# Quantum-Fluctuation-Driven Coherent Spin Dynamics in Small Condensates

Xiaoling Cui1,2, Yupeng Wang1 and Fei Zhou2 1Beijing National Laboratory for
Condensed Matter Physics and Institute of Physics,
Chinese Academy of Sciences, P. O. Box 603, Beijing 100190, China
2Department of Physics and Astronomy, The University of British Columbia,
Vancouver, B. C., Canada V6T1Z1

###### Abstract

We have studied quantum spin dynamics of small condensates of cold sodium
atoms. For a condensate initially prepared in a mean field ground state, we
show that coherent spin dynamics are purely driven by quantum fluctuations of
collective spin coordinates and can be tuned by quadratic Zeeman coupling and
magnetization. These dynamics in small condensates can be probed in a high-
finesse optical cavity where temporal behaviors of excitation spectra of a
coupled condensate-photon system reveal the time evolution of populations of
atoms at different hyperfine spin states.

Recently, single-atom detection in optical cavities has been realized in
experiments by having atoms and cavity photons in a strongly coupling
regimeMabuchi96 ; Hood98 . This remarkable achievement has been applied to
study optically transported atoms in cavitiesSauer04 ; furthermore the
coupling between a small Bose-Einstein condensate (BEC) and cavity photons and
resultant collective excitations have also been successfully
investigatedBrennecke07 . The sensitivity that a cavity-based atom detector
has, together with a translating optical lattice which can effectively
transport ultra cold atoms from a magnetic-optical trap to a cavity make it
possible to study the physics of small BECs. Especially, this potentially
opens the door to explore coherent dynamics of ultra-cold atoms in relatively
small condensates. The physics of BECs of small numbers of atoms can be
qualitatively different from the physics of big condensates and represents a
new domain of cold-atom research. In small condensates, various intrinsic
beyond-mean-field dynamics can be relevant within an experimentally accessible
time scale. These new physical phenomena however have been quite difficult to
study using the standard absorption-imaging approach to cold atoms because of
relatively fewer atoms are involved in small condensates. Cavity
electrodynamics in a strong coupling regime and high sensitivities to intra-
cavity atoms on the other hand are ideal for investigating small condensates
where the beyond-mean-field dynamics are mostly visible. In this letter, we
focus on the basic concepts of beyond-mean-field coherent spin dynamics in
BECs with typically a few tens to a few hundreds of atoms and detailed
analysis of detecting these fascinating properties of small condensates in
optical cavities with high-finesse. Research on this subject could
substantially advance our understanding of the nature of quantum-fluctuation
dynamicsSong08 , in this particular case, dynamics purely driven by
fluctuations with wavelengths of the size of condensates. Secondly, results
obtained can help to better recognize limitations of precise measurements of
various interaction constants based on mean-field coherent dynamicsChang04 .
Thirdly, our results should shed some light on the feasibility of
investigating fluctuation dynamics of small condensates using optical cavities
and also pave the way for future studies of dynamics of coupled small
condensates.

To understand spin dynamics of a small condensate, we first study the
evolution of a condensate of $N$ hyperfine spin-one sodium atoms which is
initially prepared in a mean field ground state,

$\displaystyle|{\bf n}\rangle=\frac{({\bf
n}\cdot\psi^{{\dagger}})^{N}}{\sqrt{N!}}|0\rangle.$ (1)

Here ${\bf n}$ is a unit director and three components of $\psi^{\dagger}$,
$\psi^{\dagger}_{\alpha}$, $\alpha=x,y,z$ are creation operators for three
spin-one states, $|x\rangle=(|1\rangle-|-1\rangle)/\sqrt{2}$,
$|y\rangle=(|1\rangle+|-1\rangle)/i\sqrt{2}$ and $|z\rangle=|0\rangle$
respectively. And in this representation,
$S_{\alpha}=-i\epsilon_{\alpha\beta\gamma}\psi^{\dagger}_{\beta}\psi_{\gamma}$
is the total spin operator. States in Eq.1 with ${\bf n}={\bf e}_{z}$ minimize
the interaction energy of the following Hamiltonian for spin-one atoms in the
presence of a quadratic Zeeman coupling along the $z$-direction,

$H=\frac{c_{2}}{N}\mathbf{S}^{2}+q(\psi^{\dagger}_{x}\psi_{x}+\psi^{\dagger}_{y}\psi_{y}).$
(2)

Here $c_{2}$ is a spin interaction constant and $q$ is the quadratic Zeeman
couplingHo98 ; Ohmi98 ; Law98 ; Stenger98 . Mean field ground states are
stationary solutions to the multi-component Gross-Pitaevskii equations for
spin-one atoms and dynamics of these initial states demonstrated below are
therefore a beyond-mean field phenomenon. When deriving Eq.2 for a trapped
condensate, we assume that spin dynamics are described by a single mode, i.e.
$\psi_{\alpha}({\bf r},t)=\sqrt{\rho({\bf r})}\psi_{\alpha}(t)$; for a small
condensate of less than one thousand weakly interacting atoms, this
approximation is always valid. $c_{2}$ is typically a few nano kelvin for
sodium atoms; $q=(\mu_{B}B)^{2}/(4\Delta_{hf})$ and the hyperfine splitting is
$\Delta_{hf}=(2\pi)1.77GHz$ ($\mu_{B}$ is the Bohr magneton and $\hbar$ is set
to be unity).

To illustrate the nature of non-mean-field dynamics and crucial role played by
quantum fluctuations, we expand the full Hamiltonian in Eq.2 around a mean
field ground state. In the lowest order expansion, we approximate
$\psi^{\dagger}\approx\sqrt{N}{\bf e_{z}}+\psi^{\dagger}_{x}{\bf
e}_{x}+\psi^{\dagger}_{y}{\bf e}_{y}$, and $\psi^{\dagger}_{x,y}$ are much
less than $\sqrt{N}$; the Hamiltonian then can be expressed in terms of the
bilinear terms

$H_{B}=\sum_{\alpha=x,y}\frac{q+4c_{2}}{2N}P_{\alpha}^{2}+\frac{qN}{2}\theta_{\alpha}^{2}+...$
(3)

where for $\alpha=x,y$,
$\theta_{\alpha}=\frac{1}{\sqrt{2N}}(\psi^{{\dagger}}_{\alpha}+\psi_{\alpha})$
and $P_{\alpha}=i\sqrt{\frac{N}{2}}(\psi^{{\dagger}}_{\alpha}-\psi_{\alpha})$
are pairs of conjugate operators which satisfy the usual commutation relations
$[\theta_{\alpha},P_{\beta}]=i\delta_{\alpha,\beta}$. Semiclassically,
collective coordinates $\theta_{\alpha}$, $\alpha=x,y$ represent projections
of $\psi^{{\dagger}}$ or order parameter ${\bf n}$ in the $xy$ plane, and
$P_{x(y)}\sim S_{y(x)}$ is the spin projection along the $y(x)$-direction. The
bilinear Hamiltonian is equivalent to a harmonic oscillator moving along the
direction of $\theta_{x,y}$ with a mass $m_{eff}=\frac{N}{q+4c_{2}}$, a
harmonic oscillator frequency $\omega=\sqrt{q(q+4c_{2})}$ and effective spring
constant $qN$; the mass at $q=0$ is induced by scattering between atoms. The
excitation spectrum is $E_{n}=(n+1/2)\omega$, $n=0,1,2...$ . When $q=0$, the
Hamiltonian describes a particle moving in a free space.

Figure 1: a) (color online) Time evolution of $N_{0}(t)$, the atom population
at $|1,0\rangle$ state for different quadratic Zeeman coupling $q$. Initially,
all $N=200$ atoms occupy $|1,0\rangle$ state which corresponds to a mean field
ground state. The inset is $t_{c}$, the time for the first dip in the $q=0$
data, as a function of the number of atoms $N$. b) Time evolution of
$N_{0}(t)$ for different magnetization $m$ (here $q=0$). All initial states
are again mean-field ground states for given $m$. Inset is for $m=0.98$. In
this and Fig.2,3, $c_{2}=(2\pi)50Hz$.

In the ground state of the bilinear Hamiltonian of Eq.3,
$\langle\theta_{\alpha}\rangle$ $=\langle P_{\alpha}\rangle=0$ and ${\bf n}$
and $\langle{\bf S}\rangle$ have no projections in the $xy$ plane. However,
quantum fluctuations of $\theta_{x,y}$-coordinates in the ground state can be
estimated as $\langle\theta_{\alpha}\theta_{\alpha}\rangle$
$=\frac{1}{2N}\sqrt{\frac{q+4c_{2}}{q}}$. This is a measure of how strongly
${\bf n}$ fluctuates in the $xy$-plane. As expected, these quantum
fluctuations are substantial only when $q$ is small and are suppressed by a
quadratic Zeeman field which effectively pins the order parameter along the
$z$-direction. A direct calculation also shows that the amplitude of quantum
fluctuations $\langle\theta^{2}_{\alpha}\rangle_{MF}$ in the mean field ground
state defined in Eq.1 is $1/2N$. This indicates that the mean field ground
state is a good approximation only when $q\gg 4c_{2}$. On the other hand, as
$q$ decreases and the effective spring constant gets smaller, the deviation
becomes more and more severe. When $q$ approaches zero, quantum fluctuations
$\theta_{\alpha}$ in the harmonic oscillator ground state become divergent
implying that the mean field ground state is no longer a good approximation.

Indeed, the the energy of mean field ground state is
$E_{MF}=\frac{q}{2}+c_{2}$ which is much higher than $\frac{1}{2}\omega$ when
$q\ll c_{2}$; such a state corresponds to a highly excited wave packet,
because of a relatively narrow spread along $\theta_{\alpha}$-directions and
consequently an enormous kinetic energy associated with momenta $P_{\alpha}$.
We therefore expect that dynamics in this limit could dramatically differ from
a stationary solution. Since the total number of atoms $N$ is equal to
$\sum_{\alpha}\psi^{{\dagger}}_{\alpha}\psi_{\alpha}$, the population of atoms
at $|z\rangle$ (or $|1,0\rangle$) state
$N_{0}=\langle\psi^{\dagger}_{z}\psi_{z}\rangle$ is directly related to
quantum fluctuations of $\theta_{\alpha}$ and $P_{\alpha}$,

$\displaystyle
N_{0}=N+{1}-\sum_{\alpha}\left(\frac{N}{2}\langle\theta_{\alpha}^{2}\rangle+\frac{1}{2N}\langle
P_{\alpha}^{2}\rangle\right).$ (4)

Eq.4 shows that the time evolution of $N_{0}(t)$ is effectively driven by
quantum fluctuations in $\theta_{\alpha}$ and $P_{\alpha}$; a study of
$N_{0}(t)$ probes underlying quantum- fluctuation dynamics.

For an initial state prepared in a mean field ground state with ${\bf n}={\bf
e}_{z}$ where all atoms condense in $|1,0\rangle$ state, one finds that
$\langle\theta^{2}_{\alpha}\rangle=\frac{1}{2N}$ and
$\langle{P^{2}_{\alpha}}\rangle=\frac{N}{2}$. The evolution of such a
symmetric Gaussian wave packet subject to the bilinear Hamiltonian can be
solved exactly using the standard theory for harmonic oscillators. The wave
packet will remain to be a Gaussian one with the width oscillating as a
function of time. Qualitatively, because of the symmetry, only harmonic states
with even-parity are involved in dynamics and therefore the oscillation
frequency is $2\omega$. Furthermore during oscillations, the kinetic energy
stored in initial wave packets is converted into the potential one and vice
versa. Especially when $q\ll c_{2}$, oscillations are driven by the enormous
initial kinetic energy associated with $P_{\alpha}$; the oscillation amplitude
can be estimated by equaling the total energy $E_{MF}$ to the potential energy
which leads to $\langle\theta_{\alpha}^{2}\rangle$ $\sim c_{2}/(Nq)$. A
straightforward calculation yields the time dependence of
$\langle\theta^{2}_{\alpha}\rangle$ and $\langle P^{2}_{\alpha}\rangle$ that
leads to

$\frac{N_{0}}{N}=1-\frac{8c_{2}^{2}}{q(q+4c_{2})N}\sin^{2}wt.$ (5)

The oscillating term in Eq.5 shows the deviation from the stationary behavior
due to quantum fluctuations in $\theta_{\alpha}$-coordinates. The deviation is
insignificant when $q$ is not too small; however when $q$ is of the order of
$c_{2}/N$, we expect that the non-mean field dynamics becomes very visible.
Note that the approach outlined here neglects all higher order anharmonic
interactions and therefore is only valid when the relative amplitude of
fluctuations is small; that is when $q\gg c_{2}/N$.

When $q$ approaches zero, the short time dynamics following the bilinear
Hamiltonian is equivalent to a particle of a mass $m_{eff}=N/4c_{2}$ that is
initially localized within a spread $\langle\theta_{\alpha}^{2}\rangle=1/2N$
having a ballistic expansion with a typical velocity give as ${\langle
v^{2}_{\alpha}\rangle}=8c^{2}_{2}/N$. The time dependence of spread
$\langle\theta^{2}_{\alpha}\rangle$ therefore is $1/2N+(8c^{2}_{2}/N)t^{2}$.
So at $t\sim\sqrt{N}/c_{2}$, the number of atoms not occupying the initially
prepared $|1,0\rangle$ state becomes of order of $N$. This limit was first
addressed by Law et al. in the context of four-wave-mixing theoryLaw98 , and
also in early worksZhou01 ; Diener06 ; to describe the physics after this
characteristic time scale requires analysis of full quantum dynamics. This
time scale however becomes quite long for a few million atoms which makes it
difficult to observe quantum dynamics in large condensates.

In the following, we are going to present our numerical results on dynamics
and focus on its dependence on quadratic Zeeman coupling $q$ and magnetization
$m$. For a condensate of $N=200$ atoms, we numerically integrate the time-
dependent N-body Schrodinger equation of the quantum Hamiltonian in Eq.2. The
time evolution of $N_{0}$ driven by quantum fluctuations is shown in Fig.1a).
As $q$ increases far beyond $0.2c_{2}$, $N_{0}$ oscillates as a function of
time with frequency $2\omega$ and the amplitude of oscillations decreases; the
damping is not visible over tens of oscillations. When $q$ is below
$0.2c_{2}$, anharmonic effects become substantial and oscillations are no
longer perfect; when $q=c_{2}/40$, oscillations are strongly damped after a
few cycles and revived afterwards. For $q=0$, $N_{0}$ drops to a minimum of
about $0.38N$ when $t=t_{c}=0.53\sqrt{N}/c_{2}$ and remains to be a constant
before reviving to be $0.8N$ at about $10t_{c}$. For sodium atoms with a
typically density $2\times 10^{14}cm^{-3}$, $c_{2}=(2\pi)50Hz$; $t_{c}=23.8ms$
for $N=200$ and increases to a few seconds when $N$ reaches $2\times 10^{6}$.

We have also studied the quantum dynamics of a mean field condensate with a
finite magnetization along the $z$-direction, ${\bf m}=m{\bf e}_{z}$. States
which minimize the mean field energy of the Hamiltonian in Eq.2 with $q=0$ are

$\displaystyle|m\rangle=\frac{[(\cos\eta{\bf e}_{x}+i\sin\eta{\bf
e}_{y})\cdot\psi^{\dagger}]^{N}}{\sqrt{N!}}|0\rangle$ (6)

where $\sin 2\eta=m$, $m(\in[-1,1])$ is the normalized magnetization. By
expanding the Hamiltonian around these mean field states, one obtains a
harmonic oscillator Hamiltonian defined in terms of conjugate operators,
$\theta_{z}=\frac{1}{\sqrt{2N}}(\psi^{{\dagger}}_{z}+\psi_{z})$ and
$P_{z}=i\sqrt{\frac{N}{2}}(\psi^{{\dagger}}_{z}-\psi_{z})$. The effective mass
is $m_{eff}=\frac{N}{2c_{2}(1+\sqrt{1-m^{2}})}$ and the harmonic oscillator
frequency $\omega=2|m|c_{2}$. States shown in Eq.6 have a narrow width along
the direction of $\theta_{z}$, $\langle\theta^{2}_{z}\rangle=1/2N$ and
therefore carry large conjugate momenta $P_{z}$; the corresponding large
kinetic energy drives a unique non-mean field quantum spin dynamics. The
harmonic expansion again is only valid when $m$ is large and fluctuations are
weak. Simulations of the full Hamiltonian have been carried out in this case;
in Fig.1b), we show the time dependence of $N_{0}(t)$ for different
magnetization. Only when $m$ is close to unity, weakly damped oscillations are
observed.

Figure 2: (a,b,c) (color online) Eigenfrequencies $\Delta_{p}$ as a function
of $t$ for different detuning $\Delta_{c}$ when the relative population at
state $|1,0\rangle$, $\rho_{0}(t)$ evolves. At $t=0$, all atoms occupy
$|1,0\rangle$ state and $N=200$. d) $\Delta_{p}$ as a function of time $t$ for
$\Delta_{c}=0$.

We propose a method to probe quantum spin dynamics of a small condensate of
spin-one sodium atoms using cavity quantum electrodynamics. For a Bose-
Einstein condensate with N atoms coupled to a quantized field of a cavity, a
single cavity photon can coherently interact with atoms which leads to a
collective coupling of $g\sqrt{N}$Tavis68 . In experimentsSauer04 ;
Brennecke07 , atoms are transported into a cavity via a moving optical
lattice; excitations are measured by individual recordings of cavity
transmission when frequencies of an external probe light are scanned. Here we
consider a multi-component BEC coupled to a single cavity mode; the
eigenfrequencies of the coupled system uniquely depend on populations at three
hyperfine states. By measuring the energy spectrum of this coupled system, one
obtains temporal behaviors of atom populations at different states.

We restrict ourselves to excitations which involve a single cavity photon
interacting with atoms in a BEC. We study atomic transitions from
$3S_{\frac{1}{2}}\rightarrow 3P_{\frac{1}{2}}$ in sodium atoms. The
Hamiltonian consists the following terms,

$\displaystyle{H}_{cavity}$ $\displaystyle=$ $\displaystyle\sum_{i}\hbar
w_{g_{i}}\hat{g}_{i}^{{\dagger}}\hat{g}_{i}+\sum_{j}\hbar
w_{e_{j}}\hat{e}_{j}^{{\dagger}}\hat{e}_{j}+\sum_{p}\hbar
w_{c}\hat{c}_{p}^{{\dagger}}\hat{c}_{p}$ (7)
$\displaystyle-i\hbar\sum_{p}\sum_{i,j}g_{ij}^{p}\hat{e}_{j}^{{\dagger}}\hat{c}_{p}\hat{g}_{i}+h.c.,$

where $i$ labels three states $|F=1,m_{F}=0,\pm 1\rangle$ in
$3S_{\frac{1}{2}}$ orbital and $j$ eight states
$|F^{\prime}=1,m_{F^{\prime}}\rangle$, $|F^{\prime}=2,m_{F^{\prime}}\rangle$
in $3P_{\frac{1}{2}}$ orbital. $\hat{g}_{i}^{{\dagger}}$ and
$\hat{e}_{j}^{{\dagger}}$ create atoms in one of $3S_{\frac{1}{2}}$ and
$3P_{\frac{1}{2}}$ states respectively with corresponding frequencies
$w_{g_{i}}$,$w_{e_{j}}$. $\hat{c}_{p}^{{\dagger}}$ creates a photon with
frequency $w_{c}$ and polarization $p$ in the cavity mode.
$g_{ij}^{p}(=D_{ij}^{p}\sqrt{\hbar w_{c}/2\epsilon_{0}V})$ is the coupling
strength for a transition $i\rightarrow j$ driven by a cavity photon with
polarization $p$, which depends on the dipole matrix element $D_{ij}^{p}$, the
effective mode volume $V$.

Figure 3: Eigenfrequencies $\Delta_{p}$ as a function of time $t$ driven by
dynamics of population $\rho_{0}(t)$ in the presence of quadratic Zeeman
coupling $q=0.05c_{2}$ (or $B=95mG$). Again at $t=0$, all atoms occupy
$|1,0\rangle$ state and $N=200$, $\Delta_{c}=0$.

For simplicity, we set the energy of $3S_{\frac{1}{2}}$ states to be zero,
i.e. $w_{g_{i}}=0$; the energy of excited states is
$w_{e}^{1,2}=w_{a}\pm\Delta$ with $2\Delta$ being the hyperfine splitting
between $F^{\prime}=2$ and $F^{\prime}=1$ states. For atomic transitions
induced by left-circularly($\sigma^{+}$) polarized photons, the selection rule
is $\Delta F=0,\pm 1,\Delta m_{F}=1$. In a cavity, a state with a cavity
photon (labeled as 1c),$N_{m_{F}}$ atoms at $3S_{\frac{1}{2}}$
$|1,m_{F}\rangle$ states and no atoms at excited states (labeled as $0_{j}$)
is expressed as $|1_{c};N_{1},N_{0},N_{-1};0_{j}\rangle$; it is coupled to the
following states with one of atoms excited to $3P_{\frac{1}{2}}$ states
(labeled $1_{F^{\prime}}$) and no cavity photons (as $0_{c}$),
$|0_{c};N_{1}-1,N_{0},N_{-1};1_{F^{\prime}=2}\rangle$,
$|0_{c};N_{1},N_{0}-1,N_{-1};1_{F^{\prime}=1}\rangle$,
$|0_{c};N_{1},N_{0}-1,N_{-1};1_{F^{\prime}=2}\rangle$,
$|0_{c};N_{1},N_{0},N_{-1}-1;1_{F^{\prime}=1}\rangle$,
$|0_{c};N_{1},N_{0},N_{-1}-1;1_{F^{\prime}=2}\rangle$. We diagonalize the
Hamiltonian matrix and obtain six eigenfrequencies $\omega_{p}$ for this
coupled system. Three are $3P_{\frac{1}{2}}$ orbitals without mixing with
$3S_{\frac{1}{2}}$ states, $w_{p}=w_{a}\pm\Delta$; the other three depend on
relative populations of atoms at each spin state, $\rho_{m_{F}}=N_{m_{F}}/N$,
$m_{F}=0,\pm 1$. The latter three eigen frequencies are determined by the
eigen value equation,

$\displaystyle(\Delta_{p}-\Delta_{c})(\Delta_{p}^{2}-\Delta^{2})-\Delta_{p}Ng_{1}^{2}F_{1}-\Delta
Ng_{1}^{2}F_{2}=0.$ (8)

Here $\Delta_{p}=w_{p}-w_{a}$, $m=\rho_{+1}-\rho_{-1}$ is the normalized
magnetization, and $g_{1}$ is the coupling between $3S_{\frac{1}{2}}$
$|F=1,m_{F}=1\rangle$ and $3P_{\frac{1}{2}}$
$|F^{\prime}=2,m_{F^{\prime}}=2\rangle$ by $\sigma^{+}$ light; $F_{1}=(2+m)/3$
and $F_{2}=(1+m)/2-\rho_{0}/6$. Apparently eigenfrequencies
$\Delta_{p1,p2,p3}$ are a function of $\rho_{m_{F}}$ and therefore can be used
to probe the variation in $\rho_{0,\pm 1}$ due to coherent spin dynamics.
$\Delta_{p1,p2,p3}$ depend on a dimensionless parameter
$r=\frac{\sqrt{6}}{3}\frac{\sqrt{N}g_{1}}{\Delta}$. When detuning
$\Delta_{c}=0$ and as $r\rightarrow\infty$, $\Delta_{p1,p2,p3}$ are around
$0,\pm\frac{\sqrt{6}}{3}\sqrt{N}g_{1}$ respectively. Eigenfrequency
$\Delta_{p2}$ varies from $-3\Delta/4$ to $-\Delta/2$ when $\rho_{0}$
increases from $0$ to $1$; the variation amplitude
$\delta=\Delta_{p}(\rho_{0}=1)-\Delta_{p}(\rho_{0}=0)$ reaches a saturated
value $\Delta/4$. For sodium atoms, $\Delta=(2\pi)94.4MHz$; cavity parameters
are chosen according to Ref.Brennecke07  and $g_{1}=(2\pi)10MHz$. In Fig. (2),
we show the evolution of $\Delta_{p}$ in time for different detuning
$\Delta_{c}$ when atoms are initially prepared at state $|1,0\rangle$ of
$3S_{1/2}$. The evolution of $\Delta_{p}(t)$ which can be probed by a
$\sigma^{+}$ beam maps out population $N_{0}(t)$ driven by underlying quantum
fluctuations. In Fig. (3), we further show the time dependence of $\Delta_{p}$
due to oscillatory quantum spin dynamics for $q=0.05c_{2}$($B\approx 95mG)$,
$N=200$ and $\Delta_{c}=0$.

In conclusion, we have illustrated the nature of coherent spin dynamics driven
by quantum-fluctuations in small condensates. The time evolution of population
of atoms at different hyperfine spin states is shown to reveal intrinsic
dynamics of quantum fluctuations of order parameters and spin projections.
These dynamics can be probed by studying eigenfrequencies of a coupled
condensate-photon system in a high-finesse optical cavity available in
laboratories. We thank Gerard Milburn, Junliang Song for stimulating
discussions. This work is in part supported by NSFC, $973$-Project (China),
and NSERC (Canada), CIFAR and A. P. Sloan foundation.

## References

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  * (2) C. J. Hood et al., Phys. Rev. Lett. 80, 4157 (1998).
  * (3) J. A. Sauer et al., Phys. Rev. A 69, 051804(2004).
  * (4) F. Brennecke et al., Nature 450, 268 (2007).
  * (5) Other fluctuation-driven spin dynamics were studied in J. L. Song, F. Zhou, Phys. Rev. A 77, 033628 (2008).
  * (6) M. S. Chang et al., Nature Physics 1, 111 (2005).
  * (7) T. L. Ho, Phys. Rev. Lett. 81, 742 (1998).
  * (8) T. Ohmi, K. Machinda, J.Phys.Soc.Jpn. 67, 1822 (1998).
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# A simple parametrisation for coupled dark energy

Vitor da Fonseca∗ and Nelson J. Nunes  Instituto de Astrofísica e Ciências do
Espaço,
Faculdade de Ciências da Universidade de Lisboa,
Campo Grande, PT1749-016 Lisboa, Portugal
∗E-mail<EMAIL_ADDRESS>Tiago Barreiro  Instituto de Astrofísica e
Ciências do Espaço,
ECEO, Universidade Lusófona de Humanidades e Tecnologias,
Campo Grande, 376, 1749-024 Lisboa, Portugal

###### Abstract

We propose a phenomenological generalisation of the standard model with only
one extra degree of freedom that parametrises the evolution of a scalar field
responsible for the cosmic acceleration. The model also foresees an additional
parameter in the form of a coupling between dark energy and dark matter. This
model captures a large diversity of dark energy evolutions at low redshift and
could usefully complement common CPL parametrisations widely used. In this
context, we have been constraining the parametrisation with data from Planck
and KiDS, bringing different results between the early and late universe
observations.

###### keywords:

Quintessence, dark matter, observational cosmology.

##  1 Introduction

The discovery in 1998 that the Universe is accelerating [1, 2] was really
stunning at the time. While the source of the cosmic acceleration is yet to be
unveiled, the existence of a dark energy component has been postulated to
explain it [3, 4]. In the standard model, dark energy takes the form of a
cosmological constant $\Lambda$ which by definition possesses a constant
energy density and negative pressure that fights the attractive pull of matter
[5]. That model is, at the moment, compatible with all observations and it is
the simplest one [6], at least from a parametrisation point of view. One of
the outstanding issue in the concordance model is the so-called coincidence
problem [7]. Given how the energy densities dilute with the expansion, the
fact that the cosmological constant is catching up with matter at the present
time necessitates initial conditions at the origin that are dramatically
precise. This issue has been paving the way to study varying dark energy [8]
in order to reproduce more dynamically the current abundances.

The simplest and model independent approach is phenomenological. It consists
in parametrising the dark energy equation of state, being the pressure over
the energy density that equals $-1$ for a cosmological constant. The detection
of a time dependence would exclude a cosmological constant. Introducing the
fewest possible parameters is ideal to limit degeneracies between them. A
simple and popular parametrisation [9] is a Taylor expansion at first order
which has a very limited domain of validity. Various improvements have been
introduced, like the Chevallier-Polarski-Linder (CPL) parametrisation [10,
11]. Also, to account for the possibility of early dark energy, a
parametrisation stepwise in redshift [12] has been proposed with a larger
number of additional parameters.

Here, we explore an alternative way to parametrise dark energy by proposing to
parametrise a scalar field $\phi$ called quintessence that would be
responsible for dark energy both in the early and late Universe [13]. Instead
of setting an equation of state parameter evolution, or a scalar potential,
this method parametrises the dark energy scalar field itself. We argue that
with a single parameter we recover the generic behaviour of the previous
parametrisations. We test this approach with data from Planck and KiDS to
constrain its specific parameters, identifying contrasting results between the
early and late universe observations. This is a comprehensive work going from
the analytical solutions to parameter constraints.

##  2 Scalar field parametrisation

We consider that the cosmological fluid is composed of matter (m), i.e.
baryons (b) and cold dark matter (c), as well as the scalar field itself in a
flat Friedmann-Lemaitre-Robertson-Walker background. The scalar field is
supposed to be canonical and homogeneous at large scale [14, 15], approximated
by a perfect fluid. We choose to work with the number of e-folds, $N\equiv\ln
a$, as the time variable. In the following, the prime stands for the
derivative with respect to the number of e-foldings, and the dot the
derivative with respect to cosmic time.

By combining the Friedmann equation,

$H^{2}=\frac{\kappa^{2}}{3}\left(\rho_{b}+\rho_{c}+\rho_{\phi}\right),$ (1)

with the continuity equation of the scalar field,

$\rho_{\phi}^{\prime}+3H^{2}\phi^{\prime 2}=0,$ (2)

and the continuity equation of matter,

$\rho_{c}^{\prime}+3\rho_{c}=0,$ (3)

one obtains [16] a differential equation for the dark energy density (where we
conventionnally set $\kappa=1$),

$\frac{\rho_{\phi}^{\prime}}{\rho_{m_{0}}}=-\phi^{\prime^{2}}\left(a^{-3}+\frac{\rho_{\phi}}{\rho_{m_{0}}}\right),$
(4)

whose general solution is our starting point,

$\frac{\rho_{\phi}}{\rho_{m_{0}}}=e^{-\int_{0}^{N}dN\phi^{\prime^{2}}}\times\left[\frac{\Omega_{\phi_{0}}}{\Omega_{m_{0}}}-\int_{0}^{N}dN\phi^{\prime^{2}}e^{-3N+\int_{0}^{N}dN\phi^{\prime^{2}}}\right].$
(5)

Among possible parametrisations enabling us to solve the former equation
analytically,

$\displaystyle\phi^{\prime}=$ $\displaystyle\epsilon\phi,$ (6)
$\displaystyle\phi^{\prime}=$ $\displaystyle\sigma\phi^{-1},$ (7)
$\displaystyle\phi^{\prime}=$ $\displaystyle\lambda,$ (8)

where $\epsilon$, $\sigma$ and $\lambda$ are given constants, we explore the
simplest one in Eq. (8) [16].

The advantage of this parametrisation is three folds. Firstly, it is very
simple, as we only add one single parameter to extend the standard model. It
allows to reconstruct analytically the form of the potential [16],

$V(\phi)=A\,e^{-\frac{3}{\lambda}\phi}+B\,e^{-\lambda\phi},$ (9)

with the following mass scales,

$\displaystyle A$
$\displaystyle=\frac{3}{2}\,\frac{\lambda^{2}}{3-\lambda^{2}}H_{0}^{2}\,\Omega_{m_{0}}\,e^{\frac{3}{\lambda}\phi_{0}},$
(10) $\displaystyle B$
$\displaystyle=\frac{3}{2}\,\frac{\lambda^{2}-6}{3-\lambda^{2}}H_{0}^{2}\left(\frac{\lambda^{2}}{3}-\Omega_{\phi_{0}}\right)e^{\lambda\phi_{0}},$
(11)

where $H_{0}$ is today’s Hubble expansion rate and
$\Omega_{i_{0}}=\rho_{i_{0}}/3H_{0}^{2}$ today’s abundance parameter of
species $i$. Having a simple form of the potential allows a straightforward
implementation of the model within the existing Boltzmann code CLASS[17, 18]
that can be validated by comparing the numerical results with the expected
asymptotic behaviour of the analytic expressions. Given the small number of
parameters involved and thanks to its simplicity, our analysis permits to
effectively constrain the model parameters with current data as in the next
section.

Secondly, this scalar potential, being the sum of two exponential terms, does
alleviate the severe fine-tuning of the initial conditions as illustrated in
Fig. 1. For small values of the parameter $\lambda$, when the first
exponential term in the potential is steep enough, the scalar field energy
density is attracted and temporarily scales with the background as the
universe expands, be it dominated by radiation and then matter. Later, thanks
to the second shallow exponential, whose slope is the inverse of the first
one, the scalar field energy density eventually freezes, mimicking a
cosmological constant. By freezing, dark energy overcomes matter at late time,
enabling the acceleration of the universe that we are currently experiencing.
This well-known attractor mechanism [19] ensures that one can start from a
large domain of initial conditions for the scalar field, to reach unavoidably
the tracking solution and end up with the correct order of magnitude both
during matter and dark energy domination.

Figure 1: Energy density dilution of a cosmological constant, as well as
quintessence, matter and radiation for the scalar field parametrisation
($\lambda=0.01$).

Last but not least, despite just one extra parameter, the model covers a wide
range of possible dark energy evolution at low redshift as displayed in Fig 2.
The parametrisation is thus able to capture a large variety of possible
dynamics for dark energy. The observational data might even become precise
enough in the future for discriminating between these different evolutions.

Figure 2: Dynamics of the dark energy equation of state for given values of
the parameter $\lambda$.

As a step further, we decide to complement our parametrisation by assuming a
non-minimally coupling between dark energy and cold dark matter [20]. Again
for the sake of simplicity, we consider a conformal and constant coupling
$\beta$ that parametrises the interaction between dark matter and the scalar
field. The stress-energy tensor is jointly conserved within the dark sector in
order to ensure the covariance of the theory,

$\displaystyle\nabla_{\mu}\left(T^{(\phi)}\,{}^{\mu}_{\nu}+T^{(c)}\,{}^{\mu}_{\nu}\right)$
$\displaystyle=0,$ (12) $\displaystyle\nabla_{\mu}T^{(\phi)}\,{}^{\mu}_{\nu}$
$\displaystyle=+\beta\rho_{c}\nabla_{\mu}\phi,$ (13)
$\displaystyle\nabla_{\mu}T^{(c)}\,{}^{\mu}_{\nu}$
$\displaystyle=-\beta\rho_{c}\nabla_{\mu}\phi.$ (14)

Assuming that dark energy does not interact with baryons, the latter are
separately conserved contrary to dark matter and quintessence,

$\displaystyle\rho_{b}$ $\displaystyle=\rho_{b_{0}}~{}e^{-3N},$ (15)
$\displaystyle\rho_{c}$
$\displaystyle=\rho_{c_{0}}~{}e^{-3N+\beta\left(\phi-\phi_{0}\right)},$ (16)
$\displaystyle\rho_{\phi}^{\prime}+3H^{2}\phi^{\prime 2}$
$\displaystyle=-\beta\phi^{\prime}\rho_{c}.$ (17)

The sign of the product $\beta\phi^{\prime}$, i.e. $\beta\lambda$ in our
parametrisation, determines whether energy is being pumped from the scalar
field or from dark matter. Similarly to Ref. 16, we use the Friedmann equation
(1) with the solution we find for the scalar field energy density,

$\displaystyle\rho_{\phi}=$
$\displaystyle\left(\rho_{\phi_{0}}+\frac{\lambda^{2}+\beta\lambda}{\lambda^{2}+\beta\lambda-3}~{}\rho_{c_{0}}+\frac{\lambda^{2}}{\lambda^{2}-3}~{}\rho_{b_{0}}\right)e^{-\lambda^{2}N}$
$\displaystyle-\frac{\lambda^{2}+\beta\lambda}{\lambda^{2}+\beta\lambda-3}~{}\rho_{c_{0}}\,e^{(\beta\lambda-3)N}-\frac{\lambda^{2}}{\lambda^{2}-3}~{}\rho_{b_{0}}\,e^{-3N},$
(18)

to reconstruct, for the first time, the corresponding potential with similar
properties of the dynamical system as in the minimally coupled case,

$V(\phi)=A\,e^{\left(-\frac{3}{\lambda}+\beta\right)\phi}+B\,e^{-\lambda\phi}+C\,e^{-\frac{3}{\lambda}\phi},$
(19)

where the mass scales are now the following,

$\displaystyle A$
$\displaystyle=\frac{3}{2}\,\frac{\lambda^{2}+2\beta\lambda}{3-\lambda^{2}-\beta\lambda}~{}H_{0}^{2}\,\Omega_{c_{0}}\,e^{\left(\frac{3}{\lambda}-\beta\right)\phi_{0}},$
(20) $\displaystyle B$
$\displaystyle=\frac{6-\lambda^{2}}{2}\,H_{0}^{2}\left(1+\frac{3}{\lambda^{2}-3+\beta\lambda}\Omega_{c_{0}}+\frac{3}{\lambda^{2}-3}\Omega_{b_{0}}\right)e^{\lambda\phi_{0}},$
(21) $\displaystyle C$
$\displaystyle=\frac{3}{2}\,\frac{\lambda^{2}}{3-\lambda^{2}}~{}H_{0}^{2}\,\Omega_{b_{0}}\,e^{\frac{3}{\lambda}\phi_{0}}.$
(22)

We modify the CLASS code to accommodate that potential as well as the coupled
equations (16) and (17) to simulate the model with the two additional
parameters $\lambda$ and $\beta$. The numerical results plotted in Fig. 3 are
in line with the asymptotic solution for the dark energy equation of state
during matter domination when baryonic matter is neglected,

$w_{\phi}\approx-\frac{\beta}{\lambda+\beta}.$ (23)

One can note that today’s equation of state, $w_{0}$, does not depend on the
coupling, and drops close to $-1$ during the scalar field dominated epoch for
small values of $\lambda$,

$w_{{}_{0}}=-1+\frac{\lambda^{2}}{3\Omega_{\phi_{0}}}.$ (24)

Figure 3: Dark energy evolution in the coupled parametrisation. Beta negative
corresponds to the transfer of energy from the dark matter component to the
scalar field.

Furthermore, in order to numerically predict observables that we use to
constrain the parameters with observational datasets at the perturbative
level, we again modify CLASS to implement the coupled evolution of the scalar
fluctuations at linear level in the synchronous gauge adopting usual
conventions [21]. We modify the existing perturbed Klein-Gordon equation, as
well as the equation of motion for the dark matter density contrast
$\delta_{c}$ and velocity perturbations $\theta_{c}$ to account for the
coupling, by adding the relevant sources term in the right-hand side,

$\displaystyle\ddot{\varphi}+2\mathcal{H}\dot{\varphi}+\left(k^{2}+a^{2}V_{,\phi\phi}\right)\varphi+\frac{\dot{h}\dot{\phi}}{2}$
$\displaystyle=-\beta a^{2}\bar{\rho}_{c}\delta_{c},$ (25)
$\displaystyle\dot{\delta}_{c}+\theta_{c}+\frac{\dot{h}}{2}$
$\displaystyle=\beta\dot{\varphi},$ (26)
$\displaystyle\dot{\theta}_{c}+\mathcal{H}\theta_{c}$
$\displaystyle=\beta\left(k^{2}\varphi-\dot{\phi}\theta_{c}\right),$ (27)

where the overbar denotes the background quantities. Additionally, we adapt
the transformation from the synchronous gauge into the comoving gauge to
maintain the matter density source function $\delta_{m}$ gauge-invariant,

$\delta_{m}^{C}=\frac{\delta\rho_{m}}{\bar{\rho}_{m}}+\left(3-\frac{\beta\phi^{\prime}\bar{\rho}_{c}}{\bar{\rho}_{c}+\bar{\rho}_{b}}\right)\frac{\mathcal{H}}{k^{2}}\theta_{m}.$
(28)

In parallel, to provide for the code validation, we derive the analytic
equation of motion of the dark matter fluctuations in the Newtonian limit
during the domination of matter, precisely when the coupling makes a
difference,

$\delta^{\prime\prime}_{c}+\frac{1}{2}\left(1+3\lambda\beta\right)\delta^{\prime}_{c}-\frac{3}{2}\left(1-\frac{\lambda^{2}}{3}-\frac{\lambda\beta}{3}\right)\left(1+2\beta^{2}\right)\delta_{c}=0.$
(29)

In the left panel of Fig 4, we note that the scalar field slows the growth of
perturbations against the standard model, by decreasing the dynamical term in
the equation of motion. On the other hand, the existence of the coupling
brings competitive effects that further slow the fluctuation or, on the
contrary, accelerate them, depending on whether energy is being transferred
from the scalar field or into it. We are also able to get an analytic
expression of the growth rate $m+$ under the Newtonian approximation,

$m_{+}=\frac{1}{4}\left[-1-3\lambda\beta+\sqrt{24\left(1-\frac{\lambda^{2}}{3}-\frac{\lambda\beta}{3}\right)\left(1+2\beta^{2}\right)+\left(1+3\lambda\beta\right)^{2}}~{}\right].$
(30)

One can see in the right panel of Fig. 4 that the numerical results in the
matter era on a small scale are very much in line with it. The standard
growth, which equals $1$, is obviously recovered for vanishing values of the
parameters $\lambda$ and $\beta$.

(a)

(b)

Figure 4: $\lambda=0.3$ on scale $k=0.1~{}h$/Mpc. (a) Dark matter
fluctuations. (b) Growth rate function.

##  3 Observational constraints

We turn now our attention to the observations in order to constrain the value
of the posteriors $\lambda$ and $\beta$ through Bayesian inference with
MontePython [22, 23, 24, 25, 26, 27] and GetDist [28]. Given that it affects
the growth of dark matter perturbations, our parametrisation leaves
observational signatures, both on the CMB anisotropies and matter power
spectrum that we can numerically predict with our adapted version of CLASS and
compare with observations.

As for the CMB anisotropies, early dark energy as well as the coupling affect
the amplitude and position of the acoustic peaks as illustrated on the left
panel of Fig. 5. We use the data from the Planck 2018 likelihood [29] on the
temperature and polarisation of the CMB in the MCMC analysis. As regards
today’s linear matter power spectrum, the delay or the acceleration in the
growth of dark matter perturbations has consequences too (right panel of Fig.
5). We use the gravitational weak lensing measurements, cosmic shear, from the
KiDS-450 survey [30] in the Bayesian fitting.

(a)

(b)

Figure 5: Observables predictions for $\lambda=0.3$ (a) CMB dimensionless
angular power spectrum. (b) Linear matter power spectrum at redshift $z=0$.

The probability contours and posterior distribution we obtain in the parameter
space are displayed on Fig. 6 for the two specific degrees of freedom of the
model. On the one hand, the parameter estimation done with Planck provides an
upper bound for the posterior $\lambda$ which is compatible with a
cosmological constant, that cannot be discarded. The coupling posterior
corresponds to the energy injection from dark energy to dark matter. However,
on the other hand, weak lensing data brings a different constraint on the
posterior $\lambda$, not favouring a vanishing value. It also slightly detects
energy transfer from dark matter to dark energy.

Figure 6: Constraints on the ($\lambda$,$\beta$) posterior plane. Contours
with 68$\%$ and 95$\%$ probability.

We also undertake the same likelihood analysis with the standard model as a
benchmark to compare the goodness-of-fit of the parametrisation (Table 3). In
light of the reduced chi-square, the two models look more or less equivalent.
Nonetheless, according to the Akaike Information Criterion (AIC) [31], since
it further penalises model complexity, the favourite cosmology for both probes
is $\Lambda$CDM, this is particularly true for KiDS.

Standard $\Lambda$CDM model vs scalar field parametrisation  Planck KiDS
$\Lambda$CDM $\phi$ param. $\Lambda$CDM $\phi$ param. $\chi^{2}$ $602.6$
$599.7$ $321.5$ $321.7$ $\chi^{2}_{\rm{red}}$ $1.0$ $1.0$ $2.6$ $2.7$ $AIC$
$616.6$ $617.7$ $335.5$ $339.7$ $\Delta AIC$ $0$ $1.1$ $0$ $4.2$

##  4 Concluding remarks

Despite its simplicity, the parametrisation we propose is able to cover a
large span of dark energy evolution with a limited number of parameters.
Constraints on more complex parametrisations are usually limited by the
degeneracies between the various degrees of freedom. The model preserves the
fact that dark matter and dark energy are in comparable ratios today, while
relaxing the initial conditions thanks to the existence of tracking solutions.
It is thus capable of reproducing the evolution of the background, as well as
the CMB anisotropies and the formation of Large Scale Structures. Since the
parametrisation has a $\Lambda$CDM limit for small $\lambda$ and vanishing
coupling $\beta$, it allows a direct test of departures from the standard
model. We therefore argue that the parametrisation does help constrain the
evolution of the equation of state and the departure from $\Lambda$CDM, even
though the approach is purely phenomenological. Finally, the MCMC analysis
carried out at perturbation level identifies tensions on the posteriors
between early and late time probes. Background data is not expected to provide
stronger constraints but the upcoming Euclid data will certainly improve the
situation.

## Acknowledgements

The authors thank Giuseppe Fanizza and Miguel Zumalacarregui for their useful
comments. This research was supported by Fundação para a Ciência e a
Tecnologia (FCT) through the research grants: UID/FIS/04434/2019, PTDC/FIS-
OUT/29048/2017 (DarkRipple), COMPETE 2020:POCI-01-0145-FEDER-028987 &
FCT:PTDC/FIS-AST/28987/2017 (CosmoESPRESSO) and IF/00852/2015.

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# A Multi-modal Deformable Land-air Robot for Complex Environments

Xinyu Zhang1, Yuanhao Huang1,2, Kangyao Huang3, Xiaoyu Wang1, Dafeng Jin1,4,
Huaping Liu3 and Jun Li1 This work was supported by the National High
Technology Research and Development Program of China under Grant No.
2018YFE0204300, and the National Natural Science Foundation of China under
Grant No. U1964203, and sponsored by Meituan and Tsinghua University-Didi
Joint Research Center for Future Mobility.1The School of Vehicle and Mobility,
Tsinghua University, Beijing, P.R.China<EMAIL_ADDRESS>School of
Aviation, Inner Mongolia University of Technology, Hohhot, Inner Mongolia,
P.R.China<EMAIL_ADDRESS>Department of Computer Science and
Technology, Tsinghua University, Beijing, P.R.China. Kangyao Huang is the
corresponding author<EMAIL_ADDRESS>Automobile
Research Institute, Tsinghua University, Suzhou, P.R.China.
<EMAIL_ADDRESS>
###### Abstract

Single locomotion robots often struggle to adapt in highly variable or
uncertain environments, especially in emergencies. In this paper, a multi-
modal deformable robot is introduced that can both fly and drive.
Compatibility issues with multi-modal locomotive fusion for this hybrid land-
air robot are solved using proposed design conceptions, including power
settings, energy selection, and designs of deformable structure. The robot can
also automatically transform between land and air modes during 3D planning and
tracking. Meanwhile, we proposed a algorithms for evaluation the performance
of land-air robots. A series of comparisons and experiments were conducted to
demonstrate the robustness and reliability of the proposed structure in
complex field environments.

##  I Introduction

Field tasks such as post-disaster mapping, situational awareness and analysis,
search and rescue, and even special operations typically require robots that
can move and work in uncertain and complex environments. However, it is often
difficult for conventional single locomotion robots to adapt in these
conditions. In contrast, multi-modal robots exhibit much higher flexibility
and adaptability by combining wheels, legs, propellers, wings, fins, and
hybrid actuators to achieve motion[1]. For example, land-air robots [2, 3, 4,
5, 6] offer both flying and driving capabilities for increased spatial
flexibility while reducing the energy demands of continuous flight.

Similarly, hybrid vehicles with rotary wings have shown potential for seamless
and dynamic switching between modalities [7]. Many of these systems are
comprised of a quad-rotor hinged at the center of a cylindrical cage or a pair
of coaxial passive wheels. While this design offers several advantages for
reduced energy consumption, increased duration, and expanded coverage, dust
caused by rotor downwash effects may reduce perception capabilities while
driving [8]. Another class of land-air robots offers both active aerial and
terrestrial motion [9, 11]. Active wheels are required in these systems to
enhance ground moving performance [12, 14, 31].

Figure 1: A multi-modal deformable land-air robot.

The field of small land-air robotics has seen significant progress in recent
years, due to highly integrated electronic components [15]. Although these
robots balance athletic ability and agility, their small size limits them to
short-term reconnaissance missions. In contrast, larger land-air robots can
increase the duration of an operation and can carry heavier loads, but their
large size can be difficult to maneuver in certain terrains[12]. Therefore, a
land-air robots are difficult to have both strong flight ability and ground
mobility. Previous studies have introduced deformable mechanisms to fold
propeller to change size in the design of drones and land-air robot. However,
most of these designs are based on small platforms [16, 13], since large
deformations can limit actuator performance and restrict motion. As such, this
study aims to develop a novel land-air robot with excellent ability to fly and
drive. The proposed overall design conception in Section II can solve
compatibility issues for multi-modal locomotive fusion with an hybrid land-air
robot generally. Automatic deformable arms with fixed propellers offer
stronger flight thrust than the same size robots. On this basis, the robot can
become compact and move flexibly after folding its arms, and the it is also
the first land-air robot with automatic folding arms. A series of experiments
were conducted in buildings and fields as shown in Fig. 1. According to the
motinve and passable experiments with rich scenarios and quantitative
performance comparisons with related robots, we verify the robot’s
capabilities of flying and driving. The primary technical contributions of
this study can be summarized as follows:

  * •

Proposed a generic overall design conception of land-air robot, considering
the power settings, energy selection, lightweight design and deformable
structure. The conception ensures the land-air robot’s comprehensive movement
ability.

  * •

A land-air robot with automatic deformable arms is proposed for the first
time. Combined with the introduced design conceptions, solving the land-air
robots’ compatibility issues between flying and driving modes. Larger
propellers can generate greater thrust while reducing the plane projection
area by 79 percent, which significantly improved passing capabilities.

  * •

The proposed robot and mechanisms were verified in various simulated and
practical field scenarios. The experiments and quantitative performance
evaluation method provided strong evidences for their effectiveness and
robustness.

Figure 2: The overall structure of the proposed robot, including the physical
display and deformation advantages. (a) Key components, including flight,
ground travel, and hardware elements. (b) Flight and driving modes. (c) The
vertical projection area of the robot is reduced by 79 percent compared with
the unfolded state.

The remainder of this paper is organized as follows. Section II introduces the
design concept and corresponding analysis. Section III describes the detailed
structure and deformable mechanism controls. Section IV provides the results
of experiments and Section V concludes the paper. A video documenting
experiments conducted as part of the study is available at video.

##  II Design Concepts and Analysis

The design of the multi-modal land-air robot is not a simple combination, as
multiple components (e.g., power and electronic systems) are shared to reduce
the total weight to the extent possible. Despite this approach, moment of
inertia of the robot is higher than normal land-air robots causes by weight
and dispersed mass distribution. The compatibility problems of the motion
speed characteristic requirements and controls present significant challenges
[18]. In this section, we propose design metrics and concepts for power
settings, energy selection, and the deformable structure of the land-air
robot. The overall design and key parameters are shown in Fig. 2 and Table I.

TABLE I: Robot Specifications Width | Expanding | 1250 mm
---|---|---
| Folding | 585 mm
Depth | Expanding | 1250 mm
| Folding | 670 mm
Height | Body | 552 mm
Weight | Total | 24.62 kg
| Battery | 5.74 kg
| Flight Module | 7.61 kg
| Chassis | 7.07 kg
| Payload | 4.20 kg
Speed | Max Flying Speed | 7.47 m/s
| Max Driving Speed | 10.06 m/s
Flight Module | Flight Motor | 5 Nm
| Kv Value | 170
| Propeller | 26 inch
| Steering Gear | 8 Nm
Chassis | Kv Value | 800
| Driving Motor | 12 Nm
Battery | Lithium Battery | 48 V-30 Ah
| Duration | 21.6 min  Flight Time

###  II-A Overall Design

The robot configuration depends heavily on the requirements of specific tasks
and corresponding environments. For example, most disaster rescue operations
require robots to navigate through narrow passages, which places clear
limitations on robot width. Thus, the dimensions of the robot were limited to
within 70 centimeters in this study to allow for flexible motion and passage
through narrow spaces for rapid searching. Despite these size restrictions,
the robot is still expected to carry heavy equipment, including sensing
devices, computing units, and actuators for improved situational awareness. As
a typical example of this study, DJI ZENMUSE X7 and X5S are most commonly used
camera sensors with stabilizer, which weight 449 gram and 461 gram[19]
respectively. Similarly, ultra-wide view LiDAR sensors from Ouster weight
around 500 gram for ordinary autonomous driving use[10]. Taking these
configurations and load capacity demands into consideration, the maximum power
and take-off weight can be estimated using existing techniques [20, 21].
Hereafter, we refer to the multi-copter flight evaluation strategy to optimize
our approach, which provides a complete design and selection methodology for
estimating and maximizing flight duration [22].

###  II-B Power Setting and Flight Duration Analysis

Figure 3: Optimal flight duration analysis and power. Figure 4: Optimal flight
duration for various robot weight configurations.

The trade-off between battery weight and flight duration is one of the most
challenging aspects of the robot design process. We first evaluated flight
duration using battery capacity and electrical current modeling for the
motor/propeller system. The relationship between the Li-polymer battery’s
charge capacity and its mass is typically considered to be linear. We compared
three different brands of batteries and obtained linear models using data
fitting, as shown in Fig. 3(a). The maximum propeller size is fixed as 26
inch, we tentatively selected four power train configurations for use as
assessment samples, with brushless motors (T-Motor) varying from 135$\sim$170
KV. Corresponding data combinations were then fitted into mappings, including
variations in measured current and propulsion efficiency with thrust. All data
are open-access and available on the T-Motor website at https://uav-
cn.tmotor.com/. As shown in Fig. 3(b), the MN805S-170KV model can provide a
large thrust range covering the expected region, with a relatively high
propulsion efficiency that facilitates a higher energy efficiency for the
entire power train.

The flight duration $t$ can be estimated as follows:

$t=\frac{wm}{\kappa VI+P},P=P_{avionics}+P_{perception}+P_{deform},$ (1)

where $w$ and $m$ denote the energy density and mass of the battery,
respectively, $\kappa$ represents the number of motors in the multi-copter (a
constant), $V$ is the nominal motor voltage, $I$ is the instantaneous current,
and $P$ is the total power, consisting of power consumed by avionics systems,
sensing devices, and deformations. Onboard power consumption can be calculated
from the rated power and the mass of the robot. We set the calculated power
rating to the output power when the robot hovering and was treated as a
constant. Hovering thrust generated by the motors at rated power should be
greater than or equal to the robot’s weight. Thus, we can assume the hovering
thrust is approximately equal to the combined weight of these modules:

$T=W_{r}+W_{b}+W_{d}+W_{a}+W_{p},$ (2)

where $W_{r}$, $W_{b}$, $W_{d}$, $W_{a}$, and $W_{p}$ denote the weights of
the robot frame, battery, deforming mechanical module, avionics devices, and
payload, respectively. In some cases, increasing battery capacity can improve
flight duration as more energy is provided. Flight time, however, may decrease
if the battery is too heavy. The relationships between voltage, instantaneous
current, and thrust (acquired by curve fitting) were used to develop a mapping
relationship between flight duration and battery weight, as shown in Fig.
3(c). In this study, we strictly confined the robot’s weight about 17 kg,
excluding the battery. The results showed a robot with an MN805S-170KV power
train and a $\sim$5.7 kg battery achieved the longest hovering duration
($\sim$21 minutes). Detailed relationships between the total robot weight and
the battery weight utilizing T-Motor MN805S motors are provided in Fig. 4. The
no-fly zone represents areas in which the robot cannot fly in a given
configuration, due to excessive weight. In contrast, the lines of varying
duration show weight configurations that can achieve flight. The proposed
design is indicated by the red dot in the figure.

###  II-C Lightweight Design

Land-air robots are theoretically heavier than UAVs due to redundant actuators
used for multi-modal motion and deformation. As such, achieving a lightweight
design is extremely difficult with multi-modal robots, as there is always a
compromise between weight and performance. This subsection describes the
application of two frequently-used lightweight optimization methods: light
material selection and computer-aided structural optimization [23].

####  II-C1 Lightweight materials

We first carefully screened the selected materials, using aerospace composites
such as carbon fiber reinforced polymers (CFRPs) for the primary structure of
the robot. However, some components that frequently withstand strong
collisions and impacts require the use of metal. For example, the deformable
structures and locking mechanisms adopted aluminium alloys (7xxx series) and
high-strength steel. The differential casing and chassis cantilever were
composed of a nylon material, which effectively guaranteed the strength of the
fuselage structure. Each of these choices considered both the weight and
strength.

####  II-C2 Computer-aided structural optimization

The ANSYS finite-element and SolidWorks mechanical analysis tools were used to
continuously reduce the total weight. The main body used CFRP and was designed
as a box structure, which facilitated impact resistance and enhanced stiffness
[24]. All components were designed with a honeycomb structure to reduce the
weight. In the end, the arm deformation angle did not exceed 1.5 degree with
the maximum thrust of the propellers, and the mass of the sheet was reduced by
27%.

##  III Deformable Structure Design

Linkages[13] and putters[25] have successfully been used to actuate deformable
structures in previous studies. However, redundant components create
additional weight and can lead to more failures. As such, the proposed robot
uses servos to drive the arms directly. Meanwhile, locking mechanisms and
locking propeller motors are also highlights of the novel deformable
structure.

###  III-A Foldable Arms

A propeller’s efficiency is significantly influenced by its diameter, as more
power and thrust can be produced by larger propellers. For this reason, our
robot uses 26-inch propellers to generate more thrust. However, propellers
that are too large can cause issues for ground mobility, making it difficult
to maneuver in confined spaces. In addition, a wide mass distribution can
reduce the dynamic stability of the robot and induce vibrations [26]. As such,
we propose a new type of oblique folding design in which the front arm is
folded downward at an angle of 20 degrees and the rear arm is positioned
horizontally to avoid physical interference between the front and rear arms.
As the four arms rotate 135 degrees outward on the folding servo, the flight
module unfolds and the platform enters flight mode, forming a square standard
quad-rotor configuration.

Fig. 5(b) shows the deformable spatial motion path for the robot arm. The four
arms form a circular path around the rotation axis of the deformable
mechanism. Unlike in previous studies [13], proposed robot uses a unique motor
for the fixed propeller, shown in Fig. 5(a) and Section III.C, to maintain the
relative propeller angle in the same direction as the arm.

Figure 5: A demonstration of the automated deformable structure. (a) A fixed
propeller motor. (b) Motion paths in 3D space for the deformable structures.
(c) A display of the deformable forearm. (d) The design and function of the
locking mechanism.

Fig. 5(c) shows the deformable mechanisms of the robot arms. As the motors
begin to rotate, they generate tremendous amounts of torque at the dual-axis
steering gears on the arms, as shown in Fig. 8. The Newton-Euler equation
suggests a dynamic model for the quad-rotor aircraft can be expressed as:

$M_{b}=J_{b}\dot{\omega}_{b}+\omega_{b}\times J_{b}\omega_{b}+M_{g}+M_{d}.$
(3)

The synthesis of gyroscopic torque $M_{g}$ and aerodynamic friction torque
$M_{d}$ can then be represented by:

$M_{g}=\sum_{i=1}^{4}\overrightarrow{\omega_{b}}\times J_{b}\left[0\quad
0(-1)^{i+1}\Omega_{i}\right]^{T},$ (4)

$M_{d}=\operatorname{diag}\left(d_{\phi},d_{\theta},d_{\psi}\right)\dot{\zeta},$
(5)

where $J_{b}=\operatorname{diag}\left(d_{\phi},d_{\theta},d_{\psi}\right)$ is
the moment of inertia for each rotor and $d_{\phi}$, $d_{\theta}$, and
$d_{\psi}$ are the corresponding aerodynamic damping coefficients.

Complex deformable arms can affect the payload and reliability of land-air
robots as each rotor is fixed to the main body of the robot by an independent
arm. The propeller will then generate yaw forces during rotation, relying on
the force generated by the steering gear to maintain the position of arms.
Equations (3)-(5) suggest the moment of inertia for the arm can be expressed
as:

$\begin{array}[]{l}J_{fi}=\frac{m_{i}}{12}L_{fi}^{2}\times\cos(\gamma),\end{array}$
(6) $\begin{array}[]{l}J_{ri}=\frac{m_{i}}{12}L_{ri}^{2},\end{array}$ (7)

where $m_{i}$ is the mass of the arm and $J_{i}$ is the moment of inertia for
the arm around the axis of the steering gear. $L_{fi}$ and $L_{ri}$ indicate
the length of the front arms and the rear arms, respectively. The dynamic
equation for the deformable arm is given by:

$\operatorname{diag}\left(m_{i}\right)\left[\begin{array}[]{c}\ddot{x}\\\
\ddot{y}\\\ \ddot{z}\end{array}\right]=\left[\begin{array}[]{c}F_{xi}\\\
F_{yi}\\\ F_{zi}\end{array}\right],$ (8)

where $F_{xi},F_{yi},\text{ and }F_{zi}(i=1\ldots 4)$ are the decoupling
forces along the x-axis, the y-axis, and the z-axis at the rotating shaft of
each motor, respectively. The rotation matrix decouples these torques.

The reference coordinate is a global coordinate system. When the arm and the
rotor rotate relative to the fuselage coordinate system, the inertia tensor
for the arm will not rotate. This tensor can be expressed as:

$J_{arm,i}=R_{z}\left(\theta_{i}\right)J_{arm}R_{z}\left(\theta_{i}\right)^{T},$
(9)

where $R_{z}$ is the rotation matrix around the z-axis of the global
coordinate system. The yaw angle is $(\theta_{i}),i=(fl,fr,rl,rr)$, and
$fl,fr,rl,rr$ demonstrate front left arm, front rear arm, rear left arm, and
rear right arm, respectively. The moment of inertia for the motor does not
change as it rotates around its z-axis.

Figure 6: Simulated torque on the deformable robot structure. (a) Accelerated
motion in the x-direction. (b) Accelerated motion in the y-direction. (c) The
robot performing a wave-like motion.

###  III-B Locking Mechanism

A physical locking mechanism was included in he design to improve stability
and ensure the safety of the robot. As shown in Fig. 5(d), the machine
maintains 2 lock holes at key positions on the arms. The steering gear is then
fixed on the rack and drives the auto-locking mechanism, while controlling pin
movements by modifying draw bar rotation. These lock holes were denoted
limiter-1 and limiter-2. Limiter-1 prevented the pin from deviating off the
motion tracking course. Limiter-2 contained two key position holes,
corresponding to the expanding and folding modes of the deformable arm. This
pin could be accurately inserted into the holes by rotating the arm. This
structure increased the operating safety of the platform in flight mode and
effectively reduced the working pressure on the dual-axis steering gear.

###  III-C Fixed Propeller Motor

Unsecured propellers can cause collisions with the robot body during
deformations. To address this issue, we designed a motor to ensure the
propeller blades remained oriented in the direction of the arms. The motor
included 2 south and 2 north magnetic poles added to the inner and outer
rotors. As shown in Fig. 5(a), the propellers were fixed to the outer rotors,
which relied on the magnets to quickly reset the propeller and keep it
parallel to the arm at all times. This design prevented excessive energy loss
during propeller rotation.

##  IV Experimental Evaluation

The robustness and stability of the automated foldable arms were evaluated by
simulations and experiments as part of the study. The resulting flexibility
for land-air composite tasks was assessed using a 3D A∗ search algorithm.

###  IV-A Deformable Structure Experiments

This subsection describes the simulated validation of the proposed deformable
structure. We measured and identified the mass, material, spatial
distributions and others parameters in Tab. I for each structure. These data
were used to construct a detailed co-simulation using ADAMS and MATLAB.

####  IV-A1 Motion simulation

Defined the front direction of the initial state of the robot as the two
coordinates’ x-direction, and established the global and robot coordinate
systems with the right-hand rule. The robot then conducted three sets of
experiments to obtain torques for the deformable arm structure. Figs. 6(a) and
(b) show the deformable servos’ torque produced during motion along the x and
y axes. In both sets of experiments, the robot was tilted 15 $degrees$ towards
the x-axes and y-axes and moved along the directions. Fig. 6(c) shows torque
variations while tracking a sinusoidal path, with trajectory and attitude
changes shown in Fig. 7. Torques generated on the front arms were primarily
concentrated at 3000 $N\cdot mm$ and torques on the rear arms were near 1000
$N\cdot mm$. The torque on the deformable structure was also much smaller than
the upper limit for the servos.

Figure 7: Robot motion during sinusoidal reference tracking. (a) A top-view of
the 2D robot trajectory. (b) The robot’s attitude angle.

####  IV-A2 Thrust simulation

In the experiments, motor rotary speed was gradually increased from a
standstill, reaching a maximum speed of 8000 $rad/min$. The torque generated
on the deformable mechanism is shown in Fig. 8. A torque of 1000 $N\cdot mm$
was generated at rest, due to the inclination of the front arm. An increase in
motor speed produced maximum torque on the front and rear arms of 6706 $N\cdot
mm$ and 1711 $N\cdot mm$, respectively, far from the upper limits of the
design 7800 $N\cdot mm$.

Figure 8: The relationship between rotational speed and arm torque.

###  IV-B Multi-modal Locomotion Planning

The flexibility of the robot was evaluated using a rescue scenario in an
abandoned factory, which required the robot to search an optimal hybrid path
to reach a specified target point. A DJI phantom 4 Pro was used to map the
target area. The map was then rasterized, the obstacles were inflated, and the
raw planned path was smoothed to satisfy the dynamic limits of flying and
driving. As shown in Fig. 9, the robot originated from the center of the
factory and attempted to reach three end points along the generation paths.
The robot utilized multiple motion modes to fly over obstacles yet conserve
energy, since the destinations were relatively isolated from the starting
point. A simple rule-based algorithm was implemented based on 2D/3D A-star
path planning [28, 29]. Energy consumption was considered in the process of
heuristic algorithm design, so the robot would tend to drive on the ground
rather than flying. As such, the robot demonstrated its adaptability and
flexibility to various environments.

Figure 9: The experiments demonstrate and verify the superiority and
flexibility of the multi-modal robot in complex scenes. Figure 10: Robot
platform experiments.

###  IV-C Platform Tests

Experiments were conducted in both field and indoor environments under various
scenarios, to verify performance. The corresponding results are shown in Fig.
10 and a video is included in the link in Section I.

####  IV-C1 Mode transition

The first set of tests illustrated the transition process between flying and
driving modes (the red and blue frames in Fig. 10). The robot only required 5
seconds to complete the deformation process. This practical experiment showed
the robot could rapidly fold the arms and switch motion modes. The propellers
could also be well positioned to avoid interference during transition.

####  IV-C2 Driving in different scenarios

In this experiment, the robot performed ground motion tests on flat roads, in
unstructured scenes in the field, and in complex indoor environments. The
robot exhibited a significant advantage in moving speed on structured paved
roads, with a maximum speed reaching over 10 $m/s$. Fig. 10 (see the yellow
frame in Fig. 10) demonstrates highly flexible maneuvering on ground. In
addition, the robot was tested on loose roads and in indoor human activity
scenarios. It could easily pass through grasses, woods, and sandy ground,
crossing a slope of more than 30° without slipping. However, it is restricted
by its tire height, which made it challenging to overcome obstacles more than
100 mm in height (see the green frame in Fig. 10). The robot was also tested
in warehouses, elevators, and dark corridors, in which it performed reliably
and stably (see the orange frame in Fig. 10).

####  IV-C3 Flying tests

Flight tests were conducted outdoors, including takeoff, landing,
acceleration, emergency braking, and path tracking (see the black frame in
Fig. 10). During movement, the robot always maintained a normal posture and
the desired control. In addition, the deformation mechanism operated stably
without triggering an intervention from the locking mechanism. Some shaking
did occur during takeoff, due to ground spoilers. Meanwhile, we measured and
estimated the robot’s average power consumption during hovering and driving.
The result shows that the power consumption of driving (2840 $W$) is 54.75 %
lower than that of flying (6276 $W$).

Figure 11: Comprehensive performance quantitative evaluation of proposed
robot, past work and typical land-air robot.

###  IV-D Evaluation and Comparison

In this subsection, we compare the proposed robot with other related works.
The robot’s size, duration, payload, and movement abilities are all critical
manifestations of its performance [30]. Meanwhile, flight and ground movement
capabilities need to be comprehensively evaluated. As such, we removed unit
inconsistencies between all parameters and evaluated proposed robot with
following indexes to obtain the comprehensive performance of the land-air
robot [12, 6, 2, 31].

####  IV-D1 Comprehensive flight ability

Hovering flight duration $t_{f}$, payload $p$, and maximum tracking speed
(air) $v_{f}$ are critical indicators of the robot’s index of flight
capability $F_{e}$. They are also positively correlated with the robot’s
vertical projection area of flight state. $w_{i}$ indicates the weight of the
corresponding indicator set according to the task requirements, and this paper
uses the same weight as $1$. We use Min-Max normalization method to evaluate
the comprehensive flight capability as follow

$n(x)=\frac{x_{i}-x_{\min}}{x_{\max}-x_{\min}}$ (10)

where $x_{i}(i=1\cdots n)$ express the parameters of robot $i$. $x_{max}$ and
$x_{min}$ are the max and min values of the robots’ same parameters.

$F_{e}\left(t_{f},p,v_{f},S_{land}\right)=\sum_{i=t_{f},p,v_{f}}w_{i}\cdot
n(i)-w_{s}\cdot n\left(S_{\text{land }}\right)$ (11)

####  IV-D2 Comprehensive driving ability

Similarly, the index of comprehensive ground performance $G_{e}$ of the robot
is evaluated by the ground driving mileage $m_{d}$, the obstacle-surmountable
height $h_{o}$, and the maximum tracking speed (land) $v_{d}$.

$G_{e}\left(m_{d},h_{i},v_{d},S_{\text{air
}}\right)=\sum_{i=m_{d},h_{i},v_{d}}w_{i}\cdot n(i)-w_{s}\cdot
n\left(S_{\text{air }}\right)$ (12)

####  IV-D3 Modes switching time

Modes switching time $T_{s}$ express the time that robot switches from driving
mode to flying mode or from flying mode to driving.

####  IV-D4 Comprehensive duration

Comprehensive duration in flying mode $t_{f}$ and driving mode $m_{d}$ are
crucial for land-air robots’ ability to undertake more productive tasks. And
the duration is also strongly related to robots’ weight $G$. Therefore, the
land and air duration time $D_{e}$ is comprehensively considered and used as a
single evaluation index.

$D_{e}\left(m_{d},t_{f}\right)=\sum_{i=m_{d},t_{f}}w_{i}\cdot n(i)-w_{G}\cdot
n(G)$ (13)

The results (Fig. 11) show that our robot got higher scores on $F_{e}$ and
$G_{e}$, thus that it has relatively comprehensive land and air capabilities.
Meanwhile, the robot also got a slightly higher score on $D_{e}$.Although the
mode switching speed of the robot is slow and the energy efficiency is
slightly below than [6, 2], our design conception improve the robot’s
comprehensive abilities obviously. It should be noted that the comparison here
is only a reference for the robot’s capabilities. The parameters’ weights in
the method also need to be adjusted according to the specific task
requirements. For example, [6] can perform more efficient and safe exploration
in narrow and complex scenarios. When evaluating such robots, we can increase
the weights of flight and size. The evaluative method can also introduce
quantitative indicators such as climbing performance and security.

##  V Conclusion

This study proposed a multi-modal land-air robot with automatic deformable
arms for the first time, with the design conception which solving the
compatibility issues for the hybrid land-air robot. By estimated and
calculated the powers and energy, the robot got a suitable battery and power
settings. Meanwhile, a simple yet reliable foldable structure was used to
significantly expand the space for a flight mechanism and increase the thrust.
This approach also reduced the size of the driving mode and allowed the
platform to traverse narrow corridors. The robot was tested in cement
pavement, field and indoor, we knew that the driving consumption is 54.75 %
lower than flying consumption. Compared with other land-air robots by proposed
evaluation methods, our robot offers better comprehensive for flying ability,
as well as improved roadway driving capabilities. The robot had a good balance
of flight and driving comprehensive capabilities.

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