I think I'll call it a "Garylanche."
My last post got me to thinking about holding and liquidating prepaid Visa/Mastercard debit cards. If you're reading this, you likely know that virtually all prepaid debit cards now come with (or can be assigned) PIN numbers. Which means, or at least has meant, that one could easily liquidate them by purchasing money orders. This week we saw the demise of one of the easiest and cheapest sources of manufactured spending when Walmart ceased accepting OneVanila Prepaid Debit Cards for the purchase of money orders. Fortunately, most of us have alternative sources for liquidating the cards.
Ouch, that's nearly a 5% loss! The rate gets worse after that first $1,000 because you lose the TCB bonus (Unless you signed up for multiple Cardpool accounts).
Ugh, still a 2.5% loss. There are other ways to liquidate cards (Amazon Payments, etc.) but almost all of them are going to entail some loss. Once again, the lesson is: don't overextend.
P.S.: Cardpool limits Topcashback commissions to the first $1K sold per Cardpool account. (Hint).
There's been a great deal of distress over at Flyertalk lately due to the demise of one avenue for manufacturing spend. Wally World (W**-Mart) has recently changed their POS software in such a way that prepaid Vanilla Visa Debit cards can no longer be used to purchase money orders or to load Bluebird cards. This is a serious blow to folks whose source of prepaid debit cards is limited to CVS, since the only cards available on credit there are Vanilla cards. There are (inflated, I hope) reports of people being stuck with many thousands of dollars in cards that they can no longer easily convert into cash equivalents. There's a lesson here, of course. Risk is an unavoidable part of life, but if you're smart, you'll limit your exposure. If you decide to use any of the manufactured spending techniques, best practices call for you to liquidate the cards as fast as you accumulate them. Don't get greedy and think that there'll always be a way to liquidate the cards later.
As they said in Battlestar Galactica, "all of this has happened before and all of this will happen again." Not long ago, there was a brief opportunity to use Home Improvement Gift Cards to purchase money orders. It was clearly a glitch and not long for this world, but, nevertheless, some overenthusiastic folks loaded up on thousands of dollars worth of the cards and were caught short when the glitch was fixed. If they were smart they only lost 5% on the deal, but then, if they were smart, they wouldn't have been holding $10K+ in HIGCs. Remember: "If something can't go on forever, it won't."
One of my favorite bloggers covered Club Carlson's current promotion for a "Flash Sale" on Club Carlson points. With this Club Carlson deal you can (for a limited time) buy points at .4cpp, a substantial discount over the normal rate. It got me thinking about the value proposition of purchasing points directly from hotels/airlines versus obtaining them by manufactured spending.
As an initial matter, its rarely a good idea to purchase points prospectively, i.e. without a specific use in mind. There's always the possibility that the points could devalue before you get the chance to use them. But when you have a specific purpose in mind, deals like this one can be worthwhile. For instance: Mrs. Pointsninja have an upcoming stay at the Radisson Blu at the Zurich Airport. The non-cancellable Advanced Purchase rate is 260CHF, or about $290. But an award stay (which is fully cancellable) runs 50,000 Club Carlson points. Through the current promotion, I could purchase 50,750 points for $203. I could then use those points to book the room and effectively save $87, plus get a cancellable rate to boot! The key, of course, is that I'd be purchasing points at a cost of .4cpp, but redeeming them at a value of .58cpp.
Of course, through manufactured spending with the Club Carlson Visa, I can quite easily "purchase" points at a rate of .21cpp, meaning that I could actually get the room for $185 cheaper than the Advanced Purchase rate.
In other words, you use your credit card of choice to purchase gift cards which you then turn into money orders that you use to pay off your credit card. You'll incur some costs (approx. 1.1% of your generated spending), but so long as the points you earn are worth more than the fees you pay, you'll be in the black. Ordinary MSing can be an effective way to greatly reduce the cost of travel.
However, thanks to our good friends at American Express (otherwise know as America's most generous credit card company) you can actually generate a substantial amount of spending and earn a profit over and above the value of the points you earn. That's because Amex has been selling gift cards through their site for the last several years. And they pay referral fees to various shopping portals, who then pass part of that fee on to you. As of the date of this post, you can earn 4% cashback on purchases of American Express gift cards.
Here's a personal example. Today, I purchased $5,000.00 worth of Amex Gift cards after linking through BeFrugal.com. As of today, BeFrugal is paying (after a waiting period of around 60 days) 4%. So, I'll earn $200 cashback for purchasing the cards. I payed for the cards with my American Express Hilton Honors card, so I'll earn 15,000 Hilton Honors points. Amex had a fee-free promotion for Father's day, so I didn't incur any purchase fees. When the cards arrive, I'll take them out and use them to purchase Visa (debit) gift cards, incurring a fee of around 1% or $50 total. I'll then use the debit cards to purchase money orders for a total fee of around $3-5. Then, the money orders go back into my checking account and I pay off the original $5K charge.
So, subtracting the fees I'll pay from the rebate I'll receive, my net profit will be $145. Plus 15,000 Hilton Honors points. That's not free travel, that's better than free travel!
UPDATE: Befrugal is now down to 2.5% cashback. This THREAD on Flyertalk tracks this deal pretty thoroughly.


Traditional mud brick construction is a cheap and easy form of making all kinds of shelters, either for humans, animals or storage. The energy used to produce the mud brick comes from the sun and manpower. Mud brick has many positive qualities it is load bearing and has good insulation properties. It is also fire resistant and forgiving, meaning that if you make a mistake you can correct it by remaking it. Mud brick construction also has the cleanest ecological footprint as it returns to the earth from where it came. The materials used to make mud bricks are available all over Cyprus. This includes clay soil, chopped straw and water. It is most economical to use readily available materials rather than transport soil in from distant areas. The clay content of soil varies from region to region, as does the colour, which is dependent on minerals such as iron oxide. The colour does not affect the strength of the brick. As there is a lot of farming in Cyprus, local supplies of straw can readily be found. Old straw which is unsuitable for animals is better for mixing and cheaper, if not free. The only other requirements are a wooden mould, a shovel and a wheelbarrow. A mixing trough for the mud is desirable but not necessary.
The soil has to have 40-60% clay content to be suitable for mud brick making. This can be tested by dissolving soil in water in a clear glass or plastic bottle and letting it settle. In this way the sediment proportions can be observed, as the clay will be settling on the top. As a common rule to find the right type of soil in a village one should ask the locals, particularly the elderly. Most villages have their own local sources of clay soil as transportation was always a problem in the past. If water was found nearby the clay soil source, that was an added advantage for mixing the soil there and making the bricks on location. Once dried the bricks would have to be transported to the building site. Another solution was to bring the soil to the building site and mix and make the bricks there.
In the ancient past the Cyprus mud brick contained no straw. The discovery of adding straw into mud might have been found when the roots of the reeds where the clay soil was quarried was left in the mud and it resulted in a stronger brick. Thereafter, straw of some form has been included in mud bricks.
Going back 60 or 70 years, the straw in the mud brick was obtained from the threshing fields. The threshing of wheat and barley was done by "düven", a board spiked with pieces of flint stone that was pulled over the stalks by oxen. This was followed by "winnowing" which separated the grain from the hay. The finest hay was blown furthest away and this was used as feed for the animals. The larger and heavier straw, mostly the knobs, was closer to where the grain fell. This was called "gondila" and was used mainly in mud brick construction.
In the early 50s the first mechanized threshing machine was introduced, called "Patos". This was driven by a tractor using a flat belt to turn the mechanism. Bundles of wheat or barley would be fed into it and it would separate the grain, gondila and animal hay into three separate piles. The machinery used today threshes the hay that the combine harvester leaves behind in the field. As a result, the straw types are all mixed together. The straw pieces are shorter in length than if done in the old fashioned way. However, to create a longer straw piece which is more suitable to make the mud brick mixture, the speed of the threshing machine can be adjusted down.
Equal amounts of finely chopped straw (which is the animal feed type) and purposely made longer length straw (up to 3 cm) has been found to be the ideal proportion for a better mud mixture than using each on its own. For economic reasons, the finely chopped straw is obtained from shepherds who clean out their storage rooms of old stock as it gets stale and tainted. The stale straw's outer surface loses its shine and therefore bonds better with clay, so it is an advantage. Wheat produces a better straw for mud brick making than barley. This is because wheat straw has thicker walls, which makes for better tensile strength. Also, the inner surface of the straw (which is exposed on threshing) bonds better with the clay because it is textured.
There are no specifications as to the type of water used in making mud bricks. Sometimes even sea water was used if it was the only water locally available. Seaweed was also used instead of straw in these cases. The salt content does not affect the strength of the mud brick.
At least one but preferably two or three wooden moulds are necessary. The length, width and depth can be variable. The size commonly used in Cyprus is 43cm x 33cm x 7cm. After drying and shrinkage, this brick will measure 42cm x 32cm x 6.5cm. The side measuring 42 cm makes the thickness of the wall. Moulds can be single, double or three-in-one. The inner surface should be smooth for easy release of the mud. The multiple moulds are used by two people and have lifting handles on either end. The single mould has a groove or a protruding edge for easy lifting. Hard timber makes a better mould than a soft wood because they don't swell when wet or get out of shape. The wood is not painted or varnished.
An ordinary construction wheelbarrow managed by one individual is all that is necessary to carry the mud mixture from the place that it is mixed to where it will be placed in the mould. As new bricks have to stay undisturbed for at least 24 hours, it is most efficient to bring a load of mud to the area where the moulding is being done rather than running back and forth with a shovelful. In olden times, mud was carried to the mould in a stretcher type apparatus carried by two men. It was like a shallow trough, so kept the mixture from falling out.
A square edged shovel is all that is needed. A longer handled one will help to avoid back ache. It is necessary to shovel the mud mix into the wheelbarrow and again into the moulds.
In past times, the mixing trough was created by digging into the ground and pouring water and adding straw. The trough would get bigger and bigger as more soil was used up. In some cases, more soil would be added and worked. Another way of creating a trough was to pile up the dirt then make a pool in the middle of it and fill it with water and let it soak overnight. The next day more water and straw would be added and worked. A concrete based pool with brick walls (50cm high) can be built to carry out the mixing. There are several advantages to this. There is no water loss through absorption in the ground and the mix can last longer without drying out. A pool built for making bricks for one house project could measure 4m X 4m X50cm. This is sufficient volume of mud to make about 300 bricks. The 50cm maximum height is to allow workers to easily step in to mix the mud. The brick pool can be demolished after the project is finished.
The clay soil is placed into the pool ¾ full, keeping track of the amount in volume. This soil is soaked with water overnight. The next day more water is added and a proportional amount of straw is added gradually as it is worked. The proportions of clay and straw are eight to five. This may vary according to the clay content of the soil. Higher clay content would require more straw.
Working is done by walking in the pool with bare feet which forces the straw to mix with the clay. Rubber boots are not satisfactory as they get sucked off the feet. The water ratio is adjusted by keeping the mix workable. Experience will guide one to create the right mix. The best mix is one that is easy to place in the mould, to remove the mould from and doesn't lose its shape or crack while drying. Too runny or too hard of a mix are impossible to work with. A good mix doesn't stick to the shovel. In olden times, the quality of a mix was judged by the way it squished out between the toes while mixing. In conclusion, the best way to learn about the amount of water is by trial and error.
The clay is worked in the mould by hand, using fingers to push the clay mix into the corners and the palm is used to flatten the top. The mould is removed by pulling upwards and the fresh brick is left to dry. The mould is wet to start with and then dipped in water again before making another brick. A small handful of chopped stray is sprinkled where the mould sits to avoid the brick sticking to the soil below. Once they are firm and can be handled, about 24 hours, the mud bricks are stacked in an A-frame position for drying as the underside will dry more evenly too. During the summer months, five days is sufficient to completely dry the bricks then they can be stacked elsewhere or used to start building.
Mud bricks are most commonly laid in the stretcher bond, like any other brick (two over one over two etc.), with one difference which is, the length of the brick becomes the width of the wall. This makes it necessary to produce half a brick along the width to bond a course. This half brick should measure 42cm x by 16cm x 6.5cm.
The mortar to lay the bricks is the same mud mixture as is used to make the brick itself. The water ratio is increased to achieve a better workability for spreading it between the courses. This also aides better bonding of the bricks together. The mud brick construction usually has continuous footings which are about 50cm high from ground level. In the traditional method they are built of stone using the mud brick mortar for bonding. The footing serves as a damp proof course and avoids the erosion of the footing by running water.
The mud brick wall surface internally and externally can be finished using different methods and materials. Traditionally, the internal surface was finished by a thin application of gypsum applied by hand and rubbed with a cloth bundled up in the palm. To increase the bonding of the gypsum to the wall surface, the joints are not filled completely while laying the bricks, leaving hollow channels for the gypsum to grip onto. Externally, a thin layer of mud brick mortar is applied by hand. The wall surface is dampened for better bonding and the hollows between the courses act as gripping points for the mortar. Seasonal maintenance of the external wall is necessary to repair wind and rain erosion, usually done in the summer.
Traditionally, the materials used in roof construction in Cyprus depended on what was locally available in the area, for free or at low cost. Transportation in the past was difficult, involving mainly animal power. Also, if one didn't have a supply of materials, they had to be bought. Building materials on public land were available for a fee to the authorities or alternatively one could buy from private individuals. In the Karpaz, juniper trees were used for roof beams, while in other areas cypress was more commonly used. On top of the beams, cane was placed to support the earth fill. If cane was not available, splintered juniper wood was used instead. In some cases, "wild cane", which is much thinner, was used, and the beams were placed closer together to compensate for the lack of tensile strength in the cane.
Juniper forests once covered large parts of Cyprus but little remains of them except in the Karpaz. They are very slow growing with a dense, aromatic wood, and a tree measuring 50 cm around could be 100 years old or more. The resin of the juniper acts as a natural deterrent to insect infestation, and, provided they were not lying wet on the ground, cut juniper logs could last for a hundred years or more in the dry interior of a house. They have an excellent tensile strength which enables them to bear the load of the earth fill. The beams we used are, as much as possible, the same ones from the original roof. However, some developed rot on the ends where they have been exposed to the elements and others were damaged when part of the roof collapsed. For these damaged beams, therefore, similar replacement logs were found from other ruins of the corresponding time period. It is estimated that the average age of the trees that were felled for this roof was 75 years and the building was probably constructed 80 years ago. Their average length is three metres and average diameter is 10cm. This is a comparatively good length and size for a juniper beam. The building length internally is 7.6 m and the width is 5.4m. The presence of an arch allows for such a wide room to be created using these three metre beams. They are placed about 33 cm apart, with the thin and thick end positioned alternating. The same principle applies when the ends meet on the arch. The flat cap stones creating the eaves on this adobe building were pieces of medium density limestone rock and were mostly broken when part of the roof caved-in or were damaged in removal as they were in a fragile state. For purposes of creating a longer lasting cap stone, we manufactured them on site, using a white and grey cement mix, stone quarry sand and fine aggregate. Sixty "cap stones" were created from a mould in this manner. An additional drip groove was added to the moulded cap stone to prevent water running down the wall. Although the roof beams are mostly straight, some have forks in them and others slight curvatures and bends. In placing each beam in position, these natural variations have to be considered to create the flattest possible surface where the cane can have maximum contact with each beam below. Once a decision is made on the arrangement of the beams, small stone wedges, gypsum and a mud-straw mix will be used to lock them in position. It is not advisable to have any of the juniper logs hanging out past the external wall as exposure to the elements will reduce its longevity. In some cases where beams were left protruding, the ends were soaked in olive oil to reduce the erosion.
Cane grows in wet areas, such as the base of valleys or as wind breaks bordering citrus groves. The best time of the year to cut cane is February or March, because it is dormant and won't shrink or be vulnerable to insect infestation. Cutting is done by using large pruning shears or a small axe. Only mature cane, two years old, should be used. After cutting, the cane is dried flat on the ground for the period of a month. This drying makes it ready for cleaning, if the cane is cleaned whilst fresh then insects can penetrate the bud and infestation occurs.
The cane is cleaned with a large kitchen knife starting from the top going to the base with frequent hacking movements of the knife held flat against the cane. This removes the dried leaves from each 'nub' or joint along the cane. Once cleaned, the cane is bundled and stored in a dry place.
When the platform of the roof beams has been created, a team of workers line up across the beams to begin tying the cane. Each worker takes a cane called a 'leader' which is placed parallel to the beams from the external wall to the arch. These leaders are about a metre apart. Two additional workers are necessary to feed the canes to the tying workers. The feeders give the canes to the top of the line of work across the leaders so that the thick and thin ends of the cane alternate each time. The workers, using twine, tie the cane to the leaders, progressing to the centre of the room, which is the hip of the roof and the top of the arch. As they work they create a strong platform which they can sit and walk on.
This grass grows in large clumps on the sides of the roads and on barren land all around the island. It used to be the stuffing material for donkey saddles, from where it got its name. It is harvested with a sickle or a sharp axe and bundled up and left to dry on the ground. When the cane platform is tied onto the beams, the bundles of saddler's grass are opened and spread crossways over the cane. In other words, the grass is at right angle to the beams, not parallel. It has to be laid dense enough to create a protective layer between the cane and the mud which will go on top.
This mix is the same material as used in the formation of adobe bricks; soil and straw. The clay content of the soil used is 40 to 60 per cent. Chopped straw is mixed in equal proportions with the soil by volume. It is applied on top of the grass in 5 cm thickness and is usually left to dry before the earth fill is added.
The traditional cap stones are cut out of medium dense limestone rock. The size is usually 50 x 50 cm and the thickness is about 10 cm. There is a hump at one end of the stone which helps to stop the earth fill from falling off the edge. It also counter balances the cap stone at the edge of the wall and also creates a guide for the runoff. They overhang the wall about 20 cm. Their purpose is to protect the wall from water erosion from rainwater running off the roof. The cap stones accommodate and provide support for the rain spouts. In some cases, the rain spout is actually carved into the cap stone itself and it protrudes further from the edge line of the other cap stones.
Earth fill is of the same soil that is used in making adobe bricks and the mud layer on top of the grass. In this case it is applied dry and there is no straw mixed in it. The purpose of the earth fill is to create the right angle desired on the roof to disperse the water successfully, so it doesn't collect or run too quickly so as to create an erosion gully. The earth fill is usually 10 cm thick and is rolled with a stone roller to compact it. In our case, we are going to place a concrete layer on top of this earth fill. However, traditionally, before concrete was available, there was another thin layer of rich clay placed on top of this earth fill. It was rolled in a similar manner and had to be maintained every year before the winter. After a period of time, when it was felt that the layer of earth and mud was too thick and therefore too heavy for the timber beams to support, layers of fill were scraped back and removed and replaced with a new top layer of rich clay.
The use of concrete on top of the roof came into practice after the Second World War. A thin layer of concrete, about 5 cm thick, was placed on top of the earth fill to replace the thin layer of rich clay. It required less maintenance because it was less likely to be eroded by the weather and plants couldn't grow on it. It was formed to accommodate some rain spouts and the hump of the cap stones acted as a guide to divert the water into the spouts and off the roof. The concrete roof served the villagers as a space to dry crops such as sesame, olives, wheat, raisins, etc. It was also commonly used in the hot summer nights for sleeping.
These spouts were made of terracotta, about 40 cm long with a diameter of 8 cm on one end and 5 cm at the overhanging end. They were mounted in concrete and water from the roof was diverted towards them. Before concrete, they were held in position with a few stones stacked up on top of them. Later on, galvanized metal pipes and much later, PVC pipes were also used for rain spouts.


"CSS for Umermenu?" (#142334) / Industrial Themes, LLC.
I went ahead and pulled the trigger on Ubermenu.
Whenever you want to identify or change the style of any part of the theme, use Chrome or Firefox+Firebug and right click on the area you want to change and choose Inspect Element. That will bring up a console that shows exactly which lines of CSS in exactly which files affect that element so that you can go in and change the style how you see fit.
Specifically for Steam's existing menus it would all be near the top of the style.css file, but the method explained above will get you exact line numbers.
I stand corrected: I understand completely how to use the inspector, but don't understand which elements I should be cutting and pasting after struggling with this for more than 12 hours.
A little hand holding please...?
That might be your issue - it's not just copying and pasting CSS because the markup is likely different for the ubermenu, meaning the same CSS can't be applied. It's going to require some knowledge of CSS to get working. If you need a developer to help you I recommend Microlancer.com.


I'm always amused by the "whatever" response. It signifies that nothing is offered in argument, but that one still doesn't agree. "Don't confuse me with the facts..."
It could and probably does mean that arguing isn't worth the trouble, but then one wonders why the first message was offered. One should be prepared to defend one's assertions when they are made in a public forum.
And I tire of the "just my opinions" excuse for saying nasty and foolish things. Yes, it is your opinion, but one assumes that you have a reason other than divine enlightenment for holding that opinion, in which case we'd like to hear it. Perhaps your opinions make more sense than bad delivery suggested the first time through, and you'll persuade us on a second attempt. As I said, Joe's authority as an opinion-giver has been established by a long history of wonderful music-making and clear thinking expressed here about how tubas work. I filter through the curmudgeonly character to get at those nuggets. I'm even entertained by the curmudgeonliness.
If you read Joe's first post fairly, you'd note that he supported and presented the notion that this less expensive Meinl-Weston horn compares favorably with the more expensive horn that he sells. Isn't that what you said?
I don't pretend to have the skills to explore the capabilities of even the instruments I own, let alone the instruments you would admire. I might still disagree with your values, and I might still have valid reasons for doing so. On the other hand, I don't see Meinl-Weston or Willson conducting competency tests before selling their horns. Checks that don't bounce are usually all they require.
I'd bet a medium-sized Chinese feast that Meinl-Weston is capable of producing more Fafners than there are BBb tuba players willing to buy them. I'd further bet that of those who would buy them, a large percentage are not good enough to explore their capabilities, and maybe not even as good as you. The same could be said for the Willson BBb tuba, and really for all the CC tubas, too.
My York Master was offered for sale four times in its life. Once, a top professional bought it new. He sold it to another top pro. He sold it to a good amateur, and then I bought it. I think it's safe to say that the talent of the owners is steadily declining since its second owner. Who would have bought it if I didn't? You? Would you appreciate it more than me just because you are better? I seriously doubt it. You might make better music on it, I'll grant that. Here's my point: There is no shortage of fine BBb tubas on the market, and amateurs buying them doesn't keep them out of the hands of pros and pro wannabes who desire them. In fact, amateurs are often more interested in their histories and capabilities than are pros, because amateurs think of them as a rare ticket to sublimity, while for a pro they are a tool (and that's all they should be for a pro who will feed his family). Most pros won't even touch them or appreciate them because they are pitched in BBb. I highly doubt any pro would waste his time on the foolish pursuits and admirations I have bestowed on that silly B&M-made York stencil, despite its provenance. That's what I mean by "serious."


Are you a reliable, hard-working individual with excellent attention to detail? Two 6-month contract positions have arisen within the busy Dispatch team of a leading Pharmaceutical company in Worthing, West Sussex. Working 8am to 4pm, Monday to Friday, this will be a fascinating role to help in the shipping of life-changing medicines to people across Europe. If you have experience from a labelling, packaging, admin, mail sorting, print, dispatch or warehouse environment (or something similar), this will be a great opportunity to build on your skills and join a fascinating business. Interviews will be in mid-June and the successful candidates will start in July. You will be paid £8.50 per hour.
Working in a small, busy team of up to 7 people, you will be responsible for processing individual patient orders, printing product labels, packaging orders into jiffy bags, and preparing them for dispatch. Using your attention to detail to the full, you will ensure that all drug packets are correctly put in bags, with the right labels (printed in the department) and placed in the right areas for distribution to countries all over Europe (and some worldwide). You will use the in-house computer system to process customer orders and it is essential that you show strong attention to detail at all times.
You do not need experience in a Pharmaceutical company, but it would be an advantage to have worked in a process-driven environment (warehouse, distribution, mail sorting / post room, packaging, printing or production) where you have performed routine duties (including administration) but had to have your wits about you at all times. Above all else, we want someone who works consistently hard, who offers excellent communication skills (both written and verbal), good literacy and numeracy, patience, an honest and likeable manner and strong attention to detail. You must be available for an interview in mid-June (date to be confirmed) and must be able to start in July. You must also be able to work on up until the end of December, as this will be a very important day for the company's final dispatch operations of the year.


text:
\section{Introduction}
	\begin{figure*}[t!]
	    \centering
	    \includegraphics[width=.99\linewidth]{Fig1-modelcomparisons.pdf}
	    \caption{Visual summary of the light-matter interaction models treated here. For multilevel, TC and Dicke models, only energy levels that have a dirrect correspondence in the JCM are shown. \textbf{Left:} Standard Jaynes-Cummings model, in which a single two-level emitter couples resonantly to a single mode field, with strength $g$. The resulting anharmonic spectrum is a cornerstone of modern quantum optics, but challenging to probe experimentally: the required regime $g\gg \kappa$ is difficult to achieve, where $\kappa$ is the cavity decay rate. \textbf{Center:} Multilevel model, the focus of this paper. A quasi-degenerate multilevel emitter, with $N$ closely spaced excited states, interacts with a single mode field. When all the transitions are approximately resonant with the field, an effective coupling constant $g_{\sf eff}$, enhanced by a factor $\propto \sqrt{N}$, is established. In principle, the resulting low energy spectrum can closely approximate the JC spectrum for any value of $N$. \textbf{Right:} Tavis-Cummings and Dicke Models, in which $N\gg1$ nearly-identical two level systems interact with the same field. A boosted effective coupling constant $g_{\sf eff}\propto \sqrt{N}$ is established also in this case, but the spectrum becomes approximately harmonic as $N$ is increased.}
	    \label{Fig-summary}
	\end{figure*}
	Strongly coupled light-matter systems, described by the Jaynes-Cummings (JC) model \cite{shore1993jaynes}, are of fundamental interest in testing the quantised nature of coupled light and matter degrees of freedom in the context of cavity QED \cite{haroche2006exploring}. Importantly, JC physics requires that a \textit{single} two-level emitter interacts with a single-mode electromagnetic field. Such systems can be relevant for quantum information processing, for example in interfacing flying and static qubits \cite{reiserer_quantum_2014} or for quantum-state engineering~\cite{haroche2006exploring}. Yet strong coupling of light and a single emitter is not easy to achieve in practice, due to the large mismatch in the spatial extension of free-space photons and typical emitters (i.e. artificial or real atoms) that feature electronic transitions in the optical domain \cite{Biagioni,walther2006cavity}.

	A possible route to overcome this difficulty is to engineer an ``effective" emitter with large dipole moment. This is most commonly implemented by combining $N\gg1$ identical emitters coupled to the same field, resulting in the well-known ``Dicke enhancement" of light-matter interaction: the effective coupling strength is boosted by a factor of order $\sqrt N,$ compared to the single-emitter case \cite{torma_strong_2015,dovzhenko_lightmatter_2018,yu_strong_2019,g-aggregates}. However, the relevant theory in such systems is the many-emitter limit of the Tavis-Cummings (TC) model \cite{tavis1968exact}, whose low-energy dynamics is effectively \textit{linear,} i.e., indistinguishable from that of coupled harmonic oscillators \cite{Quesada-2012}.

\textit{The key achievement of our work is to bypass this seemingly inevitable trade-off between coupling strength and (quantum) nonlinearity: we open an innovative route to a `Dicke-like' enhancement of the coupling constant which preserves the precious quantum optical nonlinearities of the JC model.}

In this paper we indeed show how the exploitation of a multilevel emitter, with $N$ nearly degenerate excited states coupled to the same field mode, increases the effective light-matter coupling constant by a factor of oredr $\sqrt{N}$. While this is the same scaling found in the Dicke \cite{DickeOriginal} and TC models \cite{tavis1968exact} with $N$ emitters, the crucial difference is that in our case the quantum non-linearity of the interaction is preserved: we demonstrate this based on emission spectra and second-order photon autocorrelation functions. More precisely, our system can closely approximate the behaviour of a JC model, in principle, for arbitrary values of $N$. These findings, in their simplest form, are schematised in Fig~\ref{Fig-summary}.

	Additionally, we find that our predictions are robust against typical sources of decoherence (dephasing) that are common at elevated temperatures. Thus, while our results are fully general and independent of the physical implementation of the model, we anticipate that our approach can be particularly useful in the context of \textit{room temperature} strong coupling. We thus find it useful to provide some context on the current state-of-the-art of this research field, and to highlight how our research can help it move forward.

	Recently, room-temperature strong coupling with many emitters~\cite{molecular,baranov_ultrastrong_2020,Geisler-2019,moilanen_active_2018} and even single quantum emitters \cite{Santhosh,Baumberg,our_paper,leng_strong_2018,park_tip-enhanced_2019} have been experimentally demonstrated. These experiments rely on the remarkable coupling strengths achievable in plasmonic nanoresonators, thanks to the deeply sub-wavelength mode volumes of the latter. In order to relax the mode volume requirements (which are currently pushing the limits of nanofabrication techniques) and obtain more robust experimental realizations, it would be of great interest to find emitter systems with increased dipole moments. This would result in larger coupling constants while still allowing to harness the single-emitter quantum nonlinearity of the JC model (JCM) \cite{Kasprzak-2010, Fink-2008}. So far, only the ``first rung" of the JC ladder, i.e. the single-excitation regime of the model, has been experimentally detected at ambient conditions. Efforts are still ongoing to detect higher rungs of the ladder to shed light on (and harness) the quantum non-linearity of the model.

	In order to realise the JCM experimentally, great efforts have been made over the years in the fabrication of artificial atoms --- such as semiconductor quantum dots~\cite{Reithmaier} --- that are close to ideal two-level emitters. Indeed, so far the multi-level nature of such emitters has mainly been treated as a modelling nuisance, i.e. as a source of errors with respect to an idealised two-level scenario. In this context, Madsen et al.~\cite{Madsen} and our previous paper~\cite{our_paper}, have recently mentioned the possible effects of a more complicated energy level structure of the emitter. To the best of our knowledge, however, our previous work~\cite{our_paper} was the first to explicitly suggest that an emitter with multiple ($>2$) energy levels can even bring about some advantages: in the right conditions it may be used to simulate a JC system with \textit{enhanced} coupling constant. With reports of quantum dots featuring as much as 64 quasi-degenerate excited states \cite{Liu-2010,Hens-2012}, a general quantum-mechanical analysis of such a system (\textit{multilevel model} for brevity) is in order.

	The findings presented here will thus be of importance for the selection and the design of suitable emitters for nanoscale strong-coupling experiments at ambient conditions and for the interpretation of such experiments. Importantly, our approach lives independently of the contentious issues regarding the calculation of mode volumes and coupling constants in quantum (and classical) nanophotonics, see for example~\cite{NV_review,Nuttawut_quasimodes,Hughes_quasimodes}. These approaches will of course determine the numerical values of coupling constants in different experimental situations, but they do not affect the nature of our quantum optical model and the general conclusions we can draw from it.

	The paper is organised as follows. In Section~\ref{modelsection} we introduce a Hamiltonian that models an emitter with $N$ excited levels coupled to the same ground state via a single mode field. We will show that our system can be understood as a JC model which is weakly interacting with $N-1$ ``dark states" of the emitter. In Section~\ref{MEQsec} we generalise our model to an open system via a master equation. This takes into account incoherent processes such as photon loss, dephasing and emitter pumping, all of which may be required for describing realistic experimental scenarios. In Section~\ref{2sublevels} we support our general arguments with detailed analytical results for the special case $N=2$: we find that our system's emission spectrum features approximately the same resonances as a JC model, plus additional features (due to the dark states) that we fully characterize. Section \ref{numericalsection} presents numerical studies for $N=3$, comparing common experimental observables in our model with the same quantities calculated in reference JC and TC/Dicke models.
	In Section~\ref{conclusions} we draw our conclusions.
	For completeness, Appendix~\ref{auxmodels} contains some useful reminders and formulas pertaining JC, TC and coupled-oscillator models, since those are referenced throughout the paper.

	\section{Model}\label{modelsection}

	Our \textit{multilevel model} features a quantum emitter with electronic ground state $\ket G$ and a collection of excited states $\ket{e_k}$, also called \textit{sublevels}, where $k=1,2,\dots,N$. Each internal transition of the emitter, $\ket G\leftrightarrow \ket{e_k}$, is coupled to a single-mode field (cavity for brevity) with strength $g_k\in\mathbb{R}$. The typical physical system we have in mind is a multilevel quantum dot embedded in a plasmonic nanoresonator, where the cavity resonance is \textit{broadband}, in the sense that it overlaps with all the emitter transitions, and where strong field gradients may allow for light-matter couplings beyond the dipole approximation (hence the possibility of coupling a large number of levels). Yet, it is important to note that our quantum optical formalism applies to any type of emitter or resonator that satisfy our modelling assumptions. As usual the cavity mode is described via the annihilation operator $\hat a$, with $[\hat a,\hat a^\dagger]=1$.

	Taking $\hbar=1$ for convenience, the cavity resonance has average frequency (hence energy) $\omega_0$, the energy of $\ket G$ can be set to zero without loss of generality, while each state $\ket{e_k}$ has energy $\omega_0+\Delta_k$. Here $\Delta_k$ is the detuning between the $k$-th emitter transition ($\ket{e_k}\to\ket G$) and the cavity. For convenience we assume $\Delta_1\le\Delta_2\le...\le\Delta_N$, we indicate the full range of energy detunings as $\varepsilon\equiv\Delta_N-\Delta_1$ and the average detuning as $\Delta=\tfrac{1}{N}\sum_{k}\Delta_k$.  we can write the total Hamiltonian as
	\begin{align}
	H&=H_0+V,\label{Hmany}\\
	H_0&=\omega_0\hat a^\dagger \hat a+\sum_{k=1}^n(\omega_0+\Delta_k)\proj{e_k},\\
	V&=\sum_{k=1}^{N}g_k\left(\hat a\,\kebra{e_k}{G}+\hat a^\dagger\kebra{G}{e_k}\right),
	\end{align}
	where $H_0,V$ are, respectively, the free and interaction Hamiltonians, and we have neglected counter-rotating terms in the light-matter interaction. The physics of Hamiltonian~\eqref{Hmany} can in general be quite complicated, despite $H$ being number-conserving --- i.e. $H$ commutes with the number operator $\hat{\mathcal N}_{\mathrm{{tot}}}=\hat a^\dagger\hat a+\sum_k\proj{e_k}$. However, here we are interested in the special case of quasi-degenerate emitters that are near-resonant with the cavity, i.e. we assume the parameter regime $|\Delta_k|\ll g_k$. This approximation holds e.g. for colloidal quantum dots coupled to plasmonic nanoresonators. Here, $\Delta_k$ are on the order of a few meV, while the coupling strengths, $g_k$, are on the order of 100meV \cite{our_paper}. In this scenario a number of qualitative observations can be made.
	\begin{figure}[t!]
		\includegraphics[width=\linewidth]{twolevels}
		\caption{Illustration of the transformation between the bare emitter basis and the radiation basis for $N=2$ sublevels, and the resulting appearance of a boosted effective coupling. The same principles hold for larger values of $N$. \textbf{(a)} In the bare emitter basis the two levels $\ket{e_{1,2}}$ are not directly coupled to each other, while $\varepsilon$ plays the role of a relative detuning between them. Both $\ket{G}\leftrightarrow \ket{e_{1,2}}$ transitions couple to the cavity field, and we are assuming for simplicity that the associated coupling constants are the same: $g_1=g_2=g$. \textbf{(b)} In the radiation basis, the bright and dark states $\ket B, \ket D$ have the same frequency, but are coupled to each other with strength $\varepsilon/2$. The transition $\ket G\leftrightarrow \ket{B}$ is coupled to the field with enhanced strength $g_\mathrm{eff}=\sqrt2 g$, while the dark state $\ket D$ does not directly interact with the field.}\label{Figlivelli}
	\end{figure}
	\subsection{Bare emitter basis versus radiation basis}
	To aid understanding, the main ideas behind this subsection are illustrated in Fig.~\ref{Figlivelli} for the simplest case, $N=2$. We begin our study of the general $N$-sublevel model by considering the case of exact degeneracy of the excited states, $\Delta_k=\Delta\,\forall k$, or equivalently $\varepsilon=0$ in terms of the full range of energy detunings. The resulting mathematics follow in a straightforward manner from the well-known JCM. This simple starting point will set the stage to investigate small deviations from an idealized scenario, i.e. cases with small but nonzero $\varepsilon$. For $\varepsilon=0$, we find that any linear combination of the states $\ket{e_k}$ is itself a valid emitter excited state, or more precisely, an eigenstate of $H_0$ with eigenvalue $\omega_0+\Delta$. Among all the states that can be constructed in this way, of particular significance is the coherent superposition
	\begin{equation}
	\ket{B}=\frac{\sum_k g_k\,\ket{e_k}}{\sqrt{\sum_{j}g_{j}^2}},\label{bright}
	\end{equation}
	where the denominator ensures normalization. We shall refer to $\ket{B}$ as the  \textit{bright state}, or \textit{superradiant state} (mimicking the terminology of the Dicke model). State $\ket B$ can be interpreted as the only emitter excited state that couples directly to the cavity field. Indeed it is easy to check that the light-matter interaction term in Eq.~\eqref{Hmany} may be rewritten in the simple form
	\begin{align}
	V&=g_\mathrm{eff}\left(\hat a\,\kebra{B}{G}+\hat a^\dagger\kebra{G}{B}\right),\label{Vnew}
	\end{align}
	where we have defined an effective coupling constant
	\begin{equation}
	g_\mathrm{eff}=\sqrt{\sum_{k}g_{k}^2}.\label{geff}
	\end{equation}
	Hence, through an appropriate change of basis for the emitter, the $N$ degenerate transitions $\ket G\leftrightarrow \ket{e_k}$ can be replaced by a single transition $\ket G\leftrightarrow \ket{B}$, characterized by an enhanced coupling to the cavity field. Eq.~\eqref{geff} indeed shows that the effective coupling scales like $g_\mathrm{eff}\sim\sqrt N$ with the number of sublevels. This boosting of the effective light-matter coupling is analogous to what happens in the Dicke \cite{DickeOriginal} and TC \cite{tavis1968exact} models, where $N$ degenerate emitters are coupled to the same field. A crucial difference from the TC model is that Eq.~\eqref{Vnew} retains the structure of a JC interaction Hamiltonian, so that in the model studied here there is in principle no loss of anharmonicity as the number of sublevels $N$ is increased.

	To fully specify the mentioned change of basis, we can define $N-1$ further linear combinations of the excited states $\ket{e_k}$, say $\{\ket{D_1},\ket{D_2},\dots,\ket{D_{N-1}}\}$, which together with $\{\ket{G},\ket{B}\}$ form a complete orthonormal set for the emitter internal levels. When $\varepsilon=0$ each $\ket{D_k}$ is also an eigenstate of $H_0$ with eigenvalue $\omega_0+\Delta$, due to the $N$-fold degeneracy. By construction, these additional basis states \textit{do not} directly interact with the cavity field, i.e.
	\begin{equation}
	V\ket{D_k}=\bra{D_k}V=0,\qquad k=1,2,\dots,N-1,\label{nointeraction}
	\end{equation}
	and may be called \textit{subradiant states} or \textit{dark states}. In practice, explicit expressions for the dark states may be found by standard orthogonalization procedures (details not shown). The above discussion suggests that the set $\{\ket{G},\ket{B},\ket{D_1},\ket{D_2},\dots,\ket{D_{N-1}}\}$, which we will call \textit{radiation basis}, may be a more meaningful conceptual tool to study light-matter interaction in our system, as compared to the \textit{bare emitter basis} $\{\ket{G},\ket{e_1},\ket{e_2},\dots,\ket{e_N}\}$.

	For the purpose of modelling the optical properties of the system, it is now tempting to simply neglect all the dark states and retain only the two emitter levels $\ket{B},\ket G$. This is justified if there are no additional processes, coherent or incoherent, that are able to populate the dark states and/or couple them to the two levels $\ket{B},\ket{G}$. When all these conditions are satisfied, the system is \textit{exactly} described by a JCM with coupling constant $g_\mathrm{eff}$.\\

	We are now in a position to study small deviations from the ideal case. As anticipated, the most obvious deviation from pure JC physics occurs when the excited states $\ket{e_k}$ are not exactly resonant with each other (i.e. $\varepsilon\neq0$). In the radiation basis, the (small) relative detunings between the levels $\ket{e_k}$ will translate into (weak) couplings between the bright and dark states, and between the dark states themselves. In other words, $\ket{B}$ and $\ket{D_k}$ cease to be eigenstates of $H_0$ when $\varepsilon\neq0$. Note however that Eqs.~\eqref{Vnew} and \eqref{nointeraction} remain valid in general, so that the dark states may interact with the field only \textit{indirectly}, i.e. through the mediation of state $\ket{B}$. These additional interactions will bring about deviations from the well understood JC physics described above. While a general calculation of these effects would be cumbersome and beyond the scopes of this paper, the case $N=2$ sublevels is simple enough to be treated analytically and already contains all the essential ingredients in our problem --- see Section~\ref{2sublevels} below and Fig.~\ref{Figlivelli}.

	In some cases, the exact equivalence to a JC system may be lost even in the case of perfect degeneracy between the excited sublevels. This can happen if incoherent processes couple the pair $\{\ket{G},\ket{B}\}$ to the dark states. We shall see in Section~\ref{2sublevels}, for example, that emitter dephasing provides a common mechanism for interconverting bright and dark states. Yet, our analytical and numerical results suggest that the multilevel model can provide an excellent approximation to JC physics even when these imperfections are taken into account.

	\section{Master Equation}\label{MEQsec}
	\begin{figure}[t!]
		\includegraphics[width=1.05\linewidth]{Fig3_revised.png}
		\caption{Schematic of the investigated setup. The system may be excited through an incoherent pump of intensity $\Lambda$ applied to the emitter, or coherently driving of the (plasmonic) cavity with field amplitude $\Omega$ and frequency $\omega_L$. The bright state of the emitter is coupled to the cavity with the effective strength $g_\mathrm{eff}$, and the emitter sublevels are spread in frequency over a range $\varepsilon$. $\kappa$ is the loss rate of the cavity and $\gamma$ the decay rate of all the emitter sublevels, which are also subject to dephasing at rate $\gamma_d$ (not shown). All the observables discussed in this work can be measured by collecting the cavity output field.}\label{figaschema}
	\end{figure}
	In order to predict relevant observables for future experiments, we now introduce an open system generalisation of our model, via the well established Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation~\cite{Breuer}. As anticipated above, the main experimental scenario we have in mind is a quantum dot interacting with a plasmonic nanoresonator, yet the ideas presented here may be applicable to a wider variety of setups. We will include cavity photon loss as the primary channel for energy dissipation, while the decay rate of the emitter into free-space modes will be assumed negligible (however, in simulations it will remain finite for reasons of numerical stability). Such an assumption is reasonable due to the large $\beta$-factors achievable in plasmonic resonators. On the other hand, we shall assume that the emitter excited states can be broadened well beyond their radiative linewidth by additional dephasing processes -  typically thought of as a drawback of room temperature operation. We consider two ways in which energy can be supplied to the coupled system: incoherent pumping of the emitter, or a coherent driving field applied to the cavity. While the first is a common method to feed excitations into quantum dots, the second is arguably the standard avenue to test nonlinearities in a quantum optical system. For a visual summary, the main ingredients of our open system model are sketched in Fig.~\ref{figaschema}. In formulas, the following master equation will be used to model the emitter-cavity system,
	\begin{equation}
	\dot\rho=-i[H+V_\mathrm{drive},\rho]+\sum_{L\in\mathcal{J}}\frac12\left(2L\rho L^\dagger-L^\dagger L\rho-\rho L^\dagger L\right),\label{master}
	\end{equation}
	where $H$ is the Hamiltonian \eqref{Hmany}, and
	\begin{align}
	V_\mathrm{drive}=\Omega\left(\hat a e^{i\omega_Lt}+\hat a^\dagger e^{-i\omega_Lt}\right)
	\end{align}
	is an interaction term describing coherent driving of the cavity by a classical laser at frequency $\omega_L$ ($\Omega$ quantifies the driving field strength). The $L$'s appearing on the right hand side of Eq.~\eqref{master} are the \textit{jump operators}, used to model incoherent processes. They are collected in the set ${\cal J}$, which features the following elements:
	\begin{itemize}
		\item Cavity decay (photon loss) at rate $\kappa$: $L=\sqrt{\kappa}\,\hat a$
		\item Radiative decay of the emitter $\gamma$: $L=\sqrt{\gamma} \,\kebra{G}{e_k}$,
		\item Emitter dephasing $\gamma_d$: $L=\sqrt{\gamma_d}\,\ket{e_k}\bra{e_k}$,
		\item Emitter pumping of (total) rate $\Lambda$: $L=\sqrt{\frac{\Lambda}{N}} \,\kebra{e_k}{G}$,
	\end{itemize}
	where $k=1,2,\dots,N$. We assumed that the decay and dephasing rates are the same for all sublevels, and that the pump power $\Lambda$ is split symmetrically over the $N$ sublevels. We also consider equal couplings $g_k=g_\mathrm{eff}/\sqrt{N}$ for all $k$, while the detunings $\Delta_k$ are equally spaced in the interval $[\Delta-\varepsilon/2,\Delta+\varepsilon/2].$ All the assumed symmetries between the sublevels are of course an idealisation. Many generalisations of this basic model can be included in principle. However, we do not expect that these can bring about significant qualitative departures from the physics described here. Hence, in this work we confine our analysis to the above setting for simplicity and computational efficiency.

	\section{Analytical results for 2 sub-levels}\label{2sublevels}
	We here present the analytical diagonalization of Hamiltonian \eqref{Hmany} for the case $N=2$ with equal couplings $g_1=g_2=g$, as well as the associated implications for the system's open dynamics. This instance of the model, illustrated in Fig.~\ref{Figlivelli}, is particularly transparent and fully captures the essence of our study. We verified that analytical solutions are also available for the cases $N=3,4$, using similar methods, but these are significantly more complex without adding fundamentally new insights. For $N=2$ the detunings are given by $\Delta_1=\Delta-\varepsilon/2$ and $\Delta_2=\Delta+\varepsilon/2$, respectively. Furthermore there are only one bright and one dark state given by $\ket{B}=\frac{\ket{e_1}+\ket{e_2}}{\sqrt{2}}$, $\ket{D}=\frac{\ket{e_1}-\ket{e_2}}{\sqrt2}$. In the radiation basis representation, the Hamiltonian may be recast as
	\begin{align}
	H&=H_0'+V',\label{Hprime}\\
	H_0'&=\omega_0\hat a^\dagger \hat a+\left(\omega_0+\Delta\right)(\proj{B}+\proj{D}),\label{H0prime}\\
	V'&=g_\mathrm{eff}\left(\hat a\,\kebra{B}{G}+\hat a^\dagger\kebra{G}{B}\right)-\frac{\varepsilon}{2}\left(\kebra{B}{D}+\kebra{D}{B}\right),\label{Vprime}
	\end{align}
	where in this case $g_\mathrm{eff}=\sqrt2 g$. As for the general case treated in the previous section, a JC interaction describes the coupling between the two levels $\{\ket{G},\ket{B}\}$ and the cavity field. However we can also see that the relative detuning of the sub-levels, $\varepsilon$, translates into a small coupling between $\ket{B}$ and $\ket{D}$, so that even the dark state $\ket{D}$ can indirectly influence the optical behaviour of the system. The ground state of $H$ is easily spotted as $\ket{G,0}$, with eingenvalue $E=0$. Exploting excitation number conservation, i.e. $[H,\hat{\mathcal{N}}_\mathrm{tot}]=0,$ where $\hat{\mathcal{N}}_\mathrm{tot}=\hat a^\dagger\hat a+\proj{B}+\proj{D}$, one can show that the remaining eigenvalues and eigenvectors of $H$ can be found by diagonalising the $3\times 3$ matrices ($n=1,2,\dots,\infty$)
	\begin{align}
	\mathbb{H}_n=\begin{pmatrix}
	n\omega_0 &  g_\mathrm{eff} \sqrt n& 0\\
	g_\mathrm{eff} \sqrt n& n\omega_0 +\Delta & -\varepsilon/2\\
	0&-\varepsilon/2 &n\omega_0+\Delta
	\end{pmatrix},\label{Hn}
	\end{align}
	which describe the Hamiltonian restricted to the subspace
	\begin{equation}
	\left\{\hat{\mathcal{N}}_\mathrm{tot}=n\right\} \equiv \mathrm{Span}\Big\{\ket{G,n},\ket{B,n-1},\ket{D,n-1}\Big\}.\label{spanno}
	\end{equation}
	The behaviour of the eigenvalues of $\mathbb{H}_1$ is shown in Fig~\ref{lowspectrum}, as a function of the detuning $\Delta$. Even at the non-negligible ratio $\varepsilon/g_\mathrm{eff}=0.25$ used in the figure, the commonalities with a JCM spectrum are evident. Note also that the eigenvalues of $\mathbb{H}_n$ for $n\geq 2$ will follow the same qualitative structure, as a generic $\mathbb{H}_n$ may be obtained from $\mathbb{H}_1$ by a simple rescaling $g_\mathrm{eff}\to g_\mathrm{eff}\sqrt{n}$, $\omega_0\to n\omega_0$. While a general analytical diagonalization of Eq.~\eqref{Hn} is cumbersome, for the resonant case $\Delta=0$ one can obtain simple analytical expressions. For this case, the eigenvalues of $\mathbb{H}_n$ read
	\begin{align}
	E_{n,\pm}&=n\omega_0\pm g_\mathrm{eff}\sqrt{n+\frac{\varepsilon^2}{4g_\mathrm{eff}^2}},\\
	E_{n,D}&=n\omega_0.
	\end{align}
	\begin{figure}[htp]
		\includegraphics[width=\linewidth]{lowspectrum.png}
		\caption{Eigenvalues of $\mathbb H_1$ as a function of the average detuning $\Delta$, for $\varepsilon=0.25g_\mathrm{eff}$ (blue continuous lines). The anticrossing behaviour of eigenvalues $E_{1,\pm}$ around $\Delta=0$ approximates well (the maximum relative error in the plotted range being $\sim1.3\%$) the reference JCM with coupling $g_\mathrm{eff}$ (red circles). The energy $E_{1,D}$ of the quasi-dark state is instead approximately linear in the detuning and sits in between the JCM-like energies. An elementary perturbation theory calculation indeed yields $E_{n,D}\simeq\Delta\left(1 -{ \varepsilon ^2}/{4 g_\mathrm{eff}^2 n}\right)$, which agrees well with the exact value in the considered regime (relative error below $0.2\%$). Qualitatively similar results are obtained for higher excitation numbers $n\geq 2$, and in fact it is easy to see that the relative weight of $\varepsilon$ and $\Delta$ decreases with $n$.}\label{lowspectrum}
	\end{figure}
	The corresponding normalised eigenvectors $\ket{E_{n,\pm}},\ket{	E_{n,D}}$ are
	\begin{align}
	\ket{E_{n,\pm}}&=\frac{1}{\sqrt2}\left(\frac{2g_\mathrm{eff}\sqrt n}{\sqrt{4g_\mathrm{eff}^2n+\varepsilon^2}}\ket{G,n}\pm\ket{B,n-1} \right)\nonumber\\
	&-\frac{\varepsilon}{\sqrt{8g_\mathrm{eff}^2n+2\varepsilon^2}}\ket{D,n-1},\\
	\ket{E_{n,D}}&=\frac{\varepsilon}{\sqrt{4g_\mathrm{eff}^2n+\varepsilon^2}}\ket{G,n}+\frac{2g_\mathrm{eff}\sqrt n}{\sqrt{4g_\mathrm{eff}^2n+\varepsilon^2}}\ket{D,n-1}.
	\end{align}
	The above expressions suggest that in the regime $\varepsilon\ll g_\mathrm{eff}$ it is natural to identify $\{ \ket{E_{n,\pm}}, E_{n,\pm}\}$ as a weakly perturbed JC eigensystem (the latter is indeed recovered exactly for $\varepsilon\to0$). For brevity, we refer to those as the `JC model-type' eigenvalues and eigenvectors. The eigenstates $\ket{E_{n,D}}$, which instead feature the same energy levels of an empty cavity, may be understood as dressed quasi-dark states of the emitter, accompanied by approximately $n-1$ cavity photons. They indeed result from a hybridization of states $\ket{G,n}$ and $\ket{D,n-1}$, with the second state being dominant in the superposition. It is interesting to notice that the two states involved in such superposition are only coupled indirectly in the Hamiltonian, through the mediation of state $\ket{B,n-1}$.\\
	Exploiting the analytical results obtained here, it is possible to calculate the Hamiltonian dynamics of a generic initial state via standard methods. In fact, while our work focuses on the regime $\varepsilon\ll g_\mathrm{eff},$ we note that the above diagonalization procedure is valid for a general choice of the model parameters.

	\subsection{Qualitative analysis of the master equation for \texorpdfstring{$N=2$}{N=2}}\label{QualitativeN2}
	The closed expressions obtained so far are particularly helpful in analysing the open system behaviour of our multilevel model, as well as its relationship to the JC model. Since in this work we focus on the regime of low pumping (or low coherent driving) and negligible emitter decay, we can gain valuable intuition about the properties of the master equation \eqref{master} by setting $\gamma=\Lambda=\Omega=0$. First, we observe that the introduction of a photon loss rate $\kappa$ results in a finite lifetime for the excited states of $H$, such that eigenstates with $\hat{\mathcal{N}}_\mathrm{tot}=n$ will tend to decay towards states with $\hat{\mathcal{N}}_\mathrm{tot}=n-1$, until the ground state $\ket{G,0}$ is finally reached. For a given eigenstate $\ket{E_{j}}$, an estimate of the decay rate induced by photon loss can be obtained via the perturbation theory formula
	\begin{align}
	\Gamma_{j}\simeq\kappa\bra{E_j}\hat a^\dagger \hat a\ket{E_j},
	\end{align}
	which at second order in $\varepsilon$ yields ($n\geq1$)
	\begin{align}
	\Gamma_{nD}&\simeq\kappa\left(n-1+\frac{\varepsilon^2}{4ng_\mathrm{eff}^2}\right),\label{GammaD}\\
	\Gamma_{n\pm}&\simeq\kappa \left(n-\frac12-\frac{\varepsilon^2}{8ng_\mathrm{eff}^2}\right)\label{GammaPM}.
	\end{align}
	We may immediately notice that the lowest quasi-dark state, $\ket{E_{1D}}$, is long lived, while all the remaining excited states have approximately the same lifetime of either the JCM excited states or the Fock states of an empty and lossy cavity. Next, emitter dephasing will induce further broadening of the Hamiltonian energy levels, which can be estimated as
	\begin{align}
	\delta E_j\simeq\gamma_d\left(|\sprod{e_1}{E_j}|^2+|\sprod{e_2}{E_j}|^2\right).
	\end{align}
	Some straightforward algebra yields
	\begin{align}
	\delta E_{nD}&\simeq \gamma_d\left(1-\frac{\varepsilon^2}{4ng_\mathrm{eff}^2}\right),\\
	\delta E_{n\pm}&\simeq \frac{\gamma_d}{2}\left(1+\frac{\varepsilon^2}{4ng_\mathrm{eff}^2}\right),
	\end{align}
	where again we have expanded our results at second order in $\varepsilon$. In this case we can see that the quasi-dark states are the most affected, since they feature a larger overlap with the bare excited states of the emitter. Note also that the $\delta E_j$'s are not associated with a decay towards states of lower excitation number (more below). The exact broadening of the energy levels, induced by the combined effect of photon loss and emitter dephasing, can be obtained as $-2\mathrm{Im}(\tilde E_j)$, where $ \tilde E_j$ are the eigenvalues of the non-Hermitian operator
	\begin{align}
	\tilde H=H-i\frac{\kappa}{2}\hat a^\dagger \hat a-\frac{\gamma_d}{2}\sum_k\proj{e_k }.
	\end{align}
	The resulting expressions are unwieldy, however we checked via a numerical optimization that $-2\mathrm{Im}(\tilde E_j)\simeq\Gamma_j+\delta E_j$, with a relative error below $\simeq3.1\%$ in the range $0\leq\varepsilon\leq0.25g_\mathrm{eff},0\leq\kappa\leq2g_\mathrm{eff},0\leq\gamma_d\leq 2g_\mathrm{eff}$, which is more than enough for our scopes.

	In order to gain qualitative understanding of photon emission processes in our system, we next look at the (squared) matrix elements of the annihilation operator $\hat a$,
	\begin{align}
	A_{i\to f}=|\bra{E_f}\hat a\ket{E_i}|^2,
	\end{align}
	between an initial energy eigenstate $\ket{E_i}$ and a final one $\ket{E_f}$. The quantum jump approach to GKSL master equations \cite{Breuer} provides a useful interpretation for these quantities: $A_{i\to f}$ is proportional to the probability that the transition $\ket{E_i}\to\ket{E_f} $ occurs upon loss of a cavity photon [in the master equation \eqref{master}, such transitions are triggered by the term $\kappa\hat a\rho\hat a^\dagger$]. When the transition occurs, a photon is emitted into extra-cavity modes with (approximately) frequency $\omega_{fi}=E_f-E_i$ and bandwidth $\Gamma_i+\Gamma_f+\delta E_i+\delta E_f$.
	The $A_{i\to f}$ terms involving only JCM-type eigenstates do not present any particular surprises: they approximate the well-known JCM results for  $\varepsilon\ll g_\mathrm{eff}$, and converge to them for $\varepsilon\to0$. We thus omit such cases for brevity and focus only on the matrix elements involving quasi dark states, which yield the following expressions:
	\begin{align}
	A_{1D\to0}&=\frac{\varepsilon^2}{4g_\mathrm{eff}^2n+\varepsilon^2},\label{darkG}\\
	A_{nD\to n-1,D}&=n-\frac{4 g_\mathrm{eff}^2 n}{4 g_\mathrm{eff}^2 n+\varepsilon ^2},\label{darkdark}\\
	A_{nD\to n-1,\pm}&=0,\label{darkpm}\\
	A_{n\pm\to n-1,D}&=\frac{2 g_\mathrm{eff}^2 \varepsilon ^2}{\left(4 g_\mathrm{eff}^2 (n-1)+\varepsilon ^2\right) \left(4 g_\mathrm{eff}^2 n+\varepsilon ^2\right)}.\label{pmdark}
	\end{align}
	From Eqs.~\eqref{darkG}, \eqref{darkdark} and \eqref{darkpm} we notice that, once the system is in one of the quasi-dark states $\ket{E_{nD}},$ the loss of a cavity photon can only produce the quasi dark state $\ket{E_{n-1,D}}$ for $n\geq 2$, or $\ket{G,0}$ for $n=1$. This corresponds to the emission of a photon at the cavity frequency $\omega_0$, with the associated matrix element of the order $\sim n-1$. Assuming for the moment $\gamma_d=0$ (no dephasing), this implies that once the system has evolved into a quasi-dark state, it essentially reproduces the open dynamics of an empty cavity until state $\ket{1,D}$ is reached. From there the system will decay slowly towards the lowest state $\ket{G,0}$, resulting again in the emission of a photon at the cavity frequency $\omega_0$, but with a much narrower bandwidth $\Gamma_{1D}\simeq \kappa \varepsilon^2/4g_\mathrm{eff}^2$. On the other hand, from Eq.~\eqref{pmdark} we can also see a small probability that a JCM-like state decays into a quasi-dark state by loss of a cavity photon. This process results in the emission of photons at average frequency $\omega_0\pm g_\mathrm{eff}\sqrt{n}$: these exactly overlap with the JCM spectral resonances for $n=1$, while they provide novel emission peaks for $n\geq2$. Putting back finite dephasing $\gamma_d\neq0$ into the picture, we see that, in addition to introducing further broadenings of the levels, dephasing events are capable of interconverting states $\ket B$ and $\ket D$. More in detail, we find $|\bra{f}L\ket{i}|^2=\gamma_d/2$ for $L=\sqrt{\gamma_d}\proj{e_k}$ with $k=1,2$ and $i,f\in\{D,B\}$. Referring again to the quantum jump approach \cite{Breuer} this will swap the states $\ket B$ and $\ket D$ with probability $1/2$, if a dephasing event occurs.
	\\
	In summary, we have provided a detailed characterization of the processes occurring in our open system for the emblematic case $N=2,\ \Delta=\Lambda=\gamma=\Omega=0$. While the multilevel model can approximate all features of a JCM for $\varepsilon\ll g_\mathrm{eff}$, it displays additional phenomena due to the indirect interaction of dark states and cavity field. In the next section we explore these issues quantitatively via numerical simulations.

	\begin{figure*}[t!]
		\begin{center}
		\textbf{Single-excitation regime - $\kappa=g_\mathrm{eff}$}
		\includegraphics[trim=100 0 140 0,clip,width=\textwidth]{multivsJC-Spectra1.png}
		\textbf{~\\Bi-excitation regime - $\kappa=0.05g_\mathrm{eff}$}
		\includegraphics[trim=100 0 140 0,clip,width=\textwidth]{multivsJC-Spectra2.png}
		\caption{Comparison of cavity emission spectra, $S_a(\omega)$, as obtained for our multilevel model, versus reference JC and TC models. The sub-level energy spread is fixed at $\varepsilon=0.05g_\mathrm{eff}$ in all plots. Spectra are obtained under  incoherent emitter pumping, i.e. $\Lambda\neq0,\Omega=0$. The top plots in the ``single-excitation regime" are obtained for $\kappa=g_\mathrm{eff}$and  $\Lambda=0.01\kappa$, while the ``bi-excitation regime" at the bottom is for $\kappa =0.05g_\mathrm{eff}$ and $\Lambda=0.02\kappa$. Dephasing increases from left to right. Solid blue lines indicate our multi-level model, dashed red lines show the quantities of interest for a reference JC model, while green dot-dashed lines are for the TC model. The spectra of the TC and JC models are rescaled by a factor $(1-p_\mathrm{dark})$. Gray vertical lines indicate the resonances corresponding to the first (dot-dashed) and second rung (dotted) of the JC ladder.}\label{Snumerics}
		\end{center}
	\end{figure*}

	\section{Climbing The JC Ladder}\label{numericalsection}
	In the following numerical studies, we will investigate how closely the physics of our system \--- as relevant for typical quantum optics experiments \--- resembles a JCM with enhanced coupling constant $g_\mathrm{eff}\propto\sqrt{N}$. For the sake of comparison, we shall also display the behaviour of a TC model with $N$ atoms and the same effective coupling constant $g_\mathrm{eff}$ --- see Appendix~\ref{auxmodels} for further details. Simulations are done for $N=3$: this is one step up in complexity compared to the previous section, but simple enough that the resulting calculations can be handled by a standard computer.

	Our analysis focuses on two observables that are ubiquitous in both theoretical and experimental investigations of the JC model (proper definitions will be given below):
	\begin{itemize}
		\item The \textit{steady state spectrum} $S_a(\omega)$ of the cavity output field, when the system is excited via incoherent pumping of the emitter;
		\item The photon autocorrelation function $g^{(2)}(0)$, when the emitter is excited via a coherent driving field with variable detuning.
	\end{itemize}
	The first observable provides valuable information about the energy level structure of a quantum system, in particular highlighting its optically allowed transitions. Instead, the $g^{(2)}(0)$ may be seen as a witness of optical non-linearity: we will use it in our model to show that the $\sqrt N$ enhancement of the coupling constant, $g_{\mathrm{eff}}\propto\sqrt N$, does not come at the cost of ``linearising" the light-matter interaction (something that instead happens in the TC scenario as $N$ is increased, as we shall see below). To support this last point, we will indeed compare our autocorrelation functions with the $g^{(2)}(0)$ obtained for a reference ``coupled oscillator model" --- i.e. the quintessential linear system in our context (see Appendix~\ref{auxmodels} for details). Note that we are focusing only on  observables that can be measured via the cavity output field.
	The latter has indeed the same definition in all models considered here, facilitating direct comparisons between them. On the other hand, emitter observables do not have an unambiguous correspondence in the different models, as they are defined on Hilbert spaces of different dimensions.\\
	In detail, according to standard input-output theory \cite{walls2007quantum}, the portion of the cavity output field that reaches our detector, say $\hat{E}_\mathrm{out}$ (scalar for simplicity) may be expressed as a function of the cavity annihilation operator $\hat a$ as follows:
	\begin{align}
	    \hat{E}_\mathrm{out}= E_\mathrm{c}\hat a +\hat{E}_\mathrm{vac},\label{Eout}
	\end{align}
	where $E_\mathrm{c}$ is a constant depending on the details of the cavity-plus-detector setup, while the operator $\hat{E}_\mathrm{vac}$ captures all ``vacuum noise" contributions from the detector's environment. Thanks to the assumption of a vacuum environment, which is justified for optical frequencies at room temperature, $\hat{E}_\mathrm{vac}$ does not contribute to any normally ordered observable constructed from $\hat{E}_\mathrm{out}$ --- and these are precisely the types of observables employed here. \\
	In the interest of brevity, we restrict ourselves to an experimentally desirable situation in which $\Delta=0$ and the hierarchy $\varepsilon \ll  \kappa\lesssim g_\mathrm{eff}$ holds. In other words we consider a strongly coupled system where the average transition frequency of the emitter is at resonance with the cavity, while the range of  sub-level detunings is within the bare cavity bandwidth. This scenario is indeed appropriate for the case of quantum dots embedded in plasmonic nanoresonators, as anticipated in Section~\ref{modelsection}. We shall also take the driving strengths ($\Omega$ for the coherent and  $\Lambda$ for the incoherent case) as the smallest parameters in the problem. This ensures that the driving will not induce significant modifications to the intrinsic resonances of the coupled system, nor to the conditions for achieving strong coupling \cite{ValleI,ValleII}. In this setting we focus on two representative parameter regimes:
	\begin{itemize}
		\item[(i)] $g_\mathrm{eff}=\kappa$, or `single-excitation regime',
		\item[(ii)]$g_\mathrm{eff}=20\kappa$, or `bi-excitation regime'.
	\end{itemize}
	Loosely speaking, case (i) gives access to the `first rung' of the energy ladder, thus the resulting Physics should be similar to that of a pair of coupled oscillators in all models. Case (ii) instead allows us to explore the `second rung' of the ladder, where the anharmonic nature of the investigated models becomes more prominent. In both situations we shall analyse the role of emitter dephasing by considering three scenarios: no dephasing, low dephasing (i.e. one order of magnitude smaller than the cavity decay rate $\kappa)$ and high dephasing (i.e. dephasing and cavity decay rates of the same order).
	This is particularly relevant for quantum-dot implementations of our models, especially at room temperature, in which case emitter dephasing rates can be comparable to the cavity decay rate. Finally it is important note that we shall adopt a logarithmic scale in our plots, since the examined quantities display features of very different magnitudes. This implies that the weaker features displayed in our plots (e.g. the second-rung spectral resonances) may be challenging to detect in experiments.

	\subsection{Steady state spectra}
	As anticipated, our first set of numerical examples investigates the optical emission properties of our multilevel model under incoherent pumping of the emitter, as relevant in many quantum dot experiments also at ambient conditions. Specifically, we set $\Omega=0$ (no coherent driving) and $\Lambda\neq0$. In this setting we are interested in the steady state spectrum of the cavity output field, i.e.,

	\begin{align}
	S_a(\omega)&= 2\mathrm{Re}\left(\int_{0}^{\infty}\med{ \hat{E}_\mathrm{out}^\dagger(\tau)\hat{E}_\mathrm{out}(0)}_\infty e^{-i\omega\tau}{\rm d}\tau\right),\nonumber\\
	&\propto 2\mathrm{Re}\left(\int_{0}^{\infty}\med{\hat a^\dagger(\tau)\hat a(0)}_\infty e^{-i\omega\tau}{\rm d}\tau\right),
	\end{align}
	where Eq.~\eqref{Eout} has been used, the two-point correlation function $\med{\hat a^\dagger(\tau)\hat a(0)}_\infty$ is calculated via the quantum regression theorem \cite{Breuer}, and the expectation value is calculated on $\rho_\infty$, the steady state of the master equation. The latter is found by solving the linear system ${\cal L}\rho_\infty=0,\tr{\rho_\infty}=1,$ where the superoperator ${\cal L}$ is implicitly defined by rewriting the master equation \eqref{master}  as $\dot\rho={\cal L}\rho$. We perform such calculations numerically with Python, by truncating the Hilbert space of the cavity to a sufficiently high dimension to obtain numerical convergence.
	In addition to the cavity spectrum, we shall also be interested in the quantity
	\begin{align}
	p_\mathrm{dark}&=\sum_{k}\tr{\rho_\infty\proj{D_k}}\nonumber\\
	&=1-\tr{\rho_\infty(\proj{B}+\proj{G})},
	\end{align}
	i.e. the steady-state probability that the emitter has settled in a dark state (or a mixture thereof). The resulting spectra, in logarithmic scale, are plotted in Fig.~\ref{Snumerics} together with those of the reference TC and JC models (again, see Appendix~\ref{auxmodels} for details). For a fairer visual comparison between the three models we have also rescaled the spectrum of JC and TC models by a factor $1-p_\mathrm{dark}$. We indeed recall from Section~\ref{QualitativeN2} that the multilevel model cannot display JC physics once it gets ``stuck" into a dark state. \\

	In the single excitation regime, we find that all three models display the expected resonances at $\omega\simeq\omega_0\pm g_{\mathrm{eff}}$, but the multilevel model presents additional sharp features at $\omega\simeq\omega_0$. In the case of no dephasing, these resonances account for a significant portion of the total emitted energy, and correspondingly $p_\mathrm{dark}$ is close to one. We argue that the $\omega\simeq\omega_0$ features can be ascribed to the slow decay of the long-lived quasi-dark states of the single-excitation sector: referring to our discussion and notation in Section~\ref{QualitativeN2}, this involves transitions of the form $\ket{E_{1D}}\to\ket{G,0}$ (where $\ket{E_{1D}}$ can be any of the two quasi-dark states occurring for $N=3$). As dephasing is increased, going left to right in Fig.~\ref{Snumerics}, we find that the three models show increasing agreement, and at the same time $p_\mathrm{dark}$ is significantly reduced. Referring again to our analytical discussion in Section \ref{QualitativeN2}, we recall that dephasing is indeed able to swap bright and dark states. This opens a new decay channel for the quasi-dark states $\ket{E_{1D}}$: as dephasing is increased, we may reach a parameter regime where the quasi-dark states $\ket{E_{1D}}$ are more likely to decay by a process of the form $\ket{E_{1D}}\to\ket{E_{1\pm}}\to\ket{G,0}$, where the first step is due to dephasing, the second to photon emission. Hence, in such conditions JC-like features can become more prominent in the measured spectrum. At high dephasing ($\gamma_d=\kappa$), the dark-state features are effectively invisible in the emission spectrum. \\
	The bi-excitation scenario is where the multilevel model truly shines, compared to the reference TC model: we can see in the bottom of Fig.~\ref{Snumerics} that multilevel and JC models are indeed in agreement on the location of both single-excitation ($\omega\simeq\omega_0\pm g_\mathrm{eff}$, gray dot-dashed lines) and bi-excitation [$\omega\simeq\omega_0\pm (\sqrt2\pm1)g_\mathrm{eff}]$, gray dotted lines) resonances, while the bi-excitation peaks of the TC model are visibly shifted. Also in this case the multilevel model presents an additional resonance (or perhaps a collection of closely spaced resonances) at $\omega\simeq\omega_0.$ Referring again to the notation of Section~\ref{QualitativeN2}, these can be ascribed to two processes: the photons emitted in the $\ket{E_{1D}}\to\ket{G,0}$ transition, which is associated with a small decay rate, and those emitted in the ``second rung" process $\ket{E_{2D}}\to\ket{E_{1D}}$, which we recall is very similar to photon emission by an empty cavity, and hence has a singnificantly higher decay rate $\simeq \kappa$.
	We again notice that the $\omega\simeq\omega_0$ resonance can be weakened in the presence of dephasing, however it remains prominent even in the ``high dephasing" scenario. Our interpretation for this is the following: while dephasing of order $\kappa$ can suppress the $\ket{E_{1D}}\to\ket{G,0}$ process, as discussed above, the $\ket{E_{2D}}\to\ket{E_{1D}}$ decay channel remains a probable process since it is also associated with a decay rate of order $\kappa$. \\
	It is worth pointing out that if one goes beyond our symmetric pump hypothesis (populating all the emitter excited states with equal probability - see Section~\ref{MEQsec}) it may of course be possible to reduce the value of $p_\mathrm{dark}$ and hence the prominence of dark-state features in the cavity emission spectrum.

	To summarize, the results of this subsection tell us that the multilevel model is a viable system to test the characteristic $\propto \sqrt{n}g_\mathrm{eff}$ splitting of energy levels that is still the holy grail of experimental JC research. In this light, the main strength of the multilevel model is that the effective coupling $g_\mathrm{eff}$ may be significantly boosted by considering emitters with many closely spaced sublevels.

	\subsection{Photon autocorrelation function}
		\begin{figure*}[t!]
		\begin{center}
		\textbf{Single-excitation regime - $\kappa=g$}
		\includegraphics[trim=100 0 140 0,clip,width=\textwidth]{G21X.png}
		\textbf{~\\Bi-excitation regime - $\kappa=0.05g$}
		\includegraphics[trim=100 0 140 0,clip,width=\textwidth]{G22X.png}
		\caption{steady-state photon autocorrelation $g^{(2)}(0)$ for $N=3$ emitter sublevels and a sub-level energy spread fixed at $\varepsilon=0.05g_\mathrm{eff}$. We set the incoherent pump to zero and introduce a coherent cavity drive, corresponding to model parameters $\Lambda=0,\Omega=0.01g_\mathrm{eff}$. Dephasing increases from left to right as before. Solid blue lines = multi-level model; dashed red lines = reference JC model; dot-dashed green lines = reference TC model; dotted black lines = reference coupled oscillators model. For details on all the auxiliary models used here see Appendix~\ref{auxmodels}}\label{g2numerics}
	\end{center}
	\end{figure*}
	As a second step, we will look at the steady state behaviour of the zero-delay photon autocorrelation function, under weak coherent driving of the cavity. In all three models this quantity is defined as follows:
	\begin{align}
	g^{(2)}(0)&=\lim\limits_{t\to\infty}\frac{\av{\hat{E}_\mathrm{out}^\dagger \hat{E}_\mathrm{out}^\dagger \hat{E}_\mathrm{out}\hat{E}_\mathrm{out}}}{\av{\hat{E}_\mathrm{out}^\dagger \hat{E}_\mathrm{out}}^2}\nonumber\\
	&=\lim\limits_{t\to\infty}\frac{\av{\hat a^\dagger \hat a^\dagger \hat a\hat a}}{\av{\hat a^\dagger \hat a}^2},\label{g2def}
	\end{align}
	where again we made use of Eq.~\eqref{Eout} in order to obtain an expression that only features system operators. In detail, in this subsection we consider the master equation \eqref{master}, setting $\Lambda=0$, $\Omega\neq 0$, and vary the driving field frequency $\omega_L$. While the resulting master equation is explicitly time dependent, it is still possible to define a steady state by considering an interaction picture with respect to the Hamiltonian $H_L=\omega_L\hat a^\dagger \hat a+\omega_L\sum_{k=1}^n\proj{e_k}.$ This is a standard procedure in quantum optics and it amounts to replacing $V_\mathrm{drive}\to V_\mathrm{drive}^{(\mathrm{int})}=\Omega\left(\hat a +\hat a^\dagger \right)$ and
	$H\to H^{(\mathrm{int})}=H-H_L$ in the master equation~\eqref{master}. In turn, this procedure implicitly defines a time independent superoperator $\mathcal {L}^{(\mathrm{int})}$ which does admit a steady state $\rho_\infty^{(\mathrm{int})}$. Such a steady state is again found numerically with the same process described in the previous section, and it can be easily shown that it provides the correct expectation values involved in Eq.~\eqref{g2def}. The obtained autocorrelation functions for multilevel, JC and TC models are displayed in Fig.~\ref{g2numerics} as functions of the driving field detuning. We note that in this case no rescaling of the TC and JC plots is required, since the $g^{2}(0)$ is normalised by definition.

	We observe that, excluding the region $\omega\simeq\omega_0$, the multilevel model does a good job at matching the JC model predictions, capturing both the position and magnitude of resonances and dips. These observations are true both in the single- and bi-excitation regimes. In contrast, the $g^{(2)}(0)$ of the TC model displays features that are significantly less pronounced than their JC counterparts (recall that we are using a logscale), as well as being shifted in frequency in the bi-excitation regime. This is compatible with the intuition that the TC model is ``more linear" than both the JC and multilevel ones. Finally, the coupled oscillator model is of course even farther off from the JC predictions. As expected, the latter indeed displays features that are several orders of magnitude smaller than those in all the other models, plus it satisfies $g^{(2)}(0)>1$ at all times.
	As anticipated, however, the multilevel model displays additional features in the region $\omega\simeq \omega_0$: at low or no dephasing a double-peak structure can be clearly seen, likely due to processes involving the quasi-dark states (we recall that for $N=3$ there are two quasi-dark states for each value of the excitation number $n\geq 1$). As dephasing is increased the prominence of these peaks is reduced, presumably due to the introduction of additional decay channels for the quasi-dark states (see also previous Subsection). However, dephasing also results in the appearance of a central peak in the $g^{(2)}(0)$, which may be due to an increased likelihood that a pair of driving photons excite a quasi-dark state of the system, and are later emitted at the frequency of an empty cavity by a process of the form $\ket{E_{2D}}\to\ket{E_{1D}}\to\ket{G,0}$. While a more rigorous analysis of these effects would be certainly worthwhile to confirm our intuitions, we feel that it goes beyond the scopes of this work.

	As anticipated, the $g^{(2)}(0)$ is one of the most commonly adopted witnesses of nonlinearity in emitter-cavity systems such as the ones we are investigating. The results of this subsection thus confirm that the multilevel model closely matches the nonlinear signatures of JC physics, and in this respect it clearly outperforms the reference TC model with the same values of $N$ and $g_\mathrm{eff}$.

	\section{Conclusions}\label{conclusions}
	We explored a novel route to realise the strong coupling regime of light-matter interaction, exploiting a multi-level emitter coupled to a single-mode electromagnetic field. When the $N$ excited sublevels of the emitter are nearly degenerate, we showed both analytically and numerically that
	our proposed system can closely approximate many aspects of JC physics. The associated light-matter coupling constant scales as $g_\mathrm{eff}\propto\sqrt{N}$ and, crucially, the quantum nonlinearities of our model are not suppressed with increasing $N$. Indeed, we are able to observe the characteristic energy level splitting of order $\sim\sqrt{n}$, where $n$ is the number of combined (emitter + field) excitations, as well as the associated nonlinear signatures in emission spectra and photon autocorrelation functions. Moreover, we have been able to characterise the main effects associated with the additional levels of the emitter: the appearance of very interesting sharp resonances in the middle of the emission spectrum, associated with the slow decay of ``quasi dark states". While it may be possible to observe such resonances in an experiment, our analysis shows that these are quickly washed out in the presence of dephasing. Observing dark-state signatures would thus require  cryogenic temperatures.

	Our findings are particularly relevant in solid-state and plasmonic implementations of Cavity QED, where the emitters of choice are often multi-level quantum dots, and may pave the way towards the observation and exploitation of light-matter strong coupling at room temperature.
	\section{Acknowledgments}
	We thank M. G. Genoni for the useful discussion on modelling dephasing. Support by the Science Foundation Ireland (SFI) under grant 18/RP/6236 is gratefully acknowledged. TT acknowledges support from the University of Nottingham via a Nottingham Research Fellowship.

\nocite{*}

redpajama_set_name: RedPajamaArXiv


In follow up to an earlier post, Chicago police have confirmed that the man shot in broad daylight the other day while live streaming on Facebook may have been shot in retaliation of a 2009-second degree murder conviction and have expressed concerns that the shooting may have been gang related.
The man who was in Chicago to visit his mother received a phone call from a friend who arranged to meet up with the individual at the corner of Hoyne Street.
Shot several times in the face, back, stomach and legs, the convicted murderer is presently on a ventilator. The man's name has not been released.
…The mother said that shortly before the shooting Thursday, her son met up with her at a gas station not far from the corner store. He was visiting from out of town, she said. At some point, her son got a phone call from a friend and agreed to meet him around the corner on Hoyne.
Outside the store, her son began "live chatting" on his phone, streaming video of himself as he talked, the mother said.
The dramatic video starts with the man holding the camera toward himself, with the convenience store in the background. A young man is standing behind him.
Not long after the video starts, the man looks away and drops the phone. Shots ring out. The man appears to take a few steps while the shooting continues and drops the phone, but the camera continues to roll, facing skyward….


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text:
Rezvani Motors
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As a young boy, founder Ferris Rezvani aspired to become a fighter pilot just like his father. In order to develop a thrill similar to flying an F-4 fighter jet, Mr. Rezvani set out to create an extreme sports car that gave drivers the opportunity to experience the same exhilaration and g-forces behind the wheel. This is represented in the Rezvani emblem, with wings representing his roots in aviation and steering wheel and racing stripes representing the brand's dedication to driving.
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2. Beat egg in bowl. Mix in beef, pork, 2/3 cup barbecue sauce, crumbs, parsley, onion, garlic and salt. Shape scant 3/4 cup mixture into 4 x 2 x 1-inch loaf, for total of 6. Place on jelly-roll pan. Top each loaf with about 2 tablespoons of remaining barbecue sauce.
3. Bake in 350 degrees F oven 30 minutes, until internal temperature registers 160 degrees F. Let stand on pan 10 minutes.
4. Serve immediately or cool completely; refrigerate and serve cold.
Servings Per Recipe: 6; Amount Per Serving: cal. (kcal): 346, Fat, total (g): 16, chol. (mg): 114, sat. fat (g): 6, carb. (g): 25, fiber (g): , pro. (g): 24, sodium (mg): 1118, Percent Daily Values are based on a 2,000 calorie diet.


text:
Q: How can I tell if a post has been published at least once? I am working on a plugin that has a piece of meta data attached to each post. The fields are editable in a meta box on the post. This is all working fine.
I would like to prevent anyone from modifying the settings in meta box once the post has been published. By virtue of the application looking for this meta data, it doesn't make any sense for the meta data to change after publish.
So, is there any way to tell if a post has been published at least once? This way I can disable the controls in the meta box.

A: We can do this by storing the value into postmeta when post is published first time.
function save_ispublished( $post_id ) {

    if (defined('DOING_AUTOSAVE') && DOING_AUTOSAVE)
        return;

    $published_once = get_post_meta( $post_id, 'is_published', true );

    // Check if 'is_published' meta value is empty.
    if ( ! empty( $published_once ) ) {

        $published_once = 'yes';
    }

    // store is_published value when first time published.
    update_post_meta( $post_id, 'is_published', $published_once );
}
add_action( 'save_post', 'save_ispublished' );

you can check it by get the meta value.
$is_published = get_post_meta( $post_id, 'is_published', true );

if( $is_published == 'yes' ) {

    /*
    * Actions if post is already published atleast once.
    */
}

Hope this help !

A: I totally agree with @Govind Kumar except for the action uses in the example, you can use publish_post action  instead of save_post.
publish_post is an action triggered whenever a post is published, or if it is edited and the status is changed to publish. Save_post will trigger each time the post is save.
publish_post action

redpajama_set_name: RedPajamaStackExchange


text:
Details emerge on repelling of South City home invasion
By BY Examiner Staff • October 8, 2010 12:00 am
A South City man desperately protecting his family tackled a robber in his home and had him pinned to floor when cops showed, officials said.
That only begins to explain how a robbery victim held at gunpoint outside his El Camino Real residence on Aug. 21 ended up having to fight off the thug inside the home, where two children and two women had been sleeping, according to San Mateo County prosecutors.
According to court documents, the victim had been drinking at a bar on his block and walked home around 3:30 a.m. The suspect, 27-year-old Brian O'Rourke, of Burlingame, approached the victim outside his home and asked for a cigarette.
After the victim gave him a smoke, O'Rourke allegedly pulled out a .45 caliber handgun and demanded money and car keys, San Mateo County Chief Deputy District Attorney Steve Wagstaffe said.
The victim "pleaded with suspect and offered to call a taxi," Wagstaffe said. O'Rourke then allegedly walked into the victim's unlocked home, put a gun to his head and demanded jewelry, money or cell phones.
The victim "offered two cell phones but suspect refused them," Wagstaffe said.
Then, O'Rourke reportedly began making his way into another room in the house, where the victim's two children were sleeping, Wagstaffe said. Fearing for them, the victim "ran at and tackled" the suspect, grabbed the handgun and restrained him while his cousin called cops, he said.
Cops arrived and arrested O'Rourke, who was scheduled to be arraignment on charges in connection with the home invasion Friday, police said. O'Rourke remains in custody on $200,000 bail.
maldax@sfexaminer.com
Tags: Bay Area News, Crime, Crime & Courts, Local, NEP


If you're interviewing at an organisation with a smart dress code, mirror your interviewer by wearing a suit. If you're uncertain about the dress code, you should ask your recruiter, and if you're still not clear, it's always wise to over-dress rather than under-dress to show you're someone who's interested in the role and wishes to present themselves well. If you get the job, you can always align your office attire with the rest of the company, but you'll risk coming across as sloppy or even disrespectful if you don't put your best foot forward for the interview.


text:
Q: I can't place two images on the center of layout I try to place the two images in the center of layout,
but I have the extra space on the top and on the bottom of my layout.
Images are png files.

Important part of my XML is below:
                    <LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
                              android:layout_width="wrap_content"
                              android:layout_height="wrap_content"
                              android:background="@color/backDark2"
                              android:orientation="horizontal"
                        >

                <ImageView
                        android:id="@+id/imageDriverProfileFace"
                        android:layout_width="0dp"
                        android:layout_height="wrap_content"
                        android:layout_gravity="center"
                        android:layout_weight="1"
                        android:src="@drawable/driver"
                        />
                <ImageView
                        android:id="@+id/imageDriverProfileCar"
                        android:layout_width="0dp"
                        android:layout_height="wrap_content"
                        android:layout_gravity="center"
                        android:layout_weight="1"
                        android:src="@drawable/car"
                        />
                </LinearLayout>

Any ideas please how to remove mentioned extra space?
Thanks!

A: set android:adjustViewBounds="true" param to your imageview's