# Quantum Rabi interferometry of motion and radiation

Kimin Park<EMAIL_ADDRESS>Department of Optics, Palacky University,
77146 Olomouc, Czech Republic  Center for Macroscopic Quantum States (bigQ),
Department of Physics, Technical University of Denmark, Building 307,
Fysikvej, 2800 Kgs. Lyngby, Denmark  Jacob Hastrup  Center for Macroscopic
Quantum States (bigQ), Department of Physics, Technical University of Denmark,
Building 307, Fysikvej, 2800 Kgs. Lyngby, Denmark  Petr Marek  Department of
Optics, Palacky University, 77146 Olomouc, Czech Republic  Ulrik L. Andersen
Center for Macroscopic Quantum States (bigQ), Department of Physics, Technical
University of Denmark, Building 307, Fysikvej, 2800 Kgs. Lyngby, Denmark
Radim Filip  Department of Optics, Palacky University, 77146 Olomouc, Czech
Republic

(Apr 15 2022)

###### Abstract

The precise determination of a displacement of a mechanical oscillator or a
microwave field in a predetermined direction in phase space can be carried out
with trapped ions or superconducting circuits, respectively, by coupling the
oscillator with ancilla qubits. Through that coupling, the displacement
information is transferred to the qubits which are then subsequently read out.
However, unambiguous estimation of displacement in an unknown direction in the
phase space has not been attempted in such oscillator-qubit systems. Here, we
propose a hybrid oscillator-qubit interferometric setup for the unambiguous
estimation of phase space displacements in an arbitrary direction, based on
feasible Rabi interactions beyond the rotating-wave approximation. Using such
a hybrid Rabi interferometer for quantum sensing, we show that the performance
is superior to the ones attained by single-mode estimation schemes and a
conventional interferometer based on Jaynes-Cummings interactions. Moreover,
we find that the sensitivity of the Rabi interferometer is independent of the
thermal occupation of the oscillator mode, and thus cooling it to the ground
state before sensing is not required. We also perform a thorough investigation
of the effect of qubit dephasing and oscillator thermalization. We find the
interferometer to be fairly robust, outperforming different benchmark
estimation schemes even for large dephasing and thermalization.

##  I Introduction

Quantum sensing is about estimating unknown processes of interest using a
finite ensemble of probes and detectors with a sensitivity that goes beyond
the reach of classical sensing DegenRMP2017 ; GiovannettiNatPho2011Metrology ;
Sidhu2020 ; sensinghighenergyphysics2018 ; DAlessandroBook2021control .
Historically, optical probes have been at the center of this field due to the
experimental accessibility of lasers and non-classical light resources, high-
efficiency detectors, and the strong robustness of light to external noise
sources PirandolaNatPhot2018PhotSens ; pop00003 ; pop00009 ; Polino2020 .
Quantum optical interferometers exploit the interference between a probe and a
reference beam to detect weak signals DEMKOWICZDOBRZANSKI2015345 , a prominent
example being the detection of gravitational waves from black hole mergers
LigoPRL2016 ; LIGO2020 . A similar approach can be adopted for microwave
traveling waves Lang2013microwaveHOM , microwave cavity fields
Gao2018MwMemInterf , matter waves Bongs2019atominterferometer ;
Cronin2009atominterferometer , between light and atomic ensembles
PezzeRMP2018AtomEnsemble ; ChenPRL2015 , and optomechanics tsangPRL2010 ;
motazedifard2020 ; Bemani2021 .

Numerous sensing proposals use quantum non-Gaussian states as probes for
improved estimation of phase and displacement dalvit_filho_toscano_2006 ;
TerhalPRA2017GKP ; BraunRMP2018Sensing . For example, quantum displacement
sensing has been performed with Fock states and non-Gaussian superpositions of
Fock states of the phononic modes of trapped ions 2020FockDispersive ;
McCormick2019 . Similarly, for superconducting circuits, the preparation of
microwave Fock states and their superpositions has been recently mastered
Premaratne2017 ; Wang2017 ; Pfaff2017NatureConversion ; Gely2019  as well as
the preparation of non-Gaussian acoustic modes of a quartz crystal Chu2018 ,
which can be also used for measuring displacements. Superconducting transmon
has been recently used for sensing magnons Lachance-Quirion2017 ; Wolski2020 ;
Lachance-Quirion2020 , search for dark matter Dixit2020 , microwave radiometry
Wang2019 ; Kristen2019 ; Wang2021  including sensing with a non-Gaussian probe
in a Fock-state-superposition Wang2019HeisenSupercond . All these non-Gaussian
states are however non-optimal for the detection of a phase space
displacement. The optimal single-mode state for unambiguous estimation of
displacement is the Gottesman-Kitaev-Preskill state TerhalPRA2017GKP  which
however are experimentally challenging to generate. This challenge can
alternatively be mitigated by using a correlated squeezed probe and
continuous-variable measurement Park2022 .

Recently, significant progress has been made in exploiting the advanced hybrid
qubit-continuous-variable (CV) interface for quantum sensing, e.g. as it is
available for quantum dots pop00001 , color centers Schirhagl2014 , trapped
ions KienzlerPRL2016 ; Bruzewicz2019APRtrappedion ; FluhmannHomeNature2019  or
superconducting circuits Wendin2017superconducting ; GuPR2017Superconducting ;
Touzard2019 ; BlaisNatPhys2020superconducting ; SchoelkopfNature2020 ;
ClerkNatPhy2020SuperCon ; Kwon2021Superconduting ; BlaisRMP2021CircuitQED .
Amplification of a displacement through squeezing was recently tested on
trapped ions BurdScience2018SqzAmpDisp . For the phase estimation, Ramsey
interferometry Ramsey1949 ; RiehlePRL1991Ramsey ; CadoretPRL2008Ramsey ;
AriasPRL2018Ramsey  has been used specifically for metrology with trapped ions
Leibfried2004HeisenbergMultipartite  and superconducting circuits
Wang2019HeisenSupercond . In such hybrid systems, a short burst of mechanical
force and electromagnetic radiation exerted by classical fields DegenRMP2017 ;
Brownnutt2015  generate a small displacement in phase space of the weakly
coupled mechanical or microwave oscillator TerhalPRA2017GKP , which can be
estimated indirectly by coupling it to a qubit system.

The strong coupling between the oscillator and the qubit can be accurately
described by the Rabi Hamiltonian which does not invoke the rotating-wave
approximation (RWA) as is done for the conventional Jaynes-Cummings (JC)
Hamiltonian. Such a Rabi Hamiltonian has been shown to enable the preparation
of different CV quantum states Hastrup2019GKP ; Hastrup2020squeezing , the
construction of CV quantum gates ParkNJP2018 , slowing down the decoherence of
CV states Park2021 , conversion of the states between oscillators and qubits
HastrupPRL2022 , and the realization of certain phase transitions hwang15 ;
Cai2021 . These Rabi interactions also appear to be favorable to quantum
interferometry of photon scattering events, termed cat-state spectroscopy as
demonstrated on single qubits on a trapped ion crystal
HempelNatPho2013catspectroscopy ; GilmoreScience2021Displacement . This idea
has been extended to a driven interferometry of a spin-dependance force
Martinez-Garaot2018 , and to a motion echo McCormick2018 . However, a Rabi
interferometer simultaneously estimating both of the conjugate variables of
the oscillator has not been considered yet and is indispensible for
unambiguous sensing of a displacement in an arbitrary direction  in phase
space caused by an unknown mechanism TerhalPRA2017GKP ;
IvanovPhysRevA2020TwoDisplac .

In this work, we present a method for unambiguously estimating the
displacement of an oscillator when the coupling to discrete level systems is
available. This method is based on a hybrid interferometric setup in which the
oscillator is coupled through Rabi couplings NoriRabiRMP2019 ;
SolanoRabiRMP2019  only to a pair of auxiliary two-level systems. It can be
used to measure mechanical force and electromagnetic radiation by estimating
the displacement in an arbitrary direction in phase space of a single-atom
oscillator and microwave cavity mode. This hybrid interferometric scheme has
sensitivity quantified by classical Fisher information (CFI) (for the
definition, see Appendix A), that is quadratically increasing with the
coupling strength, and thus is superior to the JC interferometer which shows a
bound sensitivity. We compare the hybrid interferometric scheme to the non-
interferometric protocols using equivalent resources where the Rabi gates are
used only to prepare single-mode quantum non-Gaussian states
IvanovPRAp2015ForceSensor ; Ivanov2016 ; IvanovPRA2018  and to measure the
signal together with qubit detectors.

We also show that the unwanted interference between the non-commuting Rabi
interactions can be avoided by choosing proper Rabi coupling strengths, or be
actively cancelled through an engineered two-qubit interaction. For a proof-
of-principle experiment, we consider realistic experimental imperfections and
decoherence effects on the estimation precision, under which the superiority
of the interferometer is maintained over all levels of the considered noises.
Interestingly, the dynamical range of the interferometer is wider than the
non-interferometric setups as well. Notably, we find that even a naturally
existing thermal state probe is as useful as a pure state probe, which
alleviates the necessity of cooling the mechanical mode and preparing highly
non-classical states, and thus enhances the experimental feasibility. This
protocol can be readily implemented both in superconducting circuits
Touzard2019 ; BlaisNatPhys2020superconducting ; SchoelkopfNature2020 ;
Kwon2021Superconduting ; ClerkNatPhy2020SuperCon ; BlaisRMP2021CircuitQED  and
trapped ion systems KienzlerPRL2016 ; Bruzewicz2019APRtrappedion ;
FluhmannHomeNature2019 , and be applied to radiometry and magnon sensing
Wang2019 ; Wang2021 ; Wolski2020 ; Lachance-Quirion2020 ; Kristen2019 .

##  II Displacement estimation using Rabi interactions

An important physical quantity that needs to be estimated for precise control
is a short-time external force, which weakly displaces the mechanical motion
of trapped ion LeibfriedRMP2003 ; Biercuk2010 ; Gilmore2017 ; Affolter2020 ,
or microwave field in a cavity ritschRMP2012 ; Wang2019  at a time scale below
the decoherence time. In trapped ions and superconducting circuits, qubits
coupled to an internal cavity field can sense such a small displacement (See
Table I of DegenRMP2017  and references therein).  Here, we focus on the
estimation of the effect of force components on a single oscillator regardless
of the physical mechanisms, described by a complex parameter
$\alpha=\alpha_{\mathrm{r}}+\mathrm{i}\alpha_{\mathrm{i}}$ (with
$\alpha_{\mathrm{r,i}}\in\mathbb{R}$) of an unknown displacement of
$\hat{D}[\alpha]=\exp[\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}]$ (with field
operators $\hat{a}$ and $\hat{a}^{\dagger}$). Without the force or radiation,
we assume that the probe can also be affected by a known phase shift,
represented by a unitary operator
$\hat{\mathcal{R}}[\theta]=\exp[\mathrm{i}\theta\hat{n}]$ where
$\hat{n}=\hat{a}^{\dagger}\hat{a}$. We note that an unknown target mechanical
force generates a change in the momentum as the primary effect, while the
known phase rotation angle $\theta$ is a secondary effect caused by the
simultaneously occurring oscillator free evolution. Together, they create an
unknown displacement in phase space. The rotation effect
$\hat{\mathcal{R}}[\theta]$ can be estimated independently in advance to
calibrate the interferometer for the displacement measurement. On the other
hand, for microwave radiation, the displacement can happen simultaneously in
both the position and momentum quadratures of the electric field during free
evolution. Note that the qubit may not be directly impacted by the external
force, but only coupled to the disturbed field for the read-out, which may add
flexibility to the setup design in the choice of the qubit. In the main text,
we focus on the estimation of the arbitrary displacement, assuming that the
extra pure phase rotation has been a priori estimated. This rotation
estimation is summarized in Appendix B.

In hybrid quantum systems xiang12 ; kotler16 ; MonroeRMP2021 ; Kurizki2015 ;
Lachance-Quirion2020 , e.g. a mechanical or microwave oscillator coupled to
two-level atomic or transmonic systems, the probe and ancilla of different
nature and/or dimensions are interacting and being controlled for the
estimation. Hybrid interferometry  of such systems (as described in Appendix
C) can be studied within the RWA at resonance, i.e. exploiting JC interaction
with Hamiltonian
$H^{\mathrm{JC}}_{\mathrm{int}}=\hat{\sigma}_{+}\hat{a}+\hat{\sigma}_{-}\hat{a}^{\dagger}$
ShoreKnight1993JC ; FinkNature2008 ; schindler2013quantum ; Casanova2010JC ,
where Pauli matrices acting on the qubit space are denoted as
$\hat{\sigma}_{j}$ for $j=\mathrm{x},\mathrm{y},\mathrm{z}$ and
$\hat{\sigma}_{\pm}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=\frac{\hat{\sigma}_{\mathrm{x}}\pm
i\hat{\sigma}_{\mathrm{y}}}{2}}$. Recently, for both trapped ions
KienzlerPRL2016 ; Bruzewicz2019APRtrappedion ; FluhmannHomeNature2019  and
superconducting microwave circuits Touzard2019 ; SchoelkopfNature2020 ;
Kwon2021Superconduting  among many others, the Rabi coupling
$H_{\mathrm{int}}^{\mathrm{Rabi}}=\hat{\sigma}_{\mathrm{x}}\hat{X}_{\theta}$
containing anti-JC term
$H_{\mathrm{int}}^{\mathrm{AJC}}=\hat{\sigma}_{+}\hat{a}^{\dagger}+\hat{\sigma}_{-}\hat{a}$
has become experimentally accessible and precisely controllable, exhibiting
promising applications beyond RWA. Here
$\hat{X}_{\theta}=(\hat{a}\mathrm{e}^{\mathrm{i}\theta}+\hat{a}^{\dagger}\mathrm{e}^{-\mathrm{i}\theta})/\sqrt{2}$
is a generalized quadrature operator of the oscillator with phase $\theta$,
for example the position $\hat{X}=\hat{X}_{0}$ and the momentum
$\hat{P}=\hat{X}_{\mathrm{\mathrm{\pi}}/2}$. For optical experiments, such a
coupling SpillerNJP2006  is currently under development ParkNJP2020Rabi .

The quantum probe and ancilla states as resources for sensing are assumed here
to be initially separable for a fair assessment of the estimation power of the
setups, while correlated probe-ancilla states can be considered for future
extensions. The states of the qubit ancilla are assumed to be  easily
prepared, manipulated, and measured with high precision as is most often the
case in real experimental systems. In contrast, in the experimental platforms
of trapped ions and superconducting cavity circuits, the oscillator mode is
not directly measurable, but only indirectly through the detection of the
coupled qubit system. Non-classical states of the probes can be considered as
a realistic resource to further enhance the estimation protocol, as such
states have been prepared for both trapped ions KienzlerPRL2016 ;
Bruzewicz2019APRtrappedion ; FluhmannHomeNature2019 ; HackerCatNatPho2019  and
superconducting circuits Touzard2019 ; SchoelkopfNature2020 ;
Kwon2021Superconduting . However, for many future schemes with solid-state
optomechanical systems yin13 , the probe oscillator is typically in non-
squeezed Gaussian states at thermal equilibrium, and preparation of pure
states requires cooling of the oscillator to extremely low temperature. In
addition, the non-classical states do not bring beneficial effects to the
estimation by the interferometer, as will be explained later. Therefore, we
focus our study on readily accessible thermal probes and leave non-classical
probes for future investigation.

Figure 1:  Setups to unambiguously estimate a short-time external classical
force $\mathbf{F}$ with an unknown direction in phase space by simultaneous
estimation of parameters $\alpha_{\mathrm{r}}=\operatorname{Re}(\alpha)$ and
$\alpha_{\mathrm{i}}=\operatorname{Im}(\alpha)$ of displacement
$\hat{D}[\alpha]$ using two sets of Rabi interactions
$\hat{R}_{\mathrm{x}}[t\hat{X}]=\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}]$
and
$\hat{R}_{\mathrm{x}}[t\hat{P}]=\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{P}]$
and their inverses  for each displacement component. The same setup can be
used to independently estimate the free-evolution phase $\theta$ of rotation
in phase space $\hat{\mathcal{R}}[\theta]$ as in Appendix B. The oscillator is
initially in a thermal state $\rho^{\mathrm{th}}[\bar{n}]$ with average
excitation number $\bar{n}$, including the oscillator ground state at
vanishing excitation limit $\bar{n}\to 0$. Oscillator heating $\Gamma_{h}$
from coupling to a thermal environment is considered as a threat to all of the
estimation strategies as a practical consideration. The titles of the sub-
figures in colors match those of the curves in Fig. 2. a) A direct-measurement
setup using Rabi interactions and qubit detectors only after the target
displacement as a measurement device (Rabi detector). b) A non-interferometric
setup for a prepare-and-measure strategy where still only a local state of the
oscillator (a compass state) is experiencing the effect of the force. Reset
prepares a new state of the same two-level system. c) A Rabi interferometer
where the ancillary qubits initially prepared in the eigenstate of
$\hat{\sigma}_{\mathrm{y}}$ become entangled to the oscillator by Rabi
interactions and remains so throughout signal encoding of the displacement and
rotation, to be measured by a Rabi detector made of the inverse Rabi
interactions and a qubit detector. Here the qubit dephasing $\Gamma_{d}$ can
additionally deteriorate the estimation precision due to the non-local nature
of the protocol as the qubit ancillary mode is now participating in the
estimation.

We consider three simple and comparable protocols exploiting Rabi interactions
as depicted in Fig. 1: the direct measurement, the non-interferometric
(prepare-and-measure), and the interferometric sensing setups. For an
unambiguous estimation of the displacement, we assume that the rotation
$\hat{\mathcal{R}}[\theta]$ can be pre-estimated separately before the
displacement, as it requires a pre-displaced probe in the oscillator as
discussed in Appendix B. For the analysis, we assume the unknown target
displacement to be weak and thus consider the weak-displacement limit (WDL),
$|\alpha|\ll 1$. These protocols can equivalently be tweaked to estimate the
amplitude $|\alpha|$ and the phase $\arg(\alpha)$ simultaneously by a
different set of Rabi interactions. The direct measurement scheme in Fig. 1a
uses a single-mode probe in a thermal state at various temperatures (including
oscillator ground state) and the Rabi interactions with the qubit ancillas
only to measure the oscillator after the applied force. Here the qubit state
$\left|\psi\right\rangle_{\mathrm{A}_{1,2}}$ can be optimized for the best
performance, which is found to be the eigenstate of
$\hat{\sigma}_{\mathrm{y}}$ in the WDL. The non-interferometric setup in Fig.
1b, on the other hand, uses Rabi interactions both as a part of the
preparation of the oscillator probe state as well as a part of the detection
process. The first set of Rabi interactions prepares a superposition of
coherent states in the oscillator, while the second set of interactions with
the optimal qubit states prepared as the eigenstate of
$\hat{\sigma}_{\mathrm{y}}$ and qubit detection in the energy eigenbasis
comprises a Rabi detector. The reset prepares two-level systems for the second
set. In contrast, for the interferometer in Fig. 1c the qubit ancillas
participate in all stages of the setup from the preparation, during the signal
accumulation, and up to the measurement. We stress that the Rabi
interferometer is the main setup of interest, while the other schemes are
considered mainly for comparison, to clarify if and to what degree the
interferometer and its non-locality are crucial for the enhancement in the
performance.

The degrees of freedom in these setups comprise the probe in the oscillator
$\mathrm{C}$, and the ancillary qubits $\mathrm{A}_{j=1,2}$. The initial
ancillary qubit can be set in any state as
$\left|\psi\right\rangle_{\mathrm{A}_{j}}=c_{\mathrm{e}}\left|\mathrm{e}\right\rangle_{\mathrm{A}_{j}}+c_{\mathrm{g}}\left|\mathrm{g}\right\rangle_{\mathrm{A}_{j}}$
with qubit eigenbasis
$\\{\left|\mathrm{e}\right\rangle_{\mathrm{A}_{j}},\left|\mathrm{g}\right\rangle_{\mathrm{A}_{j}}\\}$,
where the coefficients $c_{\mathrm{e},\mathrm{g}}\in\mathbb{C}$ can be chosen
for the optimal performance of estimation at all stages, and thus are known.
The initial state of the oscillator is assumed universally to be in a
naturally existing thermal state probe
$\rho^{\mathrm{th}}[\bar{n}]=\sum_{n=0}^{\infty}\frac{\bar{n}^{n}}{(\bar{n}+1)^{n+1}}\left|n\right\rangle_{\mathrm{C}}\left\langle
n\right|$, with Fock basis $\left|n\right\rangle_{\mathrm{C}}$ and average
excitation number $\bar{n}$ representing the various levels of the initial
cooling for realistic experiments. The lowest energy state
$\rho^{\mathrm{th}}[0]=\left|0\right\rangle_{\mathrm{C}}\left\langle 0\right|$
for the microwave cavity field and the phononic oscillator is considered as a
limit when the cooling in the preparation stage is ideal.

The CFI with the hypothetical quadrature detector (denoted as $F$, defined in
(19)), and the quantum Fisher information (QFI, denoted as $Q$, defined as
(18)), of Gaussian states (e.g. vacuum/ground state) and thermal state
$\rho^{\mathrm{th}}[\bar{n}]$ set the benchmark in the estimation of single
displacement variables $\alpha_{\mathrm{r}}=\operatorname{Re}(\alpha)$ and
$\alpha_{\mathrm{i}}=\operatorname{Im}(\alpha)$ for $k=\mathrm{r},\mathrm{i}$:

$\displaystyle
Q_{k}[\rho^{\mathrm{th}}[\bar{n}];\alpha_{k}]=F^{x_{\theta}}_{k}[\rho^{\mathrm{th}}[\bar{n}];\alpha_{k}]=4/(1+2\bar{n})$
(1)

(classical benchmark) TerhalPRA2017GKP , also related to the standard quantum
limit. We see that it has a lower value when cooling is not perfect (i.e.
$\bar{n}>0$), even though the number of particles in the probe is larger than
the vacuum/oscillator ground state. In most experimental realizations, the
oscillator cannot be directly measured, and under this condition, we are
forced to use an indirect detection scheme for the estimation, which
inevitably leads to a smaller CFI than the above-stated classical benchmark.

We note that utilizing an input squeezed state in the oscillator
Hastrup2020squeezing  in the setups to be introduced can have a QFI and CFI
larger than the classical benchmark in the estimation of one displacement
component $\alpha_{\mathrm{r}}$, but smaller in the estimation of the
conjugate component $\alpha_{\mathrm{i}}$, and thus it does not help in the
simultaneous estimation of both.

###  II.1 Direct measurement

We first consider the setup in Fig. 1a which we refer to as the direct
measurement approach. In this case, Rabi couplings between the probe and the
ancillas are only applied during the actual measurement.  At the resonance
between the oscillator probe and the ancillary qubit modes, two non-commuting
unitary Rabi interactions
$\hat{R}_{\mathrm{x}}[t\hat{X}]=\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}]\equiv\hat{R}_{\mathrm{X}}$
and
$\hat{R}_{\mathrm{x}}[t\hat{P}]=\exp[\mathrm{i}t\hat{\sigma}^{\prime}_{\mathrm{x}}\hat{P}]\equiv\hat{R}_{\mathrm{P}}$
(prime denotes the second qubit) with dimensionless strength $t$ (a product of
the interaction time and the interaction strength) are applied to transfer the
information about the displacement to the qubit detector. Using this direct-
measurement approach, we can jointly estimate the components
($\alpha_{\mathrm{r}}$ and $\alpha_{\mathrm{i}}$) of the complex displacement
parameter. Each ancilla coupling corresponds to a detection module for the
estimation of $\alpha_{\mathrm{r}}$ ($\alpha_{\mathrm{i}}$) exploiting
$\hat{R}_{\mathrm{X}}$ ($\hat{R}_{\mathrm{P}}$).

We assume here no prior knowledge about the target parameters
$\alpha_{k=\mathrm{r},\mathrm{i}}$, and therefore, we use a fixed ancillary
qubit eigenstate of $\hat{\sigma}_{\mathrm{y}}$ with eigenvalue $\pm 1$:

$\displaystyle\left|\phi^{\mathrm{opt}}\right\rangle_{\mathrm{A}}=2^{-1/2}(\left|\mathrm{e}\right\rangle_{\mathrm{A}}\pm\mathrm{i}\left|\mathrm{g}\right\rangle_{\mathrm{A}})=\left|\pm_{\mathrm{i}}\right\rangle_{\mathrm{A}},$
(2)

which gives the largest values of the CFI among all qubit states in the WDL.
In a setup with a fixed qubit state, the CFI of $\alpha_{\mathrm{r}}$ by the
qubit detection in $\hat{\sigma}_{\mathrm{z}}$-eigenbasis
$\\{\left|\mathrm{e}\right\rangle_{A_{1}}\left\langle\mathrm{e}\right|,\left|\mathrm{g}\right\rangle_{A_{1}}\left\langle\mathrm{g}\right|\\}$
depends generally on the actual value of $\alpha_{\mathrm{r}}$, but approaches
its maximum in the WDL. This behavior is different from the constant behavior
of QFI using a thermal probe for all $\alpha_{k}$ as in (1), which can be
approached only with the prior knowledge of the target value $\alpha_{k}$.
When a thermal probe and the qubit state
$\left|\phi^{\mathrm{opt}}\right\rangle$ are used as the input and the
ancilla, the CFI is reduced for a non-zero $\bar{n}$ for both
$k=\mathrm{r},\mathrm{i}$:

$\displaystyle
F_{k}^{\mathrm{direct}}[\rho^{\mathrm{th}}[\bar{n}];\alpha_{k},t]=\frac{8t^{2}\cos^{2}\left(2\sqrt{2}t\alpha_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k}\right)}{\mathbb{A}_{\mathrm{th}}-\sin^{2}\left(2\sqrt{2}t\alpha_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k}\right)}$
$\displaystyle\stackrel{{\scriptstyle\alpha_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k}\ll
1}}{{\approx}}8t^{2}\left(\mathbb{A}_{\mathrm{th}}^{-1}-8t^{2}\alpha_{k}^{2}\frac{\mathbb{A}_{\mathrm{th}}-1}{\mathbb{A}_{\mathrm{th}}^{2}}\right),$
(3)

as only the thermal effect
$\mathbb{A}_{\mathrm{th}}=\mathrm{e}^{2t^{2}\left(2\bar{n}+1\right)}$ is
monotonously increasing with $\bar{n}$. The maximum value of this CFI is given
at the optimal Rabi strength $t_{\mathrm{opt}}=(4\bar{n}+2)^{-1/2}$ as

$\displaystyle
F_{k}^{\mathrm{direct}}\left[\rho^{\mathrm{th}}[\bar{n}];\alpha_{k},t_{\mathrm{opt}}\right]$
$\displaystyle\stackrel{{\scriptstyle\alpha_{k}\ll
1}}{{\approx}}\frac{4}{\mathrm{e}\left(1+2\bar{n}\right)}+\frac{16\alpha_{k}^{2}}{\mathrm{e}^{2}\left(1+2\bar{n}\right)^{2}}-\frac{16\alpha_{k}^{2}}{\mathrm{e}\left(1+2\bar{n}\right)^{2}}$
(4)

in the WDL. It establishes a practical benchmark for the following schemes.

The CFI beyond the WDL is also an important measure of performance. We remark
that the CFIs for all examples in this work do not have the odd-order term in
$\alpha_{k}$ due to the symmetry in the sign of the signal, and thus the
second-order term is the lowest order dependence on $\alpha_{k}$, as is
evidenced in (3). The rate by which a CFI changes with $\alpha_{k}$ can be
found by the normalized curvature in the WDL as

$\displaystyle\mathbb{C}=-\left.\dfrac{\partial^{2}F_{k}[\alpha_{k}]}{\partial\alpha_{k}^{2}}\right|_{\alpha_{k}=0}(2F_{k}[\alpha_{k}=0])^{-1}.$
(5)

For the CFI in (3), the rate is found to be $\mathbb{C}=8t^{2}$ for a large
Rabi strength $t\gg 1$ exhibiting a narrower dynamic range for a large $t$
regardless of $\bar{n}$. The half-width-at-half-maximum (HWHM) of CFI is a
measure of the dynamic range, given by

$\displaystyle\alpha_{k}^{\mathrm{(HWHM)}}=\frac{\operatorname{arcsec}\left[2\mathbb{A}_{\mathrm{th}}-1\right]}{4\sqrt{2}t}\stackrel{{\scriptstyle
t\gg 1}}{{\to}}\frac{\mathrm{\mathrm{\pi}}}{8\sqrt{2}t}.$ (6)

The CFI in (3) is zero at multiple values of $\alpha_{k}$, and (6) is located
at the half of the one closest to $0$ at
$\alpha_{k}=\frac{\mathrm{\mathrm{\pi}}}{4\sqrt{2}t}$. At the optimal strength
$t=t_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{opt}}$,
it becomes
$\alpha_{k}^{\mathrm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(HWHM)}}}\approx
0.45\sqrt{1+2\bar{n}}$. The product of the maximum CFI at WDL in (3) and the
square of the HWHM in (6) to be compared with other schemes as a way to show
the trade-off relation is given as
$\frac{\mathbb{A}_{\mathrm{th}}^{-1}}{4}\operatorname{arcsec}^{2}\left[2\mathbb{A}_{\mathrm{th}}-1\right]\stackrel{{\scriptstyle
t\gg
1}}{{\approx}}\frac{\mathrm{\pi}^{2}}{16}\mathrm{e}^{-2t^{2}(2\bar{n}+1)}$,
asymptotically approaching $0$.  It shows that an increase in maximum CFI is
always compensated by a smaller dynamic range. At optimal strength
$t=t_{\mathrm{opt}}$ this product is given as a constant
$\frac{\operatorname{arcsec}^{2}[2\mathrm{e}-1]}{4\mathrm{e}}$ regardless of
$\bar{n}$.

An encompassing figure of merit that reflects both the dynamical range and the
maximum precision is the average CFI. As the CFI is periodic in $\alpha_{k}$
due to the qubit nature of the detection, we can average out the CFI in a
single period of
$\alpha_{k}\in\left[-\frac{\mathrm{\pi}}{4\sqrt{2}t},\frac{\mathrm{\pi}}{4\sqrt{2}t}\right]$,
which gives

$\displaystyle
F^{\mathrm{(av)}}_{k}[\rho^{\mathrm{th}}[\bar{n}];\alpha_{k},t]=\frac{8t^{2}}{\mathbb{A}_{\mathrm{th}}+\sqrt{\mathbb{A}_{\mathrm{th}}^{2}-\mathbb{A}_{\mathrm{th}}}}.$
(7)

This is reduced by increasing $\bar{n}$ and $t$ beyond the optimal strength
$t_{\mathrm{opt^{\prime}}}[\bar{n}]$ different from $t_{\mathrm{opt}}$,
asymptotically approaching $0$, as in Fig. D.1a. These results indicate that a
non-ideal cooling significantly reduces the CFI of the direct measurement
strategy.

When the two parameters, $\alpha_{\mathrm{r}}$ and $\alpha_{\mathrm{i}}$, are
simultaneously estimated (as in Fig. 1 a), the CFI for $\alpha_{\mathrm{r}}$
is still given by (3) while for $\alpha_{\mathrm{i}}$ it acquires a modulation
factor of $\cos^{2}\left[2t^{2}\right]$:

$\displaystyle
F_{\mathrm{i}}^{\mathrm{direct}}[\rho^{\mathrm{th}}[\bar{n}];\alpha_{\mathrm{i}},t]\stackrel{{\scriptstyle
t\gg
1}}{{\rightarrow}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\cos^{2}\left[2t^{2}\right]\frac{8t^{2}\cos^{2}\left(2\sqrt{2}t\alpha_{\mathrm{i}}\right)}{\mathbb{A}_{\mathrm{th}}-\sin^{2}\left(2\sqrt{2}t\alpha_{\mathrm{i}}\right)}.}$
(8)

This modulation factor is arising from the interference of the two non-
commuting Rabi interactions yielding an additional factor $\mathrm{e}^{\pm
it^{2}/2}$ in the coefficients of the hybrid entangled state before the qubit
detection. At $t^{\mathrm{c}}=\sqrt{c\mathrm{\pi}/2}$, where $c\in\mathbb{Z}$,
the modulation factor attains its maximum value and the CFI becomes equal to
that of the independent estimation. However, due to the non-matching $t^{c}$
and $t_{\mathrm{opt}}$, the direct measurement strategy will suffer either
from interference or sub-optimal precision. Alternatively, we can actively
counteract this redundant factor with a two-qubit entangling interaction
$\exp[-\mathrm{i}t^{2}\hat{\sigma}_{\mathrm{x}}\hat{\sigma}_{\mathrm{x}}^{\prime}]$
before or after the Rabi interaction, generating a proper qubit entanglement.
This interaction can be engineered using by a geometric-phase effect of a set
of Rabi interactions as

$\displaystyle\exp[-\mathrm{i}t^{2}\hat{\sigma}_{\mathrm{x}}\hat{\sigma}_{\mathrm{x}}^{\prime}]$
$\displaystyle=\exp[-\mathrm{i}\frac{t}{\sqrt{2}}\hat{\sigma}_{\mathrm{x}}\hat{X}]\exp[-\mathrm{i}\frac{t}{\sqrt{2}}\hat{\sigma}^{\prime}_{\mathrm{x}}\hat{P}]$
$\displaystyle\times\exp[\mathrm{i}\frac{t}{\sqrt{2}}\hat{\sigma}_{\mathrm{x}}\hat{X}]\exp[\mathrm{i}\frac{t}{\sqrt{2}}\hat{\sigma}^{\prime}_{\mathrm{x}}\hat{P}]$
(9)

for any qubit-oscillator state.

###  II.2 Non-interferometric prepare-and-measure setup

An advanced non-interferometric estimation protocol is presented in Fig. 1b.
In this scheme, Rabi interactions with ancillary qubits $\mathrm{A}_{1}$ and
$\mathrm{A}_{2}$ are used both for the conditional oscillator-probe state
preparation and detection, i.e. a prepare-and-measure strategy. The individual
estimation process of the real component of the displacement parameter
$\alpha_{\mathrm{r}}$ described below is identical to the estimation process
of $\alpha_{\mathrm{i}}$ when they are estimated separately for which the Rabi
interactions of $\hat{R}_{\mathrm{P}}$ and $\hat{R}_{\mathrm{P}}^{\dagger}$
are used instead of $\hat{R}_{\mathrm{X}}$ and
$\hat{R}_{\mathrm{X}}^{\dagger}$.

The estimation process is made of three stages: preparation, encoding, and
measurement. In the preparation stage, Rabi interactions and postselection by
the subsequent qubit detectors enable the generation of coherent state
superpositions which are highly sensitive to the displacements
Zurek2001Compass ; munroPRA2001cat . The optimal state is an odd balanced
superposition of coherent states as a result of its large quantum coherence
measured by the total Wigner function negativity AlbarelliPRA2018 . For the
preparation of such probe states, we take
$\left|\mathrm{g}\right\rangle_{\mathrm{A}_{1,2}}$ as the input ancillary
states of both qubits. After the application of the Rabi interactions and the
probabilistic projection onto the qubit excited state
$\left|\mathrm{e}\right\rangle_{\mathrm{A}_{1,2}}\left\langle\mathrm{e}\right|$,
we get a superposition of four coherent states (known as a compass state
ZurekNature2001 ). If a projection onto the qubit ground state
$\left|\mathrm{g}\right\rangle_{\mathrm{A}_{1,2}}\left\langle\mathrm{g}\right|$
occurs in any one of the two-qubit detection, the estimation precision is
slightly reduced. The remaining part of the setup - encoding and detection -
is identical to the setup for the direct estimation discussed in the previous
section. Interestingly, the prepare-and-measure and direct measurement setups
require only sequential interactions with the qubit ancilla achievable by a
single ancilla and therefore give an advantage in the experimental
implementation.

The CFI for the independent estimation of the two parameters using a thermal
state probe can be found as (see Appendix H)

$\displaystyle
F_{k}^{\mathrm{PnM}}[\rho^{\mathrm{th}}[\bar{n}];\alpha_{k},t]=\frac{8t^{2}\cos^{2}\left[2\sqrt{2}t\alpha_{k}\right]}{4\mathbb{B}_{\pm}-\sin^{2}\left[2\sqrt{2}t\alpha_{k}\right]}$
$\displaystyle\stackrel{{\scriptstyle\mathrm{WDL}}}{{\approx}}\frac{2t^{2}}{\mathbb{B}_{\pm}}-\frac{4(4\mathbb{B}_{\pm}-1)t^{4}\alpha_{k}^{2}}{\mathbb{B}_{\pm}^{2}}$
(10)

where the non-monotonous thermal effect for the prepare-and-measure strategy
$\mathbb{B}_{\pm}=\left(1\pm\mathrm{e}^{-t^{2}\left(1+2\bar{n}\right)}-\mathrm{e}^{-2t^{2}\left(1+2\bar{n}\right)}\pm\mathrm{e}^{-3t^{2}\left(1+2\bar{n}\right)}\right)^{-2}\stackrel{{\scriptstyle
t\gg 1}}{{\to}}1$. Here the signs $\pm$ refer to those of the superposition of
coherent states that depend on the outcome of the qubit detection at the
preparation stage. This thermal effect can be saturating faster for a large
$\bar{n}$ compared to the ground state as
$\mathbb{B}_{\pm}\stackrel{{\scriptstyle\bar{n}\gg 1}}{{\to}}1$. The CFI in
the WDL in (10) has a non-monotonous behavior against $t$ as shown in Fig.
2a,b. An analysis of such a non-monotonous behavior is summarized in Appendix
F. We note that this CFI of the prepare-and-measure strategy in (10) has a
larger value than the maximal CFI of the direct measurement strategy for
strengths $t\gtrsim 1.21$. The average CFI of the prepare-and-measure strategy
is found to
$F^{\mathrm{(av)}}_{k}[\rho^{\mathrm{th}}[\bar{n}];\alpha_{k},t]={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}8t^{2}-\frac{4\sqrt{4\mathbb{B}_{\pm}-1}t^{2}}{\sqrt{\mathbb{B}_{\pm}}}}\stackrel{{\scriptstyle
t\gg 1}}{{\to}}4(2-\sqrt{3})t^{2}$, and is larger than the maximum average CFI
by the direct measurement strategy for $t\geq 0.879$. The normalized curvature
is
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{C}=\frac{2(4\mathbb{B}_{\pm}-1)t^{2}}{\mathbb{B}_{\pm}}\stackrel{{\scriptstyle
t\gg 1}}{{\to}}6t^{2}}$, slightly lower than what was obtained for the direct-
measurement approach, implying a wider dynamic range. The HWHM of the CFI is
given asymptotically as
$\alpha_{k}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{\mathrm{(HWHM)}}}\stackrel{{\scriptstyle
t\gg
1}}{{\approx}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{\arctan\left[\frac{2}{\sqrt{3}}\right]}{2\sqrt{2}t}}\ignorespaces$,
slightly increased compared to the direct measurement case. This implies that
the dynamical range is slightly extended as well. For small displacements, a
product of the square of HWHM of CFI and Fisher information in (10) is a
constant in the asymptotic $t$ limit given approximately as
$\frac{1}{4}\arctan^{2}\left[\frac{2}{\sqrt{3}}\right]\ignorespaces$;
therefore, they form a trade-off between the maximum CFI and the dynamical
range, in contrast to monotonously decreasing product of the direct
measurement strategy.

Interestingly, a thermal input state is not affecting the estimation protocol
by a large degree, as the asymptotic expression for the CFI for $t\gg 1$ is
given by $2t^{2}$ regardless of $\bar{n}$ (see Fig. D.1 b). This observation
implies that the oscillator does not have to be pre-cooled before the
estimation protocol to access the asymptotic quadratic trend. The dependence
of the modulation of CFI on $\alpha_{k}$ is the same as in (10), linked to the
robustness to the initial thermal noise. We can interpret this robustness as
being born out of the robustness of the qubit detectors to excess noise. This
can be partially evidenced by considering a coherent state
$\left|\beta\right\rangle$, whose CFI is asymptotically given by
$F^{\mathrm{PnM}}_{\alpha_{k}}[\left|\beta\right\rangle\left\langle\beta\right|;t]\stackrel{{\scriptstyle
t\gg 1}}{{\approx}}2t^{2}$ regardless of $\beta$. Therefore, all the
superpositions and mixtures of them have the same scaling. We also numerically
verified that the same scaling is obtained when states from the Fock basis are
randomly mixed into the probe.

The QFI for thermal states is given by (see Appendix I)

$\displaystyle Q_{k}^{\mathrm{PnM}}[\rho^{\mathrm{th}}[\bar{n}];\alpha_{k},t]$
$\displaystyle=\frac{\left(8t^{2}+4\right)\mathbb{A}_{\mathrm{th}}^{1/2}+8\bar{n}\left(\mathbb{A}_{\mathrm{th}}^{1/2}-1\right)-4}{\mathbb{A}_{\mathrm{th}}^{1/2}-1}\stackrel{{\scriptstyle\bar{n}\ll
1}}{{\approx}}4+8t^{2}+\frac{8t^{2}}{\mathrm{e}^{t^{2}}-1}.$ (11)

This is independent of $\alpha_{k}$ in contrast to the CFI, as the optimal
detector may implicitly use the knowledge about the true value of the
estimation target. Interestingly, this QFI is increasing with the average
boson number $\bar{n}$ without a bound. In addition, it has a larger value
than the QFI of the Rabi interferometer in (14). This result implies a
prospect of a better detection scheme that can surpass the Rabi
interferometer.

Now if we estimate $\alpha_{r,i}$ simultaneously, we arrive at a more
complicated form of the CFI (see (30) in Appendix H than those of the
individual component estimations in (10), as each displacement component
interferes with the estimation of the other component. The difference between
individual and simultaneous estimation is compared in Fig. H.1 of Appendix H.
We note that the CFI of $\alpha_{\mathrm{r}}$ has again an asymptotic
modulation factor $\cos^{2}[2t^{2}]$ arising from the interference of two non-
commuting Rabi interferometers, which is universal for all true values of
$\alpha_{\mathrm{r}}$. This factor again disappears at
$t^{\mathrm{c}}=\sqrt{c\mathrm{\pi}/2}$, or can be actively cancelled by (9)
or a proper qubit entangled state preparation.

###  II.3 Rabi interferometer

We finally consider the interferometric estimation scheme illustrated in Fig.
1 c), where qubit ancilla are interacting unitarily with the oscillator by
Rabi couplings before and after the unknown displacement (and/or phase
rotation) before the final qubit detection. For a fixed and known phase-space
variable, the Rabi interferometric scheme was already carried out
experimentally and thus proves its feasibility HempelNatPho2013catspectroscopy
. In what follows, we first describe the scheme for the estimation of the
individual parameters and then move on to the scheme for the simultaneous
estimation of the parameters. The inverse Rabi interaction
$\hat{R}_{\mathrm{X}}^{-1}$ or $\hat{R}_{\mathrm{P}}^{-1}$ applied before the
signals can be engineered by either $\mathrm{\pi}$-phase rotations of the
oscillator, or by using additional Rabi interactions with a strong drive at
the opposite phase as discussed in Appendix J. The total unitary
transformation of the interferometer up to the qubit detection is described as

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{U}_{\mathrm{interf}}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{R}_{\mathrm{P}}}\hat{R}_{\mathrm{X}}\hat{D}[\alpha]\hat{R}_{\mathrm{X}}^{-1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{R}_{\mathrm{P}}^{-1}}$
$\displaystyle=\hat{D}[\alpha]\hat{R}_{\mathrm{x}}[\sqrt{2}\alpha_{\mathrm{r}}t]{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{R}^{\prime}_{\mathrm{x}}[\sqrt{2}\alpha_{\mathrm{i}}t]},$
(12)

and is composed of a signal oscillator displacement and qubit rotations
$\hat{R}_{\mathrm{x}}[\phi]=\exp[\mathrm{i}\phi\hat{\sigma}_{\mathrm{x}}]$
whose angles are proportional to the displacement components
$\alpha_{\mathrm{r}}$ and $\alpha_{\mathrm{i}}$ (we used the same notation
with Rabi interaction which is a form of a controlled qubit rotation).

The CFI associated with the estimation of $\alpha_{k}$ is found to be (see
Appendix K)

$\displaystyle
F^{\mathrm{interf}}_{k}[\rho^{\mathrm{th}}[\bar{n}];\alpha_{k},t]=8t^{2},$
(13)

surpassing the prefactor of the asymptotic scaling, $2t^{2}$, of the non-
interferometric setups in (10). Note that the scaling in Eq. (13) holds
regardless of the true value of $\alpha_{k}$. This aspect overcomes all
limiting trade-offs between CFI and measurement range for the direct and
prepare-and-measure method in the absence of noise.  Moreover, it beats the
maximal CFI for the direct measurement strategy for $t=0.429$, the benchmark
of (4) at $t=0.707$, and the prepare-and-measure strategy for all $t$. It
approaches asymptotically for large $t$ the scaling of practically
inaccessible CFI in Eq. (33) that uses the quadrature detection on the
oscillator, or QFI in (14) that may require infeasible detectors. Remarkably,
the CFI in eq. (13) holds for any non-pure probe state, $\rho$, which is
initially separable from the input qubit e.g. a thermal state
$\rho^{\mathrm{th}}[\bar{n}]$.

The CFI in (13) guarantees in principle no sensitivity bias within the
dynamical range of the Rabi interferometer. When the data is finite, the
dependence on $\alpha_{k}$ appears around the values
$\alpha_{k}=\pm\frac{\pi}{4\sqrt{2}t}$ due to the statistical fluctuation or
noise, which will be more visible on Fig. 2. This independence of CFI to
$\alpha_{k}$ arises due to the implied local estimation method.  There exists
one practical limitation resulting from the periodicity of the qubit rotations
$\hat{R}_{\mathrm{x}}[\sqrt{2}\alpha_{k}t]$ over $\alpha_{k}$ and thus the
probabilities $P_{g}(\alpha_{k})$ and $P_{e}(\alpha_{k})$ if a global
estimation of arbitrary strength displacement $\alpha_{k}$ is tried. This
periodicity narrows the range
$\alpha_{k}\in[-\frac{\mathrm{\pi}}{2\sqrt{2}t},\frac{\mathrm{\pi}}{2\sqrt{2}t}]$
in which unique estimation can be performed. The higher precision in
estimation is therefore achieved at a cost of a smaller dynamical range. This
type of trade-off between precision and range of estimation is a rather common
trait in many sensing protocols (see e.g. Bonato2016 ; DegenRMP2017 ;
HerbschlebNatComm2021DynamicRange ). However, we note that estimation of the
target values beyond this range is possible by combining two setups with
slightly different coupling strengths. For an example, see Appendix M.

Only for comparison, we also compute the QFI of the interferometric scheme
which is found by considering the state of the system right after the signal
has been imposed and thus allowing for an optimal detection strategy. We find

$\displaystyle
Q_{k}^{\mathrm{interf}}[\rho^{\mathrm{th}}[\bar{n}];t]=8t^{2}+\frac{4}{1+2\bar{n}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\stackrel{{\scriptstyle
t,\bar{n}\gg 1}}{{\to}}8t^{2}}.$ (14)

which is seen to approach the CFI for $t,\bar{n}\gg 1$. The extra term in the
expression for the QFI, in addition to the quadratic scaling that also appears
in the CFI, stems from the residual information contained in the oscillator
state and which is not measured in the interferometric setup. This
contribution, however, becomes insignificant for large $t$ or large $n$. We
note that an additional Rabi detector can be added after the interferometric
setup to fill the gap between QFI and CFI partially, as the oscillator is
still displaced after the interferometer. For example, with a single
additional Rabi detector after the interferometer, we get the sum of the CFIs
of the interferometer (13) and the optimal direct measurement (3)
$F_{k}[\rho^{\mathrm{th}}[\bar{0}];\alpha_{k},t]\approx
8t^{2}+\frac{4}{\mathrm{e}}-\frac{16\alpha_{k}^{2}}{\mathrm{e}}(1-\mathrm{e}^{-1})$.

We also note that the ancillary state residing in the equator subspace of
$\hat{\sigma}_{\mathrm{x}}$-basis can be chosen freely as they all lead to the
same CFIs and QFIs. However, for certain environmental noise sources, some
choices are more beneficial than others as will be discussed in Sec. III.

The interferometric scheme can be straightforwardly extended to include many
qubit ancillas interacting independently and simultaneously through Rabi
interactions with the oscillator for enhanced precision. Using such a scheme,
the CFI can be increased to $F^{\left|\mathrm{g}\right\rangle^{\otimes
m}}_{k}[\rho^{\mathrm{th}}[0];\alpha_{k},t]=8t^{2}m$, where $m$ is the number
of qubit ancillas. It can be simply understood from the independent
probabilities for each ancilla (see (41) in Appendix K), as each joint
application of $\hat{R}_{\mathrm{x}}$ and $\hat{R}_{\mathrm{x}}^{-1}$ with
each ancilla adds $8t^{2}$ to the CFI. In Appendix L, we briefly investigate a
further extension to the scheme by including the generation of input qubit-
oscillator entangled probe using Rabi interactions. This preliminary study
shows a promising further enhancement, although a more thorough investigation
is needed for the full exploration of such a possibility. We also remark that
exploiting higher-order Rabi interactions engineered from multiple
applications of Rabi interactions can enhance the precision further as
discussed in Appendix N.

##  III Robustness of Rabi interferometry

Qubit dephasing.— Dephasing error on qubit ancilla is described by a map
$\Gamma_{\mathrm{d}}^{(p)}[\rho]=(1-\frac{p}{2})\rho+\frac{p}{2}\hat{\sigma}_{\mathrm{z}}\rho\hat{\sigma}_{\mathrm{z}}$
with dephasing parameter $p\in[0(\text{no dephasing}),1(\text{complete
dephasing})]$. It occurs simultaneously with the signal displacement
Bruzewicz2019APRtrappedion ; Kjaergaard2020SuperCondQub , thereby introducing
a degradation in the performance which can be only avoided by using error
correction. As a result of qubit dephasing, the CFI of the Rabi interferometer
changes to

$\displaystyle
F_{k}^{\mathrm{\Gamma_{d}^{(p)}}}[\rho^{\mathrm{th}}[0];\alpha_{k},t]=\frac{8t^{2}\cos^{2}\left(2\sqrt{2}t\alpha_{k}\right)}{4\mathbb{D}-\sin^{2}\left(2\sqrt{2}t\alpha_{k}\right)}$
$\displaystyle\stackrel{{\scriptstyle\alpha_{k}\ll
1}}{{\approx}}\frac{2t^{2}}{\mathbb{D}}-\frac{4(4\mathbb{D}-1)t^{4}}{\mathbb{D}^{2}}\alpha_{k}^{2}.$
(15)

where we assume that the initial qubit state is
$\left|\pm_{\mathrm{i}}\right\rangle$. The dephasing effect is given by the
parameter $\mathbb{D}=\left(2-p(1+\mathrm{e}^{-4t^{2}})\right)^{-2}$ attaining
the value $1/4$ for no dephasing and $1$ for complete dephasing in the
asymptotic limit $t\gg 1$. The maximum value of the CFI in ( 15) has an
asymptotic scaling, $2t^{2}(2-p)^{2}$, that is reduced to the scaling of
prepare-measure protocols at complete dephasing, $2t^{2}$ . Here again, the
CFI is monotonous with $t$ regardless of $p$ and $\alpha_{k}$ in the
asymptotic limit. The normalized curvature is found to be
$\mathbb{C}=\frac{2(4\mathbb{D}-1)t^{2}}{\mathbb{D}}$ which is zero for no
dephasing and attains its maximum asymptotic value of $6t^{2}$ for complete
dephasing, thereby coinciding with the curvature of the prepare-and-measure
strategy. For values of $\alpha_{k}$ beyond the WDL, the dependency of the CFI
on $t$ is non-monotonous, having multiple local optimal values for $t$ that
depend on the true value of $\alpha_{\mathrm{r}}$. This CFI has its first zero
at $\alpha_{k}=\frac{\mathrm{\pi}}{4\sqrt{2}t}$ regardless of $p$, while the
HWHM is found to be

$\displaystyle\alpha_{k}^{\mathrm{(HWHM)}}=\tan^{-1}\left[\sqrt{\frac{4\mathbb{D}}{4\mathbb{D}-1}}\right]$
$\displaystyle\stackrel{{\scriptstyle p\ll
1}}{{\approx}}\frac{\mathrm{\pi}}{4\sqrt{2}t}-\frac{\sqrt{p}\sqrt{\mathrm{e}^{-4t^{2}}+1}}{2\sqrt{2}t},$
(16)

which becomes narrower for larger values of $t$ and $p$. This dynamic range is
approximately twice as large as that of the prepare-and-measure strategy.  The
product of the CFI at WDL and the square of HWHM is given at asymptotic $t$ as
$\frac{1}{4}(2-p)^{2}\arctan^{2}\left[\frac{2}{\sqrt{4-p}\sqrt{p}}\right]$
independent of $t$, again showing a trade-off between the dynamic range and
maximum CFI for a fixed $p$. This product is larger than the prepare-and-
measure strategy for all $p$.

The average CFI is given as

$\displaystyle
F_{k}^{\mathrm{(av)}}[\rho^{\mathrm{th}}[\bar{0}];\alpha_{k},t]={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}8t^{2}-4t^{2}\frac{\sqrt{4\mathbb{D}-1}}{\sqrt{\mathbb{D}}}}$
$\displaystyle\stackrel{{\scriptstyle t\gg
1}}{{\rightarrow}}8t^{2}-4t^{2}\sqrt{(4-p)p}.$ (17)

Without dephasing, this expression reduces to $8t^{2}$ as previously found,
while at complete dephasing, it becomes $\frac{4t^{2}}{2+\sqrt{3}}$, which is
equal to the asymptotic average value of CFI for the prepare-and-measure
strategy. This equality to the prepare-and-measure strategy arises due to the
mathematical equivalence of the complete qubit dephasing and the projective
measurement onto the qubit energy eigenbasis. We note that the same equations
(15-17) hold for the thermal state probes, as the qubit dephasing only
linearly transforms the qubit detection probabilities from those in the ideal
setup, which are the same for all probe states of the oscillator.

The CFI for a single displacement component estimation by a single qubit
ancilla in (15) is different from the CFIs of joint estimation of the
displacement parameters when two-qubit ancillas are used as in Fig. 1c under
the existence of dephasing and other noises. The same modulation factor
$\cos^{2}[2t^{2}]$ is multiplied to the CFI of one of the displacement
components as in the unambiguous estimation by the prepare-and-measure
strategy (see Appendix H), which can be negated at the same specific Rabi
strengths $t_{c}=\sqrt{c\frac{\mathrm{\pi}}{2}}$ for $c\in\mathbb{Z}$.
Similarly as for other estimation methods, an active correction by (9) is
possible.

Figure 2: a) Classical Fisher information re-scaled by the quadratic scaling
of an ideal Rabi interferometer in (13) about the individual estimation of
$\alpha_{\mathrm{r}}$ (or equally $\alpha_{\mathrm{i}}$) for Rabi
interferometer (RI, blue) vs. JC interferometer (JC, black), prepare-and-
measure strategy (PnM, red), direct estimation (DM, gray) at WDL. The
oscillator in a ground state under various qubit dephasing parameter $p$,
where CFI of the interferometer is reduced to that of the prepare-and-measure
strategy at a large $t$ for a complete dephasing $p=1$. b) The relative CFI
under oscillator heating stronger than known experimental parameters
equivalent to the setup from Ballance2016 . We can see that the Rabi
interferometer surpasses the JC interferometer more than twice even at a
vanishing strength, direct Rabi measurement, and prepare-and-measure Rabi
strategies even under an infinite level of heating noises. c, d) The CFI
beyond WDL under the influence of qubit dephasing and oscillator heating at
$t=\sqrt{3\mathrm{\pi}/2}$ at which the harmful interference from the
simultaneous estimation can be avoided. The dip in the CFI common to the Rabi
interferometers and the prepare-and-measure setups correspond to the
parameters at $\alpha_{k}=\frac{\mathrm{\pi}}{4\sqrt{2}t}$ arising from the
periodicity of the qubit detection, the width of which gets narrower as $t$ is
increased. This dip appears due to the fragility of the estimation when one of
the qubit detection probability is $0$ in deal case without noise. Beyond the
values shown here the CFIs are periodic, even though the results are
inconclusive from the periodicity of the data. The JC interferometer has a
broad dynamical range, however, being unable to sense the conjugate
displacement. The CFI by the direct measurement is small on this scale and is
not shown.

In Fig. 2a, we compared the CFI of the Rabi interferometer in the WDL under
qubit dephasing, with the noise-free JC interferometer (addressed in Appendix
P), the noise-free direct-measurement, and prepare-and-measure strategies. We
find that the Rabi interferometer is superior to the ideal prepare-and-measure
strategy for all dephasing levels, while they converge asymptotically under
complete dephasing. The supremacy is more pronounced over the noise-free JC
interferometer. Even at a vanishing strength, the Rabi interferometer is twice
more precise, and the enhancement is larger for a larger strength.  For
$p\lesssim 0.7$ it is superior to the direct measurement for $t\gtrsim 0.71$
and to the JC interferometer for $t\gtrsim 1$. When considering an input
oscillator in a thermal state we attain similar comparative trends between the
different schemes as for the pure state case. The CFI beyond the WDL is shown
in Fig. 2c as a function of $\alpha_{k}$ for a fixed $t$. It is clear that in
this case, the interferometer is superior to all other methods for all levels
of dephasing $p$.

Oscillator heating.— All mechanical oscillators (e.g. trapped-ion motion
LeibfriedRMP2003 ; Affolter2020 ; Gilmore2017 ; Wang2019 ; Biercuk2010 ) are
inevitably coupled to a thermal bath and experience heating. This heating
denoted as $\Gamma_{\mathrm{h}}[\rho]$ for an arbitrary initial oscillator
density matrix $\rho$, can be described by the solution to the Lindblad
equation $\partial_{t_{\mathrm{th}}}\rho=\sum_{i=1,2}L_{\mathrm{i}}\rho
L_{\mathrm{i}}^{\dagger}-\frac{1}{2}\\{L_{\mathrm{i}}^{\dagger}L_{\mathrm{i}}\rho\\}$
with Lindblad operators
$L_{1}=\sqrt{\gamma^{\mathrm{th}}}\sqrt{\bar{n}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{\mathrm{th}}}+1}\hat{a}$
and
$L_{2}=\sqrt{\gamma^{\mathrm{th}}}\sqrt{\bar{n}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{\mathrm{th}}}}\hat{a}^{\dagger}$
Barnett2003 . The effect of a thermal channel is fully characterized by the
average number of the thermal photon in the bath
$\bar{n}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{\mathrm{th}}}$,
the heating rate $\gamma^{\mathrm{th}}$, and the heating time
$t_{\mathrm{th}}$. Here, the heating parameters for scaling are chosen as in
Ref. Ballance2016  where the oscillator ground state was gaining an additional
excitation of $\Delta_{\bar{n}}=1.1\times 10^{-4}$ in a single gate slot,
which approximately corresponds to
$\bar{n}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{\mathrm{th}}}=1$,
$\gamma^{\mathrm{th}}=0.005\equiv\gamma_{0}$, and $t_{\mathrm{th}}=0.01$.
Numerical analysis shows that the CFI of an interferometer is monotonously
decreasing with all these variables
$\bar{n}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{\mathrm{th}}}$,
$\gamma^{\mathrm{th}}$ and $t_{\mathrm{th}}$. In Fig. 2 b, we plot CFI for the
Rabi interferometer associated with different strengths for the heating
effect, and we see that this interferometer is superior to the two other
estimation strategies even under strong oscillator heating effects.

In Fig. 2d, we plot the CFI beyond the WDL under various oscillator heating
strengths, and we see the superiority of the Rabi interferometer over other
methods that are assumed to be free of any decoherence.  The analysis for the
input thermal state under the same environments shows the equivalent
superiority over other benchmarks, as for vacuum/ground state oscillator in
Fig. 2b and d despite the initial thermal noise. This robustness to thermal
noise proves a clear-cut advantage of the Rabi interferometric setup over
other alternatives. Other minor sources of noise are considered in Appendix Q.

Interestingly, the formulas (15-17) can be used to model the effect of an
oscillator heating on CFI by replacing $p$ by a heuristic parameter
$p^{\mathrm{th}}=1-\mathrm{e}^{-ct^{2}\gamma^{\mathrm{th}}}$ with $c=0.159$.
This universality of the CFI by dephasing formula arises because the noises or
interference in the joint estimation linearly mixes the qubit detection
probabilities which has the same mathematical description as the dephasing
model.

We briefly comment on the similarity of the formulas for the CFI in (3), (10),
and (15). This originates from the qubit nature of the detection module. For
example, if the probabilities of the qubit detection outcome in the excited
state is given by
$P_{\mathrm{e}}=\frac{1}{2}(1+\mathbb{E}\sin^{2}[\sqrt{2}t\alpha_{k}])$ for
$\mathbb{E}\in[-1,1]$ after a given process, the CFI is
$F_{k}[\alpha_{k}]=\frac{8t^{2}\cos^{2}\left[2\sqrt{2}t\alpha_{k}\right]}{\mathbb{E}^{-2}-\sin^{2}\left[2\sqrt{2}t\alpha_{k}\right]}$.
Now the largest value of the CFI is attained at $\mathbb{E}=\pm 1$, which
produces the same scaling for CFI with $t$ as in the ideal Rabi
interferometer. This proves the optimality of the Rabi interferometer among
the estimation setups using qubit ancillas. This is further discussed in
Appendix Q.

##  IV Discussion and outlook

In this work, we proposed a Rabi interferometric estimation protocol for the
unambiguous estimation of phase space displacements in an unknown direction of
a mechanical oscillator or an electromagnetic field, e.g. represented by the
motional state of a trapped ion KienzlerPRL2016 ; Bruzewicz2019APRtrappedion ;
FluhmannHomeNature2019  or the microwave field of a superconducting circuit
Wendin2017superconducting ; Touzard2019 ; BlaisNatPhys2020superconducting ;
SchoelkopfNature2020 ; Kwon2021Superconduting ; ClerkNatPhy2020SuperCon ;
BlaisRMP2021CircuitQED , and in future, potentially also by light
NoriRabiRMP2019 ; SolanoRabiRMP2019 . It provides a simple and robust approach
to the detection of weak forces with a sensitivity that goes beyond what is
possible with other approaches that are exploiting similar resources. It is
based on an experimentally feasible approach that has been proven in previous
experiments on Rabi interferometry HempelNatPho2013catspectroscopy ;
PenasaPRA2016 ; GilmoreScience2021Displacement , and even for small
interaction strengths, it beats the state-of-the-art interferometer based on
the standard JC coupling with a bounded CFI in which the RWA holds. We
explored the robustness of the Rabi interferometer to various realistic noise
models, and notably, we found that the thermal occupation of the probing
oscillator is not critical.

The CFI for both the interferometric and prepare-and-measure setups are
scaling quadratically with the Rabi coupling strength, while the former has a
4-fold increase in contrast to the latter. The enhancement is maintained for
all levels of qubit dephasing and oscillator heating, and thus shows an
absolute superiority of the interferometric scheme. Notably, the dynamic range
of the interferometer is limited only by the periodic nature of the qubit
detector that can be overcome and is superior to those of the other methods.
In all strategies investigated here, the harmful interference that arises in
the simultaneous estimation can be avoided by choosing proper Rabi strengths
where the performance is equal to the individual estimation.

These protocols can be further extended to the direct detection of the
oscillators together with the qubit detectors as summarized in Appendix I, or
even composite setups of Rabi interferometer and additional Rabi detector and
engineered high-order Rabi interferometers as in Appendix N. The non-
interferometric setups are benefiting the most from the detection of the
oscillator mode, and might even surpass the performance of the interferometric
setup when the noise is absent. In the future, this scheme can be further
extended to the interferometric setups exploiting entangled qubit ancillas or
qudit ancillas (a simple example has been analyzed in Appendix L). The
application of this scheme can be extended to the estimation of general
Gaussian unitary or non-unitary channels HolevoBook2019Channels . Other types
of nonlinear interactions on platforms such as optics, and
opto/electromechanics AspelmeyerRMP2014cavoptomech ;
TeufelNature2011electromech  can be used in a similar setup.

## Acknowledgment

KP and RF acknowledge Project 21-13265X of the Czech Science Foundation. PM
acknowledges the funding from the European Union’s Horizon 2020 (2014-2020)
research and innovation framework program under Grant Agreement No 731473
(ShoQC). Project ShoQC has received funding from the QuantERA ERA-NET Cofund
in Quantum Technologies implemented within the European Union’s Horizon 2020
Program. KP, JH, and ULA acknowledge the support from the Danish National
Research Foundation (DNRF142).

##  Appendix A Quantum Fisher information and classical Fisher information

To conclusively assess the sensitivity of the estimation methods by complex
setups to be considered here, we calculate the Fisher informations of them
which can predict the precision of estimation of the signals. The QFI
Paris2009  specifies the upper bound of the precision for the given probe
state for the optimal estimation strategy, whereas CFI is the asymptotically
achievable precision by a specific detection. For a density matrix
$\rho(\mathbf{\Theta})=\hat{\mathbb{U}}[\mathbf{\Theta}]\rho_{\mathrm{in}}\hat{\mathbb{U}}[\mathbf{\Theta}]^{\dagger}$
with an input state density matrix $\rho_{\mathrm{in}}$ and the unitary
evolution up to the point of the encoding of the unknown signal
$\hat{\mathbb{S}}[\mathbf{\Theta}]$ with the unknown parameters
$\mathbf{\Theta}$ written as
$\hat{\mathbb{U}}[\mathbf{\Theta}]=\hat{\mathbb{S}}[\mathbf{\Theta}]\exp[-it\hat{H}_{\mathrm{int}}]$,
the QFI about target variable $\Theta_{j}$ generated by a  Hamiltonian
$\hat{H}_{j}$ is expressed as Paris2009 ; Liu2016 :

$\displaystyle
Q_{j}[\rho_{\mathrm{in}};\Theta_{j}]=2\sum_{k,l}\frac{(\lambda_{k}-\lambda_{l})^{2}}{\lambda_{k}+\lambda_{l}}|\left\langle
k\right|\hat{H}_{j}\left|l\right\rangle|^{2},$ (18)

where $\lambda_{k}$ and $\left|k\right\rangle$ are eigenvalues and
eigenvectors of $\rho(\mathbf{\Theta})$ and the summation is only for
$\lambda_{k}+\lambda_{l}\neq 0$.

The CFI about a variable $\Theta_{j}$ in hybrid setups with a specific
discrete detector can be calculated as

$\displaystyle
F_{j}[\rho_{\mathrm{in}};\Theta_{j}]=\sum_{n}P(n;\mathbf{\Theta})^{-1}(\partial_{\Theta_{j}}P(n;\mathbf{\Theta}))^{2}.$
(19)

Here, a discrete detection setup consisting of POVM elements
$\\{\hat{\mathbf{\mathrm{\Pi}}}[n]\\}$ satisfies
$\sum_{n=1}^{N}\hat{\mathbf{\mathrm{\Pi}}}[n]=\hat{\mathbf{1}}$ where the
index $n$ represents the measurement outcomes from the discrete detectors. For
a continuous detection, integration substitutes for the summation.  The CFI
implies common assumptions of local estimation strategy, meaning it only
discerns the difference of $\mathbf{\Theta}$ and
$\mathbf{\Theta}+d\mathbf{\Theta}$. It also assumes average of infinite number
of probes, where the frequency of the data can be considered as probabilities.
The detection on many copies of the probe generates the outcome data forming
an asymptotic probability
$P(n;\mathbf{\Theta})=\operatorname{tr}[\mathbf{\mathrm{\Pi}}[n]\rho^{\prime}(\mathbf{\Theta})]$
from a final density matrix
$\rho^{\prime}(\mathbf{\Theta})=\hat{\mathbb{U}}^{\prime}[\mathbf{\Theta}]\rho_{\mathrm{in}}\hat{\mathbb{U}}^{\prime}[\mathbf{\Theta}]^{\dagger}$
where now $\hat{\mathbb{U}}^{\prime}[\mathbf{\Theta}]$ is the unitary
evolution up to the point of detection, equal to $\hat{U}_{\mathrm{interf}}$
in the interferometer in (23). The superscripts over the CFI will be used
throughout the text below to specify the setup where the estimation is
performed. A brief demonstration of the equivalence of the precision predicted
by CFI and that achievable by the maximum likelihood estimation in the
asymptotic infinite copy limit applicable to the examples in the main text
using a binary outcome detection $N=2$ is summarized in Appendix E.

We remark that the maximization of the CFI over all quantum POVM measurements
can reach the QFI Paris2009  in principle, but in general practices, the CFI
depends on many experimental parameters, such as noises, the ancillary state,
and even the true value of the target parameter. However, as we cannot fully
know the target parameter beforehand of the estimation, the choice of the
setup is inevitably based on certain assumptions about it. A brief strategy to
adapt the setup based on the partial knowledge about the target parameter was
described in Appendix G, but it is not of the main interest. Here, the main
assumption throughout the paper is the weakness of the force, i.e. the
displacement is very weak, i.e. $|\alpha|\ll 1$, as it is the region of the
largest impact and the experimental difficulty.

We note that the QFI in 14 is comparable to the QFI for the prepare-and-
measure strategy as discussed in Appendix I. This is to be expected due to the
similarity of the prepared states in the oscillator undergoing the signal
displacement when they are considered as the input states for general
measurements.

##  Appendix B Estimation of oscillator phase shift and qubit rotation

Oscillator rotation estimation and oscillator phase noise suppression The
unknown rotation $\hat{\mathcal{R}}[\theta]$ exerted by the force where
$\theta$ is the target parameter can be estimated by the Rabi interferometer
using the adjoint action in place of (12):

$\displaystyle\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}]\exp[\mathrm{i}\theta\hat{n}]\exp[-\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}]$
$\displaystyle=\exp[\mathrm{i}\theta(\hat{n}-t\hat{\sigma}_{\mathrm{x}}\hat{P}-\frac{t^{2}}{2})]\approx\exp[\mathrm{i}\theta\hat{n}]\exp[-\mathrm{i}t\theta\hat{\sigma}_{\mathrm{x}}\hat{P}].$
(20)

Besides the unknown signal rotation after the approximation, the second Rabi
operation after the final approximation represents a momentum-dependent qubit
rotation. Therefore, using a probe state with a large momentum such as
imaginary amplitude coherent state $\left|i\beta\right\rangle_{\mathrm{C}}$
can increase the qubit rotation to be estimated by a qubit detector, similarly
as in the estimation of displacement, where we can replace
$\hat{P}\to\sqrt{2}\beta$ in the last expression of (20), and thus all
properties of the displacement estimation are equally applicable. In the
absence of dephasing, the highest precision can be expected at $\theta=0$. The
CFI by Rabi interferometer under dephasing is given as

$\displaystyle F[\theta]=\frac{2\beta^{2}t^{2}}{\mathbb{D}}\ignorespaces,$
(21)

having a scaling of $8t^{2}\beta^{2}$ for $p=0$. In comparison, the optimal
prepare-and-measure strategy in the rotation estimation by the same probe
state has CFI of
$F^{\mathrm{PnM,\pm,opt}}[\theta]=\frac{2\beta^{2}t^{2}}{\mathbb{B}_{\pm}}\ignorespaces$,
again when the qubit ancilla is set as
$\left|\phi^{\mathrm{opt}}\right\rangle$. From the equivalence to the
estimation of displacement, the Rabi interferometer is superior to the
prepare-and-measure strategy in all cases of $\beta$ and $t$.

On the other hand, if we see the rotation as the noise in the estimation of
the displacement, we can efficiently suppress this oscillator phase noise by
choosing a probe state with a small average momentum of
$\langle\hat{P}\rangle\approx 0$ such as $\left|0\right\rangle_{\mathrm{C}}$,
where now $\hat{\mathcal{R}}[\theta]$ plays as a noise in the estimation of
displacement. In (20), the qubit rotation then becomes effectively $0$. This
suppression implies that displacement and rotation can be independently
estimated by our setups by choosing coherent states in the oscillator
$\left|\mathrm{i}\beta\right\rangle_{c}$ with $\beta\ll 1$ or $\beta\gg 1$.

Qubit rotation estimation and qubit rotation noise  We can consider various
scenarios where qubit rotation in an arbitrary direction is present, either as
the estimation target or as noise. For an estimation of single rotation
component $\Theta_{\mathrm{z}}$, we again get the transformation of the qubit
rotation by the Rabi interferometer:

$\displaystyle\exp[-\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}]\exp[\mathrm{i}\Theta_{\mathrm{z}}\hat{\sigma}_{\mathrm{z}}]\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}]$
$\displaystyle=\exp[\mathrm{i}\Theta_{\mathrm{z}}(\hat{\sigma}_{\mathrm{z}}\cos[2t\hat{X}]-\hat{\sigma}_{\mathrm{y}}\sin[2t\hat{X}])],$
$\displaystyle\exp[-\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}]\exp[\mathrm{i}\Theta_{\mathrm{y}}\hat{\sigma}_{\mathrm{y}}]\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}]$
$\displaystyle=\exp[\mathrm{i}\Theta_{\mathrm{y}}(\hat{\sigma}_{\mathrm{y}}\cos[2t\hat{X}]+\hat{\sigma}_{\mathrm{z}}\sin[2t\hat{X}])].$

When these qubit rotations are estimation targets, we can calculate the Fisher
informations from these equations in equivalent ways as before. In summary, we
can say that the sensitivity in $\Theta_{\mathrm{x}}$ measurement is not
affected, $\Theta_{\mathrm{y}}$ measurement deteriorated,
$\Theta_{\mathrm{z}}$ measurement is enhanced by the Rabi interferometer.

Now we can think of the qubit rotation as noises in the estimation of
displacement. In this case, the total operation is written using Eq. (B) as

$\displaystyle\exp[-\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}]\exp[\mathrm{i}\Theta_{\mathrm{z}}\hat{\sigma}_{\mathrm{z}}]\exp[\mathrm{i}\Theta_{\mathrm{y}}\hat{\sigma}_{\mathrm{y}}]D[\alpha]\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}]$
$\displaystyle=\exp[\mathrm{i}\Theta_{\mathrm{z}}(\hat{\sigma}_{\mathrm{z}}\cos[2t\hat{X}]-\hat{\sigma}_{\mathrm{y}}\sin[2t\hat{X}])]$
$\displaystyle\times\exp[\mathrm{i}\Theta_{\mathrm{y}}(\hat{\sigma}_{\mathrm{y}}\cos[2t\hat{X}]+\hat{\sigma}_{\mathrm{z}}\sin[2t\hat{X}])]D[\alpha]R[\phi].$
(22)

This relation can be again used to obtain the CFI, which is decreased from the
ideal interferometer as expected.

##  Appendix C Interferometer for the estimation of a signal

The target classical signal in quantum sensing is commonly composed of
simultaneously occurring various unitary processes,  described as
$\hat{\mathbb{S}}[\mathbf{\Theta}]=\exp[\mathrm{i}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sum_{j}}\Theta_{j}\hat{H}_{j}]$
where $\hat{H}_{j}$’s are transformation-generating Hamiltonians and
$\Theta_{j}$ are the unknown classical signal strengths. In this work, we
simplify the problem and focus on a single parameter estimation
$\hat{\mathbb{S}}[\Theta]=\exp[i\Theta\hat{H}]$. This classical signal below
the smallest decoherence time transforms a known input quantum state (probe)
into an unknown state, and the encoded information about this signal can be
drawn by processing the data from the available detection on many copies of
probes. Quantum sensing aims to optimize the estimation setups and the quantum
probes to infer such signals efficiently. In many experimental systems of
interest, the choice of the probes and setups including detectors is often
pre-imposed and thus limited. We assume here setups with probes factorized in
time where $\hat{\mathbb{S}}[\Theta]$ is occurring to ensembles of only one
probe state, leaving the consideration of correlated probes for future
investigation.

Quantum interferometers use multipartite, typically bipartite, interaction
Hamiltonian with ancillary systems which may be of different dimensions,
described by $\hat{H}_{\mathrm{int}}$ applied before and after the signal
$\hat{\mathbb{S}}[\mathbf{\Theta}]$ to exploit quantum interference for a high
precision estimation. The evolution in the symmetrical setup with an equal
strength $t$ of the pre- and post-processing  before the final detector  is
described by a unitary operation composed of sequential unitary operations

$\displaystyle\hat{U}_{\mathrm{interf}}=\exp[\mathrm{i}t\hat{H}_{\mathrm{int}}]\hat{\mathbb{S}}[\Theta]\exp[-\mathrm{i}t\hat{H}_{\mathrm{int}}].$
(23)

To explore the full power of given interferometry, we consider interactions
that are unitary and fully controllable in strength $t$.  We note that for the
absence of the signal $\Theta=0$, $\hat{U}_{\mathrm{interf}}$ in (23) is
reduced to the identity operation, and the following detection measures the
deviation from it.

##  Appendix D Estimation with thermal probes

Figure D.1: a) Average CFI by direct measurement using a thermal probe at
various $\bar{n}$ with increment $\Delta\bar{n}=0.5$ in (3). The ground state
has a larger CFI than thermal probes when direct estimation is performed. b)
The difference of CFI by a prepare-and-measure strategy in (10) between
thermal probes with various $\bar{n}$ from the ground state probe. At
$t\approx 1.2$, the thermal state shows a larger estimation precision than the
ground state, while approaching the latter in the asymptotic $t$.

In both direct measurement and prepare-and-measure strategy, using thermal
probes instead of oscillator ground state probes changes the CFI. In the
direct measurement, the approximate formula for the optimal strength and the
maximum average CFI at $\bar{n}\lesssim 3$ is given as
$(t_{\mathrm{opt^{\prime}}}[\bar{n}],F^{\mathrm{(av)}}[\rho^{\mathrm{th}}[\bar{n}];t_{\mathrm{opt^{\prime}}}])\approx(0.22+0.40*0.29^{0.71\bar{n}},0.059+0.78*3.49^{-\bar{n}^{0.69}})$
showing a decreasing average CFI by $\bar{n}$. For example at $\bar{n}=1$, the
maximum is given as $F^{\mathrm{(av)}}[t_{\mathrm{opt^{\prime}}}]=0.276$ at
$t_{\mathrm{opt^{\prime}}}=0.379$, reduced by approximately half from the
vacuum probe.

In the prepare-and-measure strategy, thermal probes can be beneficial to the
estimation at an intermediate $t$. As can be seen in Fig. D.1 b), the CFI is
increased from that of the ground state at around $t\approx 1.2$.

##  Appendix E Maximum likelihood estimation by qubit detector

Here we consider the maximum likelihood estimation (MLE) of the displacement
parameter $\alpha_{\mathrm{r}}$ using a qubit detector, which can apply to all
the protocols introduced here. We first note that the detection outcomes from
a qubit detector can be understood as a random sampling from binary outcomes
with a fixed sampling probabilities $p$ and $1-p$. With a fixed number of $n$
probes, if we repeat this sampling many times, the distribution of the number
of one outcome, e.g.
$\left|\mathrm{g}\right\rangle\left\langle\mathrm{g}\right|$, makes a binomial
distribution by definition. In addition, the outcomes with the same frequency
of data will give the same maximum likelihood function accordingly. We note
that MLE estimation is equivalent to simply estimating the sampling
probability from the frequencies. Therefore, the variance of MLE about $p$ is
given by that of a binomial distribution $\sigma_{p}^{2}=p(1-p)/n$. This
scaling in $n$ can be described by the theory of Fisher information, where the
uncertainty of an estimation method is given as $1/nF$ where $F$ is the Fisher
information LyFI2017 . Therefore, we can identify the implied Fisher
information by maximum likelihood estimation as
$F[p]^{\mathrm{(MLE)}}=1/p(1-p)$, in a single copy limit $n=1$. We note that
this can be alternatively written also as
$\sigma_{p}^{2}=(p^{-1}+(1-p)^{-1})^{-1}$ and
$F_{p}^{\mathrm{(MLE)}}=p^{-1}+(1-p)^{-1}$. Now if we change our estimation
target to $\alpha_{k}$, we can use the propagation of uncertainty as:

$\displaystyle\sigma_{\alpha_{\mathrm{r}}}^{2}\approx\sigma_{p}^{2}|\partial_{\alpha_{\mathrm{r}}}p|^{-2}=(p^{-1}+(1-p)^{-1})^{-1}|\partial_{\alpha_{\mathrm{r}}}p|^{-2}$
$\displaystyle=1/F_{k}^{\mathrm{(MLE)}}=1/F_{k},$ (24)

and we can immediately see the equivalence to the CFI.

##  Appendix F non-monotonicity of the prepare-measure strategy

In Fig. 2, we can see that there is a non-monotonous behavior vs $t$ in the
CFI by the prepare-measure strategy in (10). It shows at $t\ll 1$ the scaling
of $8t^{2}$ until it reaches the first maximum at $t^{\mathrm{max}}\approx
0.382$ with local maximum CFI $F_{\alpha_{\mathrm{r}}}\approx 0.461$, dropping
to $0$ at $t^{\mathrm{min}}\approx 0.78$. At this strength, the probabilities
of qubit detection $P_{e}=P_{g}=1/2$ are independent of either
$\alpha_{\mathrm{r}}$, or the initial qubit state. This behaviour at $t_{0}$
arises because the oscillator is maximally entangled to the qubit at this
strength, and the reduced qubit state after the oscillator mode is traced out
right after the second Rabi interaction before the qubit detection is given as
a maximally mixed qubit state

$\displaystyle\rho_{\mathrm{q}}=0.5\left|+\right\rangle\left\langle+\right|+0.5\left|-\right\rangle\left\langle-\right|$
$\displaystyle+0.5i\sin\phi\sin\theta\left|+\right\rangle\left\langle-\right|-0.5i\sin\phi\sin\theta\left|-\right\rangle\left\langle+\right|,$
(25)

where $\theta,\phi$ are parameters of initial second ancillary qubit state
represented in a Bloch sphere as
$\cos[\theta/2]e^{\mathrm{i}\phi/2}\left|\mathrm{e}\right\rangle+\sin[\theta/2]\mathrm{e}^{-\mathrm{i}\phi/2}\left|\mathrm{g}\right\rangle$.
The reduced qubit state $\rho_{\mathrm{q}}$ does not have any dependence on
$\alpha$ and thus shows zero FI, even though the full state (two head cat
entangled with qubit) has the dependence on $\alpha$. For optimal initial
qubit state $\left|\pm_{\mathrm{i}}\right\rangle$, it is given as the fully
mixed state. Beyond this strength again this strategy has an increased FI
about $\alpha_{\mathrm{r}}$ vs. $t$.

##  Appendix G Adaptive strategy for setups with varying performance

Often, the setup with a fixed architecture works better for certain ranges of
target variables than the others. In such cases, the maximum in Fisher
information can be reached asymptotically by an adaptive method. In the
adaptive estimation protocol, the setup including the input probe state is
chosen based on all the already known partial knowledge gained about the range
of target values with a subset of probe state ensembles. This partial
knowledge can be gained and accumulated by dividing the ensembles into
subensembles made of a finite number of probes. These subensembles are used
for different rounds, and the gained information in all previous rounds can be
used for the adjustment of the setup of the current round to increase the
amount of information that can be gained per each probe. Therefore, the
precision can approach the maximum value asymptotically as the uncertainty in
the estimation is thus decreased.

For example, consider a virtual estimation protocol of the displacement in
known direction $D[\alpha_{\mathrm{r}}]$, whose performance may vary depending
on the true value of $\alpha_{\mathrm{r}}$, e.g. the maximum precision is
around $\alpha_{\mathrm{r}}^{\mathrm{max}}\neq 0$. In the first round, we do
not have any pre-knowledge about $\alpha_{\mathrm{r}}$, so we are forced to
assume the true value randomly, e.g. $\alpha_{\mathrm{r}}^{(0)}=0$. Now in the
first round of estimation, we apply an auxiliary displacement
$D[\alpha_{\mathrm{r}}^{\mathrm{max}}-\alpha_{\mathrm{r}}^{(0)}]$ with a
finite number of the probes, and update $\alpha_{\mathrm{r}}^{(1)}$ with the
gained data. Now for the second round, we apply an auxiliary displacement
$D[\alpha_{\mathrm{r}}^{\mathrm{max}}-\alpha_{\mathrm{r}}^{(1)}]$ to make a
composite displacement
$D[\alpha_{\mathrm{r}}^{\mathrm{max}}-\alpha_{\mathrm{r}}^{(1)}+\alpha_{\mathrm{r}}]$,
and repeat such estimation for many rounds. The estimated value
$\alpha_{\mathrm{r}}^{(j)}$ can be dynamically updated so that asymptotically
for $j\rightarrow\infty$ when
$\alpha_{\mathrm{r}}^{(j)}\rightarrow\alpha_{\mathrm{r}}$, the composite
displacement becomes $D[\alpha_{\mathrm{r}}^{\mathrm{max}}]$ and the
uncertainty can be optimally minimized against the number of probes used and
reach the precision predicted by the maximum of the CFI.

##  Appendix H Unambiguous determination of displacement parameters by the
prepare and measure strategy and Rabi interferometer

Figure H.1:  Comparison of CFI by prepare-and-measure strategy (red) and Rabi
interferometer under qubit dephasing $p=0.1$ (blue) between an individual
estimation (solid) and simultaneous estimation (dotted and dashed) at
$\alpha_{\mathrm{r}}=\alpha_{\mathrm{i}}=0$. The qubit ancillas were set as
$\left|-_{\mathrm{i}}\right\rangle_{A_{1},A_{2}}$. We note that asymptotically
the CFI by simultaneous estimation in (30) is behaving similarly to that by
the individual estimation in (29). Here, we can see that the interference due
to the simultaneous estimation can be avoided at certain strengths $t^{c}$.
This interference can be actively cancelled using a two-qubit gate in (9).

In the main text, the individual estimation of $\alpha_{\mathrm{r}}$ is
described, which is equivalent to the estimation of $\alpha_{\mathrm{i}}$,
when everything is substituted as $X\rightarrow P$ and
$\alpha_{\mathrm{r}}\rightarrow\alpha_{\mathrm{i}}$.  The joint effect of the
second Rabi coupling after the signal with the optimal qubit ancillary state
$\left|\psi\right\rangle_{\mathrm{A}}=c_{e}\left|\mathrm{e}\right\rangle_{\mathrm{A}}+c_{g}\left|\mathrm{g}\right\rangle_{\mathrm{A}}$
and the qubit detector afterward
$\\{\left|\mathrm{e}\right\rangle_{\mathrm{A}}\left\langle\mathrm{e}\right|,\left|\mathrm{g}\right\rangle_{\mathrm{A}}\left\langle\mathrm{g}\right|\\}$
in the prepare and measure setup in Fig. 1 (b) is described in the operator
forms as

$\displaystyle\hat{O}_{e}=\prescript{}{\mathrm{A}}{\left\langle\mathrm{e}\right|}\hat{R}_{\mathrm{x}}^{\dagger}(c_{e}\left|\mathrm{e}\right\rangle_{\mathrm{A}}+c_{g}\left|\mathrm{g}\right\rangle_{\mathrm{A}})$
$\displaystyle=c_{e}\cos[tX]-\mathrm{i}c_{g}\sin[tX],$
$\displaystyle\hat{O}_{g}=\prescript{}{\mathrm{A}}{\left\langle\mathrm{g}\right|}\hat{R}_{\mathrm{x}}^{\dagger}(c_{e}\left|\mathrm{e}\right\rangle_{\mathrm{A}}+c_{g}\left|\mathrm{g}\right\rangle_{\mathrm{A}})$
$\displaystyle=c_{g}\cos[tX]-\mathrm{i}c_{e}\sin[tX].$ (26)

These operations conditionally act on the probe state based on the detector
outcome
$\left|\mathrm{e}\right\rangle_{\mathrm{A}}\left\langle\mathrm{e}\right|$ or
$\left|\mathrm{g}\right\rangle_{\mathrm{A}}\left\langle\mathrm{g}\right|$ to
give final states
$\hat{O}_{e,g}D[\alpha]\left(\left|\mathrm{i}\frac{t}{\sqrt{2}}\right\rangle_{\mathrm{C}}-\left|-\mathrm{i}\frac{t}{\sqrt{2}}\right\rangle_{\mathrm{C}}\right)$,
whose norms correspond to the probability of detection given as

$\displaystyle
P_{\mathrm{e}}=\frac{\mathrm{e}^{-\frac{3t^{2}}{2}}\left(\sinh\left[\frac{3t^{2}}{2}\right]-\cosh\left[\frac{t^{2}}{2}\right]\right)\cos\left[\theta+2\sqrt{2}t\alpha_{\mathrm{r}}\right]+1}{2},$
$\displaystyle P_{\mathrm{g}}=1-P_{\mathrm{e}},$ (27)

for $c_{\mathrm{e}}=\cos[\theta/2]\mathrm{e}^{\mathrm{i}\phi/2}$ and
$c_{\mathrm{g}}=\sin[\theta/2]\mathrm{e}^{-\mathrm{i}\phi/2}$. However, the
analytical form of CFI is complex and the optimal setting depends on the
target parameter $\alpha_{\mathrm{r}}$ and strength $t$. When the qubit state
used for the measurement is at $\left|-_{\mathrm{i}}\right\rangle_{A_{1}}$
which is found to be optimal for $\alpha_{\mathrm{r}}=0$, we obtain

$\displaystyle
F^{\mathrm{PnM},\left|-_{\mathrm{i}}\right\rangle}_{\alpha_{\mathrm{r}}}=4\left(\mathrm{e}^{t^{2}}+\mathrm{e}^{2t^{2}}-\mathrm{e}^{3t^{2}}+1\right)^{2}t^{2}\cos^{2}[2\sqrt{2}t\alpha_{\mathrm{r}}]$
$\displaystyle\times\frac{\left(\mathrm{e}^{t^{2}}+\mathrm{e}^{2t^{2}}-\mathrm{e}^{3t^{2}}+1\right)^{2}\cos[4\sqrt{2}t\alpha_{\mathrm{r}}]-2\mathrm{e}^{t^{2}}-3\mathrm{e}^{2t^{2}}+\mathrm{e}^{4t^{2}}+2\mathrm{e}^{5t^{2}}+7\mathrm{e}^{6t^{2}}-1}{\left(\left(\mathrm{e}^{t^{2}}+\mathrm{e}^{2t^{2}}-\mathrm{e}^{3t^{2}}+1\right)^{2}\sin^{2}[2\sqrt{2}t\alpha_{\mathrm{r}}]-4\mathrm{e}^{6t^{2}}\right){}^{2}}.$
(28)

When an adaptive strategy in Appendix G for optimal estimation can be used,
the CFI is reduced for $\alpha_{\mathrm{r}}\ll 1$ as

$\displaystyle F^{\mathrm{PnM,\alpha_{\mathrm{r}}\ll
1}}_{\alpha_{\mathrm{r}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=F^{\mathrm{PnM,\alpha_{\mathrm{i}}\ll
1}}_{\alpha_{\mathrm{r}}}}=2t^{2}e^{-6t^{2}}\left(\mathrm{e}^{t^{2}}+\mathrm{e}^{2t^{2}}-m^{3t^{2}}+1\right)^{2}$
(29)

in the optimal setup with the second qubit input state
$2^{-1/2}(\left|+\right\rangle-i\left|-\right\rangle)$. For $t\ll 1$, it
scales as $8t^{2}$, while for $t\gg 1$, it scales as $2t^{2}$.

Now we summarize the simultaneous estimation precision obtained in an
equivalent way in WDL:

$\displaystyle
F^{\mathrm{PnM,Simul}}_{\alpha_{\mathrm{r}}}=\frac{2\mathrm{e}^{-6t^{2}}t^{2}\left(\mathrm{e}^{4t^{2}}-\left(\mathrm{e}^{t^{2}}+2\mathrm{e}^{3t^{2}}-2\mathrm{e}^{4t^{2}}+\mathrm{e}^{5t^{2}}\right)\cos[2t^{2}]+1\right)^{2}}{\left(\mathrm{e}^{t^{2}}-\mathrm{e}^{2t^{2}}+\mathrm{e}^{t^{2}}\cos[2t^{2}]-1\right)^{2}},$
$\displaystyle
F^{\mathrm{PnM,Simul}}_{\alpha_{\mathrm{i}}}=\frac{2\mathrm{e}^{-6t^{2}}t^{2}\left(2\mathrm{e}^{3t^{2}}-\mathrm{e}^{4t^{2}}+\mathrm{e}^{5t^{2}}-2\mathrm{e}^{4t^{2}}\cos[2t^{2}]+\mathrm{e}^{t^{2}}\cos[4t^{2}]-1\right)^{2}}{\left(\mathrm{e}^{t^{2}}-\mathrm{e}^{2t^{2}}+\mathrm{e}^{t^{2}}\cos[2t^{2}]-1\right)^{2}}.$
(30)

Here, the symmetry between the estimation precision of $\alpha_{\mathrm{r}}$
and $\alpha_{\mathrm{i}}$ is broken, while asymptotically it matches Eq. (29)
for $t>2$. The CFI of $\alpha_{\mathrm{r}}$ has an asymptotic modulation
factor $\cos[2t^{2}]^{2}$ multiplied by the CFI by individual estimation.
Again, this factor is universal for all true values of $\alpha_{\mathrm{r}}$.
We briefly note that preparation of other superposition states such as
$\left|\mathrm{i}\alpha\right\rangle+\left|-\mathrm{i}\alpha\right\rangle+\left|2\mathrm{i}\alpha\right\rangle+\left|-2\mathrm{i}\alpha\right\rangle$
does not possess an enhanced CFI.

Similarly, we can calculate the simultaneous estimation of displacement
parameters by the Rabi interferometer using two ancillas as in Fig. 1 (c)
under qubit dephasing and oscillator thermal noises. First under the qubit
dephasing, the full expressions for CFIs are given in complex forms.
Interestingly, both CFIs can be very precisely described by the CFI model in
(15-17) with different dephasing levels
$p_{\mathrm{r}}=\left(2-\cos\left[4t^{2}\right]\right)p+\frac{1}{8}\left(-5+4\cos\left[4t^{2}\right]+2\cos^{2}\left[4t^{2}\right]-\cos\left[8t^{2}\right]\right)p^{2}\geq
p$ and $p_{\mathrm{i}}=p$ in place of $p$. This arises from the same reduction
factor $\cos^{2}[2t^{2}]$ in one of the CFIs. At certain Rabi strength
$t^{\mathrm{c}}=\sqrt{\mathbf{c}\mathrm{\pi}/2}$ with integer
$\mathbf{c}\in\mathbb{Z}$, $p_{\mathrm{r}}=p$ and the result is the same as
the independent estimation. Therefore, by choosing such specific strengths
$t^{\mathrm{c}}$, we can avoid the interference occurring from the
simultaneous estimation, both in Rabi interferometers, the prepare-and-
measure, and the direct measurement.  In WDL, they are given as

$\displaystyle
F^{\mathrm{RI,Simul}}_{\alpha_{\mathrm{r}}}=2\mathrm{e}^{-4t^{2}}t^{2}\left(2-p+p\cos
4t^{2}\right)^{2}\left((1-p)\cosh 2t^{2}+\sinh 2t^{2}\right)^{2},$
$\displaystyle
F^{\mathrm{RI,Simul}}_{\alpha_{\mathrm{i}}}=2\mathrm{e}^{-8t^{2}}t^{2}\left(\mathrm{e}^{4t^{2}}(-2+p)+p\cos
4t^{2}\right)^{2}.$ (31)

In Fig. H.1, they are compared, and we notice that asymptotically the CFI by
simultaneous estimation in (30) and (31) is behaving similarly to that by the
individual estimation in (29) and (15) at $t=t^{\mathrm{c}}$.

##  Appendix I Estimation with fictitious quadrature detector and quantum
Fisher information

Figure I.1:  Comparison of CFI by Rabi interferometer and a non-
interferometric setup under the same oscillator heatings $\gamma_{0}$,
$10\gamma_{0}$ and $100\gamma_{0}$ as in Fig. 2 b,d) when quadrature detector
is available (gray) and without it (black) for a) vacuum state and b) thermal
state. The Rabi interferometer is superior to the prepare-and-measure strategy
with an aligned quadrature detector at sufficient Rabi strengths. We can
observe that the enhancement over the prepare-and-measure strategy with an
aligned quadrature detector is even more prominent than vacuum.

The CFIs about $\alpha_{\mathrm{r}}$ and $\alpha_{\mathrm{i}}$ with the Rabi
detector containing hypothetical field quadrature detector are calculated to
have
$(F_{\alpha_{\mathrm{r}}}^{\mathrm{X_{0,\mathrm{\pi}/2}}}[\rho^{\mathrm{th}}[0]],F_{\alpha_{\mathrm{i}}}^{\mathrm{X_{0,\mathrm{\pi}/2}}})[\rho^{\mathrm{th}}[0]]=(4,0)$
or $(0,4)$ depending on the alignment of the phase of the quadrature detector,
as this detector is sensitive to phase. Detection of displacement in an
arbitrary direction requires an alignment of the quadrature detector to it. To
estimate both $\alpha_{\mathrm{r}}$ and $\alpha_{\mathrm{i}}$ simultaneously,
the statistical distribution of the setups into each alignment is necessary,
and the CFI is halved if we equally distribute the resources between both
estimations. For thermal states, the CFI is obtained as

$\displaystyle
F_{\alpha_{\mathrm{r}}}^{\mathrm{direct,fixed,quad}}[\rho^{\mathrm{th}}[0];t]=\frac{4}{1+2\bar{n}},$
(32)

reaching QFI but still below the classical benchmark.

If a hypothetical field quadrature detector is used in the Rabi in the
prepare-and-measure setup, the CFI can be computed as

$\displaystyle
F_{\alpha_{\mathrm{r}}}^{\mathrm{PnM,opt,quad}}[\rho^{\mathrm{th}}[0]]=\left(\frac{8}{\mathrm{e}^{t^{2}}-1}+8\right)t^{2}+4.$
(33)

This is equal to QFIs about $\alpha_{\mathrm{r}}$ and $\alpha_{\mathrm{i}}$
simultaneously  increasing monotonously by Rabi strength $t$ as
$Q_{\alpha_{\mathrm{r}}}[\rho^{\mathrm{th}}[0]]=Q_{\alpha_{\mathrm{i}}}[\rho^{\mathrm{th}}[0]]=4+8t^{2}+\frac{8t^{2}}{\mathrm{e}^{t^{2}}-1}\stackrel{{\scriptstyle
t\gg 1}}{{\rightarrow}}4+8t^{2}$. This is equal to the CFI of the input cat
state measured by a quadrature detector, and thus the measure-and-prepare
strategy does not necessitate Rabi interaction and qubit detector. In the
small $t$ limit, the odd cat state is reduced to a single photon state which
has a larger QFI than that of the vacuum: $\lim_{t\rightarrow
0}Q_{\alpha_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{r,i}}}[\rho^{\mathrm{th}}[0]]=12$.
Again, we remark that the field mode quadrature detector is necessary to
access this high sensitivity, as neither the qubit detector nor field
quadrature detector alone is not sufficient. For large $t$, it approaches
scaling by a factor of $4$ of the CFI obtained without the quadrature
detector, i.e. $t^{2}$ but with a different multiplication factor. It is in
sharp contrast to Eq. (3) as well where exponential decay suppresses the
polynomial scaling. Furthermore, it shows even larger values than the CFI by
the interferometric setup to be introduced below.  This can be partially
explained by a high average number of excitation of the odd cat states in the
oscillator, given as
$\langle\hat{n}\rangle=\frac{\left(\mathrm{e}^{t^{2}}+1\right)t^{2}}{2\left(\mathrm{e}^{t^{2}}-1\right)}$.
This high average excitation number may cause it to be vulnerable to the
noises on the oscillator, especially compared to the interferometric setup
below. Again, this CFI is accessible only by a hypothetical detector, and
without such a detector, the CFI is reduced only back to Eq. (10). In
addition, $\alpha_{\mathrm{r}}$ and $\alpha_{\mathrm{i}}$ cannot be measured
simultaneously, because the field detector should be aligned to only one of
the estimations. For the estimation of a displacement parameter about which
the quadrature detector is not sensitive (i.e. aligned to the conjugate
variable estimation), the CFI is reduced to twice the CFI of the previous case
with only qubit detector, i.e. we get scaling of $4t^{2}$ (D2’), with
optimized qubit state at $\alpha=0$. This result is interesting, as such a
detector is completely insensitive to the conjugate variable displacement
without the qubits. Both Eq. (33) and (10) can be used as a benchmark for the
estimation by the interferometric scheme in Sec III.

To understand our result (13) better, we can compare with the maximum of CFI
using virtual field quadrature detection of $\hat{X}$ after the
interferometer. The total probability density is given as

$\displaystyle
P_{\mathrm{e}}(X)=\frac{\mathrm{e}^{-\mathrm{e}^{2r}(\alpha_{\mathrm{r}}-X)^{2}}\sin^{2}\left[\sqrt{2}\alpha_{\mathrm{r}}t\right]}{\sqrt{\mathrm{\pi}}\mathrm{e}^{-r}},$
$\displaystyle
P_{\mathrm{g}}(X)=\frac{\mathrm{e}^{-\mathrm{e}^{2r}(\alpha_{\mathrm{r}}-X)^{2}}\cos^{2}\left[\sqrt{2}\alpha_{\mathrm{r}}t\right]}{\sqrt{\mathrm{\pi}}e^{-r}},$
(34)

where $X$ is the quadrature detection outcome. The CFI about
$\alpha_{\mathrm{r}}$ reaches QFI $Q_{\alpha_{\mathrm{r}}}$ regardless of the
value of $\alpha_{\mathrm{r}}$ and $\alpha_{\mathrm{i}}$. This equality to the
QFI after the first Rabi interaction implies that this detection setup is
optimal for $\alpha_{\mathrm{r}}$ estimation. We again note that for the
simultaneous estimation, this detector does not enhance the CFI of the
estimation of $\alpha_{\mathrm{i}}$. CFI at WDL with quadrature detector is
given as

$\displaystyle
F^{\mathrm{PnM,opt,quad}}_{\alpha_{\mathrm{r}}}[\rho^{\mathrm{th}}[\bar{n}]]=\frac{8\left(4t^{2}\bar{n}-\mathrm{e}^{-t^{2}\left(2\bar{n}+1\right)}+2t^{2}+1\right)}{\left(2\bar{n}+1\right)\left(2-2\mathrm{e}^{-t^{2}\left(2\bar{n}+1\right)}\right)}.$
(35)

Here again, a thermal state is not very detrimental. The asymptotic expression
is given as
$\lim_{t\rightarrow\infty}F^{\mathrm{PnM,opt,quad}}_{\alpha_{\mathrm{r}}}[\rho^{\mathrm{th}}[\bar{n}]]=8t^{2}$
for $t\gg 1$, regardless of $\bar{n}$. In the limit of $t\ll 1$, it is given
as $\lim_{t\rightarrow
0}F^{\mathrm{PnM,opt,quad}}_{\alpha_{\mathrm{r}}}[\rho^{\mathrm{th}}[\bar{n}]]=12/(1+2\bar{n})$,
where the thermal probe decreases the CFI most.

If a virtual field quadrature measurement on the oscillator
$\\{\left|X\right\rangle_{\mathrm{C}}\left\langle X\right|\\}$ in eigenbasis
of $\hat{X}$ is assumed in the Rabi interferometer, then we obtain the CFI
$F^{\mathrm{interf},x}_{\alpha_{\mathrm{r}}}[\rho^{\mathrm{th}}[0]]=8t^{2}+4$,
while the conjugate component has
$F^{\mathrm{interf},x}_{\alpha_{\mathrm{i}}}[\rho^{\mathrm{th}}[0]]=8t^{2}$.
For a thermal light, it is modified as
$F_{\alpha_{\mathrm{r}}}^{\mathrm{interf},x}[\rho^{\mathrm{th}}[\bar{n}];t]=8t^{2}+\frac{4}{1+2\bar{n}}$
and
$F_{\alpha_{\mathrm{r}}}^{\mathrm{interf},x}[\rho^{\mathrm{th}}[\bar{n}]]=8t^{2}$
with quadrature detector $\\{\left|X\right\rangle_{\mathrm{C}}\left\langle
X\right|\\}$, and vice versa for the alignment as
$\\{\left|P\right\rangle_{\mathrm{C}}\left\langle P\right|\\}$. The CFI using
simultaneous measurement of $\hat{X}$ and $\hat{P}$ with a POVM form
$\\{\mathrm{\pi}^{-1}\left|\beta\right\rangle_{\mathrm{C}}\left\langle\beta\right|\\}$
together with a qubit detector at the end of the setup is given as
$F_{\alpha_{\mathrm{r,i}}}=8t^{2}+2$, thus still not reaching but approaching
the QFIs in (11).

In Fig. H.1, we compared the Rabi interferometer and prepare-and-measure
strategy under noises. As was noted above, in the ideal channel, prepare-and-
measure has a larger CFI than Rabi interferometer. Interestingly,
interferometers can have a higher CFI even without a quadrature detector than
the prepare-and-measure strategy with a quadrature detector with sufficient
Rabi strength when the heating exists. This is partly due to a large number of
excitation in the oscillator in the prepare-and-measure strategy, which
therefore exposes a vulnerability to oscillator noises. Remarkably, the
advantage of the Rabi interferometer is more prominent in the case of the
input thermal state in the oscillator.

##  Appendix J Engineering inverse Rabi interaction using auxiliary
oscillators

The key element of an interferometer is the accessibility of an inverse
interaction after the signal. The inverse Rabi interaction can be engineered
by two methods: first by adding a $\mathrm{\pi}$-phase shift to the oscillator
to make $\hat{X}\rightarrow-\hat{X}$, or using qubit rotations as evidenced in
the following transformation equation:

$\displaystyle\exp[\mathrm{i}T\hat{\sigma}_{\mathrm{y}}]\hat{\sigma}_{\mathrm{x}}\exp[-\mathrm{i}T\hat{\sigma}_{\mathrm{y}}]=\hat{\sigma}_{\mathrm{x}}\cos[2T]+\hat{\sigma}_{\mathrm{z}}\sin[2T]$
(36)

for an arbitrary $T$, and at $T=\mathrm{\pi}/2$, it becomes
$\hat{\sigma}_{\mathrm{x}}\rightarrow-\hat{\sigma}_{\mathrm{x}}$. By this
transformation, a Rabi interaction becomes an inverse Rabi interaction
$\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}]\rightarrow\exp[-\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}]$.
These qubit rotations necessary for such a conversion can in turn be
engineered from other Rabi interactions with a strong drive:

$\displaystyle\exp[\mathrm{i}k\hat{\sigma}_{\mathrm{y}}\hat{X}^{\prime}]\left|\phi\right\rangle\left|\delta\right\rangle_{C^{\prime\prime}}\approx\exp[\mathrm{i}k\hat{\sigma}_{\mathrm{y}}\sqrt{2}\delta]\left|\phi\right\rangle\left|\delta\right\rangle_{C^{\prime\prime}},$
(37)

and at $k=\frac{\mathrm{\pi}/2}{\sqrt{2}\delta}$, we can achieve a flip.
Alternatively, using only JC coupling we have

$\displaystyle\exp[\mathrm{i}\tau\sigma_{+}a+\mathrm{i}\tau\sigma_{-}a^{\dagger}]\left|\phi\right\rangle\left|\mathrm{i}\delta\right\rangle_{C^{\prime\prime}}\approx\exp[-\mathrm{i}\tau\sigma_{\mathrm{y}}\delta]\left|\phi\right\rangle,$
(38)

and at $\tau=\frac{\mathrm{\pi}}{2\delta}$ we achieve an approximate qubit
flip. If we have an access to squeezed state in this auxiliary oscillator, we
can have an improved approximation by substituting
$\left|\delta\right\rangle_{C^{\prime\prime}}\rightarrow
D[\delta]\left|r\right\rangle_{C^{\prime\prime}}$ where $\left|r\right\rangle$
is a squeezed state. The gate fidelity is enhanced by such a substitution as
$0.962\rightarrow 0.995$ at $\mathbf{\delta}=4$ and $r=1$.

##  Appendix K Derivation of the probabilities for Rabi interferometer

This operator
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{U}_{\mathrm{interf}}}$
in (12) acts on an arbitrary input pure state
$\left|\varphi\right\rangle_{\mathrm{C}}$ in the oscillator and the initial
qubit state
$\left|\phi^{\mathrm{opt}}\right\rangle=\left|\pm_{\mathrm{i}}\right\rangle_{\mathrm{A}_{1}}$
to give the final state

$\displaystyle\left|\Psi^{\prime}\right\rangle={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{U}_{\mathrm{interf}}}\left|\varphi\right\rangle_{\mathrm{C}}\left|\pm_{\mathrm{i}}\right\rangle_{\mathrm{A}_{1}}=\hat{D}[\alpha]\left|\varphi\right\rangle_{\mathrm{C}}\stackunder[1pt]{\hat{R}_{\mathrm{x}}[-\sqrt{2}\alpha_{\mathrm{r}}t]\left|\pm_{\mathrm{i}}\right\rangle_{\mathrm{A}_{1}}}{\underbrace{\hskip
77.49976pt}_{\makebox[0.0pt][l]{\scriptsize$\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\equiv\left|\phi\right\rangle_{\mathrm{A}_{1}}$}}}$ (39)

This state $\left|\Psi^{\prime}\right\rangle$ is factorized into the local
oscillator and ancilla states, where the final qubit state
$\left|\phi\right\rangle_{\mathrm{A}_{1}}$ is obtained independent of the
initial CV state $\left|\varphi\right\rangle_{\mathrm{C}}$, due to the
geometric phase attained by the entire process. This factorization gives many
robust properties of this interferometric estimation protocol and enables it
to work on realistic noises. We also note that the output state from a mixed
state $\rho_{\mathrm{C}}$ in the oscillator is given simply as

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{U}_{\mathrm{interf}}}\rho_{\mathrm{C}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(40)

For the estimation of $\alpha_{\mathrm{r}}$, we detect with only a qubit
detector in $\hat{\sigma}_{\mathrm{z}}$-eigenbasis
$\\{\left|\mathrm{e}\right\rangle_{\mathrm{A}_{1}}\left\langle\mathrm{e}\right|,\left|\mathrm{g}\right\rangle_{\mathrm{A}_{1}}\left\langle\mathrm{g}\right|\\}$
at one arm without any detection in the oscillator to get the probabilities
for binary outcomes

$\displaystyle
P_{\mathrm{g}}(\alpha_{\mathrm{r}})=\cos^{2}[\sqrt{2}\alpha_{\mathrm{r}}t],\leavevmode\nobreak\
P_{\mathrm{e}}(\alpha_{\mathrm{r}})=\sin^{2}[\sqrt{2}\alpha_{\mathrm{r}}t].$
(41)

We can simultaneously estimate $\alpha_{\mathrm{i}}$ by a natural extension of
the setup to that with two-qubit ancillas where the second qubit ancilla
interacts with the oscillator by another type of Rabi interaction
$\hat{R}^{\prime}_{\mathrm{P}}=\exp[it^{\prime}\hat{\sigma}_{\mathrm{x}}^{\prime}\hat{P}]$
and its inverse where prime represents the second ancillary mode. We obtain
similarly the probabilities for qubit detection outcomes given as
$P_{\mathrm{g}}=\cos^{2}[\sqrt{2}t^{\prime}\alpha_{\mathrm{i}}]$ and
$P_{\mathrm{e}}=\sin^{2}[\sqrt{2}t^{\prime}\alpha_{\mathrm{i}}]$, importantly,
where the joint probabilities for estimation of $\alpha_{\mathrm{r}}$ and
$\alpha_{\mathrm{i}}$ factorizes, from which the same CFI is obtained as
$F^{\mathrm{interf}}_{\alpha_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{r,i}}}[\rho^{\mathrm{th}}[0]]=8t^{2}$.

Under the existence of qubit dephasing noise, the probability is reduced in
visibility to
$P_{\mathrm{e}}=\frac{2+\left\\{2-p\left(e^{-4t^{2}}+1\right)\right\\}\sin\left[2\sqrt{2}t\alpha_{r}\right]}{4}$,
and $P_{\mathrm{g}}=1-P_{\mathrm{e}}$. This formula was used in the derivation
of the CFI in (15).

##  Appendix L Extended Rabi interferometer with multiple qubits

Entangled qubits such as
$2^{-1/2}(\left|\mathrm{e}\right\rangle_{\mathrm{A}_{1}}\left|\mathrm{e}\right\rangle_{\mathrm{A}_{2}}+\left|\mathrm{g}\right\rangle_{\mathrm{A}_{1}}\left|\mathrm{g}\right\rangle_{\mathrm{A}_{2}})$
can make further enhancement for the Rabi interferometers over the separable
qubit ancillas. With a straightforward extension of the calculation where the
same qubit rotations $\hat{R}[-\sqrt{2}\alpha_{\mathrm{r}}t]$ are applied on
both qubits, the qubit detectors on $\sigma_{\mathrm{z}}$ eigenbasis will give
CFI $F_{\alpha_{\mathrm{r}}}=32t^{2}$, twice larger than $16t^{2}$ from simply
additive Fisher information from separable two-qubit ancillas with $m=2$. This
result implies that the entangled state of $m$ qubits may have a larger
scaling of CFI. A simple GHZ state with $m$ qubits has a linear scaling of
$16mt^{2}$ by the Rabi interferometer, calculated in the same way. These
states can be approximately generated using Rabi interactions before the
interferometer from a vacuum oscillator in the preparation stage, as

$\displaystyle\exp[\mathrm{i}\frac{\mathrm{\pi}}{4t}\sum_{j=2}^{m}\hat{\sigma}_{\mathrm{y}}^{(\mathrm{A}_{j})}\hat{P}]\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{z}}\hat{X}]\left|+\right\rangle_{\mathrm{A}_{1}}\left|+\right\rangle_{\mathrm{A}_{2}}...\left|+\right\rangle_{\mathrm{A}_{m}}\left|0\right\rangle_{\mathrm{C}}$
$\displaystyle\approx
2^{-1/2}\left|\mathrm{e}\right\rangle_{\mathrm{A}_{1}}\left|\mathrm{e}\right\rangle_{\mathrm{A}_{2}}...\left|\mathrm{e}\right\rangle_{\mathrm{A}_{m}}\left|\mathrm{i}mt/\sqrt{2}\right\rangle_{\mathrm{C}}+2^{-1/2}\left|\mathrm{g}\right\rangle_{\mathrm{A}_{1}}\left|\mathrm{g}\right\rangle_{\mathrm{A}_{2}}...\left|\mathrm{g}\right\rangle_{\mathrm{A}_{m}}\left|-\mathrm{i}mt/\sqrt{2}\right\rangle_{\mathrm{C}},$
(42)

where the approximation works well for a large $t$. The Rabi interactions
$\exp[\mathrm{i}t\sigma_{\mathrm{z}}\hat{X}]$ can be obtained from
$\exp[\mathrm{i}t\sigma_{\mathrm{x}}\hat{X}]$ by the adjoint application of
qubit rotation as in Eq. (36) setting $T=\mathrm{\pi}/4$. This result implies
that the exploration of a highly complex qubit circuit with Rabi interactions
generating broader ranges of entangled states can be profitable.

##  Appendix M Asymmetric Rabi interactions

We can check if different strengths and types of Rabi interactions before and
after the displacement may have any advantages for the estimation by the Rabi
interferometer. With the same calculation as in the main text, we obtain a CFI

$\displaystyle
F_{\alpha_{\mathrm{r}}}=\frac{8\mathrm{e}^{4tt^{\prime}}t^{\prime
2}\left(-1+\mathrm{e}^{4\mathrm{i}\sqrt{2}\alpha_{\mathrm{r}}t^{\prime}}\right){}^{2}}{2\mathrm{e}^{4t^{\prime}\left(t+\mathrm{i}\sqrt{2}\alpha_{\mathrm{r}}\right)}+\mathrm{e}^{4t^{\prime}\left(t+2\mathrm{i}\sqrt{2}\alpha_{\mathrm{r}}\right)}-4\mathrm{e}^{2\left(t^{2}+t^{\prime}\left(t^{\prime}+2\mathrm{i}\sqrt{2}\alpha_{\mathrm{r}}\right)\right)}+\mathrm{e}^{4tt^{\prime}}}.$
(43)

We can see that Eq. (43) has a dependence on the $\alpha_{\mathrm{r}}$. The
maximum value of Fisher information over $\alpha_{\mathrm{r}}$ reaches the FI
of the value $8\max[t,t^{\prime}]^{2}$, while for certain
$\alpha_{\mathrm{r}}$ it drops to $0$. A bigger difference in the strengths
makes an adaptive estimation more suitable.

On the other hand, exploiting two Rabi interferometers utilizing slightly
different strengths $t,t^{\prime}$ on two subensembles can uniquely determine
$\alpha_{\mathrm{r}}$ in all ranges non-adaptively, as there is a unique
solution of $\alpha_{\mathrm{r}}\in[-\infty,\infty]$ for
$(P_{g},P_{g}^{\prime})=(\cos^{2}[\sqrt{2}\alpha_{\mathrm{r}}t],\cos^{2}[\sqrt{2}\alpha_{\mathrm{r}}t^{\prime}])$.
If we use half of the total $n$ probes into an interferometer for duration $t$
and the remaining half into that for duration $t^{\prime}$, the CFI is
obtained as $8n(t^{2}+t^{\prime 2})/2$. This result implies that as
$t^{\prime}$ approaches $t$, the whole-range estimation is attained with the
same precision as Eq. (13).

##  Appendix N High-order Rabi interferometers

A high-order Rabi gates
$\hat{R}_{X^{(k)}}=\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}^{k}]$ for
$k>1$ can have enhancement over linear Rabi gate when used in the
interferometers. These gates can be engineered from linear Rabi gates by using
multiple of them as in ParkNJP2018 . In these high-order Rabi interferometers,
we use the following identity to obtain the transformation of the signal:

$\displaystyle\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}^{k}]\hat{D}[\alpha]\exp[-\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}^{k}]=\hat{D}[\alpha]\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\\{(\hat{X}+\sqrt{2}\alpha_{\mathrm{r}})^{k}-\hat{X}^{k}\\}]\stackrel{{\scriptstyle\alpha_{\mathrm{r}}\ll
1}}{{\approx}}\hat{D}[\alpha]\exp[it\hat{\sigma}_{\mathrm{x}}\sqrt{2}k\alpha_{\mathrm{r}}\hat{X}^{k-1}].$
(44)

Now the probabilities of qubit detector outcomes are given as

$\displaystyle
P_{g}=\prescript{}{C}{\left\langle\psi\right|}\cos[\sqrt{2}kt\hat{X}^{k-1}\alpha_{\mathrm{r}}]\left|\psi\right\rangle_{\mathrm{C}},\leavevmode\nobreak\
P_{e}=1-P_{g}.$ (45)

For vacuum/ground state in the oscillator, we obtain the following exact
expressions without approximation for CFI of lowest $k$’s (superscript denotes
the order of Rabi interactions):

$\displaystyle
F_{\alpha_{\mathrm{r}}}^{(k=2)}=\frac{256t^{4}\alpha_{\mathrm{r}}^{2}}{\mathrm{e}^{16t^{2}\alpha_{\mathrm{r}}^{2}}-1},$
$\displaystyle
F_{\alpha_{\mathrm{r}}}^{(k=3)}=-\frac{9t^{2}\left(\mathrm{i}\left(\sqrt{1-6\mathrm{i}\sqrt{2}t\alpha_{\mathrm{r}}}-\sqrt{1+6\mathrm{i}\sqrt{2}t\alpha_{\mathrm{r}}}\right)+6\sqrt{2}t\alpha_{\mathrm{r}}\left(\sqrt{1-6\mathrm{i}\sqrt{2}t\alpha_{\mathrm{r}}}+\sqrt{1+6\mathrm{i}\sqrt{2}t\alpha_{\mathrm{r}}}\right)\right){}^{2}}{\left(72t^{2}\alpha_{\mathrm{r}}^{2}+1\right){}^{2}\left(-144t^{2}\alpha_{\mathrm{r}}^{2}+\sqrt{72t^{2}\alpha_{\mathrm{r}}^{2}+1}-1\right)}.$
(46)

In the limit of $\alpha_{\mathrm{r}}\rightarrow 0$, they are reduced to
$F_{\alpha_{\mathrm{r}}\ll 1}^{(k=2)}=16t^{2}$, $F_{\alpha_{\mathrm{r}}\ll
1}^{(k=3)}=54t^{2}$, and $F_{\alpha_{\mathrm{r}}\ll 1}^{(k=4)}=240t^{2}$.
However, we note that this enhancement is present around the vicinity of a
small $\alpha_{\mathrm{r}}$ limit, and in a larger $\alpha_{\mathrm{r}}$ the
linear Rabi interferometer has a larger CFI. In contrast, a high amplitude
coherent state probe $\left|\beta\right\rangle$ with $\beta\gg 1$ can be
exploited in these high order Rabi interferometers in the same parameter
regions for better performances, with CFI scaling as
$F_{\alpha_{\mathrm{r}}}^{(k)}=2^{k+2}k^{2}t^{2}\beta^{2k-2}$.

##  Appendix O Enhancement of Rabi strengths by squeezing

We briefly note that online squeezing can further enhance the estimation
sensitivity simultaneously in both displacement components. For example, we
have

$\displaystyle
S[r]\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{X}]S[-r]=\exp[\mathrm{i}te^{2r}\hat{\sigma}_{\mathrm{x}}\hat{X}],$
$\displaystyle
S[-r]\exp[\mathrm{i}t\hat{\sigma}_{\mathrm{x}}\hat{P}]S[r]=\exp[\mathrm{i}te^{2r}\hat{\sigma}_{\mathrm{x}}\hat{P}].$
(47)

In principle, the squeezing transformation can also be provided by the Rabi
interactions. Eq. (12) implies such an amplification, as the qubit rotation
produced by the Rabi interferometer can be transferred back to the oscillator
displacement by another set of interactions. We leave again the detailed
analysis of such a strategy for future investigations.

##  Appendix P Interferometers based on other types of interactions

A JC interaction under RWA is experimentally available in much broader systems
than Rabi interactions with significantly lower difficulty in the
implementation. Overcoming an interferometer built from such JC interactions
is an important benchmark in the clarification of the role of the counter-
rotating term in Rabi interaction and architecture built from it for the
estimation. We first note that the JC interferometer does not alter the ground
state of joint qubit-oscillator system
$\left|\psi\right\rangle_{\mathrm{in}}=U_{\mathrm{JC}}\left|0\right\rangle_{\mathrm{C}}\left|\mathrm{g}\right\rangle_{\mathrm{A}}=\left|0\right\rangle_{\mathrm{C}}\left|\mathrm{g}\right\rangle_{\mathrm{A}}$
and is thus effectless in this case, and therefore an initial preparation of
the excited state in the qubit is required for the estimation beyond the
classical benchmark. The action of a JC interaction on such a qubit excited
state prepares an entangled state
$\left|\psi\right\rangle_{\mathrm{in}}=U_{\mathrm{JC}}\left|0\right\rangle_{\mathrm{C}}\left|\mathrm{e}\right\rangle_{\mathrm{A}}=\cos[\tau]\left|0\right\rangle_{\mathrm{C}}\left|\mathrm{e}\right\rangle_{\mathrm{A}}+i\sin[\tau]\left|1\right\rangle_{\mathrm{C}}\left|\mathrm{g}\right\rangle_{\mathrm{A}}$.
The QFI of this state about $\alpha_{\mathrm{r}}$ after the displacement is
given as

$\displaystyle Q_{\mathrm{\alpha_{\mathrm{r}}}}=8-4\cos[2\tau],$ (48)

having the maximum value $12$ at $\tau=\frac{\mathrm{\pi}}{2}$ at which the
single quanta $\left|1\right\rangle_{\mathrm{C}}$ in the oscillator is
prepared completely decoupled from the qubit. This QFI in (48) is observed to
be always smaller than that of the state prepared by Rabi interaction of the
same strength $t=\tau$.

We can now proceed to calculate the CFI of the JC interferometric setup with
only qubit detectors. After the unknown signal displacement and the inverse JC
interaction, the state evolves to
$\left|\psi\right\rangle_{\mathrm{out}}=U_{\mathrm{JC}}^{-1}\hat{D}[\alpha]\left|\psi\right\rangle_{\mathrm{in}}$.
Now we can detect on the qubit in the energy eigenbasis, and the CFI can be
calculated from the following probability of detection outcome in the qubit
excited state in Fock basis expansion:

$\displaystyle
P_{e}=\sum_{n}\frac{\mathrm{e}^{-|\alpha|^{2}}|\alpha|^{2n}\left\\{\tau(n-|\alpha|^{2}+1)\sin\tau\leavevmode\nobreak\
\mathrm{sinc}\left(\sqrt{n+1}\tau\right)+\cos\tau\cos\left(\sqrt{n+1}\tau\right)\right\\}^{2}}{n!}.$
(49)

This probability is dependent only on $|\alpha|$ and thus naturally more
suitable for the estimation of it, but not the phase of the displacement
parameter $\arg[\alpha]$. This limitation in the full characterization of the
displacement including the phase can be partially mitigated by exploiting a
known auxiliary $D[\beta]$ or equivalently a coherent state probe, by which
the phase relation between $\alpha$ and $\beta$ can be used. This summation in
(49), however, is difficult to be performed analytically, and we perform a
numerical simulation.

We note the equivalence of the protocols with those achieved by the dispersive
interactions available in superconducting systems vanLoock2008HybridOptics .
For example, the interferometer by dispersive interactions can be achieved
with the total operation before the detection:

$\displaystyle\exp[-\mathrm{i}\frac{\mathrm{\pi}}{2}\hat{\sigma}_{\mathrm{z}}\hat{n}_{\mathrm{C}}]\exp[-\mathrm{i}\frac{\mathrm{\pi}}{4}\hat{\sigma}^{\prime}_{\mathrm{z}}\hat{n}_{\mathrm{C}}]D_{\mathrm{C}}[\alpha]\exp[\mathrm{i}\frac{\mathrm{\pi}}{4}\hat{\sigma}^{\prime}_{\mathrm{z}}\hat{n}_{\mathrm{C}}]\exp[\mathrm{i}\frac{\mathrm{\pi}}{2}\hat{\sigma}_{\mathrm{z}}\hat{n}_{\mathrm{C}}]\left|-\right\rangle_{\mathrm{A}_{1}}\left|-\right\rangle_{\mathrm{A}_{1}}\left|\beta\right\rangle_{\mathrm{C}},$
(50)

which can be shown to have the same CFI, in a similar spirit to PenasaPRA2016
. Furthermore, if we have an access to the squeezed state in the oscillator,
we can obtain an entangled displaced squeezed state by the same set of gates.
We leave this possibility again for a future investigation.

##  Appendix Q Minor noises

Dependence of CFI on the target variable under noise We note that the Rabi
interferometer has a dependence on the target parameter $\alpha_{\mathrm{r}}$
in the precision under noise, not well explained by CFI in an ideal situation.
Let us consider the example of the exact identity channel
$\alpha_{\mathrm{r}}=0$, where the final qubit state is equal to the input
qubit state after the interferometer regardless of $t$. This can cause an
issue of indeterminacy in the calculation of CFI when a detector is detecting
in that basis, as the orthogonal state probability and its derivatives of the
probabilities by $\alpha_{\mathrm{r}}$ give $0$. For non-zero
$\alpha_{\mathrm{r}}$, a similar problem happens at specific
$t_{\mathrm{inst}}=\mathrm{\pi}/(2\sqrt{2}\alpha_{\mathrm{r}})$ when the noise
is weak. Both of these instabilities can be avoided in practice, as generally
the displace signal is weak but not strictly zero, and therefore
$t_{\mathrm{inst}}$ is much larger than experimentally available values.
Alternatively, we can avoid this issue by adopting a different input qubit
state, such as $\left|\pm_{\mathrm{i}}\right\rangle_{\mathrm{A}}$. We notice
that these instability points tend to make CFIs prone to noises, e.g.
oscillator thermal noise. This instability is impacted by any noise in the
form of the fluctuation of detection probabilities, which are smeared by
noise, thus sharply reducing CFI.

Loss on CV mode The boson loss on the CV mode is a special case of the heating
with the replacement $\bar{n}=0$, and can be described by a Lindblad operator
with $L=\sqrt{\gamma}\hat{a}$. The loss parameter can be redefined as
$\sqrt{\eta}=\mathrm{e}^{-\gamma{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t_{f}}}$,
where $t_{f}$ is the duration in time when the loss is being applied. The
effect of loss on the Rabi interferometer can be analytically calculated, and
the probabilities are obtained as

$\displaystyle
P_{e}=\frac{1}{2}\left(1-\mathrm{e}^{-2t^{2}\left(1-\sqrt{\eta}\right)}\cos[2\sqrt{2\eta}t\alpha_{\mathrm{r}}]\right),\leavevmode\nobreak\
P_{g}=1-P_{e}.$ (51)

The CFI calculated from these probabilities is given as

$\displaystyle F_{\alpha_{\mathrm{r}}}=\frac{8\eta
t^{2}\cos^{2}\left(2\sqrt{2\eta}t\alpha_{\mathrm{r}}\right)}{\mathrm{e}^{4\left(1-\sqrt{\eta}\right)t^{2}}-\sin^{2}\left(2\sqrt{2\eta}t\alpha_{\mathrm{r}}\right)}.$
(52)

This Fisher information can be rewritten as $F_{\alpha_{\mathrm{r}}}=8\eta
t^{2}\frac{B}{A-(1-B)}$ with $A=\mathrm{e}^{4t^{2}\left(1-\sqrt{\eta}\right)}$
and $B=\cos^{2}\left(2\sqrt{2\eta}t\alpha_{\mathrm{r}}\right)$, showing
monotonous behaviors in $A$ and $B$ in the range of values
$A\in[1,\infty]$,$B\in[0,1]$. For the target values which can be most
sensitively estimated satisfies $B=1$, in which case the CFI is maximized as
$F_{\alpha_{\mathrm{r}}}=8\eta\frac{t^{2}}{A}$. The maximum CFI can be found
in WDL as $\frac{8\eta}{fe}$ at optimized $t=f^{-1/2}$ for
$A=\mathrm{e}^{ft^{2}}$. Here $f=4(1-\sqrt{\eta})$.

Gaussian distribution of the displacement signal We can now consider a case
where the unknown displacement does not have a precise strength due to the
existence of noise force but is uncertain with a Gaussian distribution. This
presents one major effect, especially when an external force is weak. The
encoded state from the input state $\rho$ is then given as

$\displaystyle\rho_{\mathrm{out}}=(2\mathrm{\pi}\hat{\sigma}_{\mathrm{r}}\hat{\sigma}_{\mathrm{i}})^{-1}\int
d\alpha_{\mathrm{r}}d\alpha_{\mathrm{i}}\exp[-\frac{(\alpha_{\mathrm{r}}-\alpha_{r0})^{2}}{2{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma}_{\mathrm{r}}^{2}}-\frac{(\alpha_{\mathrm{i}}-\alpha_{i0})^{2}}{2{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(53)

Now in the Rabi interferometer for the estimation of the real displacement
parameter, the probabilities of detection are given as

$\displaystyle
P_{e}=\frac{1}{2}\left(1-\mathrm{e}^{-4\sigma_{\mathrm{r}}^{2}t^{2}}\cos\left(2\sqrt{2}\alpha_{r0}t\right)\right),\leavevmode\nobreak\
P_{g}=1-P_{e}.$ (54)

The CFI from these probabilities is given as

$\displaystyle
F_{\alpha_{\mathrm{r}}}=\frac{8t^{2}\cos^{2}\left(2\sqrt{2}\alpha_{r0}t\right)}{\mathrm{e}^{8\sigma_{\mathrm{r}}^{2}t^{2}}-\sin^{2}\left(2\sqrt{2}\alpha_{r0}t\right)}.$
(55)

We can see the correspondence with the previous result when
$\sigma_{\mathrm{r}}\rightarrow 0$. Interestingly, at strengths
$t\alpha_{r0}=\frac{\mathrm{\pi}}{2\sqrt{2}}$, $F_{\alpha_{\mathrm{r}}}=0$
regardless of $\sigma_{\mathrm{r}}$. Eq. (55) can be rewritten as
$F_{\alpha_{\mathrm{r}}}=8t^{2}\frac{B_{G}}{A_{G}-(1-B_{G})}$ with
$A_{G}=\mathrm{e}^{8\sigma_{\mathrm{r}}^{2}t^{2}}$ and
$B_{G}=\cos^{2}\left(2\sqrt{2}\alpha_{r0}t\right)$, showing a monotonous
behavior in the ranges $A_{G}\in[1,\infty],\leavevmode\nobreak\
B_{G}\in[0,1]$. For the optimal value of $\alpha_{r0}$ accessible by an
adaptive strategy, this is further reduced as
$F_{\alpha_{\mathrm{r}}}=8t^{2}A_{G}^{-1}$. Here, the maximal CFI is given as
$\frac{8}{f_{G}e}$ at $t_{G}=f_{G}^{-1/2}$, with
$f_{G}=8\sigma_{\mathrm{r}}^{2}$.

Qubit heating Qubit heating is described by a Lindblad equation
$\partial_{t}\rho=\sum_{i=1,2}L_{\mathrm{i}}\rho
L_{\mathrm{i}}^{\dagger}-\frac{1}{2}\\{L_{\mathrm{i}}^{\dagger}L_{\mathrm{i}}\rho\\}$
with $L_{1,2}=\sqrt{\gamma}\hat{\sigma}_{\mp}$, where $\gamma$ is the qubit
heating rate, not necessarily equivalent to the oscillator heating
rate$\gamma^{\mathrm{th}}$. The solution of this equation is given
analytically as

$\displaystyle\rho[t]=\begin{pmatrix}\rho_{ee}[0]\frac{1+\mathrm{e}^{-2\gamma
t}}{2}+\rho_{gg}[0]\frac{1-\mathrm{e}^{-2\gamma
t}}{2}&\rho_{ge}[0]\mathrm{e}^{-\gamma t}\\\ \rho_{eg}[0]\mathrm{e}^{-\gamma
t}&\rho_{gg}[0]\frac{1+\mathrm{e}^{-2\gamma
t}}{2}+\rho_{ee}[0]\frac{1-\mathrm{e}^{-2\gamma t}}{2}\end{pmatrix}.$ (56)

For the initial qubit density matrix
$\rho[0]=(\cos[\sqrt{2}t\alpha_{\mathrm{r}}]\left|\mathrm{e}\right\rangle+\sin[\sqrt{2}t\alpha_{\mathrm{r}}]\left|\mathrm{g}\right\rangle).h.c$,
we can obtain the probabilities from this solution and the Fisher information
for heating time $t^{\prime}$ is calculated as

$\displaystyle
F_{\alpha_{\mathrm{r}}}=\frac{8t^{2}\cos^{2}\left(2\sqrt{2}t\alpha_{\mathrm{r}}\right)}{\mathrm{e}^{4\gamma
t^{\prime}}-\sin^{2}\left(2\sqrt{2}t\alpha_{\mathrm{r}}\right)}.$ (57)

This Fisher information can be rewritten as
$F_{\alpha_{\mathrm{r}}}=8t^{2}\frac{B_{H}}{A_{H}-(1-B_{H})}$ with
$A_{H}=\mathrm{e}^{4\gamma t^{\prime}}\in[1,\infty]$ and
$B_{H}=\cos^{2}[2\sqrt{2}t\alpha_{\mathrm{r}}]\in[0,1]$, showing a monotonous
bahavior of the maximal value as $8t^{2}/A_{H}$.

Qubit relaxation Qubit relaxation is needed for the full consideration of the
qubit errors in our systems, especially in superconducting systems, even
though it is a weaker error than qubit dephasing. We can describe qubit
relaxation by spontaneous emission by solving the Lindblad master equation
with Lindblad operator $\hat{\sigma}_{-}$. This noise is impacting many
systems of trapped ions and superconducting systems, although weaker than the
dephasing or boson loss. The solution of the Lindblad equation from any
arbitrary input qubit state is given by a single parameter
$q=\mathrm{e}^{-\gamma t_{\mathrm{relax}}/2}$, where $\gamma
t_{\mathrm{relax}}$ is a dimensionless loss parameter as:
$c_{\mathrm{gg}}\left|\mathrm{g}\right\rangle\left\langle\mathrm{g}\right|+c_{\mathrm{ee}}\\{q^{2}\left|\mathrm{e}\right\rangle\left\langle\mathrm{e}\right|+(1-q^{2})\left|\mathrm{g}\right\rangle\left\langle\mathrm{g}\right|\\}+qc_{\mathrm{eg}}\left|\mathrm{e}\right\rangle\left\langle\mathrm{g}\right|+qc_{\mathrm{ge}}\left|\mathrm{g}\right\rangle\left\langle\mathrm{e}\right|$.

For the same setups as before, we simply obtain the CFI as

$\displaystyle
F_{\alpha_{\mathrm{r}}}=\frac{8t^{2}B_{\mathrm{r}}}{q^{-2}-(1-B_{\mathrm{r}})},$
(58)

where $q=\mathrm{e}^{-\gamma t_{\mathrm{relax}}/2}$ is the qubit relaxation
parameter, and $B_{\mathrm{r}}=\cos^{2}[2\sqrt{2}t\alpha_{\mathrm{r}}]$, again
showing a monotonous behavior for the zero displacement limit
$\alpha_{\mathrm{r}}=0$.

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