fragmented in the form of wires
aligned along the wave fronts. These results open the
way for the dynamic manipulation of polaritonic quantum
phases by tunable periodic potentials necessary for com-
putational schemes [24] and for the investigation of many-
body phenomena such as Josephson coupling and, for
shorter SAW wavelengths, condensate–Mott insulator tran-
sition [7]. Finally, the acoustic transport of coherent polar-
iton states can be realized in high Q-microcavities with
longer condensate coherence times.
We would like to thank F. Große for discussions as well
as M. Ho
¨
ricke, W. Seidel, and E. Wiebecke for the fabri-
cation of the samples. E. A. C.-M. would like to thank the
Deutscher Akademischer Austausch Dienst for partial fi-
nancial support.
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FIG. 4. Dependence of (a) L
y
and (b) L
x
on the pump power I
p
for different acoustic modulation amplitudes u.
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