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arXiv:quant-ph/0507137 · Physical Review A

Quantum phase-gate operation based on nonlinear optics: Full quantum analysis

Carlo Ottaviani, S. Rebić, David Vitali, P. Tombesi

Published Jan 3, 2006• 82 citations
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Abstract

We present a full quantum treatment of a five-level atomic system coupled to two quantum and two classical light fields. The two quantum fields undergo a cross-phase-modulation induced by electromagnetically induced transparency. The performance of this configuration as a two-qubit quantum phase gate for traveling single photons is examined. A trade-off between the size of the conditional phase shift and the fidelity of the gate is found. Nonetheless, a satisfactory gate performance is still found to be possible in the transient regime, corresponding to a fast gate operation.

DOI
10.1103/physreva.73.010301

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We present a full quantum treatment of a five-level atomic system coupled to two quantum and two classical light fields. The two quantum fields undergo a cross-phase-modulation induced by electromagnetically induced transparency. The performance of this configuration as a two-qubit quantum phase gate for traveling single photons is examined. A trade-off between the size of the conditional phase shift and the fidelity of the gate is found. Nonetheless, a satisfactory gate performance is still found to be possible in the transient regime, corresponding to a fast gate operation.
We analyze in detail the proposal for a two-qubit gate for travelling single-photon qubits recently presented by Ottaviani et al. [Phys. Rev. A 73, 010301(R) (2006)]. The scheme is based on an ensemble of five-level atoms coupled to two quantum and two classical light fields. The two quantum fields undergo cross-phase modulation induced by electromagnetically induced transparency. The performance of this two-qubit quantum phase gate for travelling single-photon qubits is thoroughly examined in the steady-state and transient regimes, by means of a full quantum treatment of the system dynamics. In the steady-state regime, we find a general trade-off between the size of the conditional phase shift and the fidelity of the gate operation. However, this trade-off can be bypassed in the transient regime, where a satisfactory gate operation is found to be possible, significantly reducing the gate operation time.
The interaction of a five-level atomic system involving electromagnetically induced transparency with four light fields is investigated. Two different operating regimes are considered, semi-classical and quantized, where two light fields are treated as quantum. In the semi-classical regime, the dispersive properties of the scheme are analyzed in detail, and the conditions leading to group velocity matching for two of the light fields are identified. In the quantized regime, the two quantum fields undergo cross-phase modulation induced by electromagnetically induced transparency. Based on this, a recent proposal [C. Ottaviani et al., Phys. Rev. A 73 , 010301 (R) (2006)] for a two-qubit quantum phase gate for traveling single-photon qubits is examined. An experimental scheme for verification of QPG operation is proposed.
In this dissertation, we shall focus on theoretically studying quantum nonlinear optical schemes to construct a conditional phase gate at the single-photon level. With an aim to develop analytical models, we shall carry out a rigorous quantized multimode field analysis of some of these schemes involving only the interacting field operators. More specifically, we shall first study the three-wave mixing process involving two single-photons in a second-order nonlinear medium (x(2)) under two different cases viz. when the photons are traveling with equal velocities and when they are traveling with different velocities, and explore the possibility of using them for building a conditional phase phase gate. Finally, we shall study the interaction of single-photon wavepackets with a realistic atomic system viz. an ensemble of five-level atoms, to construct a phase gate. We will particularly look at the "giant Kerr" effect in electromagnetically induced transparency to explore the possibility of using this scheme for achieving a conditional phase shift and to understand how the bandwidth gets restricted naturally in such an atomic system.
We first investigate the nonlinear optical response of a five-level tripod atomic system in electromagnetically induced transparency and then propose a 3-qubit phase gate protocol based on the fifth-order nonlinearity. The cross-phase modulation among the three weak fields induced by the fifth-order nonlinearity in electromagnetically induced transparency can produce the phase shift of order π, which can be used to realize the 3-qubit polarization phase gate. This protocol should have potential applications in quantum information processing.
We theoretically investigate the implementation of a quantum controlled-phase gate in a system constituted by a single atom inside an optical cavity, based on the electromagnetically induced transparency effect. First we show that a probe pulse can experience a $\ensuremath{\pi}$ phase shift due to the presence or absence of a classical control field. Considering the interplay of the cavity-EIT effect and the quantum memory process, we demonstrated a controlled-phase gate between two single photons. To this end, first one needs to store a (control) photon in the ground atomic states. In the following, a second (target) photon must impinge on the atom-cavity system. Depending on the atomic state, this second photon will be either transmitted or reflected, acquiring different phase shifts. This protocol can then be easily extended to multiphoton systems, i.e., keeping the control photon stored, it may induce phase shifts in several single photons, thus enabling the generation of multipartite entangled states. We explore the relevant parameter space in the atom-cavity system that allows the implementation of quantum controlled-phase gates using the recent technologies. In particular, we have found a lower bound for the cooperativity of the atom-cavity system which enables the implementation of phase shift on single photons. The induced shift on the phase of a photonic qubit and the controlled-phase gate between single photons, combined with optical devices, enable one to perform universal quantum computation.