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arXiv:0812.1626 · Physical Review A

Generating “squeezed” superpositions of coherent states using photon addition and subtraction

Petr Marek, Hyunseok Jeong, M. S. Kim

Published Dec 4, 2008• 117 citations
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Abstract

We study how photon addition and subtraction can be used to generate squeezed superpositions of coherent states (SSCSs) in free-traveling fields with high fidelities and large amplitudes. It is shown that an arbitrary $N$-photon subtraction results in the generation of a SSCS with nearly the perfect fidelity $(F>0.999)$ regardless of the number of photons subtracted. In this case, the amplitude of the SSCS increases as the number of the subtracted photons gets larger. For example, two-photon subtraction from a squeezed vacuum state of $6.1\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ can generate a SSCS of $\ensuremath{\alpha}=1.26$, while in the case of the four-photon subtraction a SSCS of a larger amplitude $\ensuremath{\alpha}=1.65$ is obtained under the same condition. When a photon is subtracted from a squeezed vacuum state and another photon is added subsequently, a SSCS with a lower fidelity $(F\ensuremath{\approx}0.96)$ yet higher amplitude $(\ensuremath{\alpha}\ensuremath{\approx}2)$ can be generated. We analyze some experimental imperfections including inefficiency of the detector used for the photon subtraction.

DOI
10.1103/physreva.78.063811

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We study how photon addition and subtraction can be used to generate squeezed superpositions of coherent states (SSCSs) in free-traveling fields with high fidelities and large amplitudes. It is shown that an arbitrary NNN-photon subtraction results in the generation of a SSCS with nearly the perfect fidelity (F>0.999)(F>0.999)(F>0.999) regardless of the number of photons subtracted. In this case, the amplitude of the SSCS increases as the number of the subtracted photons gets larger. For example, two-photon subtraction from a squeezed vacuum state of 6.1dB6.1\phantom{\rule{0.3em}{0ex}}\mathrm{dB}6.1dB can generate a SSCS of \ensuremathα=1.26\ensuremath{\alpha}=1.26\ensuremathα=1.26, while in the case of the four-photon subtraction a SSCS of a larger amplitude \ensuremathα=1.65\ensuremath{\alpha}=1.65\ensuremathα=1.65 is obtained under the same condition. When a photon is subtracted from a squeezed vacuum state and another photon is added subsequently, a SSCS with a lower fidelity (F\ensuremath≈0.96)(F\ensuremath{\approx}0.96)(F\ensuremath≈0.96) yet higher amplitude (\ensuremathα\ensuremath≈2)(\ensuremath{\alpha}\ensuremath{\approx}2)(\ensuremathα\ensuremath≈2) can be generated. We analyze some experimental imperfections including inefficiency of the detector used for the photon subtraction.
We have created heralded coherent-state superpositions (CSSs) by subtracting up to three photons from a pulse of squeezed vacuum light. To produce such CSSs at a sufficient rate, we used our high-efficiency photon-number-resolving transition edge sensor to detect the subtracted photons. This experiment is enabled by and utilizes the full photon-number-resolving capabilities of this detector. The CSS produced by three-photon subtraction had a mean-photon number of $2.{75}_{\ensuremath{-}0.24}^{+0.06}$ and a fidelity of $0.{59}_{\ensuremath{-}0.14}^{+0.04}$ with an ideal CSS. This confirms that subtracting more photons results in higher-amplitude CSSs.
We study the method of generation of states that approximate superpositions of large-amplitude coherent states (SCSs) with high fidelity in free-traveling fields. Our approach is based on the representation of an arbitrary single-mode pure state, and SCSs in particular, in terms of displaced number states with an arbitrary displacement amplitude. The proposed optical scheme is based on alternation of photon additions and displacement operators (in the general case, N photon additions and N − 1 displacements are required) with a seed coherent state to generate both even and odd displaced squeezed SCSs regardless of the parity of the used photon additions. It is shown that the optical scheme studied is sensitive to the seed coherent state if the other parameters are unchanged. Output states can approximate either even squeezed SCS or odd SCS shifted relative to each other by some value. This allows constructing a local rotation operator, in particular, the Hadamard gate, which is a mainframe element for quantum computation with coherent states. We also show that three-photon additions with two intermediate displacement operators are sufficient to generate even displaced squeezed SCS with the amplitude 1.7 and fidelity more than 0.99. The effects deteriorating the quality of output states are considered.
In this paper, the fidelity of photon subtracted (or added) squeezed vacuum state with arbitrary number of photons and squeezed cat state is derived analytically. The result shows that whether the photon is added or subtracted, the maximum fidelity increases with the increase of the change of photon number, and the amplitude of the superposition state corresponding to the maximum fidelity also increases. In addition, for the same number of subtracted or added photons, the amplitude of the superposition state corresponding to the maximum fidelity in the case of added photon is larger than in the case of subtracted photon, but the maximum fidelity in the case is smaller. Although it is more difficult to make photon added than photon subtracted the photon added can be used as an important method to obtain cat state of large amplitude.
It is shown that a linear superposition of two macroscopically distinguishable optical coherent states can be generated using a single photon source and simple all-optical operations. Weak squeezing on a single photon, beam mixing with an auxiliary coherent state, and photon detecting with imperfect threshold detectors are enough to generate a coherent state superposition in a free propagating optical field with a large coherent amplitude $(\ensuremath{\alpha}>2)$ and high fidelity (F>0.99)(F>0.99)(F>0.99). In contrast to all previous schemes to generate such a state, our scheme does not need photon number resolving measurements nor Kerr-type nonlinear interactions. Furthermore, it is robust to detection inefficiency and exhibits some resilience to photon production inefficiency.