# Real-time milli-Kelvin thermometry in a semiconductor qubit architecture

V. Champain  E-mail<EMAIL_ADDRESS>V. Schmitt  Univ. Grenoble Alpes,
CEA, Grenoble INP, IRIG-Pheliqs, Grenoble, France.  B. Bertrand  H.
Niebojewski  Univ. Grenoble Alpes, CEA, LETI, Minatec Campus, Grenoble,
France.  R. Maurand  X. Jehl  C. B. Winkelmann  S. De Franceschi  B. Brun
E-mail<EMAIL_ADDRESS>Univ. Grenoble Alpes, CEA, Grenoble INP,
IRIG-Pheliqs, Grenoble, France.

###### Abstract

We report local time-resolved thermometry in a silicon nanowire quantum dot
device designed to host a linear array of spin qubits. Using two alternative
measurement schemes based on rf reflectometry, we are able to probe either
local electron or bosonic bath temperatures with $\mu$s-scale time resolution
and a noise equivalent temperature of $3$ $\rm mK/\sqrt{\rm Hz}$. Following
the application of short microwave pulses, causing local periodic heating,
time-dependent thermometry can track the dynamics of thermal excitation and
relaxation, revealing clearly different characteristic time scales. This work
opens important prospects to investigate the out-of-equilibrium thermal
properties of semiconductor quantum electronic devices operating at very low
temperature. In particular, it may provide a powerful handle to understand
heating effects recently observed in semiconductor spin-qubit systems.

##  I Introduction

Quantum electronics embraces a large variety of devices whose functionality
relies on quantum mechanical properties such as size quantization, phase
coherence, entanglement, etc. Cryogenic operation is generally required for
the emergence of such properties and, at low temperature, the time scales for
energy exchange and thermalization processes tend to increase dramatically,
leading to metastable regimes and to the co-existence of thermal (and non-
thermal) baths strongly out of equilibrium. The understanding and control of
these thermodynamical aspects is not only mandatory for the development and
operation of bolometers, cryogenic thermometers or coolers; it can also be
particularly crucial for improving the performance and scalability of solid-
state devices for quantum sensing and quantum computing. To this aim, access
to thermometers capable of measuring local temperatures faster than the
characteristic time scales of heat exchange dynamics is of primary
importance[1].

In this work, we address this point in the context of semiconductor quantum-
dot (QD) devices, largely motivated by their prospects for scalable spin-based
quantum computing. The operation of semiconductor spin qubits requires
applying high-frequency signals that unavoidably dissipate energy into the
qubit environment due to the Joule effect and dielectric losses. This raises
severe practical issues regarding spin qubits addressability, since a
significant temperature dependence of their Larmor frequencies has recently
been reported [2]. Fast temperature changes due to unintentional heating can
lead to spin decay and dephasing [3, 4], thereby reducing gate and readout
fidelities [5]. This problem worsens for increasing numbers of qubits and
corresponding control gates, leading to a clear bottleneck for large-scale
integration. A higher operation temperature can mitigate the impact of local
heating [2] but at the price of a reduced spin coherence time [6, 7].
Therefore, understanding the modality and dynamics of heating from microwave
excitation is a necessary step to devise more efficient measures to preserve
qubit performance. To this end, access to fast time-domain thermometry
compatible with mK temperature and hundreds of mT fields can be a clear asset.

The most advanced local thermometry techniques in quantum circuits rely on
metal-superconductor junctions [8, 9, 10]. These thermometers are incompatible
with spin-based quantum processors, due to integration constrains and to their
susceptibility to magnetic fields. Semiconductor QD thermometers[10, 11, 12,
13, 14], on the other hand, do not suffer from these limitations. Moreover,
they can measure the temperature of an electronic reservoir without requiring
galvanic coupling [14, 15, 16].

Here we apply this approach to silicon QD devices. By adjusting the tunnel
coupling between the QD thermometer and a probed electronic reservoir, we are
able to preserve its sensitivity down to the base temperature (55 mK) of a
dilution refrigerator. In addition, by adopting an isolated double-dot
configuration of the QD thermometer, we extend our thermometry capabilities to
probe the local bosonic temperature. We simultaneously show that this
thermometer can perform time-domain measurements at the microsecond scale.

##  II Device and operation modes

The QD devices consist of silicon-on-insulator metal-oxide-semiconductor (MOS)
nanowires with a rectangular cross-section of 80 $\times$16 nm2. The nanowires
are covered by a set of eight parallel gates with a pitch of 80 nm, see
example in Fig. 1(a). Thanks to p-doped leads, holes are accumulated under
negatively-biased gates as schematically shown in Fig. 1(b).

Figure 1:  Device and charge configuration for two thermometry techniques. (a)
False-color tunneling electron micrograph of a representative MOS device with
a silicon nanowire channel (blue), highly doped conductive leads (yellow) and
eight parallel gates (red). Note that the gates can be considered metallic,
however the contacts on the leads and gates are made of tungsten vias, that is
superconducting at base temperature. Three different samples were measured in
this work (see Supp. Mat. for details)[23]. (b) Schematic of a 8-gate device
illustrating the two thermometry methods. Source and drain (yellow) are highly
p-doped to form metallic leads acting as hole reservoirs. Quantum dots (QDs)
can be defined along the undoped silicon channel by means of gates G1 to G8.
In the first type of thermometer (setting I), G1 defines a single QD tunnel-
coupled to the source reservoir as illustrated by the hole-energy diagram just
below. The hole reservoir is modelled by a Fermi distribution with electronic
temperature $T_{e}$, while the dot is modelled by a single level whose
linewidth is determined by the tunnel rate $\Gamma$. The detuning,
$\varepsilon$, is defined here as the difference between the QD
electrochemical potential and the Fermi level in the reservoir. In the second
thermometer (setting II), a double QD isolated from the reservoirs is defined
under G6 and G7. Both QDs are modelled by single levels with a tunnel coupling
$t$ and detuning $\varepsilon$. The corresponding energy diagram is shown just
below, with $|L\rangle$ ($|R\rangle$) representing the state localized in the
left (right) QD at large detuning, and $|+\rangle$ ($|-\rangle$) the
hybridized bonding (antibonding) state at zero detuning. The coupling to the
leads is completely suppressed by means of the two side gates G5 and G8.

We investigate two different device settings. In the first setting (I), we
define a single QD tunnel-coupled to the hole reservoir in one the two
degenerately-doped contact leads, as shown on the left hand side of Fig. 1(b).
The impedance of a lumped element LC circuit connected to the same lead is
sensitive to charge tunneling between the QD and the lead. Both dissipative
and capacitive changes can be detected through reflectometry measurements [17,
18, 19].

The second setting (II) requires biasing four consecutive gates, as shown on
the right hand side of Fig. 1(b). The two inner gates (G6, G7) define a double
QD, whereas the outer ones (G5, G8) allow fully isolating this double QD from
the remaining part of the channel and from the leads[20, 21]. Read-out is now
performed by reflectometry on an inner gate electrode (G7), with interdot
tunneling resulting in a quantum capacitance contribution [17, 18, 22].

The tank circuits used for RF-reflectometry are formed by classical surface-
mount inductors and parasitic capacitances. While different values of
inductance have been used, all resonances are around $400$ MHz, with few-to-
several MHz bandwidth (see Supp. Mat. for details)[23]. The reflectometry
measurements are performed in a weak coupling, dispersive regime, in which the
phase shift of the reflected signal is directly proportional to the frequency
shift, and thus to the quantum capacitance [28, 29, 17, 22].

##  III Quantum-dot thermometry: basic principles and experimental
implementation

In both experimental configurations, the quantum capacitance contribution is
expected to depend on temperature. In setting I, where a single level is
tunnel-coupled to a Fermi sea, the quantum capacitance reads [16, 14, 11]:
(see Supp. Mat. for details)[23]

$\displaystyle C_{q}(\varepsilon)=\alpha^{2}e^{2}\int_{-\infty}^{+\infty}$
$\displaystyle\left(\frac{1}{4k_{B}T_{e}}\cosh\left(\frac{e\alpha\varepsilon-E}{2k_{B}T_{e}}\right)^{-2}\right)$
$\displaystyle\times$
$\displaystyle\left(\frac{\hbar\Gamma}{\hbar^{2}\Gamma^{2}+E^{2}}\right)\text{d}E.$
(1)

Here $e$ is the electron charge, $k_{B}$ the Boltzmann constant, $\hbar$ the
reduced Planck constant, $\alpha$ the gate lever-arm parameter, $\varepsilon$
the detuning (in gate voltage), and $\Gamma$ the dot-lead tunnel rate. The
relevant temperature is the electronic temperature of the lead, $T_{e}$. The
quantum capacitance is therefore a convolution of two peaks: the derivative of
the Fermi distribution, broadened by $k_{B}T_{e}$, and the dot density of
states, modeled by a Lorentzian of width $\hbar\Gamma$. Thermal broadening
dominates for $\hbar\Gamma<k_{B}T_{e}$.

In setting II, hybridization of the orbitals in the left and right QDs results
in a two-level system in the canonical ensemble. The corresponding energy
diagram as a function of the gate-dependent level detuning, $\varepsilon$, is
characterized by an avoided level crossing due to the inter-dot tunnel
coupling $t$. The quantum capacitances associated to the bonding and anti-
bonding states have opposite signs since they are directly related to the
curvatures of the corresponding energy-vs-detuning relations [29]. The total
capacitance is given by the sum of the two quantum capacitances weighted by
the population of the bonding and anti-bonding states $P_{|+\rangle}$,
$P_{|-\rangle}$ (see Supp. Mat. for details)[23], i.e.: [29, 30, 11]

$C_{q}(\varepsilon)=\alpha^{2}e^{2}\frac{2t^{2}}{\left((\alpha
e\varepsilon)^{2}+4t^{2}\right)^{3/2}}\tanh\left(\frac{\left((\alpha
e\varepsilon)^{2}+4t^{2}\right)^{1/2}}{2k_{B}T_{B}}\right).$ (2)

Where $T_{B}$ stands for the Boltzmann temperature of the double dot, given by
the relative population in both states
$k_{B}T_{B}=2t/\ln{(P_{|+\rangle}/P_{|-\rangle}})$. At thermal equilibrium
with the thermal bath, i.e. in the absence of external charge drive, the
Boltzmann temperature equals the bath temperature. We notice that
$C_{q}(\varepsilon)$ is peaked at $\varepsilon=0$ with a linewidth solely
determined by the tunnel coupling $t$, and an amplitude that becomes strongly
temperature dependent in the limit $t\lesssim k_{B}T_{B}$.

Both settings are experimentally tested by thermally anchoring the device to
the mixing chamber of a dilution refrigerator with a base temperature of 55
mK, and the results are shown in Fig 2. We initially verify that the impedance
of the LC-resonator itself is temperature independent in the range of
interest. Fig. 2(a) presents the temperature dependence of the reflected
signal measured in the case of setting I (left-hand side of Fig. 1(b)). The
data are obtained after selecting the dot-lead transition with the narrowest
Coulomb peak resonance (see Supp. Mat. for details)[23]. The phase of the
reflected signal displays a peak proportional to the expected peak in
$C_{q}(\varepsilon)$ whose amplitude decreases with the mixing-chamber
temperature, $T_{MC}$, while its width increases. This is in very good
agreement with Eq. (1), which we use to fit the experimental peaks (see Supp.
Mat. for details)[23]. We find a tunnel coupling $\Gamma/2\pi=383\pm 38$ MHz,
an order of magnitude lower than $k_{B}T_{e}/h$ at base temperature, thereby
fulfilling the condition for proper temperature sensitivity.

For the case of setting II (right-hand side of Fig. 1(b)), we first load a few
holes under G6 and G7 by applying negative voltages to G6, G7 and G8. Bringing
G8 back to zero voltage results in the trapping a few charges confined to a
double QD controlled by G6 and G7. Provided the interdot tunnel coupling is
greater than the LC resonator frequency, charge transitions from one dot to
the other can be detected by RF reflectometry. In this isolated regime, dot-
lead transitions are inhibited and the interdot transitions are the only
visible spectroscopic features, extending over the entire stability diagram
[21, 20](see Supp. Mat. for details)[23]. As we follow the interdot transition
towards more negative voltages, we create deeper electrostatic quantum wells,
effectively increasing the barrier height and hence lowering $t$. We can thus
achieve a regime where the thermal population of the double-QD two-level
system becomes highly sensitive to the bath temperature $T_{B}$. Similarly to
the first configuration, we measure the interdot transitions as a function of
$T_{MC}$, the results being shown in Fig. 2(b). Fitting to Eq. 2 yields a
tunnel rate $t/h=2.03\pm 0.04$ GHz. Even though the gap due to level repulsion
is four times larger than the thermal energy at base temperature, we still
have a $2\%$ thermal occupation of the excited state. This ensures a
measurable temperature dependence of the quantum capacitance all over the
explored temperature range. Yet extending this measurement technique to even
lower temperatures would require reducing the tunnel coupling accordingly, as
discussed later. Nonetheless, we note that the relatively low tunnel coupling
achieved here enables us to operate this non-galvanic thermometer at
temperatures an order of magnitude lower than previously reported [16].

Figure 2: Experimental demonstration of the two thermometry techniques (a)
Setting I: Phase of the reflectometry signal from the tank circuit as a
function of detuning, for a dot-lead transition at different mixing chamber
(MC) temperatures. The peak amplitude is decreasing with temperature while the
peak width increases. (b) Setting II: Phase of the reflectometry signal for an
interdot transition as a function of MC temperature. The peak amplitude is
decreasing, while the width remains unchanged. (c) $T_{MC}$ dependence of the
electronic temperature, $T_{e}$, in the source reservoir as obtained from
fitting to the Eq. (1). Inset: Closeup of the low-temperature saturation. (d)
$T_{MC}$ dependence of $T_{B}$, as obtained from fitting the the Eq. (2).

The measured temperatures extracted from the fits are shown in Fig. 2(c) and
(d) as a function of $T_{MC}$ in setting I and II, respectively. At relatively
high temperature, both $T_{e}$ and $T_{B}$ follow closely the mixing chamber
temperature, except for significant deviations above $500$ mK that can be
attributed to the population of higher-energy QD levels, which is not taken
into account in our models.

Below about 100 mK, however, the two experimental settings exhibit different
behavior. In setting I, the measured electronic temperature saturates. This
trend, commonly observed in cryogenic experiments, is due to the residual
electromagnetic noise coming from the circuitry at higher temperature,
combined with the vanishing thermal coupling of electrons to their environment
[31, 32]. The inset in Fig. 2(c) shows a closeup of these data points, with a
fit to a saturation function $T_{e}=(T_{\rm MC}^{n}+T_{0}^{n})^{1/n}$,
yielding here $n=3.4$ and a saturation temperature $T_{0}=84$ mK. In the case
of an electron-phonon dominated thermalization [10], one would expect $n=5$ in
degenerately doped silicon[31]. However, the structure of the lead in our
experiment involves also other materials in close vicinity, including
metallic/superconducting vias, hindering an accurate thermal modelling.

Interestingly, no such saturation is observed in the case of setting II.
Indeed, here the isolated double-QD is not coupled to any Fermi sea and its
temperature is determined by the interaction with the local bosonic baths,
i.e. lattice phonons and photons in the electromagnetic environment.

##  IV Thermometer calibration and optimization

We now investigate the possibility of performing local thermometry in the time
domain. We focus on setting II. The peak amplitude of the phase shift at zero
detuning strongly depends on the bath temperature and it can be measured on a
microsecond time scale. This opens the possibility to monitor the time
evolution of $T_{B}$, provided thermal equilibration between the QD two-level
system and the surrounding bath occurs on a shorter time scale.

Figure 3: Thermometer calibration and optimal operation regime for setting II.
(a) Peak amplitude, $\varphi_{0}$, as a function of $T_{MC}$ in the case of
setting II. The experimental data (open circles) are fitted to the Eq. (3)
(red solid line), giving a tunnel rate $t/h=1.72\pm 0.04$ GHz. The grey dashed
line is a linear fit to the data in the $55-100$ range of highest sensitivity
(${\lvert\partial\varphi_{0}/\partial T\rvert}=1.28$ mrad/mK). (b) Calculated
noise equivalent temperature, $NET$, versus ($T_{MC}$,$t$). Iso-NET curves are
drawn in blue and yellow colors. The white dashed line highlights the
$t$-dependence of the NET minimum (i.e., the $T_{MC}$ where $\partial
NET/\partial T_{MC}=0$).

To precisely calibrate this thermometer, we measure the reflected signal phase
at zero detuning, $\varphi_{0}$, and vary the mixing-chamber temperature. The
resulting calibration curve is shown in Fig. 3(a). From Eq. (2), by taking
$\varepsilon=0$ and, based on Fig. 2(d), $T_{B}=T_{MC}$, we obtain

$\displaystyle\varphi_{0}=\kappa\times\frac{1}{2t}\tanh\left(\frac{t}{k_{B}T_{MC}}\right),$
(3)

where $\kappa$ is a proportionality coefficient that does not depend on
temperature.

The data are in good agreement with this model (red solid line in Fig. 3(a)),
where $t$ and $\kappa$ are the free parameters. We note that, in principle,
the anti-crossing gap ($2t$) in such a configuration could be independently
measured through Landau-Zener-Stückelberg interference [33, 34]. Such an
additional measurement would make this type of thermometer a primary one,
removing the need of a calibration process.

Notably, the model used for the fit in Fig. 3(a) predicts a saturation of the
peak amplitude at low temperature, i.e. when the population of the anti-
bonding state becomes negligibly small ($\lesssim 0.5\%$). In our experimental
conditions, this saturation is not observed because it occurs slightly below
the mixing-chamber base temperature. Fig. 3(a) also shows that the thermometer
sensitivity (proportional to $\lvert{\partial\varphi_{0}/\partial
T_{MC}}\rvert$) gets maximal close to the fridge base temperature, i.e. just
above the expected low-temperature saturation. The dashed grey line in Fig.
3(a) is a linear fit around the point of maximal sensitivity (55-100 mK).

To further characterize the thermometer performance and deepen our
understanding of its optimal operating conditions, we now turn to the
evaluation of the noise equivalent temperature ($NET$) and its dependence on
$t$ and $T_{MC}$. Since
$NET=\frac{S_{\varphi\varphi}}{\lvert{\partial\varphi_{0}/\partial
T_{MC}}\rvert}$, where $S_{\varphi\varphi}$ is the phase noise amplitude, we
begin by recording the phase signal at various sampling rates covering a large
frequency domain. The measured spectrum reveals a white noise floor at
$S_{\varphi\varphi}=3.9\pm 0.2\text{ mrad}/\sqrt{\text{Hz}}$, which is
consistent with the expected noise from the first amplification stage at 4 K.
This gives $NET=3.0\pm 0.2\text{ mK}/\sqrt{\text{Hz}}$ in the maximal
sensitivity, low-temperature region of Fig. 3(a). This value is expected to
change by varying the interdot tunnel coupling and the temperature range of
operation. To fully capture this dependence, implicitly coming from the
$\lvert{\partial\varphi_{0}/\partial T_{MC}}\rvert$ term, we make use of Eq.
(3) with the additional hypothesis of a dispersive coupling between the LC
resonator and the isolated double QD, which implies[28]:

$\kappa=A_{0}\left(\frac{1}{2t+f_{r}}+\frac{1}{2t-f_{r}}\right)$ (4)

where $f_{r}$ is the resonant frequency of the tank circuit and $A_{0}$ is a
constant depending only on circuit parameters , whose value can be deduced
from the fit of Fig. 3(a). The resulting $NET(T_{MC},t)$ map, extrapolated
from the data in Fig. 3(a), is shown in Fig. 3(b).

For a given tunnel coupling, the thermometer is expected to have an optimal
temperature where $NET$ is minimized. The white dashed line highlights the
$t$-dependence of this optimal operation temperature. Interestingly, reducing
the tunnel coupling at fixed temperature always results in a lower $NET$ no
matter what the temperature is. Therefore, operating at weak tunnel coupling
is generically beneficial. However, this only holds as long as the dynamics of
the LC resonator is slow with respect to interdot tunneling (adiabatic limit),
i.e. away from the divergence in Eq. (4) for $t\sim 2f_{r}$. For this reason,
the calculation in Fig. 3(b) is limited to $t\geq 1$ GHz. (Below this cutoff,
other physical mechanisms such as charge noise would also limit the
thermometer sensitivity.) For $t=1$ GHz, the thermometer NET could be as low
as $1\text{ mK}/\sqrt{\text{Hz}}$ at $30$ mK. Moreover, a better impedance
matching between the device and the LC resonator would result in a larger
$A_{0}$ and hence an even lower NET, thereby approaching and possibly
exceeding the performance of state-of-the-art metal-superconductor
thermometers [8].

##  V  Proof-of-concept: real-time thermometry

Finally, we address the use of our device for real-time thermometry. The
measured $NET$ is in principle too large to allow stochastic temperature
fluctuations to be resolved at a microsecond time scale. However, the
developed thermometry could still be applied to probe the dynamics of local
thermal exchange caused by deterministic and periodic heating events, such as
those associated with the sequences of microwave bursts typically employed in
the operation of spin qubit devices. Under these conditions, the signal to
noise ratio can be increased by averaging over multiple cycles. As a result,
the thermometry bandwidth is no longer limited by the $NET$ but by the
response time of the system itself.

We identify two fundamentally limiting bandwidths. The first is the response
time of the measurement apparatus. It would be defined here by the bandwidth
of the resonator, allowing for a time resolution of approximately $20$ ns
($7.8$ MHz). The second limiting factor is the charge relaxation time $T1$
towards a thermal state, since the measurement of the Boltzmann temperature is
only meaningful at thermal equilibrium. This happens in this system in about
15 ns (see Supp. Mat. for details)[23], whereas maximal relaxation time of
about 100 ns have been reported in similar devices[35]. DQD thermometry is
therefore robust up to a few MHz.

Figure 4: Real-time thermometry with setting II. Microwave bursts are
periodically applied to gate G5 adjacent to the double-dot thermometer. The
microwave of frequency 15 GHz and power -40 dBm is modulated by a square
signal of frequency 20 kHz, resulting in 50 $\mu$s-long bursts, with 50 $\mu$s
waiting time between each burst. The phase signal is acquired with a
synchronized lock-in at a sampling rate of $1.7$ MHz with a $400$-ns
integration time. It is averaged over $2.10^{6}$ cycles and is plotted as a
function of time (open circles). Left panel (heating) : At each cycle, the
microwave burst is turned on at $Time=0$, and the phase $\varphi_{0}$
decreases from $\varphi_{c}\sim 220$ to $\varphi_{on}\sim 200$ mrad. Right
panel (cooling): the microwave excitation is switched off at $Time=0$ and the
phase raises back to $\varphi_{on}$ with a two stage relaxation, see plain
text for details.

For a proof-of-concept demonstration of time-resolved thermometry, we
periodically induce local heating by means of microwave bursts applied to a
nearby gate, other than the gates defining the double-dot thermometer. Data
acquisition is synchronized with the applied bursts to monitor the locally
induced temperature modulation [8].

We plot the resulting phase signals versus time in Fig.4. We observe the
system switching between two phase levels $\varphi_{on}$ if the microwave is
$on$ and $\varphi_{c}$ when it is $off$. This values would translate to $75$
mK and $55$ mK respectively with the previous calibration. However when the
drive is $on$, non-adiabatic (Landau-Zener) processes can occur and populate
the excited state of the double dot. This would decrease the phase measured.
The bath temperature of the "warm" stage can then hardly be precisely known.
After the drive is turned $off$, the phase increases back to thermal
equilibrium.

The transient regimes exhibit however different characteristic time scales for
cooling and heating.

We model the time evolution after the microwave is turned on by an exponential
decay:

$\varphi_{0}(\tau)=\varphi_{on}+(\varphi_{c}-\varphi_{on})\exp\left(-\tau/\tau_{m}\right).$
(5)

Which we plot as a red solid lines in Fig.4. We find a characteristic time
$\tau_{m}=0.93\pm 0.35$ µs close to our experimental time resolution, which is
set by the lock-in measurement bandwidth.

The cooling dynamic reveals two relaxation processes, as evidenced in right
panel of Fig.4. The first relaxation corresponds to the rapid thermalization
of the the QD thermometer. This occurs on the characteristic time scale for
charge relaxation, T1, which we estimate to be about 15 ns (see Supp. Mat. for
details)[23]. Since $T1\ll\tau_{m}$, this charge-relaxation dynamics cannot be
resolved and the phase decay is again limited by the measurement bandwidth. A
single relaxation on time scale $\tau_{m}$, however, cannot capture the whole
dynamics, as evidenced by the comparison to the grey line in Fig.4. A second,
slower relaxation, with a characteristic decay time of $\tau_{b}=6.0\pm 1.5$
µs, is in fact observed. We ascribe this to the relaxation of the heat bath,
which can be revealed once the QD thermometer has thermalized with the
surrounding bath. We then propose a more complete model, including a
relaxation of the thermometer toward equilibrium with the bath at time
$\tau_{m}$,

$\varphi_{0}(\tau)=\varphi_{bath}(\tau)+(\varphi_{on}-\varphi_{bath}(\tau))\exp\left(-\tau/\tau_{m}\right),$
(6)

while the bath relaxes at time $\tau_{b}$

$\varphi_{bath}(\tau)=\varphi_{c}+(\varphi_{h}-\varphi_{c})\exp\left(-\tau/\tau_{b}\right).$
(7)

Injecting (7) in (6) leads to the red model in the right panel of Fig.4 while
$\varphi_{bath}$ is plotted with a green dashed line. We can extrapolate the
relaxation of the bath at $Time=0$ to estimate the bath temperature $\sim 62$
mK in the presence of heating, effectively smaller than $75$.

A deeper investigation (e.g. exploring the effect of varying the microwave
excitation cycles and the way they are applied) could provide important
missing information about the dissipation mechanisms and the nature of the
thermal bath in the QD environment.

Previous works on different types of Si-based spin-qubit devices have
indirectly inferred thermal relaxation times from time-resolved Larmor
frequency shifts attributed to local overheating [36, 4, 2] . As compared to
this recent literature, the relaxation times reported here are either
comparable [4] or significantly shorter [36, 2]. The discrepancy could be
ascribed to the different device structures and to the different experimental
conditions (e.g. power and duration of the microwave excitation pulses).

##  VI Conclusions

In summary we have implemented and characterised non-invasive, non-galvanic
thermometers sensing either the electronic temperature of a Fermi reservoir,
or the local bosonic temperature in a semiconductor quantum-dot device. For
the DQD thermometer we could obtain a state-of-the-art noise equivalent
temperature of $3$ $\rm mK/\sqrt{\rm Hz}$ and identify a path to even lower
noise levels, well below 1 $\rm mK/\sqrt{\rm Hz}$. By synchronizing
temperature acquisition to a periodic sequence of microwave bursts, we could
increase the signal-to-noise ratio and probe temperature variations on a
micro-second time scale. While this time-resolved QD thermometry is
demonstrated in a silicon MOS device, it could be readily reproduced in any
other semiconductor platform, such as SiGe-based heterostructures. Hence the
present work opens new experimental prospects to understand heating effects in
semiconductor quantum processors and to tackle the widely unexplored field of
experimental quantum thermodynamics in nanoelectronic systems [37].

##  VII Acknowledgements

This research has been supported by the European Union’s Horizon 2020
research, innovation program under grant agreements number 951852 (QLSI
project) and number 810504 (ERC project QuCube), the Program QuantForm-UGA n°
ANR-21-CMAQ-0003 France 2030 and by the LabEx LANEF n° ANR-10-LABX-51-01.

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$\widetilde{\partial}_{\mathrm{int}}\bar{B}\subseteq\widetilde{\partial}_{\mathrm{int}}B$.
We argue by contradiction. Assume that there exists
$x\in\widetilde{\partial}_{\mathrm{int}}\bar{B}\setminus\widetilde{\partial}_{\mathrm{int}}B$.
Since
$\widetilde{\partial}_{\mathrm{int}}\bar{B}\subseteq\bar{B}=B\cup(\cup_{H\in\mathfrak{H}_{B}}H)$
and $B$ is disjoint from $\bar{B}\setminus B$, there are two cases. Either
$x\in B\setminus\widetilde{\partial}_{\mathrm{int}}B$ or
$x\in\mathfrak{(}\cup_{H\in\mathfrak{H}_{B}}H)\setminus\widetilde{\partial}_{\mathrm{int}}B$.

For the first case assume that $x\in
B\setminus\widetilde{\partial}_{\mathrm{int}}B$. Then all its
${\mathbb{Z}}^{d}_{1}$-neighboring vertices were also in $B$ by Definition 2.4
of the interior boundary. Thus $x$ is surrounded by $B$, and hence
$x\in\bar{B}$. Similarly, the neighboring vertices are also in $\bar{B}$,
contradicting that $x\in\widetilde{\partial}_{\mathrm{int}}\bar{B}$.

For the second case assume that $x\in(\cup_{H\in\mathfrak{H}_{B}}H)$. Then $x$
was surrounded by $B$, but then also all its
${\mathbb{Z}}^{d}_{1}$-neighboring vertices were either a member of $B$ or
surrounded by $B$, contradicting again that
$x\in\widetilde{\partial}_{\mathrm{int}}\bar{B}$.

We move on to part (ii). Assume that
$\bar{B}_{1}\cap\bar{B}_{2}\neq\emptyset$, then there exists
$x\in\bar{B}_{1}\cap\bar{B}_{2}$. Since $B_{1}$ and $B_{2}$ are
${\mathbb{Z}}^{d}_{1}$-disconnected, it is excluded that $x\in B_{1}\cap
B_{2}$. Assume $x\in(\bar{B}_{2}\setminus B_{2})\cap B_{1}$. Since $x$ is
surrounded by $B_{2}$, any vertex $y\in B_{1}$ must be surrounded by $B_{2}$,
since $\|B_{1}-B_{2}\|_{1}\geq 2$ and $B_{1}$ itself is
${\mathbb{Z}}^{d}_{1}$-connected. Hence, $\bar{B}_{1}\subseteq\bar{B}_{2}$.
The argument for $x\in(\bar{B}_{1}\setminus B_{1})\cap B_{2}$ follows
analogously. Lastly, assume that there exists $x\in(\bar{B}_{1}\setminus
B_{1})\cap(\bar{B}_{2}\setminus B_{2})$. Then there exists, for some $j\geq
1$, a path $\pi=(x,x_{1},\ldots,x_{j})$ on ${\mathbb{Z}}^{d}_{1}$ such that
$x_{j}\in\partial_{\mathrm{int}}B_{1}\cup\partial_{\mathrm{int}}B_{2}$ and
$x_{\ell}\in(\bar{B}_{1}\setminus B_{1})\cap(\bar{B}_{2}\setminus B_{2})$ for
all $\ell\leq j-1$. Assume w.l.o.g. that
$x_{j}\in\partial_{\mathrm{int}}B_{1}$. Since there were no vertices from
$B_{2}$ on the path and $x$ is surrounded by $B_{2}$, it must follow that also
$x_{j}$ is surrounded by $B_{2}$. Similar to the previous case, it follows
that $\bar{B}_{1}\subseteq\bar{B}_{2}$.

Part (iii) claims that when $\bar{B}_{1}\cap\bar{B}_{2}=\emptyset$, and
initially $B_{1},B_{2}$ are $1$-disconnected, then
$\|\bar{B}_{1}-\bar{B}_{2}\|_{1}\geq 2$. By definition of the distance
$\|\cdot\|_{1}$ between sets, we have

$\|\bar{B}_{1}-\bar{B}_{2}\|_{1}=\min_{x_{1}\in\bar{B}_{1}\setminus
B_{1},x_{2}\in\bar{B}_{2}\setminus
B_{2}}\big{\\{}\|B_{1}-B_{2}\|_{1},\|x_{1}-B_{2}\|_{1},\|B_{1}-x_{2}\|_{1},\|x_{1}-x_{2}\|_{1}\big{\\}}.$
(A.1)

Each $1$-connected path from $x_{i}\in\bar{B}_{i}\setminus B_{i}$ to any
$y\notin\bar{B}_{i}$ must cross a vertex in
$\widetilde{\partial}_{\mathrm{int}}\bar{B}_{i}$. Consequently,

$\|x_{1}-B_{2}\|_{1}\geq\|x_{1}-\widetilde{\partial}_{\mathrm{int}}\bar{B}_{1}\|_{1}+\|\widetilde{\partial}_{\mathrm{int}}\bar{B}_{1}-B_{2}\|_{1}\geq\|x_{1}-\widetilde{\partial}_{\mathrm{int}}\bar{B}_{1}\|_{1}+\|\widetilde{\partial}_{\mathrm{int}}B_{1}-B_{2}\|_{1}\geq
2,$

where the second inequality follows since
$\widetilde{\partial}_{\mathrm{int}}\bar{B}_{1}\subseteq\widetilde{\partial}_{\mathrm{int}}B_{1}$
by part (i), and the third inequality since by 1-disconnectedness of $B_{1}$
and $B_{2}$ the second term on the right-hand side is at least two. The third
and fourth term in the minimum in (A.1) can be bounded similarly. It follows
that $\bar{B}_{1}$ and $\bar{B}_{2}$ are $1$-disconnected.

The fourth statement is immediate from [12, Lemma 2.1] which states that the
interior and exterior boundaries with respect to ${\mathbb{Z}}^{d}$ of any
$\ast$-connected set are $\ast$-connected as well.

We turn to the last statement, and show that for any hole $H$ of a block $B$,
$H=\bar{H}$, i.e., that $H$ does not contain holes. This is true since $H$ was
formed as a maximal $1$-connected subset of the vertices in
$\Lambda_{n}\setminus B$ surrounded by $B$, see below (2.2) in Definition 2.1.
So if there were a hole $J$ inside $H$, then $J$ must be part of $B$, which
would then contradict the $1$-connectedness of $B$, since
$\|J-\partial_{\mathrm{ext}}H\|\geq 2$ as $H$ fully surrounds $J$. The fact
that the interior and exterior boundaries of a hole are $\ast$-connected
follows now from Part (iv). ∎

###### Proof of Claim 2.6.

The proof is inspired by an argument by Deuschel and Pisztora [12, Proof of
(A.3)]. The inequalities with $(\star)$ in (2.5) follow by standard
isoperimetric inequalities, but we will also derive them below.

We will first show the bounds for $\partial_{\mathrm{int}}$ and
$\widetilde{\partial}_{\mathrm{int}}$. At the end of the proof we adjust it to
$\partial_{\mathrm{ext}}$ and $\widetilde{\partial}_{\mathrm{ext}}$. We start
by showing that there exists $\delta^{\prime}>0$ such that
$\partial_{\mathrm{int}}A\geq\delta^{\prime}\widetilde{\partial}_{\mathrm{int}}A$
for all $A\subseteq\Lambda_{n}$ of size at most $3n/4$. Fix such a set
$A\subseteq\Lambda_{n}$. We recall an inequality related to the isoperimetric
inequality by Loomis and Whitney [29, Theorem 2]. For a set
$A\subseteq\Lambda_{n}$, let $S_{i}$ denote the projection of $A$ onto the
$i$-th coordinate hyperplane. That is, for a vertex with coordinates
$x=(x_{1},\dots,x_{i-1},x_{i},x_{i+1},\dots,x_{d})$ we define
$\pi_{i}x:=(x_{1},\dots,x_{i-1},0,x_{i+1},\dots,x_{d})$. Then

$|A|^{d-1}\leq\prod_{i\in[d]}|S_{i}|.$ (A.2)

Let $i_{\star}$ be the dimension that contains the largest projected set
$S_{i_{\star}}$ (ties broken arbitrarily), so that as a result of (A.2),

$|S_{i_{\star}}|\geq|A|^{(d-1)/d}.$ (A.3)

We abbreviate $S=S_{i_{\star}}$, and write $\pi_{\star}$ for the the
$i_{\star}$-th projection. For $s\in S$ we define the preimage of $s$ as

$\pi_{\star}^{\uparrow}:=\\{y\in A:\pi_{\star}(y)=s\\}$

We now call a vertex $s\in S$ a _fiber_ if there is no vertex in
$\partial_{\mathrm{int}}A$ that projects to $s$ via $\pi_{\star}$. Formally we
define the _fibers_ of $S$ as

$F:=\\{s\in S:\nexists y\in\partial_{\mathrm{int}}A\mbox{ with
}\pi_{\star}(y)=s\\}.$ (A.4)

The pre-image of any fiber vertex does not contain any vertex of
$\partial_{\mathrm{int}}A$ within $\Lambda_{n}$, hence, it contains a full
length-$n^{1/d}$ line ${\mathcal{L}}_{s}$ connecting the two opposite faces of
$\Lambda_{n}$, with $\pi_{\star}({\mathcal{L}}_{s})=s$. This is because all
vertices that share all coordinates with $s$ except the $i_{\star}$th
coordinate, project to $s$ via $\pi_{\star}$, so $A$ must contain all of them
(the possibility of $A$ containing none of ${\mathcal{L}}_{s}$ is excluded by
assuming $s\in S$), otherwise there would be a boundary vertex of $A$ among
them. Then the pre-image of any such fiber vertex intersects the box-boundary
$\widetilde{\partial}_{\mathrm{int}}\Lambda_{n}$ in exactly $2$ vertices:

$|\pi_{\star}^{\uparrow}(s)\cap\widetilde{\partial}_{\mathrm{int}}\Lambda_{n}|=|\pi_{\star}^{\uparrow}(s)\cap\widetilde{\partial}_{\mathrm{int}}A|=2,\qquad\forall
s\in F.$

The pre-image of vertices in $F$ does not contribute to
$\partial_{\mathrm{int}}A$ by their definition in (A.4). By the same
definition, the pre-image of each vertex $z\in S\setminus F$ contains at least
one vertex in $\partial_{\mathrm{int}}A$. A similar argument as the one for
fibers shows that the pre-image of each vertex $z\in S\setminus F$ contains at
least two vertices in $\widetilde{\partial}_{\mathrm{int}}A$. Formally

$|\pi_{\star}^{\uparrow}(z)\cap\partial_{\mathrm{int}}A|\geq
1\quad\mbox{and}\quad|\pi_{\star}^{\uparrow}(z)\cap\widetilde{\partial}_{\mathrm{int}}A|\geq
2\qquad\forall z\in S\setminus F.$

The difference between the above two intersections is the number of vertices
that
$A\cap{\mathcal{L}}_{s}\cap\widetilde{\partial}_{\mathrm{int}}\Lambda_{n}$
contains. We obtain

$|\pi_{\star}^{\uparrow}(z)\cap\widetilde{\partial}_{\mathrm{int}}A|-|\pi_{\star}^{\uparrow}(z)\cap\partial_{\mathrm{int}}A|\leq
2\qquad\forall z\in S\setminus F.$

We characterize $z\in S\setminus F$ according to this difference. For
$i\in\\{0,1,2\\}$ we define

$(S\setminus F)_{i}:=\big{\\{}z\in S\setminus
F:|\pi_{\star}^{\uparrow}(z)\cap\widetilde{\partial}_{\mathrm{int}}A|-|\pi_{\star}^{\uparrow}(z)\cap\partial_{\mathrm{int}}A|=i\big{\\}}.$

Then

$\displaystyle|\widetilde{\partial}_{\mathrm{int}}A|$
$\displaystyle=\sum_{s\in
F}|\pi_{\star}^{\uparrow}(s)\cap\widetilde{\partial}_{\mathrm{int}}A|+\sum_{i\in\\{0,1,2\\}}\sum_{z\in(S\setminus
F)_{i}}|\pi_{\star}^{\uparrow}(z)\cap\widetilde{\partial}_{\mathrm{int}}A|$
$\displaystyle=2|F|+\sum_{i\in\\{0,1,2\\}}\sum_{z\in(S\setminus
F)_{i}}|\pi_{\star}^{\uparrow}(z)\cap\widetilde{\partial}_{\mathrm{int}}A|=2|F|+\sum_{i\in\\{0,1,2\\}}\sum_{z\in(S\setminus
F)_{i}}(|\pi_{\star}^{\uparrow}(z)\cap\partial_{\mathrm{int}}A|+i).$

Now we consider the ratio of the boundaries, i.e.,

$\displaystyle\frac{|\partial_{\mathrm{int}}A|}{|\widetilde{\partial}_{\mathrm{int}}A|}$
$\displaystyle=\frac{\sum_{i\in\\{0,1,2\\}}\sum_{z\in(S\setminus
F)_{i}}|\pi_{\star}^{\uparrow}(z)\cap\partial_{\mathrm{int}}A|}{2|F|+|(S\setminus
F)_{1}|+2|(S\setminus F)_{2}|+\sum_{i\in\\{0,1,2\\}}\sum_{z\in(S\setminus
F)_{i}}|\pi_{\star}^{\uparrow}(z)\cap\partial_{\mathrm{int}}A|}.$

Taking the set sizes $|(S\setminus F)_{i}|,|F|$ fixed, it is elementary to see
that the ratio is increasing in the summands of the double sum, i.e., its
minimal value is attained when all summands are minimal. Now we use that each
of the summands is at least 1, and obtain

$\frac{|\partial_{\mathrm{int}}A|}{|\widetilde{\partial}_{\mathrm{int}}A|}\geq\frac{|S\setminus
F|}{2|F|+|(S\setminus F)_{1}|+2|(S\setminus F)_{2}|+|S\setminus F|}.$

We bound the denominator from above, i.e.,

$|F|+|S\setminus F|+|F|+|(S\setminus F)_{1}|+|(S\setminus F)_{2}|+|(S\setminus
F)_{2}|\leq 3|S|,$

to obtain

$\frac{|\partial_{\mathrm{int}}A|}{|\widetilde{\partial}_{\mathrm{int}}A|}\geq\frac{|S\setminus
F|}{3|S|}.$

We focus on the ratio on the right-hand side. For each vertex $s\in F$, there
are $n^{1/d}$-many vertices in $A$ that are projected onto it (namely
${\mathcal{L}}_{s}$). Using also $|S|\geq|A|^{(d-1)/d}$ by (A.3) and
$n\geq(4/3)|A|$, we get

$|A|\geq|F|n^{1/d}=\frac{|F|}{|S|}|S|n^{1/d}\geq\frac{|F|}{|S|}|A|^{(d-1)/d}n^{1/d}\geq(4/3)^{1/d}\frac{|F|}{|S|}|A|.$

After rearranging we obtain $|F|\leq(3/4)^{1/d}|S|$, and
$|\partial_{\mathrm{int}}A|\geq\big{(}1-(3/4)^{1/d}\big{)}|\widetilde{\partial}_{\mathrm{int}}A|/3$
follows. The second inequality in (2.5) follows immediately from (A.3), since
each projected vertex corresponds to at least two boundary vertices with
respect to ${\mathbb{Z}}^{d}$, i.e.,
$|\widetilde{\partial}_{\mathrm{int}}A|\geq 2|S|\geq 2|A|^{(d-1)/d}$.

We turn to the inequality concerning $\partial_{\mathrm{ext}}A$ and
$\widetilde{\partial}_{\mathrm{ext}}A$ in (2.5). The inequality with $(\star)$
in (2.5) holds for the same reason as for $\partial_{\mathrm{int}}A$. To
obtain a lower bound on the ratio
$|\partial_{\mathrm{ext}}A|/|\widetilde{\partial}_{\mathrm{ext}}A|$, we use
that each exterior boundary vertex is within distance one from an interior
boundary vertex, which holds for both for $\partial$ and
$\widetilde{\partial}$. Since each vertex has at most $2d$ vertices within
distance one, it follows that

$|\partial_{\mathrm{ext}}A|\geq\frac{1}{2d}\cdot|\partial_{\mathrm{int}}A|\geq\frac{1}{2d}\cdot\frac{1-(3/4)^{1/d}}{3}\cdot|\widetilde{\partial}_{\mathrm{int}}A|\geq\frac{1}{2d}\cdot\frac{1-(3/4)^{1/d}}{3}\cdot\frac{1}{2d}\cdot|\widetilde{\partial}_{\mathrm{ext}}A|,$

and the proof is finished for $\delta=(2d)^{-2}\big{(}1-(3/4)^{1/d}\big{)}/3$.
∎

###### Proof of Lemma 2.7.

We first show that there exists $c_{\mathrm{pei}}^{\prime}>0$ such that for
all $x\in{\mathbb{Z}}^{d}$ and $m\in{\mathbb{N}}$

$|\\{A\subseteq\Lambda_{n}:A\ni x,|A|=m,A\mbox{ is
$\ast$-connected}\\}|\leq\exp(c_{\mathrm{pei}}^{\prime}m).$ (A.5)

Let $A$ be in the set on the left-hand side. Since $A$ is $\ast$-connected,
the induced subgraph ${\mathbb{Z}}^{d}_{\infty}[A]$ contains a spanning tree
containing $x$, which can be associated to a walk on the spanning tree (for
example, walking through the tree in depth-first order), visiting each vertex
in $A$ at most twice. Since the degree of any vertex is $3^{d}-1$ in
${\mathbb{Z}}_{\infty}^{d}$, the walk has at most $3^{d}-1$ options at each
step for its next vertex, and has length at most $2m$. This shows (A.5).

For (2.6), we observe that each set $A$ without holes ($A=\bar{A}$) can be
uniquely reconstructed from its interior boundary
$\widetilde{\partial}_{\mathrm{int}}A$ (a vertex is in
$\bar{A}\setminus\widetilde{\partial}_{\mathrm{int}}A$ iff it is surrounded by
$\widetilde{\partial}_{\mathrm{int}}A$), which is $\ast$-connected [12, Lemma
2.1]. Since we assume $|\widetilde{\partial}_{\mathrm{int}}A|=m$, the
isoperimetric inequality (2.5) ensures that $|A|\leq C_{1}m^{d/(d-1)}$ for
some $C_{1}>0$ for all $m\in{\mathbb{N}}$. This interior boundary must either
contain $x$ or surround $x$ as defined in Definition 2.4.

We claim that there is a constant $C>0$ such that
$\|x-\widetilde{\partial}_{\mathrm{int}}A\|_{2}\leq Cm^{1/(d-1)}$ for all
$x,A$ with $A\ni x$ and $A=\bar{A}$. Indeed, suppose otherwise. Then, on
${\mathbb{Z}}^{d}$, vertices in the Euclidean ball of radius $Cm^{1/(d-1)}$
around $x$ would be contained fully in $A$ (without containing a vertex of
$\widetilde{\partial}_{\mathrm{int}}A$). This would mean, for some dimension-
dependent constant $c_{d}$, that $|A|\geq c_{d}(Cm^{1/(d-1)})^{d}$, which
contradicts that $|A|\leq m^{d/(d-1)}$ by Claim 2.6 when $C$ is chosen
sufficiently large.

Hence, we may find a vertex
$y\in\widetilde{\partial}_{\mathrm{int}}A\cap\mathrm{Ball}(Cm^{1/(d-1)},x)$
where the latter set denotes the Euclidean ball of radius $Cm^{1/(d-1)}$
around $x$. Then, since $\widetilde{\partial}_{\mathrm{int}}A$ is a
$\ast$-connected set of size $m$, (A.5) ensures that the number of possible
sets $S$ that may form $\widetilde{\partial}_{\mathrm{int}}A$ is
$\exp(c_{\mathrm{pei}}m)$. Summing over the possible choices of
$y\in\mathrm{Ball}(Cm^{1/(d-1)},x)$, we arrive at

$|\\{A\in{\mathcal{A}}:A\ni
x,A=\bar{A},|\widetilde{\partial}_{\mathrm{int}}A|=m\\}|\leq
c_{d}C^{d}m^{d/(d-1)}\exp(c_{\mathrm{pei}}^{\prime}m).$

The result follows by absorbing the factor $c_{d}Cm^{d/(d-1)}$ into the
constant $c_{\mathrm{pei}}$. ∎

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