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Culture Lynsey G November 7, 2016
Interlude at Standing Rock
Between clashes with police, protesters take time to trick-or-treat and catch their breath
It's Halloween, and I'm standing on a bridge just north of where Highway 1806 passes north out of the Standing Rock Sioux reservation in North Dakota and crosses the Cannon Ball River, which empties into the Missouri River about half a mile to the east. The bridge is blocked by a burned-out car, two still-smoldering trucks with the words "Morton County" barely legible on their doors, and the remains of a torched lighting panel. Glass and ashes litter the road where earlier blockades were set alight. Over 140 people who call themselves water protectors were arrested here four days ago. It's an almost perfect post-apocalyptic scene out of a seasonally appropriate horror movie, but it's all too real.
We've joined the Oceti Sakowin (the Sioux term, meaning "the Seven Council Fires," for the Sioux Nation) Camp, situated on the banks of the Cannon Ball. The land the camp occupies is currently controlled by the Army Corps of Engineers, relatively neutral ground for the time being — as the long-escalating battle between the Dakota Access Pipeline's proponents and its protesters explodes into violence. It's an eerie time to be here, as the several hundred people in camp are regrouping between clashes with private security forces and fully armed riot police; meanwhile, they're also celebrating Halloween.
The water protectors have been here since April, blocking the oil pipeline company Energy Transfer Partners as it attempts to build the Dakota Access Pipeline, which would run nearly 1,200 miles from the Bakken oil fields in North Dakota to a river port in Patoka, Illinois. The pipeline would carry 470,000 barrels of crude oil per day at inception along a path that would take it beneath the Missouri River at Lake Oahe, just north of the Standing Rock Sioux reservation's current boundaries.
But the water protectors, with backing from Standing Rock Sioux Tribal Chairman David Archambault II, have been resisting the efforts of Energy Transfer Partners to dig in the area. They cite the likelihood of the pipeline's bursting into a waterway that provides clean water to their people — and millions of others as the Missouri flows southeast and eventually into the Mississippi — and they further invoke the fact that much of the contested land should, by rights, still belong to the Standing Rock tribe.
The pipeline was originally planned to run north of Bismarck, North Dakota's second-largest city, but residents voiced concerns over possible contamination of their drinking water. In response, Energy Transfer Partners rerouted the pipeline to cross the river a half mile north of the Standing Rock Sioux reservation, a move the tribe has taken as adding hazardous insult to historical injury.
Increasingly, the protectors' efforts have been met with violence from law enforcement and private security. Construction in the area began in August; by early September, Energy Transfer Partners had bulldozed a recently discovered ancient indigenous burial site. In response, protests increased in frequency and scope, culminating in private security's use of attack dogs against water protectors in early September. In response to outrage over the violence, the Obama administration requested that the pipeline company voluntarily halt construction until federal agencies could conduct a more thorough review of the project's proposed route, with an eye toward cultural sensitivity and environmental impact.
Energy Transfer Partners, however, ignored the request and accelerated the pace of construction, leading to more frequent, and more violent, clashes with water protectors in the ensuing months, despite an investigation by the UN and a visit from Amnesty International concerning human rights violations.
Now, on Halloween, there are plans for trick-or-treating for the children who attend school in a large green army tent in the center of camp; meanwhile, its residents are still recovering from the largest and most brutal confrontation thus far.
As construction approached the existing camp, water protectors erected the smaller 1851 Treaty Camp (named for the original Fort Laramie Treaty of 1851, which declared the land now in question part of the Great Sioux Nation — land that was never legally ceded by the tribe) nearby, directly in the path of the pipeline's construction, on Energy Transfer Partners land, in late October.
Just days after the camp's inception, on October 27, over 140 water protectors were arrested there during a police sweep. Bean bags were fired at their legs, rubber bullets at their heads and at their horses (one of which died); beatings were delivered with police batons, Tasers deployed, pepper spray and mace squirted in their faces, concussion grenades lobbed into their midst, a sound cannon aimed at their front lines, and protectors dragged out of prayer circles and sweat lodges during a sacred ceremony.
Just two days from my Halloween day at the camp, on November 2, water protectors by the dozens will wade, swim, and paddle across the Cannon Ball River to attempt to access a sacred burial site atop a nearby hill on Army Corps land. They will go there with the intention of praying, but they will met with brute force by law enforcement armed with familiar weapons. Over 100 will be treated by the camp's inundated medics for their injuries. But on Halloween, the camp is quiet as the water protectors catch their breath.
Sage Robertson, a soft-spoken man of mixed tribal ancestry whose Ojibwa name, Bizaaniigawbaw, means "Quietly Standing Still," has arrived back to camp today, his daughter's birthday. He was arrested on October 27, he tells me, on charges of engaging in a riot, conspiring to commit arson, and resisting arrest.
"We were up there at the [1851] Treaty Camp in prayer," he tells me. "Our intentions were always prayer." When hundreds of police stormed the camp, he says, "They were in full-fledged riot gear. We were unarmed." Robertson was helping to hold the front line as the police advanced, he says, "literally jumping in front of people who were being targeted by the cops." He says that at one point, he saw a young girl being grabbed by police. "I jumped in and did an angry war whoop, and I could see in the cop's face that he was scared. He let go of her."
When he saw a prayer circle being formed by women and elders, he joined by sitting down and locking arms with them. "It was the locked arms, I think, that was considered resisting arrest," he says. He was thrown to the ground on his stomach and his wrists were zip-tied so tightly that he tells me his thumb is still numb from the cut-off circulation. He was never read his rights. When he arrived at the Morton County jail for processing, he was strip-searched and then kept in a dog kennel for around an hour before being sent off to Stanton County, where he was held for three days before the camp's legal team was able to post bail. (The camp's legal defense fund had spent over $300,000 on bail for arrestees as of Halloween.) Robertson's arrest number, 227, was written on his arm in black Sharpie and is still visible, along with part of the badge number of his arresting officer.
Despite his arrest and inhumane treatment, however, Robertson is back at Oceti Sakowin Camp, and he intends to remain there. "How long is the world going to watch?" he asks. "This is where we stop it, right here. We're protecting water and life."
Even with the arrested making their way back to camp, water protectors remain on high alert. Many of those I speak to are wary of my press badge due to rumors that infiltrators from Energy Transfer Partners have posed as media in the camp, and have been responsible for violence on the part of the resistance during clashes with police. Some protectors tell us that the people who set the fire on the bridge and threw rocks during the October 27 incident weren't familiar faces.
Stuart American Horse, a Standing Rock native, remembers a car full of strangers who stayed in camp for a time as press. As they prepared to leave, he says, they yelled, "Fucking Indians!" and took off toward the police lines. A canoe was found only that morning, abandoned by the shore of the river, and many in camp suspect that a mole hired by Energy Transfer Partners was sent to find out what actions are being planned, or to incite violence at the next meeting of police and water protectors.
Heightening the stress of recent events, a small propeller plane has been noisily circling the camp continuously since before dawn on the 31st, joined eventually by two helicopters that fly low overhead. They'll continue to fly over the encampment throughout the night, edging away at the water protectors' ability to sleep, as do the bright lights that shine from the construction site. By the night of November 2, these will be joined by floodlights aimed down at the camp from the surrounding hilltops. Around midday on Halloween, official-looking SUVs and the tiny forms of their faraway occupants begin to appear on the surrounding hilltops, clearly surveilling the camp.
Late that afternoon, as I'm watching a group of young trick-or-treaters in Halloween garb, a young man named Myron tells me that he and four companions just got back from swimming across the river to a hill to the northeast, where a burial site is located. They found a cadre of police armed with AR-15s watching the camp. "We told them to get out of that sacred site," he said. "It was just four of us, and we were just in our swim trunks!" Their near-nakedness and the almost-freezing chill of the water notwithstanding, it appears to have worked — a glance at the skyline now reveals none of the SUVs or surveillance teams I saw before.
From the top of the hill at Sacred Stone Camp, situated on the south side of the river, I can see sections of pipeline laid out to the north, waiting to be put into the earth. The Dakota Access Pipeline in this area is being laid parallel to an existing natural gas pipeline called the Northern Border Pipeline, which Stuart American Horse says was constructed in this area in the 1970s, before 1978's American Indian Religious Freedom Act allowed indigenous people in the United States access to religious sites and ceremonies. "They bulldozed right through our sacred sites back then," he says, "but nobody could do anything about it" because of existing laws that prohibited indigenous people from practicing many of their most sacred rites.
American Horse grew up on the Standing Rock Sioux reservation, except for the years he spent in one of the notorious boarding schools where Native children were forced to assimilate to white culture. There, he tells me, he was beaten repeatedly. "They would take us into a little room and just beat the shit out of us," he says, squinting into the sun. But aside from these early memories, he says, he lived a bucolic childhood on his family's land near where the current conflicts are taking place.
"I have so many memories of this land, of swimming in the river, of hunting, of picking sage," he says. "I'm fighting for my kids and their kids. They deserve to live that way, to swim and to hunt." American Horse has been one of few native water protectors who's happy to speak to the press, hoping that he can get the word out to the world. "If you write anything," he tells me, "write that we need people here, on the ground. They keep arresting people and taking them off to jail, but they keep coming back. But we need more people here to fight for the water."
Since the events of October 27, new water protectors have been arriving from around the country and the world, eager to help. Sage Robertson tells me that in the few days he's been gone since his arrest, the camp has already grown considerably. Many of the allies I speak to at the camp have arrived in the past several days, galvanized to action by the violence they saw in the news.
Awareness of the events at Standing Rock, though slow on the uptake, has spread like wildfire in late October and early November. Millions "checked in" to the Standing Rock reservation on Facebook in an effort to show solidarity with the cause and to confuse any police surveillance efforts. Protesters in New York City shut down Grand Central Station before marching on the headquarters of major banks. National media has begun to take notice. And President Obama has personally addressed the situation, saying that his administration is investigating possible ways to reroute the pipeline. Neil Young has released a new song, "Indian Givers," about the situation, as have numerous younger artists like Tech N9ne and RedCloud.
In the meantime, hope and passion are sustaining the water protectors as they face down the long, freezing winter months on the prairies of North Dakota, where temperatures drop well below zero regularly.
The camp's art tent is busy making banners, posters, and silkscreens to send out with water protectors to the front lines. Francisco Ormaza, a student who's heading up the art-making process for the camp, says, "The media tries to dehumanize us, but when we have beautiful art, it shows our humanity. Art shows the heart of a human"—even when that human is being beaten by police batons.
The numerous kitchens around the camp provide three healthy meals a day to everyone at camp who needs food, free of charge. The medical and herbal healing tents continue to operate, along with the school. While I'm there, a truck arrives bearing 200 army surplus sleeping bags, which I help to unload along with dozens of other allies from around the country and the world. Meanwhile, a dome is being constructed as a central meeting place for the camp as winter sets in, and training is being offered on the construction of yurts, since several dozen are expected to arrive shortly. Occasional shouts of "Mni wiconi!" ("Water is life!") break out and echo around the camp.
As the trick-or-treaters finish their rounds and night closes in, the bodies of two bison are brought into camp from the tribe's nearby bison range. After elders have sung and prayed over them, water protectors go to work butchering them. An elder woman arrives on the scene as the organs are being divvied up, just in time to be sure that everything is treated properly, according to tradition. Nothing will go to waste, and the meat will be distributed to the hundreds of water protectors, fueling them in the difficult days to come.
The gruesome yet awesome spectacle of the animals' splayed ribcages, their innards strewn across the ground, steaming into the dark night of the new moon, is appropriate for All Hallow's Eve, but the singing, praying, and drumming that go on well into the wee hours of the morning are balm to campers' nerves. The pending violence of November 2 is still a thing of the future, but plans are already being laid for what will come next, and the camp is standing strong.
Justice Fire, a bipolar-suicide-prevention counselor from New Jersey, calls the energy of the camp "soul-evolving." "I've been an activist my whole life," she says, "and when I was here last time, I felt like this camp is an example of what [humans] could be."
Dan Hein, an ally from Columbus, Ohio, who was arrested on October 10 but who came straight back to camp and has been helping ever since, tells me, "I'm here because I think this is the beginning of the new world. And I think the new world will be led by indigenous people."
Lynsey G
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McIver mocked for blaming NDP over McDonald's self-service kiosks
February 18, 2016 Economy, Front Page
Alberta Tory interim leader Ric McIver has become a target of derision after blaming Alberta's NDP government's minimum wage policy for McDonald's installing touchscreen self-service order kiosks across the province.
"4 less AB jobs due to #NDP minimum wage policy," the Calgary-Hays MLA tweeted with a photo of some kiosks.
4 less AB jobs due to #NDP minimum wage policy. #ableg pic.twitter.com/Z2v8z7qWJT
— Ric McIver (@RicMcIver) February 17, 2016
"Actually I covered this launch for CTV," CTV Edmonton's Kim Taylor tweeted. "McDonald's says they hired 10-15 MORE per store: greeters and table service."
"Roving gangs of meth-withdrawl former rig workers maraud large swaths of the countryside, while unemployed hordes shiver in the wintery desolation of the province's former large cities, now depopulated," Redditor philwalkerp wrote sarcastically. "Once-proud oil barons tear their robes in frustration at the crushing and oppressive regime's policies responsible for driving the price of oil to record low levels. Refugees flood into BC and Newfoundland. Hope is lost. I know this because I've been reading Conservative blogs."
"Oddly enough, eThor – a world leader in this technology – was founded right here in Calgary," angel investor Brian F. Singh tweeted.
The NDP government plans to raise Alberta's minimum wage to $15 an hour by 2018.
"By keeping Alberta's minimum wage as low as possible, many people and their families had no choice but to resort to food banks and social support programs to make ends meet," Labour Minister Lori Sigurdson said last month. "Paying people a decent minimum wage will translate into a better life."
McIver, a staunch supporter of the temporary foreign worker program, vehemently opposes increasing the minimum wage claiming that Premier Rachel Notley is driven by ideology and dogma into doing so.
"The NDP's plan to increase the minimum wage to $15/hr, which is about one and a half times of what it is now, is completely ideologically and dogmatically driven," McIver said. "It will not do what they say it will do, in fact it will probably do the exact opposite of what they say it will do."
ableg, alberta, mciver, notley
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Comoran Vice-President Ikililou Dhoinine voting. Mr Dhoinine won most of the votes in Moheli's first election round.
afrol News, 11 November - Three candidates for the Comoros Union presidency elections on Sunday have emerged to compete in a second round in December. Ruling party candidate Ikililou Dhoinine received most votes in the first round.
The Independent National Electoral Commission of Comoros (CENI) today announced the provisional results of Sunday's elections. Mr Dhoinine gained 26.97 percent of the votes; Mohamed Said Fazul came in second with 21.65 percent. Bianrifi Tarmidi came in third, receiving 10.90 percent of votes.
According to local and international election observers, the first election round went along in an orderly and well organised way. The poll marks the end of a period with political turmoil in Comoros, starting as outgoing President Ahmed Abdallah Sambi unilaterally declared an extension of his presidential term.
Following decades of political instability due to coups and secessionist movements, the three islands forming the Comoros Union introduced a complicated electoral system, granting each of the islands Gran Comore, Anjouan and Moheli a rotating access to the union presidency.
At this junction, the island of Moheli - the smallest in the union - was set to take over the presidency. Therefore, the first round of the elections was only held on that island's 21,429 voters.
The three leading candidates emerging from the elections on Moheli Island are to meet voters from all the three islands in a run-off poll on 26 December. The one gaining most votes at that occasion - even if not reaching 50 percent - will become the next President of the Comoros Union.
However, the opposition was split among seven candidates in Moheli, with Mr Fazul gaining almost as many votes as the ruling party candidate. With only two opposition candidates standing in the run-off, Mr Fazul may have a great chance of uniting opposition votes, which are widely expected to be in a majority in Comoros.
Outgoing President Sambi from Anjouan island, also a Muslim religious leader, managed to deepen the impoverished archipelago's relations with the Arab world, also achieving giant development aid pledges from the Gulf states. During his presidency, the Comoran economy noted significant progress.
President Sambi however also was noted for an authoritarian style, jeopardising the fragile inter-island power-sharing agreement and as such the political stability in Comoros. During his presidency, Union authorities managed to override the autonomous governments of the three islands, transferring powers to the central authority.
His term at the presidency officially ended on 26 March this year, but he extended it for one year through a referendum. The referendum was boycotted on Moheli island, which demanded a Mohelian taking on the presidency, in accordance with the rotation principle.
Massive protests on Moheli, threats of yet another secessionist movement and pressure from the African Union (AU) led to President Sambi backing down. According to the agreement that led to Sunday's election on Moheli, President Sambi is to hand over the Comoran presidency to a Mohelian "no later than the second half of January 2011."


Our Seagull magnet brings a bit of seaside right inside to brighten up any kitchen. Crafted from sturdy stone resin and featuring a lovely, folksy painted beach scene, the Seagull magnet will draw the eye of any bird lover, beach goer or fan of the shore. No friend of the outdoors can call his or her collection of fridge magnets complete without one of these colorful 2 x 2 inch Seagull magnets holding up a note, grocery list or child's drawing.


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The Horn News
Storms unearth hidden bones on New York island (creepy!)
Storms and the tides are unearthing the long-hidden bones of Hart Island, creating eerie scenes of skulls, femurs and collarbones on this sliver of land where New York City's destitute dead have for 150 years been sent off to be unceremoniously buried and forgotten.
After photos of exposed bones began turning up in news reports, forensic anthropologists from the city medical examiner's office went out last week and collected 174 human bones that they carefully cataloged, including six skulls, six jawbones, 31 leg bones and 16 pelvises. Small red flags dotted spots along the rocky shoreline where some remains were found.
"When I hear about the erosion, I always think, 'Are the bones his? Could any of them be his?'" asked Carol DiMedio whose grandfather Luigi Roma was buried on the island after dying from tuberculosis in 1933.
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Advocates for Hart Island say the bones are a jarring sign that it's long past time for improvements to be made. In addition to stepping up a $13.2 million federal project to repair erosion caused by 2012′s Hurricane Sandy and other storms, they want the 101-acre island in the Long Island Sound to be turned into a park and historic site, even if it continues to be used as a burial ground for the city's poor and nameless dead.
"These are New Yorkers," City Council member Mark Levine said of the remains. "These are human beings who were largely marginalized and forgotten in life, they were people who died homeless or destitute, victims of contagious disease, the AIDS crisis. And we're victimizing them again in their final resting place."
As many as 1 million souls lie buried on Hart Island, purchased by the city in 1868 as land for a workhouse for wayward boys and a potter's field. Over the decades, it housed a Civil War prison, an asylum, a tuberculosis hospital, a jail and a missile base. All the while, New York continued ferrying bodies there that went unclaimed at the city morgue.
The island, managed by the city's Department of Correction for more than a century, has never been kept up like a traditional cemetery with manicured lawns or even headstones.
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Prisoners dig the long trenches used to bury the dead. Adults go into the ground in pine boxes, stacked on top of one another. Babies are placed in shoebox-sized containers. Around 1,000 people are buried there each year.
The internments take place out of public view and access to the island is limited to trips, once per month, that must be reserved in advance. Only people with loved ones buried on the island are allowed to visit the graves . Others are restricted to a visitors' gazebo near the ferry dock.
In 2012, one of the island's embankments was stripped of earth by Sandy's floodwaters, exposing a Civil War-era cistern. Meanwhile, more storms have battered the island, further eroding it. Work on the federal project to repair the erosion wasn't scheduled to begin until 2020 but it has been moved up a year.
"We are doing everything we can to limit future exposures and expedite the firming up of the shoreline," said Jason Kersten a correction spokesman.
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In the meantime, city officials say an archaeologist will visit at least once a month to remove and bury any exposed remains.
Unearthed remains have been a problem before. A March 1985 report by the city's sanitation department found bones strewn on the island — including a skull on the beach.
"It was explained to us that this was a rather common thing since the city has been burying bodies there for almost 80 years and the water has caused some erosion at older burial spots," according to the report obtained by island advocate Melinda Hunt.
On a recent photography trip around the island on a chartered boat, Hunt pointed out the places along the northern shore where the bones were, and where she has seen them before.
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"They came to clean this up, but it isn't the first time and it won't be the last," said Hunt, creator of the Hart Island Project , who has been documenting conditions on the island since 1991.
DiMedio, 61, said she wishes the uncovered bones would be tested for DNA evidence in order to help other families locate missing loved ones.
She had done research since she was a young girl to try to find out where her grandfather was buried. And when she finally found the answer, she didn't want to tell her ailing, elderly mother where her father had been laid to rest.
"When I found him … I lied and I said he was buried in a beautiful place, with blue water and blue skies and lots of trees and green grass and seagulls above," DiMedio said. "I didn't have the heart to tell her there's this grim place called Hart Island."
The Associated Press contributed to this article.
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\section{Introduction}\label{sec:intro}

Understanding the impact of correlations on many-body quantum systems constitutes
a major prerequisite for designing and controlling quantum matter. In these regards,
ultracold atoms provide a highly flexible platform due to their isolatedness
from an environment
and their controllability, which allows for realizing a great variety of physical situations
(variable trap geometries and interactions)
and thereby for studying many-body effects in a systematic, unprecedented manner \cite{Pethick_Smith2008,many_body_physics_bloch,atoms_lattice_quantum_sim__book2012}.
At the same time, this flexibility constitutes a challenge for theoretical
approaches due to the wide range of particle numbers and interaction strengths involved.

While there has been tremendous progress in developing highly advanced wavefunction
propagation method such as the time-dependent tensor-network approaches \cite{DMRG_at_age_of_MPS_Schollwoeck2010,out_of_equil_dynamics_with_MPS_Carr_NJP_2012}
or the Multi-Configuration Time-Dependent Hartree method for Bosons (MCTDHB) \cite{MCTDHB_PRA08},
describing an ultracold gas of thousands or even hundred-thousands of atoms with the
inherent correlations taken accurately into account is not within the immediate reach of methods aiming at the full
many-body wavefunction. In such situations, an effective description of relevant subsystems of the complete
many-body system appears to be much more appropriate. Here, we consider such a
description of reduced density operators corresponding to few-particle subsystems of an
ultracold bosonic
quantum gas by means of the appropriately truncated Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations of motion
in its quantum version
\cite{bogoliubov65,born_green47,kirkwood46,yvon57,qua_kin_theo_bonitz}.

While there are numerous theoretical works on the BBGKY hierarchy and
its truncation (see the references in \cite{BBGKY_1} as well as e.g.\ \cite{qua_kin_theo_bonitz}
for an overview), the literature on the accuracy and stability
of the truncated BBGKY equations of motion (EOM) in dependence on the truncation order by
explicit numerical simulations
is - to the best of our knowledge - limited \cite{akbari_challenges_2012,jeffcoat_n-representability-driven_2014,schuck14,schuck17,lackner_propagating_2015,lackner_high-harmonic_2017,elliott_density-matrix_2016}.
Actually, most studies deal with fermions (for bosons, see \cite{kira_excitation_2014,kira_coherent_2015,kira_hyperbolic_2015}
 as well the BBGKY-related approaches
\cite{kohler_microscopic_2002,kohler_microscopic_2003,vardi_bose-einstein_2001,anglin_dynamics_2001,tikhonenkov_quantum_2007,trimborn_decay_2011,witthaut_beyond_2011,kordas_dissipative_2015-1})
and are based on the truncation of the BBGKY hierarchy after the second order.

For this reason, the purpose of this work is to comprehensively study the accuracy of
BBGKY simulations of the non-equilibrium dynamics of finite ultracold bosonic
ensembles when systematically increasing the truncation order in the framework
of a cluster-expansion based closure approximation. While the immediately preceding work \cite{BBGKY_1} covers
the underlying
theory and its efficient formulation, which allows for going to large truncation orders, we are
here solely concerned with evaluating its accuracy and stability by numerical applications in two physically very transparent
scenarios. The results for various truncation orders $\bar o$
are compared to  the MCTDHB simulations for the full $N$-particle wavefunction.

The first scenario is concerned with the tunneling dynamics
of bosonic atoms in a double-well potential. Treating
the system in the lowest-band tight-binding approximation allows us to
go to large truncation orders without the need of dynamically optimizing the
single-particle basis via the corresponding MCTDHB EOM \cite{MCTDHB_PRA08}.
Thereby, we probe
solely effects stemming from the truncation of the BBGKY hierarchy. Here, we
find the short-time dynamics to be excellently described by the BBGKY hierarchy
and the accuracy to increase monotonously with increasing truncation order.
For longer times, strong deviations are observed, which are linked to $N$-particle
correlations becoming dominant as well as exponential instabilities of the
truncated BBGKY EOM resulting in unphysical solutions.
The phenomenology of these instabilities is analyzed in detail and we show that the
minimal-invasive correction scheme for the BBGKY EOM (truncated at
order $\bar o=2$) \cite{BBGKY_1} can stabilize these EOM indeed.

In the second scenario,
we consider a harmonically trapped bosonic ensemble subjected to an interaction
quench from the non-interacting regime to finite inter-particle repulsion.
Here, we use the fully coupled system of the BBGKY EOM (6)
of \cite{BBGKY_1}
and the MCTDHB EOM (5)
of \cite{BBGKY_1} for the time-dependent, variationally optimized single-particle basis.
For low excitation energies, we find the system to be highly accurately described by the
truncated BBGKY approach. For higher excitation energies, however, exponential instabilities
again occur. Also in this case, we can stabilize the BBGKY EOM truncated at $\bar o=2$ by
our EOM correction scheme and obtain reasonably accurate results for longer times.

This work is organized as follows. While the main text is focused
on the applications, namely the tunneling scenario in Section \ref{sec:BH_dimer} and the
interaction-quench scenario in Section \ref{sec:breath}, technical comments on the numerical integration
of the truncated BBGKY EOM are provided in Appendix \ref{app:integration}.
Finally, we conclude in Section \ref{sec:concl}.
\section{Tunneling dynamics in a Bose-Hubbard dimer}\label{sec:BH_dimer}

In this scenario, we assume that $N$ bosonic atoms are loaded into an effectively
one-dimensional double-well potential. Preparing the system in an initial state featuring a
particle-number imbalance with a left and the right well allows for studying
the tunneling dynamics of such a many-body system, which has been subject of
numerous studies covering both mean-field \cite{raghavan_coherent_1999,smerzi_quantum_1997} and many-body calculations
taking correlations into account \cite{milburn_quantum_1997,exact_quantum_dynamics_Cederbaum_PRL2009,sakmann_quantum_2010,univ_frag_dw,gertjerenken_beyond-mean-field_2013}. Effects unraveled in such a realization
of a bosonic Josephson junction cover macroscopic tunneling and self-trapping \cite{raghavan_coherent_1999,smerzi_quantum_1997,direct_obs_boson_JJ}
as well a decay of tunneling oscillations due to the dephasing of populated
many-body eigenstates of the post-quench Hamiltonian \cite{milburn_quantum_1997,exact_quantum_dynamics_Cederbaum_PRL2009,sakmann_quantum_2010,univ_frag_dw,gertjerenken_beyond-mean-field_2013}.

For sufficiently deep wells, the microscopic many-body Hamiltonian of this
system can be well
approximated by a two-site Bose-Hubbard Hamiltonian within the
lowest-band tight-binding approximation
\begin{align}
\label{eq:BH_hamilt}
 \hat H = &-J(\ad{L}\a{R}+\ad{R}\a{L})\\\nonumber
 &+\frac{U}{2}\big[\hat n_L(\hat n_L-1)
 +\hat n_R(\hat n_R-1)
 \big]
\end{align}
where $\a{L/R}$ annihilates a boson in the lowest-band Wannier state localized in
the left/right well and $\hat n_{i}\equiv\ad{i}\a{i}$ denotes the corresponding
occupation-number operator of the site $i\in\{L,R\}$. The first
term in \eqref{eq:BH_hamilt} describes tunneling between the two wells
weighted with the hopping amplitude $J>0$. The second term refers to
on-site interaction of strength $U$ and stems from the
short-range van-der-Waals interaction between the atoms. For convenience,
we take the hopping amplitude as our energy scale and state times
in units of $1/J$.

The Bose-Hubbard dimer features an almost trivial computational complexity since
the full many-body wavefunction depends only on $C^N_2=N+1$ complex-valued
coefficients such that the corresponding time-dependent Schr\"odinger equation can be
numerically exactly solved for very large atom numbers. So there is no
need for an alternative computational approach here. On the other hand,
this system can serve as a good playground for analyzing the properties of the
truncated BBGKY approach because (i) the corresponding numerically exact solution
is available and (ii) we can easily represent RDMs of large order without using
a dynamically optimized truncated single-particle basis. This allows for
systematically investigating the accuracy of our results solely in dependence on
the truncation order $\bar o$.

In the following,
we consider the initial state $|\Psi_0\rangle=|N,0\rangle$ with all atoms located in the
left well and focus on the tunneling regime by setting the dimensionless interaction parameter
$\Lambda=U(N-1)/(2J)$ to $0.1$, i.e.\ well below the critical value $\Lambda_{\rm crit}=2$
for self-trapping \cite{raghavan_coherent_1999,smerzi_quantum_1997}. In the weak interaction
regime $\Lambda\ll1$, beyond mean-field effects such as
the aforementioned collapse of tunneling oscillation \cite{milburn_quantum_1997} and the universal formation of
a two-fold fragmented condensate out of a single condensate \cite{univ_frag_dw}
are expected to
play a significant role after the time-scale $t_{\rm mf}\approx\sqrt{2N+1}/(J\Lambda)$,
the so-called quantum break time
\cite{gertjerenken_beyond-mean-field_2013}.

Most of the following calculations deal with $N=10$ atoms such that $t_{\rm mf} \approx 46/J$.
For comparison, we also increase $N$ to $100$ atoms while keeping
$\Lambda$ constant, which results in a longer quantum break time of $t_{\rm mf}  \approx 142/J$.
We analyze the accuracy of the truncated BBGKY hierarchy approach in three steps. First,
we inspect the particle-number imbalance, a highly-integrated quantity
characterizing the tunneling dynamics, second turn to the eigenvalues of the lowest-order RDMs, which constitute a highly
sensitive measure for correlations, and third compare the whole
lowest order RDMs to the corresponding exact results. For a deeper interpretation
of these findings, we finally analyze the exact results for the
whole $N$-particle wavefunction as well as for the corresponding $o$-particle
correlations. Finally, we investigate the performance of the
correction strategies outlined in Section V B
of \cite{BBGKY_1}.

\subsection{Particle-number imbalance}

\begin{figure}[t]
\centering
 \includegraphics[width=0.495\textwidth]{imbalance_N10_no_purif.pdf}
 \caption{(color online) Time evolution of the particle-number imbalance $(N_L-N_R)/N$
 with $N_i\equiv\langle\hat n_i\rangle$, $i\in\{L,R\}$ for various
 truncation orders $\bar o$. Inset: numerically exact solution of the many-body Schr\"odinger
 equation. Parameters: $N=10$ atoms located initially in the left
 well, dimensionless interaction parameter $\Lambda=0.1$.}
 \label{fig:BH_imbalance}
 \end{figure}

In order to study the tunneling dynamics, the imbalance of the particle
numbers between the left and right well, $[\langle\hat n_L\rangle-\langle\hat n_R\rangle]/N$,
is depicted in Figure \ref{fig:BH_imbalance} for $N=10$ atoms.
Focusing first on the inset, which shows the numerically exact results,
we see the expected collapse of tunneling oscillations due to a dephasing
of the populated post-quench Hamiltonian eigenstates. Indeed, this collapse
happens on the time-scale $t_{\rm mf} \approx 46/J$, while a
corresponding Gross-Pitaevskii
mean-field simulation would reveal undamped tunneling oscillations (not shown,
see e.g.\ \cite{gertjerenken_beyond-mean-field_2013}). After $t\sim 200/J$, a
revival of the tunneling oscillations emerges in the numerically exact
calculation (not shown).

Turning now to the truncated BBGKY approach, we see that all truncation orders
$\bar o\geq2$ give good results for the first $\sim 8$ tunneling oscillations.
Thereafter, the $\bar o=2$ curves departs from both the exact
and the higher truncation-order results, and features a premature maximal
suppression of tunneling oscillations at $t\sim50/J$. In
the subsequent premature revival of tunneling oscillations unphysical
values $|\langle\hat n_L\rangle-\langle\hat n_R\rangle|/N>1$ are reached
at about $t=100/J$, indicating a lack of $1$-RDM representability.

These findings suggest that higher-order correlations than $\hat c_2$ (see Section IV B 3
of \cite{BBGKY_1} for the definition) play a
significant role. Increasing the truncation order $\bar o$ stepwise up to
the maximally reasonable order $\bar o=N-1=9$,
we clearly see that the accuracy of our results improves systematically.
The larger $\bar o$ is, the more accurate is the collapse of the tunneling
oscillations described. However, all non-trivial truncations $\bar o<N$
predict a premature revival of the tunneling oscillations, which goes
hand and in hand with a maximal suppression of the tunneling-oscillation
amplitude to small but noticeable values (while the exact results
do not feature noticeable oscillations at the corresponding times).
We note that for $2<\bar o<10$ the simulations suffer from drastic instabilities
of the EOM,
being discussed in the subsequent Section, such that
we had to stop them after a certain time. This is why the corresponding
curves in Figure \ref{fig:BH_imbalance} are not provided for the
whole range of depicted times.

\subsection{Natural populations}
 \begin{figure}[t]
   \includegraphics[width=0.495\textwidth]{eval_o1_N10_no_purif.pdf}
   \caption{(color online) Natural populations of the $1$-RDM for various
   truncation orders $\bar o$. Inset: numerically exact solution of the many-body Schr\"odinger
 equation. Parameters: same as in Figure \ref{fig:BH_imbalance}.}
   \label{fig:BH_npop1}
\end{figure}
 \begin{figure}[t]
   \includegraphics[width=0.495\textwidth]{eval_o2_N10_no_purif.pdf}
   \caption{(color online) Natural populations of the $2$-RDM for various
   truncation orders $\bar o$. Inset: numerically exact solution of the many-body Schr\"odinger
 equation. Parameters: same as in Figure \ref{fig:BH_imbalance}.}
   \label{fig:BH_npop2}
\end{figure}

Next we analyze the eigenvalues $\lambda^{(o)}_i$  of the $o$-RDM, called natural populations\footnote{The terms 'natural population' and
'natural orbital' have originally been introduced for the 1-RDM only \cite{loewdin_norb55}
but are employed for all orders in this work.} (NPs) in the following and
start with the NPs of the $1$-RDM NPs in Figure
\ref{fig:BH_npop1}, which can diagnose beyond mean-field behavior.
The numerically exact results (see corresponding inset)
reveal dynamical quantum depletion leading to
a two-fold fragmented condensate for $t\gtrsim 80/J$ with almost equal population
of the corresponding NOs, $\lambda_1^{(1)}\approx0.5\approx\lambda_2^{(1)}$ (see also e.g.\
\cite{univ_frag_dw}). Strikingly fast oscillations in these NPs emerge and decay
around $t\sim140/J$, which we can connect to the periodical emergence and decay
of a NOON state of the total system (see below).

The corresponding results of the truncated BBGKY approach feature a similar
dependence on the truncation order $\bar o$ as the particle-number imbalance does.
While the $\bar o=2$ prediction starts to deviate noticeably from the exact results already
for $t\gtrsim25/J$, we obtain trustworthy results for a longer time, the larger
$\bar o$ is chosen. In particular, the truncated BBGKY approach can accurately
determine the achieved mean degree of fragmentation
(see $\bar o=8,9$ results at $t\sim100/J$). Even for the largest truncation order $\bar o=9$, however,
the truncated BBGKY simulations
predict a premature and very fast
revival of condensation (i.e. $\lambda_1^{(1)}\approx 1$), while
this process starts only after $t\sim 200/J$ in the exact calculation and happens
more slowly (not shown).
Most importantly, this unphysical fast re-condensation overshoots the
range of valid NPs such that the $1$-RDM ceases to be positive semi-definite, indicating
an exponential-like instability of the EOM.

While we have so far only studied the prediction of the truncated BBGKY approach
for one-particle properties,
we now inspect the NPs of the $2$-RDM in Figure
\ref{fig:BH_npop2}, also called
natural geminal populations \cite{rdm_Sakmann_PRA2008}.
The exact dynamics (see the inset) features two important aspects, which we have
also observed for the NPs of higher-order RDMs (not shown). (i)
The dominant NP $\lambda_1^{(2)}$ first loses weight in favor for
the other NPs. (ii) At about $t\sim140/J$, all NPs are suppressed except for
$\lambda_1^{(2)}\approx 0.5\approx\lambda_2^{(2)}$. Having observed
the latter feature for the NPs of all orders $o\in\{1,...,9\}$, we may conclude that
in this stage of the dynamics a subsystem of $o$ particles occupies approximately
only two $o$-particle states with almost equal probabilities. As we will see below,
this finding is caused by the periodical emergence and decay
of a NOON state of the total system which is discussed below.

Turning now to the predictions of the truncated BBGKY approach, we
see again a systematic improvement of accuracy with increasing truncation
order $\bar o$. The maximal time for which the highest truncation order
$\bar o=9$ gives reliable results, however, has reduced from $t\sim110/J$
for the $1$-RDM NPs (see Figure \ref{fig:BH_npop1}) to
$t\sim70/J$ for the $2$-RDM NPs (see Figure \ref{fig:BH_npop2}). Thereafter,
the largest NP $\lambda_1^{(2)}$ is well described until $t\sim130/J$,
while the other two NPs already show strong deviations: it seems that the emergence of
the feature (ii) discussed above happens premature, name at about $t\sim120/J$.
Furthermore, we also witness the exponential-like instabilities leading to
$2$-RDM NPs outside the interval $[0,1]$.

In order to analyze how this unphysical behavior emerges, we depict the first
time $t_{\rm neg}(o)$ when the lowest $o$-RDM NP $\lambda^{(o)}_{o+1}$
is smaller than the threshold $\epsilon=-10^{-10}$ for various
$o$ and different truncation orders $\bar o$ in Figure \ref{fig:BH_neg_t} a).
For fixed truncation order $\bar o$, $t_{\rm neg}(o)$ decreases
with increasing order $o$. This means that the representability defect
of $\hat\rho_o$ lacking positive semi-definiteness starts at the
truncation order $o=\bar o$ and propagates then successively to
lower orders due to coupling via the collision integral. For most
orders $o$, we moreover find that $t_{\rm neg}(o)$ increases with
increasing truncation order $\bar o$, which fits to the above findings regarding
enhanced accuracy for larger $\bar o$ (exceptions occur at order $o=1,2$  in particular
for $\bar o=2$).

Increasing the number of atoms to $N=100$ while keeping the dimensionless interaction parameter
$\Lambda=0.1$ constant, we again find a monotonous decrease of $t_{\rm neg}(o)$ with increasing $o$
for fixed truncation order $\bar o$ (see Figure \ref{fig:BH_neg_t} b)). This confirms
the above finding that the lack of positivity successively propagates from higher to lower orders.
In contrast to the $N=10$ case, we only find an enhancement of $t_{\rm neg}(o)$ with increasing
$\bar o$ for orders $o\geq6$. In particular, we see that the largest truncation order considered,
$\bar o=12$, features the smallest $t_{\rm neg}(o=2)$. It is
quite possible that the an ``enhancement'' of non-linearity with increasing truncation
order $\bar o$ (note that the applied closure approximation, cf.\ Section IV B 3
of \cite{BBGKY_1}, is a polynomial of degree $(\bar o+1)$
in $\hat\rho_1$ and of degree $\lfloor(\bar o+1) /o\rfloor$ in the cluster $\hat c_o$) is the
reason why the BBGKY EOM are more prone to these instabilities for larger $\bar o$.

Having compared so far only certain aspects of $o$-particle properties,
we finally aim at comparing the prediction of the truncated BBGKY approach for
the whole $o$-RDM to the exact results.

\begin{figure*}[t]
 \includegraphics[width=0.495\textwidth]{negTimes_no_purif_N10.pdf}
 \includegraphics[width=0.495\textwidth]{negTimes_no_purif_N100.pdf}
 \caption{(color online) First time $t_{\rm neg}(o)$ when the lowest $o$-RDM NP is smaller than $\epsilon=-10^{-10}$
 in dependence on $o$ for various truncation orders $\bar o$.
 a) Same parameters as in Figure \ref{fig:BH_imbalance}. b) Same as a) but for the atom number $N$ increased to $100$
 while keeping the interaction parameter $\Lambda=0.1$ constant. The results for $\bar o=2$ are not plotted in b)
 and read $t_{\rm neg}(1)\approx216/J$ as well as  $t_{\rm neg}(2)\approx168/J$. }
 \label{fig:BH_neg_t}
\end{figure*}

\subsection{Reduced density operators}

\begin{figure*}[t]
 \includegraphics[width=0.495\textwidth]{rdm_comp_N10_no_purif.pdf}
  \includegraphics[width=0.495\textwidth]{rdm_comp_N100_no_purif.pdf}
 \caption{(color online) Time evolution of the trace distance $\mathcal{D}(\hat\rho_o^{\rm tr},\hat\rho_o^{\rm ex})$
 between the exact result and the truncated BBGKY prediction for the $o$-RDM ($o=1,...,4$) and various truncation
 orders $\bar o$. The dotted horizontal lines at unity ordinate value indicate the upper bound for the trace distance
 between two density operators (see main text).
 a) Same parameters as in Figure \ref{fig:BH_imbalance}. b) Same as a) but for the atom number $N$ increased to $100$
 while keeping the interaction parameter $\Lambda=0.1$ constant.}
 \label{fig:BH_rdm_comp}
\end{figure*}

For this purpose, we
take the trace distance
$\mathcal{D}(\hat\rho_o^{\rm tr},\hat\rho_o^{\rm ex})\equiv||\hat\rho_o^{\rm tr}-\hat\rho_o^{\rm ex}||_1/2$
\cite{nielsen_chuang_book} as a measure for deviations between the
truncated BBGKY prediction for the $o$-RDM denoted by $\hat\rho_o^{\rm tr}$ and the
numerically exact result $\hat\rho_o^{\rm ex}$. Here, $||\cdot||_1$ refers to the trace-class norm (also
called Schatten-1 norm) being defined as $||\hat A||_1\equiv {\rm tr}(\sqrt{\hat A^\dagger\hat A})$
for any trace-class operator $\hat A$. For hermitian operators $\hat A$, $||\hat A||_1$
equals the sum of absolute values of $\hat A$'s eigenvalues.
By means of $\mathcal{D}(\hat\rho_o^{\rm tr},\hat\rho_o^{\rm ex})$,
one can estimate the maximal difference for the expectation values of any (trace-class)
$o$-body observable $\hat A_o$ as follows $| {\rm tr}(\hat A_o\hat\rho_o^{\rm tr})- {\rm tr}(\hat A_o\hat\rho_o^{\rm ex})|
\leq 2||\hat A_o||_1\,\mathcal{D}(\hat\rho_o^{\rm tr},\hat\rho_o^{\rm ex})$, whose
proof is reviewed in Appendix \ref{app:exp_value_bound}. Moreover,
given that its arguments are density operators (i.e. hermitian, positive semi-definite
and trace one), the trace-distance is bounded by
$\mathcal{D}(\hat\rho_o^{\rm tr},\hat\rho_o^{\rm ex})\in[0,1]$ and
can be interpreted as the
probability that these two quantum states can be distinguished by the outcome of a single measurement
\cite{nielsen_chuang_book}.

In Figure \ref{fig:BH_rdm_comp}, we depict $\mathcal{D}(\hat\rho_o^{\rm tr},\hat\rho_o^{\rm ex})$
for the orders $o=1,...,4$ and various truncation orders $\bar o$, where subfigures a) and b) refer
to the $N=10$ and $N=100$ case with the same interaction parameter
$\Lambda=0.1$, respectively. For fixed truncation order $\bar o$, we clearly see that the accuracy of the
truncated BBGKY prediction for the $o$-RDM decreases with increasing order $o$. Up to a
certain time, which depends on the order $o$, we moreover find $\mathcal{D}(\hat\rho_o^{\rm tr},\hat\rho_o^{\rm ex})$
to decrease with increasing truncation order $\bar o$.

The instabilities of the truncated BBGKY EOM manifest themselves in the trace distant exceeding
its upper bound $\mathcal{D}(\hat\rho_o^{\rm tr},\hat\rho_o^{\rm ex})\leq 1$ for density operators,
implying that $\hat\rho_o^{\rm tr}$ lacks to have trace one or to be positive semi-definite. Since
the conservation of the initial RDM trace is ensured by the truncated BBGKY approach, violations
of $ {\rm tr}(\hat\rho_o^{\rm tr})=1$ can at most occur numerically if the system gets deep into the
exponential-like instability (where we observe the truncated BBGKY EOM to become stiff
such that the integrator has a hard time). Thus, exceeding the upper bound on the trace distance
is connected to a lack of positive semi-definiteness and can be observed to
happen earlier for increasing order $o$ and fixed truncation order $\bar o$.

For the case of $N=100$ atoms (see Figure \ref{fig:BH_rdm_comp} b)), we observe the additional
particularity that in the vicinity of $t\sim63/J$ the accuracy of the truncated BBGKY prediction
for the $o$-RDM does not depend on the truncation order $\bar o$, which happens slightly earlier
for larger $o$. Before this point, a systematic increase of accuracy is observed for increasing
truncation order $\bar o$. Thereafter, lower truncation orders give (slightly) better results than
higher ones. Furthermore, while in the $N=10$ case one-body properties (such as e.g.\ the particle-number
imbalance) can be described with reasonable accuracy up to $t\sim2\, t_{\rm mf}$ (when the collapse
of tunneling oscillations has already taken place), the instabilities hinders us to obtain
accurate results for $t$ larger than about $0.56\, t_{\rm mf}$ in the case of $N=100$ (at this time,
the tunneling oscillation amplitude is still significant).

\subsection{Many-body state and $o$-particle correlations: exact results}

\begin{figure*}[t]
 \includegraphics[width=0.7\textwidth]{cluster_s1norm_N10_plus_ns_no_truncation.pdf}
 \caption{(color online) a) Time evolution of the cluster's trace-class norm $||\hat c_o||_1$
 for all orders $o$, obtained from the numerically exact solution of the
 time-dependent Schr\"odinger equation. Inset: zoom into early time dynamics. b) and c) Probability
 to find $n$ atoms in the right and $(N-n)$ atoms in the left well, $|\langle N-n,n|\Psi_t\rangle|^2$, versus time
 for two characteristic stages of the dynamics. Parameters: same as in Figure \ref{fig:BH_imbalance}.}
  \label{fig:BH_cluster_dynamics}
\end{figure*}

In order to obtain physical insights in the above findings, we finally come back
to the numerically exact results for $N=10$ and measure the strength of $o$-particle correlations (see the
definition given in Section IV B 3
of \cite{BBGKY_1})
in terms of $||\hat c_o||_1$ in
Figure \ref{fig:BH_cluster_dynamics} a). From the inset, we infer that the correlations
initially build up in a hierarchical manner. First, only two-particle correlations  start to play a role,
then three-particle correlations and so on. This hierarchy in $||\hat c_o||_1$ holds, however, only until
$t\sim8/J$, when the ordering of the $||\hat c_o||_1$'s with respect to the
order $o$ starts to become reversed. After a certain point, $N$-particle correlations become the most dominant
ones.
This holds in particular in the vicinity of $t\sim140/J$, where we have observed fast oscillations
in the NPs $\lambda^{(1)}_{1/2}$ and found for all orders $o=1,...,9$ that
the RDMs feature approximately only two finite NPs $\lambda^{(o)}_{1}\approx0.5\approx\lambda^{(o)}_{2}$.
At this stage of the dynamics, all clusters $\hat c_o$ of odd order $o$ are strongly suppressed.

For connecting the above findings regarding $o$-particle correlations to the full many-body state,
we depict in the Subfigures \ref{fig:BH_cluster_dynamics} b)
and c) the probability $|\langle N-n,n|\Psi_t\rangle|^2$ of finding $n$ atoms in the right and $(N-n)$ atoms in the left well.
For the early dynamics, we witness how the system
becomes delocalized in the Fock space such
that the tunneling oscillations become suppressed (Subfigure \ref{fig:BH_cluster_dynamics} b)).
At later times, around
$t\sim140/J$, we, however, find the system to periodically oscillate between
a NOON state $(|N,0\rangle+e^{i\theta}|0,N\rangle)/\sqrt{2}$ (with some phase $\theta\in\mathbb{R}$)
and some broad distribution being approximately symmetric with respect to its maximum
at about $n=5$ (Subfigure \ref{fig:BH_cluster_dynamics} c)). Due to this approximate symmetry of the
distribution around $n=5$, the particle-number imbalance approximately equals
$[\langle\hat n_L\rangle-\langle\hat n_R\rangle]/N\approx0.5$, i.e.\ tunneling
oscillations are still suppressed. This approximate symmetry moreover leads to
a doubling of the oscillation frequency compared to the initial tunneling-oscillation frequency,
which is most probably linked to the fast oscillations in  $\lambda^{(1)}_{1/2}$.
Finally, one can analytically show  that
the $n$-RDM of the above mentioned NOON state reads $\hat\rho_n=(|n,0\rangle\!\langle n,0|+|0,n\rangle\!\langle 0,n|)/2$,
meaning that the state of an $n$-particle subsystem is an incoherent statistical mixture with
all particles residing in the left (right) well with probability $0.5$.
Thereby, we can directly connect the fact that the RDMs of all orders feature approximately only two finite
NPs of approximately equal value to the underlying many-body state. Coming back to the findings for  $||\hat c_o||_1$
of Figure \ref{fig:BH_cluster_dynamics} a), we may conclude that a NOON state
leads to strong high-order correlations $\hat c_o$ such that
truncating the BBGKY hierarchy by means of the applied cluster expansion
cannot be expected to give accurate results.

In summary, we have seen following.  (i) While the truncated BBGKY approach gives highly accurate
results for short times with controllable accuracy via the truncation order $\bar o$,
 the BBGKY approach
shows deviations at longer times. (ii) Exponential-like instabilities, induced
by the non-linear truncation approximation, propagate form high to low orders
and lead to unphysical results at a certain point.
(iii) $o$-particle correlations arise very fast
in this tunneling scenario and soon cease to be in decreasing order with respect
to $o$. (iv) The system evolves into a NOON state being dominated by $N$-particle
correlations.

There appear to be at least two plausible causes why the BBGKY approach fails at a certain point:
First, the number of terms in the cluster expansion
(23)
of \cite{BBGKY_1} drastically increases with the order $o$,
which implies that clusters should decay fast for a
controllable approximation. For example, at the largest truncation order
considered above, $\bar o=12$, the truncation approximation $\hat\rho_{13}^{\rm appr}$
already involves $100$ terms. Our findings (iii) and (iv), however,
might indicate that this system is not suitable for
a truncation based on the $o$-particle correlations defined in
Section IV B 3
of \cite{BBGKY_1}. Other truncation approximations
might by more suitable.

Second, the exponential-like instabilities, being connected to a
lack of representability, might be the main cause for the failure of the
BBGKY EOM at longer times. This hypothesis is supported by the
fast break-down of the BBGKY approach in the $N=100$ case for the truncation
order $\bar o=12$. For this reason, we analyze next the performance of the
correction strategies outlined in Section V B
of \cite{BBGKY_1}.

\subsection{Performance of the correction algorithms}

\begin{figure*}[t]
 \includegraphics[width=0.9\textwidth]{eval_purif_schemes_N10_damp10.pdf}
 \caption{(color online) Comparison of the correction
 strategies outlined in Section V B
 of \cite{BBGKY_1} (for the BBGKY hierarchy
 truncated at $\bar o=2$). Left column:
 Time evolution of the NPs $\lambda^{(2)}_i$. Right column [except for (d)]:
 Time evolution of the $\hat K_2$ eigenvalues $\xi_i$. First row: Truncated BBGKY
 results without correction versus exact ones. Second row: Truncated BBGKY
 results with iterative minimal-invasive purification of the $2$-RDM after each
 $\Delta t=0.1/J$ (maximal number of iterations: $500$). Third row:
  Truncated BBGKY results with minimal-invasive correction of the $2$-RDM EOM
  (damping rate of negative eigenvalues: $\eta=10J$). Insets of (c.1) and (c.2):
  close-ups showing the imposed exponential damping of negative eigenvalues.
  For both correction strategies, eigenvalues are regarded as negative if they
  are smaller than the threshold $\epsilon=-10^{-10}$. Subfigure (d): number
  of ZVODE integrator \cite{zvode} steps per write-out time-step $\Delta t$.
  Parameters: same as in Figure \ref{fig:BH_imbalance}.}
  \label{fig:BH_correction_alg}
\end{figure*}

In the following, we first focus on the correction algorithms applied
to the BBGKY hierarchy truncated at $\bar o=2$. Thereafter, we comment on
the performance of these algorithms if extended to larger truncation orders
by means of a corresponding ansatz for the correction operator (see Section V B
of \cite{BBGKY_1}).

Figure \ref{fig:BH_correction_alg} depicts the time evolution of the NPs $\lambda^{(2)}_i$
and the $\hat K_2$ eigenvalues $\xi_i$ (see the definition given in Section V A
of \cite{BBGKY_1}) for the truncated BBGKY results without correction,
with the iterative minimal invasive purification of the $2$-RDM and the minimal
invasive correction of the $2$-RDM EOM in comparison to the exact results.
Apparently, all cases deviate significantly from the exact results after
$t\gtrsim17/J$ so that we shall concentrate here solely on the stabilization
performance of the correction algorithms.

Inspecting first the uncorrected
results [Figures \ref{fig:BH_correction_alg} (a.1), (a.2)], we observe that
the $K$-condition\footnote{We call this condition $K$- instead of $G$-condition, as commonly done, in order to stress that we operate
with the original, more restrictive definition of the one-particle-one-hole RDM of \cite{garrod_1964}.} (i.e.\ $\hat K_2\geq0$) diagnoses earlier a lack of representability
compared to the $D$-condition (i.e.\ $\hat\rho_2\geq0$). For both
operators, the falling of an eigenvalue below zero is accompanied by an avoided
crossing which involves the next-larger eigenvalue (this is hardly visible
in the case of the $\hat K_2$ eigenvalue where the avoided crossing happens at
about $t\sim71.5/J$). In fact, we have observed that level-repulsion ``pushes'' eigenvalues below zero
in various other situations (see also Section
\ref{sec:breath}).

Now turning to the minimal invasive correction algorithms based on the $2$-norm minimization
of the correction operator $\hat{\mathcal{C}}_2$,
we set the threshold $\epsilon$ below
which an eigenvalue is regarded as negative to $-10^{-10}$. Let us first inspect the dimensionality
of the optimization problem underlying both our purification algorithm
of the $2$-RDM and the correction algorithm of its EOM (see Appendix H
of \cite{BBGKY_1} for the details).
The bosonic hermitian correction operator $\hat{\mathcal{C}}_2$ can be parametrized by
$m^2(m+1)^2/4=9$ real-valued parameters. Requiring $\hat{\mathcal{C}}_2$ to be contraction-free
and energy-conserving imposes $m^2+1=5$ constraints such that the system of linear
equations corresponding to the constraints is underdetermined as long
as the numbers of negative $\hat\rho_2$ eigenvalues $d$ and negative $\hat K_2$ eigenvalues
$d'$ obey $d+d'<4$.

Figure  \ref{fig:BH_correction_alg} (b.1) and (b.2) depict the results if
the iterative minimal-invasive purification algorithm is applied after each $\Delta t=0.1/J$.
Clearly, we see that this correction
algorithm induces strong noise in the $\hat K_2$ eigenvalues when the smallest
eigenvalue $\xi_i$ has reached significant negative values in the
uncorrected BBGKY calculation [see subfigure (a.2)]. Actually, after $t=86.5/J$, the
iterative purification algorithm fails to converge after the maximal number of $500$ steps.
Thus, this iterative scheme
fails to prevent that smallest eigenvalue is pushed
to negative values due to level repulsion.

In a certain sense, we may view the iterative purification algorithm of the $2$-RDM
as being based on a fixed stepsize as well as perturbative. In each iteration step namely,
we update $\hat\rho_2(t)
\rightarrow\hat\rho_2(t)+\hat{\mathcal{C}}_2$ with $\hat{\mathcal{C}}_2$
shifting negative eigenvalues to zero in first-order perturbation theory. In the correction
algorithm for the $2$-RDM EOM, we effectively allow for variable update stepsizes by
coupling the correction scheme to the integration of the EOM, i.e.\ to the
employed integrator ZVODE \cite{zvode} featuring adaptive stepsizes. Moreover, by imposing constraints
on the time-derivative of negative eigenvalues, we realize a non-perturbative correction
scheme.

This can nicely be inferred from the insets of Figure \ref{fig:BH_correction_alg} (c.1) and (c.2)
showing a close-up of slightly negative eigenvalues. These are exponentially damped to zero, namely
as e.g.\ $\xi_i(t+\tau)=\xi_i(t)\exp[-\eta\tau]$ for $t$ and $\tau$ such that $\xi_i(t+\tau)<\epsilon$,
with the chosen damping constant $\eta=10J$. As a consequence, the truncated BBGKY EOM
becomes stabilized and we have observed that the $D$- and $K$-representability condition
are fulfilled to a good approximation for at least $t\leq1000/J$ (times
later than $t=150/J$ not shown in Figure \ref{fig:BH_correction_alg}).
When enforcing negative eigenvalues to be damped to zero, one might
fear that eigenvalues accumulate in the range $[\epsilon,0]$. This, however, is not the
case as shown in the insets of Figure \ref{fig:BH_correction_alg} (c.1) and (c.2)
because no constraint on the time-derivative of an eigenvalue is enforced if its value exceeds the threshold $\epsilon$
such that the (corrected) EOM may lift this eigenvalue above zero. We finally
remark that the number of integrator steps per $\Delta t$
significantly increases
in the vicinity of avoided crossings of
$\hat\rho_2$ or $\hat K_2$ eigenvalues close to zero
[see Figure
\ref{fig:BH_correction_alg} (d)]. This finding confirms the non-perturbative, adaptive
nature of the EOM correction algorithm and at the same time highlights
the significance of controlling such avoided crossings for a successful stabilization of the
truncated BBGKY EOM.

Without showing additional graphical illustrations, let us now briefly comment on the behavior of the correction algorithms for truncation
orders $\bar o>2$, using the Mazziotti ansatz \cite{mazziotti_purification_2002} for the correction operators $\hat{\mathcal{C}}_o$
on orders $o>2$ (see Section V B
of \cite{BBGKY_1}).
Focusing first on the RDM purification, we have observed that $\hat\rho_{\bar o}$
can be kept positive semi-definite up to a few tens $1/J$ longer (compared to the uncorrected case) before
this iterative correction algorithm fails to converge after $500$ steps. Due
to the losing of positive semi-definiteness in decreasing sequence
with respect to the RDM order (see Figure \ref{fig:BH_neg_t}),
we found for $\bar o\geq 4 $ that also  $\hat\rho_{\bar o-1}\geq0$
is valid for somewhat longer times compared to the
uncorrected case. Unfortunately, however, this correction scheme fails to converge so
early that it does not improve the timescale, on which
the most important RDMs for making predictions for ultracold quantum gas
experiments, namely $\hat\rho_1$ and $\hat\rho_2$, obey the considered representability
conditions.

Extending the EOM correction scheme to higher truncation orders $\bar o>2$
by means of the Mazziotti ansatz for the higher-order correction operators unfortunately
proved to be quite unsuccessful. This failure manifests itself in an enormous increase
of integrator steps per $\Delta t$, i.e.\ the EOM becoming stiff, in combination
with the quadratic optimization problem for determining $\hat{\mathcal{C}}_2$ having
no solution, i.e.\ constraints contradicting one another. Unfortunately, we
cannot tell whether the latter is a fundamental problem or whether it is
only induced by the EOM to become stiff
due to an inappropriate ansatz of $\hat{\mathcal{C}}_o$ for $o>2$, potentially
leading to integration errors.

To sum up, while we can successively stabilize the BBGKY EOM  truncated at
order $\bar o=2$ by enforcing the $D$- and $K$-representability condition
via a minimal invasive correction of the $2$-RDM EOM, the issue of
higher-order correlations becoming dominant after a certain time remains unsolved
in this example. Since this tunneling scenario might well be
unsuitable for a closure approximation based on neglecting certain few-particle correlations,
we now turn to an example, where a BEC becomes only slightly depleted in the course of the
quantum dynamics.
\section{Interaction-quench induced breathing dynamics of harmonically trapped bosons}\label{sec:breath}

In this application, we are concerned with collective excitations of
$N$ ultracold bosons confined to a quasi one-dimensional harmonic trap. In harmonic
oscillator units (HO units), the corresponding Hamiltonian reads
\begin{align}
 \hat H = \sum_{i=1}^N\frac{\hat p_i^2+\hat x_i^2}{2}+g\sum_{1\leq i<j\leq N}\delta(\hat x_i-\hat x_j)
\end{align}
where we model the short-range van-der-Waals interaction by the contact potential \cite{Pethick_Smith2008}
of strength $g$. Initially, we assume all atoms to reside in the ground state of
the single-particle Hamiltonian, i.e.\ a Gaussian orbital, which
is the exact many-body ground state in the absence of interactions. Then, the interaction strength
is instantaneously quenched to $g=0.2$ such that the ideal BEC becomes slightly
depleted and its density performs breathing oscillations, i.e.\ expands and contracts periodically.
This so-called breathing mode has been investigated theoretically as well as experimentally in different settings (see e.g.\ \cite{Stringari2002,Moritz2003,bauch_2009,Bonitz2014,Schmitz2013,Tschischik2013,Bouchoule2014} for single-component
systems
and e.g.\ \cite{pyzh_spectral_2017} for mixtures), and measuring its frequency proves to be useful
for characterizing the interaction regime \cite{Haller2009}.

\begin{figure}[t]
 \includegraphics[width=0.495\textwidth]{breath_density_exact.pdf}
 \caption{(color online) Time-evolution of the reduced one-body density $\rho_1(x;t)=\langle x|\hat \rho_1(t)| x\rangle$
 for $N=10$ [subfigure (a)] and $N=30$ [subfigure (b)] bosons quenched from the non-interacting ground-state
 to a contact-interaction strength of $g=0.2$. These results are obtained by MCTDHB simulations with $m=4$ dynamically
 optimized SPFs.}
  \label{fig:breath_dens_ex}
\end{figure}

Before we discuss the results of the truncated BBGKY approach, let us first
inspect the results of MCTDHB simulations with $m=4$ dynamically optimized SPFs,
which shall serve as the reference in the following. For representing the SPFs,
a harmonic discrete variable representation \cite{dvr_and_their_utilization_Light_Carrington_2000,MCTDH_BJMW2000} with $n=256$ ($n=320$) grid points
is employed for case of $N=10$ ($N=30$) particles.
In Figure \ref{fig:breath_dens_ex},
we depict the time evolution of the reduced one-body density, i.e.\ the diagonal
of the $1$-RDM in position representation  $\rho_1(x;t)=\langle x|\hat \rho_1(t)| x\rangle$,
for $N=10$ and $N=30$ bosons. In both cases, we clearly see that the atomic density
periodically expands and contracts. Since the interaction quench leads
to a more than three times larger interaction energy per particle of the ensemble of $N=30$ atoms
compared to $N=10$ (at $t=0$), the density of the former expands much further
into the outer parts of the trap. In contrast to this, the density of the $N=10$ atom
ensemble seems to stay Gaussian (with a time-dependent width) to a good approximation,
indicating that we operate in the linear-response regime here. We note
that the quench leads to only a slight quantum depletion of at most $3\%$ in both cases (see below).

In the following, we first show that the truncated BBGKY approach leads to stable
results in the $N=10$ case, whose accuracy can be systematically improved by
increasing $\bar o$. Thereafter, we turn to the $N=30$ case where
we again encounter instabilities of the EOM and thus apply correction algorithms.
We stress that for both
cases we  operate with $m=4$ dynamically optimized SPFs,
solving the truncated BBGKY EOM coupled to the MCTDHB EOM for the SPFs,
which is in contrast to the Bose-Hubbard tunneling scenario of Section
\ref{sec:BH_dimer}.

\subsection{Breathing dynamics of $N=10$ bosons}

In Figure \ref{fig:breathing_NP2_N10}, we show the time-evolution of the
$2$-RDM NPs for various truncation orders. Focusing first on the MCTDHB results,
we see that correlations (in the sense of deviations from a Gross-Pitaevskii mean-field
state where on all orders $o$
there is only one finite NP $\lambda_1^{(o)}=1$ and all other NPs vanish) repeatedly emerge and decay. The deviations from the NP distribution
of a Gross-Pitaevskii mean-field state is approximately most pronounced when the
density is most spread-out and become almost negligible when the density has approximately recovered
its initial width [see Figure \ref{fig:breath_dens_ex} a)].

While the truncated BBGKY results for $\bar o=2$ feature significant deviations
from the MCTDHB results, the results drastically improve when going to $\bar o=3$
and become practically indistinguishable from the MCTDHB results already
at the truncation order $\bar o=4$. Actually, convergence of the $1$-RDM NPs
$\lambda^{(1)}_i$ is reached even at $\bar o=3$ (not shown). Coming back to $\bar o=2$,
we point out that the $2$-RDM quickly becomes indefinite where small negative eigenvalues
are in particular pushed further to larger negative values when the density
contracts to its initial width and small but positive NPs approach zero. As in
the case of the above tunneling scenario, we interpret this finding as ``induced'' by
level repulsion. Upon increasing the truncation order, we see that the $2$-RDM
stays positive semi-definite on the considered time-interval, which is a nice
example for how increasing the accuracy of the closure approximation
also stabilizes the truncated BBGKY EOM.

For a systematic comparison, we next compare the trace-class distance
$\mathcal{D}(\hat\rho_o^{\rm tr},\hat\rho_o^{\rm ex})$
between the truncated BBGKY result for the $o$-RDM, $\hat\rho_o^{\rm tr}$, and
the corresponding MCTDHB result, $\hat\rho_o^{\rm ex}$, in Figure
\ref{fig:breathing_rdm_comp} a). We remark that although the SPFs of the truncated BBGKY approach
obey the same EOM (5)
of \cite{BBGKY_1} as the
dynamically optimized SPFs of the MCTDHB method, we cannot expect these two
sets of SPFs to coincide because the $1$- and $2$-RDM entering
the SPF EOM differ in general, which has to be taken into account
when calculating $\mathcal{D}(\hat\rho_o^{\rm tr},\hat\rho_o^{\rm ex})$.
In stark contrast to the tunneling scenario, we see that the accuracy of the
truncated BBGKY results for the $1$- and $2$-RDM
systematically improves upon increasing $\bar o$ for all considered times.

Finally, we quantify the strength of few-particle correlations
in terms of $||\hat c_o||_1$, as extracted from the $\bar o=7$ calculation [see
Figure \ref{fig:breathing_rdm_comp} a)]. Here, we see that the correlations
stay bounded on the considered time interval and are ordered in a
clear hierarchy, i.e.\ $||\hat c_{o+1}||_1(t)<||\hat c_{o}||_1(t)$.
Apparently, these are ideal working conditions for the truncated BBGKY approach.

 \begin{figure}[t]
   \includegraphics[width=0.495\textwidth]{breath_eval_o2_N10_no_purif.pdf}
   \caption{(color online) Natural populations of the $2$-RDM for
   the truncation orders $\bar o=2$ [a)], $\bar o=3$ [b)] and $\bar o=4$ [c)]
   in comparison to the \mbox{MCTDHB} results. Parameters: $N=10$ atoms, post-quench interaction
   strength $g=0.2$, $m=4$ dynamically
   optimized SPFs.}
   \label{fig:breathing_NP2_N10}
\end{figure}

\begin{figure*}[t]
 \includegraphics[width=0.495\textwidth]{breath_rdm_comp_N10_no_purif.pdf}
  \includegraphics[width=0.495\textwidth]{breath_rdm_comp_N30_no_purif.pdf}
 \caption{(color online)
 First and second row:
 Time evolution of the trace distance $\mathcal{D}(\hat\rho_o^{\rm tr},\hat\rho_o^{\rm ex})$
 between the MCTDHB and the truncated BBGKY prediction for the $o$-RDM ($o=1,2$) and various truncation
 orders $\bar o$. The dotted horizontal line at unity ordinate value indicate the upper bound for the trace distance
 between two density operators. Third row: Time-evolution of the cluster's trace-class norm $||\hat c_o||_1$
 for $o=1,...,7$ obtained from the data of the $\bar o=7$ simulations. Left column: $N=10$ atoms. Right columns: $N=30$.
 Otherwise, same parameters as in Figure \ref{fig:breathing_NP2_N10}.}
 \label{fig:breathing_rdm_comp}
\end{figure*}

\subsection{Breathing dynamics of $N=30$ bosons}
Next, let us increase the quench-induced excitation energy per particle by more than
a factor of three when going to $N=30$ bosons and keeping the post-quench
interaction strength $g=0.2$ the same. Similarly to the tunneling scenario,
we first inspect the natural populations, then compare lowest order RDMs and
finally evaluate the performance of the correction algorithms under discussion.

\subsubsection{Natural populations}

\begin{figure*}[t]
 \includegraphics[width=0.9\textwidth]{breath_evals_N30_no_purif.pdf}
 \caption{(color online) Top (bottom) row: Time-evolution of the $1$-RDM ($2$-RDM) NPs
 for various truncation orders $\bar o$ in comparison to the MCTDHB results.
 We note that the ordinates are broken into two parts for covering the whole
 range of relevant values. In some cases, this leads to discontinuous curves, see
 e.g.\ the $\bar o=2$ curve in (a.2).
 Number
 of bosons: $N=30$.
  Otherwise, same parameters as in Figure \ref{fig:breathing_NP2_N10}.}
  \label{fig:breathing_NPs_N30}
\end{figure*}

In Figure \ref{fig:breathing_NPs_N30}, we show the NPs of the $1$- and $2$-RDM for various
truncation orders $\bar o$ in comparison
to the MCTDHB results. Similarly to the $N=10$ case, we see how the NP distributions as obtained
from MCTDHB
oscillates between the characteristics of the Gross-Pitaevskii mean-field state
and a (slightly) correlated one, which is approximately synchronized to the strongest
contraction and expansion of the density, respectively [see Figure
\ref{fig:breath_dens_ex} b)]. In contrast to the former case, however,
we can converge the NPs to the MCTDHB results upon increasing the
truncation order $\bar o$ only for times $t\lesssim 5$ HO units.
For all considered truncation
orders, we witness an exponential-like instability in the $2$-RDM NPs resulting in large negative
eigenvalues while the $1$-RDM stays positive semi-definite for the
considered time-span. Fixing $\bar o$, we have observed also for this scenario that
the lack of positive semi-definiteness of the $o$-RDMs  happens in
decreasing sequence with respect to the order $o$ (not shown).
Moreover,
these instabilities in the $2$-RDM NPs seem to be triggered by small positive NPs approaching zero
from above, namely when the density approximately shrinks to its initial width,
see e.g.\ Figure \ref{fig:breathing_NPs_N30} (c.2).
Finally, we have observed for the case $\bar o=2$ that increasing the number of SPFs from $m=4$ to $m=8$
slightly enhances the time-scale on which the instability of the $2$-RDM NPs
takes place (not shown). This finding is reasonable since the
projector $(\mathds{1}-\hat{\mathbb{P}})$, occurring in the SPF EOM (5)
of \cite{BBGKY_1},
projects onto a smaller subspace when increasing $m$ such that the impact
of the non-linearity in the SPF EOM is effectively reduced.

\subsubsection{Reduced density operators}
Comparing the BBGKY prediction for the complete $1$- and $2$-RDM with the corresponding MCTDHB results in terms of the trace-class distance in Figure \ref{fig:breathing_rdm_comp} b),
we see that deviations emerge much faster as compared to the $N=10$ case.
 At longer times, we also
observe that the accuracy of the BBGKY results does not monotonously increase anymore
with increasing $\bar o$. Moreover, the above mentioned instabilities
also partly manifest themselves in $\mathcal{D}(\hat\rho_2^{\rm tr},\hat\rho_2^{\rm ex})$
attaining unphysical values above unity. Finally, we also
depict $||\hat c_o||_1$ as a measure for correlations in Figure
\ref{fig:breathing_rdm_comp} b). While the correlations
are hierarchically ordered in decreasing sequence with respect to the order $o$ up
to $t\sim 2.7$ HO units, this ordering becomes  reversed later on.
This finding, however, is not conclusive, i.e.\ might be unphysical and
related to the observed instability, since the values of $||\hat c_o||_1$ have been extracted
from the BBGKY data with $\bar o=7$ (in contrast to the tunneling
scenario where the numerically exact $\hat c_o$ have been used).

At this point, we shall remark that we expect a much better agreement for the
$N=30$ case when quenching to much lower interaction
strengths $g\ll0.2$ and thereby reducing the overall excitation energy.

\subsubsection{Performance of the correction algorithms}
\begin{figure*}[t]
 \includegraphics[width=0.9\textwidth]{breath_eval_purif_schemes_N30_damp10.pdf}
 \caption{(color online)
 Comparison of the correction
 strategies outlined in Section V B
 of \cite{BBGKY_1} (for the BBGKY hierarchy
 truncated at $\bar o=2$), i.e.\ same as Figure \ref{fig:BH_correction_alg}
 but for the interaction-quench scenario with $N=30$ bosons. Parameters:
 threshold $\epsilon=-10^{-10}$, damping constant $\eta=10$ HO units, write-out
 time-step $\Delta t=0.05$ HO units, maximal number of iterations: $500$.
  Otherwise, same parameters as in Figure \ref{fig:breathing_NP2_N10}.}
  \label{fig:breathing_correction_alg}
\end{figure*}

Here, we again focus mainly on the performance of the correction algorithms
applied to the $\bar o=2$ BBGKY approach and comment later on larger truncation
orders. In Figure \ref{fig:breathing_correction_alg}, we depict the spectrum\footnote{
We note that the $\hat K_2$ spectra at $t=0$ in the tunneling and the interaction-quench
scenario differ although the system is initially a fully condensed BEC in both cases
(see Figures \ref{fig:BH_correction_alg} and \ref{fig:breathing_correction_alg}).
This is due to the fact that the $\hat K_2$ spectrum is sensitive to the total number
of SPFs $m$ even if not all of them are occupied.} of
$\hat K_2$ as well as a close-up to the $2$-RDM spectrum in the vicinity of zero
for the uncorrected BBGKY approach, the minimal invasive RDM purification algorithm
and the minimal invasive EOM correction algorithm. In the minimization problem
underlying both correction algorithms, we have to find the optimal
$\hat{\mathcal{C}}_2$ which depends on $m^2(m+1)^2/4=100$ real-valued parameters.
Being contraction-free and energy conserving leads to $m^2+1=17$ constraints.
Moreover, $\hat{\mathcal{C}}_2$ has to obey the parity symmetry of our problem
imposing $m^4/8+m^3/4=48$ further constraints (see Appendix H
of \cite{BBGKY_1}). Thereby,
our system of linear constraints remains underdetermined as long
as the number $d$ of negative $\hat\rho_2$ eigenvalues and number $d'$ of negative
$\hat K_2$ eigenvalues obey $d+d'<35$.

In Figure \ref{fig:breathing_correction_alg}, we see that the minimal-invasive
RDM purification algorithm clearly suppresses significant negative eigenvalues until
$t\sim2.5$ HO units. Thereafter, noticeably negative eigenvalue emerge but stay
bounded from below until $t\sim6$ HO units when a drastic
instability kicks in. Thus, this iterative algorithm soon fails to converge
after the maximal number of $500$ iteration steps. In order to understand
the deeper reason of this failure, we have analyzed the spectrum
of the updated operators $\hat\rho_2(t)+\alpha\,\hat{\mathcal{C}}_2$
and $\hat K_2(t)+\alpha\,\hat{\Delta}_2$
for $\alpha\in[0,1]$ and the first few iteration steps at such an instant in time (not shown). Thereby, we
have found that while the tangent on a negative eigenvalue (with respect to $\alpha$)
indeed crosses zero as imposed by our constraints, level repulsion
with other (in most cases negative) eigenvalues often hinders this negative eigenvalue
to significantly move towards zero. We cannot rigorously prove that this is indeed
the only mechanism for the breakdown of this iterative purification algorithm, of course.

Yet at least, this finding gives a useful hint why our non-perturbative, adaptive approach,
 the minimal invasive correction scheme of the $2$-RDM EOM, gives very
 stable results [see Figure \ref{fig:breathing_correction_alg} (c.1) and (c.2)].
 Actually, we observe that the $D$- and $K$-conditions are fulfilled to
 a good approximation much longer, namely for at least $t\leq 36$ HO unit (not shown).
 From Figure \ref{fig:breathing_correction_alg} (d), we furthermore infer that the integrator
 variably adapts its step-size, but in contrast to the Bose-Hubbard tunneling scenario
 no systematic enhancement of integrator steps is observed when $\hat\rho_2$ or
 $\hat K_2$ eigenvalues avoid each other in the vicinity of zero. Apparently, the
 though stabilized result features noticeable deviations from the MCTDHB
 results for the respective eigenvalues.
 Yet, we find that the overall accuracy of the $\bar o=2$ results for the $1$-
 and $2$-RDM as measured by the trace-class distance is systematically
 improved for most times by correcting the $2$-RDM EOM,
 as one can infer from Figure \ref{fig:breathing_rdm_comp} b).

 In order to judge the accuracy of the EOM-corrected $\bar o=2$ simulation
more descriptively, we depict the deviations of its prediction for the
reduced one-body density from the MCTDHB results in Figure \ref{fig:breathing_dens_comp_N30}.
Note that this plot covers a longer time-span compared to the previous ones.
As expected, we find that the deviations increase in time. Compared to the absolute values of
the density, these deviations are, however, small and, most importantly, somewhat
smaller than the deviations of corresponding Gross-Pitaevskii mean-field simulation from
the $m=4$ MCTDHB results (not shown). Finally, let us connect
the errors in the one-body density to the errors measured by the
trace-class distance $\mathcal{D}(\hat\rho_1^{\rm tr},\hat\rho_1^{\rm ex})$
as depicted in Figure \ref{fig:breathing_rdm_comp} b). For this purpose, we note that
the density at position $x$ can be expressed as the expectation value
of the one-body observable $\hat A_1=|x\rangle\!\langle x|$ featuring $||\hat A_1||_1=1$.
Thereby, the inequality of Appendix \ref{app:exp_value_bound}
gives $|\rho_1^{\rm tr}(x;t)-\rho_1^{\rm ex}(x;t)|\leq 2
\mathcal{D}(\hat\rho_1^{\rm tr}(t),\hat\rho_1^{\rm ex}(t))$,
which is consistent with the results depicted in Figure \ref{fig:breathing_dens_comp_N30}.

Going to higher truncation orders by making the Mazziotti ansatz for the
corresponding higher-order correction operators $\hat{\mathcal{C}}_o$ unfortunately does
not improve the BBGKY results, as already observed in the tunneling scenario.
While the iterative RDM purification
scheme fails to prevent the instabilities, we observe the same obstacle
for the EOM correction algorithm as previously encountered, namely the optimization problem
at order $o=2$ lacking a solution (results not shown). Yet due to the very promising
results of the EOM correction algorithm when truncating the BBGKY hierarchy at order
$\bar o=2$, we believe that extending the EOM correction algorithm to higher orders
without employing the Mazziotti ansatz for the correction operator
is a highly promising direction to go.

\begin{figure}[t]
 \includegraphics[width=0.495\textwidth]{breath_density_N30_comp_o2_eom_corr.pdf}
 \caption{(color online) Subfigure a): Time-evolution of the reduced one-body density $\rho_1^{\rm ex}(x;t)$
 as obtained from \mbox{MCTDHB} on a longer time-scale.
 Subfigure b): Absolute deviation $\rho_1^{\rm tr}(x;t)-\rho_1^{\rm ex}(x;t)$
 of the reduced one-body density $\rho_1^{\rm tr}(x;t)$
 as obtained from the BBGKY approach truncated at $\bar o=2$ and
 stabilized by the minimal-invasive EOM correction algorithm with
  $\epsilon=-10^{-10}$ and $\eta=10$ HO units. Number of bosons: $N=30$.
  Otherwise, same parameters as in Figure \ref{fig:breathing_NP2_N10}.}
  \label{fig:breathing_dens_comp_N30}
\end{figure}
\section{Conclusions}\label{sec:concl}
While we have provided the theoretical framework of the truncated BBGKY hierarchy for
simulating the quantum dynamics of few-particle reduced density operators of
an ultracold bosonic many-body system in the immediately preceding work \cite{BBGKY_1},
we have provided here extensive applications.  The central aim of this work is to systematically study
the impact of the  BBGKY hierarchy truncation order $\bar o$ on the numerically obtained results
where the highly efficient formulation of the theory in \cite{BBGKY_1} allows us for going
to truncation orders as high as $\bar o=12$. For this purpose, we have
considered both a tunneling scenario where a Bose-Einstein condensate fragments into
a two-fold fragmented, highly entangled state and thus strong correlations emerge, and
an interaction-quench scenario where a Bose-Einstein condensate becomes weakly depleted
when performing collective oscillations. While the former scenario is based on the Bose-Hubbard
dimer model where we may use
time-independent Wannier states for representing the reduced density operators (RDMs), we
solve in the second scenario the truncated BBGKY hierarchy as derived from the variational Multi-Configuration Time-Dependent Hartree method
for Bosons (MCTDHB) \cite{BBGKY_1,MCTDHB_PRA08} and thereby employ dynamically
optimized single-particle functions for efficiently representing the RDMs.
The truncated BBGKY results are comprehensively evaluated by inspecting a variety
of observables, covering descriptive ones such as the atomic density but also the eigenvalues
of the low-order RDMs and their trace-class distances
to the corresponding numerically exact and \mbox{MCTDHB} results in the first and second scenario,
respectively.

In all applications, we have found that the short-time dynamics can be highly accurately
described by the truncated BBGKY approach, where the accuracy of the results systematically
improves with increasing truncation order $\bar o$. At longer times, the BBGKY gives also excellent
results with controllable accuracy for the interaction-quench scenario and not too high
excitation energies. For higher excitations energies as well as in the tunneling scenario,
however, we obtain only faithful results up to a certain time even if we
truncate the hierarchy at the highest non-trivial order $\bar o=N-1$ with $N$ denoting
the total number of particles. After that time, the accuracy does not monotonously improve
with increasing truncation order and instabilities emerge, manifesting
themselves in an exponential-like decrease of RDM eigenvalues to negative values.
By inspecting the exact numerical results for the tunneling scenario,
we find that few-particle correlations on all orders quickly
play a significant role and eventually $N$-particle correlations dominate
because the total system evolves into a NOON-state. This finding indicates
that the long-time physics of this scenario prevents to use a truncation approximation
which is based on neglecting $(\bar o+1)$-particle correlations.

Nevertheless, it is important to clearly separate the stability properties of the
truncated BBGKY equations of motion (EOM) from accuracy issues because (i) it is not
desirable to have a highly accurate theory which is exponentially unstable under slight e.g.\
numerical perturbations and (ii) also a not highly accurate truncation approximation
may give useful, sufficiently accurate results for low-dimensional observables such as the
density if the EOM are sufficiently stable. In order to disentangle stability issues
from the accuracy of the results, we have performed an in-depth analysis of the observed exponential-like
instabilities. In all examples, we witness that the instability sets in at the truncation
order $\bar o$ and then propagates down to lower orders meaning that $o$-particle RDMs lack
to be positive semi-definite in decreasing sequence with respect to the order $o$. The time until
which the highest-order RDM, $\hat \rho_{\bar{o}}$, stays positive
semi-definite only gradually increases with the truncation order $\bar o$.
Upon increasing $\bar o$, the most important RDMs for calculating relevant observables for ultracold systems, $\hat\rho_1$ and $\hat\rho_2$,
are found to stay a bit longer positive semi-definite in some examples, in
others, however, the opposite holds. That the instabilities set in more rapidly with increasing
truncation order $\bar o$ might by explained by the fact that the cluster-expansion
based truncation approximation (see Section I B 3
of \cite{BBGKY_1}) constitutes a polynomial of degree $(\bar o+1)$ in $\hat\rho_1$
and of degree $\lfloor(\bar o+1) /o\rfloor$ in the cluster $\hat c_o$. Thereby,
the non-linearity of the EOM is effectively enhanced for increasing $\bar o$.
Moreover, the instabilities often take
place when the lowest RDM eigenvalues undergo an avoided crossing in the vicinity of zero.

Ultimately, these instabilities emerge due to the applied truncation
approximation, which conserves the RDM compatibility and energy as well as respects
symmetries such as the parity invariance if existent, but does not ensure
that important necessary representability conditions such as the positive
semi-definiteness of $\hat\rho_o$ and of the one-particle-one-hole RDM
are fulfilled \cite{BBGKY_1}. In order to enforce such representability conditions
and stabilize the EOM for a given truncation order $\bar o$ (independently of the
accuracy of the results), we have evaluated the two novel correction algorithms
developed in \cite{BBGKY_1}. Both strategies aim at ensuring the positive semi-definiteness
of $\hat\rho_2$ and the (modified) one-particle-one-whole RDM as defined in \cite{garrod_1964}
in a minimal-invasive and energy-conserving way by solving an optimization problem.
For higher orders, the Mazziotti ansatz \cite{mazziotti_purification_2002}
for the correction operator is chosen. The first strategy corrects the solution
of the truncated BBGKY EOM after each small time-step $\Delta t$ by
iteratively adding a correction operator which raises negative eigenvalues to zero
in first order perturbation theory. Unfortunately, this iterative algorithm does not
lead to a significant stabilization over a reasonable time-span and soon fails to
converge most likely due to level-repulsion of the lowest eigenvalues which cannot
be overcome by a perturbative approach. Yet the second approach, which corrects the truncated BBGKY EOM themselves in a
minimal-invasive, energy-conserving way such that negative eigenvalues are exponentially
damped to zero, has proven to stabilize the dynamics for long times if applied to the truncation order $\bar o=2$.
This highlights that a successful stabilization of the dynamics requires a non-perturbative control of
the low-lying spectrum of e.g.\ $\hat\rho_2$. Moreover, the long-time results of the EOM-corrected BBGKY approach for various
observables are also qualitatively correct in the case of the
interaction-quench scenario. Yet
extending the EOM-correction algorithm to
higher orders by the Mazziotti ansatz has been found to lead to stiff EOM or the optimization
problem at order $o=2$ lacking a solution.

Due to the success for $\bar o=2$, however,
we believe that research on how to properly and efficiently extend the EOM-correction
algorithm to higher orders constitutes a highly promising route for the future,
which can result in a robust BBGKY approach with controllable accuracy for
broad applications. A second future line of research should address how to
improve the truncation approximation of the BBGKY hierarchy for bosonic
systems with a definite number of particles, where the existing literature
is rather sparse compared to the fermionic case.
We note here that all results of this work rely on the chosen definition of few-particle correlations,
of course.
One novel
approach for the truncation might involve to use machine-learning techniques such as artificial
neural networks
for deducing the unknown collision integral at the truncation order in
combination with an EOM correction algorithm for stabilizing the dynamics.
In any case, the theoretical framework and the highly efficient formulation
of the theory as provided in \cite{BBGKY_1} together with the deep
empirical insights of this work serve as an important step
in the direction of establishing the BBGKY approach as a simulation
tool for ultracold atomic systems.

\begin{acknowledgments}
The authors thank Iva B\u{r}ezinov\'a, Joachim Burgd\"orfer, David A.\ Mazziotti,
Hans-Dieter Meyer, Angel Rubio, Johannes M.\ Schurer and Jan Stockhofe for fruitful discussions.
This work has been supported by the excellence cluster ``The Hamburg Centre for Ultrafast Imaging
- Structure, Dynamics and Control of Matter at the Atomic Scale'' of the Deutsche Forschungsgemeinschaft.
\end{acknowledgments}

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