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arXiv:quant-ph/0510053 · Journal of Physics B Atomic Molecular and Optical Physics

Generating continuous variable quantum codewords in the near-field atomic lithography

Stefano Pirandola, Stefano Mancini, David Vitali, Paolo Tombesi

Published Feb 1, 2006• 33 citations
arXivHTMLPDF

Abstract

Recently, D. Gottesman et al. [Phys. Rev. A 64, 012310 (2001)] showed how to encode a qubit into a continuous variable quantum system. This encoding was realized by using non-normalizable quantum codewords, which therefore can only be approximated in any real physical setup. Here we show how a neutral atom, falling through an optical cavity and interacting with a single mode of the intracavity electromagnetic field, can be used to safely encode a qubit into its external degrees of freedom. In fact, the localization induced by a homodyne detection of the cavity field is able to project the near-field atomic motional state into an approximate quantum codeword. The performance of this encoding process is then analyzed by evaluating the intrinsic errors induced in the recovery process by the approximated form of the generated codeword.

DOI
10.1088/0953-4075/39/4/023

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Recently, D. Gottesman et al. [Phys. Rev. A 64, 012310 (2001)] showed how to encode a qubit into a continuous variable quantum system. This encoding was realized by using non-normalizable quantum codewords, which therefore can only be approximated in any real physical setup. Here we show how a neutral atom, falling through an optical cavity and interacting with a single mode of the intracavity electromagnetic field, can be used to safely encode a qubit into its external degrees of freedom. In fact, the localization induced by a homodyne detection of the cavity field is able to project the near-field atomic motional state into an approximate quantum codeword. The performance of this encoding process is then analyzed by evaluating the intrinsic errors induced in the recovery process by the approximated form of the generated codeword.
Recently it has been proposed to construct quantum error-correcting codes that embed a finite-dimensional Hilbert space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables [D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, 012310 (2001)]. The main difficulty of this continuous variable encoding relies on the physical generation of the quantum codewords. We show that ponderomotive interaction suffices to this end. As a matter of fact, this kind of interaction between a system and a meter causes a frequency change on the meter proportional to the position quadrature of the system. Then, a phase measurement of the meter leaves the system in an eigenstate of the stabilizer generators, provided that system and meter's initial states are suitably prepared. Here we show how to implement this interaction using either travelling or trapped atoms, and how the encoding can be performed on their motional degrees of freedom. The robustness of the codes with respect to the various experimental imperfections is then carefully analyzed.
Most quantum computation schemes propose encoding qubits in two-level systems. Others exploit the use of an infinite-dimensional system. In "Encoding a qubit in an oscillator" [Phys. Rev. A 64, 012310 (2001)], Gottesman, Kitaev, and Preskill (GKP) combined these approaches when they proposed a fault-tolerant quantum computation scheme in which a qubit is encoded in the continuous position and momentum degrees of freedom of an oscillator. One advantage of this scheme is that it can be performed by use of relatively simple linear optical devices, squeezing, and homodyne detection. However, we lack a practical method to prepare the initial GKP states. Here we propose the generation of an approximate GKP state by using superpositions of optical coherent states (sometimes called "Schrödinger cat states"), squeezing, linear optical devices, and homodyne detection.
We show how a qubit can be fault-tolerantly encoded in the infinite-dimensional Hilbert space of an optical mode. The scheme is efficient and realizable with present technologies. In fact, it involves two travelling optical modes coupled by a cross-Kerr interaction, initially prepared in coherent states, one of which is much more intense than the other. At the exit of the Kerr medium, the weak mode is subject to a homodyne measurement and a quantum codeword is conditionally generated in the quantum fluctuations of the intense mode.
Quantum information can be encoded in a scalable manner over the continuous variables (CV) that are the canonically conjugate quantum amplitudes of the electromagnetic field, which are mathematical equivalents of the position and momentum of the quantum harmonic oscillator. Previous results in our group in Charlottesville and the groups of Akira Furusawa in Tokyo, of Nicolas Treps in Paris, and of Ping-Koy Lam in Canberra, have shown that CV quantum information is massively scalable. In addition, Nicolas Menicucci in Melbourne has shown that there exists a fault tolerance threshold for CV quantum computing with the Gottesman-Kitaev-Preskill qubit error encoding. Here, we report on our exploration of the extension of scalable CV encoding by way of intramode, rather than intermode, squeezing and entanglement. As is well known, single-mode squeezing can only be considered along with the "quasi-mode" approximation which assimilates the modes of the optical parametric oscillator cavity used in the experiment to delta functions. As Hans Bachor and Tim Ralph noted, when one ambles beyond this approximation, single-mode squeezing can be described as intramode two-mode squeezing. Here, we aim at generalizing this situation to the case of multipartite entangled cluster states.
Encoding quantum information in continuous variables is intrinsically faulty. Nevertheless, redundant qubits can be used for error correction, as proposed by Gottesman, Kitaev and Preskill in Phys. Rev. A \textbf{64} 012310, (2001). We show how to experimentally implement this encoding using time-frequency continuous degrees of freedom of photon pairs produced by spontaneous parametric down conversion. We experimentally illustrate our results using an integrated AlGaAs photon pairs source. We show how single qubit gates can be implemented and finally propose a theoretical scheme for correcting errors in a circuit-like and in a measurement-based architecture.
Schrödinger cat states, representing coherent superpositions of macroscopically distinguishable states, are indispensable nonclassical resources for continuous-variable quantum information processing. Existing generation protocols typically rely on strong nonlinear interactions, complicated control techniques, or engineered dissipation, posing challenges for experimental implementation. Here, we propose a simple measurement-based protocol for generating Schrödinger cat states in a cavity-QED system by combining coherent driving, dispersive atom--cavity interactions, and atomic postselection. The atom--cavity interaction establishes coherent correlations between the atomic and photonic degrees of freedom, while the subsequent atomic postselection projects the cavity field onto a non-Gaussian superposition state with pronounced Wigner negativity. Numerical simulations based on the Lindblad master equation show that the generated Schrödinger cat states remain robust against moderate cavity dissipation. Our results demonstrate that conditional atomic measurements provide an effective and experimentally accessible approach for preparing nonclassical cavity states without relying on strong optical nonlinearities or engineered dissipation.