Study to improve the performance of interferometer with 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  (?).

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.

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).

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  (?).

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  (?, ?, ?).

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  (?).

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.

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.

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:

(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.

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  (?, ?).