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Chinese Optics Letters
Vol. 13, Issue 3, 030801 (2015)

New fractional entangling transform and its quantum mechanical correspondence

Shuang Xu, Liyun Hu^*, and Jiehui Huang

Author Affiliations

Center for Quantum Science and Technology, Jiangxi Normal University, Nanchang 330022, China

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DOI: 10.3788/COL201513.030801 Cite this Article Set citation alerts

Shuang Xu, Liyun Hu, Jiehui Huang. New fractional entangling transform and its quantum mechanical correspondence[J]. Chinese Optics Letters, 2015, 13(3): 030801 Copy Citation Text

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Abstract

In this Letter, a new fractional entangling transformation (FrET) is proposed, which is generated in the entangled state representation by a unitary operator

\exp {i θ (a b^{} + a^{} b)}

where

a (b)

is the Bosonic annihilate operator. The operator is actually an entangled one in quantum optics and differs evidently from the separable operator,

\exp {i θ (a^{} a + b^{} b)}

, of complex fractional Fourier transformation. The additivity property is proved by employing the entangled state representation and quantum mechanical version of the FrET. As an application, the FrET of a two-mode number state is derived directly by using the quantum version of the FrET, which is related to Hermite polynomials.

F_{θ} [f (x)] = \int_{- \infty}^{\infty} K_{θ} (x, y) f (x) d x,

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K_{θ} (x, y) = \sqrt{\frac{e^{i (\frac{π}{2} - θ)}}{2 π \sin θ}} \exp {- i \frac{x^{2} + y^{2}}{2 \tan θ} + \frac{i x y}{\sin θ}} .

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S_{θ} = \int_{- \infty}^{\infty} d x d y | y 〉 K_{θ} (x, y) 〈 x | ≕ \exp {(e^{i θ} - 1) a^{†} a} : = \exp {i θ a^{†} a},

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F_{θ} [f (x)] = \int_{- \infty}^{\infty} 〈 y | S_{θ} | x 〉 〈 x | f 〉 d x = 〈 y | S_{θ} | f 〉,

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\int_{- \infty}^{\infty} d x | x 〉 〈 x | = 1 .

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F_{θ} [f] (η^{'}) = \int_{- \infty}^{\infty} \frac{d^{2} η}{π} K_{C} (η^{'}, η) f (η), K_{C} (η^{'}, η) = \frac{e^{i (θ - \frac{π}{2})}}{2 \sin θ} \exp [\frac{i ({| η^{'} |}^{2} + {| η |}^{2})}{2 \tan θ} - \frac{i (η^{' *} η + η^{*} η^{'})}{2 \sin θ}] .

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\int_{- \infty}^{\infty} \frac{d^{2} η}{π} | η 〉 〈 η | = 1 .

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| η 〉 = \exp [- \frac{1}{2} {| η |}^{2} + η a^{†} - η^{*} b^{†} + a^{†} b^{†}] | 00 〉 .

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F_{θ} [f] (η^{'}) = 〈 η^{'} | \exp [- i θ (a^{†} a + b^{†} b)] | f 〉 .

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K_{C} (η^{'}, η) = \frac{1}{2 \sin θ} \exp [\frac{i (η^{' 2} + η^{2} + η^{' * 2} + η^{* 2})}{4 \tan θ} - \frac{i (η η^{'} + η^{*} η^{' *})}{2 \sin θ}],

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U_{θ} = \int_{- \infty}^{\infty} \frac{d^{2} η d^{2} η^{'}}{π^{2}} | η^{'} 〉 K_{C} (η^{'}, η) 〈 η | .

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U_{θ} = \frac{1}{2 \sin θ} \int \frac{d^{2} η d^{2} η^{'}}{π^{2}} : \exp {- \frac{1}{2} ({| η^{'} |}^{2} + {| η |}^{2}) + η^{'} a^{†} - η^{' *} b^{†} + η^{*} a - η b + \frac{i (η^{2} + η^{* 2} + η^{' 2} + η^{' * 2})}{4 \tan θ} - \frac{i (η η^{'} + η^{*} η^{' *})}{2 \sin θ} + a b + a^{†} b^{†} - a^{†} a - b^{†} b} ≕ \exp [(\cos θ - 1) (a^{†} a + b^{†} b) + i (a b^{†} + a^{†} b) \sin θ] :

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\int \frac{d^{2} z}{π} \exp (- ζ {| z |}^{2} + ξ z + η z^{*} + f z^{2} + g z^{* 2}) = \frac{1}{\sqrt{ζ^{2} - 4 f g}} \exp [\frac{ζ ξ η + ξ^{2} g + η^{2} f}{ζ^{2} - 4 f g}] .

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U_{θ} a U_{θ}^{†} = {\frac{\partial}{\partial τ} \exp [(a \cos θ - i b \sin θ) τ] |}_{τ = 0} = a \cos θ - i b \sin θ, U_{θ} b U_{θ}^{†} = b \cos θ - i a \sin θ,

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\frac{\partial}{\partial θ} U_{θ} = i (b^{†} U_{θ} a + a^{†} U_{θ} b) \cos θ - (a^{†} U_{θ} a + b^{†} U_{θ} b) \sin θ = [i (b^{†} U_{θ} a U_{θ}^{†} + a^{†} U_{θ} b U_{θ}^{†}) \cos θ - (a^{†} U_{θ} a U_{θ}^{†} + b^{†} U_{θ} b U_{θ}^{†}) \sin θ] U_{θ} = i (a b^{†} + a^{†} b) U_{θ},

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U_{θ} = \exp {i θ (a b^{†} + a^{†} b)} .

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F_{θ} [f] (η^{'}) = \int_{- \infty}^{\infty} \frac{d^{2} η}{π} K_{C} (η^{'}, η) f (η) = \int_{- \infty}^{\infty} \frac{d^{2} η}{π} 〈 η^{'} | U_{θ} | η 〉 〈 η | | f 〉 = 〈 η^{'} | \exp {i θ (a b^{†} + a^{†} b)} | f 〉,

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F_{θ + α} [f (η)] = 〈 η^{'} | \exp {i (θ + α) (a b^{†} + a^{†} b)} | f 〉 = \int_{- \infty}^{\infty} \frac{d^{2} η^{''}}{π} 〈 η^{'} | \exp {i θ (a b^{†} + a^{†} b)} | η^{''} 〉 \times \int_{- \infty}^{\infty} \frac{d^{2} η}{π} 〈 η^{''} | \exp {i α (a b^{†} + a^{†} b)} | η 〉 f (η) = \int_{- \infty}^{\infty} \frac{d^{2} η^{''}}{π} 〈 η^{'} | \exp {i θ (a b^{†} + a^{†} b)} | η^{''} 〉 F_{α} [f (η)] = F_{θ} F_{α} [f (η)] .

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f (η) = {\frac{e^{- \frac{1}{2} {| η |}^{2}}}{\sqrt{m! n!}} \frac{\partial^{m + n}}{\partial α^{m} \partial β^{n}} \exp [η^{*} α - η β + α β] |}_{α, β = 0} = \frac{i^{m + n} e^{- \frac{1}{2} {| η |}^{2}}}{\sqrt{m! n!}} H_{m, n} (- i η^{*}, i η),

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H_{m, n} (x, y) = {\frac{\partial^{m + n}}{\partial t^{m} \partial τ^{n}} \exp [- t τ + t x + τ y] |}_{t = τ = 0} .

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U_{θ} | m, n 〉 = \frac{1}{\sqrt{m! n!}} {(a^{†} \cos θ + i b^{†} \sin θ)}^{m} {(b^{†} \cos θ + i a^{†} \sin θ)}^{n} | 00 〉 = {\frac{1}{\sqrt{m! n!}} \frac{\partial^{m + n}}{\partial α^{m} \partial β^{n}} e^{a^{†} (α \cos θ + i β \sin θ) + b^{†} (i α \sin θ + β \cos θ)} | 00 〉 |}_{α, β = 0} = {\frac{1}{\sqrt{m! n!}} \frac{\partial^{m + n}}{\partial α^{m} \partial β^{n}} | \bar{α}, \bar{β} 〉 |}_{α, β = 0},

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F_{θ} [f (η)] = 〈 η^{'} | U_{θ} | m, n 〉 = {\frac{e^{- \frac{1}{2} {| η^{'} |}^{2}}}{\sqrt{m! n!}} \frac{\partial^{m + n}}{\partial α^{m} \partial β^{n}} \exp [η^{' *} \bar{α} - η^{'} \bar{β} + \bar{α} \bar{β}] |}_{α, β = 0} = \sum_{l = 0}^{\min (m, n)} {(i \sqrt{\frac{i \sin 2 θ}{2}})}^{m + n - 2 l} \frac{\sqrt{n! m!} e^{- \frac{1}{2} {| η^{'} |}^{2}} \cos^{l} 2 θ}{4^{l} l! (n - l)! (m - l)!} \times H_{m - l} (\frac{\bar{η}}{i \sqrt{2 i \sin 2 θ}}) H_{n - l} (\frac{{\bar{η}}^{*}}{i \sqrt{2 i \sin 2 θ}}),

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H_{n} (x) = {\frac{\partial^{n}}{\partial t^{n}} \exp (2 x t - t^{2}) |}_{t = 0}, \frac{d}{d x^{l}} H_{n} (x) = \frac{2^{l} n!}{(n - l)!} H_{n - l} (x) .

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Shuang Xu, Liyun Hu, Jiehui Huang. New fractional entangling transform and its quantum mechanical correspondence[J]. Chinese Optics Letters, 2015, 13(3): 030801

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