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As a kind of quantum phenomenon, Hong-Ou-Mandel (HOM) interference is more robust against phase noise. Because of this feature, robust quantum holography emerges, through which wave function of interested photon can be retrieved according to HOM interference pattern. For better understanding and developing this method, we derive a theoretical framework of robust HOM holography. In the quantum holography scheme, test state and reference state interfere at beam splitter (BS). Then, degree of freedom (DOF) resolved detections (such as spatial resolved detection, temporal resolved detection or spectrum resolved detection) are used at the BS output ports, respectively. Based on the single photon detection results, the DOF resolved coincidence counts are postselected, producing interference patterns. The information of the test states is retrieved from the patterns. According to different test states and reference states, four combinations are analysed, including measuring the wave function of single photon state by using standard single photon state or coherent state and measuring the wave function of coherent state through using standard single photon state or coherent state. In all cases, information of the test states is reflected in normalized second-order correlation function or interference patterns in similar forms. Specially, the wave function of test states can be directly retrieved from the interference patterns, with no complex algorithm required. Besides, phase noise from environment has no influence on this kind quantum holography. Comparison between traditional holography and quantum holography is made and analysed.
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Keywords:
- quantum holography /
- Hong-Ou-Mandel interference /
- normalized second-order correlation function /
- robustness
[1] Hong C K, Ou Z Y, Mandel L 1987 Phys. Rev. Lett. 59 2044
Google Scholar
[2] Mandel L 1999 Rev. Mod. Phys. 71 S274
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[19] Jin R B, Gerrits T, Fujiwara M, Wakabayashi R, et al. 2015 Opt. Express 23 28836
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 Hong-Ou-Mandel干涉装置图 (a)一般Hong-Ou-Mandel装置图. 量子态分别入射分束器的两个输入口
$ {\rm{a}}_1 $ 和$ {\rm{a}}_2 $ , 发生干涉后, 通过两个单光子探测器在输出口$ {\rm{b}}_1 $ 和$ {\rm{b}}_2 $ 进行测量. 根据单光子探测结果后选择出符合测量结果. (b)基于Hong-Ou-Mandel干涉的量子全息. 两个输入的量子态发生干涉后, 在输出口$ {\rm{b}}_1 $ 和$ {\rm{b}}_2 $ 做自由度内可分辨的单光子测量, 并通过后选择筛选出符合测量结果Figure 1. Setup of Hong-Ou-Mandel interference: (a) General Hong-Ou-Mandel setup. The two input states enter input ports of beam splitter (BS)
$ {\rm{a}}_1 $ and$ {\rm{a}}_2 $ , respectively. After passing through the BS, the photon is recorded by single photon detectors at output ports${\rm{ b}}_1 $ and$ {\rm{b}}_2 $ , respectively. According to the single photon detection results, the coincidence counts are postselected. (b) Quantum holography based on Hong-Ou-Mandel interference. After interference at the BS, degree of freedom (DOF) resolved detections are applied at the output ports$ {\rm{b}}_1 $ and$ {\rm{b}}_2 $ , respectively. According to the single photon detection results, the DOF resolved coincidence counts are postselected
图 2 不同待测态和参考态下的归一化二阶关联函数值分布 (a)相位; (b)振幅; (c)利用单光子态测量单光子态波函数的全息图; (d)利用相干态测量单光子态波函数的全息图; (e)利用单光子态测量相干态波函数的全息图; (f)利用相位随机相干态测量相干态波函数的全息图
Figure 2. Ideal normalized second-order correlation function of different quantum states: (a) Phase; (b) amplitude; (c) hologram of single photon state wave function measurement using single photon state; (d) hologram of single photon state wave function measurement using coherent state; (e) hologram of coherent state wave function measurement using single photon state; (f) hologram of coherent state wave function measurement using random phase coherent state.
表 1 Hong-Ou-Mandel量子全息的4种情况
Table 1. Four combinations in Hong-Ou-Mandel Holography.
待测态 参考态 归一化的二阶关联函数 $\left| {\psi}_{\text{test}} \right > =\displaystyle\int{{\psi}_{\text{test}}(q)}\left|{1}_{q}\right > {\rm{d}}q $ $\left| { \psi}_{ \text{ref} } \right > =\displaystyle\int { { \psi }_{ \text{ref} }(q) } \left| { 1 }_{ q } \right > \text{d}q $ $\dfrac{T^2\left(q_1\right)+T^2\left(q_2\right)-2 T\left(q_1\right) T\left(q_2\right) \cos \left[\phi\left(q_1\right)-\phi\left(q_2\right)\right]}{\left[T^2\left(q_1\right)+1\right]\left[T^2\left(q_2\right)+1\right]} $ $\left| {\psi}_{\text{test}} \right > =\displaystyle\int{{\psi}_{\text{test}}(q)}\left|{1}_{q}\right > {\rm{d}}q $ $\left| { \psi }_{ \text{ref} } \right > = \displaystyle\prod\nolimits_{ q }^{ }{ \left| { \alpha_ \text{ref} (q) } \right > } $ $1-\dfrac { T^{2}(q_{1})T^{2}(q_{2})+2 T(q_{1})T(q_{2})\cos \left[\phi\left(q_1\right)-\phi\left(q_2\right)\right]}{ ({T^{2}(q_{1})+1})(T^{2}(q_{2})+1)} $ $\left| { \psi }_{ \text{test} } \right > = \displaystyle\prod\nolimits _{ q }^{ }{ \left| { \beta_ \text{test} (q) } \right > } $ $\left| { \psi}_{ \text{ref} } \right > =\displaystyle\int { { \psi }_{ \text{ref} }(q) } \left| { 1 }_{ q } \right > \text{d}q $ $1-\dfrac { 2 T(q_{1})T(q_{2})\cos \left[\phi\left(q_1\right)-\phi\left(q_2\right)\right]+1}{ ({T^{2}(q_{1})+1})(T^{2}(q_{2})+1)} $ $\left| { \psi }_{ \text{test} } \right > = \displaystyle\prod\nolimits_{ q }^{ }{ \left| { \beta_ \text{test} (q) } \right > } $ $\left| { \psi }_{ \text{ref} } \right > =\displaystyle\prod\nolimits _{q }^{ }{ \left| { \alpha_ \text{ref} (q) } \right > } $ $1-\dfrac { 2 T({ q }_{ 1 })T({ q }_{ 2 }) \text{cos}[\phi ({ q }_{ 1 })-\phi ({ q }_{ 2 })]}{ ({ { T }^{ 2 }({ q }_{ 1 }) }+1)({ { T }^{ 2 }({ q }_{ 2 }) }+1) } $ 
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[1] Hong C K, Ou Z Y, Mandel L 1987 Phys. Rev. Lett. 59 2044
Google Scholar
[2] Mandel L 1999 Rev. Mod. Phys. 71 S274
Google Scholar
[3] Kaplan A E K , Krajewska C J, Proppe A H, et al. 2023 Nat. Photonics 17 775
[4] Lopes R, Imanaliev A, Aspect A, Cheneau M, Boiron D, Westbrook C I 2015 Nature 520 66
Google Scholar
[5] Kobayashi T, Ikuta R, Yasui S, et al. 2016 Nat. Photonics 10 441
Google Scholar
[6] Lo H K, Curty M, Qi B 2012 Phys. Rev. Lett. 108 130503
Google Scholar
[7] Liu Y, Chen T Y, Wang L J, et al. 2013 Phys. Rev. Lett. 111 130502
Google Scholar
[8] Tang Y L, Yin H L, Chen S J, et al. 2014 Phys. Rev. Lett. 113 190501
Google Scholar
[9] Duan L M, Lukin M D, Cirac J I, Zoller P 2001 Nature 414 413
Google Scholar
[10] Lvovsky A I, Sanders B C, Tittel W 2009 Nat. Photonics 3 706
Google Scholar
[11] Boto A N, Kok P, Abrams D S, Braunstein S L, Williams C P, Dowling J P 2000 Phys. Rev. Lett. 85 2733
Google Scholar
[12] Giovannetti V, Lloyd S, Maccone L 2011 Nat. Photonics 5 222
Google Scholar
[13] Edamatsu K, Shimizu R, Itoh T 2002 Phys. Rev. Lett. 89 213601
Google Scholar
[14] Chrapkiewicz R, Jachura M, Banaszek K, Wasilewski W 2016 Nat. Photonics 10 576
Google Scholar
[15] Xu Y K, Sun S H, Liu W T, Liu J Y, Chen P X 2019 Phys. Rev. A 100 042317
Google Scholar
[16] Thiel V, Davis A, Sun K, D’Ornellas P, Jin X M, Smith B 2020 Opt. Express 28 19515
[17] Orre V V, Goldschmidt E A, Deshpande A, et al. 2019 Phys. Rev. Lett. 123 123603
Google Scholar
[18] Thekkadath G S, Bell B A, Patel R B, Kim M S, Walmsley I A 2022 Phys. Rev. Lett. 128 023601
Google Scholar
[19] Jin R B, Gerrits T, Fujiwara M, Wakabayashi R, et al. 2015 Opt. Express 23 28836
Google Scholar
[20] Qin Z Z, Prasad A S, Brannan T, MacRae A, Lezama A, Lvovsky A I 2015 Light Sci. Appl. 4 e298
Google Scholar
[21] Lvovsky A I, Raymer M G 2009 Rev. Mod. Phys. 81 299
Google Scholar
[22] Anis A, Lvovsky A I 2012 New J. Phys. 14 105021
Google Scholar
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