# Characterizing ultra-narrow momentum of atoms by standing-wave light-pulse
sequences

Shuyu Zhou<EMAIL_ADDRESS>Shanghai Institute of Optics and Fine Mechanics,
Chinese Academy of Sciences, Shanghai 201800, China  Center of Materials
Science and Optoelectronics Engineering, University of Chinese Academy of
Sciences, Beijing 100049, China  Chen Chen  Shanghai Institute of Optics and
Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China  Center of
Materials Science and Optoelectronics Engineering, University of Chinese
Academy of Sciences, Beijing 100049, China  Bowen Xu  Shanghai Institute of
Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
Center of Materials Science and Optoelectronics Engineering, University of
Chinese Academy of Sciences, Beijing 100049, China  Angang Liang  Shanghai
Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai
201800, China  Center of Materials Science and Optoelectronics Engineering,
University of Chinese Academy of Sciences, Beijing 100049, China  Ying Wang
School of Science, Jiangsu University of Science and Technology, Zhenjiang
212003, China  Bin Wang  Shanghai Institute of Optics and Fine Mechanics,
Chinese Academy of Sciences, Shanghai 201800, China

###### Abstract

We propose a method to characterize the ultra-narrow momentum distribution of
atomic gases by employing a standing-wave light-pulse sequences beam-splitter.
The mechanism of beam splitting is analyzed in detail, and the influence of a
finite-width momentum distribution on the population of each diffraction order
is given. The temperature of ultracold atomic gases can be calibrated by
measuring the ratio of population in different diffraction orders after double
standing-wave light-pulses. We obtain analytical expressions for two typical
cases, and demonstrate phase space evolution in the whole process by using the
Wigner function. This method is valid for both classical atomic gas and Bose-
Einstein condensates, and it is suited for temperature measurement on the
space ultra-cold atomic physics platform, in which the ultra-narrow momentum
distribution of atomic gas is on the order of $100pK$ or even lower.

ultracold atomic gases, temperature measurement, one-dimensional optical
lattice

##  I I. Introduction

Atomic samples with ultralow kinetic energy provide a favorable experimental
condition for basic research and high-tech applications such as testing
quantum mechanics at macroscopic scales [1, 2], preparing the exotic phases of
matter [3, 4, 5, 6], exploring few-body physics [7, 8, 9], and developing
high-precision atomic interferometer [10, 11]. In principle, ultracold atomic
samples with kinetic energy of $100pK$ or less can be obtained by adiabatic
decompression [12, 13] or delta kick cooling [14, 15, 16]. In the microgravity
environment of the international space station, the kinetic energy equivalent
temperature of Bose-Einstein condensates (BEC) was reduced to a minimum of
about $230pK$ after adiabatic expansion [17]. Recently, by combining an
interaction-driven quadrupole-mode excitation of a BEC with a magnetic lens,
the total internal kinetic energy of a BEC in three dimensions was lowered to
$38pK$ [18].
Common approaches of measuring temperatures or momentum distributions of
ultracold atoms include the time-of-flight (TOF) method [17, 18] and Bragg
spectroscopy [19, 20]. The atomic knife-edge method has also been used for
measuring narrow momentum distributions [21]. Nevertheless, when we consider a
realistic ultracold atomic experimental device, which is limited by the space
station, we will find all above methods have their own weaknesses to measure
such low temperature. We discuss this problem in detail in Section V.
The finite momentum width of atoms will affect the contrast of the
interference fringes [22, 23]. Based on the two-photon Raman transition
method, temperature measurements have been demonstrated for a cold atom
ensemble [24] and a single atom [25]. However, this scheme also requires
precise control of the laser phase and intensity, and because Raman
transitions involve different internal states, it is susceptible to the
effects of magnetic field inhomogeneity and fluctuation.
In this paper, we suggest using the method of standing-wave light-pulse
sequences [26, 27] to measure the temperature of ultracold atoms or the
kinetic energy equivalent temperature of BECs. The method we propose is well
suited for ultracold atoms or BECs with kinetic energy below $1nK$, and can
even be lower than $100pK$. This approach is easy to realize since it is based
on the common one-dimensional optical lattice and only needs appropriate
timing control.
This paper is organized as follows. In Sec. II, we discuss the mechanism of
double-pulse beam splitting and establish the theoretical framework. Section
III considers the effect of a finite width momentum distribution and delivers
the analytical expressions of the population of diffraction orders verse the
momentum width under two typical cases. The phase space evolution in the beam-
splitting process is presented in Secs. IV., and the physical image of the
diffractive order intensity change caused by the momentum broadening is
discussed. In sec.V, we compare our proposed method with other methods for
measuring temperature of ultracold atoms, discuss the effects due to
interaction between atoms and vibrations in the space station, and illustrate
the advantage of shorter wavelengths of optical lattice laser in experimental
implementation. Finally, a brief summary is given in Sec. VI.

##  II II. Mechanism of double-pulse beam-splitting

In this section we discuss the mechanism of double-pulse beam splitting. The
potential of the one-dimensional optical lattice is

$V\left(x\right)=2\hbar\Omega_{m}\cos^{2}\left({k_{0}x}\right)=sE_{r}\cos^{2}\left({k_{0}x}\right).$
(1)

where $E_{r}{\rm{=}}\hbar\omega_{r}$ is the recoil energy and
$\omega_{r}=\frac{{\hbar k_{0}^{2}}}{{2m}}$ is the angular frequency
corresponding to the photon recoil energy, $m$ is the mass of the atom and
$k_{0}$ is the mode of the wave vector of the light field.
The method of beam-splitting is described as two standing wave square pulses
of amplitude $\Omega_{m}=2\sqrt{2}\omega_{r}$, both of pulse width
$\tau_{1}=\frac{{\left({2n_{1}+1}\right)\pi}}{{4\sqrt{2}\omega_{r}}}$ and the
interval time $\tau_{2}=\frac{{\left({2n_{2}+1}\right)\pi}}{{4\omega_{r}}}$,
as shown in Fig.1(a) [26].

Figure 1: Schematic diagram of double standing wave pulse beam splitting. (a)
The sequence of the two-pulse. (b) The Bloch sphere interpretation of the
matter-wave evolution. (a)(b) is similar to Fig. 1 in Ref. [26]. (c) Evolution
of the state vector in the Bloch sphere when the trap depth of the optical
lattice is slightly higher than the optimal value.

In Ref. [26], the motion of the atom was described by the coupled Raman-Nath
equations (RNEs) and the RNEs was cut off to include only the $n=0$ and $\pm
1$ diffraction orders. The $n=\pm 1$ diffraction order forms a symmetric state
$\left|{C_{+}}\right\rangle=\frac{1}{{\sqrt{2}}}\left({\left|{C_{+2}}\right\rangle+\left|{C_{-2}}\right\rangle}\right)]$,
which together with the zero-momentum state $\left|{C_{0}}\right\rangle$ forms
a two-state system. The evolution process of the double-pulse beam splitting
can be expressed intuitively by the rotation of the state vector on a Bloch
sphere, as shown in Fig.1(b). On the Bloch sphere, the state vector rotates
around the Rabi vector $\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\Omega}}{\rm{=}}\left({\sqrt{2}\Omega_{m},0,4\omega_{r}}\right)]$,
which is caused by the standing wave light pulse. The state vector rotating
from the $-Z$ axis to the $X$ axis or from the $-X$ axis to the $Z$ axis is
equivalent to a Hadmard gate operation, while the rotation of the state vector
from the $X$ axis to the $-X$ axis in the free evolutionary time is equivalent
to the function of a $\pi$ phase gate. It is well known that the effect of
sequential operation process of a Hadmard gate, a $\pi$ phase gate and a
Hadmard gate is equivalent to the non-gate operation, or bit reversal. Thus,
the atoms are initially in the zero-momentum state
$\left|{C_{0}}\right\rangle$, and after this sequence of operations will
evolve to the symmetric state
$\left|{C_{+}}\right\rangle=\frac{1}{{\sqrt{2}}}\left({\left|{C_{+2}}\right\rangle+\left|{C_{-2}}\right\rangle}\right)]$.
The above physical picture is clear, but there are two remaining questions.
The first question is: can we regard the light pulse as doing phase imprinting
on atoms? The reason we ask this question is that in this scheme, the
individual pulses are only tens of microseconds long, and the movement of
atoms during this time can probably be ignored. Usually, in this case, we
could think of the light pulse as doing phase imprinting on atoms.
The shortest time between two standing wave pulses is

$\tau_{2}=\frac{\pi}{{4\omega_{r}}}=\frac{\pi}{4}\frac{{2m}}{{\hbar
k_{0}^{2}}}=\frac{m}{{2\hbar
k_{0}}}\frac{\pi}{{k_{0}}}=\frac{d}{{v_{recoil}}},$ (2)

here
$d{{=\lambda}\mathord{\left/{\vphantom{{=\lambda}2}}\right.\kern-1.2pt}2}$ is
the grating constant. Interestly, the Talbot length for the secondary Talbot
image is $z_{T}=\frac{{d^{2}}}{\lambda}$, so the time it takes for light to
travel this distance is

$\tau_{T}=\frac{{z_{T}}}{c}=\frac{1}{c}\frac{d}{\lambda}d=\frac{1}{{c\sin\theta_{1}}}d=\frac{d}{{v_{recoil}}}.$
(3)

Therefore, the minimum propagation time in the double-pulse beam splitting is
the same expression of as the time required for light to propagate from the
grating to the Talbot length for the secondary Talbot image position [28, 29].
Can we consider the double-pulse beam splitting as doing phase imprinting
twice and the traveling distance between the two pulses is the Talbot length
for the secondary Talbot image? We will prove that this perception is
incorrect.
If the initial wavefront is

$A\left(x\right)=\sum\limits_{n=-\infty}^{\infty}{C_{n}}e^{\frac{{i2\pi
nx}}{d}},$ (4)

here $d$ is the grating constant. After traveling the distance z, the wave
front is [29]

$E\left(x\right)\propto\sum\limits_{n=-\infty}^{\infty}{C_{n}}e^{-i\frac{{\pi\lambda
n^{2}z}}{{d^{2}}}}e^{i\frac{{2\pi nx}}{d}}.$ (5)

We assume that we start with a plane wave of unit amplitude, and add a phase-
only modulation $e^{iz\cos\frac{{2\pi x}}{d}}$, then the output wavefront is
the same as the amplitude transmittance

$T\left(x\right)=\exp\left[{iz\cos\frac{{2\pi
x}}{d}}\right]=\sum\limits_{n=-\infty}^{\infty}{i^{n}}J_{n}\left(z\right)e^{i\frac{{2\pi
nx}}{d}}{\rm{=}}B{\rm{+}}C.$ (6)

Here we denote the sum of the even terms as $B$ and the sum of the odd terms
as $C$.
According to Eq. (5), after the wave propagate a distance $d$, the wavefront
is

$E\left(x\right)\propto\sum\limits_{n=-\infty}^{\infty}{i^{n}}\left({-1}\right)^{n^{2}}J_{n}\left(z\right)e^{i\frac{{2\pi
nx}}{d}}.$ (7)

Compared with
$\sum\limits_{n=-\infty}^{\infty}{i^{n}}J_{n}\left(z\right)e^{i\frac{{2\pi
nx}}{d}}$,
$\sum\limits_{n=-\infty}^{\infty}{i^{n}}\left({-1}\right)^{n^{2}}J_{n}\left(z\right)e^{i\frac{{2\pi
nx}}{d}}$ is characterized by the fact that even terms do not change, and odd
terms are multiplied by $-1$. So we have

$E\left(x\right)\propto B-C.$ (8)

Then we do the phase - only modulation $e^{iz\cos\frac{{2\pi x}}{d}}$ again,
the final wavefront is

$E\left(x\right)T\left(x\right)\propto B^{2}-C^{2}=1.$ (9)

So the outgoing wave is still a plane wave. This indicates that double-pulse
beam splitting cannot be regarded as double phase imprinting of a thin
grating.
Then the second question rises: whether the standing wave pulse can be treated
as a Hadmard gate under more strict conditions, rather than make an
approximation including only the lowest-order Raman-Nath equations?
To answer this question, we use the Bloch wave band method to deal with the
effect of square pulse on atoms, which is also the approach to studying the
Kapitza-Dirac Scattering. Bloch wave functions $\psi_{n,q}\left(x\right)$ are
the products of a plane wave $e^{iqk_{0}x}$ and a function
$u_{n,q}\left(x\right)$ with the periodicity of the optical lattice

$\psi_{n,q}\left(x\right)=e^{iqk_{0}x}u_{n,q}\left(x\right),$ (10)

where

$u_{n,q}\left(x\right)=\sum\limits_{l=-\infty}^{\infty}{c_{l,n,q}}e^{i2lk_{0}x}.$
(11)

To deal with the evolution in optical lattices, the key is to find
coefficients $c_{l,n,q}$ and eigenenergies $\varepsilon_{n,q}$ of
$u_{n,q}\left(x\right)$. We can get them by numerically solving the matrix
equation [30]

$\sum\limits_{l=-\infty}^{\infty}{H_{l,l^{\prime}}\cdot
c_{l,n,q}}=\varepsilon_{n,q}c_{l,n,q^{\prime}},$ (12)

where

$H_{l,l^{\prime}}=\left\\{{\begin{array}[]{*{20}c}{\left({2l+q}\right)^{2}+{s\mathord{\left/{\vphantom{s{2\begin{array}[]{*{20}c}{}\hfil&{if}&{}\hfil&{l=l^{\prime}}\\\
\end{array}}}}\right.\kern-1.2pt}{2\begin{array}[]{*{20}c}{}\hfil&{if}&{}\hfil&{l=l^{\prime}}\\\
\end{array}}}}\\\
{-{s\mathord{\left/{\vphantom{s{4\begin{array}[]{*{20}c}{}\hfil&{if}&{}\hfil&{\left|{l-l^{\prime}}\right|=1}\\\
\end{array}}}}\right.\kern-1.2pt}{4\begin{array}[]{*{20}c}{}\hfil&{if}&{}\hfil&{\left|{l-l^{\prime}}\right|=1}\\\
\end{array}}}}\\\ {0\begin{array}[]{*{20}c}{}\hfil&{else}&{}\hfil&{}\hfil\\\
\end{array}}\\\ \end{array}}\right.{\rm{.}}$ (13)

When atoms are suddenly loaded into an optical lattice, they are projected
onto superpositions of different Bloch bands. For the case where $q=0$ and the
potential-well depth of the optical lattice is relatively shallow, for example
when $s<10$, only bands $0$ and $2$ have significant populations, and the
components with momentum $0$ and $\pm 2\hbar k_{0}$ account for the majority
of these bands [31]. In our actual calculation, we truncated the equation from
$l=-20$ to $l=20$, so the calculation accuracy is high enough. Our calculation
strategy is to increase the well depth from a small value, until the minimum
of the population of zero momentum part at the first oscillation period
approaches $0.5$. The population of $+2\hbar k_{0}$ and $-2\hbar k_{0}$ parts
at this time will be very close to $0.25$, while the population of the other
momentum components can be ignored. Here we choose the population of zero
momentum part equaling $0.5$ at the minimum point of the first oscilltion
period, which suggests that such a pulse does indeed rotate the state vector
from $-Z$ to $X$ axes and a subsequent standing wave pulse of the same length
can return the atom to its initial zero-momentum state. Therefore, this
standing wave square optical pulse can realize a Hadamard gate operation well.
For example, if the wavelength of the lattice beam is $780nm$ and the initial
momentum is zero, when $s=5.5$ and the pulse width $\tau_{1}=24.8\mu s$, the
population of the zero momentum part is $0.4984$, and the population of
$+2\hbar k_{0}$ and $-2\hbar k_{0}$ momentum part are both $0.2490$, which is
close to a good Hadamard gate operation.
If the initial momentum is not strictly zero, but a small momentum $q\hbar
k_{0}$ ($\left|q\right|<<1$), the standing wave square optical pulse can still
perform an Hadamard gate operation approximately. For example, if the
wavelength of the lattice beam is $532nm$ and the initial momentum is
$0.03\hbar k_{0}$, when $s=5.5$ and the pulse width $\tau_{1}=11.7\mu s$, the
populations of order zero and order $\pm 1$ diffraction are $0.4996$, $0.2391$
and $0.2576$, respectively. The phase difference between order zero and order
$\pm 1$ diffraction varies by only $0.05$ radian compared to the case where
the initial momentum is zero. Besides, a subsequent standing wave pulse of the
same length still can return the atom almost completely to its initial zero-
momentum state, as showed in Fig. 2.

Figure 2: Population of momentum components as a function of lattice pulse
time. The blue, red, and green lines correspond to populations with momentum
$0\hbar k_{0}$, $2\hbar k_{0}$, and $-2\hbar k_{0}$ parts, respectively.

We would like to discuss the slight difference between the optimization in
Ref. [26, 27] and our results. In Ref. [26], the amplitude of the double
square pulse is fixed at $\Omega_{m}=2\sqrt{2}\omega_{r}$, which corresponds
to $s=5.66$ and is slightly above our set point. After optimization, the pulse
width $\tau_{1}$ is shorter than $\frac{\pi}{{4\sqrt{2}\omega_{r}}}$ and the
interval time $\tau_{2}$ is longer than $\frac{\pi}{{4\omega_{r}}}$. Here we
will give an intuitive physical images to understand these results. The state
vector rotates around the Rabi vector $\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\Omega}}{\rm{=}}\left({\sqrt{2}\Omega_{m},0,4\omega_{r}}\right)$
is an approximate physical model by truncating the Raman-Nath equations to the
lowest-order 3 equations.According to this model, if we set
$\Omega_{m}=2\sqrt{2}\omega_{r}$ and the pulse length
$\tau_{1}=\frac{\pi}{{4\sqrt{2}\omega_{r}}}$, the state vector starting from
the $-Z$ axis should stop at the $X$ axis, which is the top of the rotating
track. Nevertheless, we calculated the population evolution of the part with
zero momentum by using the Bloch wave band method with the same parameter. The
result showed the population of zero momentum part is less than $0.5$ at the
minimum point of the first oscillation period, which means the top of the
rotating track is higher than the $X-Y$ plane, as shown in Fig. 1(c). If we
want to stop the state vector on the $X-Y$ plane, the pulse length $\tau_{1}$
should be relatively shorter, which also brings in an additional phase
$\Delta\phi$. In order for the next light pulse to rotate the state vector to
the $Z$ axis, the phase change corresponding to the free evolution is not
$\pi$ but $\pi{\rm{+}}2\Delta\phi$, so it takes a longer time.
Then we deduced the evolution matrix corresponding to free propagation and
presented the whole theoretical framework. We can choose the non-coupled basis
vectors
$\left\\{{\left|{C_{0}}\right\rangle,\left|{C_{+2}}\right\rangle,\left|{C_{-2}}\right\rangle}\right\\}$,
where $\left|{C_{0}}\right\rangle$, $\left|{C_{+2}}\right\rangle$ and
$\left|{C_{-2}}\right\rangle$ represent states with momentum $q\hbar k_{0}$,
$\left({2+q}\right)\hbar k_{0}$ and $\left({-2+q}\right)\hbar k_{0}$,
respectively. In the momentum representation, their wave functions are
$\delta\left({p^{\prime}-\hbar k}\right)$,
$\delta\left({p^{\prime}-\left({\hbar k-2\hbar k_{0}}\right)}\right)$ and
$\delta\left({p^{\prime}-\left({\hbar k+2\hbar k_{0}}\right)}\right)$. We can
also choose the coupled basis vectors
$\left\\{{\left|{C_{0}}\right\rangle,\left|{C_{+}}\right\rangle,\left|{C_{-}}\right\rangle}\right\\}$,
where
$\left|{C_{+}}\right\rangle=\frac{1}{{\sqrt{2}}}\left({\left|{C_{+2}}\right\rangle+\left|{C_{-2}}\right\rangle}\right)$
and
$\left|{C_{-}}\right\rangle=\frac{1}{{\sqrt{2}}}\left({\left|{C_{+2}}\right\rangle-\left|{C_{-2}}\right\rangle}\right)$.
However, in order to make operations similar to the previous Hadmard gate -
phase gate - Hadmard gate, we prefer to do it on a coupled basis vectors.
After propagation, we take the phase change of state
$\left|{C_{0}}\right\rangle$ as the reference zero, then the phase change of
$\left|{C_{+2}}\right\rangle$ is $\varphi_{+2}$, and the phase change of
$\left|{C_{-2}}\right\rangle$ is $\varphi_{-2}$. The state
$\left|{C_{+2}}\right\rangle$ has higher energy than
$\left|{C_{0}}\right\rangle$ by

$\Delta
E_{+2}=\frac{{\hbar^{2}\left({2+q}\right)^{2}k_{0}^{2}}}{{2m}}-\frac{{\hbar^{2}q^{2}k_{0}^{2}}}{{2m}}=\frac{{\left({2+2q}\right)\hbar^{2}k_{0}^{2}}}{m},$
(14)

so the phase difference between $\left|{C_{+2}}\right\rangle$ and
$\left|{C_{0}}\right\rangle$ is

$\varphi_{+2}=\frac{{\Delta
E_{+2}\tau_{2}}}{\hbar}=\frac{{\left({2+2q}\right)\hbar
k_{0}^{2}\tau_{2}}}{m},$ (15)

where $\tau_{2}$ is the time of free evolution between two pulses. We can set
$\varphi_{+2}=\theta+\Delta\theta$, where $\theta=\frac{{2\hbar
k_{0}^{2}\tau_{2}}}{m}$ is the phase difference corresponding $q=0$, and
$\Delta\theta=\frac{{2q\hbar k_{0}^{2}\tau_{2}}}{m}$ is the contribution from
$q$. The phase difference between $\left|{C_{-2}}\right\rangle$ and
$\left|{C_{0}}\right\rangle$ is

$\varphi_{-2}=\frac{{\Delta
E_{-2}\tau_{2}}}{\hbar}=\frac{{\left({2-2q}\right)\hbar
k_{0}^{2}\tau_{2}}}{m}.$ (16)

Similarly, $\varphi_{-2}$ can be expressed as
$\varphi_{-2}=\theta-\Delta\theta$. We know that the momentum eigenstate
satisfies

$\Phi\left({k,t}\right)=\Phi\left({k,0}\right)\exp\left[{{{-iEt}\mathord{\left/{\vphantom{{-iEt}\hbar}}\right.\kern-1.2pt}\hbar}}\right]=\Phi\left({k,0}\right)\exp\left[{-i\varphi}\right].$
(17)

Then, under the representation with
$\left\\{{\left|{C_{0}}\right\rangle,\left|{C_{+2}}\right\rangle,\left|{C_{-2}}\right\rangle}\right\\}$
as base kets, the evolution matrix of free propagation is

$F=\left[{\begin{array}[]{*{20}c}1&0&0\\\ 0&{e^{-i\varphi_{+2}}}&0\\\
0&0&{e^{-i\varphi_{-2}}}\\\
\end{array}}\right]=\left[{\begin{array}[]{*{20}c}1&0&0\\\
0&{e^{-i\left({\theta+\Delta\theta}\right)}}&0\\\
0&0&{e^{-i\left({\theta-\Delta\theta}\right)}}\\\ \end{array}}\right].$ (18)

If we chose base kets as
$\left\\{{\left|{C_{0}}\right\rangle,\left|{C_{+}}\right\rangle,\left|{C_{-}}\right\rangle}\right\\}$,
the evolution matrix is $P=S^{+}FS$, where representation transformation
matrix

$S=S^{+}=S^{-1}=\left[{\begin{array}[]{*{20}c}1&0&0\\\
0&{\frac{1}{{\sqrt{2}}}}&{\frac{1}{{\sqrt{2}}}}\\\
0&{\frac{1}{{\sqrt{2}}}}&{-\frac{1}{{\sqrt{2}}}}\\\ \end{array}}\right].$ (19)

Therefore, the evolution matrix of free propagation under the representation
with base kets
$\left\\{{\left|{C_{0}}\right\rangle,\left|{C_{+2}}\right\rangle,\left|{C_{-2}}\right\rangle}\right\\}$
is

$P=\left[{\begin{array}[]{*{20}c}1&0&0\\\
0&{e^{-i\theta}\cos\Delta\theta}&{-ie^{-i\theta}\sin\Delta\theta}\\\
0&{-ie^{-i\theta}\sin\Delta\theta}&{e^{-i\theta}\cos\Delta\theta}\\\
\end{array}}\right].$ (20)

Next, we will examine the matrix corresponding to the standing wave square
pulse. By selecting the appropriate parameters, we make the evolution matrix
corresponding to the pulse under the representation of
$\left\\{{\left|{C_{0}}\right\rangle,\left|{C_{+}}\right\rangle,\left|{C_{-}}\right\rangle}\right\\}$
be

$H=\left[{\begin{array}[]{*{20}c}{\frac{1}{{\sqrt{2}}}}&{\frac{1}{{\sqrt{2}}}}&0\\\
{\frac{1}{{\sqrt{2}}}}&{-\frac{1}{{\sqrt{2}}}}&0\\\ 0&0&1\\\
\end{array}}\right].$ (21)

This matrix implies that the symmetric basis vectors
$\left|{C_{0}}\right\rangle$ and $\left|{C_{+}}\right\rangle$ are decoupled
from the asymmetric basis vector $\left|{C_{-}}\right\rangle$, and a Hadamard
gate operation is performed for $\left|{C_{0}}\right\rangle$ and
$\left|{C_{+}}\right\rangle$.
Thus, the unitary transformation corresponding to the complete impulse - free
evolution - impulse ($HPH$) is

$HPH=\left[{\begin{array}[]{*{20}c}{\frac{{1+e^{-i\theta}\cos\theta}}{2}}&{\frac{{1-e^{-i\theta}\cos\theta}}{2}}&{-\frac{{ie^{-i\theta}\sin\theta}}{{\sqrt{2}}}}\\\
{\frac{{1-e^{-i\theta}\cos\theta}}{2}}&{\frac{{1+e^{-i\theta}\cos\theta}}{2}}&{\frac{{ie^{-i\theta}\sin\theta}}{{\sqrt{2}}}}\\\
{-\frac{{ie^{-i\theta}\sin\theta}}{{\sqrt{2}}}}&{\frac{{ie^{-i\theta}\sin\theta}}{{\sqrt{2}}}}&{e^{-i\theta}\cos\theta}\\\
\end{array}}\right].$ (22)

If the momentum of the initial state is $q\hbar k_{0}$, then

$HPH\left[{\begin{array}[]{*{20}c}1\\\ 0\\\ 0\\\
\end{array}}\right]=\left[{\begin{array}[]{*{20}c}{\frac{{1+e^{-i\theta}\cos\Delta\theta}}{2}}\\\
{\frac{{1-e^{-i\theta}\cos\Delta\theta}}{2}}\\\
{-\frac{{ie^{-i\theta}\sin\Delta\theta}}{{\sqrt{2}}}}\\\ \end{array}}\right].$
(23)

For the case of $\theta=\left({2N+1}\right)\pi$ and $q=0$, the final output is
$\left|{C_{+}}\right\rangle$, this is exactly the case of Ref. [26].

##  III III. The effect of finite momentum width

Next, we will consider the effect of a finite width momentum distribution. Eq.
(23) already gives the final state under the representation of
$\left\\{{\left|{C_{0}}\right\rangle,\left|{C_{+}}\right\rangle,\left|{C_{-}}\right\rangle}\right\\}$
when the initial state momentum is $q\hbar k_{0}$. Using the representation
transformation formula, we can obtain the final state output under
representation of
$\left\\{{\left|{C_{0}}\right\rangle,\left|{C_{+2}}\right\rangle,\left|{C_{-2}}\right\rangle}\right\\}$
is

$\left[{\begin{array}[]{*{20}c}1&0&0\\\
0&{\frac{1}{{\sqrt{2}}}}&{\frac{1}{{\sqrt{2}}}}\\\
0&{\frac{1}{{\sqrt{2}}}}&{-\frac{1}{{\sqrt{2}}}}\\\
\end{array}}\right]\left[{\begin{array}[]{*{20}c}{\frac{{1+e^{-i\theta}\cos\Delta\theta}}{2}}\\\
{\frac{{1-e^{-i\theta}\cos\Delta\theta}}{2}}\\\
{-\frac{{ie^{-i\theta}\sin\Delta\theta}}{{\sqrt{2}}}}\\\
\end{array}}\right]=\left[{\begin{array}[]{*{20}c}{\frac{{1+e^{-i\theta}\cos\Delta\theta}}{2}}\\\
{\frac{{\left({1-e^{-i\theta}\cos\Delta\theta}\right)-ie^{-i\theta}\sqrt{2}\sin\Delta\theta}}{{2\sqrt{2}}}}\\\
{\frac{{\left({1-e^{-i\theta}\cos\Delta\theta}\right)+ie^{-i\theta}\sqrt{2}\sin\theta}}{{2\sqrt{2}}}}\\\
\end{array}}\right].$ (24)

The wave function under the momentum representation can be expanded as

$\Phi\left(p\right)=\int{\Phi\left({p^{\prime}}\right)}\delta\left({p^{\prime}-p}\right)dp^{\prime},$
(25)

where $\delta\left({p^{\prime}-p}\right)=\delta\left({p-p^{\prime}}\right)$ is
the momentum eigenstate with the eigenvalue of $p$, and the expansion
coefficient is $\Phi\left({p^{\prime}}\right)$. If the initial wave function
is

$\Phi_{ini}\left({p=\hbar
k}\right)=\int{\Phi\left({p^{\prime}}\right)}\delta\left({p^{\prime}-\hbar
k}\right)dp^{\prime},$ (26)

after $HPH$ evolution, $\delta\left({p^{\prime}-q\hbar k_{0}}\right)$ evolves
into

$\begin{array}[]{l}\frac{{1+e^{-i\theta}\cos\Delta\theta}}{2}\delta\left({p^{\prime}-\hbar
k}\right)+\frac{{\left({1-e^{-i\theta}\cos\Delta\theta}\right)-ie^{-i\theta}\sqrt{2}\sin\Delta\theta}}{{2\sqrt{2}}}\delta\left({p^{\prime}-\left({\hbar
k-2\hbar k_{0}}\right)}\right)\\\
+\frac{{\left({1-e^{-i\theta}\cos\Delta\theta}\right)+ie^{-i\theta}\sqrt{2}\sin\Delta\theta}}{{2\sqrt{2}}}\delta\left({p^{\prime}-\left({\hbar
k+2\hbar k_{0}}\right)}\right)\\\ \end{array}.$ (27)

So the final wave function is

$\Phi_{fin}\left({p=\hbar
k}\right)=\int{\Phi\left({p^{\prime}}\right)}\left[\begin{array}[]{l}\frac{{1+e^{-i\theta}\cos\Delta\theta}}{2}\delta\left({p^{\prime}-\hbar
k}\right)+\frac{{\left({1-e^{-i\theta}\cos\Delta\theta}\right)-ie^{-i\theta}\sqrt{2}\sin\Delta\theta}}{{2\sqrt{2}}}\delta\left({p^{\prime}-\left({\hbar
k-2\hbar k_{0}}\right)}\right)\\\
+\frac{{\left({1-e^{-i\theta}\cos\Delta\theta}\right)+ie^{-i\theta}\sqrt{2}\sin\Delta\theta}}{{2\sqrt{2}}}\delta\left({p^{\prime}-\left({\hbar
k+2\hbar k_{0}}\right)}\right)\\\ \end{array}\right]dp^{\prime}.$ (28)

If $\left|q\right|<<1$, the central zero-momentum term and the $\pm 2\hbar
k_{0}$ terms are separated after a time-of-flight expansion in free space. The
information about the momentum distribution can be obtained by statistical
population of the terms.
As an example, we take the wave function of the initial state as a least
uncertain Gaussian wave packet

$\Phi_{ini}\left(k\right)=\frac{1}{{\alpha^{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}\pi^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}}e^{-\frac{{k^{2}}}{{2\alpha^{2}}}}.$
(29)

If we set $k=qk_{0}$ and $\alpha=q_{m}k_{0}$, the width of the initial wave
packet in momentum space in units $k_{0}$ is equal to $q_{m}$, the normalized
initial state wave function in terms of $q$ is

$\Phi_{ini}\left(q\right)=\frac{1}{{q_{m}^{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}\pi^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}}e^{-\frac{{q^{2}}}{{2q_{m}^{2}}}}.$
(30)

The final wave function is

$\begin{array}[]{l}\Phi_{final}\left(k\right)=\frac{1}{{\alpha^{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}\pi^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}}e^{-\frac{{k^{2}}}{{2\alpha^{2}}}}\cdot\frac{{1+e^{-i\theta}\cos\Delta\theta}}{2}+\frac{1}{{\alpha^{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}\pi^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}}e^{-\frac{{\left({k-2k_{0}}\right)^{2}}}{{2\alpha^{2}}}}\cdot\frac{{\left({1-e^{-i\theta}\cos\Delta\theta}\right)-ie^{-i\theta}\sqrt{2}\sin\Delta\theta}}{{2\sqrt{2}}}\\\
+\frac{1}{{\alpha^{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}\pi^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}}e^{-\frac{{\left({k+2k_{0}}\right)^{2}}}{{2\alpha^{2}}}}\cdot\frac{{\left({1-e^{-i\theta}\cos\Delta\theta}\right)+ie^{-i\theta}\sqrt{2}\sin\Delta\theta}}{{2\sqrt{2}}}\\\
\end{array}.$ (31)

After sufficient free flight, the three parts of central momentum $0\hbar
k_{0}$, $2\hbar k_{0}$, and $-2\hbar k_{0}$ separate in space, with no
interference terms. Therefore, the optical depth of these three parts can be
experimentally integrated respectively in space, so as to obtain the
proportion of these three parts. Wherein, the modulus square of the central
zero-momentum term is

$\frac{1}{{\alpha\sqrt{\pi}}}e^{-\frac{{k^{2}}}{{\alpha^{2}}}}\cdot\left({\frac{{1+e^{-i\theta}\cos\Delta\theta}}{2}}\right)\left({\frac{{1+e^{i\theta}\cos\Delta\theta}}{2}}\right)=\frac{1}{{\alpha\sqrt{\pi}}}e^{-\frac{{k^{2}}}{{\alpha^{2}}}}\left({\frac{3}{8}+\frac{{\cos\theta}}{2}\cos\Delta\theta+\frac{{\cos
2\Delta\theta}}{8}}\right).$ (32)

Then we integrate with respect to $k=qk_{0}$, and notice that
$\Delta\theta=\frac{{2q\hbar k_{0}^{2}\tau_{2}}}{m}$ contains the integral
variable. For simplicity, we define a dimensionless time $t_{2}=\frac{{2\hbar
k_{0}^{2}\tau_{2}}}{{m\pi}}$, then $\Delta\theta=t_{2}\pi q$ and
$\theta=t_{2}\pi$. The integral is

$\int_{-\infty}^{+\infty}{\frac{1}{{\alpha\sqrt{\pi}}}e^{-\frac{{k^{2}}}{{\alpha^{2}}}}\left({\frac{3}{8}+\frac{{\cos\theta}}{2}\cos\Delta\theta+\frac{{\cos
2\Delta\theta}}{8}}\right)}dk=\frac{3}{8}+\frac{1}{2}\cos\left({t_{2}\pi}\right)e^{-\frac{{\pi^{2}q_{m}^{2}t_{2}^{2}}}{4}}+\frac{1}{8}e^{-\pi^{2}q_{m}^{2}t_{2}^{2}}.$
(33)

Similarly, we get that the population of the momentum part of $2\hbar k_{0}$
and $-2\hbar k_{0}$ both are
$\frac{5}{{16}}-\frac{1}{4}\cos\left({t_{2}\pi}\right)e^{-\frac{{\pi^{2}q_{m}^{2}t_{2}^{2}}}{4}}-\frac{1}{{16}}e^{-\pi^{2}q_{m}^{2}t_{2}^{2}}$.
When $t_{2}=2N+1$, where $N$ is an integer greater than or equal to zero, the
ratio of central zero-momentum to $2\hbar k_{0}$ or $-2\hbar k_{0}$ part is

$Ratio={{\left({\frac{3}{8}-\frac{1}{2}e^{-\frac{{\pi^{2}q_{m}^{2}t_{2}^{2}}}{4}}+\frac{1}{8}e^{-\pi^{2}q_{m}^{2}t_{2}^{2}}}\right)}\mathord{\left/{\vphantom{{\left({\frac{3}{8}-\frac{1}{2}e^{-\frac{{\pi^{2}q_{m}^{2}t_{2}^{2}}}{4}}+\frac{1}{8}e^{-\pi^{2}q_{m}^{2}t_{2}^{2}}}\right)}{\left({\frac{5}{{16}}+\frac{1}{4}e^{-\frac{{\pi^{2}q_{m}^{2}t_{2}^{2}}}{4}}-\frac{1}{{16}}e^{-\pi^{2}q_{m}^{2}t_{2}^{2}}}\right)}}}\right.\kern-1.2pt}{\left({\frac{5}{{16}}+\frac{1}{4}e^{-\frac{{\pi^{2}q_{m}^{2}t_{2}^{2}}}{4}}-\frac{1}{{16}}e^{-\pi^{2}q_{m}^{2}t_{2}^{2}}}\right)}}.$
(34)

Although we have discussed the Gaussian wave packet of a single atom above,
this method is still suitable for measuring the temperature of the ultracold
atomic ensemble. Each atom in the ultracold atomic ensemble is treated as a
Gaussian wave packet [25, 32], and the width of the wave packet in the
momentum space is determined by temperature
$q_{m}^{2}=\frac{{2k_{B}Tm}}{{k_{0}^{2}\hbar^{2}}}$. The different initial
positions of each atom will introduce a different initial phase on the
momentum wave packet, but this initial phase will not affect the population of
each diffraction order after $HPH$ evolution. As long as the flight time after
$HPH$ evolution is sufficient, the diffraction order of the ultracold atomic
cloud can be experimentally distinguished.
Fig. 3(a) shows the evolution of the population of zero momentum term with the
dimensionless time $t_{2}$ when the temperature is $300pK$. Fig. 3(b) is the
curve of the population ratio verse temperature when $t_{2}=31$. In the above
calculation, we set the wavelength of the optical lattice laser as $532nm$, so
the actual interval time $\tau_{2}$ is about $0.477ms$. Therefore, even if the
whole beamsplitting process of the dual standing wave pulse is less than
$1ms$, it is capable of detecting temperatures as low as $100pK$ in principle.

Figure 3: The effect of finite momentum width. (a) Temporal evolution of the
population of zero-momentum term with the dimensionless time $t_{2}$ when the
temperature is $300pK$. (b) The population ratio of $0\hbar k_{0}$ to $2\hbar
k_{0}$ ($-2\hbar k_{0}$) verse temperature of the Gaussian wave packet when
the dimensionless time $t_{2}=31$. (c) Temporal evolution of the population of
zero-momentum term with the dimensionless time $t_{2}$ when the kinetic energy
equivalent temperature of the BEC is $300pK$. (d) The population ratio of
$0\hbar k_{0}$ to $2\hbar k_{0}$ ($-2\hbar k_{0}$) verse the kinetic energy
equivalent temperature of the BEC when the dimensionless time $t_{2}=31$.

Then we take another example of a BEC released from a trap. Under the Thomas-
Fermi approximation, the density distribution of a BEC in the trap is [33]

$n\left({\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{r}}}\right)=\frac{{\mu-V\left({\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{r}}}\right)}}{{U_{0}}},$
(35)

Where $\mu$ is the chemical potential and $U_{0}$ is the effective interaction
between two atoms. In an anisotropic three-dimensional harmonic-oscillator
potential
$V\left({x,y,z}\right)=\frac{m}{2}\left({\omega_{x}^{2}x^{2}+\omega_{y}^{2}y^{2}+\omega_{z}^{2}z^{2}}\right)$,
the density distribution of the BEC is

$n\left({x,y,z}\right)=n\left({0,0,0}\right)\max\left[{1-\frac{{x^{2}}}{{R_{x}^{2}}}-\frac{{y^{2}}}{{R_{y}^{2}}}-\frac{{z^{2}}}{{R_{z}^{2}}},0}\right],$
(36)

where $R_{i}=\sqrt{\frac{{2\mu}}{{m\omega_{i}^{2}}}}$ $\left({i=x,y,z}\right)$
is the half-lengths of the trapped condensate. After released from the trap,
the BEC evolves as a rescaling of its parabolic shape and the local velocity
of the expanding cloud is [34, 35]

$v_{i}\left({\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{r}},t}\right)=r_{i}\frac{{\dot{\lambda}_{i}\left(t\right)}}{{\lambda_{i}\left(t\right)}},$
(37)

where
$\lambda_{i}={{R_{i}\left(t\right)}\mathord{\left/{\vphantom{{R_{i}\left(t\right)}{R_{i}}}}\right.\kern-1.2pt}{R_{i}}}\left(0\right)$
are the scaling factors. Therefore, if we selectively detect the region of
$y\approx 0$ and $z\approx 0$ along the $X-$ axis, which can be achieved
experimentally with a thin optical pumping beam, then we will find that the
momentum distribution is also parabolic shape. So the initial wave function of
the BEC in this region can be approximately expressed as

$\Phi_{ini}\left({p_{x}}\right)=Ae^{i\varphi\left({p_{x}}\right)}\sqrt{1-\frac{{p_{x}^{2}}}{{p_{m}^{2}}}},$
(38)

here $p_{x}=q\hbar k_{0}$, $p_{m}=q_{m}\hbar k_{0}$ and $-p_{m}\leq p_{x}\leq
p_{m}$. The wave function after $HPH$ evolution is

$\begin{array}[]{l}\Phi_{final}\left({p_{x}=\hbar
k}\right)=Ae^{i\varphi\left({\hbar k}\right)}\sqrt{1-\frac{{\left({\hbar
k}\right)^{2}}}{{p_{m}^{2}}}}\cdot\frac{{1+e^{-i\theta}\cos\Delta\theta}}{2}\\\
+Ae^{i\varphi\left({\hbar k-2\hbar k_{0}}\right)}\sqrt{1-\frac{{\left({\hbar
k-2\hbar
k_{0}}\right)^{2}}}{{p_{m}^{2}}}}\cdot\frac{{\left({1-e^{-i\theta}\cos\Delta\theta}\right)-ie^{-i\theta}\sqrt{2}\sin\Delta\theta}}{{2\sqrt{2}}}\\\
+Ae^{i\varphi\left({\hbar k+2\hbar k_{0}}\right)}\sqrt{1-\frac{{\left({\hbar
k+2\hbar
k_{0}}\right)^{2}}}{{p_{m}^{2}}}}\cdot\frac{{\left({1-e^{-i\theta}\cos\Delta\theta}\right)+ie^{-i\theta}\sqrt{2}\sin\Delta\theta}}{{2\sqrt{2}}}\\\
\end{array},$ (39)

where the wave function of the zero-momentum term is
$Ae^{i\varphi\left({q\hbar
k_{0}}\right)}\sqrt{1-\frac{{q^{2}}}{{q_{m}^{2}}}}\cdot\frac{{1+e^{-i\theta}\cos\Delta\theta}}{2}$
and its population is
$\int\limits_{-q_{m}}^{q_{m}}B\left({1-\frac{{q^{2}}}{{q_{m}^{2}}}}\right)\cdot\frac{{1+e^{-i\theta}\cos\Delta\theta}}{2}\cdot\frac{{1+e^{i\theta}\cos\Delta\theta}}{2}dq$.
Since the population of all atoms is unit, the normalization coefficient is
$B=\frac{3}{{4q_{m}}}$, so the population of the zero momentum term is

$\begin{array}[]{l}\int\limits_{-q_{m}}^{q_{m}}{\frac{3}{{4q_{m}}}}\left({1-\frac{{q^{2}}}{{q_{m}^{2}}}}\right)\cdot\frac{{1+e^{-i\theta}\cos\Delta\theta}}{2}\cdot\frac{{1+e^{i\theta}\cos\Delta\theta}}{2}dq\\\
=\frac{3}{8}+\frac{3}{{2\pi^{3}t_{2}^{3}q_{m}^{3}}}\cos\left({\pi
t_{2}}\right)\cdot\sin\left({\pi
t_{2}q_{m}}\right)-\frac{3}{{2\pi^{2}t_{2}^{2}q_{m}^{2}}}\cos\left({\pi
t_{2}}\right)\cdot\cos\left({\pi t_{2}q_{m}}\right)\\\
+\frac{3}{{64\pi^{3}t_{2}^{3}q_{m}^{3}}}\sin\left({2\pi
t_{2}q_{m}}\right)-\frac{3}{{32\pi^{2}t_{2}^{2}q_{m}^{2}}}\cos\left({2\pi
t_{2}q_{m}}\right)\\\ \end{array}.$ (40)

Here $q_{m}^{2}=\frac{{5k_{B}Tm}}{{k_{0}^{2}\hbar^{2}}}$ is determined by the
kinetic energy equivalent temperature $T$.
We still set the wavelength of the optical lattice laser as $532nm$. Fig.3(c)
shows the evolution of the population of zero momentum term with dimensionless
time $t_{2}$ when the equivalent temperature of the BEC is $300pK$. And
similarly, we got the curve of the population ratio verse equivalent
temperature when $t_{2}=31$, as shown in Fig.3 (d). Comparing this curve with
that of Gaussian wave packet (Fig.3(b)), we found there is an obvious
deviation when the population ratio is bigger than $0.5$. Therefore, in the
absence of prior information on the momentum distribution, it is best to limit
the experimental test results to a population ratio less than $0.5$.

##  IV IV. Phase space evolution of double-pulse beam-splitting process

A quasi-probability distribution in phase space can be obtained by using the
Wigner function [36, 37, 38], which provides a convenient physical
interpretation for the double pulse beam splitting process. The Wigner
function constructed based on the momentum representation wave function is
[39]

$W\left({\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{x}},\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{k}}}\right)=\int{e^{i\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{p}}\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{x}}}\phi\left({-\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{k}}-\frac{{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{p}}}}{2}}\right)}\phi^{*}\left({-\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{k}}+\frac{{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{p}}}}{2}}\right)d\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{p}},$ (41)

where $\phi^{*}$ is the complex conjugate of $\phi$,
$\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{p}}$ in the formula is
actually the same dimensional as $\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{k}}$.
We take a one-dimensional minimum uncertainty coherent state packet as an
example

$\Phi\left(k\right)=\frac{1}{{\sqrt{\alpha}\pi^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}}e^{-\frac{{k^{2}}}{{2\alpha^{2}}}}.$
(42)

After the first pulse, the momentum space wave function is

$\Phi\left(k\right)=\frac{1}{{\sqrt{2\alpha}\pi^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}}e^{-\frac{{k^{2}}}{{2\alpha^{2}}}}+\frac{1}{{2\sqrt{\alpha}\pi^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}}e^{-\frac{{\left({k+2k_{0}}\right)^{2}}}{{2\alpha^{2}}}}\frac{1}{{2\sqrt{\alpha}\pi^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}}e^{-\frac{{\left({k-2k_{0}}\right)^{2}}}{{2\alpha^{2}}}},$
(43)

and the Wigner function is

$\begin{array}[]{l}W\left({x,k}\right)=e^{-\frac{{k^{2}}}{{\alpha^{2}}}}e^{-\alpha^{2}x^{2}}+\frac{1}{2}e^{-\frac{{\left({k-2k_{0}}\right)^{2}}}{{\alpha^{2}}}}e^{-\alpha^{2}x^{2}}+\frac{1}{2}e^{-\frac{{\left({k+2k_{0}}\right)^{2}}}{{\alpha^{2}}}}e^{-\alpha^{2}x^{2}}\\\
+\cos\left({4k_{0}x}\right)e^{-\frac{{k^{2}}}{{\alpha^{2}}}}e^{-\alpha^{2}x^{2}}+\sqrt{2}\cos\left({2k_{0}x}\right)e^{-\frac{{\left({k-k_{0}}\right)^{2}}}{{\alpha^{2}}}}e^{-\alpha^{2}x^{2}}\\\
+\sqrt{2}\cos\left({2k_{0}x}\right)e^{-\frac{{\left({k+k_{0}}\right)^{2}}}{{\alpha^{2}}}}e^{-\alpha^{2}x^{2}}\\\
\end{array}.$ (44)

The result shows that after the first square pulse, there are three regions
with central momentum $0\hbar k_{0}$ and $\pm 2\hbar k_{0}$. In addition,
there are three fringe regions associated with the interference term, two of
which are located in the middle of $0\hbar k_{0}$ and $\pm 2\hbar k_{0}$
momentum regions respectively, and one overlaps with the region of zero
momentum. In Fig.4(a), we give an example. The initial state is Gaussian wave
packet with temperature $30nK$. The phase determined by the optical lattice in
coordinate space varies periodically with the lattice constant
$d={\lambda\mathord{\left/{\vphantom{\lambda 2}}\right.\kern-1.2pt}2}$, where
$\lambda=532nm$. The fringes in the interference region are vertical after the
first pulse, indicating that the phases of the different momentum states are
consistent at this time.

Figure 4: The evolution of phase space distribution during the double standing
wave pulse beam splitting. (a) The phase space distribution just after the
first light pulse. (b)The phase space distribution after free evolution with
the dimensionless time $t_{2}=5$. (c) The phase space distribution just after
the second light pulse. (d) The momentum distribution after the second light
pulse.

When the external potential is zero, the evolution of Wigner function follows

$W\left({\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{x}},\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{k}},t}\right)=W\left({\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{x}}-\frac{{\hbar\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{k}}}}{m}\delta
t,\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{k}},t_{0}}\right),$ (45)

here $\delta t=t-t_{0}$. After free evolution, regions with central momentum
of $0\hbar k_{0}$ and $\pm 2\hbar k_{0}$ tend to separate and the fringes of
the interference region tilt. The fringe inclination in the interference
region indicates that different momentum states have different phases.
Interestingly, through the free evolution of the Wigner function in phase
space, we can clearly see the corresponding relationship between the phase
difference and the positions of different momentum states. Here we give an
example when the dimensionless time $t_{2}=5$, the three parts with momentum
$0\hbar k_{0}$ and $\pm 2\hbar k_{0}$ completely separate in space, as shown
in Fig. 4(b). The tilt of fringe in the interference regions indicates that
the phases of different momentum components due to finite temperature are
different after free evolution. Since the fringe inclination is so much, the
projection of fringes in the coordinate space becomes smoothing and the
periodic variation disappears, that means the coherence between the three
parts is hidden. When the second square pulse is applied, the population of
component with zero central momentum is simply the sum of zero central
momentums coming from three parts under the action of this standing wave
pulse. If we consider the population in the zero momentum after the second
pulse verse the interval time $\tau_{2}$ as an interference fringe, then the
contrast of the interference fringe reduces to zero after this time. This is
consistent with our common understanding that interference does not occur once
wave packets do not overlap. Fig. 4(c) shows the phase space distribution
after the second pulse. After integrating the coordinates, we got the momentum
distribution at this time, as shown in Fig. 4(d).

Figure 5: The evolution of phase space distribution during the double standing
wave pulse beam-splitting when the initial Gaussian wave packet expands before
the first pulse. (a) The phase space distribution just after the first light
pulse. (b)The phase space distribution after free evolution with the
dimensionless time $t_{2}=5$. (c) The phase space distribution just after the
second light pulse. (d) The momentum distribution after the second light
pulse.

Nevertheless, we want to point out that the overlap of wave packets does not
guarantee that the interference fringe contrast will not drop to zero. As an
example, we consider the case where the least uncertain Gaussian wave packet
diffuses freely for the dimensionless time $t_{{\rm{diff}}}=20$ before
applying the first square pulse, and other experimental conditions are the
same as those in Fig. 4. Just after the first pulse, the phase space
distribution is shown in Fig. 5(a). After the same free evolution time, the
three wave packets with different momentum are not entirely separated in
space, and the inclination of the fringes does not make the projection in the
coordinate space disappearing, as shown in Fig. 5(b). However, for any two
adjacent groups of fringes, the distance between fringe centers in coordinate
positions after free evolution is not equal to the optical lattice constant
$d$, so there is a phase shift between the fringes. If we divide and
superimpose the projection with the lattice constant $d$, the distribution is
still smooth without fringe. Fig. 5(c) shows the phase space distribution
after the second pulse, which is different from Fig. 4(c). But the momentum
distribution, as shown in Fig. 5(d), is the same as that in Fig. 4(d). As we
emphasized earlier, the final momentum distribution depends on the initial
momentum distribution and is independent of the spatial distribution.
Therefore, whether the wave packet expands before the first pulse does not
affect the measurement result. Previous studies have shown that the coherent
length of matter waves does not change when the wave packet expands [40]. Our
results are consistent with this conclusion.

##  V V. Discussion

We would like to discuss the defects of existing methods for measuring the
temperature of ultra-cold atoms, as well as the unique advantages of our
proposed approach.
The commonly used TOF method requires a free expansion time of hundreds of
milliseconds to measure a temperature on the order of $100pK$. For ground
systems, this is simply not possible due to gravity. Long-term TOF can be
achieved in microgravity environments, but many experimental measurements are
still required for fitting [17]. In addition, if there is a weak interaction
between atoms, such as BECs, the interaction potential energy convert to
kinetic energy continuously during a long period of expansion, so the TOF
method is not actually suitable for measuring the change in the momentum width
over time after being released from the trap.
Bragg Spectroscopy is another commonly used method to obtain the distribution
of atomic velocities [19, 20]. To measure a narrow velocity distribution, the
frequency stability of Bragg beams with tens of Hz is necessary. For example,
when the temperature of ${}^{87}{\rm{Rb}}$ atomic gas is about $100pK$, the
corresponding velocity width is about $0.2mm/s$. If we set the resolution as
$0.02mm/s$, the frequency difference between two counter-propagating Bragg
beams would correspond to about $50Hz$. The simplest way to create such Bragg
beam splitters is to employ two acousto-optic modulators with phase-stable
radio frequency drives. However, the laser system of the ultra-cold atomic
experiment device on the Chinese space Station separates from the experiment
chamber [41], and optical fibers connected in the middle, which will bring
additional phase noise due to the influence of the environment [42].
Therefore, a low noise phase-locked loop (PLL) with fiber output sampling is
required. Due to resource constraints, the current system is not equipped with
such a PLL. Moreover, a series of experiments are required to obtain the Bragg
spectrum. If a single continuous scan is used, the continuous scanning time
will also reach about $100ms$. The optical depth of the atomic cloud
diffracted by such long time scanning is very low and the signal-to-noise
ratio is poor.
The atom-optics knife-edge technique is more suitable for measuring the
momentum widths of atoms in waveguides [21]. Due to the need to generate a
potential barrier in the order of micron width, optical systems with large
numerical apertures are required, which is difficult to achieve for some
experimental systems due to the geometrical size restrictions. This method
also requires a series of experiments and then fitting the results.
We also looked at other measurement methods, such as detection of the change
of atomic densities in a bucket region and the spin gradient thermometry. The
former can do nearly nondestructive temperature measurements of cold atomic
ensembles, but the measurement range is hard to extend below the order of
nanokelvin [43]. The latter in principle allows measurement of temperatures
down to $50pK$, but can only be used under certain conditions [44]. In recent
years, quantum thermometry has aroused great interest in researchers, which
also has the potential to measure sub-nK temperature [45, 46, 47]. However, no
experiment so far has demonstrated its practical feasibility in ultracold
atoms.
Compared with the above methods, our approach is simple in structure and
timing control. For many cold atom experimental setups, one-dimensional
optical lattices already exist, so the measurements can be made with only the
appropriate timing. Since we measure the ratio of the population in each
momentum state, the fluctuation of atom number only has very slight effect.
In dealing with the case of BECs, we ignore the effects of atomic
interactions. The interaction between atoms in evolution provides an
additional phase. We use the chemical potential of $100pK$ as an example to
estimate the additional phase. Assuming the laser wavelength of the optical
lattice is $532nm$, the free flight of about 10ms after the double pulse
process can completely separate the center zero momentum part from the $\pm
2\hbar k_{0}$ momentum part in space. The maximum value of phase imprinting
after all the evolution time is $\frac{{k_{B}T\tau}}{\hbar}\approx
0.157\approx 0.05\pi$ and the actual inhomogeneous phase is less than this
value. This shows it is a reasonable approximation to ignore the interactions
between atoms.
We also estimate the effect of the optical lattice vibration during the actual
measurement period [48]. In the space station, the acceleration can be reduced
to no more than a magnitude of
$10^{-4}{m\mathord{\left/{\vphantom{m{s^{2}}}}\right.\kern-1.2pt}{s^{2}}}$
(RMS) at full frequency after vibration isolation [49]. In the double-pulse
beam-splitting period, the position of the optical lattice drifts in the order
of nanometer, which is much smaller than the lattice constant d of the optical
lattice. Thus, this method is valid to measure the momentum distribution of
ultra-cold atoms approximately $100pK$, even after taking into account the
actual vibrations on the space station.
In the experimental implementation, the shorter wavelength of optical lattice
laser is more advantageous, which will bring three benefits: 1) At the same
temperature, the shorter the wavelength of the optical lattice laser, the
smaller the random momentum of the atom relative to the recoil momentum of the
optical lattice. Therefore, the standing wave pulse can be better approximated
as a Hadamard operation. 2) The optical lattice transfers more recoil momentum
to the atoms, so that the complete separation of the wave packet takes less
time and the interaction between the atoms causes less impact. 3) The required
interval $\tau_{2}$ is shorter and the measurement results are less sensitive
to vibration.
Recently, we did a simulation experiment on an optical system. We also measure
the momentum width of BEC in the optical trap by the above method. Preliminary
experimental results demonstrate that the scheme is viable and can be used to
measure ultra-narrow momentum distribution.

##  VI VI. Conclusion

We have proposed a technique for characterizing ultralow-velocity width of
atomic gases, which requires only a one-dimensional optical lattice and simple
timing control. Since what we need to measure is the ratio of population of
each momentum state, the effect of atomic number fluctuation is not
significant. And, in principle, a single shot can obtain the outcome. The
scheme has the ability to measure ultra-cold atomic samples at temperatures
even lower than $100pK$, providing temperature calibration for cold atomic
physics experiments on the space station.

##  VII Acknowledgments

###### Acknowledgements.

We thank Prof. Changhe Zhou for helpful discussions and Ms. Xuemei Jia for her
help. This work was supported by the National Key Research and Development
Program of China (No. 2021YFA0718303) and the National Natural Science
Foundation of China under Grant No.11674338, No.U1730126 and No.11547024, and
funds provided by Technology and Engineering Center for Space Utilization,
Chinese Academy of Sciences.

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