# Study to improve the performance of interferometer with ultra-cold atoms

Xiangyu Dong1, Shengjie Jin1, Hongmian Shui1, Peng Peng1,
Xiaoji Zhou1,⋆
1State Key Laboratory of Advanced Optical Communication System and Network,
Department of Electronics, Peking University, Beijing 100871, China

> Ultra-cold atoms provide ideal platforms for interferometry. The macroscopic
> matter-wave property of ultra-cold atoms leads to large coherent length and
> long coherent time, which enable high accuracy and sensitivity to
> measurement. Here, we review our efforts to improve the performance of the
> interferometer. We demonstrate a shortcut method for manipulating ultra-cold
> atoms in an optical lattice. Compared with traditional ones, this shortcut
> method can reduce manipulation time by up to three orders of magnitude. We
> construct a matter-wave Ramsey interferometer for trapped motional quantum
> states and significantly increase its coherence time by one order of
> magnitude with an echo technique based on this method. Efforts have also
> been made to enhance the resolution by multimode scheme. Application of a
> noise-resilient multi-component interferometer shows that increasing the
> number of paths could sharpen the peaks in the time-domain interference
> fringes, which leads to a resolution nearly twice compared with that of a
> conventional double-path two-mode interferometer. With the shortcut method
> mentioned above, improvement of the momentum resolution could also be
> fulfilled, which leads to atomic momentum patterns less than 0.6 $\hbar
> k_{L}$. To identify and remove systematic noises, we introduce the methods
> based on the principal component analysis (PCA) that reduce the noise in
> detection close to the $1/\sqrt{2}$ of the photon-shot noise and separate
> and identify or even eliminate noises. Furthermore, we give a proposal to
> measure precisely the local gravity acceleration within a few centimeters
> based on our study of ultracold atoms in precision measurements.

Keywords: Precision Measurement, Ultra-cold atoms, Atomic interferometer,
Gravity measurements

PACS: 42.50.Dv, 67.10.Ba, 07.60.Ly, 91.10.Pp

##  1 Introduction

Precision measurement is the cornerstone of the development of modern physics.
Atom-based precision measurement is an important part. In recent years, ultra-
cold atoms have attracted extensive interest in different fields (?, ?, ?, ?),
because of their macroscopic matter-wave property (?, ?). This property will
lead to large coherent length and long coherent time, which enable high fringe
contrast (?, ?, ?). Hence, with these advantages, ultra-cold atoms provide
ideal stages for precision measurement (?, ?) and have numerous applications
(?, ?, ?, ?, ?, ?, ?, ?), ranging from inertia measurements (?, ?, ?) to
precision time keeping (?, ?). For example, a Bose-Einstein condensate is used
as an atomic source for a high precision sensor, which is released into free
fall for up to 750 ms and probed with a $T=130$ ms Mach-Zehnder atom
interferometer based on Bragg transitions (?). A trapped geometry is realized
to probe gravity by holding ultra-cold cesium atoms for 20 seconds (?), which
suppresses the phase variance due to vibrations by three to four orders of
magnitude, overcoming the dominant noise source in atom-interferometric
gravimeters. With ultra-cold atoms in an optical cavity, the detection of weak
force can achieve a sensitivity of 42 $\rm{yN}/\sqrt{\rm{Hz}}$, which is a
factor of 4 above the standard quantum limit (?).

With the superiority mentioned above, it is obvious to consider an
interferometer by ultra-cold atoms to further precision measurements.
Interferometers with atoms propagating in free fall are ideally suited for
inertia measurements (?, ?, ?, ?, ?, ?, ?). Meanwhile, with atoms held in
tight traps or guides, they are better to measure weak localized interactions.
For example, a direct measurement to the Casimir-Polder force is performed by
I. Carusotto et al. in 2005, which is as large as $10^{-4}$ gravity (?).
However, ultra-cold atoms still get some imperfections needed to be surmounted
when combining with interferometry. The macroscopic matter-wave property is
generated simultaneously with non-linear atom-atom interactions. Phase
diffusion caused by interactions limits the coherence time, and ultimately
restricts the sensitivity of interferometers. Besides the interrogation time,
the momentum splitting as well as the path number also has an impact on the
sensitivity. It has been demonstrated by experiments of multipath
interferometers that, interferometric fringes can be sharpened due to the
higher-harmonic phase contributions of the multiple energetically equidistant
Zeeman states (?, ?), whereas a decrease in the average number of atoms per
path causes a greater susceptibility to shot noise. Equilibrium between these
parameters could lead to an optimal resolution. In addition, we should also
pay attention to the signal analysis procedure as the interferometric
information is mainly extracted from the signal detected. The resulting
resolution severely relies on the probing system.

In this review, we mainly introduce our experimental developments that study
these fundamental and important issues to improve the performance of
interferometer with ultra-cold atoms. The main developments are concentrated
in three aspects: increasing coherent time, using multimode scheme and
reducing systematic noises.

A. Enhanced resolution by increasing coherent time. We introduce an effective
and fast (few microseconds) methods, for manipulating ultra-cold atoms in an
optical lattice (OL), which can be used to construct the atomic interferometer
and increase the coherent time to finally get a higher resolution. This
shortcut loading method is a designed pulse sequence, which can be used for
preparing and manipulating arbitrary pure states and superposition states.
Another advantage of this method is that the manipulation time is much shorter
than traditional methods (100 ms$\to$100 $\rm{\mu s}$). Based on this shortcut
method, we constructed an echo-Ramsey interferometer (RI) with motional Bloch
states (at zero quasi-momentum on S- and D-bands of an OL) (?). Thanks to the
rapidity of shortcut methods, more time could be used for the RI process. We
identified the mechanisms that reduced the RI contrast, and greatly increased
the coherent time (1.3 ms $\to$14.5 ms) by a quantum echo process, which
eliminated the influence of contrast attenuation mechanisms mostly.

B. Enhanced resolution by multimode scheme. Several efforts have been made to
avoiding the decays of interferometric resolution because of the experimental
noises. We demonstrated that the improvement of the phase resolution could be
accomplished by a noise-resilient multi-component interferometric scheme. With
the relative phase of different components remaining stable, increasing the
number of paths could sharpen the peaks in the interference fringes, which led
to a resolution nearly twice compared with that of a conventional double-path
two-mode interferometer. Moreover, improvement of the momentum resolution was
fulfilled with optical lattice pulses. We got results of atomic momentum
patterns with intervals less than the double recoil momentum. The momentum
pattern exhibited 10 main peaks.

C. Enhanced resolution by removing the systematic noise. The method to
identify and remove systematic noises for ultra-cold atoms is also introduced
in this paper. For improving the quality of absorption image, which is the
basic detection result in ultra-cold atoms experiments, we developed an
optimized fringe removal algorithm (OFRA), making the noise close to the
theoretical limit as $1/\sqrt{2}$ of the photon-shot noise. Besides, for the
absorption images after preprocessing by OFRA, we applied the principal
component analysis to successfully separate and identify noises from different
origins of leading contribution, which helped to reduce or even eliminate
noises via corresponding data processing procedures.

Furthermore, based on our study of ultracold atoms in precision measurements,
we demonstrated a scheme for potential compact gravimeter with ultra-cold
atoms in a small displacement.

The text structure is as follows. In Sec.2, a shortcut method manipulating
ultra-cold atoms in an optical lattice and an Echo-Ramsey interferometry with
motional quantum states are introduced, which can increase the coherent time.
In Secs.3, we prove that the resolution can be increased using a double-path
multimode interferometer with spinor Bose-Einstein condensates (BECs) or an
optical pulse, both of which can be classified into multimode scheme. In
Secs.4, methods for identifying and reducing the systematic noises for ultra-
cold atoms are demonstrated. Finally, we give a proposal on gravimeter with
ultra-cold atoms in Secs.5.

##  2 Enhanced resolution by increasing coherent time

Figure 1: Schematic diagram of the shortcut method (take ground state
preparation as an example). (a) At the beginning, the BECs are formed in a
weak harmonic trap. (b) Time sequence of shortcut method. (c) Mapping the
shortcut process onto the Bloch sphere. The track $A\to C\to|S\rangle$ and
track $A\to B\to E\to M\to|S\rangle$ represent one pulse and two pulses
shortcut process, respectively. (d) After this shortcut process, the desired
states of an 3D optical lattice are prepared. (e) Band structure of 1D OL with
different quasi-momentum $q$ when $V_{0}=10\;E_{r}$. Reproduced with
permission from Ref. (?).

The macroscopic coherent properties of ultra-cold atoms (?, ?, ?, ?, ?, ?, ?,
?, ?, ?, ?) are conducive to precise measurement. To make full use of the
advantages of ultra-cold atomic coherence properties, one method is to reduce
the manipulation time and another is to suppress the attenuation of coherence.

Firstly, we demonstrated a shortcut process for manipulating BECs trapped in
an OL (?, ?, ?). By optimizing the parameters of the pulses, which constitute
the sequence of the shortcut process, We can get extremely high fidelity and
robustness for manipulating BECs into the desired states, including the ground
state, excited states, and superposition states of a one, two or three-
dimensional OLs. Another advantage of this method is that the manipulation
time is much shorter than that in traditional methods (100 ms$\to$100 $\rm{\mu
s}$).

This shortcut is composed of optical lattice pulses and intervals that are
imposed on the system before the lattice is switched on. The time durations
and intervals in the sequence are optimized to transfer the initial state to
the target state with high fidelity. This shortcut procedure can be completed
in several tens of microseconds, which is shorter than the traditional method
(usually hundreds of milliseconds). It can be applied to the fast manipulation
of the superposition of Bloch states.

Then, based on this method, we constructed an echo-Ramsey interferometer (RI)
with motional Bloch states (at zero quasi-momentum on S- and D-bands of an OL)
(?). The key to realizing a RI is to design effective $\pi$\- and $\pi/2$
pulses, which can be obtained by the shortcut method (?, ?, ?). Thanks to the
rapidity of shortcut methods, more time can be used for the RI process. We
identified the mechanisms that reduced the RI contrast, and greatly increased
the coherent time (1.3 ms $\to$14.5 ms) by a quantum echo process, which
eliminated the influence of most contrast attenuation mechanisms.

###  2.1 Shortcut manipulating ultra-cold atoms

Efficient and fast manipulation of BECs in OLs can be used for precise
measurements, such as constructing atom-based interferometers and increasing
the coherent time of these interferometers. Here we demonstrate an effective
and fast (around 100 $\rm{\mu s}$) method for manipulating BECs from an
arbitrary initial state to a desired OL state. This shortcut method is a
designed pulse sequence, in which the parameters, such as duration and
interval of each step, are optimized to maximize fidelity and robustness of
the final state. With this shortcut method, the pure Bloch states with even or
odd parity and superposition states of OLs can be prepared and manipulated. In
addition, the idea can be extended to the case of two- or three-dimensional
OLs. This method has been verified by experiments many times and is very
consistent with the theoretical analysis (?, ?, ?, ?, ?, ?, ?, ?).

We used the simplest one-dimensional standing wave OL to demonstrate the
design principle of this method. The OL potential is
$V_{OL}(x)=V_{0}\cos^{2}{kx}$, where $V_{0}$ is used to characterize the depth
of the OL.

Supposing that the target state $\left|{\psi_{a}}\right\rangle$ is in the OL
with depth $V_{0}$, $m$-step preloading sequence has been applied on the
initial state $\left|\psi_{i}\right\rangle$. The final states
$\left|{\psi_{f}}\right\rangle$ is given by

$\centering|\psi_{f}\rangle=\prod_{j=1}^{m}\hat{Q}_{j}|\psi_{i}\rangle,\@add@centering$
(1)

where $\hat{Q}_{j}=e^{-i\hat{H}_{j}t_{j}}$ is the evolution operator of the
$j$th step. By maximizing the fidelity

$Fidelity=|\langle\psi_{a}|\psi_{f}\rangle|^{2},$ (2)

we can get the optimal parameters $\hat{H}_{j}$ and $t_{j}$.

This preprocess is called a shortcut method, which can be used for loading
atoms into different bands of an optical lattice. For example, the shortcut
loading ultra-cold atoms into S-band in a one-dimensional optical lattice is
shown in Fig. 1.

By setting different initial state and target state, different time sequences
can be designed to manipulate atoms, to build different interferometers, which
greatly saves the coherent time. Based on this shortcut method, we can prepare
exotic quantum states (?, ?, ?) and construct interferometer with motional
quantum states of ultra-cold atoms (?).

###  2.2 Increasing coherent time in a Ramsey interferometry with motional
Bloch states of ultra-cold atoms

Suppressing the decoherence mechanism in the atomic interferometer is
beneficial for increasing coherence time and improving the measurement
accuracy. Here we demonstrated an echo method can increase the coherent time
for Ramsey interferometry with motional Bloch states (at zero quasi-momentum
on S- and D-bands of an OL) of ultra-cold atoms (?). The RI can be applied to
the measurement of quantum many-body effects. The key challenge for the
construction of this RI is to achieve $\pi$\- and $\pi/2$-pulses, because
there is no selection rule for Bloch states of OLs. The $\pi$\- or
$\pi/2$-pulse sequences can be obtained by the shortcut method (?, ?, ?, ?,
?), which precisely and rapidly manipulates the superposition of BECs at the
zero quasi-momentum on the 1st and 3rd Bloch bands. Retaining the OL, we
observed the interference between states and measured the decay of coherent
oscillations.

We identified the mechanisms that reduced the RI contrast: thermal
fluctuations, laser intensity fluctuation, transverse expansion induced by
atomic interaction, and the nonuniform OL depth. Then, we greatly increased
the coherent time (1.3 ms $\to$14.5 ms) by a quantum echo process, which
eliminated the influence of most contrast attenuation mechanisms.

####  2.2.1 Ramsey interferometer in an optical lattice

This RI starts from BECs of ${}^{87}\rm{Rb}$ at the temperature 50 nK similar
to our previous work (?, ?, ?, ?, ?, ?, ?, ?, ?). Then a 1D standing wave OL
is formed (Fig. 2). After a shortcut sequence, the BEC is transferred into the
ground band of the OL, denoted as $\phi_{S,0}$.

Figure 2: Experimental configuration for a Ramsey interferometer in a
$V_{0}=10\;E\mathrm{r}$ lattice: (a) The BEC is divided into discrete pancakes
in $yz$ plane by an 1-dimensional optical lattice along $x$ axis with a
lattice constant $d=426$ nm. (b) Band energies for the S-band and the D-band.
(c) Time sequences for the Ramsey interferometry. The atoms are first loaded
into the S band of OL, followed by the RI sequence: $\pi/2$ pulse, holding
time $t_{OL}$, and the second $\pi/2$ pulse. Finally band mapping is used to
detect the atom number in the different bands. (d) The used pulse sequences
designed by an optimised shortcut method. Reproduced with permission from Ref.
(?). Figure 3:  (a) Change of $p_{\mathrm{D}}$, the population of atoms in the
D-band, over time $t_{OL}$ with temperature $T=50$ nK. (b) Influence of
different mechanisms on the RI. (c) Characteristic time $\tau$ for the
different number of $\pi$ pulse $n$ and different temperatures. The circles,
squares, and diamonds represent the experimental results and lines are fitting
curves. Reproduced with permission from Ref. (?).

Fig. 2(b) illustrates that the RI is constructed with Bloch states
$\phi_{i,q}$, which includes the ground band $|S\rangle$, the third band
$|D\rangle$, and their superposition state
$\psi=a_{S}|S\rangle+a_{D}|D\rangle$ (denoted as $\binom{a_{S}}{a_{D}}$). It
is difficult to realize the interferometer with this pseudo-spin system,
because there is no selection rule for Bloch states of OLs. However, thanks to
the existence of a coherent, macroscopic matter-wave, a $\pi/2$-pulse for BECs
in an OL can be obtained, where the Bloch states $|S\rangle$ and $|D\rangle$
are to be manipulated to $|\psi_{1}\rangle$=$(|S\rangle+|D\rangle)/\sqrt{2}$
and $|\psi_{2}\rangle$=$(-|S\rangle+|D\rangle)/\sqrt{2}$, respectively. Fig.
2(d) shows the $\pi/2$ pulse we used for the RI (?, ?, ?) with fidelities of
$98.5\%$ and $98.0\%$ respectively (?, ?, ?, ?).

Fig. 2(c) illustrates the whole process of RI. First, the BEC is transferred
to $|S\rangle$ of an OL. Then, a $\pi/2$-pulse $\hat{L}(\pi/2)$ is applied to
atoms, where $\hat{L}(\pi/2)\binom{1}{0}=\frac{1}{\sqrt{2}}\binom{1}{1}$.
After holding time $t_{OL}$ and another $\pi/2$-pulse, the state is:

$\psi_{f}=\hat{L}(\pi/2)\hat{Q}(t_{OL})\hat{L}(\pi/2)\psi_{i},$ (3)

where
$\hat{L}(\alpha)=(\cos\frac{\alpha}{2}-i\sin\frac{\alpha}{2})\hat{\sigma}_{y}$.
The operator $\hat{Q}(t_{OL})=(\cos\omega t+i\sin\omega t)\hat{\sigma}_{z}$
($\omega$ corresponds to the energy gap between S and D bands at zeros quasi-
momentum).

Fig. 3(a) depicts the results $p_{D}(t_{OL})=N_{D}/(N_{S}+N_{D})$ at different
$t_{OL}$ for the RI process ($\hat{R}(\pi/2)-\hat{U}(t_{OL})-\hat{R}(\pi/2)$).
$N_{\mathrm{S}}$ ($N_{\mathrm{D}}$) represents the number of atoms in S-band
(D-band). The period of the oscillation of $p_{D}$ is $41.1\pm 1.0\;\mu s$.
This period related to the band gap and the theoretical value is $40.8\;\mu
s$. From Fig. 3(a), we can see that the amplitude, or the contrast
$C(t_{OL})$, decreases with the increase of $t_{OL}$, where

$p_{D}(t_{OL})=[1+C(t_{OL})\cos(\omega t_{OL}+\phi)]/2.$ (4)

We defined a characteristic time $\tau$, which corresponds to the time when
the $C(t_{OL})$ decreases to $1/e$. Temperature can affect the length of
$\tau$.

####  2.2.2 Contrast decay mechanisms

To improve RI’s coherent time and performance, we should analyze the
mechanisms that cause RI signal attenuation. By solving the Gross-Pitaevskii
equation(GPE), which considers the mechanism that may lead to decay, we can
get the process of contrast decay in theory. In Fig. 3(b), The following
mechanisms are introduced in turn: the effect of the imperfection of the
$\pi/2$ pulse (brown dashed line), inhomogeneity of laser wavefront (blue
dotted line), the transverse expansion caused by the many-body interaction
(blue dashed line), laser intensity fluctuation (the dash-dotted line), and
the thermal fluctuations (the orange solid line). Fig. 3(b) illustrates that
the theoretical (the orange solid line) and experimental (black dots) curves
of the final result are very consistent.

####  2.2.3 An echo-Ramsey interferometer with motional Bloch states of BECs

In order to extend the coherence time $\tau$, we proposed a quantum echo
method. The echo process refers to a designed $\pi$ pulse ($\hat{L}(\pi)$)
that flips the atomic populations of the two bands. So the evolution operator
of Echo-RI is
$\hat{L}(\pi/2)[\hat{Q}(t_{OL}/2n)\hat{L}(\pi)\hat{Q}(t_{OL}/2n)]^{n}\hat{L}(\pi/2)$,
where $n$ is the number of the $\pi$ pulse inserted between the two $\pi/2$
pulses.

Table 1: The effects for the contrast decay. Decay factor | Beam inhomogeneity | Echo recovery
---|---|---
$Dephasing$ | Momentum dispersion | Yes
$Collision$ | Unbalance of population | Yes
$Decoherence$ | fluctuation | No

Fig. 3(c) illustrates the characteristic time $\tau$ for different $n$ and
temperatures. And the effects for the contrast decay are listed in Table. 1.
It can be seen from Fig. 3(c) that the interferometer with the longest
characteristic time (14.5 ms) was obtained when $n\geq 6$ and $T=50$ nK.

##  3 Enhanced resolution by multimode scheme

As an essential indicator, the resolution evaluates the performance of
interferometers. The resolution is theoretically restricted to shot-noise
limit, or sub-shot noise limit (?, ?), however, it will decay easily due to
other experimental noises, with those upper limits beyond reach. Therefore, we
have made several efforts to increase the resolution in practice. Improvement
of the phase resolution was accomplished by a noise-resilient multi-component
interferometric scheme. With the relative phase of different components
remaining stable, increasing the number of paths could sharpen the peaks in
the interference fringes, which leads to a resolution nearly twice compared
with that of a conventional double-path two-mode interferometer with hardly
any attenuation in visibility. Moreover, improvement of the momentum
resolution is fulfilled with optical lattice pulses. Under the condition of 10
$\rm{E_{R}}$ OL depth, atomic momentum patterns with interval less than the
double recoil momentum can be achieved, exhibiting 10 main peaks,
respectively, where the minimum one we have given was 0.6 $\hbar k_{L}$. The
demonstration of these techniques is shown in the next four subsections.

###  3.1 Time evolution of two-component Bose-Einstein condensates with a
coupling drive

For the multicomponent interferometer, it is necessary to study the
interference characteristics of multi-component ultra-cold atoms. Here we
introduced a basic method to deal with this problem, which simulates the time
evolution of the relative phase in two-component Bose-Einstein condensates
with a coupling drive (?).

We considered a two-component Bose-Einstein condensate system with weak
nonlinear interatomic interactions and coupling drive. In the formalism of the
second quantization, the Hamiltonian of such a system can be written as

$\hat{H}=\hat{H}_{1}+\hat{H}_{2}+\hat{H}_{int}+\hat{H}_{driv},$ (5)
$\hat{H}_{i}=\int
dx\Psi_{i}^{\dagger}(x)[-\frac{\hbar^{2}}{2m}\nabla^{2}+V_{i}(x)+U_{i}(x)\Psi_{i}^{\dagger}(x)\Psi_{i}(x)]\Psi_{i}(x),$
(6) $\hat{H}_{int}=U_{12}\int
dx\Psi_{1}^{\dagger}(x)\Psi_{2}^{\dagger}(x)\Psi_{1}(x)\Psi_{2}(x),$ (7)
$\hat{H}_{driv}=\int
dx[\Psi_{1}^{\dagger}(x)\Psi_{2}(x)e^{i\omega_{rf}t}+\Psi_{1}(x)\Psi_{2}^{\dagger}(x)e^{-i\omega_{rf}t}],$
(8)

where $i=1$ and $2$.

Then the interference between two BEC’s is

$I(t)=\frac{1}{2}N+\frac{1}{2}(N_{1}-N_{2})\cos{\omega_{rf}t}+\frac{1}{2}e^{-A(t)}\sin{\omega_{rf}t}\mathcal{R}(t).$
(9)

Previous analysis can be used to simulate the time evolution of the relative
phase in two-component Bose-Einstein condensates with a coupling drive, as
well as to study the interference of multi-component ultra-cold atoms. This
simulation would help to construct a multimode interferometer of a spinor BEC
(see Subsecs. 3.3).

###  3.2 Parallel multicomponent interferometer with a spinor Bose-Einstein
condensate

Figure 4:  (a) One typical interference picture. These spatial interference
fringes come from the five sub-magnetic states of $\left|F=2\right\rangle$
hyperfine level. (b1-5) Density distributions corresponding to different sub-
magnetic components respectively, where the points are the experimental data
and the curves are fitting results according to the empirical expression (?,
?, ?). (c) Average of 15 consecutive experimental shots with a visibility
reduction to zero for the chosen state $\left|m_{F}=-1\right\rangle$.
Reproduced with permission from Ref. (?).  Figure 5:  (a1)-(a4) Histograms of
relative phases distributions
$(\phi_{2}-\phi_{1},\phi_{-2}-\phi_{-1},\phi_{2}-\phi_{-2},and\;\phi_{1}-\phi_{-1})$
respectively. These relative phases show good reproducibility, for the first
two are concentrated at about $0^{o}$, while the latter two are concentrated
at about $180^{o}$ in 61 consecutive experimental shots; (b) Relative phase
distributions of 41 consecutive experimental shots with $t_{0}=3.6$ ms.
Distributions of relative phases $\phi_{2}-\phi_{1}$ and $\phi_{1}-\phi_{-1}$
are shown in (b1) and (b2); (c)When $t_{0}=3.5$ ms, distributions of relative
phases $\phi_{2}-\phi_{1}$ and $\phi_{1}-\phi_{-1}$ are shown in (c1) and
(c2). The polar plots of relative phase vs visibility (shown as angle vs
radius) are shown as these insets, respectively, where the value of visibility
is an average of the visibility involved in calculation. Reproduced with
permission from Ref. (?).

Revealing the wave-particle duality, Young’s double-slit interference
experiment plays a critical role in the foundation of modern physics. Other
than quantum mechanical particles such as photons or electrons which had been
proved in this stunning achievement, ultra-cold atoms with long coherent time
have got the potential of precision measurements when utilizing this
interferometric structure. Here we have demonstrated a parallel multi-state
interferometer structure (?) in a higher spin atom system (?, ?, ?), which was
achieved by using our spin-2 BEC of ${}^{87}\rm{Rb}$ atoms.

The experimental scheme is described as following. After the manufacture of
Bose-Einstein condensates in an optical-magnetic dipole trap, we switched off
the optical harmonic trap and populate the condensates from
$\left|F=2,m_{F}=2\right\rangle$ state to $\left|m_{F}=2\right\rangle$ and
$\left|m_{F}=1\right\rangle$ sub-magnetic level equally. After the evolution
in a gradient magnetic field for time $t_{1}$, these two wave packets were
converted again into multiple $m_{F}$ states ($m_{F}=\pm 2,\pm 1,0$) as our
spin states, leading to the so-called parallel path. All these states were
allowed to evolve for another period time $t_{2}$, then the time-of-flight
(TOF) stage $t_{3}$ for absorption imaging. Spatial interference fringes had
been observed in all the spin channels. Here, we used the technique of spin
projection with Majorana transition (?, ?, ?) by switching off the magnetic
field pulses nonadiabatically to translate the atoms into different Zeeman
sublevels. The spatial separation of atom cloud in different Zeeman states was
reached by Stern-Gerlach momentum splitting in the gradient magnetic field.

A typical picture after 26 ms TOF is shown in Fig. 4(a). Fig. 4(b1-b5) are the
density distributions for each interference fringes. To reach the maximal
visibility, we studied the correlation between the interference fringes’
visibility and the time interval applying Stern-Gerlach process. Though
separated partially, the interfering wave pockets must overlap in a sort of
way. The optimal visibility was about 0.6, corresponding to
$t_{1}=210\;\rm{\mu s}$ and $t_{2}=1300\;\rm{\mu s}$. We also measured the
fringe frequencies of different components, which exhibited a weak dependence
on $m_{F}$.

Special attention is required in Fig. 4(c). After an average of 15 consecutive
CCD shots in repeated experiments, the interference fringe almost disappeared
for the chosen state $\left|m_{F}=-1\right\rangle$. This result manifested the
phase difference between the two copies of each component in every
experimental run is evenly distributed. The poor phase repeatability could be
attributed to uncontrollable phase accumulation in Majorana transitions.

However, the relative phase across the spin components remained the same after
more than 60 continuous experiments, just as Fig. 5(a) illustrates.
Furthermore, evidence has been spotted that the relative phase can be
controlled by changing the time $t_{0}$ before the first Majorana transition,
as shown in Fig. 5(b)(c), paving a way towards noise-resilient multicomponent
parallel interferometer or multi-pointer interferometric clocks (?).

###  3.3 Implementation of a double-path multimode interferometer using a
spinor Bose-Einstein condensate

Figure 6:  (a1-a3) Single-shot spatial interference pattern with five
interference modes after TOF = 26 ms. Fringes of each mode are (a1)in phase
(a2)partially in phase (a3)complementary in space. (b1-b3) Black points are
the experimental data by integrating the image in panels (a1-a3) along the z
direction. Red solid lines are fitted by Thomas-Fermi Distribution (?).
Visibilities are 0.55, 0.24, and 0.05, respectively. (c) Schematic of the
spatial interference image. $\Delta\phi(T_{d},T_{N})$ is the relative phase
between adjacent mode fringes. The fringe in each color represents the
interference between the two wave packets of a single mode. Reproduced with
permission from Ref. (?).  Figure 7:  Dependence of the visibility on the
number of modes N and initial relative phase $\phi_{m_{F}}$ of the same mode
in two paths. (a) Dependence on N in a situation that $\phi_{m_{F}}$ are all
zero. FWHM of the N-mode fringe is 2/N times that of the two-mode fringe. (b)
Dependence on $\phi_{m_{F}}$ using $N=4$ as an example. The green dashed line,
red solid line, and purple dotted line show the fringes with
$(\phi_{1},\phi_{2},\phi_{3},\phi_{4})=(0,0,0,0)$, $(0,0,\pi,\pi)$, and
$(0.7\pi,0.2\pi,0.5\pi,\pi)$, respectively. Reproduced with permission from
Ref. (?).

The experiment described above was achieved by Stern-Gerlach momentum
splitting, separating the wave pockets in different spin states or Zeeman sub-
magnetic states in space. The conclusion that relative phases across the spin
components remain stable gives us an inspiration to carry on the double-path
multimode matter wave interferometer scheme. With the number of paths
increased, it will suppress the noise and improve the resolution (?, ?, ?, ?,
?, ?) compared with the conventional double-path single-mode structure. The
results show that resolution of the phase measurements is increased nearly
twice in time domain interferometric fringes (?).

The experimental procedure is similar to the previous one. The major
difference lies in the splitting stage, during the optical harmonic trap
participating in the preparation of the condensates is not going to switch off
until the TOF stage, thus the Stern-Gerlach process in the gradient magnetic
field mentioned above cannot significantly split the wave packets. With
different momentum atomic clouds are spatially separated only for tens of
nanometers, approximately 1% of the BEC size, thus well overlapped (?). As a
result, multi-modes from two paths will interfere in one region instead of
five. Another difference lies in the second spin projection with non-adiabatic
Majorana transition. Here we replace it with a radio frequency pulse for its
higher efficiency as a 1 to 5 beam splitter, although that we still use it to
transfer the initial condensates into $\left|m_{F}=2\right\rangle$ and
$\left|m_{F}=1\right\rangle$ sub-magnetic levels. The performance of Majorana
transition is better than RF pulse as a 1 to 2 beam splitter. There are also
some changes with experimental parameters that count a little and we would not
discuss them here.

Hence the global view of our interferometer is as follows: The magnetic
sublevels are considered as modes in the interferometer, each has its own
different phase evolution rates in gradient magnetic field. The double path
configuration is made up of Majorana transition as well as the evolution of
the first two $m_{F}$ superposed states during time $T_{d}$, makes up (path I,
path II). RF pulse leading to the multiple $m_{F}$ superposed states together
with their evolution in time $T_{N}$ forms the multi-modes configuration.
During the TOF stage, atomic clouds expand and interfere with each other.
Owing to the different state-dependent phase evolution rate
$\omega_{m_{F}}^{(I,II)}$, the absorption image shows something more than
spatial interference fringes, which is a periodic dependence of the visibility
on phase evolution time as the function $V_{N}(T_{d},T_{N})$. We refer to it
as the time domain interference.

Fig. 6(a) shows a group of absorption images with various combinations of
$T_{d}$, $T_{N}$. The observed fringe is a superposition of the interference
fringes of different modes. Consequently, the visibility depends on the
relative phase $\Delta\phi(T_{d},T_{N})$ between the interference fringes of
each mode [Fig. 6(c)] and can also be modulated.

By carefully analyzing with expression
$V_{N}(T_{d},T_{N})=\langle\Psi^{(I)}|\Psi^{(II)}\rangle$[34], we can acquire
the expression of the relative phase between two adjacent components:

$\displaystyle\begin{split}\Delta\phi(T_{d},T_{N})&=(\Delta\phi_{m_{F}}-\Delta\theta_{m_{F}-1})\\\
&=\Delta\omega T_{N}+\Delta\theta\end{split}$ (10)

where $\Delta\omega$ is the relative phase evolution rate between the two
paths, $\Delta\theta$ is the relative initial phase introduced through the
double path stage $T_{d}$. Yet we have already demonstrated that the
visibility $V_{N}$ is modulated with the period $2\pi/\Delta\omega$ along with
how the time domain fringe emerges theoretically.

A remarkable feature of the multi-modes interferometer is the enhancement of
resolution, which is defined as (fringe period)/(full width at half maximum).
We have investigated the resolution of the time domain fringe experimentally
and theoretically. It can be influenced by parameters like modes number and
initial phase, which is $R(N,\phi_{m_{F}})$. $\phi_{m_{F}}$ refers to the
initial relative phase of $m_{F}$ states accumulated in double paths $T_{d}$.

Fig. 7 is the numerical results considering an arbitrary number of modes. Fig.
7(a) is under the condition that the phases $\phi_{m_{F}}$ are all the same
for any modes. In that case, if we denote $\Delta\omega T_{N}=2n\pi/N$, then
the visibility achieves $V_{N}=1$ when n is the multiple of N and a major peak
is observed in this case. A remarkable feature of our interferometer is the
enhancement of resolution by $N/2$ times without any changes in visibility nor
periodical time. It is the harmonics that cause the peak width to decrease
with the number of modes increasing in this case (?). Fig. 7(b) indicates
$\phi_{m_{F}}$ varies from mode to mode for comparison. Neither the maximum
visibility $V_{N}=1$ nor the minimum could be reached. Meanwhile, the time
domain fringe shows more than one main peak in one period. Therefore, the
initial phase $\phi_{m_{F}}$ needs to be well controlled to achieve the
highest possible visibility and clear interference fringe in the time domain.

We also experimentally study the time domain fringes. The experimental data
(not depicted here) coincides with the numerical results of Fig. 7(b) red
line, testifying its superiority to the resolution of the phase measurement.
Moreover, the relative phase evolution rate $\Delta\omega$ can be controlled
by adjusting the difference between the two paths accumulated in $T_{d}$ stage
(?, ?, ?). With enhanced resolution, the sensitivity of interferometric
measurements of physical observables can also be improved by properly
assigning measurable quantities to the relative phase between two paths, as
long as the modes do not interact with each other (?, ?, ?).

###  3.4 Atomic momentum patterns with narrower interval

Figure 8:  (a) Shortcut method for loading atoms: (a1) after the first two
pulses and the $30$ ms holding time in the OL and the harmonic trap, the state
becomes the superposition of the Bloch states in S-band with quasi-momenta
taking the values throughout the FBZ, and is denoted by
$\left|{\psi\left(0\right)}\right\rangle$. Then $1$ms band mapping is added.
(a2) The single pulse acted on the superposed state
$\left|{\psi\left(0\right)}\right\rangle$. (b) the superposed Bloch states of
S-band spreading in the FBZ (black circles). The top Patterns in (c) and (d)
are the TOF images in experiments. The lower part of (c) and (d) depicts the
atomic distributions in experiments (red circles) and theoretical simulations
(blue solid lines). There are seven peaks in (c) and ten peaks in (d) with
$q=\pm 3\;\hbar k_{L}$. Reproduced with permission from Ref. (?).

For ultra-cold atoms used in precise measurement, improving the precision of
momentum manipulation is also conducive to improving the measurement
resolution. The method to get atomic momentum patterns with narrower interval
has been proposed and verified by experiments (?). Here we applied the
shortcut pulse to realize the atomic momentum distribution with high
resolutions for a superposed Bloch states spreading in the ground band of an
OL.

While difficult to prepare this superposition of Bloch states, it can be
overcome by the shortcut method. First, the atoms are loaded in the
superposition of S- and D-bands $(|S,q=0\rangle+|D,q=0\rangle)/\sqrt{2}$,
where $q$ is the quasi-momentum. Fig. 8(a1) depicts the loading sequence. The
atoms in S and D bands Collision between atoms in S and D bands will cause the
atoms to gradually transfer to S band with non-zero quasi-momentum. After $30$
ms, as shown in Fig. 8(b), atoms cover the entire ground band from $q=-\hbar
k_{L}$ to $\hbar k_{L}$.

The momentum distribution of the initial state is a Gaussian-like shape. After
an OL standing-wave pulse, which is similar to that in the shortcut process,
the different patterns with the narrower interval can be obtained. The
standing-wave pulse sequence is shown in Fig. 8(a2). Fig. 8(c) and (d) show
the different designs for patterns of multi modes with various numbers of
peaks under OL depth 10 $E_{r}$, where the top figures are the absorption
images after the pulse and a $25$ ms TOF. The red circles are the experimental
results of the atomic distribution along the x-axis from the TOF images. These
results are very close to the numerical simulation result (blue lines). For
the numerical simulation, we can get the initial superposition of states by
fitting the experimental distributions (Fig. 8(b)). Fig. 8(c) and (d) depict
the atomic momentum distribution with resolutions of $0.87\;\hbar k_{L}$
interval (seven peaks within $q=\pm 3\;\hbar k_{L}$) and $0.6\;\hbar k_{L}$
(ten peaks within $q=\pm 3\;\hbar k_{L}$) interval, respectively.

The superposed states with different quasi-momenta in the ground band cause
the narrow interval (far less than double recoil momentum) between peaks,
which is useful to improve the resolution of atom interferometer (?).

##  4 Enhanced resolution by removing the systematic noise

Noise identification as well as removal is crucial when extracting useful
information in ultra-cold atoms absorption imaging. In general concept,
systematic noises of cold atom experiments originate from two sources, one is
the process of detection, such as optical absorption imaging; the other is the
procedure of experiments, such as the instability of experimental parameters.
Here we provided an OFRA scheme, reducing the noise to a level near the
theoretical limit as $1/\sqrt{2}$ of the photo-shot noise. When applying the
PCA, we found that the noise origins, which mainly come from the fluctuations
of atom number and spatial positions, much fewer than the data dimensions of
TOF absorption images. These images belong to BECs in one dimensional optical
lattice, where the data dimension is actually the number of image pixels. If
the raw TOF data can be preprocessed with normalization and adaptive region
extraction methods, these noises can be remarkably attenuated or even wiped
out. PCA of the preprocessed data exhibits a more subtle noises structure.
When we compare the practical results with the numerical simulations, the few
dominant noise components reveal a strong correlation with the experimental
parameters. These encouraging results prove that the OFRA as well as PCA can
be a promising tool for analysis in interferometry with higher precision (?,
?).

###  4.1 Optimized fringe removal algorithm for absorption images

Optical absorption imaging is an important detection technique to obtain
information from matter waves experiments. By comparing the recorded detection
light field with the light field in the presence of absorption, we can easily
attain the atoms’ spatial distribution. However, due to the inevitable
differences between two recorded light field distributions, detection noises
are unavoidable.

Figure 9: Comparison between ordinary method and OFRA method. The integral of
the atomic distribution in the red box in (a1) and (b1) correspond to (a2) and
(b2). The atomic distribution (blue dots) is fitted by a bi-modal function to
extract the temperature of atoms, which is shown in (c). Reproduced with
permission from Ref. (?).

Therefore, we have demonstrated an OFRA scheme to generate an ideal reference
light field. With the algorithm, noise generated by the light field difference
could be eliminated, leading to a noise close to the theoretical limit (?).
The OFRA scheme is based on the PCA, we confirmed its validity by experiments
of triangular optical lattices. The experimental configuration has been
described in our prior work (?, ?). When the experiment was in process, the
depth of the lattice was adiabatically raised to a final value, followed by a
hold time of 20 ms to keep the atoms in the lattice potential before the
optical absorption imaging. There are several parameters to characterize the
triangle lattice system (?), among which the visibility, the condensate
fraction, and the temperature matter. Fig. 9 (a2) and Fig. 9 (b2) are bimodal
fitting to the scattering peaks by summing up the atomic distribution within
the red box in the direction perpendicular to the center. Here Fig. 9 (a)
stands for the common way of calculation and 9(b) for the OFRA. The bi-modal
curve is composed of two parts: a Gaussian distribution for the thermal
component and an inverse parabola curve for the condensed atoms. For each
part, the column densities along the imaging axis can be written as

$\displaystyle n_{th}(x)$ $\displaystyle=$
$\displaystyle\frac{n_{th}(0)}{g_{2}(1)}g_{2}[\exp(-(x-x_{0})^{2}/\sigma_{T}^{2})],$
(11) $\displaystyle n_{c}(x)$ $\displaystyle=$ $\displaystyle
n_{c}(0)\max[1-\frac{(x-x_{0})^{2}}{\chi^{2}}].$

In the formula there are 5 parameters accounting for the bi-mode fitting, the
amplitude of two components $n_{th}(0)$ and $n_{c}(0)$, the width of two
components $\sigma_{T}$, $\chi$ and the center position $x_{0}$ of the atomic
cloud. The Bose function is defined as $g_{j}(z)=\sum_{i}z^{i}/i^{j}$. In
practice, we performed the least-squares fitting of $n_{th}(x)+n_{c}(x)$ to
the real distribution obtained from the imaging. From the fit, we can get the
atom number and width of the two components separately. Note that the
measurement in Fig. 9 (9c) is performed at different lattice depths. For each
lattice depth, 30 experiments have been performed to acquire the statistical
results. The temperature is given as $T=1/2M\sigma_{T}^{2}/t_{TOF}^{2}/k_{B}$,
where $M$ described the atom mass and $t_{TOF}$ width of the thermal part (?).

For the number of condensed atoms, the fitting outcome is less affected by the
fringe shown in Fig. 9 (a). Whereas the influence of the fringe on the fitting
of temperature is much more evident. The temperature is proportional to the
width of the Gaussian distribution $\sigma_{T}$ as mentioned above. Fig. 9(c)
shows the temperature extracted from the TOF absorption images with and
without the OFRA separately, namely Fig. 9(a1) and 9(b1). Fig. 9(a2) and 9(b2)
are the corresponding integrated one-dimensional atomic distributions for each
method. Fig. 9 depicts that the temperature we get with the common way of
calculation has a large error of 400 nK, extraordinary higher than the initial
BEC temperature of 90 nK. The turning on the procedure of lattice potential
would indeed lead to a limited heating effect, nevertheless the proportion of
condensed atom should be reduced significantly considering our system has been
heated up by 4 times. This is still not consistent with the observation.
However, the temperature is measured with much small variance at a much
reasonable value if we dive into the OFRA scheme. For example, the measured
temperature is 123.5 nK for a lattice depth of $V=4\;E_{r}$, with 183.9 nK for
$V=9\;E_{r}$. Comparison between these two results illustrates that only by
using the fringe removal algorithm we can get a reliable result, especially in
the case of small atom numbers when fitting physical quantities such as the
temperature.

In conclusion, with this algorithm, we can measure parameters with higher
contradiction to the conventional methods. The OFRA scheme is easy to
implement in absorption imaging-based matter-wave experiments as well. There
is no need to do any changes to the experimental system, only some algorithmic
modifications matter.

###  4.2 Extraction and identification of noise patterns for ultracold atoms
in an optical lattice

Furthermore, on the basis of the absorption images after preprocessing by
OFRA, the PCA method is used to identify the external noise fluctuation of the
system caused by the imperfection of the experimental system. The noise can be
reduced or even eliminated by the corresponding data processing program. It
makes the task more difficult that these external systematic noises are often
coupled, covered by nonlinear effects and a large number of pixels. PCA
provides a good method to solve this problem (?, ?, ?, ?, ?).

PCA can decompose the fluctuations in the experimental data into eigenmodes
and provide an opportunity to separate the noises from different sources. For
BEC in a one-dimensional OL, it was proved that PCA could be applied to the
TOF images, where it successfully separated and recognizing noises from
different main contribution sources, and reduced or even eliminated noises by
data processing programs (?).

The purpose of PCA is to use the smallest set of orthogonal vectors, called
principal components (PCs) to approximate the variations of data while
preserving the information of datasets as much as possible. The PCs correspond
to the fluctuations of the experimental system, which can help to distinguish
the main features of fluctuations. In the experimental system of BECs, the
data are usually TOF images. A specific TOF image $A_{i}$ can be represented
by the sum of the average value of the images and its fluctuation:

$\displaystyle A_{i}=\bar{A}+\sum\varepsilon_{ij}P_{j},$ (12)

Here $P_{j}$ is the different eigenmodes of fluctuation, $\varepsilon_{ij}$ is
the weight of the eigenmodes $P_{j}$.

Taking the BEC experiment (?, ?, ?, ?) in an OL as an example, we demonstrated
the protocol of PCA for extraction noise. A TOF image for ultra-cold atoms in
experiments can be represented as a $h\times w$ matrix. The PCA progress,
shown in Fig. 10, will be applied to the images:

Figure 10: Process diagram of PCA method. (a) Transform the $h\times w$ matrix
(raw images) into a $1\times hw$ vector, denoted by $A_{i}$. And stack these
vectors together. (b) Calculate the mean vector
$\bar{A}=\frac{1}{n}\sum_{i=1}^{n}A_{i}$ and the fluctuations
$\delta_{i}=A_{i}-\bar{A}$. Leaving only the fluctuation term in the matrix.
(c) Stack $\delta_{i}$ together to form a matrix
$X=\left[\delta_{1},\delta_{2},\cdots,\delta_{n}\right]$. Then the covariance
matrix $S$ is obtained by $S=\frac{1}{n-1}X\cdot X^{T}$. (d) Decompose
covariance matrix so that ${{V}^{-1}{S}{V}={D}}$ where D is a diagonal matrix.
(e) Transform eigenvectors of interest back to a new TOF image. Reproduced
with permission from Ref. (?).

(1) Transform the $h\times w$ matrix into a $1\times hw$ vector, denoted by
$A_{i}$.

(2) Express $A_{i}$ as the sum of the average value $\bar{A}$ and the
fluctuation $\delta_{i}$, where $\bar{A}=\frac{1}{n}\sum_{i=1}^{n}A_{i}$ and
$\delta_{i}=A_{i}-\bar{A}$.

(3) Stack $\delta_{i}$ together to form a matrix
$X=\left[\delta_{1},\delta_{2},\cdots,\delta_{n}\right]$. Then the covariance
matrix $S$ is obtained by $S=\frac{1}{n-1}X\cdot X^{T}$.

(4) Decompose covariance matrix with ${{V}^{-1}{S}{V}={D}}$, where $V$ is the
matrix of eigenvectors.

Figure 11: PCA results of the TOF images. (a) Example of a raw TOF image.
(b)-(f) correspond to the fluctuation in atom number (b), atom position
(c)(d), peak width (e) and normal phase fraction (f), respectively. (a2),
(b2), (d2), (e2), and (f2) are the integrated results of atom distributions
along x direction. (c2) is for the atom distribution along z direction. The
blue lines are the experimental results, and the orange lines are the
simulation results. Reproduced with permission from Ref. (?).

Fig. 11 shows the PCA results of the TOF images. Fig. 11(b)-(f) correspond to
the first to fifth PCs, respectively.

The first PC is from the fluctuation in the atom number. The normalization
process can be applied to reducing the fluctuation in the atomic number.
Because for a macroscopic wave function
$\Psi{\left(\textbf{r}\right)}=\sqrt{N}\phi{\left(\textbf{r}\right)}$ (?), we
usually concentrate on the relative density distribution, instead of the
$\sqrt{N}$. After the normalization, the impact of this PC becomes very small.

The second and third PCs correspond to the position fluctuations along the
$z$\- and $x$-directions, respectively. The fluctuation in spatial position of
the TOF images originates from the vibration of the system structure, such as
the OL potential, trapping potential, and imaging system. We used a dynamic
extraction method to eliminate the fluctuation in spatial position. We chose a
region whose center is also the center of the density distribution. We first
set a criterion to determine the center of the density distribution in the
extraction area, and then used this center as the center of the new area to
extract the new one. We repeated this process until the region to be extracted
becomes stable.

The fourth PC is from the fluctuation in the width of the Bragg peaks in the
TOF images. The final PC shown in Fig. 11(f1) comes from the normal phase
fraction fluctuation.

By studying the first five feature images, we have identified the physical
origins of several PCs leading to the main contributions. We numerically
simulated this understanding using the GPE with external fluctuation terms,
and got very consistent results (?). It is helpful to understand the physical
origins of PCs in designing a pretreatment to reduce or even eliminate
fluctuations in atom number, spatial position and other sources. Even in the
absence of any knowledge of the system, the PCA method is very effective to
analyze the noise, so that it can be applied to interferometers with higher
precision (?, ?).

##  5 Proposal on gravity measurements

Figure 12: Schematic of the experimental protocol. ${}^{87}\rm{Rb}$ atoms are
evaporation cooled as a Bose-Einstein Condensate in the
$\left|F=2,m_{F}=0\right\rangle$ initial state. However, they are transferred
to $\left|F=1,m_{F}=0\right\rangle$ as the two arms of interferometry,
followed by $P$ sequence of accelerate optical lattice pulse to maintain the
atoms against gravity: When atoms fall to a velocity of $q\times v_{Recoil}$,
they acquire a velocity of $2q\times v_{Recoil}$ upwards. The delay
$T_{Bloch}$ is chosen as $T_{Bloch}=2qv_{Recoil}/g$ to eliminate the fall
caused by gravity. Still, the probe beam should consist a laser light resonant
with the $F=1$ ground state to upper levels, thus we can take absorption
photos of $F=1$ population for analysis.

Inertia measurements (?, ?, ?, ?, ?, ?, ?), especially those for gravity
acceleration $g$, have always drawn lots of attention. Until now, the
performance of atom interferometry has reached a sensitivity of $8\times
10^{-9}$ at 1 second (?, ?), pushing forward the determination of the
Newtonian gravitational constant G (?, ?) or the verification of equivalence
principle (?, ?). Yet the bulky size of these quantum sensors strictly
restricts their application for on-site measurements. Therefore, based on our
previous study of ultra-cold atoms in precision measurements, we intend to
precisely measure the local gravity acceleration with our ${}^{87}\rm{Rb}$
Bose-Einstein Condensates in a small displacement. Note that this conception
bases on previous research of Perrier Cladé (?).

Fig. 12 illustrates the protocol of this BEC gravimeter. Instead of the Mach-
Zender method $(\pi/2-\pi-\pi/2)$ widely used in free-falling or atomic
fountain gravimeters, we utilize the Ramsey-Bordé approach by two pairs of
$\pi/2$ Raman pulses to get a small volume. Application of the Doppler-
sensitive Raman beam rather than the 6.8 GHz microwave field provides a far
more efficient way to realize larger momentum splitting, which will
significantly boost the interference resolution. Raman light pulse can also
attain the effects of velocity distribution. Consequently, the first pair of
$\pi/2$ pulses selects the initial velocity while the second pair can measure
the final distribution. It should be noticed that right after the velocity
selection step, a cleaning light pulse resonant with the $D_{2}$ line will
shine on the condensate to clear away atoms remaining in $F=2$ state, leading
to the two arms in interferometry.

The critical feature lies in the evolving stage between the two pairs of
$\pi/2$ pulses. By periodically inverting the velocity, the two arms shall
replicate parabolic trajectory in a confined volume, without any decreasing of
interrogation time. Choosing appropriate parameters, displacement of ultra-
cold atoms can be limited in a few centimeters, at least 1 order less than
that of a Mach-Zender gravimeter. This assumption is accomplished by a
succession of Bloch oscillation (BO) in a pulsed accelerated optical lattice
which transfers many photon recoils to the condensates (?, ?, ?, ?). Here the
application of ultra-cold atoms instead of optical molasses (?) makes it more
efficient when loading the atoms in the first Brillouin zone adiabatically,
owing to their wave function consistency and narrow velocity distribution.
This pulsed accelerated optical lattice should be manufactured along the
direction of gravity with higher lattice depth, to minimize the effect of
Landau-Zener Tunneling loss.

To deduce the value of g, we may scan the evolving time between the two pairs
of $\pi/2$ pulses, keeping the Raman frequency of each pair of $\pi/2$ pulses
fixed. When the time interval equals $2Pqv_{Recoil}/g$, where $P$ is the
number of pulses and $q$ is the number of recoil velocities($v_{Recoil}$)
obtained by a single pulse (shown in Fig. 12), the two arms are in phase and
the value of g can be extrapolated. Here the absorption image is used to
extract the interference information, due to the number of atoms being one
order less than that obtained by the conventional method.

We also give a qualitative analysis of this compact BEC gravimeter. Besides
its enormous potential in transportable instruments, prospective sensitivity
maybe even better. This encouraging outlook can be attributed to a longer
coherent time of ultra-cold atoms where the phase shift scales quadratically.
The smaller range of movement possesses other superiorities. Systematic errors
stemming from the gradients of residual magnetic fields and light fields
become negligible, especially for Gouy phase and wave-front aberrations (?, ?,
?). Furthermore, the value of the gravity acceleration is averaged over a
smaller height compared with the Mach-Zender ones. Finally, this vertical
Bloch oscillation technique offers a remarkable ability to coherently and
efficiently transfer photon momenta (?), though decoherence induced by the
inhomogeneity of the optical lattice must be taken into consideration.

In conclusion, we believe that this compact BEC gravimeter will have a
sensitivity of a few tens of ${\mu}$Gal at least. The falling distance will be
no more than 2 centimeters. Further improvement should be possible by
performing atom chip-assisted BEC preparation as well as the interaction-
suppressing mechanism (?). Gravity measurements with sub-${\mu}$Gal accuracies
in miniaturized, robust devices are sure to come in the future.

##  6 Conclusion

In summary, we review our recent experimental developments on the performance
of interferometer with ultra-cold atoms. First, we demonstrated a method for
effective preparation of a BEC in different bands of an optical lattice within
a few tens of microseconds, reducing the loading time by up to three orders of
magnitude as compared to adiabatic loading. Along with this shortcut method, a
Ramsey interferometer with band echo technique is employed to atoms within an
OL, enormously extend the coherence time by one order of magnitude. Efforts to
boost the resolution with multimode scheme is made as well. Application of a
noise-resilient multi-component interferometric scheme shows that increasing
the number of paths could sharpen the peaks in the time-domain interference
fringes, which leads to a resolution nearly twice compared with that of a
conventional double-path two-mode interferometer. We can somehow boost the
momentum resolution meanwhile. The patterns in the momentum space have got an
interval far less than the double recoil momentum, where the narrowest one is
given as $0.6\;\hbar k_{L}$. However, these advancements are inseparable with
our endeavor to optimize data analysis based on the PCA. Extrinsic systematic
noise for absorption imaging can be reduced efficiently. A scheme for
potential compact gravimeter with ultra-cold atoms has been proposed. We
believe it will tremendously shrink the size of a practical on-site
instrument, promoting another widely used quantum-based technique.

## Acknowledgment

This work is supported by the National Basic Research Program of China (Grant
No. 2016YFA0301501), the National Natural Science Foundation of China (Grants
No. 61727819, No. 11934002, No. 91736208, and No. 11920101004), and the
Project funded by China Postdoctoral Science Foundation.

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