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arXiv:0907.4900 · Journal of Modern Optics

Transformation design and nonlinear Hamiltonians

Thomas Brougham, Goce Chadzitaskos, Igor Jex

Published Aug 10, 2009• 2 citations
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Abstract

We study a class of nonlinear Hamiltonians, with applications in quantum optics. The interaction terms of these Hamiltonians are generated by taking a linear combination of powers of a simple ‘beam splitter’ Hamiltonian. The entanglement properties of the eigenstates are studied. Finally, we show how to use this class of Hamiltonians to perform special tasks such as conditional state swapping, which can be used to generate optical cat states and to sort photons.

DOI
10.1080/09500340903194633

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We study a class of nonlinear Hamiltonians, with applications in quantum optics. The interaction terms of these Hamiltonians are generated by taking a linear combination of powers of a simple ‘beam splitter’ Hamiltonian. The entanglement properties of the eigenstates are studied. Finally, we show how to use this class of Hamiltonians to perform special tasks such as conditional state swapping, which can be used to generate optical cat states and to sort photons.
The generation of arbitrary single-mode quantum states from the vacuum by alternate coherent displacement and photon adding as well as the measurement of the overlap of a signal with an arbitrarily chosen quantum state are studied. With regard to implementations, the transformation of the quantum state of a traveling optical field at an array of beam splitters is considered, using conditional measurement. Allowing for arbitrary quantum states of both the input reference modes and the output reference modes on which the measurements are performed, the setup is described within the concept of two-port non-unitary transformation, and the overall non-unitary transformation operator is derived. It is shown to be a product of operators, where each operator is assigned to one of the beam splitters and can be expressed in terms of an s-ordered operator product, with s being determined by the beam splitter transmittance or reflectance. As an example we discuss the generation of and overlap measurement with Schroedinger-cat-like 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.
Using conditional measurement on a beam splitter, we study the transformation of the quantum state of the signal mode within the concept of two-port non-unitary transformation. Allowing for arbitrary quantum states of both the input reference mode and the output reference mode on which the measurement is performed, we show that the non-unitary transformation operator can be given as an sss-ordered operator product, where the value of sss is entirely determined by the absolute value of the beam splitter reflectance (or transmittance). The formalism generalizes previously obtained results that can be recovered by simple specification of the non-unitary transformation operator. As an application, we consider the generation of Schrödinger-cat-like states. An extension to mixed states and imperfect detection is outlined.
Entangled states represent fascinating elements of quantum mechanics and quantum technology. Their generation is an interesting problem combining many techniques developed in quantum optics. Using algebraic properties of special transforms we derive the explicit form of the effective Hamiltonian that enables the generation of highly nonclassical states of the GHZ type. Such Hamiltonians belong to the higher order Hamiltonians often discussed in nonlinear quantum optics. We derive the form of the effective Hamiltonian that enables the generation of highly nonclassical states of the GHZ type.
We reelaborate on a general method for diagonalizing a wide class of nonlinear Hamiltonians describing different quantum optical models. This method makes use of a nonlinear deformation of the usual su(2) algebra and when some physical parameter, dictated by the particular model under consideration, becomes small, it gives a diagonal effective Hamiltonian that describes correctly the dynamics for arbitrary states and long times. We extend the technique to NNN-level atomic systems interacting with quantum fields, finding the corresponding effective Hamiltonians when the condition of kkk-photon resonance is fulfilled.