Low-entropy states of neutral atoms in polarization-synthesized optical lattices

We create low-entropy states of neutral atoms by utilizing a conceptually new optical-lattice technique that relies on a high-precision, high-bandwidth synthesis of light polarization. Polarization-synthesized optical lattices provide two fully controllable optical lattice potentials, each of them confining only atoms in either one of the two long-lived hyperfine states. By employing one lattice as the storage register and the other one as the shift register, we provide a proof of concept using four atoms that selected regions of the periodic potential can be filled with one particle per site. We expect that our results can be scaled up to thousands of atoms by employing a atom-sorting algorithm with logarithmic complexity, which is enabled by polarization-synthesized optical lattices. Vibrational entropy is subsequently removed by sideband cooling methods. Our results pave the way for a bottom-up approach to creating ultralow-entropy states of a many-body system.

We create low-entropy states of neutral atoms by utilizing a conceptually new optical-lattice technique that relies on a high-precision, high-bandwidth synthesis of light polarization. Polarization-synthesized optical lattices provide two fully controllable optical lattice potentials, each of them confining only atoms in either one of the two long-lived hyperfine states. By employing one lattice as the storage register and the other one as the shift register, we provide a proof of concept using four atoms that selected regions of the periodic potential can be filled with one particle per site. We expect that our results can be scaled up to thousands of atoms by employing an atom-sorting algorithm with logarithmic complexity, which is enabled by polarizationsynthesized optical lattices. Vibrational entropy is subsequently removed by sideband cooling methods. Our results pave the way for a bottom-up approach to creating ultralow-entropy states of a many-body system. DOI: 10.1103/PhysRevLett.118.065302 Introduction.-Compared to other quantum systems, ultracold atoms in optical lattice potentials stand out for being naturally scalable to a large number of particles. They offer thousands of sites, arranged in periodic arrays, in which quantum particles such as atoms can be confined and manipulated [1]. The idea of employing the myriad of sites available as a well-controlled Hilbert space has influenced modern research frontiers ranging from quantum metrology [2], quantum information processing [3][4][5][6][7][8], discrete-time quantum walks [9], up to quantum simulations of strongly correlated condensed-matter systems [10][11][12] with single lattice-site resolution [13,14]. Substantial experimental effort has recently been devoted to creating low-entropy states of atoms in the lattice, with each site being occupied by an integer number of atoms. Low-entropy states play an essential role in a host of quantum applications including the creation of highly entangled cluster states for quantum information processing [15], investigation of Hong-Ou-Mandel-like quantum correlations in many-body systems [16,17], and the quantum simulation of quantum spin liquids in frustrated systems [18,19].
To date, the standard approach to generate low-entropy states in optical lattices relies on a Mott insulator phase [10,11]. This is denoted as a top-down approach since ultracold atoms, due to interactions, self-organize in domains with integer filling factors. Other approaches [20] relying only on laser cooling techniques have recently demonstrated filling factors beyond the one-half limit imposed by inelastic light-assisted collisions [21,22], though without providing a fully deterministic method. In contrast, a bottom-up approach generating arbitrary low-entropy states from individual atoms has long been desired [3,4], yet never been experimentally realized. In this Letter, we demonstrate a bottom-up approach to generate arbitrary atom patterns, including unity filling of lattice sites, in a one-dimensional (1D) optical lattice. Inspired by the seminal work by Jaksch et al. [3] proposing spin-dependent optical lattices to control individual atoms' positions, our work realizes the atomsorting scheme proposed by Weiss et al. [23]. The experimental challenge consists in developing spin-dependent optical lattices able to shift atoms by any amount of lattice sites conditioned to their spin state. Previous implementations [24,25] of spin-dependent optical lattices were limited to only relative displacements of the two spin components and to relative shift distances of one site at most. To overcome these limitations, we have devised a scheme for spin-dependent transport based on a high precision, large bandwidth synthesizer of polarization states of light. Hence, we refer to our new implementation of spin-dependent optical potentials as polarization-synthesized (PS) optical lattices. PS optical lattices allow us to reposition individual atoms with a precision of 1 Å, reducing thereby the positional entropy of a randomly distributed ensemble to virtually zero. This is in stark contrast to the atom-sorting technique formerly demonstrated by our group [26], whose positioning precision was limited to about five sites. In addition, the novel approach requires no postselection, which has limited the success rates in earlier efforts to create ordered patterns from a thermal ensemble [27,28]. Very recently, similar results [29,30] have been obtained using atoms in movable arrays of optical tweezers, which appear particularly suited to exploit Rydberg interactions [31], as it suffices for them to sit at well localized positions spaced by relatively large distances [32,33]; in comparison, PS optical lattices, besides allowing matter waves to delocalize in space in highly stable optical potentials [34], also permit parallel sorting of atoms through the control of two spin species. Relying on that, we identify a sorting algorithm with logarithmic complexity, which allows in two dimensions a dramatic speed-up promising unity filling with thousands of atoms. Moreover, PS optical lattices directly allow one to investigate the dynamics of spinor quantum particles [35][36][37].
Atom sorting.-The principal result of this work is shown in Fig. 1: four 133 Cs atoms from a dilute thermal ensemble are rearranged into a predefined, ordered distribution inside a 1D optical lattice. The atom-sorting procedure works akin to Maxwell's demon. In essence, an automated feedback-based experimental setup acquires the initial location of atoms through fluorescence imaging with single site precision, and it uses this information to subsequently shift the atoms, one by one, to form the desired pattern. As illustrated in Fig. 1(a), two spatially overlapped optical lattices, with identical lattice constant (π=k L ) are used to sort atoms. The first lattice is kept fixed, serving as a storage register for atoms in the hyperfine state j↑i ¼ jF ¼ 4; m F ¼ 4i, while the other one is mobile, providing a shift register for atoms in j↓i ¼ jF ¼ 3; m F ¼ 3i. A digitally programmable polarization synthesizer, as will be detailed later, gives us full independent control of both lattice depths U s (s ∈ f↑; ↓g) as well as of the lattice positions x s ðtÞ, which are varied in time to shift the atoms. We choose deep lattices, of the order of a thousand recoil energies, to allow fast transport on the time scale of ten microseconds, while preventing intersite tunneling. For each atom we intend to reposition, we flip its spin, j↑i → j↓i, using a positionresolved microwave pulse [28,38], which transfers it into the shift register. Once the atom is repositioned by translating the shift register, it is transferred back into the storage register through optical pumping. The fluorescence images in Figs. 1(b)-1(e) show the final distribution of atoms for four different target patterns, including the unity filling of a region of the lattice. Between fluorescence images, several sorting operations are carried out with no need to continuously monitor the positions of atoms. If errors are detected in the final distribution (e.g., imperfect spin flips, wrong position reconstruction, atom losses), a feedback control system attempts to correct them. Experimental apparatus.-A small ensemble of cesium atoms is captured from the background vapor into a magnetooptical trap, and transferred into a 1D optical lattice produced by linearly polarized light at the wavelength λ L ¼ 2π=k L ¼ 866 nm. The lifetime of atoms due to collisions with background gas is about 360 s. The lattice depth is chosen equal for both spin species, U 0 ↑ ¼ U 0 ↓ ≈ 75 μK, and significantly larger than the atoms' temperature, which is about 8 μK after molasses cooling. The loading procedure is adjusted to spread the atoms along the lattice with an average separation of around 20 lattice sites. Optical pumping initializes atoms in j↑i state with > 99% efficiency using a σ þ -polarized laser. To detect the atoms' positions, we acquire fluorescence images with 1 s illumination time using an electron-multiplying CCD camera. We employ a superresolution-microscopy technique [39] to resolve in real time the individual atoms beyond the diffraction limit of around four sites, as can be seen comparing Figs. 1(c) and 1(d). The local addressing of individual atoms is achieved through spectrally narrow Gaussian-shaped microwave pulses (7 kHz rms width) in combination with a weak magnetic field gradient (11.6 G/cm) along the direction of the optical lattice [28]. For the sorting procedure, we select atoms isolated by more than 20 sites to ensure a probability < 1% that local addressing pulses spin-flip a neighboring atom. We choose adiabatic, sinusoidal ramps to transport the addressed atoms in the shift register in approximately 1 ms, 065302-2 much shorter than the longitudinal spin relaxation time of 100 ms due to inelastic scattering of the lattice photons. We pump atoms back into the storage register by 2-ms-long optical pumping. Atoms in excess are removed from the lattice by first spin-flipping the sorted atoms into the shift register, and by subsequently applying a resonant pulse with the F ¼ 4 → F 0 ¼ 5 transition, pushing atoms in the j↑i state out of the optical lattice, while not affecting atoms in the j↓i state. Our deep optical lattice, superimposed with a bluedetuned doughnut-shaped trap, yields longitudinal and transverse trapping frequencies of about ω ‖ ≈ 2π × 110 kHz and ω ⊥ ≈ 2π × 20 kHz, well above the recoil frequency 2π × 2 kHz of cesium. The large trapping frequencies allow us to employ microwave [40] and Raman [41] sideband cooling to cool atoms in the longitudinal and radial directions, respectively, achieving a ground state occupation of 99% along the axial direction and of 80% along each of the two radial directions.
Polarization-synthesized optical lattices.-The key elements in realizing the spin-dependent optical-lattice potentials shown in Eqs. (1) and (2) are two superimposed, yet independently controllable optical standing waves with opposite circular polarizations, σ þ and σ − . For both standing waves we choose a so-called magic wavelength λ L of cesium, allowing atoms in j↑i and j↓i states to be trapped in the maximum-intensity regions of the σ þ -and σ − -polarized light field, respectively [42]. Such a wavelength exists because of the different ac vector polarizability of the two internal states [43], and was already employed in earlier implementations of spin-dependent optical lattices (e.g., Refs. [24,25]). However, these implementations permitted only relative displacements and, most importantly, maximum shift distances of one lattice site, thereby precluding the possibility of sorting randomly distributed atoms into predefined patterns. In contrast, PS optical lattices entirely overcome these limitations by relying on two fully independent optical standing waves. In order to create the standing waves, we let two copropagating laser beams with opposite circular polarization each interfere with a linearly polarized, counterpropagating beam, as illustrated in Fig. 2(a). We employ an optical fiber to ensure that the resulting standing waves are perfectly matched to the same transverse mode, and thereby that atoms in both spin states, j↑i and j↓i, experience an identical transverse potential. Transverse-mode filtering is essential to ensure long spin-coherence times for spectrally narrow coherent pulses (e.g., spin flips for single-atom addressing), or else thermal atoms would undergo inhomogeneous spin dephasing in a few microseconds due to a strong differential light shift [44].
While in the transverse directions the two standing waves are perfectly overlapped, they are free to slide with respect to each other in the lattice direction. The position of each standing wave must be controlled with interferometric precision to ensure that atoms are shifted with single-site precision, and that no motional excitation is created when atoms are transferred between the storage and shift registers [42]. We achieve this by employing two independent optical phase-locked loops (PLLs) that actively stabilize the phases of each circularly polarized beam, ϕ ↑ and ϕ ↓ , with respect to a common optical reference beam. As shown in Fig. 2(b), each optical phase ϕ s is referenced to a low-phase-noise rf reference signal (DDS). Varying the phase of the rf signals according to a digitally programmed profile allows us to independently steer ϕ s , and thereby the position of the respective optical potential U s : A measurement of the relative phase noise Δϕ ¼ ϕ ↑ − ϕ ↓ yields an uncertainty of 0.1° [46], which translates into a jittering of the relative position of Δx ¼ 1.20 Å. This is more than 2 orders of magnitude smaller than the extent of the atomic wave function in the vibrational ground state (20 nm) [47]. Moreover, to attest the reliability of the spindependent transport operations, we shifted spin-polarized atoms using a single transport operation, chosen to be about 1 ms long, over a distance varying from a few to one hundred sites. We measure a success rate of 97.4(3)%, nearly independent of the distance. We attribute the remaining unsuccessful events to optical pumping errors (0.4%), spin flips during transport (0.6%), and position reconstruction errors (1.6%). This dramatically differs from previously reported transport efficiencies, decreasing exponentially with the transport distance. Even with the best reported efficiency of 99% per shift operation [48], transport efficiency over 20 sites had never exceeded 0.99 2·20 ≈ 67%. Thousands of atoms.-While low-entropy states comprising as few as four atoms already suffice to study a wide range of few-body phenomena [16,17,49,50], it is important to discuss how the atom-sorting scheme based on PS optical lattices can be extended to much larger numbers of atoms, thus providing a bottom-up pathway to many-body physics. Compared to recent results [29,30] demonstrating the sorting of 50 atoms with optical tweezers, fewer atoms are sorted in Fig. 1 because of the limited addressing resolution (20 sites) of the present apparatus (see Supplemental Material [51]). However, we expect that with a higher addressing resolution [52] PS optical lattices allow sorting even thousands of atoms into arbitrary target patterns.
To that purpose, we propose a new atom-sorting algorithm based on PS optical lattices, which rearranges N atoms using a number of operations of the order of log N. The logarithmic complexity is enabled by two properties unique to PS optical lattices, namely, their ability to shift atoms (1) spin dependently and (2) by any arbitrary number of sites; moreover, its logarithmic complexity also holds for two-dimensional (2D) PS optical lattices, which have been recently proposed in Ref. [53]. Hitherto, the best algorithm [54] for sorting atoms in a 2D array requires a larger number of operations of order N 1=2 , because only onelattice-site shifts are used instead of property (2).
The PS-optical-lattice atom-sorting (PSOLAS) algorithm proposed here iterates four steps: Step 1 identifies, among the atoms not yet sorted, patterns of atoms that best match the distribution of defects, i.e., the empty sites to be filled with one atom; this step requires acquiring the positions of the atoms through a fluorescence image [55].
Step 2 transfers the identified atoms from the storage register into the shift register; this step can be performed in parallel by optically addressing atoms using a spatial light modulator [56,57] or serially using a beam deflector [29,58].
Step 3 shifts the whole pattern of selected atoms, in parallel, to fill the defects; this step is the crucial one, which is enabled by the properties (1) and (2) of PS optical lattices.
Step 4 transfers the shifted atoms into the storage register by optical pumping. An illustrative demonstration of the PSOLAS algorithm to fill a square region of a 2D optical lattice is provided in the Supplemental Material [51].
Since the duration of each step is independent from the number of sorted atoms for a broad parameter range, the overall time required by PSOLAS is determined by the number of iterations. The latter is estimated by considering that each iteration fills, on average, a fraction α of the defects in the target pattern, where α denotes the initial filling probability of a lattice site. In reality, because the algorithm searches for the best matching pattern of atoms to shift, the fraction of filled defects per iteration is generally higher than α. Hence, the number of defects after n iterations amounts to less than Nð1 − αÞ 1þn , meaning that to attain a number of defect of the order of Oð1Þ, about Oðlog NÞ iterations are required. To validate this scaling law under realistic conditions, we carried out Monte Carlo simulations in two scenarios representing (A) conservative and (B) state-of-theart conditions; the conditions of both scenarios are derived from individual results demonstrated either in our or other laboratories. In both scenarios, PSOLAS aims to fill a square target pattern of 31 × 31 sites by "tapping" into the atoms stored in a larger region of 100 × 100 sites. Scenarios (A) and (B) rely on 80% and 95% [58] single-site addressing efficiency, and the filling probability α is chosen equal to 40% and 60% [20], respectively (all parameters are summarized in the Supplemental Material [51]). As shown in Fig. 3, we find that even in scenario (A), about 1000 atoms can be sorted in a time of about 1 s.
Conclusions.-In this Letter, we demonstrated a bottomup approach to the generation of low-entropy states of ultracold atoms in optical lattices. Our work demonstrates that arbitrary filling patterns with virtually zero entropy can be realized experimentally. The key to our sorting procedure is the development of PS optical lattices, which provide us with a new set of operations for the control of atoms depending on their spin orientation. Presently, the entropy of our prepared states is limited by the vibrational entropy [59] due to the limited efficiency (60%) of the sideband cooling into the three-dimensional vibrational ground state. A tighter optical confinement of atoms shall enable significantly higher efficiencies. The construction of a 2D PS optical lattice is underway [53] in a new experimental apparatus, which additionally features a high optical resolution objective lens for optical single-site addressing [52]; this should allow us to demonstrate PSOLAS with thousands of atoms.
Supplemental Material for: Low-entropy states of neutral atoms in polarization-synthesized optical lattices

SI. LOCAL ADDRESSING OF INDIVIDUAL ATOMS
The microwave addressing resolution of 20 lattice sites given in the main text limits the maximum number of atoms that we can sort in a low-entropy state. Our imaging system's field of view spans over 160 lattice sites, which corresponds to 8 atoms separated by 20 sites. However, the situation of 8 atoms equally separated is extremely improbable since atoms are initially distributed in random positions. Thus, with the experimental apparatus presently at hand we are limited to between 4 and 6 atoms.
With the same experimental apparatus, we have demonstrated in the past much higher addressing resolutions, by working with magnetic field gradients about a factor 20 higher [28]. The technical challenge using high magnetic field gradients resides in the long time (more than 100 ms) required to ramp up or down the magnetic field, during which off-resonant scattering of lattice photons can occur (T 1 ≈ 100 ms). This is not a problem for atoms sorted in distinct lattice sites, since they can be reinitialized into the motional ground state by sideband cooling. However, it poses a limitation for sorting two or more ground-state-cooled atoms into the same lattice site, since the subsequent application of sideband cooling would lead to highly detrimental losses by light-assisted collisions instead of reinitializing atoms into the ground state. Hence, for our atom-sorting demonstration, we opted for a rather weak magnetic field gradient, which enables in the future the study of intriguing many-body effects.
Moreover, switching from magnetic field gradients to tightly focused laser beams employing a high-NA objective lens [52] should allow us to achieve local addressing with single-lattice-site resolution.

SII. MONTE CARLO SIMULATIONS OF PSOLAS
We carried out Monte Carlo simulations of the PSoptical-lattice atom-sorting (PSOLAS) algorithm using the parameters reported in Tab. S1 for two different scenarios based on either conservative or state-of-the-art conditions. As mentioned in the main text, we have chosen the conditions of both scenarios based on individual results obtained either in our or other laboratories.
We are currently setting up a two-dimensional (2D) polarization-synthesized (PS) optical lattice experimental apparatus, which is designed to reach the experimental conditions described in the state-of-the-art scenario [53]. A key component of our new experimental apparatus is a custom-built objective lens with high numerical aperture NA=0.92 [52]. The objective lens achieves a diffraction limited resolution of 460 nm, compared to the lattice constant of 612 nm, and increases the photon collection efficiency by a factor 30 compared to that obtained with the apparatus described in the main text. Consequently, we expect a significant improvement in the localization precision, allowing us to determine the lattice position of individual atoms in few tens of milliseconds. Furthermore, the high-NA objective lens should enable us to address individual atoms with high efficiency using tightly focused laser beams, which induces a differential light shift for the addressed atoms [58]. It was recently demonstrated that the efficiency of this technique can exceed 99 % [8]. Thus, we assumed in our Monte Carlo simulations an addressing efficiency of 80 % and 95 % for the two scenarios. We additionally accounted for an addressing crosstalk of 5 % or 10 %, which leads to unwanted spin flips of neighboring atoms. Moreover, the region of 100×100 sites considered in the Monte Carlo simulations corresponds to the field of view of our high-NA objective lens.
The initial filling probability of the lattice is determined by the density of atoms that are initially captured in the magneto-optical trap and by the efficiency of the so-called parity projection caused by light-assisted collisions. In our current implementation we found initial filling probabilities to vary between 40 % and 65 %. Notably, Fung et al. [20] show that initial filling probabilities exceeding 80 % can be realized using an additional blue-detuned laser beam while loading atoms from the magneto-optical trap into the optical lattice.
With the experimental apparatus described in the main text, we achieve storage times in the range of 360 s due to the ultra-high vacuum conditions realized in the "science" cell, where the partial vacuum pressure of Cs atoms is very low. With the new 2D experimental apparatus, we expect to achieve storage times of the order of one minute.
The chosen duration of optical pumping (10 ms) and that of a lattice shift operation ( addressing, the addressing duration depends on how fast one can steer the laser beam generating the local differential light shift and the duration of the subsequently applied microwave pulse itself. Previous demonstrations of the addressing scheme achieved local differential light shifts of ≈ 60 kHz [58]. In this case, microwave pulses with a duration of 30 µs have the required frequency resolution to ensure low addressing crosstalk. As a conservative assumption, we take the current microwave pulse duration of 150 µs, which correspond to smaller AC starkshifts. In both scenarios, the beam steering duration can be neglected due to the high bandwidth of acousto-optic modulators (steering time < 1 µs). Furthermore, we note that spatial light modulators [56,57] (e.g., liquid crystal on silicon operated in amplitude modulation mode, with refresh rates of 10 Hz) could also enable parallel addressing of hundreds of atoms by imaging the square grid of pixels onto the square two-dimensional optical lattice.
For the complexity estimate of PSOLAS, we neglected the computational time required to process, e.g., atom positions, which can be efficiently performed by a CPU or in parallel by a dedicated FPGA chip. This computational time is much shorter than the other time scales.

SIII. ILLUSTRATION OF PSOLAS
In Fig. S1, we provide a graphical illustration of the working principle of PSOLAS algorithm under the conservative conditions (see Tab. S1) to accompany the description given in the main text. For the chosen example, we use PSOLAS to create unity filling in a square target pattern of 31 × 31 lattice sites, repositioning atoms taken from within a region of 100 × 100 lattice sites. The three images in the figure are Monte Carlo-simulated atom distributions at the initial time, after three iterations of the algorithm, and at the completion of the algorithm. The initial distribution, see Fig. S1(a), is a random distribution of atoms with an initial filling probability of 40 %. PSOLAS chooses the initially densest region of 31 × 31 lattice sites as the target region, which is indicated by a green rectangle in the figure. The atom distribution after three iterations, see Fig. S1(b), shows a highly increased density in the target region, whereas a visible depletion in the other square regions indicated by numbers 1, 2, 3, identified by PSOLAS as those regions with atom patterns best matching the distribution of the remaining holes of the target region. In Fig. S1(c), we show the final distribution with perfect unity filling in the target region after 12 iterations of PSOLAS algorithm.