Designing Arbitrary One-dimensional Potentials on an Atom Chip

: We use laser light shaped by a digital micro-mirror device to realize arbitrary optical dipole potentials for one-dimensional (1D) degenerate Bose gases of 87 Rb trapped on an atom chip. Superposing optical and magnetic potentials combines the high ﬂexibility of optical dipole traps with the advantages of magnetic trapping, such as eﬀective evaporative cooling and the application of radio-frequency dressed state potentials. As applications, we present a 160 µm long box-like potential with a central tuneable barrier, a box-like potential with a sinusoidally modulated bottom and a linear conﬁning potential. These potentials provide new tools to investigate the dynamics of 1D quantum systems and will allow us to address exciting questions in quantum thermodynamics and quantum simulations


Introduction
Ensembles of ultra cold atoms are an ideal and versatile tool to investigate a plethora of physical phenomena. Their near-perfect isolation from the environment, paired with the availability of powerful techniques to probe and manipulate them, allows for the observation of coherent quantum evolution over long times with unprecedented detail [1,2]. When building such model systems, a flexible and precise control over the confining potential often represents a key ingredient to access new phenomena. The realization of homogeneous systems, for example, can unveil effects otherwise masked by an inhomogeneous density distribution [3][4][5], which allowed for the observation of recurrences in our isolated quantum many-body system [6]. Full local and dynamic control of the potential landscape can allow for the controlled excitation of collective modes [7], the realization of sonic black holes [8,9] or exploring highly entangled topological states [10]. This flexibility is often achieved through optical dipole potentials [11] of shaped light fields realized by spatial light modulators (SLMs) [12][13][14][15].
Another versatile technique for trapping and manipulating cold neutral atoms are micro-traps generated by atom chips [16,17]. In a number of experiments a large variety of potential configurations have been realized using combinations of static magnetic [18,19], electric [20], optical [21] and radio-frequency (RF) fields [22]. Atom chips are well suited to create precisely controlled, steep and robust confinements, in particular highly elongated trapping configurations. In such traps, the atoms are tightly confined along two spatial dimensions and loosely confined arXiv:1908.01563v1 [cond-mat.quant-gas] 5 Aug 2019 Atom chip Imaging optics DMD pattern Atomic cloud x z y Fig. 1. Combining optical dipole potentials with the magnetic confinement created by current carrying micro-wires of an atom chip (not to scale). The DMD pattern is imaged onto the plane of the atomic cloud trapped below the atom chip. The controllable light field can be used to modify and 'correct' the trapping potential created by the atom chip. In the remainder of the manuscript, the x-axis is oriented along the propagation direction of the dipole light. The y-axis points towards gravity and the z-axis is oriented along the longitudinal direction of the atom chip trap. along the third dimension, realizing a quasi one-dimensional (1D) configuration (see Fig. 1). In our setup, the tight transverse confinement is controlled by the current flowing in a single micro-wire and a homogeneous external bias field. The weak longitudinal confinement, on the other had, is generated by additional current carrying wires on the chip. The detailed shape of the longitudinal confinement is influenced by imperfections in the current flow [23] which can be measured by magnetic field microscopy [24][25][26][27].
In this paper we demonstrate that the spatial structure of the longitudinal trapping potential and its imperfections can be 'corrected' by applying an additional structured dipole potential to the existing atom chip trap, which maintains the transverse trapping and manipulation. A far blue detuned light field will create a conservative repulsive potential for the atoms. Applying the structured light field from a direction orthogonal to the 1D chip trap allows to create a nearly arbitrary modification of the potential landscape along the 1D trap.
This setup preserves the advantages of the tight magnetic trap in the transverse direction, such as the ability to deform the trap into a double-well potential using RF dressed state potentials, while greatly improving the control over the longitudinal 1D confinement. We use a digital micro-mirror device (DMD) to shape arbitrary intensity profiles necessary for realizing a large variety of 1D optical dipole potentials.

Basic Design
The potential created by a light field interacting with induced dipole moments of neutral atoms is proportional to the intensity of the light field and the sign of the detuning, V dip (r) ∝ sgn(ω−ω 0 )I(r) [11]. Here ω 0 is the optical transition frequency, ω and I(r) are driving frequency and intensity of the light field, respectively. The sign of the detuning ∆ = ω − ω 0 determines whether the potential is repulsive (∆ > 0, blue-detuned) or attractive (∆ < 0, red-detuned). Fig. 2 shows an example how the right intensity distribution can be used to create a box-like potential by correcting a rough harmonic confinement through the addition of blue-or red-detuned dipole potentials.
The spatial structure of optical dipole potentials is determined by the spatial structure of the light beam. As such, beam-shaping techniques based on different types of SLMs can be exploited to modulate not only the intensity but also the phase of light fields. We use a DMD which is a reflective SLM offering fast and flexible intensity modulation of light fields.
Creating a box-like potential (green) by superposing blue-or red-detuned light fields (∆ > 0 and ∆ < 0 respectively) with a rough harmonic potential (black). The intensity profiles needed to achieve the target potential is shown for each case.
DMDs are powerful devices in terms of controllability and speed of performance, but ultimately, they can only produce pixelated binary patterns. To create gray-scale images from binary DMD patterns, time and/or spatial averaging has to be applied.
Imaging the 2D binary pattern of a DMD results in the 2D intensity distribution I(y, z) on the image plane (plane of atoms). Consider a 1D Bose gas extending along the z-axis in a highly anisotropic harmonic trap (ω ⊥ /ω z 1). For typical experimental values of ω ⊥ , the transverse width of such an atomic cloud is much smaller than the resolution of commonly used optical imaging systems. For example, in our experiment with ω ⊥ ∼ 2π × 2 kHz, the width of the Gaussian ground state wave function is a ⊥ = /(Mω ⊥ ) ∼ 250 nm. Thus, assuming that the cloud is located at y cloud , only a tiny slice from the image, I(z) ≡ I(y = y cloud , z), is relevant for shaping the longitudinal potential. In the rest of the manuscript, I(z) refers to a 1D intensity profile in the longitudinal direction at (y = y cloud , z).
To achieve a fine intensity resolution in the projected light pattern we must implement spatial averaging of the binary DMD pattern. This can be accomplished by designing the imaging system in such a way, that through large demagnification, many binary pixels contribute to a single diffraction limited spot in the image plane (see Fig. 3). In our case of controlling the longitudinal potential of a 1D system, the demagnification has an additional advantage: the DMD pixels contribute to the intensity at y cloud depending on their vertical distance from the central axis, and therefore we obtain an improved intensity resolution at y = y cloud .
An additional knob of control is spatial filtering: manipulating the Fourier components of light affects the optical resolution of the system. Blocking high frequency components worsens the optical resolution, which ultimately extends the contribution range of a single pixel in the binary pattern of the DMD. In the case of 1D systems, only the spatial resolution along the longitudinal direction is relevant. Reducing the numerical aperture along the perpendicular direction will leave the resolution in the longitudinal direction unchanged. But more pixels from the perpendicular direction will contribute to the 1D intensity. This improves the tuneability of the potential at y = y cloud . The price to pay for performing spatial filtering is a loss in overall intensity.
Beyond spatial averaging fast intensity modulation by the DMD can be exploited to extend the dynamical range of the created potential. This modulation has to be fast compared to the P P Image plane Expanding the dynamical range of intensity using the concept of spatial averaging and spatial filtering. The images of single DMD pixels are shown in the image plane. For a diffraction limited spot much samller than the pixel size, the DMD pixels are well resolved in the image plane, and of the five pixels switched on (shown in different colors), only one (red) would contribute to the intensity of point P (P ). If the image is blurred due to the finite imaging resolution, the pixels are imaged as a broad, overlapping spots. In this case all pixels within the orange circle (ellipse) contribute to the intensity of point P (P ). In (a), the point spread function (PSF) of the imaging system is symmetric and three of the five active pixels (red, blue, green) contribute substantially to the intensity at point P. In (b), horizontal-pass spatial filtering broadens the PSF in the vertical direction, which leads to an increased number of pixels contributing to the intensity at point P .
dynamical timescales of the trapped atoms, such that only a time-averaged potential is seen by the atoms [28]. An alternative to imaging the DMD directly is to employ it in the Fourier plane of an imaging system as a binary filter to modulate both phase and amplitude of the light field [29]. However, if phase modulation is not necessary, direct imaging turns out to be more straightforward.

Experimental Implementation
In our experiment, we realize 1D degenerate Bose gases of 87 Rb atoms with an atom chip [19]. Current carrying micro-fabricated wires on the chip surface along with an external bias field create a highly anisotropic Ioffe-Pritchard type trap with approximately 3D harmonic confinement: being the mass of a single atom, ω ⊥ the transverse frequency and ω z the longitudinal frequency of the trap. Typical parameters are ω z ∼ 2π × 10 Hz and ω ⊥ ∼ 2π × 2 kHz. Furthermore, the cigar shaped cloud can be transversely split into two by applying RF fields [22].
Absorption imaging is used to obtain projected 2D atomic density distributions, ρ(y k , z k ), in time of flight (TOF) after switching off the trap [30]. The spatial grid (y k , z k ) correspond to camera pixels in the plane of atoms. Density profiles along the longitudinal axis of the 1D system, ρ(z k ), are extracted by integrating over the y-direction. The scheme of our absorption imaging system (AIS) is shown in Fig. 4. The projected size of the camera pixel in the object plane of the AIS (plane of atoms) is 1.05 µm. The influence of the imaging resolution of the AIS onto the measured 1D profiles can be approximately modeled by a convolution with a Gaussian point spread function with σ AIS = 2.1 µm [31].
The small transverse width of the cloud renders direct in situ imaging not practical.  longitudinal confinement suppresses the longitudinal expansion and allows us to measure the longitudinal density profile at short TOFs (between 1 and 3 ms) with negligible influence of the expansion. For cold samples, in situ thermal fluctuations are also negligible at this short TOFs [32][33][34].
Although a variety of trapping geometries can be produced using RF dressed state potentials, once the atom chip is designed, adjusting the longitudinal confinement is greatly limited. Moreover, there exists potential roughness due to imperfections of micro-wires on the chip [23]. Compensating for this potential roughness without an additional potential is possible in principle, but requires special chip designs [35]. Adding an optical dipole potential allows to correct the potential roughness and shape the the longitudinal trapping potential in a very flexible way.
To create the dipole potentials we use a λ dip = 660 nm laser (opus660, Laser Quantum) which provides 1.5 W of spatially single mode light, blue-detuned with respect to the D line of 87 Rb (λ D 2 = 780 nm, λ D 1 = 795 nm). For our purpose, which is primarily correcting an existing potential and creating potentials with steep walls in 1D, a blue-detuned light field is advantageous [11]. The choice of λ dip is mainly motivated by the anti-reflection coating of the existing optical elements in the experiment (see Fig. 4) which favours wavelengths ranging from 650 to 1150 nm. Being far-detuned from the resonance guaranties that for U dip / ∼ 2π × 1 kHz the scattering rate is ∼ 2π × 0.1 mHz. This leads to only one spontaneous scattering per second for an ensemble of 10 4 atoms, causing negligible atom loss and heating.
We apply the concepts discussed in Sec. 2 by imaging DMD patterns directly onto the plane of atoms. Our DMD chip is a DLP9500 from Texas Instruments with 1080 × 1920 micro-mirrors, each 10.8 × 10.8 µm. The refresh rate for the whole array is 17 kHz, which increases to 50 kHz when the number of active rows is limited to 50. The DMD chip is a part of the V-9501 module by ViALUX. This module also contains a V4395 main board which includes a completely configured high-speed FPGA logic and firmware.
The optical setup imaging DMD patterns onto the plane of atoms is shown in Figure 4. It consists of two demagnification stages: first, a 4f system is used to perform the horizontal-pass spatial filtering using an adjustable rectangular slit. Secondly, a plano-convex lens along with the objective of the AIS images the DMD pattern onto the plane of atoms.
The DMD chip is placed at the focal point of the first lens and the distance between lens 1 and lens 2 is the sum of the two focal lengths, f 1 + f 2 = 30 cm + 10 cm = 40 cm. The width of the rectangular slit sitting in the Fourier plane of the system can be adjusted independently in vertical and horizontal direction from 0 to 12 mm. The image formed by the first two lenses is further demagnified by lens 3 and the objective of the AIS. The focus of the whole system is adjusted by lens 3 which is mounted on a motorized stage. This is necessary such that the AIS and the dipole light can be focused independently. The total magnification of the DMD imaging system is 1/25.9 and it has a minimum spot size (Airy diameter) of 4 µm in the plane of atoms. This implies that more than 70 DMD pixels contribute to the Airy disk at a single point. The DMD chip is illuminated by a Gaussian beam whose horizontal extension is elongated by a cylindrical telescope. This is done to match the available intensity to the size of the cloud and thereby efficiently use the available power. The polarization of the dipole potential beam is also set such that the beam gets reflected in the PBS cube in the second demagnification stage with minimal loss.

Real-Time Pattern Optimization
With the optical setup described in the previous section, an area of interest (AOI) on the DMD chip with n y × n z pixels (typically 20 × 500 pixels for a 200 µm long 1D trap) will be imaged onto the plane of atoms. With these pixels, 2 (n y × n z ) different patterns can be created. The challenge is to find a pattern which produces a 1D optical dipole potential which results in the desired target potential V T (z) when superposed with the magnetic confinement of the atom chip. In principle, we can calculate the 2D image of DMD pattern numerically and design the pattern to obtain desired potentials. However, due to interferences the actual dipole potential is sensitive to precise alignment of optical elements and imperfections in the setup and is difficult to predict precisely in a real world setting. Hence, an iterative pattern optimization is used to obtain the best possible reconstruction of the target potential.
In the experiment, we measure density profiles ρ(z k ). Thus, the ultimate goal of our pattern optimization process is to find a DMD pattern which creates a 1D density profile ρ(z k ) as close as possible to a simulated target density ρ T (z k ). The target density is obtained by calculating the ground state of a 1D non-polynomial Schrödinger equation (NPSE) [36] with the external target potential V T (z k ).
Before starting the pattern optimization, a calibration is performed to infer the corresponding position in the plane of atoms for the different DMD pixels in horizontal and vertical directions separately. For horizontal (longitudinal) calibration, a number of DMD columns are turned on at different positions. Their position is then detected on the 1D density and used to derive a linear map. To perform vertical calibration, a simple barrier is created by a thin super pixel in the center of the atomic cloud (typically 3 rows with 30 to 50 pixels each). Shifting this super pixel vertically changes the barrier height. At the point where this height is maximum, the center of the super pixel is selected as the central row of the DMD pattern. This row corresponds to the position of the elongated atomic cloud (y = y cloud ) in the plane of atoms and has the maximum contribution to I(z). The vertical calibration also defines the vertical AOI, the area outside where the DMD pixels do not contribute to I(z).
The process of optimizing the DMD pattern starts by choosing an initial pattern p 0 . This pattern can be either totally black (all DMD pixels are off), a pattern that was calculated to correct the actual potential V(z k ) estimated from a measurement of ρ(z k ), or a pattern from a previous optimization processes. In each iteration i, with a DMD pattern p i , the average 1D density profile,ρ i (z k ), and the mean atom number,N i , are obtained from n absorption pictures. The mean atom numberN i is then used to calculate ρ i T (z k ). To estimate how well the measured densityρ i (z k ) and the calculated target density ρ i T (z k ) agree, we calculate the normalized root-mean-square deviation, in each iteration for a specific region from z j to z l and compare it with a target value, T . Note that there is a lower limit for T which is given by the noise in the density measurement: T should not be smaller than the average of the standard error of the mean density, σρ, in the same region. We terminate the optimization process, when i RMS < T . In that case, the final DMD pattern is p i . If i RMS ≥ T a new pattern p i+1 is generated using an update procedure discussed below and then the process is repeated.
In the DMD pattern update procedure, the states of individual pixels are altered based on the difference betweenρ i (z k ) and ρ i T (z k ). In regions where the measured density is higher than the target density more pixels are switched on, thus yielding a locally higher potential. Conversely, by switching pixels off or shifting active pixels away from the center, the optical dipole potential in the corresponding region is decreased resulting in higher densities.
In Fig. 5, the image of seventeen rows of the DMD chip is shown in the plane of atoms, where the zeroth row is the central row discussed before. The orange dashed lines represent the camera pixels mapped in the plane of atoms, defining the spatial grid points z k .
For each grid point z k , the corresponding region on the DMD remains unchanged if |ρ i (z k ) − ρ i T (z k )| < σ iρ (z k ) where σ iρ (z k ) is the standard error of the mean density, represented by the error bars in this figure. In our example here, this is the case for z 1 , z 6 , z 7 , z 8 , z 11 , z 12 and z 13 . In the Plane of atoms other cases where |ρ i (z k ) − ρ i T (z k )| > σ iρ (z k ), the magnitude of the total deviation determines how many pixels corresponding to grid point z k will be changed.
, the potential has to be increased by turning additional pixels on. To chose these, first the distances of the outermost active pixels for the columns involved are compared together to select the column for which this distance is the smallest. Consider three columns involved in the region defined by z 10 : the maximum distance of an active pixel from the central row is only 4 for the column in the left. However, for the central column and for the column in the right this number is 5. So in this case the left column will be chosen. In the selected column, if the outermost pixel is located below the center, the pixel above the center which is one row further away will be turned on, if the outermost pixel is above the center, then the pixel with the same distance but below the center will be turned on. For z 10 , the pixel marked with the black dot will be turned on. For z 9 , the pixel marked by the orange dot will be turned on.
Ifρ i (z k ) + σ iρ (z k ) < ρ i T (z k ), the potential has to be decreased by shifting the outermost pixel away from the center in each region. For example in the region corresponding to z 3 (z 5 ) the pixel in the row number −5 (−6) will be shifted to the row number 6 (7) in the same column, as marked in the Fig. 5. If the selected pixel is in the border of AOI it will be simply turned off.    Fig. 8. Evolution of RMS for three different cases (lines are guides to the eye): a box-like potential with length L = 160 µm (black), two box-like potentials separated by a barrier (brown), a box-like potential with a sinusoidally modulated bottom (green) and a V-shaped potential (pink). All initial and final densities and potentials are plotted in Fig. 6 (a), (b), (c) and (d).

Results
We used the technique described in Sec. 4 to realize a variety of 1D trapping potentials including a box-like potential, two box-like potentials separated with barriers with different heights, a box-like potential with sinusoidal modulations and a V-shaped potential. Results are shown in Fig. 6 and 7.
In these examples, the optimization process started with different initial DMD patterns. In the first and the fourth example ( Fig. 6 (a) and (d)), the optimization started with all DMD pixels turned off. Thus the initial density profiles (potential) plotted in Fig. 6 (a) and (d) (orange) represent the 1D magnetic potential. In the examples presented in Fig. 6 (b) and (c) we started with a previously optimized DMD pattern.
The initial and final potentials, plotted in Fig. 6, are calculated using the NPSE in the limit where the kinetic energy of motion in z-direction is neglected (Thomas-Fermi approximation), Here, µ is the chemical potential (only an offset), a s is the s-wave scattering length, g = 4π 2 a s /M is the 3D scattering amplitude and a ⊥ is the oscillator length mentioned in Sec. 2.
For the examples presented here, the number of experimental realizations were varied in each iteration depending on the normalized root-mean-squared deviation of the previous step. This helps speed up the optimization process in the beginning where only a rough measurement of the density is needed. At the initial stages of the process, only n = 2 shots were used to calculate the averaged 1D density and the mean atom number. At the more advanced stages of the optimization this number was increased to n = 5 and n = 10 accordingly. For all measurements, the optical dipole trap was turned on during the evaporative cooling in a transverse double-well. Absorption pictures are taken after 2 ms TOF.
In Fig. 8, the normalized root-mean-squared deviation in each iteration, i RMS , is plotted for different examples. This figure suggests that the number of iterations needed to fulfill the termination criterion depends on the initial potential. Given the fact that each experimental realization takes less than 30 s, the total optimization time for these examples was between 60 to 135 minutes.

Conclusion
We have implemented a robust and versatile way to control the longitudinal confinement of 1D Bose gases created on an atom chip. This is accomplished by modifying the longitudinal potential landscape defined by the magnetic fields by adding an optical dipole potential that is locally controlled by a DMD. We have furthermore demonstrated an autonomous optimization procedure to achieve any given longitudinal potential.
Our system opens up many new possibilities for experiments with 1D quantum systems. A boxlike potential allows to create homogeneous systems, additional barriers will allow longitudinal splitting and tunneling. Switching between patterns will allow quench experiments. For example, imprinting a density modulation corresponding to the density quadrature of specific modes (or superposition of modes) will allow to precisely excite them and look at their evolution and decay. Furthermore, the DMD patterns can be changed faster than the typical time scales of the 1D quantum gas. Therefore we expect to be able to implement time varying potentials which will be essential to implement and explore thermal quantum machines with 1D quantum fields [37].