Atomic array rearrangement path planning method using spatial light modulator tensor

Abstract

Provided is an atomic array rearrangement path planning method using a spatial light modulator tensor, comprising: obtaining a source distribution and a target distribution; constructing a crosstalk weighted cost matrix based on a displacement vector from a source position to a target position and a matrix function quantifying the self displacement cost of the optical trap in the spatial light modulator tensor; constructing a kernel function matrix under an entropy regularization framework based on the crosstalk weighted cost matrix; executing an entropy regularization optimal transport solving algorithm based on the kernel function matrix, the source distribution and the target distribution to obtain an optimal transport coupling matrix; and determining an optimal allocation mapping and an optimal transport path from the optimal transport coupling matrix by constructing a time optimal atomic transport problem under a spatial light modulator crosstalk metric. In this way, by quantifying the anisotropic structure of the spatial light modulator optical crosstalk into the transport cost matrix and performing source-target pairing and geodesic path planning based thereon, the accuracy of the atomic array rearrangement path planning can be improved.

Description

Technical Field

[0001] This application relates to the field of quantum computing technology, and more specifically, to a method for rearranging atomic array paths using spatial light modulator tensors, electronic devices, and computer-readable storage media. Background Technology

[0002] Neutral atoms (such as or Quantum computing is one of the most promising quantum computing routes. Its core advantages are: the same type of atoms have natural uniformity (no manufacturing tolerances and parameter dispersion), the hyperfine ground state coherence time is extremely long (up to 10 seconds, far exceeding the operation time of typical quantum gates), and it allows the realization of high-fidelity fully connected two-qubit gates between two qubits by physically moving atomic qubits.

[0003] This technology captures individual cold atoms with optical tweezers and achieves quantum entanglement gates through Rydberg excitation. Typically, a large-scale programmable optical tweezers array is realized using a spatial light modulator (SLM). The SLM modulates the phase of the incident laser light through individual liquid crystal pixels on a liquid crystal panel. After Fourier transform by a lens, an optical tweezers array is formed on the focal plane, thereby fixing individual cold atoms in individual optical traps by the optical tweezers array.

[0004] During the loading of cold atoms into optical traps constructed from optical tweezers arrays, the probability of each optical trap being occupied by a single atom follows an independent Bernoulli distribution (typical occupancy rate) due to atomic number density fluctuations and photo-induced collision blocking mechanisms. This means that for a quantum array, for example, containing 100 physical bits, the probability of forming a completely defect-free target array after natural random loading is extremely low (approximately). ).

[0005] Therefore, before each quantum computation, an atomic array rearrangement mechanism must be introduced. This involves identifying the current defect state through camera fluorescence imaging, implementing motion planning using low-level control algorithms, and frequently refreshing the hologram to drive the SLM to change the optical trap positions in real time, physically transforming the scattered atoms into a specified defect-free target configuration. Here, the SLM (e.g., comprising 1024×1024 pixels) can simultaneously control the real-time movement of atoms trapped in multiple optical traps by applying a phase delay to each liquid crystal pixel.

[0006] When planning the path for rearranging an atomic array, it is necessary to determine the target position to which each atom moves and the trajectory along which each atom moves. Currently, the standard Sinkhorn algorithm is mainly used to solve the entropy-regularized optimal transport problem for path planning. Its cost matrix is ​​the square of the isotropic Euclidean distance, which assumes that the physical cost of moving the same distance in any direction in a two-dimensional plane is the same. Therefore, after determining the atom-target position pairing, each atom is made to move along the Euclidean straight line from the source position to the target position at a uniform or trapezoidal speed profile. Furthermore, the WGS algorithm is used to recalculate the hologram frame by frame to track the instantaneous position of each optical sink.

[0007] However, in reality, crosstalk exists between optical traps. That is, the effect of moving one optical trap on the other optical traps (crosstalk) is not uniform and cannot be ignored in all directions, which makes the straight path not the optimal trajectory.

[0008] Therefore, it is desirable to provide an atomic array rearrangement path planning scheme based on crosstalk between optical traps. Summary of the Invention

[0009] This application provides an atomic array rearrangement path planning method using spatial light modulator tensors. By introducing spatial light modulator tensors to quantize and encode the anisotropic structure of optical crosstalk of spatial light modulators into the transmission cost matrix, and performing source-target position pairing and geodesic path planning based on this matrix, the accuracy of atomic array rearrangement path planning can be improved.

[0010] According to one aspect of this application, an atomic array rearrangement path planning method using a spatial light modulator tensor is provided, comprising: determining a source position set and a target position set based on image detection results of atoms to be rearranged and the rearrangement target, and constructing their discrete probability distributions to obtain a source distribution and a target distribution; constructing a crosstalk weighted cost matrix based on the displacement vector from the source position to the target position and a matrix function in the spatial light modulator tensor that quantizes the displacement cost of the optical trap itself; determining an entropy regularization parameter based on the error, and constructing a kernel function matrix under the entropy regularization framework based on the crosstalk weighted cost matrix; performing an entropy regularization optimal transmission solution algorithm on the source distribution and the target distribution based on the kernel function matrix to obtain an optimal transmission coupling matrix; and determining an optimal allocation mapping and an optimal transmission path from the optimal transmission coupling matrix by constructing a time-optimal atomic transmission problem under the spatial light modulator crosstalk metric.

[0011] In the above-mentioned atomic array rearrangement path planning method using spatial light modulator tensors, constructing the discrete probability distributions of the source position set and the target position set includes: constructing the source distribution corresponding to the source position set and the target distribution corresponding to the target position set into a uniform discrete probability distribution.

[0012] In the above-mentioned atomic array rearrangement path planning method using spatial light modulator tensors, constructing the discrete probability distributions of the source position set and the target position set includes: constructing the source distribution corresponding to the source position set and the target distribution corresponding to the target position set into non-uniform discrete probability distributions, wherein the non-uniform source distribution is determined based on the fluorescence image signal-to-noise ratio or occupancy probability weighting of each source light trap.

[0013] In the aforementioned atomic array rearrangement path planning method using spatial light modulator tensors, constructing a crosstalk weighted cost matrix based on the displacement vector from the source position to the target position and the matrix function in the spatial light modulator tensor that quantizes the displacement cost of the optical trap itself includes: multiplying the row vector of the displacement vector from the source position to the target position by the matrix function in the spatial light modulator tensor that quantizes the displacement cost of the optical trap itself corresponding to the source position, and then multiplying it by the column vector of the displacement vector from the source position to the target position to obtain the anisotropic transmission cost; and combining all the anisotropic transmission costs corresponding to the pairings between the source position and the target position into the crosstalk weighted cost matrix.

[0014] In the above-mentioned atomic array rearrangement path planning method using spatial light modulator tensors, determining the entropy regularization parameter based on the error includes: determining that the entropy regularization parameter is less than or equal to the quotient obtained by dividing the error, which serves as the upper bound of the allowable matching position error, by the natural logarithm of the total number of optical traps.

[0015] In the above-described atomic array rearrangement path planning method using spatial light modulator tensors, the upper bound of the allowable matching position error is 0.01 times the optical trap spacing.

[0016] In the above-mentioned atomic array rearrangement path planning method using spatial light modulator tensors, constructing the kernel function matrix under the entropy regularization framework based on the crosstalk weighted cost matrix includes: dividing the matrix elements of the crosstalk weighted cost matrix by the entropy regularization parameter and taking the negative number as the exponent of the natural constant to obtain the corresponding matrix elements of the kernel function matrix.

[0017] In the above-described atomic array rearrangement path planning method using spatial light modulator tensors, the entropy-regularized optimal transmission solution algorithm is executed based on the kernel function matrix to obtain the optimal transmission coupling matrix. This includes: solving the entropy-regularized optimal transmission problem using the entropy-regularized optimal transmission solution algorithm based on the kernel function matrix, the source distribution, the target distribution, and the transmission coupling matrix representing the transmission quality from the source to the target to obtain the optimal transmission coupling matrix.

[0018] In the above-described atomic array rearrangement path planning method using spatial light modulator tensors, the optimal transmission coupling matrix is ​​obtained by solving the entropy-regularized optimal transmission problem using an entropy-regularized optimal transmission solution algorithm. This includes: initializing the dual row scaling initial vector and column scaling initial vector as all-one vectors; performing Sinkhorn alternating scaling iterations starting from the row scaling initial vector and the column scaling initial vector through the kernel function matrix; stopping the iteration in response to the marginal residuals of the iteration satisfying a predetermined condition or reaching the maximum number of iterations to obtain the optimal row scaling vector and the optimal column scaling vector; and constructing the optimal transmission coupling matrix using the optimal row scaling vector, the optimal column scaling vector, and the kernel function matrix.

[0019] In the above-described atomic array rearrangement path planning method using spatial light modulator tensors, performing Sinkhorn alternating scaling iterations from the initial row scaling vector and the initial column scaling vector through the kernel function matrix includes: multiplying the kernel function matrix with the column scaling vector of the previous iteration and adding a numerical underflow protection constant to obtain a first sum vector, and then dividing the source distribution by the first sum vector to obtain the row scaling vector of the current iteration; and multiplying the transpose of the kernel function matrix with the row scaling vector of the current iteration and adding a numerical underflow protection constant to obtain a second sum vector, and then dividing the target distribution by the second sum vector to obtain the column scaling vector of the current iteration.

[0020] In the above-described atomic array rearrangement path planning method using spatial light modulator tensors, constructing the optimal transmission coupling matrix using the optimal row scaling vector, the optimal column scaling vector, and the kernel function matrix includes: determining the matrix elements of the optimal transmission coupling matrix as vector elements of the optimal row scaling vector, and the product structure of the vector elements of the optimal column scaling vector and the matrix elements of the kernel function matrix, based on the KKT first-order optimality condition of the matrix elements of the transmission coupling matrix.

[0021] In the above-mentioned atomic array rearrangement path planning method using spatial light modulator tensors, determining the optimal allocation mapping and optimal transmission path from the optimal transmission coupling matrix by constructing a time-optimal atomic transmission problem under the spatial light modulator crosstalk metric includes: deriving the optimal velocity field analytical structure by constructing a time-optimal atomic transmission problem under the spatial light modulator crosstalk metric based on the optimal transmission of the spatial light modulator.

[0022] In the aforementioned method for reordering the atomic array path planning using spatial light modulator tensors, deriving the optimal velocity field analytical structure by constructing a time-optimal atomic transport problem under a spatial light modulator crosstalk metric based on optimal spatial light modulator transport includes: constructing a time-optimal atomic transport problem under a spatial light modulator crosstalk metric and introducing momentum density for convexity; introducing Lagrange multipliers as dual potential functions to construct a Lagrange functional; and taking variational steps on the momentum density based on the Lagrange functional to derive the optimal velocity field analytical structure.

[0023] In the above-mentioned atomic array rearrangement path planning method using spatial light modulator tensors, determining the optimal allocation mapping and optimal transmission path from the optimal transmission coupling matrix by constructing a time-optimal atomic transmission problem under the spatial light modulator crosstalk metric includes: obtaining the optimal transmission path from the endpoint coordinates by backward integration with respect to the spatial light modulator tensor based on the HJB equation corresponding to the optimal velocity field analytical structure.

[0024] In the above-mentioned atomic array rearrangement path planning method using spatial light modulator tensors, the spatial light modulator tensor can optionally be generalized as the optical field modulator crosstalk metric tensor for acousto-optic deflectors or a spatial light modulator + acousto-optic deflector hybrid system.

[0025] According to another aspect of this application, an electronic device is provided, including a processor coupled to a memory, the processor being configured to execute a computer program stored in the memory, such that the electronic device performs the atomic array rearrangement path planning method using spatial light modulator tensors as described above.

[0026] According to another aspect of this application, a computer-readable storage medium is provided, the computer-readable storage medium storing a computer program that, when the computer program is run, implements the atomic array rearrangement path planning method using spatial light modulator tensors as described above.

[0027] The atomic array rearrangement path planning method using spatial light modulator tensors provided in this application improves the accuracy of atomic array rearrangement path planning by introducing spatial light modulator tensors to quantize and encode the anisotropic structure of optical crosstalk of spatial light modulators into the transmission cost matrix, and performing source-target position pairing and geodesic path planning based on this matrix. Attached Figure Description

[0028] Various other advantages and benefits of this application will become apparent to those skilled in the art upon reading the detailed description of the preferred embodiments below. The accompanying drawings are for illustrative purposes only and are not intended to limit the scope of this application. It is obvious that the drawings described below are merely some embodiments of this application, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort. Furthermore, the same reference numerals denote the same parts throughout the drawings.

[0029] Figure 1 The illustration shows a schematic flowchart of an atomic array rearrangement path planning method using spatial light modulator tensors according to an embodiment of this application.

[0030] Figure 2 The illustration shows a schematic flowchart of the generation process of the spatial light modulator tensor in the atomic array rearrangement path planning method using the spatial light modulator tensor according to an embodiment of the present application.

[0031] Figure 3 The illustration shows a schematic flowchart of the Sinkhorn process in the atomic array rearrangement path planning method using spatial light modulator tensors according to an embodiment of this application.

[0032] Figure 4 The illustration shows a schematic flowchart of the process of deriving the optimal velocity field analytical structure in the atomic array rearrangement path planning method using spatial light modulator tensors according to an embodiment of this application.

[0033] Figure 5 A schematic block diagram of an electronic device according to an embodiment of this application is shown. Detailed Implementation

[0034] Hereinafter, exemplary embodiments according to this application will be described in detail with reference to the accompanying drawings. Obviously, the described embodiments are merely some embodiments of this application, and not all embodiments of this application. It should be understood that this application is not limited to the exemplary embodiments described herein.

[0035] Figure 1 The illustration shows a schematic flowchart of an atomic array rearrangement path planning method using spatial light modulator tensors according to an embodiment of this application. Figure 1 As shown, the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application includes the following steps.

[0036] S110, based on the image detection results of the atoms to be rearranged in the atomic array and the rearrangement target, determine the source position set and the target position set, and construct their discrete probability distributions to obtain the source distribution and the target distribution.

[0037] When performing atomic array rearrangement, the atoms to be rearranged are first detected, for example, using sCMOS fluorescence imaging, to identify the positions of the optical traps currently occupied by the atoms, which are denoted as the source position set.

[0038] in This represents the total number of optical traps currently occupied by atoms. Source optical trap Two-dimensional coordinates.

[0039] Simultaneously, based on the rearrangement target, i.e. the target quantum circuit configuration, the target positions of the required atoms in the target defect-free array are determined, denoted as the target position set:

[0040] in The number of atoms in the target array satisfies ,and For target location Two-dimensional coordinates. Here, if... Then, you can choose a subset of the source atoms to participate in the transport, such as the nearest neighbor. Individual source atoms, or a greedy selection based on crosstalk cost. That is, in the embodiments of this application, including... In the case of [the case], further processing includes source atom selection and / or optional processing of untransported source atoms.

[0041] Then, a discrete probability distribution is constructed, where both the source and target distributions are uniform in form:

[0042] in It is the first source distribution Each element represents the source optical trap. Prior probability weights for the selected transport. It is the first of the target distributions Each element represents the target location. The weights that need to be filled.

[0043] Here, the uniform form of the source distribution and the target distribution indicates that all source atoms are considered with equal probability, and all target positions must be filled with equal priority. Those skilled in the art will understand that in the embodiments of this application, the priorities of source atoms and target positions can also be set, meaning that the source distribution and target distribution are not uniform.

[0044] Then, smoothing filters are applied to the source and target distributions, which are discrete probability distributions, such as convolution smoothing based on Gaussian kernels and spatial normalization, to obtain the initial atomic probability density distribution, for example, denoted as... and the probability density distribution of the target atom, for example denoted as In other words, in the embodiments of this application, when it is necessary to generate paths based on continuous HJB equations, kernel smoothing can be performed on discrete distributions, while when only discrete pairing is required, discrete distributions can be directly used to solve Sinkhorn-G.

[0045] In the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application, constructing the discrete probability distributions of the source location set and the target location set includes: constructing the source distribution corresponding to the source location set and the target distribution corresponding to the target location set into a uniform discrete probability distribution.

[0046] Step S120: Construct a crosstalk weighted cost matrix based on the displacement vector from the source position to the target position and the matrix function of the quantization optical trap displacement cost in the spatial light modulator tensor.

[0047] Here, for the source location set and the target location set, regarding each source-target pair... ,in and The displacement vector is calculated as follows:

[0048] here, Indicates source optical trap To the target location displacement vector (unit: m).

[0049] Then, the metric of crosstalk between optical traps is determined, which is achieved by a matrix function in the spatial light modulator tensor that quantifies the displacement cost of the optical trap itself, such as the diagonal block elements of the spatial light modulator tensor. The generation process of the spatial light modulator tensor will be described in detail below.

[0050] As described above, optical tweezers capture individual cold atoms by forming a bright spot with a laser beam highly focused by a lens, thereby locking the cold atom at the focal point through a dipole force generated by the gradient direction towards the focal point due to uneven light intensity. The cold atom can then move with the laser beam. Meanwhile, spatial light modulators utilize their liquid crystal panels (e.g., 1 pixel, of which Each liquid crystal pixel modulates the phase of light passing through it by changing the phase of the hologram. For example, the coordinates of each pixel are represented as... The hologram composed of the applied phase delay is then represented as: , In other words, the hologram is essentially a phase diagram.

[0051] Thus, the input coherent laser beam, after being phase-modulated by the spatial light modulator, passes through a focal length of... Focusing lenses, such as Fourier lenses, are superimposed on the focal plane to form an array of optical tweezers consisting of independent optical tweezers. This array holds atoms within the optical traps of a corresponding optical trap array, allowing the tweezers to move them between the traps. Therefore, for coordinates in the optical trap array on the focal plane... There is a complex amplitude distribution That is, the complex amplitude field of the focal plane, is expressed as:

[0052] in It is the laser wavelength, for example, 399nm. Thus, the first [missing information] in the optical trap target array... The location of the optical trap, for example, denoted as The strength at that point is Therefore, in order to achieve the target amplitude for each optical trap, for example denoted as Holograms are needed. Make In this embodiment, the WGS (Weighted Gerchberg-Saxton) algorithm is used to solve for the hologram. This is a phase recovery problem, and the WGS algorithm is essentially a steady-state (fixed-point) iteration, which will be explained in detail below.

[0053] In a dense array of atoms, each atom is held in an optical trap by its own optical tweezers. However, since all the optical tweezers share the same diffraction field generated by the spatial light modulator based on the hologram, moving one of the optical tweezers requires the spatial light modulator to recalculate the entire light field, thus shifting the positions and intensities of all the other optical tweezers. In other words, changes in the atom's position affect the equilibrium state of the entire hologram. Therefore, it is desirable to accurately represent how much the movement of a particular atom will disturb the other atoms, thereby determining the new equilibrium state of the hologram.

[0054] Figure 2 The illustration shows a schematic flowchart of the spatial light modulator tensor generation process in the atomic array rearrangement path planning method using spatial light modulator tensors according to an embodiment of this application. Figure 2 As shown, the generation process of the spatial light modulator tensor in the atomic array rearrangement path planning method using the spatial light modulator tensor according to the embodiments of this application includes the following steps.

[0055] S121, construct an initial pixel plane complex amplitude field on the pixel plane of the spatial light modulator based on the target illumination parameters, and perform WGS single-step iteration using the WGS single-step operator on the initial pixel plane complex amplitude field to obtain the pixel plane complex amplitude field of the corresponding iteration round.

[0056] Based on this, the target illumination parameters are first obtained. Here, the target illumination parameters include the set of target amplitudes of each optical trap in the pre-configured target array. (including the first) The target amplitude of each optical trap ), the set of positions of each optical trap (including the first) The position of each optical trap Initial hologram Uniformity convergence accuracy threshold The weights of the initial states of each optical trap (including the first one). (weight of each optical trap), and laser wavelength Focal length of the front Fourier lens .

[0057] Here, the weights of the initial states of each optical trap are denoted as... And the first The weights of the optical traps are denoted as follows: .

[0058] Based on the initial hologram The normalized initial pixel plane complex amplitude field (i.e., the pure phase field with a constant modulus of 1) required for constructing a single-step WGS iteration is expressed as:

[0059] in This represents the complex amplitude field of the SLM pixel plane during the 0th (initial) iteration. ( ,here (This refers to the side length of an SLM LCD panel, expressed in pixels.) Two-dimensional pixel coordinates of an SLM LCD panel ( , The initial phase distribution is the input. The imaginary unit, The natural index. In the following text, the complex amplitude field of the SLM pixel plane is referred to as the pixel plane complex amplitude field.

[0060] Mapping the physical positions of each optical trap in the target array on the focal plane to the spatial frequency coordinates required for diffraction calculations is expressed as:

[0061] in For the first The spatial frequency coordinates (or simply frequencies) corresponding to each optical trap. For the target array on the focal plane, the first The spatial coordinates of each optical trap (or simply, position). For a specific light wave (such as a laser), The focal length of the front Fourier lens. It is a phase period constant.

[0062] In the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application, constructing the initial pixel plane complex amplitude field on the plane of the spatial light modulator based on target illumination parameters includes: obtaining target illumination parameters, which include: a set of target amplitudes of each optical trap in a pre-configured target array; a set of positions of each optical trap; a set of weights for an initial hologram, a uniformity convergence accuracy threshold, and the iterative initial state of each optical trap; and the laser wavelength and the focal length of the pre-Fourier lens.

[0063] In the above-mentioned atomic array rearrangement path planning method using spatial light modulator tensors, constructing the initial pixel plane complex amplitude field on the pixel plane of the spatial light modulator based on the target illumination parameters includes: multiplying the initial hologram based on the liquid crystal pixel coordinates of the liquid crystal panel of the spatial light modulator with the imaginary unit and using the result as the exponent of the natural constant to obtain the initial pixel plane complex amplitude field.

[0064] In the above-mentioned atomic array rearrangement path planning method using spatial light modulator tensors, constructing the initial pixel plane complex amplitude field on the plane of the spatial light modulator based on the target illumination parameters includes: multiplying the position of each optical trap in the target array on the focal plane of the spatial light modulator by a phase period constant and then dividing by the laser wavelength and the focal length of the pre-Fourier lens to obtain the frequency of each optical trap.

[0065] In the generation of the spatial light modulator tensor, the WGS single-step iteration includes the following steps.

[0066] Divide the current pixel plane complex amplitude field by its magnitude to obtain the phase-processed pixel plane complex amplitude field. That is, after obtaining the initial pixel plane complex amplitude field, perform WGS single-step iterations using the WGS single-step operator on the initial pixel plane complex amplitude field. During the WGS single-step iteration, for example in the... In the nth iteration, first for the current iteration, i.e., the nth iteration... The pixel plane complex amplitude field at the next iteration Performing phase conversion decouples non-physical amplitude fluctuations (corresponding to the hardware properties of pure phase SLM), represented as:

[0067] in For the first The phase-modulated complex amplitude field of the pixel plane in the next iteration (the modulus is always 1). For the first The pixel plane complex amplitude field of the SLM in the next iteration. Let be the modulus of the complex amplitude field in the pixel plane. This essentially expresses the hardware phase modulation property of a pure phase spatial light modulator, since the actual SLM only changes the spatial phase delay of the incident laser. This does not change the amplitude of the laser. Since the calculated result of the complex amplitude field of the pixel plane obtained in each step of the WGS single-step iteration during the forward-inverse Fourier iteration of the wave field is a complex number containing both amplitude and phase, it can be obtained by dividing by its own modulus. This decouples non-physical amplitude oscillations and fluctuations, retaining only pure phase information to achieve holographic phase recovery.

[0068] A forward Fourier transform is performed on the phased pixel plane complex amplitude field to obtain the focal plane complex amplitude field. Here, the phased pixel plane complex amplitude field... Performing a forward Fourier transform to simulate lens diffraction projection onto the focal plane is represented as:

[0069] That For the first The complex amplitude field of the focal plane on the focal plane in the next iteration. For the positive Fourier transform operator, The position on the focal plane (represented as spatial coordinates). For the position on the corresponding focal plane The frequency (represented as spatial frequency coordinates), for Summation represents the position of all pixels traversed on the SLM plane. They came to seek peace.

[0070] At the location of each optical trap in the target array, the complex amplitude field of the focal plane on the focal plane is divided by its modulus and then multiplied by the target amplitude and the weight of the current round to obtain the amplitude-constrained complex amplitude field of the focal plane. That is, at the location of each optical trap in the target array... An amplitude constraint is applied at this point, and the actual diffraction phase is recombined with the target weighted amplitude, which is expressed as: For the first The complex amplitude field of the focal plane after amplitude constraint in the next iteration For the first The iteration of the ... Weights of the optical trap For the first The target amplitude of the optical trap, For the first Position of the next iteration Complex amplitude (including amplitude and phase) at the focal plane. Its modulus length.

[0071] An inverse Fourier transform is performed on the amplitude-constrained focal plane complex amplitude field to obtain the pixel plane complex amplitude field for the next iteration. Here, an inverse Fourier transform is performed on the amplitude-constrained focal plane complex amplitude field to obtain the next iteration, i.e., the [number missing]th iteration. The pixel plane complex amplitude field on the SLM pixel plane in the next iteration is expressed as:

[0072] in It is the total number of optical traps in the target array.

[0073] In summary, the complete WGS single-step operator is represented as:

[0074] Those skilled in the art will understand that the pixel plane complex amplitude field and the focal plane complex amplitude field are actually tensors composed of complex numbers containing amplitude and phase at each position. That is, the pixel plane complex amplitude field is composed of the pixel plane complex amplitude corresponding to each position of the pixel plane, while the focal plane complex amplitude field is composed of the focal plane complex amplitude corresponding to each position of the focal plane, i.e., each optical trap position.

[0075] In the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application, performing WGS single-step iteration using the WGS single-step operator on the initial pixel plane complex amplitude field to obtain the pixel plane complex amplitude field of the corresponding iteration round includes: dividing the pixel plane complex amplitude field of the current round by its magnitude to obtain a phased pixel plane complex amplitude field; performing a forward Fourier transform on the phased pixel plane complex amplitude field to obtain a focal plane complex amplitude field on the focal plane; at the position of each optical trap in the target array, dividing the focal plane complex amplitude field by its magnitude and multiplying it by the target amplitude and the weight of the current round to obtain an amplitude-constrained focal plane complex amplitude field; and performing an inverse Fourier transform on the amplitude-constrained focal plane complex amplitude field to obtain the pixel plane complex amplitude field of the next round.

[0076] Additionally, it can be done in the current round, i.e., the [number]th round. In the next iteration, the optical trap position is extracted. The actual diffraction amplitude of each optical trap in the complex amplitude of the focal plane at that location And based on the gain negative feedback law, adaptive weight updates are performed (if the actual weight is too weak, compensation is amplified; if the weight is too strong, suppression is performed), as follows:

[0077] in and The first , No. During the first iteration The weights of each optical trap.

[0078] In other words, due to the inherent long-range spatial interference crosstalk of the Fourier diffraction integral, the photon energy at each point in the optical tweezers array is difficult to be uniform (resulting in uneven brightness and causing random photothermal escape of the central atom). This can be addressed by monitoring the ideal value. (i.e., the first) (ideal target amplitude of each optical trap) and current actual value The ratio (which is the first) After the first simulation iteration of forward injection, at the specified position The complex amplitude field of diffracted light intensity generated by actual interference (amplitude value), if the actual light intensity at a certain optical trap is too weak (ratio) If the calculation weight of that point is increased, it will be compensated for and amplified; if it is too strong, it will be suppressed, so that the actual amplitude of each optical trap tends to its target amplitude. This ensures that all neutral atoms on the array are in a deep potential well with highly uniform energy, meaning the light intensity uniformity meets the convergence criterion. .

[0079] In the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application, performing WGS single-step iterations using the WGS single-step operator on the initial pixel plane complex amplitude field to obtain the pixel plane complex amplitude field of the corresponding iteration round further includes: dividing the target amplitude of each optical well by the modulus of the focal plane complex amplitude at the position of each optical well and multiplying by the weight of each optical well in the current round to obtain the weight of each optical well in the next round.

[0080] Step S122: Obtain the light intensity uniformity index by dividing the standard deviation of the mode length of the complex amplitude of the focal plane of each optical trap in the target array by the mean, and determine the steady-state complex amplitude field of the pixel plane in response to the light intensity uniformity index being less than the uniformity convergence accuracy threshold.

[0081] Here, the discrete distribution of the actual light intensity of all optical traps in the target array is statistically analyzed, and a scalar index of light intensity uniformity is calculated. ,in, Let be the standard deviation of the mode length (i.e., amplitude) of the complex amplitude at the focal plane of each optical trap. The average amplitude of each optical trap is expressed as follows:

[0082] The scalar index of light intensity uniformity With the preset uniformity convergence accuracy threshold In comparison, the preset uniformity convergence accuracy threshold is used here. In practice, it is usually taken as 0.1%.

[0083] If the scalar index of light intensity uniformity Greater than the preset uniformity convergence accuracy threshold Then continue to update the pixel plane complex amplitude field of the SLM. and the weights of each optical trap Otherwise, the weight of the optical trap is determined. Full convergence, i.e., the weights at this point It has stabilized and can be considered a parameter rather than a dynamic variable. In practice, approximately 100 iterations are typically sufficient to achieve the desired scalar index of light intensity uniformity. Less than the preset uniformity convergence accuracy threshold .

[0084] In this embodiment of the application, the complete state of WGS consists of two types of variables, represented as follows:

[0085] in A hologram of SLM. The weight set consists of the weights of each optical trap, for example, represented as a weight vector. When treated as parameters rather than dynamic variables, the state simplifies to And converges to the steady-state complex amplitude field of the pixel plane. , can be represented as:

[0086] here, This is a complete WGS single-step operator (including phase transformation, forward transformation, amplitude forcing, and inverse transformation). It is the set of locations of all optical traps in the target array.

[0087] At this point, the complete state of WGS after convergence. That is, the fully convergent state variables of WGS Approximately equal to a stable hologram after convergence The state approximately degenerates to contain only phase, which indicates that after repeated oscillations along the optical path, the entire optical diffraction coherent network reaches an energy-phase equilibrium. In other words, if the hologram at this point... The input is given to the SLM, and after being diffracted by a real lens through parallel light, it hits the focal plane and can achieve the preset... The required complex amplitude distribution of the focal plane is accurately generated. Then, the WGS single-step operator is executed again in this state. Since the input and output are identical, no further drift in the complex amplitude field occurs. Therefore, the steady-state complex amplitude field of the pixel plane... That is, WGS single-step operator The fixed point. This is also the basis for subsequent treatment of atomic position shifts ( The premise that the measurement of crosstalk in the optical field caused by changes is valid is based on the following assumptions.

[0088] Here, those skilled in the art will understand that weights Sufficient convergence can be viewed as an assumption of parameter simplification, if the weights... If convergence is not achieved, a complete block Jacobian matrix is ​​required. That is, the derivation of the steady-state Jacobian matrix described below assumes that the weights have fully converged, as illustrated above. Otherwise, it is necessary to compute the complete block Jacobian matrix corresponding to the non-converged weights as an optional extension that can be derived from this application.

[0089] Step S123: Calculate the partial derivative of the WGS single-step operator with respect to the steady-state complex amplitude field of the pixel plane to obtain the steady-state Jacobian matrix.

[0090] In this embodiment of the application, the WGS single-step operator Including pixel plane complex amplitude field Length of the module Since the calculation involves both complex conjugate and non-holomorphic mappings, its derivative can be processed using Wirtinger calculus to calculate the partial derivative of the WGS single-step operator with respect to the steady-state complex amplitude field of the pixel plane. Of course, those skilled in the art will understand that other methods can also be used to calculate the partial derivative of the WGS single-step operator with respect to the steady-state complex amplitude field of the pixel plane, such as converting the differentiation of complex functions into a method of differentiating the mapping over a two-dimensional real space.

[0091] In the process of generating the spatial light modulator tensor, obtaining the steady-state Jacobian matrix using Wirtinger calculus includes the following steps.

[0092] The phased pixel-plane steady-state complex amplitude field is constructed based on the phased operation in the WGS single-step operator, and the Jacobian coefficients of each optical trap corresponding to the phased pixel-plane steady-state complex amplitude field are obtained. Here, for the pixel-plane steady-state complex amplitude field... Phase conversion is performed to decouple the amplitude, and only the phase is retained to obtain the phase-converted pixel plane steady-state complex amplitude field. :

[0093] Then, the phased pixel plane steady-state complex amplitude field is... Perform Fourier transform and then apply it to the spatial frequency coordinates of each optical trap. Take the modulus at point 1 to obtain the first... Focal plane amplitude of each optical trap :

[0094] Based on the Weight of each optical trap Target amplitude Amplitude of focal plane The first rank required to obtain the subsequent rank-k expansion Jacobian coefficient of each optical trap (Eliminating the (Explicit dependence of amplitude)

[0095] A focal plane tangent space projection operator is constructed based on the complex perturbation vector acting on the phased pixel plane steady-state complex amplitude field to project the complex perturbation vector onto the imaginary part direction of the focal plane. Here, for the pixel plane steady-state complex amplitude field... Wirtinger calculus respectively for and The partial derivative is expressed as:

[0096] in Represents phase mapping Here, the phase mapping pairs and It is simultaneously dependent and not holomorphic, therefore the standard complex derivative does not exist. The two Wirtinger partial derivatives can give its complete first-order linear response, avoiding the omission of the antiholomorphic term.

[0097] Then, the two partial derivatives are applied to the first-order complex perturbation vector. The combined effects of these operators can be simplified to a projection operator that retains only the tangential (imaginary) phase component while filtering out the radial amplitude component, expressed as:

[0098] in This represents the operation of taking the imaginary part of a complex number. Represents the pixel coordinates applied to the SLM. The complex perturbation vector at that point. Here, for the complex perturbation vector... Its function is to have That is, only the imaginary components perpendicular to the local phase of a small perturbation on the SLM will change the pure phase field, while the radial amplitude component is filtered out by the pure phase constraint. Thus, the focal plane tangent space projection operator is obtained. , it put along The direction is projected using the imaginary part, where It is the first The phase of each optical trap.

[0099] That is, in the steady-state complex amplitude field of the pixel plane fixed point (Right now ) at, to Applying a perturbation, denoted as the first-order complex perturbation vector. ,Depend on Phased pixel plane steady-state complex amplitude field exist The disturbance at that point is:

[0100] in This indicates the operation of extracting the real part of a complex number.

[0101] Define the pixel plane tangent space projection operator (It projects the complex perturbation vector onto a pure virtual direction orthogonal to the existing phase on the pixel plane), resulting in:

[0102] in It is a normalized perturbation field that retains only the pure phase variation component after projection is applied. Position on the SLM pixel plane The applied complex perturbation vector, It is the steady-state complex amplitude field of the pixel plane (as a reference field). It is the operator for extracting the imaginary part of a complex number, and It is the imaginary unit.

[0103] Then perform a forward Fourier transform, that is:

[0104] Amplitude-forced linearization is used for steady-state complex amplitude fields at the focal plane. At this point, the phase-normalized perturbation is calculated as follows:

[0105] in This indicates that the perturbation at the focal plane is also projected onto a phase perpendicular to its convergence. The imaginary (tangential) direction.

[0106] Therefore, the above pixel plane tangent space projection operator With the projection of the imaginary part of the focal plane here Combined, and due to the setting of the perturbation by the position of the SLM pixel plane The injection and projection operators are defined according to the phase distribution at the starting point, thus obtaining the focal plane tangent space projection operator. This indicates that the complex perturbation vector can only propagate backward in the component along the phase tangent direction, thus filtering out the radial component.

[0107] The steady-state Jacobian matrix is ​​obtained by multiplying the Jacobian coefficients of each optical trap, the tangent space projection operator, and the plane wave factor used for plane wave modulation. That is, based on the forward Fourier transform. perturbation When propagated to the focal plane, it will appear at the target frequency point in the Fourier integral. Introduce a base factor at [location]. The updated complex amplitude field of the focal plane is reflected back to the pixel plane of the SLM through an inverse Fourier transform, i.e. This allows the inverse transform to be calculated from the focal plane. Each optical trap returns to the pixel of the SLM. At this point, the inverse phase factor is introduced. This forms the plane wave factor used for plane wave modulation. In other words, plane wave modulation configures each optical trap with its frequency. Determined plane wave factor , used to indicate the first Coordinates of each optical trap in the SLM plane and Phase transfer between them.

[0108] Then, the Jacobian coefficient of each optical trap is... The tangent space projection operator and the plane wave factor Multiplying them together yields the matrix elements of the steady-state Jacobian matrix:

[0109] in, These are the matrix elements of the steady-state Jacobian matrix, representing coordinates. When a disturbance occurs at a location, it causes the coordinates to... The differential rate of change of the complex amplitude, It is the total number of optical traps in the target array. Indicates the first Each optical trap in spatial frequency coordinates The interference plane wave factor under the coordinate difference is used to... Spatial phase transfer is performed on top. Used to locate the source point Disturbance at the location Projected towards the The characteristic imaginary part direction of each optical trap It is a one-dimensional composite dimension reduction projection operator.

[0110] This is equivalent to applying the linearity of the forward and inverse Fourier transforms, along with the linearization of amplitude constraints, to each optical trap position. The local perturbation at a single point is represented by the Jacobian-Vector Product (JVP) backpropagation, and the interference plane wave factor terms of all optical traps are combined to obtain the explicit elements of the expansion of the Jacobian matrix describing the crosstalk quotient of the perturbation at any pixel on the SLM plane to other coordinate nodes on the entire SLM plane, i.e., the above. .

[0111] Therefore, the obtained steady-state Jacobian matrix accurately represents WGS at the microscopic level, where the steady-state Jacobian matrix is ​​essentially a matrix with a rank of maximum value. The coupling response of the optical system is not only modulated by the ratio of weight to amplitude, but its energy is also dependent on the plane wave substrate across the entire plane, as part of the operator superposition of (i.e., the total number of optical traps in the target array). The divergent interference precisely reveals the complex superposition and leakage entanglement of each optical trap field at the holographic spatial frequency.

[0112] Therefore, those skilled in the art will understand that calculating the partial derivative of the WGS single-step operator with respect to the steady-state complex amplitude field of the pixel plane by applying a small perturbation to the steady-state complex amplitude field of the pixel plane using Wirtinger calculus is essentially based on the WGS single-step operator, through the steady-state complex amplitude field of the pixel plane... fixed point (Right now ) at, to Apply micro-perturbation The matrix elements of the steady-state Jacobian matrix are obtained through the phase transformation, forward Fourier transform, amplitude forcing, and inverse Fourier transform process of the WGS single-step operator.

[0113] That is, by Phased pixel plane steady-state complex amplitude field exist The disturbance at that point is:

[0114] Based on linear forward Fourier transform:

[0115] Based on linear amplitude forcing, in At this point, the phase-normalized perturbation is:

[0116] Finally, based on the linear inverse Fourier transform and by combining the results of the four steps, we obtain:

[0117] That is, the focal plane tangent space projection operator corresponds to the inverse part of the phase transformation and amplitude forcing steps, while the Jacobian coefficient and plane wave factor correspond to the forward Fourier transform, the amplitude part of the amplitude forcing step, and the inverse Fourier transform.

[0118] In the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application, calculating the partial derivative of the WGS single-step operator with respect to the steady-state complex amplitude field of the pixel plane to obtain the steady-state Jacobian matrix includes: calculating the partial derivative of the WGS single-step operator with respect to the steady-state complex amplitude field of the pixel plane by applying a micro-perturbation to the steady-state complex amplitude field of the pixel plane using Wirtinger calculus to obtain the steady-state Jacobian matrix.

[0119] Furthermore, in the aforementioned atomic array rearrangement path planning method using spatial light modulator tensors, calculating the partial derivative of the WGS single-step operator with respect to the steady-state complex amplitude field of the pixel plane by applying a micro-perturbation to the steady-state complex amplitude field of the pixel plane using Wirtinger calculus to obtain the steady-state Jacobian matrix includes: constructing a phased steady-state complex amplitude field of the pixel plane based on the phased operation in the WGS single-step operator and obtaining the Jacobian coefficients of each optical trap corresponding to the phased steady-state complex amplitude field of the pixel plane; constructing a focal plane tangent space projection operator for projecting the complex perturbation vector onto the imaginary part direction of the focal plane based on the complex perturbation vector acting on the phased steady-state complex amplitude field of the pixel plane; and multiplying the Jacobian coefficients of each optical trap, the focal plane tangent space projection operator, and the plane wave factor for plane wave modulation to obtain the steady-state Jacobian matrix.

[0120] In this embodiment of the application, since the total pixel size of the SLM is extremely large (e.g. The storage of the steady-state Jacobian matrix may require a large amount of storage space, therefore sparsification is desired. Here, the physical oscillation characteristics of the steady-state Jacobian matrix are considered because of the plane wave factor. Interference with distance exhibits antiphase cancellation (i.e., when...) Time oscillations superimpose and cancel each other out. The crosstalk characteristic width (characterized by the aperture diffraction limit of the SLM) is an optical property. When the physical lattice point extends beyond a specific nearest-neighbor diffraction range boundary, the crosstalk value rapidly decays and degrades. Therefore, it is possible to construct a probe vector that is non-zero only within the neighborhood of the optical trap. Thus, utilizing the rank... Structure, only needs to be detected sparse elements, of which It is half the side length of the neighborhood, meaning the neighborhood is the target optical trap. Grid points. Among them, the crosstalk feature width. Represented as:

[0121] in It is the laser wavelength. It is the focal length of the Fourier lens, and This is the effective illumination aperture size of the spatial light modulator (e.g., the product of the number of pixels and the pixel pitch, on the order of millimeters), which determines the angular resolution limit of SLM diffraction. It is important to note that here... This is not the minimum center-to-center distance (minimum optical trap spacing) between adjacent optical traps in the target optical trap array; the order of magnitude of the minimum optical trap spacing is approximately... For example, in With a crosstalk characteristic width of approximately 8.19 mm, the crosstalk characteristic width is calculated. Approximately 9.74 .

[0122] Specifically, nearest neighbor detection can be performed per light trap using torch.autograd.functional.jvp ( Grid-based sparse elements are stored in cuSPARSE CSR format (approximately 24MB) to avoid explicit storage. A dense matrix, in which Here, torch.autograd.functional.jvp computes the Jacobian-vector product to probe the vector. (in the neighborhood) Grid processing 1) Column-by-column detection The non-zero element is represented as:

[0123] in It is the amplitude of an infinitesimal perturbation (take the limit after differentiation). It is obtained directly using the directional derivative. With probe vector The product does not require explicit construction. For dense matrices, by probing each local basis vector, the nearest non-zero columns of the Jacobian can be extracted, reducing the computational cost from... It drops to a level proportional to bandwidth.

[0124] In the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application, after obtaining the steady-state Jacobian matrix, it further includes: probing the non-zero elements of the steady-state Jacobian matrix column by column with probe vectors that are non-zero only in the neighborhood of the light trap to extract the non-zero elements of the steady-state Jacobian matrix to obtain a sparsified steady-state Jacobian matrix.

[0125] Step S124: Calculate the partial derivatives of the WGS single-step operator with respect to the positions of all optical traps to obtain the position partial derivative matrix.

[0126] In the process of generating the spatial light modulator tensor, the calculation of the position partial derivative matrix includes the following steps.

[0127] The phased steady-state complex amplitude field of the pixel plane is modulated pixel by pixel using the horizontal and vertical coordinates of the pixel plane of the SLM to obtain the horizontal coordinate modulation field and the vertical coordinate modulation field.

[0128] That is, for the phased pixel plane steady-state complex amplitude field The positional partial derivative originates from the plane wave factor. For the position of the optical trap Factors lost during differentiation ,in Since the SLM's pixel coordinates are constants, the pixel coordinates of the SLM need to be used as the multiplicative operator. and It acts on the phased field. Therefore, using the horizontal coordinate... with vertical coordinate Pixel-by-pixel modulation of phase-shifted pixel plane steady-state complex amplitude field The transverse coordinate modulation field is obtained. and longitudinal coordinate modulation field :

[0129] In other words, the differential theorem of the Fourier transform states that differentiating the frequency domain with respect to frequency is equivalent to multiplying the spatial domain by the coordinates. (Optical trap position) By frequency Entering the phase of a plane wave, taking its derivative yields the coordinate factor, therefore using coordinates... and Modulated pixel plane steady-state complex amplitude field This transforms position sensitivity into an object that can be directly calculated using Fourier transform.

[0130] Fourier transform is performed on the transverse and longitudinal modulation fields, and samples are taken at the frequency of each optical trap to obtain the transverse and longitudinal modulation spectrum sample values ​​of each optical trap.

[0131] That is, the frequency of each optical trap on the focal plane. It is achieved by using the phased pixel plane steady-state complex amplitude field The frequency of each optical trap, obtained by Fourier transform after performing Fourier transforms on the transverse and longitudinal coordinate modulation fields respectively. The above transformation results are sampled to obtain the transverse and longitudinal component modulation spectrum sample values ​​for each optical trap:

[0132] in It is a two-dimensional continuous (or discrete) forward Fourier transform integral operator, which can transform the coordinate modulation field to the focal plane and sample it at the frequency of the target optical trap. The resulting complex value is proportional to the direction and amplitude of the first-order change of the complex amplitude field of the focal plane when the optical trap is moved, and is the frequency domain carrier of position sensitivity.

[0133] The transverse and longitudinal component modulation spectrum samples of each optical trap are multiplied by a frequency-coordinate mapping constant to combine the positional partial derivatives of each optical trap along the transverse and longitudinal directions.

[0134] That is, the sampled values ​​of the transverse component modulation spectrum of each optical trap. and longitudinal component modulation spectrum sampling value Multiply by the frequency-coordinate mapping constant (This is a physical conversion of coordinate modulation equivalent to frequency domain differentiation), and the position partial derivatives of each optical trap along the transverse and longitudinal directions are obtained by combining them:

[0135] The position of each optical trap Represented as This is to deconstruct the local spatial degrees of freedom in the horizontal and vertical directions. Here, in order to correct for the dependency simplification assumption (the ordinary multiplication factor)... Algorithm errors, such as those caused by replacing the integral weighting term, lead to a systematic overestimation of crosstalk strength. Expanding along the Fourier kernel phase angle of the true wavefront, when the... When an optical trap attempts to physically offset itself, a chained partial derivative operation is performed within the originally smooth integral transform network. Furthermore, the spatial matrix scale dimension coordinates (i.e., the horizontal coordinates) of the SLM plane of the liquid crystal panel are... with vertical coordinate The direct weighted operator is injected into the integral kernel to calculate the flux distortion partial derivatives of each optical trap caused by the shift drive, which is expressed as follows:

[0136] here, It is the WGS single-step operator with respect to the first step in the focal plane. The horizontal coordinates of each optical trap The lateral partial derivative component represents the rate of change of the entire projection field induced by a very small lateral physical movement of the optical trap. As described above, the global optical coupling constant multiplier derived from the derivative of the exponentially transferred phase factor of the far-field Fourier diffraction integral can be represented.

[0137] In other words, when performing partial derivative calculations on the target optical trap, if the frequency domain spatial constant components (such as spatial frequency coordinates) are simply used... While directly extracting the common multiplier simplifies computation, it neglects the practical realities of large-aperture planar light emission in SLMs. The calculation of the partial derivative of the WGS single-step operator with respect to the light trap position considers the core factors causing changes in the reconstruction domain due to focus movement, which depends on the solid plane coordinate distribution of the liquid crystal pixels in the source-end holographic liquid crystal panel. and For steady-state phase field After non-uniform aperture-scale weighted modulation, the spatial spectrum abrupt change that occurs at a specific wave vector accurately reproduces the spatial derivative law followed by the light path when the optical trap undergoes forced motion.

[0138] The position partial derivative components of each optical trap along the lateral and longitudinal directions are flattened into column vectors of complex amplitude fields and spliced ​​according to the lateral and longitudinal coordinates of the optical trap to obtain the position partial derivative matrix.

[0139] In other words, after traversing all the optical traps, the partial derivatives of the positions along the lateral and longitudinal axes corresponding to each individual optical trap are summarized to generate a set of components for the discrete sites. Here, since the motion of each optical trap includes lateral and longitudinal axes (usually denoted as second-order degrees of freedom, i.e., each optical trap has...), the partial derivatives of the positions along the lateral and longitudinal axes are summarized to generate a set of components for the discrete sites. shaft and The vector behavior of the axis translation state space is thus normalized and configured into a high-dimensional coherent tensor. For example, setting the shape mapping boundary of the macroscopic array (e.g., the total number of optical traps is...) The total number of degrees of freedom for overall displacement is The positional partial derivative components of each optical trap along the lateral and longitudinal directions are flattened into column vectors of the complex amplitude field of the pixel plane, and then... By sequentially concatenating the horizontal and vertical coordinates, we obtain the position partial derivative matrix representing how the minute movements of all optical trap positions are linearly mapped to the changes in the complex amplitude field of the pixel plane on the SLM plane.

[0140] In the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application, calculating the partial derivatives of the WGS single-step operator with respect to the positions of all optical traps to obtain the position partial derivative matrix includes: modulating the phased pixel plane steady-state complex amplitude field pixel by pixel with the lateral and longitudinal coordinates of the pixel plane of the SLM to obtain a lateral coordinate modulation field and a longitudinal coordinate modulation field; performing Fourier transform on the lateral coordinate modulation field and the longitudinal coordinate modulation field and sampling at the frequency of each optical trap to obtain the lateral component modulation spectrum sample value and the longitudinal component modulation spectrum sample value of each optical trap; multiplying the lateral component modulation spectrum sample value and the longitudinal component modulation spectrum sample value of each optical trap by a frequency-coordinate mapping constant to combine the position partial derivative components of each optical trap along the lateral and longitudinal directions; and flattening the position partial derivative components of each optical trap along the lateral and longitudinal directions into column vectors of the pixel plane complex amplitude field and concatenating them according to the lateral and longitudinal coordinate order of the optical traps to obtain the position partial derivative matrix.

[0141] Step S125: Based on the implicit function representation of the fixed point condition of the WGS single-step operator, the steady-state Jacobian matrix and the position partial derivative matrix are combined with the total differential of the optical trap position set to obtain the spatial light modulator tensor.

[0142] The process of generating the spatial light modulator tensor involves the following steps.

[0143] Calculate the implicit functional representation of the fixed-point condition of the WGS single-step operator as the total differential with respect to the set of optical trap positions.

[0144] As mentioned above, in atomic rearrangement, the set of optical trap positions As time progresses, the spatial light modulator tensor expectation metric... tiny changes At that time, the pixel plane steady-state complex amplitude field How it changes, including the set of optical trap locations. Abbreviated as:

[0145] When the WGS iteration converges, the steady-state complex amplitude field of the pixel plane is obtained. Therefore, when the WGS single-step operator is at a fixed point, it is in the steady-state complex amplitude field of the pixel plane. This state can be described as the WGS single-step operator at a fixed point. Then the fixed-point condition of the WGS single-step operator can be written as an implicit function:

[0146] Set of optical trap locations The total differential is obtained by the implicit function theorem:

[0147] The amplitude sensitivity matrix is ​​obtained from the total differential representation based on the steady-state Jacobian matrix. That is, the steady-state Jacobian matrix is ​​expressed as:

[0148] From the above total differential expression, we obtain:

[0149] in It is a multidimensional diagonal reference identity matrix, obtained by subtracting the inverse of the steady-state Jacobian matrix in the WGS convergence region. As a crosstalk amplifier, it acts on the position partial derivative matrix That is, the positional partial derivative matrix. This only describes the direct effect of the moving optical trap on the hologram. However, since WGS iterates to a fixed point, this initial perturbation will propagate and superimpose repeatedly through the iteration of the WGS single-step operator, forming an infinite series. The geometric series converges when the spectral radius is less than 1. This is called a crosstalk amplifier. The essence of optical crosstalk is amplifying a single direct effect into a convergent total effect; moving one optical trap will cause leakage through the diffraction field, affecting all optical traps. Therefore, As an amplitude sensitivity matrix, it can be denoted, for example, as... .

[0150] Furthermore, the above equation holds true if WGS converges, and the necessary and sufficient condition for WGS convergence is that the WGS single-step operator... At steady state (i.e., at a fixed point), the Jacobian satisfies the spectral radius. In other words, can WGS converge to a fixed point in a single iteration? It depends on the WGS single-step operator. The first derivative at this fixed point is the aforementioned steady-state Jacobian matrix. Therefore, spectral radius This means that when the hologram is subjected to a small external perturbation, the error will decrease geometrically in subsequent iterations of WGS (with the spectral radius as the attenuation factor), and the WGS iterative algorithm will spontaneously reconverge to the fixed point. Conversely, if... If the perturbation diverges, the iteration will fail to converge.

[0151] The spatial light modulator tensor is obtained by taking the Gram matrix of the amplitude sensitivity matrix. The spatial light modulator tensor measures the change in the steady-state hologram caused by the optical trap displacement.

[0152] Based on the amplitude sensitivity matrix Construct the spatial light modulator tensor , represented as:

[0153] in This represents the Hermitian conjugate transpose, which is obtained by transposing the matrix and then finding its conjugate over the complex domain. In other words, it involves taking the amplitude sensitivity matrix. Gram matrix This yields the spatial light modulator tensor, which measures the change in the hologram caused by the displacement of the optical trap. Its matrix size is (Each optical trap has) and Two degrees of freedom). Block element. Indicates the move of the first The and the first The inner product (i.e., coupling) between the holographic changes caused by each coordinate component.

[0154] The spatial light modulator tensor is divided into blocks according to the optical trap number. Can be written as indivual sub-block The scalar coupling matrix used for visualization can be defined as:

[0155] That is, because each optical trap has and Two degrees of freedom, therefore Middle , The coupling of the optical trap is Sub-blocks. By taking their Frobenius norm and compressing the directional degrees of freedom into a single scalar, we obtain the scalar coupling matrix between optical traps. This is used to visualize crosstalk distribution.

[0156] according to It can be seen The change in the steady-state hologram caused by the optical trap displacement (i.e., the change in the hologram when WGS iteration converges to a fixed point) was accurately measured, and its block elements... Quantization of the moving optical trap After a small step, and through infinite iterations of WGS, the optical trap finally converges. The perturbation intensity generated by the amplitude, i.e. the WGS optical crosstalk coupling intensity between the two optical traps.

[0157] In the atomic array rearrangement path planning method using a spatial light modulator tensor according to an embodiment of this application, the method of obtaining the spatial light modulator tensor by combining the steady-state Jacobian matrix and the positional partial derivative matrix with respect to the total differential of the implicit functional representation of the fixed-point condition of the WGS single-step operator with respect to the optical trap position set includes: calculating the total differential of the implicit functional representation of the fixed-point condition of the WGS single-step operator with respect to the optical trap position set; obtaining the amplitude sensitivity matrix from the total differential representation based on the steady-state Jacobian matrix; and obtaining the spatial light modulator tensor by taking the Gram matrix of the amplitude sensitivity matrix, wherein the spatial light modulator tensor measures the change in the steady-state hologram caused by the optical trap displacement.

[0158] Due to the spatial light modulator tensor It is a Gram matrix, which is naturally Hermitian positive semidefinite and can be directly used as a Riemannian metric. Here, the spatial light modulator tensor can be verified. The positive semidefiniteness and positive definiteness of the property. Let... , , It is a Gram matrix, and ,but (Hermitian) and (This holds true for all semi-positive definite values).

[0159] and, , When the column is full Therefore, it is correct. That is to say, (Strict positive definiteness if and only if) When the column rank is full, that is, the optical trap configuration is non-degenerate and In typical parameters ( The positive definiteness holds under the condition that the optical traps are non-overlapping. Here, the necessary and sufficient condition for WGS convergence is that the steady-state Jacobian matrix of the WGS single-step operator satisfies the spectral radius. This can be used to satisfy the WGS iterative convergence, and it is also the tensor of the spatial light modulator. The necessary and sufficient condition for the positive definiteness of .

[0160] In the atomic array rearrangement path planning method using the spatial light modulator tensor according to the embodiments of this application, after obtaining the spatial light modulator tensor, it further includes: determining the positive semidefiniteness of the spatial light modulator tensor based on the Gram matrix representation of the amplitude sensitivity matrix based on the spatial light modulator tensor tensor.

[0161] In the atomic array rearrangement path planning method using a spatial light modulator tensor according to an embodiment of this application, after obtaining the spatial light modulator tensor, the method further includes: determining the positive definiteness of the spatial light modulator tensor in response to the full column rank of the amplitude sensitivity matrix.

[0162] In the atomic array rearrangement path planning method using a spatial light modulator tensor according to an embodiment of this application, after obtaining the spatial light modulator tensor, the method further includes: determining the positive definiteness of the spatial light modulator tensor in response to the spectral radius of the steady-state Jacobian matrix being less than one.

[0163] Furthermore, in order to enhance the density Spatial light modulator tensor Storage efficiency, for the spatial light modulator tensor Perform sparsification, for example Sparsification is performed to convert it into a CSR sparse matrix that can be stored and computed in real time on the GPU.

[0164] Here, the spatial light modulator tensor is subjected to... The process of sparsification to obtain a sparsified spatial light modulator tensor includes the following steps.

[0165] Obtain the block elements of the spatial light modulator tensor as the inner product between the corresponding columns of the amplitude sensitivity matrix, and the position partial derivative matrix.

[0166] For amplitude sensitivity matrix The spatial light modulator tensor block elements Defined as the amplitude sensitivity matrix The inner product between the corresponding columns:

[0167] in For the spatial light modulator tensor The Block elements and These are the amplitude sensitivity matrices. The and the List, These are the spatial coordinates of the optical trap. For complex conjugate, This is the inner product. As mentioned above, block elements... It is the first move and the The inner product of the steady-state hologram changes caused by each optical trap is used to quantify the optical crosstalk coupling strength between the two optical traps after complete convergence by WGS.

[0168] Furthermore, the positional partial derivative matrix is ​​obtained to obtain the WGS single-step operator with respect to the first... The position of each optical trap The partial derivatives, here, are obtained by using the differential theorem of the Fourier transform to convert the differentiation with respect to position into a function of pixel coordinates. and Weighted values ​​are then subjected to a Fourier transform, for example, for pixel coordinates. Represented as:

[0169] here, It is a WGS single-step operator. It is the first The position (i.e., spatial coordinates) of each optical trap. For the phased pixel plane steady-state complex amplitude field, These are the horizontal and vertical coordinates of the SLM pixel (in units of length). The wavelength of the laser. Let be the focal length of the Fourier lens. For Fourier transform, The spatial frequency coordinates of the optical trap. The unit is the imaginary unit. That is, moving the optical trap position is equivalent to shifting the sampling point in the frequency domain. According to the Fourier differential theorem, its sensitivity is determined by the pixel coordinates. The spectrum of the weighted, phase-modulated complex amplitude field of the pixel plane is given. It should be noted that the weighting factor is the pixel coordinate. (Length dimension), not spatial frequency The two have different dimensions and cannot be used interchangeably.

[0170] The positional partial derivative matrix is ​​simplified using the iterative initial state and the uniform rectangular aperture to obtain the simplified amplitude sensitivity matrix, and the simplified block elements of the spatial light modulator tensor of the iterative initial state, expressed as the aperture integral, are obtained based on the simplified amplitude sensitivity matrix.

[0171] In the initial state of the iteration Approximation to neglect the steady-state Jacobian matrix ,Right now And assuming a uniform rectangular aperture The position partial derivative is simplified to a single plane wave multiplied by the first-order moment of the aperture, for example, for pixel coordinates. Represented as: The first of the amplitude sensitivity matrix List in the initial state of the iteration Approximation, For the first Complex weights of each optical trap ( As weight, For the target amplitude, For the first (phase of each optical trap) For rectangular aperture at frequency The first moment at the point (approximately constant under uniform illumination). This is the SLM aperture characteristic function. That is, under conditions of weak crosstalk and uniform illumination, the sensitivity of the optical trap displacement can degenerate to a plane wave multiplied by the first-order aperture moment. For pixel coordinates... The quantities are the same.

[0172] Then, substituting the approximate amplitude column into the inner product representation of the block elements of the spatial light modulator tensor, the corresponding columns of the amplitude sensitivity matrix are combined in two components to obtain the iterative initial state represented by the aperture integral. Block elements: Initial state of the iteration Approximate block elements The complex weighted conjugate product of the two optical traps. The pixel coordinates are squared and weighted (from two coordinate weightings). The frequency difference between the two optical traps is summed and iterated over all pixels of the SLM. Here, the approximate block element is determined by the sum of phase differences weighted by the squares of the aperture coordinates, which is a weighted sum of the squares of the pixel coordinates. This can accurately reflect the modulation of crosstalk by the spatial distribution of the SLM aperture.

[0173] The simplified block elements of the spatial light modulator tensor are summed using Dirichlet kernels and subjected to large pixel number continuum to obtain the crosstalk intensity between the two optical traps. Coupling rate.

[0174] The iterative initial state The approximate block elements are summed along a finite number of pixels in one dimension (geometric series) in a closed form as the Dirichlet kernel, i.e., a finite DFT summation:

[0175] in For the number of pixels in one dimension, For pixel index. for Directional frequency difference, The pixel pitch is given, and the exponential factor is the linear phase. That is, the sum of the phase differences under a finite aperture is exactly equal to the Dirichlet (periodic) phase difference. The core is the intrinsic structure of finite pixel diffraction.

[0176] when Sometimes, Then the Dirichlet nucleus is approximately equal to... function:

[0177] in For normalization Function, denominator This is the linearized result of the denominator for a small-angle approximation. That is, when the number of pixels is sufficiently large and the diffraction angle is sufficiently small, the Dirichlet kernel and the continuous aperture... The diffraction point diffusion functions are consistent.

[0178] Therefore, substituting the relationship between frequency difference and optical trap spacing, for the transverse... and longitudinal Multiplying the squares of the two dimensions by taking their modulo, we obtain the principal order of the scalar coupling matrix. Decay law:

[0179] in: Scalar coupling matrix element ( (Frobenius norm of sub-blocks) It is the aperture integral constant. , The distance between the two optical traps, For crosstalk feature width, This refers to the effective illumination aperture size of the spatial light modulator (determined by the number of pixels and pixel pitch of the spatial light modulator), rather than the minimum spacing between the optical traps. This indicates that the crosstalk strength between the two optical traps follows a two-dimensional... Attenuation law, crosstalk characteristic width It is determined by the diffraction limit (effective aperture of the SLM, not the optical trap spacing). Specifically, the condition for its validity is: a uniform rectangular aperture ( (constant), weight convergence ( It is a constant. ), Approximation (ignoring the steady-state Jacobian matrix), Furthermore, while simultaneously containing , and In the case of component light, non-uniform illumination or large-angle diffraction does not degenerate into However, under uniform illumination and small-angle diffraction conditions, The approximate accuracy is better than 1%.

[0180] Using the above The far-end integral rate of the coupling rate is used to estimate the truncation bandwidth, and the spatial light modulator tensor is bandwidth-truncated with the truncation bandwidth to obtain the sparsified spatial light modulator tensor.

[0181] use Far-end integral law estimation of truncated bandwidth The remaining Frobenius energy is used to select a bandwidth that meets the required accuracy:

[0182] Here, bandwidth is truncated. It is based on the crosstalk characteristic width The number of grid points in units can be seen Tail energy varies with bandwidth by The rate decay ensures that only a small number of nearest neighbors are needed to capture the vast majority of crosstalk energy.

[0183] Therefore, the bandwidth of the spatial light modulator tensor can be truncated to retain only the bandwidth required for the modulator. Non-zero elements in the neighborhood of the integer are set to zero, and the rest are set to zero (e.g., ...). (Capturing 99.98% of the energy), thereby compressing and storing the sparsed spatial light modulator tensor in cuSPARSE CSR format (e.g., approximately 24 MB).

[0184] Table 1 below shows the numerical comparison of sparsification of the spatial light modulator tensor (where atoms are...). 171 Yb, , optical trap spacing )

[0185] Table 1

[0186] Here, in terms of bandwidth Under the level-1 approximation, the approximation error is And energy capture %. Here, level-1 is approximately equivalent to reserving a preset bandwidth. Set the nearest neighboring elements in the block to zero and set the rest of the elements to zero.

[0187] The atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application further includes: performing... Sparsification is used to obtain a sparsified spatial light modulator tensor.

[0188] Furthermore, in the aforementioned atomic array rearrangement path planning method using spatial light modulator tensors, the spatial light modulator tensor is subjected to... The sparsification process to obtain a sparsified spatial light modulator tensor includes: obtaining the block elements of the spatial light modulator tensor represented by the inner product between corresponding columns of the amplitude sensitivity matrix and the position partial derivative matrix; simplifying the position partial derivative matrix with an iterative initial state and a uniform rectangular aperture to obtain a simplified amplitude sensitivity matrix, and obtaining the simplified block elements of the spatial light modulator tensor in the iterative initial state represented by the aperture integral based on the simplified amplitude sensitivity matrix; performing Dirichlet kernel summation and large pixel number continuum on the simplified block elements of the spatial light modulator tensor to obtain the crosstalk intensity between the two optical traps. Coupling rate; and, using the above The far-end integral rate of the coupling rate is used to estimate the truncation bandwidth, and the spatial light modulator tensor is bandwidth-truncated with the truncation bandwidth to obtain the sparsified spatial light modulator tensor.

[0189] Here, the spatial light modulator tensor can be used as a Riemannian metric for atomic array rearrangement path planning. That is, the generalized kinetic energy of the optical trap moving at a predetermined speed can be represented based on the spatial light modulator tensor, and the geodesic along this metric gives the optimal transmission path under SLM crosstalk constraints. Furthermore, the spatial light modulator tensor can be used as a Fisher-like metric for hologram generation. That is, under the Poisson photon noise model, the block elements of the spatial light modulator tensor can be interpreted as Fisher-like information about the focal plane intensity distribution with respect to the optical trap position. It should be noted that this Fisher-like metric is an approximate correspondence of the leading-order in the sense of the complete noise and observation model, rather than a strict Fisher information equivalence. In addition, the spatial light modulator tensor can also be used as a preconditioner for complex amplitude domain gradient flow (CDGF), where the tensor inverse matrix of the spatial light modulator tensor is... As the natural precondition matrix for the complex amplitude domain gradient flow, it can guarantee that each correction is made along the optimal descent direction (natural gradient).

[0190] In this embodiment of the application, for the spatial light modulator tensor It can be obtained through pre-calibration, simulation calculation, experimental measurement or generated by tensor generation module, and this application is not intended to impose any special restrictions on it.

[0191] Therefore, the matrix function in the spatial light modulator tensor that quantizes the displacement cost of the optical trap itself, such as the optical trap, is used. The diagonal block elements of the corresponding spatial light modulator tensor For each source-target pair Calculate the cost of anisotropic transport:

[0192] in This indicates transpose. Source optical trap To the target location The transmission cost, weighted by the spatial light modulator tensor, quantifies the transfer of atoms from... Move to At that time, the perturbation (in the square norm sense) to all other fixed points of the optical trap hologram caused by the SLM diffraction optical structure. This is related to the Euclidean cost, for example, expressed as... different, It is anisotropic, that is, along Moving along the direction of large eigenvalues ​​(high crosstalk direction) is more costly, while moving along the direction of small eigenvalues ​​(low crosstalk direction) is less costly.

[0193] Here, optical trap The diagonal block elements of the corresponding spatial light modulator tensor Can quantize optical traps The unidirectional effect of self-displacement on the hologram is based on the correct input of the optimal transmission cost matrix of the spatial light modulator. Furthermore, the spatial light modulator tensor also includes cross blocks. ( The cost matrix is ​​used to quantify the coupling between optical traps. If the cross-blocks are included in the cost matrix construction, the optimal transmission problem degenerates into a quadratic assignment problem (QAP), which is generally an NP-hard problem. Standard entropy-regularized optimal transmission solution algorithms (such as Sinkhorn-type algorithms) cannot solve it directly. Therefore, in this embodiment, the path planning preferably uses only the diagonal blocks of the spatial light modulator tensor to construct the cost matrix. The global joint optimization of cross-block coupling can be implemented using a quadratic assignment problem approximation algorithm, the alternating direction multiplier method (ADMM), or mean field relaxation as optional extended implementation methods.

[0194] By iterating through all Source-target pairing The obtained anisotropic transmission costs can then be organized into a complete crosstalk weighted cost matrix. .

[0195] Here, the matrix function in the spatial light modulator tensor that quantizes the displacement cost of the optical trap itself, excluding the diagonal block elements of the spatial light modulator tensor, is used in addition to the diagonal block elements of the spatial light modulator tensor. In addition, it can also include diagonal block elements. The eigenvalues ​​are diagonalized (e.g., still an anisotropic measure after rotation to the principal axis direction); or the 2×2 block is replaced with Frobenius norm scalarization, represented as:

[0196] Or a full matrix (Including cross-coupling blocks, crosstalk is more accurate but computationally more expensive); or based on Analytical approximation direct calculation Without relying on numerical values .

[0197] That is, in the first tensor of the spatial light modulator Each optical trap corresponds to a 2×2 diagonal block element. Based on this, the matrix function representing the displacement cost of the optical trap itself in the spatial light modulator tensor can be obtained by interpolating multiple sub-blocks in the neighborhood of the source location. For example, it can be denoted as... Furthermore, the matrix function that quantizes the displacement cost of the optical trap itself in the spatial light modulator tensor can be obtained by using the Schur complement or locally effective metric that includes the cross-block of the source optical trap neighborhood.

[0198] In the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application, constructing a crosstalk weighted cost matrix based on the displacement vector from the source position to the target position and the matrix function in the spatial light modulator tensor that quantizes the displacement cost of the optical trap itself includes: multiplying the row vector of the displacement vector from the source position to the target position by the matrix function in the spatial light modulator tensor that quantizes the displacement cost of the optical trap itself corresponding to the source position, and then multiplying it by the column vector of the displacement vector from the source position to the target position to obtain anisotropic transmission cost; and combining all the anisotropic transmission cost values ​​corresponding to the pairing of the source position and the target position into the crosstalk weighted cost matrix.

[0199] Step S130: Determine the entropy regularization parameters based on the error, and construct the kernel function matrix under the entropy regularization framework based on the crosstalk weighted cost matrix.

[0200] Here, the entropy regularization parameter is determined based on the error and expressed as:

[0201] in It is the entropy regularization parameter (for) A scalar (of which) is used to control the smoothness and approximation accuracy of the optimal transport solution. It is the error, that is, the upper bound of the allowable matching position error (typically). The unit is m, (where the optical trap spacing is...) It is the total number of optical traps, and This represents the natural logarithm of the total number of optical traps.

[0202] Furthermore, those skilled in the art will understand that, given the maximum feasible value under the error constraint, the following can be obtained: This is a preferred embodiment.

[0203] Here, the entropy regularization parameter The smaller the value, the closer the transmission scheme is to the exact optimal discrete allocation, but the slower the convergence speed; while The larger the value, the smoother the solution, but the further it deviates from the optimal solution. Determining the entropy regularization parameter based on the error ensures that the matching error introduced by regularization does not exceed the upper bound of the allowable matching position error. In practice, it can also be... Values 1%–5% of the order of magnitude.

[0204] Based on each element of the crosstalk weighted cost matrix, i.e. Construct the corresponding matrix elements of the kernel function matrix under the entropy regularization framework, i.e. Represented as:

[0205] here, This indicates that, within the entropy regularization framework, the source optical trap... Transmitted to the target location The prior probability weights. Pairing kernel function values ​​are large for low-cost (low-crosstalk transport paths), while pairing kernel function values ​​decay exponentially to near zero for high-cost (high-crosstalk transport paths), thus naturally suppressing the allocation of high-crosstalk directions.

[0206] In the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application, determining the entropy regularization parameter based on the error includes: determining that the entropy regularization parameter is less than or equal to the quotient obtained by dividing the error, which serves as the upper bound of the allowable matching position error, by the natural logarithm of the total number of optical traps.

[0207] In the above-described atomic array rearrangement path planning method using spatial light modulator tensors, the upper bound of the allowable matching position error is 0.01 times the optical trap spacing.

[0208] In the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application, constructing a kernel function matrix under the entropy regularization framework based on the crosstalk weighted cost matrix includes: dividing the matrix elements of the crosstalk weighted cost matrix by the entropy regularization parameter and taking the negative number as the exponent of the natural constant to obtain the corresponding matrix elements of the kernel function matrix.

[0209] Step S140: Based on the kernel function matrix, the source distribution and the target distribution are subjected to an entropy regularization optimal transmission solution algorithm to obtain the optimal transmission coupling matrix.

[0210] In this embodiment of the application, the entropy-regularized optimal transport solution algorithm may be, for example, log-domainSinkhorn (a numerically stable version). (Superior to the standard Sinkhorn algorithm when extremely small), Greenkhorn algorithm (a greedy variant of row and column scaling), Each step, not The proposed algorithms include: Network Simplex (exact optimal transport, no regularization), Earth Mover's Distance (EMD) general solution, and Dykstra's algorithm (equivalent to Sinkhorn but with more stable convergence). The following explanation will use the Sinkhorn iteration as an example to illustrate the entropy-regularized optimal transport solution algorithm.

[0211] Here, the kernel function matrix Represented as Source distribution Target distribution The optimal transmission coupling matrix is ​​obtained by solving the entropy-regularized optimal transmission problem, and is expressed as:

[0212] in It is the entropy regularization Wasserstein-2 squared distance under the spatial light modulator tensor metric. It is a transmission coupling matrix, and its matrix elements are... Indicates from the source Transmit to target The transmission quality.

[0213]

[0214] It satisfies both action and behavior. , column sum is The coupled set. And The divergence term, as a regularization term, is expressed as:

[0215] Here, solving the entropy-regularized optimal transmission problem yields the atom source-target allocation scheme with the minimum total transmission cost under the crosstalk metric of the spatial light modulator tensor, where entropy regularization guarantees the uniqueness of the solution and the Sinkhorn product structure.

[0216] In the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application, the entropy-regularized optimal transmission solution algorithm is executed based on the kernel function matrix to obtain the optimal transmission coupling matrix. This includes: based on the kernel function matrix, the source distribution, the target distribution, and the transmission coupling matrix representing the transmission quality from the source to the target, the entropy-regularized optimal transmission solution algorithm is used to solve the entropy-regularized optimal transmission problem to obtain the optimal transmission coupling matrix.

[0217] Figure 3 The illustration shows a schematic flowchart of the Sinkhorn process in an atomic array rearrangement path planning method using spatial light modulator tensors according to an embodiment of this application. Figure 3 As shown, the Sinkhorn process in the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application includes the following steps.

[0218] Step S141: Initialize the dual row scaling initial vector and column scaling initial vector to all-one vectors. Here, the dual row scaling initial vector... and column scaling initial vector They are represented as follows:

[0219] They are all all-one vectors.

[0220] Step S142: Starting from the row scaling initial vector and the column scaling initial vector, perform Sinkhorn alternating scaling iterations through the kernel function matrix. Wherein, for the... In each iteration, the row scaling is updated as follows:

[0221] Column scaling updated to: , They are the first The row scaling vector and column scaling vector of each iteration correspond to the Lagrange multipliers of the source and target margin constraints, respectively. It is a numerical underflow protection constant to prevent division by zero errors, and the division is an element-by-element Hadamard division.

[0222] Here, the row scaling vector The update makes the row margin of the current transmission plan exactly equal to the source distribution. Column scaling vector The update makes the column margin exactly equal to the target distribution. Alternating execution causes the two constraints to be satisfied synchronously.

[0223] Step S143: In response to the marginal residuals of the iteration satisfying a predetermined condition or reaching the maximum number of iterations, stop the iteration to obtain the optimal row scaling vector and the optimal column scaling vector.

[0224] Here, the marginal residual of the iteration satisfying the predetermined condition can be expressed as:

[0225] in It is the convergence tolerance (e.g., the value). ).

[0226] Step S144: Construct the optimal transmission coupling matrix using the optimal row scaling vector, the optimal column scaling vector, and the kernel function matrix.

[0227] Here, the matrix elements of the optimal transmission coupling matrix The corresponding vector value is represented by the optimal row scaling vector. The corresponding matrix elements of the kernel function matrix and the corresponding vector values ​​of the optimal column scaling vector The product of these can be expressed as:

[0228] Its equivalent matrix is ​​represented as:

[0229] here, Indicates source optical trap Assigned to the target location The optimal transmission quality, the optimal transmission coupling matrix In the spatial light modulator tensor The probabilistic representation of the atom-target allocation scheme that minimizes the total transmission cost with entropy regularization under anisotropic crosstalk constraints. Its product structure. Strictly guaranteed by the first-order optimality conditions of KKT (Karush-Kuhn-Tucker), that is, for The first-order optimality condition of KKT is expressed as:

[0230] Solution:

[0231] make , This yields the product structure. .

[0232] In the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application, obtaining the optimal transmission coupling matrix by solving the entropy-regularized optimal transmission problem using an entropy-regularized optimal transmission solution algorithm includes: initializing the dual row scaling initial vector and column scaling initial vector as all-one vectors; performing Sinkhorn alternating scaling iterations starting from the row scaling initial vector and the column scaling initial vector through the kernel function matrix; stopping the iteration in response to the marginal residuals of the iteration satisfying a predetermined condition or reaching the maximum number of iterations to obtain the optimal row scaling vector and the optimal column scaling vector; and constructing the optimal transmission coupling matrix using the optimal row scaling vector, the optimal column scaling vector, and the kernel function matrix.

[0233] In the above-described atomic array rearrangement path planning method using spatial light modulator tensors, performing Sinkhorn alternating scaling iterations from the initial row scaling vector and the initial column scaling vector through the kernel function matrix includes: multiplying the kernel function matrix with the column scaling vector of the previous iteration and adding a numerical underflow protection constant to obtain a first sum vector, and then dividing the source distribution by the first sum vector to obtain the row scaling vector of the current iteration; and multiplying the transpose of the kernel function matrix with the row scaling vector of the current iteration and adding a numerical underflow protection constant to obtain a second sum vector, and then dividing the target distribution by the second sum vector to obtain the column scaling vector of the current iteration.

[0234] In the above-described atomic array rearrangement path planning method using spatial light modulator tensors, constructing the optimal transmission coupling matrix using the optimal row scaling vector, the optimal column scaling vector, and the kernel function matrix includes: determining the matrix elements of the optimal transmission coupling matrix as vector elements of the optimal row scaling vector, and the product structure of the vector elements of the optimal column scaling vector and the matrix elements of the kernel function matrix, based on the KKT first-order optimality condition of the matrix elements of the transmission coupling matrix.

[0235] Step S150 involves determining the optimal allocation map and optimal transmission path from the optimal transmission coupling matrix by constructing a time-optimal atom transmission problem under the spatial optical modulator crosstalk metric. Here, the optimal transmission coupling matrix is ​​first used as the basis for... The optimal allocation mapping is determined by performing a greedy matching algorithm, i.e.:

[0236] Among them, the optimal allocation mapping Indicate each target location The corresponding source location index, Indicates in The Select the row index corresponding to the largest element in the column. This allows us to determine the optimal transmission coupling matrix, which serves as the soft probability allocation matrix. Extract the unique source location with the highest probability for each target location to obtain the spatial light modulator tensor. The optimal deterministic one-to-one allocation scheme under the crosstalk metric.

[0237] Among them, when Multiple columns share the same When multiple targets prefer the same source, considering that the entropy regularization of Sinkhorn guarantees... The elements are strictly positive and dispersed in practice. Very few conflicts (probability) If a conflict occurs, it can be resolved, for example, by using an augmenting path.

[0238] Then, based on the time-optimal atom transport problem under the crosstalk metric of spatial light modulators, the analytical structure for determining the optimal velocity field is derived. Figure 4 The illustration shows a schematic flowchart of the process of deriving the optimal velocity field analytical structure in the atomic array rearrangement path planning method using spatial light modulator tensors according to an embodiment of this application. Figure 4 As shown, the process of deriving the optimal velocity field analytical structure in the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application includes the following steps.

[0239] Step S151: Construct the time-optimal atom transport problem under the crosstalk metric of spatial light modulators and introduce momentum density for convexity. In this embodiment, the SCMTAT (SLM-Crosstalk-Metric Time-optimal Atom Transport) problem is constructed based on SLM-OT (SLM-Optimal Transport). Each optical trap is considered as a continuous density field. The SCMTAT problem is then expressed as:

[0240] in It is the atomic probability density distribution, i.e., the density field, characterizing time. In position Atom probability distribution density (scalar field) at , Satisfy normalization ); It is a velocity field, representing time. In position The velocity of the atoms at that location (two-dimensional vector field, unit) m / ms); It is the optimal transport time, which is the shortest time to complete the transport of all atoms from the source position to the target position (decision variable, unit: ms).

[0241] It is a continuity equation (mass conservation constraint) that describes the time evolution of the density field under velocity field transport. It is the initial probability density distribution ( (Time), from the source location set Sure; It is the terminal density distribution ( (Time), from the set of target locations Sure; yes The square norm of the velocity is used to replace the Euclidean norm.

[0242] That is, by finding the density field Velocity field and minimum transmission time This allows the atoms to be moved from their initial distribution. Transport to target distribution The transmission time is the shortest, while simultaneously satisfying the continuity equation (mass conservation) and the spatial light modulator tensor. The metric constraints. The difference between SLM-OT and Euclidean OT is that SLM-OT uses... (Isotropic) replaced with (Anisotropic, including crosstalk coupling), it can be seen that It degenerates to the standard case.

[0243] Furthermore, momentum density variables are introduced. Then the continuity equation becomes That is, about The linear form of the kinetic energy density is given by the perspective function:

[0244] in It is a perspective function, representing a perspective function. , The crosstalk weighted kinetic energy density of the variables; It is momentum density, which is the product of density and velocity (two-dimensional vector field, unit). m / ms); It is momentum in Square norm under metric (scalar, ).

[0245] The perspective function has joint convexity, that is, if ,but about Joint convex, and in It is strictly convex. That is, through variable substitution. The original non-convex problem (because) middle (Containing ratios) transformed into , This is a joint convex problem with variables. The convexity of the perspective function guarantees the existence of a globally unique optimal solution, eliminating the possibility of multiple local extrema, making the first-order conditions obtained in the subsequent variational derivation not only necessary but also sufficient.

[0246] The proof process is as follows: and ,set up Then we have:

[0247] in These are convex combination coefficients. , and There are two arbitrary feasible points. The first inequality uses... The parallelogram inequality, and the second inequality uses The convexity inequality guarantees that the cost of linear interpolation between any two feasible transmission schemes does not exceed the linear interpolation of the costs of the two schemes, and the variational first-order condition is the global optimum. Furthermore, the joint convexity of the perspective function is used to guarantee that the subsequent variational problem has a globally unique optimum (without local extremum traps).

[0248] Step S152: Lagrange multipliers, acting as dual potential functions, are introduced to construct the Lagrange functional. Here, the Lagrange multipliers... Used to relax the constraints of the continuity equation Lagrange multipliers Its physical role is that of a potential function, which imposes a penalty on the degree of violation of the continuity equation in the space-time domain.

[0249] The objective functional (the integral of the kinetic cost in the space-time domain) is combined with the constraint Lagrange multiplier penalty term, and integration by parts is performed. The derivative is transferred to and Above, construct the Lagrange functional, expressed as:

[0250] in, It is a Lagrange functional (the objective function of the saddle point problem), regarding Take the smallest, about Take the maximum value. The first term of the integrand. For crosstalk-weighted kinetic energy density, the second and third terms This represents the dual penalty contribution to the continuity equation constraints. Furthermore, the boundary terms are the boundary contributions generated by integration by parts, determined by the initial conditions. and terminal conditions Determine (known quantities, not involved in variational calculus). Specifically, terminal conditions. In It is the target location set { } as the target density for support The Kantorovich potential can be obtained from the optimal column scaling vector output by the Sinkhorn algorithm. Logarithmic Interpolation is obtained.

[0251] Here, the Lagrange functional transforms the constrained optimization into an unconstrained saddle point problem. The introduction of this allows the continuity equation constraints to be internalized into the objective function, i.e., when When it is exactly the optimal function, for about The unconstrained minimum is equivalent to the solution of the original constrained problem, which is the basis for subsequent work. and The mathematical basis for taking the variational form.

[0252] Step S153: Based on the Lagrange functional, take the variational equation of the momentum density to derive the optimal velocity field analytical structure. That is, based on the aforementioned Lagrange functional, perform variational analysis on the... Take the first-order variation and set it to zero (as a necessary condition for first-order optimality), i.e. ,get:

[0253] in It is a Lagrange functional Regarding momentum density The functional derivative of represents the force balance condition for optimal transmission, that is, at the optimal solution, the kinetic energy gradient ( This can be understood as "inertia" and force (which can be understood as "inertia" and force). (i.e., "driving force") precise balance.

[0254] Solve the above equation algebraically by multiplying both sides on the left. The optimal momentum is obtained:

[0255] This indicates that the optimal momentum is along The direction is not along the gradient direction itself, but rather the direction after rotation and stretching of the inverse metric tensor, reflecting that in anisotropic media, the force and motion directions are not necessarily parallel.

[0256] Depend on Considering the physical direction of the optimal velocity (movement in the direction of decreasing potential function, taking the negative sign), the optimal velocity field is obtained:

[0257] That is, in In the crosstalk metric space, the optimal direction of motion is not the direction of the negative gradient of the potential function (the Euclidean steepest descent direction), but rather the direction of the negative gradient along the path of the potential function. The direction after linear transformation, i.e., the Riemannian manifold The Riemann gradient direction. This allows atoms to automatically avoid directions of high crosstalk and move along curved geodesics instead of straight lines.

[0258] In the atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application, determining the optimal allocation mapping and optimal transmission path from the optimal transmission coupling matrix by constructing a time-optimal atomic transmission problem under spatial light modulator crosstalk metric includes: deriving the optimal velocity field analytical structure by constructing a time-optimal atomic transmission problem under spatial light modulator crosstalk metric based on optimal transmission of spatial light modulator.

[0259] In the aforementioned method for reordering the atomic array path planning using spatial light modulator tensors, deriving the optimal velocity field analytical structure by constructing a time-optimal atomic transport problem under a spatial light modulator crosstalk metric based on optimal spatial light modulator transport includes: constructing a time-optimal atomic transport problem under a spatial light modulator crosstalk metric and introducing momentum density for convexity; introducing Lagrange multipliers as dual potential functions to construct a Lagrange functional; and taking variational steps on the momentum density based on the Lagrange functional to derive the optimal velocity field analytical structure.

[0260] Furthermore, the HJB equation can be derived and its well-posedness confirmed by taking variational measures of density based on the Lagrange functional.

[0261] Here, the above Lagrange functional pair Take the first-order variation and set it to zero, i.e. Substitute ,get:

[0262] This established a precise relationship between the kinetic energy density at the optimal solution and the time evolution of the potential function, namely, the kinetic energy per unit mass is exactly equal to the negative of the time decay rate of the potential function.

[0263] Substitute the optimal momentum Then we have:

[0264] because The cancellation of factors indicates that the HJB equation is completely determined by the potential function. Self-determined, with density field Decoupling. This allows for independent solutions. (First solve HJB), then from and the recovery of the continuity equation .

[0265] Potential function The equations that satisfy the backward HJB (Hamilton-Jacobi-Bellman) equation are:

[0266] In other words, the HJB equation describes the optimal cost (potential function). The backward propagation pattern over time. Propagating from the terminal condition back to the initial moment, the decay rate at each spatiotemporal point is equal to the Hamiltonian function value. The only difference between this equation and the classical Euclidean HJB is that it uses the isotropic norm. Replace with anisotropic norm All physical information for crosstalk measurement is obtained through The code is incorporated into the equation.

[0267] In summary, SLM-OT can be represented as: set up exist The condition is positive definite and Lipschitz continuous, the SCMTAT has a smooth solution, and the optimal velocity field satisfies the following conditions: ,in It is the unique viscous solution to the following backward HJB equation (Crandall-Lions significance):

[0268] in:

[0269] Degradation verification: Let ,but The HJB equation degenerates into:

[0270] The Hamilton-Jacobi equation in Benamou-Brenier (2000) was accurately restored, verifying the correctness of the degradation path.

[0271] And by the Crandall-Lions (1992) theorem, in Under the condition of positive definite Lipschitz continuity, the HJB equations in There is a unique viscous solution. This ensures the determinism of the optimal velocity field, meaning that for a given initial and final density distribution, the optimal transmission path is unique.

[0272] Therefore, SLM-OT proves through the above rigorous variational derivation that: Under this metric, the total cost of optimal transmission is exactly equal to the cost along the path. The kinetic energy of the geodesic integral, i.e. the crosstalk weighted cost matrix, is the exact correspondence of the continuous optimal control problem after discretization.

[0273] Thus, the optimal velocity field is determined from the optimal allocation map, thereby determining the optimal transmission path. That is, after obtaining the start-point coordinates, end-point coordinates, and optimal velocity field from the optimal allocation map, the HJB corresponding to the optimal velocity field can be obtained. Furthermore, based on the HJB equation, the optimal transmission path can be determined from the end-point coordinates via... By performing backward integration with respect to the spatial light modulator tensor, the optimal path for each step is determined, thereby determining the optimal transmission path and achieving path planning.

[0274] Here, the solution based on the HJB equation generally includes two typical methods: one is discretization using the finite difference / characteristic line method, and the other is based on the optimal transmission coupling matrix. After directly interpolating the velocity field, perform integration of the ordinary differential equation (ODE) (thus eliminating the need for explicit PDE solution). For example, in practice, one can... The source-target pair corresponding to the row with the highest weight is directly connected, so that... Perform Euler integration by directional stepping, with a step size of, for example, 1% of the light trap spacing, thus eliminating the need for an explicit PDE mesh.

[0275] Furthermore, the optimal path is obtained by backward integration from the endpoint coordinates using the tensor of the spatial light modulator. When the changes are minor or when high real-time requirements are needed, it can also be directly from Maximum probability pairing of line segments (without solving HJB, greedy approximation, Runge-Kutta 4th order integral and geodesic tracing) (along... The curve of the Riemannian manifold), but preferably still the HJB / geodetic / ODE integral, the engineering approximation is the straight line or spline path after the maximum probability pairing.

[0276] That is, in the atomic array rearrangement path planning method using the spatial light modulator tensor according to the embodiments of this application, determining the optimal allocation mapping and the optimal transmission path from the optimal transmission coupling matrix by constructing the time-optimal atomic transmission problem under the spatial light modulator crosstalk metric includes: obtaining the optimal transmission path from the endpoint coordinates by backward integration with respect to the spatial light modulator tensor based on the HJB equation corresponding to the optimal velocity field analytical structure.

[0277] Furthermore, for very large-scale arrays (e.g.) If the atomic array rearrangement needs to be decomposed into multiple time frames, with each frame planning a subset of transport tasks (multi-step serial rearrangement), then the atomic array rearrangement path planning method using spatial light modulator tensors according to the embodiments of this application iteratively applies the multi-step serial rearrangement scenario, that is, each step uses the density distribution output from the previous step as the basis for the next step. enter.

[0278] As mentioned above, the diffractive optical system of a spatial light modulator inherently exhibits direction-dependent optical crosstalk, which contradicts the isotropic assumption and leads to the following technical problems: I. Inconsistency between cost metric and physical cost: Spatial light modulators generate holographic phase maps using the WGS iterative algorithm. When a certain optical trap is displaced, this displacement propagates throughout the entire holographic phase map via the implicit function relationship of the WGS fixed point, thus affecting the amplitude and phase of all other optical traps. Furthermore, this crosstalk propagation exhibits strong anisotropy: the holographic perturbation caused by moving an optical trap along certain directions of the spatial light modulator pixel array (e.g., the diffraction order sub-dense direction) is much greater than that caused by moving it along other directions. If Euclidean distance is used... In return, this direction dependence is ignored. Since two transport paths of the same Euclidean length may cause drastically different crosstalk amplitudes in a real spatial optical modulator system, if the same cost is imposed on both, it means that the allocation scheme may assign atoms to target locations that need to be moved along the direction of high crosstalk.

[0279] II. Straight-line paths are not globally optimal: Under anisotropic crosstalk metric, the shortest path between two points is no longer a Euclidean straight line, but a curved geodesic (bypassing high crosstalk regions). If a straight-line path is used for transport, atoms will inevitably cross high crosstalk regions during movement, causing significant phase jumps in the holograms between frames, leading to the following chain of problems: the optical trap potential depth will experience undesirable fluctuations during transport, increasing the probability of atom escape (loss); the positions of adjacent optical traps will drift, affecting the fidelity of stationary atoms that are not involved in the transport; to suppress the instability caused by crosstalk, the transport speed needs to be reduced or the number of intermediate transition frames increased, resulting in a longer total rearrangement time.

[0280] Third, there is a lack of unified optimality guarantee for allocation and execution: If allocation planning and path execution are treated as two independent stages, and Euclidean optimal transport is used to calculate allocation first, and then straight path execution is used, then because the crosstalk constraints in path execution are not considered in the allocation stage, the two stages cannot be jointly optimized, and there is a lack of a unified framework to guarantee the global optimality from allocation to trajectory.

[0281] The atomic array rearrangement path planning method using spatial light modulator tensors according to embodiments of this application introduces spatial light modulator tensors to quantize and encode the anisotropic structure of optical crosstalk of spatial light modulators into the transmission cost matrix, and performs source-target position pairing and geodesic path planning based on this matrix, achieving the following technical effects: I. Precise Alignment of Cost Metric and Physical Crosstalk: The spatial light modulator tensor is rigorously derived from the Jacobian structure of the WGS algorithm. Therefore, the length of the optical trap displacement under the spatial light modulator tensor metric is exactly equal to the magnitude of the change in the hologram fixed point caused by this displacement. By substituting the diagonal self-coupled sub-blocks of this tensor into the cost matrix, the allocation scheme can automatically favor transmission paths along low crosstalk directions and reject transmission paths along high crosstalk directions when making source and target pairing decisions. Thus, compared to Euclidean cost allocation, the spatial light modulator tensor-weighted allocation significantly reduces the cumulative hologram perturbation during transport, reducing the risk of inter-frame phase jumps and atom loss.

[0282] II. Significantly Reduced Overall Rearrangement Time: Because pairing using the spatial light modulator tensor naturally avoids pairing in high-crosstalk directions, and the optimal velocity field given by SLM-OT performs transport along curved geodesics (bypassing high-crosstalk regions), atoms can move at higher speeds without exciting excessive crosstalk. For example, in the total number of optical traps... Crosstalk feature width optical trap spacing Under simulation conditions, experiments show that, compared to the standard Euclidean scheme, the atomic array rearrangement path planning method using spatial light modulator tensors according to the embodiments of this application can reduce the theoretical rearrangement time by about 19.4% by reducing crosstalk cost, improving pairing rationality, and shortening path planning. Furthermore, since the crosstalk weighted cost matrix is ​​substantially different from the Euclidean cost matrix, the source-target allocation scheme of about 10% to 20% of the atoms has changed compared to the standard Euclidean optimal transmission scheme, indicating that the method is not merely a fine-tuning of the existing allocation scheme.

[0283] III. No increase in algorithm complexity: The atomic array rearrangement path planning method using spatial light modulator tensors according to the embodiments of this application has the same algorithm structure as the standard Sinkhorn algorithm (alternating row and column scaling iteration), only the calculation method of the cost matrix is ​​different. Replace with The complexity of each iteration remains at O(n). (Single matrix-vector product), GPU parallelism is fully preserved, without the need to introduce additional optimizers or iterative layers.

[0284] IV. Pairing and trajectory execution are in the same Maintaining consistency under metrics: SLM-OT proves through rigorous variational derivation (Lagrange multipliers, HJB equations, uniqueness of viscous solutions) that the crosstalk weighted cost matrix used in this application is a legitimate discretization (non-heuristic approximation) of the continuous optimal control problem, and that the trajectory execution after pairing has a uniquely determined optimal velocity field (given by the viscous solution of the HJB equations). This forms a closed link with consistent optimality guarantee from pairing to trajectory execution.

[0285] According to an embodiment of this application, an electronic device is further provided, the electronic device including a processor coupled to a memory, the processor being configured to execute a computer program stored in the memory, such that the electronic device performs the atomic array rearrangement path planning method using spatial light modulator tensors according to an embodiment of this application as described above.

[0286] Figure 5 A schematic block diagram of an electronic device according to an embodiment of this application is shown. Figure 5 As shown, an electronic device 200 according to an embodiment of this application includes a coupled processor 210 and a memory 220.

[0287] According to an embodiment of this application, a computer program product is further provided, the computer program product comprising: computer program code, which, when the computer program code is run on a computer, causes the computer to execute the atomic array rearrangement path planning method using spatial light modulator tensors according to an embodiment of this application as described above.

[0288] According to an embodiment of this application, a computer-readable storage medium is further provided, the computer-readable medium storing program code that, when run on a computer, causes the computer to perform the atomic array rearrangement path planning method using spatial light modulator tensors according to an embodiment of this application as described above.

[0289] The terms “component,” “module,” “system,” etc., used in this specification are used to refer to computer-related entities, hardware, firmware, combinations of hardware and software, software, or software in execution. For example, a component can be, but is not limited to, a process running on a processor, a processor, an object, an executable file, an execution thread, a program, and / or a computer. As illustrated, applications running on computing devices and computing devices can both be components. One or more components may reside in a process and / or an execution thread, and components may be located on a single computer and / or distributed among two or more computers. Furthermore, these components can be executed from various computer-readable media on which various data structures are stored. Components can communicate, for example, via local and / or remote processes based on signals having one or more data packets (e.g., data from two components interacting with another component between a local system, a distributed system, and / or a network, such as the Internet interacting with other systems via signals).

[0290] Those skilled in the art will recognize that the various illustrative logical blocks and steps described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementations should not be considered beyond the scope of this application.

[0291] Those skilled in the art will understand that, for the sake of convenience and brevity, the specific working processes of the systems, devices, and units described above can be referred to the corresponding processes in the foregoing method embodiments, and will not be repeated here.

[0292] In the several embodiments provided in this application, it should be understood that the disclosed systems, apparatuses, and methods can be implemented in other ways. For example, the apparatus embodiments described above are merely illustrative; for instance, the division of units is only a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the coupling or direct coupling or communication connection shown or discussed may be through some interfaces; the indirect coupling or communication connection between apparatuses or units may be electrical, mechanical, or other forms.

[0293] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment according to actual needs.

[0294] In addition, the functional units in the various embodiments of this application can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit.

[0295] If the aforementioned functions are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, or a portion of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this application. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0296] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A method for rearranging atomic array paths using spatial light modulator tensors, characterized in that, include: Based on the image detection results of the atoms to be rearranged in the atomic array and the rearranged target, the source position set and the target position set are determined, and their discrete probability distributions are constructed to obtain the source distribution and the target distribution. A crosstalk weighted cost matrix is ​​constructed based on the displacement vector from the source position to the target position and the matrix function that quantizes the displacement cost of the optical trap itself in the spatial light modulator tensor. The entropy regularization parameters are determined based on the error, and the kernel function matrix under the entropy regularization framework is constructed based on the crosstalk weighted cost matrix. Based on the kernel function matrix, the source distribution and the target distribution are subjected to an entropy regularization optimal transmission solution algorithm to obtain the optimal transmission coupling matrix; as well as The optimal allocation mapping and optimal transmission path are determined from the optimal transmission coupling matrix by constructing a time-optimal atom transmission problem under the spatial optical modulator crosstalk metric.

2. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 1, characterized in that, Constructing the discrete probability distributions of the source location set and the target location set includes: The source distribution corresponding to the source location set and the target distribution corresponding to the target location set are constructed into a uniform discrete probability distribution.

3. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 1, characterized in that, Constructing the discrete probability distributions of the source location set and the target location set includes: The source distribution corresponding to the source location set and the target distribution corresponding to the target location set are constructed into a non-uniform discrete probability distribution. The non-uniform source distribution is determined based on the fluorescence image signal-to-noise ratio or occupancy probability weighted for each source optical trap.

4. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 1, characterized in that, The crosstalk weighted cost matrix is ​​constructed based on the displacement vector from the source position to the target position and the matrix function that quantizes the displacement cost of the optical trap itself in the spatial light modulator tensor, including: Multiply the row vector of the displacement vector from the source position to the target position by the matrix function representing the displacement cost of the quantized optical trap in the spatial light modulator tensor corresponding to the source position, and then multiply by the column vector of the displacement vector from the source position to the target position to obtain the anisotropic transmission cost; and, The anisotropic transmission costs corresponding to all pairs of the source location and the target location are combined into the crosstalk weighted cost matrix.

5. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 1, characterized in that, The entropy regularization parameters determined based on error include: The entropy regularization parameter is determined to be less than or equal to the quotient obtained by dividing the error, which serves as the upper bound of the allowable matching position error, by the natural logarithm of the total number of optical traps.

6. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 5, characterized in that, The upper bound of the allowable matching position error is 0.01 times the optical trap spacing.

7. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 1, characterized in that, The kernel function matrix constructed based on the crosstalk weighted cost matrix under the entropy regularization framework includes: The corresponding matrix elements of the kernel function matrix are obtained by dividing the matrix elements of the crosstalk weighted cost matrix by the entropy regularization parameter and taking the negative number as the exponent of the natural constant.

8. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 1, characterized in that, Based on the kernel function matrix, the source distribution and the target distribution are subjected to an entropy-regularized optimal transmission solution algorithm to obtain the optimal transmission coupling matrix, including: Based on the kernel function matrix, the source distribution, the target distribution, and the transmission coupling matrix representing the transmission quality from the source to the target, the optimal transmission coupling matrix is ​​obtained by solving the entropy-regularized optimal transmission problem using the entropy-regularized optimal transmission solution algorithm.

9. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 8, characterized in that, The optimal transmission coupling matrix is ​​obtained by solving the entropy-regularized optimal transmission problem using an entropy-regularized optimal transmission solution algorithm, which includes: Initialize the row scaling initial vector and column scaling initial vector of the dual as all-one vectors; Starting from the row scaling initial vector and the column scaling initial vector, perform Sinkhorn alternating scaling iterations through the kernel function matrix; The iteration stops when the marginal residuals satisfy a predetermined condition or the maximum number of iterations is reached to obtain the optimal row scaling vector and the optimal column scaling vector; and, The optimal transmission coupling matrix is ​​constructed by multiplying the optimal row scaling vector, the optimal column scaling vector, and the kernel function matrix.

10. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 9, characterized in that, Performing a Sinkhorn alternating scaling iteration from the kernel function matrix, starting with the row scaling initial vector and the column scaling initial vector, includes: The kernel function matrix is ​​multiplied by the column scaling vector of the previous iteration, and then an underflow protection constant is added to obtain the first sum vector. The source distribution is then divided by the first sum vector to obtain the row scaling vector of the current iteration. The transpose of the kernel function matrix is ​​multiplied by the row scaling vector of the current iteration round, and then a numerical underflow protection constant is added to obtain the second sum vector. The target distribution is then divided by the second sum vector to obtain the column scaling vector of the current iteration round.

11. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 10, characterized in that, Constructing the optimal transmission coupling matrix using the optimal row scaling vector, the optimal column scaling vector, and the kernel function matrix includes: The optimal transmission coupling matrix is ​​determined by the KKT first-order optimality condition of the matrix elements of the transmission coupling matrix. The matrix elements of the optimal row scaling vector are the vector elements of the optimal column scaling vector and the matrix elements of the kernel function matrix.

12. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 1, characterized in that, Determining the optimal allocation mapping and optimal transmission path from the optimal transmission coupling matrix by constructing a time-optimal atom transmission problem under a spatial optical modulator crosstalk metric includes: The optimal velocity field analytical structure is derived by solving the time-optimal atom transport problem under the spatial light modulator crosstalk metric based on the optimal transport of the spatial light modulator.

13. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 12, characterized in that, The optimal velocity field analytical structure is derived by solving the time-optimal atom transport problem under the spatial light modulator crosstalk metric based on optimal spatial light modulator transport, including: We construct the time-optimal atom transport problem under the crosstalk metric of spatial light modulators and introduce momentum density for convexity; We construct the Lagrange functional by introducing Lagrange multipliers as dual potential functions; and, The optimal velocity field analytical structure is derived by taking variational steps on the momentum density using the Lagrange functional.

14. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 13, characterized in that, Determining the optimal allocation mapping and optimal transmission path from the optimal transmission coupling matrix by constructing a time-optimal atom transmission problem under a spatial optical modulator crosstalk metric includes: Based on the HJB equation corresponding to the optimal velocity field analytical structure, the optimal transmission path is obtained by back-integrating from the endpoint coordinates with respect to the tensor of the spatial light modulator.

15. The atomic array rearrangement path planning method using spatial light modulator tensors as described in claim 1, characterized in that, The spatial light modulator tensor can optionally be generalized as a crosstalk metric tensor for optical field modulators in acousto-optic deflectors or a hybrid system of spatial light modulators and acousto-optic deflectors.

16. An electronic device, characterized in that, The device includes a processor coupled to a memory, the processor being configured to execute a computer program stored in the memory, such that the electronic device performs the atomic array rearrangement path planning method using a spatial light modulator tensor as described in any one of claims 1 to 15.

17. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed, implements the atomic array rearrangement path planning method using spatial light modulator tensors as described in any one of claims 1 to 15.

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