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David Lodda
Profession: Social worker
Registration Number: SW33426
Hearing Type: Review Hearing
Date and Time of hearing: 09:00 27/04/2018 End: 11:30 27/04/2018
Location: Health and Care Professions Council, 405 Kennington Road, London, SE11 4PT
Panel: Conduct and Competence Committee
Outcome: Struck off
Please note that the decision can take up to 5 working days to be uploaded onto the HCPTS website. Please contact one of our Hearings Team Managers via tsteam@hcpts-uk.org or +44 (0)808 164 3084 if you require any further information.
Allegation Finding Order Notes Hearing History
This is the second review of a Suspension Order imposed on 8 August 2016 which came into effect on 5 September 2016. That suspension order was imposed for a period of 12 months, and was extended at the first review on 4 August 2017 for a period of 9 months.
This order is due to expire on 5 June 2018.
Allegation:
Whilst registered as a Social Worker and during the course of your employment as a Social Worker with Ealing Council:
1) You did not pass on information and/or give handover notes relating to complex cases to your Team Manager prior to taking annual leave which commenced on 20 August 2013;
2) In relation to Service User A, you did not in a timely manner or at all:
a) make adequate records on Frameworki of your conduct of this case between approximately July - August 2013;
b) progress a safeguarding investigation in relation to Service User A's possessions/ property;
c) complete an overview assessment and/or a mental capacity assessment;
d) obtain and / or record on Frameworki a copy of the section 2 document;
e) oversee a "blitz clean" process of Service User's A property to be completed prior his discharge from hospital;
3) In relation to Service User B, after an assessment on or around 5 August 2013 you did not:
a) record any or any adequate follow up actions on Frameworki;
b) undertake and/or complete any or any adequate follow up actions;
4) In relation to Service User C, following an assessment on or around 3rd July 2013 you did not:
b) take steps to progress this case or record any or any adequate information regarding the progress in this case;
5) In relation to Service User D, after an assessment and visit on 22 July 2013, you did not:
a) record adequately or at all the assessment on 22 July 2013;
b) record adequately or at all any follow up action you took as a result of the assessment on 22 July 2013;
c) take any or any appropriate follow up action arising out of your visit on 22 July 2013;
6) In relation to Service User E, whom you visited on 2 August 2013 in order to conduct an assessment, you did not:
a) record adequately or at all the assessment;
b) (Not found proved)
c) set up a support plan for the Service User;
7) In relation to Service User F, upon having the case assigned to you, you did not:
a) record adequately or at all any visits that you undertook; (Not found proved)
b) record adequately or at all any other actions that you took in relation to this Service User;
8) In relation to Service User G, whose case was allocated to you on 14 February 2013, you failed to:
a) (Not foundproved)
b) (Not foundproved)
9) In relation to Service User H, you did not:
a) arrange in a timely manner or at all for Service User H to be jointly assessed with the Community Psychiatric Nurse;
b) visit Service User H in a timely manner or at all;
c) keep adequate records;
10) In relation to Service User I, following a home visit on 18 July 2013 you did not:
a) record adequately or at all the follow up actions on this case and/or the outcome of any police investigation;
b) arrange in a timely manner or at all strategy meeting;
c) devise and/or implement an interim protection plan;
d) discuss and/or manage and/or adequately record the management of the self-harm incident on 29 July 2013;
11) In relation to Service User J, after a visit on 5 August 2013, you did not:
a) complete and/or send for authorisation the assessment of Service User J that you undertook; or
b) ensure that this matter was presented to the risk panel before funding ran out;
c) undertake a financial assessment and/or MCA in a timely manner or at all;
12) In relation to Service User K, after the case was assigned to you in August 2013, you did not:
a) complete the assessment;
13) In relation to Service User L, after an assessment on or around 5 July 2013, you did not:
a) record the assessment on Frameworki in a timely fashion or at all; and/or
b) complete the assessment in a timely fashion;
14) In relation to Service User M, after a visit on 3 July 2013, you did not:
a) undertake any or any appropriate follow up work;
b) complete the assessment in a timely fashion or at all;
c) progress a request for the cash budget;
15) In relation to Service User N, after a visit on 11 July 2013, you did not:
a) record adequately or at all the involvement that you had with this case until 29 October 2013;
16) In relation to Service User O, after an assessment on 25 June 2013, you did not complete the assessment document and/or a carer's assessment in a timely manner or at all;
17) In relation to Service User P, after completing an assessment on 28 June 2013, you did not:
a) produce at all and/or produce an adequate plan;
18) In relation to Service User Q, after a visit on 15 July 2013, you did not:
a) set up a package of care/contact (POC); and/or
19) In relation to Service User R, you did not:
a) follow up on a safeguarding concern in a timely manner; and/or
b) ensure that an interim protection plan was in place prior to any strategy meeting;
20) The matters described in paragraphs 1 to 19 constitute misconduct and/or lack of competence;
21) By reason of your misconduct and / or lack of competence your fitness to practise is impaired.
Preliminary Matters
Service of Notice
1. The Panel had sight of information that Notice of today's hearing was sent to the Registrant's registered address by first class post on 27 March 2018. The Panel accepted the advice of the Legal Assessor and concluded that service had been effected in accordance with Rules 3 and 6 of the Health and Care Professions Council (Conduct and Competence Committee) (Procedure) Rules 2003 ("the Rules").
Proceeding in Absence
2. Ms Rapu on behalf of the Council applied for the Panel to proceed in the absence of the Registrant. Ms Rapu referred to a letter from the Registrant dated 20 April 2018 which states that he would not be attending today. Ms Rapu submitted that it is in the public interest to proceed in the Registrant's absence.
3. The Panel accepted the advice of the Legal Assessor and took account of the HCPTS Practice Note entitled "Proceeding in the Absence of the Registrant". The Panel took into account the Registrant's position in his letter dated 20 April 2018 namely that he has decided not to attend the hearing. The Panel decided that he has voluntarily waived his right to attend. The Panel took into account that if it proceeds today, there is likely to be some disadvantage to the Registrant. However, in the context of his voluntary waiver, and weighing up the Registrant's interests with the public interest, the Panel was satisfied that it is in the public interest and in the interests of justice to proceed expeditiously with the hearing today.
4. The Registrant is a Social Worker. He commenced his employment with Ealing Council ("the Council") on 18 February 2008 in the Adult Services Team. He worked as a Care Manager in the Acton Older People's Team ("the Team"). His responsibilities included carrying out and reviewing client and carer assessments in order to establish their needs and then agreeing with the service user (and, where
appropriate, their carers) as to how those needs could best be met. The Registrant's work included commissioning services, the provision of support and the investigation and management of safeguarding issues to ensure that service users were protected from abuse. He also took part in a duty rota in the team.
5. The Registrant was on annual leave from 20 August 2013 until 9 September 2013. He became ill and was on sick leave from 10 September 2013 until his return to work on 18 October 2013. During the Registrant's absence the team received a number of enquiries from relatives and carers about progress in the cases of four service users. Subsequently, three complaints were received from relatives and carers about the slow progress of cases. These were passed on to one of the Joint Heads of Older People who directed that managers should review the case records on Frameworki, the client record-keeping system. A subsequent review of 18 cases allocated to the Registrant between June 2013 and August 2013 revealed that a number of cases allocated to the Registrant had not been actioned. In others, assessments had not been completed. Three of the cases involved adult safeguarding issues. The Council's policy in relation to such matters was that they should be actioned as a priority within three days of allocation.
6. The Council commenced a disciplinary investigation and a disciplinary hearing was convened on 10 April 2014. On 5 March 2014 the matter was referred to the HCPC.
7. The Registrant attended the substantive HCPC hearing which concluded on 8 August 2016. That panel found the Registrant's fitness to practise to be impaired on the grounds of misconduct and imposed a
12 month Suspension Order. At the first review of that Order on 4 August 2017, which the Registrant also attended, the reviewing Panel concluded that impairment remained and that the Suspension Order should extend to a further 9 months. That Panel stated in its decision as follows:
"A future reviewing panel would expect the Registrant to provide the following:
a) A reflective piece demonstrating that the Registrant has reflected on his failings as identified by the previous panel and has developed meaningful insight on the impact on service users and colleagues;
b) Evidence of training and learning in relation to his failings;
c) Evidence of continuing professional development relevant to social work practise;
d) References and/or testimonials from those who have employed him in either a paid or unpaid capacity. It would be particularly helpful if the Registrant was able to provide such testimonials from those able to comment on his skills relevant to social work."
8. The Panel heard the submissions of Ms Rapu that the Registrant's fitness to practise remains impaired on the basis that the Registrant presents an ongoing risk of harm to the public, as well as on the basis that a finding of impairment is necessary to uphold the public interest.
9. The Panel took into account the letter from the Registrant dated 20 April 2018 sent for the purposes of today's review, and which states:
"1. My conscience informs me that it is not ethical for me to write a reflective piece on something I know I did not do, and having to include an apology for it.
2. At this point in time, I'm engaged in interests other than social work, which I will be furthering. The two reasons stated here form the basis for my rationale for not attending the meeting on Friday 27th April 2018. I hasten to express my gratitude to you for the opportunity."
10. The Panel was aware that it was undertaking a comprehensive review of the Order in light of the current circumstances. It was mindful that it is considering firstly the question of whether the Registrant remains impaired. The Panel was aware that impairment is a matter for its own independent judgment, and took into account the HCPTS Practice Note entitled "Finding that Fitness to Practise is 'Impaired'".
11. The Panel noted that at the first review, the Registrant's reflection was assessed as not being "meaningful", and that his insight was neither "meaningful" nor "well-developed". Further, the previous reviewing Panel was of the view that he had not sufficiently taken steps to focus learning and development to address the misconduct found proved. The Panel took into account that the Registrant has not fulfilled any of the suggestions of the previous Panel in respect of further evidence
which he might have provided to the Panel today and as a result, is no further forward today with regard to his insight and remediation than he was at the first review hearing. The Panel today, in addition, formed the view from the Registrant's letter dated 20 April 2018 that the Registrant clearly does not accept the regulatory findings against him, and therefore evidence before the Panel of insight into his failings was absent.
12. As a result, the Panel today was of the view that in the absence of sufficient remediation and the absence of evidence of insight, there remains a real risk of harm to service users and the public.
13. The Panel was also of the view that in light of such an ongoing risk to the public, the need to maintain public confidence in the profession as well as to uphold proper professional standards would be undermined if a finding of impairment were not made in the particular circumstances of this case.
14. The Panel therefore determined that the Registrant's fitness to practise remains impaired.
15. The Panel next considered what sanction, if any, should be imposed. Ms Rapu submitted that the matter of sanction is for the decision of the Panel but submitted that no conditions of practice could be formulated which are workable, appropriate and verifiable. Ms Rapu further submitted that only Suspension or a Striking Off Order would maintain public confidence and protect the public.
16. The Panel accepted the advice of the Legal Assessor and took into account the Indicative Sanctions Policy. The Panel was aware that sanction is a matter for the Panel's own independent judgment and that any sanction must be proportionate, and that the purpose of sanction is to uphold the public interest, which includes protection of the public.
17. The Panel first considered taking no action. However, the Panel considered that this was neither proportionate nor appropriate considering the ongoing risk to the public which would not be addressed if there were no restriction on the Registrant's practice. Nor would such an outcome be sufficient to maintain public confidence. The Panel considered and rejected a Caution Order for the same reasons.
18. As noted by the previous panel the Registrant has not worked as a social worker since 2014 and in his letter dated 20 April 2018 he states "that he is engaged in interests other than social work". The Panel also noted that in this letter he makes no expression of interest in returning
to the social work profession. When these factors are combined with the Registrant's lack of activity in responding to the previous panel's suggestions this leads the current Panel to conclude that a Conditions of Practice Order is neither practicable nor workable.
19. The Panel considered that whilst imposing a further period of suspension would protect the public in the short term it would not uphold the public interest given the Registrant's stated position. This is on the basis that the Registrant's written response dated 20 April 2018 confirms his unwillingness to remedy his failings. He has not addressed any of the issues required to be addressed by the previous reviewing Panel, nor has he expressed any commitment to developing his social work knowledge and skills in a tangible way. The Panel also notes that this failure is despite having two Suspension Orders imposed in the past.
20. The Panel was therefore of the view that the only proportionate Order sufficient to protect the public and maintain the public interest is a Striking Off Order. In this regard, the Panel considered para. 48 of the Indicative Sanctions Policy which states as follows:
"Striking off should be used where there is no other way to protect the public, for example, where there is a lack of insight, continuing problems or denial. A registrant's inability or unwillingness to resolve matters will suggest that a lower sanction may not be appropriate."
21. The Panel is of the view that there is a lack of insight shown by the Registrant, clear denial of wrongdoing, and an unwillingness to remediate.
22. The Panel did consider the effect a Striking Off Order will have on the Registrant, namely removing his right to practise, and considered proportionality. However, in light of the position taken by the Registrant and the lack of evidence of insight and remediation, the Panel was satisfied that the need to protect the public and maintain public confidence in the profession outweighed the Registrant's interests.
23. The Panel therefore decided to impose a Striking Off Order.
That the Registrar is directed to strike the name of Mr David Lodda from the Register on the date that this Order comes into effect.
The Order imposed today will apply from 5 June 2018.
Hearing History
History of Hearings for David Lodda
Hearing type
Outcomes / Status
27/04/2018 Conduct and Competence Committee Review Hearing Struck off
04/08/2017 Conduct and Competence Committee Review Hearing Suspended
08/08/2016 Conduct and Competence Committee Final Hearing Suspended
© 2020 Health & Care Professions Tribunal Service
The Health and Care Professions Tribunal is the fitness to practise adjudication service of the Health and Care Professions Council.
405 Kennington Road, London SE11 4PT +44 (0)808 164 3084
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Vape Tax Question: What to Tax?
(INDIANAPOLIS) – A tax on vaping is still moving at the statehouse, but senators are trying to decide exactly what to tax.
The House approved a tax based on the amount of e-liquid in a cartridge. Health advocates say that tax of four to 12 cents a cartridge is a good start, but too low. Bryan Hannon with Tobacco Free Indiana says Juul, by far the dominant player in the vaping industry, uses little liquid and would barely be touched. He's urging senators to set the tax at 24%of the price, about what the tax on traditional cigarettes adds up to. That would push the tax as much as 24 times higher.
John McCullough, president of Evansville e-liquid manufacturer Vapor Bank, says the tax is unfair for the opposite reason. He says it should be based on the amount of nicotine in the product. He argues it's unfair for his product to face a higher tax than cigarettes with a third more nicotine.


text:
Recording facility fosters S African musical talent
Johannesburg's Downtown Music Hub is a government initiative offering high-quality recording facilities at a low cost.
by Tania Page
Johannesburg - The Downtown Music Hub in Johannesburg is a government initiative, offering high-quality recording facilities at a low cost.
The building housing the music hub has been associated with the music industry since the 1970s.
The government bought it seven years ago with the idea to make world-class recording facilities accessible to the general public by setting the prices low. The renovations are not quite finished but the studios are busy with budding artists.
Artists like Christopher Manana, who works in a factory at night, has been honing his passion for maskandi Zulu folk music at the Music Hub.
Chola Makgamathe, CEO of the hub, says the music centre strives to keep its price low to develop artists.
"From its inception it has always had a developmental agenda, and access has always been a core feature," he said.
"Our aim is really to keep our prices at a level where anyone can come through and record their music."

redpajama_set_name: RedPajamaCommonCrawl


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Maria Francesca Ticchi
Maria Francesca Ticchi (1887-1922), Capuchin Poor Clare from Mercatello
On the afternoon of November 23, 2020, the Holy Father authorized the signing of the decree super virtutibus for Mother Maria Francesca Ticchi, a Capuchin Poor Clare from the Monastery of Mercatello.
The Venerable Servant of God, Maria Francesca Ticchi was born on April 23, 1887 in Belforte all'Isauro (in Pesaro e Urbino), second child of Giuseppe and Anna Gostoli. Baptized the following day with the names Clementina Adelaide Cesira, she grew up in a serene and hard-working family environment, helping with the management of the house and with the care of her brothers -- one in particular whom she carried around for so long in her arms, that a deformation was caused in the bones of her hip.
Once a year, Clementina's family used to visit the monastery of the Capuchin nuns of Mercatello sul Metauro, built on the birthplace of the Capuchin mystic Saint Veronica Giuliani. It was during one of these visits, on August 1, 1899, that the Servant of God felt the call to cloistered life. At the age of 15, with the consent of her parents, she entered as a proband chorister on November 24, 1902.
Due to a particular and distressing situation inside the monastery, the proband period for Clementina lasted for three and a half years.
Before taking the habit, the young candidate was sent home for about twenty days where she found the usual affection, but also a certain reluctance, especially in her father, to see her enter the monastery – as was so much talked about.
She returned to the monastery on March 6, 1906, and after having overcome an acute episode of rheumatoid arthritis, on June 21, 1906 she was finally able to begin her probationary period by taking the name of Sister Maria Francesca. She made her first profession on July 9, 1910 and three years later on July 9, 1910, her perpetual profession.
In the monastery, the Servant of God lived a life of total dedication to the Lord, of faithful observance to the Rule, to prayer, to poverty, and to service and charity towards her sisters. She first had charge of the dispensary and then, from 1914 until her death, held the position of novice mistress, a service in which she was appreciated for her wisdom, motherhood and balance.
In the winter of 1907, pleurisy developed, and a relapse occurred in the winter of 1908. She was finally diagnosed with pulmonary tuberculosis. From 1915, rheumatoid arthritis was complicated by painful ulcerations in the knee and leg. In July 1915, she fell ill with typhus and bronchitis and in 1916, the tuberculous manifested itself as bone tuberculosis which affected some vertebrae. By her early twenties, due to vertebral deformation, the Servant of God appeared bent over like an old woman.
An attempt was made to intervene with some therapeutic remedies but without any apparent results. In early 1922, she was diagnosed with pulmonary tuberculosis. With great patience, striving not to be a burden to the sisters and trying to fulfill all her religious duties, the Servant of God offered herself in a spirit of reparation and in union with the Lord.
The condition of the Servant of God worsened during the night of June 19, 1922, yet she remained always alert and smiling. Having received the last sacraments, she died on June 20, 1922 and was buried in the cemetery of Mercatello.
Last modified on Wednesday, 02 December 2020 07:15
The Analecta in Digital Format
Changes in the General Curia
Advent Day of Recollection
Laurentianum 2021 – special issue


Lyrics Depot is your source of lyrics to If You're In A Pop Group You'll End Up Paying A Fortune Practicing At Peter Practice's Practice Place by Toy Dolls. Please check back for more Toy Dolls lyrics.
Do you want some crisps and pop?
money's gone to Peter's head.


Hannah V Warren teaches writing, poetics, and literature courses. She serves as a co-editor for KU's literary journal, and her works have recently appeared in Whiskey Island, Bear Review, Longleaf, and Spirit's Tincture. She writes and paints and tends her plants in Lawrence, Kansas.


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Appears In  APIGroup

Field        | Description
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groupVersion
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version
 *string* | version specifies the version in the form of "version". This is to save the clients the trouble of splitting the GroupVersion.

redpajama_set_name: RedPajamaGithub


98 03 ford ranger 60 40 bench seat black charcoal car seat. saddle blanket bench seat cover bench design for your ideas. 1998 2001 ford ranger xlt xcab front high back 60 40 split. 1990 used ford f 150 f 150 xlt lariat at online. 1990 ford f150 future classics. 1956 chevy truck seat cover velcromag. 1998 2001 ford ranger xlt xcab custom fit seat covers cl. prepping a cab and mounting custom bucket seats hot rod. 87 91 bench seat cup holder ford truck enthusiasts forums. 2009 f150 xlt arm rest cupholder removal for clazzio seat.


text:
Home » Collections » Productions database
General collections archive
Special collections archive
Productions database
Ancient performances database
Dulwich College, London
Productions by this company
Casina (1864) Dulwich College (London, Greater London, England)
Heautontimorumenos (1873) Dulwich College (London, Greater London, England)
The Knights (1873) Dulwich College (London, Greater London, England)
Vespae (1874) Dulwich College (London, Greater London, England)
Frogs (1876) Dulwich College (London, Greater London, England)
Cistellaria (1882) Dulwich College (London, Greater London, England)
Acharnians (1884) Dulwich College (London, Greater London, England)
Ajax (1885) Dulwich College (London, Greater London, England)
Knights (1886) Dulwich College (London, Greater London, England)
Aulularia (1890) Dulwich College (London, Greater London, England)
Captivi (1892) Dulwich College (London, Greater London, England)
Clouds (1894) Dulwich College (London, Greater London, England)
The Frogs (1895) Dulwich College (London, Greater London, England)
Birds (1897) Dulwich College (London, Greater London, England)
The Clouds (1914) Dulwich College (London, Greater London, England)
The Birds (1926) Dulwich College (London, Greater London, England)
Wasps (1933) Dulwich College (London, Greater London, England)
Alcestis (1938) Dulwich College (London, Greater London, England)
Ion (1945) Dulwich College (London, Greater London, England)
Orestes (1993) Dulwich College (London, Greater London, England)
The Bacchae (1995) Dulwich College (London, Greater London, England)
All content © APGRD 1996 - present


text:
This is where the theory ends. Real leadership and transformational change are born here.
About Todd Leader
Thank you for the book launch!
I would like to express my sincere gratitude to all who were able to attend last night's official launch of my book, "It's Not About Us; The Secret to Transforming the Mental Health and Addiction System in Canada", at Saint Mary's University. I was truly flattered and humbled by the standing-room-only crowd, and by the many very kind and generous comments people made. I would also like to thank all those who were unable to make it but sent me well-wishes. Thank you to Lindsey MacIntosh (CBU Alumni Association) for being in attendance to support this CBU grad.
This has been a work of passion that has found its way to being published, thanks to the amazing support and guidance of Lisa Pomfrey-Talbot and Cathydia Press. I am so grateful for the generousity of time and words, shown by my guest speakers, Sheldon MacLeod (News 95.7 Radio), Dr. Robert Summerby-Murray (President of Saint Mary's University), Dr. Marc Patry (Chair, Psychology Department at SMU), Dr. Peter Vaughan (Nova Scotia Deputy Minister of Health and Wellness), and my wonderful friend Laurel Walker (mental health advocate).
Special thanks to Dr. Adam Sarty, Danielle Boudreau, and Courtney Merriam, with the Faculty of Science at Saint Mary's University, and Liz Galbraith with the Change Lab Action Research Institute (CLARI), for planning, organizing, and hosting such a wonderful event.
Thanks as well to all the SMU students (my favourite people) who came out. I was hounoured by your support, and what I know to be your interest in changing the world:)
Finally, thanks to my family members who were able to attend. Your presence, and your support through the process, meant more than I can say.
Suicides - Real change needed in Labrador mental health system
Review of new book "It's Not About Us", by Sara Ramsay
Social Workers' Frustrations
Coming in October...a call to action
Copyright © 2016. Leader Development.
Site by WiNiTh


I had a birthday, so, some books and a CD came my way.
The Big Book of Sherlock Holmes Stories edited by Otto Penzler [Pantheon Books 2015 hardcover, new] – mystery short story collection, Sherlock Holmes. The cover says this is "the biggest collection of Sherlock Holmes stories ever assembled". There are 82 stories within, though some are parodies and certainly not all of them are in the mold of the originals, in terms of time, place or method. The recent MX Book of New Sherlock Holmes Stories contains 61 stories but are all set in Victorian England with proper characters Doyle could very well have used. So two different types of collections, but a whopping total of stories to read, or in the case of this book, some to reread.
Captain's Memories by Henry Melton [Wire Rim Books 2014 trade paper, new] – science fiction short stories. Collection of 28 short stories, mostly in the Project Saga, tales of terraforming in the solar system. Stories range from 1997 to 2014.
The Cape Verdean Blues Horace Silver Quintet plus J.J. Johnson [Blue Note CD] – jazz, originally recorded October 1965. I picked up on this from a post on Todd Mason's blog. Really fine sessions. Ive listened to it several times already.
This entry was posted in Books & Reading, Music, Mystery. Bookmark the permalink.
Richard, I'd like to lay my hands on "The Big Book of Sherlock Holmes Stories." I will see if I can get a physical copy.
Ah, now you're talking. These look good. Let us now how you like the Melton because that one sounds like something I would try.
You're Star Trek willies must be up.
Prashant, it was published Tuesday, so will be available whenever that would translate to your locale.
Nice Birthday presents! I'll probably pick up a copy of that Sherlock Holmes tome. Music always goes great with a Birthday! And Todd Mason makes some excellent recommendations!
He does indeed, George. I figured you'd get this "Big Book" since you have gotten the others Penzler has done.


This Spyro Compost Mixing Tool is made of heavy duty steel and available in both 37" (steel) and 30" (plastic) sizes. It has a soft but firm, large T-Handle grips for easy handling. The large fold-down "wings" are designed for better mixing of compost. Handle is 9"W X 1-1-4" Diameter. Wings on bothe models are 4.5" long and made of solid steel. Includes a hook for easy storage on a wall. One Year Ltd. Warranty.
This Spyro Compost Mixing Tool is made of heavy duty steel and available in both 37" (steel) and 30" (plastic) sizes. It has a soft but firm, large T-Handle grips for easy handling. The large fold-down "wings" are designed for better mixing of compost. Handle is 9"W X 1-1-4" Diameter. Wings on both models are 4.5" long and made of solid steel. Includes a hook for easy storage on a wall. One Year Ltd. Warranty.
NOTE: 30" Plastic version is available as a promotional item with some of our composters. Ask about this offer!


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\begin{document}

\title{ Quantum Optics in Maxwell's Fish Eye Lens with Single Atoms and Photons}

\author{J. Perczel}
\affiliation{Physics Department, Massachusetts Institute of Technology, Cambridge, MA 02139, USA}
\affiliation{Physics Department, Harvard University, Cambridge,
MA 02138, USA}

\author{P. K\'{o}m\'{a}r}
\affiliation{Physics Department, Harvard University, Cambridge,
MA 02138, USA}

\author{M. D. Lukin}
\affiliation{Physics Department, Harvard University, Cambridge,
MA 02138, USA}

\date{\today}

\bigskip
\bigskip
\bigskip
\begin{abstract}

We investigate the quantum optical properties of Maxwell's two-dimensional fish eye lens at the single-photon and single-atom level. We show that such a system mediates effectively infinite-range dipole-dipole interactions between atomic qubits, which can be used to entangle multiple pairs of distant qubits. We find that the rate of the photon exchange between two atoms, which are detuned from the cavity resonances, is well described by a model, where the photon is focused to a diffraction-limited area during absorption. We consider the effect of losses on the system and study the fidelity of the entangling operation via dipole-dipole interaction. We derive our results analytically using perturbation theory and the Born-Markov approximation and then confirm their validity by numerical simulations. We also discuss how the two-dimensional Maxwell's fish eye lens could be realized experimentally using transformational plasmon optics.

\end{abstract}

\pacs{
		42.50.Ex
 		03.67.Bg
		42.50.Dv

	}
\maketitle

\section{Introduction}

Maxwell's two-dimensional fish eye is an optical lens with remarkable imaging properties.
Light emitted from {\it any} point inside the lens refocuses at the antipodal point on the opposite side of the lens. Since J. C. Maxwell's original work that studied ray optics inside the lens \cite{Maxwell1854}, the properties of the fish eye have been analyzed in a variety contexts, including electromagnetic waves \cite{Tai1958,Rosu1994}, scalar waves \cite{Greenwood1999}, quantum mechanics \cite{Makowski2009} and supersymmetry \cite{Rosu1996}.

More recently, it was proposed that Maxwell's fish eye lens may have the ability to perfectly refocus electromagnetic waves emerging from a point source \cite{Leonhardt2009,Leonhardt2010b,Ma2011}, thereby overcoming the diffraction limit \cite{Born1999}. The idea of perfect imaging with Maxwell's fish eye has generated vigorous debate \cite{Leonhardt2010,Leonhardt2010a,Leonhardt2011b,Tyc2011,Blaikie2010,Merlin2010,Blaikie2011,Kinsler2010,Gonzalez2011,Quevedo-Teruel2012,Tyc2014,Gonzalez2012,Xu2012,Ma2013,Sun2010,Sun2010b,Kinsler2011,Leonhardt2011,Pazynin2012,Liu2013,Tyc2014,Alonso2015,Horsley2015,He2015,Rosenblatt2017,Minano2011,Minano2014,Leonhardt2015,Leonhardt2015a}
. It has focused on how the presence of a point-like detector, placed at the focus point, changes the image formed and whether perfect imaging is an artifact of the detector. On the one hand, it has been argued that the presence of the detector, which can absorb the incoming radiation, is necessary to form a perfect image \cite{Leonhardt2009,Leonhardt2010b,Ma2011,Leonhardt2010,Leonhardt2010a,Leonhardt2011b,Tyc2011}. On the other hand, concerns have been raised that the detector itself would contribute electromagnetic waves to the image formed, giving rise to the apparent subwavelength focus point \cite{Blaikie2010,Tyc2011,Merlin2010,Blaikie2011}. Subsequently, the discussion about perfect imaging has shifted to finding a simple and realistic model for such detectors \cite{Kinsler2010,Gonzalez2011,Quevedo-Teruel2012,Tyc2014,Gonzalez2012,Xu2012,Ma2013}. More recently, it was suggested that perfect imaging may be possible when operating very close to the resonances of the fish eye lens \cite{Minano2011,Minano2014,Leonhardt2015,Leonhardt2015a}.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=8.5cm]{./01.png}
\end{center}
\caption{(color online) Light rays propagating within the infinite 2D fish eye lens trace out perfect circles (dashed red lines). If a mirror of radius $R_0$ is introduced (black circle), the trajectories remain closed (solid red lines). All light rays emerging from an arbitrary point within the lens (green dot) refocus at the antipodal point (blue star). The color code and the inset show the spatial variation of the refractive index as a function of the radius, where we assume that $n_0=1$ in \refeq{FEn}. For $r>R_0$ the refractive index of the fish eye dips below 1.}
\label{geometry}
\end{figure}

In this paper, we study the imaging properties of Maxwell's two-dimensional (2D) fish eye lens at the single-photon level using single atoms. In particular, we assume that both the source and the detector of the photon are individual atoms and thus no ambiguity arises regarding their fundamental properties. One atom, initially in its excited state, emits the photon and the second atom, initially in its ground state, absorbs the photon, storing it in a metastable state for fluorescent readout. This is conceptually the simplest model for a source and a detector \cite{Leonhardt2015}

We model the 2D lens as an effective photonic cavity filled with an inhomogeneous dielectric material and solve for the atom-photon dynamics inside the lens. Since the rate of photon exchange between the atoms is set by the local electric field strength, the atomic dynamics is a sensitive indicator of the electric field distribution of the photon during absorption. In particular, we find the the photon exchange rate between the two atoms, which are detuned from the cavity resonances, is well described by a simple model, which assumes that the photon is focused to a diffraction-limited area during absorption.

\begin{figure*}[!]
\includegraphics[width=16.5cm]{./02_dd_range.png}
\caption{(color online) (a) Schematic depiction of the two dipoles embedded in the fish eye cavity, which is surrounded by mirrors on all sides. (b) Spectrum of the cavity $(\omega_l=\sqrt{l(l+1)}c/(R_0n_0)$, $l=1,2,3\dots)$ in the absence ($\kappa = 0$) and the presence ($\kappa \neq 0$) of losses. The atomic resonant frequency $\omega_0$ is tuned between two resonances of the cavity. (c) Strength of the dipole-dipole interaction $\delta\omega({\mathbf r}_1,{\mathbf r}_2)/\Gamma_0$ between two atoms for four different lens radii: (i) $R_0=4.93\lambda$, (ii) $R_0=8.11\lambda$, (iii) $R_0=11.3\lambda$ and (iv) $R_0=14.48\lambda$, assuming lens thickness $b=\lambda/10$ and $\Gamma_0=d_z^2\omega_0^3/(3\pi \varepsilon_0\hbar c^3)$. The lens radii are chosen such that the transition frequency of the atoms $\omega_0=2\pi c/\lambda$ lies halfway between the resonances of the lens ($l =1,2,3\dots$). In particular, we chose the order parameters (i) $\nu=30.5$, (ii) $\nu=50.5$, (iii) $\nu=70.5$ and (iv) $\nu=90.5$, where $\nu=\frac{1}{2}(\sqrt{16\pi^2(R_0n_0/\lambda)^2+1}+1)$ and $n_0=1$. The atom on the left is positioned exactly $\lambda$ away from the mirror, whereas the position of the second atom is sweeped. The strength of the interaction peaks $\lambda$ away from the opposite mirror surface with a height that is independent of the radius of the lens and the interatomic distance. (d) Enlarged view of the dipole-dipole interaction near the antipodal point, showing that the width of the peak is approximately $\lambda/2$.}
\label{range}
\end{figure*}

We also analyze the capabilities of the fish eye to enhance the interaction between distant atoms. In particular, we show that the dipole-dipole interaction mediated by the fish eye lens is effectively infinite in range. This infinite-range interaction is a consequence of the unique focusing properties of the fish eye lens and is analogous to the infinite-range interactions mediated by quasi-1D waveguides, which have been the subject of extensive research in recent years in the context of hollow \cite{Gomez-Medina2001,Shahmoon2013}, plasmonic \cite{Chang2006,Gonzalez-Tudela2011,Dzsotjan2010}, microwave \cite{Lalumiere2013,vanLoo2013,Inomata2014} and dielectric \cite{Domokos2002,Horak2003,Klimov2004,LeKien2006,Chang2012,Douglas2015} waveguides. Within this model, we quantitatively evaluate entangling operations and discuss a realistic experimental realization.

This paper is organized as follows. In Section \ref{formalism} we discuss the general formalism behind our work and derive the dipole-dipole interaction mediated by the lens between atoms. In Section \ref{results} we discuss the entanglement of atoms within the lens. In Section \ref{realization} we discuss a possible physical realization of the 2D fish eye using transformational plasmon optics.  Key insights of our work are summarized in Section \ref{conclusion}.

\section{General Formalism}\label{formalism}

In this section we describe the general formalism behind our calculations for exploring the quantum optical properties of the system and calculate the dipole-dipole interaction between atoms placed inside the lens.

\subsection{Maxwell's Fish Eye Lens}

The two-dimensional fish eye lens is a dielectric medium of infinite size with refractive index \cite{Leonhardt2010b}
\begin{equation}\label{FEn}
n({\mathbf r})=\frac{2n_0}{1+(r/R_0)^2},
\end{equation}
where $r=\sqrt{x^2+y^2}$, $R_0$ is the natural length scale of the problem and $n_0\geq 1$ can be chosen arbitrarily. We assume $n_0=1$ for all numerical calculations in this paper. In the limit of geometric optics, light rays propagate in perfect circles (Fig.~\ref{geometry}, dashed circles). All rays emitted from a single point inside the lens ultimately meet at the antipodal point. For $|{\mathbf r}|>R_0$ the refractive index varies between ${n_0}$ and 0, which is difficult to achieve in practice. Thus the lens is modified by placing a mirror around the circle of radius $|\mathbf{r}|=R_0$ (black circle in Fig.~\ref{geometry}). In the presence of the mirror the trajectories still remain closed (solid red lines in Fig.~\ref{geometry}) \cite{Leonhardt2009}.

The 2D fish eye can be realized for {\it electromagnetic waves} in a thin disk of radius $R_0$ with a dielectric material of radially varying refractive index given in Eq.~\eqref{FEn}, which is constant along the $\hat{z}$ direction. If all surfaces of the disc are surrounded with mirrors, the TE modes with the lowest frequency will realize the ideal dynamics of the 2D fisheye \cite{Ma2011,Liu2013}. To ensure that a source radiating at frequency $\omega_0$ excites only the lowest TE modes of the lens, the thickness of the disk, $b$, is chosen such that the relevant optical frequency of the source is much smaller than $\pi c/b$, where c is the speed of light in vacuum \cite{Justice2006,Liu2013}. Later, we consider a realistic realization of the two-dimensional fish eye with surface plasmons, where the transverse confinement arises naturally from the confinement of the plasmons to the metal-dielectric interface \cite{Liu2010,Zentgraf2011}.

\subsection{Hamiltonian}

We model the atoms as two-level systems with ground and excited states denoted by $\ket{g}$ and $\ket{e}$, respectively. The Hamitonian describing the evolution of the system composed of the two atoms and the fish eye modes is given by
\begin{equation}\label{Hamiltonian}
H=H_\text{atom}+H_\text{field}+V,
\end{equation}
where the atoms evolve according to ${H_\text{atom}=\hbar\omega_0\sum_{i=1,2}\Ket{e_i}\Bra{e_i}}$ and the evolution of the electromagnetic field is described by $H_\text{field}=\sum_{l,m}\hbar\omega_l a\+_{l,m}a_{l,m}$, where $a_{l,m}$ is the annihilation operator of an eigenmode of the lens labelled by $(l,m)$. The interaction of the two atoms with the electromagnetic field is given by $V=-\sum_{i=1,2}\mathbf{d}_i\cdot\mathbf{E}({\mathbf r}_i)$, where $\mathbf{d}_i=d_z(\sigma^\dagger_i+\sigma_i)\hat{z}$ with $\sigma_i=\Ket{g_i}\Bra{e_i}$ and
$d_z$ is the $z$-component of the dipole moment of the $e\to g$ transition of the atom, $\mathbf{E}({\mathbf r}_i)$ is the electric field operator at position ${\mathbf r}_i$ within the lens, and we neglect variations of the field over the size of the atoms. The two atoms are positioned at ${\mathbf r}_1$ and ${\mathbf r}_2$ (see Fig.~\ref{range}(a)). Note that \refeq{Hamiltonian} describes a closed lossless system composed of the lens and the two atoms with no coupling to free-space modes. Later we will consider how photon loss from the fish eye modes affects our results.

\subsection{Quantization in the Fish Eye Lens}

We follow the quantization scheme of Glauber and Lewenstein \cite{Glauber1991} to write down the expression for the quantized electromagnetic field $\mathbf{E}({\mathbf r}_i)$ of the lens
\begin{equation}\label{Efield}
\mathbf{E}({\mathbf r}_i)=i\sum_{l,m}\bigg(\frac{\hbar\omega_{l}}{2\varepsilon_0}\bigg)^{1/2}[a_{l,m}\mathbf{f}_{l,m}({\mathbf r}_i)-a_{l,m}^{\dagger}\mathbf{f}^*_{l,m}({\mathbf r}_i)],
\end{equation}
where $\mathbf{f}_{l,m}$ are the classical eigenmodes of the cavity that are solutions of the wave equation
\begin{equation}\label{waveEquation}
n({\mathbf r})^2\,\frac{\omega_{l,m}^2}{c^2}\mathbf f_{l,m}({\mathbf r}) - \nabla\times\left[\nabla\times \mathbf f_{m,l}({\mathbf r})\right]=0,
\end{equation}
subject to the transversality condition
\begin{equation}\label{transversal}
\nabla\cdot \left[n({\mathbf r})^2\,\mathbf f_{l,m}({\mathbf r})\right]=0,
\end{equation}
together with the boundary condition that ${\mathbf f_{l,m}\cdot \hat z=\mathbf f_{l,m}\cdot \hat \phi=0}$ at ${|{\mathbf r}|=R_0}$ due to the presence of the mirror. The position-dependent refractive index $n({\mathbf r})$ is given by \refeq{FEn}. The solutions of \refeq{waveEquation} and \refeq{transversal} can be chosen to form an orthonormal set satisfying
\begin{equation}
\int_\mathcal{V}d^3r\; n({\mathbf r})^2\,  \mathbf{f}_{l,m}({\mathbf r})\cdot \mathbf{f}_{l',m'}^*({\mathbf r}) = \delta_{ll'}\delta_{mm'},
\end{equation}
where the integral is performed over the quantization volume $\mathcal{V}$.

Solving these equations, the lowest TE modes of the fish eye take the following form
\begin{equation}\label{modes}
\mathbf{f}_{l,m}(r,\phi)=\sqrt{\frac{2}{bR_0^2n_0^2}}Y_l^m\bigg(\text{arccos}\bigg(\frac{|{\mathbf r}|^2-R_0^2}{|{\mathbf r}|^2+R_0^2}\bigg),\phi\bigg)\hat{z},
\end{equation}
where $Y^m_l(\theta,\phi)$ are the spherical harmonic functions, ${\phi=\arccos(x/|{\mathbf r}|)}$ is the azimuthal angle associated with position ${\mathbf r}$ and the eigenfrequencies are ${\omega_l=c\sqrt{l(l+1)}/(R_0n_0)}$. The modes $\mathbf{f}_{l,m}$ are labelled with the rescaled wavenumber $l=1,2,3\ldots$ and the angular momentum index $m$, where ${m(l)=-(l-1),-(l-3),...\,,(l-1)}$ is enforced by the boundary condition $\mathbf{f}_{l,m}(R_0,\phi) = 0$. The discrete spectrum of the fish eye is schematically shown in Fig.~\ref{range}(b). The number of degenerate states increases linearly with $l$, since $\sum
_{m(l)}1=l$.

\subsection{Photon transfer between two atoms via dipole-dipole interaction}

In this section, we investigate the resonant transfer of a photon between two atoms via the dipole-dipole interaction, the strength of which we denote by $\delta\omega$.

In quantum optics, the most fundamental model for photon emission and detection assumes that one atom is initially in its excited state $\ket{e_1}$, while the second atom is in its ground state $\ket{g_2}$. When the system evolves coherently in time, the excited atom (virtually) emits the photon and after time $t_{\text{int}}\sim \pi/(2\delta\omega)$ the second atom fully absorbs the photon as its atomic population is transferred to the excited state $\ket{e_2}$ \cite{Milonni1974,Dzsotjan2010}.

Furthermore, by making use of additional metastable states $\ket{s_i}$ with $i=1,2$ (see \reffig{levelstructure}) that only couple to $\ket{e_i}$ via the time-dependent classical control pulse $\Omega_i(t)$ (such that $\Omega_i \gg \delta\omega$), the photon transfer can be performed in a controlled, realistic scheme \cite{Pellizzari1995,Cirac1997,Chang2007}.
In particular, by adjusting $\Omega_1(t)$ and $\Omega_2(t)$, the photon transfer can be initiated via the excitation of $\ket{e_1}$ and, as the photon is reabsorbed, the atomic population of the second atom can be transferred to the metastable state $\ket{s_2}$. Then, by switching off $\Omega_2(t)$, reemission into the cavity can be prevented. From the metastable state the photon can be read out using standard fluorescence techniques \cite{Fuhrmanek2011,Gibbons2011}. This completes the detection of the photon.

In a standard quantum optical setting, the dipole-dipole interaction between two atoms with level spacing $\omega_0$ between ground $\ket{g_i}$ and excited states $\ket{e_i}$ in any environment can be expressed in terms of the classical Green's function components $G_{\alpha\beta}({\mathbf r}_1,{\mathbf r}_2,\omega_0)$ (with $\alpha,\beta = x,y,z$) through the following expression \cite{Dung1998,Dung2002,Scheel2008,Buhmann2012}
\begin{equation}\label{dd}
\delta\omega({\mathbf r}_1,{\mathbf r}_2)=\frac{d_z^2\omega_0^2}{\hbar\varepsilon_0c^2}\text{Re}\{G_{zz}({\mathbf r}_1,{\mathbf r}_2,\omega_0)\},
\end{equation}
where we assume that the two atoms are located at ${\mathbf r}_1$ and ${\mathbf r}_2$ and their dipole moments $d_z$ are oriented along the $z$-axis. Note that the real (imaginary) part of the Green's function $G_{zz}({\mathbf r}_1,{\mathbf r}_2,\omega_0)$ has the simple interpretation of being the $z$-component of the in-phase (out-of-phase) component of the electric field generated at position ${\mathbf r}_2$ within the lens due to the presence of a $z$-oriented point-like dipole at position ${\mathbf r}_1$ radiating at frequency $\omega_0$.\\

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.45 \textwidth]{./07_level_structure.png}
\end{center}
\caption{(color online) Schematic depiction of a realistic scheme for the photon transfer between the two atoms. The first atom emits the photon, while the second atom fully absorbs it. By applying classical time-dependent control pulses $\Omega_1(t)$ and $\Omega_2(t)$, the transfer can be initiated and the photon can be captured in the metastable state of the second atom, from which the photon can be read out using fluorescence techniques.}
\label{levelstructure}
\end{figure}

We note that, when the classical Green's function of a problem is analytically known, it is typically a simple matter to evaluate \refeq{dd} and find the dipole-dipole interaction between atoms. However, for the fish eye there is debate about what Green's function correctly describes the imaging process. The subtlety of the issue arises from the fact that the fish eye, which models the closed surface of the sphere, is inherently a closed system from which radiation cannot escape in the absence of losses and detectors \cite{Leonhardt2009,Leonhardt2010b,Ma2011}. As mentioned previously, the accurate mathematical modeling of detectors has been a key focus of the discussion regarding perfect imaging \cite{Blaikie2010,Leonhardt2010,Kinsler2010,Leonhardt2010a,Sun2010,Sun2010b,Merlin2010,Blaikie2011,Kinsler2011,Tyc2011,Gonzalez2011,Leonhardt2011,Quevedo-Teruel2012,Pazynin2012,Gonzalez2012,Xu2012,Ma2013,Liu2013,Tyc2014,Alonso2015,Horsley2015,Leonhardt2015,Leonhardt2015a,He2015,Rosenblatt2017}.

Here, since we model both the `source' of the radiation and the `detector' as atoms, the exact expression for the dipole-dipole interaction can be obtained from the standard quantum optical master equation \cite{Gross1982}, where no ambiguity arises in the derivation of the results. Furthermore, as we show below, the expression obtained for the dipole-dipole interaction from the master equation exactly matches one of the two Green's functions discussed extensively in the fish eye literature, allowing us to directly use \refeq{dd}, which substantially simplifies numerical calculations.

The quantum optical master equation in the Born-Markov approximation, which governs the evolution of the atoms inside the lens, takes the following form in the interaction picture \cite{Gross1982}
\begin{eqnarray} \label{ME}
\frac{d{\tilde \rho}}{dt}=-\frac{1}{\hbar^2}\int_0^\infty \!\!\!d\tau\, \text{Tr}
\left[\tilde V(t),\left[\tilde V(t-\tau),\tilde \rho (t)\otimes \ket{0}\bra{0}\right]\right]\!\!,\quad
\end{eqnarray}
where $\ket{0}\bra{0}$ is a projector onto the vacuum state of the lens (i.e. no photons in the lens) and the trace is implied over all photonic Fock states of the lens, $\sum_n\bra{n}...\ket{n}$, and
\begin{equation}\label{MEfirst}
\tilde \rho(t)=e^{iH_\text{atom}t/\hbar}\rho(t)e^{-iH_\text{atom}t/\hbar},
\end{equation}
and
\begin{equation}
\tilde V(t)=e^{i[H_\text{atom}+H_\text{field}]t/\hbar}V(t)e^{-i[H_\text{atom}+H_\text{field}]t/\hbar}.
\end{equation}
In \refeq{ME}, the Born approximation was performed by writing the density matrix for the system in the form $\tilde \rho (t-\tau)\otimes \ket{0}\bra{0}$, which amounts to neglecting correlations between the atoms and the electromagnetic modes of the lens  \cite{Gross1982}. The Markov approximation was made by replacing $\tilde \rho(t-\tau)$ by $\tilde \rho (t)$, which is based on the assumption that the atom-field correlation time is negligibly short compared to the time scale on which the system evolves \cite{Gross1982}. The Markov approximation allowed us to self-consistently extend to infinity the upper limit of the integration with respect to $d\tau$. We confirm the validity of the Born-Markov approximation in Section \ref{BornMarkov}.

After performing the trace over the modes of the fish eye lens, we need to evaluate the following standard integral
\begin{equation}\label{integralDP}
\int_0^\infty d\tau e^{-i(\omega_l\mp \omega_0)\tau}=\pi\delta(\omega_0\mp\omega_l)\pm iP\frac{1}{\omega_0\mp\omega_l},
\end{equation}
where $\delta(x)$ stands for the Dirac delta and $Pf(x)$ denotes the principal value component of the function $f(x)$. Since the spectrum of the fish eye modes (which act as the reservoir for the atoms) is discreet, the Dirac delta and the principal value do not contribute away from resonances and we may simply replace the right-hand side of \refeq{integralDP} with $\pm i/(\omega_0\mp\omega_l)$. More specifically, in the absence of any mechanism for photon loss that would broaden the energy levels, the atoms experience no spontaneous decay or cooperative emission when their transition frequency does not coincide with the resonant frequencies of the lens. The master equation then describes the fully coherent, lossless evolution of the atoms and takes the form
\begin{equation}
\frac{d\rho}{dt}=\frac{1}{i\hbar}[H_\text{at},\rho]-i\!\!\!\sum\limits_{\substack{i,j=1,2\\i\neq j}} \delta\omega({\mathbf r}_i,{\mathbf r}_j)\,[\sigma_i^\dagger\sigma_j,\rho],
\end{equation}
where the dipole-dipole interaction between the atoms is given by
\begin{equation}\label{sumdd}
\delta\omega(\mathbf{r}_i,{\mathbf r}_j)  =  \frac{d_z^2}{\hbar\varepsilon_0}\sum_{l,m}\frac{\omega_l^2}{\omega_l^2-\omega_0^2} \,f_{l,m}^*(\mathbf{r}_i)f_{l,m}(\mathbf{r}_j),
\end{equation}
where the fish eye modes $f_{l,m}({\mathbf r})$ are given by \refeq{modes} and the summation runs over all eigenmodes of the fish eye. Given the summation over an infinite number of modes, it is difficult to work directly with the expression given in \refeq{sumdd} and it is desirable to replace it with a simple, closed-form expression.

As shown in Appendix \ref{transformation}, the right-hand side of \refeq{sumdd} can indeed be replaced by an expression of the same form as \refeq{dd} using a Green's function, where the Green's function is given by the following expression
\begin{eqnarray} \label{analytic}
G_{zz}({\mathbf r}_1, {\mathbf r}_2, \omega) =-\frac{P_\nu(\xi(\alpha_1, \alpha_2)) -P_\nu(\xi(\alpha_1, 1/\alpha_2^\ast))}{4b\sin(\pi \nu)},\quad\;\;\;
\end{eqnarray}
where $P_\nu$ is the Legendre function of (non-integer) order ${\nu  = \frac{1}{2}(\sqrt{16\pi^2\,( R_0n_0/\lambda)^2  + 1}- 1)}$. Note that the order parameter $\nu$ depends on the atom frequency $\omega_0$ through the free-space wavelength $\lambda=2\pi c/\omega_0$ and the order parameters with integer values ($\nu = 1,2,3\dots$) correspond to the resonances of the lens. We have also defined ${\xi(\alpha_1, \alpha_2) = (|\zeta(\alpha_1, \alpha_2)|^2-1)/(|\zeta(\alpha_1, \alpha_2)|^2 + 1)}$ and ${\zeta(\alpha_1, \alpha_2) = (\alpha_1 -
\alpha_2)/(\alpha_1\alpha_2^\ast + 1)}$, with ${\alpha_j =\frac{r_j}{R_0} e^{i\phi_j}}$, where $(r_j,\phi_j)$ are the cylindrical coordinates of the positions of the two atoms ($j=1,2$) within the lens. In Eq.~\eqref{analytic} the second term on the right hand side accounts for the presence of the mirror at $|{\mathbf r}|=R_0$, ensuring that the electric field goes to zero \cite{Leonhardt2009}. This Green's function was first derived in Ref.~\cite{Leonhardt2011b}, and is obtained from the canonical equation of the dyadic Green's function in the presence of a single source term \cite{Leonhardt2009,Leonhardt2010a,Ma2011}. This Green's function has been used previously to describe the static electric field distribution inside the lens for the case when a diffraction-limited image forms at the antipodal point in the presence of a classical source and in the absence of a `drain' \cite{Blaikie2010,Leonhardt2010a}.

Using \refeq{dd} and \refeq{analytic}, the dipole-dipole interaction can be calculated in a straightforward manner within the lens. In Fig.~\ref{range}(c) we plot the strength of the dipole-dipole interaction between two atoms. The position of the first atom is fixed exactly one wavelength away from the mirror and the position of the second atom is varied across the lens. We plot the interaction strength for four different radii of the fish eye. As Fig.~\ref{range}(c) shows, the strength of the dipole-dipole interaction peaks at the antipodal point, exactly one wavelength away from the mirror.

As noted at the start of this section, in quantum optics the strength of the dipole-dipole interaction sets the rate at which a photon can be resonantly transferred from one atom to the other. Physically, this exchange rate depends on the strength of the photon field at the location of the second atom that absorbs the photon. In general, the smaller the volume the photon is focused to, the larger the field strength gets. Thus, the dipole-dipole exchange rate depends sensitively on the area the photon is focused to. Fig.~\ref{range}(d) provides an enlarged view that shows the dipole-dipole interaction rate -- and thus the electric field strength -- experienced by the second atom near the antipodal point \footnote{Note that since the Green's function in \refeq{analytic} is purely real, the dipole-dipole interaction is simply proportional to the electric field strength.}. The width of the peak is approximately $\lambda/2$, suggesting that the photon is focused to a diffraction-limited area at the location of the second atom.
These results for the rate of photon transfer are numerically confirmed in Section \ref{BornMarkov}.

Fig.~\ref{range}(c) also shows that the height of the peak remains constant as the radius of the fish eye and, therefore, the distance between the two atoms is increased. The photon emitted by an atom anywhere within the cavity gets refocused at the antipodal point regardless of the size of the lens. Such infinite range dipole-dipole interaction is a well-known feature of quasi-1D waveguides \cite{Domokos2002,Horak2003,Klimov2004,LeKien2006,Chang2012,Douglas2015,Chang2006,Gonzalez-Tudela2011,Dzsotjan2010,Gomez-Medina2001,Shahmoon2013,Lalumiere2013,vanLoo2013,Inomata2014}. Intuitively, the 2D fish eye lens acts as quasi-1D system due to the fact that the lens mimics the propagation of light on the surface of a sphere \cite{Leonhardt2010b}. Just as in 1D light is confined to propagate along a single axis without dispersion, the same way light emitted from a point on the 2D surface of a sphere is constrained to propagate along the geodesics of the sphere and refocuses at the antipodal point without any dispersion.

The functional form of the dipole-dipole interaction can also be understood analytically by considering the asymptotic behavior of the Green's function near the source and image points. In particular, note that the source and image points in the lens correspond to $\xi(\alpha_1, \alpha_2)=-1$ and $\xi(\alpha_1, 1/\alpha_2^\ast)=+1$ respectively \cite{Leonhardt2009}. As $\xi \to -1$ we obtain the asymptotic expansion  \cite{Leonhardt2009,Erdelyi1953}
\begin{eqnarray} \label{expansionAtSource}
P_\nu(\xi)  \to \frac{\sin(\nu \pi)}{\pi} \Bigg[&&\log\left(\frac{1+\xi}{2}\right)+F(\nu)\Bigg],
\end{eqnarray}
where we have defined the function
\begin{eqnarray}
F(\nu)=\gamma+2\psi(\nu+1)+\pi \cot(\nu \pi).
\end{eqnarray}
Here $\gamma$ is Euler's constant and $\psi$ is the digamma function. In addition, when $\xi \to 1$  we obtain the asymptotics $P_\nu(\xi)\to 1$ \cite{Leonhardt2009,Erdelyi1953}. Thus, near the source point the first term dominates in \refeq{analytic} and a logarithmic divergence is formed. In contrast, near the image point, the second term dominates and
we can analytically approximate the Green's function as
\begin{equation}\label{GreenApproxNew}
G_{zz}\approx -\frac{1}{4b\sin(\pi\nu)}.
\end{equation}
This shows that the absolute value of the Green's function is maximized when the frequency falls half-way between two resonances such that $\nu  = m+0.5$, where $m\in \mathbb N$. Furthermore, this expression also shows that the height of the peak at a given frequency only depends on the transverse confinement of the modes $b$ and is independent of the lens radius $R_0$. Finally, we note that \refeq{GreenApproxNew} also shows that the dipole-dipole interaction is independent of where we place the atoms within the lens as long as they are situated at antipodal points.

\subsection{Spontaneous and cooperative decay of atoms}\label{spontAndCoop}

In all calculations so far, we assumed that the fish eye lens is completely isolated from its surrounding environment and the photon cannot leak out of the cavity. Here, we next consider the situation when the lifetime of the eigenmodes of the fish eye are finite e.g. due to the imperfection of the mirrors and dissipation in the dielectrics. We account for the gradual loss of photons from the fish eye modes by modifying the Hamiltonian in \refeq{Hamiltonian} with a non-Hermitian term of the following form
\begin{equation}\label{modeLosses}
H_\text{field}=\sum_{l,m}\hbar(\omega_l-i\kappa) a\+_{l,m}a_{l,m},
\end{equation}
where $2\kappa$ sets the rate of decay from the modes, which is assumed to be frequency-independent in the range of interest. The decay of the cavity modes broadens the discrete energy levels of the fish eye, creating a continuous spectrum, as shown schematically in Fig.~\ref{range}(b).

With this modification, we can re-derive the master equation from \refeq{MEfirst}. We evaluate the following integral
\begin{equation}
\int_0^\infty d\tau e^{-i(\omega_l\mp \omega_0)\tau}e^{-\kappa\tau}=\frac{1}{i(\omega_l\mp\omega_0)+\kappa},
\end{equation}
and after neglecting the off-resonant decay terms \cite{Gross1982} we obtain the master equation in the following form
\begin{eqnarray}
&&\frac{d\rho}{dt}=\frac{1}{i\hbar}[H_\text{at},\rho]-i\!\!\!\sum\limits_{\substack{i,j=1,2\\i\neq j}} \delta\omega({\mathbf r}_i,{\mathbf r}_j)\,[\sigma_i^\dagger\sigma_j,\rho]\nonumber\\
&&\qquad\quad-\sum\limits_{\substack{i,j=1,2\\i\neq j}}\Gamma ({\mathbf r}_i,{\mathbf r}_j)\left(\sigma_i\rho\sigma_j^\dagger-\frac{1}{2}\left\{\sigma_i^\dagger\sigma_j,\rho\right\}\right)\!,\quad
\end{eqnarray}
where the rate of decay is given by
\begin{eqnarray} \label{Gcoop}
\Gamma(\mathbf{r}_i,{\mathbf r}_j)=\frac{d_z^2}{\hbar\varepsilon_0}\sum_{l,m} \kappa\,f_{l,m}^*(\mathbf{r}_i)f_{l,m}(\mathbf{r}_j) (L_l^++L_l^-),\quad\;\;\;
\end{eqnarray}
and the modified dipole-dipole interaction is given by
\begin{eqnarray} \label{lossdd}
\delta\omega(\mathbf{r}_1,{\mathbf r}_2) = \frac{d_z^2}{2\hbar\varepsilon_0}\sum_{l,m}\omega_l \,f_{l,m}^*(\mathbf{r}_1)f_{l,m}(\mathbf{r}_2)\big(D_l^++D_l^-\big),\qquad
\end{eqnarray}
where we have defined
\begin{eqnarray}
L_l^\pm=\frac{\mp\omega_l}{\kappa^2+(\omega_l\pm\omega_0)^2}\;\; \text{and}\;\; D_l^\pm=\frac{\omega_l\pm\omega_0}{\kappa^2+(\omega_l\pm\omega_0)^2}.\quad\;\;
\end{eqnarray}
Since we are now including losses in the system, the excited states of the two atoms can irreversibly decay into the eigenmodes of the lens and leave the cavity, leading to non-zero single atom decay $\gamma({\mathbf r}_i)=\Gamma({\mathbf r}_i,{\mathbf r}_i)$ (with ${i=1,2}$) and cooperative decay $\gamma_\text{coop}({\mathbf r}_1,{\mathbf r}_2)=\Gamma({\mathbf r}_1,{\mathbf r}_2)$. The single atom decay $\gamma$ describes how quickly an excitation decays from state $\ket{e}$ of an individual atom to the fish eye modes, whereas the cooperative decay $\gamma_\text{coop}$ governs the coherent joint emission of the two atoms into the modes leading to super ($\gamma+\gamma_\text{coop}$) and subradiant decay ($\gamma-\gamma_\text{coop}$) of the symmetric and anti-symmetric superpositions of the two atoms, respectively \cite{Gross1982}.

As for the lossless case, it is desirable to find closed-form expressions to replace the expressions that involve infinite summations on the right-hand side of \refeq{Gcoop} and \refeq{lossdd}. As shown in Appendix \ref{transformation}, the decay rates and the dipole-dipole interaction can be expressed using the Green's function of \refeq{analytic} in the following form
\begin{eqnarray} \label{acoopMain}
\Gamma(\mathbf{r}_i,{\mathbf r}_j)\!=\!\frac{2d_z^2}{\hbar\varepsilon_0 c^2}\text{Im}\{(\omega_0+i\kappa)^2G_{zz}({\mathbf r}_i,{\mathbf r}_j,\omega_0+i\kappa) \},\quad\;\;
\end{eqnarray}
and
\begin{eqnarray} \label{addMain}
\delta\omega(\mathbf{r}_i,{\mathbf r}_j)\!=\!\frac{d_z^2}{\hbar\varepsilon_0c^2}\text{Re}\{(\omega_0+i\kappa)^2G_{zz}({\mathbf r}_i,{\mathbf r}_j,\omega_0+i\kappa)\}.\;\quad\;
\end{eqnarray}
These simple, analytic expressions provide a convenient way to calculate the quantum optical properties of atoms inside the lossy fish eye lens and to study the atomic dynamics.

We also note that when $\kappa\ll \omega_0$, Eq.~\eqref{acoopMain} and Eq.~\eqref{addMain} can be approximated as
\begin{eqnarray} \label{acoopMainApprox}
\Gamma(\mathbf{r}_i,{\mathbf r}_j)\approx\frac{2d_z^2\omega_0^2}{\hbar\varepsilon_0 c^2}\text{Im}\{G_{zz}({\mathbf r}_i,{\mathbf r}_j,\omega_0+i\kappa) \},\quad
\end{eqnarray}
and
\begin{eqnarray} \label{addMainApprox}
\delta\omega(\mathbf{r}_i,{\mathbf r}_j)\approx\frac{d_z^2\omega_0^2}{\hbar\varepsilon_0c^2}\text{Re}\{G_{zz}({\mathbf r}_i,{\mathbf r}_j,\omega_0+i\kappa)\}.\quad
\end{eqnarray}
Eq.~\eqref{acoopMainApprox} and Eq.~\eqref{addMainApprox} suggest an alternative way of accounting for the loss of photons from the modes of the fish eye. In particular, it can be shown (see Appendix \ref{transformation}) that ${G_{zz}({\mathbf r}_i,{\mathbf r}_j,\omega_0+i\kappa)}$ is the Green's function of the fish eye lens with the following complex refractive index
\begin{equation}
\tilde n({\mathbf r})=n({\mathbf r})(1+i\alpha),
\end{equation}
where
\begin{equation}
\alpha=\kappa/\omega_0,
\end{equation}
and $n({\mathbf r})$ is given by \refeq{FEn}. Therefore, the loss of photons from the modes of the fish eye can also be thought to arise from material absorption in the dielectric \cite{Leonhardt2009}. This is a key observation, which allows us to associate a $\kappa$ value with material absorption and, therefore, treat all losses that contribute to photon decay from the fish eye modes in a unified manner. In particular, even if different loss processes are present, e.g. material absorption and leakage through the mirror, we can still associate a $\kappa$ value with each of these processes and calculate the total decay rate via
\begin{equation}
\kappa_\text{total} = \kappa_\text{abs}+\kappa_\text{mirror},
\end{equation}
which can be substituted into \refeq{acoopMainApprox} and \refeq{addMainApprox} to calculate the relevant atomic properties in the lossy lens. This will be particularly useful when we consider a possible physical realizations of the fish eye lens with plasmons (see Section \ref{realization}).

Furthermore, we can also find how $\nu$, $\Gamma$ and $\delta\omega$ scale with $\alpha$ for system parameters of interest. First, we note that ${16\pi^2(R_0/\lambda)^2\gg 1}$, whenever ${\lambda\lesssim R_0}$. Assuming $\alpha \ll 1$, to first order in $\alpha$ we find that
\begin{equation}\label{nuApprox}
\nu \approx \frac{2\pi R_0}{\lambda} (1+i\alpha).
\end{equation}
Assuming that $\text{Re}[\nu]=m+0.5$ with $m\in \mathbb N$ (which corresponds to tuning the atomic frequency between two resonances), from \refeq{GreenApproxNew} we obtain that, to lowest order in $\alpha$, the following approximation holds at the image point (${\mathbf r}_1=-{\mathbf r}_2$)
\begin{eqnarray}
G_{zz}({\mathbf r},-{\mathbf r},\omega_0+i\kappa)&\approx& -\frac{1}{4b\sin(\pi\nu)}\nonumber\\
 &\approx& \mp \frac{1}{4b(1+(2\pi^2R_0\alpha/\lambda)^2)},
\end{eqnarray}
where the choice of sign $\mp$ depends on whether $m$ is even or odd. This is a purely real quantity and, therefore, from \refeq{acoopMainApprox} and \refeq{addMainApprox} we find that the cooperative decay is given by
\begin{eqnarray} \label{coopscale}
\gamma_\text{coop}=\Gamma(\mathbf{r},-{\mathbf r})\approx 0,
\end{eqnarray}
and the dipole-dipole interaction takes the form
\begin{eqnarray} \label{ddscale}
\delta\omega(\mathbf{r},-{\mathbf r})\approx \mp\frac{d_z^2\omega_0^2}{\hbar\varepsilon_0c^2} \frac{1}{4b(1+(2\pi^2R_0\alpha/\lambda)^2)}.\quad
\end{eqnarray}

Finally, we can find the single atom decay rate $\gamma$ by substituting ${\mathbf r}_i={\mathbf r}_j$ into \refeq{acoopMainApprox} and substituting  \refeq{expansionAtSource} and \refeq{nuApprox} into \refeq{analytic}. We find that to leading order in $\alpha$ the following approximation holds
\begin{equation}\label{decayscale}
\gamma=\Gamma(\mathbf{r},{\mathbf r})\approx \frac{d_z^2\omega_0^2}{\hbar\varepsilon_0c^2}\frac{\pi^2R_0\alpha}{ b \lambda}.
\end{equation}

\section{Entanglement of atoms}\label{results}

Structures that mediate long-range dipole-dipole interactions are of significant interest in quantum information processing, as such interactions make it possible to entangle \cite{Shahmoon2013} and perform deterministic phase gates between distant atoms \cite{Dzsotjan2010}. In this section, we characterize the potential of the fish eye to entangle distant atomic quits. We focus on the simple case of a single excitation being exchanged between two atoms due to the dipole-dipole interaction. In what follows, for simplicity we assume that the two atoms are located at antipodal points (i.e. $|{\mathbf r}_1|=|{\mathbf r}_2|$ and $\phi_1=\phi_2+\pi$) and, therefore, $\gamma=\Gamma({\mathbf r}_1,{\mathbf r}_1)=\Gamma({\mathbf r}_2,{\mathbf r}_2)$.

In the absence of a driving field, the no-jump evolution of the system can be described by a non-Hermitian effective Hamiltonian of the form \cite{Brennen2000}
\begin{eqnarray} \label{nonHermitian}
H_0&=&(\hbar\omega_{0}-i\gamma)\ket{e_1,e_2}\bra{e_1,e_2}\nonumber\\
&+&\left(\delta\omega-i(\gamma+\gamma_\text{coop})/2\right)\ket{+}\bra{+}\nonumber\\
&+&\left(-\delta\omega-i(\gamma-\gamma_\text{coop})/2\right)\ket{-}\bra{-},
\end{eqnarray}
where we have defined ${\ket{\pm}=(\ket{e_1,g_2}\pm\ket{g_1,e_2})/\sqrt{2}}$, and recall from the previous section that $\gamma_\text{coop}=\Gamma({\mathbf r}_1,{\mathbf r}_2)$ and $\delta\omega({\mathbf r}_1,{\mathbf r}_2)$ stand for the cooperative decay and dipole-dipole interaction of the atoms, respectively.
Note that the overall decrease of population in \refeq{nonHermitian} due to the non-Hermitian terms reflects the gradual loss of the photonic excitation from the cavity.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=8.5cm]{./05_photon_trajectory.png}
\end{center}
\caption{(color online) Excitation probability of two atoms within the cavity as a function of time. Initially, atom 1 is excited and atom 2 is in its ground state. As the system evolves, the two atoms repeatedly exchange a photon via the dipole-dipole interaction. The photon gradually decays from the cavity modes, leaving the atoms in their ground states. A fully entangled state with maximal fidelity is formed at  $ t=\pi/(4\delta\omega)$ (see arrow). The plot was obtained for $R_0=3.34\lambda$ with a cavity loss rate of $\alpha = \kappa/\omega_0=5\times 10^{-4}$, assuming that the two atoms are located at two antipodal points within the lens such that $|{\mathbf r}_1|=|{\mathbf r}_2|=0.27R_0$ and $\phi_1=\phi_2+\pi$.   }
\label{trajectory}
\end{figure}

Assuming that at $t=0$ the two atoms are in the state ${\ket{\psi(0)}=\big|e_1,g_2\big>=(\ket{+}+\ket{-})/\sqrt{2}}$, the time evolution of the atomic wavefunction is governed by
\begin{eqnarray}
\ket{\psi(t)}&=&\frac{1}{\sqrt{2}}\Big(e^{-i[\delta\omega-\frac{i}{2}(\gamma+\gamma_\text{coop})]t}\ket{+}\nonumber\\
&&\qquad\qquad\quad+e^{-i[-\delta\omega-\frac{i}{2}(\gamma-\gamma_\text{coop})]t}\ket{-}\Big),
\end{eqnarray}
which, upon substitution, yields
\begin{equation}
\ket{\psi(t)}=C_+(t)\ket{e_1,g_2}+C_-(t)\ket{g_1,e_2},
\end{equation}
where
\begin{equation}
\left|C_\pm(t)\right|^2=\frac{e^{-\gamma t}}{2}\left[\text{cosh}(\gamma_\text{coop} t)\pm\text{cos}(2\delta\omega t)\right].
\end{equation}
The expressions $\left|C_+\right|^2$ and $\left|C_-\right|^2$ give the excitation probability of atom 1 and atom 2, respectively, as a function of time. In \reffig{trajectory} we plot the excitation probability of the two atoms as a function of time. As the plots shows, the photon is coherently exchanged a number of times between the two atoms before it gradually decays from the cavity modes.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=8.5cm]{./03_fidelity.png}
\end{center}
\caption{(color online) Error $(1-F)$ of the entangling operation between two qubits located at two antipodal points within the lens ($|{\mathbf r}_1|=|{\mathbf r}_2|=0.27R_0$ and $\phi_1=\phi_2+\pi$). (a) Error of the entangling operation as a function of the cavity loss rate $\alpha=\kappa/\omega_0$ for four different lens sizes $R_0\in\{1.75,3.34,8.11,14.5\}\lambda$, where the $R_0/\lambda$ ratio was chosen such that ${\text{Re}(\nu)=\frac{1}{2}(\sqrt{16\pi^2(R_0/\lambda)^2+1}+1)\in \{10.5,20.5,50.5,90.5\}}$. The error increases as the losses and lens radii increase. (b) Error of the entangling operation for a fixed loss rate $\alpha=5\times 10^{-4}$ as a function of the detuning $\Delta\nu=\text{Re}(\nu)-\nu_\text{center}$, where $\nu_\text{center}\in\{10.5,20.5,50.5,90.5\}$. Error is plotted for the same four lens radii as in (a). The error increases with radius and as the frequency approaches one of the resonances. Analytic (numerical) results are shown with solid (dotted) lines. Good agreement is obtained between analytic and numerical data, confirming the validity of the Born-Markov analysis.}
\label{compare}
\end{figure}

During time evolution, the state $\ket{\psi(t)}$ will have maximal overlap with the maximally entangled state ${\ket{\xi}=\left(\ket{e_1,g_1}-i\ket{g_1,e_1}\right)/\sqrt{2}}$ when $\left|C_+(t)\right|=\left|C_-(t)\right|$, which happens when $2\delta\omega t\approx\frac{\pi}{2}+m\pi$, where $m\in \mathbb{Z}$. Since in the presence of losses the fidelity decreases over time, we choose $m=0$.  Thus, the time needed to reach the maximal overlap with the entangled state is $t_0=\pi/(4\delta\omega)$  (see arrow in \reffig{trajectory}) and the maximum fidelity of the entanglement operation will be
\begin{eqnarray} \label{fidelityE}
\mathbf{{\it F}}=\big|\langle\xi\ket{\psi(t_0)}\big|^2
=\text{exp}\left(-\frac{\pi}{4}\left|\frac{\gamma}{\delta\omega}\right|\right)\text{cosh}\left(\frac{\pi}{4}\left|\frac{\gamma_\text{coop}}{\delta\omega}\right|\right)\!.\qquad
\end{eqnarray}
Eq.~\eqref{fidelityE} gives a simple, analytic expression for the fidelity of the entangling operation in terms of $\gamma=\Gamma({\mathbf r}_i,{\mathbf r}_i)$, $\gamma_\text{coop}=\Gamma_\text{coop}({\mathbf r}_i,{\mathbf r}_j)$ and $\delta\omega({\mathbf r}_i,{\mathbf r}_j)$, which can be evaluated analytically through \refeq{acoopMain} and \refeq{addMain}. Here, the key figure of merit is the ratio $\beta=\delta\omega/(\gamma+\gamma_\text{coop})$. If the frequency of the atoms is chosen to lie half-way between two resonances of the fish eye (see Fig.~\ref{range}(b)), the single atom decay $\gamma$ and the cooperative decay $\gamma_\text{coop}$ are small and the dipole-dipole interaction dominates \cite{Brennen2000}. Intuitively, in the absence of losses ($\gamma = \gamma_\text{coop}=0$), the fidelity of the entangling operation is 1.

In Fig.~\ref{compare}(a) we plot the error in the entangling operation $(1-F)$ for four different lens radii as a function of $\alpha$, where $\alpha=\kappa/\omega_0=1/Q$ is the inverse of the cavity Q-factor, characterizing the ratio of the lifetime of the eigenmodes of the lens to the frequency of the excitation. For all lens sizes, the position of the two atoms is fixed at two antipodal points such that $|{\mathbf r}_1|=|{\mathbf r}_2|=0.27R_0$ and $\phi_1=\phi_2+\pi$.
The ratio of the lens radius to the transition wavelength ($R_0/\lambda$) was chosen such that the real part of the order parameter $\nu=\frac{1}{2}(\sqrt{16\pi^2(R_0/\lambda)^2+1}+1)$ associated with the atomic frequency falls half-way between two resonances of the fish eye for all four lens radii (i.e. $\text{Re}(\nu)=q+0.5$ with $q\in\{10,20,50,90\}$, where note that for $\text{Re}(\nu)=1,2,3\dots$ the transition frequency $\omega_0$ is resonant with one of the eigenenergies $\omega_l$ of the lens). Clearly, the error increases with increasing $\alpha$ and increasing $R_0$ (i.e. increasing interatomic distance). The maximal value of the error is $0.5$, which is reached when $\beta$ becomes so small that the initial state has the highest fidelity ($F=|\langle\xi\ket{\psi(0)}|^2=0.5$).

Fig.~\ref{compare}(b) shows the error for a fixed value of ${\alpha=5\times 10^{-4}}$ for the same four lens radii as in (a) and the same antipodal atomic positions. The error is now plotted as a function of the detuning $\Delta\nu=(\text{Re}(\nu)-\nu_\text{center})$, where $\nu_\text{center}=q+0.5$ with $q\in\{10,20,50,90\}$. Clearly, the error is minimal half-way between the resonances and increases as the frequency approaches the resonances.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=8.5cm]{./06_fidelity_scaling.png}
\end{center}
\caption{(color online) Maximum fidelity of the entanglement operation as a function of the lens radius. The fidelity was evaluated at discrete values of $R_0/\lambda$ that correspond to tuning the atomic frequency half-way between two resonances, i.e. ${\text{Re}[\nu]=m+0.5}$, where $m$ is an integer. The red line marked with circles was obtained from the exact analytical expression in \refeq{fidelityE}, whereas the blue line marked with squares was obtained from the approximate expression in \refeq{fidelityApprox}. Good agreement is obtained between the two curves. The loss rate was assumed to be $\alpha = \kappa/\omega_0=5\times 10^{-4}$.   }
\label{fidelity_scaling}
\end{figure}

To gain further insight, we assume that the atomic frequencies lie between two resonances of the lens and obtain the scaling of the fidelity with system parameters by substituting \refeq{coopscale}, \refeq{ddscale} and \refeq{decayscale} into \refeq{fidelityE}. We obtain the following simple expression
\begin{equation}\label{fidelityApprox}
F = e^{-\pi^3R_0\alpha/\lambda}.
\end{equation}
In Fig.~\ref{fidelity_scaling} we plot the fidelity of the entangling operation as a function of the lens radius using both the exact expression in \refeq{fidelityE} and the analytic approximation in \refeq{fidelityApprox}. Very good agreement is observed between the two curves.

Finally, we note that the fish eye lens could be used for entangling many pairs of atoms simultaneously. As the radius of the fish eye is increased, the dipole-dipole interaction at all points further than $\lambda/2$ away from the antipodal point monotonically decreases. For lens radii with $R_0>5\lambda$, the dipole-dipole interaction at the antipodal point is an order of magnitude larger than anywhere else in the cavity (see Fig.~\ref{range}(c)). Thus, by placing numerous pairs of atoms into the cavity simultaneously, they can be entangled pairwise, without substantial interaction between the different pairs.

\section{Validity of the Born-Markov approximation}\label{BornMarkov}

In our derivation of \refeq{sumdd}, \refeq{Gcoop} and \refeq{lossdd} we made use of the Born-Markov approximation, which presupposes that the environment is large and the correlation time of the environment is very short compared to the evolution of the atomic states \cite{Gross1982}. Since in our formalism the role of the `environment' is played by the modes of the finite cavity, the validity of these assumptions needs to be evaluated carefully.

In order to verify the validity of the above results, we numerically solve the Sch\"odinger equation, where the Hamiltonian is given by Eq.~\eqref{Hamiltonian} together with the non-Hermitian term introduced in \refeq{modeLosses}. The form of $V$ is considerably simplified when the two atoms are placed at two antipodal points within the lens such that $|{\mathbf r}_1|=|{\mathbf r}_2|$ and $\phi_1=\phi_2+\pi$. In this case the in-phase combination of the atomic dipole moments ($d_z(\sigma_1+\sigma_2)/\sqrt{2}$ + h.c.) only couples to the odd modes ($l=1,3,5\ldots$) and the out-of-phase combination of the dipole moments ($d_z(\sigma_1-\sigma_2)/\sqrt{2}$ + h.c.) only couples to the even modes $(l=2,4,6\ldots)$ of the fish eye (see Appendix \ref{numericalSchroedinger}). This reduces the size of the Hilbert space, making it possible to efficiently simulate the system while including a large number of the eigenmodes of the lens with frequencies close to $\omega_0$. We further restrict the Hilbert space to have at most a single excitation in the system.

We numerically determine the time-evolution, starting from the state $\ket{\psi(0)}=\ket{e_1,g_2}$ via the operator $U(t) = \exp[-iH t/\hbar]$. To obtain the maximum fidelity of the entangling operation, the overlap of the time-evolved atomic state is calculated with the maximally entangled state $(\ket{e,g}-i\ket{g,e})/\sqrt{2}$. In Figs.~\ref{compare}(a) and \ref{compare}(b) we plot the numerically obtained values for the error $(1-F)$ (dotted lines) for different lens radii as a function of losses and atom frequencies, respectively. Even though the analytical results were derived using the Born-Markov approximation and neglecting retardation \cite{Milonni1974}, good agreement is obtained between the analytic results and numerical data. This confirms the validity of the analytical formalism described in previous sections.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=8cm]{./04_realization.png}
\end{center}
\caption{(color online) Physical realization of the fish eye lens using transformation plasmon optics. (a) Effective refractive index ${n(d)=
\text{Re}\{\tilde n(d)\}}$ created as a function of the height of the dielectric $d$ deposited on the silver surface. The inset shows the material losses ${\chi(d)=\text{Im}\{\tilde n(d)\}}$ as a function of the dielectric height $d$. (b) Schematic depiction of the plasmonic fish eye lens. The two emitters are embedded in the dielectric. The height of the dielectric varies across the lens, which creates the effective refractive index distribution of the fish eye lens. The lens is surrounded by mirrors from all sides (the front part of the mirror has been removed to show the interior).   }
\label{realizationFig}
\end{figure}

\section{Possible experimental realization}\label{realization}

A promising way to realize the fish eye lens is via transformational plasmon optics \cite{Liu2010,Zentgraf2011}. The idea behind this approach is to engineer an effective refractive index distribution for surface plasmon polaritons by depositing a layer of high-index dielectric on top a 2D silver surface (see Fig.~\ref{realizationFig}). By varying the height of the dielectric layer on the surface, the effective refractive index seen by the plasmons can be changed. In particular, when there is no dielectric on top of the silver, the effective refractive index seen by the plasmons is close to 1, whereas in the presence of a thick dielectric layer, the effective plasmonic refractive index will be close to the refractive index of the dielectric itself. Through this experimental technique, complex spatially-varying refractive index profiles can be engineered \cite{Liu2010}. Crucially, the behavior of plasmons in a plasmonic lens with a particular refractive index profile closely mimics the predicted behavior of classical light rays in the corresponding 2D lens. This correspondence between 2D classical lenses and quasi-2D plasmonic lenses was theoretically established in Ref.~\cite{Liu2010} and experimentally confirmed for the nanoscale Luneburg and Eaton lenses \cite{Zentgraf2011}.

We expect that the plasmonic version of the nanoscale fish eye lens could be experimentally realized analogously to the Luneburg and Eaton lenses. A dielectric layer of varying height could be deposited on a flat silver surface while the lens is surrounded by a circular mirror (see Fig.~\ref{realizationFig}(b)). To explore the quantum optical properties of the fish eye, atom-like color defects in diamond could be used as quantum emitters. Subwavelength positioning and coherent manipulation of such color defects has been experimentally demonstrated previously \cite{Dolde2013,Sipahigil2014,Kolkowitz2015,Iwasaki2015}. Recently, the entanglement of two silicon-vacancy (SiV) color defects inside a nanoscale cavity was also demonstrated \cite{Sipahigil2016}.

For illustration, we provide here an estimate of the entanglement fidelity of two atoms inside a particular example of a plasmonic fish eye lens. We assume that the lens operates at $406.706$THz, which is the zero-phonon resonance of SiVs corresponding to a vacuum wavelength of $\lambda_\text{SiV}=737$nm. Furthermore, we assume that the lens has a radius of $R_0=1.749\,\lambda_\text{SiV}$, which ensures that the SiV resonance falls between two resonant modes of cavity ($\text{Re}(\nu)=10.5$). We also assume that the flat silver substrate is made of single-crystal silver \cite{High2015}, which at the SiV resonance frequency has a permittivity of $\epsilon_m=-25.23+0.589i$ and gives rise to plasmonic propagation distances on the order of $\sim 160\, \lambda_\text{Siv}$. It is also assumed that there is a thin ($\sim 10-15$nm) diamond layer  on top of the metal that has two SiVs implanted at two antipodal points such that $|{\mathbf r}_1|=|{\mathbf r}_2|=0.27R_0$ and $\phi_1=\phi_2+\pi$, as schematically shown in Fig.~\ref{realizationFig}(b). Due to their proximity to the silver surface, the two $\hat z$-polarized emitters will couple strongly to the surface plasmons, which are tightly confined to the metal-dielectric interface.

The spatially varying refractive index $n({\mathbf r})$ of the fish (\refeq{FEn} with $n_0=1$) could be experimentally realized by depositing a dielectric of permittivity $\epsilon_d=3.6$ on top of thin diamond layer. By varying the height of the dielectric between 0 and 200nm, the effective refractive index seen by the plasmons can be varied between 1 and 2.
The refractive index of the dielectric ($n_d=\sqrt{\epsilon_d}=1.9$) was chosen such that the effective index can reach 2, but a dielectric with even higher index (such as diamond with $\epsilon_\text{diamond}=5.76$) was avoided to ensure that the plasmons are not confined unnecessarily tightly to the silver surface, which would give rise to significantly higher ohmic losses.

The direct relationship between the height of the dielectric layer $d$ and the resulting (complex) refractive index ${\tilde n}(d)=n(d)+i\chi(d)$ can be obtained from the following implicit equation \cite{Liu2010}
\begin{equation}
\text{tanh}(k_d\epsilon_dd)=-\frac{k_\text{air}k_d+k_dk_m}{k_d^2+k_\text{air}k_m},
\end{equation}
where
\begin{equation}
k_\text{air}=\sqrt{({\tilde n}k_0)^2-k_0^2},
\end{equation}
\begin{equation}
k_{d}=\frac{\sqrt{({\tilde n}k_0)^2-\varepsilon_{d}k_0^2}}{\varepsilon_{d}},
\end{equation}
and
\begin{equation}
k_{m}=\frac{\sqrt{({\tilde n}k_0)^2-\varepsilon_{m}k_0^2}}{\varepsilon_{m}},
\end{equation}
where $k_0=2\pi/\lambda_\text{SiV}$ and in our calculation we ignored, for simplicity, the presence of the diamond layer, as it does not significantly modify the effective index seen by the plasmons as long as the diamond layer is much thiner than the transverse confinement of the plasmons, which is on the order of a wavelength.

Fig.~\ref{realizationFig}(a) shows the real part $n(d)$ and imaginary part $\chi(d)$ (inset) of the complex refractive index ${\tilde n}(d)$ seen by the plasmons as the thickness of the dielectric $d$ is varied. The effective refractive index increases monotonically with the thickness of the dielectric layer. Since the refractive index of the fish eye increases radially inward (see \refeq{FEn}), the dielectric layer in the fish eye lens takes a conical shape as shown in Fig.~\ref{realizationFig}(b).

From the imaginary part of the effective refractive index $\chi(d)$, we can estimate the average photon loss rate due to ohmic losses via the relation ${\kappa_\text{abs}({\mathbf r})/\omega_0=\chi({\mathbf r})/n({\mathbf r})}$ (see Section \ref{spontAndCoop}). Since this loss rate varies significantly across the lens, we numerically average $\chi({\mathbf r})/n({\mathbf r})$ over the radius of the lens and obtain the averaged quantity $\overline{\kappa_\text{abs}({\mathbf r})}/\omega_0\approx 3\times 10^{-3}$. This is the leading order contribution to the photon loss.

Photons can also be lost from the lens by leaking out through the mirror. Assuming that the reflectivity of the mirror is $r^2$, we can estimate the loss rate $\kappa_\text{mirror}/\omega_0$. In the absence of other loss mechanisms, the photon would bounce off the mirror $\sim1/t^2$ times before being lost, where $t^2=1-r^2$. The time interval between two bounces is approximately $(2R_0)/(c/\bar{n})$, where $R_0$ is the radius of the lens, $c$ is the speed of light in vacuum and $\bar{n}$ is the average index of refraction in the lens. Thus the lifetime of the photon due to the finite mirror reflectivity is
\begin{equation}
\tau_\text{mirror}\sim\frac{1}{\kappa_\text{mirror}}\sim\frac{2R_0}{c/\bar{n}}\frac{1}{t^2}.
\end{equation}
Making the conservative estimate that $r^2=0.95$, we obtain the following loss rate
\begin{equation}
\frac{\kappa_\text{mirror}}{\omega_0}\sim \frac{1}{4\pi}\frac{1}{\bar{n}}\frac{\lambda_0 t^2}{R_0}\sim 4\times 10^{-4},
\end{equation}
where we have used $\overline{n({\mathbf r})}= 1.57$, which is obtained by numerically averaging the refractive index over the radius of the lens. Note that this shows that the losses due to the finite reflectivity of the mirror are an order of magnitude smaller than the ohmic losses.

Next, we consider emission into free space $\gamma_0$. In the close proximity of a metal surface, the rate of decay of the emitter into plasmonic modes $\gamma$ can significantly exceed the rate of emission into free-space modes ${\gamma_0=d_z^2\omega_0^3/(3\pi\varepsilon_0\hbar c^3)}$ \cite{Chang2006,Francs2016}. Here, we take the Purcell factor to be $\eta=\gamma/\gamma_0\approx 3$, which is the approximate value for a z-oriented dipole 10-15 nm away from a flat silver surface emitting radiation at 737nm \footnote{We calculate the Purcell factor exactly near a flat metal surface by evaluating ${\gamma_\text{pl}=2d_z^2\omega_0^2/(\hbar\varepsilon_0c^2)\text{Im}\{G_{zz}\}}$, where $G_{zz}$ is the exact Green's function of a z-oriented dipole near the surface \cite{Sipe1987}}. Furthermore, we also make the conservative estimate that the emission to free space is reduced by a factor of two due to the presence of the silver surface \cite{Arcari2014}. In order to account for the presence of this additional decay channel, we need to make the replacement $\gamma\to \gamma+\gamma_0/2$ in \refeq{nonHermitian}, and thus \refeq{fidelityApprox} becomes
\begin{equation}\label{fidelityApproxNew}
F = e^{-\pi^3\left(1+\frac{1}{2\eta}\right)\frac{R_0}{\lambda}\alpha}\;.
\end{equation}
Note that this equation holds only if the atomic frequencies fall half-way between two resonances and the atoms are placed at two antipodal points anywhere in the lens.
Substituting $R_0/\lambda_\text{SiV}=1.749$, $\alpha={(\overline{\kappa_\text{abs}}+\kappa_\text{mirror})/\omega_0}=3.4\times 10^{-3}$  and $\eta=3$ into \refeq{fidelityApproxNew}, we obtain that the fidelity of the entangling operation would be approximately $F=80\%$. We note that this fidelity could be further improved by utilizing the adiabatic passage of a dark state in a Raman scheme \cite{Pellizzari1995}.

\section{Conclusion}\label{conclusion}

In conclusion, we have investigated the single-photon dynamics of atoms inside the fish eye lens. We demonstrated that the lens mediates long-range interactions between distant emitters. The dipole-dipole interaction has an infinite range, limited only by the decay rate of the cavity modes. Furthermore, our results show that the fish eye focuses a single photon to a diffraction-limited area during the exchange of a photon between two antipodal atoms, whose frequency is tuned between two resonances of the cavity. We derived closed-form expressions for the decay rates and dipole-dipole interaction of atoms in the presence of losses and studied the fidelity of entangling operations. We confirmed the validity of our analysis, which relied on the Born-Markov approximation, by numerically solving the Schr\"odinger equation. Finally, we proposed a possible realization for the fish eye lens using tranformational plasmon optics and silicon-vacancy centers that could open up the fish eye for practical applications.

\section{Acknowledgements}

We thank U. Leonhardt, E. Shahmoon, M. Mezei, S. Horsley, K. Lalumi\`ere, W. M. R. Simpson, A. Levy, S. Choi, D. Wild, and M. Kan\'{a}sz-Nagy for stimulating discussions. We acknowledge funding from the MIT-Harvard CUA, NSF and the Hungary Initiatives Foundation.

\onecolumngrid

\section*{Appendix}

redpajama_set_name: RedPajamaArXiv