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arXiv:quant-ph/9905085

Conditional quantum state engineering at beam splitter arrays

Clausen, J., Mohammed Dakna, L. Knoell, D.‐G. Welsch

Published May 25, 1999• 3 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/9905085

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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.
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.
We show that conditional output measurement on a beam splitter may be used to produce photon-added states for a large class of signal-mode quantum states, such as thermal states, coherent states, squeezed states, displaced photon-number states, and coherent phase states. Combining a mode prepared in such a state and a mode prepared in a photon-number state, the state of the mode in one of the output channels of the beam splitter ``collapses'' to a photon-added state, provided that no photons are detected in the other output channel. We present analytical and numerical results, with special emphasis on photon-added coherent and squeezed vacuum states. In particular, we show that adding photons to a squeezed vacuum yields superpositions of quantum states which show all the typical features of Schrödinger-cat-like states.
Preparation of Schrödinger-cat-like states via conditional output measurement on a beam splitter is studied. In the scheme, a mode prepared in a squeezed vacuum is mixed with a mode prepared in a Fock state and photocounting is performed in one of the output channels of the beam splitter. In this way the mode in the other output channel is prepared in a Schrödinger-cat-like state that is either a photon-subtracted or a photon-added Jacobi polynomial squeezed vacuum state, depending upon the difference between the number of photons in the input Fock state and the number of photons in the output Fock state onto which it is projected. Two possible photocounting schemes are considered, and the problem of monitoring cats that are ``hidden'' in a statistical mixture of states is studied.
A scheme is proposed for implementing a desired transformation of the unknown quantum state of a traveling optical multimode signal. The operator describing this transformation is built up as a polynomial in products of the mode operators. There is no limitation to unitary operators realized by the scheme, the implementation is conditional however, i.e., the desired operator can be realized only with a certain success probability which depends on the respective operator and the input state. While the scheme requires the preparation of single-photon and coherent states, it exclusively makes use of beam splitters and zero- and single-photon detections. The setup consists of a number of stations, each controlling one of the signal modes. The stations share an auxiliary entangled state which has been prepared beforehand. This state is then entangled with each of the signal modes and a measurement is performed of each of the auxiliary modes afterwards. In this way, the stations can be spatially separated from one another, similar to a multimode generalization of a quantum teleportation setup. Some examples of application such as the realization of cross-Kerr nonlinearities, multidirectional quantum teleportation (i.e., multimode mirrors), and the preparation of multimode Greenberger-Horne-Zeilinger- and Einstein-Podolsky-Rosen-like states are considered.
A scheme for generating Schr\"odinger-cat-like states of a single-mode optical field by means of conditional measurement is proposed. Feeding a squeezed vacuum into a beam splitter and counting the photons in one of the output channels, the conditional states in the other output channel exhibit a number of properties that are very similar to those of superpositions of two coherent states with opposite phases. We present analytical and numerical results for the photon-number and quadrature-component distributions of the conditional states and their Wigner and Husimi functions. Further, we discuss the effect of realistic photocounting on the states.