Retrieving the phase of a complex-valued field from the measurements of its amplitude is a crucial problem with a wide range of applications in microscopy and ultracold atomic physics. In particular, obtaining an accurate and efficient solution to this problem is a key step in shaping laser beams for trapping atoms in optical tweezer arrays and applying high-fidelity entangling gates on a neutral atom quantum computer. Current approaches to this problem fail to converge on the optimal solution due to a phenomenon known as vortex formation. In this work, we present an efficient optimization algorithm using Optimal Transport. Our approach completely bypasses the creation of phase vortices and allows for a state-of-the-art solution both in terms of accuracy and efficiency. Furthermore, we show a deep theoretical connection between the Optimal Transport plan and the ray-optics limit of the Wigner distribution of the unknown complex-valued field, and show that our method can be used to retrieve the phase-space transformation of any unknown quadratic phase system. Finally, we reinterpret this problem in the modern quantum learning framework. The techniques we develop provide both useful intuition and practical tools for advancing the frontiers of phase retrieval and laser beam shaping.
Retrieving the phase of a complex-valued field from the measurements of its amplitude is a crucial problem with a wide range of applications in microscopy and ultracold atomic physics. In particular, obtaining an accurate and efficient solution to this problem is a key step in shaping laser beams for trapping atoms in optical tweezer arrays and applying high-fidelity entangling gates on a neutral atom quantum computer. Current approaches to this problem fail to converge on the optimal solution due to a phenomenon known as vortex formation. In this work, we present an efficient optimization algorithm using Optimal Transport. Our approach completely bypasses the creation of phase vortices and allows for a state-of-the-art solution both in terms of accuracy and efficiency. Furthermore, we show a deep theoretical connection between the Optimal Transport plan and the ray-optics limit of the Wigner distribution of the unknown complex-valued field, and show that our method can be used to retrieve the phase-space transformation of any unknown quadratic phase system. Finally, we reinterpret this problem in the modern quantum learning framework. The techniques we develop provide both useful intuition and practical tools for advancing the frontiers of phase retrieval and laser beam shaping.
The development of new laser beam shaping methods is important in a variety of fields within optics, atomic physics and biophotonics. Spatial light modulators offer a highly versatile method of time-dependent beam shaping, based on imprinting a phase profile onto an incident laser beam that determines the intensity in the trapping plane laser field. The calculation of the required phase is a well-known inverse problem, which can be tackled with different approaches. Our method based on conjugate gradient minimisation [1] not only allows the calculation of smooth and accurate intensity profiles suitable for trapping cold atoms, but can also be used to generate multi-wavelength traps [2] and for simultaneous control over both the intensity and the phase of the light [3], with exceptionally high reconstruction fidelity. References: [1] T Harte, et al., "Conjugate gradient minimisation approach to generating holographic traps for ultracold atoms" Opt. Express 22, 26548 (2014) [2] D Bowman, et al., "Multi-wavelength holography with a single spatial light modulator for ultracold atom experiments" Opt. Express 23, 8365 (2015) [3] D Bowman, et al., "High-fidelity phase and amplitude control of phase-only computer generated holograms using conjugate gradient minimisation" Opt. Express 25, 11692 (2017)
Optical phase spaces represent fields of any spatial coherence and are typically measured through phase-retrieval methods involving a computational inversion, optical interference, or a resolution-limiting lenslet array. Recently, a weak-values technique demonstrated that a beam's Dirac phase space is proportional to the measurable complex weak value, regardless of coherence. These direct measurements require raster scanning through all position-polarization couplings, limiting their dimensionality to less than 100 000 [C. Bamber and J. S. Lundeen, Phys. Rev. Lett. 112, 070405 (2014)]. We circumvent these limitations using compressive sensing, a numerical protocol that allows us to undersample, yet efficiently measure, high-dimensional phase spaces. We also propose an improved technique that allows us to directly measure phase spaces with high spatial resolution with scalable frequency resolution. With this method, we are able to easily and rapidly measure a 1.07-billion-dimensional phase space. The distributions are numerically propagated to an object in the beam path, with excellent agreement for coherent and partially coherent sources. This protocol has broad implications in quantum research, signal processing, and imaging, including the recovery of Fourier amplitudes in any dimension with linear algorithmic solutions and ultra-high-dimensional phase-space imaging.
We present an optimal control procedure for the non-adiabatic transport of ultracold neutral thermal atoms in optical tweezers arranged in a one-dimensional array, with focus on reaching minimal transfer time. The particle dynamics are modeled first using a classical approach through the Liouville equation and second through the quantum Wigner equation to include quantum effects. Both methods account for typical experimental noise described as stochastic effects through Fokker-Planck terms. The optimal control process is initialized with a trajectory computed for a single classical particle and determines the phase-space path that minimizes transport time and ensures high transport fidelity to the target trap. This approach provides the fastest and most efficient method for relocating atoms from an initial configuration to a desired target arrangement, minimizing time and energy costs while ensuring high fidelity. Such an approach may be highly valuable to initialize large atom arrays for quantum simulation or computation experiments.
Abstract High-precision preparation of holographic optical tweezers, which is a critical requirement for neutral atom quantum computing platforms, faces the challenge of system aberrations that degrade trap performance. In this paper, we propose a simple and efficient aberration correction algorithm for wavefront-sensorless optical systems, based on correcting the relative lateral shift of diffracted light across spatial regions. Applied to a cold atomic system, this technique demonstrates active correction of aberrations up to several wavelengths, reducing them to a peak-to-valley difference of 0.19(3) λ . The diffraction spot quality and optical trap depth are significantly improved after aberration correction. Combined with subsequent array optimization, the technique enables the generation of uniform optical trap arrays with enhanced atom-loading efficiency. The method requires no prior wavefront information or additional hardware, utilizing only an SLM and a standard CCD camera. Its computational efficiency and hardware simplicity make it broadly applicable to fields such as laser lithography and microscopy.
Quantum systems entanglement in continuous variables has been receiving increasing attention from the scientific community. The transverse spatial shape of photons (i.e. transverse position and momentum degree of freedom) is such a continuous variable systems, which due to its ease of control is prone to be used for testing some of the new physical predictions that continuous variables quantum systems can offer, as the improvement of the security of quantum key distribution cryptography and the efficiency enhancement of quantum communication protocols. Following this line, the mean goal of this Thesis has been the characterization and measurement of the spatial shape of photons. Another outstanding problem in Quantum Optics Information Technology (QOIT) is the coherent and reversible storage of photons states in matter, i.e. the necessity of a quantum memory. Theoretical and experimental results about this topic have been showed.
Below we summarize the more important contributions of this Thesis:
Demonstration about the non-validity of the OAM transformation rule FORr highly focused light beams: We have shown from a theoretical and experimental point of view that the OAM transformation rule is not valid for highly focused light beams, since Dove prisms inherently introduce astigmatism, and therefore further OAM changes. We have analyzed how light beams with a well defined value of the OAM per photon, after traversing the Dove prism, are transformed into a superposition of states with well defined OAM. The violation of the rule, turns out to be more important for highly focused light beams. We have provided a quantitative study of the properties of the Dove prism, and we have verified experimentally the validity of the theoretical results in a series of experiments with a commercially available Dove prism.
Presentation of a new method to measure the amplitude and phase of Laguerre-Gaussian like beams: A method to measure the amplitude an phase of Laguerre-Gaussian like beams have been presented, from a theoretical and experimental point of view. This method is based on a small variation of the spiral phase interferometry technique, which allows to avoid some technical problems that can be found in the reconstruction of finite sized beams and beams with phase singularities. A few examples of the use of our technique for the characterization of complex beams have been presented.
Description of the transverse spatial shape of bipartite entangled photons generated in Spontaneous Parametric Down Conversion (SPDC) using the Wigner function formulism: We have show some theoretical and experimental results regarding the spatial Wigner function of entangled paired photons generated in SPDC. Concretely, we have studied the remote preparation of one of the photons of the pair, for the photon being in a Gaussian beam and in a superposition of two modes. Also, we have analyzed and measured the case of the Wigner function of only one of the two photons of the pair, which shows that one photon of the pair is in a statistical mixture of modes. Finally, an experiment to completely measure the spatial state of the two photons was proposed, and also analytical expressions for the results one would expect in this case were given.
Investigation about the storage of spatial light polarization grating into the Zeeman ground states of cold cesium atoms: We have investigated theoretically and experimentally the dynamics of light grating stored in electromagnetically induced transparent (EIT) medium associated with a degenerate two-level system. The dependence of the stored light grating with the intensities of the incident and reading beams has been investigated. Bragg diffraction into the stored grating has been employed to probe the dynamics under different conditions showing reasonable results in agreement with the theoretical model developed. Moreover, we also have observed the coherent evolution of the stored grating in the presence of an applied magnetic field, which allows collapses and revivals of the stored grating, which is associated with the Larmor precession of the induced grating around the applied magnetic field.
We think that the results of this Thesis are an important step in order to explore the quantum capabilities of the transverse spatial shape of photons and can encourage going further in this line of research. Moreover, the investigation on the dynamics of light grating stored in an EIT medium is of considerable importance for a better understanding of the coherent memory for multidimensional state spaces. Thus, a further and more detailed investigation of the topics presented here should be of special interest for the scientific community.
Optimal transport methods have recently established state of the art accuracy and efficiency for holographic laser beam shaping. However, use of such methods is hindered by severe O(N2) memory and O(N2) time requirements for large scale input or output images with N total pixels. Here we leverage the dual formulation of the optimal transport problem and the separable structure of the cost to implement algorithms with greatly reduced O(N) memory and O(NlogN) to O(N3/2) time complexity. These algorithms are parallelizable and can solve megapixel-scale beam shaping problems in tens of seconds on a CPU or seconds on a GPU.