Apparatus and method for laser beam shaping using a spatial light modulator

Abstract

The present disclosure relates to a method for characterizing an optical system. In some aspects, such a method comprises modulating an input laser beam with a spatial light modulator, SLM, preferably essentially placed in a Fourier plane of the optical system with respect to a target region of the optical system by applying a modulation mask to the SLM. The mask is configured for splitting a portion of the input laser beam into a first part and a second part of the modulated laser beam, wherein the second part of the modulated laser beam diverges with respect to the first part of the modulated laser beam. The divergence angle of the modulated laser beam is small enough such that the second part at least partially interferes with the first part in the target region, thereby generating a self-interference pattern for the modulated laser beam. At least a part of the generated self-interference pattern of the modulated laser beam is measured in the target region of the optical system and the optical system is characterized based on the measured interference pattern in the target region. The present disclosure also relates to methods and apparatuses for laser beam shaping and quantum computing.

Description

September 24, 2024Max-Planck-Gesellschaft zur FörderungM175165WO ANE / Bmnder Wissenschaften e.V. & Ludwig-Maximilians-Universität München APPARATUS AND METHOD FOR LASER BEAM SHAPING USING ASPATIAL LIGHT MODULATOR FIELD OF INVENTION

[0001] The present disclosure relates to methods, apparatuses and computerprograms for characterizing an optical system using a spatial light modulator and using the characterization to shape a laser beam fast and accurately. Aspects of the presentdisclosure may benefit laser-based techniques spanning a wide range of applicationscenarios, including trapping or manipulating of atoms or ions, e.g. for quantumcomputing), medicine, material processing and astrophysics.INTRODUCTION

[0002] Laser-based devices are central to a wide range of technologies includingquantum computing, quantum simulation, quantum communication, quantum metrology, lithography, material processing, medical treatment, and astrophysicalmeasurements. Essentially, such applications of laser-based devices typically requireoptics for guiding and shaping laser beams to implement a respective desired function.For example, realizing a neutral atom quantum computer, typically includes loading aplurality of neutral atoms into a plurality of optical trapping potentials, e.g. formed byoptical tweezer arrays or optical lattices or combinations thereof. In such an application scenario, shaping a laser beam to create a plurality of optical tweezer traps, mightconstitute a key step in building a quantum computing device capable of executingquantum algorithms using neutral atoms to implement qubits of a quantum register.Here, the quality of the optical tweezer traps (e.g., determined, inter alia, by the shapeand / or the homogeneity of the electromagnetic field trapping the neutral atoms) isimportant for preparing defect-free quantum registers and realizing high-fidelityquantum gates, e.g., by optically controlling internal states of the qubits. Further, suchapplications typically require changing the shape of the laser beam dynamically andarbitrarily, e.g., for initializing and maintaining the quantum register, as described in applicant’s own PCT / EP2024 / 053102, incorporated herein in its entirety.

[0003] For such a purpose, spatial light modulators, SLMs, have been employed,as they allow to modulate an input laser beam rapidly and accurately in aprogrammable manner, e.g., to generate an array of optical tweezer traps. Examples forSLMs include digital micromirror devices, liquid crystal modulators and similar devicesknown in the art. Such SLMs are typically placed in an image plane or in a Fourierplane of an optical system configured to generate an optical trap array or similardesired electric field distributions in a target region in order to spatially modulate theelectric field of the laser beam in a controllable manner, such that, for instance, themodulated laser beam acquires a desired shape in the target region (e.g., viaholographic interference) of the optical system. As the laser beam propagates throughthe optical system, it is subject to imperfections of the optical components, causingoptical aberrations which are difficult to know a priori. As a consequence, accuratelaser beam shaping typically requires obtaining knowledge about the actual opticalsystem and its aberrations to ensure that the obtained electric field distribution in thetarget area matches the desired distribution as accurately as possible.

[0004] In this context, for example, P. Zupancic et al. Ultra-precise holographicbeam shaping for microscopic quantum control, Optics Express 24, 13881-13893,present a method for characterizing an optical system and shaping a laser beam byplacing a digital micromirror device in the Fourier plane of an optical system. In thisprotocol, the input laser beam and the SLM are used to generate two parallel sub-beams emerging from two distinct locations, e.g., patches, within the aperture of theSLM, with a controllable phase shift between the two sub-beams. Aberrations of theoptical system corresponding to the active pair of patches in the SLM aperture are thenmeasured by detecting a phase-dependent interference pattern as a function of thephase shift. Repeating this measurement many times for essentially all patches in theSLM aperture, can then be used to characterize aberrations of the optical systemaffecting the electric field distribution in a target region. Such a characterization of theoptical system can then be taken into account when modulating the SLM in a way suchthat a specific electric field distribution is generated in the target region of the opticalsystem, therefore compensating, at least in part, some of the optical aberrations.SUMMARY

[0005] For many laser beam shaping applications, it is important that the beamshaping protocol is accurate and fast and can be executed at high speed, possiblyallowing to compensate for the aberrations in the optical system essentially in real-time. For instance, in the context of quantum computing with neutral atoms, such aprotocol would allow to realize dynamically configurable tweezer traps for implementing e.g., inter alia, resorting algorithms or quantum gate operations.

[0006] As known in the art, in an optical system, the electric fields of the laserbeam in the Fourier plane and an image plane in a target region can be related througha Fourier Transform FT:(1 )where (^, ^) and (^, ^) are the Cartesian coordinates in the image plane and the Fourierplane, respectively, ^(^, ^) and ^(^, ^) are the complex electric fields of the laser beamin the image plane and the Fourier plane, respectively. Further, ^ denotes the effectivefocal length of the optical setup which essentially focusses the modulated laser beamonto the image plane, and ^ is the wavelength of the laser emitting at least partiallycoherent light. FT designates the (spatial) Fourier Transform over the coordinates(^, ^) evaluated at the coordinates (^, ^). In the following, Eq. 1 may also be referred toas the forward propagator. A SLM placed in the Fourier plane causes a modulation ofthe laser beam’s wavefront described by ^^ ^(^,^), where the phase ^(^, ^) correspondsto a modulation mask applied by the SLM. Generally, the laser beam may be subject tooptical aberrations caused by, for instance, imperfections in the optical components (including the SLM itself) which are part of the optical system and the laser beam’spropagation path. Their impact is captured in the above formula by the complex fieldwhere ^(^, ^) denotes a spatially dependent attenuation of the laser beam and ^(^, ^)is a spatially dependent distortion of the laser beam’s wavefront. A characterization ofthe optical system may then comprise obtaining ^(^, ^) and ^(^, ^) in order to, forinstance, accurately compute the electric field ^(^, ^) in the target region and to makepredictions about the shape of the modulated laser beam in the image plane.

[0007] The complex field ^(^, ^), describing both optical aberrations and theattenuation of the beam intensity according to Eq. 2, may be subject to short- and / orlong-term drifts, e.g., caused by temperature dependent mechanical stress, such thatthe optical system needs to be re-characterized at regular intervals. Thus, the overallefficiency of a device, the functionality and efficiency of which relies on laser beamshaping, may be limited by how fast the optical characterization can be performed andhow fast the SLM can be controlled in order to apply a modulation mask to the laserbeam which leads to the desired laser beam shape in the target region of the opticalsystem. Generally, controlling the SLM may comprise determining the modulationmask required to obtain a certain laser beam shape in the target region of the opticalsystem.

[0008] For example, in the context of quantum computing with neutral atoms inoptical tweezer traps, laser beam shaping using an SLM may be employed to generate aplurality of optical tweezer traps. The plurality of optical tweezer traps may be used totrap the neutral atoms implementing the qubits which may be part of a quantumregister. Further, laser beam shaping may be employed to optically address an arbitrarysubset of the plurality of trapped particles, comprising, for instance, controlling theirinternal and / or motional state. Fast updates of the laser beam shape may be used t0shift the position of an arbitrary subset of the plurality of optical tweezer traps andmove the trapped particles therein, e.g., to repopulate defects in the quantum register.These techniques may be used for resorting an arbitrary subset of the plurality oftrapped particles and initializing the quantum register with high-fidelity and / or forimplementing quantum gates. Further, if the target region comprises a plane in whichthe neutral atoms are trapped, obtaining information about the laser beam at the position of the atoms using the atoms is difficult and costly, such that it is desirable tokeep the number of measurements needed for (re-)characterization of the opticalsystem as small as possible. Thus, there is a perpetual need for fast and accurate beamshaping and / or characterization of optical systems while overcoming at least some ofthe discussed limitations.

[0009] To address such and similar problems, the present disclosure provides, ina first aspect, a method for characterizing an optical system, comprising modulating aninput laser beam with a SLM (e.g. a liquid crystal modulator or a digital mirror device,etc.) preferably essentially placed in a Fourier plane of the optical system with respectto a target region of the optical system by applying a modulation mask to the SLM forsplitting at least a portion of the input laser beam into a first part and a second part ofthe modulated laser beam. The modulation mask is applied such that the second part ofthe modulated laser beam diverges with respect to the first part of the modulated laserbeam, wherein a divergence angle of the modulated laser beam is small enough such that the second part of the modulated laser beam at least partially interferes with the first part of the modulated laser beam in the target region, thereby generating a self- interference pattern for the modulated laser beam. Next, at least a part of the generated self-interference pattern of the modulated laser beam in the target region of the opticalsystem is measured and the optical system is characterized based on the measured self-interference pattern in the target region (for further details, see the description of Fig. 1to Fig. 8 below).

[0010] As discussed below, the purpose of the optical system may be to generatea plurality of optical tweezer traps (see, for example, Fig. 2) for a quantum computingdevice, by shaping the input laser beam, such that the shape of the laser beam in thetarget region, where the neutral atoms are located, corresponds to a plurality (e.g. aregular grid) of points. For this purpose, the optical system may comprise a SLM placedessentially in the Fourier plane of the optical system to modulate the input beam usingan applied modulation mask. “Essentially in the Fourier plane” may comprise positionswhich are not exactly in the Fourier plane, but possibly somewhere between the lasersource and a focusing optics (see, for example, Fig. 1). For example, for a Gaussianbeam profile, the SLM may be placed within the Rayleigh length of the Gaussian beamwith respect to the Fourier plane. The SLM may be followed by a lens or a combinationof lenses forming an objective and / or imaging setup which focusses the modulatedlaser beam onto a region where particles are to be trapped inside a vacuum chamber.The particles may be atoms, for example, alkali-metal atoms (auch as Rb or Cs),alkaline-earth-metal atoms (such as Sr) or lanthanide atoms (such as Yb). The particlesmay also be ions or molecules. The system may include additional optical componentssuch as lenses, mirrors or other reflective surfaces as well as transmittable volumes ofvarious materials, shapes and optical properties. The input laser beam might be of anyoptical wavelength in the ultraviolet, visible or near-infrared range compatible with thespecifications of at least some of the optical elements in the path (e.g., including theSLM), and the internal level structure of the trapped atoms. The characterization of theoptical system may include the measurement of aberrations and the attenuations in thebeam profile, induced by imperfections in the optical components of the optical system.

[0011] In a typical implementation, the splitting of the input laser beam can beachieved by applying two different partial modulation masks in two differentsubregions of the SLM (see Fig. 3 for a variety of examples). As a consequence, at leastsome part of the input laser beam may be modulated according to a first partial modulation mask(^, ^) in the first region of the SLM and at least another part of thelaser beam may be modulated according to a partial modulation mask ^ ^^ (^, ^) in thesecond region of the SLM, where ^(^, ^) = ^^(^, ^) + ^^(^, ^). The two subregions ofthe SLM may correspond to a periodic pattern of alternating stripes, where everysecond stripe belongs to the first subregion and every other stripe belongs to the second subregion.

[0012] The two subregions together may cover a significant part of the usableSLM area, typically above 90% or close to 100% of the usable aperture of the SLM, suchthat essentially the entire usable SLM area is used to modulate the laser beam. Bothpartial modulation masks may be chosen, such that the amount of phase by which the input laser beam is shifted, depends spatially on the position inside the active area of the SLM, e.g. linearly, or quadratically. The spatial phase variation of the first partial modulation mask and the spatial phase variation of the second partial modulationmask may differ from each other, leading to, as known to the skilled expert, a non-zerodivergence angle between the two parts of the laser beam after the SLM(see, forexample, Fig. 1). In addition to the amount of angular deflection, the two parts of themodulated laser beam after the SLM may also differ by a phase shift Δ^ imposed by thecorresponding partial modulation masks in the two subregions of the SLM. The pair ofdiverging laser beam parts may then be turned into a pair of parallel propagating laserbeam parts, displaced by a beam displacement vector of length ^′, which at leastpartially interfere in the target region of the optical system (see, for example, Fig.1).This may be achieved, for example, by a lens or a combination of optical elements, suchas an objective, etc.

[0013] The self-interference pattern, i.e. the modulus squared of the electric fieldof the laser beam |^(^, ^)|^ in the target region of the optical system, can then bemeasured using a spatially sensitive photodetector such as a camera. In some implementations, the target region might correspond to the vacuum region in which neutral atoms or ions are trapped. In that case, a placement of the camera sensor in thetarget region may be impossible and / or too difficult, and the neutral atoms themselvesmay be used for measuring the spatially varying self-interference pattern. This may beachieved by measuring one or an arbitrary combination of the following observables:fluorescence imaging of the plurality of trapped particles, Rabi oscillations between anarbitrary pair of states |1> and |2>, performing Ramsey spectroscopy on an arbitrarypair of states |1> and |2> (for details see, for example, Fig. 13), light shifts affecting thetransition between an arbitrary pair of states |1> and |2> which are at least partiallyinduced by the beam shaped light modulated by the SLM. In essence, the exemplarymethods listed above can allow to use the plurality of trapped particles for measuringthe spatially varying light intensity in the target region, such that a camera is notneeded in the target region of the optical system.

[0014] In a possible implementation of Ramsey spectroscopy (see Fig. 13), eachparticle in the plurality of particles trapped in the target region of the optical system may be prepared in a quantum superposition |1> + |2> of a first state |1> and a secondstate |2> by a π / 2 pulse. Following such a π / 2 pulse, the plurality of particles may beexposed to detuned addressing light, which may be the laser beam modulated by the SLM. During such an exposure each particle may pick up a different quantum phase, which depends on the intensity of the detuned addressing light at its position. A secondπ / 2 pulse may then map the quantum phase of the superposition into the quantummechanical ^^-basis belonging to|1> and |2>. Repeated measurements of the quantum states |1> and |2> may then reveal the intensity of the detuned addressing light at the respective positions of the trapped particles.

[0015] In a possible implementation of Rabi oscillations, each particle in theplurality of particles trapped in the target region of the optical system may be prepared in a qubit state |1>. Next, a laser pulse of controllable duration, resonantly coupling tothe transition between the qubit states |1> and |2>, may be applied to the plurality of trapped particles. The measurement of Rabi oscillations may then comprise measuring the probability of finding each qubit in the qubit state |1> and / or |2> for various pulse durations. The presence of a self-interference pattern may lead to local variations of the optical potential and affect the amplitude and the frequency of the detected Rabi oscillations for individual trapped particles. The local intensity of the self-interference pattern at the position of each particle may then be determined based on the shape of the Rabi oscillation detected for the trapped particle.

[0016] In some implementations, the self-interference pattern in the targetregion of the optical system can be measured for a plurality of different SLM maskscorresponding to a plurality of different phase shifts Δ^ between the two parts of theinterfering laser beam (see, for example, Fig. 4). For such a purpose, the phase of onlyone of the two partial modulation masks may be altered between subsequentmeasurements. Alternatively, the phases of both partial modulation masks may be altered between the measurements by different amounts. The plurality ofmeasurements may allow to observe the spatial profile of the self-interference patternas a function of the phase shift Δ^ between the interfering parts of the laser beam and / or to derive the characterization of the optical system (see, for example, Fig. 5), asdescribed below.

[0017] The measured interference pattern may only contain information aboutthe wavefront in a direction of a beam displacement induced by the phase gradient inthe respective subregion of the SLM (for example, the x-axis if the two interfering partsof the laser beam are displaced along that axis). Thus, the measured self-interferencepattern does not contain information about the shape of the wavefront along theorthogonal direction (e.g., the y-axis, following the example above). Thus, there may beat least two measurement sequences involving two linearly independent displacements.In some implementations, the displacement vectors may be chosen orthogonal, e.g.(^^ , 0) and (0, ^′), where ^^ and ^′ denote the displacement in the x- and the y-direction,respectively. Determining the displacements ^’ and ^’ may comprise applying each ofthe two partial modulation masks to the entire SLM area (rather than to just the first orthe second region, respectively) and measuring the position of the modulated beam inthe target region. Then, the displacements ^’ and ^’ can be obtained from comparingthe two measured beam positions.

[0018] Such a plurality of self-interference pattern measurements may then beused to reconstruct the wavefront and the intensity of the laser beam in the targetregion, i.e. obtaining arg (^(^, ^)). Such a reconstruction may comprise fitting themeasured intensity of the self-interference pattern at each detector point (^, ^) in thetarget region as a function of the phase shift Δ^ with a cosine-shaped function (see, forexample, Fig. 5) and extracting the phase of the cosine Δχ^′,^′(^, ^). In order for thefitting procedure to be successful, at least two, preferably at least three measurementswith different phase shifts Δ^^ , Δ^^ , Δ^^,… may be necessary. In addition to suchmeasurement sequences, at least one additional measurement of the laser beamintensity |^(^, ^)|^ in the target region may be taken in absence of any SLMmodulation, possibly for determining ^(^, ^) and ^(^, ^) in Eq.~(2). The minimumnumber of measurements may be six.

[0019] The reconstruction of the wavefront arg (^(^, ^)) may also comprisecomputing the wavefront phase map based on the measured cosine phases Δχ^′,^′(^, ^).Such a computation is based on that each fitted cosine phase Δχ^′,^′(^, ^) belonging to apoint (^, ^) in the target region relates the phases at two points of the wavefront in thedetector plane separated by a displacement vector (^′, ^′), i.e. Δ^^^,^^(^, ^) = Φ(^, ^) −Φ(^ − ^^ , ^ − ^^), where Φ(^, ^) = arg^^(^, ^)^ is the phase of the wavefront at position(^, ^) in the target plane (for details see e.g. L. Huang et al., Comparison of two-dimensional integration methods for shape reconstruction from gradient data, OpticsAnd Lasers In Engineering 64, 1–11). The reconstructed wavefront may then allow toobtain the aberrated wavefront arg^^(^, ^)^ + ^(^, ^) and the attenuated beam profile|^(^, ^)| ^ by evaluating the inverted form of Eq. 1.

[0020] An advantage of this method compared to the state of the art is that theelectric field (including wavefront and amplitude) in the target region of the opticalsystem may be characterized at a possibly large number of points (^, ^) in the targetregion simultaneously, corresponding to the positions where the electric field of thelaser is detected. Since possibly only at least six measurements are necessary, themethod may serve as a faster way for fully characterizing the optical system asdescribed above. Such a characterization may allow to compensate for long-term driftsof the optical system related to unstable external conditions, e.g. mechanical stressinduced by temperature instabilities, without causing downtime.

[0021] A further aspect of the present disclosure is related to a method forshaping a laser beam by a SLM (e.g. a digital mirror device or a liquid crystalmodulator), essentially placed in a Fourier plane of an optical system with respect to atarget region (e.g. the image plane) of the optical system. The method comprisesobtaining a self-interferometry-based characterization of the optical system anddetermining, based on the self-interferometry-based characterization of the opticalsystem, a new modulation mask for the SLM. Further, the method comprises shapingthe laser beam according to the target shape by applying the determined mask to the SLM.

[0022] Once the optical system is characterized and the aberration- andattenuation-induced contribution, i.e. A(u, v), to the optical propagator in Eq. (1) isknown, it may be possible to employ a computer program to implement the opticalpropagator and simulate the optical system. In particular, the simulation may compriseevaluating Eq. (1) and / or predicting the electric field in the target region based on themodulation mask that is applied to the SLM. In turn, the simulation may be employedto find a modulation mask ^(^, ^) which, if applied to the SLM, produces a desiredelectric field ^^(^, ^) in the target region of the optical system. For such a purpose, thesimulation may be part of an inverse Fourier transform algorithm, IFTA, (see, forexample, Fig. 14), which may be used to determine the modulation mask ^(^, ^) SLMbelonging to an electric field yielding the desired intensity distribution |^^(^, ^)|^byiteratively minimizing the difference |^^(^, ^)|^- |^(^, ^)|^ between the |^^(^, ^)|^andthe detected or simulated laser beam shape |^(^, ^)|^ in the target region of the opticalsystem. Such a protocol amounts to solving a form of the so-called phase retrievalproblem, which seeks to reconstruct a complex valued signal from the magnitude of its Fourier transform.

[0023] In some implementations, the target beam shape may be updated at lowor high rate in order to dynamically change the shape of the laser beam in the target region of the optical system. Any incremental change of the target beam shape may require determining a new modulation mask for the SLM according to the methoddescribed above. For instance, if the laser beam shaping method is employed togenerate a plurality of tweezer traps for neutral atoms in the context of quantumcomputing, continuously updating the beam shape, possibly at a high rate, may allow toimplement smooth movements of an arbitrary subset of tweezer traps. Suchmovements may assist in preparing high-quality quantum registers by creating aregular array of tweezer traps each of which is occupied by a single neutral atom. Due tothe statistical nature of loading optical tweezers from a magneto-optical trap, not alloptical tweezer traps may be occupied by a single neutral atom after the loadingprocess. Yet, a high-quality initial state may be achieved by resorting the tweezer trapsin a way that only the occupied tweezer traps remain, possibly forming a fully occupiedarray.

[0024] In some implementations, also in the context of quantum computing withneutral atoms, the laser beam may not be used for trapping but to optically address anarbitrary subset of particles trapped in a plurality of tweezer traps and / or in opticallattices. In such a case, the trapping may be achieved by a different laser. Opticallyaddressing may refer to controlling the internal and / or motional state of the trapped particles. By shaping the laser beam in accordance with the method above, it may be possible to address an arbitrary subset of the plurality of trapped particles. Dynamicallyupdating the shape of the laser beam at a high rate may allow to address differentarbitrary subsets of the plurality of trapped faster in quick succession. Such an abilitymay benefit the implementation of for example high-fidelity quantum gate operations,which may rely on a sequence of addressing operations that are fast compared to the coherence time of the particles.

[0025] The present disclosure is also related to a method for shaping a laserbeam with a spatial light modulator, SLM, placed essentially in a Fourier plane of an optical system, comprising obtaining a self-interferometry based characterization of the optical system, and a desired target shape of the laser beam in a target region of the optical system. The method further comprises determining, based on the self-interferometry-based characterization of the optical system and the desired targetshape, a modulation mask for the SLM, and shaping the laser beam by applying thedetermined modulation mask to the SLM. Determining the modulation mask to beapplied to the SLM comprises simulating, based on the obtained characterization of theoptical system, a propagation of the laser beam in the optical system, possibly using anon-uniform fast Fourier transform, NUFFT., sometimes also referred to as NUDFT orNDFT in the art.

[0026] The simulation of the optical system using NUFFT may comprisecalculating the forward propagator in Eq. (1) (or the corresponding backwardpropagator) and / or making predictions about the electric field anywhere in the opticalsystem, including in the target region of the optical system based on the modulationmask of the SLM. Additionally, it may be utilized as part of the process to determine anew modulation mask for the SLM, based on the optical characterization and the desired target shape, as described above. In the latter case, NUFFT may be integratedinto at least part of the IFTA to iteratively obtain the modulation mask ^(^, ^) requiredto minimize the difference |^ ^ ^^(^, ^)| - |^(^, ^)| between the desired target shape|^^(^, ^)|^ and the detected or simulated laser beam shape |^(^, ^)|^ in the targetregion of the optical system.

[0027] As known in the art, NUFFT refers to a type of Fourier transform, whereinthe input or output function s not sampled at equally distant points in real and / orFourier space, see, for example, Bagchi, S., Mitra, S.K., “The Nonuniform DiscreteFourier Transform” in Marvasti, F. (eds), “Nonuniform Sampling. InformationTechnology: Transmission, Processing, and Storage” Springer (2001). The applicationof NUFFT in the method described above may reduce the computational complexityand the memory requirements of solving the phase retrieval problem, as describedherein, by factors in the range 10 to 1000 compared to the current state of the art.Given the parameters of commercial SLMs and the high spatial resolution that is required in the atomic volume for most beam shaping applications, memoryrequirements may be reduced from tens of gigabytes for the current state of the art tothe megabyte or even kilobyte range, when the method is utilized for e.g. the generationof optical tweezers, as described above. In essence, NUFFT may enable efficientpredictions about the laser beam only for an arbitrary set of points in the target region, as opposed to e.g. an array of points of a fixed shape which may span the full extent ofthe target region. Such predictions may contribute to a significant speed advantage, asin some implementations, not all parts of the laser beam in the target region may berelevant to particular application scenarios and require to be shaped accurately. Thus,NUFFT may allow (but is not limited to) effectively concentrating computational powerinto the parts of the laser beam in the target region which is relevant to the respectiveapplication. For instance, for the generation of optical tweezer traps, the laser beam inthe target region is shaped, such that the intensity of the electric field in the targetregion is mostly zero except in close vicinity of the tweezer traps (see, for example, Fig.7). The reduced computation times and memory requirements may also enable faster calculations by the above factors compared to the current state of the art for in- sequence beam shaping in neutral atom-based or ion-based or molecule-basedquantum computers (see, for example, Fig. 12) / quantum simulators / atomic clocks,including dynamic re-configuration of atomic / molecular registers or arrays of resonant excitation beams.

[0028] Further details of the apparatuses and methods described above and arelated computer program are discussed in the following with reference to exemplaryimplementations illustrated by the drawings. The foregoing paragraphs broadlyoutlines the features and technical advantages of examples in accordance with thepresent disclosure in order that the detailed description that follows may be better understood. Additional features and advantages will be described hereinafter. Theconception and specific examples disclosed may be readily utilized as a basis formodifying or designing other structures for carrying out the same purposes of the present disclosure. Characteristics of the concepts disclosed herein, both their organization and method of operation, together with associated advantages will be better understood from the following description when considered in connection with the accompanying figures. Each of the figures is provided for the purposes of illustration and description, and not as a definition of the limits of the claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Fig. 1 illustrates an exemplary setup for characterizing an optical systemand for laser beam shaping using a SLM arranged essentially in a Fourier plane of theoptical system according to an exemplary implementation of the present disclosure;

[0030] Fig. 2 illustrates an exemplary beam shaping setup for generating aplurality of optical tweezer traps using an objective and a SLM arranged essentially inthe Fourier plane according to aspects of the present disclosure;

[0031] Fig. 3 shows exemplary SLM modulation masks for splitting the inputlaser beam up in two diverging parts. Such masks may be used in / or may be part of a possible implementation of the present disclosure;

[0032] Fig. 4 shows exemplary measurements of the self-interference patterns inthe target region of the optical system and a corresponding plurality of SLM modulation masks which may be used in / or may be part of a possible implementation of the present disclosure;

[0033] Fig. 5 illustrates a fitting procedure for characterizing the optical systembased on a plurality of self-interference pattern measurements which may be used in / or may be part of a possible implementation of the present disclosure;

[0034] Fig. 6 illustrates an exemplary simulation of the optical system based ona characterization of the optical system according to aspects of the present disclosure;

[0035] Fig. 7 illustrates a possible implementation of laser beam shaping forgenerating an array of optical traps according to aspects of the present disclosure;

[0036] Fig. 8 illustrates a method for characterizing an optical system accordingto aspects of the present disclosure, e.g., by using an apparatus as described herein;

[0037] Fig. 9 illustrates a method for shaping a laser beam by a SLM accordingto aspects of the present disclosure, e.g., by using an apparatus as described herein;

[0038] Fig. 10 illustrates a method for quantum computing according to aspectsof the present disclosure, e.g., by using an apparatus as described herein;

[0039] Fig. 11 illustrates a method for shaping a laser beam by a SLM involvingNUFFT, according to aspects of the present disclosure, e.g., using an apparatusdescribed herein.

[0040] Fig. 12 shows a block diagram of an exemplary quantum computingdevice according to a possible implementation of the present disclosure.

[0041] Fig. 13 shows a sequence diagram for a possible Ramsey sequenceaccording to a possible implementation of the present disclosure, e.g., using anapparatus described herein.

[0042] Fig. 14 shows a block diagram for a possible implementation of an IFTAaccording to a possible implementation of the present disclosure, e.g., using an apparatus described herein.

[0043] Fig. 15 shows another exemplary SLM modulation mask profile forsplitting the input laser beam up in two diverging parts. Such as mask may be used in / or may be part of a possible implementation of the present disclosure;DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

[0044] Various aspects of the present disclosure are described in more detailhereinafter with reference to the accompanying drawings. The present disclosure may,however, be implemented in many different forms and should not be construed as limited to any specific structure or function presented herein. Rather, these aspects are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the present disclosure to those skilled in the art. Based on the teachings herein one skilled in the art should appreciate that the scope of the present disclosure is intended to cover any aspect of the present disclosure disclosed herein, whether implemented independently of or combined with any other aspect of the presentdisclosure. For example, an apparatus, a device or a system may be implemented, or amethod may be practiced using any number of the aspects set forth herein. In addition,the scope of the present disclosure is intended to cover such a device, apparatus, systemor method which is practiced using other structure, functionality, or structure and functionality in addition to or other than the various aspects of the present disclosure set forth herein. Any aspect of the present disclosure disclosed herein may be implemented by one or more elements of a claim. While specific feature combinations are described in the following with respect to certain aspects of the present disclosure, it is to be understood that not all features of the discussed examples must be present forrealizing the technical advantages of the devices, apparatuses, systems, methods andcomputer programs disclosed herein. Disclosed aspects may be modified by combining certain features of one aspect with one or more features of other aspects. A skilled person will understand that features, steps, components and / or functional elements of one aspect can be combined with compatible features, steps, components and / or functional elements of any other aspect of the present disclosure.

[0045] Several aspects of trapping and manipulating (e.g., imaging, gateoperations, spectroscopy, etc.) of neutral atoms will now be presented with reference tovarious devices, apparatuses, systems and methods that are described in the followingdetailed description and illustrated in the accompanying drawings by various blocks,modules, components, circuits, steps, processes, algorithms, and / or the like (collectively referred to as “elements”). These elements may be implemented using hardware, software, or combinations thereof. Whether such elements are implemented as hardware and / or software depends upon the particular application and designconstraints imposed on the overall system. Further, the apparatuses and methodsdisclosed herein can be part of complex quantum technology systems such as neutralatom quantum computers (see, for example, Fig. 12), quantum simulation systems, andquantum metrology systems, such as optical lattice clocks. The skilled person will appreciate that in the following several components of such systems, such as lasersources, imaging and measurement devices, system / experiment control and timingunits etc. are not explicitly described.

[0046] FIG. 1 illustrates an exemplary setup for characterizing an optical setupand laser beam shaping using a SLM in the Fourier plane. In a possibleimplementation, a laser beam 110 (e.g. emitted by a laser device) may propagatethrough the active area of an SLM 120 which can be, e.g., a digital mirror device or aliquid crystal modulator. The SLM 120 modulates the beam, which may involvechanging its wavefront. The modulated beam then propagates through focusing opticalelements 140 (e.g. an objective) which may focus the beam into the target region of theoptical system 150 which could correspond to an image plane. The modulation maskapplied to the SLM 120 may be configured, such that at least a portion of the input laserbeam is split into a first part 130a and a second part 130b of the modulated laser beam;and such that the second part 130b of the modulated laser beam diverges with respectto the first part 130a of the modulated laser beam, wherein a divergence angle of themodulated laser beam may be small enough, such that the second part 130b of themodulated laser beam at least partially interferes with the first part 130a of themodulated laser beam in the target region. The dashed lines in Fig. 1 indicate thecenters of the two parts of the laser beam. In the target region 150, the interfering twoparts of the laser beam may be separated by a distance ^′. This protocol may generate aself-interference pattern for the modulated laser beam in the target region 150 of theoptical system. Next, at least a part of the self-interference pattern of the modulatedlaser beam may be measured by a detector in the target region 150. In someimplementations, the detector may be a camera device, in other implementations the target region may be inaccessible to camera devices and / or trapped particles such asneutral atoms, ions or molecules may be used for detecting the interference pattern (seee.g., Fig. 13).

[0047] In some implementations, the SLM may be followed by a lens or acombination of lenses forming an objective and / or imaging setup which focuses themodulated laser beam onto a plane where particles are to be trapped inside a vacuumchamber. The system may include additional optical components such additional lenses, mirrors or other reflective surfaces as well as transmittable volumes of various materials, shapes and optical properties. The input laser beam might be of any opticalwavelength ^ in the ultraviolet, visible or near-infrared range compatible with thespecifications of at least some of the optical elements in the path (e.g., including theSLM), and the internal level structure of the trapped atoms. In some implementations,the focusing optics 140 may be characterized by an effective focal length ^ and / orgenerate an image plane in the target region which is located effectively a distance ^after the focusing optics. The detector in the target region 150 may be characterized byand / or a plurality of sampling points at which the intensity |^(^, ^)|^ of the electricfield ^(^, ^) in the target region of the optical system is detected. The maximumdistance ^ between adjacent sampling points may follow the condition given by Nyquistsampling theorem, i.e., ^ <^^ ^, where L denotes the spatial extent of the SLM. In otherimplementations, the SLM may follow a lens of the optical system as known in the art (cf. J. Goodman, Introduction to Fourier Optics, Roberts; 3rd edition (22 April 2016)).

[0048] FIG. 2 illustrates an exemplary beam shaping setup for generating opticaltweezer traps using an objective and a SLM, such as a liquid crystal light modulator or adigital micromirror device, possibly when the characterization of the system (includingaberrations and attenuations) is already known. The aberrations and attenuations maybe caused by imperfections in the optical components and their optical coatings thatare part of the optical system comprised in the setup as well as their reflection andtransmission properties. The input laser beam 220a, generated by a laser device 210,interacts with a SLM 230 and, as a consequence, is modulated. The modulated laserbeam 220b then passes through an objective 240 which focusses the modulated laserbeam onto the target region 250 of the optical system. The modulation mask applied tothe SLM may be chosen such that the laser beam in the image plane 250 generates aplurality of optical tweezer traps 260, possibly while compensating for at least some ofthe aberrations in the imaging system. Each of the traps is designed to trap a singleparticle 270. The image plane may be located inside a vacuum chamber. The SLMmodulation mask may be updated at a high rate, such that the fast sequence of targetbeam shapes in the target region creates a smooth movement of an arbitrary subset ofoptical tweezer traps for shifting the position of an arbitrary subset of the plurality particles. Each SLM modulation mask of such a sequence may be chosen in such a way, that at least some of the aberrations are compensated, leading to a higher quality of the shape of the laser beam in the image plane 250. The sequence of modulation masks may be used to create a dynamically configurable plurality of tweezer traps, wherein any arbitrary subset of the plurality of trapped particles can be smoothly moved withinthe target region. Such a scenario may benefit the implementation of quantum gates forquantum computing. It may also benefit the initialization of a quantum register of aquantum computer (see, for example, Fig. 12), wherein the quantum register comprisesa plurality of trapped particles such as neutral atoms, ions or molecules.

[0049] The setup may also be used for characterizing the optical systemcontained in the setup, as described above in the context of Fig. 1. In such a scenario,the plurality of trapped particles in the image plane, possibly trapped in a plurality oftweezer traps generated by a different input beam (not shown) may be used as adetector for detecting the electric field of the laser beam in the image plane. This can beachieved by, e.g., among other methods, performing a Ramsey spectroscopy sequence,as described below (see Fig. 12 for details). In some implementation, the plurality ofoptical tweezer traps may be generated by a different setup and the shaped laser beammay be used for addressing the particles 270 in the plurality of optical tweezer traps260 for changing their internal and / or motional state or for inducing local light shifts.

[0050] FIG. 3 shows four exemplary SLM modulation masks which may lead toa splitting of the input laser beam into two diverging parts. In each of the panels a-d thetop part illustrates how the SLM area may be divided into two subregions. For clarity,the partial modulation mask in the first subregion is colored white and the partial modulation mask in the second region is colored black. The bottom part in each of thepanels a-d shows one possible way of applying phase gradients to the modulation maskin the top part, such that at least some of the input laser beam is split up into two partswhich diverge under a small deflection angle. Such a way could comprise adding a spatial linear phase gradient to one of the two subregions in the direction of the desiredbeam splitting, while possibly keeping the phase in the other subregion spatiallyconstant. Alternatively, both subregions could have different linear phase gradients.The four possible ways of dividing the SLM area into two subregions shown in thisfigure are:Panel a) a periodic vertical stripe pattern;Panel b) a regular checkerboard pattern;Panel c) a horizontal stripe pattern with randomly varying stripe widths; andPanel d) a substantially random pattern.In some implementations, the SLM modulation mask used for splitting the input beam may not cover the entire area of the SLM.

[0051] FIG. 4 shows a possible plurality of measurements of the self-interference patterns in the target region of the optical system and a corresponding plurality of SLM modulation masks.Panels a-c show a possible set of three modulation masks ^(^, ^) applied to the SLMfor splitting the input laser beam into two diverging parts. The modulation mask 410consists of a periodic stripe pattern in the u-direction of the Fourier plane(corresponding to the x-direction of the image plane). Every second stripe belongs tothe first subregion and every other stripe to the second subregion. While the firstpartial modulation mask in the first subregion is characterized by a linear phasegradient in u-direction, leading to a beam displacement of ^′ in the target region of theoptical system. The second partial modulation mask in the second subregion may havea phase that is constant in the entire SLM area, lacking a spatial gradient. The firstpartial modulation mask is the same in all the modulation masks part of the set, while the phase of the second partial modulation mask is set to a different value in eachmodulation mask part of the set, leading to different phase shifts Δ^ between the firstand the second partial modulation mask.Panels d-f show a possible set of three modulation masks ^(^, ^) applied to the SLMfor splitting the input laser beam into two diverging parts, in analogy with panels a-c. The modulation mask consists of a periodic stripe pattern in the v-direction of the Fourier plane (corresponding to the y-direction of the image plane), leading to a beamdisplacement of ^′ in the target region of the optical system.Panels g-i show a possible set of three self-interference patterns 420, corresponding to|^(^, ^)|^, measured in the target region of the optical system and belonging to the SLMmodulation masks shown in the panels a-c.Panels j-l show a possible set of three self-interference patterns |^(^, ^)|^ measured inthe target region of the optical system and belonging to the SLM modulation masks shown in the panels d-f.

[0052] FIG. 5 illustrates the fitting procedure for characterizing the opticalsystem based on a plurality of interference pattern measurements. This fittingprocedure is carried out for each sampling point (^, ^) of the detector in the targetregion of the optical system. The intensity |^(^, ^)|^ measured at a sampling point (^, ^)is shown as a function of the phase shift between the first and the second partial modulation mask that is varied in accordance with the set of self-interference patternsillustrated in Fig. 4. The data is then fitted using a cosine function (black curve) and thefit allows to extract the phase of the cosine Δχ^^,^^(^, ^), belonging to a beamdisplacement of ^′ in the x-direction and ^′ in the y-direction in the target region of theoptical system. The cosine phases then allow to reconstruct the wavefront of the laserbeam in the detector plane in the target region of the optical system along the directionof the displacement, as described above.

[0053] FIG. 6 illustrates a possible simulation of the optical system.Panels a and c show a possible electric field ^(^, ^) in the Fourier plane of the opticalsystem, e.g. at the position of the SLM. Panel a shows the amplitude |^(^, ^)| of the fieldand panel c shows the corresponding wavefront arg (^(^, ^)). In this example, thesimulation of the optical system comprises computing the electric field ^(^, ^) in thetarget region of the optical system according to Eq.1.Panels b and d show the computed electric field ^(^, ^) in the target region of theoptical system. Panel b shows the amplitude |^(^, ^)| of that field and panel c shows thecorresponding wavefront arg (^(^, ^)).In this example, the electric field in the target region of the optical system, shown in panels b and d, may correspond to a regular array of optical tweezer traps which may bepart of a quantum computing device. The wavefront of the electric field in the Fourierplane of the optical system arg (^(^, ^)) may correspond to the modulation mask whichis applied to the SLM.

[0054] FIG. 7 illustrates a possible implementation of laser beam shaping.Panel a shows the measured intensity |^(^, ^)|^ of the electric field in the image planeof the optical system after a plurality of iterations of the IFTA protocol (see Fig. 14). Inthis example, the modulation mask of the SLM has been configured for the purpose of creating a plurality of tweezer traps in the target region of the optical system.Panel b shows the predicted intensity |^(^, ^)|^ of the electric field in the image planeof the optical system according to Eq.1.Panel c shows the difference between the measured and the predicted intensity frompanel a and b, respectively. The small range of the values in this plot may suggest that the optical system has been characterized well, such that the simulation of the optical system using Eq.1 gives accurate results.Panel d shows the absolute difference between the measured and the predictedintensity, as shown in panel c.

[0055] FIG. 8 illustrates a method 800 for characterizing an optical system. Step810 modulates an input laser beam with a spatial light modulator, SLM, preferablyessentially placed in a Fourier plane of the optical system with respect to a target regionof the optical system. Step 820 applies a modulation mask to the SLM for splitting atleast a portion of the input laser beam into a first part and a second part of the modulated laser beam, wherein the second part of the modulated laser beam diverges with respect to the first part of the modulated laser beam, wherein a divergence angle of the modulated laser beam is small enough such that the second part at least partially interferes with the first part in the target region thereby generating a self-interferencepattern for the modulated laser beam. Step 830 measures at least a part of thegenerated self-interference pattern of the modulated laser beam in the target region ofthe optical system. Step 840 characterizes the optical system based on the measuredinterference pattern in the target region.

[0056] FIG. 9 illustrates a method 900 for shaping a laser beam by a spatial lightmodulator, SLM, preferably essentially placed in a Fourier plane of an optical system with respect to a target region of the optical system, comprising obtaining 910 a self- interferometry-based characterization of the optical system, and a desired target shapeof the laser beam in a target region of the optical system. The method further comprisesdetermining 920, based on the self-interferometry-based characterization of the opticalsystem and the desired target shape, a modulation mask for the SLM, and shaping(930) the laser beam according to the target shape by applying the determined modulation mask to the SLM.

[0057] FIG. 10 illustrates a method 1000 for quantum computing. Step 1010obtains, from a remote user device via a network, a set of instructions for performingthe set of quantum gate operations on the selected subset of the trapped particle qubitsof the quantum register. Step 1020 traps and manipulates a plurality of particles in anarray of optical traps (e.g., tweezer traps or sites of an optical lattice) by performing themethod of any of the claims 19-22. Step 1030 performs a set of quantum gateoperations of a quantum computing algorithm on a selected subset of trapped particlequbits by manipulating an internal and / or a motional state of the selected subset oftrapped particle qubits. Step 1040 determines a result of the quantum computingalgorithm by measuring a state of the selected subset of trapped particle qubits. Andstep 1050 outputs data corresponding to the result of the quantum computingalgorithm. In some implementations, the method may be used to as part of a quantumcomputer, an exemplary implementation of which is shown in Fig. 12. Step 1030 mayinvolve shaping a laser beam using the methods disclosed herein.

[0058] FIG. 11 illustrates a method 1100 for shaping a laser beam with a spatiallight modulator, SLM, placed preferably essentially in a Fourier plane of an opticalsystem, comprising a desired target shape of the laser beam in a target region of theoptical system and, optionally, a characterization of the optical system. The methodfurther comprises determining 1120, based on the desired target shape of the laser beam in the target region of the optical system and the characterization of the optical system, a modulation mask to be applied to the SLM for modulating the laser beam, wherein determining the modulation mask comprises simulating, based on the obtained characterization of the optical system, a propagation of the laser beam in theoptical system using a non-uniform fast Fourier transform, NUFFT and shaping 1130the laser beam by applying the determined modulation mask to the SLM.

[0059] FIG. 12 shows a typical implementation of a quantum computer 1200comprising a quantum register 1210, e.g. formed by a plurality of trapped particles, anda quantum gate laser system 1220. In some implementations, the quantum computercan be controlled by a (remote) user device 1260, possibly via a network 1250. In someimplementations, the quantum gate laser system 1220 may be configured to createplurality of trapped particles and / or manipulate and / or cause a controlled quantum state evolution of one or more atomic objects within the quantum register 1210. For example, the quantum gate laser system 1220 may comprise one or more lasers, whichprovide one or more laser beams to atomic objects (such as neutral atoms, molecules orions) in the quantum register 1210. In some implementations, some of the laser beamsmay be shaped by performing the method of either claims 11-14 and / or of claims 15-18.

[0060] In some implementations, the qubit state readout system 1230 may beconfigured to collect and / or detect photons generated by qubits (e.g., during readingprocedures). The optics collection system may comprise one or more optical elements (e.g., lenses, mirrors, waveguides, fiber optics cables, and / or the like) and one or more photodetectors. In various embodiments, the photodetectors may be photodiodes, photomultipliers, charge-coupled device (CCD) sensors, complementary metal oxide semiconductor (CMOS) sensors, Micro-Electro-Mechanical Systems (MEMS) sensors,single-photon avalanche diode, SPAD, sensors and / or other photodetectors that aresensitive to light at an expected fluorescence wavelength of the qubits of the quantum computer. In various embodiments, the detectors may be in electronic communication with the processing and control circuitry 1240.

[0061] In some implementations, the user device 1260 is configured to allow auser to provide input to the quantum computer 1200 and receive, view, and / or store the output from the quantum computer 1200. The user device may be in communication with the processing and control circuitry 1240 of the quantum computer 1200 via one or more wired or wireless networks 1250 and / or via direct wired and / or wireless communications. In an example embodiment, the user device 1260 may translate, configure, format, and / or the like information / data, quantum computing algorithms and / or circuits, and / or the like into a computing language, executable instructions, command sets, and / or the like that the processing and control circuitry 1240 can understand and / or implement.

[0062] In some implementations, the processing and control circuitry 1240 maybe configured to control, inter alia, the quantum gate laser system 1220 and / or thequbit state readout system 1230. For example, the processing and control circuitry 1240may be configured to cause a controlled evolution of quantum states of one or moreatomic objects within the quantum register 1210 to execute a quantum circuit and / oralgorithm. For example, the processing and control circuitry 1240 may cause a readingprocedure, possibly as part of executing a quantum circuit and / or algorithm. In variousembodiments, the atomic objects confined within the quantum register 1210 are usedas qubits of the quantum computer 1200.

[0063] FIG. 13 shows a sequence diagram for Ramsey spectroscopy which maybe utilized for measuring the intensity of the electric field in the target region of theoptical system using particles including but not limited to neutral atoms, ions or molecules. A possible implementation of this sequence is described above.

[0064] FIG. 14 shows a block diagram illustrating a typical IFTA which may beemployed for solving the hase retrieval problem as part of a method for laser beamshaping with an SLM essentially placed in the Fourier plane. Without limiting thegenerality of the algorithm, we may assume that the wavefront of the input beam is flat and characterized by a global phase of zero. One possible goal of such an algorithmcould be to obtain a modulation mask ^(^, ^), which, if applied to the SLM, maymodulate the electric field of the laser beam in the Fourier plane ^(^, ^), such that inthe target region of the optical system a desired electrical field ^(^, ^) is generated, seeEq. 1. In this figure ^(^, ^) may refer to the modulated electric field essentially in theFourier plane of the optical system, i.e. ^(^, ^) = ^^(^, ^)where ^^(^, ^) is theamplitude of the input laser beam which may be assumed to have a flat wavefront. Insome implementation of the algorithm 1410, first an initial guess for ^(^, ^) is madeand used as the wavefront for the electric field in the Fourier plane of the opticalsystem. Further 1420, the intensity of the electric field in the Fourier plane may be setto the measured intensity of the input beam. The intensity of the input beam may be measured at the position of the SLM using a detector device, such as a camera. It may also be obtained from the beam characterization method described above, possiblyincluding the attenuation ^(^, ^). Next 1430, the electric field in the target region of theoptical system ^(^, ^) may be simulated according to Eq. 1, based on the electric field inthe Fourier plane ^(^, ^) and, possibly, the characterization of the optical system. . Insome implementations, the electric field in the target region of the optical system^(^, ^) may be measured. The result of the simulation or measurement may then becompared with the desired electric field ^^(^, ^). If the difference 1440 between thesimulated and desired electric field is considered sufficiently small, then ^(^, ^) =arg(S(u, v)) can be considered the desired modulation mask, as illustrated in step 1450.Otherwise, another iteration of the algorithm may be necessary and the absolute of the electric field in the target region of the optical system could be constrained to the absolute of the desired electric field, as shown in step 1455. Then, using anothersimulation which may involve an inverted form of Eq. 1 with an inverse Fouriertransformation, a new version of ^(^, ^) may be computed from ^(^, ^), see 1460,before the algorithm may fall back to step 1420.

[0065] In some implementations, the simulation of the optical system in step1430 and / or step 1460 and / or other parts of the algorithm may utilize NUFFT. Insuch a scenario, the simulation of the optical system and, therefore, the algorithmwhich comprises such simulations, may benefit from the smaller computational andmemory requirements associated with using NUFFT.

[0066] Fig. 15 shows another exemplary SLM modulation mask profile forsplitting the input laser beam up in two diverging parts. Fig. 15 shows the phase profile(along the displacement vector) of a possible modulation mask to be applied to the SLM in order to split the input laser beam into a first part and a second part. In contrast to the masks shown in Fig.4, the profile shown may not be divided into two distinct subregions.

[0067] The foregoing disclosure provides illustration and description but is notintended to be exhaustive or to limit the aspects to the precise form disclosed. Modifications and variations may be made in light of the above disclosure or may be acquired from practice of the aspects. As used herein, the term component is intended to be broadly construed as hardware, firmware, or a combination of hardware and software. As used herein, a processor is implemented in hardware, firmware, or a combination of hardware and software.

[0068] It will be apparent that systems and / or methods described herein may beimplemented in different forms of hardware, firmware, or a combination of hardware and software. The actual specialized control hardware or software code used to implement these systems and / or methods is not limiting of the aspects. Thus, the operation and behavior of the systems and / or methods were described herein without reference to specific software code—it being understood that software and hardware can be designed to implement the systems and / or methods based on the description herein.

[0069] Even though particular combinations of features are recited in the claimsand / or disclosed in the specification, these combinations are not intended to limit the disclosure of various aspects. In fact, many of these features may be combined in ways not specifically recited in the claims and / or disclosed in the specification. Althougheach dependent claim listed below may directly depend on only one claim, the disclosure of various aspects includes each dependent claim in combination with every other claim in the claim set. A phrase referring to “at least one of” a list of items refers to any combination of those items, including single members. As an example, “at least one of: a, b, or c” is intended to cover a, b, c, a-b, a-c, b-c, and a-b-c, as well as any combination with multiples of the same element (e.g., a-a, a-a-a, a-a-b, a-a-c, a-b-b, a- c-c, b-b, b-b-b, b-b-c, c-c, and c-c-c or any other ordering of a, b, and c).

[0070] No element, act, or instruction used herein should be construed as criticalor essential unless explicitly described as such. Also, as used herein, the articles “a” and “an” are intended to include one or more items, and may be used interchangeably with “one or more.” Furthermore, as used herein, the terms “set” and “group” are intended to include one or more items (e.g., related items, unrelated items, a combination of related and unrelated items, and / or the like), and may be used interchangeably with “one or more.” Where only one item is intended, the phrase “only one” or similar language is used. Also, as used herein, the terms “has,” “have,” “having,” and / or the like are intended to be open-ended terms.

[0071] As used herein, the phrase “based on” shall not be construed as areference to a closed set of information, one or more conditions, one or more factors, or the like. In other words, the phrase “based on A” (where “A” may be information, a condition, a factor, or the like) shall be construed as “based at least on A” unless specifically recited differently.

[0072] As used herein, the term “or” is an inclusive “or” unless limiting languageis used relative to the alternatives listed. For example, reference to “X being based on A or B” shall be construed as including within its scope X being based on A, X being based on B, and X being based on A and B. In this regard, reference to “X being based on A or B” refers to “at least one of A or B” or “one or more of A or B” due to “or” being inclusive. Similarly, reference to “X being based on A, B, or C” shall be construed as including within its scope X being based on A, X being based on B, X being based on C, X being based on A and B, X being based on A and C, X being based on B and C, and X being based on A, B, and C. In this regard, reference to “X being based on A, B, or C” refers to “at least one of A, B, or C” or “one or more of A, B, or C” due to “or” beinginclusive. As an example of limiting language, reference to “X being based on only one of A or B” shall be construed as including within its scope X being based on A as well as X being based on B, but not X being based on A and B.

[0073] Further, process diagrams such as Fig. 8, Fig. 9 and Fig. 10, Fig. 11 andFig. 14 do not necessarily indicate a particular order or sequence of steps. For example,steps may also be performed in a different order or, if hardware capabilities allow it, simultaneously, without deviating from the scope of the present disclosure.

Claims

1September 24, 2024 Max-Planck-Gesellschaft zur Förderung M175165WO ANE / Bmn der Wissenschaften e.V. & Ludwig- Maximilians-Universität München Claims 1. Method (800) for characterizing an optical system, comprising:modulating (810) an input laser beam (110) with a spatial light modulator, SLM,(120) preferably essentially placed in a Fourier plane of the optical system with respectto a target region (150) of the optical system byapplying (820) a modulation mask (410) to the SLM for splitting at least aportion of the input laser beam into a first part (130a) and a second part (130b) of themodulated laser beam, wherein the second part of the modulated laser beam divergeswith respect to the first part of the modulated laser beam, wherein a divergence angle ofthe modulated laser beam is small enough such that the second part at least partiallyinterferes with the first part in the target region, thereby generating a self-interferencepattern (420) for the modulated laser beam;measuring (830) at least a part of the generated self-interference pattern of themodulated laser beam in the target region of the optical system; andcharacterizing (840) the optical system based on the measured interference pattern in the target region.

2. Method for characterizing an optical system according to claim 1, wherein themodulation mask applied to the SLM comprises: a first partial mask applied in a first region of the SLM; and a second partial mask applied in a second region of the SLM; such that the modulation of the laser beam in the first region and in the second regioncauses the splitting of the laser beam into the first and the second part.

23. Method for characterizing an optical system according to claim 2,wherein the first region of the SLM comprises multiple subregions spatiallyalternating with corresponding subregions of the second region of the SLM; andwherein the first partial modulation mask induces a spatially varying modulation in thefirst region of the SLM; andthe second partial modulation mask induces a different spatially varyingmodulation in the second region of the SLM.

4. Method for characterizing an optical system according to claim 3, wherein thesubregions of the first and the second region alternate periodically.

5. Method for characterizing an optical system according to claim 3 or 4, whereinthe total area of both subregions amounts to at least 50%, to at least 60%, to at least70%, to at least 80%, or to at least 90% of a total effective usable area of the SLM.

6. Method for characterizing an optical system according to any of the claims 1 to 5,wherein measuring the self-interference pattern in the target region of the optical system comprises: measuring a plurality of different self-interference patterns induced by applyinga plurality of different SLM modulation masks generating a plurality of correspondingdifferent phase shifts between the interfering two parts of the laser beam.

7. Method for characterizing an optical system according to claims 1 to 6, whereinmeasuring the self-interference pattern comprises: measuring at least two, preferably at least three self-interference patterns createdby applying varying phase shifts in a first direction; and3 measuring at least two, preferably at least three self-interference patterns createdby applying varying phase shifts in a second direction, not parallel with respect to thefirst direction.

8. Method for characterizing an optical system according to any of claims 1 to 7,wherein characterizing the optical system comprises: determining an optical propagator function and / or an optical convolution kernelfor the optical system based on the measured interference pattern.

9. Method for characterizing an optical system according to any of the claims 1 to 8,wherein measuring the self-interference pattern in the target region of the optical system comprises: measuring the self-interference pattern with a spatial resolution below adiffraction limit associated with an effective aperture of the SLM, a wavelength of theinput laser beam, and a focal length which relates the Fourier plane of the opticalsystem to the target region.

10. Method for characterizing an optical system according to any of the claims 1 to 9,wherein characterizing the optical system comprises characterizing one or more opticalaberrations of the optical system.

11. Method (900) for shaping a laser beam by a spatial light modulator, SLM,preferably essentially placed in a Fourier plane of an optical system with respect to atarget region of the optical system, comprising: obtaining (910) a self-interferometry-based characterization of the opticalsystem, and a desired target shape of the laser beam in a target region of the opticalsystem; determining (920), based on the self-interferometry-based characterization ofthe optical system and the desired target shape, a modulation mask for the SLM; and4 shaping (930) the laser beam according to the target shape by applying thedetermined modulation mask to the SLM.

12. Method for shaping a laser beam according to claim 11, wherein obtaining theself-interferometry-based characterization of the optical system comprisescharacterizing the optical system by performing the steps of the method of any of theclaims 1 to 10.

13. Method for shaping a laser beam according to claim 11 or 12, further comprising:determining a difference between the shaped laser beam at the target region of the optical system and the desired target shape of the laser beam; and modifying the SLM modulation mask based on the self-interferometry-basedcharacterization of the optical system and the determined difference between theshaped laser beam at the target region of the optical system and the desired target shape of the laser beam.

14. Method for shaping a laser beam according to any of claims 11 to 13, furthercomprising: obtaining an updated desired target shape of the laser beam; and updating the SLM modulation mask based on the self-interferometry-basedcharacterization of the optical system and the updated desired target shape of the laser beam.

15. Method (1100) for shaping a laser beam with a spatial light modulator, SLM,placed preferably essentially in a Fourier plane of an optical system, comprising:obtaining (1110) a desired target shape of the laser beam in a target region of theoptical system and, optionally, a characterization of the optical system;5 determining (1120), based on the desired target shape of the laser beam in thetarget region of the optical system and, optionally, based on the characterization of theoptical system, a modulation mask to be applied to the SLM for modulating the laserbeam, wherein determining the modulation mask comprises simulating, based on theobtained characterization of the optical system, a propagation of the laser beam in theoptical system using a non-uniform fast Fourier transform, NUFFT; andshaping (1130) the laser beam by applying the determined modulation mask tothe SLM.

16. Method for shaping a laser beam according to claim 15, wherein obtaining thecharacterization of the optical system comprises characterizing the optical system byperforming the method of any of claims 1 - 10.

17. Method for shaping a laser beam according to claim 15 or 16, wherein using theNUFFT to determine the modulation mask comprises:using the NUFFT for calculating a phase and an intensity of the laser beam for aplurality of arbitrarily arranged points in the optical system, possibly in the targetregion and / or in the Fourier plane of the optical system.

18. Method for shaping a laser beam according to any of claims 15 to 17, furthercomprising: obtaining an updated desired target shape of the laser beam in the target region;and determining a corresponding updated SLM modulation mask using the NUFFT.

19. Method for trapping and / or manipulating a plurality of particles in a targetregion of an optical system, comprising:6 trapping and / or manipulating the plurality of particles in the target region ofthe optical system using a laser beam shaped by performing the method of any ofclaims 11 to 14 and / or by performing the method of any of claims 15 to 18.

20. Method of claim 19, wherein trapping and / or manipulating the plurality ofparticles in the target region of the optical system comprises:generating a plurality of optical traps for the plurality of particles in the targetregion by shaping a first input laser beam by performing the method of any of claims 11to 14 and / or by performing the method of any of claims 15 to 18; and / ormanipulating an internal and / or a motional state of a selected subset of theplurality of particles by shaping a second input laser beam by performing the method ofany of claims 11 to 14 and / or by performing the method of any of claims 15 to 18.

21. Method of claim 20, further comprising:dynamically updating a position of a subset of the plurality of optical traps,preferably moving the position of the subset of the plurality of optical traps alongarbitrary trajectories within the target region, by dynamically updating the modulationmask applied to the SLM.

22. Method of any of claims 19 to 21, wherein measuring the self-interferencepattern using the subset of the trapped particles comprises: generating a plurality of optical traps for the plurality of particles in the targetregion; andtrapping the plurality of particles in the plurality of optical traps; anddetermining an intensity of the self-interference pattern in the target region ofthe optical system at the positions of the trapped particles based on performing aRamsey sequence.

723. Method (1000) for quantum computing, comprising:trapping (1020) and / or manipulating (1020) a plurality of particles in an arrayof optical traps forming a quantum register of trapped particle qubits by performing themethod of any of the claims 19 to 22.

24. Method of claim 23, further comprising:performing (1030) a set of quantum gate operations of a quantum computingalgorithm on a selected subset of trapped particle qubits by manipulating an internaland / or a motional state of the selected subset of trapped particle qubits; and determining (1040) a result of the quantum computing algorithm by measuring astate of the selected subset of trapped particle qubits.

25. Method for quantum computing of claim 24, further comprising:obtaining (1010) a set of instructions for performing the set of quantum gateoperations on the selected subset of the trapped particle qubits of the quantum register;and outputting (1050) data corresponding to the result of the quantum computing algorithm.

26. Method for quantum computing of claims 24 and 25,wherein obtaining the set of instructions for performing the set of quantum gateoperations of the quantum computing algorithm, comprises obtaining the set ofinstructions from a remote user device via a network interface; and / orwherein outputting the data corresponding to the result of the quantumcomputing algorithm comprises sending, to the remote user device via the network8 interface, the data corresponding to the result of the quantum computing algorithm; and / or wherein performing the set of quantum gate operations on the selected subset of trapped particle qubits comprises manipulating an internal and / or a motional state ofthe selected subset of trapped particle qubits by performing the method of any of claims19 to 22.

27. Method for quantum computing according to any of claims 23 to 26, whereinmeasuring the generated self-interference pattern of the modulated laser beam in thetarget region of the optical system comprises:measuring the self-interference pattern using a subset of the trapped particles.

28. Method for laser-based material processing, comprising processing the materialwith a laser beam shaped by performing the steps of the method of any of claims 11 to14 and / or by performing the steps of the method of any of claims 15 to 18.

29. Method for laser-based medical treatment, comprising illuminating a biologicalsample or a biological material with a laser beam shaped by performing the steps of the method of any of claims 11 to 14 and / or by performing the steps of the method of any of claims 15 to 18.

30. Trapped particle quantum computing apparatus (1200) comprising means forexecuting the method of any of the claims 23 to 27.

31. Trapped particle quantum computing apparatus according to claim 30, furthercomprising: aprocessing and control unit (1240); and9 aquantum register (1210); anda quantum gate laser system (1220), operably connected to the control unit, and,possibly, configured to generate and / or manipulate the quantum register.

32. Trapped particle quantum computing apparatus according to claims 30-31, further comprising: a particle state readout system (1230), operably connected to the control unit, and configured for detecting, optionally via internal state selective imaging, a state ofthe quantum register; and / ora network interface, operably connected to or integrated into the control unit,and configured for receiving, from a remote user device via a network, data indicativeof the set of instructions of the quantum algorithm and / or for sending to the remoteuser device (1250), via the network (1260), data indicative of a result of the quantumalgorithm.

33. Computer program for carrying out the method of any of the claims 1 to 18 whenexecuted by a data and signal processing device configured for controlling an SLM,controlling an interference pattern measurement apparatus, and for characterizing theoptical system based on an interference pattern measurement.

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