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Chinese Optics Letters
Vol. 19, Issue 1, 012701 (2021)

Photons can hide where they have been | Editors' Pick

Qizhang Yuan^* and Xunli Feng^**

Author Affiliations

Department of Physics, Mathematics and Science College of Shanghai Normal University, Shanghai 200234, China

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DOI: 10.3788/COL202119.012701 Cite this Article Set citation alerts

Qizhang Yuan, Xunli Feng. Photons can hide where they have been[J]. Chinese Optics Letters, 2021, 19(1): 012701 Copy Citation Text

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Abstract

Recently, the nested Mach–Zehnder interferometer [Phys. Rev. Lett. 111, 240402 (2013)] was modified by adding Dove prisms in a paper [Quantum Stud.: Math. Found. 2, 255 (2015)], and an interesting result is that, after the Dove prisms were inserted, a signal at the first mirror of the nested interferometer was obtained. But, according to the former original paper, the photons have never been present near that mirror. In this work, we interpret this result naturally by resorting to the three-path interference method. Moreover, we find that even though the photons have been somewhere, they can hide the trace of being there.

Keywords

multi-path interference quantum foundations

| ψ_{BS 1} 〉 = \frac{1}{\sqrt{3}} | L 〉 + i \sqrt{\frac{2}{3}} | U 〉,

(1)

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| ψ_{iBS 1} 〉 = \frac{1}{\sqrt{3}} | L 〉 + i \frac{e^{i φ_{e}}}{\sqrt{3}} | l 〉 - \frac{e^{i φ_{e}}}{\sqrt{3}} | u 〉,

(2)

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| ψ_{Dove} 〉 = \frac{1}{\sqrt{3}} | L 〉 + i \frac{e^{i φ_{e}}}{\sqrt{3}} | l 〉 - \frac{e^{i {\tilde{φ}}_{e}}}{\sqrt{3}} | u 〉,

(3)

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| ψ_{A B C} 〉 = \frac{e^{i φ_{c}}}{\sqrt{3}} | L 〉 + i \frac{e^{i (φ_{b} + φ_{e})}}{\sqrt{3}} | l 〉 - \frac{e^{i (φ_{a} + {\tilde{φ}}_{e})}}{\sqrt{3}} | u 〉 .

(4)

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| ψ_{i BS 2} 〉 = \frac{e^{i φ_{c}}}{\sqrt{3}} | L 〉 + i \frac{1}{\sqrt{6}} [e^{- \frac{i χ}{2}} e^{i (φ_{b} + φ_{e})} - e^{\frac{i χ}{2}} e^{i (φ_{a} + {\tilde{φ}}_{e})}] | U' 〉 - \frac{1}{\sqrt{6}} [e^{- \frac{i χ}{2}} e^{i (φ_{b} + φ_{e})} + e^{\frac{i χ}{2}} e^{i (φ_{a} + {\tilde{φ}}_{e})}] | out 1 〉 .

(5)

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| ψ_{BS 2} 〉 = \frac{1}{3} [\begin{matrix} e^{i ϕ / 2} e^{i φ_{c}} - e^{- i ϕ / 2} e^{- i χ / 2} e^{i (φ_{b} + φ_{e} + φ_{f})} \end{matrix} \begin{matrix} - e^{- i ϕ / 2} e^{i χ / 2} e^{i (φ_{a} + {\tilde{φ}}_{e} + φ_{f})} \end{matrix}] | D 〉 + \frac{i}{3} [\begin{matrix} \sqrt{2} e^{i ϕ / 2} e^{i φ_{c}} \end{matrix} \begin{matrix} + \frac{e^{- i ϕ / 2} e^{- i χ / 2}}{\sqrt{2}} e^{i (φ_{b} + φ_{e} + φ_{f})} - \frac{e^{- i ϕ / 2} e^{i χ / 2}}{\sqrt{2}} e^{i (φ_{a} + {\tilde{φ}}_{e} + φ_{f})} \end{matrix}] | out 2 〉 - \frac{1}{\sqrt{6}} [\begin{matrix} e^{- i χ / 2} e^{i (φ_{b} + φ_{e})} \end{matrix} \begin{array}{l} + e^{i χ / 2} e^{i (φ_{a} + {\tilde{φ}}_{e})} \end{array}] | out 1 〉,

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P_{D} = \frac{1}{9} {| \begin{matrix} e^{i ϕ} e^{i φ_{c}} \end{matrix} \begin{matrix} - e^{- i ϕ / 2} e^{- i χ / 2} e^{i (φ_{b} + φ_{e} + φ_{f})} - e^{- i ϕ / 2} e^{i χ / 2} e^{i (φ_{a} + {\tilde{φ}}_{e} + φ_{f})} \end{matrix} |}^{2} .

(7)

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P_{D} = \frac{1}{9} {| e^{i φ_{c}} - e^{i (φ_{b} + φ_{e} + φ_{f})} + e^{i (φ_{a} + {\tilde{φ}}_{e} + φ_{f})} |}^{2} \approx \frac{1}{9} [1 - 2 I m (φ_{a} - φ_{b} + φ_{c} - φ_{e} + {\tilde{φ}}_{e})] .

(8)

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P_{D} = \frac{1}{9} [1 - 2 I m (φ_{a} - φ_{b} + φ_{c} - 2 φ_{e})] .

(9)

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P_{D} = \frac{1}{9} {| - i e^{i φ_{c}} - e^{i (φ_{b} + φ_{e} + φ_{f})} - i e^{i (φ_{a} + {\tilde{φ}}_{e} + φ_{f})} |}^{2} \approx \frac{1}{9} [9 - 6 I m (φ_{a} + φ_{b} + φ_{c} + φ_{e} + {\tilde{φ}}_{e} + 2 φ_{f})] .

(10)

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P_{D} = \frac{1}{9} [9 - 6 I m (φ_{a} + φ_{b} + φ_{c} + 2 φ_{f})] \approx 1.

(11)

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P_{D} = \frac{1}{9} [9 - 6 I m (φ_{a} + φ_{b} + φ_{c} + 2 φ_{e} + 2 φ_{f})],

(12)

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Qizhang Yuan, Xunli Feng. Photons can hide where they have been[J]. Chinese Optics Letters, 2021, 19(1): 012701

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