1. Introduction

Cold atom systems provide a unique opportunity to obtain various information with high precision from an interacting many-body system, which can help us gaining physical understanding of strongly correlated systems. For instance, analysis of spatial noise correlations in the time-of-flight (TOF) images can reveal density or spin correlations for atoms loaded in optical lattices (OLs) [1, 2], or pairing correlation in a Fermi superfluid [2, 3]. Measurements of in-situ density fluctuations have revealed the Pauli blocking effect [4], provided information of spin or density susceptibility of strong interacting gases [5], and verified the scaling law of a critical state in two-dimensional Bose gas [6].

To reveal quantum correlation and fluctuation effects in an ultracold atomic gas, a central task is to separate the extrinsic noises due to the imperfect experimental setup and state preparation from the intrinsic sources of quantum or thermal fluctuations. These noises are often coupled, masked by nonlinear effects, and buried in a large number of pixels, which make the task even harder. Principal component analysis (PCA) provides a great approach for solving this problem [7–12]. Nowadays this method is being widely used in computer science to help reducing data dimensions and find internal structure of massive high dimensional data. For a similar purpose, we use PCA to analyze time-of-flight (TOF) images of Bose-Einstein condensates (BECs) in one-dimensional (1D) OLs., where data dimensions are as many as the number of image pixels but the noise origins are much fewer. The application of PCA on the raw TOF data suggests that the leading noise sources in our experiment are fluctuations of atom number and spatial position. By preprocessing the raw data with normalization and adaptive region extraction methods, we can significantly reduce or even eliminate these two noises. As a result, PCA of the preprocessed data reveals more subtle structure of noises. We attribute the few dominant noise components with their corresponding physical origins, and compare experimental results with numerical simulations using physical parameters determined by experiments.

The aim of PCA is to approximate the variations of a dataset using a minimal group of orthogonal vectors called principal components (PCs) while preserving information in the dataset as much as possible. With the help of eigenvalues given by PCA, we can distinguish and focus on main features of variations in a dataset. As PCA is applied to experimental data, PCs acquire their physical meanings apart from their original concepts in mathematics. In our system, experimental data are time-of-flight images and PCs are effectively eigenmodes of fluctuations in our experiments. Thus, the total fluctuation is a linear combination of these eigenmodes, and the result of a specific TOF image denoted by A_i can be represented by the average over images plus its fluctuation, namely $A_i=\overline{A}+\sum {\epsilon }_{ij}{P}_j$. Here, P_j denotes different eigenmodes of fluctuation, the absolute value of the coefficient ${\epsilon }_{ij}$ describes how much P_j contributes to the i-th measurementA_i, and its sign indicates in which way P_j influences A_i.

Because of the linearity of PCA, the eigenmodes associated with different PCs have a one-to-one correspondence to different sources of fluctuations in a linear system. For a nonlinear system, such as an interacting many-body quantum system, where variations are masked by nonlinear interaction effects, the eigenmodes of PCA are in general nonlinear combinations of various noises. However, as we will demonstrated below, the nonlinearity in the present system turns out to be sufficiently small, such that different noises can be decomposed efficiently using PCA.

The remainder of this manuscript is organized as follows. In Sec. 2, we briefly introduce the experimental setup. The protocols of the PCA method is explained in Sec. 3, and then implemented for BECs both in the absence and presence of OL in Secs. 4 and 5, respectively. By comparing with numerical simulation, we identify the physical origins of up to five dominant PCs in the noise of TOF images. Finally, we discuss some remarks of the PCA method and summarize in Sec. 6.

2. Experimental setup

The system we used here is similar to the one in our previous experiments [13–15], which is a hybrid trap composed of a quadrupole magnetic trap and an optical dipole trap, as shown in Fig. 1. Our BEC setup is as follows. A BEC of about $N_0=2\times {10}^5{}^{87}$Rb atoms in the $|F=2,m_F=2>$ state is first prepared in the trap with frequencies ${\omega }_x=2\pi \times$ 28Hz, ${\omega }_y=2\pi \times$ 50 Hz and ${\omega }_z=2\pi \times 60$ Hz. The initial temperature of BEC is about $90nK\sim 110nK$ and the initial BEC fraction is around 55%. Within 40 ms the BEC is adiabatically loaded into a 1D OL along the x-direction. The lattice wavelength is $852nm$, and the height can be tuned within a range from $6E_R$ to $21E_R$, where $E_R={\hslash }^2{k}_L^2/2m$ is the photon recoil energy. After 35 ms we turn off the harmonic trap and the OL simultaneously to release the BEC. The absorption images are taken in the x

–z plane upon 31ms of free expansion with the size of each CCD pixel $6.8\mu$m$\times 6.8\mu$m. Here, we use strong saturated near-resonance imaging laser to obtain the density distribution of atomic gas of high density, and calibrate the imaging system to validate the TOF measurement [16]. Since absorption imaging is destructive, atoms have to be prepared repeatedly, resulting in variations from shot to shot. In our experiments, we took about 40 images at each lattice depth. We also studied 100 TOF images of BEC without OL as a preliminary experiment. For each raw TOF image, we implement an optimized fringe removal algorithm (OFRA) to eliminate the background interference of imaging light [17]. As a result, the residue fringes are much weaker in magnitude and can be easily discriminated from signals for the few leading principal components.

 

Fig. 1 (a) Experimental setup. The harmonic trap is composed of potentials produced by the quadruple coil and the Gaussian beam. The direction of the optical lattice (OL) is roughly the same as the Gaussian beam and they overlap atthe position of the atomic cloud. (b) Trap time sequence. BEC is prepared in a harmonic trap first. Then OL starts to ramp up adiabatically at t = 0 ms and reaches the configured depth in 40 ms. After a holding time of 35 ms, both the harmonic trap andthe OL are turned off at the same time. Absorption images are taken in the x-z plane upon 31ms of free expansion.

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3. Methods of data analysis

3.1. Protocol of PCA

The size of the area where atoms reside (about 100 pixels × 50 pixels) is usually much smaller than the size of raw TOF images (1317 × 1035) given by the CCD camera in our experiments. Hence we need to extract a $h\times w$ region of interest (denoted by $\mathcal{A}$ below) before we apply PCA. We attempted to apply PCA to TOF images directly, in which case PCA gave patterns of probe light interference fringes and treated atom distributions as noises. $\mathcal{A}$ should be as small as possible to reduce noises, but fully cover the area where atoms reside. In our experiment the size of $\mathcal{A}$ is 27 × 121 that comes from a theoretical estimation of the atom cloud size. However, we’d like to emphasize that our method is not sensitive to the size of $\mathcal{A}$. We changed $\mathcal{A}$’s size to 35 × 130. The patterns didn’t show much difference. Details of the method are described below and illustrated in Fig. 2:

 

Fig. 2 PCA protocols. (a) Transform two-dimensional regions of interest into column vectors A_i and stack them together. (b) Subtract the mean vector from A_i and keep fluctuations ${\delta }_i=A_i-\overline{A}$. (c) Construct covariance matrix $S=\frac{1}{n-1}X\cdot X^T$. (d) Decompose covariance matrix so that $V^{-1}SV=D$ where D is a diagonal matrix. (e) Transform eigenvectors of interest back to two-dimensional regions to reconstruct images.

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Vectorize image For the i-th image, retrieve the region of interest and transform it into a d-dimensional vector A_i, where $d=h\times w$.

Decompose A_i Denote $A_i=\overline{A}+{\delta }_i$, where $\overline{A}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{A}_i$ is the average of all n vectors, and ${\delta }_i=A_i-\overline{A}$ is the fluctuation.

Construct covariance matrix Stack δ_i together to form a matrix $X=\left[{\delta }_1,{\delta }_2,\cdots ,{\delta }_n\right]$. Then the covariance matrix S is obtained by $S=\frac{1}{n-1}X\cdot X^T$.

Decompose covariance matrix Compute the matrix V of eigenvectors which diagonalizes the covariance matrix with $V^{-1}SV=D$.

Reconstruct images If needed, reshape those $d\times 1$ eigenvectors of interest back to $h\times w$ matrices to reconstruct feature images.

We use the Scree graph method [11] to determine the number of PCs to be retained. Specifically, we plot the eigenvalues in descending order and determine the turning point from which the curve flattens. The eigenvalues above the turning point are of significance and about to be retained.

3.2. Preprocessing method

To identify the physical origins of the PCs, we also preprocess the raw TOF data to eliminate the fluctuations of atom number and spatial position, and compare the new outcome of PCA with the original ones. The preprocessing methods are listed as follows.

Normalization

Normalization is designed to reduce atom number fluctuation in the region of interest. In TOF images, the atom number is determined as

As weakly interacting BECs in our experiments can be described by a macroscopic wave function $\mathrm{\Psi }\left(r\right)=\sqrt{N}\varphi \left(r\right)$ [18], our interest is the density distribution of the normalized wave function ${\left|\varphi \left(r\right)\right|}^2$ instead of the prefactor N. So it is safe to normalize the TOF data so that the pixel values in a region of interest are summed up to unity. By doing so, densities in different TOF images fall into the same range and become comparable.

To normalize the raw TOF data, we first eliminate the bias. In principle, the pixel values at positions where no atom is present should be zero. However, there may exist a finite signal as a bias in real experiments. One method to eliminate this noise is by simply taking the bias as the minimal value of pixels. A slightly more complicated but more robust method that is used in our experiment is taking the mean value of pixels where there is no atom as a uniform background noise, and then subtracting it from all pixels. After the elimination of bias, we can normalize the pixel values using

Adaptive region extraction

Another fluctuation of TOF images is the shift of the cloud position, which may be induced by experimental misalignments of trapping potential, optical lattice potential, or imaging camera. Adaptive region extraction is designed to select a region whose center is also the center of density distribution. We first set a criterion to determine the center of density distribution within an extracted region, then use the center as a new region center to extract a new region. We iterate this procedure until the region to be extracted becomes stable.

Under the same experiment condition, the cloud position fluctuations are within a few pixels that is much smaller than $\mathcal{A}$’s size. Therefore, instead of searching region centers in the whole TOF image, we can choose a initial region center manually and apply it to all TOF images. Consequently, we are able to use a fixed region size in our iterative procedure. We also note that in general the coordinates of the cloud center are not integers. While simply rounding them to integers may cause artificial anisotropy in our images, we use interpolation to estimate the pixel values with non-integer coordinates.

For the choice of criterion, a simple method is to set the pixel with maximal value as the center of density distribution. This procedure works well in most cases provided that the CCD pixel noises are small enough. In the following discussion, however, we use a slightly generalized criterion which determines the center by a weighted mean of all pixels in the region, where the weight is chosen to be the pixel values.

4. PCA analysis in the absence of OL

As a preliminary experiment, we first analyze TOF absorption images of a 31ms free-expanding BEC released from the harmonic trap, whose fluctuation modes should be relatively simpler. Figure 3(a) shows percentages of the total variance associated with the first 10 PCs. The green dashed line is a smoothed line that connects these ten points, from which we can easily tell the turning point resides between the third and fourth PCs and the critical value for retaining PCs is around 5%. So we reconstruct the feature images corresponding to the first three primary PCs, which are shown in Figs. 4(a)–4(c).

Figure 4(a) is similar to the original absorbing image, which corresponds to atom number fluctuations. Figures 4(b) and 4(c), whose percentages are of the same order of magnitude, reflect the position uncertainty of the BEC along two inter-perpendicular directions in x − z plane. They together represent atom cloud position fluctuations in x − z plane. Their origin should be some mechanical effects such as a shift of the magnetic trap position or drift of the CCD camera, both of which can cause a position deviation of BEC in TOF images. Intuitively we may expect position shift directions in Figs. 4(b) and 4(c) to align to x and z axes respectively. However, physical origin of Figs. 4(b) and 4(c) mentioned above should cause isotropic position fluctuations in x − z plane, in which case the result of PCA also has this symmetry and the patterns in Figs. 4(b) and 4(c) can be jointly rotated by an arbitrary angle in the x − z plane. In realistic experiments, noises in our dataset make position fluctuations slightly anisotropic, and PCA chooses a bias direction for a specific dataset as shown in Figs. 4(b) and 4(c). By applying the method to other datasets, we do observe very same patterns as in Figs. 4(b) and 4(c) along different directions while the eigenvalues remain roughly the same. We’d like to emphasize that the result is different in presence of the optical lattice, as is discussed in section 5.1.

 

Fig. 3 The magnitude (right axis) and percentage ratio (left axis) of the eigenvalues associated to the first 10 PCs before (a) and after (b) preprocessing. The ratio is defined as the percentage of corresponding eigenvalue out of the summation of all eigenvalues. The green dashed line is a smoothed line that connects the first ten points to help find the turning point. The gray dashed line indicates the threshold for distinguishing important PCs. In our experiments, the threshold is 5%. PCs of interest are highlighted with different colors. Note that the first three PCs are significantly reduced in magnitude by preprocessing. Meanwhile, a new PC depicted by Fig. 4(d) appears after preprocessing as other leading noises sources are strongly suppressed.

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Fig. 4 Reconstructed feature images. (a) represents the fluctuation of atom number. (b) and (c) correspond to spatial fluctuations along two inter-perpendicular directions in x-z plane. They together represent atom cloud position fluctuations in x-z plane. (d) The new PC after preprocessing is characterized by a peak wrapped by a dip ring.

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To validate our attribution, we preprocess data using methods introduced in Sec. 3.2 to eliminate the number and position fluctuations, which respectively correspond to the feature images of PCs shown in Fig. 4(a) – 4(c). We then apply PCA to the resulting data, and plot the percentages and eigenvalues of the first 10 leading PCs in Fig. 3(b). As compared to the outcome without preprocessing, there are only two PCs left above the critical line (gray line in Fig. 3(b)), with a combined contribution of $>90\%$. We reconstruct feature images of the first four PCs, and find that PCs whose patterns are similar to Figs. 4(a) – 4(c) now take the first, third and fourth places. The eigenvalue of the PC depicted in Fig. 4(a) decreases by 36%, from 1.846 × 10⁻⁵ to 1.180×10⁻⁵, even if its ratio becomes larger because other noises are suppressed stronger. The eigenvalues and ratios of PCs depicted in Figs. 4(b) and 4(c) are below the critical line at present, ready to be ignored in our analysis. A new PC takes the second place after preprocessing, corresponding to the variation of the transversal radius of the cloud as depicted in Fig. 4(d), which may be attributed to the breathing mode of BEC [8].

From this result, we can conclude that the atom number fluctuations are effectively reduced and the position fluctuations are nearly eliminated by our preprocessing. We stress that if one employs a simpler but coarse version of adaptive region extraction where the pixel coordinates is always rounded to integers, although the PCs associated with position fluctuations can still be significantly reduced in magnitude, they remain to be leading PCs as the truncation errors are relevant if the size of our CCD pixels are larger than the real spatial shifts.

5. PCA analysis of BEC in OL

We now apply PCA to analyze TOF images of BECs in an OL. In Sec. 5.1, we present the first five leading PCs and their corresponding feature images for a typical optical lattice depth of $15E_R$. The experimental results are in good quantitative agreement with numerical simulation as discussed in Sec. 5.2. Finally, we discuss in Sec. 5.3 the variation of the leading PCs with optical lattice depth.

5.1. PCA results of BEC in OL

We analyze TOF images of BECs loaded in an OL with depth of $15E_R$ with the PCA method introduced above, and use the Scree graph method to obtain the feature images of the first five leading PCs as shown in the top panels of Fig. 5(b) – 5(f). To see clearly the variation patterns of these images, we also integrate over the vertical (horizontal) dimension for Figs. 5(b) and 5(d) – 5(f) (Fig. 5(c)) by summing up pixel values, and show the columnar density by solid blue lines in the corresponding bottom panels. In the top panel of Fig. 5(a), we present a typical example of TOF image before preprocessing, while the blue line in the bottom shows the three interference peaks clearly.

The first PC, denoted by P₁ (Fig. 5(b1)), has the same pattern as the atom number fluctuation PC discussed in Sec. 4, except for the symmetric side peaks caused by the presence of OL. Indeed, even in the case of high lattice depth where the atoms residing on different lattice sites form a local quasi-condensate while the system as a whole does not possess long-range phase coherence, the side peaks are still present as a consequence of short-range correlation [19]. If we normalize the data, this PC will disappear or become significantly less important.

The second (P₂) and third (P₃) PCs as depicted in Figs. 5(c1) and 5(d1) show clear patterns of position fluctuation along the z- and x-directions, respectively. Different from the case without OL where directions in these two patterns are random if we run PCA on part of our data, directions of P₂ and P₃ always align to z and x axes and the eigenvalue of P₃ is larger than that of P₂ constantly. This result suggests that there are causes not existing in the absence of OL that can enhance position fluctuations along x direction, as is discussed in section 5.2. After preprocessing the images with normalization and adaptive region extraction, a PCA on the resulting data gives a leading PC of number fluctuation mode as in the case without OL, while the position fluctuation modes almost disappear. Meanwhile, we also note that the absolute strength of the number fluctuation, i.e., the eigenvalue of the corresonding mode after preprocessing, is larger than that in the case without OL, indicating that the number fluctuations of BEC in OLs can not be normalized as effectively as in the case without OL.

The feature image of the forth PC P₄ as shown in Fig. 5(e1) is similar to that of the number fluctuation mode P₁, but with two negative dips accompanying the interference peaks. As we will see in the next subsection, this pattern reflects fluctuations of the width of each peak in the TOF image.

The fifth PC (P₅ as in Fig. 5(f1)) is featured by a central dip and two side dips with an overall Gaussian profile. The profile strongly suggests an intimate relation to fluctuations of the normal fluid fraction. The presence of dips can be understood by noticing that within the constraint of atomic number conservation, the increase of thermal atomic number is accompanied by a decrease of condensation fraction, which in turn leads to a reduced visibility of interference pattern.

Before concluding this subsection, we emphasize that the patterns of P₄ and P₅ are very small variations which can strongly couple with background noises. It is very difficult to distinguish these noises by conventional analysis on the TOF images. As a comparison, PCA works very effectively to extract these information.

 

Fig. 5 PCA results of BEC in OL. (a) Raw TOF image. (b) Atom number fluctuation. (c) and (d) Position fluctuations. (e) Peak width fluctuation. (f) Normal phase fraction fluctuation. We integrate the results of (a₁), (b₁), (d₁), (e₁), (f₁) vertically and (c₁) horizontally, to obtain the columnar integral as depicted by blue lines in the bottom parts of (a-f). Orange lines are simulation results. The unit of horizontal axes in (a₂), (b₂), (d₂), (e₂), (f₂) and the vertical axis in (c₂) are μm. The units of the columnar (col.) and row summations in all bottom panels are arbitrary and kept fixed within each individual panel since PCA always normalizes the resulting eigenvectors.

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5.2. Numerical simulation and comparison with experiments

To validate our previous attribution of physical origins to different PCs, we perform a numerical simulation for BEC in an OL using time-split spectral algorithm (TSSP) algorithm [20, 21]. We first calculate the ground state wave function of the BEC by solving the conventional Gross-Pitaevskii equation (GPE) within the potential generated by the OL and the magneto-optical hybrid trap. The wave function then undergoes a free expansion of 31ms governed by the time-dependent GPE. At the end, a Gaussian envelop is added to the density distribution to simulate the excited fraction, which can not be described properly with the GPE. The reason we use a Gaussian distribution to describe non-condensed particles is because the velocities of these atoms obey a Maxwell distribution after release.

To incorporate the fluctuation effects in our simulation, we extract the thermal fraction and the temperature of each shot by a bimodal fit of the raw TOF images. From Fig. 6, we find that the temperature T is about $100nK$ and the fluctuations of thermal atom fraction $P_{\text{ex}}\equiv N_{\text{thermal}}/N_{\text{tot}}$ is about $15\sim 20\%$ at each lattice depth respectively, where fluctuations is defined as the range of P_ex divided by the average of P_ex, such as $\left(66\%-56\%\right)/61\%\approx 16.4\%$ at $6Er$ or $\left(74\%-63\%\right)/68.5\%\approx 16.1\%$ at $9Er$. We stress that these quantities, together with the average values of T and P_ex for different lattice depth, are all directly measured from experiments with no fitting parameters.

 

Fig. 6 Statistical results of experiments. Each asterisk in the figure corresponds to one experimental shot. (a) Statistical results of temperature. (b) Statistical results of thermal atom fraction, which is defined as N_thermal/N_tot. The orange lines in panel (a) and (b) indicate the variation of the corresponding quantity. The variation is defined as the standard deviation divided by the mean.

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We consider separately the tilt of OL, the fraction of excited atoms, and the width variation of interference peaks induced by defocusing effect, and compare the corresponding numerical results with experimental outcome of P₃, P₄, and P₅, respectively. In Figs. 5(d2), 5(e2) and 5(f2), blue lines are experimental results and orange lines are simulation results. The details of our simulation results are as follows.

Misalignment of the optical lattice

As shown in Fig. 5(d), a translational shift along the x-direction is mainly manifested by P₃. In the absence of OL, the eigenvalue of P₃ is of the same order in magnitude as that of P₂, which characterizes position fluctuations along the z-direction (see Fig. 3). In an OL, however, P₃ becomes more significant for most experimental realizations. This strongly indicates that there must be some effects related to the OL contribute to P₃, in addition to the spatial shifts of the magnetic trap, the BEC and the camera.

We attribute this fluctuation to a misalignment of the optical lattice from the horizontal axis. Consider an atomic cloud initially trapped in a global harmonic potential with minimum at $\left(x_0,y_0,z_0\right)$. If the OL is perfectly aligned along the x-direction, the potential minimum is still at the original position and the center of mass of the BEC remains fixed. However, if the OL is slightly tilted towards the vertical z-axis, to the first order we can approximate the lattice as a perfectly aligned OL and a linear potential with energy gradient along the x-direction. As a result, the minimum of the total potential will be shifted to a now position $\left(x_0^{\text{'}},y_0,z_0\right)$. The shift of energy minimum then corresponds to an external push of the atomic cloud, and leads to a finite velocity of the cloud.

For the specific configuration discussed here, the lattice is ramped up at time t = 0. Suppose the lattice is tilted at an angle θ towards the z-direction, the linear potential can be represented as $V_{\text{tilt}}\left(x,t\right)=V_{\text{ol}}\left(t\right)\cos \theta x/a$, where $V_{\text{ol}}\left(t\right)$ is the lattice height at time t, and a denotes the lattice constant. The force exerted by this linear potential on an atom can be written as $f=-V_{\text{ol}}\left(t\right)\cos \theta /a$. Thus, at the end of the ramping process, the total velocity gained by the atom is

In our numerical simulation, we consider a random tilting with an angle of no more than $\pm 2.5\times {10}^{-5}$ rad. We find a good agreement between the simulation and the experimental result, as shown in Fig. 5(d2). This observation suggests that the PCA method can reveal very small misalignment of the OL, which may have application in detecting microgravity. Although the sensitivity reported here is about two orders of magnitude smaller than the uncertainty of ${10}^{-7}$ reached by measuring the 5th harmonic of Bloch oscillation of ${ }^{88}$Sr atoms in tilted optical lattices [22], our scheme can be easily implemented with ${ }^{87}$Rb atoms in a simpler experimental setup with conventional TOF techniques.

Width variation caused by defocusing effect

The fourth principal component P₄ represents width fluctuations of three momentum peaks, which we think are induced by fluctuations of interaction energy. The GPE is a nonlinear equation with an interaction potential term $N{U}_0{\left|\psi \left(r,t\right)\right|}^2\psi \left(r,t\right)$. This repulsive interaction between atoms tends to broaden the atomic distribution in both spatial and momentum space, resulting a defocusing effect with wider peak widths in TOF images [21]. Thus, the variation of atom number N can induce fluctuations to TOF signals, which can not be fully eliminated by a normalization of the BEC wave function. Another factor one needs to take into account is that while reducing three-dimensional GPE to 1D GPE, we have to assume a distribution along the y

- and z-directions to reduce ${\left|\psi \left(r,t\right)\right|}^2$ to ${\left|\psi \left(x,t\right)\right|}^2$. This means the density distributions along the y- and z-directions still have an influence on peak width along the x-direction. Fluctuation in these transversal directions as shown in Fig. 4(d) for the case without OL is hence another source of noise.

Fraction of excited atoms

Another quantity that presents sizable fluctuations from shot to shot is the fraction of non-condensed atoms. For a BEC in a 1D optical lattice discussed here, such kind of fluctuations can be induced by the variations of temperature or effective interaction strength, as the results of imperfect system preparation and loading processes [23]. Specifically, since each lattice site corresponds to a pancake-shaped cloud with a few tens or hundreds of atoms, the effective interaction strength U₀ is sensitively dependent on the number of particles because of the broadening effect of Wannier function [24]. We emphasize that the interaction induced fluctuations are quantum effect which can persist even at zero temperature. We numerically simulate the fifth principal component P₅ one would obtain by considering excitation fraction fluctuations under a constraint of total particle number conservation, and observe a quantitative agreement with the experimental observation of P₅ as depicted in Fig. 5(f1). Note that the normalization of the solution of GPE should be imposed after a Gaussian fluctuation is added.

5.3. Variations with OL depth

We now turn to the PCA results for OLs of different depths. To quantify the trend of variation of PCs, we define a quantity

From Fig. 7(a), we notice that both ${\gamma }_{P_1}$ and ${\gamma }_{P_2+P_3}$ decrease with increasing lattice depth, while ${\gamma }_{P_4}$ and ${\gamma }_{P_5}$ exhibit an opposite dependence. These trends are qualitatively consistent with our attribution of physical origins of noises, considering the fact that the fluctuation of excited fraction (P₄) and the interaction effect (P₅) become more severe as OL gets deeper. In fact, we numerically simulate peak width variation under density distribution fluctuation at different lattice depths with all other conditions fixed. As shown in Fig. 7(b), ${\gamma }_{P_4}$ indeed grows when approaching the quantum phase transition from a coherent state to a number squeezed state with increasing OL depth [25]. In our experiment, the transition takes place at ∼24 E_R as can be extracted from the interference pattern of TOF images, which is consistent with the fluctuation results of Fig. 7. In fact, one would naturally expect that fluctuation of interference between different lattice sites will be significantly enhanced near the phase transition point.

 

Fig. 7 (a) Trends of PCs as lattice depth increases in experiments. We add up portions of P₂ and P₃ because they have the same physical origin. (b) The trend of ${\gamma }_{P_4}$ from numerical simulation. The red crossings are results of configurations we chose to simulate and the blue line is a smoothed line that connects our results. In our simulation, we use the values extracted from TOF images for the mean and fluctuation of thermal atom fraction and temperature.

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6. Summary and final remarks

Before concluding, we emphasize that the eigenvectors given by PCA are uncorrelated but not necessarily independent with respect to physical noises. In other words, one single PC can reflect a combination of more than one physical origins, although in most cases only one source is dominating and can be clearly distinguished. In this paper, we demonstrate that PCA can be readily implemented to obtain accurate physical interpretations for noises in complex many-body systems. As a comparison, independent component analysis (ICA), a method based on PCA, can reach optimized and parameterized independence between basis vectors [12]. But it requires a clear understanding about the sources of noise and their effects on the system to properly set criteria of independence. An inappropriate parameter setting could lead to severe wrongful analysis. This kind of misleading is absent in PCA as it is a standard, non-parametric method, requiring no prior knowledge of the system.

In summary, we introduce a method of PCA to analyze noise in TOF images of a BEC in a 1D OL. By investigating the corresponding feature images, we identify the physical origins associated to a few PCs of leading contribution. This understanding is then confirmed by a numerical simulation of a GPE with external sources of fluctuations. In particular, we can extract not only classical fluctuations such as a small tilt angle of the optical lattice laser, but also fluctuations of excitation fraction induced by finite temperature and interaction. Both factors are very weak effects that can not be extracted by conventional investigation of interference patterns. Based on the knowledge of the physical origins of leading PCs, we also design a preprocessing method to significantly reduce or even eliminate fluctuations of atom number and spatial position. As the method does not require any knowledge of the system properties, it has the potential to be applied to other quantum states and quantum phase transitions, such as the number squeezed states, which are left for future investigations. Besides, the PCA method could also be a convenient and powerful analysis tool in aspects including interferometers with higher precision [12, 26], measurement of microgravity, and high precision level meters.