††thanks: D. Hu and L.X. Niu contributed equally to this work††thanks: D. Hu
and L.X. Niu contributed equally to this work

# Echo-Ramsey Interferometry with Motional Quantum States

D. Hu  L. X. Niu  S. J. Jin  X. Z. Chen  School of Electronics Engineering and
Computer Science, Peking University, Beijing 100871, China  G. J. Dong
Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan,
Shanxi 030006, China  State Key Laboratory of Precision Spectroscopy, East
China Normal University, Shanghai, 200062, China  J. Schmiedmayer
<EMAIL_ADDRESS>Vienna Center for Quantum Science and Technology,
Atominstitut, TU Wien, Stadionallee 2, 1020 Vienna, Austria  X. J. Zhou
<EMAIL_ADDRESS>School of Electronics Engineering and Computer Science,
Peking University, Beijing 100871, China  Collaborative Innovation Center of
Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China

###### Abstract

Ramsey interferometers (RIs) using internal electronic or nuclear states find
wide applications in science and engineering. We develop a matter wave Ramsey
interferometer for _motional_ quantum states exploiting the S- and D-bands of
an optical lattice and identify the different de-phasing and de-coherence
mechanisms. We implement a band echo technique, employing repeated
$\pi$-pulses. This suppresses the de-phasing evolution and significantly
increase the coherence time of the motional state interferometer by one order
of magnitude. We identify thermal fluctuations as the main mechanism for the
remaining decay contrast. Our demonstration of an echo-Ramsey interferometer
with motional quantum states in an optical lattice has potential application
in the study of quantum many body lattice dynamics, and motional qubits
manipulation.

subject areas

###### pacs:

pacs

Ramsey interferometers (RIs) using internal electronic or nuclear states
Ramsey  already have played an important role in accurate quantum state
engineering and quantum metrology, such as in nuclear magnetic resonance NMR ,
atomic clocks Aclock , quantum information Qinf  and quantum simulation Qsimu
. In general, echo techniques are used in RIs to suppress dephasing for
significantly increasing the coherent time Schmiedmayer1 ; nmr2 ; Bo . Even
though motional states of particles are on the same footing as internal states
in current quantum technologies, i.e., motional qubits Neg , motional logic
gates Lee ; Wineland ; Briegel , motional quantum error correction Steinbach ,
quantum entanglement and coherent control Sangouard ; Zhuang , quantum
metrology with nonclassical motional states Dalvit , and communication
multiplexing Muga , conventional echo-RIs rarely exploit the quantum
interference of external center-of-mass motional states.

Recently, a RI with motional states of a Bose-Einstein condensate (BEC)
trapped in an anharmonic potential has been demonstrated with $92\%$ contrast
for several cycles Schmiedmayer2 . This proof-of-principle experiment holds
great promises for studying quantum many-body physics out of equilibrium,
quantum metrology with non-classical motional states and quantum information
processing with motional qubits. Recently this new technique has been used to
investigate decoherence and relaxation dynamics cohe , or to measure the
phononic Lamb shift Lamb .

Ultracold atoms trapped in an optical lattice (OL) is an ideal test platform
for studying quantum many-body dynamics Bloch , and have also been widely used
as a high-precision metrology tool Derevianko . Conventionally, these atoms
are prepared in the lowest band of the OL. Over last few years, there is an
increasing experimental and theoretical interest to prepare atoms in higher
bands Bloch2 ; Schori , opening a new way to simulate exotic orbital physics
in strongly correlated matter with rich degrees of freedom, i.e., the
formation of multi-flavor systems flav , supersolid quantum phases in cubic
lattices Larson , quantum stripe ordering in triangular lattices stripe  and
Wigner crystallization in honeycomb lattices Wig .

In this paper, we demonstrate an echo-RI with motional quantum states (MQS) of
atoms employing the S- and D-bands in an OL. Due to the lack of selection
rules for lattice band transition, a key challenge for constructing this RI is
to realize $\pi$\- and $\pi/2$-pulses analogous to those in conventional RIs.
Using a shortcut loading method FF ; Chen , we have designed sequences of
optical pulses Liu ; Zhai , analogous to a $\pi$\- or $\pi/2$-pulse, to
efficiently prepare a superposition of atoms in S- and D-band states in the OL
at zero quasi-momentum with high fidelity within tens of microseconds, which
is much shorter than the characteristic time scales of the decay process.
Keeping the lattice on, we observe state interference and measure the decay of
the coherent oscillations.

We have identified the mechanisms leading to the RI contrast reduction as
following: the nonuniform optical lattice depth, interaction-induced
transverse expansion after loading the atoms from the harmonic trap into the
optical lattice, laser intensity fluctuation and thermal fluctuations at
finite temperature. We then implement a matter-wave band echo technique to
significantly suppress all the contrast decay effects except for quantum and
thermal fluctuations, increasing the coherence time to 14.5 ms compared to 1.3
ms without echo at condensate temperature of 50 nK and $10E_{\mathrm{r}}$
lattice depth.

##  I A Ramsey interferometer within an optical lattice

###  I.1 Experimental implementation

Our experiment start with a BEC of ${}^{87}Rb$ prepared in a hybrid trap
formed by a single-beam optical dipole trap with wavelength 1064 nm and a
quadruple magnetic trap (Details are presented in our former works Wang ; Hu
). A nearly pure condensate of about $1.0\times 10^{5}$ atoms at the
temperature 50 nK is achieved with the harmonic trapping frequencies
$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times$(24, 48, 58) Hz, respectively.
The system’s temperature can be controlled by the evaporative cooling process
(see appendix). After preparation of the condensate, a one dimensional optical
lattice is formed by two counter-propagating laser beams with wavelength 852
nm resulting in a lattice constant $d=426$ nm along $x$ axis, as shown in Fig.
1(a). Using a shortcut control method Liu , the BEC is loaded into the lowest
lattice band with the quasi-momentum $q=0$ within a few microseconds with
nearly $100\%$ fidelity. The interaction energy of the condensate in the
ground state of OL with a lattice depth of $10E_{\mathrm{r}}$ is 1.2 kHz
($E_{\mathrm{r}}=(\hbar k)^{2}/2m$ is the recoil energy, $k=2\pi/426$ nm-1 is
the wave vector associated with the lattice and $m$ is the atomic mass).

Figure 1: Experimental configuration for a Ramsey Interferometer in a
$V_{0}=10E\mathrm{r}$ 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) Wave-
functions for the S-band and the D-band. At zero quasi-momentum, the widths of
the bands are $0.08E_{\mathrm{r}}$ and $3.57E_{\mathrm{r}}$, respectively. The
band gap between S and D is $7.77E_{\mathrm{r}}$. d) 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. e) The used pulse sequences designed by an optimised
shortcut method.

In the experiment, we construct the Ramsey interferometer with Bloch states
$\phi_{i,q}$ with energy $\epsilon_{i,q}$. We use the lowest band $\phi_{S,0}$
and the second excited band $\phi_{D,0}$ at quasi-momentum $q=0$, denoted as
$|S\rangle,|D\rangle$ respectively in the following, to form a superposition
state $\psi=a_{S}|S\rangle+a_{D}|D\rangle$. The two bands of S and D are
considered a _two-state_ system (spin-1/2 system), i.e., the two states can be
expressed as $\binom{a_{S}}{a_{D}}$. The scheme for the band energy and the
superposition states are shown in Fig. 1(b) and Fig. 1(c), respectively, with
the lattice depth $V_{0}=10.0E_{\mathrm{r}}$. During the lattice pulse
sequence, a accusto-optic modulator controlled by an RF switch is used to
switch on and off the lattice potential (light) quickly.

Manipulating the pseudo-spin system has its own challenges: unlike
conventional RI where selection rules can be used to prepare population in two
sates, the lattice band transition, similar to transition for vibration states
in molecules mole , has no selection rules. Thus a $\pi/2$-pulse analogous to
those in conventional RIs had to be numerically designed, so that the atoms in
S-band and D-band are to be transferred into the target states
$|\psi_{1}\rangle$=$(|S\rangle+|D\rangle)/\sqrt{2}$ and
$|\psi_{2}\rangle$=$(-|S\rangle+|D\rangle)/\sqrt{2}$, respectively. Extending
our optimal control method Liu ; Zhai , we construct a $\pi/2$ pulse from a
sequence of lattice pulses with special selected timing and duration as shown
in Fig. 1(e). We achieve a fidelity of $98.5\%$ and $98.0\%$, for atoms
initially on the S- and D-band respectively. (See appendix).

The full time sequence for RI is shown in Fig. 1(d). First the atoms in the
harmonic trap are transferred into the S band by a fast loading process, then
the first pulse $\hat{R}(\pi/2)$ is applied to prepare an initial
superposition state
$\hat{R}(\pi/2)\binom{1}{0}=\frac{1}{\sqrt{2}}\binom{1}{1}$. The final state,
after evolution in the OL for time $t_{OL}$ and a second $\pi/2$ pulse can be
expressed as:

$\psi_{f}=\hat{R}(\pi/2)\hat{U}(t_{OL})\hat{R}(\pi/2)\psi_{i},$ (1)

with
$\hat{R}(\alpha)=(\cos\frac{\alpha}{2}-i\sin\frac{\alpha}{2})\hat{\sigma}_{y}$,
and the evolution operator $\hat{U}(t)=(\cos\omega t+i\sin\omega
t)\hat{\sigma}_{z}$, here $\omega=(\epsilon_{D,0}-\epsilon_{S,0})/\hbar$ is
the frequency difference between two states, and $\hat{\sigma}$ the Pauli
matrix, $\alpha$ denotes rotation angle around an axis in the Bloch sphere.

We then apply band mapping Bloch2  to read out the state of the RI. For band
mapping the lattice depth is exponentially-ramped down in the form
$e^{-t/\eta}$ with a characteristic decay time $\eta=100\mu s$ for a total
length of 500 $\mu$s. Then absorption imaging is used to measure the
population in the different bands after 31 ms time of flight (TOF) (see
appendix). The atoms originally occupying $|S\rangle$ state populate a narrow
Gaussian distribution around $0\hbar k$. The atoms originally occupying the
$|D\rangle$ are detected at $\pm 2\hbar k$ in the side zone. The total atom
number we detected consists of condensed atoms and thermal atoms. From a
bimodal fitting to each peak Schori , we can quantitatively determine the
population of condensate, and we note the condensed atom number as
$N_{\mathrm{S}}$ ($N_{\mathrm{D}}$) for S-band (D-band) (see appendix).

###  I.2 Ramsey Interferometer in an optical lattice

Experimental results on the time evolution of the atom population in the
D-band $p_{D}(t_{OL})=N_{D}/(N_{S}+N_{D})$ with evolution time in the lattice
$t_{OL}$ are shown in Fig. 2(a) for the time sequence
$\hat{R}(\pi/2)-\hat{U}(t_{OL})-\hat{R}(\pi/2)$, $V_{0}=10E_{\mathrm{r}}$ and
$T=50$ nK. Each solid point with error bar is the mean of three measurements,
shown as red circles. The red solid line is a fit of an damped oscillation
with a period of $41.1\pm 1.0\mu s$ which is consistent to the reciprocal of
the band gap energy of about $40.8\mu s$. The details of the early RI
oscillation for $t_{OL}<100\mu s$ is shown in Fig. 2(b), displaying a nearly
perfect oscillation between the S-band and the D-band with amplitude close to
1. At $P_{1}$ nearly all the atoms are transmitted from the S-band to the
D-band by two $\pi/2$ pulses which can be seen as a $\pi$ pulse. At $P_{2}$,
the two $\pi/2$ pulses offset each other due to phase evolution of two states
in the lattice, and the atoms are transferred to the S band. However, when
$t_{OL}$ gets longer, the oscillation amplitude decays, as shown in Fig. 2(a).
The contrast $C(t_{OL})$ at $t_{OL}$ can be obtained by fitting the amplitude
of oscillation $p_{D}(t_{OL})$ in Fig. 2(a) with

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

Fig. 2(c) shows the measured contrast decay versus time $t_{OL}$ for the
different initial temperatures of condensates, where each contrast value is
fitted from about 20 experimental points with a time step 5$\mu$s and each
point is the mean value of three measurements. The error-bars are given by
$95\%$ confidence bounds. The horizontal dashed line in Fig. 2(c) indicates
the contrast drops to the value of $1/e$, which is used to define a coherence
time $\tau$. This time decreases from 1.3 ms to 0.8 ms when the temperature
increases from 50 nK to 180 nK.

Figure 2: Signal for a Ramsey-interferometer between the S- and D-bands for
$V_{0}=10E_{\mathrm{r}}$. a) The oscillation of the population of atoms in the
D-band $p_{\mathrm{D}}$ (initial temperature $T=50$ nK). The dots with
errorbars are the mean of three measurements, while single measurement is
shown as red circle, and the red solid line is the corresponding fitting curve
to the mean value. The inserts show typical time of flight pictures after band
mapping. b) Detail of the oscillation for $t_{OL}<100$ $\mu$s (shown as black
dots). Insets: The TOF images $P_{1}$ at 0 $\mu$s and $P_{2}$ at 20 $\mu$s,
also the images $P_{3}$ and $P_{4}$ around $1.5ms$. c) The decay of the
interferometer contrast $C(t_{OL})$ for different temperatures. The black
dots, red squares and green diamonds corresponding to 50 nK, 110 nK and 180
nK, respectively, with the corresponding solid lines a guide to eye. The
horizontal dashed line indicates the contrast drops to the value of $1/e$ that
gives the coherence time $\tau$.

###  I.3 Contrast decay mechanisms

In order to improve the performance of the RI, we now investigate the
mechanisms that lead to RI signal attenuation. We first study the effect of
imperfect design of the $\pi/2$ pulse on the fidelity by solving Schrödinger
equation with an uniform lattice potential, and the unbalanced population
between the S- and D-bands (see appendix). The numerical result (brown dashed
line in Fig. 3(a)) shows that the imperfectly designing of $\pi/2$ pulses and
the unbalanced population have negligible effects on the contrast decay during
the evolution in lattice potential.

In our further theoretical analysis, we replace the ideal lattice potential by
a non-uniform potential distribution to account for the Gaussian beam in the
radial direction, as shown in Fig. 1(a), and include the quasi-momentum
distribution for the condensate distributed in the harmonic trap dep . These
two effects result in inhomogeneous broadening of the transition frequency
$\omega$ between the S- and D-bands, and lead to to de-phasing and contrast
decay. By solving the zero-temperature Gross-Pitaevskii equation(GPE) with the
real inhomogeneous potential Tdamping  (see appendix), we obtained the
contrast as shown by the blue dotted line in Fig. 3(a).

The atom-atom interaction leads to transverse expansion after the fast (non-
adiabatic) loading of the atoms from the harmonic trap into the 1-d optical
lattice. This expansion leads to a significant reduction on contrast (blue
dashed line in Fig. 3(a)).

Moreover, adding a $0.1\%$ laser intensity fluctuation further reduces the
contrast as shown in the purple dash-dotted line in Fig. 3(a). Quantum
fluctuations at zero temperature is further added using a truncated Wigner
method (see appendix) and the related results shown in green dashed line
nearly overlap with the dash-dotted line. In our experiment, the atom number
is high enough such that the influence by the quantum fluctuation is not
significant.

Finally, we take into account the thermal fluctuations using a finite
temperature truncated Wigner calcualtion (see appendix). The result is shown
in the orange solid line in Fig. 3(a), agreeing well with the experimental
results shown in black dots.

Figure 3: Contribution to contrast decay for $\pi/2$-$\pi/2$ RI. a) The
experimental contrast and the theoretical calculation for
$V_{0}=10E_{\mathrm{r}}$. b) The coherence time versus the different depths
$V_{0}$. Components as described in the text.

The relation of the coherence time to lattice depth is presented in Fig. 3(b).
When the lattice depth increases, the confinement in x axis gets tighter,
accordingly, the effects of the expansion in radial direction will increase
due to the increased interaction energy. In addition, the non-uniformity of
the lattice potential (the difference of the lattice potential from center to
edge) also increases with the increasing of lattice depth. Due to these two
effects, the coherence time $\tau$ decreases with increasing lattice depth.
The theoretical curve in Fig. 3(b) including all the decay mechanisms (orange
solid line) fits well with an experimental measurement.

##  II An Echo-Ramsey Interferometer with motional quantum states of atoms

To further improve the contrast of the RI with MQS, we develop a matter-wave
band echo technique. A $\pi$ pulse is designed which swaps the atom population
in $|S\rangle$ and $|D\rangle$ band. The pulse schema of our echo-RI scheme is
shown in Fig. 4. In between the initial and final $\pi/2$ pulses we insert $n$
identical $\pi$ pulses. A single $\pi$ pulse is realised with the operation
$\hat{U}(t_{OL}/2n)\hat{R}(\pi)\hat{U}(t_{OL}/2n)$. The full Echo-Ramsey pulse
sequence is then:
$\hat{R}(\pi/2)[\hat{U}(t_{OL}/2n)\hat{R}(\pi)\hat{U}(t_{OL}/2n)]^{n}\hat{R}(\pi/2)$.
The contrast decay $C(t_{OL})$ of a $4\times\pi$ pulse echo-RI versus the
holding time at various temperatures between 50 nK and 180 nK is shown in Fig.
5(a). The solid lines are fits with an exponential function which allow to
extract the coherence time $\tau$.

Figure 4:  Scheme for an echo-Ramsey interferometer: Between two $\pi/2$
pulses, $n\times\pi$ pulse are inserted. Band mapping is implemented after the
second $\pi/2$ pulse to detect the population in the different bands.

The relationship between the coherence time $\tau$ and the number of $\pi$
pulses $n$ is shown in Fig. 5(b) for three different temperatures ($T$=50, 110
and 180 nK). The zero-temperature theoretical results of the coherence time
can again be obtained by solving the GPE including the non-uniform lattice
potential and the radial expansion, as shown by the circles in Fig. 5(b). When
further taking into account of the laser intensity fluctuation, the calculated
coherence time is shown with the cross points. After implementing one $\pi$
pulse, the two theoretical points merger together, which shows that the
intensity fluctuation can be well suppressed with one $\pi$ pulse; however, in
order to eliminate the effects of non-uniform lattice potential and transverse
expansion, more echo pulses are needed; after applying $n=6$ echo pulses, the
theoretical curves get nearly flat, showing that the two effects are well
suppressed. There remains a significant discrepancy between the experimental
and the zero temperature theoretical calcualtions, which we attribute to
thermal fluctuations. Lowering the condensate temperature, the thermal
fluctuations get weaker, and the coherence time can be greatly increased with
multiple echo-pulses. The measured coherence time $\tau$ for the RI at $T=50$
nK is 14.5 ms, which is one order of magnitude longer than the case without
echo-pulses.

Figure 5: Echo-Ramsey Interferometer: a) Contrast $C$ vs holding time $t_{OL}$
in the OL after using four $\pi$ pulses. Solid curves are a fit with an
exponential function to extract the coherence time $\tau$. b) Coherence time
$\tau$ vs the number of applied $\pi$ pulse $n$. The circle and cross are
theoretical results at zero temperature with or without intensity fluctuations
respectively. Experimental results and fitting curves with parameter $\beta$
for the different temperatures are also shown. c) Experimental (red dots)
decay rate $\beta$ induced by thermal fluctuations $vs$ temperature. The
error-bars give a 95$\%$ confidence interval. The dashed line illustrates that
the decay rate $\beta$ is proportion to the temperature.

We will now look at the remaining fluctuations that cannot be suppressed by
the echo technique: We fit the experimental contrast with
$C_{0}(t_{OL})\exp[-\beta t_{OL}]$, where $C_{0}$ is the calculated contrast
for zero temperature and the decay rate $\beta$ characterises the additional
decay. For different temperatures, the fitted values of $\beta$ are shown in
Fig. 5(c) with red dots, showing a linear dependence on the temperature. This
is a strong indication that the remaining decay of the interference contrast
of the echo-Ramsey interferometer is caused by thermal fluctuations. To
confirm this we conducted finite temperature truncated Winger calcualtions
Temperature ; Temperature2  (see appendix)

Figure 6: a) The contrast decay rate from thermal effect $\beta$ versus the
different lattice depth $V_{0}$ with six $\pi$ pulses and $T$ = 50 nK with the
experimental results (solid points) and theoretical calculation at zero
temperature (square points). b) Typical TOF images for different depths and
the red square shows atom on other momentum which causes an increasing of
decay rate in experiment.

Finally we look at the performance of the echo-RI for different lattice depth.
As shown in Fig. 6(a) the contrast decay rate strongly depends on the lattice
depth for a condensate at 50 nK and using a sequence with six $\pi$ pulses.
The decay rate is small at medium optical lattice depth, and it becomes bigger
for the shallow ($<$5$E_{\mathrm{r}}$) and deep lattices
($>$10$E_{\mathrm{r}}$). For the case of $V_{0}=4E_{\mathrm{r}}$, the
separation between D and P bands is small and it is hard to form a close cycle
between S- and D-band without any transition to P-band in practice. We should
note that for the two pulse RI, transition to other bands are not so evident
due to the short lattice time. When the echo technique is applied, the
coherence time is much increased, and transition to other bands induced by
atom-atom interaction become important. This can be seen in Fig. 6(b) where
the momentum distribution for 4$E_{\mathrm{r}}$ has components (marked with a
red square) different from $\pm 2\hbar k$.

When the lattice potential gets deeper, the band separation increases, and the
coherence time increases accordingly. However, when the lattice depth is more
than 10$E_{\mathrm{r}}$, the coherence time decreases quickly. In fact the
probability for atoms to be excited to higher momentum states by the pulse is
proportional to the lattice depth. For a very deep lattice the atoms can
easily obtain higher momenta YYZ . In Fig. 6(b) the momentum distributions
clearly show the components at $4\hbar k$ for 20 $E_{\mathrm{r}}$. Since the
Rabi transition is designed for the transition between the S- and D-band at
zero quasi-momentum, the higher states will lead to a reduced pulse fidelity
and a faster decay of the contrast.

##  III Summary and Discussion

In summary, we have demonstrated a Ramsey interferometer for atoms within an
OL. We further employ a matter wave band echo technique to significantly
enhance the coherence time by one order of magnitude. We have identified the
mechanisms leading to the contrast decay for the RI signal. The contrast decay
is closely related to nonuniform optical lattice, laser intensity fluctuation
and interaction-induced transverse expansion, and finite temperature dynamics
of a BEC. All except for the effects of finite temperature can be suppressed
by a matter-wave band echo sequence. Thus the damping from thermal
fluctuations is well uncovered in this way.

So far, the $\pi$ pulses and $\pi/2$ pulses for echo-RI using MQS are designed
based on zero temperature single atom dynamics. In furure developments, it
would be interesting to unveil quantum many body dynamics in OL, like loop
structures and swallowtails nband1 ; nband2 ; nband3 . These deliberate
quantum control technologies could be applied in quantum information and
precise measurements based on MQS of atoms.

## Acknowledgement:

We thank Z. K. Chen for the calculation of pulse time sequences and
discussion. Thank P. Zhang, Q. Zhou, and B. Wu for helpful discussion. This
work is supported by the National Key Research and Development Program of
China (Grant No. 2016YFA0301501), and the NSFC (Grants No.11334001,
No.61475007 and No. 61727819). G. J. Dong acknowledges the support by the
National Science Foundation of China (Grants No. 11574085 and No. 91536218),
and 111 Project ( B12024). JS acknowledges support by the European Research
Council, ERC-AdG _QuantumRelax_.

##  Appendix A Design of Pi/2- and Pi-pulses

In a 1D optical lattice along x-direction, the eigen-states of the atom, i.e.
the Bloch states, can be expressed as the superposition of momentum states

$|i,q\rangle=\sum_{l=-\infty}^{+\infty}c_{i,l}(q)|2l\hbar k+q\rangle,$

where $|2l\hbar k+q\rangle$ is the basis in momentum space, $i$ is the band
index and $q$ is the quasi-momentum, $c_{i,l}(q)$ is the superposition
coefficient with $l=0,\pm 1,\pm 2,...$. While the target state is considered
to be the superposition of Bloch states,

$\psi=\sum_{i}A_{i}|i,q\rangle=\sum_{i}A_{i}\sum_{l}c_{i,l}(q)|2l\hbar
k+q\rangle$

with $A_{i}$ is the superposition coefficient.

Varying the pulse parameters, we numerically find a time sequence

$\\{t_{1}=0,t_{2},t_{3},...\\}$

of optical lattice pulses

$V_{latt}(t)=V_{0}\ \mathrm{for}\ (t_{2i-1}<t<t_{2i})$ $V_{latt}(t)=0\
\mathrm{for}\ (t_{2i}<t<t_{2i+1})$

that creates a $\frac{\pi}{2}$ pulse in the Ramsey interferometer, i.e., after
this operation, the atoms on S-band and D-band are transferred into target
states $(|S\rangle+|D\rangle)/\sqrt{2}$ and $(-|S\rangle+|D\rangle)/\sqrt{2}$,
respectively. In our designed pulse sequences, both the laser intensity and
the duration can be changed. In the real experiment, we keep the lattice depth
constant and only change the time lattice on and off for experimental
simplicity without losing of fidelity.

The evolution operator for ${t_{2i-1}<t<t_{2i}}$ is

$\hat{U}_{\mathrm{on}}(t_{2i}-t_{2i-1})=\exp\\{-\frac{i}{\hbar}\int_{0}^{t_{2i}-t_{2i-1}}[\frac{p^{2}}{2m}+V_{0}\cos^{2}(kx)]dt\\}$

and

$\hat{U}_{\mathrm{off}}(t_{2i+1}-t_{2i})=\exp(-\frac{i}{\hbar}\int_{0}^{t_{2i+1}-t_{2i}}\frac{p^{2}}{2m}dt)$

for the free evolution operator at ${t_{2i}<t<t_{2i+1}}$, where $p$ and $m$
are respectively the momentum and mass of a single atom. After applying the
pulse sequence, the final states $|\psi_{f_{1}}\rangle$ and
$|\psi_{f_{2}}\rangle$ can be written as

$\displaystyle|\psi_{f_{1}}\rangle$ $\displaystyle=$
$\displaystyle\prod_{i}\hat{U}_{\mathrm{on}}(t_{2i}-t_{2i-1})\cdot\hat{U}_{\mathrm{off}}(t_{2i+1}-t_{2i})\cdot|S\rangle$
(3) $\displaystyle|\psi_{f_{2}}\rangle$ $\displaystyle=$
$\displaystyle\prod_{j}\hat{U}_{\mathrm{on}}(t_{2i}-t_{2i-1})\cdot\hat{U}_{\mathrm{off}}(t_{2i+1}-t_{2i})\cdot|D\rangle.$
(4)

By changing the parameters of the pulse sequence, one can maximise the two
fidelities $F_{1}$=$|\langle\psi_{A_{1}}|\psi_{f_{1}}\rangle|^{2}$ and
$F_{2}$=$|\langle\psi_{A_{2}}|\psi_{f_{2}}\rangle|^{2}$. For
$V_{0}=10E_{\mathrm{r}}$ the time sequence for $\pi/2$ pulse as
$\\{0,56.2,84.2,106.8,129.9\\}(\mu$s) with $F_{1}=96.9\%,F_{2}=97.9\%$.

The method for designing $\pi$ pulse is similar to the case of $\pi/2$ pulse,
but with other target states. After the sequence of optical lattice pules, the
atoms should be transferred from $|S\rangle$ or $|D\rangle$ into the target
states $|D\rangle$ or $-|S\rangle$, respectively. For $V_{0}=10E_{\mathrm{r}}$
the time sequence for $\pi$ pulse as $\\{0,49.2,101.7,123.8,150.2\\}(\mu$s),
and the transfer of $|S\rangle$ and $|D\rangle$ to the target states are with
the fidelity $98.5\%$ and $98.0\%$, respectively.

During design of the pulse sequences, we have neglected atom-atom interaction
because the duration of pulses is short enough and atom-atom interaction
strength in our system is weak Liu . Numerical results of the GPE with atom-
atom interaction show that the interaction leads to a change of fidelity less
than $1\%$ compared with our designed time sequence.

##  Appendix B Data analysis

After the band mapping, we release the condensate from the trap and let it
expand freely for 31 ms to perform time of flight (TOF) imaging. The TOF image
shows the atoms from the S band distributed in the center $0\hbar k$ while
atoms from the D-band are distributed on the side zone $\pm 2\hbar k$. We
apply an algorithm to remove the background fringes fringe , and integrate the
atom distribution in $y$-direction of the TOF image. Then the atomic
distribution is fitted with a bimodal function

$\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1,2,3}G_{i}F_{th}(e^{-(x-x_{i})^{2}/\sigma^{2}})+$
(5)
$\displaystyle+\sum\limits_{i=1,2,3}H_{i}(\max[1-(x-x_{i})^{2}/\chi_{i}^{2},0])^{2}$

with a Gaussian thermal distribution $F_{th}$ and a Thomas Fermi like the
condensate. $i=1,2,3$ corresponds to the three atom clouds for $-2\hbar k$,
$0\hbar k$, and $2\hbar k$, respectively. We also add a restriction to all six
amplitude terms $H_{i}$ and $G_{i}$ as $H_{1}/G_{1}=H_{3}/G_{3}$. From the
amplitude of each component $H_{i}$ and the width of condensate part
$\chi_{i}$, we can determine the population of condensate in the S-band
$N_{S}$ and in the D-band $N_{D}$, and further calculate the population of the
D-band as $p_{D}(t_{OL})=N_{D}/(N_{S}+N_{D})$. The experimental contrast at a
certain time is given by fitting the measured amplitude of oscillation $p_{D}$
for 100$\mu s$ (about 2.5 periods) with a time step 5$\upmu$s using a cosine
function, where the errorbar is given by the fitting error with 95$\%$
confidence bounds.

##  Appendix C Calculation of contrast decay at zero temperature within an
echo-Ramsey interferometer

###  C.1 Contrast decay induced by atom-atom interaction

When atoms collide in the excite band they can decay to a lower band
Spielman06 ; Bucker11 . We calculate the collision decay of atom population in
different bands using a simple rate equation model Zhai ,

$\frac{dN_{S(D)}(t)}{dt}=-K_{S(D)}N_{S(D)}(t)^{2},$ (6)

where the factor $K_{S(D)}$ is given by summation of cross section of the two-
atom inelastic collision from different channels. Solving Eq.(6), the
population from the D-band $N_{D}(t)$ decays as $1/(1+t/\tau_{D})$, with the
time constant of atom decay from the D-band given by $1/\tau_{D}=K_{D}N_{D}$.
$N_{S}(t)$ can similarly be given from the collision rate of atoms between the
S- and the D-bands.

We measure the collisional decay rate of atoms in different bands by band
mapping without the final $\pi/2$ pulse and get $\tau_{D}=1.9\pm 0.3$ ms,
$\tau_{S}=5.1\pm 0.8$ ms.

During the collision decay, the potential energy of the D-band is transferred
to kinetic energy in radial direction. This energy is large enough so that the
majority of the decayed atoms will not be counted in for the condensed peaks.
After the population decay atom number $N_{S}(t)$ and $N_{D}(t)$ is unbalanced
which leads to a reduction of the contrast by a factor
$\frac{2\sqrt{N_{S}N_{D}}}{N_{S}+N_{D}}$. From the measured $N_{S}$ and
$N_{D}$ we find that the influence of this population unbalance to the
calculated contrast $C_{0}(t_{OL})$ is less than $1\%$ for an experiment with
$t_{OL}<2$ ms as shown as the brown dashed line in Fig.3(a).

Applying the repeated $\pi$-pulses the coherence time is much longer than this
decay time: (_i_) The $\pi$-pulses reverse the population in the S- and
D-bands, and therefore prevent a strong imbalance. (_ii_) In the data analysis
extract the remaining condensed part by bimodal fitting. Both together
dramatically reduce the effect of collisional decal on the contrast of the
observed interference between the remaining coherent parts, and the
collisional decay time does not limit the coherence time of our interferometer
signal, as long as the remained population in condensed part is large enough
for detection.

###  C.2 The contrast decay induced by the quasi-momentum distribution and the
nonuniform of optical lattice potential in radial direction

For these we have to turn to numerical calculations taking the real potentials
and their variation into account. Considering that the trapping frequencies in
$y,z$ direction are very similar in our system, we can use use a mean field
model (GPE) in cylindrical coordinates to describe the system at zero
temperature. The evolution of the wave function $\Phi(\textbf{r},t)$ is
governed by

$i\hbar\frac{\partial}{\partial
t}\Phi(\textbf{r},t)=[-\frac{\hbar^{2}}{2m}\triangle+V_{\mathrm{ext}}+NU_{0}|\Phi(\textbf{r},t)|^{2}]\Phi(\textbf{r},t),$
(7)

where $\textbf{r}=(x,r,\theta)$, $V_{\mathrm{ext}}$ is the external potential,
and $U_{0}=\frac{4\pi\hbar^{2}a_{s}}{m}$ is the interaction term with $a_{s}$
the scattering length.

In our case the radial part of wavefunction is in the ground state and uniform
in $\theta$ coordinate. The lattice potential itself depends on the radial
position $r$. We first neglect the kinetic term in radial direction and
separate the wavefunction into radial part and axial part as
$\Phi(\textbf{r},t)=\frac{1}{\sqrt{2\pi}}\psi(x,t)\phi(r)$, then Eq.(7) can be
simplified to the following 1d GPE at a certain value $r=r_{i}$ as

$i\hbar\frac{\partial}{\partial
t}\psi(x,t)=[-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}+V_{\mathrm{ext,r_{i}}}(x,t)+\rho_{r_{i}}U_{0}|\psi(x,t)|^{2}]\psi(x,t)$
(8)

$V_{\mathrm{ext,r_{i}}}$ is the combination of both harmonic potential
$V_{\mathrm{trap}}=\frac{m}{2}\omega_{x}^{2}x^{2}$ and lattice potential
$V_{\mathrm{latt},r_{i}}=V_{0}Q(t)e^{-2r_{i}^{2}/w_{\mathrm{latt}}^{2}}\cos^{2}(kx)$
where $Q(t)$ takes value 0 or 1 depending on the time sequence, and
$w_{\mathrm{latt}}$ is the waist of lattice laser.
$\rho_{r_{i}}=\frac{\exp[{-m\omega_{r}r_{i}^{2}/\hbar}]}{\sqrt{\pi\hbar/m\omega_{r}}}N$
is the linear density. We solve Eq.(8) to get the $\psi(x,t)$ and the
population in the D-band $p_{D_{i}}$ for position $r_{i}$. We then calculate
30 different radial positions and take their weighted average according to the
atom number distribution $\sum\limits_{i}(2\pi r_{i}\rho_{r_{i}}\cdot
p_{D_{i}})/\sum\limits_{i}(2\pi r_{i}\rho_{r_{i}})$. Finally the oscillation
amplitude of the average $p_{D}$ is fitted to get a contrast as shown in
dotted line in Fig.3(a).

In this simulation, we consider the wave-function’s distribution instead of
using a single atom model, thus the influence of quasi-momentum distribution
is also included automatically.

###  C.3 Contrast decay induced by radial motion of the condensate

In the above calculation, we consider the distribution of lattice potential in
radial direction, however, the radial size of he wavefunction also changes
with time. To consider this effect we need to estimate the expansion speed of
the atom cloud in the transverse direction.

We can take each site of the lattice as a small independent BEC with about
$1.5\times 10^{3}$ atoms (about 65 lattice sites), and calculate how it
spreads after the trapping potential changed during switch on and loading into
the lattice. When the lattice is turned on, the trapping frequency in x
direction increases to $20kHz$ for 10$E_{\mathrm{r}}$ which is much larger
than the harmonic trap of $2\pi\times 24Hz$, this sudden increase of trap
frequency induces the spread in the radial direction, which can be calculated
as Castin ,

$\ddot{\lambda_{x}}=\frac{\omega_{x}^{2}(0)}{\lambda_{x}^{2}\lambda_{r}^{2}}-\omega_{x}^{2}(t)\lambda_{x},\
\ \
\ddot{\lambda_{r}}=\frac{\omega_{r}^{2}(0)}{\lambda_{x}\lambda_{r}^{3}}-\omega_{r}^{2}(t)\lambda_{r},$
(9)

where $\omega_{x},\omega_{r}=\sqrt{\omega_{y}\omega_{z}}$ are the frequency of
the effective trapping potential in $x$ and $r$ direction for the small BECs
in each lattice site, respectively, and $\lambda_{r}=r_{i}(t)/r_{i}(0)$ is the
expansion with $r_{i}(0)$ the initially radial position. Using this time
dependent radial expansion we then follow a procedure like above to calculate
the average $p_{D}$ and contrast as shown in dashed line in Fig.3(a) of the
main text.

###  C.4 Contrast decay induced by laser intensity fluctuations

The laser intensity in our experiment fluctuates by about $0.1\%$ during the
holding time. The laser intensity changes are slow and do not cause excitation
between the different bands. Simulations like above are then repeated with
different laser intensities sampled from the measured fluctuations. The
averaged result is shown in Fig.3 of the main text with the purple dash-dotted
line.

###  C.5 Calculation of contrast decay induced by thermal fluctuations

To calculate that contrast decay induced by thermal fluctuations we apply the
finite-temperature truncated Wigner method Temperature . We solve the 1d GPE
with a stochastic initial wavefunction
$\psi^{\prime}(x)=\psi+\sum_{j}\psi_{j}$, where $\psi$ is the zero-temperature
condensate wave-function within the one-dimensional optical lattice,
$\psi_{j}$ corresponding to the thermal fluctuations that is given by
$\psi_{j}=A(r_{0},\theta_{0})[u_{j}(x)\beta_{j}-v_{j}^{*}(x)\beta_{j}^{*}]$,
where $A(r_{0},\theta_{0})=\sqrt{n_{0}(0,r,\theta)/\iint
n_{0}(0,r,\theta)r\mathrm{d}r\mathrm{d}\theta}$, with the condensation density
$n_{0}(x,r,\theta)$. For simplicity and without loss of generality, we choose
$r_{0}=0,\theta_{0}=0$. Here $\beta_{j}$ is a complex number with random phase
which satisfies $\beta_{j}^{*}\beta_{j}=\bar{N}_{j}+1/2$, where the quantum
fluctuations are included by the $1/2$ term. in which
$\bar{N}_{j}=\sum_{i}1/(e^{E_{ji}/k_{B}T}-1)$ and $E_{ji}=E_{j}+\Lambda_{i}$
is the summation of the energy of the j-th Bogoliubov mode $E_{j}$ plus the
i-th eigen-energy $\Lambda_{i}$ of the radial harmonic potential. $u_{j}$ and
$v_{j}$ are the $j$-th Bogoliubov modes solved from the one-dimensional
Bogoliubov de Gennes equation as Temperature ,

$\displaystyle[H_{sp}+2U_{0}n_{0}-\mu]u_{j}-U_{0}n_{0}v_{j}=E_{j}u_{j},$ (10)
$\displaystyle-[H_{sp}+2U_{0}n_{0}-\mu]v_{j}+U_{0}n_{0}u_{j}=E_{j}v_{j},$ (11)

with
$H_{sp}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+\frac{1}{2}m\omega_{x}^{2}x^{2}$.

In the calculation we use 300 excitation modes and repeat our simulation 15
times with different stochastic initial states. From the wavefunctions and the
population distributions, we then obtain the contrast $C(t_{OL})$, calculated
at the different holding times $t_{OL}$, which can then be compared to the
experiment. In a separated calculation, we also calculate for the quantum
fluctuations by taking $\bar{N}_{j}=0$, the result is shown in Fig. 3(a) of
the main text with green dashed line, which is close to the result without
quantum fluctuations, where the coherent time is only reduced by $0.2\%$.

## References

  * (1) N. F. Ramsey, A molecular beam resonance method with separated oscillating fields, _Phys. Rev._ 78, 695 (1950).
  * (2) T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, Nuclear magnetic resonance spectroscopy on a (5-Nanometer)3 sample volume, _Science_ 339, 561 (2013).
  * (3) G. Santarelli, Ph. Laurent, P. Lemonde, A. Clairon, A. G. Mann, S. Chang, A. N. Luiten, and C. Salomon, Quantum projection noise in an atomic fountain: a high stability cesium frequency standard, _Phys. Rev. Lett._ 82, 4619 (1999).
  * (4) X. Maître, E. Hagley, G. Nogues, C. Wunderlich, P. Goy, M. Brune, J. M. Raimond, and S. Haroche, Quantum memory with a single photon in a cavity, _Phys. Rev. Lett._ 79, 769 (1997).
  * (5) M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, Atom-chip-based generation of entanglement for quantum metrology, _Nature_ 464, 1170 (2010).
  * (6) A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, Optics and interferometry with atoms and molecules, _Rev. Mod. Phys_. 81, 1051(2009).
  * (7) S. M. Zhang, B. H. Meier, and R. R. Ernst, Polarization echoes in NMR, _Phys. Rev. Lett._ 69, 2149 (1992).
  * (8) B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Observation of dipolar spin-exchange interactions with lattice-confined polar molecules, _Nature_ 501,521 (2013).
  * (9) A. Negretti, P. Treutlein, and T. Calarco, Quantum computing implementations with neutral particles, _Quant. Inf. Proc._ 10, 721(2011).
  * (10) P. J. Lee, K. A. Brickman, L. Deslauriers, P. C. Haljan, L. M. Duan and C. Monroe, Phase control of trapped ion quantum gates, _J. Opt. B: Quant. Semiclass. Opt_. 7, S371(2005).
  * (11) D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, Experimental issues in coherent quantum-state manipulation of trapped atomic ions, _J. Res. Nat. Inst. Stand. Technol._ 103, 259 (1998).
  * (12) H. J. Briegel, T. Calarco, D. Jaksch, J. I. Cirac, and P. Zoller, Quantum computing with neutral atoms, _J. Mod. Opt_. 47, 415 (2000).
  * (13) J. Steinbach, and J. Twamley, Motional quantum error correction, _J. Mod. Opt_. 47, 453 (2000).
  * (14) N. Sangouard, C. Simon, H. Riedmatten, and N. Gisin, Quantum repeaters based on atomic ensembles and linear optics, _Rev. Mod. Phys_. 83, 33 (2011).
  * (15) C. Zhuang, C. R. Paul, X. Liu, S. Maneshi, L.S. Cruz, and A.M. Steinberg, Coherent control of population transfer between vibrational states in an optical lattice via two-path quantum interference, _Phys. Rev. Lett._ 111, 233002 (2013).
  * (16) D. Dalvit, R. Filho and F. Toscano, Quantum metrology at the Heisenberg limit with ion trap motional compass states, _New J. Phys_. 8, 276 (2006).
  * (17) S. M. Garaot, E. Torrontegui, X. Chen, M. Modugno, D. G. Odelin, S. Y. Tseng, and J. G. Muga, Vibrational mode multiplexing of ultracold atoms, _Phys. Rev. Lett_. 111, 213001 (2013).
  * (18) S. van Frank, A. Negretti, T. Berrada, S. Montangero, J. F. Schaff,T. Schumm,T. Calarco, and J. Schmiedmayer, Interferometry with non-classical motional states of a Bose-Einstein condensate, _Nat. Comm_. 5, 4009 (2014).
  * (19) R. Scelle, T. Rentrop, A. Trautmann, T. Schuster, and M. K. Oberthaler, Motional Coherence of Fermions Immersed in a Bose Gas, _Phys. Rev. Lett._ 111, 070401 (2013).
  * (20) T. Rentrop, A. Trautmann, F. A. Olivares, F. Jendrzejewski, A. Komnik and M. K. Oberthaler, Observation of the Phononic Lamb Shift with a Synthetic Vacuum, _Phys. Rev. X_ 6, 041041 (2016)
  * (21) I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases _Rev. Mod. Phys_. 80, 885 (2008).
  * (22) A. Derevianko, H. Katori, Colloquium: Physics of optical lattice clocks, _Rev. Mod. Phys_. 83, 331 (2011).
  * (23) T. Müller, S. Fölling, A. Widera, and I. Bloch, State preparation and dynamics of ultracold atoms in higher lattice orbitals, _Phys. Rev. Lett._ 99, 200405 (2007).
  * (24) C. Schori, T. Stöferle, H. Moritz, M. Köhl, and T. Esslinger, Excitations of a superfluid in a three-dimensional optical lattice, _Phys. Rev. Lett._ 93, 240402 (2004).
  * (25) J. S. Krauser, J. Heinze, N. Fläschner, S. Götze, O. Jürgensen, D. S. Lühmann, C. Becker, and K. Sengstock, Coherent multi-flavour spin dynamics in a fermionic quantum gas, _Nat. Phys._ 8, 813 (2012).
  * (26) F. Pinheiro, G. M. Bruun, J. P. Martikainen, and J. Larson, XYZ quantum Heisenberg models with p-orbital bosons, _Phys. Rev. Lett_. 111, 205302 (2013).
  * (27) C. Wu, W. V. Liu, J. Moore, and S. D. Sarma, Quantum stripe ordering in optical lattices, _Phys. Rev. Lett._ 97, 190406 (2006).
  * (28) C. Wu, D. Bergman, L. Balents, and S. D. Sarma, Flat bands and Wigner crystallization in the honeycomb optical lattice, _Phys. Rev. Lett._ 99, 070401 (2007).
  * (29) S. Masuda, K. Nakamura, and A. D. Campo, High-fidelity rapid ground-state loading of an ultracold gas into an optical lattice, _Phys. Rev. Lett_. 113, 063003 (2014).
  * (30) X. Chen, A. Ruschhaupt, S. Schmidt, A. D. Campo, D. Gusery-Odelin, and J. G. Muga, Fast optimal frictionless atom cooling in harmonic traps: Shortcut to adiabaticity, _Phys. Rev. Lett._ 104, 063002 (2010).
  * (31) X. X. Liu, X. J. Zhou, W. Xiong, T. Vogt, and X. Z. Chen, Rapid nonadiabatic loading in an optical lattice, _Phys. Rev. A_ 83, 063402 (2011).
  * (32) I. B. Spielman, P. R. Johnson, J. H. Huckans, C. D. Fertig, S. L. Rolston, W. D. Phillips, and J. V. Porto, Collisional de-excitation in a quasi-two-dimensional degenerate bosonic gas, _Phys. Rev. A_ 73, 020702(R) (2006)
  * (33) R. Bücker, J. Grond, S. Manz, T. Berrada, T. Betz, C. Koller, U. Hohenester, T. Schumm, A. Perrin and J. Schmiedmayer, Twin Atom Beams, _Nature Physics_ 7, 608 (2011)
  * (34) Y. Y. Zhai, X. G. Yue, Y. J. Wu, X. Z. Chen, P. Zhang, and X. J. Zhou, Effective preparation and collisional decay of atomic condensates in excited bands of an optical lattice, _Phys. Rev. A_ 87, 063638 (2013).
  * (35) Z. K. Wang, B. G. Yang, D. Hu, X. Z. Chen, H. W. Xiong, B. Wu, and X. J. Zhou, Observation of quantum dynamical oscillations of ultracold atoms in the F and D bands of an optical lattice, _Phys. Rev. A_ 94, 033624 (2016).
  * (36) D. Hu, L. X. Niu, B. G. Yang, X. Z. Chen, B. Wu, H. W. Xiong, and X. J. Zhou, Long-time nonlinear dynamical evolution for P-band ultracold atoms in an optical lattice, _Phys. Rev. A_ 92, 043614 (2015).
  * (37) K. Kneipp, Y. Wang, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, Population pumping of excited vibrational states by spontaneous surface-enhanced Raman scattering, _Phys. Rev. Lett._ 76, 2444 (1996).
  * (38) M. Horikoshi and K. Nakagawa, Suppression of dephasing due to a trapping potential and atom-atom interactions in a trapped-condensate interferometer, _Phys. Rev. Lett_. 99, 180401 (2007).
  * (39) F. Dalfovo, S. Giorgini, Lev P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, _Rev. Mod. Phys_. 71, 463 (1999).
  * (40) R. G. Scott, D. A. W. Hutchinson, T. E. Judd, and T. M. Fromhold, Quantifying finite-temperature effects in atom-chip interferometry of Bose-Einstein condensates, _Phys. Rev. A_ 79, 063624 (2009).
  * (41) L. Isella, and J. Ruostekoski, Nonadiabatic dynamics of a Bose-Einstein condensate in an optical lattice, _Phys. Rev. A_ 72, 011601(R) (2005).
  * (42) Y. Zhai, P. Zhang, X. Chen, G. Dong, and X. Zhou, Bragg diffraction of a matter wave driven by a pulsed nonuniform magnetic field, _Phys. Rev. A_ 88, 053629 (2013).
  * (43) K. Li, L. Deng, E. W. Hagley, M. G. Payne, and M. S. Zhan, Matter wave self-imaging by atomic center-of-mass motion induced interference, _Phys Rev Lett._ 101, 250401 (2008).
  * (44) S. Yoon, F. Dalfovo, and G. Watanabe, Swallowtail band structure of the superfluid Fermi gas in an optical lattice, _Phys. Rev. Lett._ 107, 270404 (2011).
  * (45) B. T. Seaman, L. D. Carr, and M. J. Holland, Period doubling, two-color lattices, and the growth of swallowtails in Bose-Einstein condensates, _Phys. Rev. A_ 72, 033602 (2005).
  * (46) S. B. Koller, E. A. Goldschmidt, R. C. Brown, R. Wyllie, R. M. Wilson, and J. V. Porto, Nonlinear looped band structure of Bose-Einstein condensates in an optical lattice, _Phys. Rev. A_ 94,, 063634 (2016).
  * (47) C. F. Ockeloen, A. F. Tauschinsky, R. J. C. Spreeuw, and S. Whitlock, Detection of small atom numbers through image processing, _Phys. Rev. A_ 82, 061606(R)(2010).
  * (48) Y. Castin and R. Dum, Bose-Einstein Condensates in time dependent traps, _Phys. Rev. Lett._ 77, 5315 (1996).