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Acta Photonica Sinica
Vol. 49, Issue 10, 1027001 (2020)

Quantum Theory of Optical Fractional Fourier Transform

Ke ZHANG¹, Lan-lan LI¹, Hai-jun YU¹, Jian-ming DU¹, and Hong-yi FAN^2、*

Author Affiliations

¹School of Electronic Engineering，Huainan Normal University，Huainan，Anhui 232038，China

²Department of Material Science and Engineering，University of Science and Technology of China，Hefei 230026，China

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DOI: 10.3788/gzxb20204910.1027001 Cite this Article

Ke ZHANG, Lan-lan LI, Hai-jun YU, Jian-ming DU, Hong-yi FAN. Quantum Theory of Optical Fractional Fourier Transform[J]. Acta Photonica Sinica, 2020, 49(10): 1027001 Copy Citation Text

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Abstract

The aim of this paper is to find out the operator for generating fractional Fourier transform in Hermitian polynomial theory with the operator as the argument， and to incorporate fractional Fourier transform into quantum theory. The role of coordinate-momentum exchanging operator is explored in playing FFrT's addition rule. In the whole derivation the generalized generating function formula of operator Hermitian polynomials and the integration method within ordered product of operators are used. The core of operator Hermite polynomial theory is the operator identity

H_{n} (Q) = : {(2 Q)}^{n} :

， which turns the operator of complex special function into power series in normal ordering， as a result，this greatly simplifies calculations.

Keywords

Coordinate-momentum exchanging operator Fractional Fourier transformation Integration method within ordered product of operators Operator Hermite polynomial theory

［ Q, P ］ = i ℏ

（1）

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e^{2 λ q - λ^{2}} = \sum_{n = 0}^{\infty} \frac{λ^{n}}{n!} H_{n} (q)

（2）

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H_{n} (q) = \frac{d^{n}}{λ^{n}} {e^{2 λ q - λ^{2}}|}_{λ = 0}

（3）

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e^{2 λ Q - λ^{2}} = \sum_{n = 0}^{\infty} \frac{λ^{n}}{n!} H_{n} (Q)

（4）

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e^{2 λ Q - λ^{2}} = e^{\sqrt[]{2} λ (a + a^{†}) - λ^{2}} = : e^{\sqrt[]{2} λ (a + a^{†})} : = : e^{2 λ Q} : = \sum_{n = 0}^{\infty} : \frac{2 λ Q}{n!} :

（5）

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H_{n} (Q) = : {(2 Q)}^{n} :

（6）

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\int_{- \infty}^{\infty} d q |q〉 〈q| = \int_{- \infty}^{\infty} \frac{d q}{\sqrt[]{π}} : e^{- {(q - Q)}^{2}} : = 1

（7）

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H_{n} (Q) = \int_{- \infty}^{\infty} d q H_{n} (q) |q〉 〈q| = \int_{- \infty}^{\infty} \frac{d q}{\sqrt[]{π}} H_{n} (q) : e^{- {(q - Q)}^{2}} : = : {(2 Q)}^{n} :

（8）

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\int_{- \infty}^{\infty} \frac{d x}{\sqrt[]{π}} H_{n} (x) e^{- {(x - y)}^{2}} = {(2 y)}^{n}

（9）

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\sum_{n = 0}^{\infty} \frac{t^{n}}{2^{n} n!} H_{n} (q) H_{n} (p) = \frac{1}{\sqrt[]{1 - t^{2}}} e x p [\frac{2 t p q - t^{2} (p^{2} + q^{2})}{1 - t^{2}}]

（10）

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\sum_{n = 0}^{\infty} \frac{t^{n}}{2^{n} n!} H_{n} (Q) H_{n} (p) = \sum_{n = 0}^{\infty} \frac{t^{n}}{n!} : Q^{n} : H_{n} (p) = : e^{2 p t Q - t^{2} Q^{2}} :

（11）

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\begin{array}{l} e^{- f Q^{2} + g Q} = \int_{- \infty}^{\infty} d q e^{- f q^{2} + g q} |q〉 〈q| = \int_{- \infty}^{\infty} \frac{d q}{\sqrt[]{π}} : {e^{- {(q - Q)}^{2} -}}^{f q^{2} + g q} : = \\ \sqrt[]{\frac{1}{f + 1}} : e x p [\frac{- f Q^{2} + g Q}{1 + f} + \frac{g^{2}}{4 (1 + f)}] : \end{array}

（12）

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\sum_{n = 0}^{\infty} \frac{t^{n}}{2^{n} n!} H_{n} (Q) H_{n} (p) = \frac{1}{\sqrt[]{1 - t^{2}}} e x p [\frac{2 p t Q - t^{2} (p^{2} + Q^{2})}{1 - t^{2}}]

（13）

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f (p) = 〈p | f〉 = \int_{- \infty}^{\infty} 〈p| q〉 〈q| f〉 d q = \frac{e^{- i p q}}{\sqrt[]{2 π}} f (p) d q

（14）

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F_{α} [f] (p) = \frac{e^{i (α / 2 - π / 4)}}{\sqrt[]{2 π s i n α}} e x p [\frac{i}{2} (\frac{p^{2} + q^{2}}{t a n α} - \frac{2 p q}{s i n α})] f (p) d q

（15）

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\frac{e^{i (α / 2 - π / 4)}}{\sqrt[]{2 π s i n α}} e x p [\frac{i}{2} (\frac{p^{2} + q^{2}}{t a n α} - \frac{2 p q}{s i n α})] = 〈p| e^{i (\frac{π}{2} - α) a^{†} a} |q〉

（16）

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\sum_{n = 0}^{\infty} \frac{t^{n}}{2^{n} n!} H_{n} (p) H_{n} (q) = \frac{1}{\sqrt[]{1 - t^{2}}} e x p [\frac{2 t p q - t^{2} (p^{2} + q^{2})}{1 - t^{2}}]

（17）

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\begin{array}{l} \sum_{n = 0}^{\infty} \frac{e^{- i α n}}{2^{n} n!} H_{n} (p) H_{n} (q) = \frac{e^{i α / 2}}{\sqrt[]{e^{i α} - e^{- i α}}} e x p [\frac{2 p q}{e^{i α} - e^{- i α}} - \frac{e^{- 2 i α} (p^{2} + q^{2})}{1 - e^{- 2 i α}}] = \\ \frac{e^{i α / 2}}{\sqrt[]{2 i s i n α}} e x p [\frac{2 p q}{2 i s i n α} - \frac{(e^{- 2 i α} - 1) (p^{2} + q^{2}) + (p^{2} + q^{2})}{1 - e^{- 2 i α}}] = \\ \frac{e^{i α / 2}}{\sqrt[]{2 i s i n α}} e^{(p^{2} + q^{2})} e x p [\frac{2 p q}{2 i s i n α} - \frac{(p^{2} + q^{2}) (c o s α + i s i n α)}{2 i s i n α}] = \\ \frac{e^{i α / 2}}{\sqrt[]{2 i s i n α}} e^{(p^{2} + q^{2}) / 2} e x p [\frac{2 p q}{2 i s i n α} - \frac{p^{2} + q^{2}}{2 i t a n α}] = \\ \frac{e^{i α / 2}}{\sqrt[]{2 i s i n α}} e^{(p^{2} + q^{2}) / 2} e x p [\frac{i}{2} (\frac{p^{2} + q^{2}}{t a n α} - \frac{2 p q}{s i n α})] \end{array}

（18）

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K_{α} (p, q) \equiv \frac{e^{i (α / 2 - π / 4)}}{\sqrt[]{2 π s i n α}} e x p [\frac{i}{2} (\frac{p^{2} + q^{2}}{t a n α} - \frac{2 p q}{s i n α})] = e^{- (p^{2} + q^{2}) / 2} π^{- 1 / 2} \sum_{n = 0}^{\infty} \frac{{(- i)}^{n} e^{i (\frac{π}{2} - α) n}}{2^{n} n!} H_{n} (p) H_{n} (q)

（19）

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〈q| n〉 = e^{- q^{2} / 2} \frac{H_{n} (q)}{\sqrt[]{\sqrt[]{π} 2^{n} n!}} = 〈n| q〉

（20）

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|q〉 〈q| = \frac{1}{\sqrt[]{π}} e^{- q^{2}} : e^{2 q Q - Q^{2}} : = \frac{1}{\sqrt[]{π}} e^{- q^{2}} \sum_{n = 0}^{\infty} : \frac{Q^{n}}{n!} H_{n} (q) :

（21）

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\begin{array}{l} 〈m |q〉 〈q| 0〉 = \frac{1}{\sqrt[]{π}} e^{- q^{2}} \sum_{n = 0}^{\infty} \frac{H_{n} (q)}{n!} 〈m| : {(\frac{a + a^{†}}{\sqrt[]{2}})}^{n} : |0〉 = \frac{1}{\sqrt[]{π}} e^{- q^{2}} \sum_{n = 0}^{\infty} \frac{H_{n} (q)}{\sqrt[]{2^{n}} n!} 〈m| : a^{† n} : |0〉 = \\ \frac{1}{\sqrt[]{π}} e^{- q^{2}} \sum_{n = 0}^{\infty} \frac{H_{n} (q)}{\sqrt[]{2^{n} n!}} 〈m| n〉 = \frac{1}{\sqrt[]{π}} e^{- q^{2}} \frac{H_{m} (q)}{\sqrt[]{2^{m} m!}} \end{array}

（22）

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〈p| n〉 = π^{- 1 / 4} e^{- p^{2} / 2} \frac{{(- i)}^{n} H_{n} (q)}{\sqrt[]{2^{n} n!}}

（23）

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K_{α} (p, q) = \sum_{n = 0}^{\infty} e^{i (\frac{π}{2} - α) n} 〈p| n〉 〈n| q〉 = 〈p| e^{i (\frac{π}{2} - α) a^{†} a} \sum_{n = 0}^{\infty} |n〉 〈n| q〉 = 〈p| e^{i (\frac{π}{2} - α) a^{†} a} |q〉

（24）

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F_{α} [f] (p) = \int_{- \infty}^{+ \infty} K_{α} (p, q) f (p) d q = 〈p| e^{i (\frac{π}{2} - α) a^{†} a} |f〉

（25）

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\begin{array}{l} (F_{α} \circ F_{β}) [f] (p) \equiv \int_{- \infty}^{+ \infty} K_{α} (p, p') d q' \int_{- \infty}^{+ \infty} K_{β} (p', q) f (p) d q = \\ \int_{- \infty}^{\infty} 〈p| e^{i (\frac{π}{2} - α) a^{†} a} |q |_{q = p'}〉 d q' 〈p'| e^{i (\frac{π}{2} - β) a^{†} a} |f〉 \end{array}

（26）

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|q〉 = \frac{1}{π^{1 / 4}} e x p [- \frac{q^{2}}{2} + \sqrt[]{2} q a^{†} - \frac{a^{† 2}}{2}] |0〉

（27）

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|p〉 = \frac{1}{π^{1 / 4}} e x p [- \frac{p^{2}}{2} + \sqrt[]{2} i p a^{†} - \frac{a^{† 2}}{2}] |0〉

（28）

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|0〉 〈0| = : e^{- a^{†} a} :

（29）

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\begin{array}{l} \int_{- \infty}^{\infty} |q |_{q = p'}〉 d q' 〈p'| = {\int_{- \infty}^{\infty} d q |q〉 〈p'|}_{p' = q} = \\ \int \frac{d q}{π^{1 / 2}} e x p [- \frac{q^{2}}{2} + \sqrt[]{2} q a^{†} - \frac{a^{† 2}}{2}] |0〉 〈0| e x p [- \frac{q^{2}}{2} - \sqrt[]{2} q a^{†} + \frac{a^{2}}{2}] = \\ \int \frac{d q}{\sqrt[]{π}} : e x p [- q^{2} + \sqrt[]{2} q (a^{†} - i a) - \frac{a^{† 2} - a^{2}}{2} - a^{†} a] : = \\ : e x p [- (i + 1) a^{†} a] : = e x p [- \frac{i π}{2} a^{†} a] \end{array}

（30）

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e^{λ a^{†} a} = : e x p [(e^{λ} - 1) a^{†} a] :

（31）

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\int_{- \infty}^{\infty} |q |_{q = p'}〉 d q' 〈p'| = e x p [- \frac{i π}{2} a^{†} a]

（32）

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\{\begin{array}{l} |q' |_{q' = p}〉 = e^{i π a^{†} a / 2} |p〉 \\ 〈p| e^{i π a^{†} a / 2} = 〈q'|_{x' = p}| \end{array}

（33）

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\begin{array}{l} (F_{α} \circ F_{β}) [f] (p) = \int_{- \infty}^{\infty} 〈p| e^{- i α a^{†} a} ||p'〉 d p' 〈p'| e^{i (\frac{π}{2} - β) a^{†} a} |f〉 = \\ 〈p| e^{i (\frac{π}{2} - β) a^{†} a} |f〉 = F_{α + β} [f] (p) \end{array}

（34）

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Ke ZHANG, Lan-lan LI, Hai-jun YU, Jian-ming DU, Hong-yi FAN. Quantum Theory of Optical Fractional Fourier Transform[J]. Acta Photonica Sinica, 2020, 49(10): 1027001

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