Much of the richness in nature arises due to the connection between classical and quantum mechanics. In advanced science, the tools of quantum mechanics was not only applied in microscopic description but also found its efficacy in classical phenomena, broadening the fundamental scientific frontier. A pioneering inspiration is substituting Fock state with structured spatial modes to reconstruct a novel Hilbert space. Based on this idea, here we propose the classical analogy of squeezed coherent state for the first time, deriving classical wave-packets by applying squeezed and displacement operators on free space structured modes. Such a generalized structured light not only creates new degrees of freedom into structured light, including tunable squeezed degree and displacement degree but also exhibits direct correlation between quadrature operator space and real space. Versatile generalized classical squeezed states could be experimentally generated by a simple large-aperture off-axis-pumped solid-state laser. On account of its tunablity, we initially put forward a blueprint using classical structured light, an analogy of squeezed states to realize super-resolution imaging, providing an alternative way to beat diffraction limit as well as opening an original page for subsequent applications of high-dimensional structured light, such as high-sensitive measurement and ultra-precise optical manipulation.
Much of the richness in nature arises due to the connection between classical and quantum mechanics. In advanced science, the tools of quantum mechanics was not only applied in microscopic description but also found its efficacy in classical phenomena, broadening the fundamental scientific frontier. A pioneering inspiration is substituting Fock state with structured spatial modes to reconstruct a novel Hilbert space. Based on this idea, here we propose the classical analogy of squeezed coherent state for the first time, deriving classical wave-packets by applying squeezed and displacement operators on free space structured modes. Such a generalized structured light not only creates new degrees of freedom into structured light, including tunable squeezed degree and displacement degree but also exhibits direct correlation between quadrature operator space and real space. Versatile generalized classical squeezed states could be experimentally generated by a simple large-aperture off-axis-pumped solid-state laser. On account of its tunablity, we initially put forward a blueprint using classical structured light, an analogy of squeezed states to realize super-resolution imaging, providing an alternative way to beat diffraction limit as well as opening an original page for subsequent applications of high-dimensional structured light, such as high-sensitive measurement and ultra-precise optical manipulation.
Control of structured light is of great importance to explore fundamental physical effects and extend practical scientific applications, which has been advanced by accepting methods of quantum optics - many classical analogies of exotic quantum states were designed using structured modes. However, the prevailing quantum-like structured modes are limited by discrete states where the mode index is analog to the photon number state. Yet, beyond discrete states, there is a broad range of quantum states to be explored in the field of structured light -- continuous-variable (CV) states. As a typical example of CV states, squeezed state plays a prominent role in high-sensitivity interferometry and gravitational wave detection. In this work, we bring together two seemingly disparate branches of physics, namely, classical structured light and quantum squeezed state. We propose the structured light analogy of squeezed state (SLASS), which can break the spatial limit following the process of surpassing the standard quantum limit (SQL) with quantum squeezed states. This work paves the way for adopting methods from CV quantum states into structured light, opening new research directions of CV entanglement, teleportation, classical and quantum informatics of structured light in the future.
Quantum-classical analogy refers to the fact that quantum and classical phenomena share a similar mathematical description and exhibit analogous behaviors, which has advanced optical frontiers. In this work, the mechanism of the squeezed state is incorporated into classical structured light, constructing a novel kind of structured light, termed the squeezed geometric beam, that is, the squeezed coherent state of geometric beams, and revealing a novel quantum-classical analogy between the noise of quantum squeezed state and the beam size of squeezed geometric beams, both evolving with the same squeezing parameter. The presented method provides an inspirational extensible framework to reveal more structured light analogies of quantum states. In addition, it is demonstrated in theory and experiment that thanks to the unveiled squeezing mechanism, the squeezed geometric beam as an information carrier can effectively lower the modal crosstalk, thus boosting the communication capacity of spatial mode multiplexing.
Structured light has become topical of late, where controlling light in all its degrees of freedom has offered novel states of light long predicted, enhanced functionality in applications, and a modern toolbox for probing fundamental science. Structuring light as single photons and entangled states allows the spatial modes of light to be used to encode a large alphabet, accessing high dimensional Hilbert spaces for fundamental tests of quantum mechanics and improved quantum information processing tasks. In this tutorial, we outline the basic concepts of high dimensional quantum states expressed in a basis of spatial modes (structured light) and explain how to create, control, and detect such quantum states in the laboratory with a focus on transverse spatial modes such as the orbital angular momentum and pixel (position) modes. Finally, we highlight some example applications of such quantum structured light, from communications to imaging.
We use the spatial degree of freedom of light modes to construct optical analogs of generalized quantum coherent states for Hermite- and Laguerre--Gauss modes. Our optical analogs preserve the statistical properties of their quantum counterparts, encoded in their amplitude and phase distributions. We explore three basic symmetries that provide generalized displaced, rotated, and squeezed coherent states. Given the substantial interest in squeezed states for probing matter, we believe that the optical analogs introduced here have significant implications for optical sensing. Specifically, the single-particle nature of our spatial modes makes them robust candidates for sensing photosensitive materials. Overall, our approach opens the door to optical metrology and sensing protocols that mimic those already existing in the quantum realm, and facilitates further exploration of the quantum state zoo through classical optical analogs.
Quantum–classical analogy reveals the shared mathematical framework and dynamical physical behaviors between quantum and classical systems. This work reviews structured light as a versatile platform for realizing such analogies. Theoretical mappings between quantum states and optical modes are summarized, followed by representative cases including entanglement state, cat state, coherent state, and squeezed state, where quantum features are emulated through tailored amplitude, phase, and polarization structures. Applications in quantum-inspired communication, teleportation, random walks, and billiard are further discussed. This work provides a systematic overview of the connections between quantum phenomena and classical optics and offers methodological guidance for future photonic implementations.
The emergence and development of quantum optics as a discipline in the latter half of the twentieth century has established photons as one of the most promising candidates for future technologies in computing, metrology, communication, and imaging. In imaging and sensing, a series of recent theoretical works and proof-of-concept experiments have shown that by examining the mode structure of photons, one can build optical systems with resolutions below the Rayleigh–Abbe limit. Building upon these developments, in one of our works, we investigate the use of higher-order Hermite–Gauss (HG) modes for sensing optically induced transverse displacements. We show that projective measurements onto neighboring modes are optimal and sufficient in the small-displacement regime, and that the sensitivity—quantified by the Fisher information—scales linearly with the order of the mode used. We generalize the calculations to arbitrary displacements and varying degrees of coherence between the displaced modes, and perform a proof-of-concept experiment with higher-order HG modes, obtaining an order-of-magnitude enhancement in sensitivity compared to the fundamental mode. This approach enables enhanced displacement sensitivity with minimal measurements and highlights the potential of structured light for ultrasensitive displacement sensing, particularly for birefringence measurements with broadband or low-coherence light sources. We also consider displacement sensing in the light field originating from a biphoton source, which possesses rich high-dimensional correlations in its biphoton mode structure. We show that the Fisher information is proportional to the degree of entanglement, or equivalently, its Schmidt rank. In addition to sensing, photons structured in their transverse degrees of freedom also provide a convenient toolkit for quantum computing and communication due to the high dimensionality of their Hilbert space. In one of the works presented, we introduce a technique for nonlocal transfer of high-dimensional computational results using spatial correlations in photon pairs. We expect these findings to be useful in future photonic quantum networks, where computational resources are distributed and users can access the results of complex computations via a centrally located quantum processor. We also experimentally implement the protocol in down-converted photon pairs and demonstrate the transfer of unitaries in one-dimensional and two-dimensional transverse momentum spaces. In another work, we address the noise inherent in quantum computing circuits and simulate the interaction between a qubit and its environment by manipulating transverse momentum modes. This study, performed by fabricating three liquid-crystal metasurfaces, provides a practical solution for quantum state purification, demonstrating a versatile approach for simulating open-qubit dynamics. Combining the transverse spatial structure with polarization yields the so-called vector modes of photons. Structuring two-qubit interference, or the Hong–Ou–Mandel (HOM) effect, offers significant potential to extend the applications of such modes beyond the classical regime. We demonstrate the simultaneous generation of all four Bell states in polarization by exploiting HOM interference of vector modes. These results represent a significant step toward manipulating HOM interference within structured photons, offering promising applications in high-dimensional quantum information processing, quantum communication, and sensing. On the algorithmic side of photonic information processing, we propose a quantum pattern recognition algorithm to identify images obtained from quantum imaging techniques with correlated photons—such as “interaction-free” or “ghost” imaging. Interfacing these imaging techniques with quantum pattern recognition promises a significantly improved image acquisition and identification system, with potential applications in medical diagnostics and biological imaging.