# Dynamics of the fractional quantum Hall edge probed by stroboscope
measurements of trions

Akinori Kamiyama  Department of Physics, Tohoku University, Sendai 980-8578,
Japan  Masahiro Matsuura  Department of Physics, Tohoku University, Sendai
980-8578, Japan  John N. Moore  Department of Physics, Tohoku University,
Sendai 980-8578, Japan  Takaaki Mano  National Institute for Materials
Science, Tsukuba, Ibaraki 305-0047, Japan  Naokazu Shibata  Department of
Physics, Tohoku University, Sendai 980-8578, Japan  Go Yusa<EMAIL_ADDRESS>Department of Physics, Tohoku University, Sendai 980-8578, Japan

###### Abstract

By using observations from pump-probe stroboscopic confocal microscopy and
spectroscopy, we demonstrate the dynamics of trions and the fractional quantum
Hall edge on the order of $\sim 1$ ps. The propagation of the quantum Hall
edge state excited by a voltage pulse is detected as a temporal change in
reflectance in the downstream edge probed by optical pulses synchronized with
the voltage pulse. The temporal resolution of such stroboscopic pump-probe
measurements is as fast as the duration time of the probe pulse ($\sim 1$ ps).
This ultra-fast stroboscope measurement enables us to distinguish between the
normal mode of edge excitation, known as the edge magneto-plasmon or charge
density wave, and other high-energy non-linear excitations. This is the only
experimental method available to study the ultra-fast dynamics of quantum Hall
edges, and makes it possible to derive the metric tensor $g_{\mu\nu}$ of the
$(1+1)=2$-dimensional curved spacetime in quantum universe and black hole
analogs implemented in the quantum Hall edge.

††preprint: AIP/123-QED

In general, a topological material comprises a gapped bulk and a conducting
edge Thouless  _et al._ (1982); Wen (1995); Hasan and Kane (2010). The quantum
Hall (QH) state, which forms when electrons constrained in two dimensions (2D)
are exposed to a strong perpendicular magnetic field $B$, was the earliest
example of such topological insulators. When the Landau-level filling factor
$\nu=\frac{h}{eB}n_{e}$, which is the ratio of electron density $n_{e}$ and
the flux quanta density $n_{\phi}=\frac{eB}{h}$, is an integer or rational
fraction, the bulk of the system develops a gap to form integer Klitzing,
Dorda, and Pepper (1980) and fractional QH (FQH) states Tsui, Stormer, and
Gossard (1982). Here $h$ and $e$ stand for the Planck constant and elementary
charge, respectively. The excitation of the edge, also known as the edge
magneto-plasmons (EMPs) or charge density wave Wassermeier  _et al._ (1990);
Aleiner and Glazman (1994), has chirality and propagates
unidirectionallyAshoori  _et al._ (1992); Ernst  _et al._ (1997); Kamata  _et
al._ (2014); Hashisaka  _et al._ (2017) because $B$ breaks the time-reversal
symmetry. The excitation of the edge has sparked new interest due to both its
unique prospective applications and fundamental physics Bid  _et al._ (2010);
Sabo  _et al._ (2017); Nakamura  _et al._ (2020); Bartolomei  _et al._ (2020).
Topics include flying qubit quantum computation Yamamoto  _et al._ (2012);
Yang  _et al._ (2016); Dlaska, Vermersch, and Zoller (2017); Elman, Bartlett,
and Doherty (2017); Shimizu  _et al._ (2020), quantum energy
teleportationYusa, Izumida, and Hotta (2011); Matsuura  _et al._ (2018);
Hotta, Matsumoto, and Yusa (2014), and quantum field simulators which simulate
the quantum universe and black holes in $(1+1)=2$-dimensional spacetime Hotta,
Matsumoto, and Yusa (2014); Hotta  _et al._ (2022). Here, $(1+1)$ represents
each spatial and temporal dimension as one. It is necessary to learn about the
dynamics of the QH edge to create a simulator for a toy model of an expanding
universe. In cosmological language, learning about the QH edge dynamics
corresponds to obtaining a tensor field of the metric $g_{\mu\nu}$ Hotta  _et
al._ (2022) which describes how the spacetime is curved.

Figure 1: (a) Confocal stroboscopic microscopy and spectroscopy experimental
setup diagram. Yellow and orange lines and arrows, respectively, designate the
pump and probe pathways. Here PD, AOM, and PM fiber denote photo detector,
acoustic optical modulator, and polarization maintaining fiber. Black lines
denote coaxial cables. The edge of the mesa is defined as $y=0$, and the
center of the excitation gate is specified as $x=0$. The width of the
excitation gate along the $x$-axis is $\sim 5$ $\mu$m. (b) and (c) The typical
spectra of (b) pump and (c) probe pulses measured by the monochromator and the
CCD.

Scanning stroboscopic photoluminescence (PL) spectroscopy was recently
successful in imaging the propagating QH edge in the $\nu=1/3$ FQH regime in
real-space and real-timeKamiyama  _et al._ (2022). The excitation of the edge
was probed using the PL intensity and energy of trions, which are the bound
states of two electrons and a holeWójs, Quinn, and Hawrylak (2000); Yusa,
Shtrikman, and Bar-Joseph (2001). Although the lifetime of trions in the FQH
regime has not been reported, it is expected to be comparable to the lifetime
of trions in $B$, which is on the order of $\sim 100$ psFinkelstein  _et al._
(1998). Because the PL signal for the stroboscopic measurement is averaged
across time the time resolution of the stroboscopic PL was anticipated to be
limited to the order of $\sim 100$ psKamiyama  _et al._ (2022). In this paper,
by analyzing the lifetimes of trions in the QH regime using time-resolved PL
measurement and pump-probe reflection measurements, we present a method to
enhance the time resolution and show excitation of the edge different from the
Tomonaga-Luttinger liquid type of excitationYoshioka (2002).

The experimental setup consists of room and low-temperature regions (Fig.
$1$). A $\sim 100$-fs mode-locked Ti:sapphire laser with a repetition rate of
$76$ MHz splits into the pump and probe paths. To reduce wavelength dispersion
in polarization-maintaining (PM) fibers, we stretch the pulse duration to
$\sim 1$ ps using a 4f pulse shaperWeiner (2000); Anghel  _et al._ (2016)
consisting of a slit and two sets of $1200$ /mm-gratings and lenses. The
typical spectra of pump and probe lasers have $\sim 3$-meV full width at half
maximum (FWHM) (Figs. 1(b) and 1(c)). A part of the pump pulse is introduced
to a fast photo detector (PD) in the room-temperature region to synchronize
with the pulse generator used to excite the QH edge by a voltage pulse. The
pump pulse is introduced into a PM fiber combined with another PM fiber
introduced to the low-temperature region by a combiner. For pump-probe
measurement, the pump pulse is chopped by the optical chopper. The probe pulse
is passed through an acoustic optical modulator (AOM) to control the power of
the laser. By adjusting the 4f pulse shaper’s slit, the peak photon energy of
the probe pulse is tuned such that it resonates with the tron’s PL peak
energy. The probe pulse is introduced to a PM fiber and illuminates the sample
through the low-temperature optics. Both pump and probe travel through the
same PM fiber and have the same focal position on the sample without
noticeable extension of the pulse duration, enabling us to attain high space
and time resolution.

Figure 2: (a) Microscopic PL spectra obtained in the QH bulk region of
$\nu=2/3$ at $V_{\mathrm{b}}=-0.35$ V plotted as a function of time at $B=9$ T
and a temperature $T\sim 50$ mK. The streak camera uses a color-coded system
to display the photon counts. (b) Time-resolved PL intensity for singlet (red)
integrated over energy between $1532.326$ and $1532.580$ eV and for triplet
(blue) integrated between $1533.361$ and $1533.842$ eV (offset for clarity).
Fitting functions for $If(t-t_{0},\tau_{\mathrm{d}})$ are represented by the
black and gray lines. (c) PL spectra that have been integrated over a time
range of $-0.12$ to $0.87$ ns.

The same PM fiber, or a multi-mode fiber, are used to gather both the PL and
reflected light. The PM fiber is split by another combiner, and one of the PM
fibers can be selectively connected to one of the two detection systems. The
first system is for the stroboscopic PL measurementKamiyama  _et al._ (2022)
and for the time- and energy-resolved measurements using the streak camera.
The system comprises a monochromator outfitted with a streak camera and
charge-coupled device (CCD) detector. The time resolution of the streak camera
itself is $\sim 20$ ps, but that of the total system can be larger than $20$
ps because the optical path in the monochromator is not optimized Schiller and
Alfano (1980). A PD connected to a lock-in amplifier makes up the second
system. This system is used for stroboscopic pump-probe reflection
measurement, which we will explain in detail later. By switching the coupling
of the PM and multi-mode fibers at room temperature, the streak camera,
stroboscopic PL, and pump-probe reflection measurements can be chosen (see
gray dotted lines in Fig. $1$(a)).

The wafer used for the experiment is a $15$-nm GaAs/AlGaAs quantum well (QW)
(Fig. $1$(a)) grown on an $n+$ substrate, which functions as a back gate to
control $n_{e}$ by applying a back gate voltage $V_{\mathrm{b}}$. Chemical
etching is used to define the sample edge. A front gate electrode made with
titanium and gold functions as an excitation gate, and is connected to the
pulse generator. The low-temperature optics consists of fiber collimators,
beam splitters, a polarizer, and an objective lens with the numerical aperture
of 0.55 Kamiyama  _et al._ (2022); Hayakawa, Muraki, and Yusa (2013); Moore
_et al._ (2016, 2017, 2018). Piezoelectric stages can move the sample in the
$x$-, $y$-, and $z$-directions relative to the low-temperature optics in a low
temperature ($40$ mK) and high $B(\leq 14$ T) environment without heating the
sample.

We begin by displaying time-resolved microscopic PL spectra (Fig. $2$). PM
fiber is used to transmit light from the pump laser onto the sample in a
region away from the sample edge (a QH bulk region). Using the same PM fiber,
the streak camera attached to the monochromator with confocal geometry detects
PL from this region (Fig. $1$(a)). $\nu$ is set to be $2/3$. The PL spectrum
consists of two peaks originating from singlet and triplet trionsWójs, Quinn,
and Hawrylak (2000); Yusa, Shtrikman, and Bar-Joseph (2001). The rate equation
describes the dynamics of the trion population undergoing generation and
recombination. Although the laser pulse resembles a delta function in time,
the response of the illumination and detection parts of the system is
broadened. To extract the lifetime of trions, we calculate the convolution of
the population decay function and the impulse response of the optical system.

$\displaystyle f(t,\tau)$
$\displaystyle=\int_{-\infty}^{\infty}N(t^{\prime},\tau)I_{\mathrm{sys}}(t-t^{\prime})dt^{\prime}$
(1a) $\displaystyle N(t,\tau)$
$\displaystyle=\left\\{\begin{array}[]{cc}0&(t<0)\\\
e^{-\frac{t}{\tau}}&(0\leq t)\end{array}\right.$ (1d) $\displaystyle
I_{\mathrm{sys}}(t)$
$\displaystyle=\frac{1}{\sqrt{2\pi}t_{\mathrm{s}}}\exp{\left(-\frac{t^{2}}{2t_{\mathrm{s}}^{2}}\right)},\
$ (1e)

where $N(t,\tau)$ is the number of trions with a lifetime $\tau$, and
$I_{\mathrm{sys}}(t)$ is the impulse response of the system with a full width
at half maximum (FWHM) of $2t_{\mathrm{s}}\sqrt{2\mathrm{ln}2}$. The
$t_{\mathrm{s}}$ corresponds to the standard deviation. The following function
containing the error function $\mathrm{erf}(t)$ can be used to compute the
convolution.

$\displaystyle f(t,\tau)$
$\displaystyle=\frac{1}{2}\left(\mathrm{erf}\left(\frac{t\tau-
t_{\mathrm{s}}^{2}}{\sqrt{2}t_{\mathrm{s}}\tau}\right)+1\right)\exp\left(\frac{-2t\tau+t_{\mathrm{s}}^{2}}{2\tau^{2}}\right)$
(2a) $\displaystyle\mathrm{erf}(t)$
$\displaystyle=\frac{2}{\sqrt{\pi}}\int_{0}^{t}e^{-x^{2}}dx.\ $ (2b)

The time dependence of the PL can be fitted by
$If(t-t_{0},\tau_{\mathrm{d}})$, where $I$ is a coefficient and $t_{0}$ is the
time when the pump pulse arrives at the sample. $\tau_{\mathrm{d}}$, the decay
time for the singlet and triplet, is respectively $361$ and $214$ ps (Fig.
2(b)). The speed of the edge is experimentally reported as on the order of
$10^{4-5}$ m$/$sKamata  _et al._ (2014); Hashisaka  _et al._ (2017); Matsuura
_et al._ (2018); Kamiyama  _et al._ (2022). Therefore, the edge excitation
propagates $2\sim 30$ $\mu$m within $200\sim 300$ ps. However, the time
resolution of the stroboscopic PL measurement is limited by the duration of
the PL intensity peak which is mostly determined by the 200 to 300 ps decay of
the trion population. This means that the time resolution needed to fully
capture the dynamics of edge propagation is insufficient.

Figure 3: (a) Reflectance change $\Delta R$ as a function of the time
difference $\Delta t_{\mathrm{probe}}$ between the pump and probe pulses (red
line) at $\nu=2/3$ ($B=11.5$ T and $T=60$ mK). Their fitting curves (dashed
blue line: one-$\tau_{\mathrm{d}}$ model and black line:
two-$\tau_{\mathrm{d}}$ model) are calculated (see body text). The measurement
position is $x=20$ $\mu$m. The photon energy of the pump and probe pulse is
1.56–1.57 and $1.53$ eV, and the laser power of the pump and probe pulses is
$\sim$200–400 and $\sim 100$ nW, respectively. Both the pump and probe pulses
have a $\sim 1$ ps duration. The integrated time per data point is $40$ s. (b)
Magnified view of Fig. 3(a) around $\Delta t_{\mathrm{probe}}=0$. Measurement
data is plotted in red circles.

Owning to the Kramers–Kronig relation, reflectance measurement of light which
resonates with trions is equivalent to transmittance measurement, and provides
the relaxation time of trions. Therefore, we conduct a measurement of the
pump-probe reflectance as a probe in place of the PL measurement to enhance
the time resolution of the space- and time-resolved measurement. In addition
to the pump pulse, we introduce a probe pulse. The probe pulse’s peak energy
is set so that it will resonate with the trion’s PL peak ($1.53$ eV). The
intensity of the pump pulse is modulated by a chopper with a frequency of $1$
kHz, and the modulation component of probe pulse intensity reflected from the
sample is detected by a lock-in amplifier connected to the PD (Fig. 1(a)). The
long pass filter ($<1.54$ eV) in front of the PD blocks the pump light. The
time difference $\Delta t_{\mathrm{probe}}$ between the pump and probe pulses
can be changed by the delay stage.

We measure the reflectance change $\Delta R$ as a function of $\Delta
t_{\mathrm{probe}}$ at $\nu=2/3$ close to the sample edge ($x=20$, $y=1$
$\mu$m). $\Delta R$ increases rapidly by as much as $~{}0.08\%$ at $\sim 20$
ps; then it decays gradually (Fig. $3$). To fit the $\Delta R$ dependence on
$\Delta t_{\mathrm{probe}}$, we used one-$\tau_{\mathrm{d}}$ and
two-$\tau_{\mathrm{d}}$ models which presuppose one and two decay processes
for trions, respectively. The fitting function for the one-$\tau_{\mathrm{d}}$
model is $-If(\Delta t_{\mathrm{probe}}-t_{0},\tau_{\mathrm{r}})+If(\Delta
t_{\mathrm{probe}}-t_{0},\tau_{\mathrm{d}})$ and that for the
two-$\tau_{\mathrm{d}}$ model is $-(I_{1}+I_{2})f(\Delta
t_{\mathrm{probe}}-t_{0},\tau_{\mathrm{r}})+I_{1}f(\Delta
t_{\mathrm{probe}}-t_{0},\tau_{\mathrm{d1}})+I_{2}f(\Delta
t_{\mathrm{probe}}-t_{0},\tau_{\mathrm{d2}})$. Here, we consider a rise time
$\tau_{\mathrm{r}}$, which corresponds to the relaxation time from initially
high-energy states of photo-excited electrons and holes to the trion state.
This is important because the time resolution of our pump-probe reflectance
measurement is comparable to $\tau_{\mathrm{r}}<\sim 10$ ps (Fig. 3(b)). The
optimized parameters for the one-$\tau_{\mathrm{d}}$ model are
$\tau_{\mathrm{r}}=5.2$ ps, $\tau_{\mathrm{d}}=359$ ps, and $I=0.0828$ %
(dashed blue curves in Fig. $3$). Compared to the one-$\tau_{\mathrm{d}}$
model, the two-$\tau_{\mathrm{d}}$ model with $\tau_{\mathrm{r}}=8.0$ ps,
$\tau_{\mathrm{d1}}=179$ ps, $\tau_{\mathrm{d2}}=2018$ ps, $I_{1}=0.0732$ %,
and $I_{2}=0.0225$ % (black curves) clearly fits better than the
one-$\tau_{\mathrm{d}}$ model. Here, $\tau_{\mathrm{d1}}$ can be attributed to
the relaxation time of trions. $\tau_{\mathrm{d2}}$ may be related to the spin
decay time Anghel  _et al._ (2016), but further investigation is required for
more details.

Around $\Delta t_{\mathrm{probe}}=0$, there is a measurable change in $\Delta
R$ within the $2$-ps time step of our measurement (see Fig. $3$(b)). This fact
shows that the pump-probe reflectance measurement’s time resolution has been
enhanced to the same order as the laser pulse duration time ($\sim 1$ ps).

We integrate the pump-probe reflectance measurement into the stroboscopic
measurement, where the electrical excitation of the edge by voltage pulses and
subsequent detection by optical light pulses are coordinated. In this
stroboscopic measurement, the population of trions is probed within the
duration of the probe pulse ($\sim 1$ ps). The pulse generator electrically
controls $\Delta t$, which is the time difference between the pump and voltage
pulses. $\Delta t_{\mathrm{probe}}$ is fixed to $20$ ps such that $\Delta R$
becomes maximum (Fig. $3$(b)). The voltage pulse is a square pulse with a
$150$ mV amplitude and a $0.5$ ns duration (Fig. $4$(a)). In Fig. $4$(b) we
show $\Delta R$ as a function of $\Delta t$ at several points of varying $x$
and fixed $y$.

Figure 4: (a) Voltage waveform applied to the excitation gate to excite the
edge. An oscilloscope and pulse generator were connected to measure it.
Typical rise and fall time is $\sim 80$ ps for $20-80\%$ (b) Reflectance
change $\Delta R$ (solid lines for the left $y$-axis) and PL intensity (red
dotted line for the right $y$-axis) as a function of $\Delta t$ (offset for
clarity). The PL intensity was measured at $x=10$ $\mu$m, and the $\Delta R$
was measured at $x=5$, $10$, $15$, and $20$ $\mu$m. For all measurements the
position of $y$ was $1$ $\mu$m. The integration time is $400$ s per point for
$\Delta R$. (c) $x$ as a function of the $\Delta t$ value where dips are
observed in the traces of Fig. $4$(b). The horizontal bars represent $50\%$
changes in $\Delta R$.

At $x=10$ $\mu$m, small and large dips are observed around $\Delta t=0.1$ and
$0.6$ ns labeled as A and B, respectively(Fig. $4$(b)). Similar dips are seen
in the PL intensity, but the PL dips are wider and more overlapping (dashed
red curve in Fig. $4$(b)). This can be attributed to the difference in the
time resolution. In stroboscopic PL measurement, the time resolution is
limited by the lifetime of trions excited by the pump pulse laser. In
contrast, the time resolution of the stroboscopic pump-probe reflection
measurement is the duration of the probe pulse, and is unrelated to the trion
lifetime. Moreover, because the transient of the PL intensity is temporally
asymmetric, the broadening of the intensity in the stroboscopic PL measurement
is also asymmetric: in Fig. 4(b) at $t<0$ ns, $\Delta R$ is constant before
the voltage pulse is applied, whereas the PL intensity already starts to
change. Therefore, the time resolution of the stroboscopic pump-probe
reflection measurement is superior to that of the stroboscopic PL measurement.

The $\Delta t$ at which the dips A and B appear increases as $x$ increases. In
the case of dip A, $\Delta t$ at which dip A appears is proportional to $x$
and the slope of $x$ v.s. $\Delta t$ (red dotted line in Fig. 4(c)) is $\sim
1\times 10^{5}$ m/s, which is in good agreement with the edge excitation’s
propagation speedKamata  _et al._ (2014). The amplitude of dip A is about
$0.01\%$ and does not decrease as $x$ increases. On the other hand, $\Delta t$
at which dip B appears behaves differently: the dip at $x=20$ $\mu$m is
noticeably smaller than that close to the excitation gate at $x=5$ $\mu$m.
Furthermore, $\Delta t$ at which the dip B appears is not proportional to $x$.
These findings imply that in addition to the typical form of edge excitation
known to occur in the Tomonaga-Luttinger liquid, there may also be alternative
excitation mechanisms present close to the excitation gate. To discuss the
details of this excitation, further experiments are required.

In summary, we reported on the dynamics of the FQH edge probed by stroboscopic
PL and pump-probe measurements. While the time resolution of stroboscopic
pump-probe measurement can be as short as the duration of the probe pulse
($\sim 1$ ps), the time resolution of stroboscopic PL measurement is
constrained by the radiative recombination time of trions. Such an ultra-fast
stroboscope measurement enables us to distinguish the normal mode of edge
excitation from other higher energy non-linear excitations. The metric tensor
$g_{\mu\nu}$ of the $(1+1)=2$-dimensional spacetime in the quantum universe
simulator implemented in QH edges can be obtained via a direct adaptation of
this method.

The authors are grateful to T. Fujisawa and M. Hotta for fruitful discussions.
This work is supported by a Grant-in-Aid for Scientific Research (Grants Nos.
19H05603, 21F21016, 21J14386, and 21H05188) from the Ministry of Education,
Culture, Sports, Science, and Technology (MEXT), Japan.

Data availability statement: The data that support the findings of this study
are available upon reasonable request from the author.

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