Research progress of measurement of propagators in path integrals

Tian Li-Man; Wen Yong-Li; Wang Yun-Fei; Zhang Shan-Chao; Li Jian-Feng; Du Jing-Song; Yan Hui; Zhu Shi-Liang

doi:10.7498/aps.72.20230902

Abstract

The propagator plays a central role in path integral theory and therefore has significant value in various fields of modern quantum physics, where path integral representations can be used. However, owing to the fact that it has not been directly measured in experiment, progress of experimental studies of quantum systems based on path integral representations has been seriously limited. Recently, we proposed a propagator measurement scheme based on the direct measurement of the wave function and successfully performed the first experimental measurement of the propagator by using a single photon experiment. Furthermore, in this study, the quantum principle of least action is demonstrated for the first time. This research successfully addresses the technical challenges of path integral experimental studies. In this work, we review the research progress in this field, including a brief introduction to the basic concepts and research progress of direct wave function measurement, and a detailed description of the theoretical model, experimental design, and experimental results of propagator measurement. Finally, we introduce an important application example, which can serve as the experimental demonstration of the quantum principle of least action through propagator measurement. The research progress of propagator measurement reviewed in this work will provide important references for future experimental studies by using this method.

Keywords:

Authors and contacts

1.
Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of Education), School of Physics, South China Normal University, Guangzhou 510006, China
2.
Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,South China Normal University, Guangzhou 510006, China
3.
Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research Institute for Physics, South China Normal University, Guangzhou 510006, China

Corresponding author: Wen Yong-Li, ylwen@m.scnu.edu.cn

Funds: Project supported by the Regional Joint Funds of Basic and Applied Basic Research of Guangdong Province, China (Grant No. 2022A1515110921).

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References

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Cited By

图 1 路径积分示意图及测量方法 (a) 传播子, 粒子从$ (x_a, t_a) $ 传播到$ (x_{\rm{m}}, t) $; (b) 测量传播子的量子线路. 为了测量$ K(x_{\rm{m}}, t; $$ x_a, t_a) $, 首先需要制备量子态$ |\psi\rangle $和指针$ |0\rangle $. 然后, 在$ t_a $时刻, 耦合操作$ {\hat{U}} $ 将系统的位置波函数与指针进行耦合. 演化之后, 将系统投影到t时刻的位置$ x_{\rm{m}} $上, 从指针读出$ K(x_{\rm{m}}, t;x_a, t_a) $

Figure 1. Path integration diagram and measurement method: (a) Propagators: the propagation from $ (x_a, t_a) $ to $ (x_{\rm{m}}, t) $; (b) protocol to measure the propagators. To measure $ K(x_{\rm{m}}, t;x_a, t_a) $, the quantum state $ |\psi\rangle $ and pointer $ |0\rangle $ are prepared first. Then, at time $ t_a $, a coupling operation $ {\hat{U}} $ couples the positional wave function of the probe system with the pointer. After the evolution, the system is projected to position $ x_{\rm{m}} $ at t, and $ K(x_{\rm{m}}, t;x_a, t_a) $ can be read out from the pointer

DownLoad: Full-Size Img PowerPoint

图 2 实验示意图 (a)演化和探测区域; (b1)初态制备; (b2)系统位置波函数与指针耦合; (b3)态的演化; (b4)位置的后选择与指针的读出; (b5)波函数的测量

Figure 2. Schematic diagram of the experiment: (a) The evolution and detection region; (b1) initial state preparation; (b2) coupling of wave function and pointer; (b3) state evolution; (b4) post-selection of position and the readout of the pointer; (b5) measurement of the wave function

DownLoad: Full-Size Img PowerPoint

图 3 $ t=20 $ mm/c (c为真空中的光速)的$ K^{\prime\prime}(x, t;0, 0) $和波函数$ \psi(x, 0) $的实验测量结果. 蓝色方块和红色圆点分别展示了$ K^{\prime\prime}(x, t;0, 0) $的实部和虚部. 绿色菱形和紫色三角形分别展示了$ \psi(x, 0) $的实部与虚部

Figure 3. Measured wave function of single photon and $ K^{\prime\prime}(x, t=20{\rm{\; mm/{\rm{c}}}};0, 0) $. The green diamond and purple triangle represent the real and imaginary parts of the wave function, respectively. The red dot and the blue square show the real and imaginary parts of the $ K^{\prime\prime}(x, t= $$ 20{\rm{\; mm/{\rm{c}}}};0, 0) $ at $ t=20 $ mm/c

DownLoad: Full-Size Img PowerPoint

图 4 $ K_{\rm{f}}(x, t;x_a, t_a) $以x和t为变量的理论与实验结果. 纵轴是演化时间t, 横轴是横向位置x, 颜色代表实部或者虚部的幅度大小. 对演化时间t 每间隔1 mm/c从$ 0 $测量到30 mm/c, 横向位置每间隔8 μm从$ -0.2 $ mm测量到0.2 mm

Figure 4. Theoretical and experimental results of $ K_{\rm{f}}(x_b, t_b;x_a, t_a) $ over time. The vertical axis is the evolution time t, the horizontal axis is the transverse position x. The colormaps represent the magnitude of the real or imaginary part of $ K_{\rm{f}}(x, t;x_a, t_a) $. Evolution time is measured from $ 0 $ to 30 mm/c at 1 mm/c interval. Transverse positions are measured from $ -0.2 $ mm to 0.2 mm at 8 μm intervals.

DownLoad: Full-Size Img PowerPoint

图 5 在自由空间中测到的传播子红线和蓝线是理论计算的传播子实部和虚部的结果, 红色圆圈和蓝色方块是实验测到的数据

Figure 5. Measured propagators of photons in free space. The red and blue lines are the results of theoretical calculations of the real and imaginary parts of the propagators, and the red circles and blue squares are experimentally measured data.

DownLoad: Full-Size Img PowerPoint

图 6 由最小作用量原理得到的光子自由演化时的经典路径. 实线展示的是理论计算所得的自由演化的经典路径, 离散点为通过传播子的实验数据求得的自由演化的经典路径位置

Figure 6. The classical path of photons in free evolution derived from the principle of least action. The solid line shows the classical path in free evolution obtained from the theoretical calculation, and the discrete point is the classical path position in free evolution obtained from the experimental data of the propagator.

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图 7 在谐振子势场中的经典路径. 实线展示的是理论计算所得的在谐振势中的经典路径, 离散点为通过传播子的实验数据求得的在谐振势中的经典路径位置

Figure 7. Classical trajectories in the harmonic potential. The solid line shows the classical path in the harmonic potential obtained by theoretical calculation, and the discrete point is the classical path position in the harmonic potential obtained by the experimental data of the propagator

DownLoad: Full-Size Img PowerPoint

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[2]	Feynman R P 2005 Feynman’s Thesis—A New Approach to Quantum Theory (Singapore: World Scientific Publishing Co Pte Ltd) pp1–69
[3]	Feynman R P, Hibbs A R, Styer D F 2010 Quantum Mechanics and Path Integrals (Chelmsford: Courier Corporation) pp1–384
[4]	Wen Y L, Wang Y F, Tian L M, Zhang S C, Li J F, Du J S, Yan H, Zhu S L 2023 Nat. Photonics 17 717 Google Scholar
[5]	Von Neumann J 2018 Mathematical Foundations of Quantum Mechanics (New Edition) (Princeton: Princeton university press) pp271–288
[6]	Aharonov Y, Albert D Z, Vaidman L 1988 Phys. Rev. Lett. 60 1351 Google Scholar
[7]	Lundeen J S, Sutherland B, Patel A, Stewart C, Bamber C 2011 Nature 474 188 Google Scholar
[8]	Hofheinz M, Wang H, Ansmann M, Bialczak R C, Lucero E, Neeley M, O’Connell A D, Sank D, Wenner J, MartinisJ M, ClelandA N 2009 Nature 459 546 Google Scholar
[9]	Cramer M, Plenio M B, Flammia S T, Somma R, Gross D, Bartlett S D, Landon-Cardinal O, Poulin D, Liu Y K 2010 Nat. Commun. 1 149 Google Scholar
[10]	Ritchie N W M, Story J G, Hulet R G 1991 Phys. Rev. Lett. 66 1107 Google Scholar
[11]	Hosten O, Kwiat P 2008 Science 319 787 Google Scholar
[12]	Starling D J, Dixon P B, Jordan A N, Howell J C 2009 Phys. Rev. A 80 041803 Google Scholar
[13]	Zhou L, Turek Y, Sun C P, Nori F 2013 Phys. Rev. A 88 053815 Google Scholar
[14]	Zhou X X, Xiao Z C, Luo H L, Wen S C 2012 Phys. Rev. A 85 043809 Google Scholar
[15]	Rodenburg B, Mirhosseini M, Magaña-Loaiza O S, Boyd R W 2014 JOSA B 3 1 Google Scholar
[16]	Magaña-Loaiza O S, Mirhosseini M, Rodenburg B, Boyd R W 2014 Phys. Rev. Lett. 112 200401 Google Scholar
[17]	Starling D J, Dixon P B, Williams N S, Jordan A N, Howell J C 2010 Phys. Rev. A 82 011802 Google Scholar
[18]	Xu X Y, Kedem Y, Sun K, Vaidman L, Li C F, Guo G C 2013 Phys. Rev. Lett. 111 033604 Google Scholar
[19]	Starling D J, Dixon P B, Jordan A N, Howell J C 2010 Phys. Rev. A 82 063822 Google Scholar
[20]	Brunner N, Simon C 2010 Phys. Rev. Lett. 105 010405 Google Scholar
[21]	Strübi G, Bruder C 2013 Phys. Rev. Lett. 110 083605 Google Scholar
[22]	Hallaji M, Feizpour A, Dmochowski G, Sinclair J, Steinberg A M 2017 Nat. Phys. 13 540 Google Scholar
[23]	Du S J, Peng Y G, Feng H R, Han F, Yang L W, Zheng Y J 2020 Chin. Phys. B 29 074202 Google Scholar
[24]	Zhang C Y, Fang M F 2021 Chin. Phys. B 30 010303 Google Scholar
[25]	Dressel J, Malik M, Miatto F M, Jordan A N, Boyd R W 2014 Rev. Mod. Phys. 86 307 Google Scholar
[26]	Pfeifer M, Fischer P 2011 Opt. Express 19 16508 Google Scholar
[27]	Qiu J D, Li Z X, Xie L G, Luo L, He Y, Ren C L, Zhang Z Y, Du J L 2021 Chin. Phys. B 30 064216 Google Scholar
[28]	Fang S Z, Dai Y, Jiang Q W, Tan H T, Li G X, Wu Q L 2021 Chin. Phys. B 30 060601 Google Scholar
[29]	Lundeen J S, Resch K J 2005 Phys. Lett. A 334 337 Google Scholar
[30]	Salvail J Z, Agnew M, Johnson A S, Bolduc E, Leach J, Boyd R W 2013 Nat. Photonics 7 316 Google Scholar
[31]	Malik M, Mirhosseini M, Lavery M P J, Leach J, Padgett M J, Boyd R W 2014 Nat. Commun. 5 3115 Google Scholar
[32]	Thekkadath G S, Giner L, Chalich Y, Horton M J, Banker J, Lundeen J S 2016 Phys. Rev. Lett. 117 120401 Google Scholar
[33]	Xiao Y, Kedem Y, Xu J S, Li C F, Guo G C 2017 Opt. Express 25 14463 Google Scholar
[34]	Xiao Y, Wiseman H M, Xu J S, Kedem Y, Li C F, Guo G C 2019 Sci. Adv. 5 eaav9547 Google Scholar
[35]	Cohen E, Pollak E 2018 Phys. Rev. A 98 042112 Google Scholar
[36]	Bolduc E, Gariepy G, Leach J 2016 Nat. Commun. 7 10439 Google Scholar
[37]	Pan W W, Xu X Y, Kedem Y, Wang Q Q, Chen Z, Jan M, Sun K, Xu J S, Han Y J, Li C F, Guo G C 2019 Phys. Rev. Lett. 123 150402 Google Scholar
[38]	Chen M C, Li Y, Liu R Z, Wu D, Su Z E, Wang X L, Li L, Liu N L, Lu C Y, Pan J W 2021 Phys. Rev. Lett. 127 030402 Google Scholar
[39]	Kobayashi H, Tamate S, Nakanishi T, Sugiyama K, Kitano M 2010 Phys. Rev. A 81 012104 Google Scholar
[40]	Shi Z M, Mirhosseini M, Margiewicz J, Malik M, Rivera F, Zhu Z Y, Boyd R W 2015 Optica 2 388 Google Scholar
[41]	Maccone L, Rusconi C C 2014 Phys. Rev. A 89 022122 Google Scholar
[42]	Mirhosseini M, Magaña-Loaiza O S, Hashemi Rafsanjani S M, Boyd R W 2014 Phys. Rev. Lett. 113 090402 Google Scholar
[43]	Xu L, Xu H C, Jiang T, Xu F X, Zheng K M, Wang B, Zhang A N, Zhang L J 2021 Phys. Rev. Lett. 127 180401 Google Scholar
[44]	Piacentini F, Avella A, Levi M P, Gramegna M, Brida G, Degiovanni I P, Cohen E, Lussana R, Villa F, Tosi A, Zappa F, Genovese M 2016 Phys. Rev. Lett. 117 170402 Google Scholar
[45]	Shojaee E, Jackson C S, Riofrío C A, Kalev A, Deutsch I H 2018 Phys. Rev. Lett. 112 130404 Google Scholar
[46]	Bamber C, Lundeen J S 2014 Phys. Rev. Lett. 112 070405 Google Scholar
[47]	Wu S J 2013 Sci. Rep. 3 1193 Google Scholar
[48]	Kocsis S, Braverman B, Ravets S, Stevens M J, Mirin R P, Shalm L K, Steinberg A M 2011 Science 332 1170 Google Scholar
[49]	Wen Y L, Zhang S C, Yan H, Zhu S L 2022 Chin. Phys. B 31 034206 Google Scholar
[50]	温永立, 张善超, 颜辉, 朱诗亮 2021 物理学报 70 110301 Google Scholar Wen Y L, Zhang S C, Yan H, Zhu S L 2021 Acta Phys. Sin. 70 110301 Google Scholar
[51]	Romito A, Gefen Y 2014 Phys. Rev. B 90 085417 Google Scholar
[52]	Vallone G, Dequal D 2016 Phys. Rev. Lett. 116 040502 Google Scholar
[53]	Zhang S C, Zhou Y R, Mei Y F, Liao K Y, Wen Y L, Li J F, Zhang X D, Du S W, Yan H, Zhu S L 2019 Phys. Rev. Lett. 123 190402 Google Scholar
[54]	Zhang C R, Hu M J, Hou Z B, Tang J F, Zhu J, Xiang G Y, Li C F, Guo G C, Zhang Y S 2020 Phys. Rev. A 101 012119 Google Scholar
[55]	Zhang C R, Hu M J, Xiang G Y, Zhang Y S, Li C F, Guo G C 2020 Chin. Phys. Lett. 37 080301 Google Scholar
[56]	Zhou Y Y, Zhao J, Hay D, McGonagle K, Boyd R W, Shi Z M 2021 Phys. Rev. Lett. 127 040402 Google Scholar
[57]	Yang M, Xiao Y, Liao Y W, Liu Z H, Xu X Y, Xu J S, Li C F, Guo G C 2020 Laser Photonics Rev. 14 1900251 Google Scholar
[58]	Calderaro L, Foletto G, Dequal D, Villoresi P, Vallone G 2018 Phys. Rev. Lett. 121 230501 Google Scholar
[59]	Knarr S H, Lum D J, Schneeloch J, Howell J C 2018 Phys. Rev. A 98 023854 Google Scholar
[60]	Pan Y M, Zhang J, Cohen E, Wu C W, Chen P X, Davidson N 2020 Nat. Phys. 16 1206 Google Scholar
[61]	Rojo A, Bloch A 2018 The Principle of Least Action: History and Physics (Cambridg: Cambridge University Press) pp1–266
[62]	朱诗亮, 温永立, 颜辉 2023 物理 52 502 Google Scholar Zhu S L, Wen Y L, Yan H 2023 Physics 52 502 Google Scholar
[63]	Salières P, Carré B, Le Déroff L, Grasbon F, Paulus G G, Walther H, Kopold R, Becker W, Milosevic D B, Sanpera A, Lewenstein M 2001 Science 292 902 Google Scholar
[64]	Gibbons G W, Hawking S W, Perry M J 1978 Nucl. Phys. B 138 141 Google Scholar
[65]	Fujikawa K 1979 Phys. Rev. Lett. 42 1195 Google Scholar
[66]	Ord G N, Gualtieri J A 2002 Phys. Rev. Lett. 89 250403 Google Scholar
[67]	Caldeira A O, Leggett A J 1983 Physica A 121 587 Google Scholar
[68]	Mühlbacher L, Rabani E 2008 Phys. Rev. Lett. 100 176403 Google Scholar
[69]	Pollock E L, Ceperley D M 1984 Phys. Rev. B 30 2555 Google Scholar
[70]	Reisenberger M P, Rovelli C 1997 Phys. Rev. D 56 3490 Google Scholar
[71]	Hartle J B, Hawking S W 1983 Phys. Rev. D 28 2960 Google Scholar
[72]	Ashtekar A, Campiglia M, Henderson A 2010 Phys. Rev. D 82 124043 Google Scholar
[73]	Fujikawa K 1982 Phys. Rev. D 25 2584 Google Scholar
[74]	Hartle J B, Hawking S W 1976 Phys. Rev. D 13 2188 Google Scholar

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Vol. 72, Issue 20 - 2023-10-20

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