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arXiv:1804.08531 · Physical Review A

Two-photon self-Kerr nonlinearities for quantum computing and quantum optics

Joshua Combes, Daniel J. Brod

Published Dec 10, 2018• 22 citations
arXivHTMLPDF

Abstract

The self-Kerr interaction is an optical nonlinearity that produces a phase shift proportional to the square of the number of photons in the field. At present, many proposals use nonlinearities to generate photon-photon interactions. For propagating fields these interactions result in undesirable features such as spectral correlation between the photons. Here we engineer a discrete network composed of cross-Kerr interaction regions to simulate a self-Kerr medium. The medium has effective long-range interactions implemented in a physically local way. We compute the one- and two-photon $S$ matrices for fields propagating in this medium. From these scattering matrices we show that our proposal leads to a high-fidelity photon-photon gate. In the limit where the number of nodes in the network tends to infinity, the medium approximates a perfect self-Kerr interaction in the one- and two-photon regime.

DOI
10.1103/physreva.98.062313

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The self-Kerr interaction is an optical nonlinearity that produces a phase shift proportional to the square of the number of photons in the field. At present, many proposals use nonlinearities to generate photon-photon interactions. For propagating fields these interactions result in undesirable features such as spectral correlation between the photons. Here we engineer a discrete network composed of cross-Kerr interaction regions to simulate a self-Kerr medium. The medium has effective long-range interactions implemented in a physically local way. We compute the one- and two-photon SSS matrices for fields propagating in this medium. From these scattering matrices we show that our proposal leads to a high-fidelity photon-photon gate. In the limit where the number of nodes in the network tends to infinity, the medium approximates a perfect self-Kerr interaction in the one- and two-photon regime.
A cross-Kerr interaction produces a phase shift on two modes of light proportional to the number of photons in both modes and is sometimes called cross-phase modulation. Cross-Kerr nonlinearities have many applications in classical and quantum nonlinear optics, including the possibility of a deterministic and all-optical controlled-phase gate. We calculate the one- and two-photon SSS matrices for fields propagating in a medium where the cross-Kerr interaction is spatially distributed at discrete interaction sites comprised of atoms. For the interactions considered, we analyze the cases where the photons copropagate and counterpropagate through the medium and give a physical interpretation to the differences between the two cases. Finally, we obtain the SSS matrix in the limit of infinitely long chains, showing that it corresponds to a perfect controlled-phase operation.
We show that the optical Kerr effect can be used to construct a quantum phase gate. It is well known from quantum nondemolition techniques that, as two photon field modes pass through a Kerr medium, the phase of each mode will be shifted, and the size of the phase shift will depend on the number of photons in both modes. We discuss the Hamiltonian responsible for this effect and show how this can produce an effective photon-photon interaction which corresponds to the quantum phase gate operating on two qubits each of which is represented by the elliptical polarization of one photon field. We discuss decoherence and losses and suggest some methods for dealing with them.
According to idealized models, a strong Kerr non-linearity may be used to build optical quantum gates for optical quantum information processing by inducing conditional phase shifts on quantum states. Recently, Shapiro (PRA 73, 062305 (2006)) argued that for a Kerr medium with non-instantaneous but fast response, essentially no phase shift is induced on two-single-photon input states, and thus a quantum gate build from such a medium cannot work. Here we show that a fast response Kerr medium induces some but very little phase shifts on a two-single-photon input state, and it is insufficient for high fidelity quantum computation. We point out that this is caused by the medium imparting spectral entanglement to the input photons. We further show that a way to circumvent this problem and achieve a high fidelity gate, is to engineer the dispersion properties of the medium to give a dominant spectral effect over the non-instantaneous response, in addition to satisfying a phase matching condition.
Abstract We have studied the photon statistics of a nondegenerate two-photon laser with a medium that has both self- and cross-Kerr interactions as well as intensity-dependent atom–field coupling in the Hamiltonian. The density matrix equation of the atoms and two-mode cavity fields was derived using the standard Scully–Lamb approach and numerically solved. Under certain conditions, the photon distribution function equation was derived and solved analytically. The first few moments of photon numbers were calculated in the region above the lasing threshold. Nonclassicality of the fields was investigated by means of the Mandel Q parameter, second-order correlation functions and the Cauchy–Schwarz inequality. The results are analyzed for a range of finite Kerr parameters. Intensity-dependent coupling suppresses the nonclassicality of the field. It pushes the statistics of the field towards super-Poissonian.The self- and cross-Kerr interactions have significant roles in the nonclassicality of the two-photon cavity field and its statistics. These results are interesting for the field of quantum nonlinear optics and useful in quantum communications.
Summary form only given. Achieving photon-photon interactions is one of the main objectives of the quantum information technology community, since these interactions provide the basis for deterministic quantum gates with high fidelity. In the last few years new milestones in creating phase-shifts [1] and quantum gates[2] have been achieved. Researching resource-moderate experimental realizations for the future development of this field provides a scalable framework for devices and opportunities for deployment in quantum communication networks.We highlight photon-photon interactions in unexplored physical regimes that are outside the standard approach of single atom QED or Rydberg transitions. Building on the first complete quantum optical characterization of an engineered free space phase shift triggered via atomic Kerr nonlinearities [3], we characterize novel architectures using the interaction of single photons in both a room temperature atomic vapor and cold atomic ensembles in miniaturized cavity QED setups. These different systems provide very compelling pathways to achieving photonphoton interactions.
A possible two-qubit gate for optical quantum computing is the parity gate based on the weak Kerr effect. Two photonic qubits modulate the phase of a coherent state, and a quadrature measurement of the coherent state reveals the parity of the two qubits without destroying the photons. This can be used to create so-called cluster states, a universal resource for quantum computing. Here, the effect of self-phase modulation on the parity gate is studied, introducing generating functions for the Wigner function of a modulated coherent state. For materials with non-electromagnetically-induced-transparency-based Kerr nonlinearities, there is typically a self-phase modulation that is half the magnitude of the cross-phase modulation. Therefore, this effect cannot be ignored. It is shown that for a large class of physical implementations of the phase modulation, the quadrature measurement cannot distinguish between odd and even parity. Consequently, weak nonlinear parity gates must be implemented with physical systems where the self-phase modulation is negligible.