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arXiv:0710.0980 · Physical Review A

Preparation of entangled states of two photons in several spatial modes

Karel Lemr, Jaromı́r Fiurášek

Published Feb 5, 2008• 18 citations
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Abstract

We describe a protocol capable of preparing an arbitrary state of two photons in several spatial modes using pairs of photons generated by spontaneous parametric down conversion, linear optical elements, and single-photon detectors or postselection. The protocol involves unitary and nonunitary transformations realizable by beam splitters and phase shifters. Nonunitary transformations are implemented by attenuation filters. The protocol contains several optimization capabilities with the goal of improving overall probability of its success. We also show how entangled two-photon states required for quantum computing with linear optics can be prepared using a very simple and feasible scheme.

DOI
10.1103/physreva.77.023802

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We describe a protocol capable of preparing an arbitrary state of two photons in several spatial modes using pairs of photons generated by spontaneous parametric down conversion, linear optical elements, and single-photon detectors or postselection. The protocol involves unitary and nonunitary transformations realizable by beam splitters and phase shifters. Nonunitary transformations are implemented by attenuation filters. The protocol contains several optimization capabilities with the goal of improving overall probability of its success. We also show how entangled two-photon states required for quantum computing with linear optics can be prepared using a very simple and feasible scheme.
Advancing quantum information processing is predicated on the preparation of ever-larger multiqubit states. Photonic realizations of such states may be achieved by increasing the number of photons populating the state or the number of qubits encoded per photon. Typical approaches to the latter strategy utilize distinct degrees of freedom of the photon field. We present here an approach that encodes two qubits per photon in the spatial parity of its transverse spatial profile. Simple linear optical devices transform each parity qubit separately or the two qubits jointly. Furthermore, we demonstrate that entangled photon pairs produced by spontaneous parametric down-conversion may be used to prepare arbitrary four-qubit states through sculpting the spatial profile of the classical optical pump. Two examples are highlighted---the preparation of two-photon four-qubit Greenberger-Horne-Zeilinger and WWW states, whose encoding in a photon pair has thus far eluded other approaches.
Two experimental arrangements consisting of coupled spontaneous parametric down-converters with type-I phase matching pumped simultaneously by a powerful optical field in a coherent state through a balanced beam splitter and linear optical elements are proposed for conditional preparation of macroscopic entangled states in output pumping modes of the studied system. Successful generation of the macroscopic entangled state in the pumping modes is unambiguously heralded by coincident detection of two photons in the generated signal and idler modes of the system. We calculate the amount of entanglement and probabilities of successfully observing the X (2) macroscopic entangled states in the total wavefunction. We show that the proposed schemes can be used to obtain a new type of macroscopic entangled states.
An approach to generating the entangled photonic states $|{\ensuremath{\Psi}}_{1},{\ensuremath{\Psi}}_{2}\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|{\ensuremath{\Psi}}_{2},{\ensuremath{\Psi}}_{1}\ensuremath{\rangle}$ from two arbitrary states $|{\ensuremath{\Psi}}_{1}\ensuremath{\rangle}$ and $|{\ensuremath{\Psi}}_{2}\ensuremath{\rangle}$ is proposed. The protocol is implemented by the conditionally induced beam-splitter coupling, which leads to the selective swapping between two photonic modes. Such coupling occurs in a quantum system prepared in the superposition of two ground states with only one of them being involved in the swapping. All the entangled states in the category, such as entangled pairs of coherent states or Fock states (NOON states), can be efficiently produced in the same way by this method.
Nowadays, the nonlinear optical process of spontaneous parametric down-conversion is considered as the canonical approach for creating entangled-photon pairs. We consider three pairs of entangled photons emitted by the parametric down-conversion source, and introduce a setup for evolving these photons based on linear optics, which is composed of several polarizing beam splitters, beam splitters, and half wave plates. By using the parametric down-conversion source and the setup, we carefully design an efficient scheme for preparing six-photon hyperentangled states in both the polarization and the spatial degrees of freedom. Because we use almost all possible behaviors of the three pairs of entangled photons, the present scheme is efficient for creating six-photon hyperentangled states. Next, in the regime of weak nonlinearity we design a quantum nondemolition detection to distinguish the two cases of photons in two special spatial modes. It is worth pointing out that our scheme is much easier to realize, since the strength of the nonlinearities in the process of quantum nondemolition detection can be restricted to the scalable orders of magnitude in practicality.
Discretizing transverse linear momentum of photon pairs, generated by spontaneous parametric down conversion (SPDC), is one of the simplest methods for producing bipartite entangled states in high dimensions. So far, it has been employed only to prepare states in dimensions $m\ifmmode\times\else\texttimes\fi{}m$. In this work, we study the generalization for engineering entangled states in dimensions $m\ifmmode\times\else\texttimes\fi{}n$. Our approach relies on the manipulation of the pump beam transverse profile and the phase-matching function of the SPDC process to prepare, behind an mmm- and nnn-slit aperture, different $m\ifmmode\times\else\texttimes\fi{}n$ spatial entangled states. We demonstrate the technique experimentally for some $2\ifmmode\times\else\texttimes\fi{}3$ states. Compared to previous approaches in producing $2\ifmmode\times\else\texttimes\fi{}n$ photonic entanglement, which require either more than two photons or hybrid entanglement, our scheme is less demanding and simpler since we employ only two photons and a single degree of freedom to encode the states.