[Paper Review] Spectral Effects of Fast Response Cross Kerr Non-Linearity on Quantum Gate

Patrick M. Leung, Timothy C. Ralph+2 more|ArXiv.org|Oct 16, 2008

Spectroscopy and Quantum Chemical Studies1 references3 citations

TL;DR

This paper demonstrates that fast-response cross-Kerr nonlinearities induce only negligible phase shifts on two-photon input states due to spectral entanglement, rendering them ineffective for high-fidelity quantum gates. The authors show that engineering the medium's dispersion to dominate over non-instantaneous response, combined with phase matching, can suppress spectral entanglement and enable high-fidelity quantum gate operation.

ABSTRACT

According to idealized models, a strong Kerr non-linearity may be used to build optical quantum gates for optical quantum information processing by inducing conditional phase shifts on quantum states. Recently, Shapiro (PRA 73, 062305 (2006)) argued that for a Kerr medium with non-instantaneous but fast response, essentially no phase shift is induced on two-single-photon input states, and thus a quantum gate build from such a medium cannot work. Here we show that a fast response Kerr medium induces some but very little phase shifts on a two-single-photon input state, and it is insufficient for high fidelity quantum computation. We point out that this is caused by the medium imparting spectral entanglement to the input photons. We further show that a way to circumvent this problem and achieve a high fidelity gate, is to engineer the dispersion properties of the medium to give a dominant spectral effect over the non-instantaneous response, in addition to satisfying a phase matching condition.

Motivation & Objective

• To investigate the spectral effects of fast-response cross-Kerr nonlinearities on two-photon quantum states in optical quantum gates.
• To resolve the contradiction between idealized models (which predict high-fidelity gates) and Shapiro's claim (that fast-response media induce negligible phase shifts).
• To identify spectral entanglement as the root cause of reduced phase shifts in non-instantaneous Kerr media.
• To propose a solution via engineered dispersion and phase matching to restore high-fidelity gate operation.

Proposed method

• Model the spectral response of a χ(3) cross-Kerr medium with non-instantaneous but fast response using time-ordered Dyson series expansion.
• Derive the unitary evolution operator for the interaction Hamiltonian, accounting for time-ordering due to non-commuting interactions at different times.
• Analyze the output state for a separable two-photon input state, showing that spectral entanglement arises from non-instantaneous response.
• Introduce dispersion into the medium model to dominate over non-instantaneous response, leading to time-commuting interaction Hamiltonians.
• Derive a phase matching condition that ensures constructive interference and suppresses spectral entanglement.
• Use delta-function constraints and sinc functions to analyze interference terms and show cancellation of non-commuting contributions when dispersion dominates.

Experimental results

Research questions

• RQ1 Why do fast-response cross-Kerr nonlinearities fail to induce sufficient phase shifts on two-single-photon states despite strong interaction strength?
• RQ2 What is the role of spectral entanglement in degrading the fidelity of quantum gates based on non-instantaneous Kerr media?
• RQ3 Can dispersion engineering mitigate the spectral entanglement problem and restore high-fidelity gate operation?
• RQ4 What phase matching condition must be satisfied to ensure dominant spectral effects over non-instantaneous response in a Kerr medium?
• RQ5 Under what conditions does the interaction Hamiltonian commute at different times, enabling a simple unitary evolution?

Key findings

• A fast-response cross-Kerr medium induces only negligible phase shifts on two-single-photon input states due to spectral entanglement, contradicting idealized models.
• The spectral entanglement arises because the non-instantaneous response correlates the frequencies of the two photons, transforming a separable input state into a spectrally entangled output.
• When dispersion dominates over non-instantaneous response, the interaction Hamiltonian commutes at different times, simplifying the unitary evolution to a time-ordered exponential without higher-order corrections.
• The condition for time-commuting interaction is achieved when the dispersion-induced phase shifts dominate, leading to cancellation of non-commuting terms in the Hamiltonian commutator.
• A phase matching condition is derived that ensures constructive interference of the spectral components, minimizing spectral entanglement and maximizing gate fidelity.
• With proper dispersion engineering and phase matching, the gate fidelity can be restored to high levels, making fast-response Kerr media viable for optical quantum computation.