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arXiv:1009.2390 · Physical Review A

Quantum state engineering by a coherent superposition of photon subtraction and addition

Su‐Yong Lee, Hyunchul Nha

Published Nov 15, 2010• 166 citations
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Abstract

We study a coherent superposition $t\mathrm{a\ifmmode \hat{}\else \^{}\fi{}}+r{\mathrm{a\ifmmode \hat{}\else \^{}\fi{}}}^{\ifmmode\dagger\else\textdagger\fi{}}$ of field annihilation and creation operators acting on continuous variable systems and propose its application for quantum state engineering. Specifically, it is investigated how the superposed operation transforms a classical state to a nonclassical one, together with emerging nonclassical effects. We also propose an experimental scheme to implement this elementary coherent operation and discuss its usefulness to produce an arbitrary superposition of number states involving up to two photons.

DOI
10.1103/physreva.82.053812

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We study a coherent superposition ta\ifmmode^\elseˆ\fi+ra\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fit\mathrm{a\ifmmode \hat{}\else \^{}\fi{}}+r{\mathrm{a\ifmmode \hat{}\else \^{}\fi{}}}^{\ifmmode\dagger\else\textdagger\fi{}}ta\ifmmode^\elseˆ\fi+ra\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fi of field annihilation and creation operators acting on continuous variable systems and propose its application for quantum state engineering. Specifically, it is investigated how the superposed operation transforms a classical state to a nonclassical one, together with emerging nonclassical effects. We also propose an experimental scheme to implement this elementary coherent operation and discuss its usefulness to produce an arbitrary superposition of number states involving up to two photons.
We investigate the practical scheme of engineering a qubit of two optical continuous-variable states, based on squeezed vacuum and coherently displaced photon subtraction. A fully multimode model is developed to describe a continuous wave (cw) setting of the experiment, which has been successfully implemented to generate highly pure non-classical states. The model agrees well with the experimental results and the numerical results show that the intrinsic imperfection due to the cw nature of the source is relatively small. Our photon subtraction technique could also be useful for preparing optical Schrödinger's cat-like states in arbitrary superposition, which is a basic qubit resource for linear optics coherent state quantum computing.
We consider an experimentally realizable scheme for manipulating quantum states using a general superposition of products of field annihilation ( ) and creation ( ) operators of the type ( ), with s 2 + t 2 = 1. Such an operation, when applied on states with classical features, is shown to introduce strong nonclassicality. We quantify the generated degree of nonclassicality by the negative volume of Wigner distribution in the phase space and investigate two other observable nonclassical features, sub-Poissonian statistics and squeezing. We find that the operation introduces negativity in the Wigner distribution of an input coherent state and changes the Gaussianity of an input thermal state. This provides the possibility of engineering quantum states with specific nonclassical features.
We propose an experiment to directly prove the commutation relation between bosonic annihilation and creation operators, based on the recent experimental success in single-photon subtraction and addition. We devise a single-photon interferometer to realize coherent superpositions of two sequences of photon addition and subtraction. Depending on the interference outcome, the commutation relation is directly proven or a highly nonclassical state is produced. Experimental imperfections are assessed to show that the realization of the scheme is highly feasible.
We investigate how the entanglement properties of a two-mode state can be improved by performing a coherent superposition operation $t\mathrm{a\ifmmode \hat{}\else \^{}\fi{}}+r{\mathrm{a\ifmmode \hat{}\else \^{}\fi{}}}^{\ifmmode\dagger\else\textdagger\fi{}}$ of photon subtraction and addition, proposed by Lee and Nha [Phys. Rev. A 82, 053812 (2010)], on each mode. We show that the degree of entanglement, the Einstein-Podolsky-Rosen--type correlation, and the performance of quantum teleportation can be all enhanced for the output state when the coherent operation is applied to a two-mode squeezed state. The effects of the coherent operation are more prominent than those of the mere photon subtraction $\mathrm{a\ifmmode \hat{}\else \^{}\fi{}}$ and the addition ${\mathrm{a\ifmmode \hat{}\else \^{}\fi{}}}^{\ifmmode\dagger\else\textdagger\fi{}}$ particularly in the small-squeezing regime, whereas the optimal operation becomes the photon subtraction (case of r=0r=0r=0) in the large-squeezing regime.
Based on the coherent state (S1) and the operator [Formula: see text], we induce other three quantum states (here we abbreviate them as S2, S3 and S4). S2 is obtained by operating the operator on S1 directly. S3 is an orthogonal state of S1 constructed from the orthogonalizer relevant with that operator. S4 is a continuous-variable (CV) qubit state superposed from S1 and S3. We study and compare the mathematical and physical properties of such four quantum states. We demonstrate some statistical properties for S1–S4, including the mean photon number (MPN), anti-bunching effect, quadrate squeezing, photon number distribution, Husimi Q-function and Wigner function. The numerical results show some interesting non-classical characters for such states. It is worthy to note that the photon-added coherent state introduced by Agarwal and Tara is only a special case of our considered states.