# Broadband Multidimensional Variational Measurement with Non-Symmetric
Coupling

Aleksandr A. Movsisian  Faculty of Physics, M.V. Lomonosov Moscow State
University, Leninskie Gory, Moscow 119991, Russia  Albert I. Nazmiev  Faculty
of Physics, M.V. Lomonosov Moscow State University, Leninskie Gory, Moscow
119991, Russia  Andrey B. Matsko  Jet Propulsion Laboratory, California
Institute of Technology, 4800 Oak Grove Drive, Pasadena, California
91109-8099, USA  Sergey P. Vyatchanin  Faculty of Physics, M.V. Lomonosov
Moscow State University, Leninskie Gory, Moscow 119991, Russia  Quantum
Technology Centre, M.V. Lomonosov Moscow State University, Leninskie Gory,
Moscow 119991, Russia  Faculty of Physics, Branch of M.V. Lomonosov Moscow
State University in Baku,
1 Universitet street, Baku, AZ1144, Azerbaijan

(August 28, 2024)

###### Abstract

A broadband multidimensional variational measurement allows overcoming the
Standard Quantum Limit (SQL) of a classical mechanical force detection for a
mechanical oscillator. In this measurement quantum back action, which perturbs
the evolution of a mechanical oscillator, can be completely removed in a broad
detection frequency band after post-processing. The measurement is performed
by optical pumping of the central optical mode and analyzing the light
escaping the two other optical modes, which have the frequency separation with
the central mode equal to the mechanical frequency. To realize such a scheme
in practice one either needs to use a very long optical interferometer or
should utilize optical modes belonging to different mode families. In the
second case the modes have different geometries and their coupling with the
mechanical mode is not identical. Here we analyze a general case of the non-
symmetric measurement scheme, in which the coupling strengths with the light
modes are not equal to each other, and take into account optical losses. We
found that the back action can be completely excluded from the measurement
result in the case of the asymmetric lossless system. The nonzero loss limits
the sensitivity. An experimental implementation of the proposed scheme is
discussed.

##  I Introduction

Optical transducers are used to characterize mechanical motion, which changes
a quadrature amplitude of the probe light. The sensitivity of the measurement
can be extremely high. For instance, it was demonstrated that optical
gravitational wave detectors are capable of measuring the relative mechanical
displacement much smaller than a proton size [1, 2, 3, 4, 5, 6, 7]. The search
for better detection methods is still actual nowadays. In this paper we study
theoretically a realistic optical detection scheme that can potentially
improve sensitivity of the optical detection of a mechanical force.

The optical detector of a classical force involves optical and mechanical
subsystems. The thermal fluctuations in the mechanical probe system (Nyquist
noise) and quantum noise of the meter introduce fundamental limits of the
measurement sensitivity. The impact of the thermal noise can be considerably
reduced if one measures a variation of the position during a time period that
is much smaller than the mechanical ring down time [8, 9]. However, one cannot
avoid the thermal noise that is spectrally overlapping with the signal and
coming to the system from the same channel as the signal. The quantum optical
noise restricts sensitivity by the well-known standard quantum limit (SQL) [8,
9]. SQL is a consequence of noncommutativity between the optical probe noise
and the quantum back action noise associated with the ponderomotive action of
the probe light on the mechanical system. However there are several ways to
overcome it.

In this work we study the optimization of the measurement technique involving
a few optical frequency harmonics. This method allows beating SQL. For
instance, a dichromatic optical probe may lead to observation of such
phenomena as negative radiation pressure [10, 11] and optical quadrature-
dependent quantum back action evasion [12].

A multidimensional variational measurement [13] is one of methods that involve
a few optical harmonics. Advantage of the measurement is its relatively broad
bandwidth. In such a measurement the force of interest acts on a mechanical
oscillator. The mechanical oscillator is coupled to a system with three
optical modes, whose frequencies $\omega_{\pm},\ \omega_{0}$ are separated by
the mechanical frequency $\omega_{m}$ so that
$\omega_{\pm}=\omega_{0}\pm\omega_{m}$. The measurement is performed by
optical pumping of the central optical mode $\omega_{0}$ and measuring the
light escaping the two other modes $\omega_{\pm}$. Detection of optimal
quadrature components of the output waves of modes $\omega_{\pm}$ separately
provides two channels registration. It allows to detect back action and to
remove it completely in broad band from the measured data via post processing.

Previous studies of the multidimensional variational measurements involved
exclusively ideal and symmetrical systems. However, practical realization of
the idea calls for usage of a not ideal configuration. To prove practical
validity of the measurement scheme we here further investigate the sensitivity
of broadband multidimensional variational measurement in the presence of
optical losses. We also consider the non-symmetric case when the relaxation
rates as well as couplings of the optical modes $\omega_{\pm}$ with probe
mechanical oscillator are not equal to each other. We found that it is still
possible to suppress the back action in the case of zero loss and the
asymmetry present. The loss limits the sensitivity, but the SQL still can be
beaten. The result proves robustness of the measurement technique. We also
discuss the feasibility of separation of the optical modes of interest
escaping the mechanical resonator. This particular problem is hard because the
reasonable frequency separation between the modes in the optical triplet is
less than a MHz.

##  II Physical Model

Figure 1: Frequencies $\omega_{0},\ \omega_{\pm}$ of three optical modes in
the cavity are separated by the frequency $\omega_{m}$ of mechanical
oscillator. Optical modes are coupled to the mechanical oscillator via optical
pressure. The relaxation rates of the optical modes are smaller than
mechanical frequency $\gamma,\ \gamma_{\pm}\ll\omega_{m}$. The middle mode
with frequency $\omega_{0}$ is resonantly pumped. The outputs of modes
$\omega_{\pm}$ are detected separately.

Let us consider an optical cavity with a triplet of optical modes
characterized with resonant frequencies $\omega_{-},\ \omega_{0},\
\omega_{+}$, separated by eigen frequency $\omega_{m}$ of mechanical
oscillator (Fig. 1). We assume that the relaxation rates of the optical modes
are different and characterized by the full width at the half maxima (FWHM)
equal to

$\displaystyle 2\gamma_{\pm}$ $\displaystyle=2(\gamma_{0\pm}+\gamma_{e\pm}),$
(1a) $\displaystyle 2\gamma$ $\displaystyle=2\gamma_{0}+\gamma_{e},$ (1b)

for modes $\omega_{\pm}$ and for $\omega_{0}$, respectively. Here
$\gamma_{0\pm},\ \gamma_{0}$ characterize effective transmittance of the input
mirror and $\gamma_{e},\ \gamma_{e\pm}$ stand for optical losses of the
cavity. We assume that the relaxation rates $\gamma_{e},\ \gamma_{e\pm}$ are
small if compared with the transmission ones

$\displaystyle\gamma_{e},\ \gamma_{e\pm}\ll\gamma_{0},\ \gamma_{0\pm}.$ (2)

The mechanical relaxation rate $\gamma_{m}$ is small in comparison with the
optical one. We also assume that the optical triplet is symmetric and matched
with the mechanical oscillator. The conditions of the resolved side band
interaction and the condition of optical loss smallness are also valid.

$\displaystyle\omega_{\pm}=\omega_{0}\pm\omega_{m},\quad\gamma_{m}\ll\gamma\ll\omega_{m}.\quad$
(3)

There are two external actions on the opto-mechanical system. The central mode
with frequency $\omega_{0}$ is resonantly pumped and a classical force of
interest is acting on the mechanical mass. The optical modes $\omega_{\pm}$
are not pumped. Mass of mechanical oscillator is a movable end mirror, which
provides coupling with optical modes. We detect the output of optical modes
$\omega_{\pm}$ to measure the force (Fig. 1). The quadrature components of the
sideband modes are measured separately by balanced homodyne detectors with
corresponding local oscillators having frequencies coincident with the
frequencies of the sideband modes $\omega_{\pm}$. The signal is inferred by
post-processing of the linear combination of the measured results.

The generalized Hamiltonian describing the system can be presented in form

$\displaystyle H$ $\displaystyle=H_{0}+H_{\text{int}}+H_{s}+$ (4a)
$\displaystyle+H_{T0}+H_{\gamma_{0}}+H_{Te}+H_{\gamma_{e}}+H_{T,\,m}+H_{\gamma_{m}},$
$\displaystyle H_{0}$
$\displaystyle=\hslash\omega_{+}\hat{c}_{+}^{\dagger}\hat{c}_{+}+\hslash\omega_{0}\hat{c}_{0}^{\dagger}\hat{c}_{0}+$
(4b)
$\displaystyle\qquad+\hslash\omega_{-}\hat{c}_{-}^{\dagger}\hat{c}_{-}+\hslash\omega_{m}\hat{d}^{\dagger}\hat{d},$
$\displaystyle H_{\text{int}}$
$\displaystyle=\frac{\hslash}{i}\left(\left[\eta_{-}\hat{c}_{0}^{\dagger}\hat{c}_{-}+\eta_{+}\hat{c}_{+}^{\dagger}\hat{c}_{0}\right]\hat{d}-\right.$
(4c)
$\displaystyle\qquad-\left.\left[\eta^{*}_{-}\hat{c}_{0}\hat{c}_{-}^{\dagger}+\eta^{*}_{+}\hat{c}_{+}\hat{c}_{0}^{\dagger}\right]\hat{d}^{\dagger}\right),$
$\displaystyle\hat{H}_{s}$
$\displaystyle=-F_{s}x_{0}\left(\hat{d}+\hat{d}^{\dagger}\right).$ (4d)

Here $\hslash$ is Plank constant. $H_{0}$ describes energies of the optical
modes and the mechanical oscillator, $\hat{c}_{0},\ \hat{c}^{\dagger}_{0},\
\hat{c}_{\pm},\ \hat{c}^{\dagger}_{\pm}$ are annihilation and creation
operators for the corresponding optical modes, $\hat{d},\ \hat{d}^{\dagger}$
are annihilation and creation operators of the mechanical oscillator. The
operator of coordinate $\hat{x}$ of the mechanical oscillator is presented in
form

$\displaystyle\hat{x}=x_{0}\left(\hat{d}+\hat{d}^{\dagger}\right),\quad
x_{0}=\sqrt{\frac{\hslash}{2m\omega_{m}}},$ (5)

where $m$ is the mass of the oscillator. $H_{\text{int}}$ is the interaction
Hamiltonian written in the rotation wave approximation for the optical and
mechanical modes. 111 The canonical interaction Hamiltonian
$H_{\text{int}}\sim(E_{0}+E_{+}+E_{-})^{2}x$, where $E_{0},E_{-},E_{+}$ are
electric fields of modes $0,-,+$ on surface of mirror (mass of oscillator),
can be transformed into form (4c) after omitting fast oscillating terms. We
here consider non symmetric interaction and introduce different coupling
constants $\eta_{\pm}\simeq x_{0}\omega_{0}/L$, where $L$ is the length of the
cavity. $H_{s}$ is a part of Hamiltonian describing signal force $F_{s}$.
$H_{T0}$ is the Hamiltonian describing the outer environment (regular and
fluctuational fields incident on the input mirror) and $H_{\gamma_{0}}$ is the
Hamiltonian of the coupling between the outer environment and the optical
modes, resulting in decay rate $\gamma_{0}$; $H_{Te}$ and $H_{\gamma_{e}}$
describe optical losses. The pump is also included into $H_{\gamma_{0}}$.
Similarly, $H_{T,\,m}$ is the thermal bath Hamiltonian and $H_{\gamma_{m}}$ is
the Hamiltonian describing coupling between the environment and the mechanical
oscillator resulting in a decay rate $\gamma_{m}$.

##  III Analysis

###  III.1 Basic equations

We denote the values of the input and output optical amplitudes as
$\hat{a}_{\pm,\,0}$ and $\hat{b}_{\pm,\,0}$, respectively. Using the
Hamiltonian (4) we derive the equations of motion for the intracavity slow
amplitudes of fields.

$\displaystyle\dot{\hat{c}}_{0}+\gamma\hat{c}_{0}$
$\displaystyle=\eta_{+}^{*}\hat{c}_{+}\hat{d}^{\dagger}-\eta_{-}\hat{c}_{-}\hat{d}+$
(6a)
$\displaystyle\qquad+\sqrt{2\gamma_{0}}\,\hat{a}_{0}+\sqrt{2\gamma_{e}}\,\hat{e}_{0},$
$\displaystyle\dot{\hat{c}}_{+}+\gamma_{+}\hat{c}_{+}$
$\displaystyle=-\eta_{+}\hat{c}_{0}\hat{d}+\sqrt{2\gamma_{0+}}\,\hat{a}_{+}+\sqrt{2\gamma_{e+}}\hat{e}_{+},$
(6b) $\displaystyle\dot{\hat{c}}_{-}+\gamma_{-}\hat{c}_{-}$
$\displaystyle=\eta_{-}^{*}\hat{c}_{0}\hat{d}^{\dagger}+\sqrt{2\gamma_{0-}}\,\hat{a}_{-}+\sqrt{2\gamma_{e-}}\,\hat{e}_{-},$
$\displaystyle\dot{\hat{d}}+\gamma_{m}\hat{d}$
$\displaystyle=\eta^{*}_{-}\hat{c}_{0}\hat{c}_{-}^{\dagger}+\eta^{*}_{+}\hat{c}_{0}^{\dagger}\hat{c}_{+}+\sqrt{2\gamma_{m}}\,\hat{q}+f_{s}.$
(6c)

Here operators $\hat{e}_{\pm}$ describe quantum fluctuations due to optical
losses, see definitions (1), $\hat{q}$ is the fluctuation force acting on
mechanical oscillator, and $f_{s}$ is the signal force. The temporal structure
of the force is defined in what follows.

The operators $\hat{a}_{\pm},\ \hat{e}_{\pm},\ \hat{q}$ are characterized by
the following commutators and correlators

$\displaystyle\left[\hat{a}_{\pm}(t),\hat{a}_{\pm}^{\dagger}(t^{\prime})\right]$
$\displaystyle=\left\langle\hat{a}_{\pm}(t)\,\hat{a}_{\pm}^{\dagger}(t^{\prime})\right\rangle=\delta(t-t^{\prime}),$
(7a)
$\displaystyle\left[\hat{e}_{\pm}(t),\hat{e}_{\pm}^{\dagger}(t^{\prime})\right]$
$\displaystyle=\left\langle\hat{e}_{\pm}(t)\,\hat{e}_{\pm}^{\dagger}(t^{\prime})\right\rangle=\delta(t-t^{\prime}),$
(7b) $\displaystyle\left[\hat{q}(t),\hat{q}^{\dagger}(t^{\prime})\right]$
$\displaystyle=\delta(t-t^{\prime}),$ (7c)
$\displaystyle\left\langle\hat{q}(t)\hat{q}^{\dagger}(t^{\prime})\right\rangle$
$\displaystyle=(2n_{T}+1)\,\delta(t-t^{\prime}),$ (7d) $\displaystyle
n_{T}=\left(e^{\hslash\omega_{m}/\kappa_{B}T}-1\right)^{-1}.$ (7e)

Here $\langle\dots\rangle$ stands for the ensemble averaging. $n_{T}$ is the
thermal number of mechanical quanta, $\kappa_{B}$ is the Boltzmann
constant,and $T$ is the ambient temperature.

The input-output relations connecting the incident ($\hat{a}_{\pm}$) and
intracavity ($\hat{c}_{\pm}$) amplitudes with output ($\hat{b}_{\pm}$)
amplitudes are

$\displaystyle\hat{b}_{\pm}=-\hat{a}_{\pm}+\sqrt{2\gamma_{0\pm}}\hat{c}_{\pm}.$
(8)

It is convenient to separate the expectation values of the wave amplitudes at
frequency $\omega_{0}$ (described by block letters) as well as its fluctuation
part (described by small letters) and assume that the fluctuations are small:

$\displaystyle\hat{c}_{0}$ $\displaystyle\Rightarrow C_{0}+c_{0},\quad$ (9)

here $C_{0}$ stands for the expectation value of the field amplitude in the
central optical mode and $c_{0}$ represent the quantum fluctuations of the
field confined in the mode, $|C_{0}|^{2}\gg\langle
c_{0}^{\dagger}c_{0}\rangle$. Similar expressions can be written for the
sideband optical modes as well as the mechanical mode. The normalization of
the amplitudes is selected so that $\hslash\omega_{0}|A_{0}|^{2}$ describes
the optical power of incident wave [14].

We assume in what follows that the expectation amplitudes are real, same as
the coupling constants $\eta_{\pm}$

$\displaystyle A_{0}=A_{0}^{*},\quad
C_{0}=C_{0}^{*}=\sqrt{\frac{2}{\gamma_{0}}}A_{0},\quad\eta_{\pm}=\eta_{\pm}^{*}.$
(10)

The Fourier transform of operators, for example, $\hat{a}_{\pm}$ is defined as
follows

$\displaystyle\hat{a}_{\pm}(t)$
$\displaystyle=\int_{-\infty}^{\infty}a_{\pm}(\Omega)\,e^{-i\Omega
t}\,\frac{d\Omega}{2\pi}.$ (11a) For operators $a_{\pm}(\Omega)$ the following
commutators and correlators are valid:
$\displaystyle\left[a_{\pm}(\Omega),a_{\pm}^{\dagger}(\Omega^{\prime})\right]$
$\displaystyle=2\pi\,\delta(\Omega-\Omega^{\prime}),$ (11b)
$\displaystyle\left\langle
a_{\pm}(\Omega)a_{\pm}^{\dagger}(\Omega^{\prime})\right\rangle$
$\displaystyle=2\pi\,\delta(\Omega-\Omega^{\prime})$ (11c)

Similar expressions can be written for the other noise operators
($\hat{e}_{\pm},\ \hat{q}$).

Substituting (9) and (10) into the equations of motion (6) and keeping only
terms of first order of smallness, we obtain

$\displaystyle(\gamma_{+}-i\Omega)c_{+}(\Omega)$
$\displaystyle=-\eta_{+}C_{0}d(\Omega)+$ (12a)
$\displaystyle+\sqrt{2\gamma_{0+}}\,a_{+}(\Omega)+\sqrt{2\gamma_{e+}}e_{+}(\Omega),$
$\displaystyle(\gamma_{-}-i\Omega)c_{-}(\Omega)$
$\displaystyle=\eta_{+}C_{0}d^{\dagger}(-\Omega)+$ (12b)
$\displaystyle+\sqrt{2\gamma_{0-}}\,a_{-}(\Omega)+\sqrt{2\gamma_{e-}}\,e_{-}(\Omega),$
$\displaystyle(\gamma_{m}-i\Omega)d(\Omega)$
$\displaystyle=C_{0}\big{[}\eta_{-}c_{-}^{\dagger}(-\Omega)+\eta_{+}c_{+}(\Omega)\big{]}+$
(12c) $\displaystyle\qquad+\sqrt{2\gamma_{m}}\,\hat{q}(\Omega)+f_{s}(\Omega).$

Outputs $b_{\pm}$ around frequencies $\omega_{\pm}$ have to be detected
separately. We also see that fluctuation waves around $\omega_{0}$ do not
influence on field components in the vicinity of frequencies $\omega_{\pm}$
and the first equation (6a) does not couple with others, so we omit it in the
further consideration.

We assume that the signal force is a resonant square pulse acting during time
interval $\tau$ ($\omega_{m}\tau\gg 1$):

$\displaystyle F_{S}(t)$ $\displaystyle=F_{s0}\sin(\omega_{m}t+\psi_{f})=$
(13) $\displaystyle=$ $\displaystyle
i\left(F_{s}(t)e^{-i\omega_{m}t}-F_{s}^{*}(t)e^{i\omega_{m}t}\right),\quad-\frac{\tau}{2}<t<\frac{\tau}{2},$
$\displaystyle f_{s}(\Omega)$
$\displaystyle=\frac{F_{s}(\Omega)}{\sqrt{2\hslash\omega_{m}m}},$ (14)
$\displaystyle f_{s0}(\Omega)$
$\displaystyle=\frac{F_{s0}(\Omega)}{\sqrt{2\hslash\omega_{m}m}}=2f_{s}(\Omega).$
(15)

where $F_{s}(\Omega)\neq F_{s}^{*}(-\Omega)$ is the Fourier amplitude of
$F_{s}(t)$.

Let us introduce quadrature amplitudes of amplitude and phase

$\displaystyle a_{\pm a}$
$\displaystyle=\frac{a_{\pm}(\Omega)+a_{\pm}^{\dagger}(-\Omega)}{\sqrt{2}}\,,$
(16a) $\displaystyle a_{\pm\phi}$
$\displaystyle=\frac{a_{\pm}(\Omega)-a_{\pm}^{\dagger}(-\Omega)}{i\sqrt{2}}\,.$
(16b)

Using (12) we obtain equations for amplitude quadratures

$\displaystyle(\gamma_{+}-i\Omega)c_{+a}$
$\displaystyle+\eta_{+}C_{0}d_{a}=\sqrt{2\gamma_{0+}}{a}_{+a}+$ (17a)
$\displaystyle+\sqrt{2\gamma_{e+}}e_{+a},$
$\displaystyle(\gamma_{-}-i\Omega)c_{-a}$
$\displaystyle-\eta_{-}C_{0}d_{a}=\sqrt{2\gamma_{-}}{a}_{-a}+$ (17b)
$\displaystyle+\sqrt{2\gamma_{e-}}e_{-a},$
$\displaystyle(\gamma_{m}-i\Omega)d_{a}$ $\displaystyle-
C_{0}\Big{(}\eta_{+}c_{+a}+\eta_{-}c_{-a}\Big{)}=$ (17c)
$\displaystyle=\sqrt{2\gamma_{m}}q_{a}+f_{s\,a},$

and for phase quadratures

$\displaystyle(\gamma_{+}-i\Omega)c_{+\phi}$
$\displaystyle+\eta_{+}C_{0}d_{\phi}=\sqrt{2\gamma_{0+}}{a}_{+\phi}+$ (18a)
$\displaystyle+\sqrt{2\gamma_{e+}}e_{+\phi},$
$\displaystyle(\gamma_{-}-i\Omega)c_{-\phi}$
$\displaystyle+\eta_{-}C_{0}d_{\phi}=\sqrt{2\gamma_{-}}{a}_{-\phi}+$ (18b)
$\displaystyle+\sqrt{2\gamma_{e-}}e_{-\phi},$
$\displaystyle(\gamma_{m}-i\Omega)d_{\phi}$ $\displaystyle-
C_{0}\Big{(}\eta_{+}c_{+\phi}-\eta_{-}c_{-\phi}\Big{)}=$ (18c)
$\displaystyle=\sqrt{2\gamma_{m}}q_{\phi}+f_{s\,\phi}.$

Set (17) for amplitude quadratures does not depend on set (18) for phase
quadratures, so we analyze the set for amplitude quadratures only.

###  III.2 Quantum noise spectral density

Detailed derivation of the expressions for the output quadratures is presented
in Appendix B. Here we analyze the resultant expressions and derive expression
for the spectral density of the quantum noise of the meter.

Inequality of $\eta_{\pm}$ and $\gamma_{\pm}$ introduces classical dynamic
back action manifesting by the optical damping rate $G$

$\displaystyle
G=\left[\frac{\eta_{+}^{2}}{\gamma_{+}-i\Omega}-\frac{\eta_{-}^{2}}{\gamma_{-}-i\Omega}\right]C_{0}^{2}.$
(19)

The complete mechanical relaxation rate $\Gamma_{m}=\gamma_{m}+G$ is a sum of
the intrinsic relaxation rate $\gamma_{m}$ and the introduced optical damping
$G$.

Assuming that we are able to measure $b_{+a}$ and $b_{-a}$ separately (the
quadratures are not commuting Hermitian operators), present their weighted sum
$\Sigma$ as

$\displaystyle\Sigma=z_{+}\,\frac{b_{+a}\big{(}\Gamma_{m}-i\Omega\big{)}}{A_{+}\big{(}z_{-}-z_{+}\big{)}}+z_{-}\,\frac{b_{-a}\big{(}\Gamma_{m}-i\Omega\big{)}}{A_{-}\big{(}z_{-}-z_{+}\big{)}},$
(20)

introducing coefficients

$\displaystyle A_{\pm}$
$\displaystyle=\frac{\sqrt{2\gamma_{0\pm}}\,\eta_{\pm}C_{0}}{(\gamma_{\pm}-i\Omega)},\
B_{e\pm}=\frac{\sqrt{2\gamma_{0\pm}}(\Gamma_{m}-i\Omega)}{\eta_{\pm}C_{0}},$
(21a) $\displaystyle B_{\pm}$
$\displaystyle=\frac{(\gamma_{0\pm}-\gamma_{e\pm}+i\Omega)(\Gamma_{m}-i\Omega)}{\sqrt{2\gamma_{0\pm}}\,\eta_{\pm}C_{0}},$
(21b) $\displaystyle Y_{\pm}=\frac{A_{\pm}}{B_{\pm}},\quad
Y_{e\pm}=\frac{A_{\pm}}{B_{e\pm}}$ (21c)

The complex functions $z_{\pm}(\Omega)$, depending on spectral frequency
$\Omega$, are free post-processing parameters, which should be selected in an
optimal way. Due to homogeneity of the weighted sum $\Sigma(\lambda
z_{+},\lambda z_{-})=\lambda\Sigma(z_{+},z_{-})$, we can introduce one
parameter $y$ instead of two parameters $z_{\pm}$ in the optimization task,

$y+\frac{1}{2}=\frac{z_{-}}{z_{-}-z_{+}},\quad
y-\frac{1}{2}=\frac{z_{+}}{z_{-}-z_{+}}.$ (22)

Assuming that the input fields $a_{\pm}$ are in the vacuum states and using
(50), we write for the quantum noise spectral density:

$\displaystyle S_{f}$ $\displaystyle=S_{qu}+S_{T},$ (23a) $\displaystyle
S_{qu}$
$\displaystyle=|B_{+}|^{2}\left|y-\frac{1}{2}+Y_{+}\right|^{2}+|B_{-}|^{2}\left|y+\frac{1}{2}+Y_{-}\right|^{2}+$
$\displaystyle+\frac{\gamma_{e+}}{\gamma_{0+}}|B_{e+}|^{2}\left|y-\frac{1}{2}+Y_{e+}\right|^{2}+$
(23b)
$\displaystyle+\frac{\gamma_{e-}}{\gamma_{0-}}|B_{e-}|^{2}\left|y+\frac{1}{2}+Y_{e-}\right|^{2},$
(23c) $\displaystyle S_{T}$
$\displaystyle=2\gamma_{m}\left(n_{T}+\frac{1}{2}\right).$ (23d)

Here $S_{qu}$ relates to quantum noise, $S_{T}$ stands for the thermal noise
in (23), the terms (23b), (23c) appear due to the optical losses. The first
two terms in $S_{qu}$ describe main optical noise, values $Y_{\pm}$ are
generated due to back action.

Let us assume that the condition [8, 9] of smallness of thermal noise compared
with SQL

$\displaystyle B=\frac{n_{T}\omega_{m}\tau}{Q}\ll 1$ (24)

is fulfilled ($Q$ is a mechanical quality factor). The main requirement for
this is a large ring down time $\gamma_{m}^{-1}$ and a fast interrogation time
$\tau$, i.e. $\gamma_{m}\tau\ll 1$. For the parameters listed in Table 1
factor $B\simeq 0.2$.

###  III.3 Symmetric system

In the simplest case of lossless ($\gamma_{e\pm}=0$, terms (23b), (23c)
disappear) and symmetric ($B_{+}=B_{-}\equiv B,\ Y_{+}=Y_{-}\equiv Y$) system
the back action can be excluded completely by optimally choosing
$y_{\text{opt}}=-Y$:

$\displaystyle S_{qu}|_{\text{noLoss}}^{\text{sym}}$
$\displaystyle=|B|^{2}/2=\frac{\gamma_{m}^{2}+\Omega^{2}}{\mathcal{K}},\quad
y_{\text{opt}}=-Y,$ (25)
$\displaystyle\mathcal{K}=\frac{4\gamma\eta^{2}C_{0}^{2}}{\gamma^{2}+\Omega^{2}}.$
(26)

In this case back action is completely removed and spectral density
monotonically decreases with power ($\sim\mathcal{K}$) and can be made smaller
$S_{SQL}=2\sqrt{\gamma_{m}^{2}+\Omega^{2}}$ (38) [12] (see reminder on SQL in
Appendix A).

###  III.4 Measurement optimization

In general case in order to find the minimum of $S_{qu}$ we introduce the
optimal complex function $y_{\text{opt}}$

$\displaystyle y_{\text{opt}}$
$\displaystyle=\frac{\left|B_{+}\right|^{2}}{\mathbb{B}^{2}}\left(\frac{1}{2}-Y_{+}\right)-\frac{\left|B_{-}\right|^{2}}{\mathbb{B}^{2}}\left(\frac{1}{2}+Y_{-}\right)+\Delta_{e},$
(27a) $\displaystyle\mathbb{B}^{2}$
$\displaystyle=\left|B_{+}\right|^{2}+\left|B_{-}\right|^{2}+\mathbb{B}_{e}^{2},$
(27b) $\displaystyle\mathbb{B}_{e}^{2}$
$\displaystyle=\frac{\gamma_{e+}}{\gamma_{0+}}|B_{e+}|^{2}+\frac{\gamma_{e-}}{\gamma_{0-}}|B_{e-}|^{2}\,,$
(27c) $\displaystyle\Delta_{e}$
$\displaystyle=\left[\frac{\gamma_{e+}}{\gamma_{0+}}\right]\frac{\left|B_{e+}\right|^{2}}{\mathbb{B}^{2}}\left(\frac{1}{2}-Y_{e+}\right)-$
(27d)
$\displaystyle\quad-\left[\frac{\gamma_{e-}}{\gamma_{0-}}\right]\frac{\left|B_{e-}\right|^{2}}{\mathbb{B}^{2}}\left(\frac{1}{2}+Y_{e-}\right)\,.$
(27e)

It should be substituted into (23) to find the minimum achievable noise. In
what follows we consider a few realizations of the measurement systems and
find the optimal measurement sensitivity for those cases.

Figure 2: Plots of ratio $R=S_{qu}/S_{SQL}$, of spectral densities without
optical losses for symmetric (28) (with $\gamma_{+}=\gamma_{-},\
\eta_{+}=\eta_{-}$) and non-symmetric cases (28) as a function of spectral
frequency $\Omega$, $\tau$ is the time of signal force action, other
parameters are taken from Table 1. Plot marked as $P\to 10P$ corresponds to
the non-symmetric case with 10 times increased pump $P_{in}$. Table 1:
Parameters of a mechanical oscillator (SiN membrane) and optical cavity, used
for estimates. Membrane
---
Mass, $m$ | 50 |  $10^{-9}$ g
Frequency, $\omega_{m}/2\pi$ | 350 |  $10^{3}$ Hz
Quality factor $Q=\frac{\omega_{m}}{2\gamma_{m}}$ | $10^{9}$ |
Temperature, $T$ | 20 | K∘
Thermal phonons number, $n_{T}$ | $1.2\cdot 10^{6}$ |
Time of signal force $\tau=30\cdot\frac{2\pi}{\omega_{m}}$ | $0.84\cdot 10^{-3}$ | sec
Cavity
Length of cavity | 10 | cm
Bandwidth , $\gamma_{0}+\gamma_{e}$ | 2.3 $10^{5}$ | s-1
$\gamma_{e},\ \gamma_{e\pm}$ | 2.3 $10^{3}$ | s-1
$\gamma_{\pm}$ | $(1\mp 1\%)\gamma_{0}$ |
Couplings $\eta_{\pm}$ | $\frac{\omega_{0}x_{0}}{L}(1\pm 3\%)$ |
Wave length, $\lambda=2\pi c/\omega_{0}$ | 1.55 |  $10^{-6}$ m
Input power $P_{in}$ | 1 |  $10^{-6}$ W

####  III.4.1 Non-symmetric case without optical losses

For the non-symmetric case ($\eta_{+}\neq\eta_{-}$) and for the absence of
optical losses ($\gamma_{e\pm}=0$) the formulas are more compact

$\displaystyle S_{qu}|_{\text{noLoss}}^{\text{non-sym}}$
$\displaystyle=\frac{\left|B_{+}\right|^{2}\left|B_{-}\right|^{2}}{\left|B_{+}\right|^{2}+\left|B_{-}\right|^{2}}\left|1-Y_{+}+Y_{-}\right|^{2},$
(28) $\displaystyle
y_{\text{opt}}=\frac{\left|B_{+}\right|^{2}}{\mathfrak{B}^{2}}$
$\displaystyle\left(\frac{1}{2}-Y_{+}\right)-\frac{\left|B_{-}\right|^{2}}{\mathfrak{B}^{2}}\left(\frac{1}{2}+Y_{-}\right),$
(29) Here
$\displaystyle\mathfrak{B}^{2}=\left|B_{+}\right|^{2}+\left|B_{-}\right|^{2}$
(30)

We see that in non-symmetric case back action can be excluded only partially
(term $(Y_{+}-Y_{-})$ in (28)).

In nearly-resonant case, $\gamma_{\pm}\gg\Omega$, formula (28) can be
simplified using definitions (19) and (21a)

$\displaystyle S_{qu}|_{\text{noLoss}}^{\text{non-sym}}$
$\displaystyle\underbrace{\simeq}_{\gamma_{\pm}\gg\Omega}\frac{1}{2G_{+}}\left(\big{(}\gamma_{m}-G)^{2}+\Omega^{2}\right),$
(31) $\displaystyle G_{+}=\frac{\eta_{+}^{2}C_{0}^{2}}{\gamma_{+}}$
$\displaystyle+\frac{\eta_{-}^{2}C_{0}^{2}}{\gamma_{-}},\quad
G\simeq\frac{\eta_{+}^{2}C_{0}^{2}}{\gamma_{+}}-\frac{\eta_{-}^{2}C_{0}^{2}}{\gamma_{-}}.$
(32)

We see that the introduced optical damping $G$ can be zero even in the non-
symmetric case, if $\eta_{+}^{2}/\gamma_{+}=\eta_{-}^{2}/\gamma_{-}$. The
spectral density practically coincides with (25). It means that in the
lossless asymmetric measurement system the back action can be completely
excluded (at corresponding choice of $Y_{\pm}$). The normalized spectral
density $S_{qu}$ monotonically decreases with the optical power increase. This
is an important finding of our research.

A non-zero value $G$ of optical damping characterizes back action, which
partially degrades sensitivity. For condition $\gamma_{m}\ll G,\ \Omega$ we
have

$\displaystyle S_{qu}|_{\text{noLoss}}^{\text{non-sym}}$
$\displaystyle=\frac{G^{2}+\Omega^{2}}{2G_{+}}\geq\left|\frac{G\Omega}{G_{+}}\right|<S_{SQL}\simeq
2|\Omega|.$ (33)

So while in the non-symmetric case one can surpass SQL, the sensitivity
becomes limited.

Figure 3: Plots of ratio $R=S_{qu}/S_{SQL}$, of spectral densities with
optical losses for non-symmetric cases (23) as a function of spectral
frequency $\Omega$. For the plot marked by "NonSymLo" the parameters are taken
from Table 1 (the dashed line "NonSymNoLo" correspond to same parameters but
zero optical losses). Plots "NonSymLo2" and "NonSymNoLo2" corresponds to 10
times larger input pump $P_{in}$. $\tau$ is the time of signal force action.

####  III.4.2 Non-symmetric case with optical losses

We studied the most general case of the sensor will loss numerically. using
the expressions (23) and considering the optimal function $y_{\text{opt}}$
(27b). The plots of normalized spectral densities are presented in Fig. 3 for
the non-symmetric case with losses. The numerical parameters listed in Table 1
were utilized. The dashed lines correspond to the same parameters but zero
loss.

Naturally, the sensitivity degrades rather significantly because of the loss
present. However, for the 1% attenuation in the cavity the observed
sensitivity improvement is still significant.

Figure 4: Plots of ratio $R=S_{qu}/S_{SQL}$, of spectral densities for the
symmetric case with optical losses. For plot marked by "SymLoss" the
parameters are taken from Table 1 (the dashed line "SymNoLo" corresponds to
the same parameters but zero optical losses). Plots "SymLoss2" and "SymNoLo2"
correspond to 10 times larger input pump $P_{in}$. $\tau$ is the time of
signal force action.

####  III.4.3 Symmetric case with optical losses

Optical losses restrict sensitivity even in the symmetric case. Corresponding
plots are presented in Fig 4. We see that even small losses (about only 1%
from losses through the input mirror) limit the sensitivity. The reason is the
uncorrelated back action resulting from the additional noise due to optical
losses which can not be completely subtracted.

Figure 5: Higher order modes in a Fabry-Perot cavity can be used as modes
$\omega_{0},\ \omega_{\pm}$, their distribution functions $\Psi_{00}(u),\
\Psi_{01}(u),\ \Psi_{02}(u)$) are shown in the top row, $u$ is one of the
transverse coordinates.

##  IV Physical realization

There are a few practical problems with experimental realization of the
proposed measurement scheme. Those include i) difficulty to create a triplet
with equidistant separation of modes $\omega_{\pm}$ from $\omega_{0}$; ii)
achieve the value of separation $|\omega_{0}-\omega_{\pm}|$ to be equal to the
mechanical frequency $\omega_{m}$ about 1…100 MHz; iii) separate the optical
harmonics without loss. It is rather a small frequency difference in the
optical domain. Realization of such a narrow and lossless optical filter is a
difficult experimental task.

We can use optical modes belonging to different geometrical mode families and
corresponding geometrical mode sorters [15, 16] to realize the proposed
system.

Let us consider higher order modes (HOM) in a Fabry-Perot cavity. The
frequency range between these modes can be made much less than the free
spectral range (FSR) of the cavity by properly selecting the radii of
curvature of mirrors. Fig. 5 illustrates such a possibility. Distribution
functions of the modes $\Psi_{00}(u),\ \Psi_{01}(u),\ \Psi_{02}(u)$) are shown
in the top row, where $u$ is one of the transverse coordinates, distributions
over the other transverse coordinate $v$ are assumed to be Gaussian.

These optical modes can interact with the dipole elastic mode of the mirror
(membrane). Such a mechanical mode has geometrical distribution $\Psi_{m}$
illustrated by Fig. 5. The generalized force acting on the elastic mode is
defined by the overlap integral. In other words, the coupling constants
$\eta_{\pm}$ is proportional to

$\displaystyle\eta_{+}$
$\displaystyle\sim\int\Psi_{01}(u)\,\Psi_{02}(u)\,\Psi_{m}(u)\,du,$ (34)
$\displaystyle\eta_{-}$
$\displaystyle\sim\int\Psi_{01}(u)\,\Psi_{00}(u)\,\Psi_{m}(u)\,du$ (35)

While the overlap integral is not zero, the absolute values of $\eta_{+}$ and
$\eta_{-}$ can be different. As we have shown above, this asymmetry does not
prevent the BAE measurement.

Next step is to separate the modes leaving the cavity in space. The output
light consists of radiation from all three optical modes. Theoretically, a
mode sorter can do it perfectly, and experimental efficiency exceeding 80 %
was recently confirmed [15]. Therefore, the modes can be separated, reshaped
to Gaussian beams and optimally detected to realize the proposed measurement.

##  V Conclusion

We have investigated a broadband multidimensional variation measurement of a
force acting on a mechanical oscillator taking into account asymmetry of
interaction as well as optical loss. We found that the asymmetry does not
prevent the complete removal of the quantum back action from the measurement
result. The back action compensation is not complete in the presence of
optical losses, it restricts the value of back action correlation in the two
measurement channels and, hence, sensitivity.

###### Acknowledgements.

The research of SPV has been supported by Theoretical Physics and Mathematics
Advancement Foundation “BASIS” (Contract No. 22-1-1-47-1), by the
Interdisciplinary Scientific and Educational School of Moscow University
‘‘Fundamental and Applied Space Research’’ and by the TAPIR GIFT MSU Support
of the California Institute of Technology. The reported here research
performed by ABM was carried out at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and
Space Administration (80NM0018D0004). This document has LIGO number P2400323.
.

##  Appendix A Standard Quantum Limit

Here we recall details on the derivation of SQL for the force acting on the
mechanical oscillator. Let us assume that we measure the phase quadrature of
the resonant light reflected from the movable end mirror of a Fabry-Perot
cavity.

The single-sided power spectral density, corresponding to the unity signal to
noise ratio, recalculated to the quadrature $f_{s\,a}$ of the normalized
signal force (13, 14), acting during time $\tau$ is

$\displaystyle S_{fa}$
$\displaystyle=2\gamma_{m}\left(n_{T}+\frac{1}{2}\right)+\frac{\gamma_{m}^{2}+\Omega^{2}}{\mathcal{K}}+\mathcal{K}\geq$
(36) $\displaystyle\geq 2\gamma_{m}\left(n_{T}+\frac{1}{2}\right)+S_{SQL},$
(37) $\displaystyle S_{SQL}=2\sqrt{\gamma_{m}^{2}+\Omega^{2}}$ (38)

Here parameter $\mathcal{K}$ is proportional to pump power, $\Omega$ is
spectral frequency. Below we put angle $\psi_{f}=0$ in (13).

The approximate condition the force detection is

$\displaystyle f_{s\,a}$ $\displaystyle=\frac{F_{s0}}{\sqrt{2}\sqrt{2\hslash
m\omega_{m}}}\geq\sqrt{\int_{0}^{2\pi/\tau}S_{f}(\Omega)\,\frac{d\Omega}{2\pi}}=$
(39) $\displaystyle=$
$\displaystyle\sqrt{\left[2\gamma_{m}\left(n_{T}+\frac{1}{2}\right)+\frac{\gamma_{m}^{2}+\frac{1}{3}\left[\frac{2\pi}{\tau}\right]^{2}}{\mathcal{K}}+\mathcal{K}\right]\frac{1}{\tau}}$
(40)

We should consider the case of short time $\tau$

$\gamma_{m}\tau\ll 1$ (41)

since in the opposite case of large $\tau$ the thermal limit restricts the
sensitivity. Optimizing the measurement with respect of $\mathcal{K}$ and
taking into account of (41), we obtain the minimum detectable force $F_{s0}$
amplitude

$\displaystyle\frac{F_{s0}^{2}}{4\hslash m\omega_{m}}$ $\displaystyle\geq
2\gamma_{m}\left(n_{T}+\frac{1}{2}\right)\,\frac{1}{\tau}+\frac{2}{\sqrt{3}}\frac{2\pi}{\tau^{2}}$
(42)

Here first term here describes the thermal limit whereas the second one
introduces SQL:

$\displaystyle F_{s0}^{SQL}=\frac{4}{\tau}\sqrt{\frac{\pi\hslash
m\omega_{m}}{\sqrt{3}}}$ (43)

This formula is valid with an accuracy of a constant multiplier of about unity
due to the approximation of the force envelope shape (39).

We can use (38) instead of (36) and obtain

$\displaystyle\frac{\tilde{F}_{s0}^{2}}{4\hslash m\omega_{m}}$
$\displaystyle\geq
2\gamma_{m}\left(n_{T}+\frac{1}{2}\right)\frac{1}{\tau}+\frac{4\pi}{\tau^{2}}$
(44)

The second terms in (42) and in (44) differ only by a multiplier of about
unity. Therefore, we can use spectral density (38) for SQL characterization in
the frequency domain.

##  Appendix B Derivation of the output quadrature

We start from the equations 17 for the amplitude quadratures

$\displaystyle(\gamma_{+}-i\Omega)c_{+a}$
$\displaystyle+\eta_{+}C_{0}d_{a}=\sqrt{2\gamma_{0+}}{a}_{+a}+$ (45a)
$\displaystyle+\sqrt{2\gamma_{e+}}e_{+a},$
$\displaystyle(\gamma_{-}-i\Omega)c_{-a}$
$\displaystyle-\eta_{-}C_{0}d_{a}=\sqrt{2\gamma_{-}}{a}_{-a}+$ (45b)
$\displaystyle+\sqrt{2\gamma_{e-}}e_{-a},$
$\displaystyle(\gamma_{m}-i\Omega)d_{a}$ $\displaystyle-
C_{0}\Big{(}\eta_{+}c_{+a}+\eta_{-}c_{-a}\Big{)}=$ (45c)
$\displaystyle=\sqrt{2\gamma_{m}}q_{a}+f_{s\,a},$

From (45a, 45b) we find the back action term
$\sim(\eta_{+}c_{+a}+\eta_{-}c_{-a})$ and substitute it into (45c) to derive

$\displaystyle\left(\Gamma_{m}-i\Omega\frac{}{}\right)$ $\displaystyle
d_{a}=C_{0}\frac{\eta_{+}(\sqrt{2\gamma_{+}}a_{+a}+\sqrt{2\gamma_{e}}e_{+a})}{\gamma_{+}-i\Omega}+$
$\displaystyle+$ $\displaystyle
C_{0}\frac{\eta_{-}(\sqrt{2\gamma_{-}}a_{-a}+\sqrt{2\gamma_{e}}e_{-a})}{\gamma_{-}-i\Omega}$
(46a) $\displaystyle+\sqrt{2\gamma_{m}}q_{a}+f_{s\,a},$ (46b)
$\displaystyle\Gamma_{m}=\gamma_{m}+G,$ $\displaystyle\quad
G\equiv\left[\frac{\eta_{+}^{2}}{\gamma_{+}-i\Omega}-\frac{\eta_{-}^{2}}{\gamma_{-}-i\Omega}\right]C_{0}^{2}$
(46c)

The optical damping rate $G$ appears due to non-symmetry of $\eta_{\pm}$ and
$\gamma_{\pm}$.

Using the input-output relations

$\hat{b}_{\pm}=-\hat{a}_{\pm}+\sqrt{2\gamma_{0\pm}}\hat{c}_{\pm}$ (47)

We find the output amplitude quadratures and write them in form

$\displaystyle b_{+a}$
$\displaystyle=\xi_{+}\,{a}_{+a}+\mu_{+}e_{+a}-\frac{\sqrt{2\gamma_{0+}}\,\eta_{+}}{\gamma_{+}-i\Omega}\,C_{0}d_{a},$
(48a) $\displaystyle b_{-a}$
$\displaystyle=\xi_{-}\,{a}_{-a}+\mu_{-}e_{-a}+\frac{\sqrt{2\gamma_{0-}}\,\eta_{-}}{\gamma_{-}-i\Omega}\,C_{0}d_{a},$
(48b) $\displaystyle\xi_{\pm}=$
$\displaystyle\frac{\gamma_{0\pm}-\gamma_{e\pm}+i\Omega}{\gamma_{0+}+\gamma_{e\pm}-i\Omega},\quad\mu_{\pm}=\frac{2\sqrt{\gamma_{0\pm}\gamma_{e\pm}}}{\gamma_{0\pm}+\gamma_{e\pm}-i\Omega}$
(48c)

After substitution of (46) into (48) we obtain

$\displaystyle b_{+a}$
$\displaystyle=\frac{A_{+}}{\Gamma_{m}-i\Omega}\left\\{\left(B_{+}-A_{+}\right)a_{+a}-A_{-}\,a_{-a}+\frac{}{}\right.$
(49a)
$\displaystyle+\sqrt{\frac{\gamma_{e+}}{\gamma_{0+}}}\left(B_{e+}-A_{+}\right)e_{+a}-\sqrt{\frac{\gamma_{e-}}{\gamma_{0-}}}A_{-}e_{-a}$
(49b)
$\displaystyle\quad-\left.\left(\sqrt{2\gamma_{m}}q_{a}+f_{s\,a}\right)\right\\}$
(49c) $\displaystyle b_{-a}$
$\displaystyle=\frac{A_{-}}{\Gamma_{m}-i\Omega}\left\\{\left(B_{-}+A_{-}\right)a_{-a}+A_{+}\,a_{+a}+\frac{}{}\right.$
(49d)
$\displaystyle+\sqrt{\frac{\gamma_{e-}}{\gamma_{0-}}}\left(B_{e-}+A_{-}\right)e_{-a}+\sqrt{\frac{\gamma_{e+}}{\gamma_{0+}}}A_{+}e_{+a}$
(49e)
$\displaystyle\quad+\left.\left(\sqrt{2\gamma_{m}}q_{a}+f_{s\,a}\right)\right\\}$
(49f) $\displaystyle B_{\pm}$
$\displaystyle=\frac{(\gamma_{0\pm}-\gamma_{e\pm}+i\Omega)(\Gamma_{m}-i\Omega)}{\sqrt{2\gamma_{0\pm}}\,\eta_{\pm}C_{0}},$
(49g) $\displaystyle A_{\pm}$
$\displaystyle=\frac{\sqrt{2\gamma_{0\pm}}\,\eta_{\pm}C_{0}}{(\gamma_{\pm}-i\Omega)},\
B_{e\pm}=\frac{\sqrt{2\gamma_{0\pm}}(\Gamma_{m}-i\Omega)}{\eta_{\pm}C_{0}}.$
(49h)

We measure $b_{+a}$ and $b_{-a}$ separately and take its weighted sum to
perform the measurement. Let consider $\Sigma$ to be a measured linear
combination ofthe amplitude quadratures

$\displaystyle\Sigma=z_{+}\,\frac{b_{+a}\big{(}\Gamma_{m}-i\Omega\big{)}}{A_{+}\big{(}z_{-}-z_{+}\big{)}}+z_{-}\,\frac{b_{-a}\big{(}\Gamma_{m}-i\Omega\big{)}}{A_{-}\big{(}z_{-}-z_{+}\big{)}}$
(50)

where $z_{\pm}(\Omega)$ are complex functions depending on spectral frequency
$\Omega$, which should be chosen for post-processing in an optimal way. The
parameter $\Sigma$ is chosen so that the normalized signal term
$\big{(}f_{s\,a}\big{)}$ has coefficient of unity.

$\displaystyle\begin{split}\Sigma&=B_{+}\left[y-\frac{1}{2}+Y_{+}\right]a_{+a}+B_{-}\left[y+\frac{1}{2}+Y_{-}\right]a_{-a}+\\\
&+\sqrt{\frac{\gamma_{e+}}{\gamma_{0+}}}B_{e+}\left[y-\frac{1}{2}+Y_{e+}\right]e_{+a}+\\\
+&\sqrt{\frac{\gamma_{e-}}{\gamma_{0-}}}B_{e-}\left[y+\frac{1}{2}+Y_{e-}\right]e_{-a}+\sqrt{2\gamma_{m}}q_{a}+f_{s\,a},\end{split}$
(51a) $\displaystyle
y+\frac{1}{2}=\frac{Z_{-}}{Z_{-}-Z_{+}},\quad-\frac{1}{2}+y=\frac{Z_{+}}{Z_{-}-Z_{+}}$
(51b) $\displaystyle Y_{\pm}=\frac{A_{\pm}}{B_{\pm}},\quad
Y_{e\pm}=\frac{A_{\pm}}{B_{e\pm}}$ (51c)

Here we introduced one optimization parameter $y$ instead of two parameters
$z_{\pm}$.

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aainstitutetext: Max-Planck-Institut für Physik, Föhringer Ring 6, 80805
München, Germanybbinstitutetext: DESY, Notkestraße 85, 22607 Hamburg,
Germanyccinstitutetext: Institute for Theoretical Physics, Utrecht University
Princetonplein 5, 3584 CC Utrecht, The Netherlands

# Generalised Known Kinematics (GKK)
An Approach for Kinematic Observables in Pair Production Events with Decays
Involving Invisible Particles

Thomas M. G. Kraetzschmar,111Corresponding author. a Fabian M. Krinner a
Marvin Pfaff b Navid K. Rad b Armine Rostomyan c Lorenz Schlechter a Frank
Simon<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Many analyses in high energy physics are limited due to missing kinematic
information of known invisible particles in the detector, for example
neutrinos. The undetected particle carries away momentum and energy
information, preventing the full reconstruction of such an event. In this
paper, we present a method to handle this missing information, referred to as
the Generalised Known Kinematics (GKK) approach. It is based on constructing
event-by-event probability density distributions that describe the physically
allowed kinematics of an event. For GKK we take into account the available
kinematic information and constraints given by the assumed final state.
Summing these event-wise distributions over large data sets allows the
determination of parameters that influence the event kinematics, such as
particle masses, which are otherwise obscured by the missing information on
the invisible final-state particles. The method is demonstrated in simulation
studies with $\tau^{+}\tau^{-}$ events in $e^{+}e^{-}$ collisions at the
$\mathchar 28935\relax$(4S) resonance, presenting a new, promising approach
for the measurement of the $\tau$ lepton mass.

##  1 Introduction

$e^{+}e^{-}$-colliders provide an ideal environment for precision
measurements. This is because of the well known initial state and, compared to
hadron colliders, simpler final-state kinematics. However, there is a class of
events where the final state kinematics cannot be fully reconstructed. This is
caused by the existence of invisible particles, most commonly neutrinos $\nu$,
that escape detection which lead to a degradation of measurement precision. In
this work we discuss a new method to mitigate the problem of invisible
particles in particle-pair events. We present the method using the example of
$\tau$-pair production in $e^{+}e^{-}$ annihilation data,
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$. However, the method can be
generalised to any kind of similar problem statement.

We consider the case in which the $\tau$ leptons are boosted and their decay
products are well separated in two opposite hemispheres. For $\tau$ leptons a
decay is of the general form,

$\tau\rightarrow nV+mI,$ (1)

with a number of $n$ visible daughters $V$ and $m$ invisible daughters $I$.
There is always at least one invisible particle $I$ involved. The existence of
invisible particles $mI$ in the final state prevent us from a straight forward
determination of observables, which rely on the full kinematic information of
the $\tau$. An example for such an observable is the mass of the $\tau$, which
would be accessible via the invariant mass of the final state.

There are several approaches to tackle this problem. For example, ARGUS
proposed an approximate method to determine the $\tau$-mass from its
$\tau\rightarrow 3\pi\nu_{\tau}$ decay mode ARGUS:1992chv . Using energy and
momentum conservation ARGUS derives a $\tau$ pseudomass by neglecting the
neutrino mass and by approximating that the flight direction of the $3\pi$
(hadronic) system is the flight direction of the $\tau$. Then, the pseudomass
distribution exhibits a sharp threshold behaviour in the region close to the
$\tau$ mass. The $\tau$-mass is then measured by determining the endpoint of
the $\tau$-pseudomass distribution. In principle, this method can be
generalised to other $\tau$-decay that fulfils $n\geq 3$ and $m=1$.

Another possible solution has been developed by the CLEO collaboration
CLEO:1993vmo . In the CLEO approach, a special class of events with one
invisible particle $I$ is considered. Kinematic considerations show that the
three-momentum of the $\tau$ lies on a cone around the three-momentum vector
of the respective visible system $V$. CLEO reconstructs the $\tau$-pair
kinematics from constrains given by the decay cones on both sides around the
three-momentum of the visible system. $I$ is considered to be mass-less. They
use the simplifying assumption that the plane defined by the the three-
momentum vector of $V$ and of $I$ is equal for both decays of the $\tau$-pair.

The method we propose avoids this simplification, and is valid for $\tau$-pair
events with at least one $\tau$ lepton decaying with one invisible particle
$I$. We define this event to be of type $\tau_{h}$ (an hadronic $\tau$-decay)

$\tau_{h}\rightarrow n_{h}V_{h}+I_{h}.$ (2)

with $n_{h}\geq 1$, $V_{h}$ being hadrons, and $I_{h}$ the tau-neutrino
$\nu_{\tau}$ Here, we derive the probability distribution of $I$ and calculate
the observable of interest, with a set of candidates drawn from the
probability distribution of $I$, rather than a single value. That means that
instead of obtaining a single value per event we get a distribution of values.
The sum of all distributions for all events accumulates to a new distribution,
a limiting distribution. The limiting distribution allows to determine the
observable of interest without the bias of the missing particles. We call this
approach Generalised Known Kinematics (GKK).

##  2 Concept

For the measurement of any quantity the optimal observable is an uniform
minimum-variance unbiased estimator of a statistic. The idea of GKK, as
developed in this article, is inspired by the concept of such an estimator.
For this concept to work, it is necessary that a statistic $T$, used to
determine the estimator of interest, contains all available information
mood1974introduction . However, in particle pair events with invisible
particles, kinematic information is lost. Due to physical constraints, we are
able to recover some of the lost information and determine a statistic
$T^{\prime}$ which is complete with regard to the kinematic information.

Let us consider a sample of $\tau$-pair events, with two hemispheres
$\tau_{1}$ and $\tau_{2}$, of which we want to measure an observable dependent
on the momentum-spectrum of the visible daughters in the rest-frame of
$\tau_{2}$.

We can identify particle-pair events by reconstructing one of the two
particles undergoing a well known decay. In this case we demand the $\tau_{1}$
hemisphere to be a type $\tau_{h}$ decay, an hadronic $\tau$-decay, which we
will call tag-side from hereon. Reconstruction of the tag-side allows to study
the properties of the second particle $\tau_{2}$, the so called signal-side,
without introducing a reconstruction bias.

We simplify the $\tau_{1}$ decay into a two body decay, with the invisible
particle $I_{1}$ and the visible-daughters-system $V^{1}_{eff}$ by combining
all $n_{1}V_{1}$ daughters into an effective particle
$V_{\text{eff}}^{1}=\bigoplus_{i=1}^{n_{1}}V_{1}^{i}$, with it’s four-momentum
$p^{\mu}_{V_{\text{eff}}^{1}}$ given by the set of four-momentum vectors
$[p^{\mu}_{V_{1}^{1}},...,p^{\mu}_{V_{1}^{n_{1}}}]$:

$p^{\mu}_{V_{\text{eff}}^{1}}=\sum_{i=1}^{n_{1}}p^{\mu}_{V_{1}^{i}}\,.$ (3)

The missing information of $I_{1}$ is translated into a probability
distribution function for the $\tau_{1}$ momentum vector using energy-momentum
conservation and the isotropic distribution of the decay of $I_{1}$ in the
rest-frame of $\tau_{1}$. This results in a cone shaped momentum distribution
for $I_{1}$ around the $\tau_{1}$ momentum in the center-of-mass system of the
event. Now, we can turn the argument around and constrain the $\tau_{1}$
momentum direction on a cone around the $V^{1}_{eff}$ direction, as shown in
Figure 1.

As a first step, the $\tau_{1}$-energy $E_{\tau_{1}}$ is determined. In
$\tau^{+}\tau^{-}$-events, it can be approximated as half of the centre of
mass beam-energy $\sqrt{s}$:

$E_{\tau_{1}}\approx\frac{\sqrt{s}}{2}.$ (4)

This approximation also allows to determine the magnitude of the
$\tau_{1}$-momentum in the center-of-mass system via the known $\tau$-mass,
$m_{\tau}$. Doing so, we can derive the angle $\theta$ between the
$V^{1}_{eff}$-momentum $\vec{p}_{V^{1}_{eff}}$ and the $\tau_{1}$-momentum
$\vec{p}_{\tau_{1}}$ by using the law of cosines222$\theta$ is given by
$\cos(\theta)$ because in the polar coordinates it is confined between 0 and
1.:

$cos(\theta)=\frac{\vec{p}_{\tau_{1}}\cdot\vec{p}_{V^{1}_{eff}}}{|\vec{p}_{\tau_{1}}||\vec{p}_{V^{1}_{eff}}|}.$
(5)

The three-momentum of $\tau_{1}$ can be deconstructed into a parallel
($\vec{p}_{\tau_{1},||}$) and perpendicular ($\vec{p}_{\tau_{1},\perp}$)
component with respect to the $V^{1}_{eff}$ momentum direction (see Figure 1).

Figure 1: Visualisation of the GKK method. The true $\tau_{1}$ momentum-vector
$\vec{p}_{\tau_{1}}$ is shown in black. $\theta$ indicates the angle between
the $\vec{p}_{\tau_{1}}$ and $\vec{p}_{V^{1}_{eff}}$, the resulting momentum
vector sum for all visible daughters $V^{1}$ is shown in yellow. The invisible
particle $I_{1}$ involved in the decay leads to an angular offset, which we
parameterise by $\theta$ and $\varphi$. The red dashed lines indicate the
components of the $\tau$-momentum. $\vec{p}_{\tau,\parallel}$ is parallel to
$\vec{p}_{V^{1}_{eff}}$, $\vec{p}_{\tau,\perp}$ is orthogonal. The unknown
component is then translated into the direction of $\vec{p}_{\tau,\perp}$,
parameterised by the azimuth angle $\varphi$.

Constraining $\tau_{1}$ on a cone around $V^{1}_{eff}$ allows us to
parameterize $\vec{p}_{\tau_{1}}$ such that the unknown direction is expressed
in terms of the azimuth angle $\varphi$ in a cylindrical coordinate system
parallel to $V^{1}_{eff}$.

We know that each possible $\vec{p}_{\tau_{1}}$ on the cone is equally
probable, so the $\vec{p}_{\tau_{1}}$-distribution can be sampled by stepping
through the equally distributed $\varphi$, providing a set of equally probable
candidates for the $\tau_{1}$ momentum. This is the concept of GKK.

The finite detector resolution can cause this approach to yield
$\vec{p}_{\tau_{1}}$-momentum candidates which deviate considerably from the
actual $\vec{p}_{\tau_{1}}$ of the event. This makes it worthwhile to add
further physical constraints. In the case of particle pair events, we can
utilise the signal side for these constraints.

First, we consider a special case and generalise afterwards. In the case of
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events we can have events where both
$\tau_{1}$ and $\tau_{2}$ decay hadronically. In this case, we can reconstruct
the $\tau_{2}$ momentum in the same way as done for $\tau_{1}$ with the
corresponding angles $\theta^{\prime}$ and $\varphi^{\prime}$. This results in
two momentum cones, as depicted in Figure 2.

Figure 2: Visualisation of the GKK method in a special case, in which both
taus are of type $\tau_{h}$. Here, we depict a $\tau^{+}\tau^{-}$ event in the
Centre of Mass System (CMS). The true $\tau$-momentum-vectors are shown in
black. The tau-pair events are divided into a $\tau_{1}$\- and a
$\tau_{2}$-side. $\theta$ and $\theta^{\prime}$ indicate the angle between the
true $\tau$ momentum vector and the sum of all reconstructed daughters on the
tag and signal side, depicted in yellow and blue respectively. The
$\tau_{i}$-momentum can be deconstructed into two components indicated in red.
A parallel component $\vec{p}_{\tau_{i},\parallel}$ to the daughter momentum
$\vec{p}_{V^{i}_{eff},\parallel}$, and one that is perpendicular
$\vec{p}_{\tau_{i},\perp}$. The unknown direction of $\vec{p}_{\tau_{i}}$ is
translated into the direction of $\vec{p}_{\tau_{i},\perp}$. The direction of
$\vec{p}_{\tau_{i},\perp}$ can be parametrized by the azimuth angles $\varphi$
and $\varphi^{\prime}$, for $\tau_{1}$ and $\tau_{2}$ respectively.

If the momenta of $V^{1}_{eff}$ and $V^{2}_{eff}$ were perfectly known, we
could reconstruct the $\tau$-momentum by inverting the momentum on one of the
two sides and look for the momentum-vectors that fulfil the requirements
imposed by energy and momentum conservation. These are that both $\tau$
momenta must be back-to-back in the center-of-mass frame and lie on their
respective cones. In general, this leads to two solutions. In the extreme
cases, we obtain either one solution (the cones touch each other) or infinite
(the cones are on top of each other).

Finite detector resolution smears the reconstructed values. This implies that
we do not have a perfect knowledge of $\vec{p}_{V^{i}_{eff}}$, with $i=1,2$.
This causes the cones to not perfectly align but to overlap, as indicated in
Figure 3.

Figure 3: Sketch of detector smearing effects on the reconstructed
$\tau$-momentum-candidates distributed on a cone. Here, the special case of a
$\tau^{+}\tau^{-}$ event is considered where both taus are of type $\tau_{h}$.
In an ideal case, indicated by the solid-lined cones denoted as “True Cones”,
we expect the $\tau$-cones to touch each other just at one point. The
idealised case enables the determination of the true $\tau$-momentum from the
common point of the $\tau$-cones. However, due to detector smearing, the cones
can be wider and misaligned. This case is indicated as “Smeared Cones” with
dashed-lined cones. The smeared cones do not touch, but overlap, giving rise
to a physically sound range, indicated in red

The overlap is caused because information is lost not only due to $I_{i}$ but
also due to the detector smearing, leading to an overestimation of angles
$\theta$ and $\theta^{\prime}$ and a slight misalignment of the cones. Instead
of looking for a single $\vec{p}_{\tau}$-candidate which is in both statistics
of $\tau_{1}$ and $\tau_{2}$, we are thus looking for those
$\vec{p}_{\tau_{1}}$-candidates which are on or within the
$\vec{p}_{\tau_{2}}$-cone, as indicated in Figure 3.

This approach can be generalised to the case in which only $\tau_{1}$ decays
is of type $\tau_{h}$, and $\tau_{2}$ has an unspecified number of invisible
particles $m_{2}I_{2}$. In this case, we are not able to determine a cone of
$\tau_{2}$-momentum-candidates. Instead, we can give a constraint to the
momentum-candidate-cone of $\tau_{1}$ by maximising the $\theta^{\prime}$ to a
$\theta^{\prime}_{\text{max}}$ and defining a maximised $\tau_{2}$-momentum-
cone, as shown on the right side of Figure 4. How $\theta^{\prime}$ is
maximised is discussed in Section 3. We constrain the
$\vec{p}_{\tau_{1}}$-cone by demanding that it has to be within or on the cone
of $\vec{p}_{\tau_{2}}$-candidates given by $\theta^{\prime}_{\text{max}}$,
indicated on the left of Figure 4. This method basically rejects all
$\vec{p}_{\tau_{1}}$-candidates outside the momentum constraints of the event.
From now on, events which pass the momentum constraint are referred to as
physical candidates, whereas rejected candidates are referred to as non-
physical. As a further refinement step, we can restrict the sampling range of
$\varphi$ to the range given by the physical candidates $\varphi_{new}$ and
redo the sampling, giving more weight to physical $\varphi$-ranges.

Figure 4: On the left, the construction of the maximised cone, using
$\theta_{max}$ is displayed. We do this in the case of two or more invisible
particles in the $\tau_{2}$-decay, because the missing information prevents us
from constructing the $\tau_{2}$-candidate cone. Instead, we can construct a
cone with the maximum possible deviation of the $\tau_{2}$-momentum and the
momentum of all visible $\tau_{2}$-daughters. We can map this cone on a
2D-plain, as well as the cone of the $\tau_{1}$, which by definition has to be
of type $\tau_{h}$. The “Max-Cone” gives us the physical area, as defined in
the text, constraining the $\tau_{1}$-cone to a certain range $\varphi_{new}$.
We can sample this physical $\varphi$-range again, resulting in a narrower
probability distribution of the $\vec{p}_{\tau_{1}}$.

##  3 Mathematical Description

After we presented the concept of the GKK method in Section 2, we are now
going to discuss the implementation in more detail. We condensed the
$\tau_{1}$-decay visible and invisible decay partners into an effective two
particle decay problem if $\tau_{1}$ is of type $\tau_{h}$. Here, we express
the invisible daughter’s $I_{1}$ four-momentum $p^{\mu}_{I_{1}}$ as

$p^{\mu}_{I_{1}}=p^{\mu}_{\tau_{1}}-p^{\mu}_{V_{\text{eff}}^{1}}.$ (6)

We can determine the angle $\theta$, displayed in Figure 1, by calculating the
mass of $I_{1}$, $m_{I_{1}}$ with the sum $p^{\mu}_{I_{1}}\,p_{\mu,I_{1}}$,
and the scalar product $p^{\mu}_{\tau_{1}}\cdot p^{\mu}_{V_{\text{eff}}^{1}}$.

First, we derive

$m_{I_{1}}^{2}=m_{\tau_{1}}^{2}+m_{V^{1}_{\text{eff}}}^{2}-2(p^{\mu}_{\tau_{1}}\cdot
p^{\mu}_{V_{\text{eff}}^{1}})$ (7)

to replace $p^{\mu}_{\tau_{1}}\cdot p^{\mu}_{V_{\text{eff}}^{1}}$ in

$p^{\mu}_{\tau_{1}}\cdot
p^{\mu}_{V_{\text{eff}}^{1}}=E_{\tau_{1}}E_{V^{1}_{\text{eff}}}-\cos(\theta)|\vec{p}_{\tau_{1}}||\vec{p}_{V^{1}_{\text{eff}}}|,$
(8)

which we solve for $\cos(\theta)$, resulting in

$\cos(\theta)=\frac{2\,E_{\tau_{1}}E_{V^{1}_{\text{eff}}}+m_{I_{1}}^{2}-(m_{\tau_{1}}^{2}+m_{V^{1}_{\text{eff}}}^{2})}{2~{}|\vec{p}_{\tau_{1}}||\vec{p}_{V^{1}_{\text{eff}}}|}.$
(9)

We can now study how to get $\theta_{\text{max}}$. To maximise $\theta$, we
have to minimise $\cos(\theta)$. All components of Equation (9) are given by
the detected event, except for $m_{I_{1}}$. So we minimise $\cos(\theta)$ by
setting $m_{I_{1}}=0$.

The expression derived from hereon are dependent on the reference frame, thus
the following considerations are only valid in the centre of mass frame and we
consider all appearing quantities to be given in the centre of mass frame. The
$\tau_{1}$ momentum vector $\vec{p}_{\tau_{1}}$ is decomposed as

$\vec{p}_{\tau_{1}}=\vec{p}_{\tau_{1},\parallel}+\vec{p}_{\tau_{1},\perp}\;.$
(10)

With the parallel component of $\vec{p}_{\tau_{1}}$ to
$\vec{p}_{V_{\text{eff}}^{1}}$

$\vec{p}_{\tau_{1},\parallel}=|\vec{p}_{\tau_{1}}|\cos(\theta)\,\vec{n}^{\,V^{1}_{\text{eff}}}_{\parallel},$
(11)

and the orthogonal component of $\vec{p}_{\tau_{1}}$ to
$\vec{p}_{V_{\text{eff}}^{1}}$

$\vec{p}_{\tau_{1},\perp}=|\vec{p}_{\tau_{1}}|\sin(\theta)\,\vec{n}^{\,V^{1}_{\text{eff}}}_{\perp},$
(12)

we get

$\vec{p}_{\tau_{1}}=|\vec{p}_{\tau_{1}}|\cos(\theta)\,\vec{n}^{\,V^{1}_{eff}}_{\parallel}+|\vec{p}_{\tau_{1}}|\sin(\theta)\,\vec{n}^{\,V^{1}_{\text{eff}}}_{\perp}.$
(13)

The angle $\theta$ is given by Equation (9),
$\vec{n}^{\,V^{1}_{\text{eff}}}_{\parallel}$ is a unit vector in the direction
of $\vec{p}_{V^{1}_{\text{eff}}}$, and $\vec{n}^{\,V^{1}_{eff}}_{\perp}$
perpendicular to $\vec{p}_{V^{1}_{\text{eff}}}$. We estimate the absolute
value of the $\tau_{1}$ momentum $|\vec{p}_{\tau_{1}}|$ as

$|\vec{p}_{\tau_{1}}|=\sqrt{\big{(}E_{\tau_{1}}\big{)}^{2}-m_{\tau_{1}}^{2}},$
(14)

using Approximation (4) and the decay-topology. This leaves only one unknown,
the unit vector $\vec{n}^{\,V^{1}_{eff}}_{\perp}$. We can define a basis with
respect to $\vec{p}_{V^{1}_{eff}}$, with a parallel basis vector

$\vec{e}^{~{}V^{1}_{\text{eff}}}_{\parallel}=\begin{pmatrix}0\\\ 0\\\
1\end{pmatrix}$ (15)

and the orthogonal basis vector

$\vec{e}^{~{}V_{\text{eff}}^{1}}_{\perp}=\begin{pmatrix}cos(\varphi)\\\
sin(\varphi)\\\ 0\end{pmatrix}.$ (16)

In this basis, the basis vectors and the unit vectors of the combined tag
particles are identical. So, by determining the basis transformation from the
detector basis to $\vec{e}^{\,V^{1}_{\text{eff}}}_{\parallel}$, presented in
Equation (17), we get an expression of
$\vec{n}^{\,V^{1}_{\text{eff}}}_{\perp}$, shown in Equation (18), which
incorporates the ambiguity of $\vec{p}_{\tau_{1}}$.

$D_{y}(\rho)\cdot
D_{z}(\xi)\cdot\vec{n}^{\,V^{1}_{\text{eff}}}_{\parallel}=\vec{e}_{\parallel}^{\,V^{1}_{\text{eff}}}$
(17) $\vec{n}^{\,V^{1}_{\text{eff}}}_{\perp}=D_{z}(\rho)^{\mathrm{T}}\cdot
D_{y}(\xi)^{\mathrm{T}}\cdot\vec{e}_{\perp}^{\,V^{1}_{\text{eff}}}$ (18)

Here, $D_{y}(\rho)$ and $D_{z}(\xi)$ are the rotation matrices around the
$y$\- and $z$-axis of the detector respectively. They are defined as

$D_{y}(\rho)=\begin{pmatrix}\cos(\rho)&0&-\sin(\rho)\\\ 0&1&0\\\
\sin(\rho)&0&\cos(\rho)\end{pmatrix}$ (19)

and

$D_{z}(\rho)=\begin{pmatrix}\cos(\xi)&\sin(\xi)&0\\\ -\sin(\xi)&\cos(\xi)&0\\\
0&0&1\end{pmatrix}.$ (20)

The angles $\rho$ and $\xi$ are the two polar angles of the detector which
rotate $\vec{n}^{V^{1}_{eff}}_{\parallel}$ into the basis
$\vec{e}^{~{}V^{1}_{\text{eff}}}_{\parallel}$. By combing above results, we
obtain the expression

$\vec{n}_{\perp}^{\,V^{1}_{\text{eff}}}=\begin{pmatrix}\cos(\xi)\cos(\rho)\cos(\varphi)-\sin(\xi)\sin(\varphi)\\\
\sin(\xi)\cos(\rho)\cos(\varphi)-\cos(\xi)\sin(\varphi)\\\
-\sin(\rho)\cos(\varphi)\end{pmatrix}.$ (21)

With the general expression of $\vec{p}_{\tau_{1}}$ and the knowledge of the
distribution function $f(\varphi)=\text{const.}$ we can sample
$\vec{p}_{\tau_{1}}$. We obtain a statistic of size $N$ for each event, being
$[^{1}p^{\mu}_{\tau_{1}},...,^{n}p^{\mu}_{\tau_{1}}]$. By sampling the tag
momentum with $f(\varphi)$ we obtain a statistic independent of $\varphi$, so
the statistic is only dependent on the momentum of $V_{\text{eff}}^{1}$.

As discussed in our example, we can use the calculated statistic of the
$\tau_{1}$-momenta to determine a momentum statistic of the
$\tau_{2}$-daughters in the rest-frame of their mother particle. The resulting
distribution function of the rest-frame momentum $p^{\star}(\varphi)$ cannot
be inverted, as a closed form is not known to the authors. In principle, we
expect that the limiting distribution can be described analytically, which
could be part of future studies of the GKK method. For the rest of this work
we consider purely numeric approaches.

##  4 Results

In order to illustrate the capabilities of the GKK method in
$\tau^{+}\tau^{-}$ events, we consider the momentum spectrum of a
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}$ decay, the signal-side, in the
$\tau$-rest frame. As described in Section 2, we first obtain a set of
$\vec{p}_{\tau}$ candidates of the tag-side $\tau$, considering the decay mode
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$. We use the set of
$\vec{p}_{\tau}$ to boost the signal-side $\pi^{-}$-momentum, $\vec{p}_{\pi}$,
into the rest-frame of the signal $\tau$. We denote the momentum of the
signal-side in the $\tau$-rest-frame as $p^{\tau}_{\pi}$. In the process
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}$, we expect a peak at
$p^{\tau}_{\pi}=\frac{{\rm\,m}_{\tau}-m_{\pi}}{2}$. First, we illustrate in
Figure 5 how the resulting GKK limiting distribution (in short GKK-
distribution) is formed. We do this by stacking the
$p_{\pi}^{\tau}$-distributions of each event. As the number of events
increases, a limiting distribution emerges, which should only depend on the
parameter of interest (in our special case $p^{\tau}_{\pi}$). Already with 25
events a clear peak emerges around the expected $p^{\tau}_{\pi}\approx
0.82\mathrm{\,Ge\kern-1.00006ptV}$.

Figure 5: Step by step formation of the GKK limiting distribution for the
$\tau\rightarrow\pi\nu$ decay. Here, for every $\tau^{+}\tau^{-}$ event, we
calculate a set of 1000 possible $\pi$-momentum candidates. An example
distribution for one event is given by the blue curve. By stacking one event
on top of the next the limiting distribution emerges for large enough number
of events, as indicated by the distributions, step by step, in the plot.

If there are enough events and the input-mass is the true mass, as is the case
in Figure 6a, a sharp peak emerges at the expected momentum.

###  4.1 GKK: A new Method to Measure the $\tau$-Mass

In order to use the GKK-method for $\tau$-mass measurements, we need to
understand the behaviour of the GKK-method. We studied specifically the
influence of the $\tau$-mass input, which is needed to calculate the GKK-
distribution. This was done by considering the input masses
$m_{\tau_{\text{input}}}$, which deviated by $\Delta m$ from the mass value
$m_{\tau_{\text{PDG}}}=1776.86\pm 0.12$ MeV ParticleDataGroup:2020ssz  used in
the event generation:

$\Delta m=m_{\tau_{\text{input}}}-m_{\tau_{\text{PDG}}}.$ (22)

The influence of $\Delta m$ is illustrated in Figure 6a. Here, we compare the
resulting limiting-distribution for 3 different $\Delta
m=[0,0.05,0.1]~{}\mathrm{\,Ge\kern-1.00006ptV}$. In order to quantify the
influence of $\Delta m$, we also calculate the Full Width Half Maximum (FWHM)
of the resulting limiting-distributions. This quantity can be understood as a
measure of spread or smearing of the distribution. The relation of FWHM versus
$\Delta m$ is shown in the scatter-plot of Figure 6b.

(a)

(b)

Figure 6:  Smearing effect of the GKK-distribution for different deviations of
the mass-hypothesis used for the calculation of the GKK-distribution. (a) GKK
distribution for the $\pi$-momentum of the $\tau\rightarrow\pi\nu$ decay for
$3\times 10^{5}$ events in the $\tau$-rest-frame. The effect of $\Delta m$ is
demonstrated for the GKK-distribution. (b) Full Width Half Maximum (FWHM) for
different $\Delta m$. Here, Toy MC denotes simulation data.

We observe two important aspects of the GKK-distribution from Figure 6a.
First, the true/expected momentum $p_{\pi,exp}^{\tau}$ is within the GKK-
distributions peak-region. We define the peak-region to be within the two
exponentially decreasing flanks of the distribution. Second, a simple peak-
search for the determination of the $\tau$-mass will not yield reliable
results, since a miss match of the assumed and true value of the mass, $\Delta
m$, leads to a distortion of the peak shape. For non-zero $\Delta m$ the
maximum of the distribution does not represent the true $p_{\pi,exp}^{\tau}$.
Instead, we observe that the width of the distribution increases with
increasing $\Delta m$.

This is further quantified in Figure 6b, which shows the Full Width Half
Maximum, FWHM, of the GKK-distribution for different values of $\Delta m$. We
calculate the FWHM of the distribution numerically. Here, we see that the FWHM
depends linearly on $\Delta m$ for both negative and positive $\Delta m$
values. We do not observe any significant differences in the behaviour of a
positive or negative $\Delta m$. This means that it is possible to determine
the $\tau$-mass by two methods. One is by simply minimising the FWHM in a
numerical approach with different input masses. The second is to utilise the
linear behaviour of the FWHM vs. $\Delta m$. Here, we can fit a linear
function for both the negative and positive $\Delta m$ distributions and
determine the true $\tau$-mass as the intersection point of the two linear
fits.

We interpret the behaviour as follows: the $\tau$-mass is used for calculating
the boost in the $\tau$-rest-frame and for determining the candidate-cone of
the $\tau$-momenta. In both cases, a mismatch between the true and the input
values leads to a smearing effect, thus allowing us to measure the $\tau$-mass
by selecting the input value which minimises the width of the distribution.

The new method we propose uses the information of all available events of the
chosen decay topology. This is in contrast to the ARGUS- and CLEO-method for
measuring the $\tau$-mass ARGUS:1992chv ; CLEO:1993vmo . In the case of the
ARGUS method, only the subsample of all events close to the endpoint of a
distribution is used. In the case of the CLEO method only a fraction of events
contributing to the peak position are used.

Today, the ARGUS-method is used at $e^{+}e^{-}$-colliders well above the
$\tau^{+}\tau^{-}$-production threshold Belle:2006qqw ; BaBar:2009qmj . It has
worse precision than the one used by the BES III collaboration exploring the
$\tau^{+}\tau^{-}$-production threshold Anashin:2007zz ; BESIII:2014srs . We
see potential that the GKK-method will improve the precision of the
$\tau$-mass measurment, compared to the ARGUS-method, significantly.

##  5 Conclusions

We have presented a new method for the determination of observables in
particle-pair events. We demonstrated the method on an explicit example, being
$\tau^{+}\tau^{-}$ events in $e^{+}e^{-}$ collisions. We showed that inferring
physical constraints for missing information leads to a probability
distribution which can be treated numerically and seems to show properties of
an unbiased best estimator. We demonstrated the example of the $\tau$-mass
measurement as a concrete use case of the GKK-method. Moreover, this method
can also be used for the determination of the mass of a new physics particle
in $\tau$-events. Here, we turn around the argument. Instead of two fixed
daughter-particle masses we replace one with the known mother-particle mass.
The peak position then determines the unknown daughter-particle mass, making
the GKK method suitable for the search for a invisible, massive particle.

We showed, that the GKK-method can lead to an improvement in parameter
estimation. In the case of the $\tau$-mass, there is clear linear behaviour
which has the potential to be utilised for a precise measurement. Further
studies which evaluate the performance in a real detector environment are
necessary to evaluate the possible improvements over present-day techniques.

Furthermore, there are other properties of the GKK method that remain to be
studied. We expect that the limiting distribution of the GKK distribution we
considered can be described analytically. A possible function family could be
the Asymmetric Generalised Gaussian Family of Distributions 778737 . With an
analytic description, we believe that the parameter estimation and smearing
could be described more reliably than with the numeric approach used until
now.

## References

  * [1] H. Albrecht et al.  A Measurement of the tau mass.  Phys. Lett. B, 292:221–228, 1992.
  * [2] R. Ballest et al.  A Measurement of the tau lepton mass.  Phys. Rev. D, 47:R3671–R3675, 1993.
  * [3] A.M.F. Mood, F.A. Graybill, and D.C. Boes.  Introduction to the Theory of Statistics.  International Student edition. McGraw-Hill, 1974.
  * [4] P. A. Zyla et al.  Review of Particle Physics.  PTEP, 2020(8):083C01, 2020.
  * [5] Kazuo Abe et al.  Measurement of the mass of the tau-lepton and an upper limit on the mass difference between tau+ and tau-.  Phys. Rev. Lett., 99:011801, 2007.
  * [6] Bernard Aubert et al.  Measurements of the tau mass and the mass difference of the tau+ and tau- at BABAR.  Phys. Rev. D, 80:092005, 2009.
  * [7] V. V. Anashin et al.  Measurement of the tau lepton mass at the KEDR detector.  JETP Lett., 85:347–352, 2007.
  * [8] M. Ablikim et al.  Precision measurement of the mass of the $\tau$ lepton.  Phys. Rev. D, 90(1):012001, 2014.
  * [9] J.-Y. Lee and A.K. Nandi.  Maximum likelihood parameter estimation of the asymmetric generalised gaussian family of distributions.  In Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics. SPW-HOS ’99, pages 255–258, 1999.


# User Study for Improving Tools for Bible Translation

Joel Mathew, Ulf Hermjakob
Information Sciences Institute, University of Southern California
{joel<EMAIL_ADDRESS>
###### Abstract

Technology has increasingly become an integral part of the Bible translation
process. Over time, both the translation process and relevant technology have
evolved greatly. More recently, the field of Natural Language Processing (NLP)
has made great progress in solving some problems previously thought
impenetrable. Through this study we endeavor to better understand and
communicate about a segment of the current landscape of the Bible translation
process as it relates to technology and identify pertinent issues. We conduct
several interviews with individuals working in different levels of the Bible
translation process from multiple organizations to identify gaps and
bottlenecks where technology (including recent advances in AI) could
potentially play a pivotal role in reducing translation time and improving
overall quality.

##  1 Introduction

Technology has played a key role in accelerating the Bible translation process
by automating and assisting with time-consuming tasks. Today, multiple
software tools have been developed that cater to users from varied backgrounds
and following different translation processes. It is assumed that Bible
translation being done today will rely on some technology to support their
work. Routine tasks like editing and search are well studied and implemented
in various tools available to translators.

The field of Natural Language Processing (NLP) is a field within the study of
Artificial Intelligence (AI) that focuses on solving problem that relate to
languages and text. The past decade has witnessed multiple points of
inflection that dramatically improved the capability of NLP systems. We also
observe that popular software tools used on the field for Bible translation do
not take advantage of newer technology. Though this could be attributed to the
lack of expertise in the development teams, we think there is still a bigger
problem of clearly identifying use-cases and recommending changes in the
translation process to effectively employ newer technology.

Arguably, technical teams that have developed and released applications on the
field would have completed some form of user study to identify pain-points and
areas where such a software tool could help. The conclusions of such studies,
however, maybe colored by the technology stack and translation process or
organization familiar to these teams. Moreover, the authors are unaware of any
concerted efforts to share and collaborate on such learnings.

We attempt to tackle these issues through this study and communicate our
findings. We consider our work as an important first step towards better
understanding and communicating about the problem space and thereby encourage
further participation from the Bible translation community at large. Our
contributions in this work are:

  1. 1.

We conduct interviews with multiple individuals involved with Bible
translation in different language projects.

  2. 2.

We report our preliminary findings and make recommendations for further
investigation.

In the following sections we describe our interviewing process, discuss our
observations and finally share our conclusions.

##  2 Interviews

Taking interviews is a quick and effective means to gather a breadth of
information and identify important areas for further investigation. It maybe
difficult to make the case that a study is thoroughly systematic if it only
relies on interviews. Instead of making such a claim, we consider this study
an important first step in the right direction.

Overall, we conduct 7 hour-long interviews involving 16 individuals over
video-conferencing. We record111Only for internal circulation; cannot be
shared. the interviews and keep detailed notes of our interactions. Most of
our interviewees work on Bible translation projects in India (Central, East
and North) though a few work on projects in Nepal and East Asia222Place names
have been obscured due to the sensitivity of the work.. One consultant
currently works on developing resources for translators in English for
subsequent translation.

Interviewee Role | Frequency
---|---
Mother Tongue Translator (MTT) | 3
Quality Checker (QC) | 4
Consultant-in-Training (CIT) | 4
Consultant | 3
Project Manager/Coordinator (PM) | 2
Table 1: Self-reported roles of interviewees surveyed for this study.

Before the start of each interview, we re-shared our intent (as described in
this document) and ensured anonimity of the interviewees. Moreover, we
clarified our position as interested third-parties and that any candid
feedback about their organizations or tools would be kept private (especially
the meeting recordings). Though we kept the tone of the interviews casual so
as to elicit genuine and open responses, we were largely guided by a Question
Bank A.1 developed to keep us grounded to the topic at hand which we share as
an appendix of this report.

##  3 Observations and Discussion

###  3.1 Technology Friendliness

It was observed that there were translation team members who were introduced
to a computer (laptop) for the first time just three months prior to project
start along with individuals who considered themselves experienced and
technology savvy. There were also individuals educated until high-school
working along with Ph.Ds, having decades of experience and expertise on the
field. Backgrounds of the translation team members can vary considerably. Any
tool or technology that is designed to interact with translation teams would
benefit from designing applications that are targeted and marketed for a
subset of the users with specific backgrounds. One-size may not fit anyone
well. Moreover, assuming most of the remaining languages yet to be translated
are spoken by minorities (when compared to LWCs333Languages of wider
communication), there is a bigger demand for tools with simpler user
interfaces (UIs) that do not assume significant background knowledge from the
user.

###  3.2 Improved Resources

A frequent request was for better resources in related languages other than
LWCs to quickly identify context and produce accurate translations. Consultant
notes were reported to be helpful though sometimes were only available in
English which limited their accessibility. Moreover, it was not unusual for
the teams to look up relevant resources available in multiple other languages
(based on their ability to read) to make translation decisions for the more
difficult passages.

Finding accurate meaning and context for words or phrases (including figures-
of-speech) unfamiliar to the translators was reported as difficult and as a
time-sink. Dictionaries were helpful though seeing relevant pictures would be
better (e.g. for the word ephod). There are, however, legitimate instances
where this process could take time. Especially, if there did not exist an
equivalent term in the target language. e.g. ’yeast’ was unknown to one of the
target language communities.

There is a need for a resource that lists cultural references in the Bible
with their location and significance. e.g. Exchanging slippers to make a
covenant in the book of Ruth. It was noted that having access to good
resources empower the lingual church to take ownership and more responsibility
of the translation process.

In terms of integration with tools, offline access was reported as important.
Tools that unobtrusively show only the parts of the reference resources that
are relevant to the verse/word/phrase at-hand without requiring the user to
switch screens were considered more helpful. Being able to quickly see all the
translations of an exact word or phrase in related languages (those known to
the translator) would greatly speed-up the drafting process while improving
quality. Automated script conversion could be used to make more resources
accessible for the translator.

###  3.3 Translation Consistency

There did not seem to exist a good strategy around maintaining consistency
across Biblical books. An effort is made by limiting the number of people
working on a book at a time, reading through drafts repeatedly and using the
same list of key Biblical terms (limited scope) across books. Even when
checking based on the key Biblical terms list, the current tools did not
automatically account for inflections and required the users to manually list
all possible inflections (e.g. affixes) as separate entries which makes
maintaining the list laborious and unwieldy.

We recommend using available NLP techniques to automatically flag inconsistent
usage of terms based on context and inflections, just in time. Such a feature
could also tackle another class of difficult-to-spot issues: inconsistent
usage of tense and singular/plural form between the source and target text.
Moreover, suggesting semantically similar words (e.g. God and Lord) during
drafting was a stated need. Consultants seem to rely on their experience and
knowledge to spot inconsistencies in the text and would appreciate a tool to
aid them.

###  3.4 Spell Checking

Spell checking is done by manually reading over all words used in the drafted
text and fixing incorrect spellings case-by-case. General guidance for
translators is to develop the dictionary in the target language before
starting the translation process though most teams do not seem to follow this
advice.

Though the list of words are automatically generated in the tool, we see this
as an area that can be improved using modern techniques: it should be possible
to flag suspicious spellings during drafting without requiring users to do
this post hoc (e.g. after drafting a verse). Moreover, current tools show
spelling suggestions based on a manually curated list of accepted words
(dictionary). This can be augmented using statistical techniques to
incrementally improve the quality of the suggestions based on the availability
of the drafted corpus.

###  3.5 Bottleneck Stages

Multiple translators reported that the initial ’Drafting’ (including ’Peer
Checking’) and ’Exegesis’ stages are the most time consuming since they
involve deep thinking to keep the text faithful to the source. The time taken
in subsequent stages were reported to be dependent on the quality of the
output of these initial stages. A consultant noted that achieving clarity for
the New Testament and both clarity and accuracy for the Old Testament take
time.

We recommend incorporating machine translation suggestions during the drafting
stage, especially at the word/phrase level. Though starting with a machine
translated first draft could be useful, it was reported that these are
counterproductive and slow the work down when the machine generated output is
of poorer quality. Moreover, highly specific (at the word/phrase level) just-
in-time reference resources for translators would improve the both speed and
quality of the initial draft.

###  3.6 Backtranslation

As a translation project progresses the backtranslations become more
important. This is especially true for consultants who usually rely on them to
catch issues and assess accuracy. Having more than one version of the
backtranslation was reported as being helpful to better grasp the text. The
advances in Neural Machine Translation makes it possible today to bootstrap
reasonable quality of backtranslations using limited training data.
Integrating such capabilities into tools would speed up the review process.

###  3.7 Fonts and Inputs

Typing the text in the target language has been noted as a time-consuming task
(e.g. selecting diacritics from a list, etc.), despite using software tools
like Google Input Tools444https://www.google.com/inputtools and Phonetic. Most
of the translators did not learn to type in their scripts directly citing a
sharp learning curve. Some tools did not handle complex fonts correctly (e.g.
for search and replace). Also, it was noted that there is a tendency to use
hard-copies of drafts for checking and revising even though software tools
facilitate this directly on the computer. This points to the importance of
features that generate quick readable/printable formats from the drafts.

###  3.8 Tool Complexity

Complex tools (in terms of user interface and features) discourage users from
trying newer or more advanced features due to fear of losing or messing up the
data. It was also noted that strict tool policies (e.g. parallel passages) can
sometimes confuse users and cause them to make changes in the text just to
make the tool happy rather than reasoning through the text. This may also
point to the need of having just-in-time documentation and guides
(demonstrations) for tools.

###  3.9 Community Checking

It was noted that there did not seem to be much structure for the ’Community
Checking’ stage. This could be improved by stating concrete requirements for
the number and types of participants along with simple methodologies to elicit
meaningful feedback from the community (listeners).

###  3.10 Translation Progression

It was observed that the general high-level progression for Bible translation
is from New to Old Testament. Though there maybe exceptions, this information
is useful in designing tools that leverage incremental learning. e.g.
Automated machine translation suggestions that are intentionally biased to
learn contextually similar words found both in the New and Old Testaments.

###  3.11 Quality Checking: Mental Modal

There was an interesting perspective shared by an experienced consultant which
we believe is helpful in thinking about the role of automated techniques for
quality checking Bible translations: it is possible to think of quality
checking being comprised of objective and subjective parts. The former is
based on fixed patterns that a machine could learn and where technology could
provide automated tooling for. Whereas it should provide support for the
latter which has to do with the specificities of the target culture and
language. Moreover, it maybe argued that a significant portion of the quality
checking process is objective since the original languages (Hebrew, Greek)
remain constant.

##  4 Conclusion

We survey multiple individuals through interviews who are working in different
roles in the Bible translation process from various parts of the world and
share our observations and make relevant recommendations for tool developers.
Overall, we find multiple areas where the current state of technology could
significantly improve the speed and quality of the Bible translation process.

##  Appendix A Appendix

###  A.1 Interview Question Bank

####  A.1.1 Personal

  * •

What is your name and where are you based?

  * •

Which project(s) do you work in?

  * •

What is your educational background?

  * •

What languages do you speak?

  * •

How did you get involved with Bible translation and for how long?

####  A.1.2 Technical

  * •

Do you own a smartphone or use a computer outside of work? If so, for how
long?

  * •

Which software applications do you use for work? On what device?

  * •

What are three things you like most about the applications you use?

  * •

What are the three biggest issues you faced in the applications?

  * •

What feature/app do you think would really help you speed up your work?

  * •

Do you have a wish list for the tools you use?

  * •

Have you used AutographaMT?

    * –

If so, which language did you use it for? What was your experience like?

    * –

What could be done better?

  * •

Have you used Paratext? If so, did you need training for it- how long?

    * –

How easy/difficult do you find it?

    * –

What 3 features do you like the most?

    * –

What are 3 things do you like the least?

####  A.1.3 Translation Related

  * •

Explain in detail all the steps involved in your work.

    * –

Include applications/tools you use and how you interact with them.

    * –

How much time does each step take? How can each step be done faster?

  * •

Explain the progression of the work and what the process is after the current
step.

  * •

What are you most concerned about when working on each step?

  * •

What tools/improvements do you wish for in the current work you are doing?

  * •

Would you be willing to share your screen and show us an example of your work?

  * •

What do you enjoy most about this work? What do you dislike?

####  A.1.4 Feedback

  * •

What were your thoughts when you were asked about this interview?

  * •

How could we have helped you be better prepared for this interview?

  * •

Would you be willing to do another interview, if required?

  * •

What did you like and what could have been done better about this interview?

  * •

Are there others you think who could also provide relevant information? Who
and why?