# Atom walking in a traveling-wave light

Wenxi Lai<EMAIL_ADDRESS>School of Applied Science, Beijing Information
Science and Technology University, Beijing 100192, China

###### Abstract

In this paper, we investigate mechanical motion of ultra-slow single atoms
considering each atom is coherently coupled to a traveling-wave light. The
main noise in this system is originated from Doppler broadening due to the
continuous momentum distribution in atom wave packet. Here, it is proved that
the Doppler broadening could be effectively suppressed in strong coupling
regime. Under the coherent coupling, individual neutral atoms periodically
walk in a definite direction. Direction of the motion depends on occupation of
the atom in its two internal states related to the optical transition, since
the atom would be affected by attractive or repulsive forces depending on the
internal states. It is analogous to the electric force acting on negatively or
positively charged particles. We explain them with spin-orbit coupling of
atoms which is hidden in our Hamiltonian. These results have potential
applications for the construction of future atomic devices.

###### pacs:

37.10.Vz, 32.90.+a, 3.75.-b, 42.50.Ct

Control and manipulation of atomic gas have wide applications in many areas.
Recently, it attracts great attentions of researchers, such as atomtronic
transistors Pepino ; Fuechsle ; Daley ; Anderson , atomic circuits Lee ; Chow
; Compagno , atomtronic SQUID Aghamalyan ; Ryu ; Jezek , quantum flux qubits
Aghamalyan2 ; Safaei , atomic batteries Zozulya ; Caliga2 ; Lai ,
interferometries Parazzoli ; Palmer ; Lachmann , atomic clocks Sortais ;
Takamoto , precision measurements Ido ; Chapurin  and topological materials
Liu ; Potirniche .

These applications benefit from cold atom technologies which remove heat of
atomic gas optically and electromagnetically Davis ; Wieman ; Maunz . Atoms in
gases move more freely than they in solids. Therefore, to study behavior of
individual atoms in a cold atom system is very important. It is also
fundamental for the development of single atom devices Walthe ; Huesmann ;
Bianchet ; Hetet . Our present interest is mainly concentrated on mechanical
motion of individual atoms under external fields which is at the hard of
atomic transport and related applications mentioned above.

In free-space with infinite volume, a single atom could be described by a
plane wave with definite momentum. However, in practical materials or devices,
each atom should be described by a wave packet localized in a finite volume
Lachmann ; Gattobigio . The finite volume naturally leads to momentum
uncertainty according to the Heisenberg uncertainty principle, i.e., $\Delta
x\Delta p\sim h$. In atom-light interactions, the momentum uncertainty of wave
packets would cause Doppler broadening to atom transition levels, which
seriously influences the efficiency of control and detection of atoms Uys ;
Chang . Previously, some researchers have proposed intensity correlation
method to remove Doppler broadening in atoms and molecules Merlin .

Figure 1: (Color on line) (a) Schematic illustration of a single atom wave
packet that coupled to a traveling wave of light propagating along the $x$
coordinate. (b) Energy structure of the atom-light coupling in which Doppler
effect induced energy broadening due to the continuous momentum distribution
in wave packet has been considered.

Motivated by the analogous properties between atomtronics and electronics
Pepino ; Fuechsle ; Daley ; Anderson ; Lee ; Chow ; Compagno ; Aghamalyan ;
Ryu ; Jezek , in this paper, we study the motion of single atoms under a
monochromatic traveling-wave light and try to exploit similarities between
charged particles in electric field and neutral atoms in optical fields. To
this end, here we consider ultra-slow atoms and a clean optical field. Much
earlier researches on the mechanical effect of light mostly involve
spontaneous emission or standing wave lasers to achieve large acceleration of
atoms Scully ; Foot . Here, taking advantage of long coherent lifetime in
clock transitions of alkaline-earth atoms Wall , we consider pure coherent
coupling between an atom wave packet and a traveling-wave light, which allow
us to neglect limited lifetime of atoms. Although, Doppler shift induced
noises broaden energy levels of electronic states in atom wave packet, the
Doppler broadening induced noise could be suppressed in strong coupling regime
as described later.

We will show that the coherent couplings make periodic oscillation of atom
internal states transmit to mechanical oscillation of the atom center of mass
under the drive of optical field. As a result, an individual atom could
periodically walk in the light. Similar to charged particles in electric
field, two-level atoms in the ground state and in the excited state move in
opposite directions respectively due to the action of a light field. This
behavior can be understood with the frame of spin-orbit coupling in cold atom
systems Wall ; Livi ; Kolkowitz . Indeed, in the strong coupling and low
momentum regime, Dirac cone could be predicted in energy structure of atom
dressed states, which reflects the topologically protected locking between
atom momentum and internal electronic states. Recently, spin-orbit couplings
and their topological properties in cold atom systems are widely exploited
with Raman transitions Huang ; Meng ; Curiel ; Zhang . In the configurations
of Raman transitions, sub-spaces generated through dark states play important
role for the realization of coherent atom-light couplings Zhang .

The model in which a two-level atom wave packet interacting with a traveling-
wave light is conceptually shown in Fig. 1 (a). The wave packet would be
described by the wave function $|\Psi(t)\rangle=\int
dp\sum_{n}\Psi_{n}(p,t)|n,p\rangle$ which satisfies the Schrödinger equation
$i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=\textbf{H}|\Psi(t)\rangle$
with the total Hamiltonian
$\textbf{H}=\textbf{H}_{e}+\textbf{H}_{K}+\textbf{H}_{I}$. In the Hamiltonian,
the first term $\textbf{H}_{e}=\sum_{n}\hbar\omega_{n}|n\rangle\langle n|$
represents quantized energy of electronic motion around nuclear with
eigenvalue $\hbar\omega_{n}$ and eigenstate $|n\rangle$. The second term
describes kinetic energy of the atom wave packet in the momentum
representation,

$\displaystyle\textbf{H}_{K}=\int dp\frac{p^{2}}{2M}|p\rangle\langle p|,$ (1)

where $M$ is atom mass and $p$ represents momentum of the atom center of mass.
As an example of the two-level system, we consider Ytterbium (173Yb) atom with
mass number $172.938208$ Sansonetti . It has a clock transition 1S0
$\rightarrow$ 3P0 that characterized by long lifetime ($\sim$ $20$ s) under
the resonant wave length $578$ nm Livi . The states 1S0 and 3P0 are
corresponding to the ground $n=0$ and excited state $n=1$, respectively. The
traveling-wave light is described by the electric wave function
$\vec{\textbf{E}}(\textbf{x},t)=\vec{e}_{z}E_{0}cos(\omega t-k\textbf{x})$ in
which x has been regarded as an operator. Then, the interaction term of the
atom-light coupling could be given by,

$\displaystyle\textbf{H}_{I}=-\frac{\hbar}{2}\int dp(\Omega e^{-i\omega
t}|1,p+\hbar k\rangle\langle 0,p|+H.c.),$ (2)

where the Rabi frequency is $\Omega=\mu E_{0}/\hbar$ with dipole moment of the
atom $\mu=\langle 0|e\textbf{z}|1\rangle=|\mu|e^{i\phi}$ Scully . The initial
phase would be taken to be $\phi=0$ for convenience of discussion. In
derivation of Eq.(2), the relation of momentum displacement operator $e^{\pm
ik\textbf{x}}|p\rangle=|p\pm\hbar k\rangle$ has been used Meystre .

To solve the wave function $|\Psi(t)\rangle$, the original Schrödinger
equation can be written into interaction picture
$i\hbar\frac{\partial}{\partial
t}|\varphi(t)\rangle=\textbf{V}|\varphi(t)\rangle$ with the wave function
$|\varphi(t)\rangle=e^{i\textbf{H}_{0}t/\hbar}|\Psi(t)\rangle$ and time
independent Hamiltonian
$\textbf{V}=e^{i\textbf{H}_{0}t/\hbar}(\textbf{H}-\textbf{H}_{0})e^{-i\textbf{H}_{0}t/\hbar}$,
where the unitary transformation is generated from the matrix
$\textbf{H}_{0}/\hbar=\omega_{0}|0\rangle\langle
0|+(\omega_{0}+\omega)|1\rangle\langle 1|$. The wave function
$|\varphi(t)\rangle$ can be expanded with probability distribution functions
as $|\varphi(t)\rangle=\int dp\sum_{n}\varphi_{n}(p,t)|n,p\rangle$. For a
definite momentum $p$, the Schrödinger equation in interaction picture would
be reduced into a differential equation of $2\times 2$ matrix in the sub space
of dressed states $\\{|0,p\rangle,|1,p+\hbar k\rangle\\}$ as

$\displaystyle i\left[\begin{array}[]{cccc}\dot{\varphi}_{0,p}(t)\\\
\dot{\varphi}_{1,p+\hbar
k}(t)\end{array}\right]=\left[\begin{array}[]{cccc}\omega_{p}&-\frac{\Omega}{2}\\\
-\frac{\Omega}{2}&\Delta+\omega_{p+\hbar
k}\end{array}\right]\left[\begin{array}[]{cccc}\varphi_{0,p}(t)\\\
\varphi_{1,p+\hbar k}(t)\end{array}\right],$ (9)

where $\omega_{p}=\frac{p^{2}}{2M\hbar}$, $\omega_{p+\hbar k}=\frac{(p+\hbar
k)^{2}}{2M\hbar}$ and the detuning $\Delta=\omega_{1}-\omega_{0}-\omega$.
Eq.(9) can be easily diagonalized and the eigenfrequencies are
$W_{0,1}=\frac{1}{2}(\Delta+\omega_{p+\hbar k}+\omega_{p}\mp\Sigma)$, in which
the energy gap is $\Sigma=\sqrt{\delta^{2}+\Omega^{2}}$ and
$\delta=\Delta+\omega_{p+\hbar k}-\omega_{p}$. Solutions of Eq.(9) could be
written as

$\displaystyle\varphi_{0}(p,t)=(A_{0}e^{i\frac{\Sigma
t}{2}}+B_{0}e^{-i\frac{\Sigma t}{2}})e^{-\frac{i}{2}(\Delta+\omega_{p+\hbar
k}+\omega_{p})t},$ (10) $\displaystyle\varphi_{1}(p+\hbar
k,t)=(A_{1}e^{i\frac{\Sigma t}{2}}+B_{1}e^{-i\frac{\Sigma
t}{2}})e^{-\frac{i}{2}(\Delta+\omega_{p+\hbar k}+\omega_{p})t},$ (11)

where the coefficients are

$\displaystyle
A_{0}=\frac{1}{2\Sigma}[(\Sigma+\delta)\varphi_{0}(p,0)+\Omega\varphi_{1}(p+\hbar
k,0)],$ $\displaystyle
B_{0}=\frac{1}{2\Sigma}[(\Sigma-\delta)\varphi_{0}(p,0)-\Omega\varphi_{1}(p+\hbar
k,0)],$ $\displaystyle
A_{1}=\frac{1}{2\Sigma}[(\Sigma-\delta)\varphi_{1}(p+\hbar
k,0)+\Omega\varphi_{0}(p,0)]$

and

$\displaystyle B_{1}=\frac{1}{2\Sigma}[(\Sigma+\delta)\varphi_{1}(p+\hbar
k,0)-\Omega\varphi_{0}(p,0)].$

Before the atom interacts with light, at $t=0$, we suppose the initial wave
packet could be expanded with the eigenstates $|n,p\rangle$ of free evolution
system, namely $|\varphi(0)\rangle=\int
dp\sum_{n}\varphi_{n}(p,0)|n,p\rangle$. In the initial state, each probability
amplitude is supposed to be a Gaussian function
$\varphi_{n}(p,0)=\frac{C_{n}}{\pi^{1/4}\sqrt{\Pi}}e^{-(p-p_{c})^{2}/(2\Pi^{2})}$
with the coefficient $C_{n}$ for atom electronic state $|n\rangle$. $p_{c}$
represents center of the wave packet and $\Pi$ indicates width of the wave
packet in momentum space.

Figure 2: (Color on line) (a) $\omega_{p}$ and $\Delta+\omega_{p+k}$ are
energy levels in the interaction picture with distance $\delta$. The dashed
lines denote energy structure of $\Delta$, $\omega_{B}$ and $\omega_{D}$. (b)
Comparison between eigenfrequencies and the original transition levels.
$\Sigma$ is distance between two eigenfrequencies. (c) Wave packet
distribution in momentum space at initial time $t=0$. (d) Wave packet
distribution at time $t=T/2$. (e) Displacement $\langle x(t)\rangle$ and
kinetic energy $\langle H_{k}\rangle$ of the atom as a function of time
corresponding to the wave packet distribution in (e) and (d). The parameters
here are $\Delta=0$, $\Omega=2\pi\times 2$ kHz, $\Pi=\hbar k$ and $p_{c}=0$.

In the interaction picture, the modified detuning
$\delta=\Delta+\omega_{p+\hbar k}-\omega_{p}$ can be rewritten into the form
$\delta=\Delta+\omega_{B}+\omega_{D}$ which is sum of the detuning $\Delta$,
the back action (recoil energy) shift $\omega_{B}=\frac{\hbar k^{2}}{2M}$ and
the Doppler shift $\omega_{D}=\frac{Pk}{M}$ as illustrated in Fig. 2 (a).
Corresponding back action energy is denoted as
$\varepsilon_{B}=\hbar\omega_{B}$. Since the Doppler shift is proportional to
momentum of the wave packet which has continuous distribution in momentum
space, the optical transition levels would have the Doppler broadening
$\Delta\omega_{D}=\frac{2\Pi k}{M}$ related to the width $\Pi$ of the wave
packet (see Fig. 1 (b)). In the modified detuning $\delta$, the back action
shift can be removed by taking a detuning $\Delta=-\omega_{B}$. However, the
Doppler shift $\omega_{D}$ can’t be removed simply by choosing a detuning
$\Delta$. Because the Doppler shift can be continues values within the Doppler
broadening $\Delta\omega_{D}=\frac{2\Pi k}{M}$ which is proportional to width
$\Pi$ of the wave packet.

From the point of view of eigenfrequencies shown in Fig. 2 (b), the energy gap
$\Sigma=\sqrt{\delta^{2}+\Omega^{2}}$ has two regimes, the weak coupling
regime $\delta\gg\Omega$ and the strong coupling regime $\delta\ll\Omega$. In
the weak coupling regime, for example, we take $\Omega=2\pi\times 1$ kHz in
Fig. 2 (c), (d) and (e). Initially, internal state of the atom is in the
ground state $C_{0}=1$ and corresponding wave packet is supposed to be a
Gaussian function as plotted in Fig. 2 (c). After the atom-light interaction
begins, as illustrated in Fig. 2 (d), momentum distribution of the wave packet
would change along with the atom transition from the ground state to the
excited state. It induces change in the averaged momentum of the atom and so
on the corresponding kinetic energy as shown in Fig. 2 (e). However, the
coupling strength $\Omega$ is very small in the weak coupling regime. As a
result, the modified Rabi frequency $\Sigma=\sqrt{\delta^{2}+\Omega^{2}}$
remarkably depends on momentum $p$ due to $\delta$ is related to the momentum.
After a longer time $t$, the momentum distribution in the excited state
$\varphi_{1}(p,t)$ would spread more widely. Then, the wave packet could be
regarded as many parts with different possible values of momentum. All these
parts of the wave packet involved in the resonant interaction would oscillate
with different Rabi frequencies $\Sigma$. Therefore, the averaged momentum of
the atom (reflected by the kinetic energy in Fig. 2 (e)) increases in a
definite direction at the beginning and then tends to decay into a stationary
value because of the counterbalance between those resonant transitions.
Therefore, in the weak coupling regime, atom waking is hard to be observed
although displacement of the atom increases.

Figure 3: (Color on line) (a) The energy gap $\Sigma$ as a function of the
Rabi frequency $\Omega$ and atom momentum $p$. (b) Illustration of strong
coupling and weak coupling regimes based on Eq.(13) with the detuning
$\Delta=-\omega_{B}$. Displacement, kinetic energy of the atom as a function
of time and corresponding change in the probability distribution functions
with the initial state $C_{0}=1$ in (c)-(e) and with the initial state
$C_{1}=1$ in (f)-(h). The inset figure in (f) shows optical forces acting on
the atom for different initial states. The common parameters are
$\Omega=2\pi\times 1$ MHz, $\Pi=\hbar k$, $p_{c}=0$ and $\Delta=0$.

Now, let us analyze how to identify the weak and strong coupling regimes and
discuss suppression of the Doppler broadening in the latter situation.
Substituting $\delta=\Delta+\omega_{B}+\omega_{D}$ into the relation
$\Sigma=\sqrt{\delta^{2}+\Omega^{2}}$, one can deduce the following equation
directly,

$\displaystyle\frac{(\frac{p}{\hbar
k}+\frac{\Delta}{2\omega_{B}}+\frac{1}{2})^{2}}{(\frac{\Sigma}{2\omega_{B}})^{2}}+\frac{\Omega^{2}}{\Sigma^{2}}=1.$
(12)

This elliptic equation could be used to analyze the two different coupling
regimes. Fig. 3 (a) reveals the weak and strong coupling regimes are relative.
For larger momentum range $-\Pi<p+\frac{\Delta}{2\omega_{B}}+\frac{1}{2}<\Pi$,
a larger Rabi frequency $\Omega$ is required to satisfy the strong coupling
regime where $\delta\ll\Omega$ and $\Sigma\approx\Omega$. In this case, the
Doppler shift included in $\delta$ becomes negligible small (see Fig. 4 (a)).
This is important for the following result of atom walking. To find a
quantitative relation between the width $\Pi$ of a wave packet and the Rabi
frequency $\Omega$ for the strong coupling, an infinitely small parameter
$\epsilon$ could be defined that $\Sigma=\Omega(1+\epsilon)$. It guarantees
that $\Sigma$ is bigger than $\Omega$ but can be close to $\Omega$ infinitely.
Then considering Eq.(12) the criteria of the Rabi frequency $\Omega$ that
satisfies the strong coupling regime can be found from the equation,

$\displaystyle\Omega=\frac{\sqrt{2}\omega_{B}}{\sqrt{\epsilon}}(\frac{\Pi}{\hbar
k}+\frac{\Delta}{2\omega_{B}}+\frac{1}{2}).$ (13)

In Fig. 3 (b), range of the Rabi frequency $\Omega$ given by Eq.(13) for
different $\epsilon$ has been plotted for the detuning $\Delta=-\omega_{B}$.
Obviously, when $\epsilon$ is small enough the strong coupling should be
realized.

In Fig. 3 (c-h), the Rabi frequency $\Omega=2\pi\times 1$ MHz and the width
$\Pi=\hbar k$ satisfy Eq.(13) with parameter $\epsilon=0.001$ as revealed in
Fig. 3(b). In this case, almost fully periodic motions of the atom appear in
Fig. 3 (c) and (f). Expectation value of the wave packet momentum has been
calculated through the formula $\langle
p(t)\rangle=\langle\varphi(t)|p|\varphi(t)\rangle$. In the strong coupling
regime $\delta\ll\Omega$, it has the simple form,

$\displaystyle\langle p(t)\rangle=p_{c}+(C_{0}^{2}-C_{1}^{2})\hbar
ksin^{2}(\frac{\Omega t}{2}).$ (14)

Consequently, kinetic energy of the atom could be written as $\langle
H_{K}\rangle=\frac{\langle p(t)\rangle^{2}}{2M}$. From the momentum $\langle
p(t)\rangle$ one can obtain displacement of the atom,

$\displaystyle\langle x(t)\rangle=\langle x(0)\rangle+\frac{2p_{c}\upsilon
t}{\hbar k}+(C_{0}^{2}-C_{1}^{2})\upsilon(t-\frac{sin(\frac{2\pi
t}{T})}{2\pi/T}),$ (15)

where the velocity $\upsilon=\frac{\Lambda}{T}$ is ratio of the step length
$\Lambda=\frac{\pi\hbar k}{M\Omega}$ and the periodicity
$T=\frac{2\pi}{\Omega}$. The step length $\Lambda$ of atom walking is related
to momentum $\hbar k$ of a photon, atom mass $M$ and Rabi frequency. It can be
tuned arbitrarily with the Rabi frequency for a given atom. Change in the
momentum gives rise to optical force acting on the atom,

$\displaystyle\langle F(t)\rangle=(C_{0}^{2}-C_{1}^{2})F_{0}sin(\Omega t),$
(16)

where, $F_{0}=\frac{1}{2}\hbar k\Omega$ is the force amplitude.

Figure 4: (a) The principle of strong coupling in the system. (b) Illustration
of spin-orbit coupling during the atom-photon interactions. (c) Walking of
atom in the opposite directions due its different initial internal states
$|0\rangle$ and $|1\rangle$.

Eq.(15) reveals that atoms in the ground state $|0\rangle$ and the excited
state $|1\rangle$, which are initially static, would move in opposite
directions respectively under the action of a propagating light. This
phenomenon is analogous to the positively and negatively charged particles
move in opposite directions respectively under the action of an electric
field. It is originated from the oscillating light forces have phase
difference of $\pi$ with respect to the two internal states of atom (see Fig.
3 (f)). Periodic motion of the atom in opposite directions shown in Fig. 3 (c)
and (f) could be understood with the model of stimulated absorption and
stimulated emission in Fig. 4 (b). Different from the selected transition of
wave packet in Fig. 2 (d), wave packet in the strong coupling regime entirety
transits from one state to the other state synchronously as illustrated in
Fig. 3 (d), (e), (g) and (h). This synchronous transitions of wave packet lead
to periodic walking of the atom with step length $\Lambda$ which is
schematically shown in Fig. 4 (c). Kinetic energy of the wave packet is
oscillating between $0$ and value of the back action energy as plotted in Fig.
3 (c) and (f). It reveals that kinetic energy of the atom comes from the back
action effect.

Now, in the following three paragraphs, let us summaries relations between the
atom-light coupling, the Doppler shift and atom walking. In the model, it is
clear that the optical field has a very narrow frequency with negligible
width. At the same, the atom wave packet has continuous spectrum of momentum
and width of the spectrum can not be neglected. It means the atom moves at
continuously different velocities. Therefore, when the atom encounters a
monochromatic light, Doppler shifts with continuous different values occur to
their energy levels. The atom would observe the light with wide frequency
spectrum, or equivalently, the light couples to the atom with a remarkable
level width. This is the Doppler broadening mentioned above. In this case, the
atom is equivalently interacting with a noisy environment which is
characterized by a continuous energy spectrum. This is why the atom damping
occurs in Fig. 2 (e).

Without considering the Doppler shift, when a two-level atom resonantly
interacts with a monochromatic light, the atom would oscillate with the
definite Rabi frequency $\Omega$ Scully ; Meystre . Considering Doppler shift,
but supposing the atom has only definite momentum $p$, resonant atom-light
coupling would enable the atom oscillate with the new Rabi frequency $\Sigma$.
The Doppler shift is included in the modified frequency $\Sigma$. When both
Doppler shift and momentum spread of the atom wave packet are considered, the
new Rabi frequency $\Sigma$ has continuously different values related to
momentum $p$. It reveals the atom (or wave function) would oscillate with
different Rabi frequency $\Sigma$. What we see in classical world is the
decays happen to the oscillation of atom occupation and atom walking.

However, the decays would be removed when we takes strong coupling. There are
two regimes for the atom-light interaction. The situation shown in Fig. 2
(c)-(e) is in the weak coupling regime, the situation shown in Fig. 3 (c)-(h)
is in the strong coupling regime. In Fig. 2 (c), atom is initially ($t=0$) set
in the ground state. After half periodicity ($t=T/2$), only part of the
momentum spectrum is changed. It indicates the atom oscillates at different
frequencies as mentioned above. In other words, the momentum oscillation is
not synchronized. In the strong coupling regime, all continuously distributed
momentums oscillate at the same steps as illustrated in Fig. 3 (d) and (e) or
(g) and (h). It is because, in the strong coupling regime, Rabi frequency is
set to be much larger than any Doppler shift, $\Omega\gg\omega_{D}$, so that
one reaches the approximate relation $\Sigma\approx\Omega$. In this case, the
new Rabi frequency $\Sigma$ almost does not rely on the momentum distribution.
As a result, the coherent oscillation of atom occupation and momentum center
of mass would be observed.

Figure 5: (Color on line) Atom walking along the positive direction of $x$
coordinate in real space, in which the atom is initially in its ground state.
The corresponding parameters are $\Omega=2\pi\times 1$ MHz, $\Pi=50\hbar k$,
$p_{c}=0$ and $\Delta=0$.

Next, we show atom walking in position-space. Position-space wave function of
the wave packet could be written as
$|\psi(x,t)\rangle=\psi_{0}(x,t)|0\rangle+\psi_{1}(x,t)|1\rangle$, where the
amplitudes are obtained through the Fourier transformation $\psi_{0}(x,t)=\int
dp\varphi_{0}(p,t)\langle x|p\rangle$ and $\psi_{1}(x,t)=\int
dp\varphi_{1}(p,t)\langle x|p\rangle$. The amplitudes $\varphi_{0}(p,t)$ and
$\varphi_{1}(p,t)$ in momentum space are given in Eqs.(10) and (11). With
simple derivations one can find that Fourier transformation of the Gaussian
wave function is also a Gaussian function. Fig. 5 show the wave packet in
position space moves periodically with respect to transition of the atom
between the two internal states $|0\rangle$ and $|1\rangle$. Indeed, the wave
packet modes corresponding to the ground state (pink) and the excited state
(blue) appear alternatively with the periodicity $T$ along the $x$ coordinate.
If the atom is initially set in its excited state, it would walk in the
opposite direction as illustrated in Fig. 6. The wave function evolutions in
Fig. 5 and Fig. 6 are equivalent to the lines of displacement $\langle
x(t)\rangle$ plotted in Fig. 3 (c) and (f), respectively.

Figure 6: (Color on line) Atom walking along the negative direction of $x$
coordinate in real space, in which the atom is initially in its excited state.
The corresponding parameters are $\Omega=2\pi\times 1$ MHz, $\Pi=50\hbar k$,
$p_{c}=0$ and $\Delta=0$.

The walking process is similar to a longitudinal wave whose wave length is
$\Lambda$ and frequency is $f=T^{-1}$. Through the concept of longitudinal
wave, we reproduce the speed of atom walking given in Eq.(15), i.e.,
$\upsilon=f\Lambda$. Actually, it leads to a Rabi frequency independent result
$\upsilon=\frac{\hbar k}{2M}$. The speed $\upsilon$ represents average
velocity of the atom walking which is exactly half of the recoil velocity
obtained in the theory of atom cooling Foot . For the transition 1S0
$\rightarrow$ 3P0 of the 173Yb atom, the step length can be estimated to be
$\Lambda=198.47$ nm, $19.85$ nm and $1.98$ nm for different values of the Rabi
frequency $\Omega=2\pi\times 0.01$ MHz, $2\pi\times 0.1$ MHz and $2\pi\times
1$ MHz, respectively. In this case, speed of the wave packet should be
$\upsilon=1.98\times 10^{6}$ nm/s.

The formula of velocity $\frac{\hbar k}{2M}$ reveals that atoms of different
chemical elements or isotopes have different velocities along the direction of
light propagation. The velocity which depends on mass and transition frequency
of an atom could be used to separate and purify materials in principle.
Details of the design for material purifications can be found in Ref. Lai2 .
It is should be noted that we just considered single atom process in the
present configuration. For many-atom systems, our model is suitable to
describe low density atomic gas where atom-atom interaction becomes negligible
small Gattobigio2 ; Couvert .

Near the end, to further understand feature of the atom walking, let us
discuss spin-orbit coupling effect and related properties included in Eq.(9).
We move the origin of momentum coordinate to be $p=p_{x}-\frac{\hbar k}{2}$ in
momentum space. In this case, one will find that Eq.(9) becomes,

$\displaystyle i\left[\begin{array}[]{cccc}\dot{\varphi}_{0,p_{x}-\frac{\hbar
k}{2}}\\\ \dot{\varphi}_{1,p_{x}+\frac{\hbar
k}{2}}\end{array}\right]=\left[\begin{array}[]{cccc}\omega_{p_{x}-\frac{\hbar
k}{2}}&-\frac{\Omega}{2}\\\ -\frac{\Omega}{2}&\omega_{p_{x}+\frac{\hbar
k}{2}}\end{array}\right]\left[\begin{array}[]{cccc}\varphi_{0,p_{x}-\frac{\hbar
k}{2}}\\\ \varphi_{1,p_{x}+\frac{\hbar k}{2}}\end{array}\right],$ (23)

where, we have set the detuning $\Delta=0$ for symmetry of this equation. The
Hamiltonian of the Schrödinger equation (23) could be written in the form,

$\displaystyle V(p_{x})=-\upsilon p_{x}\sigma_{z}+\alpha\sigma_{x}+\beta I,$
(24)

where $\upsilon=\frac{\hbar k}{2M}$ denotes the mean velocity of an atom
exactly given in Eq.(15). The other two coefficients are
$\alpha=-\frac{\hbar\Omega}{2}$ and
$\beta=\frac{p_{x}^{2}}{2M}+\frac{\hbar^{2}k^{2}}{8M}$, respectively.
Additionally, the identity matrix and the Pauli matrices represent pseudospin
of the atom and they are defined to be

$\displaystyle I=\left[\begin{array}[]{cccc}1&0\\\
0&1\end{array}\right],\sigma_{x}=\left[\begin{array}[]{cccc}0&1\\\
1&0\end{array}\right],\sigma_{z}=\left[\begin{array}[]{cccc}1&0\\\
0&-1\end{array}\right].$ (31)

The first term $-\upsilon p_{x}\sigma_{z}$ on the right side of Eq. (24)
represents spin-orbit coupling of our system. When momentum is close to zero,
$p_{x}\rightarrow 0$, the higher term $p_{x}^{2}$ decreases faster than the
lower term $p_{x}$ in Eq. (24). Then, this low momentum area enables us to
write the Hamiltonian as

$\displaystyle V(p_{x})\approx-\upsilon
p_{x}\sigma_{z}+\alpha\sigma_{x}+\frac{\hbar^{2}k^{2}}{8M}I,$ (32)

which is linearly proportional to the momentum $p_{x}$ and called Dirac like
Hamiltonian. The spin-orbit coupling and the Dirac cone could be related to
topological edge states as exploited in many recent researches on atom gases
Huang ; Meng ; Curiel ; Zhang ; Cooper . The Dirac cone in our system appears
in Fig. 2 (b) near the point $p=-\hbar k/2$ ($p_{x}=0$). The area near this
position ($p_{x}=0$) is namely the strong coupling regime. In this regime,
atoms move in different directions just depending on their internal states
which is similar to the particle behavior in topological edge states
Parappurath . The second term $\alpha\sigma_{x}$ in Eq. (32) reflects the
energy corresponding to Zeeman field which induces Zeeman splitting and opens
degeneracy at the Dirac point Meng ; Curiel . Splitting of the Dirac point is
shown in Fig. 2 (b). The last term is back action energy as mentioned above.

In conclusions, internal state dependent periodic motion of a single atom wave
packet under the action of a traveling-wave light is demonstrated
theoretically with analytic solutions of Schrödinger equation. There are two
key conditions for observation of the atom walking. One of the conditions is
that atoms should have long lifetime optical transition levels, the lifetime
should be at least longer than one periodicity of the atom walking. The other
condition comes from the strong coupling of atom-light interaction, which
suppresses the noise originated from Doppler broadening. The atom walking
mainly has following properties. First, step length and time periodicity of
the walking can be tuned arbitrarily by tuning the Rabi frequency. Second,
direction of the atom walking is determined by initial state of the atom
internal state. Therefore, the atom walking here is deterministic, it is
different from the quantum random walk. Third, the atom walking has the
properties of spin-orbit coupling. Forth, the highest kinetic energy of the
atom during the walking process is identical to the back action energy. We
believe the results in this work are important for the development of
atomtronic technology.

###### Acknowledgements.

This work was supported by R & D Program of Beijing Municipal Education
Commission (KM202011232017).

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