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arXiv:quant-ph/0405121 · Physical Review A

Efficiencies for the single-mode operation of a quantum optical nonlinear shift gate

Kunihiro Kojima, Holger Friedrich Hofmann, Shigeki Takeuchi, Keiji Sasaki

Published Jul 23, 2004• 25 citations
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Abstract

We investigate the single-mode operation of a quantum optical nonlinear $\ensuremath{\pi}$ phase-shift gate implemented by a single two-level atom in one-dimensional free space. Since the single-mode property of the input photons at the atom is not preserved in the interaction at the atom, we analyze the efficiency of single-mode operation that can still be achieved. We show how the input pulse shape can be optimized to obtain high efficiencies for the nonlinear single-mode operation. With this analysis, we obtain an optimal single-mode transmittance per photon of 78% for the successful nonlinear $\ensuremath{\pi}$ phase-shift operation.

DOI
10.1103/physreva.70.013810

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We investigate the single-mode operation of a quantum optical nonlinear \ensuremathπ\ensuremath{\pi}\ensuremathπ phase-shift gate implemented by a single two-level atom in one-dimensional free space. Since the single-mode property of the input photons at the atom is not preserved in the interaction at the atom, we analyze the efficiency of single-mode operation that can still be achieved. We show how the input pulse shape can be optimized to obtain high efficiencies for the nonlinear single-mode operation. With this analysis, we obtain an optimal single-mode transmittance per photon of 78% for the successful nonlinear \ensuremathπ\ensuremath{\pi}\ensuremathπ phase-shift operation.
Ideally, strong nonlinearities could be used to implement quantum gates for photonic qubits by well-controlled two-photon interactions. However, the dependence of the nonlinear interaction on frequency and time makes it difficult to preserve a coherent pulse shape that could justify a single-mode model for the time-frequency degree of freedom of the photons. In this paper, we analyze the problem of temporal multimode effects by considering the pulse shape of the average output field obtained from a coherent input pulse. It is shown that a significant part of the two-photon state transformation can be derived from this semiclassical description of the optical nonlinearity. The effect of a nonlinear system on a two-photon state can then be determined from the density-matrix dynamics of the coherently driven system using input-output theory. As an example, the resonant nonlinearity of a single two-level atom is characterized. The results indicate that the most efficient nonlinear effect may not be the widely studied single-mode phase shift, but the transfer of one of the photons to an orthogonal mode distinguished by its temporal and spectral properties.
We investigate the effects of decoherence on the nonlinear phase shift obtained from a one-dimensional atom implemented by a one-sided microcavity. The one-dimensional atom is realized by coupling a single atom confined in the cavity to the one-dimensional input-output fields of the cavity. The nonlinear optical response obtained from the one-dimensional atom can then be analyzed by using the optical Bloch equations. It is shown that the nonlinear phase shift depends on the coupling efficiency η, which is defined by the ratio of the coherent atom-field coupling rate and the total dipole relaxation rate. In particular, a maximal phase shift of π is obtained on resonance for coupling efficiencies η larger than one half, while the maximal phase shifts for coupling efficiencies η smaller than one half are obtained off resonance and are always smaller than π/2.
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.
A quantum emitter interacting with photons in a single optical-mode constitutes a one-dimensional atom. A coherent and efficiently coupled one-dimensional atom provides a large nonlinearity, enabling photonic quantum gates. Achieving a high coupling efficiency (β factor) and low dephasing is challenging. Here, we use a semiconductor quantum dot in an open microcavity as an implementation of a one-dimensional atom. With a weak laser input, we achieve an extinction of 99.2% in transmission and a concomitant bunching in the photon statistics of g^{(2)}(0)=587, showcasing the reflection of the single-photon component and the transmission of the multi-photon components of the coherent input. The tunable nature of the microcavity allows β to be adjusted and gives control over the photon statistics-from strong bunching to antibunching-and the phase of the transmitted photons. We obtain excellent agreement between experiment and theory by going beyond the single-mode Jaynes-Cummings model. Our results pave the way towards the creation of exotic photonic states and two-photon phase gates.
The nonlinear optical response obtained from a model system of a quantum phase gate is investigated. The model system consists of a thin infinite atomic layer of two-level atoms placed in front of a perfect reflecting mirror. The optical response obtained from the model system is semiclassically analyzed using the finite difference time domain method with the optical Bloch equations. It is shown that a nonlinear phase shift of $\ensuremath{\pi}$ is achieved when the atomic layer is placed at an antinode of the input field. This result is consistent with the theoretical result obtained from a one-dimensional atom model [H. F. Hofmann, K. Kojima, S. Takeuchi, and K. Sasaki, J. Opt. B: Quantum Semiclassical Opt. 5, 218 (2003)]. The dependence of the nonlinear phase shift on the position of the atomic layer is also studied in detail.
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.