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# Rabi oscillations of a quantum dot exciton coupled to acoustic phonons:
coherence and population readout

Daniel Wigger1  Christian Schneider2  Stefan Gerhardt2  Martin Kamp2  Sven
Höfling2,3  Tilmann Kuhn1  and Jacek Kasprzak4  1 Institut für
Festkörpertheorie, Universität Münster, 48149 Münster, Germany
2 Technische Physik, Universität Würzburg, 97074 Würzburg, Germany
3 SUPA, School of Physics and Astronomy, University of St. Andrews, St.
Andrews KY16 9SS, UK
4 Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble,
France<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

While the advanced coherent control of qubits is now routinely carried out in
low frequency (GHz) systems like single spins, it is far more challenging to
achieve for two-level systems in the optical domain. This is because the
latter evolve typically in the THz range, calling for tools of ultrafast,
coherent, nonlinear optics. Using four-wave mixing micro-spectroscopy, we here
measure the optically driven dynamics of a single exciton quantum state
confined in a semiconductor quantum dot. In a combined experimental and
theoretical approach, we reveal the intrinsic Rabi oscillation dynamics by
monitoring both central exciton quantities, i.e., its occupation and the
microscopic coherence, as resolved by the four-wave mixing technique. In the
frequency domain this oscillation generates the Autler-Townes splitting of the
light-exciton dressed states, directly seen in the four-wave mixing spectra.
We further demonstrate that the coupling to acoustic phonons strongly
influences the FWM dynamics on the picosecond timescale, because it leads to
transitions between the dressed states.

††journal: optica††articletype: Research Article

##  1 INTRODUCTION

The coherent control of individual spins and excitons in quantum dots (QDs)
has been the scope of semiconductor quantum optics since many years. To unveil
the potential lying in QDs and other single photon emitters for high-speed
optical quantum technology, it is urgent to bring the manipulation schemes to
perfection. This is nowadays approached using individual spins in electrically
defined [1] and epitaxial [2, 3] QDs, as well as in color centers in diamond
[4, 5], which is conditioned by setting up robust spin-photon interfacing.
This difficulty can be mitigated using QD excitons: because of their direct
coupling to light, the related coherent control protocols operate on the
picosecond (ps) timescale [6]. The context of ultrafast control points to the
subject of our work, focusing on optically driven coherent dynamics of the
exciton state efficiently coupled to longitudinal acoustic (LA) phonons of the
GaAs host matrix. Previous research has shown that by observing Ramsey fringes
[7, 8] the quantum state of single localized excitons can be navigated on the
Bloch sphere by varying the delay between two laser pulses with ps duration.
It has also been demonstrated that by increasing the applied laser power the
final exciton occupation after a single pulse can be adjusted [9, 10] by
performing Rabi rotations. Excitations with chirped laser pulses have been
used to realize robust state preparations via the so called rapid adiabatic
passage effect [11, 12, 13, 14, 15, 16, 17]. A pump-probe experiment [18] and
time-resolved resonance fluorescence [19] have been used to study the
optically driven Rabi oscillation dynamics of the QD exciton occupation. Rabi
oscillations have also been investigated in larger systems, i.e., in QD
ensembles [20, 21, 22], quantum wells [23, 24] and quantum dash ensembles
[25]. Note, that these Rabi oscillations have to be differentiated from Rabi
rotations [9, 26, 27, 10, 28, 1, 30, 31]. The oscillations directly correspond
to the dynamical aspect of the genuine Rabi problem [32]. Rabi rotations
instead reflect the final exciton state and thus lack direct visibility of the
temporal interplay between exciton and phonons.

We here propose and realize an original approach to monitor single exciton’s
Rabi oscillations in the time domain. Namely, we use the delay dependence of
four-wave mixing (FWM) signals to probe the dynamics of both central quantum
mechanical quantities of a single QD exciton, i.e., its occupation and its
microscopic coherence, which represent the entire quantum state or its Bloch
vector. Note that combined with the time-dependence of the emitted signal,
this delay dependence can be used to generate two-dimensional spectra. These
provide a rich playground to study spectral features of coupled and uncoupled
quantum systems [33, 34, 35, 36, 37, 38, 39]. Here, however, our focus is on
the temporal evolution of the exciton quantum state. Our dual insight into
Rabi oscillation dynamics of a single QD exciton represents a novel aspect and
is enabled by the FWM methodology. In a combined experimental and theoretical
study we investigate the dynamics of both quantities in FWM signals. When
increasing the intensities of the exciting laser pulses, oscillations on the
ps time scale build up during the optical driving. The measured oscillations
directly reflect Rabi oscillations of the Bloch vector of an individual QD
exciton. In exciton ensembles the isolation of a single Bloch vector has so
far not been realized, because one always deals with a bunch of Bloch vectors
undergoing different dynamics. Carrying out this study in the ultimate limit
of a single QD exciton allows for an investigation of the fundamental
interplay between the optically driven two-level system and its coupling to
the host lattice via the exciton-phonon interaction.

##  2 EXPERIMENT AND THEORY

To measure FWM on individual QDs, we employ a three-beam heterodyne spectral
interferometry setup optimized for the near-infrared spectral range [38] as
schematically depicted in Fig. 1. The FWM is driven using a Ti:Sapphire
femtosecond laser. The QD exciton is excited by Gaussian laser pulses, formed
by a passive pulse shaper based on a diffraction grating. The pulse train is:
i) split into three beams ${\cal E}_{i}$ ($i={1,2,3}$) and a reference beam
${\cal E}_{\rm R}$, ii) the beams ${\cal E}_{i}$ are individually phase-
modulated using acousto-optic modulators (AOMs) operating at distinct radio-
frequencies $\Omega_{i}$ around 80 MHz, iii) the pulses ${\cal E}_{1,2,3}$ are
delayed with respect to one another by $\tau_{12}$ and $\tau_{23}$, iv) their
intensities $P_{i}\sim|{\cal E}_{i}|^{2}$, which are varied to scan the pulse
areas $\theta_{i}$, are controlled via fixed radio-frequency drivers. After
steps i)–iv) ${\cal E}_{1,2,3}$ propagate co-linearly and are focussed on the
sample surface. The pulse shaper is also used to geometrically compensate the
first-order chirp, attaining a focal spot close to the Fourier limit. FWM is
retrieved in reflectance (note that, for the sake of readability, Fig. 1(a)
shows a transmission configuration) by locking ${\cal E}_{\rm R}$ at the
specific heterodyne frequency ($\Omega_{\rm FWM}=2\Omega_{2}-\Omega_{1}$ or
$\Omega_{3}+\Omega_{2}-\Omega_{1}$) and interfering it with the signal beam
from the sample (red in Fig. 1(a)), which is phase-modulated with the same
heterodyne frequency as ${\cal E}_{\rm R}$. The FWM signal is retrieved by
subtracting the intensities in channels A and B from each other, attaining the
shot-noise limited detection. ${\cal E}_{\rm R}$ is also used to generate
background free spectral interference on a CCD camera installed at the output
of an imaging spectrometer.

Figure 1: Schematic picture of the experiment and the theory. (a) Experimental
setup. Acousto-optical modulators (AOMs) label the excitation pulses with
radio frequencies $\Omega_{i}$. The signal beam after the sample is mixed in a
AOM with the reference beam to generate the stationary spectral interference
at the CCD camera, corresponding to the FWM signal. Note that the actual
experiment is working in reflection from the sample. To keep the picture as
simple as possible the figure shows transmission geometry. (b, c) Schematic
picture of the theoretical simulation of the exciton state on the Bloch
sphere. After each pulse with pulse area $\theta_{i}$ the state is filtered
with respect to phase factor $\varphi_{i}$ corresponding to the final FWM
phase $\varphi_{\rm FWM}$. (b) For two-pulse FWM where $\tau_{12}$ is scanned
and (c) for three-pulse FWM where $\tau_{23}$ changes.

In these experiments, the ${\cal E}_{1,2,3}$ pulse durations, hereafter
denoted as $\tau$, arriving at the QD position are in a few hundred fs up to
ten ps range. Therefore, the theoretical treatment of the optical excitation
has to go beyond the computationally handy ultrashort-pulse limit, where
analytic expressions for the FWM signals can be derived [40, 38]. To simulate
the FWM signals of the optically driven QD we model the QD exciton as a two-
level system coupled to LA phonons via the deformation potential mechanism
[41]. The equations of motion for the occupation and the polarization are then
obtained with the well established correlation-expansion approach [42]. The
full set of equations of motion are given in Ref. [43]. Practically, to
compute the FWM signal we drive the QD exciton with Gaussian laser pulses

$\mathcal{E}_{i}(t)=\frac{\hbar}{2\sqrt{2\pi}}\frac{\theta_{i}}{\mu\tau}\exp\left[-\frac{1}{2}\left(\frac{t-t_{i}}{\tau}\right)^{2}-{\rm
i}\omega_{\rm L}t+{\rm i}\varphi_{i}\right]\ ,$

in general with $i=1,\,2,\,3$, the pulse areas $\theta_{i}$, the dipole moment
$\mu$ and the pulse duration $\tau$ at times $t_{i}$. The central energy is
$\hbar\omega_{\rm L}$, which is chosen to be in resonance with the exciton
transition. We label each pulse with a phase $\varphi_{i}$ with $i=1,2$ for
two-pulse and with $i=1,2,3$ for three-pulse FWM. These phases $\varphi_{i}$
correspond to the radio-frequency shifts $\Omega_{i}$ in our experimental
heterodyne detection scheme. The FWM signal $S_{\rm FWM}$ is then isolated by
filtering the resulting microscopic polarization
$p=\left<\left|g\right>\left<x\right|\right>$ with respect to the FWM phases
$\varphi_{\rm FWM}=2\varphi_{2}-\varphi_{1}$ or $\varphi_{\rm
FWM}=\varphi_{3}+\varphi_{2}-\varphi_{1}$ resulting in the FWM polarization
$p_{\rm FWM}\sim S_{\rm FWM}$. Here, $\left|g\right>$ denotes the ground state
and $\left|x\right>$ the exciton state. Note that this filtering of the signal
with respect to the phases directly mimics the heterodyne detection scheme. In
particular, although both in experiment and in theory the lowest order
contribution to the FWM signal is of the order $\chi^{(3)}$, the signals are
not limited to this order. Instead they include all nonlinear odd orders of
the susceptibility $\chi$. This allows us to directly resolve Rabi
oscillations both in experiment and in theory. By varying the delay
$\tau_{12}=t_{2}-t_{1}$ in two-pulse FWM and $\tau_{23}=t_{3}-t_{2}$ in three-
pulse FWM we probe the coherence and population dynamics, respectively. This
can be seen from the Bloch sphere (spanned by the real and imaginary part of
$p$ and the exciton occupation $f=\left<\left|x\right>\left<x\right|\right>$)
schematics in Figs. 1(b, c). Due to the excitation with phase-modulated laser
pulse train, for each repetition, the Bloch vector switches from its initial
(grey) to its final state (black). The curved arrows represent the pulse areas
of each driving pulse. The successive selection of the FWM phase after each
pulse results in the final FWM polarization (orange arrow). In the experiment,
optical heterodyning allows to probe a FWM representation of the Bloch vector
dynamics in its rotating frame. Changing the delay $\tau_{12}$ ($\tau_{23}$)
probes the dynamics of the polarization (population) in the case of two-pulse
(three-pulse) FWM, as can be seen from the selected Bloch vector (dotted black
line and blue arrow) directly before $\tau_{12}$ ($\tau_{23}$). The
simulations reaching delay times over 100 ps are computationally demanding,
which makes finding appropriate system parameters like pulse areas, pulse
duration and QD geometry quite challenging. Note, that we add a
phenomenological long-time dephasing rate $\beta$ for the polarization and a
spontaneous decay rate $\gamma$ for the exciton occupation to the model when
simulating dynamics in the 100 ps range.

##  3 RESULTS

We have recently shown that the pulse areas $\theta_{i}$ of the driving laser
fields in FWM experiments have a strong influence on the system’s dynamics
between the pulses. For instance, the visibility of quantum beats, arising due
to the splittings in the excitonic level structure of a QD system, depends on
the pulse area [38]. Furthermore, in a two-pulse FWM experiment the strength
of the phonon-induced dephasing (PID) becomes more pronounced when increasing
$\theta_{1}$ towards $\pi$ [1].

The pulse area dependence of the exciton quantum state is rather obvious, when
considering the dynamics during the interaction with the laser field, i.e.,
Rabi oscillations take place. When keeping $\tau$ fixed and increasing $P_{i}$
and therefore $\theta_{i}$, the exciton is excited and de-excited during the
pulse more often. In the picture of the Bloch vector, it performs more
rotations, which means that the rotation speed, i.e., the Rabi frequency
$\Omega_{\rm R}$, increases. Additionally, the coupling between exciton
dynamics and phonons depends on the instantaneous Rabi frequency and the
phonon spectral density $J(\omega_{\rm ph})$. For the deformation potential
coupling between the QD exciton and LA phonons $J(\omega_{\rm ph})$ scales
like $\omega_{\rm ph}^{3}$ for small phonon frequencies $\omega_{\rm ph}$
[44]. For typical self-assembled InGaAs/GaAs QDs with sizes in the range of a
few nm, the spectral density forms a broad maximum at $\omega_{\rm ph,0}$ in
the range of a few ps-1, i.e., a few meV [45]. The upper cut-off frequency is
roughly given by $\omega_{\rm ph,max}=2c/a$, where $c$ is the sound velocity
and $a$ the localization length of the exciton. Therefore the strongest
interaction between exciton and LA phonons lies in the range of $\Omega_{\rm
R}\approx\omega_{\rm ph,0}$.

In the limit of ultrafast laser pulse excitation, in the range of $\tau\approx
100$ fs, we have shown that simulations in the delta-pulse limit yield a
satisfactory agreement with experiments[38, 46]. In this limit, the properties
of the excited phonons only depend on the final occupation of the exciton
state [47]. Therefore, to sense the PID effects related with the variation of
the pulse area, one needs to work with longer pulses. To this aim, we
spectrally shape the initial laser beam, setting durations of $\tau\simeq 300$
fs. To enhance the influence of LA phonons, we set the temperature to $T=23$
K, increasing the phonon occupation of modes in the range of 7 meV by a factor
of $7\times 10^{6}$ with respect to 4.2 K, while keeping sufficiently long
dephasing of the zero-phonon line [48, 46]. $T=23$ K is also considered in the
simulations. We first present the results obtained on a neutral exciton in an
InAs QD embedded in a planar cavity, exhibiting a low quality factor
$Q_{\rm{planar}}\simeq 1.7\times 10^{2}$, as recently employed in Refs. [6,
38, 1]. The layer of annealed and capped InAs QDs (density $2\times 10^{9}$
cm-2) is placed in the center of a GaAs spacer. The spectrum of the driving
laser pulses is tuned to cover the ground state to exciton transition as shown
in Fig. S1(a) in the Supplementary Material. Also the exciton to biexciton
transition is slightly covered by the tail of the pulse spectrum, generating
small signals for negative delays via the two-photon coherence (see Fig.
S1(a)). A strong influence from the biexciton state would also lead to a
beating for positive delays due to the biexciton binding energy [38]. Because
the pulse spectrum has only little overlap with the exciton to biexciton
transition, we do not resolve this beating for positive delays. Therefore, we
conclude that the influence of the biexciton for the signals at positive delay
times is negligible and neglect the biexciton state in our study, restricting
it to a two-level system. In Figs. 2(a) and 2(b), we plot the two-pulse FWM
amplitude as a function of $\tau_{12}$. The FWM phase is given by
$\varphi_{\rm FWM}=2\varphi_{2}-\varphi_{1}$, meaning that the first pulse
creates a coherence and the second pulse converts this coherence into the FWM
signal (see Fig. 1(b)). We show results for four different pulse areas
$\theta_{1}$ at a fixed second pulse area of $\theta_{2}=\pi$. Panel 2(a)
presents the measurement and panel 2(b) the respective simulation results for
small to large pulse areas from bottom to top. Note, that all curves are
normalized to unity. The pulse areas in the calculations are listed in the
plot next to the respective graph. For the geometry of the QD we choose
slightly different parameters than in Ref. [1]. Here, the electron and hole
localization lengths are $a_{\rm e}=7$ nm and $a_{\rm h}=1.5$ nm,
respectively. These parameters give the best agreement between the measured
data in Fig. 2(a) and the simulations in 2(b). Note, that the dimensions of
the exciton $a_{\rm e}$ and $a_{\rm h}$ are a spherical representations of the
exciton, which leads to the same physical results as a lens-shaped model [49].
Therefore, these sizes must not be seen as the real size of the exciton wave
function.

Figure 2: Normalized two-pulse FWM generated with sub-ps pulses on a QD
exciton embedded in a low-Q cavity. (a, b) FWM amplitude as a function of the
delay $\tau_{12}$ for increasing pulse area of the first pulse from bottom to
top. The FWM delay dependence probes the optically driven evolution of the
exciton polarization, performing Rabi oscillations for excitations with high
pulse areas. (a) Experiment, the $P_{1}$ impinging the sample surface are
$(0.14,\,1.1,\,2.3,\,2.7)$ µW, $P_{2}=1$ µW, temperature $T=23$ K. The dots
indicate the noise level. (b) Theory, solid lines with phonon coupling dotted
lines without phonons. $\theta_{1}$ as given in the plot. The circles mark
maxima of the FWM signal. (c, d) Schematic pictures of the FWM signal on the
Bloch sphere. (c) For small pulse areas $\theta_{1}$, (d) for pulse areas
$\theta_{1}$ exceeding $\pi$.

We start the discussion with the smallest measured pulse area at the bottom in
Fig. 2(a) (yellow). The FWM amplitude builds up around $\tau_{12}=0$ and
reaches a maximum at around 0.5 ps (marked by the circle). After that, the
signal slightly decays within the next 3 ps. This behavior is well reproduced
by the simulation in Fig. 2(b) including the coupling to LA phonons (solid)
for $\theta_{1}=0.4\pi$. The drop of the signal within the first few ps
results from the PID effect. Together, the phonons and the exciton in the QD
form a new equilibrium state, the acoustic polaron. This polaron is
accompanied by a static lattice displacement in the vicinity of the QD [50].
When the exciton, and therefore the polaron, is created faster than the
typical timescale of the involved phonons, a phonon wave packet is emitted.
This results in an irreversible loss of exciton coherence [51, 52, 46]. To
emphasize that the signal drop results from the phonon coupling, we also
provide simulations neglecting the exciton-phonon interaction, resulting in
the dashed yellow curve in Fig. 2(b), visibly lacking the initial decay due to
the PID effect.

Going over to the next larger pulse area in Fig. 2(a) (green), we clearly see
that the drop of the measured FWM signal is significantly increased and
reaches an almost stationary value within less than 1 ps. Additionally, the
circle shows that the maximum of the signal shifts to shorter delays. These
findings are consistent with the simulation for $\theta_{1}=1.0\pi$ in panel
(b) (solid green). However, the calculated FWM signal without the exciton-
phonon coupling (dashed green) goes exactly to zero after $\tau_{12}=1$ ps. To
explain this we take a look at the schematic Bloch sphere picture in Fig.
2(c), where the green arrow visualizes the movement of the Bloch vector during
the first laser pulse excitation. When the first pulse area is small, i.e.,
$\theta_{1}<\pi/2$, the microscopic polarization $p$ is maximal at the end of
the pulse. Therefore, also the two-pulse FWM signal is maximal for small
positive delays, i.e., after the first pulse. This is the case for the dashed
yellow curve. However, for pulse areas $\pi/2<\theta_{1}<\pi$ the Bloch vector
crosses the equator of the Bloch sphere, i.e., the state of maximum coherence,
during the pulse. For the special case of a $\pi$-pulse, the final
polarization is zero, leading also to a vanishing FWM signal. Already here we
see that the dynamics of the FWM signal resembles more involved movements of
the Bloch vector during the action of the first laser pulse. The full model,
including the coupling to LA phonons, is shown as solid green line in Fig.
2(b). We see that the stationary value of the curve strongly deviates from the
dashed line. This mismatch is again a consequence of the PID effect. Going
from a $0.4\pi$ pulse to a $\pi$-pulse excitation the dephasing influence of
the phonon coupling should significantly increase because the final exciton
occupation $f$ is approximately twice as large [53, 45]. Comparing the solid
lines with the dashed lines of the same color (yellow and green), we see that
this is clearly the case. A stronger excitation leads to a larger phonon wave
packet and therefore to stronger dephasing.

When increasing the pulse area further, in the second curve from the top in
Fig. 2(a) (blue), the signal drop decreases again and the maximum of the
signal also goes back to longer $\tau_{12}\approx 0.5$ ps as marked by the
circle. We find the best agreement with the experiment for the simulation with
$\theta_{1}=1.5\pi$ in panel (b) (solid blue). The dynamics of the simulated
signals, both with and without the exciton-phonon interaction, look very much
like the $\theta_{1}=0.4\pi$ case (yellow). The only significant difference of
the blue curve is an additional minimum at $\tau_{12}=0$ ps. This is again a
result of the Bloch vector dynamics. We are now dealing with the
$\theta_{1}>\pi$ case, schematically shown in Fig. 2(d). The Bloch vector
crosses the north pole of the sphere during the pulse, which leads to null
polarization and in consequence to a vanishing FWM signal. After 1 ps the
solid blue line in panel (b) is again governed by the PID drop of the signal.
The additional minimum of the FWM signal around $\tau_{12}=0$ ps is obviously
too unpronounced to be clearly resolved in the experiment in panel (a). When
comparing the simulation with and without phonon coupling we find that the
drop of the signal, i.e., the PID effect, gets weaker with respect to the
pulse area discussed above. This is in line with the previous explanation of
the strength of the PID effect. For this pulse area the final exciton
occupation is smaller than for the $\pi$-pulse, which leads to weaker
dephasing.

When we move to the largest considered pulse area at the top (violet) a clear
minimum shows up in the measured signal in Fig. 2(a). This is in excellent
agreement with the simulation for $\theta_{1}=1.8\pi$ in panel (b) (solid
violet). Here, the signal minimum is significantly more pronounced than in the
case considered before. This is an instructive demonstration of the optically
driven Rabi oscillations of the exciton state. The PID drop after
$\tau_{12}=0.5$ ps is of a comparable strength as for the $1.5\pi$ case in
blue.

From the pulse area series in Figs. 2(a) and 2(b) we find that the effect of
the PID and therefore the influence of the exciton-phonon interaction changes
measurably with the pulse area $\theta_{1}$ of the first driving laser pulse.
For large positive delays, i.e. $\tau_{12}>\tau$, pulse one arrives first,
whilst for large negative delays pulse two arrives first. Therefore, around
$\tau_{12}=0$ both pulses overlap. The additional dynamics evolving around
$\tau_{12}=0$ happen during the presence of the first pulse. Because the
$\tau_{12}$-dependence of the two-pulse FWM signal represents the dynamics of
the coherence, i.e., the microscopic polarization of the exciton $p$, the
resolved oscillations for the largest considered pulse area stem from
optically driven Rabi oscillations in the time domain.

It was shown that the exciton-phonon interaction leads to more involved
movements of the Bloch vector during optical driving. The phonon lead to i)
dephasing, i.e., shrinking of the Bloch vector length and ii) a mixture of
real and imaginary part of the polarization [54], i.e., a movement out of the
Rabi oscillation plane of the Bloch vector. For simplicity, the illustrating
pictures in Figs. 2(c) and 2(d) do not take these complexities into account.
Instead, they capture well the origin of the Rabi oscillations observable in
the FWM delay dynamics. Following this proof of principle demonstration, we
now generalize this approach using a more suited photonic nanostructure.

Figure 3: Micro-pillar cavity system. (a) Scanning electron microscopy image
of an exemplary micro-pillar cavity system with a diameter of 1.8 µm and a
height around 10 µm. (b) FWM spectra for varying temperatures, demonstrating
operation in the weak-coupling regime: exciton and cavity resonances shift in
energy and cross at $T\approx 27$ K. (c) Schematic picture of the driving
laser pulses and the measured FWM dynamics. The effective pulse duration
$\tau$ is increased by a factor of 30 inside the cavity. Green trace is the
measured time-resolved FWM field (vertical logarithmic scale) illustrating its
build up owing to the high Q-factor.

Shifting the focus from the PID phenomenon to the Rabi oscillations, we aim to
drive the QD with laser pulses that are as long as possible, to stretch the
dynamics in time. In parallel, we want to reach preferably large pulse areas,
to generate as many Rabi periods as possible. Our FWM methodology relies on
the detection of spectrally-resolved interference between the QD emission and
the reference ${\cal E}_{\rm R}$, requiring a bandwidth well beyond the
spectrometer resolution, in practice over 0.1 meV. As such, the experiment is
limited to pulse durations $\tau$ in the few ps range. To overcome this issue,
we switch to a micro-pillar cavity [55, 56] with a diameter of 1.8 µm and a
height around 10 µm, as depicted in Fig. 3(a), which yields a two orders of
magnitude higher quality factor of $Q_{\rm{pillar}}\simeq 2.4\times 10^{4}$.
This structure, containing a layer of InAs QDs at the antinode of the cavity
mode, allows us to access the photonic structure with sub-ps pulses (namely,
$\tau\simeq 0.4$ ps). See Supplementary Material for spectra of spectral
interference and laser pulses. We tightly focus them onto the top facet of the
pillar and construct spectral interferences with ${\cal E}_{\rm R}$, placed 12
µm apart at the auxiliary pillar [56]. Crucially, due to the high quality
factor the optical field reaches the QD in the cavity in a retarded way. This
stretches the pulses to durations of over 10 ps, as schematically shown in
Fig. 3(b). Figure 3(c) shows a temperature scan of the FWM spectrum of the QD-
cavity system (a corresponding scan of the photoluminescence intensity is
shown in the Supplementary Material). The resonances of exciton and cavity are
marked in the picture. Around $T=27$ K both resonances cross, which shows that
the coupled QD-cavity system operates in the weak coupling regime [55, 57,
58]. The possibility of adjusting the detuning very precisely makes this
system prototypical to demonstrate the Purcell effect [2, 3]. We do so by
applying a three-pulse FWM measurement for different detunings between the
cavity mode and the exciton transition. The results are shown in the
Supplementary Material.

The enhancement of the intra-cavity field in the pillar structure allows for
reaching larger $\theta_{i}$ with respect to the low Q-factor planar cavity
explored in Fig. 2. In the following, we choose the temperature to $T=27$ K,
setting the exciton transition to be approximately in resonance with the
cavity mode as marked by the red arrows in Fig. 3(c). Working at these
elevated temperatures results in a significant increase of the LA phonon
influence than at 5 K [46]. In all simulations we choose the temperature to 25
K.

The results for the two-pulse FWM study are shown in Fig. 4 with the
measurements in panel (a) and the simulations in panel (b). All curves are
normalized to their respective maximum. We consider four different pulse
areas, increasing from bottom to top. The second pulse area is fixed in the
simulations to $\theta_{2}=\pi$. The pulse duration $\tau$ and the long time
dephasing rate $\beta$ are fitted to the smallest pulse area and we found the
best agreement for $\tau=12$ ps and $\beta=0.01$/ps. For the simulations we
choose the same QD geometry as for the calculations shown in Fig. 2.

For the smallest considered pulse area the measured FWM signal in Fig. 4(a)
just forms one maximum around $\tau_{12}=25$ ps and decays for longer delays
single exponentially with $\beta$. These dynamics are well reproduced by the
simulation in Fig. 4(b). Additionally, there is nearly no difference between
the calculation with and without exciton-phonon interaction (solid and
dashed). This indicates that the optically driven dynamics are to slow to
induce the PID drop of the signal. In other words, the polaron creation
follows the exciton occupation adiabatically and no phonon wave packet is
emitted [45]. We can instructively describe the optical excitation process
using the laser-exciton dressed state picture [61]. These states then have an
energy splitting $\Delta_{\rm AT}$, the so called Autler-Townes (AT) splitting
[62], which is proportional to the instantaneous laser field amplitude. While
in the exciton basis the LA phonons only couple to the exciton state, in the
presence of a light field they lead to transitions between the dressed states.
For small pulse areas, i.e. $\theta_{1}\ll\pi$, $\Delta_{\rm AT}$ is
correspondingly small, in the µeV range, hence in the energy range where the
phonon spectral density is negligible [1]. Therefore, the phonons do not lead
to efficient transitions between the dressed states, which allows them to
evolve adiabatically.

Figure 4: Normalized two-pulse FWM with $\tau\simeq 12$ ps pulses in a micro-
pillar cavity. (a, b) FWM amplitude as a function of the delay $\tau_{12}$ for
increasing pulse area of the first pulse from bottom to top. (a) Experiment,
$P_{1}=(0.02,\,0.24,\,0.35,\,0.55)$ µW, $P_{2}=0.08$ µW. (b) Theory with
$\theta_{1}$ as given in the plot and $\theta_{2}=\pi$. Solid/dotted lines
with/without coupling to phonons. (c, d) FWM spectral amplitudes as a function
of excitation power for $\tau_{12}=0$ illustrating emergence of the Autler-
Townes splitting with increasing $\theta_{1}$. Experiment in (c) against
$\sqrt{P_{1}}$ and theory in (d) against $\theta_{1}$.

The appearance of a splitting between the dressed states with increasing field
strength can be seen in Figs. 4(c) and 4(d). There, we show the FWM spectrum
centered around the exciton transition energy $E_{\rm x}$. In the measurement
in panel (c) it is plotted against $\sqrt{P_{1}}\sim\theta_{1}$, which is
proportional to the field amplitude. The corresponding simulation is presented
in panel (d) and shows a good agreement with the measurement. The spectra are
taken for $\tau_{12}=0$. While for small pulse areas the spectrum is dominated
by a single line at the exciton energy, it splits according to the AT
splitting of the dressed states reaching values of approximately 0.1 meV in
panel (c). Note, that the AT splitting is time dependent for pulsed
excitations. Therefore, we are seeing a time integrated version of the
splitting between the dressed states in the FWM spectra. However, $\Delta_{\rm
AT}$ is the direct spectral domain translation of the Rabi oscillations in the
FWM signal. In agreement between the dynamics depicted in Figs. 4(a) and 4(b)
and the spectral behavior in Figs. 4(c) and 4(d), we find clear signatures of
the Rabi oscillations from pulse areas of $\theta_{1}\approx 1.5\pi$ onward.
We additionally find, that the line at positive detuning is stronger than the
other one. This stems from a slight mismatch between the energies of the
driving laser pulses and the exciton transition of $\hbar\omega_{\rm L}-E_{\rm
x}=-0.05$ meV, which was considered in the simulation in Fig. 4(d). This
detuning agrees well with the separation between the exciton and cavity line
in Fig. 3(b) at 27 K. Only for the case of exact resonance between the driving
field and the transition energy the dressed states are equally occupied [61].
As we find here, already detunings in the few µeV range result in a
significant mismatch of the two lines in the FWM signal. To check that the AT
splitting is driven by the laser pulses we performed the same measurement and
simulation as in Figs. 4(c) and 4(d) but with a large delay of $\tau_{12}=30$
ps to reduce the overlap of the pulses. Here, no AT splitting was found (see
Supplementary Material).

Comparing the second pulse area in Fig. 4(a) (green) to the yellow curve, the
maximum of the signal splits into two local maxima, with one moving to smaller
and one moving to larger delays $\tau_{12}$. The maxima are marked by the
circles. In the simulation in Fig. 4(b) we find the best agreement for
$\theta_{1}=1.4\pi$ (solid green line) where we already find clear signatures
of the Rabi oscillations during the interaction with the first laser pulse.
Here two distinct maxima of almost equal height are formed. Compared to the
simulation without the exciton-phonon interaction (dashed green line) we find
a significant deviation for the second maximum already for this pulse area.
Without the coupling to phonons, the FWM signal drops very rapidly and the
second maximum is much smaller than the first one. There are two main effects
enhancing the discrepancy between the two calculations: i) For these pulse
areas we have shown that $\Delta_{\rm AT}$ attains the 0.1 meV range. This
allows for more efficient phonon assisted transitions between the dressed
states and therefore to strong dephasing during the first laser pulse. ii) The
exciton-phonon interaction leads to a renormalization of the pulse areas [26,
63, 64, 65]. This makes a direct comparison of the pulse areas with and
without phonon coupling difficult.

Stepping to the next larger pulse area in Fig. 4(a), i.e., from the green to
the blue curve, does not alter the dynamics of the FWM signal significantly.
We basically find a slightly larger difference in the height of the two maxima
in the signal. This is reproduced by the modeled signal in panel (b). Here the
minimum between the two maxima is more pronounced as in the experiment, which
we already found in a similar way for the shorter pulses in Fig. 2. The
deviation between the simulation with and without phonon coupling remains
remarkable because of the reasons pointed out earlier.

The most significant difference for the signal dynamics is found for the
largest considered pulse area in the experiment, which is shown as violet
curve in Fig. 4(a). It forms a double peak structure within the first 25 ps
followed by a minor maximum around $\tau_{12}=50$ ps. After that the signal is
basically null. The very same behavior is found in the simulation with
$\theta_{1}=2.0\pi$ in Fig. 4(b): Two strong and narrow peaks around
$\tau_{12}=0$ ps are followed by a small maximum at $\tau_{12}\approx 50$ ps.
Together with the stunning agreement with the simulated curve in panel (b)
(solid violet), this impressively shows that multiple Rabi oscillations are
resolved in the coherence dynamics of the two-pulse FWM experiment. Without
considering the coupling to phonons the model gives the dashed line in Fig.
4(b). Here also three maxima build up, but the relative heights of the second
and third maximum do not agree with the measured curve in panel (a) at all.
This shows that coupling to phonons has a strong impact on the optically
driven dynamics of the exciton quantum state.

Figure 5: Normalized three-pulse FWM with $\tau=12$ ps pulses in a micro-
pillar cavity. (a, b) Schematic pictures of the FWM signal on the Bloch
sphere. (a) For small pulse areas $\theta_{2}$, (b) for pulse areas
$\theta_{2}$ exceeding $\pi$. Solid/dotted lines with/without coupling to
phonons. (c, d) FWM amplitude as function of the delay $\tau_{23}$ for
increasing pulse area of the second pulse from bottom to top. The FWM delay
dependence probes the optically driven evolution of the exciton occupation,
performing Rabi oscillations for excitations with high pulse areas. (c)
Experiment, $T=27$ K, $\tau_{12}=10$ ps, $P_{1}=0.05$ µW, $P_{3}=0.1$ µW,
$P_{2}=(0.13,\,0.3,\,0.56,\,1)$ µW. (d) Theory with $\theta_{1}=0.2\pi$,
$\theta_{3}=0.4\pi$ and $\theta_{2}$ as given in the plot.

The FWM technique allows to use different pulse sequences that lead to
different microscopic quantities determining the FWM signal. So far we have
studied the exciton polarization $p$ by employing a two-pulse FWM experiment
with $\varphi_{\rm FWM}=2\varphi_{2}-\varphi_{1}$. We now turn to a three-
pulse excitation with $\varphi_{\rm FWM}=\varphi_{3}+\varphi_{2}-\varphi_{1}$
where - in a two level system - the FWM signal carries information about the
occupation of the exciton state $f$. A schematic picture of the Bloch sphere
for this experiment is shown in Figs. 5(a) and 5(b) (see also Fig. 1(c)). The
first pulse drives a microscopic coherence. The second pulse converts this
coherence into an occupation of the exciton, which is then turned into the FWM
signal by the third pulse. A larger occupation of the exciton state $f$ will
accordingly lead to a stronger FWM amplitude. The idea is to fix the first and
third pulse area to small values $\theta_{1},\,\theta_{3}\lesssim\pi/2$ and
change the second area $\theta_{2}$. This will then result in different
occupations $f$ and therefore different signal strengths. For large pulse
areas $\theta_{2}$ the three-pulse FWM signal will resolve the optically
driven Rabi oscillations projected on the exciton occupation $f$.

Figures 5(c) and 5(d) present the normalized experimental and theoretical
results, respectively, for different pulse areas $\theta_{2}$ increasing from
bottom to top. The FWM amplitude is plotted as a function of the delay
$\tau_{23}$ to investigate the exciton population dynamics. The first delay is
fixed to $\tau_{12}=10$ ps to enable the build up of the intra-cavity field.
We see for the smallest pulse area at the bottom (yellow) that the dynamics of
the three-pulse FWM amplitude forms a single broad maximum. For long delays
the signal is dominated by the spontaneous decay. The decay rate was chosen to
$\gamma=0.004$/ps to give the best fit to the measurement. We found the best
agreement between measurement and simulation for $\theta_{2}=0.8\pi$. The
other pulse areas are chosen to $\theta_{1}=0.2\pi$ and $\theta_{3}=0.4\pi$ in
all simulations.

Comparing experiment and theory we find an excellent agreement for each
considered $\theta_{2}$. With increasing pulse strength, the first maximum
shifts to shorter delays $\tau_{23}$, whilst at longer $\tau_{23}$ additional
maxima emerge, as marked by the circles. These dynamics now happen during the
interaction with the second laser pulse. Therefore the signal is dominated by
the Rabi oscillations of the exciton occupation $f$. For the two largest pulse
areas (blue and violet) up to three distinct maxima can be found in the
signal. When comparing the simulations including the exciton-phonon coupling
(solid) with those omitting it (dashed), we find clear qualitative
differences. In the $\theta_{2}=1.2\pi$ case (green) the maxima have a
significant mismatch in their delay $\tau_{23}$ of approximately 25 ps.
Another striking example is the largest pulse area at the top, where the full
model including the phonon coupling develops multiple modulations, while the
uncoupled calculation yields a single broad peak. One obvious reason for this
stronger discrepancy between the model with and without phonons in Fig. 5(b)
compared to the results in Fig. 4(b) is the fact that the former one includes
three pulses, while the latter one only two. Therefore, the impact of
dephasing and pulse area renormalization come into play for one more pulse,
further enhancing their influence. Additionally, we reach higher pulse areas,
which results in general in a more efficient exciton-phonon interaction.

Figure 6: Entire complex FWM signal. Real and imaginary part of the FWM signal
$S_{\rm FWM}$. The delays are color-coded, the dotted lines show simulations
without phonon coupling and the solid lines with phonon coupling. (a, b) For
2-pulse FWM. The corresponding dynamics of the FWM amplitude is given in
Figure 4b. (c, d) For 3-pulse FWM. The corresponding dynamics of the FWM
amplitude is given in Figure 5d.

The FWM signal $S_{\rm FWM}$ is a complex quantity with real and imaginary
part or amplitude and phase. This feature has e.g. been used to distinguish
between different coupling situations in few level systems [35]. We here use
it to further illustrate the dynamics of the Rabi oscillations and especially
the influence of the exciton-phonon coupling. In Fig. 6 we plot real and
imaginary parts of the simulated FWM signals and color-code the delay to
visualize the dynamics of the signals. The arrow heads indicate the direction
of the time evolution. In principle, panels (a, b) and (c, d) are more
sophisticated representations of the data shown in Figs. 4(b) and 5(d),
respectively. In panels (a) and (b) we show the smallest and largest pulse
area for the two-pulse FWM case and in panels (c) and (d) the smallest and
largest pulse areas for the three-pulse FWM case. The remaining four
considered pulse areas are given in the Supplementary Material. The
simulations without exciton-phonon coupling are shown as dotted lines. These
curves are restricted to the vertical axis, i.e., the FWM signal is purely
imaginary. When the full model is considered (solid lines) real and imaginary
parts of the FWM signals get mixed and the curves in Fig. 6 show very involved
dynamics. While for the small pulse area examples in panels (a) and (c) the
signals are governed by a single loop, in the high area cases in panels (b)
and (d) the signals perform multiple loops. These loops represent the Rabi
oscillations, which we also found in the dynamics of the FWM amplitudes in
Figs. 4 and 5. This way of analyzing the FWM signals exploits the full
potential of the method in an appealing fashion.

##  4 CONCLUSIONS

In summary, we have investigated the optically driven dynamics of the full
quantum state of a single QD exciton in the presence of efficient coupling to
acoustic phonons. By driving the system with sub- and super-ps laser pulses
and applying two- and three-pulse FWM techniques, we could isolate the
dynamics of the microscopic polarization and the exciton occupation,
respectively. By this we showed that for large pulse areas involved dynamics
of the Bloch vector in the form of Rabi oscillations take place. Especially,
the coupling to LA phonons plays a decisive role for the optical control of
the QD exciton: On the one hand, for short laser pulses ($\tau<1$ ps) phonons
lead to the loss of coherence after the first laser pulse, due to the emission
of phonon wave packets. On the other hand, for $\tau\simeq 10$ ps the
dephasing happens already during the first pulse and it therefore strongly
depends on the applied pulse area. Only the accurate theoretical description
of the exciton-phonon interaction allowed for a quantitative analysis and
comprehension of the measured data. In the spectral domain, the observed Rabi
oscillations represent the Autler-Townes splitting, which we clearly observed
in the FWM spectra.

We thus find that FWM is a versatile technique to study optically driven
dynamics in many different aspects not only in single QDs but also in any
other potential isolated quantum systems like spins of QD trions [1, 2, 3] or
single defect centers in insulators [4, 5]. However, the timescales of spins
are by a factor of 10 – 100 slower than our optically active exciton
transition, which renders the investigation in the optical range far more
challenging.

We have demonstrated that FWM spectroscopy of single emitters can be used to
study the optically driven dynamics of the quantum state and at the same time
works in a regime where the coupling to phonons is rather strong. This makes
this technique also promising to investigate optical transitions in localized
excitons in atomically thin structures like transition metal dichalcogenides
and hexagonal boron nitride or color centers in insulators. These systems
stand out due to their functionality at elevated temperatures making them an
up-and-coming platform for quantum applications.

##  5 ACKNOWLEDGEMENTS

We acknowledge the financial support by the European Research Council (ERC)
Starting Grant PICSEN (grant no. 306387). We further thank Sebastian Lüker for
support with the implementation of the phonon coupled model.

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## Supplemental Material

## 1\. Pulse spectra

Figure S1 shows four wave mixing (FWM) spectral interferograms (orange and
green) of the low-Q cavity system studied in the first part of the paper in
(a) and the weakly coupled high-Q micro-pillar cavity system in (b), which is
studied in the second part. Spectra of the respective reference pulses are
shown in blue. The spectral width of the laser in (a) is matched to the cavity
mode. Extra dips in the laser spectrum in (b) are caused by the strong water
absorption within that spectral range. Spectral and temporal shapes of the
excitation attaining the quantum dot layer in (b) is defined, not by the
external excitation laser, but by the spectrally narrow cavity mode (around 50
µeV), governing the intracavity field and strongly spectrally filtering the
impinging laser.

Figure S1: Exemplary FWM spectral interferograms (orange and green traces) and
spectra of the excitation laser (blue traces) measured on the low-Q planar
micro-cavity (a) and high-Q micro-pillar (b). In (a) the exciton to biexciton
transition is marked by (BE).

## 2\. Photoluminescence spectra

Figure S2 shows photoluminescence (PL) spectra of the quantum dot micro-pillar
cavity system considered in the main text for varying temperature. With this
we confirm that the exciton and the cavity mode do not form an anticrossing
around their resonance. This proves that we are operating the system in the
weak coupling regime. A corresponding temperature scan of the FWM signal is
given in Fig. 2(b) in the main text.

Figure S2: Temperature scan of the photoluminescence spectra.

## 3\. Autler-Townes splitting

In the main text in Figs. 3(c) and 3(d) we show two-pulse FWM spectra for
different pulse areas $\theta_{1}$ at the pulse delay $\tau_{12}=0$. There we
show that for large pulse areas the exciton transition splits into the two
dressed states separated by the Autler-Townes (AT) splitting. In Figs. S3(a)
and S3(b) corresponding spectra are shown for a delay of $\tau_{12}=30$ ps. In
this case the overlap of the two driving laser pulses is small and the spectra
do not show the AT splitting any longer. The simulation in panel (b) is in
good agreement with the measurement in panel (a). This proves that the dressed
states can only be resolved during the interaction with the first laser pulse.
Following the signal intensity from small pulse areas to large ones we see
that it forms two maxima. These are Rabi coherence rotations as we have
discussed them in Ref. [1].

Figure S3: Absence of the Autler-Townes splitting for large pulse delays. FWM
spectra for different pulse areas at a delay of $\tau_{12}=30$ ps. (a)
Measurement and (b) simulation.

## 4\. Purcell effect

By measuring the three-pulse FWM signal in the micropillar-cavity system
introduced in the main text, we can probe the radiative lifetime of the
exciton state. In the main text we have shown that for small applied pulse
areas the dynamics as a function of the delay $\tau_{23}$ is governed by the
radiative decay of the excited state population. The signal can be fitted by a
single exponential. In Fig. S4(a) we show the measured FWM signal dynamics as
colored solid lines. The corresponding exponential fits are given as green
dashed lines. The temperature of the system increases from bottom to top
trespassing the crossing in Fig. S2. Together with the detuning curves in Fig.
S2 we can transfer the temperature of the system into a detuning between the
cavity mode and the exciton transition. The fitted decay rates from Fig. S4(a)
are plotted against the detuning in panel (b). We find the strongest radiative
decay rate in the vicinity of the resonance, i.e., for vanishing detuning. For
increasing negative and positive detunings the decay rate decreases rapidly.
The reason for this result is the Purcell effect[2, 3] which gives rise to a
decreasing radiative lifetime when the emitter is coupled stronger to the
optical mode due to an increased density of photon states.

Figure S4: Measurement of the Purcell effect. (a) Three-pulse FWM signal
dynamics as a function of the delay $\tau_{23}$. The different temperatures,
i.e., different detunings, increase from bottom to top. (b) Fitted decay rate
as a function of the detuning between the cavity mode and the transition
energy.

## 5\. Pulse area calibration

In the main text we compare the dynamics of measured FWM signals for specific
laser pulse powers with simulations, where certain pulse areas are considered
to produce the best agreement with the experiment. In Fig. S5 we plot the
fitted pulse areas $\theta_{i}$ as a function of the square-root of the
applied laser power $\sqrt{P_{i}}$, which is proportional to the field
amplitude and therefore also to the pulse area in the experiment. Note, that
for the results in Fig. 1 in the main text we used another QD sample than for
the results in Figs. 4 and 5. For the uncertainties of the applied pulse power
we assume 10% of $P_{i}$ for the data from the micro-pillar cavity (Figs. 4
and 5) and 5% of $P_{1}$ for the planar cavity (Fig. 1). The dotted lines in
the plot are linear fits to the data points and serve as a guide to the eye.
We find that the dependence between the field amplitudes in the experiment and
the pulse areas in the theory follows the expected linear trend. The slope of
the Fig. 4 and Fig. 5 data is similar, while one for Fig. 2 is flatter. This
is in line with the smaller Q-factor of the planar cavity used for Fig. 2.

Figure S5: Pulse areas calibration. Fitted pulse areas as function of applied
field amplitude from the Figures in the main text as given in the plot.

## 6\. Complex FWM signal

Figure S6 shows the real and imaginary part of the FWM signals with the color
coded delays $\tau_{12}$ and $\tau_{23}$. In Fig. 6 in the main text we show
the smallest and the largest considered pulse areas in the micropillar-cavity
system. Fig. S6 completes the considered pulse areas as it shows the remaining
intermediate pulse areas from Figs. 4(b) and 5(d) from the main text. We see
that by increasing the pulse areas an increasing number of loops appears in
the plots. These loops represent the Rabi oscillations of the Bloch vector
during the interaction with the laser pulses.

Figure S6:  Entire complex FWM signal. Real and imaginary part of the FWM
signal $S_{\text{FWM}}$. The delays are color-coded, the dotted lines show
simulations without phonon coupling and the solid lines with phonon coupling.
(a, b) For two-pulse FWM. The corresponding dynamics of the FWM amplitude is
given in Fig. 4(b) in the main text. (c, d) For three-pulse FWM. The
corresponding dynamics of the FWM amplitude is given in Fig. 5(d) in the main
text.

## References

  * [1] D. Wigger, Q. Mermillod, T. Jakubczyk, F. Fras, S. Le-Denmat, D. E. Reiter, S. Höfling, M. Kamp, G. Nogues, C. Schneider, T. Kuhn, and J. Kapsrzak, “Exploring coherence of individual excitons in InAs quantum dots embedded in natural photonic defects: influence of the excitation intensity,” Phys. Rev. B 96, 165311 (2017).
  * [2] E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
  * [3] J. M. Gerard and B. Gayral, “InAs quantum dots: artificial atoms for solid-state cavity-quantum electrodynamics,” Physica E Low Dimens Syst Nanostruct 9, 131–139 (2001).


# Experimental investigation of Wigner’s reaction matrix for irregular graphs
with absorption

Oleh Hul1, Oleg Tymoshchuk1, Szymon Bauch1, Peter M. Koch2, and Leszek Sirko1
1Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46,
02-668 Warszawa, Poland
2Department of Physics and Astronomy, State University of New York, Stony
Brook, NY 11794-3800 USA

(September 22, 2005)

###### Abstract

We use tetrahedral microwave networks consisting of coaxial cables and
attenuators connected by $T$-joints to make an experimental study of Wigner’s
reaction $K$ matrix for irregular graphs in the presence of absorption. From
measurements of the scattering matrix $S$ for each realization of the
microwave network we obtain distributions of the imaginary and real parts of
$K$. Our experimental results are in good agreement with theoretical
predictions.

###### pacs:

05.45.Mt,03.65.Nk

Pauling introduced quantum graphs of connected one-dimensional wires almost
seven decades ago Pauling . Kuhn used the same idea a decade later Kuhn  to
describe organic molecules by free electron models. Quantum graphs can be
considered as idealizations of physical networks in the limit where the
lengths of the wires greatly exceed their widths; this corresponds to assuming
that propagating waves remain in a single transverse mode. Among the systems
modeled by quantum graphs are, e.g., electromagnetic optical waveguides Flesia
; Mitra , mesoscopic systems Imry ; Kowal  , quantum wires Ivchenko ; Sanchez
and excitation of fractons in fractal structures Avishai ; Nakayama . Recent
work has shown that quantum graphs provide an excellent system for studies of
quantum chaos Kottossmilansky ; Kottos ; Prlkottos ; Zyczkowski ; Kus ; Tanner
; Kottosphyse ; Kottosphysa ; Gaspard ; Blumel ; Hul2004 . Quantum graphs with
external leads (antennas) have been analyzed in detail in Kottosphyse ;
Kottosphysa . Quantum graphs with absorption, a more realistic but more
complicated system, have been studied numerically in Hul2004 , but until now
there have been no experimental studies of the effect of absorption.

This paper presents results of our experimental study of distributions of
Wigner’s reaction matrix Akguc2001  (often called in the literature just the
$K$ matrix Fyodorov2004 ) for microwave networks that correspond to graphs
with time reversal symmetry ($\beta=1$ symmetry class of random matrix theory
Mehta ) in the presence of absorption. For the case of an experiment having a
single-channel antenna, the $K$ matrix and scattering matrix $S$ are related
by

$S=\frac{1-iK}{1+iK}.$ (1)

The function $Z=iK$ has direct physical meaning as the electrical impedance,
which has been recently measured in a microwave cavity experiment Anlage2005 .
For the one-channel case the $S$ matrix can be parameterized as

$S=\sqrt{R}e^{i\theta},$ (2)

where $R$ is the reflection coefficient and $\theta$ is the phase.

After seminal work of López, Mello and Seligman Lopez1981  came theoretical
studies of the properties of statistical distributions of the $S$ matrix with
direct processes and imperfect coupling Doron1992 ; Brouwer1995 ; Savin2001 .
A recent experiment investigated the distribution of the $S$ matrix for
chaotic microwave cavities with absorption Kuhl2005 . The distribution $P(R)$
of the reflection coefficient $R$ in Eq. (2), at the beginning investigated in
the strong absorption limit Kogan , has been recently known for any
dimensionless absorption strength $\gamma=2\pi\Gamma/\Delta$, where $\Gamma$
is the absorption width and $\Delta$ is the mean level spacing. For systems
with time reversal symmetry ($\beta=1$) Méndez-Sánchez et al. Sanchez2003
studied $P(R)$ experimentally, and Savin et al. Savin2005  found an exact
formula for $P(R)$. For systems violating time reversal symmetry ($\beta=2$),
Beenakker and Brouwer Beenakker2001  calculated $P(R)$ for the case of a
perfectly coupled, single-channel lead.

In our experiment we simulate quantum graphs with microwave networks. The
analogy between them is based on the Schrödinger equation for the former being
equivalent to the telegraph equation for the latter Hul2004 . We call them
microwave graphs. Measurements of the scattering matrix for them were
stimulated by Blumel88  and the pioneering measurements in Doron90 .

A simple microwave graph, the tetrahedral case, consists of six coaxial cables
(bonds) that meet three-at-a-time at $N=4$ different $T$-joints (vertices).
Each coaxial cable consists of an inner conductor with radius $r_{1}$
separated from a concentric outer conductor with inner radius $r_{2}$ by a
homogeneous, non-magnetic material with dielectric constant $\varepsilon$. The
fundamental $TEM$ mode that propagates (the so-called Lecher wave) down to
zero frequency exists because the cross section of the cable is doubly
connected (Jones, , p. 253). For frequencies $\nu$ below the onset of the TE11
mode in a coaxial cable, which propagates above
$\nu_{c}\simeq\frac{c}{\pi(r_{1}+r_{2})\sqrt{\varepsilon}}$ Jones , the cable
is single mode: only the TEM mode propagates. For SMA-RG-402 coaxial cable,
which has $r_{1}=0.05$ cm, $r_{2}=0.15$ cm, and $\varepsilon\simeq 2.08$
(teflon dielectric), single-mode propagation occurs below 32.9 GHz.

An (ideal) microwave graph with no aborption and no leads to the outside world
is a closed (bound) system. The presence of absorption and/or leads to the
outside world creates an open system. Because the coaxial cables are lossy, we
may vary absorption in the microwave graphs by changing the length of
cable(s), Hul2004 , by adding one or more (coaxial) microwave attenuators, or
by changing the coupling to the outside world.

Figure 1: A diagram of experimental setup used to measure the scattering
matrix $S$ of tetrahedral microwave graphs with absorption. Absorption in the
graphs was varied by changing the attenuator. The vector network analyzer used
for all measurements was an HP model 8722D.

Figure 1 shows our experimental setup for measurements of the single-channel
scattering matrix $S$ for tetrahedral microwave graphs. We used a Hewlett-
Packard model 8722D microwave vector network analyzer to measure the
scattering matrix $S$ of such graphs in two different frequency windows, viz.,
3.5–7.5 GHz and 12-16 GHz. As the figure shows, at one of the four vertices we
used a 4-joint connector to couple the microwave graph to the vector network
analyzer via a single-channel lead realized with an HP model 85133-60017, low-
loss, flexible microwave cable; the other three vertices consisted of
$T$-joints. The plane of calibration in the measurements was at the entrance
to the 4-joint connector. Note that the microwave graph in Fig. 1 has a
microwave attenuator in one of its bonds.

To investigate the distributions of imaginary and real parts of the $K$ matrix
we measured the scattering matrix $S$ for $184$ different realizations of
tetrahedral microwave graphs having a microwave (SMA) attenuator in one of the
bonds. For each graph realization, which was obtained either by the
replacement of the bonds or putting an attenuator to a different bond, the
scattering matrix $S$ was measured in 1601 equally spaced steps. The total
optical lengths of the microwave graphs, including joints and the single
attenuator, was 196.2 cm when a 3 dB, 6 dB, or 20 dB attenuator was used,
whereas it was 197.4 cm when the 10 dB attenuator was used. To avoid
degeneracy of eigenvalues in the graphs, we chose optical lengths for the
bonds that were not simply commensurable.

Figure 2: Panels (a) and (b) show, respectively, the modulus $|S|$ and the
phase $\theta$ of the scattering matrix $S$ measured for the graph (see Fig.
1) with absorption parameter $\gamma=3.6$ (see the text) over the frequency
range 12 - 13.4 GHz and with use of a 3 dB attenuator. Panels (c) and (d) show
corresponding measurements for a graph with $\gamma=6.8$ over the same
frequency range and with use of a 20 dB attenuator. The total optical length,
196.2 cm, of both microwave graphs was the same, including joints and the
attenuator.

Figure 2 shows the modulus $|S|$ and the phase $\theta$ of the scattering
matrix $S$ of a tetrahedral graph with $\gamma=3.6$ (in panels (a) and (b),
obtained with use of a 3 dB attenuator) and one with $\gamma=6.8$ (in panels
(c) and (d), obtained with use of a 20 dB attenuator); both cases cover the
frequency range 12–13.4 GHz. The lengths of corresponding bonds in the two
graphs were the same. Direct processes are present in the scattering because
the microwave vector analyzer was connected to the graphs by the 4-joint
connector. For each, individual realization of the graph we may estimate them
from the average value of the scattering matrix $\langle S\rangle$. Our
measurements averaged over all realizations of microwave graphs gave $\langle
S\rangle_{av}\simeq 0.47\pm 0.03+i(-0.01\pm 0.04)$, where $i=\sqrt{-1}$. The
experimental value for $|\langle S\rangle_{av}|\simeq 0.47$ is close to a
theoretical estimate Kottosphyse ; Kottosphysa  for the modulus of the vertex
reflection amplitude $|\rho|=0.5$ for a 4-joint connector with Neumann
boundary conditions,

$\rho=\frac{2}{n_{v}}-1,$ (3)

where $n_{v}=4$ is the number of bonds meeting at the vertex in question.

Equation (1) holds for systems with absorption but without direct processes.
The case of imperfect coupling $|\langle S\rangle|>0$ and direct processes
present can be mapped onto that of perfect one Fyodorov2004  by making the
following parametrization,

$S_{0}=\frac{S-|\langle S\rangle|}{1-|\langle S\rangle|S},$ (4)

where $S_{0}$ is the scattering matrix of a graph for the perfect-coupling
case (no direct processes present).

For systems with time reversal symmetry ($\beta=1$ in equations below), the
distributions $P(v)$ of the imaginary and $P(u)$ of the real parts of the $K$
matrix Fyodorov2004  are given by the following interpolation formulas:

$P(v)=\frac{N_{\beta}e^{-a}}{\pi\sqrt{2\gamma}v^{3/2}}(A[K_{0}(a)+K_{1}(a)]a+\sqrt{\pi}Be^{-a}),$
(5)

and

$P(u)=\frac{N_{\beta}e^{-\gamma/4}}{2\pi\bar{u}}[\frac{A}{2}\sqrt{\frac{\gamma}{4}}D(\frac{\bar{u}}{2})+BK_{1}(\frac{\gamma\bar{u}}{4})],$
(6)

where $-v=\textrm{Im}\,K<0$ and $u=\textrm{Re}\,K$ are, respectively, the
imaginary and real parts of the $K$ matrix. The normalization constant is
$N_{\beta}=\alpha\left(A\Gamma(\beta/2+1,\alpha)+Be^{-\alpha}\right)^{-1}$,
where $\alpha=\gamma\beta/2$,
$\Gamma(x,\alpha)=\int_{\alpha}^{\infty}dtt^{x-1}e^{-t}$ is the upper
incomplete Gamma function, $A=e^{\alpha}-1$ and $B=1+\alpha-e^{\alpha}$. In
Eq. (5) the variable $a=\frac{\gamma}{16}(\sqrt{v}+1/\sqrt{v})^{2}$ and
$K_{0}$, $K_{1}$ are MacDonald functions. In Eq. (6)
$D(z)=\int_{0}^{\infty}dq\sqrt{1+z(q+q^{-1})}e^{-\gamma z(q+q^{-1})/4}$ and
$\bar{u}=\sqrt{u^{2}+1}$.

Figure 3: Experimental distribution $P(v)$ of the imaginary part of the $K$
matrix at different values of the mean absorption parameter:
$\bar{\gamma}=3.8$ (open circles), $\bar{\gamma}=5.2$ (full circles) and
$\bar{\gamma}=6.7$ (open triangles), respectively. Each corresponding
theoretical distribution $P(v)$ evaluated from Eq. (5) is also shown:
$\gamma=3.8$ (dotted line), $\gamma=5.2$ (solid line), and $\gamma=6.7$
(dashed line), respectively.

Figure 3 shows experimental distributions $P(v)$ for three mean values of the
parameter $\bar{\gamma}$, viz., 3.8, 5.2, and 6.7. The distribution for
$\bar{\gamma}=3.8$ is obtained by averaging over 69 realizations of microwave
graphs having $\gamma$ within the window $[3.5,\,4.1]$. The distribution for
$\bar{\gamma}=5.2$ is obtained by averaging over 60 realizations of microwave
graphs having $\gamma$ within the window $[4.7,\,5.6]$. The distribution for
$\bar{\gamma}=6.7$ is obtained by averaging over 55 realizations of microwave
graphs having $\gamma$ within the window $[6.3,\,7.1]$. We estimated the
experimental values of the $\gamma$ parameter by adjusting the theoretical
mean reflection coefficient $\langle R\rangle_{th}$ to the experimental one
$\langle R_{0}\rangle=\langle S_{0}S_{0}^{{\dagger}}\rangle$, where

$\langle R\rangle_{th}=\int_{0}^{1}dRRP(R).$ (7)

We also applied the following interpolation formula Kuhl2005  for the
distribution $P(R)$:

$P(R)=N_{\beta}\frac{e^{-\frac{\alpha}{1-R}}}{(1-R)^{2+\beta/2}}[A\alpha^{\beta/2}+B(1-R)^{\beta/2}].$
(8)

We offer the following comment on the validity of the Eq. (8). We used it
instead of exact formulas (12-14) recently presented in Savin2005 , which may
be used to find the distribution $P(R)$, because Eq. (8) is sufficiently
accurate (see Fig. 1 in Savin2005 ) while allowing for much faster numerical
calculations.

Figure 3 also presents for comparison with each experimental distribution
$P(v)$ (symbols) the corresponding numerical distribution (lines) evaluated
from Eq. (5). We see that the experimental distribution $P(v)$ at
$\bar{\gamma}=3.8$ and at 5.2 agree well with their theoretical counterparts.
However, the comparison for $\bar{\gamma}=6.7$ shows some discrepancies,
particularly in the range $0.3<v<0.8$.

Figure 4: Experimental distribution $P(u)$ of the real part of the $K$ matrix
at different values of the mean absorption parameter: $\bar{\gamma}=3.8$ (open
circles), $\bar{\gamma}=5.2$ (full circles) and $\bar{\gamma}=6.7$ (open
triangles), respectively. Each corresponding theoretical distribution $P(u)$
evaluated from Eq. (6) is also shown: $\gamma=3.8$ (dotted line), $\gamma=5.2$
(solid line), and $\gamma=6.7$ (dashed line), respectively.

We may use measurements of the distribution $P(u)$ of the real part of
Wigner’s reaction matrix for an imporant and natural consistency check on our
determination of $\gamma$. Figure 4 compares experimental and theoretical
$P(u)$ distributions at the aforementioned values of $\bar{\gamma}$, viz.,
3.8, 5.2, and 6.7. Though each case shows good overall agreement between
experimental and theoretical results, for all three cases the middle
($-0.25<u<0.25$) of the theoretical distribution is slightly higher than its
experimental counterpart. According to the definition of the $K$ matrix (see
Eq. (1)), such behavior of the experimental distribution $P(u)$ suggests a
deficit of small values of $|\textrm{Im}\,S_{0}|$. We do not yet know the
origin of this deficit.

Though there are the small discrepancies we have mentioned, the good overall
agreement between experimental and theoretical results justifies a posteriori
the procedure we have used to determine the experimental values of $\gamma$.

The distributions $P(v)$ and $P(u)$ of imaginary and real parts of Wigner’s
reaction matrix may be also found using the alternative approach described in
Anlage2005 ; Anlage2005b . In these papers the radiation impedance approach
was developed and used to obtaining the distributions of real and imaginary
parts of the normalized impedance

$Z=\frac{\textrm{Re }Z_{c}+i(\textrm{Im }Z_{c}-\textrm{Im }Z_{r})}{\textrm{Re
}Z_{r}}$ (9)

of a chaotic microwave cavity, where $Z_{c(r)}=Z_{0}(1+S_{c(r)})/(1-S_{c(r)})$
is the cavity (radiation) impedance expressed by the cavity (radiation)
scattering matrix $S_{c(r)}$ and $Z_{0}$ is the characteristic impedance of
the transmission line. The radiation impedance $Z_{r}$ is the impedance seen
at the input of the coupling structure for the same coupling geometry, but
with the sidewalls removed to infinity. This interesting approach is
especially useful in the studies of microwave systems, in which, in general,
both the system and radiation impedances are measurable. However, it is not
obvious how to use in practice this approach in the case of quantum systems.

We used this alternative approach to find distributions $P(v)$ and $P(u)$ of
imaginary and real parts of Wigner’s reaction matrix for irregular tetrahedral
microwave graphs. Wigner’s reaction matrix can be simply expressed by the
normalized impedance $K=-iZ$. The radiation impedance $Z_{r}$ was found
experimentally by measuring in two different frequency windows, viz., 3.5–7.5
GHz and 12-16 GHz of the scattering matrix $S_{r}$ of the 4-joint connector
with three joints terminated by 50 $\Omega$ terminators.

Figure 5: Experimental distribution $P(v)$ of the imaginary part of the $K$
matrix at different values of the mean absorption parameter:
$\bar{\gamma}=3.8$ (open circles), $\bar{\gamma}=5.2$ (full circles) and
$\bar{\gamma}=6.7$ (open triangles), respectively, calculated using the
radiation impedance approach Anlage2005 ; Anlage2005b . Each corresponding
theoretical distribution $P(v)$ evaluated from Eq. (5) is also shown:
$\gamma=3.8$ (dotted line), $\gamma=5.2$ (solid line), and $\gamma=6.7$
(dashed line), respectively.

Figure 6: Experimental distribution $P(u)$ of the real part of the $K$ matrix
at different values of the mean absorption parameter: $\bar{\gamma}=3.8$ (open
circles), $\bar{\gamma}=5.2$ (full circles) and $\bar{\gamma}=6.7$ (open
triangles), respectively, calculated using the radiation impedance approach
Anlage2005 ; Anlage2005b . Each corresponding theoretical distribution $P(u)$
evaluated from Eq. (6) is also shown: $\gamma=3.8$ (dotted line), $\gamma=5.2$
(solid line), and $\gamma=6.7$ (dashed line), respectively.

In Fig. 5 and Fig. 6 we show the distributions $P(v)$ and $P(u)$ calculated
using the radiation impedance approach Anlage2005 ; Anlage2005b . As in the
case of the scattering matrix approach, the experimental distributions are
obtained at three values of the parameter $\bar{\gamma}=3.8$, 5.2 and 6.7.
Figure 5 shows that the distribution $P(v)$ of the imaginary part of Wigner’s
reaction matrix for $\bar{\gamma}=5.2$ is in good agreement with the
theoretical prediction Fyodorov2004 . However, for $\bar{\gamma}=3.8$ and 6.7
the theoretical results are slightly higher than the experimental ones, what
is especially noticeable at the peaks of the distributions. The experimental
distribution $P(u)$ of the real part of Wigner’s reaction matrix presented in
Figure 6 at three values of the parameter $\bar{\gamma}=3.8$, 5.2 and 6.7
displays a very good agreement with the theoretical result. The comparison of
Figures 3 and 5 and Figures 4 and 6 show that the distributions $P(v)$ and
$P(u)$ evaluated by means of the radiation impedance approach are at the peaks
slightly higher than the ones obtained by the scattering matrix approach, what
may suggest that the influence of the phase of $S_{r}$ on the distributions is
not negligible Anlage2005b .

In summary, using the scattering matrix approach and the radiation impedance
approach we have measured distributions $P(v)$ and $P(u)$ of imaginary and
real parts of Wigner’s reaction matrix for irregular tetrahedral microwave
graphs consisting of SMA cables, connectors, and attenuators. Use of different
attenuators allowed us to vary absorption in the graphs in a controlled,
quantitative way. For the case of time reversal symmetry ($\beta=1$), the
experimental results for $P(v)$ and $P(u)$ calculated for both approaches at
the same three values of the mean parameter $\bar{\gamma}$ are in good overall
agreement with theoretical predictions.

Acknowledgments: This work was supported by KBN grant No. 2 P03B 047 24 and an
equipment grant from ONR(DURIP).

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