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Physical Review A

Single-photon Kerr nonlinearities do not help quantum computation

Jeffrey H. Shapiro

Published Jun 7, 2006• 311 citations
Publisher

Abstract

By embedding an atom capable of electromagnetically induced transparency inside an appropriate photonic-crystal microcavity it may become possible to realize an optical nonlinearity that can impart a $\ensuremath{\pi}$-rad-peak phase shift in response to a single-photon excitation. Such a device, if it operated at high fidelity, would then complete a universal gate set for all-optical quantum computation. It is shown here that the causal, noninstantaneous behavior of any ${\ensuremath{\chi}}^{(3)}$ nonlinearity is enough to preclude such a high-fidelity operation. In particular, when a single-photon-sensitive ${\ensuremath{\chi}}^{(3)}$ nonlinearity has a response time that is much shorter than the duration of the quantum computer's single-photon pulses, essentially no overall phase shift is imparted to these pulses by cross-phase modulation. Conversely, when this nonlinearity has a response time that is much longer than this pulse duration a single-photon pulse can induce a $\ensuremath{\pi}$-rad overall phase shift through cross-phase modulation, but the phase noise injected by the causal, noninstantaneous response function precludes this from being a high-fidelity operation.

DOI
10.1103/physreva.73.062305

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By embedding an atom capable of electromagnetically induced transparency inside an appropriate photonic-crystal microcavity it may become possible to realize an optical nonlinearity that can impart a \ensuremathπ\ensuremath{\pi}\ensuremathπ-rad-peak phase shift in response to a single-photon excitation. Such a device, if it operated at high fidelity, would then complete a universal gate set for all-optical quantum computation. It is shown here that the causal, noninstantaneous behavior of any \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3) nonlinearity is enough to preclude such a high-fidelity operation. In particular, when a single-photon-sensitive \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3) nonlinearity has a response time that is much shorter than the duration of the quantum computer's single-photon pulses, essentially no overall phase shift is imparted to these pulses by cross-phase modulation. Conversely, when this nonlinearity has a response time that is much longer than this pulse duration a single-photon pulse can induce a \ensuremathπ\ensuremath{\pi}\ensuremathπ-rad overall phase shift through cross-phase modulation, but the phase noise injected by the causal, noninstantaneous response function precludes this from being a high-fidelity operation.
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.
In third order nonlinear mediums, the intensity of one optical field can modify the phase of another optical field. This is known as cross-phase modulation (XPM) and is important for Quantum Information Processing (QIP) as phase shifts of single-photons that exceed π rad can be used to implement two-qubit quantum logic gates. For a conventional XPM scheme, the Kerr nonlinearity responsible for XPM is too small for appreciable phase shifts to be achieved with low intensity optical fields. However, Electromagnetically Induced Transparency (EIT) can increase this nonlinearity by several orders of magnitude. This thesis discusses the experimental construction of a laser-atomic system capable of EIT modulated XPM. This includes construction of a rubidium magneto-optical trap (RbMOT), characterization of this RbMOT, demonstration of EIT in the RbMOT, and a system capable of detecting optical phase shifts produced by EIT. The talk will conclude with an outlook on further implementations to the experiment that will increase this nonlinearity such that phase shifts at the single-photon level are possible. | 76 pages
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.
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.