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arXiv:quant-ph/0303004 · Physical Review A

Realization of multimode operators with passive linear optics and photodetection

J. Clausen, L. Knöll, D.‐G. Welsch

Published Oct 22, 2003• 10 citations
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Abstract

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.

DOI
10.1103/physreva.68.043822

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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.
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.
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.
We propose a nearly perfect optical scheme for the quantum teleportation of entangled coherent states using optical devices such as nonlinear Kerr media, beam splitters, phase shifters, and photon detectors. Different from those previous schemes, our scheme needs only ``yes'' or `no' measurements of the photon number of the related modes, i.e. nonzero- and zero-photon measurements, while in previous schemes one has to exactly identify the even or odd parity character of the photon numbers detected by detectors.
We suggest implementation of quantum teleportation protocol of unknown qubit beyond Bell states formalism. Hybrid entangled state composed of coherent components that belong to Alice and dual-rail single photon at Bob disposal is used. Nonlinear effect on the teleported state is realized due to peculiarity of interaction of coherent components with discrete variable state on a beam splitter. Bob performs unitary transformation after receiving the appropriate classical information. The protocol is nearly deterministic but the output state is subject to amplitude distortion which reduces the fidelity of the protocol. Final problem is reduced to finding a way to get rid of amplitude-distorting factor to restore the original qubit. Some strategies to increase the fidelity of the protocol are used. We show the teleportation can be implemented with arbitrary settings. Our approach is applicable to different basic number states of original unknown qubit. The scheme is realized by linear optics methods and requires an irreducible number of optical elements.
In this thesis work, after a brief introduction of some basic concepts of quantum optics, some experimental techniques useful for the manipulation of quantum states of light are presented. In particular, I focused on two transformations operated at the single-photon level, like the addition and the subtraction of a single photon from a travelling light beam. Both these methods share a probabilistic approach, that is their implementation has a non unitary success probability. Anyway, the detection of an additional photon in an ancillary mode exactly announces when the operation has been actually implemented. That's why this type of implementation is generally called heralded. Starting from these basic operations, a way to implement some of their superpositions is then analyzed. In the second part of this thesis, I describe two state-engineering experiments based on these fundamental operations. The first one realizes a so-called universal orthogonalizer, an universal strategy that, given an arbitrary input state, can be used to produce at the output an orthogonal one. The adjective universal refers to the fact that the procedure works equally well for arbitrary input fields. The other experiment uses coherent superpositions of basic quantum operations to emulate a strong Kerr nonlinearity at the few-photon level, and could be used to implement new logic gates for quantum computation. The third part of this thesis is dedicated to a different experimental approach, in some sense complementary to state engineering: mode engineering. Indeed, while in state engineering the purpose is the detection of an unknown state after its manipulation, in mode engineering experiments the perspective is completely reversed: the final state is assumed to be known and what has to be inferred are its spatio-temporal features. By following this alternative experimental approach, the interaction of broadband single photons with resonant atomic vapours is analyzed by observing the traces left on the photon temporal shape.
We present two linear optical schemes using nonideal photodetectors to demonstrate the inseparability of WWW-type NNN-partite entangled states containing only a single photon. First, we show that the pairwise entanglement of arbitrary two modes chosen from NNN optical modes can be detected using the method proposed by Nha and Kim [Phys. Rev. A 74, 012317 (2006)], thereby suggesting a full inseparability among NNN parties. In particular, this scheme is found to succeed for any nonzero quantum efficiency of photodetectors. Second, we consider a quantum teleportation network using linear optics without auxiliary modes. The conditional teleportation can be optimized by a suitable choice of the transmittance of the beam splitter in the Bell measurement. Specifically, we identify the conditions under which maximum fidelity larger than the classical bound 2/3 is achieved only in cooperation with other parties. We also investigate the case of on-off photodetectors that cannot discriminate between the number of detected photons.