# Robust Phase metrology with hybrid quantum interferometers against particle
losses

X. N. Feng  Information Quantum Technology Laboratory, International
Cooperation Research Center of China Communication and Sensor Networks for
Modern Transportation, School of Information Science and Technology, Southwest
Jiaotong University, Chengdu 610031, China  D. He  Information Quantum
Technology Laboratory, International Cooperation Research Center of China
Communication and Sensor Networks for Modern Transportation, School of
Information Science and Technology, Southwest Jiaotong University, Chengdu
610031, China  L. F. Wei111E-mail<EMAIL_ADDRESS>Information Quantum
Technology Laboratory, International Cooperation Research Center of China
Communication and Sensor Networks for Modern Transportation, School of
Information Science and Technology, Southwest Jiaotong University, Chengdu
610031, China

###### Abstract

Entanglement is an important quantum resource to achieve high sensitive
quantum metrology. However, the rapid decoherence of quantum entangled states,
due to the unavoidable environment noise, result in practically the unwanted
sharp drop of the measurement sensitivity. To overcome such a difficulty, here
we propose a spin-oscillator hybrid quantum interferometer to achieve the
desirable precise estimation of the parameter encoded in the vibrations of the
oscillator. Differing from the conventional two-mode quantum interferometers
input by the two-mode NOON state or entangled coherent states (ECS), whose
achievable sensitivities are strongly limited by the decoherence of the
entangled vibrational states, we demonstrate that the present interferometer,
input by a spin-dependent two-mode entangled state, possesses a manifest
advantage, i.e., the measurement sensitivity of the estimated parameter is not
influenced by the decoherence from the spin-oscillator entanglement. This is
because that, by applying a spin-oscillator disentangled operation, the
information of the estimated parameter encoded originally in the vibrational
degrees can be effectively transferred into the spin degree and then can be
sensitively estimated by the precise spin-state population measurements. As
consequence, the proposed hybrid quantum interferometer possesses a manifest
robustness against the particle losses of the vibrational modes.
Interestingly, the achieved phase measurement sensitivity can still surpass
the SQL obviously, even if relatively large number of particle loss occurs in
one of the two modes. The potential application of the proposed spin-
oscillator hybrid quantum interferometer is also discussed.

##  I Introduction

In recent years, quantum metrology utilizing various quantum resources to beat
the standard quantum limit (SQL) has become one of the hot topics in quantum
information processing. Typically , quantum entanglement, as an important
quantum resource, has been widely used to improve the measurement sensitivity
of quantum interferometer. For example, if the Mach-Zender interferometer
(MZI) is input by the NOON state, the achieved phase measurement sensitivity
can reach the Heisenberg limit (HL) [2, 3, 1] . In fact, quantum
interferometers with NOON state inputs have been widely used to implement the
super-resolution quantum lithography [1], quantum microscopy [3] and the bio-
sensing [5, 4] etc. However, a practical problem for these applications is
that, the NOON state is inherently very fragile in various noise environments.
Typically, with the loss of the photons the coherence of the interferometer
decreases rapidly, leading to the achieved measurement sensitivity decays
quickly [6, 7, 8, 9, 10].

To overcome such a practical problem, a series of proposals have been proposed
[7] to the MZIs typically including the squeezed state inputs [11, 12, 13, 14,
15, 16]), rather than the NOON state, to improve the robustness again the
photon loss. However, the generation and manipulation of the strong squeezing
states is still a great challenge for the current experimental technology.
Therefore, the MZI input by either the ECS or the Schrödinger-cat-states has
become a hot research topic recently [7]. Although these states are indeed
more robust than the NOON state with respect to the noise of particle loss [7,
8], the off-diagonal elements of the density matrix for these states, which
provide the quantum enhancement of sensitivity beyond the SQL, still decays
exponentially rapidly with the loss of particles [6]. Therefore, it is still a
challenge to achieve the high metrologic sensitivity far beyond the SQL with
these states especially for large particles number. In addition, a detector
with particle number resolved is usually required to achieve the high
sensitivity detection beyond SQL. This is also a big challenge especially when
the number of particles is large. Therefore, designing the novel quantum
interferometer that is robust to the typical noise of particle loss as well as
easy for readout is still an important issue.

Recently, the spin-oscillator hybrid quantum interferometer, a natural
generalization of the MZI, have attracted much attention for quantum metrology
[21, 19, 20, 17, 18]. Compared with the conventional two-mode interferometers,
the spin-oscillator hybrid interferometers can well combine the advantages of
the spin system and the oscillator one; the quantum harmonic oscillator is
very sensitive to the changes of external environmental parameters, and thus
they can be used as the ideal probes to achieve sensitive detection; the spin
states are relatively easier to be manipulated, and thus the spin degrees of
freedom can be used to achieve the high fidelity manipulation and readout of
the oscillator states. In fact, these spin-oscillator hybrid quantum systems
have been experimentally realized with various platforms, typically e.g., the
trapped ions [22, 23], superconducting circuits [20, 21], optomechanics [24]
and the diamond color centers [18, 19, 20] etc. In particular, in a trapped
ion system, the coupling between the spin (i.e., the internal atomic levels)
of the ion and its external vibration can be achieved by applying the
designable laser pulses [25]. The parameter information encoded in the
external state of the ion (such as the displacement amplitude caused by an
external weak force) can be indirectly measured by probing the population of
the ionic internal spin states [23, 26]. Physically, the detection sensitivity
of the spin state is limited by its deccoherence, due to the interaction with
the harmonic oscillator [27]. It was shown that, the spin decoherence of spin-
oscillator interferometers can be effectively suppressed by squeezing the
vacuum fluctuation of the oscillator [22, 28] . Furthermore, Ref. [23] showed
that, by applying a reverse action to the harmonic oscillator before the spin
state detection, the inter- and external states of the trapped ions can be
completely decoupled. As a consequence, the fidelity of the spin-state
detection can be enhanced significantly.

A natural question is, are these suppression methods of the spin decoherence
also applicable to the spin-oscillator hybrid interferometer with a two-
dimensional oscillator? More interestingly, whether the practical precision
measurements are robust against the unavoidable environmental noise? In the
following we will give the positive answers to these questions. Takes a spin-
oscillator hybrid quantum interferometer with a two-dimensional quantum
harmonic oscillator as an example, we investigate in this paper how to achieve
the desirable precise measurement of the rotation parameter $\Omega$ with a
spin-dependent entangled state. Here, the two-dimensional harmonic oscillator
is equivalent to a conventional two-mode interferometer, which is used to
encode the information of the rotation parameter ($\Omega$); the spin is used
as an auxiliary degree of freedom to generate the desirable input state of the
interferometer and the readout of the parameter information by performing the
spin population detection. With certain spin-dependent entangled states as
input, the spin-oscillator entanglement can be completely decoupled by
applying the proper spin-oscillator inverse operations, yielding the
information encoded in the two-dimensional oscillator states are transformed
as the relative phase of the spin states. Therefore, by performing the high
fidelity spin-state detection, the high sensitive measurement of the parameter
can be achieved. More importantly, compared with the conventional two-mode
interferometer input with NOON state and ECS [23], we demonstrate that the
proposed hybrid quantum interferometer is more robust to the noise induced by
particles loss of the oscillator.

The paper is organized as follows. In Sec. II, we introduce a general model of
the hybrid spin-oscillator interferometer input with a general spin-dependent
entangled state, to implement the parameter estimation. In Sec. III, we
investigate to improve the measurement sensitivity of the interferometer by
inputting a special NOON-like spin-dependent entangled state. The specific
measurement of parameter $\Omega$ through the spin projective measurement is
also discussed. In Sec. IV, the performance of the interferometer is
investigated when the particle loss occur in the vibrating mode of oscillator
is considered. In Sec. V, we summarize our results and discuss their
experimental feasibilities.

##  II Spin-two-mode-oscillator hybrid quantum interferometer

We consider a hybrid quantum interferometer system [21, 19, 20, 17, 18], as
shown schematically in Figure. 1, wherein a 1/2-spin is coupled to a two-
dimensional harmonic oscillator. The parameter $\Omega$ to be estimated is
encoded in the two-mode oscillator by the designed time-evolution with the
time-independent Hamiltonian $\hat{H}=\Omega\hat{H}_{0}$, with $\hat{H}_{0}$
being the Hamiltonian of the oscillator. The spin is not only used to generate
the desired spin-dependent entanglement of the hybrid quantum system, but also
served as a detector to read out the information for the parameter
estimations.

Figure 1: (Color online)A hybrid spin-oscillator quantum system (a) is used to
implement the spin-oscillator quantum interferometer (b) with input state
$|\Psi_{i}\rangle=|N\rangle_{a}|0\rangle_{b}\otimes|+\rangle$. A controlled-
phase-gate (CPG) $e^{i\hat{a}^{\dagger}\hat{a}\sigma_{z}/2}$ and reverse CPG
$e^{-i\hat{a}^{\dagger}\hat{a}\sigma_{z}/2}$ is applied before and after the
phase accumulation $e^{i\Omega TJ_{y}}$ in the two-mode harmonic oscillator.
$R_{a}$ and $R_{b}$ are the particle loss rates of the mode a and mode b,
respectively. Finally, the information of the estimated parameter $\Omega$ is
extracted by the projective measurement on the spin.

Without loss of generality, we assume that the hybrid quantum interferometer
is prepared in the initial state (i.e, the input state of the interferometer):

$|\Psi_{i}\rangle=\frac{1}{\sqrt{2}}\Big{[}|\uparrow\rangle|\psi_{1}\rangle+|\downarrow\rangle|\psi_{2}\rangle\Big{]},$
(1)

where $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ denote the quantum states of
the two-dimensional harmonic oscillator, $|\downarrow\rangle$ and
$|\uparrow\rangle$ are the spin up and down states of the spin. Obviously, if
$|\psi_{1}\rangle=|\psi_{2}\rangle$, then the harmonic oscillator state and
the spin state are separable and unentangled. However, when
$|\psi_{1}\rangle\neq\psi_{2}\rangle$, $|\Psi_{i}\rangle$ becomes a spin-
dependent entangled state.

Supposed that the parameter $\Omega$ is encoded by the Hamiltonian
$\hat{H}=\Omega\hat{H_{0}}$ of the system, which would evolve from the initial
state $|\Psi_{i}\rangle$ to the output state: $|\Psi(\Omega)\rangle=e^{i\Omega
T\hat{H}_{0}}|\Psi_{i}\rangle$ with $T$ being the free evolution time of the
system. As a consequence, the precise measurement of the parameter $\Omega$
can be realized by detecting the output state of the interferometer. According
to the Cramer-Rao formula [29], the measurement sensitivity theoretically
satisfies the inequality:

$\Delta\Omega\geqslant\frac{1}{\sqrt{nF^{Q}(|\Psi_{i}\rangle)}}.$ (2)

Here, $n$ represents the number of measurements and $F^{Q}(|\Psi_{i}\rangle)$
is the QFI of the state $|\Psi_{i}\rangle$, which can be calculated as:

$\begin{split}F^{Q}(|\Psi_{i}\rangle)&=4\Big{(}\langle\Psi_{i}|\hat{H}_{0}^{2}|\Psi_{i}\rangle-|\langle\Psi_{i}|\hat{H}_{0}|\Psi_{i}\rangle|^{2}\Big{)}\\\
&=F^{Q}_{O}(|\Psi_{i}\rangle)+F^{Q}_{E}(|\Psi_{i}\rangle),\end{split}$ (3)

with
$F^{Q}_{O}(|\Psi_{i}\rangle)=2\Big{[}(\Delta\hat{H}_{0|\psi_{1}\rangle})^{2}+(\Delta\hat{H}_{0|\psi_{2}\rangle})^{2}\Big{]}$
and
$F^{Q}_{E}(|\Psi_{i}\rangle)=\Big{|}\langle\psi_{1}|\hat{H}_{0}|\psi_{1}\rangle-\langle\psi_{2}|\hat{H}_{0}|\psi_{2}\rangle\Big{|}^{2}$,
also
$(\Delta\hat{H}_{0|\psi_{j}\rangle})^{2}=\langle\psi_{j}|\hat{H}_{0}^{2}|\psi_{j}\rangle-\langle\psi_{j}|\hat{H}_{0}|\psi_{j}\rangle^{2}$,
$j=1,2$. Obviously, for $|\psi_{1}\rangle=|\psi_{2}\rangle$, we have
$F^{Q}(|\Psi_{i}\rangle)=4(\Delta\hat{H}_{0|\psi_{1}\rangle})^{2}$. This is
result of the conventional two-mode quantum interferometer without spin
entanglement. Interestingly, if $|\psi_{1}\rangle\neq|\psi_{2}\rangle$, the
QFI of the state $|\Psi_{i}\rangle$ might increase, and thus the sensitivity
of the parameter estimation can be improved. Specifically, suppose that [27]
$|\psi_{1}\rangle=|i\alpha\rangle_{a}|\beta\rangle_{b}$ and
$|\psi_{2}\rangle=|-i\alpha\rangle_{a}|\beta\rangle_{b}$, with $|\pm
i\alpha\rangle_{a}$ and $|\beta\rangle_{b}$ being respectively the coherent
states of the mode a and moed-b, we have
$(\Delta\hat{H}_{0|\psi_{j}\rangle})^{2}=|\alpha|^{2}+|\beta|^{2}$ $(j=1,2)$
and
$\Big{[}\langle\psi_{1}|\hat{H}_{0}|\psi_{1}\rangle-\langle\psi_{2}|\hat{H}_{0}|\psi_{2}\rangle\Big{]}^{2}=4\Re(\alpha\beta)^{2}$,
with
$\hat{H}_{0}=J_{y}\equiv(\hat{a}^{\dagger}\hat{b}-\hat{a}\hat{b}^{\dagger})/2i$.
In particular, when $\alpha=|\alpha|=\beta\gg 1$, the contribution of
$F^{Q}_{E}(|\Psi_{i}\rangle)$ to QFI is much larger than that of
$F^{Q}_{O}(|\Psi_{i}\rangle)$. In this case, the sensitivity of parameter
estimation is mainly determined by the spin-oscillator entanglement.

As we know, with the NOON state or ECS as input the conventional two-mode
interferometer without spin dependent can also achieve the precise
measurements approaching the HL [2, 7]. However, the single photon resolved
detectors are usually required to achieve the desired sensitivity. For the
present system, it is still a big challenge to realize the resolvable
detection of the vibrational phonon numbers. To avoid such a difficulty,
transferring the estimated parameters encoded in the harmonic oscillator
states to the spin states, and then detecting it by the spin projection
measurement is particularly desired. Basing on this idea, the precise
measurement of parameter $\Omega$ encoded in the harmonic oscillator state can
be accomplished by the following steps:

i) Let the spin-oscillator interferometer prepared in a spin-dependent
entangled state undergos a dynamic evolution $\hat{U}(\Omega)$, i.e.,
$|\Psi_{i}\rangle\rightarrow|\psi(\Omega)\rangle=\hat{U}(\Omega)|\Psi_{i}\rangle$
for implementing the parameter encoding;

ii) Apply a global operation $\hat{F}=\exp(i\eta\hat{S}\otimes\hat{A})$ (with
$\eta$ being the interaction strength between the spin and oscillator) on the
state $|\psi(\Omega)\rangle$ for transferring the parameter into the spin
state. Here, $\hat{S}$ and $\hat{A}$ represent the operators acting on the
spin state and the harmonic oscillator state, respectively. After such an
operation, the state $|\psi(\Omega)\rangle$ of interferometer is evolved into:

$|\psi_{f}\rangle=\hat{F}|\psi(\Omega)\rangle,$ (4)

iii) Perform the spin projection measurements after a $\pi/2$-pulse operation
on the spin. The probability of the spin at the state
$|\downarrow\rangle(|\uparrow\rangle)$ is obtained as:

$P_{\downarrow}(\Omega)=\langle\psi(\Omega)|\hat{F}^{\dagger}|\downarrow\rangle\langle\downarrow|\otimes
I|\hat{F}|\psi(\Omega)\rangle,$ (5)

with $I$ being the identity operator of oscillator,
$|\uparrow\rangle(|\downarrow\rangle)$ being the eigenstate of Pauli operator
$\sigma_{z}$ with eigenvalue 1 (-1), and
$P_{\uparrow}(\Omega)=1-P_{\downarrow}(\Omega)$. Specifically, if
$\hat{S}=\sigma_{z}$, then the equation (5) can be specifically expressed as:

$P_{\downarrow}(\Omega)=\frac{1}{2}\Big{[}1+\langle\psi_{2}|e^{-i\Omega
TH_{0}}e^{2i\eta\hat{A}}e^{i\Omega TH_{0}}|\psi_{1}\rangle\Big{]}.\\\ $ (6)

Obviously, if $\eta=0$ or $[\hat{A},\hat{H}_{0}]=0$, the spin measurement
$P_{\downarrow}(\Omega)$ does not contain any information of the parameter
$\Omega$, and thus such a measurement is invalid. While, for
$[\hat{H}_{0},\hat{A}]\neq 0$, the result of the spin measurement must be a
function of the parameter $\Omega$. Consequently, according to the error
propagation formula, the sensitivity of the parameter estimation can be
expressed as:

$\Delta\Omega=\frac{\sqrt{P_{\downarrow}(\Omega)-P^{2}_{\downarrow}(\Omega)}}{d|P_{\downarrow}(\Omega)|/d\Omega}.$
(7)

This indicates that, the sensitivity of the parameter estimation is closely
related to the variation of $|P_{\downarrow}(\Omega)|$ with respect to the
estimated parameter $\Omega$. For example, if
$|\langle\psi_{2}^{\prime}|\psi_{1}^{\prime}\rangle|\neq 1$ with
$|\psi^{\prime}_{j}\rangle=e^{i(-1)^{j+1}\eta\hat{A}}e^{i\Omega
T\hat{H}_{0}}|\psi_{j}\rangle$ $(j=1,2)$, the value of
$|P_{\downarrow}(\Omega)|$ may decay with the increase of $\Omega T$, yielding
the decrease of measurement sensitivity. Typically, as shown in Ref. [27]
that, if the spin-dependent cat state is utilized to measure the parameter
$\Omega$, the spin measurement result

$\tilde{P}_{\downarrow}(\Omega)=\frac{1}{2}\Big{[}1+e^{2i\alpha\beta\sin(\Omega
T)}e^{-8\alpha^{2}\sin^{2}(\Omega T/2)}\Big{]},\\\ $ (8)

decays exponentially with the increase of $\Omega T\leq\pi$, due to the spin
state decoherence. Above, $\alpha$ and $\beta$ are the coherent displacement
amplitudes of mode a and mode b, respectively. Obviously, in this case the
high sensitivity of the parameter estimation can only be achieved for certain
$\Omega T$ satisfying the condition $8\alpha^{2}\sin^{2}(\Omega T/2)\ll 1$,
such as $\Omega T\ll 1$. Alternatively, in the following we prove that, by
inputting a proper state to the interferometer and then performing the
suitable quantum operations, the above spin decoherent effect induced by spin-
oscillator entanglement can be effectively avoided. As a consequence, the
desired precise measurements of the arbitrary $\Omega T$ parameter can be
achieved with the proposed spin-oscillator hybrid quantum interferometer.

##  III Quantum metrology by using the hybrid interferometer with NOON-like
state inputs

From Eq. (6) we can see that, the result of the spin state population
measurement is related to the initial state, dynamic evolution, and the spin-
oscillator interaction in the interferometer. Physically, the spin-oscillator
entanglement might lead to the spin decoherence, which decreases the
sensitivity of spin state population measurement. Recently, Gilmore et al.
[23] has demonstrated an effective approach to avoid such a limit. For the
weak force measurements with the one-dimensional vibration of a trapped ion
crystal, they showed that the influence of the spin-oscillator entanglement
induced decoherence on the population measurement can be effectively avoided
by applying a inverse spin-dependent disentangled operation. In this section,
we generalize such an idea to the proposed two-dimensional oscillator by
similarly performing an inverse spin-oscillator operation to disentangle the
spin and oscillator before the spin state population measurement.

For a typical parameter estimation model, wherein the dynamical evolution of
the two-mode oscillator is described by the Hamiltonian
$\hat{H}_{0}=J_{y}=(\hat{a}^{\dagger}\hat{b}-\hat{a}\hat{b}^{\dagger})/2i$,
suppose the hybrid interferometer is input with a spin-dependent entangled
state

$\begin{split}|\Psi_{h}\rangle&=\hat{C}_{a}(\frac{\pi}{2})e^{i\frac{\pi}{2}J_{y}}|N,0\rangle|+\rangle\\\
&=\frac{(-i)^{N}}{\sqrt{2^{N+1}N!}}\Big{[}(\hat{a}^{\dagger}+i\hat{b}^{\dagger})^{N}|0_{a},0_{b}\rangle|\downarrow\rangle\\\
&+(-1)^{N}(\hat{a}^{\dagger}-i\hat{b}^{\dagger})^{N}|0_{a},0_{b}\rangle|\uparrow\rangle\Big{]},\\\
\end{split}$ (9)

with $|N,0\rangle=|N\rangle_{a}|0\rangle_{b}$ and
$|\pm\rangle=(|\uparrow\rangle\pm|\downarrow\rangle)/\sqrt{2}$ being the Fock
state of the oscillator and the spin superposition state, respectively. Above,
$\hat{C}_{a}(\pi/2)=e^{i\pi\hat{a}^{\dagger}\hat{a}\sigma_{z}/2}$ is a
controlled-phase-gate (CPG) [30, 31, 32], acting on mode a of the oscillator,
to generate the desired entangled state. The estimated parameter $\Omega$ is
encoded into the hybrid quantum interferometer under a free evolution:

$|\Psi(\Omega)\rangle=e^{i\Omega TJ_{y}}|\Psi_{h}\rangle.$ (10)

The achievable sensitivity of its estimation is determined by the QFI of the
output state $|\Psi(\Omega)\rangle$ of the interferometer. With Eq. (3), we
have: $\Delta J_{y|\varphi_{k}\rangle}=0$ and
$\Big{|}\langle\varphi_{1}|J_{y}|\varphi_{1}\rangle-\langle\varphi_{2}|J_{y}|\varphi_{2}\rangle\Big{|}^{2}=N^{2}$
with
$|\varphi_{k}\rangle=e^{(-1)^{k+1}i\pi\hat{a}^{\dagger}\hat{a}/2}e^{i\pi/2J_{y}}|N,0\rangle$
($k=1,2$), and thus $F^{Q}(|\Psi(\Omega)\rangle)=N^{2}$. According to the
Cramer-Rao inequality [29], the sensitivity of the estimated parameter can
approach the HL. To implement the desired parameter estimation by using the
spin population measurement, we apply an inverse CPG
$\hat{C}^{\dagger}_{a}(\pi)$ to the output state, and obtain

$\begin{split}|\Psi_{f}\rangle&=\hat{C}^{\dagger}_{a}(\pi/2)|\Psi(\Omega)\rangle\\\
&=e^{-i\pi\hat{a}^{\dagger}\hat{a}\sigma_{z}/2}e^{i\theta
J_{y}}e^{i\pi\hat{a}^{\dagger}\hat{a}\sigma_{z}/2}e^{i\pi
J_{y}/2}|N,0\rangle|+\rangle\\\
&=\frac{1}{\sqrt{2}}(e^{iN\theta/2}|\downarrow\rangle+e^{-iN\theta/2}|\uparrow\rangle)e^{i\pi
J_{y}/2}|N,0\rangle,\end{split}$ (11)

by which the parameter information is transferred into the relative phase of
the spin state. Specifically, the population of the spin state
$|\downarrow(\uparrow)\rangle$ reads:

$P_{\downarrow(\uparrow)}(\theta)=\frac{1}{2}\Big{[}1\pm\cos(N\theta)\Big{]},$
(12)

which implies that the sensitivity to estimate the parameter $\Omega$ can be
expressed as

$\Delta\Omega=\frac{1}{NT}.$ (13)

Eqs. (12) and (13) show clearly that, the spin population oscillates rapidly
with the phase $\theta=\Omega T$ and the higher sensitivity of the parameter
estimation can be achieved by using the Fock state $|N\rangle$ with the larger
phonon number of mode a.

Noted that, although the conventional two-mode interferometer with the NOON
state input could also be utilized to implement the precise measurement
approaching the HL, the required phonon number-resolvable detection is a
challenge to be realized in practice. More importantly, the sensitivity
reduces significantly with the loss of photon number, as the input NOON state
is very fragile in the realistic environment. In the next section we show
that, the interferometer proposed above is very robust against the phonon loss
under certain condition, alternatively.

##  IV Robustness for the ”Particle” losses

Physically, an entangled quantum system is usually fragile and its coherence
losses easily under the disturbance of the environmental noise [7]. Due to
this decoherence, the sensitivity of the parameter estimation achieved by the
interferometer with entangled state input would decreases. For example, the
photon loss affects significantly the sensitivity of the typical two-mode
interferometers [7, 10]. For the spin-oscillator hybrid quantum interferometer
proposed here, the vibrational phonon loss of the oscillator is also one of
the main noises and lead to the decoherence of the hybrid quantum system. This
will significantly reduce the achievable sensitivity of the parameter
estimation.

Physically, the particle (e.g., phonon) loss of the spin-oscillator hybrid
interferometer can be described by a beam-splitter [9], wherein the noisy
environment can be represented as a thermal bath in vacuum state
$|0_{E}\rangle$. As a consequence, the input state (9) of the hybrid quantum
interferometer should be replaced as:

$\begin{split}|\Psi\rangle_{l}&=\exp[\eta_{a}(\hat{a}^{\dagger}\hat{c}-\hat{a}\hat{c}^{\dagger})]\exp[\eta_{b}(\hat{b}^{\dagger}\hat{e}-\hat{b}\hat{e}^{\dagger})]|\Psi_{h}\rangle|0_{E}\rangle,\end{split}$
(14)

where $\hat{c}^{\dagger}(\hat{c})$ and $\hat{e}^{\dagger}(\hat{e})$ are the
creation (annihilation) operators of environment interacting with the mode a
and mode b of the oscillator, respectively. $R_{a}=\sin^{2}(\eta_{a})$ and
$R_{b}=\sin^{2}(\eta_{b})$ are the corresponding loss rates. Considering these
particle losses, the equation (14) can be rewritten as (see Appendix A )

$|\Psi\rangle_{l}=\frac{\hat{C}_{a}(\pi/2)e^{i\pi
J_{y}/2}}{\sqrt{N!}}(u\hat{a}^{\dagger}+v\hat{b}^{\dagger}+p\hat{e}+q\hat{c}\sigma_{z})^{N}|0_{a}0_{b},0_{E}\rangle|+\rangle,$
(15)

where $u=(\cos(\eta_{a})+\cos(\eta_{b}))/2$,
$v=(\cos(\eta_{a})-\cos(\eta_{b}))/2$, $p=i\sin(\eta_{a})/\sqrt{2}$, and
$q=\sin(\eta_{b})/\sqrt{2}$. After tracing the wave function of the
environment and the harmonic oscillator, the reduced density matrix of the
spin state is obtained as (see Appendix B for detail)

$\rho_{L}=\frac{1}{2}\Big{(}|\downarrow\rangle\langle\downarrow|+|\uparrow\rangle\langle\uparrow|\Big{)}+R|\downarrow\rangle\langle\uparrow|+R^{*}|\uparrow\rangle\langle\downarrow|,$
(16)

It is seen that the information of the parameter $\theta=\Omega T$ is related
to the off-diagonal element

$R=(u^{2}e^{i\theta}+v^{2}e^{-i\theta}+p^{2}-|q|^{2})^{N},$ (17)

of the above reduced density matrix. Similar to the Ramsey interference, after
a $\pi/2$-pulse operation the measured result of the spin state population is:

$P^{L}_{\downarrow}(\theta)=\frac{1}{2}\Big{[}1+|R|\cos(\phi(\theta))\Big{]},\,\phi(\theta)=\arg(R),$
(18)

Obviously, with the error propagation formula, the estimated sensitivity of
the $\theta$-parameter can be expressed as

$\begin{split}\Delta\theta=\frac{\sqrt{P^{L}_{\downarrow}-(P^{L}_{\downarrow})^{2}}}{|dP^{L}_{\downarrow}/d\theta|}\approx\frac{\sqrt{1-|R|^{2}\cos^{2}(\phi(\theta))}}{|R||\sin(\phi(\theta))d\phi(\theta)/d\theta|}.\end{split}$
(19)

Figure 2: (Color online) The non-diagonal element of the reduced matrix of
spin $|R|$ versus the phase $\theta$ for the particle loss occurring only in
one mode (mode b (the left subfigure) or mode a(the right subfigure)) with
different loss rates.

It is seen from Eq. (19) that the sensitivity $\Delta\theta$ is mainly
determined by the value of the off-diagonal element $R$ (i.e., the value of
$|R|$ and $\phi(\theta)=\arg(R)$). Obviously, when $|R|=1$ the spin state
$\rho_{L}$ of Eq. (16) is a pure state, and the sensitivity of Eq.
$(\ref{dtheta_l})$ depends only on the derivation $d\phi(\theta)/d\theta$ and
becomes $\Delta\theta\approx 1/|d\phi(\theta)/d\theta|$. While $|R|<1$, the
state $\rho_{L}$ becomes a mixed state, and the estimation sensitivity
$\Delta\theta$ decreases.

Figure 3: (Color online) The non-diagonal element $|R|$ of the reduced density
matrix of spin versus the phase $\theta$ under the different particle loss
rates of mode-a. Here, the loss rate for the mode b is fixed as: $R_{b}=0.5$,
for $N=25$.

Here the value of $|R|$ are related to the loss rate $R_{a}$, $R_{b}$ and the
value of $\theta$ as shown in Eq. (17). Fig. 2 shows how the value of the off-
diagonal element for the spin state varies with the phase $\theta$ for single
mode loss (either mode a or mode b). It is shown that, the value of
$|R|\approx 1$ is achieved around $\theta=0$ (or $\theta=\pi$) as long as the
particles loss only occurs in mode b with $R_{a}=0$ (or mode a for $R_{b}=0$).
Though with increase of loss rate the value of $|R|$ decays more rapidly with
$\theta$, $|R|\approx 1$ can still be achieved. This implies that the high
sensitivity can still be achieved in this case as shown below. While the
particle loss occur in both modes the value of $|R|(\leq 1)$ also drop quickly
and can not achieve one again for any $\theta$ as shown in Fig. 3. Given
$\theta$ is precisely determined, the value of $\Omega$ can be estimated
finally with the sensitivity being $\Delta\Omega=\Delta\theta/T$ for a fixed
evolution time $T$.

Noted that, the influence of different loss rates for two modes on the
estimated sensitivity of the phase achieved by the conventional two-mode
interferometers have been widely investigated [6, 7, 33, 34, 35, 36]. For
example, the Refs [6, 33] have proven that in a two-mode interferometer with
either NOON state or ECS input, the sensitivity of the parameter estimation is
mainly determined by the larger loss rate of the two mode. Therefore, as long
as one mode loses relatively large particle number, the achieved sensitivity
of the estimation would rapidly decrease regardless of whether the other mode
occurs the particle loss. Alternatively, with the hybrid quantum
interferometer proposed here, the optimal sensitivity achieved by using this
hybrid system is mainly determined by the smaller loss rate since the value of
$|R|$ mainly depends on the smaller loss rate shown in Fig. 3.

Figure 4: (Color online) The sensitivity of the phase estimation,
characterized by $1/\Delta\theta$, changes with the phase $\theta$ for various
loss rates of the mode-a. Here, $R_{b}=0$ for $N=25$.

Fig. 4 shows how the variation of the phase measurement sensitivity,
characterized by $1/\Delta\theta$ in y-axis, changes with the phase $\theta$
for different loss rates. Obviously, the larger the value of $1/\Delta\theta$
corresponds to the higher achievable sensitivity of the phase estimation. It
can be seen that, if particle loss only occur in mode b, the optimal
sensitivity of the phase estimation can be achieved at $\theta\approx 0$.
Typically, even if the loss rate of the mode b is relatively large,
approaching to such as $50\%$, the optimal sensitivity still obviously
surpasses the SQL.

Figure 5: (Color online) The sensitivity, characterized by $1/\Delta\theta$,
achieved by the present hybrid interferometry (marked by its input state
$|\Psi_{h}\rangle$) compared with that demonstrated by the convenient
interferometer (typically in Ref. [34]) marked as $|\Psi_{E}\rangle$ in the
figure. Here, we assume that the particle loss occurs only in mode b with the
decay rate $R_{b}$.

To show more clearly the loss robustness of the present hybrid interferometer
input by $|\Psi_{h}\rangle$, let us compare its reachable sensitivity with
that typically demonstrated with the conventional two-mode interferometer
input by the ECS [34]. Fig. 5 shows that, if the particle loss occurs only in
one mode with the same rate, the sensitivity achieved by the present hybrid
interferometer is much higher than that achieved by the conventional two-mode
interferometer. Furthermore, even for the relatively small loss rate, such as
$R=0.1$, the sensitivity with ECS quickly decreases to the SQL, while the
sensitivity with $|\Psi_{h}\rangle$ is still significantly high approaching to
the HL. Even if the loss rate is relatively large such as $R_{b}=0.5$, the
sensitivity with $|\Psi_{h}\rangle$ is still about seven times larger than the
SQL, typically for $N=100$ and $\theta=0.01$.

##  V Conclusions and Discussions

In summary, an entangled hybrid quantum interferometer is proposed to
implement the quantum precision measurement of the parameters encoded in a
two-dimensional oscillator. We showed that, the parameter wanted to be
estimated can be completely transferred into the spin state of the
interferometer by using the spin-oscillator interaction. After a free
evolution, another spin-oscillator disentangled operation is applied to
transfer the information into the spin state for the spin-state population
measurement. The significant advantage of this precision measurement scheme
is, the desired spin state population measurements can avoid the influence
from the spin-oscillator entanglement. Therefore, the achievable sensitivity
of the parameter estimation can be significantly high, even if one of the
vibrational mode of the oscillator exists the dissipated due to the particle
loss. By comparing with the conventional spin-free two-mode bosonic
interferometers, wherein the loss of the particles in any mode would lead to
the rapid decrease of the sensitivity, we showed that the spin-oscillator
hybrid quantum interferometer input with certain spin-dependent entangled
state proposed here is more robust to the particle losses.

Given a series of experimental platforms (typically such as the well-known
cooled trapping ions, the vibrational spin ensembles and also the electrons
trapped on liquid Helium, etc., [37, 38, 39, 40, 8]) have demonstrated the
interaction between the two-level atoms (spins) and the two-dimensional
bosonic vibrations, and also the successful preparations of the desired
vibrational Fock states [41, 42], we believe that the quantum metrology with
the proposed hybrid interferometers is feasible. Also, the robustness of the
proposal interferometer against the particle losses in the vibrations of the
two modes can also be discussed similarly.

##  VI Acknowledgments

This work is partially supported by the National Natural Science Foundation of
China (NSFC) under Grant No. 11974290, and the National Key Research and
Development Program of China (NKRDC) under Grant No. 2021YFA0718803.

##  Appendix A Derivation of the Eqs. (15)

In this appendix, we provide the relevant derivations by considering the
effects of the population losses on measurement accuracy. Following in the
conventional two-mode quantum interferometer, the particle loss can be
described by the beam splitters of the mode a and mode b. Therefore, with the
particle number loss, the input state of the interferometer can be written as:

$\begin{split}|\Psi\rangle_{l}&=\exp[\sqrt{2}\eta_{a}(\hat{a}^{\dagger}\hat{c}-\hat{a}\hat{c}^{\dagger})]\exp[\sqrt{2}\eta_{b}(\hat{b}^{\dagger}\hat{e}-\hat{b}\hat{e}^{\dagger})]|\Psi_{h}\rangle\\\
&=\hat{C}_{a}(\pi/2)e^{i\pi/2J_{y}}\exp\Big{[}i\eta_{a}\sigma_{z}\big{(}(\hat{a}^{\dagger}+\hat{b}^{\dagger})\hat{c}+h.c.\big{)}\\\
&+\eta_{b}\big{(}(\hat{b}^{\dagger}-\hat{a}^{\dagger})\hat{e}-h.c.\big{)}\Big{]}|N,0\rangle|0_{E}\rangle|+\rangle,\end{split}$
(20)

where $R_{k}=\sin^{2}(\eta_{k})$ $(k=a,b)$ represents the loss rate of the
particles of the mode a and mode b, respectively.
$\hat{C}_{a}(\pi/2)=\exp(i\pi\hat{a}^{\dagger}\hat{a}\sigma_{z}/2)$ with
$|0_{E}\rangle=|0_{c},0_{e}\rangle$ being the vacuum modes of the environment.
Using the Baker-Hausdorff formula [43],

$e^{\hat{A}}\hat{B}e^{-\hat{A}}=\sum^{\infty}_{n=0}\frac{1}{n!}[\hat{A^{(n)}},\hat{B}],$
(21)

with $[\hat{A}^{(n)},\hat{B}]=[\hat{A},[\hat{A}^{(n-1)},\hat{B}]]$ and
$[\hat{A}^{(0)},\hat{B}]=\hat{B}$, and letting
$\hat{A}=i\eta_{a}\sigma_{z}[(\hat{a}^{\dagger}+\hat{b}^{\dagger})\hat{c}+h.c.]+\eta_{b}[(\hat{b}^{\dagger}-\hat{a}^{\dagger})\hat{e}-h.c.]$,
$\hat{B}=\hat{a}^{\dagger}$, we have

$\begin{split}[\hat{A}^{(2n+1)},\hat{a}^{\dagger}]&=2^{n}(i\eta_{a}\sigma_{z})^{2n+1}\hat{c}^{\dagger}+(-1)^{n}2^{n}\eta_{b}^{2n+1}\hat{e}^{\dagger}\\\
[\hat{A}^{(2n)},\hat{a}^{\dagger}]&=2^{n-1}(i\eta_{a}\sigma_{z})^{2n}(\hat{a}^{\dagger}+\hat{b}^{\dagger})+(-1)^{n-1}2^{n-1}\eta_{b}^{2n}(\hat{b}^{\dagger}-\hat{a}^{\dagger}).\end{split}$
(22)

Consequently, we have

$\begin{split}&e^{\hat{A}}\hat{a}^{\dagger}e^{-\hat{A}}=\sum_{n=0}^{\infty}\frac{1}{n!}[\hat{A}^{(n)},\hat{a}^{\dagger}]\\\
=&\sum_{k=0}\frac{(\sqrt{2}i\eta_{a}\sigma_{z})^{2k}}{2(2k)!}(\hat{a}^{\dagger}+\hat{b}^{\dagger})+\sum_{k=0}\frac{(-1)^{k+1}(\sqrt{2}\eta_{b})^{2k}}{2(2k)!}(\hat{b}^{\dagger}-\hat{a}^{\dagger})\\\
&+\sum_{k=0}\frac{(\sqrt{2}i\eta_{a}\sigma_{z})^{2k+1}}{\sqrt{2}(2k+1)!}\hat{c}^{\dagger}+\sum_{k=0}\frac{(-1)^{k}(\sqrt{2}\eta_{b})^{2k+1}}{\sqrt{2}(2k+1)!}\hat{e}^{\dagger}\\\
=&\frac{1}{2}\Big{[}\cos(\sqrt{2}\eta_{a})(\hat{a}^{\dagger}+\hat{b}^{\dagger})-\cos(\sqrt{2}\eta_{b})(\hat{b}^{\dagger}-\hat{a}^{\dagger})\Big{]}\\\
&+\frac{1}{\sqrt{2}}i\sin(\sqrt{2}\eta_{a})\sigma_{z}\hat{c}^{\dagger}+\frac{1}{\sqrt{2}}\sin(\sqrt{2}\eta_{b})\hat{e}^{\dagger}\\\
=&\alpha\hat{a}^{\dagger}+\beta\hat{b}^{\dagger}+p\hat{e}^{\dagger}+q\hat{c}^{\dagger}\sigma_{z},\end{split}$
(23)

where $\alpha=(\cos(\sqrt{2}\eta_{a})+\cos(\sqrt{2}\eta_{b}))/2$,
$\beta=(\cos(\sqrt{2}\eta_{a})-\cos(\sqrt{2}\eta_{b}))/2$,
$q=i\sin(\sqrt{2}\eta_{a})/\sqrt{2}$ and $p=\sin(\sqrt{2}\eta_{b})/\sqrt{2}$.
By Noting that
$e^{\hat{A}}|N,0\rangle|0_{E}\rangle=\Big{[}e^{\hat{A}}\hat{a}^{\dagger}e^{\hat{A}}\Big{]}^{N}/\sqrt{N!}|\Theta\rangle$
with $|\Theta\rangle=|0_{a},0_{b}\rangle|0_{E}\rangle$ being the vacuum of the
oscillator system and the environment and combining the above Eq. (23), we get

$\begin{split}\widetilde{|\Psi\rangle}_{l}=&\exp(\hat{A})|N,0\rangle|0_{E}\rangle|+\rangle\\\
=&\frac{1}{\sqrt{N!}}(\alpha\hat{a}^{\dagger}+\beta\hat{b}^{\dagger}+p\hat{e}^{\dagger}+q\hat{c}^{\dagger}\sigma_{z})^{N}|\Theta\rangle|+\rangle\\\
=&\frac{1}{\sqrt{N!}}\sum_{k=0}^{N}C_{N}^{k}\hat{S}^{N-k}(p\hat{e}^{\dagger}+q\hat{c}^{\dagger}\sigma_{z})^{k}|\Theta\rangle|+\rangle\\\
=&\frac{1}{\sqrt{2N!}}\sum_{k=0}^{N}C_{N}^{k}\hat{S}^{N-k}(p\hat{e}^{\dagger}+q\hat{c}^{\dagger})^{k}|\Theta\rangle|\downarrow\rangle\\\
&+\frac{1}{\sqrt{2N!}}\sum_{k=0}^{N}C_{N}^{k}\hat{S}^{N-k}(p\hat{e}^{\dagger}-q\hat{c}^{\dagger})^{k}|\Theta\rangle|\uparrow\rangle\\\
\equiv&\hat{D}|\Theta\rangle|\downarrow\rangle+\hat{U}|\Theta\rangle|\uparrow\rangle,\end{split}$
(24)

where $\hat{S}=\alpha\hat{a}^{\dagger}+\beta\hat{b}^{\dagger}$ and

$\begin{split}|\Psi\rangle_{l}=&\hat{C}_{a}(\pi/2)e^{i\pi/2J_{y}}\widetilde{|\Psi\rangle}_{l}\\\
=&\frac{\hat{C}_{a}(\pi/2)e^{i\pi
J_{y}/2}}{\sqrt{N!}}(u\hat{a}^{\dagger}+v\hat{b}^{\dagger}+p\hat{e}+q\hat{c}\sigma_{z})^{N}|0_{a}0_{b},0_{E}\rangle|+\rangle.\end{split}$
(25)

Such Eq. (15) is obtained.

##  Appendix B Derivation of the Eq. (16) of the reduced density matrix of
spin under particle loss

The state $\widetilde{|\Psi\rangle}_{l}$ can be also represented as a density
matrix:

$\begin{split}\hat{\rho}=&\hat{D}|\Theta\rangle\langle\Theta|\hat{D}^{\dagger}\rho_{\downarrow\downarrow}+\hat{U}|\Theta\rangle\langle\Theta|\hat{U}^{\dagger}\rho_{\uparrow\uparrow}\\\
&+\hat{D}|\Theta\rangle\langle\Theta|\hat{U}^{\dagger}\rho_{\downarrow\uparrow}+\hat{U}|\Theta\rangle\langle\Theta|\hat{D}^{\dagger}\rho_{\uparrow\downarrow},\end{split}$
(26)

where
$\rho_{\downarrow\downarrow}=|\downarrow\rangle\langle\downarrow|$,$\rho_{\uparrow\uparrow}=|\uparrow\rangle\langle\uparrow|$
and
$\rho_{\downarrow\uparrow}=|\uparrow\rangle\langle\uparrow|=\rho^{*}_{\uparrow\downarrow}$
Tracing out the environment variables, we get

$\begin{split}\hat{\rho}_{1}=&\hat{\rho}_{00}\rho_{\downarrow\downarrow}+\hat{\rho}_{11}\rho_{\uparrow\uparrow}\\\
&+\hat{\rho}_{01}\rho_{\downarrow\uparrow}+\hat{\rho}_{10}\rho_{\uparrow\downarrow}\end{split}$
(27)

where $|0_{s}\rangle=|0_{a},0_{b}\rangle$ is the ground state of the two-
dimensional oscillator, and also

$\begin{split}\hat{\rho}_{00}&=Tr_{E}\Big{[}\hat{D}|\Theta\rangle\langle\Theta|\hat{D}^{\dagger}\Big{]}\\\
&=\sum_{n=0,k=0}^{N}\frac{\hat{S}^{N-k}\text{Tr}_{E}\Big{[}(p\hat{e}^{\dagger}+q\hat{c}^{\dagger})^{k}|\Theta\rangle\langle\Theta|(p^{*}\hat{e}+q^{*}\hat{c})^{n}\Big{]}\hat{S}^{\dagger(N-n)}}{2N!}\\\
&=\sum_{k=0}^{N}\frac{\hat{S}^{N-k}\text{Tr}_{E}\Big{[}(p\hat{e}^{\dagger}+q\hat{c}^{\dagger})^{k}|\Theta\rangle\langle\Theta|(p^{*}\hat{e}+q^{*}\hat{c})^{k}\Big{]}\hat{S}^{\dagger(N-k)}}{2N!}\\\
&=\sum_{k=0}^{N}\sum_{m=0}^{k}\frac{(C_{k}^{m})^{2}|p|^{2m}|q|^{2k-2m}m!(k-m)!\hat{S}^{N-k}|0_{s}\rangle\langle
0_{s}|\hat{S}^{\dagger(N-k)}}{2N!}\\\
&=\frac{1}{2N!}\sum_{k=0}^{N}k!(|p|^{2}+|q|^{2})^{k}\hat{S}^{N-k}|0_{s}\rangle\langle
0_{s}|\hat{S}^{\dagger(N-k)},\end{split}$ (28)

and

$\begin{split}\hat{\rho}_{01}&=Tr_{E}\Big{[}\hat{D}|\Theta\rangle\langle\Theta|\hat{U}^{\dagger}\Big{]}\\\
&=\sum_{k=0,n=0}^{N}\frac{\hat{S}^{N-k}\text{Tr}_{E}\Big{[}(p\hat{e}^{\dagger}+q\hat{c}^{\dagger})^{k}|\Theta\rangle\langle\Theta|(p^{*}\hat{e}-q^{*}\hat{c})^{n}\Big{]}\hat{S}^{\dagger(N-n)}}{2N!}\\\
&=\sum_{k=0}^{N}\frac{\hat{S}^{N-k}Tr_{E}\Big{[}(p\hat{e}^{\dagger}+q\hat{c}^{\dagger})^{k}|\Theta\rangle\langle\Theta|(p^{*}\hat{e}-q^{*}\hat{c})^{k}\Big{]}\hat{S}^{\dagger(N-k)}}{2N!}\\\
&=\sum_{k=0}^{N}\sum_{m=0}^{k}\frac{(C_{k}^{m})^{2}|p|^{2m}(-|q|^{2})^{k-m}m!(k-m)!\hat{S}^{N-k}|0_{s}\rangle\langle
0_{s}|\hat{S}^{\dagger(N-k)}}{2N!}\\\
&=\frac{1}{2N!}\sum_{k=0}^{N}k!(|p|^{2}-|q|^{2})^{k}\hat{S}^{N-k}|0_{s}\rangle\langle
0_{s}|\hat{S}^{\dagger(N-k)}.\end{split}$ (29)

After the phase shift and reverse CPG operations, the output state of the
hybrid system can be represented as the density matrix:

$\hat{\rho}_{L}=e^{i\theta J_{z}\sigma_{z}}\rho_{1}e^{-i\theta
J_{z}\sigma_{z}}.$ (30)

Furthermore, by tracing out the harmonic oscillator variables, the reduced
density matrix of the spin state is finally obtained as:

$\hat{\rho}_{s}=\rho_{00}|\downarrow\rangle\langle\downarrow|+\rho_{11}|\uparrow\rangle\langle\uparrow|+R|\downarrow\rangle\langle\uparrow|+R^{*}|\uparrow\rangle\langle\downarrow|,$
(31)

with

$\begin{split}\rho_{00}&=Tr(\hat{\rho}_{00})\\\
&=\sum_{k=0}^{N}\sum_{l=0}^{N-k}\frac{(C_{N-k}^{l})^{2}|\alpha_{1}|^{2l}|\beta_{1}|^{2N-2k-2l}l!(N\\!-\\!k\\!-\\!l)!k!(p^{2}+|q|^{2})^{k}}{2N!}\\\
&=\frac{1}{2N!}\sum_{k=0}^{N}(|\alpha_{1}|^{2}+|\beta_{1}|^{2})^{N-k}(p^{2}+|q|^{2})^{k}(N-k)!k!=\frac{1}{2},\end{split}$
(32)

and

$\begin{split}R&=Tr(\hat{\rho}_{01})\\\
&=\sum_{k=0}^{N}\sum_{j=0}^{N-k}\frac{(C_{N-k}^{j})^{2}[C_{N}^{k}]^{2}\alpha_{1}^{2j}\beta_{1}^{2N-2k-2j}j!(N\\!-\\!k\\!-\\!j)!k!(p^{2}-|q|^{2})^{k}}{2N!}\\\
&=\frac{1}{2N!}\sum_{k=0}[C_{N}^{k}]^{2}(N-k)!k!(p^{2}-|q|^{2})^{k}(\alpha_{1}^{2}+\beta_{1}^{2})^{N-k}\\\
&=\frac{1}{2}(\alpha_{1}^{2}+\beta_{1}^{2}+p^{2}-|q|^{2})^{N}.\end{split}$
(33)

Above $\alpha_{1}=\alpha e^{i\theta/2}$,$\beta_{1}=\beta e^{-i\theta/2}$, and
$\rho_{11}=1-\rho_{00}=1/2$, . Therefore, the reduced density matrix of the
spin state is :

$\hat{\rho}_{s}=\frac{1}{2}(|\downarrow\rangle\langle\downarrow|+|\uparrow\rangle\langle\uparrow|)+\rho_{01}|\downarrow\rangle\langle\uparrow|+\rho_{10}|\uparrow\rangle\langle\downarrow|.$
(34)

Thus Eq. (16) can be obtained.

## References

  * [1] A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
  * [2] T. Nagata, R. Okamoto, J. L. $O^{\prime}$Brien, K. Sasaki, and S. Takeuchi, Science 316, 726 (2007).
  * [3] Y. Israel, S. Rosen, and Y. Silberberg, Phys. Rev. Lett. 112, 103604 (2014).
  * [4] P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, Phys. Rev. Lett. 104, 103602 (2010).
  * [5] M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, Nat. Photonics 7, 229 (2013).
  * [6] Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, Phys. Rev. A 88, 043832 (2013).
  * [7] J. Joo, W. J. Munro, and T. P. Spiller, Phys. Rev. Lett. 107, 083601 (2011).
  * [8] J. Huang, X. Qin, H. Zhong, Y. Ke, and C. Lee, Scientific Reports 5, (2015).
  * [9] C. Oh, S. Y. Lee, H. Nha, and H. Jeong, Phys. Rev. A 96, 062304 (2017).
  * [10] J. P. Dowling, Contemporary Physics 49, 125 (2008).
  * [11] C. M. Caves, Phys. Rev. D 23, 1693 (1981).
  * [12] L. Pezz$\grave{e}$, and A. Smerzi, Phys. Rev. Lett. 100, 073601 (2008).
  * [13] J. Ma, X. Wang, C.P. Sun, and F. Nori, Phys. Rep. 509, 89-165 (2011).
  * [14] D. Li, C. H. Yuan , Z. Y. Ou, and W. Zhang, New J. Phys. 16, 073020 (2014).
  * [15] X. Zuo, Z. Yan, Y. Feng, J. Ma, X. Jia , C. Xie, and K. Peng, Phys. Rev. Lett. 124, 173602 (2020).
  * [16] S. Liu, Y. Lou, and J. Jing, Phys. Rev. Lett. 123, 113602 (2019).
  * [17] P. Ovartchaiyapong, K. W. Lee, B. A. Myers, and A. C. B. Jayich, Nat. Commun. 5, 4429 (2014).
  * [18] D. Lee, K. W. Lee, J. V. Cady, P. Ovartchaiyapong, and A. C. B. Jayich, J. Opt. 19, 033001 (2017).
  * [19] D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, Nature 430, 329 (2004).
  * [20] S. Kolkowitz, A. C. B. Jayich, Q. P. Unterreithmeier, S. D. Bennett, P. Rabl, J. G. E. Harris, and M. D. Lukin, Science 335, 1603 (2012).
  * [21] J. M. Pirkkalainen, S. U. Cho, J. Li, G. S. Paraoanu, P. J. Hakonen, and M. A. Sillanpää, Nature 494, 211 (2013).
  * [22] H. Y. Lo, D. Kienzler, L. D. Clercq, M. Marinelli, V. Negnevitsky, B. C. Keitch, J. P. Home, Nature 521, 336 (2015).
  * [23] K. A. Gilmore, M. Affolter, R. J. L. Swan, D. Barberena5, E. Jordan, A. M. Rey, J. J. Bollinger, Science 373, 673 (2021).
  * [24] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
  * [25] J. Mizrahi, C. Senko, B. Neyenhuis, K. G. Johnson, W. C. Campbell, C.W. S. Conover, and C. Monroe, Phys. Rev. Lett. 110, 203001 (2013).
  * [26] P. A. Ivanov, K. Singer, N. V. Vitanov, and D. Porras, Phys. Rev. Applied 4, 054007 (2015).
  * [27] W. C. Campbell, and P. Hamilton, J. Phys. B: At. Mol. Opt. Phys. 50, 064002 (2017).
  * [28] X. N. Feng, J. Huang, L. F. Wei, and C. Lee, Phys. Rev. A 106, 022413 (2022).
  * [29] S. L. Braunstein, and C. M.Caves, Phys. Rev. Lett. 72, 3439 (1994).
  * [30] K. Nemoto, and W. J. Munro, Phys. Rev. Lett. 93, 250502 (2004).
  * [31] M. Heuck, K. Jacobs, and D. R. Englund, Phys. Rev. Lett. 124, 160501 (2020).
  * [32] Y. Sun, P. Xu, P. X. Chen, and L. Liu, Phys. Rev. Applied 13, 024059 (2020).
  * [33] R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Phys.Rev. A 80, 013825 (2009).
  * [34] S. Y. Lee, Y. S. Ihn, and Z. Kim, Phys.Rev. A 101, 012332 (2020).
  * [35] M. Kacprowicz, R. Demkowicz-Dobrzaski, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Nat. Photonics. 4, 357 (2010).
  * [36] A. E. Ulanov, I. A. Fedorov, D. Sychev, P. Grangier, and A. I. Lvovsky, Nat. Commun. 7, 11925 (2016).
  * [37] D. Marcos, M. Wubs, J. M. Taylor, R. Aguado, M. D. Lukin, and A. S. Sørensen, Phys. Rev. Lett. 105, 210501 (2010).
  * [38] X. Zhu, et al., Nature 478, 221 (2011).
  * [39] D. A. Gangloff, G. Éthier-Majcher, C. Lang, E. V. Denning, J. H. Bodey, D. M. Jackson, E. Clarke, M. Hugues, C. Le Gall, and M. Atatüre, Science 364, 62 (2019).
  * [40] F. Anders, L. Pezz$\grave{e}$, A. Smerzi, and C. Klempt, Phys. Rev. A 97, 043813 (2018).
  * [41] K. C. McCormick, J. Keller, S. C. Burd, D. J. Wineland, A. C. Wilson, and D. Leibfried, Nature 572, 86-90 (2019).
  * [42] F. Wolf, C. Shi, J. C. Heip, M. Gessner, L. Pezz$\grave{e}$, A. Smerzi, M. Schulte, K. Hammerer, and P. O. Schmidt, Nat. Commu. 10, 2929 (2019).
  * [43] I. P. Menda$\check{s}$ and D. B. Popovi$\acute{c}$, Phys. Scripta 82(4), 45007 (2010).


# the collision avoidance and the controllability for $n$ bodies in dimension
one

Chong-Kyu Han  Research institute of Mathematics
Seoul National University
1 Gwanak-ro, Gwanak-gu
Seoul 08826, Republic of Korea<EMAIL_ADDRESS>and  Donghoon Park  Department
of Mathematics
Yonsei University
50 Yonsei-Ro, Seodaemun-gu
Seoul 03722, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

We present a method of design of control systems for $n$ bodies in the real
line $\mathbb{R}^{1}$ and on the unit circle $S^{1}$, to be collision-free and
controllable. The problem reduces to designing a control-affine system in
$\mathbb{R}^{n}$ and in $n$-torus $T^{n},$ respectively, that avoids certain
obstacles. We prove the controllability of the system by showing that the
vector fields that define the control-affine system, together with their
brackets of first order, span the whole tangent space of the state space, and
then by applying the Rashevsky-Chow theorem.

###### Key words and phrases:

control-affine system, non-holonomic constraints, obstacle avoidance,
collision-free, orbits, invariant submanifolds, generalized first integrals,
controllability

###### 2020 Mathematics Subject Classification:

37C10, 57R27, 58A17, 93B05, 93C15

The first author was supported by NRF-Republic of Korea with grants
0450-20210049.

##  1\. Introduction and the statement of the main results

Let $M^{1}$ be a smooth $(C^{\infty})$ connected manifold of dimension $1$
without boundary, namely, either the real line $\mathbb{R}^{1}$ or the unit
circle $S^{1}.$ We are concerned in this paper with the design of control
systems for $n$ particles on $M^{1}$ to be collision-free and controllable,
which we shall call the control $n$ body problem on $M^{1}$.  Let

$M^{n}:=\underbrace{M^{1}\times\cdots\times M^{1}}_{n\text{ times}}$

be the product. We shall see in §1.1 that the problem turns out to be
constructing a control-affine system on $M^{n}$ with avoidance of a certain
obstacle so that for any two points of $M^{n}$ off the obstacle one is
reachable from the other by an admissible control. Thus we construct on
$M^{n}$ a driftless control-affine system

(1.1) $\dot{\bf x}=\sum_{\ell=1}^{m}\vec{f_{\ell}}({\bf x})u_{\ell},\quad{\bf
x}=(x_{1},\cdots,x_{n})\in M^{n},~{}~{}m\leq n,$

that satisfies the following conditions:

i) The motions avoid the obstacle.

ii) The Lie algebra generated by $\vec{f_{\ell}}(x),$ $\ell=1,\cdots,m,$ span
the whole tangent space $T_{x}M$ at every point $x\in M^{n}$ off the obstacle.
In terms of $n$-dimensional kinematics, this requirement is that the
constraints on the velocity are completely non-holonomic (see [1]).

Then the controllability follows from the Rashevsky-Chow theorem (Theorem
2.3). We shall present our results separately; in §1.1 for the case
$M^{1}=\mathbb{R}^{1}$ and in §1.2 for the case $M^{1}=S^{1}$, respectedly.
Since [4] was published there has been extensive study on control-affine
systems. We mention some of them; [14] and [16] for non-holonomic constraints,
[6], [7], [15], [17], and [18] for obstacle avoidance, and [5] and [19] for
other motion planning problems.

###  1.1. Control $n$ body problem on $\mathbb{R}^{1}$

We consider $n,n\geq 2,$ particles in the real line $\mathbb{R}^{1}.$ Let

(1.2) ${\bf x}(t):=(x_{1}(t),\cdots,x_{n}(t)),\quad
x_{j}<x_{j+1},~{}~{}j=1,\cdots,n-1,$

be their position at time $t.$ Once for all in this paper we fix a small
positive number $\epsilon>0.$ We shall say that (1.2) is separated by
$\epsilon>0$ if the distance between two adjacent particles is greater than
$\epsilon,$ namely,

(1.3) $x_{j}+\epsilon<x_{j+1},\quad\text{for each }j=1,\cdots,n-1.$

We want to design a control-affine system

(1.4) $\left[\begin{matrix}\dot{x}_{1}\\\ \vdots\\\
\dot{x}_{n}\end{matrix}\right]=\sum_{\ell=1}^{m}\vec{f}_{\ell}(\underbrace{x_{1},\cdots,x_{n}}_{\bf
x})u_{\ell}(t),\quad m\leq n,~{}~{}$

where $\vec{f}_{\ell}$ are smooth vector fields in $\mathbb{R}^{n},$ expressed
as $n$-dimensional column vectors, and $u_{\ell}(t)$ are piecewise smooth,
with the following requirements:

1) Collision free, that is, (1.2) is separated by $\epsilon>0$.

2) Controllability, that is, for any sequences of real numbers ${\bf
p}:=(p_{1},\cdots,p_{n})$ and ${\bf q}:=(q_{1},\cdots,q_{n})$ that are
separated by $\epsilon>0,$ there exist piecewise smooth real-valued functions
$u_{\ell}(t)$, $\ell=1,\cdots,m,$ in some interval $0\leq t\leq T,$ so that
(1.4) has a solution ${\bf x}(t)$ with $x_{j}(0)=p_{j}$ and $x_{j}(T)=q_{j}$
for each $j=1,\cdots,n.$ The terminal point ${\bf
x}(T)=(x_{1}(T),\cdots,x_{n}(T))$ of the initial value problem (1.4) with the
initial condition ${\bf x}(0)={\bf p}$ shall be denoted by $\Gamma({\bf
p},T,{\bf u}),$ where ${\bf u}(t)=(u_{1}(t),\cdots,u_{m}(t)).$

Let $\mathbb{R}^{n}_{\epsilon}$ be the set of points ${\bf
x}=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}$ that satisfy (1.3). For each
$j=1,\cdots,n-1,$ let

(1.5) $\rho_{j}({\bf x}):=x_{j+1}-x_{j}-\epsilon.$

Then

(1.6) $\mathbb{R}^{n}_{\epsilon}=\bigcap_{j=1}^{n-1}\\{\rho_{j}>0\\}.$

The requirement 1); system’s being collision-free, is equivalent to that
$\mathbb{R}^{n}_{\epsilon}$ is invariant under the flow of the vector fields
$\vec{f}_{\ell}$. Our basic observations are the following: A smooth vector
field $\vec{f}=\sum_{i=1}^{n}a_{j}({\bf x})\partial/\partial x_{i}$ on
$\mathbb{R}^{n}$ is tangent to a hypersurfaces given by $\\{\rho=0\\},$ with
$d\rho\neq 0,$ if and only if

(1.7) $(\vec{f}\rho)({\bf x)}=0,~{}~{}\quad\text{for all {\bf x} with
}\rho({\bf x})=0.$

A function $\rho$ that satisfies (1.7) is called a generalized first integral
of the vector field $\vec{f}$ (see §2.2). For a smooth function $\rho$ on
$\mathbb{R}^{n}$ let us denote by $((\rho))$ the ideal generated by $\rho$,
namely, the collection of all the smooth functions on $\mathbb{R}^{n}$ that
are divisible by $\rho$. Then (1.7) is saying that $\vec{f}\rho$ belongs to
the ideal $((\rho));$ see Lemma 2.4 for the details. So, in order to design a
control-affine system (1.4) we construct vector fields $\vec{f}_{\ell},$
$\ell=1,\cdots,m,$ on $\mathbb{R}^{n}$ so that

i) each $\vec{f}_{\ell}$ is tangent to the boundary of
$\mathbb{R}^{n}_{\epsilon}$ (collision-free),

ii) the Lie algebra generated by $\vec{f}_{1},\cdots,\vec{f}_{m}$ spans the
whole tangent space at every point of $\mathbb{R}^{n}_{\epsilon}$
(controllability).

Now we state our main results:

######  Theorem 1.1.

Given $\epsilon>0$ and a positive integer $n,n\geq 2,$ let $\rho_{j},$
$j=1,\cdots,n-1,$ be smooth real valued fuctions on $\mathbb{R}^{n}$ as in
(1.5) and $\mathbb{R}^{n}_{\epsilon}$ is the connected open subset of
$\mathbb{R}^{n}$ given by (1.6). Then there exist smooth vector fields
$\vec{f}_{1},\cdots,\vec{f}_{m},$ $m\leq n,$ on $\mathbb{R}^{n}$ having the
following properties:

i) The hypersurfaces $\rho_{j}({\bf x})=0,$ $j=1,\cdots,n-1,$ and the
submanifold $\mathbb{R}^{n}_{\epsilon}$ are invariant under the flow of the
vector fields.

ii) The control-affine system $\dot{\bf
x}=\sum_{\ell=1}^{m}\vec{f}_{\ell}({\bf x})u_{\ell}(t)$ is controllable in
$\mathbb{R}^{n}_{\epsilon},$ namely, for any points ${\bf
p}=(p_{1},\cdots,p_{n})$ and ${\bf q}=(q_{1},\cdots,q_{n})$ of
$\mathbb{R}^{n}_{\epsilon}$ there exists a control ${\bf
u}(t)=(u^{1}(t),\cdots,u^{m}(t))$ for $t$ with $0\leq t\leq T$, for some
$T>0$, such that $\Gamma({\bf p},T,{\bf u})={\bf q}.$

Such $\vec{f}_{\ell},\ell=1,\cdots,m,$ are constructible for $m$ with

(1.8) $2m\geq n+1.$

######  Example 1.2.

The case $n=2:$ Let $\rho(x_{1},x_{2})=x_{2}-x_{1}-\epsilon$ and
$\mathbb{R}^{2}_{\epsilon}=\\{\rho>0\\}$ be the open half plane. Then $m=2$ is
the smallest integer that satisfies (1.8) and the vector fields

$\displaystyle\vec{f}_{1}$ $\displaystyle:=\frac{\partial}{\partial
x_{1}}+\frac{\partial}{\partial x_{2}}$ $\displaystyle\vec{f}_{2}$
$\displaystyle:=(1+\rho)\frac{\partial}{\partial
x_{1}}+\frac{\partial}{\partial x_{2}},$

as shown in Figure 1, are linearly independent in $\mathbb{R}^{n}_{\epsilon}$.

Moreover, $\vec{f}_{j}\rho=0$ on the set $\\{\rho=0\\}$, for each $j=1,2,$
therefore, $\vec{f}_{j}$ are tangent to the zero set of $\rho$, which is the
boundary of $\mathbb{R}^{2}_{\epsilon}.$ It follows that
$\mathbb{R}^{2}_{\epsilon}$ is invariant under the flow of $\vec{f}_{j}$ and
the controllability of (1.4) follows from the Rashevsky-Chow theorem (Theorem
2.3).

Figure 1 $\;\;\overrightarrow{f_{2}}$ on the region
$\mathbb{R}^{2}_{\epsilon}$

######  Corollary 1.3.

Given $\epsilon>0$ and an integer $n$, $n\geq 2$, the control $n$ body problem
in the real line $\mathbb{R}$ with separation by $\epsilon$ is solvable by
constructing $n$-dimensional vector fields $\vec{f}_{1},\cdots,\vec{f}_{m},$
$2m\geq n+1.$ Furthermore, for each $\ell=1,\cdots,m,$ $\vec{f}_{\ell}$
depends only on $x_{\ell-1},x_{\ell},x_{m+\ell-2}$ and $x_{m+\ell-1}.$

###  1.2. Control $n$ body problem on $S^{1}$

We consider now $n,n\geq 2,$ particles moving on $S^{1}.$ Let

(1.9) $e^{ix_{1}(t)},\cdots,e^{ix_{n}(t)}$

be their position at time $t,$ where $x_{j},$ $j=1,\cdots,n,$ is the angle
measured in radian from a reference point of $S^{1}.$ Assuming
$\epsilon<2\pi/n,$ we shall say that these points are separated by $\epsilon$
if

(1.10)
$x_{1}+\epsilon<x_{2}<x_{2}+\epsilon<x_{3}<\cdots<x_{n}+\epsilon<x_{1}+2\pi.$

Let ${\bf x}:=(x_{1},\cdots,x_{n})$ and let

(1.11) $\displaystyle\rho_{j}({\bf x})$
$\displaystyle:=x_{j+1}-x_{j}-\epsilon,\quad\text{for }j=1,\cdots,n-1$
$\displaystyle\rho_{n}({\bf x})$ $\displaystyle:=x_{1}+2\pi-x_{n}-\epsilon.$

Let

(1.12) $\widetilde{\mathbb{R}}^{n}_{\epsilon}:=\\{{\bf
x}\in\mathbb{R}^{n}~{}:~{}\rho_{j}({\bf x})>0,~{}~{}j=1,\cdots,n\\},$

that is, the set of points where (1.10) holds. Now we denote by
$2\pi\mathbb{Z}$ the set of all integer multiples of $2\pi$ and consider the
$n$-torus

$T^{n}:=\mathbb{R}^{n}/(2\pi\mathbb{Z})^{n}.$

Since the projection

$p:\mathbb{R}^{n}\rightarrow T^{n}$

is a local diffeomorphism and the subset
$\widetilde{\mathbb{R}}^{n}_{\epsilon}\subset\mathbb{R}^{n}$ is open and
connected, we have

######  Proposition 1.4.

$T_{\epsilon}^{n}:=p({\widetilde{R}}^{n}_{\epsilon})$ is an open connected
subset of $T^{n}$.

In the light of Rashevsky-Chow’s theorem (Theorem 2.3) we want to construct
smooth vector fields $\vec{f}_{\ell},$ $\ell=1,\cdots,m,$ $m\leq n,$ on
$T^{n},$ that satisfy the following conditions:

i) The submanifold $T^{n}_{\epsilon}\subset T^{n}$ is invariant under the
motions subject to the control-affine system (1.4).

ii) $\vec{f}_{\ell},$ $\ell=1,\cdots,m,$ together with their Lie-brackets of
first order span the whole tangent space of $T^{n}_{\epsilon}.$

A vector field on $\mathbb{R}^{n}$

(1.13) $\vec{f}=\sum_{j=1}^{n}a_{j}({\bf x})\frac{\partial}{\partial x_{j}}$

shall be said to be periodic if each $a_{j}$ is a periodic function in each
variable $x_{1},\cdots,x_{n}$. A periodic vector field on $\mathbb{R}^{n}$
with period $2\pi$ in each variable may be regarded as a vector field on
$T^{n}.$ Thus we state our result as follows:

######  Theorem 1.5.

Given $\epsilon>0$ and a positive integer $n,n\geq 2,$ with $\epsilon<2\pi/n,$
there exist smooth vector fields $\vec{f}_{1},\cdots,\vec{f}_{m},$ $m\leq n,$
on $\mathbb{R}^{n}=\\{{\bf x}:=(x_{1},\cdots,x_{n})\\}$, which are periodic
with period $2\pi$, having the following properties:

i) The hypersurfaces $\rho_{j}({\bf x})=0,~{}~{}j=1,\cdots,n-1,$ the
hypersurface $\rho_{n}({\bf x})=0,$ and the open submanifold
$\widetilde{\mathbb{R}}^{n}_{\epsilon}$ are invariant under the flow of these
vector fields, where $\rho_{j}$ and $\rho_{n}$ are as defined by (1.11).

ii) $\vec{f}_{\ell},[\vec{f}_{\ell},\vec{f}_{k}],$ $\ell,k=1,\cdots,m,$ span
the whole tangent space of $\widetilde{\mathbb{R}}^{n}_{\epsilon}.$

Such $\vec{f_{\ell}},$ $\ell=1,\cdots,m,$ are constructible in the following
cases:

Case 1; both $n$ and $m$ are odd numbers and $2m\geq n+1.$

Case 2; $n$ is even and $2m\geq n+2.$

##  2\. preliminaries

###  2.1. Orbits and controllability

Let $M$ be a connected smooth manifold of dimension $n$ and $\mathcal{U}$ be a
set of admissible controls $u:[0,\infty)\rightarrow\mathbb{R}^{m}$, $m\leq n$.
Let

(2.1) $\dot{\bf x}:=\dfrac{d\bf x}{dt}=\sum_{j=1}^{m}\vec{f}_{\ell}({\bf
x})u_{\ell}$

be a control-affine system on $M$, where $\vec{f}_{\ell},\ell=1,\cdots,m,$ are
smooth vector fields on $M$, and ${\bf u}=(u_{1},\ldots,u_{m})\in\mathcal{U}$.
Here the control ${\bf u}(t)$ can be chosen variously, for instance, to be
piecewise continuous, measurable, smooth, and so forth. In this paper we shall
assume that $\mathcal{U}$ is the set of all piecewise smooth functions with
finitely many discontinuities. For a point ${\bf p}\in M$ let ${\bf x}(t)$ be
the solution of (2.1) subjected to the initial condition ${\bf x}(0)={\bf p}.$
The solution ${\bf x}(t)$ is continuous, piecewise smooth and uniquely
determined by the choice of ${\bf p}$ and ${\bf u}(t)$, which we shall denote
by $\Gamma({\bf p},t,{\bf u})$.

######  Definition 2.1.

A submanifold $N$ of $M$ is said to be _invariant under controls_ of (2.1) if
${\bf x}_{0}\in N$ implies that $\Gamma({\bf x}_{0},t,{\bf u})\in N$ for all
possible choices of ${\bf u}\in\mathcal{U}$ and for all $t\geq 0.$

Now we recall some basics of non-linear control systems. For other definitions
and theorems we refer the readers to our basic references [1, 4]. Consider the
set of points that are reachable from a point in some non-negative time. A
point ${\bf q}\in M$ is said to be _reachable from_ ${\bf p}$ if
$q=\Gamma({\bf p},t,{\bf u}),$ for some ${\bf u}\in\mathcal{U}$ and for some
$t\geq 0.$ The _reachable set $\mathcal{R}_{{\bf p}}$ of the control system
(2.1) from a point ${\bf p}\in M$_ is a subset of $M$ defined by

$\mathcal{R}_{{\bf p}}=\left\\{\Gamma({\bf p},t,{\bf u}):t\geq 0,~{}~{}{\bf
u}\in\mathcal{U}\right\\}.$

The control system (2.1) is said to be _controllable from ${\bf p}\in M$_ if

(2.2) $\mathcal{R}_{{\bf p}}=M$

and it is called _controllable_ if (2.2) holds for every ${\bf p}\in M$. We
define the _orbit $\mathcal{O}_{{\bf p}}$ of the control system (2.1) through
a point ${\bf p}\in M$_ to be the set of points ${\bf q}\in M$ such that
either ${\bf q}$ is reachable from ${\bf p}$ or ${\bf p}$ is reachable from
${\bf q}$. A basic theorem is the following

######  Theorem 2.2 (Nagano-Sussmann theorem, [1]).

$\mathcal{O}_{\bf p}$ is a connected immersed submanifold of $M$.

See [1] for the proof of Theorem 2.2. Now let $\mathcal{G}$ be the Lie algebra
of vector fields generated by $\vec{f}_{1},\ldots,\vec{f}_{m}.$ Consider the
vector space $\mathcal{G}({\bf p})\subset T_{\bf p}M,$ which is the linear
span at ${\bf p}\in M$ of the left iterated Lie brackets

$[\vec{f}_{i_{1}},[\vec{f}_{i_{2}},\cdots,[\vec{f}_{i_{k-1}},\vec{f}_{i_{k}}]\cdots]]$

of the vector fields $\vec{f}_{1},\ldots,\vec{f}_{m}.$ Then we have

######  Theorem 2.3 (Rashevsky-Chow theorem, [1]).

Let $M$ and $\mathcal{G}$ be as above. If $M$ is connected and
${\mathcal{G}}({\bf p})=T_{p}M$ at every point ${\bf p}\in M$, then
$\mathcal{O}_{{\bf p}}=M,$ that is (2.1) is controllable.

###  2.2. Generalized first integrals

Let $\mathcal{O}$ be a germ of smooth ($C^{\infty})$ manifold of dimension
$n$, or a small open ball of $\mathbb{R}^{n}$ centered at the origin. A set of
real-valued smooth functions $r_{1},\cdots,r_{d}$ of $\mathcal{O}$ is said to
be non-degenerate  if

$dr_{1}\wedge\cdots\wedge dr_{d}\neq 0.$

The common zeros of a non-degenerate set of $d$ real-valued functions form a
submanifold of codimension $d$.

######  Lemma 2.4.

A smooth real-valued function $\rho$ vanishes on the common zero set of a non-
degenerate system $r_{1},\cdots,r_{d}$ of smooth real-valued functions if and
only if

(2.3) $\rho=\sum_{k=1}^{d}\alpha_{k}r_{k},$

for some smooth functions $\alpha_{k}$s.

###### Proof.

Sufficiency of the condition is obvious. To show the necessity of the
condition (2.3) notice that the non-degeneracy implies that
$r_{1},\cdots,r_{d}$ can be part of coordinate system of $\mathcal{O}.$ Thus
we may assume that $r_{j}=x_{j}$ for $j=1,\cdots,d.$ We have

$\displaystyle~{}~{}\rho(x_{1},\cdots,x_{d},x_{d+1},\cdots,x_{n})-\rho(0,\cdots,0,x_{d+1},\cdots,x_{n})$
$\displaystyle=[\rho(sx_{1},\cdots,sx_{d},x_{d+1},\cdots,x_{n})]^{1}_{0}$
$\displaystyle=\int^{1}_{0}\frac{d}{ds}\rho(sx_{1},\cdots,sx_{d},x_{d+1},\cdots,x_{n})ds$
$\displaystyle=\sum^{d}_{k=1}x_{k}\int^{1}_{0}\frac{\partial\rho}{\partial
x_{k}}(sx_{1},\cdots,sx_{d},x_{d+1},\cdots,x_{n})ds.$

By setting the integral of the last line $\alpha_{k}$ and recalling
$x_{k}=r_{k},$ for $k=1,\cdots,d$ we obtain (2.3). ∎

Lemma 2.4 is a local version in smooth category of the Hilbert’s
Nullstellensatz  and the condition (2.3) is that $\rho$ is in the ideal
$((r^{1},\cdots,r^{d})).$

######  Definition 2.5.

Let $\vec{f}_{1},\cdots,\vec{f}_{m}$ be smooth vector fields on $\mathcal{O}$
that are linearly independent at every point of $\mathcal{O}.$ A real-valued
smooth function $\rho$ is a first integral  if $\rho$ is invariant under the
flow of those vector fields, namely

(2.4) $\vec{f}_{\ell}\rho=0,~{}~{}~{}\text{for all }\ell=1,\cdots,m.$

For $m$ vector fields on $n$-manifold there can be at most $(n-m)$ non-
degenerate first integrals. This is the case that the Frobenius integrability
condition holds (see [3]).

######  Definition 2.6.

A real-valued smooth function $\rho$ is a generalized first integral of
$\vec{f}_{1},\cdots,\vec{f}_{m}$ if

(2.5) ${\vec{f}}_{\ell}\rho=0,~{}~{}\text{ on }\\{\rho=0\\},~{}~{}\text{ for
all }\ell=1,\cdots,m.$

The notion of the generalized first integral was first defined in [2]. We have

######  Proposition 2.7.

Suppose that $\rho$ is a smooth real-valued function with $d\rho\neq 0$ on
$\mathcal{O}$ and $X\subset\mathcal{O}$ is the zero set of $\rho.$ Then the
following are equivalent:

i) $\rho$ is a generalized first integral of the vector fields
$\vec{f}_{1},\cdots,\vec{f}_{m}.$

ii) For each $\ell=1,\cdots,m,$ $\vec{f}_{\ell}\rho$ is divisible by $\rho,$
that is, there is a smooth function $\alpha({\bf x})$ so that

(2.6) $\vec{f}_{\ell}\rho=\alpha\rho.$

iii) For each $\ell=1,\cdots,m,$ $\vec{f}_{\ell}$ is tangent to $X.$

iv) $X$ is invariant under the flow of $\vec{f}_{\ell},\ell=1,\cdots,m.$

###### Proof.

$i)\Longrightarrow ii)$ follows from Definition 2.5 and Lemma 2.4. The other
implications are easy to prove or obvious. ∎

Using the observations of Proposition 2.7 the partial integrability of systems
of vector fields, or equivalently, the partial holonomy  of (2.1) has been
studied in [8], [9], and in [11],[12],[13],[10] as applications in finding
invariant submanifolds of vector fields.

##  3\. Proof of the theorems

###  3.1.  Proof of Theorem 1.1

We shall construct $\vec{f}_{\ell}$, $\ell=1,\cdots,m,$ for each of the
following three cases:

1) Case $m=n$. Let $\vec{f}_{\ell}$ be the $\ell$-th column of the following
$m\times m$ matrix:

(3.1)
$\left[\begin{matrix}&1&1+\rho_{1}&1+\rho_{2}&1+\rho_{3}&\cdots&&1+\rho_{m-1}\\\
&1&1&1+\rho_{2}&1+\rho_{3}&\cdots&&1+\rho_{m-1}\\\
&1&1&1&1+\rho_{3}&\cdots&&1+\rho_{m-1}\\\ &1&1&1&1&\cdots&&1+\rho_{m-1}\\\
&\vdots&\vdots&\vdots&\vdots&\cdots&&\vdots\\\
&1&1&1&1&\cdots&&1+\rho_{m-1}\\\ &1&1&1&1&\cdots&&1+\rho_{m-1}\\\
&1&1&1&1&\cdots&&1\end{matrix}\right],$

namely,

(3.2) $\displaystyle\vec{f}_{1}$
$\displaystyle=\sum_{j=1}^{n}\partial/\partial x_{j},$
$\displaystyle\vec{f}_{\ell}$
$\displaystyle=\sum_{j=1}^{\ell-1}(1+\rho_{\ell-1})\partial/\partial
x_{j}+\sum_{j=\ell}^{n}\partial/\partial x_{j},\quad\ell=2,\cdots,m.$

Then (1.5) implies that

$\displaystyle\vec{f}_{1}\rho_{j}$ $\displaystyle=0,\quad j=1,\cdots,n-1,$
$\displaystyle\vec{f}_{\ell}\rho_{\ell-1}$
$\displaystyle=-\rho_{\ell-1},\quad\ell=2,\cdots,m,$
$\displaystyle\vec{f}_{\ell}\rho_{j}$ $\displaystyle=0,~{}~{}\text{ for the
other pairs }(\ell,j).$

Therefore, by Proposition 2.7, $\vec{f}_{\ell}$ are tangent to
$\\{\rho_{j}=0\\}$, for each pair $(\ell,j).$ This implies that the boundary
of $\mathbb{R}^{n}_{\epsilon}$ is invariant, and therefore the interior of
$\mathbb{R}^{n}_{\epsilon}$ is invariant, under the motions that are subject
to the system (1.4). Furthermore, the square matrix (3.1) is invertible on
$\mathbb{R}^{n}_{\epsilon}=\bigcap_{j=1}^{n-1}\\{\rho_{j}>0\\}.$ Hence, by
Theorem 2.3 the control-affine system (1.4) is controllable.

2) Case $m<n$ and $2m=n+1$: For $\ell=1,\cdots,m,$ let $\vec{f}_{\ell}$ be the
$\ell$-th column of the following matrix:

(3.3)
$\left[\begin{matrix}&1&1+\rho_{1}&1+\rho_{2}&1+\rho_{3}&\cdots&&1+\rho_{m-1}\\\
&1&1&1+\rho_{2}&1+\rho_{3}&\cdots&&1+\rho_{m-1}\\\
&1&1&1&1+\rho_{3}&\cdots&&1+\rho_{m-1}\\\ &1&1&1&1&\cdots&&1+\rho_{m-1}\\\
&\vdots&\vdots&\vdots&\vdots&\cdots&&\vdots\\\
&1&1&1&1&\cdots&&1+\rho_{m-1}\\\ &1&1&1&1&\cdots&&1+\rho_{m-1}\\\
&1&1&1&1&\cdots&&1\\\ &&&&&&&&\\\ &&&&&&&&\\\
&1&1+x_{m}\rho_{m}&1&1&\cdots&&1\\\
&1&1+x_{m}\rho_{m}&1+x_{m+1}\rho_{m+1}&1&\cdots&&1\\\
&1&1+x_{m}\rho_{m}&1+x_{m+1}\rho_{m+1}&1+x_{m+2}\rho_{m+2}&\cdots&&1\\\
&\vdots&\vdots&\vdots&\vdots&&&\vdots\\\
&1&1+x_{m}\rho_{m}&1+x_{m+1}\rho_{m+1}&1+x_{m+2}\rho_{m+2}&\cdots&&1+x_{n-1}\rho_{n-1}\\\
\end{matrix}\right]$

thus $\vec{f}_{\ell},$ $\ell=1,\cdots,m,$ is an $n$-dimensional vector given
by

(3.4) $\displaystyle\vec{f}_{1}$
$\displaystyle=\sum_{j=1}^{n}\partial/\partial x_{j},$
$\displaystyle\vec{f}_{\ell}$
$\displaystyle=\sum_{j=1}^{\ell-1}(1+\rho_{\ell-1})\frac{\partial}{\partial
x_{j}}+\sum_{j=\ell}^{m+\ell-2}\frac{\partial}{\partial
x_{j}}+\sum_{j=m+\ell-1}^{n}(1+x_{m+\ell-2}\rho_{m+\ell-2})\frac{\partial}{\partial
x_{j}},\text{ for }\ell=2,\cdots,m.$

Then by easy computations we have

(3.5) $\displaystyle\vec{f}_{1}\rho_{j}$ $\displaystyle=0,\text{ for all
}j=1,\cdots,n-1,$ $\displaystyle\vec{f}_{\ell}\rho_{\ell-1}$
$\displaystyle=-\rho_{\ell-1},~{}~{}\text{ for all }\ell=2,\cdots,m,$
$\displaystyle\vec{f}_{\ell}\rho_{m+\ell-2}$
$\displaystyle=x_{m+\ell-2}\rho_{m+\ell-2},~{}~{}\text{ for all
}\ell=2,\cdots,m,$ $\displaystyle\vec{f}_{\ell}\rho_{j}$
$\displaystyle=0,~{}~{}\text{ for all other pairs }(\ell,j).$

Then by Proposition 2.7 and (3.5) the vector fields
$\vec{f}_{1},\cdots,\vec{f}_{m}$ are tangent to the boundary of
$\mathbb{R}^{n}_{\epsilon},$ which implies that the interior of
$\mathbb{R}^{n}_{\epsilon}$ is invariant, under the motions that are subject
to the control-affine system (1.4). On the other hand

(3.6)
$[\vec{f}_{1},\vec{f}_{\ell}]=\rho_{m+\ell-2}\sum_{j=m+\ell-1}^{n}\frac{\partial}{\partial
x_{j}},\text{ for all }\ell=2,\cdots,m,$

so that $[\vec{f}_{1},\vec{f}_{\ell}]$ is the $(\ell-1)$th column of the
following matrix:

(3.7) $\left[\begin{matrix}&0&0&0&0&\cdots&0\\\ &0&0&0&0&\cdots&0\\\
&\vdots&\vdots&\vdots&\vdots&\cdots&\vdots\\\ &0&0&0&0&\cdots&0\\\ &&&&&&&&\\\
&&&&&&&&\\\ &\rho_{m}&0&0&0&\cdots&0\\\ &\rho_{m}&\rho_{m+1}&0&0&\cdots&0\\\
&\rho_{m}&\rho_{m+1}&\rho_{m+2}&0&\cdots&0\\\
&\vdots&\vdots&\vdots&\vdots&&\vdots\\\
&\rho_{m}&\rho_{m+1}&\rho_{m+2}&\rho_{m+3}&\cdots&\rho_{n-1}\end{matrix}\right].$

Notice that the bottom part of (3.7) is a $(m-1)\times(m-1)$ square matrix,
which is invertible on $\mathbb{R}^{n}_{\epsilon}$. Thus $m$ columns of the
matrix (3.1) and $(m-1)$ columns of (3.7), so that their totality $m+(m-1)=n$
vectors are independent on $\mathbb{R}^{n}_{\epsilon}$. Now conclusion follows
from Theorem 2.3.

3) Case $2m>n+1$:  Proof is same as the previous case, except for that we need
only the columns of (3.7) up to $(n+1-m)$-th in order to span the whole
tangent space of $\mathbb{R}^{n}_{\epsilon}$.

###  3.2. Proof of Theorem 1.5

Let $\rho_{j}$ be the function on $\mathbb{R}^{n}$ as in (1.11). We define a
function $\sigma(s)$ in a real variable $s$ as in Figure 2 by

$\sigma(s):=\sin(s-\pi/2)+1.$

Figure 2 The function $\sigma(s)$

Now we define

(3.8) $\sigma_{j}({\bf x}):=\sigma(\rho_{j}({\bf x})),~{}~{}j=1,\cdots,n.$

Immediately we have the following

######  Proposition 3.1.

For each $j=1,\cdots,n,$ $\sigma_{j}({\bf x})$ is a smooth function of
$\mathbb{R}^{n}$ with the following properties:

i) $\sigma_{j}$ is periodic in each variable with period $2\pi$.

ii) For $j=1,\cdots,n-1,$ $\sigma_{j}$ depends only on $x_{j+1}$ and $x_{j}$,
and $\sigma_{n}$ depends only on $x_{1}$ and $x_{n}$.

iii) $\sigma_{j}({\bf x})>0,$ for all ${\bf
x}\in\widetilde{\mathbb{R}}^{n}_{\epsilon}.$

iv) $\sigma_{j}({\bf x})=0,$ for all ${\bf x}$ with $\rho_{j}({\bf x})=0,$
that is, $\sigma_{j}\in((\rho_{j})).$

Now we prove Theorem 1.5 for the following three cases:

1) case $m=n;$ the largest choice of $m$. Let $\vec{f}_{j}$ be the $j$-th
column of the following matrix:

(3.9)
$\left[\begin{matrix}&1&1+\sigma_{1}\sigma_{n}&1+\sigma_{2}\sigma_{n}&\cdots&1+\sigma_{n-2}\sigma_{n}&1+\sigma_{n-1}\sigma_{n}\\\
&1&1&1+\sigma_{2}\sigma_{n}&\cdots&1+\sigma_{n-2}\sigma_{n}&1+\sigma_{n-1}\sigma_{n}\\\
&1&1&1&\cdots&1+\sigma_{n-2}\sigma_{n}&1+\sigma_{n-1}\sigma_{n}\\\
&\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\\
&1&1&1&\cdots&1+\sigma_{n-2}\sigma_{n}&1+\sigma_{n-1}\sigma_{n}\\\
&1&1&1&\cdots&1&1+\sigma_{n-1}\sigma_{n}\\\ &1&1&1&\cdots&1&1\\\
\end{matrix}\right].$

By easy computation we see that

(3.10) $\displaystyle\vec{f}_{1}\rho_{j}$ $\displaystyle=0,~{}~{}\text{ for
all }j=1,\cdots,n$ $\displaystyle\vec{f}_{\ell}\rho_{\ell-1}$
$\displaystyle=-\sigma_{\ell-1}\sigma_{n},~{}~{}\text{ for }\ell=2,\cdots,m$
$\displaystyle\vec{f}_{\ell}\rho_{n}$
$\displaystyle=\sigma_{\ell-1}\sigma_{n},\quad\text{ for }\ell=2,\cdots,m$
$\displaystyle\vec{f}_{\ell}\rho_{j}$ $\displaystyle=0,~{}~{}\text{ for all
other pairs }(\ell,j).$

From the second equation of (3.10) and Proposition 3.1-(iv) we have

(3.11) $\begin{aligned} \vec{f}_{\ell}\rho_{\ell-1}&\in((\rho_{\ell-1}))\\\
\vec{f}_{\ell}\rho_{n}&\in((\rho_{n})).\end{aligned}.$

It follows from (3.10)-(3.11) that the vector fields
$\vec{f}_{1},\cdots,\vec{f}_{m}$ are tangent to the zero sets
$\\{\rho_{j}=0\\}$, which implies that the boundary of
$\widetilde{\mathbb{R}}^{n}_{\epsilon}$ is invariant, and also, the interior
of $\widetilde{R}^{n}_{\epsilon}$ is invariant under the motions subject to
the control-affine system (1.4). Furthermore, by Proposition 3.1-(iii) the
matrix (3.9) is invertible on $\widetilde{\mathbb{R}}^{n}_{\epsilon}$, which
implies that $\vec{f}_{\ell},\ell=1,\cdots,m,$ span the whole tangent space of
$\widetilde{\mathbb{R}}^{n}_{\epsilon}$.

2) Case that $n$ is odd, $m$ is odd, and $2m=n+1.$

Let us consider the following $n\times m$ matrix, whose top part is the same
as the first $m$ columns and the first $m$ rows of (3.9) and the bottom part
has $(n-m)$ rows. For $\ell=1,\cdots,m$, let $\vec{f}_{\ell}$ be the $\ell$-th
column of the following matrix:

(3.12)
$\left[\begin{matrix}1&1+\sigma_{1}\sigma_{n}&1+\sigma_{2}\sigma_{n}&1+\sigma_{3}\sigma_{n}&\cdots&1+\sigma_{m-2}\sigma_{n}&1+\sigma_{m-1}\sigma_{n}\\\
1&1&1+\sigma_{2}\sigma_{n}&1+\sigma_{3}\sigma_{n}&\cdots&1+\sigma_{m-2}\sigma_{n}&1+\sigma_{m-1}\sigma_{n}\\\
1&1&1&1+\sigma_{3}\sigma_{n}&\cdots&1+\sigma_{m-2}\sigma_{n}&1+\sigma_{m-1}\sigma_{n}\\\
1&1&1&1&\cdots&1+\sigma_{m-2}\sigma_{n}&1+\sigma_{m-1}\sigma_{n}\\\
\vdots&\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\\
1&1&1&1&\cdots&1+\sigma_{m-2}\sigma_{n}&1+\sigma_{m-1}\sigma_{n}\\\
1&1&1&1&\cdots&1&1+\sigma_{m-1}\sigma_{n}\\\ 1&1&1&1&\cdots&1&1\\\ &&&&&&\\\
&&&&&&\\\ 1&1+A_{1}&1-B_{1}&1&\cdots&1&1\\\ 1&1+B_{1}&1+A_{1}&1&\cdots&1&1\\\
1&1&1&1+A_{2}&\cdots&1&1\\\ 1&1&1&1+B_{2}&\cdots&1&1\\\
\vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\\
1&1&1&1&\cdots&1+A_{(m-1)/2}&1+B_{(m-1)/2}\\\
1&1&1&1&\cdots&1-B_{(m-1)/2}&1+A_{(m-1)/2}\\\ \end{matrix}\right].$

The bottom part of (3.12) has $2\times 2$ square submatrices

(3.13) $\left[\begin{matrix}1+A_{\lambda}&1-B_{\lambda}\\\
1+B_{\lambda}&1+A_{\lambda}\end{matrix}\right],~{}~{}~{}\quad\lambda=1,2,\cdots,(m-1)/2,$

where

(3.14) $\displaystyle A_{\lambda}$ $\displaystyle=\sin
x_{m+2(\lambda-1)}\sigma_{m+2(\lambda-1)}\sigma_{m+2(\lambda-1)+1}$
$\displaystyle B_{\lambda}$ $\displaystyle=\cos
x_{m+2(\lambda-1)}\sigma_{m+2(\lambda-1)}\sigma_{m+2(\lambda-1)+1},$

so that the last block with $\lambda=(m-1)/2$ is

$\left[\begin{matrix}1+A_{(m-1)/2}&1-B_{(m-1)/2}\\\
1+B_{(m-1)/2}&1+A_{(m-1)/2}\end{matrix}\right],$

where

$\displaystyle A_{(m-1)/2}$ $\displaystyle=\sin
x_{n-2}\sigma_{n-2}\sigma_{n-1}$ $\displaystyle B_{(m-1)/2}$
$\displaystyle=\cos x_{n-2}\sigma_{n-2}\sigma_{n-1}.$

Since $\vec{f}_{1}=\sum_{i=1}^{n}\partial/\partial x_{i}$ calculation shows
that for each $j=1,\cdots,n,$ and each $\lambda=1,\cdots,(m-1)/2$ we have

(3.15) $\displaystyle\vec{f}_{1}\sigma_{j}$
$\displaystyle=\sigma^{\prime}(\rho_{j})\vec{f}_{1}\rho_{j}=0,$
$\displaystyle\vec{f}_{1}A_{\lambda}$ $\displaystyle=\cos
x_{m+2(\lambda-1)}\sigma_{m+2(\lambda-1)}\sigma_{m+2(\lambda-1)+1}$
$\displaystyle\vec{f}_{1}B_{\lambda}$ $\displaystyle=-\sin
x_{m+2(\lambda-1)}\sigma_{m+2(\lambda-1)}\sigma_{m+2(\lambda-1)+1}.$

Therefore, expressing the brackets $[\vec{f}_{1},\vec{f}_{\ell}]$ for
$\ell=2,3,4,5$ as column vectors, we have

(3.16) $\left[\begin{matrix}0&0&0&0\\\ \vdots&\vdots&\vdots&\vdots\\\
0&0&0&0\\\ &&&\\\ \cos x_{m}\sigma_{m}\sigma_{m+1}&\sin
x_{m}\sigma_{m}\sigma_{m+1}&0&0\\\ -\sin x_{m}\sigma_{m}\sigma_{m+1}&\cos
x_{m}\sigma_{m}\sigma_{m+1}&0&0\\\ 0&0&\cos
x_{m+2}\sigma_{m+2}\sigma_{m+3}&\sin x_{m+2}\sigma_{m+2}\sigma_{m+3}\\\
0&0&-\sin x_{m+2}\sigma_{m+2}\sigma_{m+3}&\cos
x_{m+2}\sigma_{m+2}\sigma_{m+3}\\\ \vdots&\vdots&\vdots&\vdots\\\ 0&0&0&0\\\
0&0&0&0\\\ \end{matrix}\right]$

Notice that $\sigma_{j}$’s are positive valued on
$\widetilde{\mathbb{R}}^{n}_{\epsilon}=\bigcap_{i=1}^{n}\\{\rho_{i}>0\\}.$
Thus for each $\ell=2,\cdots,m,$ $[\vec{f}_{1},\vec{f}_{\ell}]$ divided by a
positive scalar, is the $(\ell-1)$-th column of the following matrix;

(3.17) $\left[\begin{matrix}0&0&0&0&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\\ 0&0&0&0&\cdots&0&0\\\ &&&&&\\\
\cos x_{m}&\sin x_{m}&0&0&\cdots&0&0\\\ -\sin x_{m}&\cos
x_{m}&0&0&\cdots&0&0\\\ 0&0&\cos x_{m+2}&\sin x_{m+2}&\cdots&0&0\\\ 0&0&-\sin
x_{m+2}&\cos x_{m+2}\\\ \vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&0&0&\cdots&\cos x_{n-3}&\sin x_{n-3}\\\ 0&0&0&0&\cdots&-\sin x_{n-3}&\cos
x_{n-3}\\\ \end{matrix}\right].$

The above $(m-1)$ vectors are linearly independent and we see that
$\vec{f}_{\ell},$ $\ell=1,\cdots,m$ and $[\vec{f}_{1},\vec{f}_{\ell}],$
${\ell}=2,\cdots,m$, span the tangent space of
$\widetilde{\mathbb{R}}^{n}_{\epsilon}$.

3) Case that $n$ is even, $m$ is even, and $2m\geq n+2:$ It is enough to
construct for the case $2m=n+2.$ Since $m$ is even and we want $\vec{f}_{1}$
to be constant vector field $\sum_{j=1}^{n}\partial/\partial x_{j}$ as in the
previous case. Since $m-1$ is odd, we can construct $(m-2)/2$ blocks of size
$2\times 2$ in the same way as in the previous case. Construction is same
except for the brackets $[\vec{f}_{1},\vec{f}_{\ell}]$ ends up with
$\ell=m-1.$

## References

  * [1] A. Agrachev and Y. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences, Control Theory and Optimization, II, vol. 87, Springer-Verlag, Berlin, 2004.
  * [2] H. Ahn and C.-K. Han, Invariant submanifolds for systems of vector fields of constant rank, Sci. China Math. 59 (2016), no. 7, 1417–1426.
  * [3] R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt and P. A. Griffiths, Exterior differential systems, Mathematical Sciences Research Institute Publications, Vol. 18, Springer-Verlag, New-York, 1991.
  * [4] R. W. Brockett, R. S. Millman and H. J. Sussmann, editors, Differential geometric control theory , Proc. Conf. at Michigan Technological U., 1982, Birkhäuser, Boston, 1983
  * [5] I. Duleba, W. Khehifi and I. Karcz,  Layer, Lie algebraic method of motion planning for nonholomomic systems , J. of the Fanklin Institute 349 (2012), 201-215
  * [6] V. Grushkovskaya and A. Zuyev, Motion planning problem with obstacle avoidance for step-2 bracket-generating system , PAMM. Proc. Appl. Math. Mech. 17 (2017), 799–800.
  * [7] V. Grushkovskaya and A. Zuyev, Obstacle avoidance problem for driftless nonlinear systems with oscillating controls , ScienceDirect, IFAC Conference paper archive 50-1 (2017), 10476–10481.
  * [8] C.-K. Han, Pfaffian systems of Frobenius type and solvability of generic overdetermined PDE systems , Symmetries and overdetermined systems of partial differential equations, IMA Vol. Math. Appl., 144, Springer-Verlag, New York (2008), 421–429.
  * [9] by same author, Generalization of the Frobenius theorem on involutivity , J. Korean Math. Soc. 46 (2009), 1087–1103
  * [10] C.-K. Han and H. Kim, Invariant submanifolds for affine control systems, Ann. Polonici Math.124-1 (2020), 61-73
  * [11] C.-K. Han and J. Park, Method of characteristics and first inegrals for overdetermined fsystem of quasi-linear PDE of first order , Sci. China Math.58-8 (2015), 1665–1676
  * [12] C.-K. Han and J. Park,  Local solvability of quasi-linear Cauchy-Riemann equations , Pacific J. Math., 265-1 (2013), 59–84
  * [13] C.-K. Han and J. Park,  Systems of quasi-linear PDE of first order with Cauchy data of higher codimensions, J. Math. Analysis and Applications, 430-1 (2015), 390–402
  * [14] F. Jean , Control of nonholonomic systems: From sub-Riemannian geometry to motion planning , Springer -Verlag, New York (2014).
  * [15] O. Khatib , Real-time obstacle avoidance for manipulators and mobile robots, International J. of Robotics Research, Vol.5-1, (1986), 90-98.
  * [16] I. Kolmanovsky and N. H. McClamroch, Developments in nonholonomic control problems, IEEE Contol systems, (1995), 20-33.
  * [17] D. O. Popa and J. T. Wen , Nonholonomic path-planning with obstacle avoidance: a path-space approach, Proc. 1996 IEEE International Conf. on Robotics and Automation, 1996\.
  * [18] G. P Roussos, D.V. Dimarogonas and K. J. Kyriakopoulos, 3D Navigation and collision avoidance for a non-holonimic vehicle in American Control Conference 2008, Washington, USA
  * [19] A. Zuyev and V. Grushkovskaya , Motion planning for control-affine systems satisfying low-order controllability conditions , International J. of Control vol 90-11 (2017), 2517–2537