E X C I TAT I O N S I N O P E N , D R I V E N - D I S S I PAT I V E
C O N D E N S AT E S C L O S E T O E Q U I L I B R I U M
davide caputo
Advisors
Dr. Daniele Sanvitto
Prof. Giuseppe Gigli
A dissertation submitted for the degree of
Doctor of Philosophy
Department of Mathematics and Physics ”Ennio De Giorgi”
University of Salento
January 2018
to my family.
Without your support none of this would take place.
P R E FA C E
In this thesis I present the main results obtained during my PhD. The core aim
of this investigation was to study the fluid of polaritons with a particular attention on the dynamics of condensation in two-dimensional systems. Starting
from this perspective we discovered that a sample with a very long polariton lifetime allows the creation of an extended dilute condensate far from the
instabilities of the exciton reservoir. We used this coherent state to study its
coherence and superfluidity properties and to shed light on the long debated
question if there is a difference between a condensate of polaritons and a laser.
Finally we applied an external magnetic field in the Voigt geometry to the
propagation of a polariton fluid in order to study its effect on the polarization
dynamics.
The thesis is organized in the following way:
The first chapter contains a general introduction on the microcavity excitonpolaritons. Here all the concepts about exciton-polaritons and polariton condensation are introduced.
In the second chapter the physical properties and the ballistic motion of a
condensate on top of a blueshifted potential are investigated.
Chapter three is a study of the processes behind the formation of an extended, dilute two-dimensional exciton-polariton condensate at the bottom of
the lower polariton branch.
In chapter four, this extended coherent state is demonstrated to be large
enough and stable to sustain a specific two-dimensional phase transition based
on the Berezinskii, Kosterlitz and Thouless (BKT) mechanism.
Chapter five includes the description of a peculiar configuration in which this
system can be used. Indeed, the phase of the condensate is twisted by fixing
its value using two external lasers. The energy injected inside the condensate
determines the creation of a barrier in the phase with a solitonic modulation of
the density and the appearance of Josephson vortices.
Finally, in chapter six we study the effects of an external magnetic field applied in the Voigt geometry, i.e. in the same plane of the cavity. We observe
in the one-dimensional case the total suppression of the spin oscillations due
to the optical spin Hall effect and, in the two-dimensional case, the formation
of an ellipse in the circular polarization pattern that gets rotated with higher
values of the magnetic field.
v
A single ray of light from a distant star falling upon the eye of a tyrant in bygone
times may have altered the course of his life, may have changed the destiny of nations,
may have transformed the surface of the globe, so intricate, so inconceivably complex
are the processes in Nature. In no way can we get such an overwhelming idea of the
grandeur of Nature than when we consider, that in accordance with the law of the
conservation of energy, throughout the Infinite, the forces are in a perfect balance, and
hence the energy of a single thought may determine the motion of a universe.
— Nikola Tesla, September, 1893
ACKNOWLEDGMENTS
First of all, I would like to thank Daniele Sanvitto. He gave me the opportunity
to learn how to do research as an experimental physicist and he has become
a source of inspiration to look forward on how to approach new, unknown
things. He possesses a special unquenchable power that he uses to encourage
the students to do better. He was not only my supervisor in the laboratory, but
also a friend to share a lot of interests and discussions about programming languages and technology stuff, always with an original and a challenging point
of view. For all these reasons, he really has my admiration.
Thanks also to Giuseppe Gigli for being my tutor and for giving me the
possibility to work in Lecce in the CNR Nanotec.
I would also like to thank Dario Ballarini because he shared with me the
experimental activity teaching me how to work with proficiency and the way
to look at the data with rigour.
Furthermore, I want to thank some special people I worked with in the
lab like Lorenzo, who was always present when a discussion against labview
started due to some malfunctioning (of course it happened a lot!), and Milena
who introduced me to the world of lenses and lasers when I arrived to the
laboratory.
A special thank to all the people from different groups in Madrid, London,
Warwick, Petersburg and Warsaw I worked with for the theoretical simulations
and interesting discussions.
In lecce I met a lot of people with which I shared wonderful moments.
Francesco (Francisco), a “serial” laugher and a considerable singer, Antonio
(Antoñete), he brought us to Viggianello, one of the most peaceful places I have
ever visited, Antonio (Lello), we spent some sundays playing the bass and we
shared a lot of burrata and special cheese, Stefano, I think the best whistle of
Italy, Daniel, the youngest student and also my neighbour in Lecce, Gianni, we
vii
spent a lot of nights discussing about physics and Barbara, she tried to introduce me to nerd comics. A special thanks also to Paolo, mainly the technician
of the labs, but also a very good friend. Without his passionate and careful help
nobody in the laboratory can work. I would also like to thank Giuseppe, for his
friendship in Lecce, for the discussions about robotics and technology and for
taking me to the airfield during summer weekends.
I also wish to thank particularly Blanquita (but It is difficult to write the
whole list of reasons to say thanks, there are so many...). Especially to inspire
me with positive thoughts tirelessly, and to fully support me, every time I
needed. You were always there for me...I will never forget.
Finally, I cannot forget my brother, I wish he achieves anything he desires,
and my parents, with their huge attention and unconditional love.
viii
CONTENTS
Abstract
1
1 introduction
3
1.1 Microcavity polaritons
3
1.1.1 Microcavity
4
1.1.2 Light matter coupling
8
1.1.3 Why using polaritons
10
1.2 Polariton condensation and lasing
11
1.2.1 Non-Resonantly pumped polariton condensates
12
1.3 Outlook of this thesis
14
2 polariton condensate dynamics in a high finesse microcavity
17
2.1 Introduction
17
2.2 The sample
18
2.3 The experimental configuration
19
2.3.1 The lifetime
20
2.3.2 The nonlinear dynamics
21
2.3.3 Polariton density
23
2.3.4 Fourier limited extension
24
2.3.5 The effect of the spot size and spatial gradient
25
2.3.6 The temperature dependence on the top condensate
27
2.4 Conclusions
28
3 formation of a macroscopic exciton-polariton condensate
31
3.1 Introduction
31
3.2 The experiment
32
3.2.1 The formation of the extended state
33
3.2.2 The time-resolved measurements of the condensation
35
3.2.3 The stimulated scattering threshold
39
3.2.4 The polariton fluid back–flow
40
3.3 The model
43
3.3.1 The hydrodynamics in the phonon bath
43
3.4 Conclusions
45
4 phase transitions in macroscopic polariton condensates
47
4.1 Introduction
47
4.1.1 Decay of correlations and BKT phase transition
47
4.1.2 BKT phase transition with microcavity exciton-polaritons
50
4.2 The experiment
53
ix
x
contents
The spatial distribution of the polariton condensate
53
Measuring the condensate phase
56
Correlations in the space domain
58
Correlations in the time domain
66
Studying both spatial and temporal correlations - condensation and lasing
69
4.2.6 The spectrum of the excitations
70
4.2.7 The linearization of the spectrum
71
4.3 Conclusions
74
5 josephson junction and vortices in a phase twisted polariton condensate
77
5.1 Introduction
77
5.2 The experiment
78
5.2.1 The application of a phase twist
78
5.2.2 The phase boundary for different condensate densities
80
5.2.3 Josephson vortices and fluxes through the barrier
82
5.3 The model
85
5.3.1 The two-dimensional phase map in the numerical simulations
85
5.4 Conclusions
86
6 polariton fluid in an external magnetic field
89
6.1 Introduction
89
6.2 The effect of the magnetic field in the one-dimensional geometry
91
6.2.1 Resonant confined one-dimensional propagation
91
6.2.2 The model for the resonant propagation
93
6.3 The effect of the magnetic field in the two-dimensional geometry
96
6.3.1 Redistribution of the top condensate polarization pattern
96
6.3.2 The model for the nonresonant propagation
97
6.3.3 The effect on the linear polarization
99
6.4 Conclusions 100
7 conclusions
101
4.2.1
4.2.2
4.2.3
4.2.4
4.2.5
ii appendix
103
8 the physical model
105
8.1 The hydrodynamical model 105
8.1.1 The expansion and relaxation of the top energy states 107
9 bkt phase transition
109
9.0.1 The influence of the high energy condensate speed 109
contents
bibliography
111
Publications 127
xi
LIST OF FIGURES
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14
Figure 15
Figure 16
Figure 17
Figure 18
Figure 19
Figure 20
Figure 21
Figure 22
Figure 23
Figure 24
Figure 25
Figure 26
Figure 27
Figure 28
Figure 29
Figure 30
Figure 31
Figure 32
Figure 33
Figure 34
Figure 35
Figure 36
xii
Single DBR and cavity stop-band
5
SEM image of a microcavity
6
Weak and strong coupling
9
Different detuning dispersion
10
Non-Resonantly pumped condensate
13
Sketch of the sample structure
18
Interferometric setup
20
Lifetime measurements
21
Non-resonant pumping power series
22
Polariton densities and blueshift
23
Variance in real and reciprocal space
24
Fourier limited emission
24
The effect of the spot dimensions
25
Spot shape effect
26
Variation of the temperature
28
Temperature and pumping power
29
Energy resolved emission
33
Pumping mechanism and interferometric setup
34
Energy resolved time emission
35
Propagation space-time map
36
Propagation against a defect
37
Waves interference
38
The condensation threshold
39
Negative polariton mass
40
The spatial selection
41
The back–flow
42
Numerical Simulations of condensate formation
44
BKT threshold
50
Spatial emission of the condensate
54
Condensate density
55
Mach Zehnder interferometer
56
Condensate Phase
57
Michelson interferometer
59
Piezo delayed sinusoidal intensity modulation and interferometer sketch
60
Two dimensional first order spatial correlations
61
Coherence decay and BKT phase transition
62
Figure 37
Figure 38
Figure 39
Figure 40
Figure 41
Figure 42
Figure 43
Figure 44
Figure 45
Figure 46
Figure 47
Figure 48
Figure 49
Figure 50
Figure 51
Figure 52
Figure 53
Figure 54
Figure 55
Figure 56
Coherence decay fitting
63
Fitting residuals analysis
64
Vortex-antivortex distribution map.
65
Coherence decay in time and BKT phase transition
66
Temporal coherence decay fitting
67
Fitting residuals analysis
68
Spatial and temporal coherence in the weak coupling
regime
69
Spatial energy interference
71
First order correlation function maps
72
Linearized excitations spectra
73
Sketch of the twisted polariton condensate
79
Two dimensional phase map
81
Spatial barrier and fluxes
83
Simulated two dimensional phase map
85
Effect of magnetic field (1D)
92
Effect of magnetic field (1D)
93
Pc patterns with B k y
96
Model of the 2D expansion of Pc in real space
98
Linear polarization patterns in the real space
99
Expanding polariton velocities 109
ACRONYMS
BEC Bose Einstein Condensate
BKT Berezinskii-Kosterlitz-Thouless
CCD Charge Coupled Device
CW Continuous Wave
DBR Distributed Bragg Reflector
FWHM Full Width Half Maximum
LASER Light Amplification by Stimulated Emission of Radiation
LJJ Long Josephson Junction
LPB Lower Polariton Branch
xiii
xiv
acronyms
MBE Molecular Beam Epitaxy
ODLRO Off Diagonal Long Range Order
OPO Optical Parametric Oscillator
OSHE Optical Spin Hall Effect
Q Quality Factor
QLRO Quasi Long Range Order
QW Quantum Well
SEM Scanning Electron Microscope
SNS Superconducting-Normal-Superconducting
SQUID Superconducting Quantum Interference Devices
SSB Spontaneous Symmetry Breaking
UPB Upper Polariton Branch
VCSEL Vertical Cavity Surface Emitting Laser
meV Milli ElectronVolt
ABSTRACT
Exciton-polaritons are quasiparticles that arise from the interaction between
light (photons) and matter (excitons, the electron-hole pairs in a semiconductor material). This half-light half-matter bosonic state inherits some properties
from both its components, as a not negligible mass (about 10−5 me being me
the electron mass), the capability to interact and high velocities of propagation.
They were observed experimentally for the first time in 1992, and since then the
interest of the community has increased especially thanks to the first report of
their Bose–Einstein condensation in 2006. This result opened a profound debate
about the existence of a “Bose–Einstein condensate” in the domain of the nonequilibrium statistical mechanics and in particular questioning the differences
with a regime of lasing.
This thesis deepens the investigation of a new type of polariton condensates,
in a sample with a long polariton lifetime and high spatial homogeneity. This
type of condensates extends in a region far from the excitation pump, therefore
without the influence of the instabilities due to the presence of the excitonic
reservoir. Thanks to this “reservoir-free” nature, we were able to demonstrate
that in such a two-dimensional macroscopic system the proliferation of bound
topological defects (vortex and anti-vortex pairs) mediates the transition from
a disordered to a quasi-ordered state, according to the Berezinskii–Kosterlitz–
Thouless (BKT) pairing mechanism. Moreover, by using a joint measurement
of the coherence decay both in the spatial and temporal domains, we observed
that this process allows polaritons to reach a true thermal equilibrium. Furthermore, with the use of a complete characterization of the spatio-temporal decay
of correlations, we could extract the linearized spectrum of the excitations of
the condensate. This “Bogoliubov” shape of the spectrum was an additional
demonstration that thermal equilibrium is achieved in the system and, even
from an energetic point of view, that superfluidity can be sustained in polariton condensates. Thereafter, we used this kind of condensate to investigate
its superfluidity stiffness (the ability to keep a continuity of phase along the
whole condensate upon a phase torsion). With this measurement we were able
to study what happens to a condensate whose phase is twisted by using two
external lasers tuned in resonance with the condensate. Besides the stiffness, in
a transition regime, we were able to observe the formation of a domain wall
between regions of homogeneous phase, with a soliton like modulation of the
density inside the barrier in the steady state regime.
Finally, we used these long living polaritons also to investigate the effect of
an external magnetic field applied in the same plane of the quantum wells
1
2
abstract
(Voigt geometry). In this case we found that it is possible to totally suppress
the spin oscillation due to the optical spin Hall effect and to observe a rotated
two-dimensional ellipse in the degree of circular polarization.
1
INTRODUCTION
1.1
microcavity polaritons
The history of polaritons started in 1958 with Hopfield. In his seminal work,
he described how the excitonic field can interact with the electromagnetic field
of a crystal structure under certain conditions, creating new eigenstates of the
system, a mixture of both light and matter [1]. This work formed the theoretical basis of the polariton investigation. These new quasi particles were initially
studied through the photoluminescence spectra in samples with CdTe embedded between two layers of Cd1−x Znx Te. Relevant differences were found between these spectra and those of the bulk material [2].
The first experimental observation of exciton polaritons in an optical microcavity appeared in a work by Weisbuch et al. in 1992 [3]. They reported, for the
first time, the realisation of the strong coupling between light and matter in a
semiconductor microcavity. This work paved the way for a plethora of studies
on these new states, opening the era of “polaritons”. Since this work, through
this new class of devices, the interaction between light and matter started to
be investigated with angle-resolved resonant excitations and by studying the
characteristic of the light emitted by the cavity [4].
Later, profound steps forward were made thanks to the experimental works
of Savvidis et al. (in a collaboration between the groups of Southampton and
Sheffield) [5] and Baumberg et al. [6]. They started the exploration of the consequences of the polariton’s bosonic character in the planar geometry, when
stimulated scattering is triggered by a laser pulse. Thanks to these contributions, it was possible to laid the foundations to create an effective ”polariton
laser”. At the same time, a clear distinction between this specific mechanism of
formation and what would be called ”polariton condensate” became fundamental. In fact, in the case of the polariton laser, the coherent phase was transferred
directly by the pump, whereas polariton condensates were pumped with the
laser tuned out-of-resonance and with the stimulated scattering prevailing over
the dissipations.
Moreover, there were a series of evidences from various groups [7, 8], and a
final work of Kasprzak et al. [9] that reported Bose–Einstein condensation of
polaritons in 2006. These results opened a meaningful debate about the adequacy of the terminology of ”Bose–Einstein condensates” (BEC) when dealing
with optical systems.
3
4
introduction
Indeed, polariton condensates were typically strongly out-of-equilibrium due
to their short lifetime (∼ 2 − 4 ps). In fact, they require a continuous pump to
regenerate the polaritons that are emitted constantly from the microcavity as
photons [10]. Moreover, in several cases, condensation takes place not in the
ground state but in an excited state at higher energy. This behaviour is fundamentally due to the interactions between polaritons. To get through this, microcavities can be designed with higher Rabi splittings by changing the active
materials or by increasing the number of QWs [10].
Finally, It was only recently that true equilibrium polariton condensates started
to be studied thanks to cavities with exceptionally long lifetimes. In fact, by using this new kind of samples, energy distribution of the particles is found to
follow the textbook Bose-Einstein distribution, as it was reported in the work
of Yongbao Sun [11] in 2017. Nevertheless, these measurements were not sufficient to be conclusive about the possibility to reach the thermal equilibrium in
polariton systems.
1.1.1
Microcavity
The microcavity is the fundamental device used in the investigation of the interactions between light and matter in a controlled environment. Foremost, a
microcavity brings the physical concept of the cavity, i.e. a way to spatially trap
something, to the micrometer scale. In that respect, a microcavity is nothing
more than an optical resonator close to, or below the length scale set by the
wavelength of the incident light.
In fact, there are two main ways to confine light: the reflection from a single
interface, as in the case of a metallic surface, and the total internal reflection
at the boundary between two dielectrics. By using the first scheme and with
the realization of periodically patterned microstructures on the scale of the
resonant light it is possible to create the distributed Bragg reflector (DBR) on
which the microcavities are based.
In Fig 1a a simulated transmission of a single DBR made of 7 pairs of T iO2
SiO2 with refractive index n1 = 2.2 and n2 = 1.5, respectively is shown. These
mirrors are designed in such a way that the contrast in the refraction index
opens a region of wavelength, with an almost zero transmission, the so called
stop-band. In this way, between 550 and 750 nm in Fig 1a the light is totally
reflected and cannot pass through the mirror.
On the other hand, instead of using a single DBR, one could pile a couple
of them (made of the same materials and layer thickness). This doubled structure, allows the confinement of light by opening a narrow range of permitted
wavelength within the stop-band of the single DBR. This is reported in Fig. 1b,
1.1 microcavity polaritons
where a narrow peak in the transmission is visible for a wavelength of about
620 nm. This range of wavelength is called cavity mode.
1.0
Transmission(T)
0.8
0.6
0.4
0.2
0.0
a
500
600
700
Energy (nm)
800
b
500
600
700
Energy (nm)
800
Figure 1: Single DBR and cavity stop-band, a, Example of a simulated single DBR stopband in the transmission. b, Example of a cavity stop-band from a simulated
transmission.
Microcavities can be classified in several different types based on their geometry. They can be spherical, pillars, planars, stripes, etc. In this work we used
only the type with a two-dimensional planar geometry. This kind of samples,
as it will be reported in detail in Sec. 2.2, consists on two DBRs embedding one
or more two-dimensional quantum wells (Fig. 2). The quantum wells inside the
cavity are the active parts confining the excitations of the material, for instance
a layer of GaAs inside two DBRs composed of GaAs/AlGaAs, as in the case of
the scanning electron microscope (SEM) measurement reported in Fig. 2.
In these systems, the length of the cavity Lc is small compared to the wavelength of the incident light (about 0.2 − 0.4 µm ) allowing the presence of only
one cavity mode inside the stop-band. Typically, Lc is an integer number of
times (j) larger than the width of one layer of DBRs, accordingly to the relation:
kz Lc = jπ.
Being a and b the thickness of the different layers inside a single DBR and
na and nb their refractive indexes, we have:
λc
(1)
4
with λc the central wavelength of the single DBR stop-band. Clearly, the
fundamental consequence of the presence of the cavity mode is the possibility,
na a = nb b =
5
6
introduction
DBR
Quantum Well
DBR
Figure 2: SEM image of a microcavity, SEM measurement of a microcavity. The two
DBR are visible with the embedded quantum well.
for a narrow range of wavelengths, to reach the embedded active material and
to interact with it.
Several parameters are important in the description and classification of the
microcavity:
1. Q-factor (Quality factor).
The quality factor of an optical cavity has the same role as in the RLC electrical circuit. It characterizes the frequency width (δωc ) of the resonant
mode (ωc ) through the ratio:
Q=
ωc
δωc
(2)
In this sense, the Q-factor is a measure of the energy decay rate as we
move from the bare cavity mode through mirrors imperfections leakage,
scattering or absorptions, and Q−1 can be viewed as the energy lost in a
single ”round-trip” by the photon inside the cavity. This is in relation with
the definition of the photon lifetime (how long the photon is confined by
the mirrors in the cavity). The photon lifetime can be defined as:
τ=
Q
ωc
(3)
2. Cavity finesse.
The cavity finesse is defined as the ratio between the free spectral range
and the spectral width, or as the frequency separation between successive
longitudinal modes of the cavity:
F=
√
∆ωc
π R
=
δωc
1−R
(4)
1.1 microcavity polaritons
3. Purcell factor and lifetime.
The fact that the emitter is embedded in the cavity structure causes changes
in the density of the optical states that bring peculiar effects. In particular, when the linewidth of the emitter is smaller than that of the cavity
mode (λc ), it can be considered as coupled with the optical modes and
the Fermi’s golden rule governs the kinetics of the emission. The ratio
between the free space lifetime (τ0 ) and the one in the material inside the
cavity (τ) is:
τ0
δλc
2 |E(r)|2
+f
= Fp
2
2
τ
3 |Emax | δλc + 4(λc − λe )2
(5)
with λe the emitter wavelength and the Fp Purcell factor given by:
Fp =
3 λ3c Q
4π2 n3 Veff
(6)
with n the refractive index of the cavity, Veff the effective volume of the
mode, E(r) the field amplitude in the cavity, Emax its maximum value
and where f is a constant characterizing the losses into leaky modes. The
ratio Q/Veff describes the capability of the active material to emit light
into the optical field in a short timescale.
4. Quantum well’s material
Depending on the material it is possible to distinguish between inorganic
and organic microcavities. Inorganic cavities, as in samples composed of
GaAs, InGaAs, lnGaAs, normally require cryogenic temperatures. In fact,
the interesting behaviours observed with polaritons do not survive at
temperatures higher than 10K. This is due to the fact that the Bohr radius
of the exciton in these materials is huge, and at higher temperatures (with
the Coulomb binding energy lower than the thermal fluctuations), the
presence of free carriers screens the excitonic Coulomb potential between
electrons and holes, destroying the excitons. The use of materials with
smaller excitons Bohr radii can overcome this limitation by allowing the
presence of a stable excitonic state at room-temperature with a larger
oscillator strength and Rabi splitting. For these reasons, there has been
a strong interest in using materials such as organic semiconductors (as
TDAF, dyes, TDBC, perovskites) as active materials in microcavities [12,
13].
7
8
introduction
1.1.2
Light matter coupling
As previously mentioned, planar microcavities are composed by two groups of
DBRs embedding one or more two-dimensional quantum wells. Typically this
active material is placed at the antinode of the optical field in such a way that
the interaction between the light trapped by the cavity structure and the matter
excitations in the quantum well are maximized.
When an electron in the active material is promoted from the valence to
the conduction band thanks to the energy transferred by a photon, it leaves
a hole in its initial band position. Depending on the material, this hole can
bind with the original electron thanks to Coulomb interactions, creating an excited electron-hole state called exciton. These objects inherit the mass from the
electron-hole pairs and consequently, they have a large mass, in comparison
with that of the photon, with an almost flat energy dispersion. By now, the exciton can interact with other trapped photons forming a hybrid quasi-particle
called exciton-polariton (which we will refer to as polariton for the sake of simplicity).
Assuming that the energy of the exciton (ω0 ) is close to the eigenfrequency
of the photonic mode of the cavity (ωc ), the polariton state can be modeled as
two coupled harmonic oscillators, the cavity and the exciton:
(ω0 − ω − iγ)(ωc − ω − iγc ) = V 2
(7)
with γc the radiative losses of the mirrors of the cavity:
γc =
1−r
r
nc
c
(LDBR + Lc )
(8)
with r the mirrors reflectivity, LDBR the DBR length and Lc the cavity length.
In Eq. 7 V is the coupling term, given by:
s
2ω0 ωLT d
V=
(9)
LDBR + d
with ωLT the longitudinal transverse splitting energy and d the period of
the refractive index modulation inside the cavity. Thus, the diagonalization of
Eq. 7 has two complex solutions in the form of:
ω1,2 =
ω0 + ωc
i
− (γ + γc )
2
2
s
ω0 − ωc 2
ω0 − γc 2 i
2
±
+V −
+ (ω0 − ωc )(γc − γ)
2
2
2
(10)
1.1 microcavity polaritons
When ω0 = ωc , the splitting in Eq. 10 is given by the quantity
It is possible to distinguish two different regimes, with:
γ − γc
2
V>
p
4V 2 − (γ − γc )2 .
(11)
Re(ω1,2)
In this condition, there is a energy splitting between the bare exciton and
the photonic mode and the system is in the so called strong coupling regime. We
can now discriminate two distinct exciton-polariton energy bands: the lower
polariton branch (LPB) and the upper polariton branch (UPB) in both reflection
and transmission spectra. The energy separation between these resonances is
referred to vacuum-field Rabi splitting (Fig. 3).
ω0
ω0
2V
a
b
ωc
ωc
Figure 3: Weak and strong coupling. Real part of the eigenfrequencies of excitonpolariton modes in the weak coupling regime a, and in the strong coupling
regime, b.
Conversely, if:
V<
γ − γc
2
(12)
the system is in the so-called weak coupling and the energy crossing between
the exciton and the cavity photon mode appears. This condition is typically
used in the vertical cavity surface-emitting laser (VCSEL).
These two regimes are shown in Fig. 3 where the real part of the polariton
modes eigenfrequency is reported as a function of the cavity energy under both
the weak and the strong coupling.
Once the strong coupling is set, a fundamental parameter that influences the
physical characteristic of polaritons is the detuning between the bare exciton
mode and the photonic one. Namely, this is the energy difference between the
exciton energy and the minimum of the cavity modes energy: δ = ωc − ω0 .
The detuning can change the ratio between the excitonic component and the
photonic one. In this way polaritons can be more interacting (higher excitonic
9
introduction
5
4
ΔE (meV)
10
3
2
1
0
a
-3
-2
-1
0
1
2
k (μm-1)
c
b
3 -3
-2
-1
0
1
k (μm-1)
2
3 -3
-2
-1
0
1
2
3
k (μm-1)
Figure 4: Different detuning dispersion, a, Energy resolved reciprocal space emission.
The red line is the lower polariton branch, The green line is the upper polariton
branch, the blue line is the cavity photonic mode and in yellow the exciton
energy. In this case the cavity is slightly negative detuned. b, The same colors
as in a for a zero detuned cavity. c, The same colors as in a for a slightly
positively detuned cavity.
part – positive detuning) or lighter (higher photonic component – negative detuning) In Fig 4a,b,c a negative, zero and positive detuned cavity are reported,
relatively to a sample with a vacuum Rabi splitting of about 3 meV. Being a
mixture of light and matter, polaritons can be described with the following
wavefunction, which includes two different contributions:
1
Ψ = √ {ΨX ± Ψc }
2
(13)
with ΨX the wavefunction relative to the exciton contribution and Ψc the one
relative to the cavity photon.
1.1.3
Why using polaritons
Microcavity exciton-polaritons represent the paradigmatic realization of an open,
driven-dissipative system. For this reason, they can be used to investigate not
only the out-of-equilibrium class of physical phenomena but also the possibility to achieve the thermal equilibrium, in particular thanks to their capability to
interact. This distinctive characteristic, related to the presence of the excitonic
component, is sizable and can be observed by measuring an energy blueshift
in the emission spectrum. Moreover, the energy detuning between the UPB
and the LPB is modifiable. This variation allows to change the ratio between
the excitonic (matter) and the photonic (light) part. Actually, it depends on
1.2 polariton condensation and lasing
the thickness between the two layers of DBR and it largely affects the physical
characteristics of polaritons like their mass and interactions strength.
Thanks to their peculiar properties polaritons are studied in different branches
of physics. A first example is with polariton condensates. The possibility to create a spontaneous coherent phase can lead to other important effects such as
the superfluidity [14–17] or the observation of a rich phenomenology of vortices
dynamics [18–22]. Besides, the ability to extract all the information about the
phase of the system, for instance directly with interferometric measurements,
allows the study of coherent structures as solitons [23–29].
Finally, with polaritons it is possible to investigate the field of topological
phases [30–35] and they are emerging also as candidates for quantum information processing and simulation [36–39].
1.2
polariton condensation and lasing
The first prediction of a BEC was with Bose [40] and Einstein [41]. In order
to introduce formally the concept of long-range order, let us consider an ideal
Bose gas with a fixed number of particles N inside a box of volume Rd where R
is the system size and d corresponds to the dimensionality. The Bose-Einstein
statistics gives us the energy distribution of bosons:
fB (k, T , µ) =
exp
1
E(k) − µ
−1
kB T
(14)
with µ the chemical potential, Kb the Boltzmann constant, T the temperature,
E(k) the energy dispersion with the ground state at 0 and k the wavevector of
the particle of dimensionality d. The −µ represents the energy needed to add
a particle to the system. It follows the normalization condition for a given total
number of particles N:
N(T , µ) =
X
fB (k, T , µ)
(15)
k
Given that we are interested in the particle density in the ground state, it is
convenient to divide the contributions of higher energy states:
N(T , µ) =
exp
X
1
+
fB (k, T , µ)
−µ
− 1 k,k6=0
kB T
(16)
It is possible to calculate the particle density by taking the thermodynamic
limit and evaluating the integral in the reciprocal space:
11
12
introduction
N(T , µ)
1
n(T , µ) = lim
= n0 +
R→∞
Rd
(2π)d
Z∞
fB (k, T , µ)dk
(17)
0
with the density of the particle in the ground state given by:
n0 (T , µ) = lim
R→∞
1
Rd
exp
1
−µ
−1
kB T
(18)
If the chemical potential µ is different from 0, the ground state density vanishes, while the contribution from higher energy states increases with µ (integral in Eq. 17). For this reason, µ increases with number of particles n. The
maximum particle density nc that can be accommodated following the Bose
distribution:
Z∞
1
nc (T ) = lim
fB (k, T )dk
(19)
µ→0 (2π)d 0
Einstein proposed that an additional increase in the particle density can provoke the new added particles to collapse into the ground state, with the density
given by:
n0 (T ) = n(T ) − nc (T )
(20)
This is a phase transition with a massive occupation of the ground state and
where the order parameter is the chemical potential µ that vanishes at the transition. However, from an experimental point of view, this macroscopic occupation of the ground state is only a qualitative evidence of the BEC, whereas the
exact determination of the transition requires a more accurate definition. For
example, a different classification of a BEC is with the use of the wavefunction
as order parameter:
ψ(r, t) =
p
n(r, t)eiφ(r,t)
(21)
where r accounts for the position, n(r, t) is the spatio-temporal density distribution and φ(r, t) is the phase. A complementary approach uses the concept of
off-diagonal long-range order (ODLRO) in the investigation of the BECs. Since
it is widely used in the study of polariton condensates, it is described in detail
in Sec. 4.1.
1.2.1
Non-Resonantly pumped polariton condensates
Non-resonantly pumped polariton condensates show rich physics due to the
spontaneous arising of a coherent phase. Indeed, when the system is pumped
1.2 polariton condensation and lasing
a
b
c
d
Figure 5: Non resonantly pumped condensate. a, Interferogram of the spatial map
of the fragmented condensate in [42]. b, Energy resolved dispersion and real
space emission of a condensate in ballistic motion [43]. c, Spatial emission of
the organic condensate in [44]. Scale bar are 5µm. d, Energy distribution of
photons in a GaAs microcavity ([45]).
with a non-resonant laser (i.e. tuned to a minimum of the reflection stop-band
of the DBRs), an exciton reservoir at higher energy is filled. From these higher
energies, mainly thanks to phonon and polariton scattering, particles can occupy the lower energy states. This relaxation process allows the population of
the entire polariton dispersion (Ch. 3). As a consequence of the energy relaxation mediated by scattering processes, the initial phase coherence imprinted
by the pumping laser is totally destroyed. This is in contrast with what happens
with resonant injection of polaritons. In this case the coherence is transferred
to the cavity directly by the pumping laser. For this reason, this scheme of injection is not suitable for the study of the mechanism behind the spontaneous
formation of a coherent state. To formally describe the non-resonant excitation,
it is possible to use a model based on the Gross-Pitaevskii equation in the
mean-field approach. This model is introduced and applied in Ch. 3.
The relation between Bose Einstein condensation and the emergence of spontaneous coherence in optical systems has been thoroughly studied in the last
13
14
introduction
decade, basically starting with condensates of exciton polaritons in semiconductor microcavities (Kasprzak et al. [9]). That being the case, both the similarities
and the differences between a lasing regime and the coherent light emitted
from a condensate have drawn the interest of the community. A particular debate was around the possibility to observe a thermalized distribution of the
particles, despite the non-equilibrium nature of polaritons.
A work of D. N. Krizhanovskii et al. [42] in 2009 evidenced the fragmentation of the coherent state into multiple macroscopically occupied spatial modes
with a strong localization inside the polaritonic potential (Fig. 5a). They used a
non-resonantly tuned pump, demonstrating that the observed disorder in the
emission was related to both the non-equilibrium character and the interactions
with the polaritonic potential.
In 2013 Steger and coworkers reported the formation of a coherent polariton
condensate flowing with a ballistic motion with a high speed [43] (Fig. 5b). They
showed the kinematic behaviour of the quasiparticles in the low density regime.
Although the presence of the ballistic propagation of a coherent flow might not
be viewed as a problem, the existence of fluxes inside the condensate tends
to mix space and time correlations as it is explained in details in Ch. 4. These
fluxes in polariton condensates are difficult to avoid because of the presence of
a wedge between the two layers of the cavity mirrors.
In a work of 2015 by K.S. Daskalakis and colleagues [44], the authors were
able to study in an organic microcavity the arise of a spontaneous coherent
phase (Fig. 5c) with a non-resonant pump. In this type of condensates the effect
of the exciton reservoir is strong and the condensation is reached far from
equilibrium.
Furthermore, in the work by Bajoni et al. [45] the authors reported the observation of a photon energy distribution that was apparently thermalized even
though the emission was from a system in the weak coupling regime (Fig. 5d).
This type of condition is considered the paradigmatic fulfilling of an out-ofequilibrium device and consequently the thermalized photon distribution appeared to be somehow impossible.
Such an active debate highlights the importance of the investigation of a nonresonant polariton condensate in a configuration free from exciton reservoir
effects and with a polariton lifetime long enough to deepen the investigation
of the thermalization processes in open, driven-dissipative systems.
1.3
outlook of this thesis
After this brief introduction the results of this work will be presented in the
following way.
1.3 outlook of this thesis
In Ch. 2 the physical properties and the ballistic motion of a non-resonantly
pumped condensate on top of the potential due to the presence of the exciton
reservoir are investigated.
Ch. 3 addresses some of the issues reported previously, describing the formation of a macroscopic, dilute, two-dimensional polariton condensate far from
the exciton reservoir and at the bottom of the lower polariton branch.
In Ch. 4 this macroscopic condensate is used to investigate the processes
behind the spontaneous emergence of a high level of coherence and to demonstrate that the thermal equilibrium can be reached with polaritons.
In Ch. 5 the phase of this polariton condensate is twisted by fixing its value
using two external lasers. The energy is resonantly injected inside the condensate and determines the creation of a barrier in the phase and the appearance
of a dark soliton in the density.
Finally, in Ch. 6 the influence of an external magnetic field applied in the
same plane of the propagation is studied on both resonantly injected polaritons and the non-resonantly pumped condensate. Some interesting effects are
described as the total suppression of the spin oscillations due to the optical spin
Hall effect and the formation of a rotated ellipses in the circular polarizations
pattern.
15
2
P O L A R I T O N C O N D E N S AT E D Y N A M I C S I N A H I G H
F I N E S S E M I C R O C AV I T Y
2.1
introduction
When a high finesse microcavity is pumped with a laser that is nonresonantly
tuned, i.e. in a minimum of the reflection stop band, a non-trivial phenomenology can be observed, spanning from long-distance propagation to condensation
in a homogenous coherent phase. In this conditions, an excitonic incoherent
reservoir is formed in the region of the sample under the pump spot. The repulsive interactions between excitons, whose mass is so large to prevent any
movement far from the injection point, cause the formation of polariton states
that are at higher energies respect to the bottom of the lower polariton branch.
The spatial configuration in which these states can be organized has a relation
with the intensity profile of the pumping spot. Therefore, the gaussian intensity
profile of the pumping laser provokes a blueshifted potential with a gaussian
shape. Above a critical threshold, on top of this gaussian potential, a highly
energetic condensate can be observed together with its fast radially expanding
cloud of ballistic polaritons [46–48].
In this chapter we investigate some of the fundamental properties of polaritons inside this condensate by using different conditions of excitation. Here
it is shown that a distinctive feature of this state is the strong localization on
top of the gaussian potential. For this reason the geometry of the potential can
strongly alter the spatial configuration of the condensate and of its expanding
cloud. Moreover, we consider the effect of the nonresonant pumping power on
the dimensions of this blueshifted condensate. We found that, by increasing
the power of the non-resonant pump, the condensate dimensions collapse to a
size that is demonstrated to be Fourier limited considering both the real and
reciprocal space extension. Furthermore, for increasing pumping powers, the
temperature of this state decreases, in contrast to what was reported in the litterature [49]. The joint effect of the excitation power on both the size and the
temperature of the condensate allows to explain the localization of the condensate on top of the potential as a result of a gradient cooling process. From this
perspective, the gradient of the gaussian potential can trigger the propagation
of polaritons proportionally to its intensity, acting in every respect as a cooling
mechanism that tends to promote only the less energetic, i.e. colder, polaritons
to remain on top of the potential. On the contrary, more energetic, i.e. hot, par-
17
polariton condensate dynamics in a high finesse microcavity
10
8
ΔE (meV)
18
Top DBR layer
(34 pairs)
6
Quantum wells
(12)
4
Bottom DBR layer
(40 pairs)
2
GaAs substrate
0
a
b
-2
0
k (μm-1)
-2
Figure 6: Sketch of the sample structure and LPB fitting. a, LPB fitting with in red
the LPB, yellow the exciton flat dispersion and in blue the cavity energy. Black
points are extracted from the experimental far field measurement. b, schematic
representation of the cavity structure.
ticles are expelled thanks to the acceleration induced by the potential gradient.
The fact that this mechanism becomes more efficient for increasing powers confirms the reduction of the temperature of this state for higher pumping regime.
2.2
the sample
The sample used in all these measurements was grown epitaxially on a GaAs
substrate in the group of L. Pfeiffer in Princeton University. For what concerns
its structure and optical properties, it is a high quality factor (Q > 100000)
GaAs/AlGaAs planar microcavity containing 12 GaAs quantum wells of 7 nm
width, grouped in 3 blocks placed at the antinode positions of the electric field
inside the cavity. This large number of quantum wells allows to reach a high
level of excitation density despite remaining far from the critical density for the
Mott transition. The resulting collective Rabi splitting is hΩ = 16 meV and the
cavity is formed by two layers of mirrors (DBR) surrounding these quantum
wells, with the top one composed of 34 pairs of AlAs/Al0.2 Ga0.8 As, and the
bottom one with 40 pairs.
In Fig 6a the experimental points extracted from the dispersion measured
exciting the sample with the laser tuned nonresonantly are reported. In red
dashed line the analytical LPB is depicted, with the cavity photon mass mC ≈
6 · 10−5 me , with me the electron mass and the cavity photon energy Ec ≈
2.3 the experimental configuration
1608 meV. In Fig. 6b a schematic representation of the sample structure is
shown with the quantum wells embedded between the two group of DBRs.
As it will be reported in details in the following section, all the properties
described above permit to achieve a long polariton lifetime inside the cavity,
whose value is of about 100 ps.
The cavity thickness slightly varies with the distance from the sample center
manifesting only a limited wedge in the center region of the wafer. The cavity
wedge acts like an effective potential, whose presence is related to the variation
in the layer thickness. The resulting change in the energy detuning (Sec. 1.1.2)
tends to accelerate or decelerate polaritons according to the propagation direction [43]. As it is shown in Ch. 4, a consequence of the limited cavity wedge is
the possibility to have a condensate with a limited velocity of propagation at
the bottom of the LPB.
Finally, in the measurements reported in this thesis we used a point on the
sample with a slightly negative detuning (δ = −2 meV) and with the exciton
energy of about: hωx =1610.2 meV.
2.3
the experimental configuration
In the experimental measurements reported in this thesis, the sample is placed
in the vacuum-chamber of a cryostat and it is kept at a temperature of about
5 K. The incoherent exciton reservoir is populated by using a non-resonantly
tuned single-mode Ti:sapphire laser in continuous wave (cw) operation, with
stabilized output wavelength and power, with the aim to reduce significantly
the fluctuations in the density of the reservoir. Moreover, to efficiently injects
carriers in the structure, the energy of the pump is chosen to coincide with
the first minimum of the reflection stop band (1686.80 meV) and, with the purpose of avoiding a possible thermal heating of the sample, the pump laser is
chopped at a frequency of 4 kHz with a duty cycle of 8%. The spot was focused
into a gaussian beam with a fullwidth half maximum (FWHM) of about 15 µm
using two lenses (L1 and L2 in Fig. 7) to collimate the spot and a photographic
objective (working distance of about 5.6 mm, Obj in Fig. 7) to focus it on the
sample surface.
The real and reciprocal space images are reconstructed by using photoluminescence measurements in reflection configuration. This leads to obtaining all
the spatial and angle resolved informations of the emission with the use, in the
detection line, of the photographic objective, a lens with a focal length of 1 m
(L3) and, for the angle resolved emission only, an additional lens with a focal
distance of 100 mm (not shown in Fig. 7).
After this reconstruction, the image is sent to a monochromator (Mono in
Fig. 7) with a grating of 1800 grooves/mm in order to resolve in energy the
19
20
polariton condensate dynamics in a high finesse microcavity
L1
L2
laser source
sample
BS
L3
Mono
Obj
CCD
Figure 7: Interferometric setup. Sketch of the experimental setup used in photoluminescence measurements, with L1 and L2 the two lenses used in order to collimate
the laser spot, BS the beam splitter dividing the excitation path from the detection one, (Obj) the photographic objective employed for the imaging of the
emission, L3 the lens focalizing the real space image on the CCD camera and
(Mono) the monochromator used to obtain the energy-resolve measurements.
emission from the sample. Finally, the monochromator is coupled with a CCD
camera detector.
2.3.1
The lifetime
In order to measure the polariton lifetime, time-resolved measurements were
performed using a Ti:sapphire laser delivering 100 fs pulses with a 82 MHz repetition rate. By using this pulsed beam we resonantly injected polaritons inside
the cavity with a fine tuning of the frequency and of the laser incidence angle,
matching the bottom of the LPB at Ek=0 = 1602.1 meV, but with a slight shift in
angle to allow the filtering out of the reflected beam. Thanks to the fact that the
resonance was set with the bottom of the LPB, the bottleneck effects–i.e. when
the radiative polariton lifetime is less than the phonon scattering rate and a
region of energies are inefficiently populated– is eliminated together with the
phonon assisted relaxation towards the lower energy, allowing a correct estimation of the polariton lifetime. In this way this parameter can be extracted
directly analysing the light emitted from the sample using a streak camera
detector to obtain the time resolved measurement. The temporal length of po-
2.3 the experimental configuration
106
Counts (a.u.)
Time(ps)
460
345
105
230
a
104
-10
0
Space (μm)
10
b
200
300
400
Time (ps)
500
Figure 8: Lifetime measurements. a, Time resolved emission taken close to k = 0 and
with the laser pump at E = 1608.04 nm. b, Cross section of the time resolved
decay emission in the region within the dashed, red lines in panel a. The
exponential decay fitting gives a decay of 100 ps (red line).
laritons inside the cavity is measurable through the exponential decay shown
in Fig. 8. From this decay of the population in time it is possible to estimate a
lifetime of ≈ 100 ps.
2.3.2
The nonlinear dynamics
With the aim to understand how a variation in the nonresonant pumping power
affects the polariton blueshift and, consequently, the spatial emission of the
quasiparticle distribution, we performed a series of measurements by increasing the power of the excitation laser. Indeed, as already mentioned, when the
sample is excited nonresonantly, the exciton reservoir is populated in a spatial
region confined inside the laser spot. Within this area the excitons interact with
each other experimenting repulsion forces. These interactions create an energy
blueshift in the same spatial position of the reservoir. With this configuration,
polaritons in that spatial position can only occupy a blueshifted energy state.
The measurements reported in Fig. 9 are obtained by taking a one-dimensional
cross section of the planar emission from the cavity centered on the excitation
spot by closing the monochromator slits. This allows to analyze what is the
energy distribution of polariton states in the excitation region. In the real space
images (left and center column) of Fig. 9, the presence of the effective gaussian
potential due to the exciton-exciton interactions is visible as a dark region (absence of polariton states at a given position and energy, delimited by the white
21
Energy (meV)
polariton condensate dynamics in a high finesse microcavity
P=10 mW
a
h
2
1
0
Energy (meV)
Energy (meV)
-
Energy (meV)
22
b
P=30 mW
e
i
c
P=50 mW
f
l
d
P=250 mW
g
m
2
1
0
2
1
0
2
1
0
-50
0
50
Space (μm)
-50
0
50
Space (μm)
-2
-1
0
1
k (μm-1)
2
Figure 9: Power series - real space as a function of energy. a, b, c, d, Non saturated
measurements of the energy resolved real space emission increasing the nonresonant pumping power. e, f, g, Saturated measurements of the energy resolved real space emission increasing the non-resonant pumping power. h, i, l,
m, Momentum as a function of energy measurements as the same power as in
the left column. Dotted white circles indicate the occupation of wavenumbers
relative to the propagation of the expanding polariton cloud.
dashed line in the figure). The shape of this potential, given by the spatialdependent blueshift, reflects the gaussian intensity profile of the excitation spot.
At low pumping powers (top row, P = 10 mW) all the emission is from the region of the laser spot with a dimension comparable to the one of the exciting
beam. Increasing the exciting power (second and third rows, P = 30 mW and
50 mW, respectively), it is possible to observe the formation of a state extremely
confined on top of the spatial potential (Fig. 9c, d), together with the cloud of
2.3 the experimental configuration
a
b
2.00
1.00
ΔE (meV)
Density (μm-2)
100
1
0.01
0.50
0.20
0.10
0.05
1
5
10
50 100
Power (mW)
1
5
10
50 100
Power (mW)
Figure 10: Polariton densities. a, Polariton densities as a function of pumping power b,
Energy blueshift for varying pumping power.
ballistic polaritons expanding far away from the excitation spot (particularly
visible in the saturated measurements at higher power , Fig. 9f, g). From the
energy-resolved measurements of the LPB dispersion (right column) at low
power reported in Fig. 9h it is possible to note that most of the contribution
comes from polaritons outside of the laser spot, where the effect of the effective
potential is negligible. Increasing the pumping power (Fig. 9i,l,m), the state
on the top of the gaussian potential appears also in the reciprocal space but
broader in k. The two strongly emitting points in k indicated with the dotted
white circles in Fig. 9m are the signature of the fast expulsion triggered by the
excitonic gaussian potential landscape, which is also acting as the main factor
determining the small size of the top energy condensate as we will see later.
2.3.3
Polariton density
The polariton density plays a central role in the relation between the energy
blueshift and the interaction constant, according to the following relation:
∆E = g · n
(22)
where ∆E is the energy blueshift, g the interaction constant and n the density.
In particular, the value of the interaction constant g is currently at the center
of an intense debate [50]. This shows the importance of a precise estimation of
the density of polaritons.
The density n can be experimentally measured using the following relation:
τ
,
(23)
Ec
where Ec is the energy of the polariton state, τ the lifetime and P the emission
power, converted from the photon counts by using the efficiency of the CCD
camera and taking into account all the losses normally present in the setup
n=P
23
polariton condensate dynamics in a high finesse microcavity
800
0.8
0.7
Counts (a.u.)
Counts (a.u.)
600
400
200
0.6
0.5
0.4
0.3
0.2
0
-30
0
Space (μm)
0.8
30
0.4
0
k (μm-1)
0.4
0.8
Figure 11: Variance in real and reciprocal space. Variance in real (a) and reciprocal (b)
space and gaussian fitting.
because of the objective, mirrors and lenses. The power dependence of the
polariton density and the condensate energy Ec are shown in Fig. 10a, b, where
the threshold is visible for a pumping power of about 35 mW (red vertical line)
followed by a slower increase above 100 mW.
2.3.4
Fourier limited extension
1.00
10
0.70
5
2
0.50
a
2
5
10 20
50 100
Power(mW)
1.5
σk2 σx2
1.50
20
Momentum FWHM (μm)
The analysis of the FWHM of the distribution in real and reciprocal space of the
condensed state emission at the center of the excitation spot reveals an interesting feature. The variances in momentum- and real-space extracted from energy
resolved measurements by using a guassian fit at the condensate energy E = Ec ,
as shown in Fig. 11 are reported in Fig. 12a. In Fig. 12b the product of the variances computed in the real and reciprocal space is shown. At a certain power
Real space FWHM (μm)
24
1.0
0.5
b
20
30
50 70 100 150
Power(mW)
Figure 12: Fourier limited emission. a, Real space FWHM (blue points) of the polariton emission at different pumping powers. Green points are the corresponding FWHM in momentum space, showing an increase for higher pumping
powers. Threshold power for condensation is marked by a vertical line. b,
Variances in momentum- and real-space products.
2.3 the experimental configuration
15 μm
a
b
c
d
Figure 13: The effect of the spot dimensions. a, Reflection of the excitation spot and (b)
related emission from the polariton condensate for a spot size of FWHM ≈
15.6 µm. c, d, the same as in a, b but with a spot size of FWHM ≈ 24.2 µm.
(P ≈ 40 mW) the condensate emission becomes Fourier-transform-limited, with
the product almost constant with a value of about 0.5, i.e. the Fourier limit. This
regime is indicated with a vertical line in Fig. 12a,b, and this Figure shows that
the condensate size stops shrinking.
As opposed to the case of homogeneous systems at equilibrium, where the
size of the condensate increases with the number of particles at fixed temperature, here the expulsion of polaritons becomes extremely efficient and distills
in real- and momentum-space a smaller polariton droplet.
2.3.5
The effect of the spot size and spatial gradient
A possible question is what are the parameters affecting the formation of the
condensate and the mechanism of expulsion from the excitation region. Consequently, in order to deepen the relation between a change in the excitation
conditions and in the emission from the condensate, we performed a serie of
25
26
polariton condensate dynamics in a high finesse microcavity
15 μm
a
b
c
d
Figure 14: Spot shape effect. Polariton condensate shapes with different elliptical pump
spots. a, Real space images of the excitation spot with vertical ellipticity and
the gradient vectors. b, Real space emission from the condensate in the same
elliptical condition. c,d, the same as in top a,b, but with a horizontal ellipticity.
measurements by varying the spot size, and its shape. In this experiments we
measured the reflection of the excitation spot and the relative condensate emission by filtering out the spot used to excite the sample. The spatial images
of the different spot sizes together with the arrows indicating the direction
and the intensity of the gradient in the two-dimensional maps are reported in
Fig. 13a,c. It is possible to note that, in this condition, the gradient is radially
oriented only, with no relevant asymmetry. Under these conditions, the condensate should keep an almost uniform shape, because of the symmetry present
in all the directions. The emission from the top condensate is shown in the
second column of the figure. It is possible to see that there are no significant
variations in the geometry of the top condensate compared to the increase in
the extension of the excitation region. Indeed when the FWHM of the pump
spot is of about 15.6 µm, the condensate FWHM is 3 µm as it is shown in the
upper row. Hovewer, if the pump spot FWHM becomes of about 24.2 µm, the
2.3 the experimental configuration
condensate FWHM still remains similar to the previous one, with a value of
2.5 µm (Fig. 13c,d). Finally, with the spot FWHM of about 37.7 µm, the condensate FWHM again is similarly to the previous values, of about 3.4 µm (not
shown here).
At this stage, with the size of the exciting spot not playing a crucial role, we
performed a series of measurements varying the spot shape. We placed a pin
hole in the far field of the excitation beam, so that by closing it a variation in
the intensity distribution of angles was produced, with a resulting change in
the spot symmetry. In Fig. 14 it is possible to see the results of this measure. In
the top row, a vertically elliptical spot is used. The vectors in the panel show
that, this time, a strong asymmetry in the gradient is present. In this case, the
condensate feels the potential gradient of the deformed gaussian landscape by
remaining confined in the region of null potential (at the center of the spot) and
by acquiring a deformed shape in the direction of the weaker gradient. On the
contrary, in the region of stronger potential variation, it is confined in a really
limited spatial extension. In the bottom row the same dynamics is depicted but
with a vertical deformation of the exciting beam. On this occasion, the deeper
gradient is in the vertical direction. Even in this situation, the condensate feels
this warped gradient, following the shape of the excitating spot.
2.3.6
The temperature dependence on the top condensate
The analysis of the temperature of the thermalized energy tail of the top energy condensate is here reported. This investigation is important because it is
possible to represent the observed localization of the condensate on top of the
gaussian potential in analogy to what is modelled in atomic systems through
the evaporative cooling model, in which a potential is used as a trap that allows
to get out only the highly energetic particles. As a result, only the less energetic
atoms remains in the trap.
In our case, the trapping mechanism acts somehow in a similar way: the
gaussian potential landscape triggers the expulsion of only enough energetic
polaritons acting as an effective energy filter as in the case of cold atoms, a sort
of gradient cooling (because this action is driven by the gradient of the potential).
With this mechanism only colder polaritons can remain within the condensate,
whereas hotter particles are expelled. In Fig. 15 the occupancy numbers as a
function of energy are reported together with the Maxwell-Boltzmann fitting
(red lines) of the thermalized part of the emission for different powers, according to:
hNi i = Ngi e−
(i −µ)
kT
(24)
27
Normalized Intensity (a.u.)
polariton condensate dynamics in a high finesse microcavity
Normalized Intensity (a.u.)
28
1
P=44 mW
T=6K
P=55 mW
T=3K
P=190 mW
T=2K
P=220 mW
T=1.8K
0.100
0.010
0.001
1
0.100
0.010
0.001
0.0
0.2
0.4
0.6
Energy (meV)
0.8
0.0
0.2
0.4
0.6
Energy (meV)
0.8
Figure 15: Temperature dependence. Intensity as a function of the energy of the polariton emission intensity at the center of the spot, with different excitation
powers. Maxwell-Boltzmann fitting (red curve) of the thermalized component
on the high energy tail. The slope of this signal is steeper increasing the densities until a saturation is reached above 50 mW.
with gi the degeneracy of the i-th energy level, µ the chemical potential, k the
Boltzmann constant, T the temperature and i the i-th energy level. The change
in the slope of the distribution confirms that a decrease of the temperature of
the condensate appears upon a certain pump power.
In Fig 16 the variation of the temperature extracted from the Maxwell-Boltzmann
fitting of the tail of the particles distribution is shown for different pumping
powers. The temperature is decreasing from 50 K and saturates to the value
of 2 K. The behaviour shown in Fig. 16 seems to confirm the gradient cooling
model, but numerical simulations making use of the Gross-Pitaevskii equation
(it is described in Ch. 3) were not able to reproduce this effect.
2.4
conclusions
In conclusions, we have shown the formation of a non resonantly pumped small
condensate on top of the gaussian potential resulting from the repulsive interactions between excitons in the reservoir within the spot region. This condensed
state is strongly affected by the spatial gradient of the non resonant laser and is
2.4 conclusions
Temperature (K)
50
10
5
1
1
5
10
50
100
Power (mW)
Figure 16: Temperature and pumping power. Temperature of the thermal fraction extracted from the fitting in Fig. 15 as a function of the non resonant pumping
power. The temperature decreases by increasing the excitation power.
not significantly altered by a variation of the spot size. This dependence seems
to be related to the mechanism triggering the expulsion of polaritons from the
center of the excitation region. Moreover the decrease of the temperature with
higher pump intensity appears to be related to the action of the gaussian potential induced by the pump spot which balistically shots the higher energetic
polaritons from inside the condensate towards regions far away from the excitation point. After this expulsion only the less energetic polaritons can survive
within the condensate.
Finally, the way in which the top energy condensed state can propagate, relax
and populate a macroscopic condensate at the bottom of the LPB is the subject
of the next chapter.
29
F O R M AT I O N O F A M A C R O S C O P I C E X C I T O N - P O L A R I T O N
C O N D E N S AT E
3.1
introduction
As reported in the previous chapter, polariton condensates have been experimentally observed in different materials, both inorganic [51–54] and organic
semiconductors [12, 55]. However, differently from their atomic counterpart,
these condensates suffer from dephasing and density fluctuations induced by
the interactions with the exciton reservoir, effectively resulting in multimode
condensates [56–58]. This is the field of exploitation of the out-of-equilibrium
statistical mechanics, whose framework provides the basis to understand what
happens when systems rely in regimes where dissipations are not negligible.
Another effect of the exciton reservoir, which is constantly feeding polaritons,
is to act as a blueshifted potential, with a resulting condensate confinement
within the region of the excitation spot [46–48, 59]. In addition of this optically
induced potential, polariton condensation is often affected by local inhomogeneities caused by imperfections or structural defects of the sample, yielding
to a fragmentation of the phase coherence [60–64]. All of the above described
aspects blur the fundamental character of the phenomenon of phase transition
by disrupting it with technical impediments. As a result, fundamental studies or technological applications such as the investigation of out-of-equilibrium
phase transitions or the implementation of simulators and related devices [65]
are hampered by these factors. One way to allow polaritons to reach regions far
from the exciton reservoir is the confinement of the fluid in one-dimensional
structures. Indeed, this geometry permits to reduce the influence of the aforementioned effects, but with the limitation of the dimensionality [66–69].
However, in two-dimensional (2D) structures the investigation of polariton
condensation is prevented by interferences from scattering potentials, effects
of laser-induced confinement and the wedge due to the variation of the microcavity thickness. One of the main consequences of these difficulties is the
hindering in the realization of an extended and uniform condensates, even in
samples with long polariton lifetimes [11, 70–72].
In this chapter we demonstrate that using a high quality 2D microcavity without spatial inhomogeneities, it is possible to create a two-dimensional macroscopical occupied state which fills a big portion of space far away from the
31
3
32
formation of a macroscopic exciton-polariton condensate
laser spot region therefore removing the problem of the influence of the exciton reservoir.
In order to simulate the formation of this state, we consider a mean field
description of the wave function ψ(r)of the condensate, given by the GrossPitaevskii equation [47]:
ih
∂ψ(r)
=
∂t
E0 −
h2 2 ih
[R[nR (r)] − γ] + V(r) + hg|ψ(r)|2 ψ(r).
∇ +
2m r
2
(25)
Where E0 is the bottom of the LPB, m is the polariton mass, γ is the decay
rate, V(r) represents the external potential (including the excitonic repulsive
energy blueshift), g is the interaction strength and R[nR (r)] accounts for the
scattering rate of the excitonic reservoir nR (r) that has its own dynamics:
ṅR (r) = P(r) − γR nR (r) − R[nR (r)]|ψ(r)|2 .
(26)
with γR the decay rate of the reservoir and P(r) the pump intensity profile.
These two equations will be used to demonstrate that the relaxation processes
based on the presence of a phonon bath are crucial to fill the condensate far
from the excitation region.
Finally, it is important to note that the high homogeneity of the sample, with
basically no localization effects (induced by defects and dislocations), and the
long radiative lifetime (∼ 100 ps), i.e. the high cavity Q-factor, that permits the
propagating polariton to relax into the ground state, allow for the formation of
a uniform and extended polariton condensate. This is also facilitated, compared
to other works in similar kind of samples, by the absence of a cavity wedge
which present a detuning-induced acceleration along preferential directions (as
explained in Sec. 2.2).
3.2
the experiment
All the experiments shown in this chapter were performed using the same setup
and procedures as in Sec. 2.3. The excitation of the sample was performed using
a nonresonant pump with a low-noise, narrow-linewidth Ti:sapphire laser with
stabilized output frequency.
With the use of photoluminescence measurements we investigated the characteristic of the emission by collecting and imaging all the light emitted from
the sample on the entrance slit of a streak camera coupled to a spectrometer in
order to measure the time-, energy-, and space-resolved polariton dynamics.
3.2 the experiment
4
1601
Energy HmEvL
Energy (meV)
3
2
1602
1
1603
0
0
b
(b)
a
20
40
60
Space (μm)
80
1604
-3
-3
-2
-2
-1
0
1
-1
0
1
K-Space (μm-1)
K- Space Hm
m- 1 L
2
2
3
3
Figure 17: Energy resolved emission. a, Energy resolved real space emission and b,
energy resolved reciprocal space emission of a one-dimensional cut passing
through the center of the excitation spot (x = 0 µm in a)
3.2.1
The formation of the extended state
The study of the energy resolved emission allows to investigate the way this extended condensate is formed. As mentioned above, thanks to the nonresonant
excitation obtained with the laser tuned at a minimum of the reflection stop
band, a high density of excitons accumulates within the region of the pumping
spot.
The exciton density under the pump spot region induces a blueshift of the polariton energy proportional to their repulsive interaction strength. This brings
to the formation of a gaussian potential landscape placed under the laser spot.
Outside the optically pumped area, further away from the blueshifted potential
the density of uncoupled excitons decreases quickly, due to the small exciton
diffusion length (2–5 µm), and the energy blueshift completely disappears. This
is shown in Fig. 17a, where the emitted two-dimensional intensity map is energy and spatially resolved (vertical and horizontal axis, respectively) along one
direction passing through the center of the excitation spot. The high-energy polaritons sitting at the top of the Gaussian potential (4 meV above the bottom
of the LPB) and formed at the center of the laser spot, expand radially outward with a large inplane wave vector (k ∼ 2 µm−1 ). This is visible in the
cross section of the lower polariton dispersion (energy distribution in momentum space) shown in Fig. 17b where two points of strong accumulation with a
high wavevector are formed at ∆E = 4 meV. The intense emission of these two
points in k-space is the signature of the propagating particles from the center
towards regions far from the excitation. The strong occupation of the lowest
energy mode (k = 0 µm−1 ), visible at the bottom of the LPB in Fig. 17b, cor-
33
34
formation of a macroscopic exciton-polariton condensate
20 μm
(a)
a
b
Figure 18: Pumping mechanism and interferometric setup. a, Sketch of polariton relaxation in space (x, y) and energy (vertical axis). b, Two-dimensional spatial
emission map at E = 0.25 meV from the bottom of the LPB.
responds to the extended condensation outside of the spot region. In addition
to the occupation of the lower state, also higher energy states along the whole
dispersion (k 6 2 µm−1 ) are populated and propagate with different velocities.
In Fig. 18a the two-dimensional sketch of the formation process described
above is reported. The (x, y) plane represents the spatial region inside the sample and the z axis increasing energies. The carriers, injected by the nonresonant pumping laser, relax quickly into excitonic states (yellow area) spatially
confined within the pumping spot region at the center of the image. Efficient
scattering from the exciton reservoir into polariton states results in a region
of high polariton density (red area) which expands radially thanks to the expulsion triggered by the exciton blueshifted gaussian potential, as described
in Sec. 2.3.2. During this radial expansion, the long lifetime allows the relaxation of polaritons into lower energy states and eventually, when the density
increases above the efficient stimulated scattering threshold, the condensation
into the ground state. Above this threshold in the excitation power, an extended
two-dimensional polariton condensate (blue area in Fig. 18a) is formed outside
of the pumped region. The two-dimensional map of the spatial emission from
the condensate at ∆E = 0.25 meV is reported in Fig. 18b. The condensate is
completely free to expand in one direction, while in the other a small confinement is provided by two misfit dislocations (dark horizontal lines in Fig. 18b),
separated by a distance that is roughly four times the FWHM of the excitation
spot (about 80 µm). Remarkably, in Fig. 18b it is possible to note that the high
spatial homogeneity of the sample allows the polariton gas to expand with a
high level of uniformity.
3.2 the experiment
Energy (meV)
4
30 ps
50 ps
70 ps
90 ps
2
0
Energy (meV)
4
2
0
0
20
40
Space ( μm)
60
0
20
40
Space ( μm)
60
Figure 19: Energy resolved time emission. Energy vs real space at different times indicated by the labels. In the figure, the photoluminescence intensity is displayed
in a false-color linear scale.
3.2.2
The time-resolved measurements of the condensation
In order to study the time evolution dynamics of the polariton population before the formation of the steady state, the stationary polariton distribution, populated through the continuous wave (CW) pump laser, is perturbed by focusing
an additional 100 fs pulsed beam on top of the laser (both lasers are tuned to
nonresonantly excite the system at the first minimum of the mirrors’ stop band).
The evolution of the additional polaritons injected by the pulse is recorded with
a time resolution of 2 ps by using a streak camera.
In Fig. 19, four temporal snapshots of the emission intensity as a function of
energy (vertical axes) and space (horizontal axes) are shown at different delay
times (30 ps, 50 ps, 70 ps and 90 ps after the arrival of the non-resonant pulse
in Fig. 19a, b, c, and d, respectively). Polaritons at high energy quickly expand
and relax into lower energy states.
An accumulation point (yellow circles in panels of Fig. 19), induced by the
pulse spot width being slightly larger than the one of the laser pumping the
35
formation of a macroscopic exciton-polariton condensate
300
3.0
a
Speed ( μm /ps )
Time (ps)
36
200
100
b
2.5
2.0
1.5
1.0
0.5
0
0
14
28
42
Space ( μm )
56
70
0.0
0.0
0.5
1.0
1.5
Energy (meV)
2.0
2.5
Figure 20: Propagation space-time map. a, Time dependence of the polariton emission
intensity at a given energy, E = 2.6 meV, perturbed by a 100 fs pulse, as a
function of the distance from the excitation spot. At this energy, polaritons
propagate with a constant velocity of 1.97 µm/ps. b, The propagation speeds
extracted by time-resolved measurements (blue dots) are compared to the polariton velocities as calculated from the LPB (green line). The red area shows
the difference between the extracted speeds. Faster expansions than expected
are observed for energies below E = 1.7 meV due to the prevailing effect of
relaxation from higher energy state.
condensate, allows the observation of a sort of "whiplash" of relaxation (yellow
dashed lines in the panels of Fig. 19), a peculiar pattern in the energy-space
map at different times in which polaritons, for increasing time from the arrival
of the pulse, populate spatial points always farer – for a fixed energy state –
thanks to the contribution of the higher energy states coming from the relaxation of the pulsed injected polaritons. Eventually, at longer times, polaritons
relax into the bottom of the LPB, as shown in Fig. 19d. The space-time maps at
each energy can be obtained by taking the cross-sections at different energies of
the original three-dimensional space-time-energy matrix. An example of such
a two-dimensional map obtained by selecting a specific energy is shown in
Fig. 20a. In this map the slope of the fitted line is the polariton velocity of a single energy state. In this particular case, with an energy of about E = 2.6 meV,
a speed of about 1.97 µm/ps can be extracted. By Repeating the same evaluation procedure for each energy of the dispersion, the expansion velocities can
be compared with the expected group velocities calculated from the carriers
1 ∂ELPB (k)
dispersion accordingly to: vg = h
as shown in Fig. 20b. The deviation
∂k
from the value measured from the bare dispersion (indicated in Fig 20b by the
red-filling region) is related to the effect of relaxation from the higher energy
states that tends to populate lower energy states at distances greater than what
is compatible considering only the group velocity at the given state. In particular this effect becomes stronger for states near the bottom of the LPB, showing
that the effect of relaxation from higher energy states becomes considerable at
lower energies. To discard the possibility that the long propagation lengths can
3.2 the experiment
a
b
50 μm
20 μm
c
Normalized polariton density
0.2
0.4
0.6
0.8
1.0
e
1.5
1.0
0.5
0.0
0.0
20 μm
Space
0.5
1.0
1.5
In-plane wavevector (μm-1)
X10-1
Intensity (a.u.)
k x
20 μm
0.0
ΔE (meV)
d
2.0
f
3
2
1
80
90
100 110
Space(μm)
120
Figure 21: Propagation against a defect. a, Two-dimensional spatial emission at a given
energy with propagation against a natural defect (E = 0.5 meV). b-d, Particular cases of interference pattern formed in front of defect with the characteristic angle of the defect induced shadow for different energy levels (E=0.1 , 0.5 ,
1.1 meV in panels b, c and d, respectively). e, Energies versus wavevector extracted from the LPB (black line) and from the periodical spatial oscillations
(blue dots). Polariton density as a function of energy is shown by the red line.
f, Intensity cross section of the emission along the black line in panel d.
be explained by a faster expansion, for example due to a significant wedge in
the cavity [43, 73], we can compare Fig. 20 with a direct estimation of the polariton wavevectors. This is obtained by an optical tomography of a region of
the sample where a single natural defect, created in the DBR heterostructure
during the growing process, breaks the spatial homogeneity of the surface. The
pumping spot, the expanding polariton fluid and the defect are clearly visible
in Fig. 21a, which shows the (x, y) spatial reconstruction of the emission at
∆E = 0.5 meV from the bottom of the bare dispersion. In Fig 21b-d the region
37
38
formation of a macroscopic exciton-polariton condensate
a
b
c
Figure 22: Waves interference. a, Simulated spatial intensity pattern of a plane wave
and of a spherical one (b). c, Simulated intensity of the superposition of these
two wave-fronts.
around the defect is magnified and filtered at different energies, showing increasing flow speeds. By using a one-dimensional cross section (black line in
Fig 21d) it is possible to extract the intensity pattern resulting from the interference between the flow that is heading for the defect and the one reflected back
by the defect itself. These fringes have a spatial period (∆x) that is related to
π
. The wavevecthe wavevector of the propagating flow accordingly to: k = ∆x
tors extracted from these interference patterns (Fig. 21f) at each energy match
perfectly with the polariton dispersion, as shown in Fig. 21e, also at the lowest energies. In Fig. 22 it is reported the spatial intensity pattern for a plane
wave (a), a spherical wave (b) with the intensity decaying radially and the interference pattern resulting from the superposition of these two waves (c). The
intereference pattern simulated in Fig. 22c is in agreement with experimental
measure of the propagation against a defect and reported in Fig. 21b.The analysis of the polariton flow reported here shows unambiguously that the rate at
which the polariton population distributes in space is the result of the sum of
two rates: the ballistic flow propagation triggered by the blueshifted potential
in the reservoir region and an additional filling rate related to relaxation processes from higher energies. The latter effect becomes important for states close
to the bottom of the LPB. Indeed, the relaxation from the higher energy states
helps in reaching distances longer than what is expected from the bare propagation of low-speed polaritons thanks to the additional amount of polariton
population that relaxes into lower energy states after propagating (Fig. 18a).
Finally, this process contributes also in crossing the condensation threshold
even at large distances far away from the excitation spot and in a region free
from the reservoir related instabilities.
3.2 the experiment
3.2.3
The stimulated scattering threshold
In this section we describe what happens increasing the power of the pump
feeding the top condensate that eventually, after propagating, relaxes into the
lower energy states and then condenses into the bottom of the LPB.
This kind of measurements allows the study of the variation in the spatial
distribution of the condensate and the determination of the critical density at
which the stimulated scattering into the lowest energy state starts to prevail
over the ordinary relaxation processes.
The efficient distribution of the energy states through the relaxation processes allows to increase nonlinearly the density of the condensed state. In
fact, by increasing the pumping power, the polariton density outside of the excitation region, at the bottom of the LPB, manifests a nonlinear increase even
for extremely low density values, as it is shown in Fig. 23a [74]. At the same
a
b
Linewidth (pm)
30
0.5
25
20
15
0.7
c
0.1
Population Ratio
Polariton Density ( μ m-2)
1
0.05
0.6
0.5
0.4
0.3
0.2
0.01
0.0
0.5
1.0
1.5
Energy (meV)
2.0
2.5
0.7
1.0
1.3
1.6
Power (P/Pth)
1.9
Figure 23: The condensation threshold. a, Polariton density at a distance of 40 µm from
the excitation spot as a function of the energy for the same pump powers
as in the panels b-c. The threshold power is Pth = 25 mW. b, Reciprocalspace linewidth as a function of power. c, Population ratio between the lowest
energy state and the whole continuum of states at 40 µm from the spot region,
showing the nonlinear increase above the condensation threshold.
39
formation of a macroscopic exciton-polariton condensate
time, the formation of the bottom condensate is marked by a narrowing of the
linewidth, as shown in Fig. 23b.
The behaviour of the density is confirmed with the analysis reported in
Fig. 23c, where the ratio between the density in the lowest energy state and
the whole expanding cloud is plotted at different excitation powers, showing a
nonlinear increase at the condensation threshold.
Finally, the low density of the macroscopic condensate helps in keeping the
effects of polariton-polariton interaction limited, reducing the dephasing of the
condensate [58] and making this configuration appealing both for the observation of the effects described in the next section and for deep investigations of
phase transition dynamics in extended two-dimensional polariton systems as
reported in Ch. 4.
3.2.4
The polariton fluid back–flow
In this section we report the observation of an interesting effect related to the
change in the sign of the polariton mass near the inflection point. In fact, the
presence of states with higher energy and strong acceleration make it possible
to study what happens when a flow of carriers with negative mass propagates
in the two-dimensional plane of the cavity. The simulated polariton dispersion
(blue line) and the relative effective polariton mass (yellow line) are reported in
Fig. 24. It is possible to note that for |k| > 2µm−1 the value of the mass obtained
from the second derivative of the dispersion becomes negative. At this critical
1.6
8
1.2
6
0.8
0.4
4
Mass
ΔE (meV)
40
0.0
2
0
-0.4
-0.8
-6
-4
-2
0
k(μm-1)
2
4
6
Figure 24: Negative polariton mass. Simulated polariton dispersion (blue line) and polariton mass in a-dimensional units evaluated from the second derivative of
the dispersion (yellow line). The parameters used are the same as reported in
Sec. 2.2.
3.2 the experiment
value of the inplane momentum a peculiar effect related to the change of sign
in the mass can arise.
As we previously described, the fast-expanding polariton fluid at higher energies is therefore acting as a pure polaritonic reservoir for the large-area condensate at k = 0 with its kinetic energy provided by the polariton blueshift
under the laser spot. This allows the fine tuning of the ballistic velocity of expelled polaritons by the control of the excitation intensity.
In fact, for higher powers, the group velocity of the expanding wavepacket
increases until the energy of the blueshifted condensate (the state propagating
on top of the gaussian potential landscape and pumping the condensate at
the bottom of the LPB) reaches the one corresponding to the inflection point
of the polariton dispersion in the region far from the excitation point. The
change of concavity in the polariton dispersion corresponds to a change of
sign in the diffusive mass, that becomes negative above this point. As recently
predicted [75], the presence of positive and negative masses can produce a
counter-propagating flux within the expanding wavepacket. This effect, caused
by the peculiar concavity of the polariton dispersion, is the result of the selfinterference between different components of the expanding wavepacket. In
that sense, by measuring the polariton emission in the two-dimensional geometry, a counter intuitive back–flow, directed towards the position of the exciting
laser, would appear in the density maps for inplane momentum values above
the inflection point. Thanks to the high degree of coherence sustained over
large distances by the expanding wavepacket, we are able to clearly observe
it in our sample. The direct manifestation of the back–flow is indeed evident
Extended condensate
Excitation region
Detected region
Selection slits
Figure 25: Spatial selection sketch. Sketch of the spatial selection with the slits (black
rectangle) used to cut the spatial emission of the excitation area (gray rectangle) and to investigate only the condensate emission far from the excitation
(yellow region).
41
formation of a macroscopic exciton-polariton condensate
if we select the emission coming from a portion of space (excluding the emission from the excitation spot region, in which the outward propagating states
tend to hide the back–flowed polaritons), and observe its density distribution
in momentum space (Fig. 25). With respect to Fig. 26a, we now select only the
emission coming from the half–space on the right of the pump spot. The measured polariton dispersion corresponding to this region is shown in Fig. 26a-c
for three different excitation densities. If |k| < k0 , where k is the expanding
wavector and k0 corresponds to the inflection point, only the portion of the dis-
a
b
c
8
Energy (meV)
42
6
4
2
0
-3
-2
d
-1 0 1
k (μm-1)
2
3
-3
-2
-1 0 1
k (μm-1)
2
3
-3
-2
-1 0 1
k (μm-1)
2
3
e
μ
20μm
Figure 26: The back–flow. a-c, Density in momentum space of the flow directed horizontally to the right. The excitation power increases from a to c. At the inflection
point, indicated by the red vertical line, the back–flow appears as a polariton
density whose k-vector is directed to the left (k > 0). d-e, Real space density
maps, taken as in Fig. 21, for expanding wavevectors below (d) and close (e)
to the inflection point.
3.3 the model
persion associated with a current directed rightwards is visible (k < 0). This is
the case of the lower pumping power shown in Fig. 26a, where only the dispersion side relative to polaritons traveling away from the region of the excitation
with a radial expansion is populated. However, for higher blueshifts, when
k ∼ k0 (Fig. 26b,c) also the opposite k-vectors, associated to a leftward propagation, become clearly visible. The effect on the spatial polariton distribution can
be directly observed also in real space thanks to the density modulations that
appear before the defect rather than behind. The shadow cone that in Fig. 26d
shows up only in the wake of the defect, is visible in Fig. 26e also in front of
the defect, revealing the presence of a polariton flux in the opposite direction.
3.3
the model
In order to investigate what is the mechanism behind the formation of this extended condensate, we performed some numerical simulations containing both
the propagation and relaxation dynamics. More in details, to reproduce the experimental observation, a theory that joins hydrodynamics and relaxation of
an expanding and relaxing condensate is developed by combining the mean
field description of the dynamics given by the Gross-Pitaevskii equation (GPE)
[47, 76], as it is reported in 3.1, with the rate equations that account for stimulated scattering due to the interaction with a phonon bath [67, 77]. A simple
differential equation that describes the steady-state polariton distribution in
both energy and space can be derived with both components of the dynamics
treated at the mean-field level [78]. The results of the calculation reproduce the
observed phenomena using the experimental parameters and confirm that an
extended polariton condensate is formed when the phonon-mediated stimulated scattering is included.
3.3.1
The hydrodynamics in the phonon bath
In simulating the processes behind the population of the lower energy state, we
are interested on the expansion and relaxation dynamics outside the excitation
spot. In fact, the excitonic interaction within the reservoir produces a repulsive
potential that can be described by V(r) and that tends to expel polaritons from
the region of the spot where they have been created. The dynamics of polaritons
outside the spot is here taken into account. As a consequence of the low density
of the condensate at these distances from the spot, it is possible to ignore the
nonlinearity. Therefore, in the region of interest, |r| > σ (σ being the size of
43
formation of a macroscopic exciton-polariton condensate
4
a
b
3
Energy (meV)
44
2
1
0
20
40
60
Space (μm)
80
20
40
60
Space (μm)
80
Figure 27: Numerical Simulations of condensate formation. Comparison between two
numerical simulations, without (a) and with (b), relaxation. This result shows
that relaxation is needed to account for the main feature observed, that is, the
formation of an extended condensed state at the bottom of the LPB.
the spot, i.e., the region where nR (r) and V(r) are zero) the hydrodynamical
description of the polariton flow can be given by:
ih
∂ψ(r)
=
∂t
E0 −
h 2 2 ih
∇ − γ ψ(r).
2m r
2
(27)
The details of the calculation developed in the model are described in the
appendix 8.1. The equation is numerically solved outside of the pumping spot,
with σ the spot radius and r = 0 the center of the spot, assuming as boundary initial condition for each energy the polariton density at r = σ extracted
experimentally, and then solving for r > σ.
The results of the calculation are shown in Fig. 27a,b. When we pump the
system without the presence of the phonon bath, it is impossible to form the
extended state at the bottom of the LPB, as visible in Fig. 27a, where the small
portion of space occupied by the state at the bottom of the band even above
the threshold is not compatible with the experimental results. On the other
hand, the presence of the phonon bath and the related phonon assisted relaxation processes allows the formation of a state whose spatial extension agrees
with what is experimentally observed (Fig. 27b). This confirms that taking into
account only the propagation of the states, without considering the phonon
related relaxation brings fundamentally to a dynamics near k = 0. In this case
the population of the lowest energy state is only based on the small polariton
3.4 conclusions
velocity and this is in contrast with the formation of the extended coherent
state that is experimentally observed. Conversely, looking at the contributions
coming from the higher energy states thanks to the thermal phonon bath, permits to observe the arising of the extended two-dimensional state (in agreement
with the experimental observation), confirming the central importance of the
relation between phonon relaxation and propagation in the constitution of a
macroscopic polariton condensate.
3.4
conclusions
In this chapter we described the spontaneous formation of an macroscopic,
two-dimensional polariton condensate at the bottom of the LPB. The dynamics of establishment and the physical properties of this extended state are investigated with photoluminescence measurements that reveal how, after non
resonant excitation, the blueshifted gaussian potential landscape formed under
the pumping spot triggers the expulsion of a ballistic flow of polaritons at the
energy of the blueshifted polaritons at the center of the gaussian spot. Furthermore, a continuum of states along the dispersion propagates far away from the
excitation region. By analyzing the distances reached by each state, it is possible to observe how they are not compatible with the simple propagation, but
rather with the sum of the bare propagation and the relaxation from the higher
energy states.
The formation of this state is described through a numerical model able
to capture the dynamics of expansion and energy relaxation. Thanks to the
simulations, the phonon nature of these processes and the crucial role of the
phonon bath in the onset of the condensate are demonstrated. Moreover, these
blueshifted states present some peculiar effects investigated for the first time
in polariton system. For instance, the back–flow effect, observed within the
spectral content of the expanding wavepacket and the relaxation towards the
bottom of the LPB, helped by the long polariton lifetime allowing to keep the
decay rate lower than the relaxation time. Thanks to this last dynamics, above
a critical density, the condensate at the lowest energy forms at k ≈ 0 µm−1 ,
covering a spatial extension larger then 0.03 mm2 in a reservoir free region, far
from the excitation spot. Our findings provide the closest realization so far of
an infinite, not confined, two-dimensional polariton condensate, not affected by
the instabilities related to the presence of the exciton reservoir and with dilute,
tunable densities.
In the next chapter we will show how, thanks to the peculiar traits of this polariton condensate, a Berenziskii–Kosterlitz–Thouless (BKT) transition at thermal equilibrium can be observed for the first time in polariton systems.
45
4
PHASE TRANSITIONS IN MACROSCOPIC POLARITON
C O N D E N S AT E S
4.1
introduction
Collective phenomena which involve the emergence of an ordered phase in
many-body systems have a tremendous relevance in almost all fields of knowledge, spanning from physics to biology and social dynamics [79, 80]. While
the physical mechanisms can be very different depending on the system considered, statistical mechanics aims at providing universal descriptions of phase
transitions on the basis of few and general parameters, the most important
ones being dimensionality and symmetry [81–83]. The spontaneous symmetry breaking of Bose–Einstein condensates (BEC) below a critical temperature
TC > 0 is a remarkable example of such a transition, with the emergence of an
extended coherence giving rise to a long range order (LRO) [84–86]. Notably,
in infinite systems with dimensionality d 6 2, true LRO cannot be established
at any finite temperature [87]. This is fundamentally due to the presence of
low-energy, long-wavelength thermal fluctuations (i.e. Goldstone modes) that
prevail in d 6 2 geometries.
However, if we accept a lower degree of order, characterised by an algebraic
decay of coherence, it is still possible to make a clear distinction between such
a quasi-long-range-ordered (QLRO) and a disordered phase in which the coherence is lost in a much faster, exponential way. Such transitions, in two dimensions (2D) and at a critical temperature TBKT > 0, are explained in the
Berezinskii–Kosterlitz–Thouless theory (BKT) by the proliferation of vortices—
the fundamental topological defects—of opposite signs [88].
4.1.1
Decay of correlations and BKT phase transition
The coherence of a system can be measured through the first order correlation
function between points in different spatial positions (r1 , r2 ) or with a temporal
delay (t1 , t2 ):
hψ∗ (r1 , t1 )ψ(r2 , t2 )i
g(1) (r1 , t1 ; r2 , t2 ) = p
hψ∗ (r1 , t1 )ψ(r1 , t1 )ihψ∗ (r2 , t2 )ψ(r2 , t2 )i
(28)
At equilibrium, the time dependence can be removed in Eq. 28 and we can
consider the particle density distribution in the Fourier space n(k)
47
48
phase transitions in macroscopic polariton condensates
Z
3
ψ(k) = (2πh)− 2 drψ(r)eikr
In this way, the numerator of Eq. 28 can be written in a simpler form:
Z
∗
−1
hψ (r1 , t1 )ψ(r2 , t2 )i = V
dkhn(k)ie−ik(r1 −r2 )
(29)
(30)
and, below the critical temperature, the density distribution can be considered as separate contributions of the ground state and the higher energy states,
as in the following:
n(k) = N0 δ(k) +
X
n(k)
(31)
k6=0
with the ground state term that dominates over the higher energy contributions. As a consequence, the numerator in the Eq.28 becomes N0 . This allows
0
the first order correlation function to converge at the constant value N
N even in
the large distance limit, r = |r1 − r2 | → ∞, the so-called off-diagonal long-range
order (ODLRO). This physical system can be altered by the presence or absence
of a potential acting as a confinement, the dimensionality (d) of the system, the
temperature (T ) and the particle interactions. Once the temperature is finite
(T > 0), the most crucial parameter in determining the type of transition that is
possible to observe is the system dimensionality (d), while interactions change
radically the effects of the topological structures (i.e. vortices, solitons).
As mentioned before, with a finite temperature and d = 2 ODLRO is not
possible. In this condition the system can keep a high level of coherence only
through the Berezinskii-Kosterlitz-Thouless (BKT) mechanism. Besides, if particle interactions are strong enough to suppress the density fluctuations the
topological defects, i.e. the vortices, start to influence significantly the spatial
coherence.
Vortices are structures with 0 particle density in their core and possess a
quantized angular momentum with a continuous phase rotation around the
core of 2πn. Vortices with unitary charge are the only type of relevant topological structures, with both charge values of n = +1 (vortex) and n = −1
(antivortex). In fact multicharged (|n| > 1) vortices tend to dissociate into single
charged vortices.
Inside a condensate two types of vortices patterns can exist: single free vortices and bound vortex-antivortex pairs. The way in which they alter the coherence of the system is completely different. While single free vortices cause a
rotation of the phase that destroys with a non local effect the overall coherence,
the total phase rotation imposed by the vortex-antivortex pairs is null. This is
because, in this second case, the total effect on the phase is the summation of
4.1 introduction
two opposite rotations, with no alteration of the coherence far from the core of
the bound pair.
It is interesting to analyze what is the effect of these structures on the thermodynamical properties of the system. Both the vortex forms increase the entropy
of the system by a quantity proportional to the logarithm of the system size.
However, while this is true for free vortices, in the bound pairs case this factor
is finite and independent on the system size. Therefore, the configuration with
bound pairs requires a less energies compared to the case with free vortices,
by considering F = E − T S, the free energy in an extended, two-dimensional
condensate. As a consequence, it is possible the identification of a critical temperature TBKT , for which, with T < TBKT we observe a proliferation of bound
pairs of vortex-antivortex – the previously mentioned BKT phase –, whereas
with T > TBKT , these pairs start to unbind sustaining the spread of single vortices and destroying the coherence [89]. When the system is in the BKT phase,
the only way in which it can loose coherence is through thermal fluctuations
due to residual phonons. Consequently, the first order correlation function decays as a power-law:
g(1) (r1 , r2 , t1 = t2 ) ≈ |r1 − r2 |−ap
(32)
with the exponent ap = (ns λ2T )−1 that depends on the superfluidity den√
sity ns and the thermal deBroglie length λT = h/ 2πmeff KB T . This peculiar
regime is called quasi-long-range-order (QLRO), because of the presence of
such a slow decay of the spatial coherence, still evident even at large distances.
In order to calculate the upper limit for the value of the critical exponent ap ,
let us consider the effect of a vortex with a core radius χ placed in the center of
a superfluid of radius R and with meff the particle effective mass. The energy
E of the free vortex can be calculated by integrating the local kinetic energy of
the superfluid:
Z
meff ns (r)
kB T
R
2 2
(v(r)) d r = ns λT
ln
(33)
E=
2
2
χ
and considering that the entropy of a single vortex is proportional to the
logarithm of the system size:
S = KB ln
πR2
πχ2
= 2Kb ln
R
χ
(34)
so the free energy of the system is:
F = E − T S = (ns λ2T − 4)
KB T
ln
2
R
χ
(35)
49
50
phase transitions in macroscopic polariton condensates
ns
res
h
T-T
d
hol
BK
1/λT
Figure 28: BKT threshold, The BKT threshold splitting the region with the formation of
bound vortex-antivortex pairs and the one with free vortices.
with the sign of Eq. 35 determined by the factor ns λ2T − 4. This relation allows
to identify the BKT threshold value of the exponent by separating the region
with F < 0 in which free vortices proliferate and F > 0 with the formation of
bound vortex-antivortex pairs, as reported in Fig. 28. From Eq. 35 the upper
value for the exponent of the power law decay in the BKT regime at thermal
equilibrium is ap = 0.25.
This theory is well established for 2D ensembles of cold atoms in thermodynamic equilibrium, where the transition is linked to the appearance of a linear
relationship between the energy and the wavevector of the excitations in the
quasi-ordered state [90].
Moreover, a joint observation of spatial and temporal decay of coherence has
never been observed in atomic systems, mainly because of technical difficulties
in measuring long-time correlations. These are important observables to bring
together because an algebraic decay, with the same exponent α, for both the
temporal and spatial correlations of the condensed state, implies a linear dispersion for the elementary excitations[91–93] and, consequently, the possibility
of achieving thermal equilibrium.
4.1.2
BKT phase transition with microcavity exciton-polaritons
Semiconductor systems, such as microcavity polaritons, appear to be ideal platforms to extend the investigation of many-body physics to the more general
scenario of phase transitions in driven-dissipative systems [94]. However, establishing if the transition can actually be governed by the same BKT process as
for equilibrium system has proven to be challenging from both the theoretical
4.1 introduction
[95–97] and experimental perspective [98–100]. Indeed, the dynamics of phase
fluctuations is strongly modified by pumping and dissipation, and the direct
measurement of their dispersion by photoluminescence and four-wave-mixing
experiments is limited by the short polariton lifetime, by the pumping-induced
noise and by the low resolution close to the energy of the condensate, fundamentally due to the intrinsic limitations in the accuracy of the experimental
measurements. Despite an algebraic decay of the coherence has recently been
experimentally observed in a condensate of polaritons generated below the
pump spot, temporal correlations could only show an exponential or at best a
gaussian decay of the coherence, which are not compatible with a BKT transition [57, 101–103]. The lack of a power-law decay of temporal correlations is a
robust argument against a true BKT transition, as will be demonstrated later
on with a straightforward counter-example of a strongly out-of-equilibrium
system. For this reason, as reported before, it has been a constant matter of
interest what is the nature of the various polariton phases, what are the observables that allow to determine a QLRO, if any, and how they compare with
equilibrium 2D condensates and with lasers [104–110]. Recently, thanks to a
new generation of samples with record polariton lifetimes, the thermalization
across the condensation threshold has been reported via constrained fitting to
Bose–Einstein distribution, suggesting a weaker effect of dissipation in these
systems [11]. However, to unravel the mechanisms that drive the transition,
and characterize its departure from the equilibrium condition, it is crucial to
measure the correlations between distant points in space and time as we move
from the disordered to the quasi-ordered regime [92, 93, 111, 112]. So far, all
attempts in this direction have been thwarted, not only because of the polariton
lifetime being much shorter than the thermalization time and the fragmentation
induced by sample inhomogeneities [113, 114], but also because of the small extents of the condensate. Indeed, earlier measurements of coherence [101, 115,
116] were limited to the small spatial extension of the exciton reservoir set by
the excitation spot, which could result in an effective trapping mechanism [117]
and finite-size effects [105].
On the other hand, it has recently been suggested that the dissipation might
in fact have an even more profound impact on the system. Indeed, collective
phase fluctuations could destroy the algebraic order at long distances, hindering the topological ordering and leading the correlations to decay in a stretched
β
exponential way (≈ ex ). This would be the establishment of the Kardar-ParisiZhang phase (KPZ) [95]. Being that peculiar scenario strongly characterized by
out-of-equilibrium dynamics, the parameter β of the stretched exponential are
also different for space (β ≈ 0.78) and time (β ≈ 0.48). The possibility to observe this distinctive phase in one-dimensional condensates of polaritons was
suggested in a seminal paper at the time we were investigating the behaviour of
51
52
phase transitions in macroscopic polariton condensates
correlations in the macroscopic two-dimensional condensate described in Ch. 3
[118], but in a sample with a limited lifetime of about 30 ps. Indeed, a later
estimates of the KPZ length-scales appeared to be unrealistic for incoherently
driven microcavities with long lifetimes as in our case, since the presence of
free topological defects strongly hampers the possibility to observe the KPZ
phase [96]. All this intense debate about the true nature of a two-dimensional
phase transition in open/driven dissipative systems and its implications on the
thermalization of the system led us to try to study a system in the thermal equilibrium and in the true BKT regime. This was supposed to be possible thanks
to the long polariton lifetime and sample homogeneity.
In this chapter, we demonstrate that using a high quality sample to form and
control a reservoir-free condensate of polaritons over a largely extended spatial
region whose formation is described in Ch. 3, we are able to observe, for the
first time in any system, the transition to a QLRO phase both in space and
time domains. Remarkably, the convergence of spatial and temporal decay of
coherence allows us to identify the connection with the classic equilibrium BKT
scenario, in which for systems with linear spectrum the exponents take exactly
the same value α 6 1/4 [93]. Stochastic simulations tuned to the experimental
conditions reproduce the experimental observations using the numerical implementation on a finite spatial grid dV of the Fokker-Planck equation[76]. The
model is the following
∇2
idψ(r, t) = −
+ g|ψ(r, t)|2− + i(γ − κ − Γ |ψ(r, t)|2− ) ψ(r, t)dt + dW
2m
(36)
with g = 0.004 µm2 meV the interaction constant, 2k = 1/101 ps−1 the inverse of the polariton lifetime, m = 3.8 × 10−5 me the polariton mass and me
the electron mass, γ the pump rate, Γ the pump saturation rate and dW the
stochastic Wiener noise with correlations
dW ∗ (r 0 , t)dW(r, t) =
γ + κ + Γ |ψ(r, t)|2−
δr,r 0 dt,
dV
where by |ψ(r, t)|− we abbreviated the following expression for the density:
|ψ(r, t)|− = |ψ(r, t)| −
1
dV
(37)
with ψ(r, t) containing both coherent (superfluid) and incoherent (normal)
polaritons. This model allows not only the extraction of the correlations both
in space and time, but also to follow the transition tracking the vortices in each
realisation of the condensate, eventually confirming the topological origin of
the transition.
4.2 the experiment
Finally, it is shown that, by using a different sample in the weak coupling
regime– the configuration with a clear out-of-equilibrium dynamics– the powerlaw decay of the first order coherence can still be found, but only in the space
domain. This demonstrates that the absence of a joint spatio-temporal measurement of correlations does not allow to be conclusive about the thermal
equilibrium achieved by the system, and therefore if the BKT regime is genuinely established. This conclusion set a net distinction between a polariton
macroscopic coherent phase and a laser.
4.2
the experiment
All the experiments reported in this chapter were performed using the same
configuration and procedures described in Sec. 3.2. In order to reconstruct the
information about the phase of the condensate, a Mach-Zehnder interferometer
is employed before the monochromator, as explained in Sec. 4.2.2. Conversely,
to investigate the spatial and temporal decay of correlations in the system a
Michelson interferometer is employed, as it is shown in Sec. 4.2.3.
4.2.1
The spatial distribution of the polariton condensate
In a perfectly homogeneous system, the density does not depend on the position and it is almost constant in the whole region occupied by the condensate.
On the contrary, once formed, the polariton condensate described in Sec. 3.2.1
is the result of the propagation of higher energy states far from the excitation
point and the relaxation processes of these states triggered by the condensate on
top of the gaussian potential landscape due to the repulsions within the exciton
reservoir. As already mentioned in Ch. 3, this mechanism of formation allows
to avoid the instabilities related to the presence of the reservoir (located under
the spot), but at the same time, produces a variation of the density with the distance from the pump spot. Indeed, near the pump a large amount of polaritons
can accumulate while others propagate with high wavevector far away from
the excitation and eventually they relax into the lower energy state (Sec. 3.2.2).
This is reported in Fig. 29a, where the spatial two-dimensional image of the
condensate is visible as extracted from photoluminescence measurements. A
selection in the far field of the emission coming from |k| < 1 µm−1 allows to
look at only the lower energy polaritons in the condensate. The excitation region under the spot is marked here with a yellow dashed circle. In Fig. 29b the
one-dimensional profiles extracted from the black dashed line in Fig. 29a are
reported for different pumping powers.
It is possible to observe that the density is decaying exponentially increasing
the distance from the pump with a decay length of about ≈ 50 µm but also that
53
phase transitions in macroscopic polariton condensates
20 μm
a
b
Intensity (a.u.)
54
103
102
101
-20
0
Space (μm)
20
Figure 29: Spatial emission of the condensate. a, Two-dimensional real-space image of
the emitted light (arbitrary intensity units in color scale) from a portion of
the condensate. To visualise only the bottom energy state in two-dimensional
images, the emission coming from |k| < 1 µm−1 has been selected in the far
field to avoid the contribution of higher energy polaritons. The yellow, dashed
circle indicates the blue-shifted region corresponding to the position of the
laser spot. b, Spatial decay profile of the intensity along the black dashed line
in a for pump powers of (110, 130, 250, 270) mW.
it does not change significantly when the excitation powers increase (as it remains an exponential decay with approximately the same length). Remarkably,
the fact that the decay length does not rely on the pump power, confirms that
the inhomogeneous density of the condensate does not alter non-local properties such as the spatial correlations, as it will be shown in Sec. (4.2.3). This
allows to simulate this type of condensates considering a homogeneous system.
In order to compare measurements and simulations, we will use the relative density respect to the stimulated scattering threshold in the lowest energy
state (dth ) evaluated at the central point in a spatial extension of about 80 µm,
i.e.the point at 0 µm in Fig. 29b. The absolute polariton density can be estimated
using the same procedure explained in Sec. 2.3.3 obtaining an estimation for
dth ≈ 0.34 pol/µm2 . This threshold at which the density reaches dth can be
determined also by measuring the narrowing of the linewidth in the reciprocal
space at the bottom of the LPB as in Sec. 3.2.3 ( see Fig. 23). Alternatively, one
can investigate the ratio between the intensity of the lowest energy state and
the states with higher energies that corresponds to the condensed polariton
fraction. This can be done by energy resolving the signal obtained through a
CCD detector coupled to a spectrometer with an energy resolution of about
10 µeV. The result of this measurement is reported in Fig. 30a. The abrupt
increase in the condensed fraction appears at the critical density dth and it coincides with the capability of the system to sustain a higher level of coherence
even for wider spatial extensions, as will be shown in the following sections.
4.2 the experiment
a
0.5
0.4
0.3
0.2
0.1
50
10
Healing length (μm)
Condensed fraction
0.6
b
9
8
7
6
5
100
150 200
250
Power (mW)
300
150
200
250
Power (mW)
300
Figure 30: Condensate density. a, Ratio between number of polaritons in the lowest energy state and the higher energy polariton population. b, Healing length of
the condensate for increasing pumping power, above the stimulated scattering threshold.
As well as the critical density at which the stimulated scattering into the
lowest energy state starts to prevail over the dissipations (dth ), a fundamental quantity to determine the effects of topological defect on the overall scale
length of the condensate is the healing length. Indeed, when the latter is comparable to the dimension of the condensate, it can be impossible to observe
the formation of vortex-antivortex pairs whose presence is fundamental for the
establishment of the BKT transition. This peculiar length is defined as the minimal spatial distance over which the condensate wave function tends to its bulk
value when a localized perturbation is applied. In the case of a vortex, for instance, its width inside the condensate is directly related to the healing length.
From this definition it follows that a precise estimation of this parameter is
mandatory because the spatial extension of the condensate has to be compared
to the length scale set by this quantity. The healing length is defined as:
q
(38)
ξ = h2 /2mgd,
with m the polariton mass, g the interaction constant and d the polariton
density in the condensate. In Fig. 30b the healing length, evaluated through
Eq. 38, is plotted as a function of excitation power. The polariton mass, obtained
by fitting the polariton dispersion, is m = 3.8 × 10−5 me , where me is the
electron mass. It is possible to note that this length remains below 10 µm and
actually it reaches 6 µm at higher pumping regimes. Therefore, it is possible
to conclude that, compared to the healing length, the size of the condensate
investigated here, i.e. 80X60µm, is sufficiently large to sustain the transition.
Moreover, the fact that the condensate extends much beyond the healing length
assures that no finite size effects can alter the temporal coherence and that
enough vortex-antivortex pairs can be created in the fluid.
55
56
phase transitions in macroscopic polariton condensates
Figure 31: Mach Zehnder interferometer, with BS1 and BS2 beam splitters, Lens1 and
Lens2 the lenses that enlarge the reference single phase valued point.
4.2.2
Measuring the condensate phase
The establishment of the BKT transition is connected with the emergence of a
spatially homogeneous phase. From a different point of view, an alteration of
the spatial homogeneity of the phase, with the presence of fluxes of particles
inside the condensate or pinned vortices, can alter dramatically the extraction
of the coherence decay in the space domain. Basically this is because the existence of such not zero velocity distributions tends to locally mix spatial and
temporal coherence. In that respect, the net amount of coherence in a certain
point is due to the sum of two contributions: the intrinsic coherence of the point
(the one we are interested in), and the time required by the particle to reach the
same spatial point inside the flux, being the former the one that is possible to
avoid with an almost stationary system. In addition, from the direct measurements of the first order correlation function, only the relative phase difference
between two points symmetrical to the autocorrelation point can be extracted
and, in general, this cannot be conclusive about the phase gradient in the fluid
connected with the presence of flows within the condensate.
In order to extract all the information on the phase of the condensate, we
have used a Mach-Zehnder interferometer as shown in Fig. 31. The emission
from the sample is first splitted (BS1) into two arms, line 1 and 2 (Fig. 31). To
measure the overall phase of the condensate, a very localised emission point of
the condensate is used as reference and made interfere with the entire area of
the condensate. In this way it is possible to measure the local spatial variation
in the phase of the condensate with the reference obtained by using two lenses
(Lens1 and Lens2) to reconstruct an expanded image containing only a single
4.2 the experiment
20 μm
a
b
3
2
2
Velocity (μm/ps)
3
Phase
1
0
-1
-2
-3 c
0
1
0
-1
-2
20
40
Space (μm)
60
-3 d
0
20
40
Space (μm)
60
Figure 32: Condensate phase. Phase in the same spatial region as in the white dashed
rectangle in Fig. 29a for d < dth b and d > dth . Color scale goes from
−π (white) to π (black). Above threshold, a uniform phase is spontaneously
formed within a wide spatial region. A phase singularity, in this case pinned
to a small defect, is also visible on the left of the figure. c, Phase profile along
the dashed-red line in b. d, Condensate velocity extracted from the gradient
of the phase in c.
spatial point with a uniform, single-valued phase. The use of the lenses allows
making this reference map huge as the entire emission image, so that it is possible to observe the interference fringes in all the condensate spatial extension.
Finally, thanks to the beam splitter (BS2), the two images interfere on the CCD
focal plane, with the resulting interferogram. From this it is possible to extract
the wavenumber content of the modulated part of the entire frame, i.e. the one
related to the interference fringes, by using a standard FFT (Fast Fourier Transform) algorithm. From this selection in the Fourier space the related real space
content is recovered by performing an anti-transform of the selected wavenum-
57
58
phase transitions in macroscopic polariton condensates
bers. The argument of this frame contains all the information about the phase
of the measured wavefunction.
In Fig. 32a,b the two dimensional spatial maps of the phase extracted with
this procedure is reported for the same region marked in Fig. 29a with the
dashed white rectangle. In Fig. 32a the case below the threshold density is
reported. It is possible to note the absence of a homogeneous phase and the
presence of vortices within this spatial region. Conversely, in the case above the
threshold (Fig. 32b) a uniform coherent phase is established in the same spatial
extension. At larger distances, a rapidly varying phase is visible. In Fig. 32c the
one-dimensional spatial variation of the phase relative to the red line in Fig. 32b
is shown and its gradient is visible in Fig. 32d. From the gradient, a mean global
velocity |v| < 0.1 µm/ps is evaluated across the whole region. This is compatible
with the wavevector of the condensate, confined with |k| < 0.1 µm−1 , as also
independently obtained from the measured polariton dispersion in the same
spatial region (see Sec. 3.2.1).
The homogenous phase measured above the threshold of condensation and
the small velocities reported in Fig. 32d allow to conclude that there are no
fluxes inside the condensate capable of altering the degree of spatial coherence.
In fact, these small velocities cannot lead to any contribution at the coherence
for distances longer than the healing length during the entire polariton lifetime.
4.2.3
Correlations in the space domain
The spatial coherence of a system can be measured through the use of the first
order correlation function g(1) (r1 , t1 ; r2 , t2 ) as reported in Sec. 4.1. From the
experimental point of view, the light emitted by the sample carries all the information about the spatio-temporal correlations of the polariton field, that can
be extracted using a Michelson interferometer setup. The image containing the
emission from the sample is obtained selecting a region of the condensate far
from the exciton reservoir, such as the one indicated by a dashed rectangle in
Fig. 29a. The frame containing the emission image is directed into the Michelson interferometer shown in Fig. 33a. Here, the image is duplicated with the
use of a beam splitter (BS) and, in one of the two arms of the interferometer,
is reflected around the central point. This one, identify by r0 in Fig. 33b, is the
autocorrelation point. This rotation is realized by using an odd number of mirrors in one arm (M), and a backreflector (R) in the other. The superposition of
two signals coming from the interferometer gives the interferogram shown in
Fig. 33b. From this pattern of fringes, it is possible to extract the value of the
first order correlation function for all the spatial points. By moving a mirror in
one arm of the interferometer with an automatised piezoelectric stage whose
minimum step size is a small fraction of the wavelength, the sinusoidal enve-
4.2 the experiment
20 μm
1
BS
0.75
r0
0.5
R
Long delay line
M
a
b
0.25
Figure 33: Michelson interferometer. a, Sketch of the Michelson interferometer used
to measure the first order correlation function. The back-reflector present on
one arm and the odd number of mirrors in the other arm (not shown here),
produce a relative rotation of about π between the two arms. b, Interferogram
obtained through the Michelson interferometer in the region of interest of the
condensate (80X60µm)
lope of the intensity versus the delay of this stage can be measured with high
precision at each point of the interferogram. This is shown in Fig. 34 where the
intensity as a function of the small stage delay is plotted in the autocorrelation
point. Then, the first order correlation function is obtained as:
|g(1) (r, −r)| = VIideal ,
(39)
p
where Iideal = (I1 + I2 )(2 I1 I2 )−1 takes into account small asymmetries between the two interferometer arms, with I1 and I2 the intensities of the two
interferometer arms, and V is the visibility of the interference fringes. The visibility V can be extracted fitting the data, as shown in Fig. 34, with the following
relation:
I(x) = I0 + A sin(ωx + φ0 )
(40)
where V =
A
I0
and φ0 is the initial phase. In this way, the first order correlation
function at equal time (τ = 0), g(1) (r, −r) (r0 = 0 is assumed), can then be
measured between any two points symmetric about r0 as a function of their
separation |2r| following the same method used in [9].
As reported in Sec. 4.1, the BKT model describes how the existence of free
vortices is advantageous respect to the free energy when the density is less than
the critical value 4/λ2T , with λt the thermal de Broglie length. But, when the
system overcomes this density regime, we obtain a complete opposite picture
and only bound pairs of vortex and anti-vortex become profitable from the
point of view of the entropy contributions.
In our case, according to the scheme of formation described in Sec. 3.2.3, by
varying the nonresonant pumping power it is possible to change the density of
59
phase transitions in macroscopic polariton condensates
A
1500
Counts (a.u.)
60
1000
500
I0
0
0.5
1.0
1.5
2.0
Delay(μm)
Figure 34: Sinusoidal intensity modulation and interferometer sketch. Sinusoidal
modulation of the intensity obtained with the sub-wavelength piezo movement. A is the amplitude of the sine and I0 the offset.
the extended condensate allowing the transition from a disordered system to an
ordered one. The two-dimensional maps of the first order correlation function
|g(1) (r, −r)| extracted from the interferogram (Fig. 33b) are shown in Fig. 35 for
increasing densities (d) of the condensate. At low densities, the spatial extent
of strong correlations is limited to a small region near the autocorrelation point
(Fig. 35a-c). However, above the threshold density dth , the emission with strong
coherence covers distances significantly larger than the healing length ξ (with
ξ < 10 µm), as shown in Fig. 35d. Indeed, thanks to the high quality of the
sample, a uniform phase extends over large distances for d > dth (Sec. 4.2.2).
For increasing densities, a higher level of coherence is sustained over a wider
spatial region of about 80 µm×60 µm (Fig. 35e). As shown in Fig. 35f, increasing
further the excitation power results in the shrinking of the spatial extension of
coherence due to an additional dephasing induced by the pump and the formation of strongly populated excited states at higher energies whose propagation
over large distances tends to destroy the coherent content of the condensate [66],
because of additional particle interactions and density fluctuations.
In order to identify the regime at which the transition occurs and to investigate the profile of the correlations decay, a detailed analysis of the onedimensional decay of coherence is here provided. The horizontal line profile of
|g(1) (x, −x)| passing through r0 , for ∆x > 0 (with ∆x ≡ 2x) is studied for increasing pumping powers (Fig. 36(a-c)). To allow a uniform description across
4.2 the experiment
g(1)(r,-r)
0
0.2
0.4
0.6
0.8
20 μm

a
c
b

d
f
e
Figure 35: Two dimensional first order spatial correlations. |g(1) (r)| at different densities, increasing from the top-left corner: d = (0.05, 0.3, 0.5, 1.3, 3, 4)dth in (a, b,
c, d, e, f,) respectively.
the transition, both power law and stretched exponential functions are used in
the fitting procedure:
|g(1) (x, −x)| = A|2x|−α
−B|2x|β
|g(1) (x, −x)| = Ae
(41)
,
(42)
with B a scale parameter for the x-axis, A 6 1 a space-independent amplitude factor, α and β the critical exponents for the power law and exponential
functional respectively. A precise extraction of these parameters is important
in order to understand the real nature of this transition, as reported in Sec. 4.
Indeed, in the KPZ scenario the coherence has to decay as a stretched exponential, whereas, in the BKT physics, this decay eventually becomes slower, in the
form of a power law.
In Fig. 36a the case in which d < dth is shown, with the coloured blue points
representing the experimental data and the relative fitting curve in black. The
decay is exponential and it is well fitted by eq. (42) with β ≈ 1. 1
1 Note that the value of |g(1) (0)| < 1 is due to the time-averaged detection that globally reduces
the visibility of the interferograms, without changing the overall decay of the correlations. Raw
data are plotted without any normalization: residual oscillations of the setup (cryostat, noise in the
optical path, etc) can reduce the maximum value of the coherence observed with time-averaged
detection. To overcome this problem, a space-independent amplitude parameter A is included in
Eq. 41 and Eq. 42. In Fig. 36 integration times of 1100 ms in a and b and 476 ms in c are used.
61
phase transitions in macroscopic polariton condensates
g(1)
0.2
0.10
0.1
0.05
0
e
b
g(1)
0.2
0.10
0.1
0.05
0
1
f
c
0.50
0.2
0.10
0.1
0.05
0
0
10
20
30
40 50
(μm)
Δx
60
70
Residuals
1
0.50
Residuals
d
a
0.50
0
10
20
30
40 50
(μm)
60
Residuals
1
g(1)
62
70
Δx
Figure 36: Coherence decay and BKT phase transition. a, b, c, Experimental spatial
decay of |g(1) (∆x)| (logarithmic scale) and corresponding fitting residuals
(linear scale) for: d = 0.1dth exponential decay (blue), d = 1.4dth stretched
exponential decay (green), d = 2.75dth power–law decay (red), respectively.
d, e, f, Spatial decay of |g(1) (∆x)| extracted from numerical simulations. Respectively, an exponential decay, a stretched exponential with β = 0.67 and a
power–law decay with α = 0.20.
Approaching d = dth , the spatial decay of g(1) becomes slower, but still
faster than a power law (Fig. 36b, green points represent the experimental data).
This intermediate transition regime is best described with a stretched exponential decay (β < 1) that becomes a power–law only at slightly higher densities
d ≈ 2.7 dth (Fig. 36c) when a high degree of spatial coherence (>50 %) extends
over distances of ≈ 50 µm. Significantly, the slow decay shown in Fig. 36c (experimental data in red) can be best characterised using an exponent α = 0.22 extracted directly from the power–law decay. In Fig. 36d-f exponential, stretched
exponential (β = 0.67) and power–law decay (with α = 0.20) extracted from the
numerical simulations are reported together with the relative fitting curves (in
black).
4.2 the experiment
●
2.0
●
β
α
●
●
β
α
α,β
1.5
1.0
0.5
0.0
a
0.0
0.5
1.0
1.5
d/dth
2.0
2.5
3.0 0.0
b
0.5
1.0
1.5
d/dth
2.0
2.5
3.0
Figure 37: Coherence decay fitting. a, Blue line: β exponent evaluated by stretchedexponential fitting of |g(1) (∆x)| in the experimental data versus the corresponding relative polariton densities. Red line: power law α exponent from
the fitting of |g(1) (∆x)| in the experimental data versus the corresponding polariton densities. b, the same as in a, but from the theoretical data. The same
colors used in Fig. 36a-c and Fig. 36d-f indicate the corresponding densities
(square markers) for the experiment and theory, respectively.
The detailed analysis of the α and β exponents extracted from the decay
fitting is shown in Fig. 37a,b int the experiment and theory respectively for
different densities (α can be extracted only for d > dth ). It is possible to observe
that for d > dth the β exponents assume values < 1 and that, for d > 1.5dth the
power–law fitting allows the extraction of the α exponent. Remarkably, in both
experiment and theory α achieves the critical value typical of the BKT transition
(α = 0.25) for d ≈ 2.7dth showing the whole behaviour of the coherence decay
across the transition into the QLRO.
It is important to mention that the distinction between the stretched exponential (with β < 1) and the power-law fitting is not trivial, in that a sufficiently
low β exponent in the stretched exponential function can be difficultly distinguished from a power-law functional behaviour. Moreover, this difference is
crucial in order to understand if the transition described above is of the type of
a KPZ crossover or a BKT one. For this reason, we performed a comprehensive
statistical analysis of the residuals distributions trying different fitting models.
This investigation, in the case of the experimental power-law decay (Fig. 36c)
is reported in Fig. 38. In fact, the accuracy of a fitting model can be verified by
making use of the residuals distribution, i.e. the difference between the fitting
predicted value and the real data value.
Indeed, the total averaged of the absolute value of the residuals alone is not
enough in order to understand if the model is the most accurate. From this
point of view it is crucial to demonstrate that no defined trend is present in the
63
phase transitions in macroscopic polariton condensates
30
a
b
Counts
Counts
20
10
0
-0.10
0.0
Residuals
20
10
0
0.10
c
0.8
Fitting model
K-S value
64
0.6
0.4
0.2
0.0
Exp.
Gauss
S-exp
Fitting model
P-Law
-0.10
0.0
Residuals
0.10
Power-law
d
Stretched exp.
Gaussian
Exponential
0.00
0.04
0.02
Residuals ABS value
Figure 38: Fitting residuals analysis. a, b Residuals histograms from the gaussian and
power law fitting of the data for the experimental spatial decay of correlations
showed in Fig. 36c (red data). c, Kolmogorov Smirnov variable values using
different fitting model fitting the same data as in a. d, Total averaged absolute
value of the residuals for the different fitting models using the same data as
in a.
residuals and this can be achieved showing that the histogram of the residuals
can be fitted with a normal distribution centered at zero. This is showed in
Fig. 38a, b in which the histograms of the residuals by fitting the data with a
gaussian (yellow) and power–law (red) models are reported. The KolmogorovSmirnov variable can test how much a distribution is closed to a zero-centered
gaussian, for the different fitting models used in Fig. 38. In our case, when the
variable is near to one, the values follow a zero centered normal distribution.
The result is reported in Fig. 38c, where the maximization of the variable for
the power–law fitting proves the better accuracy of this model respect to the
others. Moreover, the minimization of the total averaged absolute value of the
residuals showed in Fig. 38d for the power-law model is in agreement with the
use of a power-law functional in fitting the data in Fig. 36c, supporting the BKT
scenario instead of the KPZ crossover (the stretched exponential).
Moreover, an additional demonstration supporting the establishment of the
BKT transition is given by the numerical simulations. Indeed, while the vortexantivortex binding cannot be directly observed in the experiments, which average over many realisations, the numerical analysis is able to track the presence
of free vortices in each single realisation, confirming the topological origin of
4.2 the experiment
50 μm
Figure 39: Vortex-antivortex distribution map. Left, Vortices (V) in red and antivortices (AV) in black just before (top) and after (bottom) the BKT transition
with parameters as in Fig. 36e and f, respectively. Center The same as in Left
but after filtering off in two steps high momentum states to eliminate bound
pairs. Such filtering reveals the presence of free vortices. Note that there are
no free vortices when spatial and temporal coherence show algebraic decay
(bottom) but there are some free vortices in the case of stretched exponential
decay of coherence (top). The underlying colour map shows the phase profile
of the field. The background phase values range from −π to π.
the transition. This is reported in Fig. 39 where it is possible to observe that the
exponential and stretched-exponential regimes both show the presence of free
vortices (Fig. 39 top row), the number of which decreases as we move across
the transition. This is shown by filtering the states eliminating the bound pairs
of vortices in Fig. 39 in the center and right column. The total absence of free
vortices in Fig. 39 right column, bottom row, when the system shows a power
law decay of coherence, demonstrates that in the algebraically ordered state
free vortices do not survive and the pairing is complete.
Since the stretched exponential phase is always associated with the presence
of some free vortices, this supports that we are observing a BKT crossover
rather than a Kardar-Parisi-Zhang (KPZ) phase [95]. It is interesting to note
here that the KPZ physics is indeed the paradigmatic model for a genuinely
non-equilibrium phase transition and its manifestation in the optical domain
65
phase transitions in macroscopic polariton condensates
g(1)
0.2
0.10
0.1
0.05
0
e
b
g(1)
0.2
0.10
0.1
0.05
0
1
f
c
0.50
0.2
0.10
0.1
0.05
0
0
50
100
150
Time (ps)
200
Residuals
1
0.50
Residuals
d
a
0.50
0
50
100
Time (ps)
150
Residuals
1
g(1)
66
200
Figure 40: Coherence decay in time and BKT phase transition. Temporal decay of
|g(1) (∆t)| (logarithmic scale) and corresponding fitting residuals (linear scale)
for: a, d = 0.15dth Gaussian decay (yellow), b, d = 1.3dth stretched exponential decay (green), c, d = 2.7dth power-law decay (red), respectively. The black
lines are the fitting curves. Temporal decay of coherence extracted from numerical simulations. a, stretched exponential with β = 0.41, b, stretched
exponential with β = 0.27, c, power law with α = 0.20.
of polariton condensates is, as already mentioned, currently at the center of an
intense investigation [112].
In order to understand if the system can really achieve the thermal equilibrium, it is fundamental the investigation of the temporal decay of correlations.
4.2.4
Correlations in the time domain
In the classic equilibrium BKT scenario, for a system with linear dispersion
in the ordered phase, a slow algebraic decay (as Eq. 42) of the first order coherence with exactly the same power-law exponents in both space (αs ) and
time (αt ) [92, 93] is expected. At the same time non-equilibrium dissipative
driven systems, with diffusive spectrum in the ordered phase [92, 119], have
4.2 the experiment
a
2.5
●
α,β
2.0
●
β
α
b
●
●
β
α
1.5
1.0
0.5
0.0
0.0 0.5 1.0
1.5 2.0 2.5
d/dth
3.0 0.0 0.5 1.0
1.5 2.0 2.5
d/dth
3.0
Figure 41: Temporal coherence decay fitting. a, Blue line: β exponent evaluated by
stretched-exponential fitting of |g(1) (∆t)| in the experimental data versus the
corresponding relative polariton densities. Red line: power law α exponent
from the fitting of |g(1) (∆t)| in the experimental data versus the corresponding polariton densities. b, the same as in a, but from the theoretical data.
The same colors used in Fig. 40a-c and Fig. 40d-f indicate the corresponding
densities (square markers) for the experiment and theory, respectively.
been shown to still exhibit an algebraic decay of coherence but with temporal
correlations decaying two times slower then the spatial ones αt = 1/2 αs [92,
93]. In this section we analize the temporal decay of correlations in order to
compare the α exponents from the fitting of the first order correlation decay in
the space and time domains, with the aim to understand if the polariton cloud
at the bottom of the lower polariton branch can thermalize. The measurements
of the temporal decay of the coherence is realized with the same Michelson
interferometer showed in Sec. 4.2.3. As reported in Fig. 33a, one arm of the interferometer is equipped with a long delay actuator, with which it is possible to
extract the coherence of the system when a delay δt = τ (with τmax = 200ps)
is applied to one of the two arms. The spatial point of which we investigate
the coherence behaviour in time is the autocorrelation point, i.e. r0 in Fig. 33b.
This is reported in Fig. 40a-c, where the temporal coherence at the autocorrelation point |g(1) (t, t + ∆t)| is shown for three different polariton densities 2 .
Below threshold, coherence decays quickly and following a fast Gaussian slope
(β ≈ 2). At d = 1.3 dth , the temporal coherence can be best fitted by Eq. 42 with
an exponent β ≈ 0.8 (or, with a slightly worst fit, with a power law of exponent
2 Experimental measurements have been recorded by averaging over 10 acquisitions of 300 ms
(Fig. 40a,b) and 100 ms Fig. 40c. This explains why the maximum value of the first order correlation function is < 1.
67
phase transitions in macroscopic polariton condensates
a
Counts
10
5
0
b
-0.04
0.0
Residuals
c
0.6
0.4
0.2
8
4
0
0.04
0.8
12
Fitting model
Counts
15
K-S value
68
-0.04
0.0
Residuals
Power-law
0.04
d
Stretched exp.
Exponential
0.0
Exp.
S-exp
Fitting model
P-Law
0.002
0.006
0.010
Residuals ABS value
0.014
Figure 42: Fitting residuals analysis. a, b Residuals histograms from the exponential
and power law fitting of the data for the experimental temporal decay of
correlations showed in Fig. 40c (red data). c, Kolmogorov Smirnov variable
values using different fitting model fitting the same data as in a. d, Total
averaged absolute value of the residuals for the different fitting models using
the same data as in a.
α ≈ 0.57), while at d ≈ 2.7 dth , the long time behaviour clearly follows a power
law with α = 0.2. The profiles extracted from the numerical simulations are
reported in Fig. 40d-f. Here the simulation is unable to reproduce the gaussian
decay of the temporal correlations, but, at the same fraction of the threshold
density ≈ 2.7dth , the temporal correlations remain high even for long delay
time (about 200 ps), decaying with a power-law whose α ≈ 0.2 is fully compatible with the experimental findings (α < 0.25).
In Fig. 41a, the α and β exponents of Eq. 41 and Eq. 42 that best fit the
experimental data are shown across the transition. It is possible to observe that
the β exponent from the stretched exponential decreases from value > 2 to
an intermediate regime with β < 1 and then, for density of about 2dth , the
power-law fitting starts to be the more accurate, reaching a value of α ≈ 0.2. In
Fig. 41b, the α and β exponents from the experimental simulations are shown.
The important aspect in that case is the value of the α exponent at a density of
about ≈ 2.7dth , that is, as mentioned before, compatible with the experimental
obtained value (α < 0.25).
The residuals analysis proves the agreement between the experimental data
and the fitting model for the temporal decay through checking the normalization around the zero-value of the residuals distribution (Fig. 42a,b), the maxi-
4.2 the experiment
1
g(1)
a
b
0.1
10-2
0
2
4
6
Δx (μm)
8
10
0
50
100
Time(ps)
150
Figure 43: Spatial and temporal coherence in the weak coupling regime. a, Spatial
coherence showing a power-law decay with α = 0.25. b, Temporal decay of
coherence with stretched exponential fitting exponent β = 1.8.
mization of the Komogorov-Smirnov variable (Fig. 42c) and the minimization
of the total absolute value of the residuals (Fig. 42d), as explained in Sec. 4.2.3
for the decay of the spatial correlations, provided here for the cases of the power
law decay (Fig. 40c).
Crucially, also for time correlations, we obtained a power law decay with
α < 0.25, which coincides, within the experimental accuracy, with the one obtained from the spatial coherence at the corresponding density in both the
experimental and simulated results. In conclusion, the equality of the temporal and spatial extracted α, observed experimentally for the first time in any
system, demonstrates that we are observing an equilibrium BKT transition.
4.2.5
Studying both spatial and temporal correlations - condensation and lasing
In order to demonstrate the importance of the simultaneous observation of
space and time correlations for optical systems, and in general as we move from
equilibrium towards out-of-equilibrium, we analyze the coherence behaviour of
a microcavity where driven/dissipative dynamics clearly prevail.
Using a sample with a lower quality factor and less quantum wells, we induced, under high non-resonant pumping, the photon-laser regime as in a vertical cavity surface emitting laser (VCSEL) [120, 121]. Despite the fact that this
system is strongly out-of-equilibrium, it shows a power-law decay of spatial coherence with α = 0.25 (Fig. 43a) within the pumping spot region (with a radius
of about 10 µm).
Remarkably, the behaviour of spatial correlations is very similar to what obtained in Ref. [101]. However, if we analyse the behaviour of the temporal coherence, shown in Fig. 43b, we can note that it follows an almost gaussian decay
(β ≈ 2 in the stretched exponential fitting), not compatible with the algebraic
69
70
phase transitions in macroscopic polariton condensates
order characteristic of the BKT phase. This shows that a consistent decay of
time and space is necessary to evidence the BKT transition in driven/dissipative systems.
4.2.6
The spectrum of the excitations
From a fundamental point of view, the excitation spectrum of a condensate
ω(k), can shed light on the types of excitations within the superfluid, as it is
the result of the correlations in the superfluid density. As reported in different
systems [122], the excitation spectrum can exhibit a linear dependence with the
wavenumber, for certain peculiar regimes. Indeed when phonons, the collective
excitations in the fluid, provide an efficient enough mechanism to establish the
thermal equilibrium, the total effect of the processes behind the thermalization
determines an altered shape of the dispersion, in the form of:
ω(k) ≈ ceff k
(43)
with ceff roughly the velocity of a sound pulse in the fluid, i.e. a composition
of an ensemble of phonons. This linearized spectrum is commonly known as
the Bogoliubov excitation spectrum and represents the unambiguous demonstration of a non diffusive (i.e. quadratic spectrum) thermalized phase within
the fluid.
The observation in polariton systems of this peculiar shape of the excitation
spectrum is usually hidden by the parabolic dispersion of the single diffusive
particle from the bare photoluminescence inside the cavity. Despite that, in 2008
in a work by Utsunomiya et al. [98] the authors claimed that they could measure
the linearized spectrum in a sample with a small lifetime and confined under
the pump spot, directly from the emission of the condensate. On the one hand,
this observation was quite controversial mainly due to the fact that with this
small lifetime the phonon assisted mechanism behind the thermalization is not
sufficiently efficient to let the system achieve the thermal equilibrium. On the
other hand, the experimental configuration, with the condensate under the spot
was strongly affected by the reservoir instabilities that alter the equilibrium
in the condensate. Nevertheless, this work is interesting and represents the
first attempt to measure the Bogoliubov spectrum in a condensate of exciton
polaritons. In a work of 2012 by Kohnle et al. [100] they used a four wave mixing
experiment to select and amplify the weak coherent content of the excitations.
They studied the Bogoliubov character of the excitations in a system where a
lower polariton superfluid coexists with upper polaritons and they found that
the simple Bogoliubov model was not sufficient to describe their system. In
fact, the excitation spectrum was significantly modified by the presence of both
lower and upper polaritons.
4.2 the experiment
Energy (meV)
1.0
a
b
0.5
0.0
-50
0
Space (μm)
50
-50
0
Space (μm)
50
Figure 44: Spatial energy interference, a, Energy resolved interferogram for ∆t = 0. b,
The same as in a but with ∆t = 70ps.
As it will be explained in Sec. 4.2.7, in our investigation we used a technique
based on the measurements of the first order correlation function of the light
emitted by the condensate. The experimental setup is the same described in
Sec. 4.2.3, a Michelson interferometer coupled with a monochromator in order
to analyse the energy content of the signal. An example of this kind of measurement is reported in Fig. 44. A one-dimensional line of the interferogram of
the emission passing through the autocorrelation point r0 (as in Sec. 4.2.3) is
resolved in energy using the monochromator and is reported in Fig. 44a with
no relative delay time (∆t = 0) between the two arms of the interferometer. The
white dashed line marks the perfect vertical orientation of the spatial fringes,
indication of the fact that a good temporal matching is established between the
two signals coming from the interferometer. Conversely, when a time delay is
set between the two arms, this pattern changes, as it shown in Fig. 44b with
∆t = 70 ps. In this case, the point of maximum visibility in the fringes deviates from the vertical orientation according to the motion of the autocorrelation
point proportionally to the energy of the state.
4.2.7
The linearization of the spectrum
Compared to the studies reported in Sec. 4.2.6, a more direct way to extract the
spectrum of the excitations is to use the spatio-temporal decay of correlations,
as measured with the Michelson interferometer. In fact, the first order correlation function of an electromagnetic field that is uniform and at the steady
state is given by the Fourier transform of its power spectrum in momentum
space S(k, ω) [123]. From an experimental point of view, it means that from the
Fourier transform of the energy dispersion it is possible to reconstruct all the
information about the spatio-temporal correlations. This is a well known exper-
71
phase transitions in macroscopic polariton condensates
Time(ps)
150
100
50
a
0
-100
Δx
0
(μm)
100
b
-100
Δx
0
(μm)
100
c
-100
Δx
0
(μm)
100
0.25
g(1)(x0,Δt=380ps)
72
0.20
0.15
0.10
0.05
d
180
200
220
240
Power (mW)
Figure 45: First order correlation function maps. a, b, c, First order correlation function
space-time maps for increasing pumping powers. d. g(1) (x = x0 , ∆t = 360ps)
values in the autocorrelation point with the maximum temporal delay increasing the pumping power. Red, gray and green squares mark the pumping
regime for (a, b and c), respectively
imental technique used in spectral interferometric measurements. Evidently, this
relation is true also in the opposite way: from the coherent content in space and
time, by using the Fourier transform, it is possible to extract the spectrum of
the excitations only due to the coherent content of the condensate (as the excitations), by rejecting the part relative to the photoluminescence of the polariton
emission.
In our case, we can extract the spatial decay of coherence for different ∆t, by
increasing the delay time between the two arms of the interferometer. This can
be done thanks to the use of the same long delay actuator as in Sec. 4.2.4. The
4.2 the experiment
ΔE (meV)
0.2
0.1
0.0
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
k (μm-1)
Figure 46: Linearized excitations spectra. Linearized spectra of the excitations for increasing densities of the condensate. The same regimes as in Fig. 45a-c are
shown in red, yellow and green, respectively.
first order correlation function at any time frame can be evaluated using the
Fourier Transform of the interferometric signal, by selecting only the frequencies relative to the interference fringes (i.e. the one related to the fringes) and
then going back to the real space with a renormalization through the signal
content of the continuous part (more details about the technique can be found
in Ref. [124]). These maps of |g(1) (x, −x; t, t + ∆t)| are reported for different
pumping powers in Fig. 45a-c. It is possible to observe that the autocorrelation
point (i.e. the maximum of the first order correlation function that is in ∆x = 0
for ∆t = 0), shifts its position in time, because particles are flowing with a small
wavevector (|k| < 0.1 µm−1 , see Sec. 4.2.2). With higher pumping levels, the system is able to sustain a higher level of coherence, even for long temporal delay.
This is reported in Fig. 45d, in which the visibility of the experimental fringes
(i.e. the ratio between the amplitude and the offset of the intensity modulation)
is evaluated for different pumping powers holding the temporal delay fixed to
its maximum value of 360ps. This allows the observation of a clear transition
from a long temporal incoherent emission of the lower state (visibility less than
0.05) to a coherent state with a visibility of about 0.3. The red, gray and green
squares represent the pumping powers of the measurements in Fig. 45a-c.
73
74
phase transitions in macroscopic polariton condensates
By applying the mentioned above technique, it is possible to reconstruct the
spectrum of the excitations by calculating the Fourier transform of the measured |g(1) (x, −x; t, t + ∆t)|. In Fig. 46 the points from the bare parabolic polariton dispersion extracted directly from the far field of the photoluminescence are
reported (blue data). This corresponds to the case with no blueshift (δ ≈ 0 meV)
and with the system below the condensation threshold. Clearly, in this configuration no thermalization or homogeneous phase is established. The red points
show the same regime, (δ ≈ 0 meV) but evaluated using the spatio-temporal
decay of the correlations. The fact that the system is far from the thermalization
is confirmed with the superposition of the data from the parabolic dispersion
and the red points.
Increasing the non-resonant pumping power, the thermal equilibrium is established in the polariton cloud at the bottom of the band, and consequently the
spectrum of the excitations starts to be linearized, as reported with the yellow
points in Fig. 46. In this case the blueshift is δ ≈ 0.02 meV. At the highest excitation power, the spectrum of the excitations presents still a linear behaviour
but with an increased blueshift of δ ≈ 0.08 meV (green points in Fig 46).
The linearization of the spectrum of the excitations reported in this section
confirms that, thanks to their long lifetime and to the“reservoir free” nature of
the fluid, polaritons are able to achieve the thermal equilibrium.
4.3
conclusions
In this chapter we have shown that a BKT transition at the equilibrium can
be observed in an ordered macroscopic phase composed by a two-dimensional
driven/dissipative ensembles of bosonic quasiparticles. The onset of the homogeneous phase in the whole region of the condensate is measured through both
spatial and temporal correlations across the transition. Despite the collective behavior of exciton-polaritons in semiconductor microcavities lies at the interface
between equilibrium and out-of-equilibrium phase transitions, and it has been
often compared both to atomic condensates and to photon lasers, in this case
we were able to fully demonstrate the thermal equilibrium in the polariton
cloud at the bottom of the lower polariton branch. Moreover, we demonstrated
that the measurement of spatial correlations g(1) (r) alone is not sufficient to establish whether an open/dissipative system is in the BKT phase. Instead, two
distinct measures, one in time and one in space domain, are required. By applying this criterion, we observed a power-law decay of coherence with the
onset of the algebraic order at the same relative density and with comparable
exponents for both space and time correlations.
The observation of this transition was made possible thanks to the long polariton lifetime inside the sample that allows the achievement of the macro-
4.3 conclusions
scopic coherent phase in a region with low density, and far from the exciton
reservoir confined within the pumping spot region. Furthermore, the absence
of a trapping mechanism and the formation due to a free propagation and
energy relaxation, has led to avoid the influence of finite-size effects in the
temporal dynamics of the autocorrelation [93].
Simulations with stochastic equations match the experimental results and
demonstrated that the mechanism acting on the homogeneous phase crossing
the transition is of the BKT type, i.e., binding of free vortices into bound vortex–
antivortex pairs, resulting in a joint coherence build up both in space and time.
All these observations validate that polaritons can undergo phase-transitions
following the standard BKT picture, and fulfill the expected conditions of thermal equilibrium despite their driven/dissipative nature. Finally, by using the
spatio-temporal decay of correlations, we extracted the spectrum of the excitations of the condensate, showing that, by increasing the condensate density, we
can achieve a regime with a linearized spectral region, the final signature of
the establishment of the thermal equilibrium and superfluidity in this coherent
phase.
In the next chapter we will describe how this condensate responds to a phase
twist realized setting the phase through the application of two external resonant lasers. This brings the system in a peculiar state in which the bosonic
analog of a Josephson junction can be observed together with the snaking instabilities of the barrier between different domains created within the condensate.
75
5
J O S E P H S O N J U N C T I O N A N D V O RT I C E S I N A P H A S E
T W I S T E D P O L A R I T O N C O N D E N S AT E
5.1
introduction
Soliton like modulations of the density in the form of excitations have drawn
the attention of several research areas, spanning from physics to biological science. Indeed, intringuing researches emerged in terms of spatio-optical modes
inside crystals [125], peculiar structures behind the protein folding mechanism [126], multicellular movements [127] and excitations inside Bose Einstein
condensates [128]. A peculiar type of excitations in the density of a condensate
is represented by dark solitons. They can appear as defects in the background
density when the initial state of the condensate is driven into a dynamically
unstable configuration, for instance by imprinting a phase gradient in the condensate [129, 130].
An example of such an effect occurs at the interface between two superconductors divided by an insulating barrier. In this kind of physical systems it is
possible to observe the formation of a peculiar type of contact region, the socalled Josephson junction. This phenomenon can serve as a basis for important
applications such as superconducting quantum interference devices (SQUIDs)
or superconducting qubits. According to the particular geometry provided by
the system, it is possible to discriminate the short (point-like) Josephson junctions, whose analog in polariton system was investigated in the last years [131,
132], and the long Josepson junctions (LJJs), which are characterized by an interface that extends beyond the length of the contact barrier between the two layers, at least in one dimension. This elongated structure allows the observation
of some interesting phenomena like the so-called Josephson vortices. Differently
from the Abrikosov or Pearl vortices, these entities remain localized within the
barrier and are crossed by opposite transverse supercurrents in the two superconductors [133–136]. Recently, Josephson vortices were proposed to be used
as natural mobile qubits in quantum processing architectures [133].
In this chapter, we report about the observation of a LJJ by using the same
extended condensate, as it is described in Ch. 3. In this case we used this coherent state to create two different regions of homogeneous phase by setting
the phase with two lasers tuned in resonance with the condensate and placed
at the boundaries of the considered area. The junction interface extends over
several tens of microns and is created spontaneously in response to a twist of
77
78
josephson junction and vortices in a phase twisted polariton condensate
the overall phase of the condensate. Eventually, the twist results in the creation
of a dark soliton-like coherent structure, which plays the role of an insulating barrier with a reduced order parameter, in analogy to what is investigated
in superconductive systems. By increasing the particle density, the system can
achieve a regime in which the instability is reminiscent of the snaking instability of dark solitons, with the nucleation of stable Josephson vortices but in the
steady state rather than in the transient regime.
5.2
the experiment
In these measurements we investigated the phase of the condensate from the
emitted light by using a Mach-Zehnder interferometer as the one shown in
Sec. 4.2.2. As mentioned above, the two-dimensional phase φ(~r) is extracted
from the interference pattern using only the interfering part of the signal. Indeed, by using the standard Fast Fourier Transform algorithms it is possible to
select only the Fourier numbers relative to the interference modulation whose
argument is the phase of the emission.
5.2.1
The application of a phase twist
In this section we describe a possible way to split a condensate into two different domains of homogeneous phase. In the sketch of Fig. 47a, the dashed
white lines identifies the portion of the polariton condensate under consideration in this experiment. This state is created with a non-resonant pump (dark
yellow laser) as reported in Sec. 3.2.1. It is know that the phase of a polariton
condensate can be locked to that of an external laser tuned to resonance with
the energy of the condensate [60, 137]. The phase the condensate acquires is
arbitrarily chosen in each statistical realisation of the state, with the external
resonant beam acting as a seed. In this way, the phase of the condensate is
pinned to that of the laser with a minimum density perturbation.
This effect is demonstrated in Fig. 47b, where the phase of the condensate is
locked to that of a resonant external beam, focused on the bottom-left corner of
the image. In order to detect the establishment of the resonant imposed phase,
the reference phase is taken from the resonant laser and not from a point in
the condensate. In this way, only when the phase is the same of the locking
beam, a homogeneous phase can be detected, since if this is not the case, the
phase-relation between the locking laser and the condensate is washed out by
the integration over several realisation of the experiment, which is necessary to
obtain the image of the interference pattern. Note that the phase locking of the
condensate is a non-local phenomena: while the external laser is focused into a
small spot (bottom left corner in Fig. 47b) of radius r = 5 µm and it is kept at
5.2 the experiment
b
10 μm
a
c
Figure 47: a, Sketch of the twisted polariton condensate (yellow), with the non-resonant
pump in dark yellow. In blue and red is highlighted the region under consideration, with the red and blue colors representing the phases of the two
resonant lasers. b, Homogeneous phase of the condensate with the external,
phase-locking laser in the bottom-left corner of the image. In order to observe
the locking of the phase, the reference is taken from the resonant laser. c,
Measured interferogram of the condensate with a reference of constant phase
taken by expanding a single point of the condensate. The resonant beams,
partially reflected at the surface, saturate the detector at the positions of the
two spots (black circles) despite the small amount of injected polaritons.
low enough power to induce only a negligible contribution to the condensate
density, the action of the phase locking extends over much longer distances in
the polariton condensate (in a region of about 40 × 35µm), creating a domain
of uniform phase all across the whole region of interest. This peculiar non-local
phase-locking, sustained by the homogeneous phase related to the presence of
the polariton condensate, allows for phase-imprinting configurations substantially different from those used with atomic condensates [138].
In addition, it is possible to use two resonant lasers locking the phase with a
fixed phase difference (as in Fig. 47a, red and blue spot). These two lasers impose the phase at the boundaries of the considered region forcing the condensate to assume two different phases. At a certain regime the system is broken
into two separated domains with a homogeneous phase value consistent with
that of the locking laser (red and blue spatial domains in Fig. 47a).
The effect of fixing the phase of the condensate through the use of two resonant beams is shown in the experimental interferogram reported in Fig. 47c.
In this panel it possible to note the presence of a second resonant beam with
79
80
josephson junction and vortices in a phase twisted polariton condensate
a different phase at a distance of about 60 µm from the other resonant spot
(top-right corner in Fig. 47c). This scheme is different from other phase imprinting methodology [139], in the sense that in our configuration the phase is
imposed only at the edges. Consequently the condensate phase is totally free to
re-arrange in the region between the two phase-locking points. The final result
of the simultaneous presence of two concurrent beams with a phase difference
is the establishment of two phase domains that can be expected as a result of
the competing phases of the locking lasers [140]. At this point it is interesting
to investigate how the phase twisting of the condensate reacts to a variation of
its density, keeping constant the intensity of the resonant locking lasers.
5.2.2
The phase boundary for different condensate densities
The density of the extended condensate can be tuned by increasing the power
of the nonresonant pump (as it is shown in Sec. 3.2.3). In this case we performed
all these measurements without changing the intensity of the two phase-locking
beams but keeping enough resonant polaritons to fix the phase of the condensate. The distance from the nonresonant pumping region (about 60 µm),
together with the diluite density of the condensate, ensures that even at the
higher condensate densities the resonant pumps are injecting the same low
quantity of polaritons in the cavity. Furthermore, we should stress that our
photoluminescence measurements are detecting the steady state of the system,
averaging in time. This is fundamentally due to the time-integrated detection
(few ms) performed by the CCD camera. Therefore, while the microscopic evolution of the system may change in different realisations depending on the possibly slightly different initial conditions, the more stable steady state solution
is naturally captured in the experiments.
When the polariton density is below the critical density threshold for the
stimulated scattering in the lowest state of the LPB (dth = 0.4 µm−2 ), the
two resonant beams cannot produce any locking of the polariton phase, as
there is no homogeneous phase to lock, as can be seen by the absence of a
uniform phase outside of the resonant spots in Fig. 48a. On the other hand,
with the polariton density above the condensation threshold (Fig. 48b-d), the
double phase locking of the condensate is allowed. In particular, for densities
just above the threshold, the hierarchy of excitations induced by the phase imprinting (the phase difference at the boundaries of the considered domain is of
about δφ ≈ π) ends up with a neat separation between two regions of the condensate. These spatial regions are kept separated through a sinusoidal shaped
one-dimensional interface for an extension of about 30 µm as shown in Fig. 48b
when d = 1.5dth . The phase profile across the junction (green line in Fig. 48e)
shows a steep phase jump of φ ≈ π, with a corresponding depletion in the den-
5.2 the experiment
3
2
1
0
-1
-2
-3
Phase
3
c
d
0
-3
b
Normalized counts (a.u.)
a
e
f
1
0.50
0.10
0.05
0
5
10
15 20 25
Space (μm)
30 35
Figure 48: a-d Two dimensional phase of the condensate with twisted boundary condition for increasing intensities of the nonresonant pump in a region of about
40X35µm. a, Below condensation threshold (d = 0.5dth ). b, (d = 1.5dth ). c,
(d = 2.0dth ). d, (d = 2.5dth ). e, Phase profiles along the vertical, dashed-red
lines in (a, b, c, d), in blue, green, red and yellow respectively. f, Densities
profiles extracted from the interferogram corresponding to the phase map
shown in a, b, c, d, respectively, with the same color scheme and along the
same vertical cross section as those used in panel e.
sity profile as shown by the green line in Fig. 48f. This peculiar profile shows
the spontaneous appearance of a dark soliton-like structure in the same position of the barrier between the two phase domains. For increasing densities,
this solitonic junction fragments into shorter segments with smoother phase
and density profiles. This pieces of the original barrier are interrupted by vortices along the nodal line (Fig. 48c, red line in Fig. 48e and Fig. 48f). Moreover,
while the phase jump between the two domains tends to be less abrupt, at the
same time the dark soliton in the density becomes less deep (Fig. 48e,f).
This particular shape of the density reduction, less deep than in the case of
the dark soliton, is called grey soliton. The existence of this kind of excitations
is due to the fact that, when an external source of dissipation is present in the
system, dark solitons tend to persist as local minimum in the condensate density, even if as a less strong density depletion, precisely the grey soliton. The
presence of a stable spatial soliton with a wiggling nodal line, as in Fig. 48b,c,
is a remarkable topological feature and constitutes a natural realisation of a
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82
josephson junction and vortices in a phase twisted polariton condensate
long bosonic Josephson junction in polariton condensates [141]. The solitonic
structure is characterized by a strong reduction of the order parameter (the amplitude of the condensate wavefunction), which acts as an analog of the normal
barrier in the superconducting-normal-superconducting (SNS) Jospehson junction, and separates two extended regions of the condensate with well defined
phases. As for superconductors, if some particles can tunnel from one side to
the other, the junction becomes a grey soliton and the phase difference at the
junction is reduced to less than π.
Eventually, at the highest pumping powers employed in these experiments,
the junction completely collapses. Only a shallow phase gradient remains all
across the condensate (Fig. 48d, yellow line in Fig. 48e and Fig. 48f). At these
densities (d ≈ 2.5dth ), the phase stiffness is high enough to avoid jumps in the
phase and density depletions. The superfluid character of the two-dimensional
condensate is completely restored and only a macroscopic coherent phase can
survive [140].
In analogy to what was observed in atomic Bose-Einstein condensates, optical nonlinear waves and bilayer excitonic junctions [142], the formation of
a soliton is an expected solution of the imposed phase boundary condition
[143]. Usually, in these physical systems when a two-dimensional geometry is
provided, a transversal modulation instability–the so-called snaking instability–
tends to induce a transformation of dark solitons into vortex ring, vortex dipoles,
or even into more complex dynamics [144–146]. From this perspective, even if
the topological nature of the barrier could partially explain the relative robustness of this experimental realization against the noise, it remains utterly interesting the possibility to observe such a dynamical effect with a system in the
steady state configuration. In this sense the stability of such a topological excitation in a two-dimensional polariton condensate is at the same time interesting
and surprising.
5.2.3
Josephson vortices and fluxes through the barrier
From the phase gradient it is possible to extract the information about the
fluxes through the barrier between the two phase domains.
The analysis of this quantity reveals interesting features about the topology
of the solitonic barrier and is reported in Fig. 49. The spatial regions analised
here are indicated in Fig. 49a,b with dashed white rectangles. As evidenced
in Fig. 49c, the flux across the phase jump is not uniform all along the solitonic stripe. On the contrary a point is present where the sign of the flux is
even inverted, going downwards in Fig. 49c. Indeed, as a consequence of the
polariton currents across the junction, accumulation/depletion regions with
slightly increased/decreased densities are expected close to the grey soliton.
5.2 the experiment
a
b
c
d
Phase
2
1
0
-1
e
-2
0
5
Space (μm)
10
15
Figure 49: Spatial barrier and fluxes. a, b Detail of the spatial barrier in the region of
the phase jump (≈ 30 × 16µm) for power regimes as in Fig. 48b, c with the
appearance of a snaking like configuration. c, Contour plot of the phase in
the dashed-white rectangle shown in a, with the velocity streamlines indicating the overall upwards flow and the inversion point with the formation of
Josephson vortices. d, Contour plot of the phase in the dashed-white rectangle shown in b with the velocity streamlines marking the inverted currents in
nearby domains of the junction linked by Josephson vortices. e, Phase profile
along the dashed lines in d, showing the inverted currents around a Josephson vortex.
83
84
josephson junction and vortices in a phase twisted polariton condensate
If the currents are strong enough, these accumulation/depletion points, at opposite sides of the junction, can induce a counterflow, that tends to cross the
junction in the opposite direction. To explain the inversion of the current, we
have to consider that the energy stored in the twisted phase configuration can
be partially dissipated through the nucleation of vortex-antivortex pairs in the
barrier, in analogy to the formation of Josephson vortices in superconductor
Josephson junctions or in atomic Josephson junctions [133–135]. This peculiar
behaviour of the vorticity is evident in Fig. 49c. Actually, at this regime the
single nodal line is disrupted: a vortex and an antivortex pair faces up to compensate the flux going down across the breakup in the junction. This is clearly
shown in the image with, on the left and on the right, an upward flux, while
a downward flux appears at the center between the vortices, where the nodal
line is snapped.
When the density of the condensate is higher, the separation between the
vortex and the antivortex increases and the nodal line breaks also at different
points, as shown in Fig. 49b. While in Fig. 49a the overall polariton flux points
upward, at the regime shown in Fig. 49b opposite polariton currents across the
junction are present with a comparable magnitude (Fig. 49d). The solitonic features in the density and phase profiles are still visible, even if less pronounced,
but now the nodal line assumes a more complex structure. What was a single
line at lower densities, now is made up of fragments with inverted phase difference stitched by the presence of vortices at each conjunction point (Fig. 49d).
As reported in Fig. 49e, the phase step is inverted, going from φ ≈ π to φ ≈ −π
in accordance to the inverted currents in nearby domains of the junction. This
topological structure is sustained by the presence of a vortex and an anti-vortex
alternating along the nodal line of the grey soliton, allowing energy dissipation
in response to the imposed twisted-phase boundaries.
The presented phenomena bear similarities to the snaking instability of dark
solitons. This is observed in many nonlinear two-dimensional systems [147,
148]. Differently respect to our case, snaking instability in conservative systems
develops in time, with the time evolution starting with a dark soliton state
that is prepared intentionally, eg. with an accurate phase imprinting process.
The temporal dynamics in this kind of systems can be viewed as a gradual
dissipation of the excess of energy contained in the initial excited state.
In contrast, in our driven-dissipative system a stable dark soliton state appears spontaneously, simply in reponse to the application of the external resonant phase at the boundary between two regions of homogeneous phase. Moreover, instead of evolving in time, the snaking instability changes in the “energy”
domain, with the increase in the density of the condensate.
5.3 the model
Phase
2
0
-2
c
b
d
e
Normalized counts (a.u.)
a
10
1
0.1
0.01
f
-10
0
10
Space (μm)
Figure 50: a-d Simulated two dimensional phase of the condensate with twisted boundary conditions for increasing intensities of the nonresonant pump. a, Below condensation threshold (P < Pth ). b, (d = 4.5Pth ). c, (d = 6.0Pth ). d,
(d = 8.0Pth ). e, Phase profiles along the vertical, dashed-red lines in (a, b, c,
d), in blue, green, red and yellow, respectively. f, Density profiles extracted
from the interferogram corresponding to the phase map shown in a, b, c,
d, respectively, with the same color order and along the same vertical cross
section as those used in panel e.
5.3
the model
In this case, we reproduced numerically all the regimes observed in the experiments regarding the formation and annihilation of the coherent structures.
This is realized by using the complex Ginzburg-Landau equation, modified in
order to describe a system in which the exciton reservoir is spatially separated
from the exciton-polariton condensate, with the same model used in numerical
simulations reported in Ch. 4.
5.3.1
The two-dimensional phase map in the numerical simulations
The two-dimensional phase maps extracted from the numerical simulations
with two lasers imposing the phase at the boundaries of the considered region
are reported in Fig. 50a-d. As in the experiments here we show the results of
the simulations for different pumping powers of the non-resonant laser that
populates the condensate. This results reproduce qualitatively the formation of
85
86
josephson junction and vortices in a phase twisted polariton condensate
the solitonic barrier separating two domains of homogeneous phase, together
with the appearance of vortices and finally, at higher condensate densities, the
destruction of this barrier and the establishment of an almost flat phase. In
these simulations we do not assume any particular form of imposed disorder
and, as a consequence, the soliton on the barrier is not pinned to the same
position in the plane of the sample as happens in the experiment. Moreover,
it is oriented roughly perpendicularly to the line connecting the two resonant
pumping lasers. The results here reported represent the phase of the wavefunction after long time of evolution, when the initial transient effects have washed
out and a steady state is established. Through these simulations we are able
to study the influence of the disorder on the formation of the barrier and its
characterstics. In particular, we found that in the total absence of a disorder
potential, the positions of solitons and vortices are not fixed and fluctuate. For
this reason, by averaging of different realizations it is impossible to observe the
formation of the phase jump related to the presence of the barrier. However,
a weak disorder is enough to pin the topological excitations in fixed positions
on the sample plane, making possible the observation of the soliton within the
barrier. In Fig. 50e, f the cross sections of the phase and the density profiles
along the red dashed lines in Fig. 50a-d are shown. Remarkably, a density dip
(Fig. 50f) is present in the same spatial position of the phase jump in Fig. 50e as
in the experimental measurements. The shift in the position of the soliton for
different powers is due to small differences in the pinning potential.
5.4
conclusions
In conclusion, in this chapter we showed that an extended coherent state, far
from the exciton reservoir can provide a sufficiently stable configuration allowing the observation of phenomena related to the application of a non-local
phase imprinting.
We observed the phase locking mechanism through which it is possible to
directly manipulate the phase of the condensate even in a wide region of about
40X35µm. This is made possible by using an external resonant laser imposing
the phase at the boundary of the considered domain. By applying two external
resonant lasers with a different phase (of about δ ≈ π) we were able to twist
the phase of the condensate that at a certain regime breaks into two domains
of homogeneous phase. The contact region between these two domains contains intringuing physical effects as the bosonic analogue of a long Josephson
junction together with the nucleation of vortices confined inside this structure,
the so-called Josephson vortices. The presence of vortex anti-vortex pairs is
confirmed by analyzing the fluxes through the barrier. The peculiar sinusoidal
spatial pattern assumed by the interface is described as the analogue, in the en-
5.4 conclusions
ergy domain, to the temporal dynamics of the snaking instability. On the other
hand, by increasing the density of the condensate this barrier starts to collapse
into smaller fragments with higher proliferation of vortex and anti-vortex pairs
at the edges of the small fragments of the initial domain wall. The appearance
of a single homogeneous phase in the whole condensate at higher densities
confirms that the superfluid behaviour of the fluid is totally restored.
In the next chapter we will see in which way the polarization pattern can
change when the polariton fluid is immersed in an external magnetic field
applied in the same plane of the propagation.
87
6
POLARITON FLUID IN AN EXTERNAL MAGNETIC FIELD
6.1
introduction
The remarkable progresses in the control of matter-light interaction in semiconductor optical microcavities have made it possible to design a new generation
of optoelectronic devices [149–155].
Precisely in this perspective, one other peculiar property of polaritons is that
they have a spin degree of freedom inherited from the photon chirality and
an exciton spin angular momentum that shows long coherence time and the
possibility to be actively manipulated by external fields through the excitonic
component. [156]. This additional feature significantly broadens the range of
their possible applications to include what is known under the name of spinoptronics. By now, there are already several implemented concepts in the form
of such spinoptionic devices as the “Datta and Das” spin transistor [153, 157],
the polaritonic analogue of a Berry-phase interferometer [154], and the excitonpolariton spin switch [155]. In all these realizations, the main capability is the
direct control of the spin of polaritons using both internal and external factors
to affect their polarization properties.
From a fundamental point of view, the dominant effect on the polariton spin
dynamics is the optical spin Hall effect (OSHE) [158–160].
The spin Hall effect is a physical phenomenon predicted by the russian physicists Mikhail I. Dyakonov and Vladimir I. Perel in 1971. It is related to the particles transport and consists of the appearance of a spin accumulation on the
lateral sides of a sample plane in which there is a motion of electric currents.
This spin current is then perpendicular to the direction of propagation of the
carriers. In a work by Kavokin et al. [161] in 2005, the authors predicted the
existence of an exciton-polariton counterpart of this effect which lies in the separation of the spin-polarized quasiparticles in both real and momentum space.
Basically, it originates from the longitudinal-transverse (LT) splitting of exciton polariton states. The effect of this splitting can be described by an effective
magnetic field, strongly dependent on the direction of the quasi-particle propagation and its velocity. This field produces a precession of the polariton spin as
polaritons propagate in the cavity plane. It was observed experimentally for the
first time in a work by Leyder et al. [162] in 2007 in a high quality GaAs/AlGaAs
microcavity with a polariton propagation distance of about 100µm.
89
90
polariton fluid in an external magnetic field
This effect, although being an interesting phenomenon, can also bring some
problems, since polarization is a key parameter in polariton devices. The control of such an effective magnetic field, which in turn directly affects the polariton state is of fundamental importance. One possibility is the use of an external
magnetic field with the most common geometry, perpendicular to the plane of
the sample. In this configuration, also called Faraday geometry, the studies of
the influence of the magnetic field on the polariton dispersion [163], coherence
properties [164] as well as on the spin textures in excitonic [165] and polaritonic [166] systems have been reported. Hovewer, such geometry cannot be
effective on the control of the OSHE [167–170].
By contrast, the effect on exciton-polariton spin dynamics in the Voigt geometry, with the external magnetic field directed in the plane of the quantum well,
has not been investigated in polariton systems. This is because, apart of the
experimental complications, the in-plane field does not directly couple with
the polariton pseudospin and one may expect its effect to be quite minor. In
this chapter we report the possibility to obtain a significant effect of the external magnetic field in this configuration and to use it to control directly the
polariton spin transport both in a confined one-dimensional (1D) geometry
and in the whole two-dimensional (2D) plane of the cavity. In that respect, we
show the possibility to affect and even totally suppress the OSHE for polaritons
propagating in a given direction by properly choosing the magnitude and the
orientation of the applied field.
Furthermore, using a two-dimensional radially propagating polariton cloud,
we observed that the application of the external magnetic field causes a stretching of the circular pseudospin patterns in the axis normal to the direction of the
magnetic field and a contraction in the same orientation of the external field.
From the point of view of applications, the possibility to completely control the
intensity oscillations related to the presence of the OSHE allows one to remove
the residual density modulations that appear during polariton propagation and
that would be detrimental for the elaboration of the signals inside the devices
[157].
Finally, it is important to stress that such nonlinear effects were hindered
in previous samples with lower quality factor and number of quantum wells.
Indeed, in that cases the inhomogeneity in the sample led to the localization
of the condensate within the excitation spot or destroyed the propagation of
polaritons with the presence of defects.
6.2 the effect of the magnetic field in the one-dimensional geometry
6.2
6.2.1
the effect of the magnetic field in the one-dimensional geometry
Resonant confined one-dimensional propagation
With the aim to study the influence of an external magnetic field on the polariton spin distribution, we performed a first type of measurements by using a
resonant excitation scheme. We injected polariton by matching the LPB energy
with a wavevector of about 1 µm−1 and a polariton velocity of v = 1.52 µm/ps.
The results of these measurements are shown in Fig. 51.
In order to investigate the 1D case, this flow is kept confined into natural
misfits dislocations present along the [11̄0] axis of the sample. The 1D confinement is relatively shallow but allows to observe the polariton propagation up
to 400 µm from the laser injection point with a negligible spread in the perpendicular direction (Fig. 51a, b). During the propagation the spin distribution of
polaritons is modified by the presence of the magnetic field. These spin oscillations are measured by selectively detecting the emission intensity co- (σ+ ) and
cross-polarised (σ− ) with respect to the exciting laser. The results of these two
measurements are reported in Fig. 51a, b for the two different polarizations,
respectively, in the case with B = 9 T. The circular polarization degree, Pc is
then obtained as:
Pc = (Iσ+ − Iσ− )/(Iσ+ + Iσ− )
(44)
In this measurement the magnetic field, applied in the plane of the cavity, is
perpendicular to the direction of the polariton propagation, spanning a range
of intensities from 0 to 9 T.
In Fig. 51c, a schematic representation of the behaviour relative to the polariton spin is reported. The v vector indicates the propagation directed orthogonally to the magnetic field (B). Thanks to the presence of this field, external
or intrinsic, during the propagation the spin (S) redistributes changing the total degree of circular polarization. As a consequence, it assumes a sinusoidal
shape with an amplitude that exponentially decays when the distance from
the injection point gradually increases. These pronounced oscillations in the
circular degree of polarization are reported in Fig. 51d, e as a function of the
spatial coordinate. The main effect of the external magnetic field is to change
the frequency of the spatial oscillations in Pc that increases quadratically with
the intensity of B, as it is shown in Fig. 51f.
In Fig. 52 the results of the measurement performed with the sample rotated
90◦ with respect to the previous case are shown. Now, B k v and the propagation takes place in a different dislocation with a velocity of about 0.3µm/ps.
This configuration reveals an interesting polarization pattern (Fig. 52a, f). In-
91
polariton fluid in an external magnetic field
50 μm
a
c
Pol.degree
b
d
0.5
0
Pol.degree
-0.5
v
a
v
B
e
0.5
0
-0.5
100
B
σ
+
σ-
200 300 400
Space (μm)
0.040
0.038
K (μm-1)
92
0.036
0.034
0.032
0.030
f
0
2
4
Magnetic field (T)
6
8
Figure 51: Effect of magnetic field (1D). a, b, 1D propagation inside a misfit dislocation
(co-polarized and cross-polarized). c, Sketch of the experimental setup with
the external magnetic field (B) in the quantum well plane (Voigt configuration) and the polariton propagation in the same plane. d,e, Spatial dynamics
of Pc for B = 0 and 9 T, respectively. Plots starts at 50 µm from the excitation
spot, to avoid the detection of scattered light from the laser itself. Fitting function (red line): Ae−bx sin(kx + φ). f, Spatial frequency of Pc oscillations κ for
different magnetic field values with a quadratic (blue line) fitting. Propagation velocity is about 1.52 µm/ps and the external magnetic field is oriented
orthogonal to the propagation direction.
6.2 the effect of the magnetic field in the one-dimensional geometry
deed, instead of the monotonic behaviour of the spatial oscillations frequency
reported in Fig 51d, e, here, increasing the intensity of the magnetic field, the
frequency of the oscillations first decreases (up to 5 T ) and then starts to increase assuming a constant positive offset in the co-polarised component.
6.2.2
The model for the resonant propagation
Pol.degree
The observed effects can be quantitatively described within a pseudospin model
parametrizing the polariton spin density matrix through the vector Sk whose
z component describes the circular polarization degree of the particles and the
in-plane components characterize the linear polarization degree in two sets of
axes. The results of simulations using this model are reported in blue lines in
Fig. 52a-f.
50 μm
0.5
Pol.degree
h
0.0
-0.5
a
b
0.5
0.0
-0.5
c
Pol.degree
g
d
0.5
0.0
B
-0.5
e
f
100 200 300 400
Space (μm)
100 200 300 400
Space (μm)
v
σ+
B
v
σ-
Figure 52: Effect of magnetic field (1D), magnetic field parallel to the propagation a,
b, c, d, e, f, Spatial dynamics of Pc for B = (0, 4, 6, 7, 8, 9) T, respectively, and
with the external magnetic field oriented parallel to the propagation direction. Blue lines represent the result from the simulations. g, 1D propagation
inside a misfit dislocation (co-polarized). h, 1D propagation inside a misfit
dislocation (cross-polarized). Propagation velocity is about 0.3 µm/ps.
93
94
polariton fluid in an external magnetic field
The equation of motion for the pseudospin of polaritons that are ballistically
propagating in the k state can be written as in [158]:
∂Sk
+ Sk × Ωk + γs Sk = 0,
∂t
(45)
where the set of axes is such that x k [110], y k [11̄0] and z k [001], with Ωk the
effective pseudospin precession frequency and the last term accounting for the
spin relaxation processes with the rate γs . The effective precession frequency
components are:
.
Ωk,x = ∆LT (k2x − k2y ) k2 − β(B2x − B2y ),
(46a)
Ωk,y = 2∆LT kx ky / k2 − 2βBx By ,
(46b)
Ωk,z = αSz ,
(46c)
and contain contributions from the LT splitting of polariton modes, ∆LT [171],
the splitting in the linear polarization components due to the applied external magnetic field in the cavity plane βB2x,y and the so-called self-induced
Larmor precession of the polariton pseudospin due to the polariton-polariton
interactions (α is the interaction constant) treated here within the mean-field
approach [156, 172, 173]. Substantially, the in-plane components of the effective field contain the contributions quadratic in the magnetic field. The form
of these terms follows from the symmetry arguments since the quadratic combinations Bi Bj and ki kj with i, j = x, y transform in the same way as in the
isotropic approximation and the effects of C2v point symmetry of the studied
structure on the magnetic-field induced terms are disregarded. Microscopically,
the parameter β results from the magneto-induced mixing of polariton states
and dark (spin-forbidden) excitons with the additional contribution from the
diamagnetic effect, in the same way that acts for excitons in quantum wells and
quantum dots [174, 175].
Eq. 45 can be solved analytically for various configurations relevant from
an experimental point of view. We first set α = 0, i.e. we neglect polaritonpolariton interactions, and we assume that polaritons propagate along the yaxis and B is either parallel or perpendicular to their velocity. From Eqs. 46, it
follows that Ωy ≡ 0 and that:
Sz (y) = Sz0 cos (κy) e− y/`s .
Here the wavenumber of the spatial oscillation in the pseudospin is:
h
i
κ = ∆LT k2 + β(B2x − B2y ) /v,
(47)
(48)
6.2 the effect of the magnetic field in the one-dimensional geometry
and the pseudospin decay length is `s = v/γs , where v = k/m is the
polariton velocity, with m being the polariton effective mass. For positive β
and B ⊥ k (which corresponds to the experimental measurement reported in
Fig. 51), Eq. 48 predicts a monotonic increase of the spatial frequency of the
spin oscillation κ with the square of the magnetic field magnitude, B2 , in full
agreement with the result in Fig. 51f.
On the other hand, when B||k (case of Fig. 52) the dependence κ(B) is more
complicated, as it follows from Eq. 48. In this case, the behaviour of κ(B) is not
a monotonic function, as it was experimentally observed. Actually, with the
critical value of magnetic field Bc defined as:
q
Bc =
∆LT k2 /β
(49)
it follows that, when B < Bc , κ decreases and then increases with further increase in the field. Under the condition B = Bc the complete stop of oscillations
is expected due to the suppression of the LT-splitting by the external magnetic
field, similarly to the field–induced suppression of exciton anisotropic splitting
that appears in quantum dots [174, 175]. Qualitatively, this result is in agreement with the experimental data presented in Fig. 52. Indeed, while Eqs. 47
and 48 also quantitatively explain the results in Fig. 51, the linear model used
up to now is not sufficient to describe all the peculiarities of the polariton polarization dynamics shown in Fig. 52. This is because the nonlinearity due to
spin-dependent polariton-polariton interactions becomes of particular importance with B ≈ Bc . Also, in the experimental geometry with B k k, the polariton propagation velocity is relatively small, v ≈ 0.3 µm/ps, which results in the
weaker manifestation of the influence of LT-splitting. Indeed, one of the peculiarities seen in Fig. 52 and not explained through this model is the positive
offset in the Sz (y)-dependence. This effect is the manifestation of the presence
of the third component of the effective magnetic field Ωz ∝ Sz , which tilts
the pseudospin precession axis towards the z-axis and suppresses partially the
effect of the LT-splitting [166, 176].
The results of the calculations for this configuration (blue lines in Fig. 52a-f)
that take into account this correction closely match the experimental points.
Note that, as expected, the self-induced Zeeman splitting due to the polaritonpolariton interaction decreases at lower densities. Furthermore, the dynamics
of Sz (y) returns to the harmonic character at the distance of about 300 µm, as
visible at higher distances in Fig. 52a-f.
95
polariton fluid in an external magnetic field
6.3
the effect of the magnetic field in the two-dimensional geometry
Redistribution of the top condensate polarization pattern
6.3.1
In order to understand how the two-dimensional spatial distribution of the
polarization is affected by the presence of an in-plane external magnetic field,
we performed a different experiment using the condensate described in Ch. 2.
Indeed, a fundamental characteristic of this condensate is that despite it is
30 μm
0.5
0
-0.5
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
a
b
c
d
e
f
g
0
20
40
60
Space(μm)
80
100 0
20
40
60
Space(μm)
80
h
100
Frequency (μm-1)
Pol. degree
96
0.115
0.110
0.105
0.100
0.095
0.090
i
0 2 4 6 8
Magnetic field (T)
Figure 53: Pc patterns with B k y for the magnetic field intensities of (0, 7, 9) T and
the initial polarizations σ+ (a,b,c) and σ− (d,e,f). The propagation velocity
is about 1.8 µm/ps g, Cross section of Sz along the vertical directions at 0 T
(red line) and 9 T (blue line) as indicated in a and c, respectively. The fitting
function is Ae−bx sin(kx + φ). h, Same as in g but for σ− (see d and f). i,
Spatial frequency κ extracted as best fit to the data for σ+ (blue) and σ− (red)
polarized pump.
6.3 the effect of the magnetic field in the two-dimensional geometry
obtained with a nonresonant circularly polarized excitation pump, it inherits
about 40% of circular polarization from the laser [159].
The condensation density threshold, as described before, is reached in a region within the laser spot, blueshifted of about 4 meV from the bottom of the
lower polariton branch. From this central region, this polariton flow is ballistically expelled and is free to radially propagate in the plane of the cavity outside
the excitation spot region, with an acceleration due to the gradient in the potential resulting from the blueshift under the excitation spot (Sec. 2.3.2) [177].
In this peculiar configuration polaritons can reach a speed of about 2 µm/ps.
The experimental Pc for different magnetic field intensities is reported in
Fig. 53a-f. A small asymmetry in the distribution function already present at
B = 0 T is initially compensated, and then enhanced when the external magnetic field is increased, as shown in Fig. 53a-c. Indeed, the pattern was elliptical at B = 0 T with the horizontal axis greater than the vertical. However,
by increasing the applied magnetic field, this axis reverses and the ellipse gets
rotated.
Therefore, by changing the polarization degree of the exciting laser from
right-circular to left-circular, we change the relative orientation of the polariton
spin with respect to the magnetic field. In Fig. 53d-f, we show that in this
case the long eccentricity axis at B = 9 T is oriented roughly perpendicular to
that one of Fig. 53c. This is confirmed by the opposite behaviour of the spatial
frequency respect to the vertical and horizontal axis as reported in panels of
Fig. 53g,h,i.
6.3.2
The model for the nonresonant propagation
The resulting Pc of the two-dimensional polariton expansion can be described
by the same Eq. 45 applied to all k around the elastic circle of radius k. Nonresonant excitation of the incoherent exciton reservoir results in a different polariton state population dynamics with respect to the one-dimensional case with
resonant excitation. The model used to reproduce the nonresonant excitation
of polaritons is:
Np ∝ exp [Γ0 ] exp [−Γ (t) − γt]
(50)
where Γ (t) characterises the dynamical pumping of the polariton condensate
coming from the exciton reservoir, and Γ0 = Γ (0).
Fig. 54a-f illustrates the 2D expansion of the Pc theoretically predicted from
the model. The initial polarizations and the values of the magnetic field B
shown in this case correspond to those in Fig. 53a-f. In order to distinguish
the influence of different effects on the shape of the polarization pattern it is
possible to consider that in the presence of only the LT-splitting, the Pc pat-
97
98
polariton fluid in an external magnetic field
30 μm
a
b
c
d
e
f
Figure 54: Model of the 2D expansion of Pc in real space. The magnetic field B||y. Values
of B and initial polarization σ are the same as those in the experimental
panels. The values of the parameters used in the model are the following:
∆LT = 200 meV × µm2 , α = 7.5 µeV/T 2 , Bc = 6 T, δan = −αB2c . βnp0 = 70 µeV
and ` = 270 µm, `s = 9 mm.
terns in real space are rotationally invariant. Indeed, the effect of the built-in
anisotropy splitting is to squeeze the patterns in the x direction so that these become elliptical even in the absence of the external magnetic field. The presence
of linear and diagonal components of the polarization (Sx,y ) leads to a slight
tilt of the ellipse.
Moreover, the external magnetic field tends to change the spatial frequency
κ with a different effect depending on the relative direction of that of propagation, as it is shown in Fig. 53. According to Eq. 48, the absolute value of
the spatial frequency κ(ky , B) decreases with increasing B until the magnetic
field compensates the effect of both the LT and the anisotropy splitting while a
further increase of B leads to the increase of the spatial frequency. In the case
with B oriented orthogonal to the propagation, the spatial frequency κ(kx , B)
monotonically increases with higher magnetic field.
6.3 the effect of the magnetic field in the two-dimensional geometry
6.3.3
The effect on the linear polarization
A strong effect of the magnetic field is also visible when the degree of linear
polarization is investigated. The experimental results of the application of the
0.3
30 μm
0
-0.3
a
b
c
d
e
f
Figure 55: Experimental linear polarization patterns in the real space. (a)–(c) correspond
to the magnitude of the magnetic field (0, 5, 8) T, respectively; B k y. (d) –(e)
Simulation of the 2D expansion of the degree of linear polarization in real
space. The parameters in this simulation are the same as in Fig. 19.
external field are reported in Fig 55a, b, c. It is possible to note that the emission
is polarized into quadrants with a progressive focusing of the linearly polarized
content of the top energy condensate towards the center of the excitation region
together with a rotation of the entire pattern around the point at the center of
the excitation area. The same model used to simulate the effects in the circular
polarization is not able to capture the physical processes behind this effect.
In Fig. 55d, e, f the linear polarization pattern in two dimensional maps are
reported for the same spatial extension of the experimental figures and with
the same values of the external magnetic field. In this case the application of
the field does not seem to determine an effect on the linear polarization pattern,
differently from what is found in the experiments.
Probably, the explanation of this type of influence on the linear polarization
pattern requires both a deeper experimental investigation ( with the analysis of
the linear polarization pattern in the one-dimensional case with the resonant
propagation), and a more general theoretical model also able to capture the
physics behind these processes.
99
100
polariton fluid in an external magnetic field
6.4
conclusions
In this chapter we investigated how the presence of an external magnetic field
applied in the Voigt geometry can affect the polarization properties of the polariton fluid. We injected polaritons inside the cavity both resonantly and nonresonantly. With these measurements, we observed the symmetry breaking related to the presence of the magnetic field and the dependency of these effects
on the direction of B relatively to the orientation of the propagation.
In particular, here we showed the ability to suppress the optical spin Hall
effect through the tuning of this external field in the one-dimensional confined
case and when polaritons are resonantly injected in the same direction of B.
Furthermore, with the polariton fluid propagating orthogonally respect to the
B direction, it is possible to change the spin beat frequency with a quadratic
dependence on the applied field.
Besides, in the two-dimensional expansion of a condensate of polaritons
formed by a non-resonant laser, the in-plane magnetic field induces an additional polarization anisotropy in the structure that manifests itself as a deformation of the pseudospin patterns in the real space with the excitation pump
circularly polarized. The described model reproduces all these observations.
Finally, by detecting the linear polarization degree in the same configuration
as before, an effective focusing and a rotation of the linear polarization pattern
is observed but the model is not able to simulate this behaviour suggesting
how a deeper investigation of this kind of effects is necessary from both the
experimental and the theoretical perspectives.
7
CONCLUSIONS
In this thesis, we have studied the polariton quantum fluid with particular
emphasis on polariton condensation. With this investigation, we believe to have
finally shed light on the true nature of a condensate in solid state systems.
In the first part of the thesis, we described the spontaneous formation of a
macroscopic two-dimensional polariton condensate with no energy blueshift,
in the ground state. We studied the dynamics of establishment and the peculiar physical properties of this extended state. We found how a mixture of
both propagation for long distances of states with higher energies, coming
from the blueshifted gaussian potential landscape formed under the pumping spot, and phonon-assisted mechanisms of energy relaxation, allows the
onset of this homogeneous phase. Thanks to this new dynamics of formation,
above a critical density, the condensate forms at k ≈ 0 µm−1 , at the bottom
of the band. We found that it covered an exceptionally large spatial extension
of about 30000 µm2 in a “reservoir-free” region, far from the excitation spot.
We demonstrated our system to be the closest realization so far of an infinite,
not confined, two-dimensional polariton condensate. It is of the utmost importance that for the first time, such a state was observed to be not affected by
the instabilities related to the presence of the exciton reservoir and with dilute,
tunable densities. The formation of this state was made possible thanks to the
exceptionally long polariton lifetime inside the cavity and to the high level of
spatial homogeneity in the sample.
In the second part, we focused on the phase transition that is possible to observe in this new generation of polariton condensates. Normally, the collective
behaviour of exciton-polaritons in semiconductor microcavities was considered
to be at the edge between equilibrium and non-equilibrium phase transitions.
Moreover, despite the fact that polaritons have often been compared both to
atomic condensates and to photon lasers, in this investigation we were able
to demonstrate that the transition from an incoherent ensemble of states to
an ordered, homogeneous two-dimensional system can be very different from
the regime of lasing. In fact, by using a joint measurement of both the spatial
and temporal decay of correlations we were able to observe how the polariton
cloud in the condensate achieves a true thermal equilibrium at the bottom of
the lower polariton branch. We demonstrated how, at the same critical density of the condensate, a peculiar type of topological ordering was acting on
the system, the Berezinskii–Kosterlitz–Thouless (BKT) mechanism. All these
results gave us the opportunity to prove that the measurement of spatial corre-
101
102
conclusions
lations alone is not sufficient to establish whether an open/driven, dissipative
system can reach the thermal equilibrium and the BKT phase. Eventually we
demonstrated that these properties cannot be found in lasers and truly belong
to a different regime: the one of condensates.
In the final part of this thesis, we reported the possibility to manipulate the
phase of the condensate and the effect of an external magnetic field applied
directly in the plane of the propagation of a polariton fluid (Voigt geometry).
We described the mechanism of phase locking through which it is possible to
impose directly the phase of the condensate even in a wide region of about
40 × 35µm, by using an external laser tuned in resonance with the macroscopic
condensate. By applying two external resonant lasers with a different phase
(of about δ ≈ π) we were able to twist the single homogeneous phase of the
condensate. We found that at a certain regime the condensate breaks into two
domains of different constant phase. By studying in detail the contact region between these two domains, we discovered it contained intriguing physical effects
as the bosonic analogue of a long Josephson junction, together with the enucleation of vortices confined within this structure (Josephson vortices). Moreover,
the distinctive sinusoidal spatial pattern assumed by this interface can be described as the analogue, in the energy domain, of the temporal dynamics of the
snaking instabilities reported also in Fermionic systems.
Finally, we obtained interesting results by investigating the influence of an
external magnetic field applied in the Voigt geometry. In particular, we showed
how, by tuning the intensity of the field and by heading it in the same direction
of the polariton propagation (in this case the propagation was confined in a
one-dimensional misfit dislocation), it was possible to suppress the spin spatial
oscillations related to the optical spin Hall effect. Furthermore, with the polariton fluid propagating orthogonally respect to the direction of the magnetic
field, we were able to manipulate the frequency of the spin oscillations with
a quadratic dependence on the applied magnetic field. Additionally, we found
that, by using the two-dimensional expansion of a nonresonantly pumped condensate of polaritons circularly polarized, this in-plane magnetic field induces
an additional anisotropy in the polarization with a resulting deformation of the
pseudospin patterns in the real space.
As a conclusion, this work proves that polariton condensates posses unique
properties that make these systems extremely interesting to study and to work
with and paves the way to the study of phase transitions in polariton condensates with two-dimensional geometries. An example of an experimentally
unexplored field that can be investigated with this type of systems is given
by the extraction of the critical exponents of the phase transition due to the
Kibble-Zurek mechanism (KZM), with the related study of the dynamics and
formation of topological defects.
Part II
APPENDIX
8
THE PHYSICAL MODEL
8.1
the hydrodynamical model
Since the mechanism that creates these polaritons take place for |r| < σ, the
polaritons created by the pumping terms in Eq. 25 must be accounted by the
appropriate boundary condition in Eq. 27.
Let us consider now a time-dependent spectral expansion of the wavefunction as follows:
X
ψ(r, t) =
ψω (r, t)eiωt ,
(51)
ω
where we assume that ψω (r, t) evolves in time much slower than the oscillations given by eiωt . Next, we write ψω (r, t) in terms of density and phase:
p
ψω (r, t) = nω (r, t)eiφω (r,t) .
(52)
By plugging this ansatz in Eq. (27) and taking the imaginary part, we find:
ih
X
ei[ωt+φω ]
ω
−
√
∂√
∂
nω + i(ω + φω ) nω
∂t
∂t
=
√
√
√ i
h2 X i[ωt+φω ] h 2 √
e
∇ ( nω ) + 2i∇φω ∇( nω ) + i nω ∇2 φω − (∇φω )2 nω
2m ω
h X i[ωt+φω ] √
nω .
(53)
−i γ
e
2 ω
Now, we equate the terms in the sum with the same ω and take the imaginary
part of this equation to get:
√
√
∂√
h2 h √
nω = −
2i∇φω ∇( nω ) + i nω ∇2 φω − i γ nω ,
∂t
2m
2
(54)
which we can rewrite as an energy-resolved continuity equation:
∂
nω
∂t
= −∇ (nω v! ) − γnω ,
(55)
exp
where
v! =
h
∇φω .
m
(56)
105
106
the physical model
The suffix in the partial derivative accounts for the fact that this equation only
describes the hydrodynamics of expansion of the polariton fluid. We now describe the variation in nω due to the relaxation to lower energy modes mediated by phonon scattering.
A rate equation is usually written in k space, and if we take into account only
stimulated scattering, it reads as [77]:
∂
n
∂t k
=−
relax
X
Wk,k 0 nk nk 0 − (k ↔ k 0 ) .
(57)
k0
If the scattering is mediated by a phonon bath, Fermi’s golden rule gives the
following transition rate in k space [77]:
Lz (χk χk 0 ∆Ẽk,k 0 )2 2
B (qz )|De − Dh |2 × |nph (ωk 0 − ωk )|θ(∆Ẽk,k 0 − |k − k 0 |),
hρVu2 qz
(58)
where u is the longitudinal sound velocity, Lz is the quantum well width, V is
the crystal volume, ρ is the mass density of the solid, ∆Ẽk,k 0 = |ωk 0 − ωk |/hu,
qz is, for a given in plane momentum change k − k 0 , the momentum in the z
direction that mustq
be taken by the phonons for the scattering process to conWk,k 0 ≈
serve energy qz =
∆Ẽ2k,k 0 − |k − k 0 |2 , |nph (ω)| is the absolute of the phonon
density and B(q) is given by:
B(q) =
8π2
sin
Lz q(4π2 − L2z q2 )
Lz q
2
.
(59)
Since the experiment is performed on a polar symmetry, we can assume that
nk depends only on the module of k:
nk = nk ,
and we can perform the sum in k 0 in (57) going to the continuum limit
Z
Z
X
S
S
dk
dk
=
k dk dθ
→
x
y
(2π)2
(2π)2
0
k
Z
∂
S
nk
=−
k 0 Wk,k 0 nk nk 0 − (k ↔ k 0 ) dk 0 dθ
2
∂t
(2π)
relax
Then, writing:
(60)
(61)
(62)
Z 2π
Wk,k 0 =
0
dθ Wk,k 0
(63)
8.1 the hydrodynamical model
where we integrated the angular dependence
of the scattering rate, entering in
q
the expression of Wk,k 0 from qz =
∂
nk
∂t
relax
S
=−
(2π)2
Z
∆Ẽ2k,k 0 − k2 − k 0 2 + 2kk 0 cos(θ), we get:
k 0 Wk,k 0 − Wk 0 ,k nk nk 0 dk 0 .
(64)
Since the energy of the polaritons depends only on the modulus of the momentum, ωk ≈ h2 k2 /2mLP , we can write nk as a function of energy nω , and using
dω = h2 k dk/2mLP , write:
Z
∂
nω
= − Wω,ω 0 − Wω 0 ,ω nω nω 0 dω 0 ,
(65)
∂t
relax
where Wω,ω 0 = 2mLP S/(h2 4π2 )Wk(ω),k 0 (ω 0 ) . Finally, we write a joint equation for the time evolution of nω by joining both the hydrodynamics of expansion (Eq. (55)) and relaxation (Eq. (65)):
∂
dnω
=
nω
dt
∂t
+
exp
∂
nω
∂t
=
relax
1 ∂
(rnω vω ) − γnω
Zr ∂r
− Wω,ω 0 − Wω 0 ,ω nω nω 0 dω 0 ,
−
(66)
where we wrote the velocity as a vector field with radial component only v! =
vω ur . Our next approximation will be to assume that the velocity will be given
by vω ≈ (1/h)∂ωk /∂k.
8.1.1
The expansion and relaxation of the top energy states
To study the spatial profile resolved in energies, we will look now for steady
state solutions of equation (66) setting the time derivative to zero and getting
the following integro-differential equation in real space:
nω (r)
1
∂nω (r)
=−
+
− γnω (r)
∂r
r
vω
Z
− Wω,ω 0 − Wω 0 ,ω nω nω 0 dω 0 .
(67)
This we solve numerically by writing the integral again as a discrete sum:
∂nω (r)
nω (r)
1
=−
+
− γnω (r)
∂r
r
vω
#
X
−
W̃ω,ω 0 − W̃ω 0 ,ω nω nω 0 ,
(68)
ω0
107
108
the physical model
with
W̃ω,ω 0 = κ
(χω χω 0 ∆Ẽω,ω 0 )2
0
|eβ(ω −ω) − 1|
Z
B(qz )2
dθ,
qz
(69)
where ∆Ẽω,ω 0 = |ω 0 − ω|/(hu), and
κ=
2mLP S Lz
|De − Dh |∆ω 0 .
h2 (2π)2 hρVu2
(70)
9
BKT PHASE TRANSITION
The influence of the high energy condensate speed
9.0.1
As described in Chapter 2 the Gaussian blueshifted potential triggered expansion of polaritons, accelerated outwards from the injection spot, acts as an effective polariton reservoir for the condensate at the bottom of the dispersion.
It is important to take into account the velocities of the top energy condensate because it is possible to argue that the growth of the spatial coherence is
simply related to a higher velocity of the top condensate that, allowing larger
propagation distances, increases the coherence of the entire system.
a
3.0
2.5
vΜmps
3.
EmeV
b
2.
2.0
1.5
1.0
1.
0.5
0.0
0.
0
1
kΜm1 2
3
0.0
0.5
1.0
1.5
kΜm1 2.0
2.5
3.0
Figure 56: Expanding polariton velocities. a, Dispersion relation of polaritons including the central spot, the expanding polariton reservoir and the bottom state.
The blueshift corresponding to the excitation power used in Fig. 35 (a-f) is
indicated by blue dots. b, Group velocities corresponding to blue dots in a,
and to Fig. 35 (a, b, c, d, e, f,)
On the contrary, as reported in Fig. 56b the velocity of the expanding reservoir is almost constant in the range of powers considered and close to the maximum velocity achievable at this detuning. The blueshift under the laser spot
is around 3 meV, as shown in Fig. 56a, in the whole range of powers considered in Fig. 35. As a consequence, the corresponding group velocity is instead
slightly decreasing with power due to the nonparabolic curvature of the dispersion, as shown in Fig. 56b. This demonstrate that the increase of coherence in
109
110
bkt phase transition
the system is not related to the higher propagation velocity of the expanding
states, but is a genuine effect of the onset of a homogeneous phase.
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