11institutetext: INFN - Laboratori Nazionali di Frascati, via Enrico Fermi 54,
00044 Frascati (RM), Italy 22institutetext: INFN - Ferrara, via Saragat 1,
44122 Ferrara (FE), Italy 33institutetext: Department of Physics, University
of Ferrara, via Saragat 1, 44122 Ferrara (FE), Italy 44institutetext: INFN -
Firenze, via Sansone 1, 50019 Sesto Fiorentino (FI), Italy 55institutetext:
Department of physics, University of Firenze, via Sansone 1, 50019 Sesto
Fiorentino (FI), Italy 66institutetext: Physics department, FAU Erlangen-
Nürnberg, Staudtstr. 7 / B2, 91058 Erlangen, Germany

# The LHCspin project

A polarised gas target at the Large Hadron Collider

Marco Santimaria 11<EMAIL_ADDRESS>Vittore Carassiti 22 Giuseppe
Ciullo 2233 Pasquale Di Nezza 11 Paolo Lenisa 2233 Saverio Mariani 4455
Luciano Libero Pappalardo 2233 Erhard Steffens 66

###### Abstract

The goal of LHCspin is to develop, in the next few years, innovative solutions
and cutting-edge technologies to access spin physics in polarised fixed-target
collisions at high energy, exploring the unique kinematic regime offered by
LHC and exploiting new final states by means of the LHCb detector.

##  1 Introduction

The LHC delivers proton and lead beams with an energy of
$7\leavevmode\nobreak\ \rm{TeV}$ and $2.76\leavevmode\nobreak\ \rm{TeV}$ per
nucleon, respectively, with world’s highest intensity. Fixed-target proton-gas
collisions occur at a centre-of-mass energy per nucleon of up to
$115\leavevmode\nobreak\ \rm{GeV}$ in the case of a hydrogen target. The large
centre-of-mass boost offers an unprecedented opportunity to investigate
partons carrying a large fraction of the target nucleon momentum, i.e. large
Bjorken$-x$ values.

The LHCb detector Alves:2008zz  is a general-purpose forward spectrometer
specialised in detecting hadrons containing $c$ and $b$ quarks, and the only
LHC detector able to collect data in both collider and fixed-target mode. It
is fully instrumented in the $2<\eta<5$ region with a vertex locator (VELO), a
tracking system, two Cherenkov detectors, electromagnetic and hadronic
calorimeters and a muon detector.

The fixed-target physics program of LHCb is active since the installation of
the SMOG device Aaij:2014ida , enabling the injection of noble gases in the
beam pipe section crossing the VELO detector at a pressure of
$\mathcal{O}(10^{-7})\leavevmode\nobreak\ \rm{mbar}$.

With the SMOG2 upgrade LHCbCollaboration:2673690 , an openable gas storage
cell, shown in Fig. 1 (left), has been installed in 2020 in front of the VELO.

Figure 1: Left: picture of the SMOG2 storage cell after installation. Right:
simulated vertex reconstruction efficiency for simultaneous beam-gas and beam-
beam collisions LHCB-FIGURE-2019-007 .

The cell boosts the target areal density by a factor of $8$ to $35$ depending
on the injected gas species, and creates a localised beam-gas collision region
which is well detached from the proton-proton vertices. As shown in Fig. 1
(right), full tracking efficiency is expected in the beam-gas region, despite
its upstream position with respect to the VELO. Due to these developments,
fixed-target data will be collected in the upcoming Run 3 with a novel
reconstruction software allowing for the simultaneous data-taking of beam-gas
and beam-beam collisions. SMOG2 will offer a rich physics program for the Run
3 and, at the same time, will also allow to investigate the dynamics of the
novel beam-target system, setting the basis for future developments.

The LHCspin project Aidala:2019pit ; 10.21468/SciPostPhysProc.8.050  aims at
extending the fixed-target physics program in Run 4 and Run 5 with the
installation of a polarised gas target, bringing spin physics at LHC for the
first time. The experimental setup of LHCspin is discussed in Sec. 2, while a
selection of physics opportunities is presented in Sec. 3.

##  2 Experimental setup

The LHCspin experimental setup is in R&D phase and calls for the development
of a new generation polarised target. The starting point is the setup of the
HERMES experiment at DESY Airapetian:2004yf  and comprises three main
components: an Atomic Beam Source (ABS), a Target Chamber (TC) and a
diagnostic system. The ABS consists of a dissociator with a cooled nozzle, a
Stern-Gerlach apparatus to focus the wanted hyperfine states, and adiabatic
RF-transitions for setting and switching the target polarisation between
states of opposite sign. The ABS injects a beam of polarised hydrogen or
deuterium into the TC, which is located in the LHC primary vacuum. The TC
hosts a T-shaped openable storage cell, sharing the SMOG2 design, and a dipole
holding magnet ($B=300$ mT), as shown in Fig. 2.

Figure 2: Two views of the TC with the magnet coils (orange) and the iron
return yoke (blue) enclosing the storage cell. The VELO vessel and detector
box are shown in green and grey, respectively.

The diagnostic system continuously analyses gas samples drawn from the TC and
comprises a target gas analyser to detect the molecular fraction, and thus the
degree of dissociation, and a Breit-Rabi polarimeter to measure the relative
population of the injected hyperfine states. An instantaneous luminosity of
$\mathcal{O}(10^{32})\leavevmode\nobreak\ \rm{cm}^{-2}\rm{s}^{-1}$ is foreseen
for p-H collisions during Run 4.

##  3 Physics case

The physics case of LHCspin covers three main areas: exploration of the wide
physics potential offered by unpolarised gas targets, investigation of the
nucleon spin and heavy-ion collisions.

### Unpolarised gas targets

By means of the SMOG2 gas feed system, LHCspin will allow the injection of
several species of unpolarised gases: H2, D2, He, N2, O2, Ne, Ar, Kr and Xe
with negligible impact on the LHC beam lifetime. This gives an excellent
opportunity to investigate parton distribution functions (PDFs) in both
nucleons and nuclei in the large-$x$ and intermediate $Q^{2}$ regime, which is
especially affected by lack of experimental data and impact several fields
from basic QCD tests to astrophysics. For example, the large acceptance and
high reconstruction efficiency of LHCb on heavy flavour states enables the
study of gluon PDFs, which are a fundamental input for theoretical predictions
Hadjidakis:2018ifr . Searches for an intrinsic charm component in the proton
Aaij:2018ogq  and antiproton production in p-He collisions Aaij:2018svt  are
two other high-profile example measurements that have already been pioneered
at LHCb. With the large amount of data to be collected, nuclear PDFs can also
be investigated in greater detail, helping to shed light on the intriguing
anti-shadowing effect Eskola:2016oht , which is expected to be dominant in the
$x$ range covered with LHCspin.

### Spin physics

Beside standard collinear PDFs, LHCspin will offer the opportunity to probe
polarised quark and gluon distributions by means of proton collisions on
polarised hydrogen and deuterium. For example, measurements of transverse-
momentum dependent PDFs (TMDs) provide a map of parton densities in
3-dimensional momentum space. Light quark TMDs, especially in the high-$x$
regime, can be accessed by measuring transverse single spin asymmetries
(TSSAs) in Drell-Yan processes, while gluon densities, such as the gluon
Sivers function, can be probed via heavy-flavour production. Such asymmetry
can be large (Fig. 3, left) and well in reach with just 1 month of LHCspin
data, as shown in simulated $J/\psi\to\mu^{+}\mu^{-}$ events (Fig. 3, right).

Figure 3: Left: theoretical predictions for $A_{N}$ in inclusive $J/\psi$
production PhysRevD.102.094011 . Right: simulated $J/\psi\to\mu^{+}\mu^{-}$
azimuthal asymmetries with a fit curve superimposed made of two Fourier terms.

It is also attractive to go beyond a 3-dimensional description by building
observables which are sensitive to Wigner distributions Bhattacharya:2017bvs
and to measure the elusive transversity PDF, whose knowledge is currently
limited to valence quarks at the leading order Radici:2018iag , as well as its
integral, the tensor charge, which is of direct interest in constraining
physics beyond the Standard Model Courtoy:2015haa .

### Heavy-ion collisions

Thermal heavy-flavour production is negligible at the typical temperature of
few hundreds MeV of the system created in ultra-relativistic heavy-ion
collisions. Quarkonia states ($c\overline{c}$, $b\overline{b}$) are instead
produced on shorter timescales, and their energy change while traversing the
medium represents a powerful way to investigate Quark-Gluon Plasma (QGP)
properties.

LHCb capabilities allow to both cover the aforementioned charmonia and
bottomonia studies and to extend them to bottom baryons as well as exotic
probes. QGP phase diagram exploration at LHCspin can be performed with a
rapidity scan, complementing RHIC’s beam-energy scan, while flow measurements
will greatly benefit from the excellent particle identification performance of
LHCb on charged and neutral light hadrons. An interesting topic joining heavy-
ion collisions and spin physics is the dynamics of small systems which can be
probed via ellipticity measurements in lead-ion collisions on polarised
deuterons Broniowski:2019kjo .

## References

  * (1) A.A. Alves, Jr. et al. (LHCb), JINST 3, S08005 (2008)
  * (2) R. Aaij et al. (LHCb), JINST 9, P12005 (2014), 1410.0149
  * (3) Tech. rep., CERN, Geneva (2019), https://cds.cern.ch/record/2673690
  * (4) _SMOG2 Velo tracking efficiency_ (2019), https://cds.cern.ch/record/2688875
  * (5) C.A. Aidala et al. (2019), 1901.08002
  * (6) M. Santimaria, V. Carassiti, G. Ciullo, P.D. Nezza, P. Lenisa, S. Mariani, L.L. Pappalardo, E. Steffens, SciPost Phys. Proc. p. 50 (2022), 2105.10012
  * (7) A. Airapetian et al. (HERMES), Nucl. Instrum. Meth. A 540, 68 (2005), physics/0408137
  * (8) C. Hadjidakis et al. (2018), 1807.00603
  * (9) R. Aaij et al. (LHCb), Phys. Rev. Lett. 122, 132002 (2019), 1810.07907
  * (10) R. Aaij et al. (LHCb), Phys. Rev. Lett. 121, 222001 (2018), 1808.06127
  * (11) K.J. Eskola, P. Paakkinen, H. Paukkunen, C.A. Salgado, Eur. Phys. J. C 77, 163 (2017), 1612.05741
  * (12) U. D’Alesio, L. Maxia, F. Murgia, C. Pisano, S. Rajesh, Phys. Rev. D 102, 094011 (2020)
  * (13) S. Bhattacharya, A. Metz, J. Zhou, Phys. Lett. B 771, 396 (2017), [Erratum: Phys.Lett.B 810, 135866 (2020)], 1702.04387
  * (14) M. Radici, A. Bacchetta, Phys. Rev. Lett. 120, 192001 (2018), 1802.05212
  * (15) A. Courtoy, S. Baeßler, M. González-Alonso, S. Liuti, Phys. Rev. Lett. 115, 162001 (2015), 1503.06814
  * (16) W. Broniowski, P. Bożek, Phys. Rev. C 101, 024901 (2020), 1906.09045


# Boltzman optical thermometry for cryogenics

Marek Zeman  Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000
Grenoble, France  Absolut System, 3 avenue Raymond Chanas, 38320 Eybens,
France  Philippe Camus  Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut
Néel, 38000 Grenoble, France  Thierry Chanelière  Univ. Grenoble Alpes, CNRS,
Grenoble INP, Institut Néel, 38000 Grenoble, France

###### Abstract

We propose and implement an optical technique to access the local temperature
of an erbium doped crystal by probing the electron spin population under
magnetic field. We reliably extract the sample temperature in the range 2 - 7
K. We additionally discuss the suitability of our method as a primary standard
for cryogenic thermometry. By adding an auxiliary heating laser, we are able
to measure the interface conductance between the dielectric crystal and the
cold plate of the cryostat by exploring different cooling configurations.

##  I Introduction

Starting at room temperature, optical thermometry using luminescent local
probes continues to gain popularity, now significantly amplified by its
prospects in biology and healthcare. It relies on a wide range of
materials—organic molecules, semiconductors, metallic or insulating
nanoparticles with emitting impurities (lanthanides, colored diamond centers,
transition ions) Bradac  _et al._ (2020) — whose emissions are compared at
least at two wavelengths or by measuring the radiative lifetime, seeking for
the highest thermal sensitivity of the phenomenon Wang, Wolfbeis, and Meier
(2013).

Lanthanides hold a special place as they easily integrate into inert inorganic
matrices compatible with biological tissues Brites, Balabhadra, and Carlos
(2018). They also withstand high temperatures (up to $2000$ K), both to
characterize an extreme environment where inserting a thermometer is
impossible or to induce strong, localized heating of tissues Zhong  _et al._
(2018); Abbas  _et al._ (2022); Debasu  _et al._ (2013).

The recent emergence of new applications has expanded the range of parameters
to explore Harrington  _et al._ (2023). This includes cryogenic temperatures
for monitoring superconducting magnets, space applications, and cryo-electron
microscopy of biological samples for example, especially as commercial
cryomachines are now widely available, robust and efficient.

The race to ultra-low temperatures has strongly stimulated the need for
precision thermometry, covering the range from the ambient to the millikelvin
temperatures, spanning several orders of magnitude Liu  _et al._ (2015);
Mykhaylyk, Wagner, and Kraus (2017); Back  _et al._ (2020). The question of
establishing a primary or secondary standard is recurring regardless of the
application Pobell (2007); Dedyulin, Ahmed, and Machin (2022). We will also
discuss this point in our case.

We propose using an erbium-doped dielectric crystal to explore the restricted
range around 4 K, where many cryogenic devices operate. Our goal is not to
investigate a wide range but to provide an optical probe of the crystal’s
temperature. Like many cooled insulator substrates (like sapphire or silicon
oxide for example), the question of the exact sample temperature compared to
the cold plate where the reference thermometer is attached inevitably arises.
It is therefore interesting to have a local probe with a contactless
measurement.

The success of lanthanides in thermometry ultimately comes from the abundance
of levels whose energy differences are comparable to the temperature $T$ (in
units of $k_{B}T$, where $k_{B}$ is the Boltzmann constant). This intrinsic
property of electronic levels makes spectroscopic measurements sensitive to
temperature, which affects in turn the level population and modifies the
emission or the absorption spectra. We propose to extend this logic to the 4 K
temperature range using electronic spin levels whose splitting under a
magnetic field (divided by $k_{B}$) can be adjusted to a few kelvins. We then
perform an optical measurement of the spin polarisation, i.e., a ratiometric
measurement in the sense of optical thermometry.

In this regard, our method has a kinship with magnetic thermometry, where
nuclear polarisation is measured by magnetic resonance. It is also comparable
to the tremendous work on diamond NV centers Kucsko  _et al._ (2013),
sometimes called quantum thermometry because of the remarkable coherence
properties of a unique center in nanodiamond Schirhagl  _et al._ (2014);
Sotoma, Epperla, and Chang (2018); Fujiwara and Shikano (2021). In a similar
manner, we perform the optical reading of electronic spin resonance levels,
even though we are not seeking spatial resolution as compared to NVs.

We propose to use a earth-rare ion doped crystal. More specifically, using
erbium as a dopant. Erbium has an electron spin that can be measured optically
thanks to the narrow transition linewidth. The electronic spin splitting under
a moderate magnetic field is comparable to our temperature range of interest.
Erbium has a particularly large electron spin due to the multiplicity of
electrons in the 4f shell. Although its g-factor can vary from 2 to 15
typically, due to the strong anisotropy of the dipole moment, a median value
of 7 can be used, which corresponds to a gyromagnetic splitting of 100 GHz/T.
By applying a moderate local field of 0.5 T for example, the splitting will be
2.3 K (in units of $k_{B}T$). So a moderate thermal variation in this range
will therefore induce a significant Boltzman population change that can be
measured using optical methods. This range of magnetic fields is accessible by
the use of permanent magnets, enabling compact assembly that can be
transported from one platform to the another to meet the demand.

We begin by describing the optical method for measuring the electron spin of
erbium in section II. High-resolution spectroscopy is used to measure the
polarisation of the spin in the ground state and to deduce the local
temperature by applying Boltzman’s law. The sensitivity of the method will be
analysed in section III. As we shall see, it is not possible to make an
absolute temperature measurement, despite the universality of Boltzman’s law.
We will discuss the reasons for this in section IV. The method does, however,
define a secondary standard which, after calibration, can be used to measure
in-situ the temperature of the dielectric crystal. This will be applied to
measure the heat conduction from the crystal heated by an external laser beam
to the cold plate in two cryostats using different types of thermal joints
(silver lacquer, varnish, grease) as described in section V.

##  II Methods

###  II.1 Boltzman optical thermometry of the electron spin

Our method consists of measuring the spin polarisation (population) whose
splitting under magnetic field corresponds to the temperature range of
interest, around 4K in our case. As outlined in the introduction by an order-
of-magnitude calculation, earth-rare ions are well suited for this purpose.
Not only are the Zeeman splittings compatible with our objective, but above
all the spin transitions are narrow enough to be resolved by optical
spectroscopy Liu and Jacquier (2005). Without prejudging the generalisation of
our method to other dopants, we have chosen to use erbium, which has two
additional advantages in the lanthanide series. Firstly, it has a high
gyromagnetic factor, which reduces the required magnetic field. Secondly, its
transition is in the range of fibre telecommunications where DFB lasers are
spectrally narrow, widely available, small and cheap.

At low temperatures, the structure of erbium in Y2SiO5 can be represented
schematically in Fig.1 when restricted to the transition of interest. We
choose Er3+:Y2SiO5 that has the advantage to be well characterized in the
literature Böttger  _et al._ (2006). The
${}^{4}\mathrm{I}_{15/2}\rightarrow^{4}\mathrm{I}_{13/2}$ transition is
addressed here, and more specifically the two lowest energy levels of the
crystal field multiplets. The fundamental and excited levels are strongly
anisotropic electron spin doublets when the magnetic field is oriented in the
crystal frame. The structure of interest here can be summarised as four
levels, two split doublets, linked by the optical transition at 1536.48 nm.
The g-factors in the ground and excited states for our orientation are
$g_{g}=11.2$ and $g_{e}=8.2$ respectively 111We address the so-called site 1
in Böttger’s work Böttger  _et al._ (2006) and use the same orientation as in
Dajczgewand  _et al._ (2014). The laser polarisation is aligned along the so-
called D2 extinction axis to maximise the absorption.To put things in
perspective, with a typical value for the magnetic field we use, $B=184$ mT,
the splittings are respectively $\Delta_{g}=g_{g}\mu_{B}B=29.1$ GHz and
$\Delta_{e}=g_{e}\mu_{B}B=21.1$ GHz ($\mu_{B}$ is the Bohr magneton expressed
in Hz/T).

Figure 1: The ground and optically excited states split under magnetic field
(184 mT for example) into the two components of a Kramers doublet noted
$\ket{g_{-}}$,$\ket{g_{+}}$ and $\ket{e_{-}}$,$\ket{e_{+}}$ respectively. The
level scheme is restricted to the lowest lying crystal field levels. The
splittings are respectively noted $\Delta_{g}$ and $\Delta_{e}$ with the
corresponding values at 184 mT. The $\ket{g_{-}}\rightarrow\ket{e_{-}}$ and
$\ket{g_{+}}\rightarrow\ket{e_{+}}$ are called the direct transitions (solid
magenta arrows) and the $\ket{g_{-}}\rightarrow\ket{e_{+}}$ and
$\ket{g_{+}}\rightarrow\ket{e_{+}}$ are the crossed transitions (dashed
magenta arrows).

The Boltzman law applied to the ground state doublet $\ket{g_{-}}$ and
$\ket{g_{+}}$ imposes the population to be
$\displaystyle\frac{1}{1+\exp\left(-h\Delta_{g}/k_{B}T\right)}$ and
$\displaystyle\frac{\exp\left(-h\Delta_{g}/k_{B}T\right)}{1+\exp\left(-h\Delta_{g}/k_{B}T\right)}$
if the total population is normalised to one. The population of the states
scales the absorption of the different transitions that can be measured by
optical spectroscopy. The optical setup is depicted in Fig.2

Figure 2: The optical setup is composed of two lasers. We record the
absorption spectra as plotted in Fig.3 from the frequency-swept probe laser to
measure the ratiometric temperature. We also add a independent off-resonant
heating laser that will be used in section V to evaluate the sample heat load.

###  II.2 Measurement method and temperature calibration

We record the absorption spectrum through the crystal by sweeping a narrow
band laser (Velocity® External Cavity Diode Laser) around the transition of
interest. The spectrum analysis actually reveals the population balance and in
turn the temperature as illustrated in Fig. 3. The main peaks corresponds to
the direct transitions depicted in Fig.1 and are marked with magenta arrows in
Fig. 3. They are positioned at $\pm(\Delta_{g}-\Delta_{e})/2$ originating from
the ground substates $\ket{g_{\pm}}$. The absorption of the transition from
the $\ket{g_{-}}$ is the strongest because of the Boltzman law. The crossed
transitions are not visible because they are weaker and positioned at
$\pm(\Delta_{g}+\Delta_{e})/2$. So they are generally out of our laser
frequency scan except for the lowest magnetic field, 66 mT, where the crossed
transitions are marked with dashed magenta arrows in Fig. 3.

Figure 3: Absorption spectra illustrating the Boltzman law. The left column
corresponds to an increasing temperature (from top to bottom at fixed magnetic
field), and the right column when decreasing magnetic field (from top to
bottom at fixed temperature). The absorption is expressed as the natural log
of the transmission (optical depth). The black dashed lines mark the maxima of
the peaks that are captured by a Lorentzian fit (red dotted line). Each peak
is systematically flanked with a pedestal visible on its right side. It is
explained by the natural abundance (23%) of the 167Er isotope. The latter has
a zero-field hyperfine structure that cannot be simply described by a split
doublet as in Fig.1. The Lorenztian curve chosen to fit the peaks thus
represents an average shape of the absorption line.

We focus on the direct transitions whose absorption (optical depth) will be
used to evaluate the temperature. To do so, we fit a Lorentzian profile to
each peak (red dotted line in Fig. 3). This procedure convincingly capture the
maxima of the absorption sprectrum (marked by the black dashed lines in Fig.
3) that we call $A_{-}$ and $A_{+}$ for $\ket{g_{-}}\rightarrow\ket{e_{-}}$
and $\ket{g_{+}}\rightarrow\ket{e_{+}}$ respectively. If $A_{-}$ and $A_{+}$
are indeed proportional to the populations in $\ket{g_{-}}$ and $\ket{g_{+}}$,
this assumption will be discussed specifically in IV, applying the Boltzman
law allows us to define the so-called ratiometric temperature as

$T_{\mathrm{ratio}}=\frac{h\Delta_{g}}{k_{B}}\frac{1}{\log(A_{-}/A_{+})}$ (1)

Having defined this temperature, we’d like to compare it with the temperature
measured by an independent sensor (Lakeshore DT-670) in the cryostat. The
latter, a Janis SVT wet cryostat, has a variable-temperature insert (VTI)
whose temperature we vary between 2.3 K (minimum cryostat temperature) and 7
K, corresponding to our range of interest. The magnetic field is provided by a
superconducting solenoid in the insert. The field magnitude is variable, which
is useful for this calibration and analysis stage. We now compare the
ratiometric temperature $T_{\mathrm{ratio}}$ and the sensor temperature in
Fig. 4.

Figure 4: Ratiometric temperature $T_{\mathrm{ratio}}$ (blue markers) as a
function of the cryostat sensor temperature. The latter is reported as a red
dashed line for comparison. The difference cannot be explained by a poor
thermalisation of the sample as discussed in the text. We use a linear fit to
data as a calibration curve (blue dashed line) to relate the measured
$T_{\mathrm{ratio}}$ and the cryostat temperature (red dashed line). The red
markers represent the ratiometric temperature after calibration using the
linear fit. The uncertainties (vertical error bars) will be discussed
specifically in III. The horizontal error bars are due to the uncalibrated
sensor with a $\pm 0.1$ K accuracy.

The first thing to notice is that the ratiometric temperature differs
significantly from the temperature in the cryostat insert. This is not a
problem of sample thermalisation. To rule out this hypothesis, it should be
noted that the crystal is efficiently cooled by the gas in the insert.
Furthermore, we have verified that the temperature difference does not depend
on the laser intensity (as a possible source of heating) or on the waiting
time during the experimental run. We also expect the spin dynamics to be fast
at this temperature and dopant concentration Car  _et al._ (2019). The
difference in temperature plotted in Fig. 4 is actually intrinsic to the
method. The ratiometric temperature is actually not an absolute temperature,
in other words, it is not a primary standard in the sense of metrology as we
will discuss in detail in IV. We will therefore use $T_{\mathrm{ratio}}$ as a
secondary standard which must be calibrated first. Despite the temperature
difference in the cryostat, there is a linear dependence of
$T_{\mathrm{ratio}}$ in Fig. 4, which we will use as a calibration curve (blue
dashed line) for the rest of the work, namely the in-situ measurement of the
crystal temperature as we will see in section V. Before doing so, we wish to
discuss the limitations of our method in terms of uncertainties in III and
explain the difficulty of obtaining a primary standard from spectroscopic
measurements in IV.

##  III Uncertainties and sensibility of the method

In order to evaluate the measurement uncertainty that represents the limit of
our sensor accuracy, the impact of noise on the temperature prediction must be
quantified.

The signal-to-noise ratio of the transmission curve is of the order of 103 as
acquired by the oscilloscope. The fitting procedure to estimate $A_{-}$ and
$A_{+}$ further reduces the uncertainty by an order of magnitude. The error is
canonically extracted by the fitting algorithm that provides the covariance
matrix. We typically find an uncertainty of 10-4, not very visible on the
figure 4 and in any case negligible with respect to the precision of the
sensor temperature (10-2) and its accuracy (10-1 for a non-calibrated sensor).
Regardless of the detail of the fitting algorithm, the final uncertainty can
be estimated as the noise on the sampling measurement ($\sim$ 10-3) divided
the square root of the number of samples contained in the width of each peak
$A_{-}$ and $A_{+}$ ($\sim\sqrt{1000}$ in our case) which acts as an averaging
interval for the fit.

This analysis makes it possible to question the sensitivity of the sensor more
generally than under the experimental conditions chosen for our demonstration.
The aim here is to widen its scope and explore different working ranges. The
sensitivity of the measurement will depend strongly first on the magnetic
field chosen for the temperature range to be explored. In the introduction, we
have intuitively proposed to choose the magnetic field producing a level
splitting comparable to our temperature of interest. This can be quantified as
follows.

The temperature estimation comes literally from the comparison of amplitudes
$A_{-}$ and $A_{+}$. At a given temperature, if the magnetic field is too
weak, the peaks are comparable and difficult to differentiate in the
measurement noise limit. Conversely, in strong fields, peak $A_{+}$ will be
totally overwritten and even difficult to measure in the dynamic range of the
acquisition system. This compromise can be quantitatively translated by
assuming that the standard deviation of the noise on the amplitude fitting
$\sigma_{A}$ is the same for $A_{-}$ and $A_{+}$, but the signal-to-noise
ratio $\displaystyle\mathrm{SNR_{A}}=\frac{A_{-}}{\sigma_{A}}$ applies to the
highest peak $A_{-}$ only as a reference for the measurement, and in turn the
signal-to-noise ratio of the smallest peak $A_{+}$ is
$\displaystyle\mathrm{SNR_{A}}\times\frac{A_{+}}{A_{-}}$ affected by the ratio
$\displaystyle\frac{A_{+}}{A_{-}}$.

The uncertainty on the fit $\sigma_{A}$ propagates to the ratiometric
temperature defined by Eq.(1) leading to the error $\sigma_{T}$ on the
temperature given by

$\displaystyle\frac{\sigma_{T}}{T}$
$\displaystyle=\frac{1}{\mathrm{SNR_{T}}}=\frac{k_{B}T}{h\Delta_{g}}\left[1+\exp\left(\frac{h\Delta_{g}}{k_{B}T}\right)\right]\times\frac{\sigma_{A}}{A_{-}}$
(2)
$\displaystyle=S_{N}\left(\frac{\Delta_{g}}{T}\right)\times\frac{1}{\mathrm{SNR_{A}}}$

where the noise ratio
$\displaystyle\frac{\sigma_{T}}{T}=\frac{1}{\mathrm{SNR_{T}}}$ (inverse of the
signal-to-noise ratio) is linked to $\mathrm{SNR_{A}}$ by what we define as
the noise sensibility $S_{N}$

$S_{N}\left(\frac{\Delta_{g}}{T}\right)=\frac{k_{B}T}{h\Delta_{g}}\left[1+\exp\left(\frac{h\Delta_{g}}{k_{B}T}\right)\right]$
(3)

This latter quantifies the noise amplification from the raw measurement of the
amplitudes $A_{-}$ and $A_{+}$ to the estimated ratiometric temperature. This
factor varies by orders of magnitude depending on the choice of the splitting
$\Delta_{g}$ relative to the temperature of interest $T$ as illustrated in
Fig. 5 (left).

Figure 5: Left: Noise sensibility $S_{N}$ defined by uncertainty propagation
from the raw measurement in (2). We define the optimal conditions (blue square
marker) and an acceptable range (blue shaded area) in the text. The thick red
line represents the range of temperature that we have explore in Fig.4. Right:
Illustration of the acceptable parameter range for which the noise sensibility
stays below 10 for the erbium spin ($g_{g}=11.2$, blue shaded area) and a
generic electron spin ($g=2$, orange shaded area). See the text for details.

The noise sensibility $S_{N}$ strongly varies as a function of $\Delta_{g}/T$.
It reaches a minimum of $\simeq 3.59$ (blue square marker in Fig. 5, left) for
$\displaystyle\frac{h\Delta_{g}}{k_{B}T}\simeq 1.28$. This allows us to
quantify our intuition: the splitting (in units of $k_{B}T$) must be close to
the temperature of interest. At this optimum, the noise measurement is only
amplified by a factor of $3.59$ affecting the measured temperature. We propose
to define an acceptable parameter range for which the noise sensibility stays
below 10 (blue shaded region in Fig. 5, left) this corresponds to the
following interval for
$\displaystyle\frac{h\Delta_{g}}{k_{B}T}\simeq[0.23,3.54]$. This doesn’t mean
that the method cannot be used outside of this range, this simply means that
the noise sensibility will be higher. This latter could be compensated by
technical improvements to reduce the initial measurement noise and in turn to
maintain the precision on the temperature.

In Fig. 5 (left), we have also represented the range of temperature that we
have explore in Fig.4 (thick red line), namely 2.3K - 7K with $B=184$ mT. This
appears as a red thick segment. We are more or less in the acceptable
parameter range, but below the optimum. Indeed, if we had wanted to obtain the
best sensitivity at 4.2K for example, the necessary magnetic field would have
been 554 mT, which was not directly possible in our case (limited by our
superconducting solenoid).

To be more specific, we finally propose to compare the optimal magnetic field
for a given temperature of interest in Fig. 5 (right). The optimal splitting,
defined by the relation $\displaystyle\frac{h\Delta_{g}}{k_{B}T}\simeq 1.28$,
will depend on the g-factor of the ion under consideration. Thus, we derive
the temperature range that corresponds to the previously defined acceptable
range of parameters as a function of the magnetic field, for erbium with
$g_{g}=11.2$ in blue and for a generic electron spin with $g=2$ in orange. It
is shown that erbium allows to explore the cryogenic temperature range up to
about 30K with moderate magnetic field $<$1000mT (with permanent magnets for
example). Using instead a generic electron spin with $g=2$ significantly
reduces the accessible temperature range.

After discussing the precision of the method, we wish to discuss its accuracy,
in the sense that the ratiometric temperature is not an absolute temperature.

##  IV Suitability as a primary standard for thermometry

As discussed in II.2, it is not possible to use the ratiometric temperature as
a primary standard, despite the universal nature of the Boltzman’s law. We
have ruled out the question of sample thermalisation for practical reasons
because of the efficient cooling by the gas. Furthermore, the spins whose
population we are measuring have their own dynamics driven by phonons.
However, the dynamics of the spins for these dopant concentrations and at
these temperatures is relatively fast compared with the time of the experiment
Car  _et al._ (2019). So this is not an intrinsic limitation for the moment.
To explain the discrepancy between $T_{\mathrm{ratio}}$ and the cryostat
temperature, it is useful to go back over the spectroscopy experiment in
detail. The absorption coefficient depends not only on the populations that
allow us to extract the temperature, but also on the strengths of the
transitions (in the sense of the transition dipole) as a multiplicative
factor. We could have been convinced that the two direct transitions within
two Kramers doublets have the same strength, given the similarity of the wave
functions. The latter remain relatively pure as much as the level splittings
under magnetic field remain small compared with the typical energy between
crystal field levels. However, this simplifying assumption is not verified in
practice, as the following experiment shows.

To investigate the strengths of the direct transitions, for a given population
balance (given temperature), we simply rotate the incoming polarisation and
record again the absorption spectra and derive in turn the ratiometric
temperature as shown in Fig.6.

Figure 6: Top: Maxima of the absorption peaks $A_{-}$ and $A_{+}$ as pointed
out in Fig.3 when we vary the incoming polarisation for a fixed cryostat
temperature of 2.3 K. We rotation the polarisation in the (D1,D2) plane where
0∘ and 90∘ corresponds to D2 and D1 respectively. Bottom: Corresponding
ratiometric temperature $T_{\mathrm{ratio}}$ Eq.(1).

We should not be surprised by the variation of the global absorption when the
polarisation is varied. This anisotropic character is due to the monoclinic
nature of the Y2SiO5 Petit  _et al._ (2020); Liu and Jacquier (2005). What is
more surprising is that the ratio of $A_{-}$ and $A_{+}$ varies, as
demonstrated by the variation of $T_{\mathrm{ratio}}$ (Fig.6, bottom). This
fact actually shows that the strengths of the transition also depends on the
polarisation. As a consequence, the ratio $A_{-}/A_{+}$ is not rigorously
equal to the population ratio, but also contains a contribution from the
independent transition dipoles.

We could also argue that when the polarisation is at 90∘, corresponding the D1
extinction axis, $T_{\mathrm{ratio}}\simeq 2.5$ K is closer to the cryostat
temperature of 2.3 K. That is a fact, but a discrepancy persists, at the price
of a lower absorption contrast because the absorption is weaker. At the end,
$T_{\mathrm{ratio}}$ can only be used as an indicator of the temperature that
should be calibrated at given experimental conditions, namely the polarisation
and the strength of the magnetic field.

Even if our method is limited in terms of obtaining a primary standard, it
retains a strong interest that makes its originality.

##  V Evaluation of heat conduction by optical thermometry

Optical thermometry allows a local measurement of the sample temperature.
While cryogenic thermometers ensure excellent measurement accuracy on the
sample holder, the exact temperature of the sample remains an open question,
especially as the nature of the materials is very different, mostly metals or
dielectrics (insulators).

As a proof of principle, we propose to evaluate the thermal conductance
between the sample, which can be considered as a typical crystalline
dielectric, and the sample holder or the surrounding exchange gas if present.
This is an opportunity for us to test in a dry closed-cycle cryostat developed
by Absolut System for optical applications (ACE-CUBE), whose cold plate under
vacuum reaches 2.0 K thanks to an additional Joule–Thomson cooling stage.

We therefore installed a second laser dedicated to heating the sample
directly. This laser is also in the telecom range, but detuned by a few nm to
avoid disturbing the spectroscopic probe that $T_{\mathrm{ratio}}$ is
measuring (see Fig.2). It delivers up to a hundred mW. By varying its power
and measuring the ratiometric temperature, we can characterise the thermal
load on the crystal.

To illustrate this heat load measurement in section V.1, we’ll start with the
same configuration that was used to validate the method and its calibration in
section II.2. The sample is thermally contacted with silver lacquer and is
also cooled by the gas inside the wet cryostat. The experiment will be
reproduced in the dry cryostat (sample in vacuum) with different types of
joints, silver lacquer for comparison, GE varnish and Apiezon-N grease (see
V.2). In the dry cryostat, the magnetic field is produced by a permanent
magnet and varies a little depending on the exact relative position of the
crystal from 203 mT to 215 mT for different configurations, slightly higher
than the reference value of 184 mT used in II but still comparable. Despite
the compactness of the assembly, this justifies all the more the calibration
step described in section II.2 for each configuration.

###  V.1 Thermometry under external laser heating

As mentioned, we add to the previous configuration in II.2 a heating laser
that is essentially collinear and counterpropagating with the ratiometric
temperature probe beam (See Fig.2). There is no interplay between the heating
and the probe beams in the crystal since they are well separated spectrally.
Because of the large power of the heating laser, a weak parasite is observed
on the probe detector. This may influence the temperature measurement so we
subtract it before processing $T_{\mathrm{ratio}}$. We vary the heating power
and observe the dependency in Fig.7.

Figure 7: $T_{\mathrm{ratio}}$ as a function of the heating laser power. From
the calibration procedure described in II.2, we can infere the crystal
temperature $T_{\mathrm{crystal}}$ for each measurement of
$T_{\mathrm{ratio}}$.

The ratiometric temperature, as measured, has been calibrated in II.2. So from
the curve in Fig.4, we can estimate the crystal $T_{\mathrm{crystal}}$ for
each measurement of $T_{\mathrm{ratio}}$.

We can clearly see that the temperature of the crystal increases with heating.
The dependency is essentially linear with a slope of 5.1 K per watt of heating
laser. This coefficient has the unit of a thermal resistance.

This validates the method under realistic experimental conditions, but for the
moment it says nothing about the thermal cooling of the sample. The first step
is to estimate the light power actually deposited as heat. As the crystal is
very transparent, regardless of the erbium dopants, it is difficult to
estimate its absorption by comparing the incoming and outgoing powers, the
latter also including reflection losses at the interfaces. However, we were
able to measure the absorbed power in the dry cryostat, which has a
thermometer precise enough to observe a temperature rise at maximum heating
power. This indicates that around 4% of the incident power is absorbed. So we
expect that the crystal thermal resistance obtained from Fig.7 is actually 130
K per watt of absorbed laser. Finally the thermal conductance is usually
referenced to the surface in contact with the cold plate, namely 5 mm$\times$4
mm for the base of our crystal. This definition is debatable in the case of
the wet cryostat, as the free surface of the crystal is also cooled by the
gas. Since it is not possible to distinguish between the two cooling
mechanisms, and in order to be able to compare with the corresponding
measurements in the dry cryostat (sample in vacuum), we will keep the base
surface of the crystal as a reference (5 mm$\times$4 mm) and discuss this
point again in the final analysis. Under this assumption, we find a thermal
conductance of $3.9\times 10^{-2}$ W K-1 cm-2 at the crystal base.

Our aim is to make a comparison between different configurations in section
V.2, but before doing so it is worth discussing this value quantitatively. The
heat conduction across solid/solid interfaces with different kind of joints
has been discussed comparatively by Ekin (Ekin, 2006, Fig. 2.7). The range
$10^{-1}-10^{-2}$ W K-1 cm-2 seems a good order of magnitude for solder of
layered joints, without prejudging the nature of the materials in contact and
the applied pressure. In that sense, our measurement gives a correct order of
magnitude if the conductivity is ensured by the silver lacquer. On the other
hand, estimating conduction by gas is much more speculative because of
convection around the sample. However, we can give a qualitative answer by
reproducing the same measurement with the sample in a vacuum, as we shall now
see.

###  V.2 Thermal conductance evaluation

The aim here is not to push thermal conductance metrology further in terms of
precision, but rather to validate an original method under real conditions.

We are now going to explore three new configurations compared with the
previous experiment (silver lacquer thermal contact in a wet cryostat) that
serves as our reference, in the dry cryostat (sample in vacuum) with three
types of joints, silver lacquer for comparison, GE varnish and Apiezon-N
grease, and extract the thermal conduction. In order to explore a possible
thermal gradient and inhomogeneities in the crystal, we additionally move the
heating laser position away from the probe beam (defining the position 0 mm).
The results are presented in Fig.8 and they can be analysed as follows, with
three points of view.

Figure 8: Thermal conductance measured by optical thermometry in four
different configurations: wet or dry cryostat with silver lacquer, GE varnish
or Apiezon-N grease joints as abbreviated in the legend. We vary the position
of the heating laser from the probe laser referenced at 0 mm. The crystal size
along this dimension is 5 mm. The circled marker corresponds to the reference
measurement described in section V.1 (position 0 mm, Wet Silver curve).

####  V.2.1 Heating spatial dependence

The aim of measuring the spatial variation in heating is to investigate the
existence of a gradient within the crystal. The curves have a chaotic
appearance and cannot be considered neither convex nor concave systematically.
It is impossible to draw any conclusions. The curvature is probably not linked
to a variation in conductance, but more likely to a variation in absorbed
power, which we assume to be constant (4% of the incident power). The absorbed
power can vary greatly when the heating beam is translated. Several phenomena
are possible. The beam (diameter about 1 mm) touches the edge of the crystal
(width 5 mm), encounters an inclusion in the crystal or a bonding residue. It
is then difficult to predict whether the absorbed power decreases (convex
curves in Fig.8) as it is clipped by the crystal or increases (concave curves)
as it is diffused inwards. This explains the chaotic nature of the curves. In
any case, we do not observe any systematic effect of inhomogeneous heat
diffusion in the crystal. This is difficult to imagine anyway, as the typical
conductivity of dielectric materials comparable to Y2SiO5, such as sapphire or
quartz, is around $10^{3}$ W K-1m-1 (see Fig. 2.1 of Ekin Ekin (2006)) i.e. 10
W K-1 over a typical dimension of one cm. This is at least two orders of
magnitude greater than the conductance of the interface(s) ($10^{-1}-10^{-2}$
W K-1 cm-2).

####  V.2.2 Qualitative differences between thermal joints

In the end, the three bonding techniques gave similar results in terms of
conductance. A summary of the different values can be found tabulated in
Fig.9. Silver lacquer (sample in vacuum) nevertheless appears to perform less
well than Apiezon-N grease or GE varnish. We suggest that varnish should be
preferred, as its bonding is more robust than grease. With grease, we observed
sample detachment during the colddown. Varnish is therefore an excellent
compromise between thermal conductivity, ease of use and bond strength.

Cryostat | Sample in | Thermal joint | Average conductance
---|---|---|---
Wet | gas | silver lacquer | $3.5\times 10^{-2}$
Dry | vacuum | silver lacquer | $4.0\times 10^{-3}$
Dry | vacuum | GE varnish | $1.4\times 10^{-2}$
Dry | vacuum | Apiezon-N grease | $1.6\times 10^{-2}$
Figure 9: Summary of the typical conductance of the different thermal joints
(in W K-1 cm-2). The average of the conduction is taken from the different
positions plotted in Fig.8.

####  V.2.3 Solid/solid vs solid/gas conduction

To conclude, it is possible to compare the curves with silver lacquer with
sample in vacuum (orange curve in Fig.8, average conductance of $4.0\times
10^{-3}$ W K-1 cm-2) or in gas (blue curve in Fig.8, average of $3.5\times
10^{-2}$ W K-1 cm-2 ). If we assume that the quality of the contact with the
lacquer is the same in both cases, then the difference in conductance can only
be explained by the gas cooling process whose efficiency is an order of
magnitude higher.

This is interesting information that can be compared with the literature. The
conductivity of the helium gas is well known down to 2 K Barron and Nellis
(2017); Hands and Arp (1981). We will take $10^{-4}$ W K-1 cm-1 as a good
order of magnitude at 4K. To compare with the conductance measurements in
Fig.9, we need to determine the distance between the sample holder and the
crystal, as the minimal gas conductance distance. To do this, we’ll take 3 mm,
the height of the sample. We therefore expect a conductance of $3.3\times
10^{-4}\,$W K-1 cm-2. This is actually two orders of magnitude lower than what
we are measuring that is more in the range of $10^{-2}\,$W K-1 cm-2.

The cooling by the gas is therefore more efficient than the simple conduction
mechanism. It undoubtedly relies mainly on convection, which remains difficult
to evaluate outside a dedicated test bench. However, our analysis does provide
some answers in a practical user case.

##  VI Conclusion

We have implemented a new optical thermometry technique specifically dedicated
to cryogenic temperatures. It is based on spectroscopic measurements of
Boltzman populations of the erbium spin levels. Despite its very general
nature, this method does not allow an absolute temperature measurement, but
requires a calibration step. It retains the advantage of making a local
temperature measurement of the sample without additional electrical contacts.

It is this property that we are finally exploiting to make measurements of the
interface conductance with the cold cryostat plate. An auxiliary laser is used
for direct heating of the sample.

The profusion of rare earth ion levels opens up possibilities in different
temperature ranges. As mentioned in the introduction, the crystal field levels
that distribute luminescence over several peaks are already being exploited to
make room temperature measurements, with applications in biology, for example.
As we have shown, the electron spin levels are well adapted to the cryogenic
range around 4K. It would be interesting to reproduce our study for the mK
range, this time using the hyperfine structure. Although such an analysis will
inevitably raise fundamental questions of spin thermalisation, it would allow
to study a range that is increasingly used with the democratisation of
dilution cryostats and where standard thermometry remains very delicate.

## Acknowledgements

The authors acknowledge support from the French National Research Agency (ANR)
through the project MARS No. ANR-20-CE92-0041, from the Plan France 2030
project QMEMO No. ANR-22-PETQ-0010 and from the ANRT (Association nationale de
la recherche et de la technologie) with a CIFRE fellowship No. 2021/0803.

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