T-violating effect in $\nu_{\tau}(\bar{\nu}_{\tau})-$nucleon quasielastic
scattering

A. Fatima1$\star$, M. Sajjad Athar1 and S. K. Singh1

1 Department of Physics, Aligarh Muslim University, Aligarh-202002, India.

*<EMAIL_ADDRESS>
16th International Workshop on Tau Lepton Physics (TAU2021),
September 27 – October 1, 2021 10.21468/SciPostPhysProc.?

## Abstract

The production cross sections and polarization observables of the $\tau$
leptons produced in the $|\Delta S|=0$ and $1$ induced
$\nu_{\tau}(\bar{\nu}_{\tau})-N$ quasielastic scattering have been studied.
The effect of T violation, in the case of $\Delta S=0$ and 1 processes, and
the SU(3) symmetry breaking effects, in the case of $\Delta S=1$ processes, on
the total scattering cross sections as well polarization observables are
explored. Experimentally, it would be possible to observe these effects in the
forthcoming (anti)neutrino experiments like DUNE, SHiP and DsTau.

##  1 Introduction

In this work, we present a theoretical study of the production cross section
as well as the polarization observables of the $\tau$ lepton produced in the
quasielastic $\nu_{\tau}(\bar{\nu}_{\tau})-N$ scattering in the few GeV energy
region relevant to the Deep Underground Neutrino Experiment (DUNE), Search for
Hidden Particles (SHiP) and Study of tau neutrino production at the CERN-SPS
(DsTau) experiments. The $\tau$ lepton produced in $\nu_{\tau}-N$ scattering
decays to leptons and pions through the leptonic and hadronic decay modes. In
this energy region, the production cross section of $\tau$, its decay and the
characteristics of the decay products depend significantly on the $\tau$
polarization. The production cross sections and polarization of $\tau$ leptons
are calculated using various weak nucleon form factors which are determined
using symmetry properties of the weak currents in the vector and axial vector
sectors, assuming G and T invariances and SU(3) symmetry. The longitudinal and
perpendicular components of polarization lie in the plane while the transverse
component of polarization lies perpendicular to the reaction plane and is
forbidden by G- and T-invariance. We have earlier studied the effects of G and
T violation on the total cross section as well as on the polarization
observables of the final nucleon/hyperon and the lepton produced in the
quasielastic scattering of (anti)neutrinos with nucleons for both $\Delta S=0$
and $1$ processes [3, 1, 2]. In the case of $\Delta S=1$ reactions, we study
the effect of SU(3) symmetry breaking on these observables.

In Section 2, we discuss in brief the formalism for calculating the
differential as well as total scattering cross sections for quasielastic
scattering of $\nu_{\tau}(\bar{\nu}_{\tau})$ with nucleons. The polarization
components of the produced $\tau$ lepton are discussed in Section 2.2. The
SU(3) symmetry breaking effects, following the works of Faessler et al. [4]
and Schlumpf [5], are discussed in Section 3. In Section 4, we present and
discuss the results for the total cross sections and average polarizations and
Section 5 concludes the present work.

##  2 Quasielastic production of nucleons and hyperons

The $\nu_{\tau}(\bar{\nu}_{\tau})$ induced quasielastic production on the free
nucleon target are given by the reactions

$\displaystyle{\nu_{\tau}}(\bar{\nu}_{\tau})(k)+N(p)$
$\displaystyle\longrightarrow$
$\displaystyle\tau^{\mp}(k^{\prime})+N(p^{\prime}),~{}~{}~{}~{}~{}N=n,p$ (1)
$\displaystyle\bar{\nu}_{\tau}(k)+N(p)$ $\displaystyle\longrightarrow$
$\displaystyle\tau^{+}(k^{\prime})+Y(p^{\prime}),~{}~{}~{}~{}~{}Y=\Lambda,\Sigma^{0},\Sigma^{-},$
(2)

for which the transition matrix element is given by

${\cal{M}}=\frac{G_{F}}{\sqrt{2}}~{}a~{}l^{\mu}~{}J_{\mu},$ (3)

where $a=\cos\theta_{c}~{}(\sin\theta_{c})$ for $\Delta S=0~{}(1)$ processes
and the leptonic ($l^{\mu}$) and the hadronic ($J_{\mu}$) currents are defined
as

$\displaystyle l^{\mu}$ $\displaystyle=$
$\displaystyle\bar{u}(k^{\prime})\gamma^{\mu}(1\mp\gamma_{5})u(k),$ (4)
$\displaystyle J_{\mu}$ $\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})\left[\gamma_{\mu}f_{1}(Q^{2})+i\sigma_{\mu\nu}\frac{q^{\nu}}{M+M_{Y}}f_{2}(Q^{2})+\frac{2q_{\mu}}{M+M_{Y}}f_{3}(Q^{2})\right.$
(5) $\displaystyle-$
$\displaystyle\left.\gamma_{\mu}\gamma_{5}g_{1}(Q^{2})-i\sigma_{\mu\nu}\gamma_{5}\frac{q^{\nu}}{M+M_{Y}}g_{2}(Q^{2})-\frac{2q_{\mu}\gamma_{5}}{M+M_{Y}}g_{3}(Q^{2})\right]u(p).~{}~{}~{}~{}~{}$

In Eq. (4), $+(-)$ stands for $\bar{\nu}_{\tau}(\nu_{\tau})$ induced process.
In Eq. (5), $f_{1}(Q^{2})$, $f_{2}(Q^{2})$, $g_{1}(Q^{2})$ and $g_{3}(Q^{2})$
are the form factors associated with the first class currents while
$f_{3}(Q^{2})$ and $g_{2}(Q^{2})$ are the form factors associated with the
second class currents. These form factors are determined using various
symmetry properties of the weak hadronic current like G invariance, T
invariance, SU(3) symmetry, etc. The real value of $g_{2}(0)$ gives G
violation while T is conserved whereas the imaginary value of $g_{2}(0)$ gives
G violation as well as T violation.

###  2.1 Cross sections

The $Q^{2}$ distribution is written as

$\frac{d\sigma}{dQ^{2}}=\frac{G_{F}^{2}~{}a^{2}}{8\pi
M^{2}{E^{2}_{{\nu}_{\tau}(\bar{\nu}_{\tau})}}}N(Q^{2}),$ (6)

where the expression of $N(Q^{2})$ is given in Ref. [3]. The expression for
the total cross section is obtained by integrating Eq. (6) over $Q^{2}$ as:

$\sigma=\int\frac{G_{F}^{2}~{}a^{2}}{8\pi
M^{2}{E^{2}_{{\nu}_{\tau}(\bar{\nu}_{\tau})}}}N(Q^{2})~{}dQ^{2},$ (7)

with $a=\cos\theta_{c}~{}(\sin\theta_{c})$ for $\Delta S=0~{}(1)$ processes.

###  2.2 Polarization observables of the final lepton

Figure 1: (a) Momentum and polarization directions of the final lepton
produced in the reaction
${\nu}_{\tau}(k)+n(p)\longrightarrow\tau^{-}(k^{\prime})+p(p^{\prime})$. (b)
$\hat{e}_{L}^{l}$, $\hat{e}_{P}^{l}$ and $\hat{e}_{T}^{l}$ represent the
orthogonal unit vectors corresponding to the longitudinal, perpendicular and
transverse directions with respect to the momentum of the final lepton.

If one assumes the final lepton to be polarized, then the polarization
4-vector ($\zeta^{l}$; $l=\tau$) is written as

$\displaystyle\zeta^{l}=\frac{\mathrm{Tr}[\gamma^{\tau}\gamma_{5}~{}\rho_{f}(k^{\prime})]}{\mathrm{Tr}[\rho_{f}(k^{\prime})]},$
(8)

and the spin density matrix for the final lepton $\rho_{f}(k^{\prime})$ is
given by

$\rho_{f}(k^{\prime})={\cal J}^{\alpha\beta}\text{
Tr}[\Lambda(k^{\prime})\gamma_{\alpha}(1\pm\gamma_{5})\Lambda(k)\tilde{\gamma}_{\beta}(1\pm\tilde{\gamma}_{5})\Lambda(k^{\prime})],$
(9)

with $\tilde{\gamma}_{\alpha}=\gamma^{0}\gamma^{\dagger}_{\alpha}\gamma^{0}$
and $\tilde{\gamma}_{5}=\gamma^{0}\gamma^{\dagger}_{5}\gamma^{0}$.

With $\rho_{f}(k^{\prime})$ defined in Eq. (9), the expression for $\zeta^{l}$
becomes

$\zeta^{l}=\left(g^{\tau\sigma}-\frac{k^{\prime\tau}k^{\prime\sigma}}{m_{\tau}^{2}}\right)\frac{{\cal
J}^{\alpha\beta}\mathrm{Tr}\left[\gamma_{\sigma}\gamma_{5}\Lambda(k^{\prime})\gamma_{\alpha}(1\pm\gamma_{5})\Lambda(k)\tilde{\gamma}_{\beta}(1\pm\tilde{\gamma}_{5})\right]}{{\cal
J}^{\alpha\beta}\mathrm{Tr}\left[\Lambda(k^{\prime})\gamma_{\alpha}(1\pm\gamma_{5})\Lambda(k)\tilde{\gamma}_{\beta}(1\pm\tilde{\gamma}_{5})\right]},$
(10)

where $m_{\tau}$ is the mass of the $\tau$ lepton, $\Lambda(k)=\not{k}$,
$\Lambda(k^{\prime})=\not{k}+m_{\tau}$ and ${\cal J}^{\alpha\beta}$ is the
hadronic tensor.

In the laboratory frame where the initial nucleon is at rest, the polarization
vector $\vec{\zeta}$, is calculated to be a function of 3-momenta of incoming
(anti)neutrino $({\vec{k}})$ and outgoing lepton $({\vec{k}}\,^{\prime})$, and
is given as

$\vec{\zeta}=\left[{A^{l}(Q^{2})\vec{k}}+B^{l}(Q^{2}){\vec{k}}\,^{\prime}+C^{l}(Q^{2})\vec{k}\times\vec{k}^{\prime}\right],$
(11)

where the expressions of $A^{l}(Q^{2})$, $B^{l}(Q^{2})$ and $C^{l}(Q^{2})$ are
given in Ref. [3].

The polarization vector $\vec{\zeta}$, obtained from Eq. (10), can be written
as

$\vec{\zeta}=\zeta_{L}\hat{e}_{L}^{l}+\zeta_{P}\hat{e}_{P}^{l}+\zeta_{T}\hat{e}_{T}^{l},$
(12)

where $\hat{e}_{P}^{l}$, $\hat{e}_{L}^{l}$ and $\hat{e}_{T}^{l}$ are the unit
vectors corresponding to the perpendicular, longitudinal and transverse
directions (depicted in Fig. 1), which are given as

$\hat{e}_{L}^{l}=\frac{\vec{k}^{\,\prime}}{|\vec{k}^{\,\prime}|},\quad\hat{e}_{P}^{l}=\hat{e}_{L}^{l}\times\hat{e}_{T}^{l},\quad\hat{e}_{T}^{l}=\frac{\vec{k}\times\vec{k}^{\,\prime}}{|\vec{k}\times\vec{k}^{\,\prime}|},$
(13)

with $\zeta_{L,P,T}(Q^{2})=\vec{\zeta}\cdot\hat{e}_{L,P,T}^{l}$.

The longitudinal $P_{L}^{l}(Q^{2})$, perpendicular $P_{P}^{l}(Q^{2})$ and
transverse $P_{T}^{l}(Q^{2})$ components of the polarization vector in the
rest frame of the outgoing lepton are obtained as:

$\displaystyle P_{L}^{l}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{m_{\tau}}{E_{k^{\prime}}}\zeta_{L}(Q^{2})=\frac{m_{\tau}}{E_{k^{\prime}}}\frac{A^{l}(Q^{2})\vec{k}.\vec{k}^{\,\prime}+B^{l}(Q^{2})|\vec{k}^{\,\prime}|^{2}}{N(Q^{2})~{}|\vec{k}^{\,\prime}|},$
(14) $\displaystyle P_{P}^{l}(Q^{2})$ $\displaystyle=$
$\displaystyle\zeta_{L}(Q^{2})=\frac{A^{l}(Q^{2})[|\vec{k}|^{2}|\vec{k}^{\,\prime}|^{2}-(\vec{k}.\vec{k}^{\,\prime})^{2}]}{N(Q^{2})~{}|\vec{k}^{\,\prime}|~{}|\vec{k}\times\vec{k}^{\,\prime}|},$
(15) $\displaystyle P_{T}^{l}(Q^{2})$ $\displaystyle=$
$\displaystyle\zeta_{T}(Q^{2})=\frac{C^{l}(Q^{2})M[(\vec{k}.\vec{k}^{\,\prime})^{2}-|\vec{k}|^{2}|\vec{k}^{\,\prime}|^{2}]}{N(Q^{2})~{}|\vec{k}\times\vec{k}^{\,\prime}|},$
(16)

where $\frac{m_{\tau}}{E_{k^{\prime}}}$ is the Lorentz boost factor, which
appears due to the fact that we are measuring the polarization observables in
the rest frame of the $\tau$ lepton.

Figure 2: $\sigma$ (upper left panel), $P_{L}^{\tau}(E_{\nu_{\tau}})$ (upper
right panel), $P_{P}^{\tau}(E_{\nu_{\tau}})$ (lower left panel) and
$P_{T}^{\tau}(E_{\nu_{\tau}})$ (lower left panel) vs. $E_{{\nu}_{\tau}}$ for
the process $\nu_{\tau}+n\rightarrow\tau^{-}+p$, using the different values of
$g_{2}^{I}(0)$ viz. $g_{2}^{I}(0)=0$ (solid line), 1 (dashed line) and $2$
(double-dotted-dashed line).

##  3 SU(3) symmetry breaking

The SU(3) symmetry breaking effects are incorporated following the works of
Faessler et al. [4] and Schlumpf [5]. In the following, we will discuss in
brief the main features of these models.

###  3.1 Model of Faessler et al. [4]

Faessler et al. have studied the SU(3) symmetry breaking effects on
$f_{2}(Q^{2})$ and $g_{1}(Q^{2})$ form factors, using the constituent quark
model. $f_{1}(Q^{2})$ recieves no contribution from the SU(3) symmetry
breaking, at the leading order because of the Ademollo-Gatto theorem.
$g_{3}(Q^{2})$ recieves the SU(3) breaking effects via $g_{1}(Q^{2})$. In this
model, the form factors $f_{2}(Q^{2})$ and $g_{1}(Q^{2})$ are modified, at
$Q^{2}=0$ in a similar manner i.e., $f_{2}^{SU3}=g_{1}^{SU3}={\cal F}$, which
are given as:

$\displaystyle{\cal F}^{p\Lambda}(0)$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{3}{2}}\left(F+\frac{D}{3}+\frac{1}{9}\left(H_{1}-2H_{2}-3H_{3}-6H_{4}\right)\right),$
(17) $\displaystyle{\cal F}^{n\Sigma^{-}}(0)$ $\displaystyle=$ $\displaystyle
D-F-\frac{1}{3}(H_{1}+H_{3}),$ (18)

where $D$ and $F$ are the SU(3) symmetric couplings while $H_{i},i=1-4$ are
the SU(3) symmetry breaking couplings, and the value of these parameters for
$f_{2}(Q^{2})$ and $g_{1}(Q^{2})$ form factors can be found in Refs. [4, 2].

###  3.2 Model of Schlumpf [5]

Schlumpf [5] has studied SU(3) symmetry breaking effects on the vector
$f_{1}(Q^{2})$ and axial vector $g_{1}(Q^{2})$ form factors using relativistic
quark model. In this model, the modified $f_{1}(Q^{2})$ and $g_{1}(Q^{2})$
form factors are given by

$\displaystyle f_{1}^{\prime}(Q^{2})=\alpha f_{1}(Q^{2}),\qquad
g_{1}^{\prime}(Q^{2})=\beta g_{1}(Q^{2}),$ (19)

where $\alpha$=0.976 and 0.975; $\beta$=1.072 and 1.056, respectively, for
$p\longrightarrow\Lambda$ and $n\longrightarrow\Sigma^{-}$ transitions [5].

Figure 3: $\sigma$ (left panel), $P_{L}(E_{\bar{\nu}_{\tau}})$ (middle panel),
and $P_{P}(E_{\bar{\nu}_{\tau}})$ (right panel) vs $E_{\bar{\nu}_{\tau}}$ for
$\bar{\nu}_{\tau}+p\rightarrow\tau^{+}+\Lambda$ process. The calculations have
been performed using the SU(3) symmetry (solid line), the SU(3) symmetry
breaking effects parameterized by Faessler et al. (dashed line) and by
Schlumpf (dashed-dotted line).

Figure 4: $\sigma$ (left panel), $P_{L}(E_{\bar{\nu}_{\tau}})$ (middle panel),
and $P_{P}(E_{\bar{\nu}_{\tau}})$ (right panel) vs $E_{\bar{\nu}_{\tau}}$ for
$\bar{\nu}_{\tau}+n\rightarrow\tau^{+}+\Sigma^{-}$ process. The calculations
have been performed using the SU(3) symmetry (solid line), the SU(3) symmetry
breaking effects parameterized by Faessler et al. (dashed line) and by
Schlumpf (dashed-dotted line).

##  4 Results and Discussions

In Fig. 2, we have studied the effect of T violation on the total scattering
cross section $\sigma(E_{{\nu}_{\tau}})$ (Eq. (7)), as well as on the
polarization observables $P_{L}^{\tau}(E_{\nu_{\tau}})$,
$P_{P}^{\tau}(E_{\nu_{\tau}})$ and $P_{T}^{\tau}(E_{\nu_{\tau}})$, for the
process $\nu_{\tau}+n\rightarrow\tau^{-}+p$, by varying the value of
$g_{2}^{I}(0)$ in the range $0-2$. It may be observed from the figure that the
total cross section as well as the polarization observables ($P_{P}^{\tau}$
and $P_{T}^{\tau}$) are quite sensitive to the variation in the value of
$g_{2}^{I}(0)$, while the effect of $g_{2}^{I}(0)$ variation on
$P_{L}^{\tau}(E_{\nu_{\tau}})$ is small. The transverse component of
polarization is non-zero when we take the T-violating effect
($g_{2}^{I}(0)>0$) into account. Therefore, the experimental measurement of
the non-zero value of $P_{T}^{\tau}$, if any, would independently determine
the degree of T violation in weak interactions.

In Fig. 3, we have presented the results for $\sigma(E_{\bar{\nu}_{\tau}})$,
$P_{L}(E_{\bar{\nu}_{\tau}})$ and $P_{P}(E_{\bar{\nu}_{\tau}})$ vs
$E_{\bar{\nu}_{\tau}}$ for the charged current process
$\bar{\nu}_{\tau}+p\longrightarrow\tau^{+}+\Lambda$ with SU(3) symmetry as
well as when the SU(3) symmetry breaking effects are taken into account. We
find that the effect of SU(3) symmetry breaking on
$P_{P}(E_{\bar{\nu}_{\tau}})$ is quite significant, while on
$P_{L}(E_{\bar{\nu}_{\tau}})$ the effect is small and there is hardly any
effect of SU(3) symmetry breaking on $\sigma(E_{\bar{\nu}_{\tau}})$.

In Fig. 4, we have presented the results for $\sigma(E_{\bar{\nu}_{\tau}})$,
$P_{L}(E_{\bar{\nu}_{\tau}})$ and $P_{P}(E_{\bar{\nu}_{\tau}})$ vs
$E_{\bar{\nu}_{\tau}}$ for the reaction
$\bar{\nu}_{\tau}+n\longrightarrow\tau^{+}+\Sigma^{-}$ with SU(3) symmetry as
well as when the SU(3) symmetry breaking effects are taken into account. We
find that in $\Sigma$ production, there is a large variation in
$P_{L}(E_{\bar{\nu}_{\tau}})$ and $P_{P}(E_{\bar{\nu}_{\tau}})$ due to the
SU(3) symmetry breaking while the effect on $\sigma(E_{\bar{\nu}_{\tau}})$ is
small like the one observed in $\Lambda$ production. In the case of
$P_{L}(E_{\bar{\nu}_{\tau}})$, the variation is more when Faessler et al. [4]
prescription is used and the nature of dependence on SU(3) breaking effect
using Faessler et al. [4] and Schlumpf [5], prescriptions are different.

##  5 Conclusion

The effect of T violation is appreciable for $g_{2}^{I}(0)\geq 1$ in the case
of $\sigma$ as well as the polarization observables both for the neutrino as
well as antineutrino induced processes. There is a significant variation in
$P_{L}$ and $P_{P}$, on the different parameterizations of the SU(3) symmetry
breaking, while the effect of SU(3) symmetry breaking is not much in the case
of total scattering cross sections. These results are more prominent in the
case of $\Delta S=1$ processes.

## References

  * [1] A. Fatima, M. Sajjad Athar and S. K. Singh, Quasielastic production of polarized $\tau$ leptons in $\nu_{\tau}$ and $\bar{\nu}_{\tau}$ scattering from nucleons, Phys. Rev. D 102, 113009 (2020), doi:10.1103/PhysRevD.102.113009, 2010.10311.
  * [2] A. Fatima, M. Sajjad Athar and S. K. Singh, $\bar{\nu}_{\mu}$ induced quasielastic production of hyperons leading to pions, doi:10.1140/epjs/s11734-021-00272-0, 2106.14590.
  * [3] A. Fatima, M. Sajjad Athar and S. K. Singh, ‘Second class currents and T violation in quasielastic neutrino and antineutrino scattering from nucleons, Phys. Rev. D 98, 033005 (2018), doi:10.1103/PhysRevD.98.033005, 1806.08597.
  * [4] A. Faessler et al., Semileptonic decays of the light JP = 1/2+ ground state baryon octet, Phys. Rev. D 78, 094005 (2008), doi:10.1103/PhysRevD.78.094005, 0809.4159.
  * [5] F. Schlumpf, Beta decay of hyperons in a relativistic quark model, Phys. Rev. D 51, 2262 (1995), doi:10.1103/PhysRevD.51.2262, 9409272.


# Frequency Shift Algorithm: Design of a Baseband Phase Locked Loop for
Frequency-Domain Multiplexing Readout of X-ray Transition-Edge Sensor
Microcalorimeters

Paul van der Hulst<EMAIL_ADDRESS>SRON Netherlands Institute for
Space Research, Sorbonnelaan 2, 3584CA Utrecht, The Netherlands  Jan van der
Kuur  SRON Netherlands Institute for Space Research, Kapteynborg, Landleven
12, 9747 AD Groningen, The Netherlands  Ad Nieuwenhuizen  Davide Vaccaro
Hiroki Akamatsu  Patrick van Winden  Bert-Joost van Leeuwen  Jan-Willem den
Herder  SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584CA
Utrecht, The Netherlands

###### Abstract

The Transition-Edge Sensor (TES) is an extremely sensitive device which is
used to measure the energy of individual X-ray photons. For astronomical
spectrometry applications, SRON develops a Frequency Domain Multiplexing (FDM)
read-out system for kilopixel arrays of such TESs. Each TES is voltage biased
at a specific frequency in the range 1 to 5 MHz. Isolation between the
individual pixels is obtained through very narrow-band (high-Q) lithographic
LC resonators. To prevent energy resolution degradation due to intermodulation
line noise, the bias frequencies are distributed on a regular grid. The
requirements on the accuracy of the LC resonance frequency are very high. The
deviation of the resonance frequencies due to production tolerances is
significant with respect to the bandwidth, and a controller is necessary to
compensate for the LC series impedance. We present two such controllers: a
simple orthogonal proportional-integrating (PI) controller and a more complex
impedance estimator. Both controllers operate in baseband and try to make the
TES current in-phase with the bias voltage, effectively operating as phase-
locked loops (PLL). They allow off-LC-resonance operation of the TES pixels,
while preserving TES thermal response and energy resolution. Extensive
experimental results – published in a companion paper recently – with the
proposed methods, show that these controllers allow the preservation of single
pixel energy resolution in multiplexed operation.

##  I Introduction

Transition Edge Sensors (TES) are very sensitive radiation detectors used in
ground- and space-based astronomy.[1, 2] When used as bolometers, they can
sense very weak signals such as far infrared radiation.[3] For more energetic
(e.g. x-ray) photons, they can be used as microcalorimeters to measure the
energy of individual photons, which allows the study of the hot gas
distribution in the universe, and black holes.[4]

TESs operate at the edge of superconductivity, an effect which manifests
itself at cryogenic temperatures. The critical temperature for the TESs
developed at SRON is typically around $100\,\mathrm{mK}$. In order to limit
wiring complexity for a kilopixel TES array, as well as the amount of heat
produced by cold stage pre-amplifiers, and thus the required cooling power,
the readout electronics multiplex the signals of multiple sensors over a
single transmission channel. Different multiplexing strategies are available
for multiplexing of TES sensors with Superconducting QUantum Interference
Device (SQUID) amplifiers: Time Division Multiplexing (TDM),[5] Code Division
Multiplexing (CDM)[6] and Frequency Division Multiplexing (FDM).

Over the last few years, SRON has actively been developing FDM readout of TES
sensors for the X-IFU instrument on the Athena satellite.[7, 8] FDM is
currently the the backup readout method for X-IFU. The SAFARI spectrometer on
the recently canceled Spica spacecraft[3] was also planned to be equipped with
SRON FDM readout. Recently published experimental results[9] show that the new
techniques presented in this paper, raise the performance of FDM readout to a
level competitive with the current state of the art.

The FDM method biases each TES with an ac voltage of a specific frequency. LC
filters in series with each TES select the carrier for the TESs. For a carrier
signal at the LC resonance frequency, the impedance of the resonator is zero,
biasing the TES with the provided carrier voltage. The carrier signals for the
other sensors are outside the pass-band of the LC resonator and thus they are
attenuated.

Due to fabrication tolerances, the resonance frequency of the LC filters
exhibits a spread of a few kHz from the intended 100 kHz grid.[10] When each
pixel is operated at exactly its corresponding LC resonance frequency, the
carriers are distributed somewhat randomly in frequency space. Non-linearities
in the active part of the readout chain will cause intermodulation distortion.
These distortion components can – and will – end up in the signal band of some
pixels and interference between this frequency and the carrier will result in
beating, i.e. modulation of the bias power. The pixel readout signal then
becomes a function of the phase of the beating, and the resulting energy
resolution has been observed to deteriorate by a few electronvolt.[11]

For many bolometer applications the problem is insignificant because the
signal bandwidth is rather low. This reduces the chances of intermodulation
products to end up within this signal bandwidth. When it does occur, a very
small frequency shift relative to the LC filter bandwidth (which does not
significantly change the LC series impedance) is sufficient to move it out of
the signal band. For microcalorimeters and bolometers with higher thermal
bandwidth,[12] the required frequency shift to move the distortion lines out
of the pass band of the system is proportionally larger. This increases the
series impedance of the LC filter. As a result, the TES current is no longer
primarily determined by its resistance, but increasingly by the reactive
series impedance of the LC circuit, which is dominant outside the resonator
bandwidth.

Several methods have been proposed in the past to compensate reactive series
impedance. Digitally Enhanced Voltage Bias[13] compensates a fixed series
impedance of the TES, but requires prior measurement of the series impedance,
and does not use the phase information of the measured current which is needed
to separate the resistive and reactive components of the currents. Automatic
compensation of off-resonance series impedance (with an active integrator
controller) has also been proposed and demonstrated.[14, 11] The primary aim
of that solution was to maintain a low source impedance for the TES bias
source by shifting the resonance frequency of the closed loop system. The
controllers presented in this paper are based on a reformulated control goal
and have a more robust implementation in baseband. Instead of shifting the
resonance frequency, we designed an adaptive controller which provides the
required reactive voltage to compensate the off-resonance LC series impedance
under arbitrary frequency shift.

##  II System configuration

The configuration of the original TES readout system is shown in Fig. 1.

Figure 1: The FDM and BBFB system diagram for one channel, before
modification. The demodulation stage for data output is also used to generate
the baseband feedback (in red) which linearizes the SQUID. Reproduced from den
Hartog  _et al._ , J. of Low Temp. Phys., 167, 652, 2012; licensed under a
Creative Commons Attribution (CC BY) license.

The TES sensors, LC separation filters, a shunt resistor and a SQUID buffer
amplifier are located at two different temperature levels in the cryostat. The
warm (analog) electronics buffers the signals to and from the cryostat, the
latter through a Low Noise Amplifier (LNA). The ac bias and feedback signals
are effectively converted to current sources through series resistors. The
digital electronics contains generation of the individual ac bias signals,
based on a local oscillator (LO), as well as the demodulation logic to
separate the readout signals of the individual TES sensors. The BaseBand
FeedBack (BBFB) control loop[16, 17] (in red) efficiently reuses the
demodulated signals to minimize the SQUID input signal level. This feedback
loop improves linearity of the SQUID buffer stage, while at the same time it
acts as a band-limiting filter for the scientific output signals.

##  III Controller design

The conceptual model of the TES circuit is shown in Fig. 2.

Figure 2: Fundamental single pixel TES readout circuit consisting of a bias
voltage source, a TES and an LC series resonator for blocking other
frequencies

The ac voltage source $U_{\text{bias}}$ provides the bias voltage for the TES
resistance $R_{\text{TES}}$. A series LC resonant circuit in series with the
TES provides frequency selectivity. Phasor diagrams showing the voltages and
current in the circuit are shown in Fig. 3.

Figure 3: Phasor diagrams for the voltages in the fundamental circuit of Fig.
2. On-resonance (a) and off-resonance (b)

When the bias frequency is equal to the LC resonance frequency, the voltages
across the capacitor and coil cancel exactly as shown in Fig. 3a. When the
circuit is operated above resonance, the impedance of the coil increases,
while the impedance of the capacitor decreases. The voltage across the coil is
thus higher than the voltage across the capacitor. This is illustrated in the
phasor diagram of Fig. 3b. The capacitor and coil voltages are necessarily
perpendicular to the (TES-)current. The component voltages can be
geometrically constructed with the help of the dotted circle because any
triangle with the circle’s diameter as one side and the 3rd corner elsewhere
on the circle is necessarily a right triangle. The bias voltage is now no
longer across the TES because there is a voltage drop across the capacitor and
inductor. The TES voltage is reduced in amplitude and lags the bias voltage.

Apart from the effect of bias voltage reduction, the TES is also no longer
biased with a low source impedance. The output impedance of the bias voltage
source is increased with the LC impedance. This is problematic because the TES
has a very steep positive temperature coefficient. When a photon is captured,
the TES’s temperature and thus the resistance increases. The current should
now decrease, or else self-heating power of the TES increases and thermal
runaway can occur. To prevent this, the current must dominantly be determined
by the TES resistance. The output impedance of the bias source must therefore
be lower than the TES resistance to make the thermal feedback loop stable.[18]

###  III.1 Controller properties

####  III.1.1 Orthogonal output

Based on the phasor diagrams in Fig. 3, we can list some desired properties of
the controller output voltage. The controller should compensate the voltage
drop across the LC resonator when operated off-resonance. Since the LC
impedance causes an error voltage proportional to- and perpendicular to the
TES current, the compensation voltage should be

  1. 1.

proportional to the TES current,

  2. 2.

90 ° phase shifted with respect to the TES current or voltage.

The controller is inserted into the circuit in Fig. 4.

Figure 4: Fundamental circuit of the TES readout with controller (a) and
corresponding phasor diagram (b) illustrating the concept of a controller with
output orthogonal to the TES current/voltage

The controller indicated by the dashed box in Fig. 4a provides the additional
current-dependent voltage required to compensate the voltage across the LC
resonator. Fig. 4b shows the corresponding phasor diagram. The total voltage
generated by the DAC is $U_{\text{carr}}$, which is the sum of the orthogonal
signals $U_{\text{bias}}$ and $U_{\text{ctrl}}$.

####  III.1.2 Control target

Fig. 3b and Fig. 4b show the difference between uncontrolled and desired off-
resonance operation of the LC circuit respectively. Only the bias voltage and
TES current in these figures are known variables, either because they are
output or because they are measured.

The TES current is measured with a BBFB filter[16] which also demodulates the
signal as part of its operation. The output of this filter is a current
vector, described by in-phase ($I$) and out-of-phase $Q$(uadrature) orthogonal
components. The $I$ and $Q$ signals, which are originally intended for
scientific processing, can also be used to determine the phase shift between
bias voltage and TES current. When the TES current is in-phase with the bias
voltage, the $Q$ signal is 0. A non-zero value of the $Q$ signal corresponds
to a phase-shift due to off-resonance operation. The $Q$ signal is therefore a
convenient input for the controller, with an implicit set-point
$Q_{\text{ref}}=0$.

####  III.1.3 Controller dynamics

When the TES current lags the bias voltage, a leading controller voltage must
be provided. The controller thus requires negative feedback. The controlled
system should also have no steady-state error. This implies that the
controller must at least provide an integrating action. In order to improve
dynamic behavior, a proportional parallel branch is added to get a PI
controller. Finally, to reduce interference with neighboring signals in the
FDM channel due to the proportional action, the controller inputs can be
filtered with two independent 1st order low-pass filters.

###  III.2 Baseband system model

The proposed controllers will operate in baseband rather than on ac signals.
Figure 5 shows the baseband model of the LC resonator and BBFB filter.

Figure 5: Complex baseband model of the LC resonator and BBFB filter

The transformation of the LC filter and BBFB model to baseband and the
reduction of the combination to the diagram in figure 5 is detailed in
Appendix A.

The first stage is the complex transfer function of the LC filter under
frequency shift operation. The imaginary term $j\Delta\omega$ in the
denominator indicates that the filter has a single non-conjugate complex pole,
realized by a cross path between the real and imaginary signal paths of the
filter.[19] An interesting feature is that since $R$ and $L$ are the same for
each resonator in the 1 to 5 MHz range, the dynamic behavior of the LC filter
is not determined by the resonance frequency, just by the frequency shift. The
second stage in the diagram represents the BBFB filter, transformed for
complex signals. This transformation modifies the BBFB integrator gain $K$ to
$K^{\prime}=\nicefrac{{K}}{{2}}$ in the complex implementation.

The baseband model of the system models the ac amplitude and -phase response
of the pixel current which carries the power information resulting from photon
events. By removing the frequency modulation from the model, the model has
become much simpler. More importantly the derivation demonstrates that the
pixel behavior is independent of the carrier frequency. In the model, the
carrier- and LC resonance frequencies are exchanged for a single parameter
$\Delta\omega$. Only the frequency _shift_ and the bandwidths of both the LC
filter and the BBFB filter determine the photon response.

An additional benefit of the baseband model over the hf model is that
simulation time step can be up to 1000 × larger, resulting in much faster
simulation. Normal simulation of the hf model requires at least 10 intervals
per period of the ac carrier ($T_{\text{carr}}=$ 200 to 1000 ns) to get
reasonable accuracy. The photon events that are being studied are typically 3
orders of magnitude slower, in the order of 1 ms, due to the limited thermal-
and resonator bandwidths.

###  III.3 Calibration

To enhance clarity, the bias voltage and TES current are drawn in-phase for a
resistive load in the phasor diagrams of figures 3 and 4b. This is correct
when considering just the cryogenic circuit, but there is a significant
additional time delay through the signal lines to and from the cryostat, as
well as due to digital processing. Due to the narrow frequency band of a
pixel, this delay can be expressed as a single offset angle between the bias
voltage- and current phasors. The delay can therefore be compensated by adding
the inverse angle offset between the modulation and demodulation signals. With
the delay compensated, we can pretend that the bias and measurement frames are
the same. A phase angle between voltage and current detected in the processing
stage of the digital electronics is then equal to the natural phase-shift of
the LC resonator.

The delay compensation angle must be calibrated individually for each carrier.
The impedance of the cryogenic circuit is resistive on resonance, and
therefore bias voltage and measured current should also be in-phase. Assuming
that the bias voltage is in the $I$ direction and controller is deactivated,
the residual phase angle of an imperfect delay compensation projects a
fraction of the measured current on the $Q$ axis. Calibration is than a matter
of tuning the delay compensation angle until the $Q$ component in the measured
current is 0.

##  IV Q-nuller

As a proof of concept, initially the simplest possible version of the
controller was implemented. This variant – dubbed the ‘Q-nuller’ – is shown in
Fig. 6.

Figure 6: Q-nuller configuration of the control loop: the $Q$ output of the
BBFB filter is fed to the PI controller, the controller output
$U_{\text{ctrl}}$ is fed to the imaginary (orthogonal) input of the modulator.
Modulation with the carrier frequency is indicated by a complex vector
rotation, of which only the real output (the hf voltage $U_{\text{carr}}$) is
used. Reproduced from Vaccaro  _et al._ , Rev. Sci. Instrum., 92, 033103 2021,
with the permission of AIP Publishing.

It simply sends the output of the PI controller directly to the carrier
modulator. In the modulator, the bias amplitude setpoint is multiplied with a
cosine function because it is at the real input and the controller output is
multiplied with a sine function because it is at the imaginary input. This
forces a 90 ° phase shift between the bias and controller components. This
orthogonal modulation functionality was already present in the existing
modulator implementation and thus required little modification of the digital
electronics firmware.

When operating off-resonance, the voltage across the LC resonator is
proportional to the magnitude of the TES current. The output of the Q-nuller
controller only depends on the Q-component of this current. The controller can
react only when a change in current results in an associated angle error and
thus a $Q$ current. This makes it react slow to changes in current magnitude.
To be able to follow dynamic events (photon events), the controller gain must
be high, which may affect stability.

###  IV.1 Stability

Initially, only the TES readout system (LC filter and BBFB filter) with an
integrating controller will be evaluated. Further filtering can be added if
required, e.g. for stability.

The Q-nuller controller only operates on the imaginary components of the TES
current and carrier voltage. It is therefore not possible to analyze the
system using complex math. The transfer of the model in Fig. 5 must be
expanded to separate real and imaginary signal paths.

Figure 7: Baseband model of closed loop system with Q-nuller integrating
controller, where $\Re(u)$ is the bias set point and $\Im(u)$ the controller
value. The outputs $\Re(y)$ and $\Im(y)$ are the I- and Q-components of the
TES current respectively.

The readout system controlled with the Q-nuller is shown in Fig. 7. The
feedback control loop is only active on the imaginary signal path. The real
signal path has no controller and/or feedback.

For stability analysis of the closed loop, the open loop baseband transfer of
the system with Q-nuller controller needs to be evaluated:

$\displaystyle H_{\text{ol,bb,Q}}(s)$
$\displaystyle=H_{\text{LC,}\Im}(s)\times H_{\text{BBFB}}(s)\times
H_{\text{ctrl}}(s)$ (1)
$\displaystyle=\frac{\frac{1}{2L}(s+\frac{R}{2L})}{s^{2}+\frac{R}{L}s+(\frac{R}{2L})^{2}+\Delta\omega^{2}}\times\frac{K^{\prime}}{s+K^{\prime}}\times\frac{K_{\text{i}}}{s}$
(2)

where $H_{\text{LC,}\Im}(s)$ is the transfer of the LC filter in baseband from
$\Im(u)$ to $\Im(y)$ and $H_{\text{BBFB}}(s)$ and $H_{\text{ctrl}}(s)$ are the
baseband transfer of the BBFB filter and the integrating Q-nuller controller
respectively.

Stability can be evaluated through the Nyquist plot of $H_{\text{ol,bb,Q}}(s)$
which is shown in Fig. 8 for 3 different configurations.

Figure 8: Nyquist plots for a system with Q-nuller controller
($H_{\text{ol,bb,Q}}(s)$ of (2), $L=2\,\mathrm{\text{\textmu H}}$,
$R_{\text{TES}}=15\,\mathrm{m\Omega}$, $\omega_{\text{BBFB}}=2\pi\times
10\,\nicefrac{\mathrm{krad}}{\mathrm{s}}$ and controller gain
$K_{\text{i}}=500$). Curves are provided for an I-controller with
$\Delta\omega=0\,\nicefrac{\mathrm{rad}}{\mathrm{s}}$ (blue) and
$\Delta\omega=2\pi\times 1\,\nicefrac{\mathrm{krad}}{\mathrm{s}}$ (orange),
and a PI controller with $\omega_{\text{PI}}=\omega_{\text{BBFB}}$ and
$\omega_{\text{LP}}=3\times\omega_{\text{BBFB}}$(yellow). Gain margin
decreases with frequency shift, but phase margin is improved by a phase-lead
filter.

When the controller is an integrating controller as in (2), the blue line
represents on-resonance operation – or a frequency shift of 0 kHz – and the
orange line a frequency shift of 1 kHz. In both situations this system is
conditionally stable: for high frequencies, the Nyquist plot approaches the
origin from above, crossing the negative real axis between −1 and the origin.
It will become unstable when the gain is increased more than about 2.5×. In
that case the Nyquist plot will cross the negative real axis on the left from
−1 and the closed loop system will be unstable.

The system can be made unconditionally stable by adding a proportional signal
path to the integrating controller. This decreases the high frequency phase-
shift by 90 °, which causes the Nyquist plot to approach the origin from the
left. Thus the Nyquist plot no longer _crosses_ the negative real axis,
regardless of gain. The location of the PI zero is typically set at a few kHz,
between the LC filter bandwidth and the maximum expected frequency shift.
Unfortunately, the proportional controller action increases the magnitude of
out-of-band signals. The high frequency signal content interferes with
neighboring FDM channels. An additional low-pass filter with a crossover
frequency higher than the PI zero is therefore necessary. The pole of this
filter is placed above the unity gain frequency so its’ phase shift will not
affect stability. The PI zero in combination with the low-pass filter
effectively constitutes a phase-lead network added to the integrating
controller. The Nyquist plot of this situation is plotted in Fig. 8 in yellow.
The phase lead characteristic increases stability margins or allows the
controller gain to be increased. That said, the system remains only
_conditionally_ stable.

##  V Impedance estimator

Figure 9: Z-estimator configuration of the control loop: the (PI) controller
output $U_{\text{ctrl}}$ is multiplied with the $I$/$Q$ current components
before being fed to the modulator. Modulation with the carrier frequency is
indicated by a vector rotation, of which only the real output (the hf voltage
$U_{\text{carr}}$) is used.

A more elaborate implementation of the controller uses the output of the PI
controller as estimate of the LC impedance. This variant is called the
‘Z-estimator’ and is shown in Fig. 9. This estimated impedance (controller
output) is multiplied with the measured current and added to the bias voltage
with 90 ° phase shift. Thus the controller voltage is a prediction of the
voltage across the LC resonator which needs to be compensated, given the
current. The response of this controller is inherently fast due to the
multiplication with the measured current. The PI estimator itself can be very
slow since it aims to estimate the value of a constant property.

###  V.1 Stability

The Z-estimator controller is a non-linear controller because it multiplies
two signal paths. However, the estimator output should be virtually constant,
since it provides an estimate of a constant value: the off-resonance impedance
of the LC resonator. The controller gain can therefore be set very low, or
even zero after a short initialization interval. The open-loop baseband
transfer of the system with Z-estimator controller is then:

$\displaystyle H_{\text{ol,bb,Z}}(s)$ $\displaystyle=H_{\text{LC}}(s)\times
H_{\text{BBFB}}(s)\times j\hat{Z}$ (3)
$\displaystyle=\frac{\frac{1}{2L}}{s+j\Delta\omega+\frac{R}{2L}}\times\frac{K^{\prime}}{s+K^{\prime}}\times
j\hat{Z}\ .$ (4)

Here, $H_{\text{LC}}(s)$, $H_{\text{BBFB}}(s)$ and $j\hat{Z}$ are the
transfers for analytic baseband signals of the LC filter, the baseband
feedback filter and the estimated impedance respectively. The equivalent
closed-loop model for complex signals is shown in Fig. 10.

Figure 10: Baseband model for the closed loop system with Z-estimator. All
signals, including $U_{\text{bias}}$ and $I_{\text{TES}}$ are complex.

From Fig. 10 it is clear that for sufficient bandwidth of the BBFB filter, the
imaginary component in the feedback will be canceled when
$\hat{Z}=2\Delta\omega L$. When this is the case, the controller successfully
restores the system response to the on-resonance behavior.

The Nyquist plot of $H_{\text{ol,bb,Z}}(s)$ in shown in Fig. 11.

Figure 11: Nyquist diagram of the system with Z-estimator controller
($H_{\text{ol,bb,Z}}(s)$ of (4), $L=2\,\mathrm{\text{\textmu H}}$,
$R_{\text{TES}}=15\,\mathrm{m\Omega}$ and estimator value
$\hat{Z}=2\Delta\omega L$). Curves are provided for $\Delta\omega=2\pi\times
500\,\mathrm{rad/s}$ (blue) and $\Delta\omega=2\pi\times 1\,\mathrm{krad/s}$
(orange). Although the phase margin decreases for rising $\Delta\omega$, the
system is unconditionally stable for any frequency shift, provided the loop
gain $\hat{Z}\leq 2\Delta\omega L$

It shows that this system is unconditionally stable. The radius and asymmetry
of the figure depend on the BBFB filter bandwidth $K^{\prime}$ and the
frequency shift $\Delta\omega$. For increasing frequency shift, the -1 point
is approached asymptotically. Even for a ridiculously large value of e.g.
$\Delta\omega=2\pi\times 50\,\nicefrac{\mathrm{krad}}{\mathrm{s}}$ (5× the
BBFB filter bandwidth) the -1 point is still not encircled.

As with the Q-nuller controller, additional filtering of the measured current
(the BBFB filter output) is beneficial to reduce interference with neighboring
pixels. The pole must be placed above the unity gain frequency of the loop to
prevent the filter from causing instability. A crossover frequency of 25 kHz
($\nicefrac{{1}}{{4}}$ of the frequency separation between pixels in the FDM
scheme) is considered to be a safe compromise between stability and
interference.

##  VI Parallel resonator leakage

In the X-IFU FDM system, up to 40 pixels are operated in parallel, with a
carrier frequency spacing of about 100 kHz. When operated on-resonance, the
impedance at the carrier frequency is dominated by the low impedance of the
TES. The impedance of the neighbors is negligibly high. When a pixel is
operated off-resonance, the impedance of the LC resonator will increase,
depending on the frequency shift. At some point the total impedance will be
noticeably influenced by the reactive impedance of the neighbors which causes
an offset in the measured $Q$-current. This results in a calibration error of
the controller which attempts to zero the $Q$ current. This error results in
bad reproducibility when plotting the I/V curves, as illustrated in Fig. 12.

Figure 12: TES I/V curves for various frequency shifts with Q-nuller
controller. The curves diverge due to current leakage through the parallel
resonators.

###  VI.1 Carrier dependence

The effect of the neighboring resonators is a function of the carrier
frequency, or rather the resonance frequencies of the neighbors. When a
neighbor pixel is tuned at a frequency above the carrier frequency, it
presents a capacitive impedance because it is operated below resonance. When a
neighbor pixel is tuned at a frequency below the carrier frequency, it
presents an inductive impedance because it is operated above resonance. A low
frequency pixel thus has mainly capacitive impedances in parallel whereas a
high frequency pixel has mainly inductive impedances in parallel. Necessarily
the effects compensate for pixels somewhere halfway. The calculated
(imaginary) _admittance_ for each pixel during one of the experiments is shown
in Fig. 13 as function of the pixel’s LC resonance frequency.

Figure 13: The total reactive admittance in parallel to each pixel as function
of the pixel resonance frequency (assuming $L=2\,\mathrm{\text{\textmu H}}$).
The data was calculated from a list of measured resonance frequencies of the
LC filter array.

The offset in the $Q$ current resulting from this effect can be compensated by
subtracting a bias voltage dependent fraction from the measured $Q$-current.
The compensation and its impact on the controllers are detailed in the
supplemental material in appendix B.

##  VII Results

Experiments with the new controllers have been performed to test their effect
on energy resolution under frequency shift, and the behavior while
multiplexing multiple pixels. For most of the experiments, the Q-nuller with
PI controller was applied rather than the superior Z-estimator, for the
practical reasons that it was developed earlier and that it was sufficient for
the purpose of demonstrating the readout capability of FDM. The application
and optimization of the Z estimator scheme is planned for the near future.

The PI crossover zero was typically set at 2 to 3 times the pixel bandwidth as
determined by the LC filter- and thermal bandwidths. The low pass filters were
configured for a bandwidth of a 2 to 3 × $f_{\text{PI}}$. These values were
the same for all pixels within the 1 to 5 MHz range of FDM carriers, which
confirms our expectation from the baseband LC model that the dynamics of the
pixels do not depend on the modulation frequency.

Fig. 14 shows the effectiveness of the controller in providing a constant bias
voltage to the TES, irrespective of the frequency shift.

Figure 14: TES current on a typical bias point with FSA (=Q-nuller) (in)active
across $\pm 1\,\mathrm{kHz}$ frequency shift. Reproduced from Vaccaro  _et
al._ , Rev. Sci. Instrum., 92, 033103 2021, with the permission of AIP
Publishing.

When biased in the transition region, the TES behaves as an almost constant
power sink as witnessed by the approximately $\nicefrac{{1}}{{x}}$ shape of
the typical TES i/v curve.[20] Because the current increases when the voltage
decreases, it presents a dynamically negative resistance which is initially
larger than the positive impedance of the series LC. Therefore, the TES
current increases when a frequency shift is applied and no controller is
active (black circles). Due to the negative dynamic resistance of the TES, the
carrier voltage must be increased in order to decrease the current. With the
Q-nuller (the red diamonds), the carrier voltage is increased with the
frequency shift, such that the voltage across the TES is constant and
therefore the TES power, -resistance and -current are constant.

Fig. 15 shows the measured baseband $I$-current (=hf TES current amplitude) in
response to an x-ray pulse.

Figure 15: Measured TES current pulse (hf amplitude) in response to an X-ray
photon for various frequency shifts and controllers

It is clear that a controller is required, because with uncontrolled frequency
shift, the response is already much slower for a frequency shift of 200 Hz.
The curves for the $Q$-nuller (dashed) are much closer to the reference on-
resonance curve. For a frequency shift of 1 kHz, the $Q$-nuller deviates
noticeably from the reference in Fig. 15. With the Z-estimator, the response
is virtually equal to the reference response, making the Z-estimator the
better choice.

The slow response of the uncontrolled frequency shifted pixel in figure 15,
translates directly to a reduction of energy resolution, shown in figure 16.
The purpose of carrier frequency shift is to reduce the energy resolution
deterioration as reported by Akamatsu _et al._[11] Figure 16 shows that
carrier frequency shift severely compromises the energy resolution when no
additional measures are taken (black).

Figure 16: Energy resolution as function of frequency shift with inactive
(black) and active (red) controller

With active $Q$-nuller controller (in red), the energy resolution displays no
apparent deterioration. The controllers thus enable the practical application
of frequency shift.

Extensive results confirming the merits of the new controllers in x-ray
spectrometry applications are presented by Vaccaro _et al._[9] for single
pixel and in multiplexing situations. In a three-pixel multiplexing experiment
with the $Q$-nuller controller and an LC resonator bandwidth of about 1 kHz, a
frequency shift of 600 Hz did not result in a significant degradation of the
energy resolution. Multipixel stable operation was demonstrated with 22
pixels. The limitation in the number of pixels was due to the available number
of cold components and not algorithm-related. Multi-pixel operation,
preserving the single-pixel $\approx 2.6$ eV resolution of the TESs was
demonstrated for up to 20 pixels.

###  VII.1 Hardware

The digital electronics is implemented in a Field Programmable Gate Array
(FPGA) which has a range of different hardware resources such as look-up
tables for logic or memory, flip flops, etc. This makes comparing the use of
resources complicated. The Z-estimator performs better than the $Q$-nuller, at
the cost of more hardware in the FPGA. Resources for the $Q$-nuller are about
20 % lower than for the Z-estimator. The tested implementation was not
optimized for efficiency. Some parts are reused by both controllers and
additional switching logic is added. Even with these limitations, this
implementation is about half the size of the BBFB filter.

The effect of the controllers on the dynamic range of the ac bias DAC was also
investigated. It was found that for an average frequency shift of 250 Hz, the
increase in DAC voltage is only 5 %, due to the orthogonal addition of the
bias and controller voltages. The controller is therefore not expected to
introduce crosstalk at problematic levels.

##  VIII Conclusions

A new method has been developed for off-resonance operation of frequency
domain multiplexed TES pixels. The main characteristic of the method is the
use of controllers with the $Q$ current signal from the BBFB filter as error
input, and an output voltage which is orthogonal to the TES current. Because
the modulation of the controller output is shared with the bias ac voltage
generation, the phase-shift between the carrier and controller signals is not
compromised by differences in numerical delay and implementation requires only
small modifications of the existing firmware. Tuning of the controllers is
relatively simple because the controllers automatically adapt to the applied
frequency shift. The controller gain settings are not critical in general.
Only the calibration of the offset angle between modulation and demodulation
stages requires some attention but is a straightforward operation.

The parasitic off-resonance impedance presented by the other LC resonators in
the channel ($Y_{\text{par}}$) must be compensated to allow proper calibration
of the carrier phase. Right now carrier rotation and $Y_{\text{par}}$ are
fine-tuned using an iterative procedure, but better knowledge of the transfer
of the bias and feedback transmission channels should allow us to predict
these.

As reported by Akamatsu _et al._[11], the energy resolution of individual
pixels in an FDM readout channel can reduce by more than 2 eV due to
intermodulation line noise. The proposed controllers provide a scalable method
to enable carrier frequency shift, which restores the single pixel readout
performance in FDM multiplexed readout.

###### Acknowledgements.

This work is part of the research program Athena with project number
184.034.002, which is financed by the Dutch Research Council (NWO).

## Data availability

The corresponding author makes available the data presented in this paper upon
reasonable request.

##  Appendix A Frequency transformation

The behavior and stability of the controllers in closed loop with the system
needs to be evaluated. A complication is that the controller works on quasi dc
demodulated signals, while the LC resonator operates on hf signals at the
carrier frequency, which may be close to, but not equal to the LC resonance
frequency. The frequency transformation in between makes the system non-
linear. It is necessary to remove the frequency transformation to obtain a
baseband description of the LC- and BBFB filters.

###  A.1 HF model of BBFB filter

We will first determine a linear hf description of the BBFB filter including
the feedback loop. It is known that the BBFB filter behaves similar to a
narrow-band band pass filter. We will show here that the LC filter and the
baseband feedback filter are equivalent.

The BBFB filter is a hybrid structure with the forward path in baseband- and
the feedback path in the hf domain. The filter, shown in Fig. 17, consists of
an integrator enclosed between two vector rotators, of which typically only
the real in- and outputs are used.

Figure 17: HF forward signal path of the BBFB circuit: an integrator with gain
enclosed between clockwise and counterclockwise rotations

When the rotation angle is $\mp\omega_{\text{c}}t$ respectively, the complex
input experiences a frequency transformation of $\mp\omega_{\text{c}}$
respectively. The frequency shift by the rotators results in an apparent
frequency shift of the transfer function in between. A signal at the input of
frequency $+\omega_{\text{c}}$ experiences a frequency transformation to dc.
This dc signal is integrated to a ramp, and again subjected to a frequency
shift of $+\omega_{\text{c}}$ back to the original frequency. The frequency
transfer of this filter is the complex function

$\displaystyle H_{\text{BBFB,fw}}(s)=\frac{K}{s-j\omega_{\text{c}}}\ .$ (5)

Because this filter has non-real polynomial coefficients in its numerator
and/or denominator, it has a single complex pole: the pole is not accompanied
by a complex-conjugate pole. The equivalent linear realization of this filter
is shown in Fig. 18a.

Figure 18: Simplification of the HF model of the BBFB circuit. The shifted
integrator of eq. (5) is implemented as (a). Reordering exposes a 2nd order
loop without damping (b). When adding feedback in the real signal path and
ignoring the imaginary in- and output, the band-pass filter with programmable
bandwidth (c) results.

When the filter is reordered (Fig. 18b), it is evident that the system is a
2nd order feedback system. The baseband feedback loop is closed by feeding
back the real output to the real input. At the same time, the input is
multiplied by $\omega_{\text{c}}$ and the output divided by
$\omega_{\text{c}}$. The resulting closed loop system in Fig. 18c is a 2nd
order band-pass filter with Q configurable by forward gain $K$ according to
$Q=\frac{\omega_{\text{c}}}{K}$. Parameter $K=\frac{\omega_{\text{c}}}{Q}$ is
the resonance bandwidth of the baseband feedback filter. The transfer function
for this circuit for real input signals is

$\displaystyle H_{\text{BBFB,hf}}(s)$
$\displaystyle=\frac{\frac{K}{\omega_{\text{c}}^{2}}s}{\frac{s^{2}}{\omega_{\text{c}}^{2}}+\frac{K}{\omega_{\text{c}}^{2}}s+1}$
(6)
$\displaystyle=\frac{\frac{s}{Q\omega_{\text{c}}}}{\frac{s^{2}}{\omega_{\text{c}}^{2}}+\frac{s}{Q\omega_{\text{c}}}+1}$
(7)

####  A.1.1 Equivalence of BBFB and LC filters

The LC filter circuit of Fig. 2 has structure which is very similar to the
BBFB filter shown in Fig. 18c, as illustrated in Fig. 19.

Figure 19: LC filter block diagram

This is confirmed by the transfer function for the LC filter:

$\displaystyle H_{\text{LC,hf}}(s)$
$\displaystyle=\frac{1}{R_{\text{TES}}}\times\frac{sR_{\text{TES}}C}{s^{2}LC+sR_{\text{TES}}C+1}$
(8)
$\displaystyle=\frac{1}{R_{\text{TES}}}\times\frac{\frac{s}{Q_{\text{LC}}\omega_{\text{LC}}}}{\frac{s^{2}}{\omega_{\text{LC}}^{2}}+\frac{s}{Q_{\text{LC}}\omega_{\text{LC}}}+1}$
(9)

The only difference is a gain of $\nicefrac{{1}}{{R_{\text{TES}}}}$.

Because the BBFB and LC filter have the same structure, a baseband model with
rotators can be created for the LC filter. Since parameter $K$ is the
bandwidth $\frac{\omega}{Q}$, the baseband integrator gain for the LC model
can be calculated:

$\displaystyle K_{\text{LC}}$
$\displaystyle=\frac{\omega_{\text{LC}}}{Q_{\text{LC}}}$ (10)
$\displaystyle=\sqrt{\frac{1}{LC}}\times R_{\text{TES}}\sqrt{\frac{C}{L}}$
(11) $\displaystyle=\frac{R_{\text{TES}}}{L}$ (12)

The LC filter in Fig. 19 is thus equivalent to Fig. 17, with integrator gain
$K=\frac{R_{\text{TES}}}{L}$, and an additional gain of
$\frac{1}{R_{\text{TES}}}$ before the feedback.

###  A.2 Complex baseband model of BBFB filter and LC resonator

####  A.2.1 Analytic vs. real signals

Demodulation and re-modulation of the signal in the BBFB filter is synchronous
with the carrier signal. The frequency-shifted integrator of eq. (5) in the
BBFB filter amplifies the positive frequency content of the real-valued input
signal because its frequency is exactly at the pole of the BBFB filter. All
negative frequency content of the input signal is suppressed by the filter.
The complex baseband input and -output signals are therefore by approximation
_analytic_ signals, which are complex-valued signals with only positive-
frequency content. The system can thus be analyzed assuming analytic signals.

In the system however, the imaginary component of the complex signal is
ignored at the BBFB demodulation input. The remaining real-valued sinusoidal
signal is a sum of complex-conjugate (positive and negative frequency-)
components. Fig. 20 shows that when the real component of a (complex) analytic
signal is taken and the imaginary component is ignored, the magnitude of the
positive frequency components is halved. The transfer for analytic signals of
the $\Re()$ operator is thus $\nicefrac{{1}}{{2}}$.

Figure 20: Amplitude halving when reducing a complex signal to its real
component. The original analytic signal (black) is reduced to its real part
(dashed). This real signal implicitly consists of opposite-frequency
components (grey) of half the original amplitude.

In contrast to the actual system, the imaginary output of the LC filter, as
well as the imaginary BBFB feedback signals are potentially available in
mathematical models of Fig 21. When the input signal is analytic instead of
real, the system gain must be the same to obtain the same response. The
(implicit) complex to real conversion must therefore be replaced by an
explicit gain of $\nicefrac{{1}}{{2}}$. The effective BBFB gain $K^{\prime}$
is therefore: $K^{\prime}=\nicefrac{{K}}{{2}}$ in Fig. 21a through c.
Likewise, the integrator gain in the baseband LC filter model halves from
$\frac{R_{\text{TES}}}{L}$ to $\frac{R_{\text{TES}}}{2L}$ for analytic signals
(Fig. 21d).

With complex feedback, the block diagram of Fig. 21a can be reordered. The
feedback can be moved to between the rotation operations because rotations
with $\pm\omega_{\text{c}}t$ cancel (Fig. 21b). The feedback around the
integrator results in a low pass filter in Fig. 21c. Analogous, the LC filter
is transformed to the subsystem in Fig. 21d.

Figure 21: Reduction of the BBFB model with complex feedback
($K^{\prime}=\nicefrac{{K}}{{2}}$). The block model of (a) is the complex
equivalent of the implementation in the digital electronics. The complex
feedback can be moved between the rotation (b). The feedback loop between the
rotations reduces to a simple low-pass filter for BBFB (c) as well as the LC
resonator (d) due to their equivalence.

###  A.3 Pixel readout baseband model

The analytic signal models of the LC filter and the BBFB filter constituting
the readout circuit for a single pixel can now be connected to get a model for
the complete signal chain from carrier input to measured TES current output
(Fig. 22a).

Figure 22: Pixel baseband model assuming analytic signals where
$u_{\text{carr}}$ is the complex input phasor. The original model (a) can be
reduced by merging rotations (b). The remaining rotation is incorporated in
the transfer function as a frequency shift (c).

The input signal is generated by rotating the baseband carrier phasor
$u_{\text{carr}}$ with $\omega_{\text{c}}t$. The output modulator in the BBFB
filter is discarded because it is no longer used since the feedback was moved
between the rotators (Fig. 21b).

The rotations with $\pm\omega_{\text{c}}t$ and $\mp\omega_{\text{LC}}t$ before
and after the LC filter partly cancel, so the series connection reduces to
Fig. 22b. The difference between the carrier frequency and the LC resonance
frequency is the frequency shift
$\Delta\omega=\omega_{\text{c}}-\omega_{\text{LC}}$. The remaining frequency
shifts $\pm\Delta\omega$ on either side of the LC filter are incorporated in
the LC filter transfer function in Fig. 22c.

##  Appendix B Parallel resonator leakage $Y_{\text{par}}$ Compensation

The current measured by the SQUID is the TES current plus the current through
all the parallel resonators. In baseband, the TES current can be reconstructed
from the measured (SQUID) current through subtraction of the expected parallel
current:

$\displaystyle\hat{I}_{\text{TES}}$
$\displaystyle=\hat{I}_{\text{SQUID}}-\frac{\vec{U}_{\text{ctrl}}+U_{\text{bias}}}{Z_{\text{par}}}$
(13)

where $Z_{\text{par}}$ is the cumulative reactive impedance of all the other
resonators at the pixel’s modulation frequency. The required control voltage
for the Z-estimator controller is:

$\displaystyle\vec{U}_{\text{ctrl}}$ $\displaystyle=Z_{\text{LC}}\times
I_{\text{TES}}$ (14)

This constitutes an algebraic loop which we can solve for
$\hat{I}_{\text{TES}}$ by eliminating $\vec{U}_{\text{ctrl}}$:

$\displaystyle\hat{I}_{\text{TES}}$
$\displaystyle=\hat{I}_{\text{SQUID}}-\frac{Z_{\text{LC}}\times\hat{I}_{\text{TES}}}{Z_{\text{par}}}-\frac{U_{\text{bias}}}{Z_{\text{par}}}\Leftrightarrow$
(15) $\displaystyle\hat{I}_{\text{TES}}$
$\displaystyle=\frac{Z_{\text{par}}}{Z_{\text{par}}+Z_{\text{LC}}}\times\left(\hat{I}_{\text{SQUID}}-U_{\text{bias}}\times
Y_{\text{par}}\right)$ (16) $\displaystyle\hat{I}_{\text{TES}}$
$\displaystyle=\frac{Z_{\text{par}}}{Z_{\text{par}}+Z_{\text{LC}}}\times
I^{\prime}_{\text{TES}}$ (17) and thus $\displaystyle\vec{U}_{\text{ctrl}}$
$\displaystyle=\frac{Z_{\text{par}}Z_{\text{LC}}}{Z_{\text{par}}+Z_{\text{LC}}}\times
I^{\prime}_{\text{TES}}$ (18)

where $I^{\prime}_{\text{TES}}$ is just an intermediate calculation result
with no physical meaning. Using a compensation admittance $Y$ rather than an
impedance $Z$ is appropriate to model a parallel circuit such as this. It also
has significant implementation benefits. When no compensation is needed, a
value of $Y_{\text{par}}=0$ is applied instead of
$Z_{\text{par}}=\nicefrac{{1}}{{0}}$. Therefore there is little danger of
overflow in the compensation value. Also, the calculation of the compensation
value is a multiplication $U_{\text{bias}}\times Y_{\text{par}}$ rather than a
division $\nicefrac{{U_{\text{bias}}}}{{Z_{\text{par}}}}$ which would be much
more expensive to implement.

The result of
$I^{\prime}_{\text{TES}}=\hat{I}_{\text{SQUID}}-U_{\text{bias}}\times
Y_{\text{par}}$ will be used as input of the controller rather than
$\hat{I}_{\text{TES}}$. The multiplier
$\frac{Z_{\text{par}}}{Z_{\text{par}}+Z_{\text{LC}}}$ just changes the
controller gain by a small amount, which is insignificant since the controller
gain value is not critical. The output of the controller $\hat{Z}$ will be the
estimated compensation impedance

$\displaystyle\hat{Z}=\frac{Z_{\text{par}}Z_{\text{LC}}}{Z_{\text{par}}+Z_{\text{LC}}}$
(19)

which allows calculation of the controller voltage with (18), using the same
intermediate result $I^{\prime}_{\text{TES}}$. Interestingly, $\hat{Z}$ is the
parallel impedance of _all_ the LC resonators at the carrier frequency instead
of just the LC series impedance of the pixel’s TES sensor, which was the
original intention.

The exact value of ${I}_{\text{TES}}$, which is required for scientific
analysis, can now be calculated by filling in the controller voltage
calculated with (18) in (13).

$\displaystyle\hat{I}_{\text{TES}}$
$\displaystyle=\left(\hat{I}_{\text{SQUID}}-U_{\text{bias}}\times
Y_{\text{par}}\right)-\vec{U}_{\text{ctrl}}\times Y_{\text{par}}$ (20)

This way we skip explicit solving of $Z_{\text{LC}}$. This last step is
required for both the Q-nuller and the Z-estimator.

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