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Spatial Coherence Properties of One Dimensional Exciton-Polariton Condensates

1

J. Fischer,1 I. G. Savenko,2,3 M. D. Fraser,4 S. Holzinger,1 S. Brodbeck,1 M. Kamp,1 I. A. Shelykh,5,6
C. Schneider,1 and S. Höfling1,7Technische Physik, Wilhelm-Conrad-Röntgen-Research Center for Complex Material Systems, Universität Würzburg,
Am Hubland, D-97074 Würzburg, Germany
2
QCD Labs, COMP Centre of Excellence, Department of Applied Physics and Low Temperature Laboratory (OVLL),
Aalto University, P.O. Box 13500, FI-00076 Aalto, Finland
3
National Research University of Information Technologies, Mechanics and Optics (ITMO), St. Petersburg 197101, Russia
4
Quantum Optics Research Group, RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako-shi,
Saitama 351-0198, Japan
5
Science Institute, University of Iceland, Dunhagi 3, IS-107 Reykjavik, Iceland
6
Division of Physics and Applied Physics, Nanyang Technological University, 637371 Singapore, Singapore
7
SUPA, School of Physics and Astronomy, University of St. Andrews, St. Andrews KY16 9SS, United Kingdom(Received 12 April 2014; published 14 November 2014)
In this work, we combine a systematic experimental investigation of the power- and temperaturedependent evolution of the spatial coherence function, gð1Þ ðrÞ, in a one dimensional exciton-polariton
channel with a modern microscopic numerical theory based on a stochastic master equation approach.
The spatial coherence function gð1Þ ðrÞ is extracted via high-precision Michelson interferometry, whichallows us to demonstrate that in the regime of nonresonant excitation, the dependence gð1Þ ðrÞ reaches asaturation value with a plateau, which is determined by the intensity of the pump and effective temperatureof the crystal lattice. The theory, which was extended to allow for treating incoherent excitation ina stochastic frame, matches the experimental data with good qualitative and quantitative agreement.
This allows us to verify the prediction that the decay of the off-diagonal long-range order can be almost
fully suppressed in one dimensional condensate systems.
DOI: 10.1103/PhysRevLett.113.203902

PACS numbers: 42.55.Sa, 78.20.Ls, 78.55.Cr, 78.67.De

Introduction.—Exciton polaritons evolve in semiconductor microcavities as the result of strong coupling of opticaland matter modes [1]. At sufficiently low concentrations(up to ∼1011 cm−2 ) they obey bosonic statistics, and owing
to their small effective mass, which is about 105 times
smaller than the free electron mass, manifest quantum
coherent properties at surprisingly high temperatures.
Indeed, polariton condensation is observed at temperaturesof tens of Kelvin in GaAs and CdTe-based structures [2,3]and up to room temperatures in wide band-gap materials[4–6]. Having lot of similarities with conventional BoseEinstein condensation (BEC) [1], polariton condensation
reveals some important peculiarities. Differently fromcold atoms, polaritons have a finite lifetime and in orderto reach an equilibrium state their radiative decay should becompensated by a constant pumping of the system, which
can be implemented either optically or electrically [7].
Moreover, the planar microcavity exciton-polariton system
is inherently two dimensional (2D), and in accordance with
the Hohenberg-Mermin-Wagner theorem the transition to
BEC in a uniform system is only possible at zero temperature [8] for 2D as well as 1D geometries. Restriction of thesystem to a finite size, however, inhibits the excitation ofdensity and phase fluctuations permitting the formation of acondensate or quasicondensate phase with a macroscopic
coherence length [9,10].
0031-9007=14=113(20)=203902(5)There is a long-standing discussion in the literatureaddressing the following question: What should be theexperimentally verifiable criterion of the polariton condensation? It is now commonly accepted that the onset ofthe off-diagonal long-range order (ODLRO), determinedby the first-order spatial coherence function gð1Þ ðrÞ, can beconsidered as a smoking gun criterion, putting the latterfunction in the most intense focus of theoretical [11–14]
and experimental [2,15,16] research. In particular, polariton
condensation in one dimensional channels has drawn
special attention. It was demonstrated that polaritons
trapped in a channel reveal the appearance of ODLRO
in the condensation regime [16]. Moreover, the droplets of
BECs in such systems can propagate over macroscopic
distances, preserving their coherence properties andallowing for the efficient manipulation [17]. Recently, an
experimental observation of room-temperature polariton
condensation in a one dimensional ZnO channel has beenreported [18]. These technological achievements open aroute to the creation of polariton-based logic elements
and optical integrated circuits working at relatively hightemperatures [19]. Among the phases expected for a 1D
polariton gas are the Tonks gas [20] and the condensate
or quasicondensate phases [10], the former appearing
when interactions come to dominate the energy scale andthe condensate phases resulting for a sufficiently weakly

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interacting and finite-sized system. The quasicondensate,
which still exhibits some phase fluctuations, is characterized by an exponentially decaying long-range order, butwith an enhanced coherence length. Such a phase is typicalfor the weakly interacting dilute-gas atomic condensate.
The low energy of phase fluctuations in this system preventsthe formation of a true condensate phase with a plateauing
long-range order, except at the lowest temperatures [10].
The decay of spatial coherence in a low-dimensional nonequilibrium (polariton) condensate has been studied througha mean-field treatment with a stochastic noise term [21]with the result that, provided the gas is interacting, similar
behavior to that of equilibrium BEC is expected in 1D, 2D,
and 3D systems. The results that we discuss in this Letter,
however, distinctly place our 1D polariton condensate into
the true BEC-like phase where both phase and density
fluctuations are suppressed to the level that ODLRO can beobserved with a BEC-like constant plateau in the spatial
coherence at the largest distances.
On the theoretical side, for the description of the polariton
dynamics in one dimension, an approach based on the
Lindblad master equation technique has been developed[22]. Different from the approaches based on either
Gross-Pitaevskii-type equations [23,24] or semiclassicalBoltzmann equations [25–27], it allows us to account forreal space dynamics of the polariton droplets, processes ofdecoherence and energy relaxation provided by the interaction of polaritons with the thermal bath of acoustic
phonons. Moreover, the calculation of two-point correlators
necessary for the analysis of the transition between thermal
and OLDRO phases became possible [13].
Here, we demonstrate that once the intensity of the pumpreaches a threshold value, the system undergoes the transition from the thermal to the condensate phase accompaniedby the onset of ODLRO. Theoretically, this transition ismodeled using the original microscopic approach based on astochastic density matrix formalism. We investigate theinfluence of temperature and pump intensity modulationon the coherence properties of the system and reveal good
agreement between the theory and experiment.
Experiment.—The polariton channels [Fig. 1(a)] with alength of 200 μm (width 5 μm) were etched into a high-Q
AlGaAs based λ=2 microcavity with 12 GaAs quantum
wells and a Q factor exceeding 10 000 (see Supplemental
Material [28]).
First, we investigate the power-dependent emission
features of the microwire cavity via momentum resolved
spectroscopy. The microwire is mounted in a helium flow
cryostat and it is oriented parallel to the entrance slit of thespectrometer. Polaritons are injected with a nonresonant cw
laser which is tuned to the energy of the first reflection
minimum of the Bragg reflector. The Gaussian-shaped
pumping spot has a diameter of 5 μm, and the laser beam ischopped with a duty cycle of 0.1 to reduce power-induced
heating effects. As shown in Fig. 1(b), we observe a set of

(a)

(d)

(b)

(c)

(e)

FIG. 1 (color online). (a) Scanning electron microscope imageof a one dimensional polariton channel. (b) Power-dependent
Fourier-space spectrum of the microwire emission below, and(c) above the regime of polariton condensation. (d) Input-output
characteristics of the ground state emission from the wire.
(e) Polariton linewidth as a function of the pump power.

parabolic dispersions which can be attributed to the groundstate of the microwire as well as higher order lateral modes.
Each dispersion is split into a mode doublet [34]. Theenergy-momentum detuning between the lowest photon
mode EC1 ðk∥ ¼ 0Þ and the exciton energy EX ðk∥ ¼ 0Þ ofthe ground state amounts to δ¼EC1 ðk∥ ¼0Þ−EX ðk∥ ¼0Þ¼
−13meV, which is less than the Rabi splitting ERS ¼
14.5 meV. With the increase of the pump power, polaritons
condense into the ground state which is manifested by its
massive occupation [Fig. 1(c)]. The corresponding inputoutput characteristics of the power-dependent study isshown in Fig. 1(d). It features a distinct threshold which
is typical for the onset of stimulated scattering leading to thegrowth of a condensate. Above P ¼ 2.5Pth the intensity
decreases again indicating excitation power-induced heatingof the sample. At threshold, the linewidth of the polariton
emission [shown in Fig. 1(e)] significantly reduces. Thisbehavior is commonly assigned to an increased temporal
coherence of the emission in the regime of polariton lasing[35]. Further, it increases again with the increase of theparticle densities which can be attributed to dephasing
resulted from the polariton-polariton interactions.
In order to investigate the spatial coherence properties ofthe polariton channels, we measure its spatial correlation
function, gð1Þ ðr0 ; r00 Þ. We use a Michelson interferometerwith a variable path length which overlaps the real space
image of the polariton emission with its mirror image
generated by a right angle prism. A schematic drawing of
the optical setup is shown in Fig. 2(a) (it is similar to203902-2

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(b)

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Theory.—Theoretically, the first-order spatial coherence
can be determined asχðr0 ; r00 Þ
ﬃ;
gð1Þ ðr0 ; r00 Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
χðr0 ; r0 Þχðr00 ; r00 Þ

(c)

ð1Þ

where χðr0 ; r00 Þ is the single-particle density matrix of thesystem in real space:(d)χðr0 ; r00 Þ ¼ hΨ̂† ðr0 ÞΨ̂ðr00 Þi ≡ Tr½ρΨ̂† ðr0 ÞΨ̂ðr00 Þ:FIG. 2 (color online). (a) Schematic sketch of the Michelsoninterferometer setup for the determination of the spatial correlation function. (b) Typical interference pattern at an excitation
power of P ¼ 1.6Pth . (c) By variation of the distance between the
mirror and prism, the intensity of one pixel undergoes a sinefunction (blue circles). The red line shows the fitted sine fromwhich the gð1Þ value and the phase can be calculated. (d) gð1Þ map
received by fitting all the pixels of the camera image with a sinefunction above the condensation threshold at P ¼ 1.6Pth .

Ref. [36]). The overlapped real space images from thepolariton wire are combined onto a CCD camera with highspatial resolution. The light from the excitation laser isfiltered out with a long pass filter in the optical beam path.
Figure 2(b) depicts the resulting interference pattern on
the camera. By moving one arm of the interferometer, we
can extract the interferograms. From the visibility of theinterference fringes [see Fig. 2(c)], we can reconstruct thefull spatial correlation function gð1Þ ðr; −rÞ which is plottedin the color map in Fig. 2(d). It is interesting that, incontrast to the experimentally reported algebraic decay in
2D polariton systems [15,36], the spatial coherence function of a one dimensional condensate is indicated to reach aconstant plateau at large distances [16,37,38]. However, inthese reports the clarity of this behavior is either obscured
by large experimental errors resulting from the applied
double slit technique [38] or by strong oscillations in thecorrelation function related to inhomogeneities of the 1D
trap [16] and a comparably short long-range order in therange of 10 μm [37]. For the purposes of assessing thelong-range coherence behavior, the 1D polariton condensate in our experiments may roughly be characterized as afinite area but uniform 1D condensate due to its smoothsteady-state profile. It is now well appreciated that the
cw-pumped polariton condensate may be viewed as confined to an “effective trap” [39,40] within which excitations
over the entire condensate may be observed. Thus, despiteballistic expansion, the spatially extended steady-state
profile represents the system size, and we may qualitatively
apply the expected 1D correlation functions to this experimental system.

ð2Þ

Here Ψ̂† ðrÞ; Ψ̂ðrÞ are exciton-polariton field operators; ρ isa full density matrix of the closed system (polariton system
plus the environment).
In the case of a spatially homogeneous system, the firstorder coherence is only dependent on the relative distance:
gð1Þ ðr0 ; r00 Þ ¼ gð1Þ ðrÞ, where r ¼ jr0 − r00 j. In the ODLRO
phase, the correlations do not decay even at very large
distances, and thus limr→∞ gð1Þ ðrÞ ≠ 0 [41]. In order tocalculate gð1Þ ðrÞ at different temperatures and pump intensities, we will employ a closed system of dynamic equations
for the elements of the single-particle density matrix in thereciprocal space based on the Lindblad master equation
techniques and account for the processes of polaritonpolariton and polariton-phonon interactions, pumping, andfinite lifetime. The corresponding formalism was developedby us, and is described in greater detail elsewhere [22,42].
In those previous works, we treated the pulsed excitation of
the condensate simply by introducing appropriate initialconditions [22], or we considered the cw resonant injection[42]. To accurately account for the effects of incoherent
pumping, we had to extend our model by a stochastic
approach introducing the random phase in the pumping termand performing statistical averaging. This modification
represents a nontrivial and vitally important extension of
our previous works to realistically model the experimental
data. Upon finding the single-particle density matrix in thereciprocal space, the real space behavior can be found by the
Fourier transform. The theoretical formalism is summarized
in the Supplemental Material [28] and full details can befound in Refs. [13,22,42].
In the calculations, we used parameters (effective massof polaritons, the Rabi splitting energy and detuning, thetemperature for each set of data, and dimensions of thequantum wire) taken from the experimental data. The matrix
element of polariton-polariton interaction was estimatedusing the expression U ≈ 3Eb a2B =S, where Eb is the exciton
binding energy, aB is its Bohr radius, and S is the area of
the wire, which gave U ≈ 4 neV. The polariton-polariton
scattering rate was taken independent of the wave vector, forsimplicity. The polariton-phonon scattering rate is calculatedusing Eq. (7) in [28] and the parameters there. The maximum
value of the scattering rate reads W ≈ 38 neV. Pumping
powers were chosen in accordance with the experimental data.
Results and discussion.—First, we investigate the behavior of coherence with the increase of the condensate density
controlled by the intensity of the pump, P. Below the203902-3

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(a)

(a)

(b)

(b)

(c)

(c)

FIG. 3 (color online). gð1Þ ðΔrÞ for different excitation powers(a) below condensation threshold P ¼ 0.8Pth , (b) slightly above
threshold P ¼ 1.4Pth and (c) P ¼ 1.6Pth . With higher excitation
powers the central peak width around Δr ¼ 0 increases and thevalue of the gð1Þ ðΔr → ∞Þ function reaches a plateau of about
0.15 at P ¼ 1.6Pth . The red dashed lines in (a)–(c) are the resultsof our theoretic modeling.

FIG. 4 (color online). gð1Þ ðΔrÞ function for different sample
temperatures (a) T ¼ 5 K, (b) T ¼ 20 K, and (c) T ¼ 25 K at an
excitation power of P ≈ 1.8Pth . The red dashed lines are again the
results of our model. With increasing temperatures, the plateau ofgð1Þ ðΔr → ∞Þ disappears (c) T ¼ 25 K.

threshold of condensation, the coherence function gð1Þ ðrÞ
represents a resolution-limited sharp peak and a fast drop
down to the noise level, which is shown in Fig. 3(a). Afterreaching the polariton lasing threshold Pth [Fig. 3(b)], thiscentral peak starts to broaden, indicating the increase of thecoherence [Fig. 3(c)]. In thermal equilibrium, the width ofthe peak is related to the thermal de Broglie wavelength λdBof the Bose gas. Despite our system being clearly out of
equilibrium, we follow this analogy and extract an effective
de Broglie wavelength of λdB ≈ 8.5 μm well above thecondensation threshold, which is in good agreement withearlier reports [36]. More remarkable, the spatial correlation
function acquires a nearly constant plateau in the regime ofpolariton condensation, which persists over the full range ofthe microwire significantly above the noise level.
For moderate pump powers (below two times the threshold power) we observe a consistent increase of the spatial
coherence degree with the increase of polariton occupancy.
This is exemplarily demonstrated in Fig. 3(c) for a pump
power of 1.6Pth . We can quantitatively reproduce thisbehavior with our theory, as shown in Figs. 3(a)–3(c):The spatial coherence is monotonically decaying over the
ð1Þ
distance r and reaches some nonzero value g∞ for r → ∞.

This value is determined by the percentage of the coherent
fraction in the system and it increases with the pump power,
P (in case of absent strong localization effects). Thisbehavior is also in good agreement with the theory.
Figure 4 serves as a manifestation of the thermal effects’
influence on the long-range order. The figure depicts theevolution of gð1Þ ðr; −rÞ as a function of temperature at aconstant pump power of 1.8 Pth . The sample temperature isincreased from 5 to 20 K.
We observe a significant decrease of gð1Þ ðr; −rÞ with anincrease of the temperature, which strongly indicates the
detrimental influence of acoustic phonons on the spatial
coherence properties of the condensate. While for the highest
temperature the correlation function drops to the noise levelwithin the size of the pump spot in almost perfect agreementwith theory [Fig. 4(c)], the occurrence of satellite peaks[Figs. 4(b), 3(b)] indicate some persisting influence of thesample disorder [37] or a boundary mode reflection. At atemperature of T ¼ 25 K [Fig. 4(c)] the value of gð1Þ ðr; −rÞ
quenches to zero at distances r > 20 μm and the transition
from ODLRO to the thermal phase occurs.
Conclusion.—To summarize, we have investigatedtheoretically and experimentally spatial coherence propertiesof a one dimensional exciton-polariton microwire under
different nonresonant pump intensities and temperatures.

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We have compared the experimental data with calculations,
utilizing a stochastic master equation approach. It has been
demonstrated that the gð1Þ ðrÞ function has a plateau region
which is determined by the intensity of nonresonant pumping and the effective temperature of the crystal lattice. This
result places our condensate into the true BEC-like phase
where both phase and density fluctuations are suppressed.
The authors would like to thank the State of Bavaria
for financial support. I. A. S acknowledges the support ofFP7 IRSES project POLAPHEN. I. G. S. was partially
supported by the Academy of Finland through its Centreof Excellence Programs (Projects No. 250280 and
No. 251748) and the Government of the RussianFederation, Grant No. 074-U01.

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