Further author information: (Send correspondence to T.G.) E-mail:
<EMAIL_ADDRESS>
# Testing magnetic interference between TES detectors and the telescope
environment for future CMB satellite missions

Tommaso Ghigna  Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa,
Chiba 277-8583, Japan  Thuong D. Hoang  Kavli IPMU (WPI), UTIAS, The
University of Tokyo, Kashiwa, Chiba 277-8583, Japan  Takashi Hasebe  Kavli
IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
Yurika Hoshino  Saitama University, 255 Shimookubo, Sakura-ku, Saitama,
338-8570, Japan  Nobuhiko Katayama  Kavli IPMU (WPI), UTIAS, The University of
Tokyo, Kashiwa, Chiba 277-8583, Japan  Kunimoto Komatsu  Okayama University,
3-1-1, Tsushimanaka, Kita-ku, Okayama, 700-8530, Japan  Adrian Lee  Department
of Physics, University of California Berkeley, Berkeley, 94720, California,
USA  Tomotake Matsumura  Kavli IPMU (WPI), UTIAS, The University of Tokyo,
Kashiwa, Chiba 277-8583, Japan  Yuki Sakurai  Okayama University, 3-1-1,
Tsushimanaka, Kita-ku, Okayama, 700-8530, Japan  Shinya Sugiyama  Saitama
University, 255 Shimookubo, Sakura-ku, Saitama, 338-8570, Japan  Aritoki
Suzuki  Lawrence Berkeley National Laboratory, 1 Cyclotron Rd, Berkeley, CA
94720, USA  Christopher Raum  Department of Physics, University of California
Berkeley, Berkeley, 94720, California, USA  Ryota Takaku  Kavli IPMU (WPI),
UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan  Benjamin
Westbrook  Department of Physics, University of California Berkeley, Berkeley,
94720, California, USA  for the LiteBIRD Collaboration

###### Abstract

The two most common components of several upcoming CMB experiments are large
arrays of superconductive TES (Transition-Edge Sensor) detectors and
polarization modulator units, e.g. continuously-rotating Half-Wave Plates
(HWP). A high detector count is necessary to increase the instrument raw
sensitivity, however past experiments have shown that systematic effects are
becoming one of the main limiting factors to reach the sensitivity required to
detect primordial $B$-modes. Therefore, polarization modulators have become
popular in recent years to mitigate several systematic effects. Polarization
modulators based on HWP technologies require a rotating mechanism to spin the
plate and modulate the incoming polarized signal. In order to minimize heat
dissipation from the rotating mechanism, which is a stringent requirement
particularly for a space mission like LiteBIRD, we can employ a
superconductive magnetic bearing to levitate the rotor and achieve contactless
rotation. A disadvantage of this technique is the associated magnetic fields
generated by those systems. In this paper we investigate the effects on a TES
detector prototype and find no detectable $T_{c}$ variations due to an applied
constant (DC) magnetic field, and a non-zero TES response to varying (AC)
magnetic fields. We quantify a worst-case TES responsivity to the applied AC
magnetic field of $\sim 10^{5}$ pA/G, and give a preliminary interpretation of
the pick-up mechanism.

###### keywords:

CMB, Bolometer, Transition Edge Sensor, Polarization Modulator, Half-Wave
Plate

##  1 INTRODUCTION

The main scientific target of several upcoming CMB experiments [1, 2, 3, 4] is
the primordial CMB $B$-mode spectrum. If detected it will shed new light on
the physics of the early Universe, in particular with regards to inflation
[5]. From current $B$-mode upper limits [6] we know that a detection is going
to be challenging even for the most recent and advanced detector technologies.
Hence, most upcoming and future experiments will observe the sky using
multiple telescopes with focal planes made of several thousand superconductive
Transition-Edge Sensor (TES) bolometers [7] to achieve the sensitivity
required for a successful detection of this faint signal[8].

Unfortunately intrinsic detector noise is not the only limiting factor for a
successful $B$-mode detection. Atmospheric fluctuations[9] (for ground-based
experiments), Galactic foregrounds[10] and instrumental systematic effects[11,
12, 13, 14, 15] can contaminate the data and have a negative impact on the
experiment sensitivity.

A rapid modulation of the polarized sky signal has become a popular strategy
to address and overcome some of these challenges. Polarization modulation can
be achieved with a Polarization Modulation Unit (PMU) in front of the focal
plane, which in most cases consists of a Half-Wave Plate (HWP) rotated at
constant speed. This technique offers several advantages. Since the signal of
interest from the sky is polarized, by appropriately tuning the HWP rotation
speed, it is possible to modulate the signal in a frequency range free from
low-frequency atmospheric and instrumental $1/f$ noise[16]. If the HWP is the
first optical element in

Figure 1: Left: A snapshot of LiteBIRD satellite and payload with the three
telescopes for low-frequency (LFT), middle-frequency (MFT) and high-frequency
(HFT) detectors [3]. Center: A cross-sectional view of LFT where the PMU and
superconductive focal plane are highlighted[17]. Right: LFT Polarization
Modulation Unit design and components. Notice the presence of multiple
permanent magnets in the SMB and the driving motor[18].

the telescope system, instrumental polarization effects are not modulated and
can be easily separated from the sky signal. Modulation of the polarized
signal allows also for the reconstruction of the polarization Stokes $Q$ and
$U$ parameters with a single detector, reducing the impact of differential
systematic effects due to intrinsic differences in a pair of polarization
sensitive detectors[19].

In order to minimize unwanted HWP non-ideal effects it is necessary to operate
the HWP at a stable cryogenic temperature to avoid temperature fluctuations
and reduce the HWP emissivity[16, 20]. Opting for this choice requires placing
the PMU inside the receiver dewar. This imposes tight constraints on the heat
dissipation of the rotating mechanism. For the LiteBIRD space mission Low
Frequency Telescope the thermal budget allocated to the PMU is 4 mW[21]. For
this reason we are developing a contactless rotation mechanism for LFT PMU.
The prototype consists of a Superconductive Magnetic Bearing (SMB) to levitate
the rotor that supports the HWP, and a magnetic motor to spin the rotor at
constant speed[18]. In Figure 1 we show a snapshot of the LiteBIRD satellite,
payload, a cross-sectional view of LFT and the most recent PMU design.

The unfortunate byproduct of these components is a non-zero magnetic field at
the detector location on the focal plane (if not properly shielded). For
LiteBIRD, a detailed analysis of the magnetic field strength (and its possible
time-dependence given the use of a magnetic motor made of moving segmented
permanent magnets) is still being investigated. In preparation, we have begun
addressing the impact of magnetic fields on LiteBIRD TES detector prototypes
using a Helmholtz coil to generate a stable magnetic field.

In recent years some measurements of magnetic field dependent TES behaviour
have been reported, focusing in particular on the observed magnetic field
dependence of the critical temperature $T_{c}$ of AlMn superconductive
films[22]. LiteBIRD will employ similar Mn-doped aluminium TES bolometers. Our
final goal is to determine the impact of magnetic fields generated by
components of the LiteBIRD telescope (e.g. the PMU) on LiteBIRD-like
superconductive bolometers to define the best magnetic shield configuration
around the focal plane. In this paper

Leg length [$\mu$m] | Leg cross-section [$\mu$m] | AlMn area [$\mu$m2] | AlMn thickness [nm] |  $T_{c}$ [K] |  $R_{\rm n}$ [$\Omega$]
---|---|---|---|---|---
500 | 1$\times$17 | 480 | 90 | 0.220 | $0.78$
Table 1: Size and critical temperature of the TES detector tested for this
work for comparison with Vavagiakis et al[22]. The AlMn film is made with
aluminum doped with 5500 ppm of manganese by weight. AlMn size and normal
resistance are in the same range as POLARBEAR-like detectors shown in
Vavagiakis et al., which also appear to show low $T_{\rm c}$ sensitivity to an
applied DC magnetic field.

(a)

(b)

Figure 2: Left: the Helmholtz coil positioned around the open ADR cryostat at
Kavli IPMU. Center: the TES chip tested for this work. Right: two SQUID
amplifiers with their Nb can shield.

we present the system we have developed for an initial assessment of the
magnetic field sensitivity of an AlMn TES prototype. The prototype tested for
this paper came from an early fabrication run and does not fully meet LiteBIRD
specifications, in particular we measured $T_{c}\sim 220$ mK and $P_{\rm
sat}\sim 4$ pW, while $T_{c}\sim 170$ mK and $P_{\rm sat}\sim 1$ pW are the
target[23, 24] (more design and measured parameters of the TES tested are
reported in Table 1). Nevertheless, we decided to report these results given
the scarcity of data about TES magnetic field sensitivity in the literature.

##  2 DC MAGNETIC FIELD

Magnetic field sensitivity of AlMn TES critical temperature has been reported
in Vavagiakis et al.[22]. The authors tested several different TES detectors
(from ACT [25, 26] and POLARBEAR [27] experiments) of different size (AlMn
area, thickness and thermal link), normal resistance $R_{\rm n}$ and nominal
critical temperature $T_{c}$. For most of the detector tested $T_{c}$ appears
to be decreasing with the strength of the applied magnetic field. In
particular, the detectors of lower normal resistance $R_{\rm n}$ ($R_{\rm
n}\sim 0.007$ $\Omega$ for ACT-like detectors) appear to be more sensitive to
the applied magnetic field compared to those of higher resistance ($R_{\rm
n}\sim 0.8$ $\Omega$ for POLARBEAR-like detectors).

Testing such dependence is important in the design phase of a CMB experiment
because a change in $T_{c}$ can impact two fundamental detector parameters:
saturation power $P_{\rm sat}$ and the phonon noise $NEP_{G}$ that originates
in the thermally conductive legs that keep the absorber-thermistor island
isolated from the thermal bath (this is the dominant contributor to the
detector intrinsic noise). These two parameters can be obtained knowing the
temperature of the thermal bath $T_{b}$ and the detector critical temperature
$T_{c}$. If we assume the thermal conductivity of the legs to be of the form
$k_{\rm 0}T^{n}$ (where $k_{\rm 0}$ is a constant and $n$ is the carrier index
that defines the temperature dependence), we can write[28]:

$P_{\rm sat}=N\frac{A}{\ell}\frac{k_{\rm 0}}{n+1}(T_{c}^{n+1}-T_{b}^{n+1}),$
(1) $NEP^{2}_{G}=4P_{\rm
sat}k_{b}T_{b}\frac{(n+1)^{2}}{2n+3}\frac{(T_{c}/T_{b})^{2n+3}-1}{[(T_{c}/T_{b})^{n+1}-1]^{2}},$
(2)

where $N$, $A$ and $\ell$ are respectively the number, cross-section area and
length of the thermally conductive legs, and $n$ is the thermal conductivity
index. For purely phonon-mediated heat transfer the index is assumed to be
$n=3$. However values in the range $1<n<3$ have been measured for different
detector materials and designs[29].

(a)

(b)

Figure 3: Left: Saturation power $P_{\rm sat}$ vs. bolometer transition
temperature $T_{c}$ for different values of the thermal conductivity index $n$
assuming bath temperature $T_{b}=0.1$ K. Right: Phonon Noise Equivalent Power
$NEP_{G}$ vs. bolometer transition temperature $T_{c}$ for different values of
the thermal conductance index $n$ assuming bath temperature $T_{b}=0.1$ K and
fixed $P_{\rm sat}$. In both cases the curve is normalised to the expected
value for $T_{c}=0.171$ K (baseline value for LiteBIRD detectors).

Assuming LiteBIRD baseline values for $T_{b}=100$ mK and $T_{c}=171$ mK, we
have computed $P_{\rm sat}$ and $NEP_{G}$ (for fixed $P_{\rm sat}$ value) as a
function of $T_{c}$ for three different $n$ values. In Figure 3 we show the
results normalized to the expected value for $T_{c}=171$ mK. It is clear from
Figure 3 that a change of $T_{c}$ can have an impact on the effective value of
these two parameters. Specifically, a $T_{c}$ value lower than expected will
result in a reduction of both $P_{\rm sat}$ and $NEP_{G}$. While a better
noise performance can potentially be beneficial, a reduction of $P_{\rm sat}$
changes the detectors dynamic range, and could cause them to saturate more
easily when observing a bright source.

For this reason, we need to understand the effect of magnetic fields on the
critical temperature of LiteBIRD detectors to design the most appropriate
magnetic shield to avoid $T_{c}$

(a)

(b)

Figure 4: Left: Temperature vs. resistance of the TES tested for this work.
The measurement has been repeated for different applied constant magnetic
fields perpendicular to the TES chip plane. Right: Critical temperature
measured from the plot on the left at $R=0.9\times R_{\rm n}$. The errorbar
corresponds to the measured temperature accuracy $\sigma_{\rm T}=0.0003$ K.

(a)

(b)

Figure 5: Left: Voltage bias vs. TES current for different bath temperatures
$T_{b}$. For the magnetic field sensitivity measurements we fixed $T_{b}$ to
100 mK. Right: Voltage bias vs. TES resistance.

variations that could potentially alter detector properties. For this reason
we have assembled a Helmholtz coil to generate a magnetic field to test
detector prototypes.

In Sugiyama et al.[30], we reported the expected value of the DC magnetic
field produced by the PMU prototype at LiteBIRD LFT focal plane: $B\sim 0.24$
G. Since this is only a forecast and there may be other magnetic field sources
in the fully-assembled system, we tested the detector under the effect of
larger magnetic fields. In Figure 4 we show the resistance-temperature
relation measured for different applied magnetic field values in the range
$-4.22$ G $<~{}B~{}<$ $6.24$ G, for the detector in Table 1. The magnetic
field was applied perpendicularly to the AlMn surface. The $T_{c}$ value
reported for each curve corresponds to the temperature at $R$ = $0.9\times
R_{\rm n}$. In the right panel we also show the $T_{c}$ value extrapolated
from the data in the left panel. We do not detect any significant variation of
$T_{c}$ within the measured temperature accuracy.

While no significant shift in the resistance-temperature relation is visible
around the turnaround at $R\sim 0.9\times R_{\rm n}$, a small shift can be
observed deeper into the superconductive transition. This appears to be an
artifact attributed to a noise increase for stronger magnetic fields (each
measurement was repeated twice to test the reproducibility of the results).

##  3 AC MAGNETIC FIELD

Given the presence of active components, moving parts and even moving
permanent magnets[18] in the telescope system of upcoming CMB experiments (the
PMU employ a magnetic motor with segmented permanent magnets in the rotor), we
can expect time-dependent magnetic fields to be generated. While a full
assessment of the origin and magnitude at the focal plane has yet to be
evaluated for the LiteBIRD experiment, we have measured the displacement of
the rotor of the PMU prototype for LiteBIRD LFT during contactless rotation of
the rotor. In Sugiyama et al.[30] we presented the measurement and a forecast
of the expected AC magnetic field amplitude at the focal plane location due to
the rotor displacement: $B_{\rm m}\sim 3\times 10^{-5}$ G. While at present
such a small magnetic field is challenging to reliably generate and measure,
we have used the Helmholtz coil in combination with a commercial signal
generator to test the TES succeptibility to an AC magnetic field.

$V_{b}$ [$\mu$V] | 2.0 | 1.6 | 1.4 | 1.2 | 1.1 | 1.0
---|---|---|---|---|---|---
$\mathcal{L}$ | 0.32 | 2.02 | 3.04 | 6.54 | 10.95 | 27.72
Table 2: Loop-gain values estimated from time-constant measurement: $\tau_{\rm
eff}=\tau_{\rm 0}/(\mathcal{L}+1)$. The normal time-constant $\tau_{0}$ has
been estimated as $\sim 13.5$ ms. The effective time-constant is estimated
from bias-step measurements.

(a)

(b)

Figure 6: Left: AC magnetic field pick-up. The blue line shows the output for
a magnetic field shielded SQUID with open input loop. The orange line shows
the output when a TES biased in the superconductive transition is connected to
the SQUID input loop. In both cases the magnetic field amplitude is $B_{\rm
m}\sim 1.4$ G and is modulated at 2 Hz. Right: An example of the measurement
performed for this analysis. The red line shows the applied AC magnetic field,
while the blue line is the TES current due to the magnetic field pickup.

Since we expect the succeptibility of the TES detector to depend on the exact
bias conditions during the test, in Figure 5 we report a measurement of the
voltage-current relation (V-I curve) for different values of the thermal bath
$T_{b}$, as well as the voltage-resistance relation for $T_{b}=100$ mK. In
Table 2 we also report an estimate of the loop-gain $\mathcal{L}$[7] for
different voltage bias values determined from time-constant measurements
performed with a bias-step analysis[31]. These values are obtained by biasing
the TES at voltage $V_{b}$ in the range 1-2 $\mu$V and then injecting a
voltage step of 100 nV with a square-wave generator. The decaying tail is
fitted with an exponential function to estimate the thermal time constant
$\tau_{\rm eff}$. We assumed the normal time constant to be $\tau_{0}\sim
13.5$ ms as measured above the TES transition at $V_{b}=2.5$ $\mu$V. Assuming
this as the true value we can estimate the loop-gain $\mathcal{L}$ for all
bias cases from the relation $\tau_{\rm eff}=\tau_{\rm 0}/(\mathcal{L}+1)$.

(a)

(b)

Figure 7: Left: TES AC-magnetic field sensitivity for different bias
conditions and applied field frequency. In all cases the amplitude of the
magnetic field applied is $B_{\rm m}$=1.40 G. Right: Measured time-delay
between the applied AC-magnetic field and the TES response for different bias
voltage conditions $V_{b}$. Figure 8: Left: TES AC-magnetic field sensitivity
for different amplitude of the applied magnetic field $B_{\rm m}$ under the
same bias condition $V_{b}$=1.1 $\mu$V.

We read the TES with a commercial dc-SQUID amplifier in a Flux-Locked Loop
(FLL) readout scheme[32]. All SQUID amplifiers are magnetically shielded in a
niobium can (shown in Figure 2). Since a non-shielded SQUID could directly
pick-up the magnetic field produced by the Helmholtz coil, in the left plot of
Figure 6 we show the SQUID output voltage measured for the same SQUID
amplifier in two different configurations: without the TES connected at the
SQUID input (open loop) and with the TES under test connected to the SQUID
input. In both cases a sinusoidal AC magnetic field of amplitude $B_{\rm
m}=1.4$ G amplitude (at the detector location) and 2 Hz frequency is applied.
We can observe that for the open loop measurement the SQUID output does not
seem to be affected by the presence of the magnetic field, indicating that the
magnetic field pick-up is happening through the TES connected to the SQUID
input.

As an example of the type of analysis we have performed, in the right plot of
Figure 6 we show in blue the TES current $I_{\rm tes}$ calculated from the
SQUID output in the left panel (orange line) and the magnetic field produced
by the Helmholtz coil at the TES location.

We tested the detector response for all six bias cases in Table 2 for
different magnetic field amplitudes $B_{\rm m}$ and frequencies in the range
0.01-1.4 G and 1-50 Hz. Smaller and slower fields could not be generated
because of limitations of the signal generator used. We were also limited by
the readout noise. In order to increase the signal-to-noise ratio, we had to
integrate the data over several magnetic field periods (in particular for
small field amplitudes and high frequency). While in a realistic scenario we
expect smaller fields than the ones tested, this analysis gives us a first
assessment of the TES responsivity to an AC magnetic field. We plan to explore
the lower frequency regime in the future, however the frequency range 1-10 Hz
is particularly relevant in the CMB context because both LiteBIRD and SO PMUs
will be operated in this regime.

From the data we determined the TES responsivity to the applied magnetic field
by fitting the TES current and magnetic field data in the right panel of
Figure 6 with a function of the form $A\sin{2\pi f(t-t_{\rm 0})}$. Where $A$
is the amplitude, $f$ is the frequency and $t_{\rm 0}$ is a time offset. We
computed the responsivity to the applied magnetic field from the fit
parameters as $A_{\rm I}/A_{\rm B}$ where $A_{\rm I}$ is the measured TES
current amplitude and $A_{\rm B}$ is the magnetic field amplitude. We also
measured the time-constant as the time-delay between the applied magnetic
field and the detector response $|t_{\rm 0B}-t_{\rm 0I}|$.

In the left panel of Figure 7 we show the responsivity results $A_{\rm
I}/A_{\rm B}$ for all bias cases in Table 2 for an applied magnetic field of
amplitude $B_{\rm m}=1.4$ G perpendicular to the TES chip surface and for
different frequencies. We can notice that the responsivity increases as the
detector is biased deeper into the superconductive transition, while it
decreases at higher frequencies. The responsivity appears to increase by an
order of magnitude consistently at all frequencies with an increase of the
loop-gain by roughly the same magnitude.

In the right panel of Figure 7 we show the estimated time-constant which
appears to be consistently $\sim 30$ ms for most frequencies and bias
conditions. A drop is observed for $f>20$ Hz.

Figure 9: The PSD of the magnetic field $B$, TES current $I_{\rm tes}$ and a
thermometer placed next to the TES chip at 100 mK to monitor the bath
temperature $T_{b}$. The data correspond to Figure 6. We do not observe
variations of the bath temperature $T_{b}$ correlated to the 2 Hz AC magnetic
field. Hence we can exclude eddy currents produced in the TES chip copper
holder by the changing magnetic field to be responsible for bath temperature
variations. The PSDs have been normalized to the white noise level for
clarity.

This is probably just an artifact due to the magnetic field frequency
approaching the inverse of the time-constant.

In Figure 8 we report the responsivity $A_{\rm I}/A_{\rm B}$ calculated from
three different amplitudes of the applied magnetic field for the bias case
$V_{\rm b}=1.1$ $\mu$V. This is a consistency check to confirm that the
estimated responsivity does not change for different amplitudes of the applied
field, which appears to be true at least for $f<20$ Hz. Given the smaller
signal-to-noise ratio for high frequency fields, we attribute this discrepancy
to the relative noise level increase.

Finally, in Figure 9 we show the temperature stability of the bath temperature
$T_{b}$ when an AC magnetic field of amplitude $B_{\rm m}=1.4$ G and frequency
2 Hz is applied. We show this data to exclude the possibility that the pick-up
mechanism may be due to bath temperature fluctuations possibly originating
from eddy currents produced by the magnetic field in the TES chip copper
holder. It is in fact clear from the thermometer PSD that a 2 Hz component is
not present.

Another possibility to explain the observed behaviour are eddy currents in the
TES BLING (a metal layer that acts as a heat capacity to tune and slow down
the TES thermal response) that can result in heat dissipation directly in the
TES island. At present this remains an hypothesis, but we will try to address
it in the future.

From preliminary results of the same analysis performed with a magnetic filed
tangent to the TES AlMn area, we have initial indication that the responsivity
could be orientation dependent. Unfortunately during these later tests we
could not control accurately the bath temperature because of cryogenic issues,
and the data appear to be noisy. Hence, we decided not to report these results
here. However, we believe that testing the responsivity for different magnetic
field orientations can help us understanding the pick-up mechanism, therefore
we will certainly address this aspect in the future.

##  4 CONCLUSIONS

We have reported on the DC and AC magnetic field sensitivity of TES detectors.
In this work the magnetic field has always been applied perpendicularly to the
TES AlMn area. In the future we will test the effect for different magnetic
field orientations. We have found that, for the AlMn detector tested for this
work, the critical temperature does not appear to be sensitive to an applied
constant magnetic field in the range $-4.22$ G $<B<6.24$ G. This result is in
agreement previous results reported for AlMn detectors of similar normal
resistance and physical size. This is a preliminary result that needs to be
confirmed for future LiteBIRD detector prototypes.

On the other end, we have found the TES chip to be responsive to AC magnetic
fields. We need more investigation to understand the pick-up mechanism. At
present we have excluded the SQUID amplifier to be the source of the magnetic
field pick-up as shown in the left panel of Figure 6.

From the data presented here, we can not completely rule-out the possibility
that the main pick-up mechanism is an inductive coupling between the TES
biasing circuit (which is difficult to shield in the current configuration)
and the varying magnetic field. More measurements of different TES detectors
and for different magnetic field orientations may be needed to fully
understand the mechanism.

However, we can speculate that if this was the main pick-up mechanism, the use
of Digital Frequency-Domain Multiplexing (DfMux) readout system[33] (like the
one LiteBIRD will employ) could potentially help mitigating the severity of
this effect, given the presence of MHz LC filters (needed for multiplexing) in
the TES biasing circuit. In fact, such filters can prevent unwanted low-
frequency electrical signals from coupling to the TES response.

In Figure 7 we have shown how the responsivity appears to increase for low
voltage bias values (smaller TES resistance and higher loop-gain). On the
other end, the time-constant associated with the magnetic field pick-up does
not change with bias conditions and appears to be stable around 30 ms. This
value does not match the expected thermal or electrical time-constant of the
detector-readout system. We will need more investigation to understand the
pick-up mechanism.

As mentioned in Section 3, we expect AC variations of the magnetic field
generated by the rotating PMU of LiteBIRD LFT with amplitude $B_{m}\sim
3\times 10^{-5}$ G[30]. If we assume the worst case scenario responsivity (for
a detector biased at $V_{b}=1\mbox{ }\mu$V) $A_{\rm I}/A_{\rm B}\sim 10^{5}$
pA/G from Figure 7, the current response for the expected magnetic field
(assuming that this effect is linear as suggested by the results in Figure 8)
would be $\Delta I_{\rm tes}\sim 3$ pA, for a non-shielded TES. This value is
well below the current best readout noise level measured for TES-DfMux
systems[34, 35] of NEI $\sim 10\mbox{ pA}\sqrt{\mbox{Hz}}$ (noise equivalent
current). While we can expect improvements with the noise performance of this
technology in the coming years, we can speculate that such a small response is
still going to be subdominant with respect to statistical noise sources, in
particular when we take into account not only readout noise, but also the TES
intrinsic noise and the dominant photon noise. A direct comparison of
statistical noise sources with a systematic noise source is perhaps difficult
and need to be clarified, given that the effect of the latter does not average
down with longer integration time. Therefore, in the future we should focus
not only on its amplitude but we should understand its time-evolution and how
it could interact with the scan-strategy. It is likely that the signal is
going to be synchronized with the HWP rotation. If so, we should understand
which harmonic of the HWP rotation frequency is going to be dominant.

In conclusion, from the data presented here we expect this effect to be
dominated by statistical noise, once an appropriate magnetic shielding is
included to suppress the magnetic field amplitude. However, we believe that
more investigation is necessary for lower $P_{\rm sat}$ detectors (LiteBIRD-
like), to fully understand the pick-up mechanism (possibly using the same
readout of the flight module), and to understand its time-evolution and
possible synchronization with the HWP rotation.

###### Acknowledgements.

We thank all LiteBIRD collaborators for support and help. In particular
Giovanni Signorelli, Joshua Montgomery and Samantha Lynn Stever for usuful
comments and feedback on the manuscript. This work was supported by JSPS
KAKENHI Grant Numbers 22K14054 and 18KK0083 and JSPS Core-to-Core Program, A.
Advanced Research Networks. Kavli IPMU is supported by World Premier
International Research Center Initiative (WPI), MEXT, Japan. LiteBIRD (phase
A) activities are supported by the following funding sources: ISAS/JAXA, MEXT,
JSPS, KEK (Japan); CSA (Canada); CNES, CNRS, CEA (France); DFG (Germany); ASI,
INFN, INAF (Italy); RCN (Norway); AEI (Spain); SNSA, SRC (Sweden); NASA, DOE
(USA).

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# Enhancing ground state population and macroscopic coherence of room-
temperature WS2 polaritons through engineered confinement

M. Wurdack<EMAIL_ADDRESS>ARC Centre of Excellence in Future
Low-Energy Electronics Technologies and Department of Quantum Science and
Technology, Research School of Physics, The Australian National University,
Canberra, ACT 2601, Australia  E. Estrecho  ARC Centre of Excellence in Future
Low-Energy Electronics Technologies and Department of Quantum Science and
Technology, Research School of Physics, The Australian National University,
Canberra, ACT 2601, Australia  S. Todd  ARC Centre of Excellence in Future
Low-Energy Electronics Technologies and Department of Quantum Science and
Technology, Research School of Physics, The Australian National University,
Canberra, ACT 2601, Australia  C. Schneider  Institut für Physik, Carl von
Ossietzky Universität Oldenburg, Ammerländer Heerstraße 114-118, 26126
Oldenburg, Germany  A. G. Truscott  Department of Quantum Science and
Technology, Research School of Physics, The Australian National University,
Canberra, ACT 2601, Australia  E. A. Ostrovskaya<EMAIL_ADDRESS>ARC Centre of Excellence in Future Low-Energy Electronics Technologies and
Department of Quantum Science and Technology, Research School of Physics, The
Australian National University, Canberra, ACT 2601, Australia

###### Abstract

Exciton-polaritons (polaritons herein) in transition-metal dichalcogenide
monolayers have attracted significant attention due to their potential for
polariton-based optoelectronics. Many of the proposed applications rely on the
ability to trap polaritons and to reach macroscopic occupation of their ground
energy state. Here, we engineer a trap for room-temperature polaritons in an
all-dielectric optical microcavity by locally increasing the interaction
between the WS2 excitons and cavity photons. The resulting confinement
enhances the population and the first-order coherence of the polaritons in the
ground state, which is related to dramatic suppression of scattering-induced
losses and disorder-induced inhomogeneous dephasing. We also demonstrate
efficient population transfer into the trap when optically injecting free
polaritons outside of its periphery.

††preprint: topo

Introduction.– Excitons (bound electron-hole pairs) in atomically-thin layers
of transition metal dichalcogenide crystals (TMDCs) have a large binding
energy [1] and interact strongly with light [2, 3, 4]. Excitons strongly
interacting and hybridising with photons in an optical microcavity give rise
to polaritons [5, 6, 7, 8, 9, 10]. Hence, creating a local minimum in the
potential energy of either the photon or exciton component of polaritons can
lead to their spatial confinement [11]. In particular, polariton trapping by
patterning the microcavity [12], and thus modifiying the photon energy, has
enabled many proof-of-principle demonstrations of potential applications, such
as polariton lasers [13, 14], topological insulators [15], sensors [16], and
non-classical state generators [17, 18].

Recently, enhanced coherence was demonstrated for TMDC polaritons [19, 20, 21,
22], which was attributed to either strong motional narrowing [22] or the
onset of polariton condensation [19, 20, 21]. At room temperature [19, 20,
22], these observations relied on accidental traps formed during the
microcavity fabrication, arising from strain, fractures in the monolayer, or
air gaps in the cavity spacer, all of which modulate the energies of the
excitons and/or cavity photons. More sophisticated and controlled engineering
of a potential landscape for TMDC polaritons on a microchip remains a
challenge, since the monolayers are fragile and are easily damaged by many
fabrication techniques [23]. So far, such engineering has only been achieved
in an open cavity design [24].

In this work, we create a trap for WS2 polaritons by locally altering the
strength of interactions between the excitons and the photons in a
microcavity. This is achieved by stacking a small monolayer on top of a
WS2/Ga2O3 heterostructure [25], which increases the exciton-photon coupling
strength [26] and creates an effective potential well. By studying the
photoluminescence (PL) of the trapped polaritons inside the potential well in
the double-layer structure, in comparison to the free polaritons with similar
excitonic fraction in the single layer, we observe a significant increase of
the polariton population in the ground state of the trap. Power-dependent
measurements reveal redistribution of the polariton population between the
relaxation bottleneck [27, 28, 29, 30] and the low-energy states for both the
free and the trapped polaritons. Additionally, we find that spatial and
temporal first-order coherence of polaritons in the ground state of the trap
is significantly enhanced compared to the low-energy states of the free
polaritons, and reaches the values comparable to those previously linked to
the onset of bosonic condensation in TMDCs [19, 21, 20]. However, our results
strongly suggest that the enhanced coherence of polaritons is due to coupling
of the short-living excitons to the longer-lived cavity photons, which reduces
both the homogeneous decoherence and inhomogeneous dephasing [22], with the
trapping further suppressing these effects. Finally, our sample design allows
us to efficiently populate the trap with polaritons while exciting them in a
single monolayer, outside the trapping region.

Figure 1: (a) Sample design for creating a trap for WS2 polaritons by placing
a small monolayer (red dot) on top of a WS2/Ga2O3 heterostructure (orange
dot), sandwiched between DBR mirrors. (b) Calculated distribution of the light
field inside the microcavity. The different materials are represented by their
respective dielectric constants, $n$. (c) Calculated angle-resolved
transmission spectra of the structure at the position of (left) the large
monolayer and (right) the small monolayer, marked with the orange and red
dots, respectively. The energy difference, $\Delta V$, corresponds to the
effective trapping potential at the position of the small monolayer in (a).

Engineering polariton confinement.– Fast energy exchange between excitons and
cavity photons with the energies $E_{X}$ and $E_{C}$, respectively, results in
the emergence of the polariton dispersion branches [6]: $E_{\rm
LP/UP}(k)=(1/2)[E_{X}+E_{C}\pm\sqrt{E_{R}^{2}+\delta^{2}}]$. The exciton-
photon detuning, $\delta=E_{C}-E_{X}$, and the Rabi splitting, $E_{R}$, define
the excitonic and photonic fraction of the polariton through the respective
Hopfield coefficients $|X|^{2}=(1/2)[1+\delta/\sqrt{E_{R}^{2}+\delta^{2}}]$,
$|C|^{2}=1-|X|^{2}$. A faster energy exchange increases the splitting between
the polariton branches and decreases the lower polariton energy at zero in-
plane momentum $E_{\rm LP}(k_{||}=0)$. This can be achieved by placing $N$
exciton-hosting layers in an E-field maximum of the cavity with the Rabi
splitting growing as [6]: $E_{R}\propto\sqrt{N}$. Hence, when a small WS2
monolayer, with the size comparable to the polariton de Broglie wavelength, is
placed onto a larger WS2 monolayer, the increased Rabi splitting at the
double-layer position can induce an effective trapping potential comparable to
$E_{R}$ for the lower polariton. However, interlayer interaction can affect
the optical response of the monolayers [31], create interlayer excitons [32],
and weaken the coupling of the intralayer excitons to the photons. This can be
avoided by using a tunnel barrier, such as an hBN spacer [33, 26] between the
monolayers. Here, we use a new, scalable passivation material, ultrathin Ga2O3
glass, which enables reliable fabrication of large-area samples [25].

The value of $E_{\rm LP}(k_{||}=0)$ defines the effective potential energy for
lower polaritons. To estimate the potential energy difference between
polaritons in a single layer and in a double layer (see Fig. 1a) and to
account for the local elongation of the microcavity, we perform transfer
matrix calculations [6] using the linewidth and the oscillator strength of the
excitons extracted from the experiment [34]. Since the E-field has its maximum
at the centre of the microcavity (see Fig. 1b), the energy exchange between
the photons and the excitons is maximized when the monolayers are placed at
this position. Simulated angle-resolved transmission spectra for this
configuration show the anticrossing of the transmission maxima along $k_{||}$
for both single and double layers (see Fig. 1c), signalling the strong
exciton-photon coupling regime. The Rabi splitting is significantly larger in
the double layer. At this detuning, the potential energy step from the single
layer to the double layer, $\Delta V\approx 15~{}\mathrm{meV}$, is comparable
to $E_{R}\approx 25~{}\mathrm{meV}$, and is sufficient to trap polaritons.

Figure 2: (a) Microscope image of the WS2/Ga2O3/WS2 heterostructure on top of
a DBR substrate. The large monolayer spans $\sim 20~{}\mathrm{\mu m}$ in $y$
direction and $>50~{}\mathrm{\mu m}$ in $x$ direction, and the top monolayer
is $\sim 4~{}\mathrm{\mu m}$ in both directions. The scale bar is
$5~{}\mathrm{\mu m}$. (b) Reflectivity spectra of the (blue) WS2/Ga2O3 and
(red) WS2/Ga2O3/WS2 heterostructure. (c) PL image of the complete microcavity
excited with a large Gaussian laser spot (${\rm FWHM}\approx 30~{}\mathrm{\mu
m}$, $\lambda=532~{}\mathrm{nm}$). The scale bar is $5~{}\mathrm{\mu m}$. (d)
Spatially resolved PL spectrum at $k_{||}\approx 0$ along the dashed line in
(c). (e) Angle-resolved PL spectrum at $x\approx 30~{}\mathrm{\mu m}$. (f)
Polariton mode tomography in the trap ($x\approx 30~{}\mathrm{\mu m}$) at the
energies of the (1) ground, (2) first excited, (3) second excited, and (4)
continuum state $\sim 15~{}\mathrm{meV}$ above the potential well, marked with
the arrows in (e).

After fabricating the WS2/Ga2O3/WS2 heterostructure on a highly reflective
distributed Bragg reflector (DBR) substrate ($R>99.99~{}\%$ at
$\lambda=600~{}\mathrm{nm}$) [35], see Fig. 2a, the microcavity is completed
by mechanical transfer of the top DBR [36, 22]. White-light reflectivity
measurements performed on the heterostructure prior to the microcavity
assembly show a decreased reflectivity dip at the position of the double layer
(see Fig. 2b) due to the increased exciton-photon interaction. The PL map in
Fig. 2c, obtained by exciting the sample with a large continuous-wave (cw)
laser spot, shows a strong monolayer emission, with dark areas arising from
the bulk WS2 transferred together with the small monolayer. Angle-resolved PL
measurements at different positions confirm that the whole sample is operating
in the strong coupling regime [34]. In the single-layer region, the PL reveals
a free polariton dispersion (shown in [34]) with the Rabi splitting
$E_{R}\approx 28~{}\mathrm{meV}$ and a range of detuning values corresponding
to highly photonic polaritons, $|X|^{2}\ll 1$.

The spatially resolved PL spectrum along the dashed line in Fig. 2c at
$k_{||}\approx 0$ [34] allows us to trace the potential energy profile for the
polaritons in this direction (see Fig. 2d) [37, 22]. The energy gradient
arises due to a tilt of the top DBR leading to a gradient of cavity photon
energy in the $x$-direction. Between $x=27~{}\mathrm{\mu m}$ and
$33~{}\mathrm{\mu m}$, i.e. at the position of the small monolayer, the
polariton energy at $k_{||}\approx 0$ drops, signalling the formation of a
potential well with the depth of $\Delta V\approx 10~{}\mathrm{meV}$, in
agreement with the transfer matrix calculation [34]. The angle-resolved PL
spectrum around $k_{\|}=0$ is discretized at this position (see Fig. 2e). It
becomes continuous above the potential well barrier, and can be fitted to a
free polariton dispersion with $E_{R}\approx 33~{}\mathrm{meV}$, and
$\delta=-107~{}\mathrm{meV}$, corresponding to $|X|^{2}=0.022$ [38]. PL
measurements at the discrete energies in the trap (see Fig. 2f) reveal
multiple 2D-confined modes. The intensity profiles (polariton densities) are
skewed in the direction of the potential gradient, with the same asymmetry
observed in the continuum (Fig. 2f). These observations agree with simulations
of a tilted box trap that follows the shape of the small monolayer [34].

Figure 3: (a) Energy and (b) momentum-resolved PL of the (red) free and
(black) trapped polaritons. (c,d) Power-dependent intensities at low energy
and bottleneck regions for the (c) free and (d) trapped polaritons [34]. Black
dashed lines show the power-law $I=P^{i}$ ($i=1,0.7$). (e,f) Power-dependent
relative intensities of the (${\rm Int.^{LE}}$) low energy with respect to the
(${\rm Int.^{BN}}$) bottleneck emission for the (e) free and (f) trapped
polaritons, at different duty cycles corresponding to pump pulse duration of
$10~{}\mathrm{\mu s}$, $25~{}\mathrm{\mu s}$, $50~{}\mathrm{\mu s}$. The
intensities are extracted from the angle-resolved PL spectra [34].

Enhanced ground state population.– The PL spectra integrated along energy or
momentum (see Fig. 3a,b) show an order of magnitude enhancement of the
polartion population in the ground state of the trap compared to the low-
energy states of free polaritons near the trap, at the position marked in Fig.
2c. In addition, the states at $k_{||}\approx\pm 4.4~{}\mathrm{\mu m^{-1}}$
are strongly populated. This region is known as the relaxation bottleneck of
the polariton dispersion [27, 28, 29, 30], where the energy relaxation
processes are less efficient than thermalization processes at larger momenta
[28, 7, 9]. While the bottleneck plays an important role in the energy
relaxation of polaritons in quantum-well based microcavities, its role in the
TMDC-based systems remains an open question. In our sample, the pronounced
population at the bottleneck region implies relatively long polariton
lifetimes enabling them to accumulate before decaying radiatively. This allows
us to gain some insight into the energy relaxation of WS2 polaritons by
comparing the populations at the bottleneck and the low-energy region of the
polariton dispersion with increasing pump power in a quasi-cw, pulsed regime
[34].

Figures 3c,d show the power-dependent intensities of the free and trapped
polaritons in their respective bottleneck and low-energy states at a fixed
duration of the pump pulse ($50~{}\mathrm{\mu s}$). The intensity of both
regions grows sub-linearly ($I=P^{i}$, $i<1$) and saturates at higher powers,
which could be due to non-radiative power-dependent exciton losses [39].
Similar behaviour was previously attributed to Auger recombination of excitons
in monolayer WS2 [40, 41]. Before saturating, the PL intensity of the trapped
polaritons in the ground state grows as a power law with the largest exponent
$i=0.7$.

The relative populations in the low-energy states and the bottleneck region
can be quantified by the ratio of their PL intensities (see Fig. 3e,f). For
both the free and the trapped polaritons, the relative population of the low-
energy states increases with power. This redistribution most likely arises
because the population of bottleneck polaritons becomes suppressed, as their
larger excitonic fraction $|X|^{2}$ makes them more affected by excitonic
scattering and losses (e.g, Auger recombination). For the free polaritons, the
redistribution saturates at larger pump powers, suggesting that the rate of
losses in the low-energy states approaches that in the bottleneck region.
Since the saturation becomes stronger with the increasing pulse duration, it
could also be caused by scattering with long-living states, such as phonons
[42, 43] or dark excitons [44].

Interestingly, the saturation is suppressed in the trap (Fig. 3f) and the
relative occupation of the ground state increases significantly with the
growing pump pulse duration. One possible explanation for this effect is the
presence of an additional energy relaxation mechanism, e.g., polariton-
polariton scattering [45, 46], that enables the population redistribution to
successfully compete with the power-dependent saturation, leading to a large
population in the ground state (Fig. 3a,b) and enhanced ground state emission.
However, further studies are needed to confirm this.

Figure 4: (a,b) Spectrally-resolved Michelson interference maps of the (a)
free and (b) trapped polaritons [34]. The overlays are the normalized
interference fringes at the energies marked by the $x$-axis position, with the
amplitude corresponding to their visibility and $g^{(1)}(\Delta x)$ [34]. (c)
Temporal first-order coherence $g^{(1)}(\Delta\tau)$. Solid lines are the fits
accounting for inhomogeneous and homogeneous decoherence (see text).

Enhanced macroscopic coherence.– Spectrally and spatially resolved first-order
coherence $g^{(1)}(\Delta x,\Delta\tau)$ of the polaritons is measured by
using a modified Michelson interferometer [20, 22] [34]. When overlapping the
spatially resolved PL spectrum with its flipped image at $\Delta\tau=0$, we
observe clear interference fringes (see Figs. 4a,b). The normalized fringes
[34], with the magnitude proportional to $g^{(1)}(\Delta x)$, extend over a
larger region for the trapped polaritons compared to the free polaritons in
their respective low-energy states. This is expected since the spatial extent
of a trapped polariton wavefunction is larger than its thermal de Broglie
wavelength $\lambda_{th}\approx 1.36~{}\mathrm{\mu m}$, calculated with the
effective mass extracted from the fit in Fig. 2f [34]. However, the magnitude
of partial coherence at $\Delta x\approx 0$ is similar for the trapped and the
free polaritons and remains constant over a large range of pump powers [34].
The absence of correlation with the observed enhancement of the ground state
population (Fig. 3f) suggests that the enhanced partial coherence of the WS2
polaritons is inherited from their photonic component.

The temporal coherence of the polaritons is measured by applying a time delay
$\Delta\tau$ to the flipped image [34], and extracting the fringe visibility
around $\Delta x=0$, see Fig. 4c. To quantify the decay of coherence, we fit
the data to the coherence function accounting for inhomogeneous, $T_{I}$, and
homogeneous, $T_{H}$, decoherence times [53]:
$g^{(1)}(\Delta\tau)=g^{(1)}(0)\exp\left[-(\pi/2)\left(\Delta\tau/T_{I}\right)^{2}-\left|\Delta\tau\right|/{T_{H}}\right]$.
The fitting shows that the free polaritons are mainly affected by
inhomogeneous decoherence, with $T_{I}=(1084\pm 149)~{}\mathrm{fs}$ and
negligible $T_{H}$. Indeed, the low-energy states of the free polariton
continuum are affected by fluctuations of the cavity photon energy, i.e., due
to the potential gradient (see Fig. 3d), which causes linewidth broadening and
inhomogeneous dephasing. In contrast, for the trapped polaritons in their
ground state, $T_{I}=(2334\pm 166)~{}\mathrm{fs}$ and $T_{H}=(6960\pm
3747)~{}\mathrm{fs}$. The large $T_{I}$ points to the suppression of
inhomogeneous dephasing due to exciton energy fluctuations [22] and to
insensitivity of the discrete state to photon energy fluctuations. The narrow
linewidth of the trapped polariton states, Fig. 3a, is therefore mainly
governed by their lifetime, $\tau_{p}$, which can be extracted from $T_{H}$.
The resulting $\tau_{p}$ of several picoseconds exceeds the radiative lifetime
of the WS2 excitons by at least an order of magnitude [48, 49], suggesting
that it is the long lifetime of the cavity photon, $\tau_{C}$, that determines
the lifetime and hence coherence time of highly photonic polaritons in a high-
quality microcavity: $\tau_{p}^{-1}\propto|C|^{2}\tau_{C}^{-1}$. The coherence
times for both free and trapped polaritons are similar to those reported
previously [19, 20], and are comparable to $\tau_{\rm p}$. One needs to
exercise caution when interpreting these coherence times as a signature of
bosonic condensation [19, 20] because in conventional systems (e.g., quantum
wells and organic semiconductors) the condensate coherence time is much larger
than $\tau_{p}$ [50, 51, 52].

Figure 5: (a) Schematic illustration of the ring-shaped excitation scheme on
the periphery of the small monolayer. (b) Spatially resolved intensity map of
the excitation, with the marked outline of the small monolayer. (c) Angle-
resolved PL spectrum of the trap area. (d) Spectrally-resolved Michelson
interference map of the trapped polaritons and (inset) the normalized
interference fringes.

A distinct advantage of our trap design is the ability to inject free
polaritons in a single monolayer region outside the trap, e.g. with a ring-
shaped laser profile (see Fig. 5a). We do this by using an optical mask [54]
with multiple holes arranged in a ring, and imaging it onto the sample (Fig.
5b). Because the propagation distance of the free, travelling polaritons is
limited only by their lifetime [22], a fraction of them reach the trap and are
confined. The angle-resolved PL, spatially filtered at the position of the
trap, shows the free polariton emission above $E\approx 1.94~{}\mathrm{eV}$,
as well as the trapped polariton emission (see Fig. 5c) with an inhomogeneous
intensity distribution due to the cavity gradient. This demonstrates effective
energy transfer through the Ga2O3 tunnel barrier by the polaritons that are
exclusively excited on the large monolayer. Remarkably, this indirect
excitation of trapped polaritons leads to the same magnitude and extent of
spatial coherence (Fig. 5d) as that observed when the pump is focused directly
on the trap (see Fig. 4).

Conclusion.– In conclusion, we have engineered a 2D trap for room-temperature
WS2 polaritons by locally increasing the light-matter coupling in a
mechanically assembled, all-dielectric microcavity. We demonstrated that the
confinement leads to enhancement of the polariton population in the ground
state. Power-dependent measurements indicate that this enhancement is related
to more efficient energy relaxation for the trapped polaritons. Furthermore,
we found significantly enhanced macroscopic phase coherence for the trapped
polaritons compared to free polaritons, and linked this effect to suppression
of inhomogeneous dephasing. Finally, our demonstration of populating the
engineered trap indirectly, with free polaritons excited outside the trap, is
an important step towards manipulating electrically-injected polaritons on a
microchip, e.g., by using ring contacts previously employed for polariton
lasers [14].

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