Spontaneous coherence of indirect excitons in a trap

Abstract

We report on the emergence of spontaneous coherence in a gas of indirect
excitons in an electrostatic trap. At low temperatures, the exciton coherence
length becomes much larger than the thermal de Broglie wavelength and reaches
the size of the exciton cloud in the trap.

Full text

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Spontaneous coherence of indirect excitons in a trap
A.A. High,1J.R. Leonard,1M. Remeika,1L.V. Butov,1M. Hanson,2and A.C. Gossard2
1Department of Physics, University of California at San Diego, La Jolla, CA 92093-0319, USA
2Materials Department, University of California at Santa Barbara, Santa Barbara, CA 93106-5050, USA
(Dated: October 11, 2011)
We report on the emergence of spontaneous coherence in a gas of indirect excitons in an electrostatic
trap. At low temperatures, the exciton coherence length becomes much larger than the thermal
de Broglie wavelength and reaches the size of the exciton cloud in the trap.
Potential traps are an effective tool for studies of cold
atomic gases. They allow the realization and control of
atomic Bose-Einstein condensates, see [1, 2] for review.
Condensation in momentum space is equivalent to the
emergence of spontaneous coherence of matter waves [3].
Spontaneous coherence is an intensively studied feature
of atomic condensates [1, 2]. In this paper, we report on
studies of spontaneous coherence in a cold gas of indirect
excitons in an electrostatic potential trap.
Excitons are hydrogen-like electron-hole pairs at low
densities [4] and Cooper-pair-like electron-hole pairs at
high densities [5]. The bosonic nature of excitons al-
lows for condensation in momentum space at low tem-
peratures, below the temperature of quantum degener-
acy. For a typical range of parameters, the temperature
of quantum degeneracy in an exciton gas is in the range
of a few Kelvin. Although the temperature of the semi-
conductor crystal lattice can be lowered well below 1 K
in He-refrigerators, lowering the temperature of the ex-
citon gas to even a few Kelvin is challenging [6, 7]. Due
to recombination, excitons have a finite lifetime which is
too short to allow cooling to low temperatures in regu-
lar semiconductors. In order to create a cold exciton gas
with temperature close to the lattice temperature, the
exciton lifetime should be large compared to the exciton
cooling time. Besides this, the realization of a cold and
dense exciton gas requires an excitonic state to be the
ground state and have lower energy than the electron-
hole liquid [8].
These requirements can be fulfilled in a gas of indirect
excitons. An indirect exciton in coupled quantum wells
(CQW) is a bound state of an electron and a hole in sep-
arate wells (Fig. 1a). The spatial separation allows one
to control the overlap of electron and hole wavefunctions
and engineer structures with lifetimes of indirect excitons
orders of magnitude longer than those of regular excitons
[9, 10]. In our experiments, indirect excitons are created
in a GaAs/AlGaAs CQW structure (Fig. 1a). Long life-
times of the indirect excitons allow them to cool to low
temperatures within about 0.1K of the lattice tempera-
ture, which can be lowered to about 50mK in an optical
dilution refrigerator [11]. This allows the realization of a
cold exciton gas with temperature well below the temper-
ature of quantum degeneracy TdB= 2π¯ h2n/(mgkB) (in
the studied CQW, excitons have the mass m = 0.22m0,
-100 10
y (µm)

a
b
c
x (µm)
h
0
E (meV)
5
e
z
E
Laser
excitation
x
10 µm
y
-505
d
FIG. 1: (a) CQW band diagram. e, electron; h, hole. In-
direct excitons are formed by electrons and holes confined in
separated layers. (b) SEM image of electrodes forming the di-
amond trap: a diamond-shaped electrode is surrounded by a
thin wire electrode followed by an outer plane electrode. (c,d)
Simulation of exciton energy profile through the trap center
along x (c) and y (d) for Vdiamond = −2.5 V, Vwire = −2 V,
and Vplane= −2 V. The position of the laser excitation spot
is indicated by the circle in (b) and by the arrow in (c).
spin degeneracy g = 4, and TdB≈ 3K for the density per
spin n/g = 1010cm−2).
Due to the spatial separation of the electron and hole,
indirect excitons have a built-in dipole moment ed, where
d is close to the distance between the QW centers, which
allows their energy to be controlled by voltage: an elec-
tric field Fz perpendicular to the QW plane results in
the exciton energy shift E = edFz [12]. This gives an
opportunity to create in-plane potential landscapes for
excitons E(x,y) = edFz(x,y).
statically created potential landscapes include the oppor-
tunity to realize the desired potential profile as well as
the ability to control it in-situ, on a time scale shorter
than the exciton lifetime. For instance, switching off the
confining potential, like in atomic time-of-flight experi-
ments, or modulating its depth, like in atomic collective
mode experiments [1, 2] can be realized by modulating
the electrode voltage. Excitons were studied in various
electrostatic potential landscapes: ramps [13, 14], lattices
[15–18], circuit devices [19], and traps [17, 20–23].
In this work, an electrostatic trap for indirect exci-
tons is realized using a diamond-shaped electrode (Fig.
1b). The diamond trap creates a parabolic-like confin-
ing potential for excitons [23]. The CQW structure is
Advantages of electro-
arXiv:1110.1337v2 [cond-mat.quant-gas] 8 Oct 2011

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024
T (K)
68
0.07
0.15
0.22
0.29
0.36
0.44
1
2
3
4
5
6
-10010
0
1
-505
0.0
1.8
-505
0
1
Interference visibility
at shift ?x = 4?m
HWHM of exciton cloud (µm)
-100 10

x (µm)
-100 10
0
0
0
0
0
0
0
y
1K
7K
x (µm)
6K
5K
4K
3K
2K
10 µm
Tbath=
50mK
0
f
c
a
Emission (arb. units)
x (µm)
Tbath =
50mK
3K
7K
b
I1 (arb. units)
1
0
I12 (arb. units)
1.8
0
e
y (µm)
d
FIG. 2: (a,b) Emission pattern (a) and interference pattern
at shift δx = 4µm (b) for temperatures ranging from 50
mK to 8 K. (c,d) Spatial profiles of the emission pattern
in (a) along x at y = 0 (c) and along y at x = 0 (d) for
Tbath= 50mK (black), 3K (red), and 7K (blue). (e) Spatial
profiles of the interference patterns in (b) along y at the peak
of exciton emission (x = 0) for Tbath = 50mK (black) and
7K (blue). (f) Amplitude of the interference fringes Ainterf
at shift δx = 4µm averaged from 0 < x < 1.5µm (black
squares) and half-width at half-maximum (HWHM) of the
exciton emission pattern along x (blue diamonds) vs. tem-
perature. For all data Pex = 1.9µW.
grown by MBE. An n+-GaAs layer with nSi = 1018
cm−3serves as a homogeneous bottom electrode. The
top electrodes on the surface of the structure are fabri-
cated via e-beam lithography by depositing a semitrans-
parent layer of Ti (2 nm) and Pt (8 nm). The device
includes a 3.5×30µm diamond electrode, a 600 nm wide
’wire’ electrode which surrounds the diamond, and ’outer
plane’ electrode (Fig. 1b) [23]. Two 8 nm GaAs QWs
separated by a 4 nm Al0.33Ga0.67As barrier are positioned
100 nm above the n+-GaAs layer within an undoped 1
µm thick Al0.33Ga0.67As layer.
closer to the homogeneous electrode suppresses the in-
plane electric field [17], which otherwise can lead to ex-
citon dissociation. The excitons are photoexcited by a
633 nm HeNe laser. The excitation beam is focused to a
spot about 5µm in diameter on a side of the trap (Fig.
1b,c). This excitation scheme allows the photoexcited ex-
citons to cool down further when they travel towards the
trap center, thus facilitating the realization of a cold and
dense exciton gas in the trap (cooling of excitons away
from the laser excitation spot also leads to the realization
of a cold exciton gas in the inner ring in exciton emission
pattern [24, 25] and in laser-induced traps [26]).
The pattern of the first-order coherence function
g1(δx) is measured by shift-interferometry: the emission
images produced by arm 1 and 2 of the Mach-Zehnder
interferometer are shifted with respect to each other to
measure the interference between the emission of excitons
separated by δx. A similar method was used in the stud-
Positioning the CQW
ies of spontaneous coherence in a gas of indirect excitons
in exciton rings [27]. The emission beam is made parallel
by an objective inside the optical dilution refrigerator and
lenses. The emission is split between arm 1 and arm 2 of
the interferometer. The path lengths of arm 1 and arm 2
are set equal. The interfering emission images produced
by arm 1 and 2 of the interferometer are shifted relative to
each other along x to measure the interference between
the emission of excitons, which are laterally separated
by δx. After the interferometer, the emission is filtered
by an interference filter of linewidth ±5nm adjusted to
the emission wavelength of indirect excitons λ = 800nm.
The filtered signal is focused to produce an image, which
is recorded by a liquid-nitrogen cooled CCD. We mea-
sure emission intensity I1for arm 1 open, intensity I2for
arm 2 open, and intensity I12for both arms open, and
calculate Iinterf= (I12− I1− I2)/(2√I1I2). The period
of the interference fringes is set by a tilt angle between
the image planes of the two arms and g1(δx) is given by
the amplitude of the interference fringes Ainterf [28] as
detailed below.
Figure 2 presents the temperature dependence of exci-
ton emission and interference patterns. At high temper-
atures, the exciton cloud spreads over the trap resulting
in a broad spatial profile of the exciton emission (Fig.
2a,c,f). With lowering temperature, the width of the
emission pattern of the exciton cloud in the trap reduces
indicating exciton accumulation at the trap center (Fig.
2a,c,f). This is consistent with the reduction of the ther-
mal spreading of excitons over the trap. Note that studies
of atoms in traps also reveal the accumulation of atoms
at the trap center with lowering temperature [1, 2].
Figure 2b presents the temperature dependence of the
pattern of interference fringes. The corresponding tem-
perature dependence of the amplitude of the interference
fringes Ainterfaveraged from 0 < x < 1.5µm is shown in
Fig. 2f. As detailed below, Ainterf presents the coher-
ence degree of excitons. Figure 2f shows that the exciton
accumulation at the trap center is accompanied by the
enhancement of the coherence degree of excitons.
Figure 3a presents the amplitude of the interference
fringes Ainterf(δx) for different densities.
temperature is higher in the excitation spot [25, 26]. This
is consistent with low values of Ainterf at negative δx,
which correspond to the interference between the emis-
sion of a hot exciton gas in the excitation spot and ex-
citon gas in the trap center (Fig. 3a). We will consider
positive δx, which correspond to the interference between
the emission of an exciton gas in the trap center and ex-
citon gas at positive x further away from the hot laser
excitation spot.
Figure 3c presents the density dependence of the width
of the exciton emission pattern along x. At high temper-
ature T = 4.5 K, the width of the emission pattern of
the exciton cloud monotonically increases with density
(Fig. 3c). This is consistent with screening of the poten-
The exciton

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0.1
0.2
0.3
0.4
110 100
2
4
6
-6-30
shift ?x (µm)
369 12
0.0
0.2
0.4
0.6



Tbath=
50mK
4.5K
Interference visibility
at shift ?x = 4?m
Tbath=
50 mK
4.5K

HWHM of exciton cloud (µm)
Pex (µW)
c
b
Interference visibility
Pex =
0.31 µW
2.2 µW
88 µW
a
FIG. 3: (a) Amplitude of the interference fringes Ainterf(δx)
for excitons in the trap for excitation density Pex = 0.31µW
(black squares), 2.2µW (red circles), and 88µW (blue tri-
angles) at Tbath = 50mK.(b,c) Amplitude of the inter-
ference fringes Ainterf at shift δx = 4µm averaged from
0 < x < 1.5µm (b) and HWHM of the exciton emission pat-
tern along x (c) vs. Pex for Tbath= 50mK (blue circles) and
4.5K (black squares).
tial landscape in the trap by indirect excitons due to the
repulsive exciton-exciton interaction [23]. However, at
low temperature T = 50 mK, the width of the emission
pattern of the exciton cloud nonmonotonically changes
with density: the increase of density first leads to the
reduction of the cloud width, indicating the exciton ac-
cumulation at the trap center, while at higher densities,
an increase of the cloud width with density is observed
(Fig. 3c).
Figure 3b presents the density dependence of the am-
plitude of the interference fringes Ainterf. At high tem-
perature T = 4.5 K, the coherence degree is low for
all densities (Fig.3b).However, at low temperature
T = 50 mK, the coherence degree nonmonotonically
changes with density: the increase of density first leads
to a strong enhancement of Ainterf, followed by its reduc-
tion (Fig. 3c). Note that maximum Ainterfcorresponds
to minimum width of the emission pattern of the exciton
cloud in the trap (Fig. 3b,c).
Figure 4 presents the amplitude of the interference
fringes Ainterf(δx) for different temperatures.
temperatures T>
∼4K, Ainterfquickly drops with δx. As
discussed below, this behavior is expected for a classical
gas. However, extended spontaneous coherence is ob-
served at low temperatures (Fig. 4). The spatial exten-
At high
02468 10 1214
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
048
2
4
6
Interference visibility
shift ?x (µm)
Tbath=
50mK
2K
4K
8K
a
b
?? (µm)
T (K)
FIG. 4: (a) Ainterf(δx) for excitons in the trap for Tbath =
50mK (black squares), 2K (blue triangles), 4K (green cir-
cles), and 8K (red diamonds).
which the interference visibility drops e times. For all data
Pex = 1.9µW.
(b) ξ evaluated as δx at
sion of Ainterf(δx) can be characterized by a coherence
length ξ. Here, to consider all data on equal footing, we
evaluate ξ as δx at which the interference visibility drops
e times. The temperature dependence of ξ is presented in
Fig. 4b. A strong enhancement of the exciton coherence
length is observed at low temperatures. While at high
temperatures ξ is considerably smaller than the exciton
cloud width, at low temperatures the entire exciton cloud
becomes coherent (Figs. 2f and 4).
The data are discussed below. In the reported exper-
iments, the laser excitation energy exceeds the exciton
energy by about 400 meV and the laser excitation spot
is spatially separated from the interfering excitons for
positive δx. Therefore studied coherence in the exciton
gas is spontaneous coherence; it is not induced by coher-
ence of the laser excitation. [Note that for negative δx,
the interfering excitons spatially overlap with the laser
excitation spot and exciton coherence is suppressed (Fig.
3a), confirming that exciton coherence studied in this
work is not induced by the laser excitation.]
Coherence of exciton gas is imprinted on coherence of
exciton emission, which is described by the first-order
coherence function g1(δx). In turn, this function is given
by the amplitude of the interference fringes Ainterf(δx)
in ‘the ideal experiment’ with perfect spatial resolution.
In real experiments, the measured Ainterf(δx) is given by
the convolution of g1(δx) with the point-spread function
(PSF) of the optical system used in the experiment [27,
29]. The PSF width corresponds to the spatial resolution
of the optical system.
At high temperatures, the amplitude of interference
fringes quickly drops with δx and the width of Ainterf(δx)
corresponds to the PSF. This behavior is characteristic
for a classical gas, where g1(δx) drops within the ther-
mal de Broglie wavelength λdB=
0.5µm at 0.1K, below the PSF width. At low temper-
atures, extended exciton coherence with the coherence
length much larger than in a classical gas is observed
?
2π¯ h2
mT, which is about

Page 4

4
(Fig. 4). At the lowest temperatures, the observed co-
herence length in the exciton gas in the trap exceeds
λdB = 0.5µm at 0.1 K by an order of magnitude and
the entire exciton cloud in the trap becomes coherent
(Figs. 2f and 4).
Figure 4a illustrates why δx = 4µm is selected for pre-
senting coherence degree of excitons in Figs. 2 and 3.
The shift δx = 4µm is chosen to exceed both λdB and
the PSF width. At such δx, only weak coherence given
by the PSF value at δx = 4µm can be observed for a
classical gas. Higher Ainterfexceeding such background
level reveal the enhanced coherence degree of excitons.
In conclusion, we report on the emergence of sponta-
neous coherence in a gas of indirect excitons in a trap. At
low temperatures, the exciton coherence length becomes
much larger than the thermal de Broglie wavelength and
reaches the size of the exciton cloud in the trap.
We thank Michael Fogler for discussions. This work
was supported by ARO grant W911NF-08-1-0341. The
development of spectroscopy in a dilution refrigera-
tor was also supported by the DOE award DE-FG02-
07ER46449 and NSF grant 0907349.
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