# Highly sensitive single-molecule detection
in slow protein ion beams

M. Strauß1, A. Shayeghi1, M.F.X. Mauser1, P. Geyer1,
T. Kostersitz1, J. Salapa1, O. Dobrovolskiy1, S. Daly2, J. Commandeur2,
Y. Hua3, V. Köhler3, M. Mayor3, J. Benserhir4, C. Bruschini4,
E. Charbon4, M. Castaneda5, M. Gevers5, R. Gourgues5, N. Kalhor5,
A. Fognini5, and M. Arndt1∗
1University of Vienna, Faculty of Physics & VDSP & VCQ, Boltzmanngasse 5,
A-1090 Vienna
2MSVISION, Televisieweg 40, 1322 AM Almere, The Netherlands
3Department of Chemistry, University of Basel, St. Johannsring 19, CH-4056
Basel, Switzerland
4Advanced Quantum Architecture Laboratory, EPFL, Rue de la Maladière 71b,
CH-2002 Neuchâtel, Switzerland
5Single Quantum, Rotterdamseweg 394, 2629 HH, Delft, The Netherlands
∗Correspondence should be sent to<EMAIL_ADDRESS>
(June 26, 2023)

###### Abstract

The analysis of proteins in the gas phase benefits from detectors that exhibit
high efficiency and precise spatial resolution. Although modern secondary
electron multipliers already address numerous analytical requirements, new
methods are desired for macromolecules at low energy. Previous studies have
proven the sensitivity of superconducting detectors to high-energy particles
in time-of-flight mass spectrometry. Here we explore a new energy regime and
demonstrate that superconducting nanowire detectors are exceptionally well
suited for quadrupole mass spectrometry. Our detectors exhibit an outstanding
quantum yield at remarkably low impact energies. Notably, at low ion energy,
their sensitivity surpasses conventional ion detectors by three orders of
magnitude, and they offer the possibility to discriminate molecules by their
impact energy and charge. By combining these detectors into arrays, we
demonstrate low-energy ion beam profilometry, while our cryogenic electronics
pave the way for future developments of highly integrated detectors.

## Introduction

Mass spectrometry is a ubiquitous tool in the life sciences, chemistry, and
physics, allowing detection, identification, and analysis of objects ranging
in size from atoms to large biopolymers. Some of the most wide-spread
instruments, such as quadrupole (QMS) [1] and time-of-flight mass
spectrometers (TOF-MS), use detectors that rely on secondary electron
multiplication (SEM). While SEM detectors have a quantum yield of $\eta\simeq$
90 % for electrons [2], the efficiency for macromolecules depends strongly on
their velocity, mass, and structure [3, 4] and can drop to $\eta\simeq
10^{-5}$ for proteins with kinetic energies of only a few tens of electron
volts. It only becomes sizeable for impact velocities of $v>20$ km/s, i.e. an
impact energy of $40$ keV for masses beyond $20$ kDa [5]. Since more than
$80\,$% of all known proteins are found in this mass regime [6], efficient
detectors are highly desirable.

Figure 1: Quadrupole mass spectrometry with quantum detection (ESI-QMS-QD).
(A) Proteins are volatilized using electrospray ionization (ESI) and charge-
reduced in bipolar air by a corona discharge (CD). They are filtered in a
quadrupole mass selector (QMS) and fly through a radio frequency hexapole
guide (not shown) towards a quadrupole bender. This steers them to either a
time-of-flight mass spectrometer (TOF-MS), a phosphor screen (PhS), or the
superconducting single particle detector array (SSPD). A Faraday plate (FP)
can be shifted into the position of the SSPD to calibrate the incident ion
current in the limit of high flux. (B) Corona charge reduction of high mass
proteins (C) Multi-pixel array and meander structure $D_{1}$.

The detection challenge is alleviated when working with highly charged ions
since their kinetic energy is proportional to their charge q and the
accelerating electric potential $U_{acc}$. On the other hand, a low charge
state is often favorable as it reduces the complexity of the mass spectrum and
facilitates the identification of spectra lines. In that case detectors are
required which can efficiently detect slow ions. The capability to analyze
biomolecules at low velocity and long interaction time can also open
additional new analytic opportunities in combination with optical spectroscopy
and deflectometry. Here, we explore superconducting nanowires as quantum
detectors for biomolecules at energies about 100 times smaller than commonly
required for SEM detection, providing the same or even better spatial
resolution, with a potential to improve on that by more than an order of
magnitude.

Superconducting detectors are intriguing as they possess a small energy gap in
the few meV regime that allows them to be highly sensitive to low energy
quanta. This gap can be bridged by the impact energy of a particle or by the
absorption of a photon. A variety of cryogenic sensors have found
interdisciplinary application in numerous research fields in form of
bolometers [7], transition edge sensors [8, 9], kinetic inductance detectors
[10], superconducting tunneling junctions [11, 12], or superconducting
nanowire detectors (SNWDs), also known as superconducting single-particle
detectors (SSPDs) [13, 14]. All these sensors found important use cases in
photonics and a few of them have also been successfully used in TOF-MS [12],
among them transition edge sensors [8], superconducting tunnel junctions and
SSPDs.

In comparison to superconducting tunnel junctions, SSPDs combine good spatial
and temporal resolution with a working temperature above 3 K, where
substantially higher cooling power is available at moderate cost [15]. This is
important for up-scaling current prototypes to multi-pixel devices, which are
needed for most applications. SSPDs are typically etched into a
superconducting film in a meander structure on top of a silicon-based
substrate. The nanowire is driven by a bias current $I_{b}$ close to its
critical current, typically in the 10 - 100 µA range. If a particle impacts
the detector, the energy released to the superconductor locally breaks up
Cooper pairs, causing a quantum-phase transition from the superconducting to
the normal conducting state. As a result, a normal conducting hot spot is
generated. When the current around the resistive area exceeds the local
critical current density, the strip will become normal conducting along its
full width. The continuous driving of a current into the now resistive strip
leads to a fast voltage peak that triggers the signal. Thermal relaxation to
the substrate resets the detector.

SSPDs were first developed for applications in photonics [13] where response
times below 20 ps [16, 17] are possible. Detection efficiencies of up to 99 %
can be realized even at telecom wavelengths [18]. In addition, single photon
sensitivity at MIR range of 10 µm wavelengths have been demonstrated [19].
SSPDs are impact detectors but were also designed to have photon number
resolution [20]. They have found applications in the explorations of the
foundations of physics [21, 22], in quantum optics [23, 24] and quantum
information processing [18, 25], astronomy [26] and molecular science [27].
Recently, nanowire detectors have also been used in TOF-MS at high kinetic
energy [14, 15, 28, 29], always for the explorations of fast molecular beams.
However, in the low-energy regime only a single study has been reported,
without protein mass identification [30].

Here, we explore high-resolution quadrupole mass spectrometry of proteins in
low charge states, with highly efficient superconducting nanowire detector
arrays, at energies well below the requirements of conventional secondary
electron detectors. To upscale this technology, we study different detector
sizes and geometries and discuss how low-noise cryogenic electronics can
enable large multi-pixel arrays.

Our detectors are manufactured as arrays of 10 nm thick NbTiN nanowire
meanders, each of which with a filling factor of 50 % and a critical
temperature of $T_{c}\simeq$ 10 K [31]. A pulse tube cooler keeps the SSPD at
an operating temperature of 3.7 K. Detector $D_{1}$, fills an area $A_{1}$ of
20 $\times$ 20 $\mu m^{2}$ in meander form with a line width $w_{1}$ of 100 nm
and a pitch $p_{1}$ of 200 nm. These small detectors are highly sensitive,
fast, and spatially resolving at low particle energies. In order to increase
the effective detection area, a 32-pixel linear array was fabricated, where
all active pixels were connected via individual leads to the signal
conditioning and processing electronics at room temperature. Future
parallelization to hundreds of elements will require cryogenic onboard
electronics, which we here explore as chip $D_{1,b}$, augmented by one low-
noise amplifier (LNA) per pixel. The purpose of chipset $D_{1}$ is to realize
a multi pixel array with high spatial and temporal resolution. In molecular
beam research and mass spectrometry, however, total detector size is also
important. Therefore, the pixel area was increased to $A_{2}$ = 200 $\times$
200 $\mu m^{2}$, yielding a hundred-fold detection area increase. Detector
$D_{2}$ combines ten such pixels, each with a single line with $w_{2}$ = 500
nm and $p_{2}$ = 1000 nm.

## Results

### Protein Mass Spectra

A tandem mass spectrometer was extended by a cryogenic vacuum chamber which
hosts the nanowire detectors (Figure 1). Proteins are ionized using
electrospray ionization [32] and optionally charge reduced by corona discharge
[33]. The analyte ions are guided into a customized quadrupole mass selector,
which can be scanned with a resolution of $\Delta m/m\simeq 1:1000$ up to a
mass to charge ratio of m/z = 30 000 Th. A hexapole ion guide and subsequent
ion deflector steer the ions either towards a conventional TOF-MS with MCP
detector, or a phosphor screen (PhS) with a counting photo-multiplier.
Additionally, the ions can be directed towards the SSPD with the use of a
quadrupole deflector. The MCP area $A_{MCP}$= 700 $mm^{2}$ is about 100 times
larger than the ion beam and 5000 times larger than a single pixel of detector
$D_{2}$. We have therefore chosen an integration time of 120 s for the
classical channel and 1200 s for the sum of six quantum channels to achieve
comparable signal-to-noise levels.

Figure 2: Quantum and classical detection of an ESI-quadrupole mass spectrum
of high mass proteins. Charge-reduced concanavaline A (27 kDa) is detected by
(A) the quantum SSPD array $D_{2}$ at $U_{acc}$ = 190 V (B) the classical
phosphor screen with $U_{acc}$ = 10 kV.

The kinetic energy of the ions at each detector, $E_{kin}=qU_{acc}$, is
determined by their charge q and by the potential difference $U_{acc}$ of the
detector with respect to the last acceleration electrode. In TOF-MS, the
potential $U_{acc}$ reaches 13 kV, while on the phosphor screen it is limited
to $U_{acc}$ = 10 kV. In contrast, we operate our quantum detector at
$U_{acc}\simeq$ 0.1 – 0.2 kV and find that it can accept 100 times lower
impact energies than required by conventional devices. We could determine the
detection efficiency quantitatively for impact energies down to 100 eV and can
observe well resolved mass spectra down to 40 eV. We demonstrate this in
Figure 2, where we present the QMS selected mass spectrum of charge reduced
concanavaline A (27 kDa). The signal to noise ratio (SNR) of the SSPD detector
competes favorably with the conventional phosphor screen and MCP, even for a
50 times lower acceleration voltage.

Figure 3: Comparison of protein mass spectra in ESI-QMS-SSPD an ESI-TOF-MCP.
Insulin (5.7 kDa), cytochrome C (12.3 kDa) and myoglobin (17 kDa) cations are
electrosprayed and analyzed by a quadrupole mass filter with SSPD detector (A,
C) and by a time-of-flight mass spectrometer with an MCP detector (B, D). The
dense group of peaks seen in sub-figure A and B is associated with multiply
charged proteins without any charge reduction (see Suppl. Inf.), while the
lower spectra C and D are enormously simplified by charge-reduction. Here,
post-acceleration occurs at an acceleration voltage of $U_{acc}$ = 190 V and
$U_{acc}$ =13 000 V for the SSPD and the MCP, respectively.

To demonstrate high mass and low energy detection of various compounds in a
wide mass range, we show in Figure 3 (A, C) a protein mass spectrum obtained
by high-resolution quadrupole mass spectrometry combined with low energy SSPD
detection and compare it with a TOF-MS measured with an MCP detector in Figure
3 (B, D). For this we analyze a protein mix of insulin (5.7 kDa), cytochrome C
(12.3 kDa) and myoglobin (17 kDa), after electrospray ionization Figure 3 (A,
B) and charge reduction Figure 3 (C, D). The charge reduction drastically
simplifies the spectral assignment and profits from the high sensitivity of
the SSPD. Note that here the SSPD achieves a similar quantum yield per
detector area as the MCP. However, it does this at ion energy, where
conventional SEM detectors are 1000 times less efficient.

### Energy and charge dependence

Having proven SSPDs as excellent molecular impact counters for QMS-MS, we now
analyze, how they can additionally contribute to assessing the charge and
impact energy of an ion, a feature exceeding the capability of most mass
spectrometers. In Figure 4 we plot the insulin ion count rate as a function of
the bias current $I_{b}$ for a single pixel of $D_{2}$. The curves are taken
for varying charge states q and different values of the acceleration voltage
$U_{acc}$. We find that ions of the same kinetic energy $E_{kin}$ fall onto
the same curve, while ions at different kinetic energies (380 eV, 190 eV, 95
eV) can be clearly discerned by the onset of the signal as a function of the
bias current.

Figure 4: Energy-dependence of the SSPD detection efficiency. The normalized
protein count rate is shown as a function of the SSPD bias current $I_{b}$,
for insulin in three different charge states (q = +1, +2, +4) and at three
different acceleration voltages ($U_{acc}$ = 47.5, 95 V, 190 V). Three sets of
curves confirm that the normalized count rate clearly depends on the kinetic
energy $E_{kin}=q~{}U_{acc}$, but not separately on charge.

These curves suggest that under our experimental conditions an energy
resolution of 20 to 30 eV can be achieved. Our result also shows that ions of
the same mass-to-charge ratio but different charge, such as for instance a
doubly charged insulin dimer and a singly charged insulin monomer will be
distinguishable when monitoring the SSPD bias current curves in addition to
the QMS transmission. A better energy resolution will be achieved in future
device designs.

### Confirming the hot-spot model

In all bias current curves shown in Figure 4 we can determine a threshold bias
current $I_{th}$ via the intersection of the asymptotes at low and high bias
current [34]. This is shown in Figure 5A for singly charged insulin. The
dependence of $I_{th}$ on the molecular impact energy, is shown in Figure 5B.
It is well reproduced by the analytical form
$I_{th}=I_{c}(1-\gamma\sqrt{E_{kin}}/w_{2})$, which is predicted by a normal-
core hot-spot model [35]. From a fit to the data, we can deduce the critical
current $I_{c}$ = 58.1 µA, from which we learn about the underlying physical
transduction mechanism by comparing $I_{c}$ with the depairing current
$I_{dep}(T)=I_{dep}(0)\left[1-(T/T_{c})^{2}\right]^{3/2}$.

Here $I_{\text{dep}}=0.74w\Delta^{3/2}(0)/(eR_{\square}\hbar D)$, where
$\Delta(0)$ is the superconducting energy gap at $T=0\,\text{K}$, $e$ is the
electron charge, $R_{\square}$ is the sheet resistance, and $D$ is the
electron diffusion coefficient [36]. With $R_{\square}$ = 380 $\Omega$,
$T_{c}$ = 10.5 K, and D = 0.48 $cm^{2}/s$ for NbTiN [37], we obtain
$I_{dep}(0)\simeq 350\mu$A and $I_{dep}(3.7K)\simeq 285$ $\mu A$.

Figure 5: Corroboration of the hot spot detection model for proteins.
Dependence of the threshold bias current $I_{th}$ on the impact energy. (A)
$I_{th}$ is found at the intersection of the two asymptotes (straight lines)
of the normalized count rate plotted as a function of bias current $I_{b}$. We
show data for singly charged insulin impacting on a single pixel of detector
$D_{2}$. (B) $I_{th}$ as a function of the kinetic energy
$E_{kin}=q~{}U_{acc}$. The dashed line represents the analytical normal-core
hot-spot model, which fits the experiment very well.

Since the critical current amounts to only 20 % of the depairing current
$I_{c}/I_{dep}$ = 0.2, this supports a hot spot detection mechanism [14] and
excludes an alternative vortex-assisted detection model [38], as this would
require $I_{c}/I_{dep}>0.5$. The hot-spot diameter in NbTiN is here estimated
to be $D_{HS}=4\sqrt{\tau~{}t~{}h~{}D}\simeq$ 35 nm, where $\tau_{th}$ is the
thermalization time. We conclude that detector $D_{2}$ is working in the
hotspot regime.

### Momentum dependence

For particles to create a normal-conducting hotspot, they must couple to
either the charge carriers or the phonons in the nanowire. It is therefore
interesting to explore if it is possible to distinguish between the two
effects. Since for fixed kinetic energy the momentum transfer increases with
mass, $p=(2mE_{kin})^{1/2}$, we study the bias current curves for molecules of
the same kinetic energy, $E_{kin}$ = 190 eV, but different mass, momentum and
number of constituents. Figure 6 shows count rates of detector type $D_{2}$
for singly charged rhodamine 6G (C28H31N2O3Cl, $N_{atoms}=65$, MW $=479\,$Da),
bovine insulin (C254H377N65O75S6, $N_{atoms}$ = 777, MW = 5733 Da) and
myoglobin (153 AA residues, $N_{atoms}$ = 1304, MW = 16952 Da). They cover a
mass ratio of about 1:12:35, a momentum ratio of about 1:3.5:6 and a ratio of
atom numbers of about 1:12:20.

Figure 6: Mass/momentum/structure-independence of the SSPD detection
efficiency. At equal impact energy (190 eV) and charge we find the same
normalized detector response curve for molecules of vastly different structure
and complexity, specifically rhodamine 6G, insulin, and myoglobin. This
suggests that kinetic energy is of prime relevance in the transduction
mechanism, while structure, atom number, charge and mass are not. All
measurements were made using a single pixel of detector array $D_{2}$.

Here we find that neither mass nor momentum nor structural complexity seem to
play a role under these experimental conditions. The detection mechanism seems
to depend only on the impact energy. This is interesting as one may expect to
see the influence of competing energy loss channels, such as the deformation
or dissociation of a molecule during impact, which all depend on atom number
and complexity. Future experiments will explore the detection mechanism at ion
energies down to 1 eV, which were not yet accessible in our present
experiment. This composition and mass-independence is a plus and we use it as
a feature in the calibration of the absolute detector quantum yield below.

### Protein ion beam profiling

Given the high quantum efficiency of our nanowire detector for massive
particles, its good time response and spatial resolution, one can record a
spatial image of the protein beam. We do this using detector $D_{2}$ instead
of $D_{1}$ to trade resolution for higher count rates. Figure 7 shows that
this already provides well-resolved and useful images to optimize steering
optics in ion beam trajectories in mass spectrometry. A pair of X/Y-electrodes
in front of the SSPD detector are used to deflect the incident ion beam with
high reproducibility, which can be calibrated in one axis against a mechanical
translation, with 30 µm accuracy. We use it here to record a spatial profile
of $insulin^{+5}$ with impact energy $E_{kin}$ = 1 keV. Note that the ion beam
is asymmetric because of the cylindrical symmetry of the ion bender.

Figure 7: SSPD detection of a mass-selected Insulin ion beam profile. The X
position represents the mechanical shift of the SSPD array. The Y shift
quantifies the ion beam deflection in an external field (see Suppl. Inf). The
acquisition time for this profile was approximately 1 hour, using a single
pixel of detector $D_{2}$.

For beams of high flux, the resolution can be improved by using detector
$D_{1}$. However, the two SSPD detector geometries differ in absolute counts
(by a factor of about 100 in pixel area) and in dark count rate. While $D_{1}$
is sensitive to photons and must be shielded from stray radiation, when
operated at high bias current, $D_{2}$ is blind to black body radiation and
achieves a dark count rate as low as 0.022 cps. On the other hand, $D_{1}$
offers a ten-fold better spatial resolution than $D_{2}$, which is competitive
with state-of-the-art MCPs and an advantage for molecular beam profilometry
and deflectometry.

### Absolute detection quantum yield

Superconducting nanowire detectors are known to be efficient photon detectors,
but varying numbers have been reported in literature on the absolute quantum
yield for low-energy ions [39] and absolute calibration for molecules at low
kinetic energies has been a challenge. Normalized count rates as shown in
Figure 6, may lead one to assume that observing saturation is an indicator for
a detection yield of $100\,$%. However, this may be wrong for fast atomic
ions, which can penetrate the nanowire and therefore deposit only a fraction
of their energy. It could also be wrong for fast proteins if they converted
impact energy into internal vibrational excitation or even dissociation.

Figure 8: Detection quantum yield for large molecules on a SSPD and a MCP
detector. The grey and red squares show the detection quantum yield of two
different pixels of detector $D_{2}$ as a function of the impact velocity
(energy). From left to right: Vitamin $B_{12}$ at 100 eV and 200 eV, Insulin
at 600 eV and 1000 eV of kinetic energy. These molecules have different atom
numbers, masses, momenta, and energies, but they all have detection
efficiencies 2 - 3.5 orders of magnitude greater than the detection efficiency
of a MCP detector. The red and blue line show model values for multi-channel
plates based on extensive experimental studies [40].

To solve this question quantitatively we here relate the single molecule count
rate of a single pixel of $D_{2}$ to the electrical current that is measured
by a large Faraday plate that can be positioned to intercept the ion beam at
the same location as detector $D_{2}$. The detection efficiency was measured
for singly charged vitamin $B_{12}$, $insulin^{5+}$ and $insulin^{3+}$ for two
different impact velocities (energies). Mixing the molecular species and their
charge states is justified since our measurements have shown that the
detection efficiency should be mass independent and low-mass ions are favored
here because of their higher flux, which facilitates signal comparison with a
classical femto-ammeter. In order to normalize the current induced by a 3 mm
ion beam on a 1 cm Faraday plate to the SSPD signal on a 200 µm detector, we
measure the ion beam profile and relate the count rate to the beam shape and
detector area, as shown in Figure 8 (see Suppl. Inf.).

We measure a molecule detection quantum yield of $\eta\simeq 1$ within an
energy range of 100 eV to 1000 eV. This remarkable value is 2-3.5 orders of
magnitude greater than the secondary electron yield in a typical multichannel
plate for ions of the same impact velocity [40]. For a detailed discussion of
the systematic and statistical errors, see the supplementary information.

We find a similar area-normalized quantum yield also for of our small
detectors $D_{1}$. An array of $D_{1}$ can, however, outperform an array of
$D_{2}$ in dynamic range because a smaller pixel will only saturate at a
higher molecular flux. Its smaller kinetic inductance also leads to a faster
relaxation time. Both detector types $D_{1}$ and $D_{2}$ are clearly
sufficient for most typical beams in mass spectrometry and analytical
molecular beam research. The detector pixels $D_{2}$ are well adapted to
dilute, wide molecular beams both because of their larger area and low dark
count rate. Detector $D_{1}$ is an interesting alternative for applications
with very intense molecular beams in need of high temporal resolution.

### Cryogenic amplifiers towards scalable devices

One would want to substantially upscale the number of pixels for applications
with wide particle beams, for example in molecular impact correlation
experiments, or in detectors for quantum interferometry. A large area detector
such as $D_{2}$ is driven by a high bias current of $50-70\,\mu$A and
generates a sizeable voltage pulse across a $50\,\Omega$ impedance when the
SSPD becomes normal conducting. However, the better resolving detector $D_{1}$
is typically driven by $>6\,\mu$A and an impact generates only $300\,\mu$V.
Low-noise cryogenic amplifiers (LNA) are thus useful to boost the signal-to-
noise ratio and to reduce the rise time of the output pulse of the SSDP for
later on-board digitization and time-tagging. For that purpose, we have
realized an amplifier with high gain, high bandwidth, low noise, and low power
consumption. We have realized such devices using a heterojunction-bipolar
transistor (HBT) BiCMOS process.

Figure 9: Onboard cryogenic electronics. The diagram illustrates the
integration of a SiGe low-noise amplifier (LNA) with the superconducting
nanowire detector on a shared printed circuit board at $T=3.8$ K. These
detectors with onboard electronics are used to integrate larger pixel numbers,
to go to larger detection areas comparable to those of MCPs.

It exhibits good noise and speed already at room temperature and we find even
an improvement of the current gain by about 300 % at $T=3.8\,K$ (15, 41, 42).
Our device measures only $480\times 280\,\mu$m2 including the pads and it
combines an average noise temperature of $5\,$K in the frequency range of
$0.1-8.8\,$GHz with a good weak-signal linearity. To demonstrate our
integrated concept, we have joined an array of eight SSPD pixels of type
$D_{1}$ with eight of our new LNAs to realize eight detectors $D_{1b}$ as
illustrated in Figure 9 for one pixel. This system was also successfully used
in ESI-QMS spectrometry on insulin, and it will allow up-scaling to much
larger pixel numbers in the future.

## Discussion

We have demonstrated that superconducting nanowire arrays can serve as highly
efficient detectors for massive ions even at energies where conventional
detectors fail. We can record mass spectra of singly charged insulin down to
kinetic energies nearly 100 times lower than typically required for time-of-
flight mass spectrometry. This is compatible and beneficial for continuous
mass spectrometers, such as quadrupole or sector field instruments, where
resolution improves with decreasing ion energy.

The energy dependence of the SSPD threshold current $I_{th}$ makes
distinguishing n-fold charged polymers of a given mass possible, where
conventional detection methods can only infer the presence of isobaric species
of different charge by analysis of the isotope distribution. This can be best
observed at low ion energy, where the rate of increase for the threshold
current $I_{th}$ is greatest. We have demonstrated a quantum yield close to
$100\,$% for molecules with kinetic energies of $100\,$eV or higher,
independent of their structure, mass, charge or momentum.

At lower energy the yield may be influenced by effects of mass and complexity
of molecules. Molecular imaging and molecular dynamics simulations show that
these entities can deform upon surface impact (43, 44). Even though
vibrational excitations will eventually relax their energy to the cold
nanowire this process occurs on a time scale of nanoseconds, i.e., much slower
than the relaxation time of the superconducting detector. Moreover,
fragmentation at high impact energy, will lead to an energy loss that might
reduce the detection efficiency of the nanowire. Conversion of kinetic to
internal molecular energy may thus reduce the detection yield. If molecules
bounce off rather than stick to the surface, additional detection losses might
occur for the detection with a SSPD, if not enough energy is transferred to
the superconductor. Furthermore, the coating of the detector with a protein
beam could act as a buffer for incoming particles. However, to deposit a
single monolayer on the detector surface, it would need to continuously run
for a few hundreds of hours in nominal operating conditions.

While mass spectrometry and neutral molecular beam research could be served by
a 10 $\times$ 10 array of detector type $D_{2}$, even more versatile
applications with even better spatial resolution will profit from up-scaling
the pixel number and using detector array $D_{1}$. Here we analyze some
thermal limits related to on-board current sourcing, signal amplification,
readout, and cabling. Even though superconducting nanowires transport current
without loss, the bias current $I_{b}$ must be generated by sending a voltage
into a resistor. For detectors of type $D_{1}$ ($A_{1}=400~{}\mu m^{2}$), the
power per pixel can be individually set and limited to P = 6 $\mu$W if we
assume a bias current of $I_{b}\simeq 6~{}\mu$A, and a source resistance R = 1
k$\Omega$. Onboard current sourcing is thus still thermally compatible with
100 000 individually optimized pixels, i.e., an area of $40~{}mm^{2}$,
assuming a standard 1W closed cycle cooler.

Our experiments with detector type $D_{2}$ have shown that the useful sensor
area can be increased by a factor of 100 while their bias current only needs
to be increased by a factor of ten. The same cooling power budget would thus
allow sourcing all these detectors individually across a total detector area
of 400 $mm^{2}$. New developments in the fabrication process, may improve the
uniformity of the meander, that makes the detection properties of each pixel
more reproducible. This will allow to source them all with the same bias
current as one large area multi-pixel array. In this case bias current
sourcing no longer limits the 4 K heat budget, and the detector area can be
substantially increased.

If signal amplification is required, this consumes extra power. To tackle this
challenge, we have developed onboard cryogenic low-noise amplifiers in SiGe
and integrated them successfully in a QMS Mass Spectrometer. They consume less
than 5 mW per pixel, which allows realizing up to 200 individually amplified
pixels using analog electronics. A multiplexer at 1 GHz clock frequency could
sample more than 100 amplified signals, pushing the detector size again to
more than 20’000 pixels using analog electronics. Such large and well-
resolving devices will become important for atomic and molecular beam research
with low energy particles as well as for continuous mass spectrometry with
lowly charged high mass molecules.

## Methods

### Customized mass spectrometry

A Waters Q-TOF Ultima tandem mass spectrometer was customized as follows: a)
the RF electronics of the quadrupole mass filter was upgraded by MS VISION to
allow operating it up to a molecular mass to charge ratio of m/z = 30 kTh. b)
a differentially pumped vacuum chamber was inserted between the quadrupole
mass filter and the time-of-flight mass spectrometer to add a static
quadrupole bender, static ring ion guides as well as windows for optical
access. c) The ring electrodes guide the protein ions into a differentially
pumped cryogenically cold ultra-high vacuum chamber, that hosts the nanowire
detector arrays as well as a conventional Faraday plate. The entire cryostat
is mounted on a motorized translation stage that allows to move the SSPD with
30 $\mu m$ accuracy in one axis. The detectors are cooled by a Sumitomo pulse
tube cooler (RH), which achieves a cooling power of 900 mW at 4.2 K. The
system is usually operated at a minimal temperature of 3.7 K compatible with
all inputs and cabling, d) The mass spectrometer entrance region has also been
modified to allow for protein charge reduction in bipolar air, as generated by
a corona discharge.

### SSPD working point

In photonics, the bias current is typically set to $I_{b}>0.8~{}I_{c}$ to
maximize the quantum yield and sensitivity to low-energy photons. For
molecules we typically chose $I_{b}\simeq 0.5~{}I_{c}$ in $D_{1}$ to suppress
all background contamination by photons. To avoid latching, the detector
$D_{2}$ is written as a single wire, rather than as a device of concatenated
parallel wires [15, 41, 42]. For detector $D_{2}$, local amplifiers are no
longer needed because the 500 nm wide nanowire is driven with a bias current
of $I_{b}\simeq 40-60~{}\mu$A rather than by 4 - 6 $\mu A$ for $D_{1}$. A
detection event triggers a peak nearly 10 times higher than for $D_{1}$.

### Cryogenic low-noise amplifier

A single bias is used to ensure the scalability of the LNA design. Moreover,
no large inductors are used for bandwidth extension. Alternatively, the
capacitive peaking technique is used at the output buffer, making the LNA
design very compact ($480~{}\mu m\times 280~{}\mu m$ including the pads). The
cryogenic BiCMOS LNA achieves an average noise temperature of about $5\,$K in
the frequency range of $0.1\sim 8.8$ GHz with well-matched input impedance
($S11<-10~{}dB$) and sufficient gain ($>33~{}dB$), thanks to the current
resistive feedback technique and advanced HBTs used in the chip. At the
expense of the power consumption, the present design obtains good weak signal
linearity performance. Given that the signal level at the input of the LNA is
small, the linearity can be traded with power consumption.

### Calibration of the detector deflection electrode

The cold detector assembly can be vertically shifted with 30 µm accuracy and
any shift induced on the ion beam by the vertical deflection electrode can
thus be compensated and calibrated by a corresponding mechanical shift of the
detector. Since the electrodes are symmetric in the X- and Y direction, the
same calibration can be applied to derive the shift distance as a function of
applied voltage. With this method one can determine an ion beam profile and
deduce the absolute detection efficiency of the SSPD. (see Suppl. Inf.)

### Acknowledgments

We acknowledge contributions by K. Simonović and T. Sousa in parts of the mass
spectrometer setup.

### Funding

This project has received funding from the European Union’s Horizon 2020
research and innovation program under grant agreement No. 860713 as well as
partial support by the Gordon & Betty Moore Foundation within project #10771.

### Author contributions

Mass Spectrometer Design and Realization: MS, PG, MFXM, AS, SD, JS, MA
SSPD Design & Fabrication: NK, RG, MG, MC, AF
Cryogenic Electronics Design & Fabrication: JB, CB, EC
SSPD based mass spectrometry: MS, MFXM, AS, SD, PG, TK
Supervision: MM, VK, JC, MC, AF, EC, CB, MA
Writing: MA, MS wiht input by all authors.

### Competing interests

Single Quantum is an enterprise located in Delft/NL, specialized in the
manufacturing of superconducting nanowires. MS Vision is a mass spectrometry
company in Almere/NL.

### Data and materials availability

All data, code, and materials used in the analyses will be available from the
authors on request.

## Supplementary Information

### Sample preparation & source parameters

Molecule | R6G | Vit B12 | INS | CytC | Mb | Con A
---|---|---|---|---|---|---
MW (kDa) | 0.44 | 1.58 | 5.73 | 12.32 | 17.74 | 25.72
Conc. ($\mu$M) | 200 | 20 | 20 | 20 | 20 | 20
Purity | 99% | 99% |  $>25$USP | $>95\%$ | $>95\%$ | $>90\%$
Solvent | C2H3N/H2O (1:1) | H2O | H2O | H2O | H2O | C2H3N/H2O (1:1)
Table S1: Molecules and chemicals used in the experiments: Rhodamine 6G (R6G),
vitamin B12 (Vit B12), insulin bovine pancreas (INS), cytochrome C from bovine
heart (CytC), myoglobine from equine skeletal muscle (Mb) and concanavalin A
(Con A) were used as purchased from Merck. Figure S1: Calibrating the mass
spectrum of the protein mix. The protein mix consisting of myoglobin (A),
insulin (B) and cytochrome C (C) is electrosprayed and analyzed by TOF-MS. The
peaks are identified by comparing them with the peaks for the individual
proteins. The peaks are identical to those of Figure 3 in the main text.

The molecules were electrosprayed using a needle with an inner diameter of 80
µm, operated at a voltage of $3-4$ kV. The solution is driven by a Harvard
syringe pump at typically 9 µl/min. To reduce the average molecular charge, we
use a corona discharge. Nitrogen 5.0 is passed through pure ethanol at
variable temperature and sent through a flow regulator valve to a chamber
where a stainless steel needled at a voltage of - 2.5 to -3 kV induces the
discharge. To obtain a stable ion signal the discharge current is stabilized
at 15 µA. The bipolar air is sent into an aluminum channel of 3 mm inner
diameter and 80 mm length which is directly connected to the vacuum system.
This allows to reliably reduce the charge of the proteins.

### Microchannel plate detection

Calibration mass spectra are obtained using the time-of-flight (TOF)
spectrometer, with microchannel plate detector (MCP). The TOF spectrum is
calibrated using cesium iodide.

### Dynode / Phosphor screen detection

Alternatively, a system composed of conversion dynode, phosphor screen and
photomultiplier detector (PD) is used to obtain an ion signal that can be
directly compared to the SSPD signal. Our ion deflector works both in
transmission and deflection mode, enabling the ion beam to be directed either
towards the SSPD or the PD. To ensure accurate comparison, the timing is
chosen such that the quadrupole completes a full scan before the deflection
electrode is switched. This synchronized timing allows for a direct and
reliable comparison of the spectra obtained from the SSPD and the PD.

### Mass peak assignment for cytochrome C, insulin, and myoglobin mix

The SSPD spectrum can be referenced to a calibrated TOF spectrum, which we
show in Fig. S1. for a mixture of cytochrome C, insulin, and myoglobin and
each individual compound. The peaks observed both with the MCP and SSPD,
correspond to the expected compounds in Fig. S1

### Superconducting nanowire single particle detection

The superconducting nanowire detectors are connected to a driver (Single
Quantum), that sends a constant current into each pixel of the nanowire array,
in the $6-100$ µA range, and which reads out the signal across a bias-T. The
signal outputs are connected to a time-to-digital converter (TDC). To initiate
the measurement sequence, a programmable pulse generator (PPG) is used to
provide a trigger signal for the quadrupole mass selector (QMS) scan and the
TDC measurement. The PPG is necessary to synchronize the different devises
needed for the measurement process. While the TDC acquires the ion count
information received from the SSPD driver, the quadrupole mass selector
linearly scans the RF-amplitude. This scanning process enables the acquisition
of a mass spectrum using the SSPDs, as the transmission of a specific mass-to-
charge state is dependent on the RF-amplitude. Since the QMS scan is linear,
the SSPD spectrum can here be referenced to the calibrated TOF-MS spectrum.

Figure S2: Cryogenic low-noise amplifiers (LNAs) integrated with the SSPD
chip. Each SNWD is individually biased by a current drive at room temperature.
(A) shows the eight-pixel readout printed circuit board (PCB) mounted onto the
3.8 K stage, as well as the biasing lines that consist of two bias lines per
pixel and two supply wires for the LNAs, totaling 18 leads. (B)  Detector
array $D_{1}$, mounted onto the prototype PCB. (C) Wire-bonded LNA unit.

### Cryogenic assembly

The detector unit is shown in Fig. S2. The detector assembly is bolted onto
the cold head of a pulse tube cooler (Sumitomo SHI R65p) which can be cooled
to $3.7$ K ($P_{cool}=900$ mW at $4.2$ K) with all conditioning and signal
cables connected. The SSPD detector is shielded by two gold coated OFHC copper
cylinders with only a 10 mm entrance hole for molecules and light. This is
small enough to limit heating by black body radiation at room temperature to
36 mW. Visible photons can be shielded by covering all windows and their
detection can be suppressed by choosing the bias current appropriately.

The constant current driver is connected to the cryogenic stage via $20$ SMA
cables of $1.5$ m length that were thermally grounded to the 80 K, 30 K and
3.8 K stages of the pulse tube cooler. The full system with onboard low-noise
amplifiers is shown in Fig. S1. The cryogenic assembly is mounted on a
translation stage which allows us to scan the molecular beam profile and to
lift the Faraday detector into the beam.

### Calibration of the horizontal defection electrode

In order to calibrate the horizontal axis, a linear increase in voltage was
applied to the vertical deflection electrode in steps of $10$ V. At each
voltage iteration, the superconducting nanowire detector was mechanically
moved along the vertical axis. To analyze the data and determine the effective
shift as a function of voltage, a Monte Carlo method was employed. This
involved fitting a Gaussian function to the measured scan multiple times, with
random variations within the specified precision boundaries.

Figure S3: Linear position shift of the ion beam as a function of deflector
voltage, in 10V steps, for insulin at 1000 eV. The position is scanned in 200
µm steps by a linear stage with an accuracy of 30 µm to monitor the deflection
of the ion beam in one axis. The relevant errors can be found in Fig.S4.

To account for the precision of the motorized linear stage, an upper bound
error of $0.03$ mm was measured. The mean and standard deviation of the center
position, obtained from the Monte Carlo simulation, were plotted against the
deflector voltage, along with the error due to inaccuracies in the mechanical
position of the SSPD. The plot revealed a linear increase in the center
position with respect to voltage, which was consistent with the original data,
as can be seen in Fig. S3. To calculate the error of the slope, another Monte
Carlo method was employed. The deflection voltage steps were chosen to be
slightly larger than the pixel-length of the detector (200 µm). Since the
electrodes were symmetric in both the X- and Y-direction, the same calibration
can be applied to derive the shift distance in the horizontal (Fig. S4). By
utilizing this calibration method, it is possible to determine the ion beam
profile which is important in determining the absolute SSPD detection
efficiency.

Figure S4: Ion beam profilometry: Position-shift of the ion beam at the
location of the SSPD detector as a function of the deflection electrode
voltage. We show the results of the Monte Carlo simulation described above.

### Systematic errors contributing to the determination of the absolute
detection efficiency

To determine the detection quantum yield we divide the incident ion count
rate, as measured using the Faraday plane, by the SSPD count rate integrated
over the entire ion beam profile, where we consider a meander filling factor
of $50$ %. This way, we find a detection quantum yield close to $\eta\simeq
1$. Our observations merit some more detailed comments since Fig. 8 suggests
that the SSPD might detect even more molecules than expected by the direct
current measurements, but the error bar in Fig. 8 is relevant and has several
contributions:

Figure S5: Scanning electron micrograph of detector D1. Our small SSPD
detector measures $20\,$µm$\times 20\,$µm and it is connected to leads which
are growing in thickness with distance to the chip. For a small detector as
$D_{1}$, even the leads may contribute to the detection of high energy ions.
For a detector like $D_{2}$, with an area of $200\,$µm$\times 200\,$µm this
cannot contribute more than a few percent.

The protein flux is measured to vary up to 10% over one hour. If it does the
normalization will change in proportion.

The SSPD filling factor could deviate from the nominal value of $50$%, and if
the actual area were bigger, this could explain an excess quantum yield.
However, scanning electron micrographs constrain the meander area to better
than $5$%.

We estimate a small geometry correction factor to compensate for stitching
errors that occur when sampling the $3\,$mm ion beam profile with $200~{}\mu$m
patches of the shifted SSPD. The error of this factor amounts to $8\,$% of the
signal.

SSPD count rates would be enhanced if the ions could trigger a signal even
when hitting the quartz surface in between the NbTiN nanowires. However,
earlier experiments in our lab with keV atomic ions have shown that even an
insulating oxide layer of only 1-2 nm thickness suffices to reduce the SSPD
count rate by about an order of magnitude. We therefore estimate that heat
transport across the 500 nm dielectric gaps between the meander lines
contributes only on the level of 1% to the count rate, here overestimating the
yield.

One may argue that the protein has a finite diameter which must be convoluted
with the nanowire geometry to obtain an effective detector width. For insulin
or vitamin B, this contributes less than 1% on the 500 nm width.

We may need to consider the superconducting leads feeding the detector square,
as shown in Fig. S5. They fan out to 10 µm wide lines and can be a few 10 µm
long. A $3$ mm diameter ion beam will therefore hit some of these parts. We
include the femto-ammeter, which is calibrated to better than $15$ %.

Finally, the Faraday plate can only count ions which leave at least one charge
behind. If a protein hits the detector but bounces off – with deposition of
energy but not charge – the SSPD can still click, while the Faraday plates
will not measure any additional electric current.

Since all these systematic and statistical errors are independent, we
determine the net error bar by Gaussian error propagation which adds up to a
total of $\pm 20\,$%.

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††thanks: Present address: Department of Physics, University of California,
Berkeley, CA 94720, USA and Materials Sciences Division, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA.

# Enhanced Thermalization of Exciton-Polaritons in Optically Generated
Potentials

Yoseob Yoon<EMAIL_ADDRESS>Department of Chemistry, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  Jude Deschamps  Department
of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Mark Steger  National Renewable Energy Laboratory, Golden, CO, 80401, USA  Ken
W. West  Loren N. Pfeiffer  Department of Electrical Engineering, Princeton
University, Princeton, NJ 08544, USA  David W. Snoke  Department of Physics,
University of Pittsburgh, Pittsburgh, PA 15260, USA  Keith A. Nelson
<EMAIL_ADDRESS>Department of Chemistry, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA

###### Abstract

Equilibrium Bose-Einstein condensation of exciton-polaritons, demonstrated
with a long-lifetime microcavity [Phys. Rev. Lett. 118, 016602 (2017)], has
proven that driven-dissipative systems can undergo thermodynamic phase
transitions in the limit where the quasiparticle lifetime exceeds the
thermalization time. Here, we identify the role of dimensionality and
polariton interactions in determining the degree of thermalization in
optically generated traps. To distinguish the effect of trapping from
interactions and lifetimes, we measured the polariton distribution under four
nonresonant Gaussian pumps in a square geometry and compared it with polariton
distributions measured with each pump individually. We found that significant
redistribution of polaritons arises by trapping and modification of the
density of states. Surprisingly efficient polariton-polariton scattering below
the condensation threshold is evidenced by the depletion of the inflection-
point polaritons. Our work provides a deeper understanding of polariton
distributions and their interactions under various geometries of optically
generated potentials.

##  I Introduction

Hybrid light-matter quasiparticles in a quantum-well microcavity, called
exciton-polaritons (hereafter “polaritons”), arise from the collective strong
coupling between excitons and cavity photons [1, 2, 3]. Even with equal
mixtures of excitons and photons, polaritons have _photon-like_ effective
mass, allowing macroscopic quantum phenomena such as Bose-Einstein
condensation (BEC) up to room temperature [4, 5, 6, 7, 8, 9]. Meanwhile,
photon dressing reduces the exciton-exciton interaction strength only by the
Hopfield coefficient [10], which sets the polariton-polariton interaction
strength to be _exciton-like_. Along with a much narrower polariton linewidth
protected from excitonic inhomogeneous broadening [11, 12, 13], polaritons
favor various nonlinear phenomena including the polariton blockade [14, 15,
16, 17, 18] without the need to fabricate a nanoscale quantum-dot structure.

Although two principal signatures of BEC, namely, the macroscopic occupation
in the ground state and spontaneous emergence of coherence, are routinely
observed in polaritonic systems [4, 5, 6, 7, 9, 8], debates have persisted
over whether the observed phenomena can be attributed to BEC or rather to
conventional lasing [19, 20, 21]. The former is considered a thermodynamic
phase transition, while the latter is governed by dynamic rate equations,
although in both cases the underlying formation mechanism is the same—bosonic
stimulation into a single quantum state [22, 23]. Only when the polariton gas
is in thermal equilibrium can one claim that the relaxation dynamics is not
the driving force of bosonic stimulation [24]. Therefore, full thermalization
is one of the most distinguishing features of polariton BEC from its lasing
counterpart, and it was long sought in polaritonic systems [25, 26, 27, 28,
29].

One of the main challenges to reach full thermalization is a short polariton
lifetime, which results in a decay of the polariton population before thermal
equilibrium is reached. However, when polaritons have a long enough lifetime
for thermalization, they show equilibrium properties of BEC such as (1) Bose-
Einstein distribution of the state occupation number

$\mathcal{N}_{\text{BE}}(E)=\frac{1}{e^{(E-\mu)/k_{\text{B}}T}-1}$ (1)

with well-defined chemical potential $\mu$ and temperature $T$ (for trapped
condensates in quasi-2D) [30] or (2) power-law decay of spatial and temporal
coherence (for freely expanding condensates in 2D) [31].

A Bose-Einstein distribution of polaritons has been considered one of the key
criteria for equilibrium BEC [27, 28, 29, 30], although quantitatively
assessing equilibrium is experimentally not trivial [21]. Even with the
experimental demonstration of a Bose-Einstein distribution in a GaAs
microcavity [30], it is still not fully understood how polaritons thermalize
in an optically generated potential. Unlike photon condensates [32, 33] or
organic polariton condensates [9, 8] where fast vibrational relaxation of
molecules ($<$ 100 fs) results in sufficient thermalization within their
lifetimes, inorganic polariton condensates suffer from slow polariton-phonon
scattering ($\sim$10 ps), and they need both self-interactions and
interactions with an exciton reservoir for further relaxation. Thus, it is
crucial to identify the nature of polariton interactions and scattering rates
in various potential landscapes to understand the observed thermalized
distribution.

In this paper, we discuss two aspects of the thermalized distribution observed
in Ref. [30]. We first show that the population distribution is not only
determined by relaxation and loss rates, but also by an inhomogeneous pump
profile. In particular, a spatially narrow nonresonant pump can inject a pre-
thermalized polariton distribution outside the pumped region. We then show how
polaritons relax in a trapped region away from the pump, and how the
thermalization process can be enhanced through trapping and polariton
interactions. The enhancement of the thermalization rate becomes more
important in lower-quality systems, e.g., transition metal dichalcogenides in
a microcavity, where relaxation toward the ground state is insufficient and
the bottleneck effect prevails. One possible route to achieve condensation in
such systems would be to facilitate thermalization [34] so that polaritons
condense before other high-density processes such as exciton-exciton
annihilation come into play. In this sense, enhancing thermalization may allow
us to overcome the technical limits of material and cavity fabrication and to
achieve BEC in novel material systems.

##  II $\mathbf{K}$-space distribution of transported polaritons

Engineering of the potential landscape allows us to control polariton flow and
to enhance the effect of polariton interactions [35]. One way to create a
potential profile for polaritons is to optically pump free carriers and
excitons that blueshift the polariton energy through repulsive interactions.
The optically generated potential can be dynamically controlled, while
avoiding etching processes that often lead to unwanted defects. However,
complications such as the optical pump rate being coupled to the potential
height are added. This results in nontrivial polariton distributions away from
a spatially inhomogeneous pump [36, 37], where the contributing factors to the
distributions are poorly identified.

Figure 1: Degree of thermalization under various pump geometries at zero
detuning and similar pump powers below the condensation threshold. The
detection area is shown as a white dashed circle. (a) Polariton distribution
under a single Gaussian pump is highly out of equilibrium. The number of
source particles $N_{\text{s}}(V)$ from each potential energy contour $V$ that
contribute to the detected signal is proportional to the arc length, as
indicated by white curves. (b) Two Gaussian pumps yield a distribution similar
to but not the same as the sum of (a) and flipped (a). (c) Four Gaussian pumps
yield a better-thermalized distribution. (d) Polaritons are fully thermalized
in an annular pump geometry. Anomalous spike and dip of population
distribution compared to the Bose-Einstein distribution are indicated by
yellow diamonds and orange square, respectively. Distributions are plotted on
a log scale to emphasize small features. Figure 2: Simulated and measured
polariton distributions in the detection area
$\absolutevalue{\mathbf{r}}<9~{}\mu\text{m}$ (gray shaded area) away from a
single Gaussian pump beam at $(x,y)=(-16~{}\mu\text{m},0~{}\mu\text{m})$ for
three representative detunings: (a) $\hbar\delta=1~{}\text{meV}$, (b)
$\hbar\delta=-2~{}\text{meV}$, and (c) $\hbar\delta=-8~{}\text{meV}$. For all
cases, the pump beam’s full width at half maximum (FWHM) was
$2.5~{}\mu\text{m}$ and the pump power was above the condensation threshold.
The potential $V(x)$, formed by immobile very-high-$k$ excitons and cavity
gradient $6~{}\mu\text{eV}/\mu\text{m}$, has the same FWHM with that of the
pump beam ($\Delta V(x)=2.5~{}\mu\text{m}$). Normalized source-particle
distributions $N_{\text{s}}(x)$ are plotted as orange lines (the line
thickness and color darkness both represent the geometric factor $w(x)\propto
x-x_{0}$, where $x_{0}$ is the center position of the pump beam). The FWHM of
$N_{\text{s}}(x)$ are (a) $\Delta N_{\text{s}}(x)=3.3~{}\mu\text{m}$, (b)
$\Delta N_{\text{s}}(x)=2.1~{}\mu\text{m}$, and (c) $\Delta
N_{\text{s}}(x)=0.6~{}\mu\text{m}$. Panels on the right-hand side show
simulated and measured polariton distributions.

Most earlier demonstrations of polariton BEC used a large nonresonant pump
spot [4, 5, 6, 7, 8, 9]. In this configuration, polaritons condense where they
are created and the condensate barely expands. Even if the condensate expands
due to the pump-induced potential hill, it is valid to assume that the
condensate particles are predominantly created at the maximum intensity region
of the pump spot (with potential energy $V_{\text{max}}$) and then eventually
gain momentum $\hbar k_{\text{max}}=\sqrt{2m_{\text{pol}}V_{\text{max}}}$
(where $m_{\text{pol}}$ is the polariton effective mass) corresponding to
conversion of potential to kinetic energy [38, 39].

The polariton distribution below the condensation threshold, however, is more
complicated. In particular, its dependence on the pump geometry and cavity
detuning is not well understood. Figure 1 shows four different pump geometries
and observed polariton distributions at zero detuning in a GaAs microcavity
with polariton lifetime over 200 ps, demonstrated by tracking ballistically
propagating polariton wavepackets in space and time [40, 41]. The single
Gaussian pump at $(x,y)=(-16~{}\mu\text{m},0~{}\mu\text{m})$, shown in Fig.
1(a), creates polaritons that are repelled from the pump region. The polariton
distribution measured in the detection region (indicated by the white dashed
circle, $\absolutevalue{\mathbf{r}}<8~{}\mu\text{m}$) shows that the average
wavevector is $\langle k_{x}\rangle\approx 0.3~{}\mu\text{m}^{-1}$. The origin
of the backscattering peak at $k_{x}=-2.9~{}\mu\text{m}^{-1}$, indicated by
the yellow diamond, is believed to be related to the negative polariton mass
at the inflection point [42, 43]. The broad energy and momentum distributions
are maintained even above the condensation threshold, as shown in Fig. 2(a)
with similar pump and detection conditions. This behavior is different from
the distribution at photonic detuning, shown in Fig. 2(c), where most emission
is concentrated at very narrow ranges of energy and momentum (centered at
$V_{\text{max}}$ and $\hbar k_{\text{max}}$).

To describe the detuning-dependent distributions, the multicomponent nature of
polaritonic systems must be considered. We assume that the nonresonant pump
creates (1) immobile very-high-$k$ excitons and (2) slightly more mobile
inflection-point polaritons [44]. We call the latter the “source particles”
because they can directly produce low-$k$ polaritons via parametric
scattering, whereas the former particles cannot. The very-high-$k$ excitons
form a potential profile $V(\mathbf{r})$ that closely follows the pump
intensity profile $I(\mathbf{r})$ due to the small exciton diffusion length.
On the other hand, the spatial distribution of the source particles
$N_{\text{s}}(\mathbf{r})$ can be broader than that of $I(\mathbf{r})$ through
diffusion (this effect is more pronounced with a tightly focused pump beam),
while only minimally affecting the overall potential profile. Thus, there is a
mismatch between $V(\mathbf{r})$ and $N_{\text{s}}(\mathbf{r})$, which plays a
significant role in the resulting polariton distribution.

We simulated the polariton distribution by integrating contributions from each
energy contour $V_{i}$ of the pump-induced potential ($0\leq V_{i}\leq
V_{\text{max}}$). In particular, we assumed that polaritons are quasi-
thermalized at the generation point owing to the large density of high-$k$
excitons (the spatially dependent relaxation through polariton-exciton
scattering is captured by Wouters et al. [45]), i.e., $\mathcal{N}_{i}(E)$
follows the Bose-Einstein distribution including the pump-induced potential
term $V_{i}$. After ballistic transport to the detection area, the resulting
distribution $\mathcal{N}_{\text{det}}(E)$ is a weighted sum over all
$V_{i}$’s:

$\mathcal{N}_{\text{det}}(E)=\sum_{i}w_{i}\mathcal{N}_{i}(E)=\sum_{i}\frac{w_{i}}{e^{(E-V_{i}-\mu)/k_{\text{B}}T}-1},$
(2)

where the weight $w_{i}$ is proportional to the arc length of each energy
contour [white curves in Fig. 1(a)].

Figure 2 shows simulated polariton distributions assuming different diffusion
characteristics at different detunings. At an excitonic detuning [Fig. 2(a)],
the diffusion and broadening of $N_{\text{s}}(\mathbf{r})$ due to repulsive
exciton interactions lead to larger weight contributions from
$V_{i}<V_{\text{max}}$ than from $V_{i}=V_{\text{max}}$. The simulated
polariton distribution with $\Delta V(\mathbf{r})=2.5~{}\mu\text{m}$ and
$\Delta N_{\text{s}}(\mathbf{r})=3.3~{}\mu\text{m}$ shows a broad polariton
distribution, in good agreement with the measured distribution. On the other
hand, at a photonic detuning [Fig. 2(c)], bosonic stimulation starts at the
potential maximum and the majority of polaritons gain momentum $\hbar
k_{\text{max}}$ as they propagate to the detection area. The resulting narrow
energy and momentum distributions are in line with observations and
conventional interpretations with photonic polaritons [38, 39]. Without the
mismatch between $V(\mathbf{r})$ and $N_{\text{s}}(\mathbf{r})$, the broad
distribution observed at excitonic detunings could not be reproduced at any
potential height $V_{\text{max}}$ and chemical potential $\mu$.

With the two Gaussian pumps shown in Fig. 1(b), the distribution seems to be a
linear sum of two distributions generated by each pump. However, a careful
comparison between the distribution with each individual pump and that with
both pumps reveals that nonlinear relaxation processes are present. This
indicates that some redistribution of polaritons must follow simple ballistic
transport from the pump region to the detection region. Nevertheless, the
relaxation is not complete with the two pumps and a population dip at
$k_{x}\approx 0~{}\mu\text{m}^{-1}$ is observed, as indicated by the orange
square in Fig. 1(b). This is one example of the deviation from the equilibrium
Bose-Einstein distribution shown in Eq. (1), where
$\mathcal{N}_{\text{BE}}(E)$ monotonically decreases as $E$ increases. The dip
disappears when more Gaussian pumps are added, as shown in Fig. 1(c). Finally,
the backscattering population at
$\absolutevalue{k_{x}}=2.9~{}\mu\text{m}^{-1}$ is pulled down to the ground
state when the annular pump shown in Fig. 1(d) is used, and the resulting
distribution is fully thermalized, i.e., it follows Eq. (1) with well-defined
$\mu$ and $T$. These examples show that, even with long-lived polaritons, it
is important to choose an appropriate pump geometry and detection
configuration to study their equilibrium or nonequilibrium properties [30,
46].

##  III Real-space distribution of transported polaritons

The effects of pump geometry and detuning can also be observed in real space,
similar to those observed in $k$-space distributions (Figs. 1 and 2). With a
single Gaussian pump at low power, the real-space emission pattern [Fig. 3(a)]
is simply governed by the polariton propagation and expansion,
$N_{\text{pol}}(\mathbf{r})\propto(e^{-r/l_{c}})/r$, where $l_{c}$ is the
characteristic propagation length. On the other hand, at moderate pump power
just below the condensation threshold, the diffusion of source particles
becomes more pronounced by the increased potential gradient, which leads to a
halo pattern of polaritons in the real space, as shown in Fig. 3(b). The
multicomponent nature and heterogeneity in diffusion lengths can explain the
halo patterns observed at high excitation densities. This explanation may
extend to other excitonic systems where similar halo patterns are observed
[47, 48, 49]; repulsive interactions cannot solely explain the pattern if
there is only one type of particles.

Figure 3: Real-space polariton distribution with a single Gaussian pump near
zero detuning at (a) a low pump power and (b) a moderate pump power below the
condensation threshold. A line cut at $y=0~{}\mu\text{m}$ is plotted below as
a black line, and the gray shaded area is the laser intensity profile. Figure
4: Real-space polariton distribution with two Gaussian pumps above the
condensation threshold at (a) $\hbar\delta=-12~{}\text{meV}$ and (b)
$\hbar\delta=0~{}\text{meV}$. The potential profile $V(\mathbf{r})$ in each
case is illustrated schematically on the right, where the polariton
propagation directions are shown with white arrows. The distance between the
two pumps is $18.5~{}\mu\text{m}$.

With two Gaussian pump beams located at $(x,y)=(\pm
9.2~{}\mu\text{m},0~{}\mu\text{m})$, the effects of detuning show up more
dramatically in real space. At a photonic detuning [Fig. 4(a),
$\hbar\delta=-12~{}\text{meV}$], most polaritons are generated near the top of
the potential hill $V_{\text{max}}$ and expand away from the two generation
points, resulting in an interference pattern with
$k_{x}=\pm\sqrt{2m_{\text{pol}}V_{\text{max}}}/\hbar$ components, similar to
that reported by Tosi et al. [50]. In stark contrast, at a zero detuning [Fig.
4(b), $\hbar\delta=0~{}\text{meV}$], the
$k_{x}=\pm\sqrt{2m_{\text{pol}}V_{\text{max}}}/\hbar$ components disappear and
polaritons dominantly flow along the line $x=0$, perpendicular to the line
connecting two pumps ($y=0$). After analyzing the potential landscape using
energy-resolved real-space photoluminescence measurements with a $k$-space
filter (that selects only $k<0.5~{}\mu\text{m}^{-1}$), we conclude that this
peculiar pattern arises from the broadened potential profile $V(\mathbf{r})$
and even broader source-particle distribution $N_{\text{s}}(\mathbf{r})$. The
broadened potential profiles of the two pumped regions merge at the center
$(x,y)=(0~{}\mu\text{m},0~{}\mu\text{m})$, raising the local potential energy
to $V_{0}$ to form a saddle point. If the majority of polaritons are created
from the top of the potential hills, then they are still expected to retain
its wavevector components
$k_{x}=\pm\sqrt{2m_{\text{pol}}(V_{\text{max}}-V_{0})}/\hbar$ and interfere at
the center. However, an even broader $N_{\text{s}}(\mathbf{r})$ profile leads
to the generation of polaritons directly from the center, which then gain only
$k_{y}=\pm\sqrt{2m_{\text{pol}}V_{0}}/\hbar$ components. This interpretation
gives a simple explanation to the seemingly puzzling polariton behavior
observed between nonresonant pumps [51, 36].

##  IV Redistribution by trapping and interactions

To elucidate the mechanism of full thermalization in the annular trap, we
instead used four Gaussian pumps in a geometry similar to that in Fig. 1(c)
and analyzed the contributions from each pump. This allows us to deconvolute
which parts of the trapped distribution can arise from a linear sum, and to
monitor how much population is nonlinearly redistributed in the two-
dimensional $k$ space. The four pumps are labeled as left (L), right (R), up
(U), and down (D), indicating their positions relative to the origin
$(x,y)=(0,0)$. The potential formed by the four pumps can effectively trap
polaritons (even though there are four diagonal leakage channels), while each
pump alone cannot trap polaritons. Therefore, by comparing all four pumps
(trapped) and individual pumps (untrapped), the effect of trapping can be
investigated.

Figure 5: Redistribution function $R(\mathbf{k})$, defined in Eq. (3), at four
representative pump powers at detuning (a-d)
$\hbar\delta_{1}=0.6~{}\text{meV}$ and (e-h)
$\hbar\delta_{2}=-5.8~{}\text{meV}$.

We measured polariton distribution in the two-dimensional $k$ space with all
four pumps, which we denote as $N_{\text{LRUD}}(\mathbf{k})$, and compared it
with the sum of distributions measured individually with each pump, i.e.,
$\sum_{i}N_{i}(\mathbf{k})$, where $i=\text{L, R, U, and D}$. We define a
normalized redistribution function $R(\mathbf{k})$ that quantifies the extent
of the nonlinear gain or loss:

$R\equiv\frac{N_{\text{LRUD}}-\left(N_{\text{L}}+N_{\text{R}}+N_{\text{U}}+N_{\text{D}}\right)}{N_{\text{L}}+N_{\text{R}}+N_{\text{U}}+N_{\text{D}}}.$
(3)

If there are no interactions between polaritons and if the trapping potential
has no effect on the relaxation, then
$N_{\text{LRUD}}=N_{\text{L}}+N_{\text{R}}+N_{\text{U}}+N_{\text{D}}$ and the
redistribution function must be zero. If polaritons interact and exchange
momentum as described by the term
$\mathcal{H}_{\text{int}}=\hat{a}_{\mathbf{k}_{1}+\mathbf{q}}^{\dagger}\hat{a}_{\mathbf{k}_{2}-\mathbf{q}}^{\dagger}\hat{a}_{\mathbf{k}_{1}}\hat{a}_{\mathbf{k}_{2}}$,
where $\hat{a}_{\mathbf{k}}$ is the annihilation operator for a polariton at
$\mathbf{k}$, then $R$ deviates from zero. For example, $R>0$ means that a
larger number of polaritons are detected if they are trapped (compared to the
untrapped case), i.e., there is nonlinear gain by trapping. Similarly, $R<0$
means that there are less polaritons if they are trapped (compared to the
untrapped case), i.e., there is nonlinear loss by trapping.

A series of redistribution functions $R(\mathbf{k})$ with different excitation
densities at near-zero detuning ($\hbar\delta_{1}=0.6~{}\text{meV}$) is shown
in Figs. 5(a-d). Surprisingly, even at a low pump power
($P=0.15P_{\text{th}}$) well below the condensation threshold power
$P_{\text{th}}$, a sizable amount of polariton depletion ($R<0$) is observed
at $\absolutevalue{\mathbf{k}}=2.3~{}\mu\text{m}^{-1}$, where the inflection
point of the polariton dispersion is located. At this so-called “magic” angle,
the parametric polariton-polariton scattering is facilitated due to energy-
momentum matching [52]. This shows that, in the presence of the trapping
potential, polaritons can more efficiently scatter out of the inflection point
to other $k$ states, some of which are outside the numerical aperture (NA =
0.42) of the objective lens.

At intermediate powers ($0.3P_{\text{th}}<P<0.8P_{\text{th}}$), the low-energy
states ($\absolutevalue{\mathbf{k}}<1~{}\mu\text{m}^{-1}$) show $R>0$, while
the inflection-point polaritons show $R<0$. This indicates that the parametric
polariton-polariton scattering process [52] becomes more efficient in the
presence of trapping potential. It turns out that the most probable final
state for this process is not the polariton ground state but rather the
$\absolutevalue{\mathbf{k}}\approx 0.5~{}\mu\text{m}^{-1}$ state [53], which
is determined by both the shape of the dispersion and the population
distribution of high-$k$ excitons (see Fig. S2 for details).

Above the condensation threshold ($P>P_{\text{th}}$), bosonic stimulation into
the trap’s ground state at $\mathbf{k}=0$ is observed, while the stimulation
does not start in the untrapped case at the same polariton density. By forming
a trapping potential, the density of states $\mathcal{D}(E)$ is reduced so
that the occupation number $\mathcal{N}(E)$ _per state_ is larger, even though
the total polariton density $n=(1/A)\int\mathcal{D}(E)\mathcal{N}(E)dE$
remains the same, where $A$ is the system area. The bosonic stimulation
condition $\mathcal{N}(E)>1$ is more easily reached by the trapped ground
state, which is selected as the condensate state.

The trapping facilitates the redistribution of polaritons from high- to low-
energy states by providing polaritons more time to interact, as directly
visualized by Rozas et al. [54]. Freely expanding polaritons, in contrast,
have to rely only on polariton-phonon interactions to relax down to the ground
state [46]. Our finding also suggests that high-$k$ excitons (outside the
objective lens NA) are more efficiently generated in an annular trap than
without a trap. This is more evident in the low-density data shown in Fig.
5(a), where the population in the depletion region
($\absolutevalue{\mathbf{k}}\approx 2.3~{}\mu\text{m}^{-1}$) is not fully
redistributed to the lower-$k$ region but mostly to the higher-$k$ region that
are not detected. This effect may have affected the measurement of the
polariton-polariton interaction strength in Ref. [55] by promoting the
high-$k$ dark exciton generation in a large annular trap.

To investigate the role of polariton interactions, we plotted density-
dependent $R(\mathbf{k})$ integrated within the gain region
($\absolutevalue{\mathbf{k}}<0.5~{}\mu\text{m}^{-1}$) in Fig. 6(a). The
quadratic scaling of the effective redistribution rate with the polariton
density suggests that polariton-polariton or polariton-exciton scattering
processes dominate over other processes including polariton-phonon
scattering—relaxation via phonon scattering is expected to show a linear
scaling with the polariton density. The integrated $R(\mathbf{k})$ within the
depletion region
($2.1~{}\mu\text{m}^{-1}<\absolutevalue{\mathbf{k}}<2.5~{}\mu\text{m}^{-1}$),
plotted in Fig. 6(b), shows that the depletion rate saturates at higher
densities. The depletion at the inflection point of polariton dispersion
strongly supports the important role of parametric polariton-polariton
scattering during relaxation below the condensation threshold.

Figure 6: Integrated redistribution functions at detuning
$\hbar\delta_{1}=0.6~{}\text{meV}$ as a function of the polariton density in
the detection area ($n_{\text{det}}$) up to just below the condensation
threshold. (a) $R_{\text{gain}}$ is the redistribution function integrated
within the gain region ($\absolutevalue{\mathbf{k}}<0.5~{}\mu\text{m}^{-1}$).
The orange line is a quadratic fit. (b) $R_{\text{depletion}}$ is the
redistribution function integrated within the depletion region
($2.1~{}\mu\text{m}^{-1}<\absolutevalue{\mathbf{k}}<2.5~{}\mu\text{m}^{-1}$).

We varied the detuning to $\hbar\delta_{2}=-5.8~{}\text{meV}$ so that the
polariton interaction strength is weaker [Figs. 5(e-h)]. At this photonic
detuning, the gain region ($R>0$) is shifted from the origin ($\mathbf{k}=0$)
due to the cavity gradient and increased polariton mobility. The depletion
region ($R<0$) is observed at
$\absolutevalue{\mathbf{k}}=3.0~{}\mu\text{m}^{-1}$, which corresponds to the
inflection point at this detuning (see Fig. S8). Above the condensation
threshold, bosonic stimulation occurs at propagating modes with a finite
wavevector $k_{\text{p}}$, determined by the pump-induced potential height
$V_{\text{max}}$, in stark contrast to the zero-detuning case [Figs. 5(a-d)].
Weakly interacting polaritons cannot easily relax down to the ground state;
they form a nonequilibrium condensate in the excited state.

##  V Conclusions

We demonstrated enhancement of the thermalization rate in the presence of a
trapping potential and polariton interactions. The polariton redistribution is
quantified by comparing the polariton distribution under four nonresonant
Gaussian pumps in a square geometry and the distributions measured
individually with each pump. Significant depletion of inflection-point
polaritons as well as density- and detuning-dependent redistribution rates
reveal that parametric polariton-polariton scattering is responsible for the
enhanced thermalization in optically generated potentials. Our work paves the
way to overcome the bottleneck effect and to achieve polariton BEC in low-
quality microcavities. Furthermore, understanding polariton distributions and
their interactions away from the pump allows us to predict which trapped state
is dominantly injected, which determines the coupling between the pump sites
in polariton graph simulators [56, 57, 58].

###### Acknowledgements.

Y.Y., J.D., and K.A.N. were supported in part by Skoltech as part of the
Skoltech-MIT Next Generation Program. J.D. acknowledges partial support from
the NSERC-PGS program. M.S. and D.W.S. were supported by the National Science
Foundation under Grant Number DMR-2004570. K.W.W. and L.N.P. were partially
funded by the Gordon and Betty Moore Foundation through the EPiQS initiative
Grant GBMF4420 and by the National Science Foundation MRSEC Grant DMR-1420541.

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