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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," Chin. Opt. Lett. 13, 030801 (2015) 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}} .

(2)

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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},

(3)

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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}],

(10)

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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,

(14)

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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},

(15)

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

(16)

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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 〉,

(17)

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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)] .

(18)

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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} .

(20)

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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," Chin. Opt. Lett. 13, 030801 (2015)

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