Table Of ContentCondensation to a strongly correlated dark fluid of two dimensional dipolar excitons
Yotam Mazuz-Harpaz, Kobi Cohen, and Ronen Rapaport
Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel.
(Dated: January 11, 2017)
Recently we reported on the condensation of cold, electrostatically trapped dipolar excitons in
GaAs bilayer heterostructure into a new, dense and dark collective phase. Here we analyze and
discuss in detail the experimental findings and the emerging evident properties of this collective
liquid-like phase. We show that the phase transition is characterized by a sharp increase of the
numberofnon-emittingdipoles,byaclearcontractionofthefluidspatialextentintothebottomof
7 the parabolic-like trap, and by spectral narrowing. We extract the total density of the condensed
1 phasewhichwefindtobeconsistentwiththeexpecteddensityregimeofaquantumliquid. Weshow
0 that there are clear critical temperature and excitation power onsets for the phase transition and
2 that as the power further increases above the critical power, the strong darkening is reduced down
untilnocleardarkeningisobserved. Atthispointanothertransitionappearswhichweinterpretasa
n
transitiontoastronglyrepulsiveyetcorrelatede-hplasma. Basedontheexperimentalfindings,we
a
J suggestthatthephysicalmechanismthatmayberesponsibleforthetransitionisadynamicalfinal-
statestimulationofthedipolarexcitonstotheirdarkspinstates,whichhavealonglifetimeandthus
0
supporttheobservedsharpincreaseindensity. Furtherexperimentsandmodelingwillhopefullybe
1
able to unambiguously identify the physical mechanism behind these recent observations.
]
s
a I. INTRODUCTION atlowertemperatures. Theselongrangeinteractionsare
g
of Van Der Waals type and therefore they fall quickly
-
t with the distance between particles in the fluid. As a
n The understanding of the collective effects of ultra- result, the gas phase of 4He is only weakly correlated. It
a
u cold, quantum degenerate bosonic and (more lately) is far less obvious to figure out what are the correlation
q fermionic gases has advanced significantly in the last effects of a fluid of dipoles, where the interactions de-
. two decades. This, mostly due to major advances in cay slower with the particle distance. In particular, it is
t
a systems of cold atomic species, but more recently also interesting to understand the stable phases of a fluid of
m in condensed matter systems, where cold gases of elec- particleswherethislongrangeinteractionispurelyrepul-
- tronic excitations have been realized [1–3]. One unique sive. This is true both in the classical regime where the
d class of bosonic many-body systems is that of dipolar dominant kinetic energy is that of thermal motion and
n
fluids, where the combination of Bose-Einstein quantum in the quantum regime where the dominant kinetic term
o
c statistics and the long ranged dipole-dipole interaction becomes the zero-point motion of the particles. From
[ gives rise to a rich and unique fundamental physics. Un- the experimentalist point of view, it is therefore a wor-
like the case of short range interaction, where particles thy challenge to realize and study such dipolar fluid in
1
are essentially free except during instantaneous collision various systems.
v
events, dipolar particles interact with each other over
8 Indeed,suchdipolarmany-bodysystemshavebeenre-
long distances. In this sense, dipolar fluids are much
9
alizedandstudiedinsemiconductorquantum-well(QW)
5 morecorrelatedthanweaklyinteractingfluids. Thedipo-
heterostructures,whereindirectexcitonsareopticallyex-
2 larinteraction-inducedcorrelationsbetweenparticlescan
cited, for several decades now [5–19], and more recently
0 lead to very interesting effects such as pattern forma-
1. tion and instabilities even for purely classical particles they were also demonstrated in atomic physics [20, 21],
where some striking observations related to quantum
0 [4]. Even more exotic are cases where the classical corre-
dipolar correlations and condensation have already been
7 lations compete with quantum mechanical effects. This
1 usuallyhappenswheneverthequantumkineticenergyof observed [22–34]. In the last few years the field has
: further expanded to polaritonic systems [35–37], to bi-
v eachparticle(duetothemomentum-positionuncertainty
layer two-dimensional transition metal DiChalcagonide
i principle) is of the same order of magnitude as the typi-
X systems [38] and to bilayer graphene [39, 40]. All these
cal interaction energy between particles. This makes the
r center-of-mass position of particles in a dense interact- realizationsofcolddipolarfluidshavealreadyledtonew
a andexciting,sometimesunexpected,observations. These
ing fluid uncertain, leading to a significant wavefunction
observationsareuniquetosuchcorrelatedfluidicsystems
overlap between particles and signifying the onset of col-
andthereisstillagaptobridgebetweenthemultitudeof
lective Bose-Einstein quantum effects. One well known
experimental reports and a consistent theoretical frame-
systemwheresuchcompetitionprevailsiscold4Hewhere
work.
thisinterplayleadstothecondensationoftheliquidinto
a superfluid state at low enough temperatures. In 4He In this paper we focus on a system of two dimensional
thelong-rangepartoftheinteractionisattractive,which (2D) indirect excitons (IXs) in GaAs double quantum
leads to the formation of a stable liquid at low enough well (DQW) heterostructures: a unique system of 2D
temperaturesandtothetransitionintoaquantumliquid boson-like dipolar quasi-particles which are dynamically
2
Additionally, as the effective mass of an exciton is just
a fraction of the free electron mass m , the thermal De-
0
Brogliewavelengthisverylargeandquantumstatisticsis
expected to become important at temperatures as high
as several degrees Kelvin [19, 42, 43]. In the last two
dacades, the interest in cold dipolar fluids of excitons
in GaAs-based structures has been growing consistently.
The developement of new techniques for trapping and
manipulating IXs [44–69], allows a better control of ex-
citonic fluids, and quantitative analysis of experiments.
Thisledtoasignificantprogressinthefield,andtorecent
experimental observations of several very intriguing phe-
nomena. While these are not yet fully understood, they
indeedsignifythecomplexityandrichnessofthesystem,
andtheimportanceoftheinteractionsandcorrelationsin
the formation of the collective states of dipolar quantum
fluids.
FIG. 1. (a) An illustration of IXs in an electrostatic trap
in a DQW sample. Excitons are created by laser excitation
of the trap region. An electric voltage between the conduct- B. Particle correlations in a dipolar exciton gas
ingsubstrateandthemetallictopgatepolarizestheexcitons
which are confined under the gate. (b) A microscope image
Several theoretical works have shown that due to the
of the IX trap used in the experiment described in Sec. III.
combination of the properties mentioned above, the in-
It is surrounded by a guard-gate, with the bright laser spot
in its center. terplay between the interaction-induced many-body cor-
relations and quantum statistics is expected to be very
pronouncedinIXfluidsatveryfeasibletemperatureand
excited using light and that have an internal spin de- density ranges (see e.g. [43, 70–72]). In fact, strong
greesoffreedomwhichdeterminestheirdecaydynamics. particle correlations should exist under almost every re-
In recent years the experimental progress has allowed a alistic experimental conditions. At high temperatures
consistent study of cold IX fluids in a wide range of den- and low enough densities strong pair correlations in a
sities. Thestrikingobservationsclearlypointtoanintri- classical gas phase are expected, where excitons tend
cate many-body quantum effects that are yet to be fully to avoid each other due to their mutual dipolar repul-
understood. Despitethecurrentlyunansweredquestions, sion. This repulsion is balanced by the thermal kinetic
thisprogresshasalreadyopenedawindowintothecom- energyofthedipolesinthegas,resultinginatypicalav-
plex physics of interacting dipolar fluids which are cou- eragescatteringdistancer betweenthedipolesgivenby
0
pled to light in a non-trivial manner. E (r )∼k T, where E (r)(cid:39)e2d2/εr3 is the classical
dd 0 B dd
interaction between a pair of dipoles. This balance leads
to an effective depletion circle which forms around each
II. A SHORT REVIEW OF RECENT IX, whose radius r (T) increases as the temperature is
0
THEORETICAL AND EXPERIMENTAL lowered. Such pair correlations should reduce the aver-
PROGRESS
age repulsive interactions in the IX gas from the mean
field value [70–72] and they should become stronger as
A. Dipolar excitons in GaAs double quantum wells the temperature is lowered and the thermal energy of
theIXsisreduced. Thethermalfluctuationsaroundthis
An indirect exciton is a Coulomb-bound yet spatially- average value of interaction energy should also decrease
separatedpairofanelectron(e)andahole(h),typically with decreasing temperature. The effect of the increased
created optically by laser excitation in an electrically- correlations is therefore expected to be experimentally
biased DQW heterostructure [18, 19], as illustrated in manifested by the reduction of both the measured repul-
Fig 1. Due to this e-h spatial separation, IXs carry sive interaction energy of recombining IXs in an IX gas
√
largeelectricdipolemoments. Thesearedefineased/ ε with a fixed density, as well as by the reduction of their
where d is the separation between the e and the h, and linewidth[71]. Aclearexperimentalevidenceforsuchan
ε is the background dielectric constant. Typical dipole effect was reported by Shilo et al. [73] and later on by
√
lengthsforIXsared/ ε∼3-6nm,whicharemuchlarger other several works [74–76] for a macroscopic number of
than in their atomic counterparts. The spatial charge IXs, and also for the case of only few IXs in a tiny trap
separation also leads to long IX radiative lifetimes, typi- [52]. This effect will be also briefly discussed later on in
callyintherangeof100-1000ns. Asthatlifetimeismuch this paper. These observations have proven that indeed
longerthanthethermalizationtimewiththelattice[41], a fluid of IXs is fundamentally different from a weakly
IX fluids may reach a quasi thermal equilibrium with it. interacting gas of either atoms [77], excitons, or exciton-
3
polaritons[3]. Anotherinterestinglowdensitycorrelated Contrary to that, at high fluid densities, where
gas regime is expected to occure when a very dilute gas r n1/2 (cid:29) 1 occurs at temperatures that are above T ,
0 0
is cooled down to a temperature where the thermal De- a correlated, interacting, multi-particle state should ap-
Brogli wavelength of the excitons λ (T) becomes larger pear, expected to resemble a classical liquid state [72].
db
thanr . Atthispointthedipolarscatteringbetweenex- This dipolar liquid state should be characterized by a
0
citons should be described quantum-mechanically. This shortrangeorderandreducedfluctuations,andbyasig-
leads to an effective quantum depletion region which is nificant reduction of the diffusivity. The latter, due to
expected to become temperature independent [71, 72], the quenching of the mean-free-path of the excitons re-
whichinturnleadstotemperatureindependentpaircor- sulting from the multi-particle dipolar interactions. A
relationsandtemperatureindependentaveragerepulsive quantitative theory of a classical gas-liquid transition of
interaction energy. Recently we observed a transition 2Ddipolarindirectexcitons, andofthethermalandme-
from a temperature regime where the repulsive interac- chanical properties of such liquid state, if exists, is not
tion energy per exciton (also called the ”blue shift” en- yet available as far as we know. Nevertheless, a clear
ergy) decreases with decreasing T, as expected from a first-orderphasetransitionofadensefluidofexcitonsto
classicallycorrelatedgas, toatemperatureregimewhere a new phase that has a set of features of a liquid state
the blue shift becomes T-independent (with a transition was reported recently by Stern et al. [83]. In that re-
between the regimes that goes like T2) [73]. This might port as a spatially extended, high density gas of IXs was
beanindicationtosuchclassical-to-quantumcorrelation cooled down, a new phase emerged to coexist with the
crossover but further experiments are required to verify initial gas phase, with clear spatial separation between
this point. them and a clear phase boundary. The new phase had a
lower interaction energy and a smaller, more symmetric
linewidth than the gas phase, indicating stronger par-
ticle correlations and strongly reduced fluctuations, re-
C. Multi-particle correlations and collective phases
spectively. A significant reduction in diffusivity was also
reported. These properties are a strong signature of a
Even more exciting and intriguing are the possible
liquid state. A clear critical temperature (T = 4.7K)
multi-particle correlated phases of an IX fluid, which c
and a critical excitation power were also observed, with
are expected when the fluid becomes cold and dense
a sharp onset of the liquid state beyond those critical
enough. Multi-particle interactions are expected when
values. That fact, together with the coexistance of the
the depletion radius r is larger than the actual average
0 liquid and gas phases, points toward a first order phase
inter-particle distance n−1/2, i.e., when r (T)n1/2 (cid:29) 1.
0 transition. However, it is still not completely clear if
The condition for the onset of multi-paticle interactions
there was a significant discontinuity in density between
r (T)n1/2 ∼1shouldbecomparedtothecriterionofthe
0 thetwophases,asexpectedfromafirstorderphasetran-
onset of quantum degeneracy, λ (T )n1/2 ∼ 1. At low
db 0 sition. Notably,becauseinterferometricmeasurementsof
enough temperatures and low enough densities, where
the liquid state did not show any extended spatial co-
the quantum degeneracy condition is met and the multi-
herence beyond the diffraction limit, the authors have
particle interaction condition is not, a significant wave-
identified this as an indication that the liquid state is
function overlap of neighboring IXs is expected, across
classical, following the prediction by Laikhtman and Ra-
the depletion region of each particle. In such case the
paport [71]. Interestingly, it was noted in [71, 72] that
fluid should undergo a BEC transition with a significant
the onset of a classical liquid state might suppress the
condensate fraction yet observable particle correlations
onset of quantum degeneracy, with the transition to the
[43]. SeveralexperimentalworksstudyingcoldIXclouds
quantum liquid state being pushed down to lower tem-
in exciton rings [45, 46, 78] reported a fragmantetion of
peratures. This is due to the significant localization of
the ring into ordered beads with extended spatial coher-
particlesintheliquidstatethatpreventsgoodoverlapof
ence [79–81] below a critical temperature. This was in-
thewavefunctionsofadjacentparticles. Amorequantita-
terpreted as a signature for the onset of a Bose-Einstein
tivepredictionoftheonsetofquantumliquidityofanIX
condensation of the IXs on the ring. A similar onset of
gasthroughaBerezinskii–Kosterlitz–Thoulesstransition
extended spatial coherence was reported for IXs trapped
was performed using numerical Monte-Carlo simulations
inanelectrostatictrap[82]. Inthoseworksacriticaltem-
in Ref. [43, 84]. There, an opposite effect was predicted,
peraturefortheonsetofthecoherencewasreported,but
as it was calculated that the transition temperature to
no density threshold was clearly identified. This leaves
the quantum liquid state grows with increasing dipole
openthequestionofwhichtypeofaphasetransitionwas
density and with increasing particle localization.
observed(e.g. wasitafirstorasecondordertransition),
andthequestionofthedensityregimeinwhichcoherence Themulti-particlestatesdiscussedabove,quitegeneric
appeared, i.e. whether it corresponds to the dilute limit toa2Ddipolarfluidsofbosons,donothowevertakeinto
described above. Long range coherence is expected from account two major inter-related ingredients in every IX
a dilute gas of bosons in the limit of weak interaction. It fluid in GaAs structures. The first is that IXs are opti-
is yet to be fully understood whether the experimental cally created by light and have a finite lifetime. It is fi-
observations are indeed in this range. nite since they can recombine either via optical emission
4
of photons or by e and h tunneling through the opposite described above, the IXs were considered to be in full
QWbarrierstowardsthetwocontacts,creatingelectrical thermal equilibrium and to have a well defined density.
current through the sample. The typical recombination Thus, no dynamical treatment was given for the case of
lifetimes in both channels could be very different, with finite lifetime IXs under steady state excitation, where
radiativelifetimesofexcitonsbeingusuallymuchshorter the density is not conserved but it is rather determined
than tunneling times, and both strongly depend on var- from the internal dynamics of the system and the exter-
ious experimental parameters such as the applied bias, nal pumping. In that aspect the theoretical picture is
the temperature, and the density [73, 85]. This optical still incomplete, as no published theory we are aware of
generation and the two recombination channels result in has been put forward taking into account both dipolar
a fluid of IXs without a conserved number of particles, interactions, the internal spin structure, and the dynam-
and this should have a major impact on the formation icsofthesystemunderanexternalparticlepumpingand
dynamics of any collective state. internalparticlelossesthatinturndependontherelative
occupation of the different spin states. We believe that
The second missing ingredient in the theoretical pic-
in light of the results described below, such a treatment
ture above is the unique internal spin structure of exci-
is essential in order to describe the origin and nature of
tons in GaAs heterostructures, which should also have a
the ground state of the IXs fluid. This is specifically im-
major role in the formation of their collective dynamics.
portant because any decoupling between the densities of
The total spin projection of heavy-hole excitons is ±1
the dark and bright IXs should obviously have an effect
or ±2, corresponding to two radiative (”bright”) states
onthedynamicsofdecayandformationofthefluid, due
and two non-radiative (”dark”) states. In GaAs, where
to the different lifetimes of the dark and bright IXs.
photoluminescenceisthedominantdepletionmechanism
of excitons, this naturally means that the intrinsic life-
time of the dark excitons is much longer than that of
the bright ones. However, under normal circumstances,
the very fast h spin flip in the IX leads to a full ther-
malization between the bright and dark states. Since in
typical DQW structures all four spin states are also al-
most degenerate in energy (at all realistically achievable Experimentally, in [73] we reported the first observa-
temperatures), it is assumed that the dark and bright tionofanomalousdarkeningbelowacriticaltemperature
populationsareequal,andthattheypossesasingleeffec- of T ∼=2.6K, where a sharp increase of the density of an
tive lifetime, mostlydeterminedbythe radiativelifetime IX fluid in a flat-bottom wide trap occurs, indicated by
of the bright excitons (see e.g., the supplamentary infor- an increase of luminescence energy blueshift, but with
mation of Refs. [73, 76] and [85] for discussion on the a corresponding decrease of the luminescence intensity.
IX lifetime). However, due to spin-dependent exchange This was interpreted as an evidence for a possible con-
interactionbetweentheelectronsandholesinanIX,the densation of a macroscopic number of excitons into a
dark excitons are actually slightly lower in energy than dark state. The observation of darkening was later also
the bright excitons [86, 87]. As long as no quantum col- reported in IX rings [80], and some darkening was also
lective effects are in play, this energy splitting is negligi- observedintheliquidstatereportedin[83]. Whilethese
ble. However, if aBose Einstein condensationcan occur, works gave indications that a darkening transition does
ideal bosons macroscopically condense to the lowest en- occur, the full experimental picture was still missing. In
ergy state regardless of how small the gap from excited a recent work we reported on a comprehensive, quan-
states is. It was therefore predicted a few years ago that titative experiments of an IX fluid in a quasi-parabolic
the ground state of a dilute dipolar exciton fluid is in trap[76]. Inthatwork, averyclearphasetransitionwas
fact dark and that a macroscopic phase transition to a identified, in which above a critical excitation power and
dark condensate should occur [88]. This work however belowacriticaltemperature,agasofIXscondensedtoa
neglected the interactions between IXs and thus a later dense, compact phase at the center of the trap, initially
work from the same authors considered the case where made of mostly dark particles. This phase is character-
pairwise short range interactions are introduced. It was ized by a spectral narrowing and a sharp increas of the
shown that these interactions lead to a density thresh- total density, as in a highly correlated liquid. As the
old for the dark condensate, above which the condensate excitation power further increased the IXs became less
shouldbecome”gray”,astheinteractionsmixthebright dark yet still maintained their other properties, such as
and dark spin states. The bright component of the con- the nearly closely packed density and the almost fixed
densate was predicted to grow linearily with the density spatial extent and spectral linewidth. At high enough
above the threshold density until it reaches an equality exitation power the bright and dark densities approxi-
with the dark part at high densities [89]. The theory did mately equated, and another sharp transition was ob-
not calculate what should be the threshold dark density served, where the density further increased and a large
forrealisticIXsystems,norittookintoaccountanycor- spatialexpansionwasobserved. Inthefollowingwesum-
relations that arise from the spatially extended dipolar marize the observations and analysis of the experiments
interactions. Additionally, as with the previous theories reported in that work [76].
5
FIG. 2. (a) A spatial-spectral profile of the trapping potential, obtained by scanning a weak focused excitation point across
the dotted line marked in Fig 1b. The dashed black lines mark the edges of the trapping gate. The spatial expansion of the
IX cloud for different excitation powers for (b) T = 5.4K and (c) T = 1.5K. (d) The interaction energy ∆E as a function
of excitation power for the same two temperatures, below and above T . The red dashed lines mark the two critical powers
c
P ,P separating the three regimes interpreted as the gas, liquid and expanding plasma respectively. (e) The blue shaded
c1 c2
regionistheextractedtotalIXdensityrangeforn asafunctionoftheexcitationpoweratT =1.5K.Thebluecirclesdenote
tot
twicetheextractedbrightIXdensity,2n . (f)Theinteractionenergy∆E asafunctionofT foraconstantexcitationpowerof
b
308nW. (g) The IXs spectral linewidth as a function of temperature for four exemplary excitation powers. (h) The measured
fluxesofchargecarriersinthetunnelingcurrent(red)andofemittedphotons(blue). Theorangelineistheaverageofthetwo
fluxes. The purple dashed lines in (f-h) marks T =4.8K. (i) An estimate of the total IX densities at T =5.4K (red) and at
c
T =1.5K (blue). Thehightemperaturedensityisextractedastwicethemeasuredbrightdensity,assumingn (cid:39)n aboveT
d b c
while the low temperature density is extracted from ∆E, according to a liquid model [72], as explained in the text.
III. EXPERIMENTS WITH A DIPOLAR itspotentialshapeatlowIXdensitywasquasi-parabolic,
EXCITON FLUID IN A QUASI-PARABOLIC as shown in Fig. 2a. The emission of the trapped bright
ELECTROSTATIC TRAP IXswasmonitoredasthetemperature,theexternalvolt-
age of the trapping potential and the excitation power
were varied. We introduced and employed, for the first
In the experiment reported recently by Cohen et al.
time, an experimental technique of constant energy lines
[76], a trapped IX fluid was studied in an electrostatic
(CEL). In this technique the spectral position of the IXs
trap which was excited at its center by a continuous-
is retained throughout the experiment by adjusting the
wave, non-resonant laser. The trap’s shape was circular
external voltage such that it compensates for any spec-
with a diameter of 20µm, as presented in Fig. 1b, and
6
tral shifts due to varying experimental conditions (such tionintemperature,itisalsoimportanttoshowhowthe
as temperature and excitation power variations). Since transition was manifested as a function of the excitation
the single-IX properties are mostly determined by the power, which controls the rate of creation of IXs in the
local electrostatic environment of the IX, which also de- trap. Figs. 2c,d present the normalized emission profile
termine its spectral position, maintaining the exciton on of the bright excitons in the trap as a function of the ex-
a CEL helps in approximately fixing its single-particle citation power, for two temperatures, one just above T ,
c
properties. This in turn helps tremendously in the anal- and one well below it, respectively. While for the high
ysisofthedata,andinextractingthemany-bodyeffects. temperature,thewidthofthecloudgrowsmonotonically
With this method, several observations were reported. with increasing power, as expected from a mobile repul-
Firstly, when the trapped fluid was cooled under a con- sivegasofdipolarparticles,thelowtemperaturebehaves
stantexcitationpower,thedipolarinteractionenergyper very differently. The initial expansion at low powers is
particle, measured from its blue shift ∆E, initially de- followed by a significant contraction across a clear criti-
creased, indicating the increase of particle correlations, calpowerPc1 toanarrowcloudofconstantwidth. That
as was discussed above. Surprisingly, below a temper- width is unchanged over a wide range of excitation pow-
ature of Tc = 4.8K, the correlated IX gas suddenly dis- ers, up to another clear threshold power Pc2 after which
playedasharpincreaseofitsinteractionenergy,reaching the cloud suddenly expands again, all the way to the
a constant value ∆E at a lower temperature of T ∼=2K. boundaries of the trap’s gate. The contraction of the
l
This sharp rise in ∆E, plotted for one excitation power cloud above the first critical power is accompanied by a
in Fig. 2f, indicates a sharp increase of the total den- sharp increase in ∆E to its saturated value ∆El, as is
sity, as ∆E ∝ n , where n is the total local density seen in Fig. 2d. Such a sharp increase is not observed
tot tot
of the IX fluid, including both bright and dark parti- in the high temperature data, above Tc. As mentioned
cles. A very similar increase to the same final ∆E value above, the blue shift energy can be mapped to a total
l
was observed for a very wide range of excitation powers, dipolar particle density, taking into account the possible
marking that the density increases to a saturated value. rangeofcorrelations. Thisisdepictedbytheblueregion
To quantify the final density by mapping ∆El into den- in Fig. 2e, where a sharp rise to a high density value nl
sity,requiresaknowledgeofthecorrelationsbetweenthe is observed across the critical power. This value then in-
dipoles [71, 72]. However, one can obtain a range of pos- creasesveryslowlyoverawiderangeofexcitationpowers
sible final densities by taking the two limits of possible up to the second threshold. Again, all these findings are
correlations, namely the totally uncorrelated state (de- consistentwithacorrelatedmulti-particlestate: aliquid.
scribedbyameanfieldtheory)andthehighlycorrelated The strong expansion beyond Pc2 indicates a transition
state (a liquid with short range order) [76]. These two to a new, highly mobile, dense phase, most likely a cor-
limits yield the range of possible saturation densities to related e-h plasma phase, expanding due to the strong
be n =6-9×1010cm−2. Since for all values within that in-plane repulsion. The interpreted comparison between
l
rangetheconditionofamulti-particlecorrelationsiswell the total exciton densities as a function of the excitation
satisfied, namely, r (T = 2K)n1/2 (cid:29) 1, the self consis- power below and above Tc is shown in Fig. 2i, where a
0 l liquid model is assumed for the low temperature and a
tentmappingbetween∆E andnshouldbethatofaliq-
mean field model is assumed for the high temperature.
uid state, which is the highest value in the above range,
i.e., nl (cid:39) 9×1010cm−2. In fact, this value corresponds One fundamental question that arises is what mecha-
to approximately close packing of the IX wavefunctions, nism can support such a large increase of density under
(cid:104)rl(cid:105)/ab ∼ 2-3, where ab is the IX’s Bohr radius. This constant excitation conditions? If the dark and bright
density buildup was accompanied by a clear reduction IX densities are equal due to thermal equilibrium be-
of the spectral linewidth, indicating increased particle tween them, then at steady state their combined density
correlations and the appearence of short range order, as should be given by n = Gτ . Here, G is the optical
tot eff
expected from a liquid state. Again, as with the energy generation rate of IXs and τ is the effective lifetime
eff
shift, the linewidth of the emission was reduced to a sat- whichgovernsthesimultaneousdecayofbothbrightand
urated value for a wide range of excitation powers below dark populations. With the CEL method τ should
eff
thesamecriticaltemperatueTc,asshowninFig. 2g. No- stay constant, at least for a fixed temperature, and thus
tably,Itwasalsoaccompaniedbyspatialshrinkageofthe the density should be proportional to G. It is therefore
IX cloud area towards the center of the trap, indicating not clear what can cause the sharp density increase. A
either a strong suppression of the IX mobility, a strong possible solution is that the population equality between
suppression of the thermal distribution of the kinetic en- the dark and bright states is not maintained below T
c
ergy, or the onset of attractive short-range interactions, and above P . An experimental direct and independent
c1
that perhaps can occur at close packing of IXs [90]. All measurement of the bright and dark densities, n and n
b d
these findings are consistent with the interpretation of a respectively,isahardtask. However,since∆E isamea-
transition into a dense liquid state, having short-range sureofn =n +n ,andsincen canbeextractedfrom
tot b d b
order and a well-defined density which is spontaneously the collected photon flux, it is possible to deduce also
chosen by the system itself. n independently. The calibration of the bright density
d
After establishing a clear evidence for a sharp transi- from the measured PL intensity I relies on the relation
7
IV. SUMMARY, DISCUSSION, AND OUTLOOK
We can conclude that a transition to a dense phase of
dipoles is observed below a critical temperature T and
c
above a critical excitation power P . This phase tran-
c1
sition, schematically depicted in Fig. 3, is characterized
by the following signatures:
1. Short range order and close packing: Acorre-
latedgasofIXsspontaneouslyincreasesitsdensity
as the temperature is dropped below T ∼= 4.8K,
under a fixed optical excitation power. This den-
sity saturates at T ∼= 2K to a high value n ≈
l
6-9×1010cm−2 satisfying r n1/2 (cid:29) 1, that is al-
0 l
most independent on the excitation power over at
least two orders of magnitude. This sharp spon-
FIG.3. Aschematicillustrationoftheobservedstatesofan
IX fluid trapped in a quasi-parabolic potential: A repulsive taneous increase of density is accompanied by a
bright gas phase with equal populations of dark and bright strong reduction of the spectral linewidth, indicat-
IXs for T > T and a dark, spatially compact, and dense ingaphasetransitiontoadensephasewithashort-
c
liquid phase for T <Tc. See text for details. The bright and range order and, importantly, with a well defined
dark IXs are depicted by yellow and gray halos, respectively. density that is spontaneously set by the system it-
self. Remarkably, this density range corresponds
toapproximatelyclosepackingoftheIXwavefunc-
tions (cid:104)r (cid:105)/a ∼2-3.
l b
2. Spatial condensation: Thespontaneousincrease
of density is accompanied by a strong shrinkage
of the IX cloud area, indicating either a strong
suppression of the IX mobility, a strong suppres-
I =B·n /τ ,whereB isacoefficienttakingintoaccount
b b sion of the thermal distribution of the kinetic en-
many experimental factors and τ is the bright exciton
b ergy, or the onset of attractive short-range inter-
lifetimethatcanbemeasuredorcalibrated[73,85]. Fig.
actions. Furthermore, a comparison of the exper-
2e shows the extracted value of 2n as a function of the
b imental findings with a dipolar liquid model [72]
excitation power. If there was a population equality be-
indicates that the liquid phase is expected to be
tween dark and bright particles (n =n ), we would ex-
d b within the parameter range of a quantum liquid,
pect that n =n +n =2n . This is certainly not the
tot d b b having a significant quantum kinetic energy con-
case at around P , where we find n (cid:29)n , suggesting
c1 tot b tribution and the quantum degeneracy condition
n (cid:29)n and a large imbalance between dark and bright
d b λ n1/2 (cid:29)1 should be well satisfied (see details in
particles. If indeed there is a large accumulation of dark db l
Cohen et al. [76]).
excitons, which can only decay through tunneling to the
contacts, we expect an increase in the measured current
3. Macroscopic accumulation of dark particles:
through the trap below T . This is indeed the case, as
c Below T there is an evidence that the sharp
c
can be seen in Fig. 2h. In steady state the total loss
buildup of density to the liquid state is driven
rate (i.e. emitted photons+electrons in current) should
mostly by accumulation of dark particles. At the
stay constant under a constant generation rate. This is
low excitation onset of the transition to the liquid
indeed verified in Fig. 2h, where we see that below T
c state,morethan90%oftheparticlesareinthedark
thephotonratedecreasestocompensatefortheincrease
state. This darkening is diminished as the excita-
in the tunneling rate, so that total loss stays the same.
tion power is further increased, qualitatively simi-
WenotethatastheexcitationpowerincreasesaboveP
c1 lartothepredictionbyCombescotandCombescot
the degree of darkening becomes smaller, until it disap-
[89]. This observed accumulation of dark particles
pears just before P . While in the current results there
c2 might also explain how the sharp density buildup
is no direct experimental way to assign the bright and
occursaboveatinyexcitationpowerofP ∼10nW.
dark populations to the corresponding total spin projec-
We also observed that the optical darkening of the
tion of excitons ±1 or ±2, the connection to the theory
IX fluid is quantitatively compensated by an in-
put forward by Combescot and Combescot [89] is quite
creaseofthetunnelingcurrentthroughthesample,
compelling. Ofcourse,atthispointwhilewedonothave
which is the remaining depletion channel for dark
any different picture to explain the above experimental
particles that cannot recombine radiatively.
findings, we cannot rule out that such alternative does
exist. This is discussed in the next section. 4. Well defined phase boundaries: We found
8
that the dark liquid phase at low temperatures is or is it driven by quantum degeneracy, as in the
bounded by a gas phase from the low density side quantum liquid or BEC case? Is the phase transi-
andbyabright,highlymobileplasmaphaseonthe tion purely thermodynamic, or is it driven dynam-
highdensityside. Thisisaninitialpartialmapping ically by the unique coupling of the internal spin
of the phase diagram of cold dipolar IX fluids. degrees of freedom of the IX to light? Our experi-
ments indicate that dark IX spin states might play
Alltheaboverecentfindings,byotherresearchgroups animportantrole,asacondensationtodarkstates
as well as by ours, opened up a narrow window to what whicharelonglivedisatleastconsistentwithmany
seems as a complex phase diagram of cold dipolar IX of our observations in Refs. [73, 76], as well as in
fluids. As it seems, this phase diagram is the result of the work reported in [91]. More insight into the
anintricateinterplaybetweencollectivequantumeffects, natureofthedarkeningandtheconnectiontodark
interaction driven correlations, and internal spin degrees excitons is essential.
of freedom, all bound together with the dynamics of a 3. More information on the collective ground state is
dissipative system with a finite lifetime. Remarkably, IX certainly needed. A measurement of its compress-
fluidsmightcontainmany-bodyphysicsevenricherthan ibility,microscopicspatialcorrelations,momentum
initially thought. There are many fundamental open distribution, and excitation spectrum, are impor-
questions that are now awaiting an answer. Few of these tant to fully understand its nature.
questions are listed below:
4. The issue of optical coherence also calls for ad-
1. Up until now, only a fraction of the full phase dia-
ditional study. As mentioned above, while sev-
gram of cold dipolar IX fluids have been mapped.
eral works reported the onset of extended coher-
Mappingofthefull(n,T)phasediagramisanout-
ence from a cold fluid of IXs, the works reporting
standing challenge that is at the heart of under-
the liquid formation were unable to measure spa-
standing the system. Particularly interesting is to
tial coherence beyond the diffraction limit. One
get the full mapping of the gas, liquid, and plasma
possible explanation is that the coherence is hid-
phases, and of the dark liquid phase. An inter-
den in the dark part of the fluid in those experi-
esting open question is whether there are two dis-
ments. Another option is the size of the dipole. As
tinct transitions, namely one to a classical liquid
is shown in Ref. [43], the single particle density
and one to a dark quantum liquid, or rather there
matrix gets strongly localized for increasing dipole
is only one liquid state. Strikingly, the observed
size, which is expected to reduce the spatial co-
onset density T in our experiment is in almost
c herence significantly. Indeed, in the works that re-
a perfect agreement with the critical temperature
portedextendedopticalcoherenceIXswithsmaller
for the onset of the liquid phase reported in Stern
dipoles were studied compared to those reporting
et al. [83], even though the experiments were done
a liquid formation. Finally, if the liquid state is
with different samples and in different experimen-
purelyclassical,nocoherenceisexpected,butthen
talgeometries. Thatsuggeststhattheobservations
the question arises why this liquid forms in some
might be strongly related and if they indeed are, it
experimentsandnotinothers. Aconsistentexper-
is interesting to understand what determines this
imental mapping of the effect of the dipole size has
specific temperature.
not yet been done.
2. The formation mechanisms and the nature of the
groundstateofthecollectiveliquidstatesobserved Hopefully, future experiments and more comprehen-
are far from being fully understood. In particular, sive theoreis will soon elucidate the open questions that
several questions stand out: what drives the con- remain to be answered. Whatever these answers would
densation transition in which the density sponta- eventuallybe,itisclearthattheywillinvolveinteresting
neouslyincreases? Isitaclassicalphasetransition, physics of interacting particles.
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