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Why Are Healthcare Costs Exploding? See Proton Cancer Therapy
Luke Timmerman
for 20 years after they got standard chemo and radiation often suffer long-term side effects. Almost half (47 percent) of the children suffer from a loss of IQ points, heart toxicity, hearing loss, and the appearance of secondary malignant tumors. In contrast, a longstanding proton therapy site in Switzerland has reported only 7 percent of its patients suffered such long-term side effects from treatment, Laramore says.
That might sound like an intriguing benefit, but it could be misleading for any number of reasons. The gold standard of medical evidence comes from studies in which patients of similar prognosis are randomly assigned to one treatment vs. another and followed over time at multiple clinical sites. The reduced complication rate from Switzerland is the kind of finding that just might persuade a researcher to start such a well-controlled study to ask whether protons are a superior form of treatment for medulloblastoma.
Picking apart one study or another isn't the real story here. The real problem comes back to the business model, and the financial math required to make this facility pencil out. The Northwest's new proton center, like many others around the U.S., cost $152 million to build and to operate for its first year, says Annika Andrews, the president of the new center. It has 43 employees, and expects to ramp up to more than 80 at full capacity, Andrews says. When maxed out, the Seattle proton facility should be able to serve 1,400 patients per year, from a five-state regional area, running daily from 7:30 am to 10 pm.
Whenever a private, venture-backed company like ProCure and its lenders pump $152 million into a new facility, you know they have run the numbers on what it will take for this facility to make money. The basic math says it needs a lot of patients, and they each need to pay a lot. While treatment courses vary from tumor type to tumor type, and insurance reimbursement varies by region, proton therapy is typically about 40 percent more expensive than standard treatment, Andrews says. Median Medicare reimbursement for prostate cancer patients on proton therapy is about $32,428, compared with $18,575 per patient for standard radiation, a 75 percent pricing premium, according to a recent article in the Journal of the National Cancer Institute.
Those numbers say a lot about why proton centers are popping up all over the country.
"High reimbursement per case X High throughput X High volume = High profit," wrote Theodore Lawrence and Mary Feng, a pair of radiation oncologists at the University of Michigan, in the Journal of the National Cancer Institute.
Laramore showed off the new $152 million proton therapy facility in Seattle, during a recent media tour.
Studies that compare proton therapy and standard radiation for prostate cancer are limited, as it's historically been difficult to run randomized head-to-head studies with only a few active proton therapy facilities. But one study published last April in the Journal of the American Medical Association said that patients on the standard IMRT (intensity-modulated radiation therapy) had fewer gastrointestinal side effects than patients on proton therapy. Lawrence and Feng note that another study said that 6 percent of prostate cancer patients suffered genitourinary side effects six months after getting proton treatment, compared with 10 percent on standard IMRT. But that apparent advantage disappeared after 12 months.
"Proponents of proton therapy may argue that any reduction in toxicity is worthwhile. However, is this small transient difference enough to justify a 70 percent higher cost per patient?" Lawrence and Feng wrote.
OK, so the evidence on prostate cancer at this point—albeit from a limited data set—says you'll pay more for proton therapy and not necessarily get more. In fact, you might pay more and get less.
Now, let's come back to the finances. In Seattle, ProCure has been blanketing local media outlets with ads, encouraging cancer patients to consider … Next Page »
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14 responses to "Why Are Healthcare Costs Exploding? See Proton Cancer Therapy"
DV4SC says:
This article appears to be either a total set-up by its author, or, just really crummy journalism! The reality of the situation is that the treatment of cancer, when it comes to proton therapy, has become a competition between the haves and the have-nots. Really, just how objective can Swedish Hospital's Radiation Oncologists be, when their top competitors, after more than 10 years of researching and years of developmental planning, decides to arm themselves with the most precise and specific radiation tool in the world? By the way, it is well-known that Swedish Hospital had in fact had a lot more serious intentions than they shared with this author in developing a proton project to treat far more cancer cases than just pediatric cases.
The retrospective study that the author cited in JAMA last April, which was by the way published by the have-nots, really missed the boat as well. For example, it did not even take into account the super important fact that the patient population treated in the proton arm had been significantly dose-escalated (30 – 50%), for improved control responses of the cancer, over the IMRT arm. It was like comparing an apple to an orange. Yet, because there are more have-nots than haves, and a lot of powerful ones, it got published.
Also, it should be noted that Dr. George Laramore is one of the most respected Radiation Oncologists in the entire world and this author comes in and, after his brief interview, tries to make Dr. Laramore sound like some kind of a bumbling nutty professor. Dr. Laramore is world-renowned amongst his peers for his deep research and clinical experience with both x-rays and particle therapy. He clearly knows and can explain all the advantages of protons over x-rays on a case-by-case basis, for those who really want to listen.
The author forgot to mention that IMRT costs are much more to implement than its predecessor, yet nobody seems too interested in asking for the Phase III randomized studies that were done to warrant those increased costs! Why?
Because there were none!! Rather than just focusing on the $150 Million cost of developing the center, which by the way is about $5 Million a year, as it has a 30-year life, the author should have spent more time focusing on the actual
science and then he would have clearly understood why informed patients do not want to go into a blind study where one arm (the x-ray arm) will expose their normal tissues to a 30 – 70% more radiation exposure. If the author wants to make a stance on costs for the treatment of cancer, he should focus on the
pharmaceutical industry, where their costs of just two chemo-related drugs, at
several billions of dollars, costs the US healthcare system much more than all
the radiation programs (which include all proton therapy centers) combined.
Louis Memoli says:
I was going to write a rebuttal to this article but DV4SC nailed it. The folks at SCCA are outstanding individuals, both personally and professionally… I know first hand how intensely they've researched the potential of proton therapy. SCCA is exactly the type of institution that should have this technology not only to advance patient care, but for the medical research that will be forthcoming. I am especially pleased that DV4SC stood up for Dr. Laramore – one of the finest, apolitical, down-to-earth, physician scientists I've ever met. Nice job DV4SC -.I could not have said it better. And Mr. Timmerman, next time, do your homework.
omama says:
Ahh Houston, we have a problem….DV4SC/Memoli/Lenarzt have all actually missed or conveniently ignored the excellent point that Luke made. That is, healthcare costs are too high and high-tech "innovations" must be forced to deliver VALUE.
Do protons have value? Yes. Are they worth $100 or $20M? Who knows, but when a standard LINAC can be purchased in $2M range and a fancy modified one can be had for $5M, you better be able to justify the serious multiples.
I would challenge DVASC/Memoli/Lenarzt to do just that. Ok, so protons are nice for skull base tumors–based on OLD literature I might add– and have real, obvious value for little kids who will hopefully live a long time. They also have value in SIDE EFFECT and COMPLICATION reduction, no doubt. But efficacy or tumor kill? Little, maybe 10%, right DV4SC?
As for Laramore being a clinical expert, no one argued. But is Laramore an expert on health care costs? Hardly.
As for Mevion and the $20M jobs–good but not good enough. Too expensive still for real VALUE to seep out to the masses.
IMRT, you mention. Just makes Luke's point that much more poignant. As you say, IMRT is expensive and not very justifiably so. Just like using protons in patients who may not have long enough to experience the fun side effects our treatment gives them.
All the above should not be taken as a decision to say "No" to the purchasing decision or the use of protons. For certain facilities that can squeeze the value out of them, protons may be the best decision ever made (until they see the even higher dollar carbon ions–looking at you CHIBA). But the ProCures of the world selling to the neighborhood rad onc practices–please. Don't let me decide though—just let the MARKET decide.
Finally, a "set-up" or "crummy" journalism? Quite the contrary. And, DV4SC, next time pick on people your own size, not a journalist who wrote an excellent piece. If you represent the thinking of the rad onc community, and perhaps you do, therein lies our problem. Perhaps you missed the ASTRO Keynote last year–that's right, radiation oncology was the #1 biggest specialty identified in the shame of escalating health care costs. Yes, I hate Pharma too and not sure I believe the Keynote, but let's look in the mirror first.
OMAMA: I strongly suggest you read my comments once more; I think YOU missed the key points. Try to open your mind before reading and try to keep it open throughout the read. Perhaps then you might get my points. I actually displayed the fixed costs of a 4-room facility that will be operating 2-shifts per day, at $5M/year. That is equivalent to 8 Linacs, as most Rad/Onc centers that are only using Linacs simply cannot get enough patient volume to warrant a second shift, unlike protons.
Your mentioning Rad/Onc being a cost-reduction target is a joke. Every single specialty area, including Ophthalmology, Pathology, Neurosurgery, etc.. are targets!! Perhaps, you did not attend any other specialty conferences beyond Rad/Onc or you would know the real story.
You defend this journalist, but this journalist IMO did a weak interview with Dr. Laramore, put a heavy weight on competing Rad/Onc that bad-mouthed the technology without fully disclosing that his very own institution was seriously trying to get a proton center of their own, and the Author sites articles that have no real basis for comparison (like comparing apples to oranges) written by Authors who have an axe to grind.
The article raises some good questions, but does have some problems. Personally I think it is good that ProCure is partnering with SCCA. The UW physicians now have IMRT and protons (and neutrons), so they are in a good position to do head-to-head comparisons. And the SCCA is a top research organization, so I think they are the right people to be exploring and developing different treatment modalities.
One misleading fact is comparing the $150M facility to a $3M linac. The $150M is more than just the treatment equipment in one room. One could also question why if standard radiation equipment only costs 2% of the proton facility, why are the treatments costs 50% of protons? Seems like you could make money even faster with cheaper x-ray equipment even with the lower reimbursement.
lenarzt says:
No question, DV4SC nailed it and so did Louis Memoli. One wonders what Mr.
Timmerman was thinking (or his hidden agenda) when he accepted an invitation to tour the new SCCA proton facility. Surely he was aware of the cost, which is no secret and old (journalistic) news. If costs were the same as conventional X-ray treatment, there would be no debate. We can all agree less radiation is always better to spare healthy cells and tissue in reducing recurrence, reducing toxicity, and reducing side effects. When it comes to prostate cancer patients, 99 percent of proton therapy patients treated believe they made the best treatment decision for themselves in a new data analysis of outcomes and satisfaction reported by nearly 2,000 patients. Did he happen to talk with patients themselves? It's no secret that If all hospitals could afford to have proton therapy, they would all have it. Despite Timmerman's lame opinion, the future of proton therapy is bright. It will not only be a better option for cancer patients, but also one that is not necessarily more costly than conventional X-ray treatment.
Leonard Arzt, National Association for Proton Therapy
ProtonsRule says:
Out of necessity comes invention, Mevion being the perfect example in this case. The author drowns himself in the here-and-now, yet dedicates one short paragraph to the future of Proton Therapy technology? What about the wide-reaching economical benefits of innovation? What about the unwavering drive these innovators demonstrate to improve the technology, reduce costs and complexity, and enable PATIENTS to benefit from the wider deployment (at lower power consumption levels)? Hmmm…if the author interviewed a few patients who got up and walked out of each treatment feeling no worse than they entered…his opinions might change.
Whalley says:
I agree that healthcare costs are unsustainable and that societal attitudes contribute, but this article's conclusion that more product testing will improve things is naive, at best. The supply of high-cost medical services exists because the demand is there and, as history shows, extra hurdles on the supply side will likely raise costs across the board. If the root cause is attitudes on the demand side, the solution is education on the demand side.
David Simpson says:
I've read the comments in here and with all the arguments taken. I'd say proton therapy is good news and beneficial to many.
http://www.pharmacyschooling.com
BiotechStockResearch says:
Let us charge whatever we want, don't mind there are no data, and a pox on anyone who questions our motives.
Did I summarize the thread here pretty well?
DV4SC does a nice job shifting the conversation about the JAMA study. OK, let's say I grant your point about the study, after all (as Luke points out) retrospective data have any number of problems. So, DV4SC, where are your PROSPECTIVE data showing this technology works any better over the LONG TERM?
Because if we're going to be paying this much for something, we'd better be talking long term outcomes.
The SCCA are nice individuals, Louis, but that has nothing to do with anything here does it? I have no doubt they passionately believe from their anecdotal evidence this technology can save lives. The data are probably even there in medulloblastoma. But is anecdotal evidence enough to justify taxpayers paying the cost differential here for prostate, which is a good example because 80% of patients are on Medicare?
And Leonard from the industry group, you have to be aware the stat you site about patient satisfaction is pretty meaningless. Of COURSE patients are going to say they made the right choice — especially men — because otherwise they have to admit they screwed up their own healthcare choice. Guys are notoriously skewed to having higher opinions of "new" things too. And frankly, we like all sorts of things that may or may not be good for us. Your data produced via a non-scientific sample and analyzed by your paid consultant are certainly interesting enough to be hypothesis-generating, but they are meaningless without a control or at least additional analysis as a check/balance against whether patients' opinions match their consumption of related healthcare services. Good marketing piece, I'm sure. Unacceptable by any objective standard as the sole basis of justifying the additional cost.
The medical spending path we are on is unsustainable. As an analyst for publicly-traded companies, it's great when a company announces a huge selling price and is able to make that stick because the increased revenues make money for my clients. But it isn't really about that, is it? In the long run, when it bankrupts our governments ("when" not "if"), all we've done is accept short term gain for long-term pain.
Luke's point was "Where are the Data?" Nobody here has shown any data proving proton beam therapy is superior (again, with the possible exception of medulloblastoma). If it is only "just as good" then why should taxpayers be paying 70% more for "just as good"?
It's not unreasonable to ask that question. It's not unreasonable to expect an answer. And it makes me wonder that most of you are so defensive that you have to personally attack the author.
I guess that's what you're left with when, in fact, you have no data.
Biotech: We in fact Do have data!!
We had enough data from mostly two institutions, Harvard & Loma Linda, to have convinced such institutions as MD Anderson, Univ. Of Florida, University of Pennsylvania, The Mayo Clinic, St Jude's Hospital, University of Maryland, Memorial Sloan Kettering, and countless others to pursue the development of their own centers. In 1997 there was only one hospital-based proton center in the entire world, now there are more than 12 in the states and more than 30 abroad – and growing!! If you think these institutions did not thoroughly vet out this therapy before moving forward to build out these projects – think again!!
BTW, the data will continue to grow with the addition of new high-quality clinical programs that are exploring higher thresholds, in dose placement accuracy, dose escalation, and reduced toxicity studies, than have ever been attained in conventional radiotherapy.
The main reason we cannot easily conduct a double blind study between x-ray & proton is that by law we have to fully educate the patient to both arms of the study and let them decide to forgo an industry-wide accepted reduction of at least a 50% integral dose of radiation to their normal tissues – to prove an obvious point.
BDJ384 says:
Mevion and ProNova Health Solutions are not only developing cheaper, $20 million machines; they are improving proton therapy. Read up on hypofractionation which will significantly lower not only treatment time for patients but costs as well.
The innovative, early adopters are investing in proton because they understand the studies and are looking to the future. Studies have already shown that proton works better, they just haven't been done with the perfect parameters to where we can say X=Y so proton is worth it. Hospitals and private proton providers are looking 10 years down the road. The exact same argument was made against PET. It took a large effort to get PET approved and moving but look at PET now…and it's totally worth it. I'm willing to be we could find an article exactly like this one complaining about PET from years ago.
And this isn't some American "spend tons of money" thing. Europe and Asia are also investing in proton therapy now and they are doing so largely based on high level governmental analysis of its current value. In the UK they have made the argument that it will save the system as a whole money, despite its current high cost.
I understand the argument that we should have complete and convincing proof before we invest in something so expensive. My simplified response to that is the people who are in the know in the medical community see its potential and are jumping on board. They see enough proof. For example, studies have shown that increasing the gray that you can treat a tumor with will improve the effectiveness of treatment. Studies have also shown that decreasing the entry dose of radiation to areas of the body that are clean will decrease side effects and recurrences. These studies weren't done with proton so we can't say that proton would be more effective; however, proton has a negligible entry dose and can deliver more gray. Therefore, proton can more effectively treat a tumor in a way that traditional radiation will never be able to (re: entry dose). If you throw in studies that will be proving the efficacy down the road (which many believe this is inevitable-clearly) and the improvements to proton to make it cheaper, it is reasonable to expect large patient loads and profitability that come as a result of its proven effectiveness. These folks aren't investing $200 million + just because they think they can charge a little more. That's too big of a risk.
You don't take a 30+ year risk on something that could be so easily disproved by upcoming studies. You take a 30+ year, $200 million, $2+ million maintenance cost, risk because you see the future potential and have the utmost confidence in it.
If you want to pick on high health care costs, pick on the government giving drug makers exclusive rights to gouge us on drug costs…or pick on obese people…or pick on cigarettes…or pick on preventative medicine (tort reform-the cost saving benefit of which is incalculable).
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text:
\section{\label{}}

\section{Introduction}
Historically, studies on multi-component quantum fluids were conducted on mixtures of $^3$He with $^4$He \cite{Ebner1971tlt} and hydrogen with deuterium or tritium~\cite{Stwalley1976pnq}. The more recent achievements on ultracold atomic gases~\cite{Pethick2002book,Pitaevskii2016bec}, in which the interatomic interactions can be tuned by Feshbach resonances~\cite{Chin2010fri}, offer experimentalists many opportunities to create and study interacting quantum mixtures of different spin states, isotopes, or species. One situation is mixing  quantum fluids of different quantum statistics: degenerate Fermi gases (DFG) and Bose-Einstein condensates (BEC)~\cite{Truscott2001oof,Schreck2001qbe}. In these mixtures, interspecies interactions between fermions and superfluid BECs lead to rich phase diagrams (e.g.~\cite{Ludwig2011qpt}) and dynamic phenomena~\cite{Pitaevskii2016book}.

The equilibrium state of degenerate Bose-Fermi mixtures and their stability have been studied in experiments~\cite{Modugno2002coa,Ospelkaus2006idd,Lous2018pti} and also in mean-field theories, in which often the zero-temperature case is considered~\cite{Suzuki2008psa,
Marchetti2008psa}. It is a well-established fact that the constituents of a mixture undergo phase separation when the interaction is repulsive and strong, and the BEC collapses when the interaction is substantially attractive. An important phenomenon of phase separation is the formation of a thin interface between the components, and the remaining boson-fermion overlap at the interface has recently been probed by measuring three-body recombination losses in a $^{41}$K-$^6$Li mixture~\cite{Lous2018pti}. Therefore we are encouraged to investigate the detailed structure and properties of the thin interface theoretically.

Generally, collective excitations of the BEC can be described by the Gross-Pitaevskii equation (GPE)~\cite{Pitaevskii2016bec}, and the evolution of the DPG is analyzed either with kinetic equations in a semiclassical manner or via response functions of perturbations~\cite{Brack1997sp,Fetter2003qto}. The latter is rather different from superfluid fermions~\cite{Pitaevskii2016book}, which can be described by order parameter and hydrodynamic equations.
In a miscible mixture, inter-species interaction is modeled as meanfield to study elementary excitations of zero-temperature Bose-Fermi mixtures in homogeneous~\cite{Yip2001cmi} and harmonically trapped~\cite{Vichi1998qda} conditions. In the limit of full phase-separation, BEC and DFG establish pressure balance at the interface, and their collective modes in a harmonic trap have been investigated~\cite{Lazarides2008ceo,Schaeybroeck2009tps}.
There also have been efforts to understand the dynamics over a range of interaction strengths, and Ref.~\cite{Maruyama2005moa} constructed a numerical model for the time evolution of monopole oscillations in a Bose-Fermi mixture.

Experiments on the center-of-mass (COM) mode of the BEC coupled to the fermions show very weak frequency shifts on the order of a few percent~\cite{Ferrier2014amo,Delehaye2015cva,Wu2018cdo,Roy2017tem,DeSalvo2019oof}. However, our recent measurements on the radial breathing mode (RBM) frequency demonstrate a striking frequency shift of up to about 40 percent at phase separation~\cite{Lous2018pti}, whereas the frequency remain almost unchanged when bosons and fermions are mixed.

In this work, we consider the situation in our recent experiments ~\cite{Lous2018pti,Huang2019bmo}. A small BEC is immersed in a large Fermi sea and both components are trapped in an elongated harmonic trap. We first set up a mean-field model and numerically calculate the static density profiles of the Bose-Fermi mixture at zero temperature and extract the overlap between the two components at various strength of Bose-Fermi repulsion. Then we obtain analytical descriptions of the density profiles in the regimes of weak interaction and phase separation, and hence get insight into the formation and structure of the Bose-Fermi interface. We also discuss the weak finite-temperature effects on the Bose-Fermi overlap.

As an elementary example of dynamic behavior, RBM of the BEC immersed in the DFG is theoretically studied in the current work. We consider the condition of our experiments \cite{Huang2019bmo} on RBM, and the dynamics of DFG is  investigated in three different ways.
In the first case, we assume that the DFG adiabatically follows the oscillation of the BEC. Our second approach considers single fermion trajectories in the full phase-separation limit. Finally, we also perform numerical simulations for fermions using the test-particle method. Our results show that, when the Bose-Fermi $s$-wave scattering length $a_{bf}$ increases from zero, the RBM frequency $\omega$ remains almost constant until the fermions are depleted from the trap center by the BEC. Then the frequency increases dramatically across the phase separation until it levels off at the full phase-separation limit. The relation between the plateau value of $\omega$ and the number ratio $N_b/N_f$ unveils the essential role of the compressional character of the breathing mode.

In Sec.~\ref{sec:static}, we first present our mean-field model, which has been used in Refs.~\cite{Lous2018pti,Huang2019bmo}, and  study the static properties of a Bose-Fermi mixture with tunable repulsive interspecies interaction. Then we obtain analytical results for the situations of weak interaction and phase separation, and, in particular, intuitively explain the structure of the interface in immiscible Bose-Fermi  mixtures. In Sec.~\ref{sec:dynamic}, we develop three models to study the radial breathing mode of a BEC immersed in a large Fermi sea, and the frequency of RBM at different strengths of repulsion is calculated. We compare our results from different models and discuss the validity of the adiabatic-Fermi-sea approximation. In Sec.~\ref{sec:conclusions}, we summarize our results and their impacts.

\section{Static properties}\label{sec:static}
In our recent work  \cite{Lous2018pti,Huang2019bmo}, we developed a numerical mean-field model to calculate static density profiles of Bose-Fermi mixtures at zero-temperature. At finite temperatures, we use the density of BEC and fermions to calculate mean-field potential and estimate the thermal-bosons density. Here we first describe in detail this model, of which the basic ideas have been mentioned in the Supplemental Material of Ref.~\cite{Lous2018pti}. Then we calculate the Bose-Fermi overlap $\Omega$, which is an experimental observable. We further obtain analytical results for $\Omega$ at weak interaction and full phase separation. Finally, finite-temperature effects on densities are estimated.

\subsection{Numerical model}
\label{sec:static_numeric}
\begin{figure*}[ht]
\centering
\includegraphics[scale=0.8]
{densities}
\caption{The density profiles of the BEC ($n_b$) and the fermions ($n_f$) along the radial ($r$, left column) and the axial ($z$, right column) axes. The black solid, red dashed, and green dotted curves are calculated for $a_{bf}$ equals 0, 300$a_0$ and 600$a_0$, respectively.}
\label{fig:densities}
\end{figure*}

We start with the zero-temperature energy functional of the mixture, which is \cite{Imambekov2006bot}
\begin{equation}\label{eq:energy_functional}
\begin{split}
E=&\int d^3r\left[-\frac{\hbar^2}{2m_b}\psi^*\nabla^2\psi+V_b \psi^*\psi + \frac{1}{2}g_{bb}(\psi^{*}\psi)^2
\right.\\
&+\frac{1}{9}\frac{\hbar^2}{2m_f}(\nabla \sqrt{n_f})^2+V_f n_f+\frac{\hbar^2}{2m_f}\frac{3}{5}(6\pi^2)^{2/3}n_f^{5/3}\\
&\left. + g_{bf} n_b n_f \right].
\end{split}
\end{equation}
Here $\psi$ is the order parameter of the BEC, $n_f$ is the number density of the fermions, $V_b$ and $V_f$ are the corresponding trapping potentials, $g_{bb}=4\pi\hbar^2a_{bb}/m_b$ and $g_{bf}=2\pi\hbar^2a_{bf}(m_b^{-1}+m_f^{-1})$ are the boson-boson and boson-fermion coupling constants. Considering the static case, we ignore the dynamic phase of $\psi$ and have $\psi=\psi^*=\sqrt{n_b}$, where $n_b$ is the BEC number density. Consequenctly we can replace $-\psi^*\nabla^2\psi$ in Eq.~\eqref{eq:energy_functional} with $(\nabla \sqrt{n_b})^2$. The term with $\nabla$ for bosons  arise from the kinetic energy and can be ignored in the Thomas-Fermi (TF) limit. The $\nabla$ term for fermions is the leading term from the density-gradient correction, which is much smaller than other terms under the relevant experimental conditions~\cite{Lous2018pti}.

In order to obtain the static solution, we minimize the energy functional with the steepest decent method (also known as the imaginary time evolution) \cite{Press2007book}. The evolution of the densities from step $j$ to $j+1$ follows
\begin{equation}\label{eq:E_functional_one_step}
\sqrt{n}_{j+1}=\sqrt{n}_{j}- \frac{\delta E}{\delta n}\sqrt{n}_{j}\Delta \tau,
\end{equation} where $\Delta\tau$ is the step size of evolution, and its value should be large to ensure fast converging while small enough to avoid numerical instability. We normalize atom numbers after each time step under the constraints
\begin{equation}
\begin{split}\label{eq:LagrangeEq_NConstraint}
N_b&=\int n_b d^3r,\\
N_f&=\int n_f d^3r.
\end{split}
\end{equation}
The equilibrium solution of $n_b$ and $n_f$ is obtained when the algorithm converges.

The zero-temperature density profiles at $a_{bf}=$0, 300, and 600$a_0$ are plotted in Fig.~\ref{fig:densities} for our typical experimental conditions \cite{Lous2018pti} with $N_b=1.5\times 10^4$ bosons and $N_f=1.5\times 10^5$ fermions. The radial center-of-mass (COM) trapping frequency is 171 Hz (300 Hz) for bosons (fermions), and the aspect ratio of the elongated trap is $7.55$. We observe that the fermions are depleted from the trap center by the increasing $a_{bf}$ to a moderate value slightly above 300$a_0$. With stronger repulsions, the BEC is further  squeezed to form a core that is surrounded by the fermions via a thin interface.

\subsection{Overlap function}
We consider the normalized boson-boson-fermion (BBF) overlap at zero-temperature, because three-body recombination losses are dominated by BBF collisions and they can be used to probe the interface~\cite{Lous2018pti}. We ignore other three-body collisions involving two or more identical fermions, because they are suppressed at low temperatures by Pauli Blocking. Three-boson recombinations is also negligible in relevant experimental conditions, as the boson-boson scattering length is very small. The normalized BBF overlap is
\begin{equation}
\Omega=\frac{\int n_b^2n_fdV}{\int \tilde{n}_b^2\tilde{n}_fdV},
\end{equation}
where $\tilde{n}$ denotes the density profile in the absence of Bose-Fermi interaction. The overlap function from the numerical results is plotted as the black solid line (TF+B+F) in Fig.~\ref{fig:overlap}. We observe a smooth decrease of $\Omega$ as $a_{bf}$ increases.

\begin{figure}
\centering
\includegraphics[scale=1]
{overlap}
\caption{Comparison between the $\Omega$ from the numerical algorithms and the analytical models. The full numerical model produces the black solid curve (TF+B+T), applying TF approximation for both components in the numerical model gives the black dotted line (TF), and using the TF approximation only for the fermions leads to the black dashed curve (TF+B). The results from the analytical models for weak (AM\_weak, red dash-dot) and strong (AM\_strong, green dash-dot-dot) repulsive interactions are compared to the numerical results.}
\label{fig:overlap}
\end{figure}

Now we compare our full model with the Thomas-Fermi (TF) limit to study the influence from the kinetic energy of the BEC and the density gradient of the fermions. Within TF approximation, we remove $\nabla \sqrt{n_b}$ and $\nabla \sqrt{n_f}$ in Eq.~\eqref{eq:energy_functional} and obtain $\Omega$. The results are plotted as the dotted black line (TF) in Fig.~\ref{fig:overlap}. We observe that the overlap decreases drastically and vanishes near $a_{bf}=600a_0$, where phase separation takes place.

We further test the importance of $\nabla \sqrt{n_f}$ by excluding it from the full model while keeping the BEC kinetic energy term. The results are shown as the dashed black curve (TF+B) in Fig.~\ref{fig:overlap}. We find the TF+B curve being very close to the full calculation, indicating the fact that the influence from the fermion density gradient is very weak.

\subsection{Analytical results at small $a_{\rm bf}$}

We start from the mean-field TF density profiles in a harmonic trap as
\begin{align}
n_f&=n_{f0}\left(1-\frac{g_{bf} n_b}{\mu_{f0}} \right)^{3/2},\label{eq:nf_TF}\\
n_b&=n_{b0}\left(1-\frac{U_b+g_{bf} n_f}{ \mu_{b0}}\right),
\end{align} where $n_{f0}$ and $n_{b0}$ are the peak densities of the fermions and the bosons, $\mu_{f0}$ and $\mu_{b0}$ the global (position independent) chemical potentials, and $U_b$ the trapping potential of bosons. In Eq.~\eqref{eq:nf_TF}, we applied  the fermionic reservior approximation (FRA)~\cite{Lous2018pti} by ignoring any trapping potential $U_f$ for fermions and treat the fermions as a Fermi sea with a constant chemical potential. The FRA is valid as long as the BEC extents in a region much smaller than the fermion gas.

We first consider the homogeneous case, where $U_b$ vanishes. In the weak interaction regime, we perform a Taylor expansion in $g_{bf}$ and find $\delta \mu_b/\delta n_b \approx g_{bb}+g_2$, where
\begin{equation}\label{eq:g2}
g_2=-\frac{3}{2}g_{bf}^2\frac{n_f}{\mu_f},
\end{equation}
and for the leading term we have $n_f\approx n_{f0}$ and $\mu_f\approx \mu_{f0}$. This correction $g_2$ to $g_{bb}$ indicates a fermion-mediated boson-boson interaction, which has been observed in Refs.~\cite{DeSalvo2019oof,Edri2020oos} and interpreted as the long-wavelength limit of the Ruderman-Kittel-Kasuya-Yosida (RKKY)~\cite{Ruderman1954iec} interaction. In our Eq.~\eqref{eq:g2} the scaling $g_2\propto g_{bf}^2$ means the mediated interaction is a second-order mean-field effect in the adiabatic limit.

Now we consider trapped mixtures and the effects of a small $a_{bf}$ to the overlap factor $\Omega$. In this case, the fermions are slightly repelled from the trap center while the BEC is weakly compressed. Then the Taylor expansion of $\Omega$ on the small parameter $\eta=g_{bf}n_{b0}/\mu_{f0}$ leads to
\begin{equation}\label{eq:overlap_small_abf}
\Omega=1-\eta+\left(\frac{2}{11}+\frac{9d}{10}\right)\eta^2+\left(\frac{10}{429}-2d\right)\eta^3+O[\eta^4]
\end{equation}
where $d=\mu_{f0}n_{f0}/\mu_{b0} n_{b0}$ is on the order of 1 in our experimental conditions.
Because within the TF description the overlap function does not depend on the aspect ratio of the trap, we evaluate Eq.~\eqref{eq:overlap_small_abf} up to the third order and plot the result in Fig.~\ref{fig:overlap} as a red dash-dot curve. This analytical model agrees very well with the numerics in the weak interaction regime, and it only begins to deviate near the region of phase separation.
\subsection{The analytical model at large $a_{\rm bf}$}\label{sec:static_psl}
Let us discuss the opposite limit, where the $a_{bf}$ is very large and the fermions and bosons are separated. Obviously, the residual overlap in this regime is beyond the TF approximation, and we have to consider at least the kinetic energy of the BEC. We start from an infinite system without trapping potential, where we have only BEC (fermions) at $x\rightarrow\infty$ ($x\rightarrow-\infty$) and the components are separated by an interface parallel to the $y$-$z$ plane.

By applying the TF approximation to fermions and the pressure balance at the interface (see App.~\ref{app:PSL_static}), we obtain the dimensionless differential equation for the order parameter $\psi =\sqrt{n_b}$ of the BEC as
\begin{equation}
\phi = -\frac{\partial^2 \phi}{\partial X^2}+\phi^3+\frac{5\eta}{4}\left(1-\phi^2 \eta\right)^{3/2}\phi,
\end{equation}
where $\psi$ is normalized by $\phi=\psi/\sqrt{\mu_{b0}/g_{bb}}$, and $x$ by the BEC healing length $\xi_{b0}=\hbar/\sqrt{2m_b\mu_{b0}}$ as $X=x/\xi_{b0}$.
The solution with boundary condition $\phi(+\infty)=1$ and $\phi(-\infty)=0$ is given by
\begin{equation}
\frac{dX}{d\phi}=\sqrt{\frac{2}{(1-\phi^2)^2-(1-\phi^2 \eta)^{5/2}\Theta(1-\phi\sqrt{\eta})}},
\end{equation}
where $\Theta$ is the Heaviside step function.
The one-dimension BBF integral along the $x$-axis is then
\begin{equation}
\begin{split}
I_{x}&= \frac{1}{2}\int_{-\infty}^{+\infty} n_b^2n_fdx\\
&=\frac{\sqrt{2}\xi_b }{2}n_{b0}^2 n_{f0}\eta^{-5/2}\\
&\times\int_0^1du \frac{ u^4 (1-u^2)^{3/2}}{\sqrt{(1-u^2/\eta)^2-(1-u^2)^{5/2}}}
\end{split}
\end{equation}
where we used $u=\phi\sqrt{\eta}$ for simplification and the factor 1/2 results from the suppression of thermal bunching in a BEC involving two identical bosons (see also Eq.~\eqref{eq:omega_BFt}). We recognize that the overlap integral decreases proportional to $a_{bf}^{-5/2}$ at strong repulsive interactions.

To calculate $\Omega$ for a trapped mixture in the phase separated regime, we consider that $\xi_b$ is much smaller than $R_b$ and $R_f$. Then the atom number conservation and the pressure balance at the interface fix the equilibrium condition of the system, e.g.\;$R_b$,  $R_f$ and the radial position of the interface $r=\zeta$ (see App.~\ref{app:PSL_static}). We evaluated $I_x$ at the interface and multiply it with the surface of the ellipsoidal interface, whose semi-axes are ($\zeta,\zeta,A \zeta$). We then normalize the outcome with the overlap integral at zero $a_{bf}$, which is obtained from the numerical model (TF+B) while an analytical approximation is also available \cite{Lous2018pti}. The resulted $\Omega$ is plotted in Fig.~\ref{fig:overlap} as the green dash-dot-dot curve, which is consistent with the numerical model. The small discrepancy between the numerical and  analytical results is probably caused by the surface tension of the interface \cite{Schaeybroeck2008ito} and the finite value of $\xi_b/R_b$ and $\xi_b/R_f$.

We also compare this result with a simple intuitive model, which assumes that the BEC is facing a hard wall at the interface and its $n_b(x)=n_{b0}\tanh^2({x}/{\sqrt{2}\xi_{b0}})$ is suppressed to zero within its healing length $\xi_b$. At the steep and rigid mean-field potential induced by the BEC, the Fermi sea is filled up to the Fermi energy. The corresponding $I_x$ agrees with the full calculation with a deviation less than 20\% when $\eta\gg 1$.

\subsection{The thermal faction of bosons}
\begin{figure*}
\centering
\includegraphics[scale=0.8]
{thermalBosonDensity}
\caption{The density profiles of the thermal bosons ($n_t$) along the radial ($r$, left column) and the axial ($z$, right column) axes. The black solid, red dashed and green dotted curves are calculated for $a_{bf}$ equals 0$a_0$, 300$a_0$ and 600$a_0$, respectively. We have used the $n_b$ and $n_f$ in Fig.~\ref{fig:densities} to calculate the mean-field for thermal bosons.}
\label{fig:thermalBosonDensity}
\end{figure*}
In this model we assume $n_t$ to be influenced by $n_b$ and $n_f$, but not vice versa. This approximation is valid as long as we have $n_t\ll n_b$. We then calculate $n_t(r,z)$ with a Bose-Einstein distribution for thermal bosons, which is
\begin{equation}
n_t=\left(\frac{m_bk_B T}{2 \pi \hbar^2}\right)^{3/2}\text{Li}_{3/2}\left[e^{(\mu-U_b)/k_BT}\right],
\end{equation}
where the total effective potential $U_b$ for a thermal boson is $V_b+2 g_{bb}n_b+g_{bf}n_f$ and Li$_{3/2}$ is the polylogarithm function of order 3/2. The chemical potential $\mu_b$ of the BEC is obtained from the known $n_b$ and $n_f$ using $\mu_b \psi=(-\hbar^2\nabla^2/2m_b+g_{bb}n_b+g_{bf}n_f) \psi$ and $\psi=\sqrt{n_b}$. The temperature $T$ is fixed numerically to fulfil the atom number constraint $N_t=\int n_t dV$, and we finally get the density distribution $n_t$, which is shown in Fig.~\ref{fig:thermalBosonDensity}. Here the total boson number is $3\times10^4$ and the BEC fraction is 50\%. We notice that the thermal bosons form a shell-like structure at the edge of the BEC, because both the BEC and the fermions are repulsive to thermal bosons.

Although the density $n_t$ of the thermal component of the boson gas is typically almost two order of magnitude lower than the BEC density $n_b$, the significant thermal fraction (up to $\sim50\%$) of the bosons in typical experimental conditions leads to a correction to the total overlap between bosons and fermions. The generalized overlap $\Omega$ including the thermal bosons is defined as \cite{Lous2018pti}
\begin{equation}\label{eq:omega_BFt}
\Omega=\frac{\int (\frac{1}{2}n_b^2n_f+n_b n_t n_f+n_t^2n_f)dV}{\int (\frac{1}{2}\tilde{n}_b^2\tilde{n}_f+\tilde{n}_b\tilde{n}_t\tilde{n}_f+\tilde{n}_t^2\tilde{n}_f)dV},
\end{equation}
where $n_t$ and $\tilde{n}_t$ are the thermal boson density with and without the Bose-Fermi interaction.

\begin{figure}[t]
\centering
\includegraphics[scale=1]
{overlap2}
\caption{Comparison between the overlap factor $\Omega$ from the numerical model with and without including thermal bosons. The black solid line is identical to the one in Fig.~\ref{fig:overlap}, which ignores the thermal bosons. The green dashed and red dotted curves are results including thermal bosons, which are interacting either with both the BEC and the fermions (tB+tF) or with only the BEC (tB).}
\label{fig:overlap2}
\end{figure}

We calculate $\Omega$ from $n_b$, $n_f$ and $n_t$, and plot the results as the green dashed curve in Fig.~\ref{fig:overlap2}. The black solid curve in Fig.~\ref{fig:overlap2} excludes thermal bosons and is identical to the one in Fig.~\ref{fig:overlap}. If we ignore the interaction between the thermal bosons and the fermions, $\Omega$ become the red dotted curve in Fig.~\ref{fig:overlap2}. We find that the residual of $\Omega$ in the strong interaction regime has a value of a few percent, and it slowly decreases when the repulsion between the thermal bosons and the fermions is considered.

\subsection{Finite-temperature effects on fermions}
At a finite temperature the BEC is only weakly influenced by the thermal bosons, therefore we can analyze the degenerate and non-degenerate parts separately. But we cannot define a thermal or degenerate part of the fermionic gas since all the single particle orbitals are correlated. Consequently we estimate the finite temperature effects of fermions in a perturbative way, i.e.\;calculating the finite-$T$ fermion density $n_{f}$ from the $n_b$ known at zero-$T$ with a Fermi-Dirac distribution. We will discuss only the temperature effects on the overlap integral, i.e.\;the numerator in Eq.~\eqref{eq:omega_BFt}, because the denominator is fixed in our  definition of $\Omega_{\rm eff}$.

With a typical temperature of $T/T_f\approx0.12$, $n_{f}$ is about $5\%$ lower than the zero-$T$ solution at the trap center and lightly spreads out at the edge, e.g.~$r=R_f$ in the radial direction. Since our experiments have $R_f/R_b\approx8$, we ignore the inhomogeneity of $n_f$ in the BEC region and expect about $5\%$ down shift of the overlap integral.
On the other hand, the thermal boson cloud extends much wider than the BEC and is much less sensitive to the finite-$T$ correction of $n_f$.

Another effect, which also exists at zero temperature, is the unitary limit of the cross section of the Bose-Fermi scattering. The fermions have a kinetic energy on the order of the Fermi energy $E_F$, which is much larger than the kinetic energy of the bosons and leads to a reduction of the Bose-Fermi cross section. In the relevant experimental conditions we have $1/k_F\approx 4500a_0$, where $E_F=\hbar^2k_F^2/2m_f$. Since the mean kinetic energy of fermions at the trap center is $3E_F/5$, we estimate that the cross section scales proportional to $a^2/(1+0.6\times a^2k_F^2)$. Therefore the reduction of the Bose-Fermi cross section is negligible in typical experimental conditions, where $a_{bf}$ is always below about $2000a_0$.

\section{radial breathing mode}\label{sec:dynamic}
In our recent experiment on an elongated Bose-Fermi mixture~\cite{Huang2019bmo}, we have observed
a significant frequency shift of the radial breathing mode (RBM) of the BEC when phase separation takes place.
This motivates us to theoretically investigate the collective mode of a BEC immersed in a large Fermi sea at various values of $a_{bf}$. To model the dynamics of the BEC, we use the Gross-Pitaevskii equation (GPE) and include a mean-field potential $g_{bf} n_f$ formed by the fermions. For the dynamics of fermions, we utilize different models. The first model we introduce here is the adiabatic Fermi sea (AFS) model, which assumes that the fermions adapt adiabatically to the perturbations of the BEC. This model can be solved by either performing time evolutions (TE) or extracting eigenvalues (EG) from linearized equations. Then in the second model, we use the collisionless Boltzmann-Vlasov equation (BVE) to describe the fermions. The BVE is solved numerically with the test-particle method (TPM), while an analytical result is also achieved at the phase separation limit (PSL). We obtain the RBM frequency $\omega$ from different models and compare the results.

In our following calculations we approximate our elongated mixture to an axially invariant system with cylindrical symmetry. The radial plane of the model corresponds to the radial plane of the  mixture at the trap center. The static density profiles of the mixture in the radial plane are obtained from the numerical model described in Sect. \ref{sec:static_numeric}. The temperature of the system is assumed to be zero in our oscillation models.

\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]
{freq_shift}
\caption{The radial breathing mode frequency shift $\omega/\omega_0$ calculated from different models and parameters. Two sets of atom numbers are used here: Set 1 with $N_b=1.6\times10^4$ and $N_f=1.03\times10^5$ and Set 2 with  $N_b=8\times10^3$ and $N_f=1.68\times10^5$. The black (red) solid curve is given by the AFS approximation by the direct time evolution (AFS TE) method for Set 1 (Set 2). The black (red) dotted curves also includes the AFS approximation but is obtained by finding the eigen values of the linearised equations (AFS EG) for Set 1 (Set 2). The blue and pink dash-dot curves are the results of the GPE-BVE mode solved by the TPM for two sets of parameters. The outcome of the PSL model at large $a_{bf}$ values are shown as the green and yellow dashed curves for the two sets.}\label{fig:freq_shift}
\end{figure}

\subsection{Adiabatic Fermi sea approximation}\label{sec:AFSA}
If the density of fermions is high enough to support a Fermi velocity $v_F=\sqrt{2E_F/m_f}$ much larger than the sound velocity $v_s=\sqrt{g_{bb}n_b/m_b}$ of the BEC, the fermions  follow adiabatically the fluctuations of the BEC density within a mean-field picture \cite{Yip2001cmi}. This adiabatic Fermi sea (AFS) approximation is valid in relevant experimental conditions as long as the fermions are not depleted from the BEC region.

We formally write down the time-evolution equation of the BEC dressed by the adiabatic Fermi sea as
\begin{equation}\label{eq:time_evolve}
i \hbar \frac{\partial \psi}{\partial t}=-\left[\frac{\hbar^2}{2m_b}\nabla^2+V_b +g_{bb}|\psi|^2+g_{bf}n_f\right] \psi.
\end{equation}
The time-dependent fermion density $n_f=C_f(\mu_{f0}-g_{bf}|\psi|^2)^{3/2}\Theta(\mu_{f0}-g_{bf}|\psi|^2)$, where $\mu_{f0}$ is the global Fermi energy and $C_f=(2m_f/\hbar^2)^{3/2}/6\pi^2$, is calculated with the TF approximation and FRA of the fermions.

In order to numerically solve Eq.~\eqref{eq:time_evolve}, we approximate our cigar-shaped cloud with a cylindrical system, whose radial plane represents the radial plane of the mixture at the trap center. We describe $\psi$ in the radial plane by setting up a one-dimensional complex-valued grid for $\psi$ along the radial direction.

Our first way to solve Eq.~\eqref{eq:time_evolve} is numerically calculating the time evolution of $\psi$. Inspired by our experiment~\cite{Huang2019bmo}, where we excited the RBM by switching $a_{bf}$ between a small and a large value, we perform a similar process in our simulations for a given $a_{bf}$. We take the static solution $\psi_0$ at $a_{bf}-100a_0$ as the initial value of $\psi$. Then we switch to $a_{bf}$ and let $\psi$ evolve in time according to Eq.~\eqref{eq:time_evolve}, while the TF density $n_f$ at each time-step is calculated from $|\psi|^2$ at that moment. We record the averaged BEC width $\left\langle r\right\rangle=\int dr  r W(r)/\int dr W(r)$, where the weight function is $W(r)=2\pi r n(r)$, as a function of time and fit it to a cosine function to extract the oscillation frequency $\omega$.

The calculated dependence of $\omega$ on the interaction strength is plotted as solid curves in Fig.~\ref{fig:freq_shift} and marked as AFS-TE for the AFS approximation and the time-evolution method. The black solid curve corresponds to an experimental setting with a boson number of $N_b=1.6\times 10^4$ and a fermion number of $N_f=1.03\times 10^5$ (named Set 1) while the red solid curve uses $N_b=8\times 10^3$ and $N_f=1.68\times 10^5$ (Set 2). These parameters correspond to our experimental conditions~\cite{Huang2019bmo}. We use $x=1000a_0/a_{bf}$ for the horizontal axis in the plot, and the y-axis is normalized to the RBM frequency $\omega_0$ at $a_{bf}=0$. The AFS-TE model shows that the RBM frequency $\omega$ barely changes at small scattering lengths. Then it starts to increase rapidly near $x=3$, where the fermions are depleted from the trap center. Finally $\omega$ tends to reach a maximum frequency shift near around $x=1$. The numerics begin to fail when the mixture is deeply in the phase separation regime and the interface depth is comparable to the grid step-size.

\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]
{freq_q}
\caption{The RBM frequency $\omega/\omega_0$ at the PSL as a function of the boson number fraction $Q=N_b/(N_b+N_f)$. The black solid curve is calculated from the AFS-TE model. The PSL model with and without ($\sigma=0$) the surface tension effects give the green dashed and red dotted curves, respectively. We use a total atom number of $1.5\times 10^4$ and $a_{bf}=1300a_0$.}\label{fig:freq_q}
\end{figure}

We notice that the maximum shift of $\omega$ is higher when $N_f/N_b$ is larger. So we define the boson number fraction $Q=N_b/(N_b+N_f)$, fix the total atom number $N_b+N_f=1.5\times 10^5$, and calculate the RBM frequency at $a_{bf}=1300a_0$ for various $Q$ values. The results are presented as the black curves in Fig.~\ref{fig:freq_q} for a range of $Q$ that is reachable in the experiment. We find that $\omega/\omega_0$ increases faster when $Q$ is close to 0.

As an alternative numerical method, we linearize and solve Eq.~\eqref{eq:time_evolve} for small perturbation $\delta\psi$ and $\delta n_f=-(3/2)g_{bf} C_f^{2/3} n_f^{1/3} \delta (\psi^* \psi)$.
We take the ansatz $\psi=(\psi_0+ue^{-i\omega t}+v^*e^{i\omega t})e^{-i\mu_b t/\hbar}$ near the equilibrium solution $\psi_0=\psi_0^*=\sqrt{n_{b}}$
and arrive at the linearized effective GPE as
\begin{equation}\label{eq:GPE_linear}
\begin{split}
\hbar \omega u=&\left[H_{b}+(2g_{bb}+g_2)n_b\right]u+(g_{bb}+g_2)n_b v\\
-\hbar \omega v=&\left[H_{b}+(2g_{bb}+g_2)n_b\right]v+(g_{bb}+g_2)n_b u\\
\end{split}
\end{equation}
where $H_{b}=-(\hbar^2/2m_b)\nabla^2+V_b+g_{bf}n_f-\mu_b$ accounts for the kinetic energy, the boson trapping potential, the mean-field potential induced by a static Fermi, and the global chemical potential $\mu_b$ of BEC obtained from the static solution. The boson-boson interaction that is mediated by the Fermi sea is given in Eq.~\eqref{eq:g2}.

In order to numerically solve Eq.~\eqref{eq:GPE_linear} for a cylindrical system, we discretize $u(r)$ and $v(r)$ along the radial direction.
With the $n_b$ and $n_f$ obtained in Sec.~\ref{sec:static_numeric}, we obtain the eigenfrequency $\omega$ by diagonalizing Eq.~\eqref{eq:GPE_linear} in its matrix form with a regularized boundary condition. The calculated $\omega/\omega_0$ values of the lowest RBM are shown in Fig.~\ref{fig:freq_shift} as the black and red dotted curves (AFS-EG1 and AFS-EG2) for the two sets of atom numbers. Although the diagonalization method requires $n_f>0$ and becomes no longer fully valid when $a_{bf}$ is so large that fermions are depleted from the trap center, we find its results agree very well with the outcomes of the direct time evolution before numerical instabilities take over in the deeply phase-separated regime.

Finally, we discuss our Eq.~\eqref{eq:GPE_linear} in comparison with other experimental~\cite{DeSalvo2019oof} and theoretical works (e.g.~\cite{Tsurumi2000dom,Santamore2008fmi,De2014fml}) on weakly interacting Bose-Fermi mixtures. In the earlier works, a simple replacement of the boson coupling constant $g_{bb}\to g_{bb}+g_2$ transforms the dynamics from a pure BEC to a weakly interacting Bose-Fermi mixture. In our model, such a transformation is not explicit in Eq.~\eqref{eq:GPE_linear} because of the terms proportional to $2g_{bb}+g_2$. Our results recover $g_{bb}\to g_{bb}+g_2$ only if we take the limit of $g_{bf} n_b\ll \mu_f$, where $g_{bf}\delta n_f\approx g_2n_b$ is valid.
Moreover, we find a small $g_{bf}$ also induces a correction (buoyancy-like effect) $-\frac{3n_f V_f}{2\mu_f V_b}g_{bf}$ to the boson potential $V_b$ when a weak fermion potential $V_f\ll \mu_f$ is considered. In general, we expect our results to be valid also beyond the weak-interaction regime.

\begin{figure*}[th]
\centering
\includegraphics[width=2\columnwidth]
{bve_osci_3kick}
\caption{Simulated width evolution of the bosonic (blue solid) and the fermionic (red dotted) cloud, where the GPE-BVE model is used. The time $t$ is normalized to the single boson COM period $T_b$ in the trap.  The $y$-axis shows deviation of the density-averaged width $\langle r \rangle$ of the cloud from its mean value. The oscillation is excited with 3 consecutive quenches of the Bose-Fermi interaction (scheme Exct1). The panel (a) presents the oscillation at $a_{bf}=100a_0$, and panel (b) corresponds to $a_{bf}=1000a_0$.}\label{fig:bve_osci_3kick}
\end{figure*}
\subsection{Kinetic equation simulation for fermions}

To describe the dynamics of the degenerate single component Fermi gas by first principles, we utilize the Boltzmann-Vlasov equation (BVE) with vanishing collisions, i.e.
\begin{equation}\label{eq:BVE}
\partial_t f+\frac{1}{m_f}\vec{p}\cdot\nabla_r f- \frac{1}{m_f}\vec{F}_f\cdot \nabla_v f=0,
\end{equation}
where $f(\vec{p},\vec{r})$ is the fermion distribution function in the phase space, $m_f$ the mass of a single fermion. The force $\vec{F}_f$ on the fermions is given by the trapping potential $V_f(r)=m_f \omega_f^2r^2/2$ and the repulsion $g_{bf}n_b$ from the BEC.

Equation~\eqref{eq:BVE} can be solved numerically with the quasi-particle method \cite{Maruyama2005moa,Brack1997sp}, which uses a cloud of $\tilde{N}$ classical pseudo particles (test particles) with mass $m_f$ to sample the phase space density of $N$ real fermions. The kinetic equation is then simulated with the Newtonian equations of the test particles in the external potential, and the real fermion density is calculated from the test particle density with a scaling factor $N/\tilde{N}$.

In our calculations for a cylindrical system, we implement a one-dimensional spatial grid for $\psi$ and take a value of $\tilde{N}$ so that the shot noise of $n_f$ at the grid points is smaller than the thermal statistic noise at $T/T_F=0.1$.
In order to excite the collective mode, we start with the static densities at $a_{bf}=0$ and apply three consecutive quenches of the scattering length $a_{bf}$ between 0 and $700a_0$, which closely imitates the relevant experimental sequence~\cite{Huang2019bmo}.
After the excitation stage, the time evolution of the effective width $\langle r\rangle$ for the BEC and the fermions are recorded to extra the frequency.

\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]
{bve_fft}
\caption{The FFT spectrum amplitude $A$ of the oscillations shown in Fig.~\ref{fig:bve_osci_3kick}. The radial breathing mode frequency $\omega$ is normalized to its reference value $\omega_0$ at $a_{bf}=0a_0$. The panel (a) and (b) shows the results from $a_{bf}=100a_0$ and $1000a_0$ respectively.}\label{fig:bve_fft}
\end{figure}

The typical time evolution of the $\langle r\rangle$ of the BEC (the blue solid curve) and the fermions (the red dotted curve) are plotted in Fig.~\ref{fig:bve_osci_3kick}. Our calculation uses atom numbers from Set 1 (see Sec.~\ref{sec:AFSA}). And $a_{bf}=100a_0$ and $a_{bf}=1000a_0$ applies to panel (a) and (b) in Fig.~\ref{fig:bve_osci_3kick}, respectively. We recognize that the oscillation is not necessarily a simple sinusoidal wave, and it contains  possibly multiple frequency components. So we apply the fast-Fourier-transformation (FFT) to the data and obtain the power spectra, which are shown in Fig.~\ref{fig:bve_fft}.

We find that two frequency components  are important in the spectra, one is close to or slightly above $\omega/\omega_0=1$ and another is close to $\omega/\omega_0=1.8$. Keep in mind that $\omega_{f}/\omega_b=1.75$ and the BEC extents much narrower than the fermionic gas, we recognize the higher frequency component as the single fermion mode, which is only slightly changed by the small BEC. On the other hand, the main peak in the  BEC spectrum indicates that the BEC RBM is coupled to the fermions and forms a lower branch of the oscillating spectrum.

To be consistent with former analysis on the frequency within this work, we fit the oscillation of width with a cosine function and extract the frequency $\omega$. The results are plotted in Fig.~\ref{fig:freq_shift}. The blue (pink) dash-dot line shows the results for Set 1 (Set 2). The results from the TPM are consistent with the AFS model. Furthermore, we also vary the initial excitations, e.g.\;the amplitude of the oscillation, and extract $\omega$ in different time sections of the evolution. And we observe only minor changes (less than about 5$\%$) of the value of $\omega$.

\subsection{Analytical model of oscillation at the phase separation limit (PSL)}
At the phase separation limit (PSL), both the static and dynamic properties of the Bose-Fermi mixture can be studied analytically \cite{Schaeybroeck2008ito,Schaeybroeck2009tps}. Following the procedure introduced in Ref.~\cite{Schaeybroeck2009tps} for a spherical case, we investigate the RBM of the BEC in an infinitely-long cylindrical mixture, which is more relevant to experiments. Starting from the TF approximation for both components, we assume that the two components in the mixture is connected by an infinitely thin interface layer, which has a surface tension effect with coefficient $\sigma$ \cite{Schaeybroeck2008ito} (see also App. \ref{app:surface_tension}). We first obtain separately the formal solution of the BEC density and the fermion phase space distribution function $f(r,p)$, then solve the problem of the mixture by matching the boundary conditions, i.e.~flux and pressure, at the interface. The equilibrium solution of the densities in the mixture has been used in Sec.~\ref{sec:static_psl} and discussed in details in App.~\ref{app:PSL_static}. We will now find the oscillation frequency $\omega$ of the RBM in the mixture.

The collective modes of a trapped BEC in the TF limit are well understood~\cite{Pitaevskii2016book}. The radial mode of a BEC with zero angular momentum in a cylindrical system corresponds to a density fluctuation of
\begin{equation}\label{eq:2F1_cyl}
\begin{split}
&\delta n_b(r)\propto\\
&F\left(\frac{1+\sqrt{1+2\frac{\omega^2}{\omega_b^2}}}{2},\frac{1-\sqrt{1+2\frac{\omega^2}{\omega_b^2}}}{2},1,\frac{r^2}{R_b^2}\right),
\end{split}
\end{equation}
where $F$ is the hypergeometric function ${_2F_1}$, $\omega$ the frequency of the collective mode, $\omega_b$ the COM trapping frequency of bosons, and $R_b$ the Thomas-Fermi radius of the BEC.

At the interface $r=\zeta$ of the mixture there is no exchange of components. This means that the velocity of the BEC is equal to the speed $\dot{\zeta}$ of the moving interface. Consequently we have
\begin{equation}\label{eq:BC_BEC}
{\delta\zeta}=\frac{g_{bb}}{\omega^2m_b }\partial_r\delta n_b,
\end{equation} where we denote $\partial_r n_b=\partial n_b/\partial r$ for simplicity.

In the phase separation limit, the kinetic equation for fermions reduces to
\begin{equation}\label{eq:BVE_PS}
\partial_t f+\vec{v}\cdot\nabla_r f-\omega_f^2 \vec{r}\cdot \nabla_v f=0.
\end{equation}
We then apply the ansatz describing the deformation of the Fermi surface as $f=f_0+\delta(|\vec{p}|-p_f)u(r,\alpha,\beta)e^{-i\omega t}$, where $\alpha=\cos\phi$ with $\phi$ the between the momentum $\vec{p}$ and the position $\vec{r}$ in the radial plane, $\beta=\cos\theta$ with $\theta$ the angle between $\vec{p}$ and the longitudinal $z$-axis, and $p_f(r)=\sqrt{2 m_f\mu_f(r)}$ is the local Fermi momentum. Then we get the linearised kinetic equation
\begin{equation}
\begin{split}
-i\omega u +\omega_f\rho\alpha\sqrt{1-\beta^2}\partial_r u
+\omega_f(1-\alpha^2)g(r,\beta)\partial_\alpha u
\\+\omega_f\alpha\beta\sqrt{1-\beta^2}\frac{r}{\rho}\partial_{\beta}u=0,
\end{split}
\end{equation} where $\rho=(R_f^2-r^2)^{1/2}$ with $R_f$ the Thomas-Fermi radius of fermions and $g(r,\beta)=(\rho\sqrt{1-\beta^2})/r-r/(\rho\sqrt{1-\beta^2})$.
The corresponding solution has the form
\begin{equation}\label{eq:BVE_cyl_u_form}
u(r,\chi)=\mathcal{F}[r^2(R_f^2-r^2)(1-\alpha^2)(1-\beta^2)]e^{-i\omega \tau/2},
\end{equation}
where $\mathcal{F}[x]$ is an arbitrary function of $x$ and
\begin{equation}\label{eq:tau_cyl}
\tau(\zeta,\alpha,\beta)=\frac{\psi_0-\arctan[2\alpha/g(\zeta,\beta)]}{\omega_f},
\end{equation}
with $\psi_0=\pi\Theta[g(r,\beta)]$. As $\tau$ characterizes the phase of the oscillating system, we recognize it as the time for a single fermion with Fermi velocity $v_F$ to depart and then return to the interface. Equation~\eqref{eq:tau_cyl} is valid when $\alpha\in[0,1]$ and beyond that we have $\tau(-\alpha,\beta,r)=-\tau(\alpha,\beta,r)$. It is also obvious that $\tau(\alpha,-\beta,r)=\tau(\alpha,\beta,r)$.

Now we consider the non-penetration boundary condition for fermions at the interface. It requires that
\begin{equation}
\left[u(\zeta,\alpha,\beta)-u(\zeta,-\alpha,\beta)\right]e^{-i\omega t}=2 m_f \alpha\sqrt{1-\beta^2} \dot{\zeta},
\end{equation} where $\dot{\zeta}=\partial_t \zeta=-i\omega\delta\zeta e^{-i\omega t}$ is the velocity of the phase boundary. Taking this into Eq.~\eqref{eq:BVE_cyl_u_form} and \eqref{eq:tau_cyl}, the fermion perturbation is solved to be
\begin{equation}\label{eq:u_cyl}
u(\zeta,\alpha,\beta)=2m_f\alpha\sqrt{1-\beta^2}(-i\omega\delta\zeta)(1-e^{i\omega\tau})^{-1}.
\end{equation}
We may formally expand $(1-e^{i\omega\tau})^{-1}=\sum_{n=0}^\infty e^{in\omega\tau}$ and recognize that this solution $u$ is constituted of a series of elementary excitations happened at earlier times.

The condition of pressure equilibrium  at the interface is $P_b-P_f=\sigma/\zeta$, where the boson pressure is $P_b=g_{bb}n_b^2/2$ and the pressure $P_f$ of the collisionless fermions at the interface is given by the corresponding momentum flux $\Pi(r)=(1/m_f h^3)\int d^3\vec{p} \alpha^2 (1-\beta^2) p^2 f(\vec{p},r)$ in the radial direction \cite{Schaeybroeck2009tps}. Together with the boson boundary condition Eq.~\eqref{eq:BC_BEC} we obtain the equation for the oscillation frequency $\omega$ as
\begin{equation}\label{eq:BC_PSL_cyl}
\frac{\partial_r F}{F}=\frac{\omega^2 m_b\left(n_b-\frac{1}{\zeta}\frac{\partial \sigma}{\partial\mu_b}\right)}{-\frac{\sigma}{\zeta^2}+\frac{1}{\zeta}\frac{\partial \sigma}{\partial\zeta}+\frac{p_f^4 C_{\Pi}}{(2\pi\hbar)^3}-\partial_r(P_b-P_f)},
\end{equation}
where $C_{\Pi}=8\omega\int_0^{\pi/2} d\phi \int_{0}^{1} d\beta (1-\beta^2)^{3/2}\cos^3\phi
\cot\left({\omega\tau}/{2}\right)$ and all values are calculated at the interface~(see App.~\ref{app:osci_dp} for details).

Taking typical parameters from the  experiments~\cite{Huang2019bmo}, we obtain  $\omega$ and show the results as the green and orange dashed curves in Fig.~\ref{fig:freq_shift} for the two sets of atom numbers. We find the results from the PSL model is consistent with other models. The PSL model predicts a slow increase of $\omega$ at the PSL,  and gives slightly higher $\omega$ values than other models. The $Q$ dependence of the RBM frequency shift is presented in Fig.~\ref{fig:freq_q} as the green dashed curve. In order to check the influence from the surface tension effects, we also calculate $\omega$ with $\sigma=0$ and plot the outcomes as the red dash-dot curve in Fig.~\ref{fig:freq_q}. The PSL model without surface tension leads to a slightly lower frequency shift.

\section{Conclusions}\label{sec:conclusions}
We thoroughly investigated the static density profiles and the
radial breathing mode in an elongated degenerate Bose-Fermi mixture. The experimentally relevant conditions of~\cite{Lous2018pti,Huang2019bmo}, under which the size of the BEC is much smaller than the Fermi sea, have been considered. We first presented in details our mean-field numerical model, which explains the smoothing of phase separation with the kinetic energy of the BEC~\cite{Huang2019bmo}. Then we obtained analytical forms of the density profiles in the limit of weak and strong repulsion. In particular, the latter shows intuitively the structure of the interface: The BEC at the interface behaves like being blocked by a wall potential, and the fermions penetrates into the mean-field potential of the BEC edge with a depth determined by the chemical potential of the Fermi sea.

We presented three zero-temperature models for the RBM of the BEC in cylindrical mixtures, of which the results are consistent. We first describe the adiabatic Fermi sea model, which explained the significant shift of the RBM frequency observed across the phase-separation of the mixture~\cite{Huang2019bmo}. Within this model, we find the fermion-mediated interaction between bosons to be an adiabatic second-order mean-field effect, which is valid also beyond the weak-interaction regime under relevant experimental conditions.
For very large $a_{bf}$ values, another full phase-separation model considering single-fermion trajectories gives results similar to those from the AFS model. Finally, we perform test-particle simulations for the RBM, and the outcomes are almost identical to that of the AFS model for a large range of repulsion strength.

The remaining discrepancy between the observed RBM frequencies~\cite{Huang2019bmo} and the results from our zero-$T$ models, especially in the regime of full phase-separation, stimulates future experiments in more deeply cooled samples with further reduced imperfections, e.g.~anharmonicity of the trap. Comprehensive theories  including finite-$T$ effects~\cite{Liu2003fte,Grochowski2020bmo}, quantum fluctuations, and possible hydrodynamic properties at the interface are encouraged to study relevant experimental situations.

\begin{acknowledgments}
We thank B. Van Schaeybroeck, A. Lazarides and T. Maruyama for intensive discussions and providing details, especially numerical results, of their earlier works. We acknowledge and thanks the valuable discussions with M. Baranov, D. Yang, R. Bijnen, S. Watabe about the theoretical models. We acknowledge and thank the comments on the manuscript from our experimental group R. Grimm, J. Walraven, I. Fritsche, R. Lous, C. Baroni, T. W. Grogan and E. Kirilov. This work was supported by the Austrian Science Fund FWF within the Spezialforschungsbereich FoQuS (F4004-N23)
and partially the project P32153-N36.

\end{acknowledgments}

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