Conditional nonlinear operations by sequential Jaynes-Cummings interactions

Kimin Park, Petr Marek, and Radim Filip

Phys. Rev. A 94, 012332 – Published 18 July 2016

Abstract

Nonlinear operations are essential for quantum information processing. We propose a way of implementing a class of nonlinear operations by sequential application of conditional gates based on Jaynes-Cummings (JC) interaction and projective measurements. The scheme has many advantages over the previously proposed all-optical methods and can be applied in several available experimental platforms, such as cavity quantum electrodynamics, trapped ions, and others. We demonstrate performance of the approach on the example of the cubic nonlinearity. We show several different ways in which the full nonlinear operation can be decomposed into sequences of the individual gates, and we compare their performance.

Received 11 April 2016

DOI:https://doi.org/10.1103/PhysRevA.94.012332

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Quantum information processing with continuous variables Quantum information with atoms & light Trapped ions

Atoms Ions Quantum cavities

Jaynes-Cummings model

Atomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Kimin Park^*, Petr Marek, and Radim Filip

Department of Optics, Palacký University, 17. listopadu 1192/12, 77146 Olomouc, Czech Republic

^*park@optics.upol.cz

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Issue

Vol. 94, Iss. 1 — July 2016

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Images

Figure 1
(a) A schematic depiction of a single x gate based on JC interaction. The two-level system is prepared in a specific state, allowed to interact with the oscillator, and finally projected to another specific state by measurement. (b) Schematic realization of the scheme in cavity QED, where the oscillator is a field in a cavity and the two-level system is an atom traversing it. (c) Schematic realization for trapped ions. Here the continuous system is the motional state of the ion, and the two-level system is its internal energy state. The preparation, interaction, and measurement of the two-level system are all performed by a specific train of control pulses.
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Figure 2
A cross section of Wigner function $W (x = 0, p)$ for different orders of approximations (a) by the Taylor expansion of cubic gate ${\hat{O}}_{ideal} (\hat{X}; 0.2)$ , (b) by the incremental expansion, and (c) by the trigonometric expansion on vacuum state. The error of the approximation is seen as the deviation from the curve for target state. We see that the incremental expansion can support a higher order approximation, and a breakdown of approximation occurs for higher order than the Taylor expansion, and the trigonometric expansion is even better than the incremental expansion. The fidelity between the target state and each approximate state is shown as the supplementary measure of validity of approximation.
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Figure 3
Fock representations of the state $R (π / 2) e^{i t {\hat{X}}^{3}} |0〉, R (π / 2) {\hat{O}}_{Taylor}^{(n)} |0〉, R (π / 2) {\hat{O}}_{incre}^{(n)} |0〉$ , and $R (π / 2) {\hat{O}}_{trig}^{(n_{1} \otimes n_{2})} |0〉$ at $t = 0.2$ with $n = n_{1} n_{2} = 2, 6$ . The phase rotation of $R (π / 2) = i^{\hat{n}} = e^{i π \hat{n} / 2}$ was applied to show all Fock components in real numbers. Normalization was removed to show the tendency clearly. We can see that the incremental expansion shows a much closer distribution to the ideal operation than the Taylor expansion, and the trigonometric expansion is even better than the trigonometric expansion.
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Figure 4
A cross section of Wigner function $W (x = 0.424, p)$ for different methods of approximation with order 4. Again, the fidelity between the target state and each approximate state is shown as the supplementary measure of validity of approximation.
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Figure 5
Regions of reasonable fidelities for (a) Taylor expansions, (b) incremental approximation, and (c) trigonometric expansion. We can observe that the area of the regions for a higher order of approximation is smaller than that for a smaller order in the first two cases, while for the trigonometric expansion it is not visible yet at this low order. This reversal of fidelities is due to the breakdown of approximation for a high-order approximation. For a large $α$ , the boundary $t$ is smaller for all cases. The eclipsed regions are shown with dashed boundary lines.
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