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arXiv:0712.1197 · Physical Review A

Manipulation of single-photon states encoded in transverse spatial modes: Possible and impossible tasks

Gabriel F. Calvo, Antonio Picón

Published Jan 3, 2008• 9 citations
arXivHTMLPDF

Abstract

Controlled generation and manipulation of photon states encoded in their spatial degrees of freedom is a crucial ingredient in many quantum-information tasks exploiting higher-than-two dimensional encoding. Here, we prove the impossibility to arbitrarily modify $d$-level state superpositions (qudits) for $d>2$, encoded in the transverse modes of light, with optical components associated to the group of symplectic transforms (Gaussian operations). Surprisingly, we also provide an explicit construction of how non-Gaussian operations acting on mode subspaces do enable one to overcome the limit $d=2$. In addition, this set of operations realizes the full SU(3) algebra.

DOI
10.1103/physreva.77.012302

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Controlled generation and manipulation of photon states encoded in their spatial degrees of freedom is a crucial ingredient in many quantum-information tasks exploiting higher-than-two dimensional encoding. Here, we prove the impossibility to arbitrarily modify ddd-level state superpositions (qudits) for d>2d>2d>2, encoded in the transverse modes of light, with optical components associated to the group of symplectic transforms (Gaussian operations). Surprisingly, we also provide an explicit construction of how non-Gaussian operations acting on mode subspaces do enable one to overcome the limit d=2d=2d=2. In addition, this set of operations realizes the full SU(3) algebra.
Abstract Spatial modes of light directly give the most easily accessible degree of freedom that span an infinite dimensional Hilbert space. The higher dimensional spatial mode entanglement realized using spontaneous parametric down conversion (SPDC) process is generally restricted to the subspace defined by a single spatial mode in pump. Access to other modal subspaces can be realized by pumping beams carrying several easily tunable transverse modes. As a proof of principle experiment, we generate twin-photon states in an SPDC process with pump as a superposition of first order Laguerre–Gaussian (or Hermite–Gaussian) modes. We show that the generated states can be easily tuned between different subspaces by controlling the respective modal content in the pump superposition.
Qudits, d-level quantum systems, have been shown to provide a better resource for quantum key distribution and other Quantum Information protocols. It is customary to generate photonic qudits using more than one degree of freedom of the same photon. In much the same way, multi-qubit states are generated using only a pair of photons and ingenious ways to manipulate more than one degree of freedom independently. In contrast to such costly implementations in terms of quantum resources, we present the controlled generation of two copies of two-qudit states using four photons and a single degree of freedom, transverse momentum. The degree of entanglement within each pair was inferred by exploiting the availability of two copies of the same state, without the need of a full tomographic reconstruction of the states, and both highly-entangled and separable states were generated. We show theoretically that the set of states obtainable using our setup is very diverse, ranging from maximally entangled states of qudits to separable states.
Qudits have proven to be a powerful resource for quantum information processing, offering enhanced channel capacities, improved robustness to noise, and highly efficient implementations of quantum algorithms. The encoding of photonic qudits in transverse-spatial degrees of freedom has emerged as a versatile tool for quantum information processing, allowing access to a vast information capacity within a single photon. We examine recent advances in quantum optical circuits with spatially structured light, focusing particularly on top-down approaches that employ complex mode-mixing transformations in free space and fibers. We highlight circuits based on platforms such as multi-plane light conversion, complex scattering media, multi-mode, and multi-core fibers. We discuss their applications for the manipulation and measurement of multi-dimensional and multi-mode quantum states. Furthermore, we discuss how these circuits have been employed to perform multi-party operations and multi-outcome measurements, thereby opening new avenues for scalable photonic quantum information processing.
Unitary transformations are the fundamental building blocks of gates and operations in quantum information processing, allowing the complete manipulation of quantum systems in a coherent manner. In the case of photons, optical elements that can perform unitary transformations are readily available only for some degrees of freedom, e.g., wave plates for polarization. However, for high-dimensional states encoded in the transverse spatial modes of light, performing arbitrary unitary transformations remains a challenging task for both theoretical proposals and actual implementations. Following the idea of multi-plane light conversion, we show that it is possible to perform a broad variety of unitary operations at high quality by using only a few phase modulation planes. More importantly, we experimentally implement several high-dimensional quantum gates for up to five-dimensional states encoded in the full-field mode structure of photons. In particular, we realize cyclic and quantum Fourier transformations, known as Pauli X ^ -gates and Hadamard H ^ -gates, respectively, with an average visibility of more than 90%. In addition, we demonstrate near-perfect “unitarity” by means of quantum process tomography, unveiling a process purity of 99%. Last, we demonstrate the benefit of the two independent spatial degrees of freedom, i.e., azimuthal and radial, and implement a two-qubit controlled-NOT quantum operation on a single photon. Thus, our demonstrations open up new paths to implement high-dimensional quantum operations, which can be applied to various tasks in quantum communication, computation, and sensing schemes.
Quantum entanglement is arguably the cornerstone which differentiates the quantum realm from its classical counterpart. While entanglement can reside in any photonic degree of freedom, polarization permits perhaps the most straightforward manipulation due to the widespread availability of standard optical elements such as waveplates and polarizers. As a step towards a fuller exploitation of entanglement in other degrees of freedom, in this work we demonstrate control over the transverse spatial structure of light at the single-photon level. In particular we integrate in our setup all the technologies required for: (i) fibre-based photon pair generation, (ii) deterministic and broadband single-photon spatial conversion relying on a passive optical device, and (iii) single-photon transmission, while retaining transverse structure, over 400 m of few-mode fibre. In our experiment, we employ a mode selective photonic lantern multiplexer with the help of which we can convert the transverse profile of a single photon from the fundamental mode into any of the supported higher-order modes. We also achieve conversion to an incoherent or coherent addition of two user-selected higher order modes by addressing different combinations of inputs in the photonic lantern multiplexer. The coherent nature of the addition, and extraction of usable orbital angular momentum at the single-photon level, is further demonstrated by far-field diffraction through a triangular aperture. Our work could enable studies of photonic entanglement in the transverse modes of a fibre and could constitute a key resource quantum for key distribution with an alphabet of scalable dimension.