Kwatancen Tsakanin Ilimin Haske da Ilimin Barbashi Mai Caji: Duba daga Mahangar Kwantum

1. Gabatarwa

Takardar ta kafa wani kwatance mai zurfi da dorewa tsakanin ka'idojin ilimin haske da ilimin barbashi mai caji. Wannan alaƙa, wacce ta samo asali ne daga ka'idojin Fermat (ilimin haske) da Maupertuis (ilimin motsi), William Rowan Hamilton ne ya tsara ta a shekara ta 1833. Kwatancen Hamilton ya ba da damar haɓaka ilimin lantarki na zahiri a cikin shekarun 1920, wanda ya haifar da ƙirƙira irin su na'urar duban dan tayi ta lantarki. A al'ada, wannan kwatancen ya kasance a cikin yanki na ilimin haske na geometric da ilimin motsi na gargajiya. Duk da haka, zuwan ilimin kwantum da tsawon zango na de Broglie na barbashi sun gabatar da wani sabon mataki na rikitarwa—da kuma dama.

Babban jigon aikin nan shi ne cewa kwatancen ba kawai ya tsira ba, amma ya ƙaru yayin da ake matsawa zuwa bayanin kwantum. Sabbin ci gaba a cikin ka'idojin kwantum na ilimin barbashi mai caji da kuma dokokin ilimin haske na zango da ba na gargajiya ba (ilimin haske na Helmholtz da Maxwell) sun bayyana wani kwatance mai zurfi, wanda ya dogara da tsawon zango. Wannan takarda ta ba da taƙaitaccen bayani game da waɗannan ci gaban da suka yi kama da juna, tana ba da hujjar haɗa su cikin tsari ɗaya a ƙarƙashin fagen da ke tasowa na Al'amuran Kwantum na Ilimin Barbashi (QABP).

2. Tsarin Ilimin Kwantum

Wannan sashe yana bayyana sauyi daga bayanin gargajiya zuwa na kwantum a cikin ilimin barbashi.

2.1. Mahallin Tarihi da Tushen Gargajiya

Maganin gargajiya, wanda ya dogara ne akan ilimin motsi na Hamiltonian da bin layin haske na geometric, ya yi nasara sosai wajen ƙirƙira na'urori daga na'urorin duban dan tayi na lantarki zuwa na'urorin ƙara gudu. Yana ɗaukar hanyoyin barbashi kamar hasken haske a cikin wani abu mai canza ma'aunin karkatar haske. Babban aikin Busch akan aikin ruwan tabarau na maganadisu shine aikace-aikacen kai tsaye na wannan kwatancen na haske da injiniya.

2.2. Dokokin Kwantum: Schrödinger, Klein-Gordon, da Dirac

Takardar ta nuna cewa ana buƙatar wata babbar doka ta kwantum, domin duk tsarin zahiri a ainihin su kwantum ne. Hanyar ta fara ne daga ainihin lissafin ilimin kwantum:

Lissafin Schrödinger: Don barbashi marasa relativistic spin-0.
Lissafin Klein-Gordon: Don barbashi relativistic spin-0.
Lissafin Dirac: Don barbashi relativistic spin-1/2 (kamar electrons).

Manufar ita ce a samo Hamiltonians na ilimin barbashi daga waɗannan lissafin don bayyana juyin halitta na ayyukan zango (wakiltar bayyanan barbashi) ta hanyar abubuwan haske kamar quadrupoles da maganadisu masu lanƙwasa. Wannan tsari a cikinsa ya haɗa da tasirin da ya dogara da tsawon zango (diffraction, tsangwama), waɗanda ba su da kwatankwacinsu a cikin ilimin haske na geometric na gargajiya.

2.3. Dokokin da ba na Gargajiya ba: Ilimin Haske na Helmholtz da Maxwell

Don kammala kwatancen a gefen ilimin haske, marubucin ya yi nuni ga ci gaban da ya wuce ilimin haske na geometric:

Ilimin Haske na Helmholtz: Maganin ilimin haske na zango wanda ya fara daga lissafin Helmholtz $\nabla^2 E + k^2 n^2(\mathbf{r}) E = 0$, wanda shine lissafin zango na scalar don hasken monocromatic. An nuna cewa yana da kwatankwacin kusa da ka'idar kwantum da ta dogara da lissafin Klein-Gordon.
Tsarin Matrix na Ilimin Haske na Maxwell: Cikakken maganin zango na vector wanda ya dogara da lissafin Maxwell. An gabatar da shi a matsayin yana da kwatankwacin kusa da ka'idar kwantum da ta dogara da lissafin Dirac, musamman saboda yadda yake sarrafa digiri na 'yanci irin na polarization/spin.

Waɗannan "dokokin da ba na gargajiya ba" don haske sun gabatar da nasu tasirin da ya dogara da tsawon zango, don haka suna dawo da kuma zurfafa daidaito tare da ilimin barbashi mai caji na kwantum.

3. Fahimtar Jiki & Tsarin Ma'ana

Fahimtar Jiki: Babban da'awar takardar, mai ƙarfi, ita ce cewa kwatancen da ya wuce ƙarni tsakanin ilimin haske da ilimin motsi ba abin sha'awa na tarihi bane—yana da tsarin gini wanda ke auna daga tsarin gargajiya zuwa na kwantum. Khan yana jayayya cewa ba muna kallon fage biyu daban-daban masu rikice-rikice ba, amma a kan ka'ida ɗaya, haɗe-haɗe na yaduwar zango wanda ke bayyana a cikin abubuwan zahiri daban-daban (photons da electrons). Mafi mahimmancin ma'ana na zamani shine cewa gyare-gyaren kwantum masu dogaro da tsawon zango a cikin barbashi suna da kwatankwacin gwaji kai tsaye a cikin ilimin haske na zango na ci gaba. Wannan ba wasa ne kawai na ilimi ba; yana nuna cewa ƙwararrun gyaran kuskuren launi (chromatic aberration) a cikin na'urorin duban dan tayi na lantarki za a iya samun wahayi daga fasaha a cikin ƙirar kristal na photonic, da kuma akasin haka.

Tsarin Ma'ana: Hujjar ta gina da kyau: (1) Kafa kwatancen tarihi, na gargajiya (Hamilton) a matsayin ingantacce kuma mai amfani (misali, na'urar duban dan tayi ta lantarki). (2) Gano "karyewar" a cikin kwatancen da zuwan ilimin kwantum ya haifar—barbashi sun sami tsawon zango, amma ilimin haske na gargajiya ya kasance na geometric. (3) Gina wannan gibi ta hanyar gabatar da ci gaban zamani guda biyu masu kama da juna: ilimin barbashi mai caji na kwantum (wanda ke ƙara tasirin zango ga barbashi) da ilimin haske na zango da ba na gargajiya ba (Helmholtz/Maxwell, wanda ke ba da cikakkiyar ka'idar zango don haske). (4) Nuna cewa waɗannan tsare-tsare na zamani guda biyu su kansu suna da kwatankwacin (Klein-Gordon/Helmholtz, Dirac/Maxwell), don haka sun kammala kuma sun ɗaga kwatancen zuwa wani mataki mafi girma, mafi mahimmanci. Gudun yana daga haɗuwar gargajiya, ta hanyar rarrabuwar kwantum, zuwa sake haɗuwar zamani a wani mataki mafi fasaha.

4. Ƙarfi & Kurakurai: Bincike Mai Ma'ana

Ƙarfi:

Haɗin Ra'ayi: Babban ƙarfin takardar shine haɗakar da ta yi da kyau. Ta yi nasarar haɗa batutuwa masu banbanci na ci gaba (lissafin Dirac, ilimin haske na Maxwell, ilimin barbashi) cikin labari mai ma'ana. Irin wannan taswirar tsakanin fannoni yana da mahimmanci don haɓaka ƙirƙira, kamar yadda aka gani a fannoni kamar ilimin haske na topological wanda ya aro daga ilimin kayan da aka daskarara.
Mai Nufin Gaba: Ya gano daidai kuma ya ba da gudummawa ga fagen da ke tasowa na Al'amuran Kwantum na Ilimin Barbashi (QABP), yana sanya kwatancen ba a matsayin duban baya ba, amma a matsayin jagora don bincike na gaba. An tabbatar da wannan hangen nesa, kamar yadda QABP da sauran bincike a cikin barbashi masu haɗin kai suka girma sosai.
Tsarin Koyarwa: "Teburin Hamiltonians" da aka ambata (ko da yake ba a nuna shi a cikin ɓangaren da aka zayyana ba) kayan aiki ne mai ƙarfi. Yana ba da ƙamus na lissafi kai tsaye don fassara matsaloli da mafita tsakanin yankuna.

Kurakurai & Iyakoki:

Tarko na "Kwatance" da "Ainihi": Takardar wani lokaci tana haɗarin ƙara yawan kwatancen a matsayin daidaitaccen daidaito. Duk da yake tsarin lissafi na iya yin taswira, ma'auni na zahiri, tasiri masu rinjaye, da ƙayyadaddun aiki sun bambanta sosai. Tsawon zango na de Broglie na electron 100 keV yana picometers, yayin da tsawon zango na haske yana ɗaruruwan nanometers. Wannan yana nufin "tasirin zango" suna bayyana ta hanyoyi daban-daban da ƙarfi na dangi. Maganin da ya dace da wani yanki na iya zama ba zai yiwu a zahiri ba ko kuma ba shi da mahimmanci a ɗayan.
Rashin Tabbatarwa na Zahiri: A matsayin taƙaitaccen bayani/kallo, tana gabatar da tsarin ra'ayi amma ba ta ba da kadan ba ta hanyar sakamakon gwaji na zahiri ko sabbin annabce-annabce da suka samo asali daga wannan ra'ayi na haɗe-haɗe. Tana gaya mana gada ta wanzu amma ba ta nuna mana wani kaya mai mahimmanci da ke ketare ta ba. Yi kwatanta wannan da takarda kamar na CycleGAN (Zhu et al., 2017), wanda ya gabatar da sabon tsari kuma nan da nan ya nuna ƙarfinsa tare da sakamako mai jan hankali, na zahiri na fassarar hoto.
Haɗin Injiniya da ba a haɓaka ba: Tsalle daga kwatancen Hamiltonian na zahiri zuwa ƙirar na'ura mai aiki yana da girma. Takardar ba ta magance matsalolin injiniya da isasshe ba—kamar manyan filayen maganadisu da ake buƙata don mayar da hankali ga barbashi masu ƙarfi da ƙira na dielectric da ake amfani da su don haske—waɗanda ke iyakance canja wurin fasaha kai tsaye.

5. Hanyoyin Aiki & Tasirin Dabarun

Ga masu bincike da masu dabarun R&D, wannan takarda umarni ce don rushe silos.

Kafa Haɗin gwiwar Tsakanin Fannoni: Dakunan aikin da ke aiki akan gyaran kuskure (aberration) a cikin na'urar duban dan tayi ta lantarki yakamata su sami tashoshi masu aiki tare da ƙungiyoyi a cikin ilimin haske na zango na kwamfuta da ƙirar na'urar photonic. Ya kamata a tsara tarurruka a fili don haɗa waɗannan al'ummomin.
Amfani da Kayan Aikin Lissafi: Tsarin matrix don ilimin haske na Maxwell da algorithms na yaduwar kwantum suna da kwatankwacin lissafi. Ya kamata a saka hannun jari a cikin haɓaka ko daidaita ɗakunan ajiyar software (misali, gina akan dandamali kamar MEEP don photonics ko GPT don barbashi) waɗanda za su iya magance matsaloli a cikin yankuna biyu tare da ƙaramin gyara.
Mai da hankali kan "Matsayi Mai Daɗi": Maimakon tilasta kwatancen a ko'ina, gano matsalolin inda taswirar ta fi dacewa. Sarrafa Haɗin kai shine babban ɗan takara. Dabarun samar da barbashi na guguwa (vortex beams) ko yanayin kuzari na kewayawa (orbital angular momentum) a cikin haske (ta amfani da masu gyara haske na sarari) na iya ba da wahayi ga hanyoyin ƙirƙirar barbashi masu tsari, tare da aikace-aikace a cikin binciken kayan ci gaba.
Sake Bincika Na'urorin "Gargajiya" tare da Idanun Kwantum: Yi amfani da tsarin kwantum don bincika na'urorin ƙara gudu da na'urorin duban dan tayi da ke akwai. A ina ne tasirin da ya dogara da tsawon zango da aka yi watsi da shi yana iyakance aiki? Wannan na iya haifar da ingantaccen ƙira amma mai ƙima, tun kafin gina na'urori masu tushen kwantum gabaɗaya.

A taƙaice, takardar Khan ba ta zama cikakkiyar mafita ba, amma ta zama mai ƙarfi ma'auni na bincike. Ƙimarta tana cikin ci gaba da tambaya: "Mun warware wannan matsalar zango a cikin ilimin haske/barbashi; menene matsalar kwatankwacinta a ɗayan yankin, kuma shin mafitarmu ta yi taswira?" Wannan sauƙaƙan tambaya, idan aka bi ta da ƙarfi, na iya buɗe sabbin hanyoyi a cikin fannoni biyu.

6. Cikakkun Bayanai na Fasaha da Tsarin Lissafi

Zuciyar kwatancen tana cikin kamancen tsarin lissafin da ke gudanar da su da kuma "ilimin barbashi" na Hamiltonians da aka samo. Kwatancen gargajiya ya fara ne daga Hamiltonian don barbashi mai caji a cikin filayen lantarki: $$H_{cl} = \frac{1}{2m}(\mathbf{p} - q\mathbf{A})^2 + q\phi$$ wanda, a ƙarƙashin kiyasin paraxial (ƙananan kusurwa) da zaɓin da ya dace na daidaitawa tare da axis na haske (z), za a iya jefa shi cikin wani tsari mai kama da Hamiltonian na ilimin haske na geometric.

Tsalle na kwantum yana farawa da lissafi kamar lissafin Dirac don barbashi spin-1/2: $$\left[ c\boldsymbol{\alpha}\cdot(\mathbf{p} - q\mathbf{A}) + \beta mc^2 + q\phi \right]\Psi = i\hbar\frac{\partial\Psi}{\partial t}$$ Ta hanyar tsari na yau da kullun (kamar canjin Foldy-Wouthuysen ko factorization kai tsaye), ana samun ingantaccen Hamiltonian don yaduwar abubuwan aikin zango tare da z. Wannan Hamiltonian, $\hat{\mathcal{H}}_\text{opt}$, zai ƙunshi sharuɗɗan da suka dace da ikon tsawon zango na de Broglie $\lambda_\text{dB} = h/p$, wakiltar gyare-gyaren kwantum/zango. Misali, tsarin yau da kullun zai iya zama: $$\hat{\mathcal{H}}_\text{opt} = \hat{\mathcal{H}}_0 + \lambda_\text{dB}\,\hat{\mathcal{H}}_1 + \lambda_\text{dB}^2\,\hat{\mathcal{H}}_2 + \cdots$$ inda $\hat{\mathcal{H}}_0$ ya sake samar da sakamakon ilimin haske na geometric na gargajiya, kuma $\hat{\mathcal{H}}_1$, $\hat{\mathcal{H}}_2$ suna gabatar da kurakurai na kwantum (misali, diffraction).

A gefen ilimin haske, farawa daga lissafin vector Helmholtz da aka samo daga lissafin Maxwell: $$\nabla^2 \mathbf{E} + \frac{\omega^2}{c^2}n^2(\mathbf{r})\mathbf{E} = 0$$ Irin wannan tsarin paraxial yana haifar da lissafin bambance-bambancen matrix don yaduwar vector filin lantarki, inda lambar zango $k=2\pi/\lambda_\text{haske}$ ke taka rawa mai kama da $1/\lambda_\text{dB}$.

7. Tsarin Bincike: Nazarin Lamari akan Gyaran Kuskure (Aberration)

Yanayi: Gyara kuskuren siffa ($C_s$) a cikin na'urar duban dan tayi ta lantarki mai babban ƙuduri. A al'ada, $C_s$ lahani ne na geometric na ruwan tabarau na maganadisu. Ta hanyar kwantum-mechanical, yana da gudummawar da ke haɗuwa da diffraction.

Matsalar Ilimin Haske Mai Kwatankwacinsa: Gyara kuskuren siffa da diffraction a cikin na'urar duban dan tayi ta haske mai babban lambar aperture (NA) ko tsarin mayar da hankali na Laser.

Aikace-aikacen Tsarin:

Yi Taswirar Hamiltonians: Gano sharuɗɗan a cikin Hamiltonian na barbashi mai caji na kwantum $\hat{\mathcal{H}}_\text{opt}$ waɗanda suka dace da $C_s$. Nemo sharuɗɗan isomorphic na lissafi a cikin Hamiltonian matrix da aka samo daga ilimin haske na Maxwell don tsarin babban-NA.
Fassara Mafita: A cikin ilimin haske na ci gaba, $C_s$ da diffraction sau da yawa ana gyara su lokaci guda ta amfani da ilimin haske na daidaitawa (madubai masu lalacewa) ko abubuwan haske masu rarrabawa (DOEs) da faranti na lokaci. Bayanin lokaci $\Phi(\mathbf{r})$ wanda cikakkiyar na'urar gyara ta haske ke amfani da ita ana ƙididdige ta ta hanyar juyawar zango ta baya.
Daidaituwa da Gwaji: Babban fahimtar jiki shine cewa gyaran lokaci da ake buƙata $\Phi(\mathbf{r})$ yana yin taswira zuwa gyaran da ake buƙata na gaban zango na lantarki. Ba za a iya yin hakan da madubi mai lalacewa ba amma za a iya samun wahayi daga ra'ayin DOEs. Wannan ya haifar da haɓaka faranti na lokaci na lantarki da kuma, kwanan nan, ra'ayoyin don masu gyara lokaci na lantarki na shirye-shirye ta amfani da tsarin nanofabrication ko sarrafa filayen lantarki, kwatankwacin kai tsaye da masu gyara haske na sarari (SLMs) a cikin ilimin haske.

Wannan tsarin bai ba da amsa da aka shirya ba amma yana ba da hanyar tsari: ingantattun algorithms na haɗawa don holograms da aka ƙirƙira ta kwamfuta a cikin ilimin haske sun zama farkon farko don ƙirƙira na'urori masu siffantar gaban zango na lantarki.

8. Ayyukan Gaba da Hanyoyin Bincike

Hangen nesa na haɗe-haɗe yana buɗe hanyoyi masu ban sha'awa da yawa:

Binciken Barbashi Mai Iyakar Kwantum: Yin amfani da ra'ayoyi daga ilimin haske na kwantum (misali, ganowa homodyne, matsi) don auna emittance na barbashi da kaddarorin haɗin kai a iyakar Heisenberg, wanda ya wuce dabarun bincike na gargajiya.
Barbashi Masu Tsari: Ƙirƙirar barbashi na lantarki ko ion tare da kuzari na kewayawa (orbital angular momentum), bayyanannun Airy, ko yanayin Bessel—wahayi kai tsaye daga haske mai tsari—don sabbin hulɗa tare da kwayoyin halitta a cikin spectroscopy da microscopy.
Sarrafa Haɗin kai a cikin Na'urorin Ƙara Gudu: Yin amfani da ka'idojin sarrafa haɗin kai daga ilimin lissafin Laser don daidaita bayyanan gungu na barbashi akan lokutan femtosecond, mai yuwuwar inganta ingancin Laser na lantarki masu 'yanci da tsare-tsaren ƙara gudu na ci gaba.
Ilimin Barbashi na Topological: Bincika ko matakan topological da jihohin gefe masu kariya, babban jigo a cikin ilimin haske na zamani (misali, masu keɓewar topological don haske), suna da kwatankwacinsu a cikin jigilar barbashi mai caji a cikin lattices na maganadisu na lokaci-lokaci, mai yuwuwar haifar da jagororin barbashi masu ƙarfi.
Rukunin Kwaikwayo na Haɗe-haɗe: Haɓaka software na kwaikwayo na zamani wanda ke amfani da babban mai warware gama gari don yaduwar zango, mai daidaitawa don photons, electrons, ko sauran barbashi na kwantum, yana haɓaka ƙira tsakanin fannoni da sauri.

Hanyar ƙarshe ita ce zuwa cikakken haɗin Injiniyan Kwantum na Barbashi, inda duality na barbashi/zango ba cikas bane amma ma'auni ne na ƙira, ana sarrafa shi da matakin sarrafawa iri ɗaya da aka samu a cikin ilimin haske na zamani.

9. Nassoshi

Khan, S. A. (2002). Kwatancen tsakanin ilimin haske da ilimin barbashi mai caji. arXiv:physics/0210028v2.
Hawkes, P. W., & Kasper, E. (2018). Ka'idojin Ilimin Lantarki (Vol. 1-4). Academic Press. (Babban littafi akan ilimin lantarki na gargajiya).
Dragt, A. J. (1982). Ka'idar Algebraic Lie na Ilimin Haske na Geometric da Kurakurai na Ilimin Haske. Jaridar Ƙungiyar Ilimin Haske ta Amurka, 72(3), 372-379. (Mahimmin takarda akan tsarin Hamiltonian).
Zhu, J. Y., Park, T., Isola, P., & Efros, A. A. (2017). Fassarar Hotuna zuwa Hotuna marasa Haɗin gwiwa ta amfani da Cibiyoyin Adawa na Ma'ana na Zagayowar. Proceedings of the IEEE International Conference on Computer Vision (ICCV). (Misalin takarda da ke gabatar da sabon tsari tare da sakamako nan da nan, mai iya nunawa).
Rodrigues, G. M., & de Assis, A. J. (2021). Al'amuran kwantum na ilimin barbashi mai caji: bita. Jaridar Turai ta Physics D, 75(7). (Bita na zamani wanda ke nuna girma fagen).
Verbeeck, J., Tian, H., & Schattschneider, P. (2010). Samarwa da aikace-aikace na barbashi na guguwa na lantarki. Nature, 467(7313), 301-304. (Takarda mai mahimmanci ta gwaji da ta fahimci barbashi masu tsari).
Jerin Taron OAM. Al'amuran Kwantum na Ilimin Barbashi (QABP). Proceedings akwai daga Cibiyar Ƙara Gudu ta Stanford Linear (SLAC) da sauran cibiyoyin masu masaukin baki. (Jerin taron da aka ambata a cikin takarda, yana rubuta binciken da ke gudana).